Tutorial res2 dinv

Tutorial : 2-D and 3-D electrical imaging surveys

By

Dr. M.H.Loke
Copyright (1996-2004)

email : mhloke@tm.net.my
drmhloke@yahoo.com

(All rights reserved)

(Revision date : 26th July 2004)

Copyright (1996-2004) M.H.Loke

ii

Copyright and disclaimer notice

The author, M.H.Loke, retains the copyright to this set of notes. Users may print a
copy of the notes, but may not alter the contents in any way. The copyright notices
must be retained. For public distribution, prior approval by the author is required.

It is hoped that the information provided will prove useful for those carrying out 2-D
and 3-D field surveys, but the author will not assume responsibility for any damage
or loss caused by any errors in the information provided. If you find any errors,
please inform me by email and I will make every effort to correct it in the next edition.

You can download the programs mentioned in the text (RES2DMOD, RES2DINV,
RES3DMOD, RES3DINV) from the following Web site

www.geoelectrical.com

M.H.Loke
July 2004


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Table of Contents

1. Introduction to resistivity surveys 1
1.1 Basic resistivity theory 1
1.2 Electrical properties of earth materials 5
1.3 1-D resistivity surveys and inversion – applications, limitations
and pitfalls 6
1.4 Basic inverse theory 12
1.5 2-D model discretisation methods 14
2. 2-D electrical imaging surveys 16
2.1 Introduction 16
2.2 Field survey method – instrumentation and measurement procedure 16
2.3 Available field instruments 19
2.4 Pseudosection data plotting method 21
2.5 A comparison of the different electrode arrays 23
2.5.1 The Frechet derivative for a homogeneous half-space 24
2.5.2 A 1-D view of the sensitivity function - depth of investigation 25
2.5.3 A 2-D view of the sensitivity function - lateral and vertical
resolution of the different arrays 28
2.5.4 Wenner array 29
2.5.5 Dipole-dipole array 29
2.5.6 Wenner-Schlumberger array 33
2.5.7 Pole-pole array 35
2.5.8 Pole-dipole array 36
2.5.9 High-resolution electrical surveys with overlapping data levels 38
2.5.10 Summary of array types 39
2.5.11 Use of the sensitivity values for multi-channel measurements or streamers 40
3. A 2-D forward modeling program
3.1 Finite-difference and finite-element methods 42
3.2 Using the forward modeling program RES2DMOD 43
3.3 Forward modeling exercises 44
4 A 2-D inversion program 46
4.1 Introduction 46
4.2 Pre-inversion and post-inversion methods to remove bad data points 46
4.3 Selecting the proper inversion settings 49
4.4 Using the model sensitivity and uncertainty values 58
4.5 Methods to handle topography 62
4.6 Incorporating information from borehole logs and seismic surveys 64
4.7 Model refinement 68
4.8 Pitfalls in 2-D resistivity surveys and inversion 70
5 IP inversion 77
5.1 Introduction 77
5.2 The IP effect 77
5.3 IP data types 79
6 Cross-borehole imaging 82
6.1 Introduction 82
6.2 Electrode configurations for cross-borehole surveys 82
6.2.1. Two electrodes array – the pole-pole 82
6.2.2 Three electrodes array – the pole-bipole 82
6.2.3 Four electrodes array – the bipole-bipole 86


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6.3 Single borehole surveys 89
7 2-D field examples 91
7.1 Introduction 91
7.2. Landslide - Cangkat Jering, Malaysia 91
7.3 Old Tar Works - U.K. 92
7.4 Holes in clay layer - U.S.A. 92
7.5 Time-lapse water infiltration survey - U.K. 94
7.6 Pumping test, U.K. 96
7.7 Wenner Gamma array survey – Nigeria 97
7.8 Mobile underwater survey – Belgium 98
7.9 Floating electrodes survey, U.S.A. 99
8 3-D Electrical Imaging Surveys 101
8.1 Introduction to 3-D surveys 101
8.2 Array types for 3-D surveys 101
8.2.1 The pole-pole array 101
8.2.2 The pole-dipole array 104
8.2.3 The dipole-dipole, Wenner and Schlumberger arrays 104
8.2.4 Summary of array types 106
8.3 3-D roll-along techniques 110
8.4 A 3-D forward modeling program 110
8.5 3-D inversion algorithms and 3-D data sets 114
8.6 A 3-D inversion program 115
8.6 Examples of 3-D field surveys 116
8.7.1 Birmingham field test survey - U.K. 116
8.7.2 Sludge deposit – Sweden 117
8.7.3 Copper Hill – Australia 118
Acknowledgments 123
References 124


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List of Figures

Figure Page Number
1.1 The flow of current from a point current source and the
resulting potential distribution. 3
1.2 The potential distribution caused by a pair of current electrodes 1
metre apart with a current of 1 ampere and a homogeneous
half-space with resistivity of 1 Ω⋅m. 3
1.3 A conventional array with four electrodes to measure the subsurface
resistivity. 4
1.4 Common arrays used in resistivity surveys and their geometric factors. 4
1.5 The resistivity of rocks, soils and minerals. 6
1.6 The three different models used in the interpretation of resistivity
measurements. 7
1.7 A typical 1-D model used in the interpretation of resistivity
sounding data for the Wenner array. 7
1.8 A 2-D two-layer model with a low resistivity prism in the upper layer. 10
1.9 Apparent resistivity curves for a 2-D model with a lateral inhomogeneity. 11
2.1 The arrangement of electrodes for a 2-D electrical survey and the
sequence of measurements used to build up a pseudosection. 17
2.2 The use of the roll-along method to extend the area covered by a survey. 18
2.3 Sketch outline of the ABEM Lund Imaging System. 20
2.4 The Aarhus Pulled Array System. 22
2.5 The Geometrics OhmMapper system using capacitive coupled electrodes. 22
2.6 Schematic diagram of a possible mobile underwater survey system 22
2.7 The apparent resistivity pseudosections from 2-D imaging surveys
with different arrays over a rectangular prism. 23
2.8 The parameters in the sensitivity function calculation at a point
(x,y,z) within a half-space. 26
2.9 A plot of the 1-D sensitivity function. 26
2.10 2-D sensitivity sections for the Wenner array. 30
2.11 2-D sensitivity sections for the dipole-dipole array 32
2.12 Two possible different arrangements for a dipole-dipole array
measurement. 33
2.13 2-D sensitivity sections for the Wenner-Schlumberger array. 34
2.14 A comparison of the (i) electrode arrangement and (ii) pseudosection
data pattern for the Wenner and Wenner-Schlumberger arrays. 35
2.15 The pole-pole array 2-D sensitivity section. 36
2.16 The forward and reverse pole-dipole arrays. 37
2.17 The pole-dipole array 2-D sensitivity sections. 37
2.18 The apparent resistivity pseudosection for the dipole–dipole array
using overlapping data levels over a rectangular prism. 39
2.19 Cumulative sensitivity sections for different measurement configurations 41
3.1 The output from the RES2DMOD software for the
SINGLE_BLOCK.MOD 2-D model file. 43
4.1 An example of a field data set with a few bad data points. 47
4.2 Selecting the menu option to remove bad data points manually. 48


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4.3 Error distribution bar chart from a trial inversion of the Grundfor
Line 1 data set with five bad data points. 49
4.4 Different options to modify the inversion process. 50
4.5 Example of inversion results using the smoothness-constrain and
robust inversion model constrains. 51
4.6 The different options for the Jacobian matrix calculation. 52
4.7 Different methods to subdivide the subsurface into rectangular prisms
in a 2-D model. 53
4.8. The options to change the thickness of the model layers. 54
4.9 The options under the ‘Change Settings’ menu selection. 56
4.10 The dialog box to limit the model resistivity values. 57
4.11 Landfill survey example (Wenner array). 60
4.12 Landfill survey depth of investigation determination. 61
4.13 Beach survey example in Denmark. 61
4.14 Different methods to incorporate topography into a 2-D inversion model. 63
4.15 Fixing the resistivity of rectangular and triangular regions of the
inversion model. 66
4.16 The inversion model cells with fixed regions. 67
4.17 Example of an inversion model with specified sharp boundaries. 67
4.18 The effect of cell size on the model misfit for near surface inhomogeneities. 68
4.19 Synthetic model (c) used to generate test apparent resistivity data for the
pole-dipole (a) and Wenner (b) arrays. 69
4.21 Example of the use of narrower model cells with the Wenner-Schlumberger array. 71
4.22 An example of 3-D effects on a 2-D survey. 73
4.23 Thee 2-D sensitivity sections for the pole-dipole array with a dipole length of
1 meter and with (a) n=6, (b) n=12 and (c) n=18. 75
4.24 Example of apparent resistivity pseudosection with pole-dipole array with
large ‘n’ values. 76
5.1 The IP values for some rocks and minerals. 78
5.2 The Cole-Cole model. 78
6.1 The possible arrangements of the electrodes for the pole-pole
array in the cross-borehole survey and the 2-D sensitivity sections. 83
6.2 A schematic diagram of two electrodes below the surface. 84
6.3 The 2-D sensitivity pattern for various arrangements with the
pole-bipole array. 85
6.4 The 2-D sensitivity patterns for various arrangements of the
bipole-bipole array. 86
6.5 Possible measurement sequences using the bipole-bipole array. 88
6.6 Several possible bipole-bipole configurations with a single borehole. 90
6.7 A pole-bipole survey with a single borehole. 90
7.1 Landslide field example, Malaysia. 91
7.2 Industrial pollution example, U.K. 92
7.3 Mapping of holes in a clay layer, U.S.A. 93
7.4 Water infiltration mapping, U.K. 94
7.5 Time-lapse sections from the infiltration study. 95
7.6. Hoveringham pumping test, U.K. 96
7.7. Percentage relative change in the subsurface resistivity values for the
Hoveringham pumping test. 96
7.8. Use of Archie’s Law for the Hoveringham pumping test. 97
7.9 Groundwater survey, Nigeria. 98


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7.10 Underwater riverbed survey, Belgium. 99
7.11. Thames River (CT, USA) survey with floating electrodes. 100
8.1 The arrangement of the electrodes for a 3-D survey. 102
8.2 Two possible measurement sequences for a 3-D survey. 102
8.3 3-D sensitivity plots for the pole-pole array. 103
8.4. 3-D sensitivity plots for the pole-dipole array with n=1 in the form of
horizontal slices through the earth at different depths. 105
8.5 3-D sensitivity plots for the pole-dipole array with n=4 in the form of
horizontal slices through the earth at different depths. 105
3-D sensitivity plots for the dipole-dipole array with n=1 in the
8.6 of horizontal slices through the earth at different depths. 107
8.7 3-D sensitivity plots for the dipole-dipole array with n=4 in the
form of horizontal slices through the earth at different depths. 107
8.8 The 3-D sensitivity plots for the Wenner alpha array at different depths. 108
8.9. The 3-D sensitivity plots for the Wenner-Schlumberger array with
n=4 at different depths. 108
8.10 The 3-D sensitivity plots for the Wenner gamma array at different depths. 109
8.11 Using the roll-along method to survey a 10 by 10 grid with a
multi-electrode system with 50 nodes. 111
8.12 A 3-D model with 4 rectangular prisms in a 15 by 15 survey grid. 113
8.13 The models used in 3-D inversion. 115
8.14 Arrangement of electrodes in the Birmingham 3-D field survey. 117
8.15 Horizontal and vertical cross-sections of the model obtained from
the inversion of the Birmingham field survey data set. 118
8.16 The 3-D model obtained from the inversion of the Lernacken Sludge
deposit survey data set displayed as horizontal slices through the earth. 119
8.17 3-D view of the model obtained from the inversion of the Lernacken
Sludge deposit survey data set displayed with the Slicer/Dicer program. 120
8.18 Geological map of the Copper Hill area (after Chivas and Nutter). 121
8.19 Electrodes layout used for the 3-D survey of the Copper Hill area. 121
8.19 The IP model obtained from the inversion of the Copper Hill survey data set. 122


viii

List of Tables

Table Page Number
1.1 1-D inversion examples using the RES1D.EXE program. 9
2.1 The median depth of investigation (ze) for the different arrays. 27
3.1 Forward modeling examples. 44
4.1 Methods to remove bad data points. 48
4.2 Tests with different inversion options 57
4.3 Tests with different topographic modeling options. 64
4.4 Tests with option to fix the model resistivity. 66
5.1 Tests with 2-D IP inversion. 81
6.1 A few borehole inversion tests. 88
8.1 3-D forward modeling examples. 112
8.2 3-D inversion examples. 116


1

1 Introduction to resistivity surveys
1.1 Basic resistivity theory
The purpose of electrical surveys is to determine the subsurface resistivity distribution
by making measurements on the ground surface. From these measurements, the true
resistivity of the subsurface can be estimated. The ground resistivity is related to various
geological parameters such as the mineral and fluid content, porosity and degree of water
saturation in the rock. Electrical resistivity surveys have been used for many decades in
hydrogeological, mining and geotechnical investigations. More recently, it has been used for
environmental surveys.
The fundamental physical law used in resistivity surveys is Ohm’s Law that governs
the flow of current in the ground. The equation for Ohm’s Law in vector form for current
flow in a continuous medium is given by
J=σE (1.1)
where σ is the conductivity of the medium, J is the current density and E is the electric field
intensity. In practice, what is measured is the electric field potential. We note that in
geophysical surveys the medium resistivity ρ, which is equals to the reciprocal of the
ρ σ
conductivity (ρ=1/σ), is more commonly used. The relationship between the electric potential
and the field intensity is given by
E = − ∇Φ (1.2)
Combining equations (1.1) and (1.2), we get
J = − σ∇Φ (1.3)
In almost all surveys, the current sources are in the form of point sources. In this case, over
an elemental volume ∆V surrounding the a current source I, located at ( x s , y s , z s ) the
relationship between the current density and the current (Dey and Morrison 1979a) is given
by
 I 
∇.J =   δ( x − x s ) δ( y − y s ) δ( z − z s ) (1.4)
 ∆V 
where δ is the Dirac delta function. Equation (3) can then be rewritten as
 I 
− ∇ • [σ(x, y, z )∇φ( x, y, z )] =   δ( x − x s ) δ( y − y s ) δ( z − z s ) (1.5)
 ∆V 
This is the basic equation that gives the potential distribution in the ground due to a
point current source. A large number of techniques have been developed to solve this
equation. This is the “forward” modeling problem, i.e. to determine the potential that would
be observed over a given subsurface structure. Fully analytical methods have been used for
simple cases, such as a sphere in a homogenous medium or a vertical fault between two areas
each with a constant resistivity. For an arbitrary resistivity distribution, numerical techniques
are more commonly used. For the 1-D case, where the subsurface is restricted to a number of
horizontal layers, the linear filter method is commonly used (Koefoed 1979). For 2-D and 3-
D cases, the finite-difference and finite-element methods are the most versatile. In Chapter 2,
we will look at the use of a forward modeling computer program for 2-D structures.
The more complicated cases will be examined in the later sections. First, we start with
the simplest case with a homogeneous subsurface and a single point current source on the
ground surface (Figure 1.1). In this case, the current flows radially away from the source, and
the potential varies inversely with distance from the current source. The equipotential
surfaces have a hemisphere shape, and the current flow is perpendicular to the equipotential
surface. The potential in this case is given by


2

ρI
φ= (1.6)
2πr
where r is the distance of a point in the medium (including the ground surface) from the
electrode. In practice, all resistivity surveys use at least two current electrodes, a positive
current and a negative current source. Figure 1.2 show the potential distribution caused by a
pair of electrodes. The potential values have a symmetrical pattern about the vertical place at
the mid-point between the two electrodes. The potential value in the medium from such a pair
is given by
ρI  1 1 
φ=  −  (1.7)
2π  rC1 rC 2 
 
where rC1 and rC2 are distances of the point from the first and second current electrodes.
In practically all surveys, the potential difference between two points (normally on the
ground surface) is measured. A typical arrangement with 4 electrodes is shown in Figure 1.3.
The potential difference is then given by
ρI  1 1 1 1 
∆φ =  − − + 
2π  rC1P1 rC 2 P1 rC1P 2 rC 2 P 2 
(1.8)
 
The above equation gives the potential that would be measured over a homogenous half space
with a 4 electrodes array.
Actual field surveys are invariably conducted over an inhomogenous medium where
the subsurface resistivity has a 3-D distribution. The resistivity measurements are still made
by injecting current into the ground through the two current electrodes (C1 and C2 in Figure
1.3), and measuring the resulting voltage difference at two potential electrodes (P1 and P2).
From the current (I) and potential ( ∆φ ) values, an apparent resistivity (ρa) value is
calculated.
∆φ
ρa = k (1.9)
I
2π
where k=
 1 1 1 1 

r − − + 
 C1P1 rC 2 P1 rC1P 2 rC 2 P 2 

k is a geometric factor that depends on the arrangement of the four electrodes. Resistivity
measuring instruments normally give a resistance value, R = ∆φ/I, so in practice the apparent

resistivity value is calculated by
ρa = k R (1.10)
The calculated resistivity value is not the true resistivity of the subsurface, but an
“apparent” value that is the resistivity of a homogeneous ground that will give the same
resistance value for the same electrode arrangement. The relationship between the “apparent”
resistivity and the “true” resistivity is a complex relationship. To determine the true
subsurface resistivity from the apparent resistivity values is the “inversion” problem.
Methods to carry out such an inversion will be discussed in more detail at the end of this
chapter.
Figure 1.4 shows the common arrays used in resistivity surveys together with their
geometric factors. In a later section, we will examine the advantages and disadvantages of


3

some of these arrays.
There are two more electrical based methods that are closely related to the resistivity
method. They are the Induced Polarization (IP) method, and the Spectral Induced Polarization
(SIP) (also known as Complex Resistivity (CR)) method. Both methods require measuring
instruments that are more sensitive than the normal resistivity method, as well has
significantly higher currents. IP surveys are comparatively more common, particularly in
mineral exploration surveys. It is able to detect conductive minerals of very low
concentrations that might otherwise be missed by resistivity or EM surveys. Commercial SIP
surveys are comparatively rare, although it is a popular research subject. Both IP and SIP
surveys use alternating currents (in the frequency domain) of much higher frequencies than
standard resistivity surveys. Electromagnetic coupling is a serious problem in both methods.
To minimize the electromagnetic coupling, the dipole-dipole (or pole-dipole) array is
commonly used.

Figure 1.1. The flow of current from a point current source and the resulting potential
distribution.

Figure 1.2. The potential distribution caused by a pair of current electrodes 1 meter apart with
a current of 1 ampere and a homogeneous half-space with resistivity of 1 Ω⋅m.


4

Figure 1.3. A conventional array with four electrodes to measure the subsurface resistivity.

Figure 1.4. Common arrays used in resistivity surveys and their geometric factors. Note that
the dipole-dipole, pole-dipole and Wenner-Schlumberger arrays have two parameters, the
dipole length “a” and the dipole separation factor “n”. While the “n” factor is commonly an
integer value, non-integer values can also be used.


5

1.2 Electrical properties of earth materials
Electric current flows in earth materials at shallow depths through two main methods.
They are electronic conduction and electrolytic conduction. In electronic conduction, the
current flow is via free electrons, such as in metals. In electrolytic conduction, the current
flow is via the movement of ions in groundwater. In environmental and engineering surveys,
electrolytic conduction is probably the more common mechanism. Electronic conduction is
important when conductive minerals are present, such metal sulfides and graphite in mineral
surveys.
The resistivity of common rocks, soil materials and chemicals (Keller and
Frischknecht 1966, Daniels and Alberty 1966, Telford et al. 1990) is shown in Figure 1.5.
Igneous and metamorphic rocks typically have high resistivity values. The resistivity of these
rocks is greatly dependent on the degree of fracturing, and the percentage of the fractures
filled with ground water. Thus a given rock type can have a large range of resistivity, from
about 1000 to 10 million Ω⋅m, depending on whether it is wet or dry. This characteristic is
useful in the detection of fracture zones and other weathering features, such as in engineering
and groundwater surveys.
Sedimentary rocks, which are usually more porous and have higher water content,
normally have lower resistivity values compared to igneous and metamorphic rocks. The
resistivity values range from 10 to about 10000 Ω⋅m, with most values below 1000 Ω⋅m. The
resistivity values are largely dependent on the porosity of the rocks, and the salinity of the
contained water.
Unconsolidated sediments generally have even lower resistivity values than
sedimentary rocks, with values ranging from about 10 to less than 1000 Ω⋅m. The resistivity
value is dependent on the porosity (assuming all the pores are saturated) as well as the clay
content. Clayey soil normally has a lower resistivity value than sandy soil. However, note the
overlap in the resistivity values of the different classes of rocks and soils. This is because the
resistivity of a particular rock or soil sample depends on a number of factors such as the
porosity, the degree of water saturation and the concentration of dissolved salts.
The resistivity of groundwater varies from 10 to 100 Ω⋅m. depending on the
concentration of dissolved salts. Note the low resistivity (about 0.2 Ω⋅m) of seawater due to
the relatively high salt content. This makes the resistivity method an ideal technique for
mapping the saline and fresh water interface in coastal areas. One simple equation that gives
the relationship between the resistivity of a porous rock and the fluid saturation factor is
Archie’s Law. It is applicable for certain types of rocks and sediments, particularly those that
have a low clay content. The electrical conduction is assumed to be through the fluids filling
the pores of the rock. Archie's Law is given by
ρ = aρ wφ − m (1.11)
where ρ is the rock resistivity, ρw is fluid resistivity, φ is the fraction of the rock filled with
the fluid, while a and m are two empirical parameters (Keller and Frischknecht 1966). For
most rocks, a is about 1 while m is about 2. For sediments with a significant clay content,
other more complex equations have been proposed (Olivar et al. 1990).
The resistivities of several types of ores are also shown. Metallic sulfides (such as
pyrrhotite, galena and pyrite) have typically low resistivity values of less than 1 Ω⋅m. Note
that the resistivity value of a particular ore body can differ greatly from the resistivity of the
individual crystals. Other factors, such as the nature of the ore body (massive or
disseminated) have a significant effect. Note that graphitic slate have a low resistivity value,
similar to the metallic sulfides, which can give rise to problems in mineral surveys. Most
oxides, such as hematite, do not have a significantly low resistivity value. One of exceptions
is magnetite.


6

The resistivity values of several industrial contaminants are also given in Figure 1.5.
Metals, such as iron, have extremely low resistivity values. Chemicals that are strong
electrolytes, such as potassium chloride and sodium chloride, can greatly reduce the
resistivity of ground water to less than 1 Ω⋅m even at fairly low concentrations. The effect of
weak electrolytes, such as acetic acid, is comparatively smaller. Hydrocarbons, such as
xylene (6.998x1016 Ω⋅m), typically have very high resistivity values. However, in practice the
percentage of hydrocarbons in a rock or soil is usually quite small, and might not have a
significant effect on the bulk resistivity. As an example, oil sands in Figure 1.5 have the same
range of resistivity values as alluvium.

Figure 1.5. The resistivity of rocks, soils and minerals.

1.3 1-D resistivity surveys and inversions – applications, limitations and pitfalls
The resistivity method has its origin in the 1920’s due to the work of the
Schlumberger brothers. For approximately the next 60 years, for quantitative interpretation,
conventional sounding surveys (Koefoed 1979) were normally used. In this method, the
center point of the electrode array remains fixed, but the spacing between the electrodes is
increased to obtain more information about the deeper sections of the subsurface.
The measured apparent resistivity values are normally plotted on a log-log graph
paper. To interpret the data from such a survey, it is normally assumed that the subsurface
consists of horizontal layers. In this case, the subsurface resistivity changes only with depth,
but does not change in the horizontal direction. A one-dimensional model of the subsurface is
used to interpret the measurements (Figure 1.6a). Figure 1.7 shows an example of the data
from a sounding survey and a possible interpretation model. This method has given useful


7

results for geological situations (such the water-table) where the one-dimensional model is
approximately true.
The software provided, RES1D.EXE, is a simple inversion and forward modeling
program for 1-D models that consists of horizontal layers. Besides normal resistivity surveys,
the program will enable you to model IP as well as SIP data. In the software package, several
files with extensions of DAT are example data files with resistivity sounding data. Files with
the MOD extension are model files that can be used to generate synthetic data for the
inversion part of the program. As a first try, read in the file WENNER3.DAT that contains
the Wenner array sounding data for a simple 3-layer model.

Figure 1.6. The three different models used in the interpretation of resistivity measurements.

Figure 1.7. A typical 1-D model used in the interpretation of resistivity sounding data for the
Wenner array.


8

The greatest limitation of the resistivity sounding method is that it does not take into
account lateral changes in the layer resistivity. Such changes are probably the rule rather than
the exception. The failure to include the effect of such lateral changes can results in errors in
the interpreted layer resistivity and/or thickness. As an example, Figure 1.8 shows a 2-D
model where the main structure is a two-layer model with a resistivity of 10 Ω⋅m and a
thickness of 5 meters for the upper layer, while the lower layer has a resistivity of 100 Ω⋅m.
To the left of the center point of the survey line, a low resistivity prism of 1 Ω⋅m is added in
the upper layer to simulate a lateral inhomogeneity. The 2-D model has 144 electrodes that
are 1 meter apart. The apparent resistivity pseudosections for the Wenner and Schlumberger
array are also shown. For the Schlumberger array, the spacing between the potential
electrodes is fixed at 1.0 meter for the apparent resistivity values shown in the pseudosection.
The sounding curves that are obtained with conventional Wenner and Schlumberger array
sounding surveys with the mid-point at the center of the line are also shown in Figure 1.9. In
the 2-D model, the low resistivity rectangular prism extends from 5.5 to 18.5 meters to the
left of the sounding mid-point. The ideal sounding curves for both arrays for a two-layer
model (i.e. without the low resistivity prism) are also shown for comparison. For the Wenner
array, the low resistivity prism causes the apparent resistivity values in the sounding curve
(Figure 1.9a) to be too low for spacing values of 2 to 9 meters and for spacings larger than 15
meters. At spacings between 9 to 15 meters, the second potential electrode P2 crosses over
the low resistivity prism. This causes the apparent resistivity values to approach the two-layer
model sounding curve. If the apparent resistivity values from this model are interpreted using
a conventional 1-D model, the resulting model could be misleading. In this case, the sounding
data will most likely to be interpreted as a three-layer model.
The effect of the low resistivity prism on the Schlumberger array sounding curve is
slightly different. The apparent resistivity values measured with a spacing of 1 meter between
the central potential electrodes are shown by black crosses in Figure 1.9b. For electrode
spacings (which is defined as half the total length of the array for the Schlumberger array) of
less than 15 meters, the apparent resistivity values are less than that of the two-layer sounding
curve. For spacings greater than 17 meters, the apparent resistivity values tend to be too high.
This is probably because the low resistivity prism lies to the right of the C2 electrode (i.e.
outside the array) for spacings of less than 15 meters. For spacings of greater than 17 meters,
it lies between the P2 and C2 electrodes. Again, if the data is interpreted using a 1-D model,
the results could be misleading. One method that has been frequently recommended to
“remove” the effect of lateral variations with the Schlumberger array is by shifting curve
segments measured with different spacings between the central potential electrodes. The
apparent resistivity values measured with a spacing of 3 meters between the potential
electrodes are also shown in Figure 1.9b. The difference in the sounding curves with the
spacings of 1 meter and 3 meters between the potential electrodes is small, particularly for
large electrode spacings. Thus any shifting in the curve segments would not remove the
distortion in the sounding curve due to the low resistivity prism. The method of shifting the
curve segments is probably more applicable if the inhomogeneity lies between the central
potential electrodes, and probably ineffective if the inhomogeneity is beyond the largest
potential electrodes spacing used (which is the case in Figure 1.8). However, note that the
effect of the prism on the Schlumberger array sounding curve is smaller at the larger
electrode spacings compared with the Wenner array (Figure 1.9). The main reason is
probably the larger distance between the P2 and C2 electrodes in the Schlumberger array.
A more reliable method to reduce the effect of lateral variations on the sounding data
is the offset Wenner method (Barker 1978). It makes use of the property that the effect of an
inhomogeneity on the apparent resistivity value is of opposite sign if it lies between the two
potential electrodes or if it is between a potential and a current electrode. For the example


9

shown in Figure 1.8, if the low resistivity body lies in between a current and potential
electrode (the P2 and C2 electrodes in this case), the measured apparent resistivity value
would be lower. If the low resistivity body lies in between the P1 and P2 electrodes, it will
cause the apparent resistivity value to be higher. The reason for this phenomenon can be
found in the sensitivity pattern for the Wenner array (see Figure 2.10a). By taking
measurements with different positions for the mid-point of the array, the effect of the low
resistivity body can be reduced.
Another classical survey technique is the profiling method. In this case, the spacing
between the electrodes remains fixed, but the entire array is moved along a straight line. This
gives some information about lateral changes in the subsurface resistivity, but it cannot detect
vertical changes in the resistivity. Interpretation of data from profiling surveys is mainly
qualitative.
The most severe limitation of the resistivity sounding method is that horizontal (or
lateral) changes in the subsurface resistivity are commonly found. The ideal situation shown
in Figure 1.6a is rarely found in practice. As shown by the examples in Figures 1.8 and 1.9,
lateral changes in the subsurface resistivity will cause changes in the apparent resistivity
values that might be, and frequently are, misinterpreted as changes with depth in the
subsurface resistivity. In many engineering and environmental studies, the subsurface
geology is very complex where the resistivity can change rapidly over short distances. The 1-
D resistivity sounding method might not be sufficiently accurate for such situations.
To use the RES1D.EXE program for the exercises in the table below, as well as the
other programs that we shall use in the later sections, follows the usual sequence used by
Windows 98/Me/2000/NT. Click the ‘Start’ button, followed by ‘Programs’ and the look for
the RES1D folder in the list of installed programs. Alternatively, you can create a shortcut
icon on the Windows Desktop.
Exercise 1.1 : 1-D inversion examples using the RES1D.EXE program.
Data set and purpose Things to try
WENNER3.DAT – A simple (1). Read in the file, and then run the “Carry out inversion”
synthetic data file for a 3 layer step.
model.
WENN_LATERAL.DAT and (1). Read in the files, and then invert the data sets.
SCHL_LATER.DAT – (2). Compare the results with the true two-layer model (that
Wenner and Schlumberger has resistivities of 10 Ω⋅m and 100 Ω⋅m for the first and
array sounding data shown in second layers, and thickness of 5 meters for the first layer).
Figure 1.9 that are extracted
from the 2-D pseudosections.
WENOFFSET.DAT – A field (1). Read in the files, and then invert the data set.
data set collected using the
offset Wenner method.
IPTESTM.DAT – A 1-D (1). Read in the files, and then invert the data set.
sounding data file with IP
measurements as well to round
things up.

To obtain a more accurate subsurface model than is possible with a simple 1-D model,
a more complex model must be used. In a 2-D model (Figure 1.6b), the resistivity values are


10

allowed to vary in one horizontal direction (usually referred to as the x direction) but assumed
to be constant in the other horizontal (the y) direction. A more realistic model would be a
fully 3-D model (Figure 1.6c) where the resistivity values are allowed to change in all 3
directions. The use of 2-D and 3-D surveys and interpretation techniques will be examined in
detail in the following sections.

Figure 1.8. A 2-D two-layer model with a low resistivity prism in the upper layer. The
calculated apparent resistivity pseudosections for the (a) Wenner and (b) Schlumberger
arrays. (c) The 2D model. The mid-point for a conventional sounding survey is also shown.


11

Figure 1.9. Apparent resistivity sounding curves for a 2-D model with a lateral
inhomogeneity. (a) The apparent resistivity curve extracted from the 2D pseudosection for
the Wenner array. The sounding curve for a two-layer model without the low resistivity prism
is also shown by the black line curve. (b) The apparent resistivity curves extracted from the
2-D pseudosection for the Schlumberger array with a spacing of 1.0 meter (black crosses) and
3.0 meters (red crosses) between the potential electrodes. The sounding curve for a two-layer
model without the low resistivity prism is also shown.


12

1.4 Basic Inverse Theory
In geophysical inversion, we seek to find a model that gives a response that is similar
to the actual measured values. The model is an idealized mathematical representation of a
section of the earth. The model has a set of model parameters that are the physical quantities
we want to estimate from the observed data. The model response is the synthetic data that
can be calculated from the mathematical relationships defining the model for a given set of
model parameters. All inversion methods essentially try to determine a model for the
subsurface whose response agrees with the measured data subject to certain restrictions. In
the cell-based method used by the RES2DINV and RES3DINV programs, the model
parameters are the resistivity values of the model cells, while the data is the measured
apparent resistivity values. The mathematical link between the model parameters and the
model response for the 2-D and 3-D resistivity models is provided by the finite-difference
(Dey and Morrison 1979a, 1979b) or finite-element methods (Silvester and Ferrari 1990).
In all optimization methods, an initial model is modified in an iterative manner so that
the difference between the model response and the observed data values is reduced. The set
of observed data can be written as a column vector y given by
y = col( y1 , y 2 ,....., y m ) (1.12)
where m is the number of measurements. The model response f can be written in a similar
form.
f = col( f1 , f 2 ,....., f m ) (1.13)
For resistivity problems, it is a common practice to use the logarithm of the apparent
resistivity values for the observed data and model response, and the logarithm of the model
values as the model parameters. The model parameters can be represented by the following
vector
q = col(q1 , q2 ,....., qn ) (1.14)
where n is the number of model parameters. The difference between the observed data and
the model response is given by the discrepancy vector g that is defined by
g=y-f (1.15)
In the least-squares optimization method, the initial model is modified such that the
sum of squares error E of the difference between the model response and the observed data
values is minimized.
n
E = g T g = ∑ g i2 (1.16)
i =1
To reduce the above error value, the following Gauss-Newton equation is used to
determine the change in the model parameters that should reduce the sum squares error
(Lines and Treitel 1984).
J T J ∆q i = J T g (1.17)
where ∆q is the model parameter change vector, and J is the Jacobian matrix (of size m by n)
of partial derivatives. The elements of the Jacobian matrix are given by
∂f
J ij = i (1.18)
∂q j
that is the change in the ith model response due to a change in the jth model parameter. After
calculating the parameter change vector, a new model is obtained by
q k +1 = q k + ∆q k (1.19)
In practice, the simple least-squares equation (1.17) is rarely used by itself in
geophysical inversion. In some situations the matrix product J T J might be singular, and thus
the least-squares equation does not have a solution for ∆q. Another common problem is that


13

the matrix product J T J is nearly singular. This can occur if a poor initial model that is very
different from the optimum model is used. The parameter change vector calculated using
equation (1.17) can have components that are too large such that the new model calculated
with (1.19) might have values that are not realistic. One common method to avoid this
problem is the Marquardt-Levenberg modification (Lines and Treitel 1984) to the Gauss-
Newton equation that is given by
( )
J T J + λI ∆q k = J T g (1.20)
where I is the identity matrix. The factor λ is known as the Marquardt or damping factor, and
this method is also known as the ridge regression method (Inman 1975). The damping factor
effectively constrains the range of values that the components of parameter change vector can
∆q take. While the Gauss-Newton method in equation (1.17) attempts to minimize the sum of
squares of the discrepancy vector only, the Marquardt-Levenberg method modification also
minimizes a combination of the magnitude of the discrepancy vector and the parameter
change vector. This method has been successfully used in the inversion of resistivity
sounding data where the model consists of a small number of layers. For example, it was used
in the inversion of the resistivity sounding example in Figure 1.7 with three layers (i.e. five
model parameters). However when the number of model parameters is large, such as in 2D
and 3D inversion model that consist of a large number of small cells, the model produced by
this method can have an erratic resistivity distribution with spurious high or low resistivity
zones (Constable et al. 1987). To overcome this problem, the Gauss-Newton least-squares
equation is further modified so as to minimize the spatial variations in the model parameters
(i.e. the model resistivity values change in a smooth or gradual manner). This smoothness-
constrained least-squares method (Ellis and Oldenburg 1994a) has the following
mathematical form.
( )
J T J +λF ∆q k = J T g − λFq k , (1.21)
where F = α x C C x + α y C T C y + α z C T C z
T
x y z

and Cx, Cy and Cz are the smoothing matrices in the x-, y- and z-directions. αx, αy and αz are
the relative weights given to the smoothness filters in the x-, y- and z-directions. One
common form of the smoothing matrix is the first-order difference matrix (please refer to the
paper by deGroot-Hedlin and Constable, 1990 for the details) that is given by

− 1 1 0 0 .. .. .. 0
 0 −1 1 0 .. .. .. 0
 
 0 0 −1 1 0 .. .. 0
 
..
C=  (1.22)
 .. 
 
 .. 
 .. 
 

 0


Equation 1.21 also tries to minimize the square of the spatial changes, or roughness,
of the model resistivity values. It is in fact an l2 norm smoothness-constrained optimization
method. This tends to produce a model with a smooth variation of resistivity values. This
approach is acceptable if the actual subsurface resistivity varies in a smooth and gradational
manner. In some cases, the subsurface geology consists of a number of regions that are
internally almost homogeneous but with sharp boundaries between different regions. For such


14

cases, the inversion formulation in (1.21) can be modified so that it minimizes the absolute
changes in the model resistivity values (Claerbout and Muir 1973). This can sometimes give
significantly better results. Technically this is referred to as an l1 norm smoothness-
constrained optimization method, or more commonly known as a blocky inversion method. A
number of techniques can be used for such a modification. One simple method to implement
an l1 norm based optimization method using the standard least-squares formulation is the
iteratively reweighted least-squares method (Wolke and Schwetlick, 1988). The optimization
equation in (1.21) is modified to

(J T
)
J +λFR ∆q k = J T R d g − λFR q k , (1.23)
T T T
with FR = α x C R m C x + α y C R m C y + α z C R m C z
x y z

where Rd and Rm are weighting matrices introduced so that different elements of the data
misfit and model roughness vectors are given equal weights in the inversion process.
Equation (1.23) provides a general method that can be further modified if necessary to
include known information about the subsurface geology. As an example, if it is known that
the variations in the subsurface resistivity are likely to be confined to a limited zone, the
damping factor values in λ can modified (Ellis and Oldenburg 1994a) such that greater
changes are allowed in that zone.

1.5 2-D model discretisation methods
In the previous section, we have seen that the least-squares method is used to
calculate certain physical characteristics of the subsurface, the “model parameters”, from the
apparent resistivity measurements. The “model parameters” are set by the way we slice and
dice the subsurface into different regions. Figure 1.10 shows various possibilities that can be
used.
The method most commonly used in 2-D (and 3-D) interpretation is a purely cell
based model where the subsurface is subdivided into rectangular cells. The positions of the
cells are fixed and only the resistivities of cells are allowed to vary during the inversion
process. The model parameters is the resistivity of each cell. In the example shown in Figure
1.10, the model parameters are the seventy-two cell resistivity values ρ1 to ρ72.
A radically different approach is a boundary based inversion method. This method
subdivides the subsurface into different regions. The resistivity is assumed to be homogenous
within each region. The resistivity is allowed to change in an arbitrary manner across the
boundaries, and thus it useful in areas with abrupt transitions in the geology. The resistivity
of each region and the depths to the boundaries are changed by the least-squares optimisation
method so that the calculated apparent resistivity values match the observed values. The
“model parameters” for the example shown in Figure 1.10 are the two resistivity values (ρ1
and ρ2) and the depths at five points (z1 to z5) along the boundary that gives a total of seven
parameters. While this method works well for synthetic data from numerical models, for
many field data sets it can lead to unstable results with highly oscillating boundaries
(Olayinka and Yaramanci, 2000). Its greatest limitation is probably the assumption of a
constant resistivity within each region. In particular, lateral changes in the resistivity near the
surface have a very large effect on the measured apparent resistivity values. Since this model
does not take into account such lateral changes, they are often mistakenly modelled as
changes in the depths of the boundaries.
More recent efforts have been in combining the cell based and boundary based
inversion methods (Smith et al., 1999). One such method is the laterally constrained inversion
method (Auken and Christiansen, 2004). In this method, lateral changes (but not vertical


15

changes) are allowed in each region (Figure 1.10c), and abrupt transitions across the
boundaries are also allowed. The “model parameters” for the example in Figure 1.10 are then
the twenty-four resistivity values (ρ1 and ρ24) and the depths at thirteen points (z1 to z13)
along the boundary giving a total of thirty-seven parameters. Information from other sources,
such as borehole or seismic data, can be used to provide an initial estimate of the depth to the
boundary. A common situation is when the depth information is available at only one
borehole. In this case, the initial boundary is usually set a constant depth. The inversion
method then adjusts the depths at a number of points along the boundary during the inversion
process. A smoothness-constraint is applied to minimise changes in the depths between
adjacent points on the same boundary (Smith et al., 1999). This method works particularly
well where the subsurface consists of several sedimentary zones.
A further generalisation of this concept is to allow both vertical and lateral changes
within each region (as in a pure cell based model) while also allowing sharp changes across
the boundaries (Figure 1.10d). The model shown in Figure 1.10d has seventy-two resistivity
values and five depth values, giving a total of seventy-seven model parameters. This type of
discretisation is particularly useful where near-surface inhomogeneities that occur at different
depths within the top layer have a large effect on the measured apparent resistivity values.
The model cells used so far has rectangular shapes (Figures 1.10a, 1.10c and 1.10d). This is
partly due to the use of the finite-difference method in calculating the model apparent
resistivity values. A slight disadvantage is that the boundary is approximated by a series of
rectangular steps. Figure 1.10e shows a possible variation using the finite-element method
with trapezoidal cells where the edges of the cells adjacent to the boundary are adjusted so as
to conform to the true shape of the boundary.

Figure 1.10. The different models for the subsurface used in the interpretation of data from 2-
D electrical imaging surveys. (a) A purely cell based model. (b) A purely boundary based
model. (c) The laterally constrained model. (d) A combined cell based and boundary based
model with rectangular cells, and (e) with boundary conforming trapezoidal cells.


16

2 2-D electrical surveys – Data acquisition, presentation and
arrays
2.1 Introduction
We have seen that the greatest limitation of the resistivity sounding method is that it
does not take into account horizontal changes in the subsurface resistivity. A more accurate
model of the subsurface is a two-dimensional (2-D) model where the resistivity changes in
the vertical direction, as well as in the horizontal direction along the survey line. In this case,
it is assumed that resistivity does not change in the direction that is perpendicular to the
survey line. In many situations, particularly for surveys over elongated geological bodies, this
is a reasonable assumption. In theory, a 3-D resistivity survey and interpretation model
should be even more accurate. However, at the present time, 2-D surveys are the most
practical economic compromise between obtaining very accurate results and keeping the
survey costs down (Dahlin 1996). Typical 1-D resistivity sounding surveys usually involve
about 10 to 20 readings, while 2-D imaging surveys involve about 100 to 1000
measurements. In comparison, a 3-D survey might involve several thousand measurements.
The cost of a typical 2-D survey could be several times the cost of a 1-D sounding
survey, and is probably comparable with a seismic refraction survey. In many geological
situations, 2-D electrical imaging surveys can give useful results that are complementary to
the information obtained by other geophysical method. For example, seismic methods can
map undulating interfaces well, but will have difficulty (without using advanced data
processing techniques) in mapping discrete bodies such as boulders, cavities and pollution
plumes. Ground radar surveys can provide more detailed pictures but have very limited depth
penetration in areas with conductive unconsolidated sediments, such as clayey soils. Two-
dimensional electrical surveys should be used in conjunction with seismic or GPR surveys as
they provide complementary information about the subsurface.

2.2 Field survey method - instrumentation and measurement procedure
Two-dimensional electrical imaging/tomography surveys are usually carried out using
a large number of electrodes, 25 or more, connected to a multi-core cable (Griffiths and
Barker 1993). A laptop microcomputer together with an electronic switching unit is used to
automatically select the relevant four electrodes for each measurement (Figure 2.1). At
present, field techniques and equipment to carry out 2-D resistivity surveys are fairly well
developed. The necessary field equipment is commercially available from a number of
international companies. These systems typically costs from about US$15,000 upwards.
Some institutions have even constructed “home-made” manually operated switching units at
a nominal cost by using a seismic cable as the multi-core cable!
Figure 2.1 shows the typical setup for a 2-D survey with a number of electrodes along
a straight line attached to a multi-core cable. Normally a constant spacing between adjacent
electrodes is used. The multi-core cable is attached to an electronic switching unit that is
connected to a laptop computer. The sequence of measurements to take, the type of array to
use and other survey parameters (such the current to use) is normally entered into a text file
which can be read by a computer program in a laptop computer. Different resistivity meters
use different formats for the control file, so you will need to refer to the manual for your
system. After reading the control file, the computer program then automatically selects the
appropriate electrodes for each measurement. Some field systems have an in-built
microprocessor system so that a laptop computer is not needed. This could be a significant
advantage for surveys in very rugged terrain.
In a typical survey, most of the fieldwork is in laying out the cable and electrodes.


17

After that, the measurements are taken automatically and stored in the computer. Most of the
survey time is spent waiting for the resistivity meter to complete the set of measurements!
To obtain a good 2-D picture of the subsurface, the coverage of the measurements
must be 2-D as well. As an example, Figure 2.1 shows a possible sequence of measurements
for the Wenner electrode array for a system with 20 electrodes. In this example, the spacing
between adjacent electrodes is “a”. The first step is to make all the possible measurements
with the Wenner array with an electrode spacing of “1a”. For the first measurement,
electrodes number 1, 2, 3 and 4 are used. Notice that electrode 1 is used as the first current
electrode C1, electrode 2 as the first potential electrode P1, electrode 3 as the second
potential electrode P2 and electrode 4 as the second current electrode C2. For the second
measurement, electrodes number 2, 3, 4 and 5 are used for C1, P1, P2 and C2 respectively.
This is repeated down the line of electrodes until electrodes 17, 18, 19 and 20 are used for the
last measurement with “1a” spacing. For a system with 20 electrodes, note that there are 17
(20 - 3) possible measurements with “1a” spacing for the Wenner array.
After completing the sequence of measurements with “1a” spacing, the next sequence
of measurements with “2a” electrode spacing is made. First electrodes 1, 3, 5 and 7 are used
for the first measurement. The electrodes are chosen so that the spacing between adjacent
electrodes is “2a”. For the second measurement, electrodes 2, 4, 6 and 8 are used. This
process is repeated down the line until electrodes 14, 16, 18 and 20 are used for the last
measurement with spacing “2a”. For a system with 20 electrodes, note that there are 14 (20 -
2x3) possible measurements with “2a” spacing.

Figure 2.1. The arrangement of electrodes for a 2-D electrical survey and the sequence of
measurements used to build up a pseudosection.

The same process is repeated for measurements with “3a”, “4a”, “5a” and “6a”
spacings. To get the best results, the measurements in a field survey should be carried out in a
systematic manner so that, as far as possible, all the possible measurements are made. This
will affect the quality of the interpretation model obtained from the inversion of the apparent
resistivity measurements (Dahlin and Loke 1998).


18

Note that as the electrode spacing increases, the number of measurements decreases.
The number of measurements that can be obtained for each electrode spacing, for a given
number of electrodes along the survey line, depends on the type of array used. The Wenner
array gives the smallest number of possible measurements compared to the other common
arrays that are used in 2-D surveys.
The survey procedure with the pole-pole array is similar to that used for the Wenner
array. For a system with 20 electrodes, firstly 19 of measurements with a spacing of “1a” are
made, followed by 18 measurements with “2a” spacing, followed by 17 measurements with
“3a” spacing, and so on.
For the dipole-dipole, Wenner-Schlumberger and pole-dipole arrays (Figure 1.4), the
survey procedure is slightly different. As an example, for the dipole-dipole array, the
measurement usually starts with a spacing of “1a” between the C1-C2 (and also the P1-P2)
electrodes. The first sequence of measurements is made with a value of 1 for the “n” factor
(which is the ratio of the distance between the C1-P1 electrodes to the C1-C2 dipole length),
followed by “n” equals to 2 while keeping the C1-C2 dipole pair spacing fixed at “1a”. When
“n” is equals to 2, the distance of the C1 electrode from the P1 electrode is twice the C1-C2
dipole length. For subsequent measurements, the “n” spacing factor is usually increased to a
maximum value of about 6, after which accurate measurements of the potential are difficult
due to very low potential values. To increase the depth of investigation, the spacing between
the C1-C2 dipole pair is increased to “2a”, and another series of measurements with different
values of “n” is made. If necessary, this can be repeated with larger values of the spacing of
the C1-C2 (and P1-P2) dipole pairs. A similar survey technique can be used for the Wenner-
Schlumberger and pole-dipole arrays where different combinations of the “a” spacing and
“n” factor can be used.
One technique used to extend horizontally the area covered by the survey, particularly
for a system with a limited number of electrodes, is the roll-along method. After completing
the sequence of measurements, the cable is moved past one end of the line by several unit
electrode spacings. All the measurements that involve the electrodes on part of the cable that
do not overlap the original end of the survey line are repeated (Figure 2.2).

Figure 2.2. The use of the roll-along method to extend the area covered by a 2-D survey.


19

2.3 Available field instruments
Over the last 10 years, there has been a steady growth in the number of commercial
companies that offer systems for resistivity imaging surveys. The ones that I have come
across are listed below in alphabetical order together with the web site address where
available.

Manufacturer Instrument Type
Static Dynamic IP
Abem Instruments, Sweden (www.abem.se) x Y
Advanced Geohysical Instruments, USA (www.agiusa.com) x Y
Campus Geophysical Instruments, UK x Y
Geofyzika., Czech Republic (www.geofyzika.com) x
GF Instruments, Czech Republic (www.gfinstruments.cz) x Y
Geometrics, USA (www.geometrics.com) x
Geolog, Germany (www.geolog2000.de) x
IDS Scintrex, Canada (www.idsdetection.com) x Y
Iris Instruments, France (www.iris-instruments.com) x x Y
OYO, Japan x
Pasi Geophysics, Italy (www.pasigeophysics.com) x Y
MAE srl, Italy (www.mae-srl.it) x Y

Most of the above manufacturers have sub-agents in different countries, and there are
probably a few others that I have not come across. A few academic and research institutions
have designed their own systems, for example the British Geological Survey have built an
electrosatic based mobile system.
The instrument type can be divided into two broad categories, static and dynamic
systems. Most instruments are of the static type where many electrodes are connected to a
multi-electrode cable and planted into the ground during the survey. A typical static system is
the Abem Lund system shown in Figure 2.3. One common configuration is a split spread type
of cable connection to the switching unit at the center to reduce the individual cable length
and weight. The weight of a cable roll is directly proportional to the number of nodes and the
spacing between the nodes! A common spacing used for most engineering and environmental
surveys is 5 meters. Most systems come with a minimum of 28 nodes, with some system
having up to 128 nodes or more! The Lund system is a little unusual in that there are 4
individual cables. Most systems use a 2 cables arrangement. The static systems can be further
divided into two sub-categories depending on the arrangement for the switching of the
electrodes. Most of the systems house the switches in a single unit and uses a cable with
many individual wires connected to each node. Typical examples are the Abem Lund and
Campus Geopulse systems. Another arrangement is to have a small switching unit at each
electrode and a cable with the minimum number of wires. One early example is the Campus
MRT system (Griffiths and Turnbull, 1985). A more recent example is the PASI system.
There have been two new and interesting developments in the resistivity meter
systems. One is the addition of I.P. capability. The second is multi-channel measuring
systems. In such a system, a number of potential measurements can be simultaneously made
for a single pair of current electrodes. This could significantly reduce the survey time. With
certain array configurations, a single 2-D survey line could involve thousands of


20

measurements. The major part of the survey time is waiting for a single channel instrument to
complete the measurements that could take more than several hours! The IP and multi-
channel capability are relatively new developments, so you will need to check the
manufacturer's web site to get the latest information.

Figure 2.3. Sketch outline of the ABEM Lund Imaging System. Each mark on the cables
indicates an electrode position (Dahlin 1996). The cables are placed along a single line (the
sideways shift in the figure is only for clarity). This figure also shows the principle of moving
cables when using the roll-along technique. The total layout length depends on the spacing
between the nodes, but is usually between 160 meters and 800 meters.

Static systems use a large number of nodes to get a wide data coverage. In contrast,
dynamic systems use a small number of nodes but move the entire system to obtain a wide
coverage. An example of such a system designed by Aarhus University in Denmark
(Sorenson 1996) is shown in Figure 2.4. A 100 meters cable with nine heavy cylindrical
electrodes is pulled by a small vehicle. Two of the electrodes are used as current electrodes,
while six of them are used for the potential measurements and one is used as a ground
electrode. This system relies on the current being injected into the ground by direct contact,
so it can only be used in open ground, such as farmlands in Northern Europe. A Wenner-
Schlumberger type of arrangement (Figure 1.4) is used but with non-integer “n” values for
some of the measurements. Another mobile system that does not require direct contact with
the ground but uses capacitive coupling (Gerard and Tabbagh 1991, Shima et al. 1996,
Panissod et al. 1998) to induce the flow of current in the ground. This system can be used in
areas that are paved, such as roads and city areas. One such system shown in Figure 2.5 is the
Geometrics OhmMapper system where a cable with 4 to 6 electrodes attached to a measuring
unit is pulled by a single operator. The dipole-dipole type of arrangement is used but with
again with non-integer “n” values for some measurements.
One of main problems faced by mobile systems on land to get sufficient current to
flow into the ground. Direct contact systems such as the Aarhus Pulled Array System can
only be used in areas with open ground. The capacitive coupling type does not require direct
ground contact and thus can be used in many areas where normal resistivity surveying
systems cannot be used (for example in built-up areas) but has the problem of a more limited


21

depth of penetration due to the limited amount of current that can be induced into the ground
compared to direct contact systems. An underwater environment provides an almost ideal
situation for a direct contact type of mobile system since there is no problem in obtaining
good electrode contact! Figure 2.6 shows a possible arrangement for an underwater mobile
surveying system where a cable with a number of nodes is pulled along the river/lake/sea
bottom by a boat. Two of the nodes are used as current electrodes, while the rest are used as
potential electrodes. An example of such an underwater survey is described in section 7.9. If
this system is coupled with a multi-channel resistivity meter, the survey can be carried out
very rapidly. Shallow seismic reflection surveys are frequently used in rivers/lakes/marine
environments for engineering site surveys. A mobile resistivity survey might be a useful
addition in some situations, such as in seismically opaque areas. In theory, both surveys can
be carried out simultaneously to reduce costs.

2.4 Pseudosection data plotting method
To plot the data from a 2-D imaging survey, the pseudosection contouring method is
normally used. In this case, the horizontal location of the point is placed at the mid-point of
the set of electrodes used to make that measurement. The vertical location of the plotting
point is placed at a distance that is proportional to the separation between the electrodes. For
IP surveys using the dipole-dipole array, one common method is to place the plotting point at
the intersection of two lines starting from the mid-point of the C1-C2 and P1-P2 dipole pairs
with a 45° angle to the horizontal. It is important to emphasize that this is merely a plotting
convention, and it does not imply that the depth of investigation is given by the point of
intersection of the two 45° angle lines (it certainly does not imply the current flow or
isopotential lines have a 45° angle with the surface). Surprisingly, this is still a common
misconception, particularly in North America!
Another method is to place the vertical position of the plotting point at the median
depth of investigation (Edwards 1977), or pseudodepth, of the electrode array used. This
pseudodepth value is based on the sensitivity values or Frechet derivative for a homogeneous
half space. Since it appears to have some mathematical basis, this method that is used in
plotting the pseudosections in the later part of these lecture notes. The pseudosection plot
obtained by contouring the apparent resistivity values is a convenient means to display the
data.
The pseudosection gives a very approximate picture of the true subsurface resistivity
distribution. However the pseudosection gives a distorted picture of the subsurface because
the shapes of the contours depend on the type of array used as well as the true subsurface
resistivity (Figure 2.7). The pseudosection is useful as a means to present the measured
apparent resistivity values in a pictorial form, and as an initial guide for further quantitative
interpretation. One common mistake made is to try to use the pseudosection as a final picture
of the true subsurface resistivity. As Figure 2.7 shows, different arrays used to map the same
region can give rise to very different contour shapes in the pseudosection plot. Figure 2.7 also
gives you an idea of the data coverage that can be obtained with different arrays. Note that
the pole-pole array gives the widest horizontal coverage, while the coverage obtained by the
Wenner array decreases much more rapidly with increasing electrode spacing.
One useful practical application of the pseudosection plot is for picking out bad
apparent resistivity measurements. Such bad measurements usually stand out as points with
unusually high or low values.


22

pulling vehicle

c c

c
current elec.
potential elec.

Figure 2.4. The Aarhus Pulled Array System. The system shown has two current (C)
electrodes and six potential electrodes (Christensen and Sørensen 1998, Bernstone and
Dahlin 1999).

Figure 2.5. The Geometrics OhmMapper system using capacitive coupled electrodes.
(Courtesy of Geometrics Inc.)

Figure 2.6. Schematic diagram of a possible mobile underwater survey system. The cable has
two fixed current electrodes and a number of potential electrodes so that measurements can
be made at different spacings. The above arrangement uses the Wenner-Schlumberger type of
configuration. Other configurations, such as the gradient array, can also be used.


23

Figure 2.7. The apparent resistivity pseudosections from 2-D imaging surveys with different
arrays over a rectangular prism.


24

2.5 A comparison of the different electrode arrays
As shown earlier in Figure 2.7, the shape of the contours in the pseudosection
produced by the different arrays over the same structure can be very different. The arrays
most commonly used for resistivity surveys were shown in Figure 1.4. The choice of the
“best” array for a field survey depends on the type of structure to be mapped, the sensitivity
of the resistivity meter and the background noise level. In practice, the arrays that are most
commonly used for 2-D imaging surveys are the (a) Wenner, (b) dipole-dipole (c) Wenner-
Schlumberger (d) pole-pole and (d) pole-dipole. Among the characteristics of an array that
should be considered are (i) the depth of investigation, (ii) the sensitivity of the array to
vertical and horizontal changes in the subsurface resistivity, (iii) the horizontal data coverage
and (iv) the signal strength.

2.5.1 The Frechet derivative for a homogeneous half-space
The first two characteristics can be determined from the sensitivity function of the
array for a homogeneous earth model. The sensitivity function basically tells us the degree to
which a change in the resistivity of a section of the subsurface will influence the potential
measured by the array. The higher the value of the sensitivity function, the greater is the
influence of the subsurface region on the measurement. Mathematically, the sensitivity
function is given by the Frechet derivative (McGillivray and Oldenburg 1990). Consider the
simplest possible array configuration shown in Figure 2.8 with just one current located at the
origin (0,0,0) and one potential electrode located at (a,0,0), i.e. both electrodes are on the
ground surface and they are “a” meters apart.. We inject 1 ampere of current into the ground
through the C1 current electrode that results in a potential φ observed at the potential P1
electrode. Suppose we were to change the resistivity within a small volume of the ground
located at (x,y,z) by a small amount, say δρ. What would be the corresponding change in the
potential, δφ, measured at P1? It can be shown (Loke and Barker 1995) that this is given by
δρ
δφ = 2 ∫ ∇φ ⋅ ∇φ ' dτ (2.1)
ρ V
where the change in the resistivity has a constant value in a volume element dτ and zero
elsewhere. The parameter φ’ is the potential resulting from a current electrode located at the
position of the P1 potential electrode. For the special case of a homogeneous half-space, the
potential φ at a point in the half-space due to a unit current source on the surface has a
relatively simple form, which is
ρ
φ= , and similarly
(
2π x 2 + y 2 + z 2
0.5
)
ρ
φ' = (2.2)
[
2π ( x − a ) + y + z
2 2
]
2 0.5

After differentiating the above equations to obtain the divergence, and substituting into (2.1)
we get

δφ 1 x( x − a ) + y 2 + z 2
=∫ ⋅ dxdydz (2.3)
[ ] [
δρ V 4π 2 x 2 + y 2 + z 2 1.5 (x − a )2 + y 2 + z 2 ]
1.5

The 3-D Frechet derivative is then given by the term within the integral, i.e.


25

1 x( x − a ) + y 2 + z 2
F3D ( x, y, z ) =⋅ (2.4)
[ ] [ ]
4π 2 x 2 + y 2 + z 2 1.5 ( x − a )2 + y 2 + z 2 1.5
This gives the Frechet derivative or sensitivity function for the pole-pole array consisting of
just one current and one potential electrode. To obtain the Frechet derivative for a general
four electrodes array, we need to just add up the contributions from the four current-potential
pairs, just as we have done earlier for the potential in equation (1.8).

2.5.2 A 1-D view of the sensitivity function - depth of investigation
In resistivity sounding surveys, it is well known as the separation between the
electrodes is increased, the array senses the resistivity of increasingly deeper layers. One
common question is – What is a depth of investigation of an array? One quantitative means to
put a numerical value for the depth of investigation is by using the sensitivity function or
Frechet derivative of the array. In resistivity sounding surveys, the subsurface is assumed to
consist of horizontal layers. What we want to determine is the change in the potential as
measured by the array on the surface if the resistivity of a thin horizontal is changed. For a
horizontal layer, the x and y limits of the layer extends from -∞ to +∞. Thus the sensitivity
function for a thin horizontal layer is obtained by integrating the 3D sensitivity function given
in equation (2.4) in the x and y directions, i.e.
x( x − a ) + y 2 + z 2
+∞ +∞
1
F1D (z ) = 2 ∫ ∫ dxdz
[ ] [ ]
4π −∞−∞ x 2 + y 2 + z 2 1.5 ( x − a )2 + y 2 + z 2 1.5
The above equation has a simple analytical solution (Roy and Apparao 1971), which is given
by
2 z
F1D (z ) = ⋅ (2.5)
(
π a 2 + 4z 2 1.5 )
The above function is also known as the depth investigation characteristic and has been used
by many authors to determine the properties of various arrays in resistivity sounding surveys
(Edwards 1977, Barker 1991, Merrick 1997). Figure 2.9a shows a plot of this function. Note
that it starts from zero and then increases to a maximum value at a depth of about 0.35a and
then decreases asymptotically to zero. Some authors have used the maximum point as the
depth of investigation of the array. However, Edwards (1977) and Barker (1991) has shown
that a more robust estimate is the "median depth of investigation". It is the depth above which
the area under the curve is equal to half the total area under the curve. In layman's terms, the
upper section of the earth above the "median depth of investigation" has the same influence
on the measured potential as the lower section. This tells us roughly how deep we can see
with an array. This depth does not depend on the measured apparent resistivity or the
resistivity of the homogeneous earth model. It should be noted that the depths are strictly only
valid for a homogeneous earth model, but they are probably good enough for planning field
surveys. If there are large resistivity contrasts near the surface, the actual depth of
investigation could be somewhat different.
The sensitivity function for other arrays can be determine by adding up the
contributions from the appropriate four pairs of current-potential electrodes. Figure 2.9b
shows the sensitivity function plot for the Wenner (alpha) array. Note that the curve around
the maximum is narrower for the Wenner array compared with the pole-pole array. This
implies that the Wenner array has a better vertical resolution than the pole-pole array.
Table 2.1 gives the median depth of investigation for the different arrays. To
determine the maximum depth mapped by a particular survey, multiply the maximum “a”
electrode spacing, or maximum array length “L“, by the appropriate depth factor given in
Table 2. For example, if the maximum electrode “a” spacing used by the Wenner array is 100


26

meters (or maximum L 300 meters), then the maximum depth mapped is about 51 meters.
For the dipole-dipole, pole-dipole and Wenner-Schlumberger arrays, the “n” factor (Figure
1.4) must also be taken into consideration. For the arrays with four active electrodes (such as
the dipole-dipole, Wenner and Wenner-Schlumberger arrays), it is probably easier to use the
total array length “L”. As an example, if a dipole-dipole survey uses a maximum value of 10
meters for “a” and a corresponding maximum value of 6 for n, then the maximum “L” value
is 80 meters. This gives a maximum depth of investigation of 80x0.216 or about 17 meters.
Table 2.1 also includes the geometric factor for the various arrays for an "a" spacing
of 1.0 metre. The inverse of the geometric factor gives an indication of the voltage that would
be measured between the P1 and P2 potential electrodes. The ratio of this potential compared
to the Wenner alpha array is also given, for example a value of 0.01 means that the potential
is 1% of the potential measured by the Wenner alpha array with the same "a" spacing.

Figure 2.8. The parameters for the sensitivity function calculation at a point (x,y,z) within a
half-space. A pole-pole array with the current electrode at the origin and the potential
electrode “a” meters away is shown.

Figure 2.9. A plot of the 1-D sensitivity function. (a) The sensitivity function for the pole-
pole array. Note that the median depth of investigation (red arrow) is more than twice the
depth of maximum sensitivity (blue arrow). (b) The sensitivity function and median depth of
investigation for the Wenner array.


27

Table 2.1. The median depth of investigation (ze) for the different arrays (after Edwards
1977). L is the total length of the array. Note identical values of ze/a for the Wenner-
Schlumberger and pole-dipole arrays. Please refer to Figure 1.4 for the arrangement of the
electrodes for the different arrays. The geometric factor is for an "a" value of 1.0 meter.
Array type z e/a z e/L Geometric Inverse Geometric
Factor Factor (Ratio)
Wenner Alpha 0.519 0.173 6.2832 0.15915 (1.0000)
Wenner Beta 0.416 0.139 18.850 0.05305 (0.3333)
Wenner Gamma 0.594 0.198 9.4248 0.10610 (0.6667)

Dipole-dipole n=1 0.416 0.139 18.850 0.05305 (0.3333)
n=2 0.697 0.174 75.398 0.01326 (0.0833)
n=3 0.962 0.192 188.50 0.00531 (0.0333)
n=4 1.220 0.203 376.99 0.00265 (0.0166)
n=5 1.476 0.211 659.73 0.00152 (0.0096)
n=6 1.730 0.216 1055.6 0.00095 (0.0060)
n=7 1.983 0.220 1583.4 0.00063 (0.0040)
n=8 2.236 0.224 2261.9 0.00044 (0.0028)

Equatorial dipole-dipole
n=1 0.451 0.319 21.452 0.04662 (0.2929)
n=2 0.809 0.362 119.03 0.00840 (0.0528)
n=3 1.180 0.373 367.31 0.00272 (0.0171)
n=4 1.556 0.377 841.75 0.00119 (0.0075)

Wenner - Schlumberger
n=1 0.519 0.173 6.2832 0.15915 (1.0000)
n=2 0.925 0.186 18.850 0.05305 (0.3333)
n=3 1.318 0.189 37.699 0.02653 (0.1667)
n=4 1.706 0.190 62.832 0.01592 (0.1000)
n=5 2.093 0.190 94.248 0.01061 (0.0667)
n=6 2.478 0.191 131.95 0.00758 (0.0476)
n=7 2.863 0.191 175.93 0.00568 (0.0357)
n=8 3.247 0.191 226.19 0.00442 (0.0278)
n=9 3.632 0.191 282.74 0.00354 (0.0222)
n = 10 4.015 0.191 345.58 0.00289 (0.0182)

Pole-dipole n=1 0.519 12.566 0.07958 (0.5000)
n=2 0.925 37.699 0.02653 (0.1667)
n=3 1.318 75.398 0.01326 (0.0833)
n=4 1.706 125.66 0.00796 (0.0500)
n=5 2.093 188.50 0.00531 (0.0334)
n=6 2.478 263.89 0.00379 (0.0238)
n=7 2.863 351.86 0.00284 (0.0178)
n=8 3.247 452.39 0.00221 (0.0139)

Pole-Pole 0.867 6.28319 0.15915 (1.0000)


28

2.5.3 A 2-D view of the sensitivity function – lateral and vertical resolution of the
different arrays
The plot of the 1-D sensitivity function in Figure 2.9 suggests that the sensitivity of an
array to the topmost layer is very small. The plot actually gives the net contribution
calculated by summing up the contribution for all x- and y-values at the same depth, and it
hides a multitude of effects. The net contribution for the topmost strip is small only if the
ground is completely homogeneous. If it is not homogeneous, the results can be very
different.
To study the suitability of different arrays for 2-D surveys, we need to go one step
beyond the simple 1-D sensitivity function, i.e. the 2-D sensitivity function. In this case, for a
particular (x,z) location, we add up the contribution from all points for y-values ranging from
+∞ to -∞. This involves the integration of the 3-D sensitivity function in equation (2.4) with
respect to y, which is
x( x − a ) + y 2 + z 2
+∞
1
F2 D ( x, z ) = 2 ∫ dy
[ ] [ ]
4π −∞ x 2 + y 2 + z 2 1.5 ( x − a )2 + y 2 + z 2 1.5
This integral has an analytic solution (Loke and Barker 1995) that is given in terms of elliptic
integrals. The complete solution is

F2 D ( x, z ) = 2  −
[( ) ]
2  α 2 E (k ) − β 2 K (k ) γ α 2 + β 2 E (k ) − 2β 2 K (k ) 
 (2.6)
αβ 
 (
α 2 − β2 ) ( α 2 − β2 ) 2


where

k=
(α 2
− β2 )
0.5

α
for x>0.5a,
α 2 = x 2 + z 2 , β 2 = (x − a ) + z 2 , γ = xa
2

and for x<0.5a,
β 2 = x 2 + z 2 , α 2 = (x − a ) + z 2 , γ = a(x − a )
2

and for x=0.5a,
 1 3a 2 
F2 D ( x, z ) = π  3 − 5 
, with α = 0.25a 2 + z 2
 2α 16α 
As an example, Figure 2.10a shows the contour pattern for the sensitivity function of
the Wenner array. The sensitivity function shows the degree to which a change in the
resistivity of a section of the subsurface will influence the potential measured by the array.
The higher the value of the sensitivity function, the greater is the influence of the subsurface
region on the measurement. Note that for all the three arrays, the highest sensitivity values
are found near the electrodes. At larger distances from the electrodes, the contour patterns are
different for the different arrays. The difference in the contour pattern of the sensitivity
function plot helps to explain the response of the different arrays to different types of
structures.
In the following plots of the sensitivity sections, the distance between the first
electrode and the last electrode (for example the C1 and P2 in the case of the Wenner alpha
array in Figure 2.10a) is normalized to 1.0 meter. To avoid the singularities at the electrodes,
the sensitivity values are shown from a depth of 0.025 meter downwards to 1.0 meter. In all
the sensitivity section diagrams, the location of the plotting point used in the pseudosection is
marked by a small black cross.


29

2.5.4 Wenner array
This is a robust array that was popularized by the pioneering work carried by The
University of Birmingham research group (Griffiths and Turnbull 1985; Griffiths, Turnbull
and Olayinka 1990). Many of the early 2-D surveys were carried out with this array. The
"normal" Wenner array is technically the Wenner Alpha array. For a four-electrode array,
there are three possible permutations of the positions of the electrodes (Carpenter and
Habberjam 1956). In Figure 2.10a, the sensitivity plot for the Wenner Alpha array has almost
horizontal contours beneath the center of the array. Because of this property, the Wenner
array is relatively sensitive to vertical changes in the subsurface resistivity below the center
of the array. However, it is less sensitive to horizontal changes in the subsurface resistivity. In
general, the Wenner is good in resolving vertical changes (i.e. horizontal structures), but
relatively poor in detecting horizontal changes (i.e. narrow vertical structures). In Table 2.1,
the median depth of investigation for the Wenner Alpha array is approximately 0.5 times the
“a” spacing used. Compared to other arrays, the Wenner Alpha array has a moderate depth of
investigation. The signal strength is inversely proportional to the geometric factor used to
calculate the apparent resistivity value for the array (Table 2.1). The geometric factor for the
Wenner array is 2πa. This is smaller than the geometric factor for other arrays. Among the
common arrays, the Wenner array has the strongest signal strength. This can be an important
factor if the survey is carried in areas with high background noise. One disadvantage of this
array for 2-D surveys is the relatively poor horizontal coverage as the electrode spacing is
increased (Figure 2.7). This could be a problem if you use a system with a relatively small
number of electrodes.
Note that the sensitivity section shows large negative values near the surface between
the C1 and P1 electrodes, as well as between the C2 and P2 electrodes. This means that if a
small body with a higher resistivity than the background medium is placed in these negative
zones, the measured apparent resistivity value will decrease. This phenomenon is also known
as an "anomaly inversion". In comparison, if the high resistivity body is placed between the
P1 and P2 electrodes where there are large positive sensitivity values, the measured apparent
resistivity will increase. This is the basis of the offset Wenner method by Barker (1992) to
reduce the effects of lateral variations in resistivity sounding surveys.
The other two permutations of the Wenner array are the Wenner Beta and the Wenner
Gamma arrays. The Wenner Beta array is in fact a special case of the dipole-dipole array
where the spacings between the electrodes are the same. Thus this array will be discussed in
the following section under the dipole-dipole array. The Wenner Gamma array has a
relatively unusual arrangement where the current and potential electrodes are interleaved. The
sensitivity section shows that the deepest regions mapped by this array are below the two
outer electrodes (C1 and P2 in Figure 2.10c), and not below the center of the array.

2.5.5 Dipole-dipole array
This array has been, and is still, widely used in resistivity and IP surveys because of
the low EM coupling between the current and potential circuits. The arrangement of the
electrodes is shown in Figure 1.4. The spacing between the current electrodes pair, C2-C1, is
given as “a” which is the same as the distance between the potential electrodes pair P1-P2.
This array has another factor marked as “n” in Figure 1.4. This is the ratio of the distance
between the C1 and P1 electrodes to the C2-C1 (or P1-P2) dipole length “a”. For surveys
with this array, the “a” spacing is initially kept fixed at the smallest unit electrode spacing
and the “n” factor is increased from 1 to 2 to 3 until up to about 6 in order to increase the
depth of investigation.


30

Figure 2.10. 2-D sensitivity sections for the Wenner array. The sensitivity sections for the (a)
alpha, (b) beta and (c) gamma configurations.

Figure 2.11 shows the sensitivity sections for this array for "n" values ranging from 1
to 6. The largest sensitivity values are generally located between the C2-C1 dipole pair, as
well as between the P1-P2 pair. This means that this array is most sensitive to resistivity
changes below the electrodes in each dipole pair. As the "n" factor is increased, the high
sensitivity values become increasingly more concentrated beneath the C1-C2 and P1-P2
dipoles, while the sensitivity values beneath the center of the array between the C1-P1
electrodes decreases. For "n" values of greater than 2, the sensitivity values at the
pseudosection plotting becomes negligible. The sensitivity contour pattern becomes almost
vertical for "n" values greater than 2. Thus the dipole-dipole array is very sensitive to
horizontal changes in resistivity, but relatively insensitive to vertical changes in the
resistivity. That means that it is good in mapping vertical structures, such as dykes and
cavities, but relatively poor in mapping horizontal structures such as sills or sedimentary
layers. The median depth of investigation of this array depends on both the “a” spacing and
the “n” factor (Table 2.1). In general, this array has a shallower depth of investigation


31

compared to the Wenner array, for example at n=1 the depth of investigation is 0.416a
compared to 0.512a for the Wenner Alpha array. Due to the almost vertical pattern of the
sensitivity contours, the depth of investigation (which a 1-D horizontal average of the
sensitivity values) is not particularly meaningful for the dipole-dipole array for "n" values
greater than 2. From some experience with synthetic modeling and field data, the median
depth of investigation might underestimate the depth of structures sensed by this array by
about 20% to 30% for the large “n” factors. For 2-D surveys, this array has better horizontal
data coverage than the Wenner (Figure 2.7). This can be an important advantage when the
number of nodes available with the multi-electrode system is small.
One possible disadvantage of this array is the very small signal strength for large
values of the “n” factor. The voltage is inversely proportional to the cube of the “n” factor.
For the same current, the voltage measured by the resistivity meter drops by about 56 times
when “n” is increased from 1 to 6 (Table 2.1). One method to overcome this problem is to
increase the “a” spacing between the C1-C2 (and P1-P2) dipole pair to reduce the drop in the
potential when the overall length of the array is increased to increase the depth of
investigation. Figure 2.12 shows two different arrangements for the dipole-dipole array with
the same array length but with different “a” and “n” factors. The signal strength of the array
with the smaller “n” factor is about 28 times stronger than the one with the larger “n” factor.
To use this array effectively, the resistivity meter should have comparatively high
sensitivity and very good noise rejection circuitry, and there should be good contact between
the electrodes and the ground. With the proper field equipment and survey techniques, this
array has been successfully used in many areas to detect structures such as cavities where the
good horizontal resolution of this array is a major advantage.
The plotting location of the corresponding datum point (based on the median depth of
investigation) used in drawing the apparent resistivity pseudosection is also shown in Figure
2.11. Note that the pseudosection plotting point falls in an area with very low sensitivity
values for “n” values of 4 and above. For the dipole-dipole array, the regions with the high
sensitivity values are concentrated below the C1-C2 electrodes pair and below the P1-P2
electrodes pair. In effect, the dipole-dipole array gives minimal information about the
resistivity of the region surrounding the plotting point, and the distribution of the data points
in the pseudosection plot does not reflect the subsurface area mapped by the apparent
resistivity measurements. Note that if the data point is plotted at the point of intersection of
the two 45° angle lines drawn from the center of the two dipoles, it would be located at a
depth of 0.7 units in Figure 2.11d (compared with 0.19 units given by the median depth of
investigation method) where the sensitivity values are almost zero!
Loke and Barker (1996a) used an inversion model where the arrangement of the
model blocks directly follows the arrangement of the pseudosection plotting points. This
approach gives satisfactory results for the Wenner and Wenner-Schlumberger arrays where
the pseudosection point falls in an area with high sensitivity values (Figures 2.10a and 2.13).
However, it is not suitable for arrays such as the dipole-dipole and pole-dipole where the
pseudosection point falls in an area with very low sensitivity values. The RES2DINV
program uses a more sophisticated method to generate the inversion model where the
arrangement the model blocks is not tightly bound to the pseudosection.
A final minor note. In most textbooks, the electrodes for this array are arranged in a
C1-C2-P1-P2 order that will in fact give a negative apparent resistivity. The arrangement
assumed in these notes is the C2-C1-P1-P2 arrangement.


32

Figure 2.11. 2-D sensitivity sections for the dipole-dipole array. The sections with (a) n=1,
(b) n=2, (c) n=4 and (d) n=6.


33

Figure 2.12. Two possible different arrangements for a dipole-dipole array measurement.
The two arrangements have the same array length but different “a” and “n” factors resulting
in very different signal strengths.

2.5.6 Wenner-Schlumberger array
This is a new hybrid between the Wenner and Schlumberger arrays (Pazdirek and
Blaha 1996) arising out of relatively recent work with electrical imaging surveys. The
classical Schlumberger array is one of the most commonly used arrays for resistivity
sounding surveys. A digitized form of this array so that it can be used on a system with the
electrodes arranged with a constant spacing is shown in Figure 2.14b. The “n” factor for this
array is the ratio of the distance between the C1-P1 (or P2-C2) electrodes to the spacing
between the P1-P2 potential pair. Note that the Wenner array is a special case of this array
where the “n” factor is equals to 1.
Figure 2.13 shows the sensitivity pattern for this array as the "n" factor is increased
from 1 (Wenner array) to 6 (the classical Schlumberger array). The area of highest positive
sensitivity below the center of the array becomes more concentrated beneath central P1-P2
electrodes as the "n" factor is increased. Near the location of the plotting point at the median
depth of investigation, the sensitivity contours has a slight vertical curvature below the center
of the array. At n=6, the high positive sensitivity lobe beneath the P1-P2 electrodes becomes
more separated from the high positive sensitivity values near the C1 and C2 electrodes. This
means that this array is moderately sensitive to both horizontal (for low "n" values) and
vertical structures (for high "n" values). In areas where both types of geological structures are
expected, this array might be a good compromise between the Wenner and the dipole-dipole
array. The median depth of investigation for this array is about 10% larger than that for the
Wenner array for the same distance between the outer (C1 and C2) electrodes for "n" values
greater than 3. The signal strength for this array is approximately inversely proportional to
the square of the "n" value. The signal strength is weaker than that for the Wenner array, but
it is higher than the dipole-dipole array and twice that of the pole-dipole.
Figure 2.14 shows the pattern of the data points in the pseudosections for the Wenner
and Wenner-Schlumberger arrays. The Wenner-Schlumberger array has a slightly better
horizontal coverage compared with the Wenner array. For the Wenner array each deeper data
level has 3 data points less than the previous data level, while for the Wenner-Schlumberger
array there is a loss of 2 data points with each deeper data level. The horizontal data coverage
is slightly wider than the Wenner array, but narrower than that obtained with the dipole-
dipole array.


34

Figure 2.13. 2-D sensitivity sections for the Wenner-Schlumberger array. The sensitivity
sections with (a) n=1, (b) n=2, (c) n=4 and (d) n=6.


35

Figure 2.14. A comparison of the (i) electrode arrangement and (ii) pseudosection data
pattern for the Wenner and Wenner-Schlumberger arrays.

2.5.7 Pole-pole array
This array is not as commonly used as the Wenner, dipole-dipole and Schlumberger
arrays. In practice the ideal pole-pole array, with only one current and one potential electrode
(Figure 1.4d), does not exist. To approximate the pole-pole array, the second current and
potential electrodes (C2 and P2) must be placed at a distance that is more than 20 times the
maximum separation between C1 and P1 electrodes used in the survey. The effect of the C2
(and similarly for the P2) electrode is approximately proportional to the ratio of the C1-P1
distance to the C2-P1 distance. If the effects of the C2 and P2 electrodes are not taken into
account, the distance of these electrodes from the survey line must be at least 20 times the
largest C1-P1 spacing used to ensure that the error is less than 5%. In surveys where the inter-
electrode spacing along the survey line is more than a few meters, there might be practical
problems in finding suitable locations for the C2 and P2 electrodes to satisfy this
requirement. Another disadvantage of this array is that because of the large distance between
the P1 and P2 electrodes, it is can pick up a large amount of telluric noise that can severely
degrade the quality of the measurements. Thus this array is mainly used in surveys where
relatively small electrode spacings (less than a few meters) are used. It is popular in some
applications such as archaeological surveys where small electrode spacings are used. It has
also been used for 3-D surveys (Li and Oldenburg 1992).
This array has the widest horizontal coverage and the deepest depth of investigation.
However, it has the poorest resolution, which is reflected by the comparatively large spacing
between the contours in the sensitivity function plot (Figure 2.15).


36

Figure 2.15. The pole-pole array 2-D sensitivity section.

2.5.8 Pole-dipole array
The pole-dipole array also has relatively good horizontal coverage, but it has a
significantly higher signal strength compared with the dipole-dipole array and it is not as
sensitive to telluric noise as the pole-pole array. Unlike the other common arrays, the pole-
dipole array is an asymmetrical array (Figure 1.4f). Over symmetrical structures the apparent
resistivity anomalies in the pseudosection are asymmetrical (Figure 2.7d). In some situations,
the asymmetry in the measured apparent resistivity values could influence the model obtained
after inversion. One method to eliminate the effect of this asymmetry is to repeat the
measurements with the electrodes arranged in the reverse manner (Figure 2.16). By
combining the measurements with the “forward” and “reverse” pole-dipole arrays, any bias in
the model due to the asymmetrical nature of this array would be removed. However this
procedure will double the number of data points and consequently the survey time.
The sensitivity section shows that area with the greatest sensitivity lies beneath P1-P2
dipole pair, particularly for large “n” factors. For “n” values of 4 and higher, the high
positive sensitive lobe beneath the P1-P2 dipole becomes increasingly vertical. Thus, similar
to the dipole-dipole array, this array is probably more sensitive to vertical structures. Note
also the zone with negative sensitivity values between the C1 and P1 electrodes, as well as
the smaller zone of high positive values to the left of the C1 electrode.
The pole-dipole array requires a remote electrode, the C2 electrode, which must be
placed sufficiently far from the survey line. For the pole-dipole array, the effect of the C2
electrode is approximately proportional to the square of ratio of the C1-P1 distance to the C2-
P1 distance. Thus the pole-dipole array is less affected by the C2 remote electrode compared
to the pole-pole array. If the distance of the C2 electrode is more than 5 times the largest C1-
P1 distance used, the error caused by neglecting the effect of the C2 electrode is less than 5%
(the exact error also depends on the location of the P2 electrode for the particular
measurement and the subsurface resistivity distribution).


37

Figure 2.16. The forward and reverse pole-dipole arrays.

Figure 2.17. The pole-dipole array 2-D sensitivity sections. The sensitivity sections with (a)
n=1, (b) n=2, (c) n=4 and (d) n=6.


38

Due to its good horizontal coverage, this is an attractive array for multi-electrode
resistivity meter systems with a relatively small number of nodes. The signal strength is
lower compared with the Wenner and Wenner-Schlumberger arrays but higher than the
dipole-dipole array. For IP surveys, the higher signal strength (compared with the dipole-
dipole array) combined with the lower EM coupling (compared with the Wenner and
Wenner-Schlumberger arrays) due to the separation of the circuitry of the current and
potential electrodes makes this array an attractive alternative.
The signal strength for the pole-dipole array decreases with the square of the “n”
factor. While this effect is not as severe as the dipole-dipole array, it is usually not advisable
to use “n” values of greater than 8 to 10. Beyond this, the “a” spacing between the P1-P2
dipole pair should be increased to obtain a stronger signal strength. There is another
interesting effect when the ‘n’ factor is increased that is frequently not appreciated, and leads
to an interesting pitfall in field surveys. This is discussed in section 4.7.

2.5.9 High-resolution electrical surveys with overlapping data levels
In seismic reflection surveys, the common depth point method is frequently used to
improve the quality of the signals from subsurface reflectors. A similar technique can be used
to improve the data quality for resistivity/IP surveys, particularly in noisy areas. This is by
using overlapping data levels with different combinations of “a” and “n” values for the
Wenner-Schlumberger, dipole-dipole and pole-dipole arrays.
To simplify matters, let us consider the case for the Wenner-Schlumberger array with
an inter-electrode spacing of 1 meter along the survey line. A high-resolution Wenner-
Schlumberger survey will start with the “a” spacing (which is the distance between the P1-P2
potential dipole) equals to 1 meter and repeat the measurements with “n” values of 1, 2, 3 and
4. Next the “a” spacing is increased to 2 meter, and measurements with “n” equals to 1, 2, 3
and 4 are made. This process is repeated for all possible values of the “a” spacing. To be on
the safe side, the data set should contain all the possible data points for the Wenner array. The
number of data points produced by such a survey is more than twice that obtained with a
normal Wenner array survey. Thus the price of better horizontal data coverage and resolution
is an increase in the field survey time.
A Wenner array with “a” equals to 2 meters (Figure 2.14) will have a total array
length of 6 meters and a median depth of investigation of about 1.04 meters. In comparison, a
measurement made with “a” equals to 1 meter and “n” equals to 2 using the Wenner-
Schlumberger array will have a total array length of 5 meters and a slightly smaller depth of
investigation of 0.93 meter (Figure 2.14). While the depth of investigation of the two
arrangements are similar, the section of the subsurface mapped by the two arrays will be
slightly different due to the different sensitivity patterns (Figures 2.13a and 2.13b). So the
two measurements will give slightly different information about the subsurface. A
measurement with “a” equals to 1 meter and “n” equals to 3 (Figure 2.14) will have a depth
of investigation of 1.32 meters. If all the 3 combinations are used, the data set will have
measurements with pseudodepths of 0.93, 1.02 and 1.32 metres. This results in a
pseudosection with overlapping data levels.
A similar “high-resolution” survey technique can also be used with the dipole-dipole
and pole-dipole arrays by combining measurements with different “a” and “n” values to give
overlapping data levels. In particular, this technique might be useful for the dipole-dipole
array since the signal strength decreases rapidly with increasing "n" values (section 2.5.5). A
typical high-resolution dipole-dipole survey might use the following arrangement; start with a
dipole of "1a" and "n" values of 1, 2, 3, 4, 5; followed by a dipole of "2a" and "n" values of
1, 2, 3, 4, 5; and if necessary another series of measurements with a dipole of "3a" and "n"
values of 1, 2, 3, 4, 5. Measurements with the higher "n" values of over 4 would have higher


39

noise levels. However, by having such redundant measurements using the overlapping data
levels, the effect of the more noisy data points will be reduced. Figure 2.18 shows the
apparent resistivity pseudosection for single prism model using this arrangement. This
arrangement has been widely used for IP surveys with the dipole-dipole array (Edwards
1977).
In theory, it should be possible to combine measurements made with different arrays
to take advantage of the different properties of the various arrays. Although this is not a
common practice, it could conceivably give useful results in some situations. The
RES2DINV program supports the use of such mixed data sets.

2.5.10 Summary of array types
The Wenner array is an attractive choice for a survey carried out in a noisy area (due
to its high signal strength) and also if good vertical resolution is required. The dipole-dipole
array might be a more suitable choice if good horizontal resolution and data coverage is
important (assuming your resistivity meter is sufficiently sensitive and there is good ground
contact). The Wenner-Schlumberger array (with overlapping data levels) is a reasonable all-
round alternative if both good and vertical resolutions are needed, particularly if good signal
strength is also required. If you have a system with a limited number of electrodes, the pole-
dipole array with measurements in both the forward and reverse directions might be a viable
choice. For surveys with small electrode spacings and require a good horizontal coverage, the
pole-pole array might be a suitable choice.

Figure 2.18. The apparent resistivity pseudosection for the dipole–dipole array using
overlapping data levels over a rectangular prism. Values of 1 to 3 meters are used for the
dipole length ‘a’, and the dipole separation factor ‘n’ varies from 1 to 5. Compare this with
Figure 2.7c for the same model but with ‘a’ fixed at 1 meter, and “n” varying from 1 to 10. In
practice, a ‘n’ value greater than 8 would result in very noisy apparent resistivity values.


40

2.5.11 Use of the sensitivity values for multi-channel measurements or streamers
Here we will look at one example of the use of the sensitivity values. The problem
faced is at follows.

a). We have a multi-channel resistivity meter that can make 8 measurements at a single time.
b). The cable used (possibly in the form of a streamer for mobile surveys on land or in water
covered areas) has 10 nodes. Two of the nodes are fixed as the current electrodes, while the
potential measurements are made using any 2 of the remaining 8 nodes.
c). We want to get the most information possible in terms of depth and a uniform horizontal
coverage below the streamer for a single set of 8 measurements.

To study the characteristics of the array configurations, we add up the sensitivity
values for all the 8 measurements. The formula used for the cumulative sensitivity, FI, is

1 i =8
FI = ∑ F2 Di (x, z )
8 i =1
(2.7)

Note that the absolute value of the sensitivity value is used, since negative sensitivity values
also give information about the subsurface. This is also to avoid a situation where the
negative sensitivity value for one measurement cancels out the positive value for another
measurement.
Figure 2.19a shows a dipole-dipole array based measurement sequence. The two
current electrodes are set at one end of the line and the potential measurements are made
using successive pairs of the potential nodes. Note the area with the highest cumulative
sensitivity is around the first 4 electrodes. The area of deepest penetration is between the
second and third electrodes, i.e. between the current dipole and the first potential electrode.
The second configuration places the current electrodes at the ends of the line (Figure
2.19b). Almost all the measurements are made using a fixed separation of one unit electrode
spacing between the potential electrodes. The potential dipole is moved from one end of the
survey line to the other end, i.e. a gradient array type of arrangement. One advantage of this
configuration over the dipole-dipole arrangement is the larger voltage (i.e. lower noise level)
measured by the potential electrodes. The cumulative sensitivity section shows a more
uniform pattern compared to that produced by the dipole-dipole configuration. The depth of
investigation is slightly deeper below current electrodes, and slightly less below the center of
the line.
The third configuration also places the current electrodes at the ends of the lines, but
keeps the center of the potential dipoles at or near the center of the line. The first 4
measurements uses a symmetrical Wenner-Schlumberger arrangement and increases the
separation between the potential electrodes until they reach next to the current electrodes. The
second set of 4 measurements uses separations of 2 and 3 times the unit spacing for the
potential dipole pair but with a mid-point slightly to one side of the center of the line. The
cumulative sensitivity pattern has an even more uniform pattern below the line compared
with the gradient array configuration.
To get an idea of the relative depths of investigation for the 3 configurations, we use
the light blue contour (with a value of 8 units) as a guide. The expanding Wenner-
Schlumberger configuration has the deepest depth of investigation while the dipole-dipole
configuration appears to have the shallowest.


41

Figure 2.19. Cumulative sensitivity sections for different measurement configurations using
(a) a dipole-dipole sequence, (b) a moving gradient array and (c) an expanding
Wenner-Schlumberger array.


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3 A 2-D forward modeling program
3.1 Finite-difference and finite-element methods
In the forward modeling problem, the subsurface resistivity distribution is specified
and the purpose is to calculate the apparent resistivity that would be measured by a survey
over such a structure. A forward modeling subroutine is in fact also an integral part of any
inversion program since it is necessary to calculate the theoretical apparent resistivity values
for the model produced by the inversion routine to see whether is agrees with the measured
values. There are three main methods to calculate the apparent resistivity values for a
specified model. They are (i) analytical methods, (ii) boundary element methods and (iii) the
finite-difference and finite-element methods. Analytical methods are probably the most
accurate methods, but they are restricted to relative simple geometries (such as a sphere or
cylinder). Boundary element methods are more flexible, but the number of regions with
different resistivity values that is allowed is somewhat limited (usually less than 10). In
engineering and environmental surveys the subsurface can have an arbitrary resistivity
distribution, so the finite-difference and finite-element methods are usually the only viable
choice. These methods can subdivide the subsurface into thousands of cells with different
resistivity values. However, the analytical and boundary element methods are useful
independent methods that can be used to check the accuracy of the finite-difference and
finite-element methods.
In the RES2DMOD software, the user can choose the finite-difference or the finite-
element method. The subsurface is subdivided into a large number of rectangular cells
(Figure 3.1) and the user can specify the resistivity value of each cell. The finite-difference
method is based on a method described by Dey and Morrison (1979a), but with a
modification by Loke (1994) to correct for a minor inconsistency in the Dey and Morrison
discretization by area method. The finite-element method uses the standard first-order
triangular elements (Silvester and Ferrari 1990).
While our main interest is in the inversion of field data, the forward modeling
program is also useful, particularly in the planning stage of the survey. In the previous
chapter we have seen that different arrays can have sensitivity sections that are radically
different. In theory, from the sensitivity sections we can get an idea of the type of array that
will give a reasonably good response over a particular class of structures (for example a
vertical fracture zone). However, there is no substitute for a hands-on direct calculation of the
expected apparent resistivity pseudosection. Before carrying out a field survey, some
information about the shape and size of expected targets is frequently known. By trying
different arrays digitally on the computer screen, we can avoid using an array that is
unsuitable for the detection of the structures of interest. We can also have an idea of a
suitable spacing between adjacent electrodes to use, and the maximum electrode separation
needed.
In the program, the subsurface is divided into a large number of small rectangular
cells. This program is largely intended for teaching about the use of the 2-D electrical
imaging method. Hopefully, it will assist the user in choosing the appropriate array for
different geological situations or surveys. The arrays supported by this program are the
Wenner (Alpha, Beta and Gamma configurations), Wenner-Schlumberger, pole-pole, inline
dipole-dipole, pole-dipole and equatorial dipole-dipole (Edwards 1977). Each type of array
has its advantages and disadvantages. This program will hopefully help you in choosing the
"best" array for a particular survey area after carefully balancing factors such as the cost,
depth of investigation, resolution and practicality.


43

Figure 3.1. The output from the RES2DMOD software for the SINGLE_BLOCK.MOD 2-D
model file. The individual cells in the model are shown in the lower figure, while the upper
figure shows the pseudosection for the Wenner Beta (dipole-dipole with n=1) array.

3.2 Using the forward modeling program RES2DMOD
The program requires the resistivity model values to be typed in separately in a text
file. The model data format, and other details about the use of this program, can be found in
the RES2DMOD.PDF manual file. In this course, we will use several model files that are
already present in the program package to take a look at the shapes of the contours in the
pseudosections for different geological structures. By playing around with this program, you
can get a feel of the effects of array type over the amplitude, size and shape of the contours in
the pseudosection.
After starting up the RES2DMOD program, click the ‘File’ menu option on the main
menu bar. Next select the “Read data file with forward model” suboption to read in one of the
example input model files provided. The files with the forward model have a MOD
extension. As an example, read in the file SINGLE_BLOCK.MOD that has a simple model
with a rectangular prism. After reading in this file, select the “Edit/Display Model” option
followed by the “Edit model” suboption to take a look at the model. This will show you the
actual cells that make up the model.
The program also allows you to change the model interactively using the left and right
mouse buttons. To change a single cell, click it with the left mouse button. Then move the


44

cursor to one of the color boxes in the legend above the model, and click the resistivity value
you want. Press the F1 key to get information about the keys used by the program to edit the
model. Note that clicking the cells with the mouse buttons will only change the resistivity of
the cells displayed on the screen, but will not change the resistivity of the buffer cells towards
the left, right and bottom edges of the mesh. To change the resistivity of the buffer cells, you
need to use the “[“, “]” and “D” keys.
Next select the “Model Computation” option to calculate the potential values for this
model. The calculations will probably only take a few seconds, after which you should go
back to “Edit/Display Model” option. In this option, select the “Edit model” sub-option again
to see the apparent resistivity pseudosection for this model. The program will ask you to
select the type of contour intervals you wish to use. For most resistivity pseudosections
choose the ‘Logarithmic contour intervals’, while for I.P. pseudosections choose the linear
contour intervals.
To change the type of array, use the “Change Settings” sub-option. Select another
array, such as the pole-pole or dipole-dipole, and see what happens to the shape of the
contours in the pseudosection.

3.3 Forward modeling exercises
Here we will try out a few model files that are provided to get a feel of what
pseudosections look like, and the effect of different choices on the results. The RES2DMOD
program also has an option to save the calculated apparent resistivity values into the format
used by the RES2DINV inversion program. You might like to save some of the results from
this exercise into this format to use with the RES2DINV program later on. This is useful in
studying the model resolution that can be obtained over different structures using various
arrays.

Exercise 3.1. Forward modeling examples.
Model file and purpose Things to try
BLOCK_ONE.MOD (1). The default array type is the Wenner Alpha. Run the ‘Model
To see the effect of selecting computation’ option to get things started. Next use the ‘Edit model’
the array type on the shape of option to take a look at the pseudosection. Compare the shape of the
the contours in the anomaly in the pseudosection with the actual shape of the prism.
pseudosection. The model is a What is the maximum apparent resistivity value (compare it with
simple rectangular prism. the prism resistivity of 100 Ω⋅m)?
(2). Next change the array type to Wenner Beta, and take a look at
the shape of the contours. Also, what is the maximum apparent
resistivity value? Repeat with the Wenner Gamma array. Among
the 3 versions of the Wenner array, which do you think is the
‘best’?
(4). Next try the inline dipole-dipole. What is the maximum ‘n’
value you need to use in order to fully map this prism? For a current
of 10mA, what would the voltage be for this ‘n’ value when the
dipole length is 1 meter? The program also allows you to change the
‘a’ dipole length. Try changing it to 2 and see what happens.
(5). To complete things, try out the other arrays, such as the pole-
pole, pole-dipole, equatorial dipole-dipole and Wenner-
Schlumberger.
(6). Edit the model using the ‘Edit Model’ option. Try increasing
the thickness and/or width of the prism and see what happens to the
anomaly in the pseudosection. Also try changing the resistivity of
the prism to 500 Ω⋅m.


45

BLOCK_TWO2.MOD (1). The default array type is the Wenner Alpha. Run the ‘Model
In this case, we have two computation’ option to get things started.
prisms at a mean depth of 1 (2). Next, take a look at the pseudosection. Is it possible to separate
meter, and 2 meters apart. The the highs due to each prism?
two prisms are identical and at (3). Try the whole range of arrays, i.e. the 3 different versions of the
the same depth. We will try Wenner, pole-dipole, dipole-dipole, Schlumberger etc. Which
different arrays to which ones arrays are more likely to be able to resolve the two prisms?
are more likely to be able to
resolve the prisms in the
pseudosection.
BLOCKS_UP.MOD (1). The default array type is the Wenner Alpha. Run the ‘Model
Now we have two prisms, one computation’ option to get things started.
on top of another. Is it possible (2). Next, take a look at the pseudosection. Is it possible to separate
to tell them apart? the highs due to each prism?
(3). Try the whole range of arrays, i.e. the 3 different versions of the
Wenner, pole-dipole, dipole-dipole, Schlumberger etc. Which
arrays are more likely to be able to resolve the two prisms?
THICK_DIKE.MOD (1). The default array type is the Wenner Beta. Calculate the
This model has a wide low potentials and take a look at the pseudosection. Is it possible to infer
resistivity dike in high the existence of the dike from the pseudosection?
resistivity bedrock. (2). Change to the Wenner Alpha, and see what happens.
(3). Try other arrays such as the Wenner Gamma, Wenner-
Schlumberger and pole-dipole. Which arrays show a significant low
resistivity anomaly?
THIN_DIKE.MOD (1). The default array type is the Wenner Beta. Calculate the
This model has a thin dike in potentials and take a look at the pseudosection. Is it possible to infer
high resistivity bedrock. This the existence of the dike for the pseudosection?
is a more difficult target than (2). Change to the Wenner Alpha, and see what happens.
the thick dike. (3). Try other arrays such as the Wenner Gamma, Wenner-
Schlumberger and pole-dipole. Which arrays show a significant low
resistivity anomaly?
MODEILIP.MOD (1). The default array type is the dipole-dipole, which is probably
An I.P. model just to round the most commonly used array for I.P. surveys. Calculate the
things up. potential values, and take a look at the pseudosections.
(2). Another array that is sometimes used for I.P. surveys is the
pole-dipole that has the advantage of a stronger signal strength.
Check the pseudosections obtained with this array.

Now that we had some experience in creating pseudosections, it is time to have a look the
inversion program RES2DINV that will do the reverse, i.e. creating a model from the
pseudosection.


46

4 A 2-D inversion program
4.1 Introduction
After the field survey, the resistance measurements are usually reduced to apparent
resistivity values. Practically all commercial multi-electrode systems come with the computer
software to carry out this conversion. In this section, we will look at the steps involved in
converting the apparent resistivity values into a resistivity model section that can be used for
geological interpretation. I will assume that the data is already in the RES2DINV format. The
conversion program is provided together with many commercial systems. So far, the ones
that have the conversion program include Abem, AGI, Campus, Geofysika, Geometrics, Iris,
OYO, Pasi and Scintrex. If your equipment manufacturer is not on the list, please contact
them about the conversion software. The data format used by the RES2DINV program is
described in detail in the RES2DINV.PDF manual provided with the program. Please refer to
the manual for the details.
In the next section, we will look at two methods to handle bad data points. Such bad
data points should be removed before a final interpretation is made.
Due to the large variety of data sets collected over various geological environments,
no single inversion method will give the optimum results in all cases. Thus the RES2DINV
program has a number of settings that can be changed by the user to obtain results that are
closer to the known geology. The various options are also discussed in the following sections.

4.2 Pre-inversion and post-inversion methods to remove bad data points
To get a good model, the data must be of equally good quality. Bad data points fall
into two broad categories, i.e. “systematic” and “random” noise. Systematic noise is usually
caused by some sort of failure during the survey such that the reading does not represent a
true resistivity measurement. Examples include breaks in the cable, very poor ground contact
at an electrode such that sufficient current cannot be injected into the ground, forgetting to
attach the clip to the electrode, connecting the cables in the wrong direction etc. Systematic
noise is fairly easy to detect in a data set as it is usually present in limited number of readings,
and the bad values usually stick out like sore thumbs. Random noise include effects such
telluric currents that affects all the readings, and the noise can cause the readings to be lower
or higher than the equivalent noise-free readings. This noise is usually more common with
arrays such as the dipole-dipole and pole-dipole that have very large geometric factors, and
thus very small potentials for the same current compared to other arrays such as the Wenner.
It is also common with the pole-pole array due to the large distance between the P1 electrode
and the remote (and fixed) P2 electrode. This array tends tend to pick up a large amount of
telluric noise due to the large distance between the two potential electrodes.
As a general rule, before carrying out the inversion of a data set, you should first take
a look at the data as a pseudosection plot (Figure 4.1a) as well as a profile plot (Figure 4.1b).
The bad data points with “systematic” noise show up as spots with unusually low or high
values (Figure 4.1a). In profile form, they stand out from the rest and can be easily removed
manually for the data set. In the RES2DINV program, choose the ‘Edit data’ on the top menu
bar followed by the ‘Exterminate bad data points’ option (Figure 4.2). The bad data points
can be removed by clicking them with the mouse.
When the noise is of a more random nature, the noisy data points are not as obvious,
so it might not be practical to remove them manually. Also, manually picking out the bad
data points becomes impractical if there are a large number of bad data points, particularly if
the data set contains more than a thousand data points. In some cases, it is not possible to
display the data as pseudosections or profiles, such as in 3D data sets. RES2DINV (and
RES3DINV) has a general technique to remove the bad data points with minimal input from


47

the user, and can be used for practically any array and any distribution of the data points. The
main disadvantage of the method is the larger amount of computer time needed. In this
method, a preliminary inversion of the data set is first carried with all the data points. After
carrying out the trial inversion, switch to the 'Display' window in RES2DINV, and read in the
INV file containing the inversion results. After that, select the 'RMS error statistics' option
that displays the distribution of the percentage difference between the logarithms of the
measured and calculated apparent resistivity values. The error distribution is shown in the
form of a bar chart, such as in Figure 4.3. Normally, the highest bar is the one with the
smallest errors, and the heights of the bars should decrease gradually with increasing error
values. The bad data points, caused by problems such as poor ground contact at a small
number of electrodes, should have significantly higher errors than the "good" data points.

Figure 4.1. An example of a field data set with a few bad data points. The most obvious bad
datum points are located below the 300 meters and 470 meters marks. The apparent
resistivity data in (a) pseudosection form and in (b) profile form.


48

Figure 4.2. Selecting the menu option to remove bad data points manually.

Figure 4.3 shows the error distribution bar chart for the data set shown in Figure 4.1 in
that has a few bad data points. In the bar chart, almost all the data points have errors of 20
percent or less. The bad data points show up data points with errors of 60 percent and above,
which can be easily removed from the data set by moving the green cursor line to the left of
the 60% error bar. In this way the 5 bad data points are removed from this data set. For some
data sets, the error distribution might show a more complicated pattern. As a general rule,
data points with errors of 100 percent and above can usually be removed.

Exercise 4.1 : Methods to remove bad data points
GRUNDF1.DAT – An (1) Use the ‘File’ and then the ‘Read data file’ options to
example of a field data set with read in this data file. Go to the ‘Display’ windows, and then
bad data points. the ‘Display data and model sections’ option. The bad data
points should be quite obvious.
(2) Next, leave the ‘Display’ window, and then choose
‘Edit data’ on the top menu bar followed by the
‘Exterminate bad data points’ option. Pick out the bad data
points. After that save the edited data in a file. Read in this
edited data file, and then go back to the ‘Display’ window
and check the pseudosection again.
(3) After that, leave the ‘Display’ window, and then run an
inversion of the data set using the ‘Inversion’ and then the
‘Least-squares inversion’ menu options.
(4) After the inversion has finished, go the ‘Display’
window to take a look at the model. After that choose the
‘Edit data’ and then the ‘RMS error statistics’ options. Take
a look at the bar chart. Is it possible to remove more bad
data points?
(5). Try running an inversion of the data set without first
manually removing the bad data points. Then use the ‘RMS
error statistics’ option to remove them. Does this get rid of
the bad data points also?


49

Figure 4.3. Error distribution bar chart from a trial inversion of the Grundfor Line 1 data set
with five bad data points.

4.3 Selecting the proper inversion settings
Many professionals carrying out resistivity imaging surveys will likely to be field
engineers, geologists or geophysicists who might not familiar with geophysical inversion
theory. The RES2DINV program is designed to operate, as far as possible, in an automatic
and robust manner with minimal input from the user. It has a set of default parameters that
guides the inversion process. In most cases the default parameters give reasonable results.
This section describes some of the parameters the user can modify to fine-tune the inversion
process. In the program, the different options are divided into six major groups. The groups
are placed under the ‘Change Settings’ or ‘Inversion’ choices on the main menu bar. The
groups are ‘INVERSION METHODS’, ‘MODEL DISCRETIZATION’ and ‘MODEL
SENSITIVITY OPTIONS’ under the ‘Inversion’ menu choice; and ‘INVERSION
DAMPING PARAMETERS’, ‘MESH PARAMETERS’ and ‘INVERSION PROGRESS’
under the ‘Change Settings’ menu choice. Here, we will look at a few of the more important
settings that can be changed.
a). INVERSION METHODS
The problem of non-uniqueness is well known in the inversion of resistivity sounding
and other geophysical data. For the same measured data set, there is wide range of models
that can give rise to the same calculated apparent resistivity values. To narrow down the
range of possible models, normally some assumptions are made concerning the nature of the
subsurface that can be incorporated into the inversion subroutine. In almost all surveys,
something is known about the geology of the subsurface, for example whether the subsurface
bodies are expected to have gradational or sharp boundaries.
The default smoothness-constrained inversion formulation used by the RES2DINV
program is given by (please refer to section 1.4 for the details)
( )
J T J +λF ∆q =J T g . (4.1)
This formulation constrains the change in the model resistivity values, ∆q , to be smooth but
does not guarantee that the resistivity values change in a smooth manner (Oldenburg, pers.
comm.). However, this formulation has been quite popular and used by a number of


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researchers (deGroot-Hedlin and Constable 1990, Sasaki 1992).
The first option ‘Include smoothing of model resistivity’ uses a formulation that will
apply the smoothness constrain directly on the model resistivity values. This formulation is
given by
( )
J T J +λF ∆q=J T g − λFq , (4.2)
so that the model resistivity values, q, changes in a smooth manner. The next option ‘Use
combined inversion method’ attempts to combined the smoothness-constrained method as
given in (4.1) with the Marquardt-Levemberg as given in (1.20). However, the result obtained
by this combination has not been very impressive and will not be examined.
The ‘Select robust inversion’ option has proved to be much more useful. It modifies
the formulation in (4.2) so that different elements of the model parameter change and data
misfit vectors have the same magnitudes. It is given by
( )
J T J +λFR ∆q k =J T R d g − λFR q k . (4.3)
The details are described in section 1.4. This method is also known as an l1-norm or
robust or blocky inversion method, whereas the conventional smoothness-constrained least-
squares method as given in equation (4.2) is an l2-norm inversion method. The l2-norm
inversion method gives optimal results where the subsurface geology exhibits a smooth
variation, such as the diffusion boundary of a chemical plume. However, in cases where the
subsurface consists of bodies that internally homogeneous with sharp boundaries (such as an
igneous dyke), this method tends to smear out the boundaries. The l1-norm or blocky
optimisation method tends to produce models that are piecewise constant (Ellis and
Oldenburg 1994a). This might be more consistent with the known geology in some situations.

Figure 4.4. Different options to modify the inversion process.
Figure 4.5 shows the inversion results for data from a synthetic model with sharp
boundaries. In this case, the robust inversion method gives significantly better results since


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the true model consists of three homogenous regions with sharp boundaries. Many synthetic
test models are of a similar nature with sharp boundaries, so not surprisingly, results obtained
with the l2-norm smooth inversion method are not optimal for such data sets (Olayinka and
Yaramanci 2000).

Figure 4.5. Example of inversion results using the l2-norm smooth inversion and l1-norm
blocky inversion model constrains. (a) Apparent resistivity pseudosection (Wenner array) for
a synthetic test model with a faulted block (100 Ω⋅m) in the bottom-left side and a small
rectangular block (2 Ω⋅m) on the right side with a surrounding medium of 10 Ω⋅m. The
inversion models produced by (b) the conventional least-squares smoothness-constrained or
l2-norm inversion method and (c) the robust or l1-norm inversion method.

Resistivity values have a logarithmic range, possibly ranging from less than 1 to over
1000 Ω⋅m in a single data set. By using the logarithm of the resistivity values as the
parameters in the inversion process, the numerical range of the parameters can be reduced to
a linear range. However, in some situations, it is not possible to make use of the logarithm if
there are negative or zero values. This does not usually occur for normal surface surveys with
the standard arrays, but could occur in borehole surveys or if non-standard arrays are used.
The program also allows the use to use the apparent resistivity value directly as the inversion
parameter in the “Choose logarithm of apparent resistivity” option.
The setting ‘Jacobian matrix calculation’ provides three options on the method used
by the program to determine the Jacobian matrix values (please refer to equation 1.18 to see
what is the Jacobian matrix). The first version of the RES2DINV program was written in the
days of the 80386 computers, so reducing the computer time was a high priority. The quasi-
Newton method (Loke and Barker, 1996a) was used to minimize the computer time. In the
quasi-Newton method, the Jacobian matrix for a homogenous earth model (which can be
calculated analytically) was used for the first iteration and an updating method was used to


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estimate the Jacobian matrix value for subsequent iterations. This method avoids a direct
calculation of the Jacobian matrix. This is the first option shown in Figure 4.6. This method is
fast and works well with low resistivity contrasts, and it is probably useful in providing an
approximate model for cases with larger contrasts (Loke and Dahlin 2002). Subsequent
improvements in computer technology and software algorithms have significantly reduced
the time taken to directly calculate the Jacobian matrix for non-homogeneous models using
the finite-difference or finite-element methods. The user can choose to recalculate the
Jacobian matrix in every iteration. In theory, this should give the most accurate results. As a
compromise between speed and accuracy, a third option is provided where the Jacobian
matrix is recalculated in the first few iterations (where the largest changes occur), and
estimated for the later iterations. This is the default option used by the RES2DINV program.

Figure 4.6. The different options for the Jacobian matrix calculation.

The last setting ‘Type of optimization method’ provides two numerical methods to solve the
least-squares equation (as in equations 4.1 to 4.3). The default method, particularly when the
number of model parameters n is small, is the standard (or complete) Gauss-Newton method
where a direct method (Golub and van Loan 1989) is used to solve the equation. This method
produces an exact solution but as the time taken is proportional to n3, it could take a very long
time for large data set with over 5000 model cells. An alternative method, the incomplete
Gauss-Newton method, can be used for such cases. In the incomplete Gauss-Newton method,
an approximate solution of the least method is determined by using an iterative method
(Golub and van Loan 1989). The final solution obtained with this method has a difference of
about 1 to 2 percent compared to the complete Gauss-Newton method solution. By sacrificing
a small amount of accuracy in the solution to the least-squares equation, the computer time
required could be reduced by a factor of 5 to 10 times for very large models.

b). MODEL DISCRETIZATION
This set of options control the way the program subdivides the subsurface into
rectangular cells. By default, the program uses a heuristic algorithm partly based on the
position of the data points to generate the size and position of the model blocks. In the
original algorithm by Loke and Barker (1996a), the model blocks were arranged in exactly
the same manner as the data points in the pseudosection. This works well for the relatively
simple pseudosections measured at that time, but this model discretization method will break
down for pseudosections with overlapping data levels (Figure 2.18) or if there are significant
missing data points (holes) in the pseudosection. The RES2DINV program uses a more
sophisticated algorithm to subdivide the subsurface where the distribution of the model


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blocks is loosely tied to the data points in the pseudosection.

Figure 4.7. Different methods to subdivide the subsurface into rectangular prisms in a 2-D
model. Models obtained with (a) the default algorithm, (b) by allowing the number of model
cells to exceed the number of data points, (c) a model which extends to the edges of the
survey line and (d) using the sensitivity values for a homogeneous earth model.
In the default algorithm used, the depth to the deepest layer in the model is set to be
about the same as the largest depth of investigation of the data points, and the number of
model cells does not exceed the number of data points. The thickness of each deeper layer is
increased to reflect the decreasing resolution of the resistivity method with increasing depth.
In general, this produces a model where the thickness of the layers increases with depth, and
with thicker cells at the sides and in the deeper layers (Figure 4.6a). For most cases, this gives


54

an acceptable compromise. The distribution of the data points in the pseudosection is used as
a rough guide in allocating the model cells, but the model section does not rigidly follow the
pseudosection. However, the user can change the width and thickness of the cells using a
variety of options.
After reading a data file, clicking the ‘Display model blocks’ option will show the
distribution of the model cells currently used by the program, such as in Figure 4.7a. Clicking
the ‘Change thickness of layers’ option will bring up the following dialog box.

Figure 4.8. The options to change the thickness of the model layers.

Firstly, you can choose a model where the layer thickness increases moderately (10%)
or greatly (25%) with depth. If these two options are still not satisfactory, you can choose the
‘User defined model’ where you need to set the thickness of first layer and the factor to
increase the thickness with depth. In this dialog box, you can also allow the program to use a
model where the number of model cells exceeds the number of data points. This is useful to
avoid having a model with very wide cells near the bottom for data sets with very sparse data
sets. Figure 4.7b shows such an example with this option enabled.
The program will set the model layers until it reaches the maximum pseudodepth of
the data points. If you need even deeper layers, the last setting ‘Factor to increase model
depth range’ forces the program to add more layers. For example, if you choose a value of
1.25 for this factor, the program will add layers until a depth that is 25% greater than the
maximum pseudodepth in the data set.
The ‘Modify depth to layers’ option allows you to set the depth to each layer
individually. This is useful if you want a layer boundary to coincide exactly with a known
geological boundary.
In ‘Use extended model’ option, model cells of uniform thickness right up to the left
and right edges of the survey line are used (Figure 4.7c). This is probably an extreme case. As
the number of model parameters increase, the computer time needed to carry out the
inversion also increases. This can be an important consideration for very large data sets with


55

several hundred electrodes. The ‘Make sure model blocks have same widths’ option is
probably more useful. It uses a base model such as in Figure 4.7b, but avoids thicker cells at
the sides. It ensures that the cells at the sides to have the equal widths.
The ‘Reduce effect of side blocks’ option affects the calculation of the Jacobian
matrix values for the model cells located at the sides and bottom of the model section.
Normally, for a cell located at the side, the contributions by all the mesh elements associated
with the model cell are added up right to the edge of the mesh. This gives a greater weight to
the side cell compared to the interior cells. In some cases, particularly when the robust
inversion option is used, this can result in unusually a high or low resistivity value for the
side cell. This option leaves out the contribution of the mesh elements outside the limits of
the survey line to the Jacobian matrix values for the side cells.
The last two options, ’Change width of blocks’ and ‘Type of cross-borehole model’,
are only used for certain special cases, and will not be described here. Please refer to the
RES2DINV.PDF manual for the details.
c). MODEL SENSITIVITY OPTIONS
This set of options relate to the model sensitivity (Jacobian matrix) values. The
‘Display model blocks sensitivity’ option will display the sum of the absolute values of the
sensitivity values associated with the model cell. Figure 4.7d shows an example. Note the
blocks at the sides and bottom has greater sensitivity values due to the larger sizes. To avoid
this effect, the ‘Display subsurface sensitivity’ option divides the subsurface into cells of
equal size and shows the sensitivity values. This is useful to get an idea of the regions of the
subsurface “scanned” by the survey configuration used.
All the techniques used to subdivide the subsurface described in the earlier section are
based on heuristic algorithms. Figure 4.7d shows the block distribution generated by a more
quantitative approach based the sensitivity values of the model cells. This method is selected
by the ‘Generate model block’ option. This technique takes into account the information
contained in the data set concerning the resistivity of the subsurface for a homogeneous earth
model. It tries to ensure that the data sensitivity of any cell does not become too small (in
which case the data set does not have much information about the resistivity of the cell).

The next set of inversion options are grouped below the ‘Change Settings’ choice on
the main menu bar. Clicking this will bring up the list of options shown in Figure 4.9.
d). INVERSION DAMPING PARAMETERS
This set of options is related to the damping factor λ used in the least-squares
equations (4.1) to (4.3). By default, the program uses a value of 0.16 for the initial damping
factor value for the first iteration. The damping factor is decreased by about half after each
iteration. However, to avoid instability in the model values due to a damping factor value that
is too small, a minimum limit of 0.015 is used by the program.
The ‘Damping factors’ option allows you to set the initial damping factor and the
minimum limit. If the data set appears to be very noisy, and unusually high or low model
resistivity values are obtained with the default values, try using larger values for the damping
factors.
Since the model resolution decreases with depth, the program increases the damping
factor value by about 5% for each deeper layer. You can change this factor using the “Change
damping factor with depth’ option. Instead of automatically decreasing the damping factor by
half after each iteration, the ‘Optimize damping factor’ option allows the program to look for
an optimum damping factor. The time taken per iteration will be more, but fewer iterations
might be needed for the program to converge.


56

Figure 4.9. The options under the ‘Change Settings’ menu selection.
The ‘Limit range of model resistivity values’ option is intended for cases where the
default settings (without limits) produces a model with resistivity values that are too high or
too low. Selecting this option will bring up the dialog box shown in Figure 4.10. In this
example, the upper limit for is 20 times the average model resistivity value for the previous
iteration while the lower limit is 0.05 times (i.e. 1/20 times). The program uses “soft” limits
that allow the actual resistivity model values to exceed the limits to a certain degree.
However, this option will avoid extremely small or large model resistivity values that are
physically unrealistic.
The ‘Vertical/Horizontal flatness filter ratio’ option allows the user to fine-tune the
smoothness-constrain to emphasize vertical or horizontal structures in the inversion model.
Some geological bodies have a predominantly horizontal orientation (for example
sedimentary layers and sills) while others might have a vertical orientation (such as dykes and
faults). This information can be incorporated into the inversion process by setting the relative
weights given to the horizontal and vertical flatness filters. If for example the structure has a
predominantly vertical orientation, such as a dyke, the vertical flatness filter is given a greater
weight than the horizontal filter.


57

Figure 4.10. The dialog box to limit the model resistivity values.

e). MESH PARAMETERS
This set of options controls the finite-element or finite-difference mesh used by the
forward modeling subroutine. The ‘Finite mesh grid size’ option changes the mesh size in the
horizontal direction, while the ‘Mesh refinement’ option changes the mesh settings in the
vertical direction. Using a finer mesh normally increases the accuracy of the forward
modeling subroutine, particularly if very large resistivity contrasts are present.

Exercise 4.2 : Tests with different inversion options
GRUNDFOR.DAT – An (1). Read in the data set using the ‘File’ and then carry out
example of a field data with the inversion with the default model discretization. You can
smooth variation of the take a look at the way the subsurface is divided into cells
resistivity values. The purpose by selecting the ‘Display model blocks’ option.
is to see the effect of the (2). Now choose the option to ‘Allow number of model
different model discretizations. blocks to exceed data points’, and run the inversion again.
The purpose of this field Make sure to use a different name for the inversion results
survey was to map the soil file, for example GRUNFOR2.INV. Check out the
lithology (Christensen and arrangement of the cells again using the ‘Display model
Sorenson 1994). blocks’ option.
(3). Now we will reduce the width of the side cells as well.
Select the ‘Make sure model blocks have same widths’
option, and check out the arrangement of the cells. Next run
the inversion again.
(4). Finally we will use an arrangement with even more
model cells. Select the ‘Use extended model’ option, and
then run the inversion.
As the number of model cells increase, the computer time
increases. Which do you think is the best compromise
between getting a finer model and reducing the computer
time?
BLOCK_ONE.DAT – A (1). After reading in the data file, choose the ‘Include
synthetic test model. To show smoothing of model resistivity’ option to ensure inversion
advantage of robust l1 norm model is smooth. Run the inversion and see what you get.
inversion for models with You might like to display the inversion model in the form


58

sharp boundaries. of blocks rather than contours. This shows up the smearing
out of the boundaries better. Compare the maximum and
minimum model resistivity values with the true values.
(2). Next select the ‘Select robust inversion’ option, and the
enable both the robust model and data constrains. Check
out the resulting inversion model.
ODARSLOV.DAT – A field (1). To get the best results, enable both the option to
data set with a vertical dike. “Allow number of model blocks to exceed datum points”.
Make sure the option ‘Include smoothing of model
resistivity’ is on to get a maximally smooth model.
(2). Next use the robust inversion method. You should see a
dramatic change in the model.
(3). One of the undesirable side effects of the robust
inversion option is the model resistivity values at the
bottom-left and bottom-right corners can take extreme
values. To reduce this effect, select the ‘Reduce effect of
side blocks’ option, and then run the inversion again.

4.4 Using the model sensitivity and uncertainty values
The depth of investigation and sensitivity sections described in section 2.5 gives an
idea of the depth of the regions sensed by a single electrode configuration. A 2-D survey
typically has hundreds of data points collected with electrodes at different locations and
spacings. A question that frequently arises in 2-D interpretation is as follows. What are the
regions of the subsurface sensed by the survey, and what is the reliability of the results? The
first question can be easily determined, but at present there is no simple answer to the second.
The “Display blocks sensitivity” option under the “Inversion” menu will show a plot
of the sensitivity of the cells used in the inversion model. The sensitivity value is a measure
of the amount of information about the resistivity of a model block cell in the measured data
set. The higher the sensitivity value, the more reliable is the model resistivity value. In
general, the cells near the surface usually have higher sensitivity values because the
sensitivity function has very large values near the electrodes. The cells at the sides and
bottom can also have high sensitivity values due to the much larger size of these cells that are
extended to the edges of the finite-difference or finite-element mesh (the program has an
option to reduce this effect which might produce artifacts at the edges of the model). If you
had carried out an inversion of the data set before calling this option, the program will make
use of the Jacobian matrix of the last iteration. Otherwise, it will calculate the Jacobian
matrix for a homogenous earth model. Figure 4.7d shows an example of a plot of the
sensitivity section for a model.
Figure 4.11b shows the model section obtained from the inversion of a data set for a
survey to map leakage of pollutants from a landfill site (Niederleithinger, 1994). The model
sensitivity section in Figure 4.11c shows high sensitivity values near the surface with
decreasing values with depth. This is to be expected as the near surface materials have a
larger influence on the measured apparent resistivity values.
In order to assess the accuracy of the inversion model, an estimate of the reliability of
the model values is required. One possible approach is by using the model covariance matrix
(Menke 1984). This is commonly used for models that consist of a small number of
parameters (such as the 1-D model in Figure 1.7). Figure 4.11d shows the model uncertainty
values obtained from the covariance matrix method as described by Alumbaugh and Newman
(2000) where the smoothness constraint is included in the model uncertainty estimate. In this
way, the model uncertainty values are less sensitive to size of the model cells. However, the


59

uncertainty values are only meaningful if the subsurface resistivity varies in a smooth
manner, as assumed by the smoothness constraint. If the subsurface resistivity does not vary
in a smooth manner, this method is likely to underestimate the actual uncertainty. Figure
4.11e shows the maximum and minimum resistivity values of each cell at the limits of the
model uncertainty range. Features that are common to both model sections can be considered
more reliable.
A different approach is to empirically determine the depth of investigation of the data
set is by carrying out at least two inversions of the data set using different constraints.
Oldenburg and Li (1999) used the following least-squares formulation to carry out the 2-D
inversion.
( )
J T J +λFR ∆q k = J T R d g − λFR (q k − q o ) , (4.4)
with FR = α s + α x C T R m C x + α z C T R m C z .
x z

qo is a homogeneous half-space reference model and αs is an additional “self” damping factor
that has a value of about 0.0001 to 0.01 times the αx and αz damping factors. In the proposed
method, two inversions are carried out using different resistivity values for the reference
model. Typically, the second reference model has a resistivity of 10 to 100 times the first
reference model. From the model resistivity values, the following depth of investigation
(DOI) index is calculated.
m ( x, z ) − m 2 ( x, z )
R ( x, z ) = 1 (4.5)
m1r − m2 r
m1r and m2r are the resistivity of first and second reference models, m1(x,z) and m2(x,z) are
model cell resistivity obtained from the first and second inversions. R will approach a value
of zero where the inversion will produce the same cell resistivity regardless of the reference
model resistivity. In such areas, the cell resistivity is well constrained by the data. In areas
where the data do not have much information about the cell resistivity, R will approach a
value of one as the cell resistivity will be similar to the reference resistivity. Thus the model
resistivity in areas where R has small values are considered to be “reliable”, while in areas
with high R values are not reliable.
The model used to calculate the DOI index use cells that extend to the edges of the
survey line, and a depth range of about three to five times the median depth of investigation
of the largest array spacing used. This ensures that data has minimal information about the
resistivity of the cells near the bottom of the model, i.e. in theory the bottom cells have DOI
values of almost 1.0.
Figure 4.12a shows the inversion model of the landfill survey data set with a depth
range of about 3.5 times the maximum pseudodepth in the apparent resistivity pseudosection
in Figure 4.11a. The resistivity of the reference model used is obtained from the average of
the logarithms of the apparent resistivity values. Figure 4.12b shows the DOI plot calculated
after carrying out a second inversion using a reference model with 100 times the resistivity of
the first reference model. Oldenburg and Li (1999) recommended using a value of 0.1 as the
cut-off limit for the effective depth of investigation of the data set. The depth to the contour
with a DOI value of 0.1 is about 28 m. along most of the survey line. This is close to the
maximum median depth of investigation of about 25 m. for this data set. There are shallower
regions with high DOI value, particularly below the 50 m. mark. This is probably partly due
to the low resistivity plume that limits the amount of current flowing into the deeper sections
below it. The resistivity method uses the electrical current as a probing tool, and where the
current flow is limited, the amount of information provided is less. Note the regions with high
DOI values at the sides of the section. This is to be expected as the sides of the survey line
have less data points compared to the center.


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Figure 4.11. Landfill survey example (Wenner array). (a) Apparent resistivity pseudosection ,
(b) model section, (c) model sensitivity section (d) model uncertainty section, (e) minimum
and maximum resistivity sections.


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Figure 4.12. Landfill survey depth of investigation determination. (a) Model section with
extended depths and (b) the normalized DOI index section.

Figure 4.13. Beach survey example in Denmark. The figure shows the model section with
extended depths and the normalized DOI index section.


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Figure 4.13 shows the DOI plot from a survey to map the boundary between the salt
and fresh water zones across a beach in Denmark (Marescot and Loke, 2003). The profile is
perpendicular to the seashore and the electrodes 1 to 16 are under the sea water, while the rest
of the profile is above the water. Note the strong increase in DOI values on the left part of the
profile that is covered with sea water. As this section is covered with sea water, most of the
current actually flows within the sea water and does not penetrate into the subsurface. Thus
the depth of investigation there is much shallower.
The DOI method is useful in marking the regions where the model values are well
constrained by the data set, and thus greater confidence can be placed on the model resistivity
values at such regions.

4.5 Methods to handle topography
In surveys over areas with significant changes in the elevation of the ground surface,
the effect of the topography must be taken into account when carrying out an inversion of the
data set. It is now generally recognized that the traditional method of using the “correction
factors” for a homogeneous earth model (Fox et al. 1980) does not give sufficiently accurate
results if there are large resistivity variations near the surface (Tong and Yang 1990). Instead
of trying to “correct” for the effect of the topography on the measurements, the preferred
method now is to incorporate the topography into the inversion model. The RES2DINV
program has three different methods that can be used to incorporate the topography into the
inversion model (Loke 2000).
The three methods are similar in that they use a distorted finite-element mesh. In all
these methods, the surface nodes of the mesh are shifted up or down so that they match the
actual topography. In this case, the topography becomes part of the mesh and is automatically
incorporated into the inversion model. The difference between these three methods is the way
the subsurface nodes are shifted. The simplest approach, used by the first finite-element
method, is to shift all the subsurface nodes by the same amount as the surface node along the
same vertical mesh line. This is probably acceptable for cases with a small to moderate
topographic variation (Figure 4.14b).
In the second approach, the amount the subsurface nodes are shifted is reduced in an
exponential manner with depth (Figure 4.14c) such that at a sufficiently great depth the nodes
are not shifted. This comes from the expectation that the effect of the topography is reduced
or damped with depth. This produces a more pleasing section than the first finite-element
method in that every kink in the surface topography is not reproduced in all the layers. For
data sets where the topography has moderate curvature, this is probably a good and simple
method. One possible disadvantage of this method is that it sometimes produces a model with
unusually thick layers below sections where the topography curves upwards. Thus in Figure
4.14d, the model is probably slightly too thick near the middle of the line where the
topography curves upwards and too thin towards the right end of the line where the
topography curves downwards. The resulting model is partly dependent on the degree of
damping chosen by the user. A value of 0.5 to 1.0 is usually used for the topography damping
factor in the RES2DINV program. One main advantage of this method is that it can be easily
implemented, particularly for 3-D models (Holcombe and Jirack 1984).
In the third method, the inverse Schwartz-Christoffel transformation method (Spiegel
et al. 1980) is used to calculate the amount to shift the subsurface nodes (Loke 2000). Since
this method takes into account the curvature of the surface topography it can, for certain
cases, avoid some of the pitfalls of the second finite-element method and produces a more
“natural” looking model section (Figure 4.14e). For this data set, this method avoids the
bulge near the middle of the line produced by the second finite-element method with a


63

damped distorted mesh. However, in the middle part of the line, the model produced by this
method is slightly thicker that that produced by the first finite-element method with a uniform
distorted mesh.

Figure 4.14. Different methods to incorporate topography into a 2-D inversion model. (a)
Schematic diagram of a typical 2-D inversion model with no topography. A finite-element
mesh with four nodes in the horizontal direction between adjacent electrodes is normally
used. The near surface layers are also subdivided vertically by several mesh lines. Models
with a distorted grid to match the actual topography where (b) the subsurface nodes are
shifted vertically by the same amount as the surface nodes, (c) the shift in the subsurface
nodes are gradually reduced with depth or (d) rapidly reduced with depth, and (e) the model
obtained with the inverse Schwartz-Christoffel transformation method.


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Exercise 4.3 : Tests with different topographic modeling options
RATCHRO.DAT – An (1). After reading the data file, select the robust inversion
example field data with option as the burial chamber probably has a sharp contrast
significant topography. This with the soil. After that, click the ‘Display topography’
survey was conducted over a option under the ‘Topography Options’ in the Main Menu
possible ancient burial mound bar.
in Ireland (Waddell and Barton (2). Next click the ‘Type of topographic modeling’ option.
1995). Select the uniformly distorted grid method. Run the
inversion, and then switch to the ‘Display’ window to take
a look at the results. In the ‘Display’ window, choose the
‘Display sections’ followed by ‘Include topography in
model display’ options.
(3). Now, select the damped distorted grid option in the
‘Type of topographic modeling’ dialog box. Run the
inversion again, and then take a look at the model section
with the topography. Now we will reduce the width of the
side cells as well. Select the ‘Make sure model blocks have
same widths’ option, and check out the arrangement of the
cells. Next run the inversion again.
(4). Next select the inverse Schwartz-Christoffel
transformation method in the ‘Type of topographic
modeling’ dialog box. Run the inversion again, and take a
look at the model with the topography.

Which topographic modeling method do you prefer?

GLADOE2.DAT – A data set (1). If you have the time, run the same tests as you did
taken to check for leakage earlier for the RATHCRO.DAT data set.
from a dam (Dahlin pers. This is another example where the option to reduce the
comm.). The survey area has effect of the side cells makes a significant difference when
topography. the robust inversion option is used.

4.6 Incorporating information from borehole logs and seismic surveys
In some areas, information from borehole logs is available concerning the resistivity
of part of the subsurface. This program allows you to fix the resistivity of up to 1000 sections
of the subsurface. The shape of the section to be fixed must be rectangular or triangular.
Borehole logs give the resistivity of the formations along the borehole. However, some
caution must be used when incorporating the borehole log information into an inversion
model for surface measurements. Borehole measurements usually only give the resistivity of
a very limited zone near the borehole. Depending on the type of instrument used, the
borehole log generally samples the subsurface within about 1 meter from the borehole. In
contrast, the inversion model gives the average resistivity of a much larger region of the
subsurface. Thus the RES2DINV program uses a flexible method to incorporate the borehole
log information.
The format used by the input data file for RES2DINV to fix the model resistivity
values is described in detail in the RES2DINV manual. Here, we will only look at a fragment
of it to illustrate the general ideas involved. As an example, part of the example data file
MODELFIX.DAT with the resistivity fixing option is listed below.


65

2 | Number if regions to fix, put 0 if none
R | Type of first region, R for rectangular
24,0.7 | X and Z coordinates of top-left corner of rectangle
28,2.3 | X and Z coordinates of bottom-right corner of rectangle
2.0 | Resistivity value of rectangular region
2.0 | Damping factor weight
T | Type of second region, T for triangular
30,0.0 | X and Z coordinates of first corner of triangle
30,3.0 | X and Z coordinates of second corner of triangle
45,3.0 | Coordinates of third corner of triangle
10.0 | Resistivity value of triangular region
2.0 | Damping factor weight

The first item is the number of regions where the resistivity is to be specified. In the example
above, 2 regions are specified. If a value of 0 is given (default value), then there is no region
where the resistivity is specified by the user. Next, the shape of the region is given, R for
rectangular or T for triangular. If a rectangular region is specified, then the X and Z
coordinates of the top-left and bottom-right corners of the rectangle are given, as shown in
the Figure 4.15. If a triangular region is chosen, the X and Z coordinates of the 3 vertices of
the triangle must be given in an anti-clockwise order. After the coordinates of the region to be
fixed are given, the next data item is the resistivity of the region. After that, the damping
factor weight for the resistivity of the region is needed. This parameter allows you control the
degree in which the inversion subroutine can change the resistivity of the region. There is
usually some degree of uncertainty in resistivity of the region. Thus, it is advisable that the
program should be allowed (within limits) to change the resistivity of the region. If a
damping factor weight of 1.0 is used, the resistivity of the region is allowed to change to the
same extent as other regions of the subsurface model. The larger the damping factor weight is
used, the smaller is the change that is allowed in the resistivity of the "fixed" region.
Normally, a value of about 1.5 to 2.5 is used. If a relatively large value is used, for example
10.0, the change in the resistivity of the region would be very small during the inversion
process. Such a large value should only be used if the resistivity and shape of the region is
accurately known. Figure 4.16 shows the allocation of the cells in the subsurface model
together with the fixed regions for the MODELFIX.DAT data set.
Seismic refraction and seismic reflection surveys are commonly used in engineering
surveys. Both methods can give accurate and detailed profiles of the subsurface interfaces. In
some cases, a distinct and sharp transition between two layers can be mapped by the seismic
survey. This information can be used to improve the results from the inversion of a 2-D
resistivity imaging survey along the same line. The subsurface in the inversion model can be
divided into two zones, one above and one below the interface calculated from the seismic
survey. The resistivity values are constrained to vary in a smooth manner within each zone,
but an abrupt transition across the zone boundary is allowed by removing any constrain
between the resistivity values below and above the zone boundary (Smith et al. 1999).


66

Figure 4.15. Fixing the resistivity of rectangular and triangular regions of the inversion
model.

Exercise 4.4 : Tests with option to fix the model resistivity
MODELFIX.DAT – A (1). After reading the data file, select the robust inversion
synthetic data file with two option as the synthetic model has sharp boundaries. Next
fixed regions. choose the ‘Display model blocks’ option to take a look at
the locations of the fixed regions. Run the inversion of the
data set.

CLIF4_NORMAL.DAT – A (1). After reading in the CLIF4_NORMAL.DAT data file,
data set from the Clifton, select the robust inversion option, and then run an inversion
Birmingham area (Scott et al. of the data set. In this case, we have not included additional
2000) to map layers within the constrains on the inversion. You might have to use the
unconsolidated sediments. option to “Reduce effect of the side blocks” to avoid
distortions at the bottom-left and bottom-right corners of
CLIF4_LAYERS.DAT – The the inversion model.
same data set but with the
layers specified. The depths to (2). Do the same for the CLIF4_LAYERS.DAT data file. In
the layers were determined this file, the depths to two layers determined from a seismic
from a seismic refraction refraction survey have been included. Take a look the
survey. distribution of the model cells, as well as the layers
specified, using the ‘Display model blocks’ option. Run the
inversion again, and compare the results with that obtained
earlier without the layers.


67

Figure 4.16. The inversion model cells with fixed regions. The fixed regions are drawn in
purple. Note that the triangular region extends beyond the survey line.

Figure 4.17. Example of an inversion model with specified sharp boundaries. The boundaries
in the Clifton survey (Scott et al. 2000) data set is shown by the blue lines.


68

4.7 Model refinement
Normally RES2DINV uses a model where the width of the interior model cells is the
same as the unit electrode spacing (for example as in Figure 4.17). In some situations with large
resistivity variations near the ground surface, this might not be sufficiently accurate. The
model with a cell width of one unit electrode spacing has a maximum possible misfit of one-
half the electrode spacing for a near-surface inhomogeneity (Figure 4.18a). In some cases,
this misfit can cause significant distortions in the lower sections of the inversion model. The
cell size is too coarse to accurately model the anomalies due to the small near-surface
inhomogeneities. This forces the inversion program to distort the lower sections of the model
in an attempt to reduce the data misfit.
A finer model with a cell width of half the electrode spacing has a maximum misfit of
one-quarter the unit electrode spacing (Figure 4.18b), so the effect of the model cell boundary
misfit should be much less. In theory, it is possible to reduce the cell width further, but the
error due to the misfit becomes increasingly less significant. Reducing the cell increases the
number of model parameters, thus increasing the computer time and memory required.

Figure 4.18. The effect of cell size on the model misfit for near surface inhomogeneities. (a)
Model with a cell width of one unit electrode spacing. (b) A finer model with a cell width of
half the unit electrode spacing. The near surface inhomogeneities are represented by coloured
ovals.

Figure 4.19 shows a synthetic model used to illustrate the effect of the model cell
misfit. The main structure is a faulted block of 100 Ω⋅m and a rectangular prism of 1 ohm.m
in a medium of 10 Ω⋅m. A series of small near-surface high resistivity blocks with widths of
1.0, 0.75, 0.50 and 0.25 m. and resistivity of 300 Ω⋅m are placed above the faulted block. A
similar series of low resistivity blocks of 1.0 Ω⋅m are located to the left.The pole-dipole array
has the P1-P2 dipole length (“a”) fixed at 1.0 m., but with “n” factor ranging from 1 to 16.
Note the strong anomalies produced by the near-surface inhomogeneities. Note also that the
Wenner array is much less affected by the near-surface inhomogeneities. The reason lies in
the sensitivity patterns of the two arrays (compares Figures 2.10 and 2.17). For the pole-
dipole array with large “n” values, the region with the highest positive sensitivity values is
concentrated below the P1-P2 dipole pair.
Figure 4.20 shows the inversion results for the pole-dipole data set with different cell
widths. The model with a cell width of 1.0 m. (i.e. one unit electrode spacing) shows
significant distortions near the top of the faulted block as well as in the low resistivity
rectangular block (Figure 4.20b). Most of the distortions are removed in the model with a cell
width of half the unit electrode spacing (Figure 4.20c). This means the residual misfits with
widths of up to one-quarter the unit electrode spacing does not have a significant effect on the
calculated apparent resistivity values. The model with a cell width of one-quarter the unit
electrode spacing (Figure 4.20d) does not show any major improvement over the half cell
width model although in theory it should more accurately model the near surface
inhomogeneities. In fact, there is a poorer agreement with the true model in the lower part of


69

the faulted block (Figure 4.20d). Experiments with a number of data sets show that using a
cell width of one-quarter the unit electrode spacing can sometimes lead to oscillating model
resistivity values. This is probably because the data does not have sufficient information to
accurately resolve such small cells.

Figure 4.19. Synthetic model (c) used to generate test apparent resistivity data for the pole-
dipole (a) and Wenner (b) arrays.

Figure 4.21 shows an example from a survey over an underground pipe using the
Wenner-Schlumberger array. There are large resistivity variations near the surface, probably
due to stones in the topmost layer of the soil. If the effect of the near surface variations are not
accurately accounted for by the inversion model, it can lead to distortions in the lower portions
of the model as the programs attempts to reduce the data misfit by distorting the lower part of
the model. The model with a cell width of half the unit electrode spacing in Figure 4.21c
shows a slightly better fit with the measured data (i.e. lower RMS error) and a more circular
shape for the low resistivity anomaly below the 12 meters mark.
In conclusion, for most cases, using a cell width of half the unit electrode spacing
seems to give the optimum results. Using a cell width of one-third the unit spacing seems to
be beneficial only a certain cases with the pole-dipole and dipole-dipole arrays with very
large ‘n’ values (see the following section 4.8j). A cell width of one-quarter the unit spacing
sometimes leads to instability with oscillating model values, particularly in the first few


70

layers. Thus the use of a cell width of less than one-quarter the true unit electrode spacing is
not advisable.
Note that using finer cells will lead to longer inversion times, so using a width of the
half the unit electrode spacing seems to provide the best trade-off. These examples present a
strong case for using a model with a cell width of half the unit electrode spacing as the
default choice in the inversion of most data sets. It avoids the problem caused by model cells
boundary misfits if the cell is too coarse, and the increase in computer time is tolerable.
As a final note, it appears that the effect of using narrower model cells is less dramatic
in 3-D inversion. This is probably because in the 2-D model each cell extends to infinity in
the y-direction, whereas in the 3-D model the same cell is divided into a large number of
much smaller cells. Thus the effect of a single near-surface cell in the 3-D model on the
calculated apparent resistivity values is much smaller than in the equivalent 2-D model.

Figure 4.20. The effect of cell size on the pole-dipole array inversion model. Pole-dipole
array (a) The apparent resistivity pseudosection. The inversion models obtained using cells
widths of (b) one, (b) one-half and (c) one-quarter the unit electrode spacing.

4.8 Pitfalls in 2-D resistivity surveys and inversion
While 2-D resistivity surveys have made the mapping of many complex structures
possible, caution must still be exercised in interpreting the results from the data. Below are
some of the common pitfalls.
(a). Incorrect use of the dipole-dipole array. This is still a surprisingly common problem.
There are two common mistakes in the use of this array. The first is to assume that the depth
of investigation is at the point of intersection of the two 45° diagonals projected from the


71

dipoles. This greatly overestimates the depth of investigation. For example, for the case
where the dipole separation factor “n” is equals to 6, the point of intersection is about 3 times
the median depth of investigation (see Table 2.1). The second common mistake is to
monotonically increase the “n” factor, while keeping the dipole length “a” fixed, in an effort
to increase the depth of investigation. This usually results in very noisy and unusable data,
with negative apparent resistivity values in some cases, for “n” values of greater than 8. To
solve this problem the “n” value should not exceed 6, and the method of overlapping data
levels (section 2.5.6) with different “a” dipole lengths can be used.
(b). Poor electrode ground contact. This problem arises in stony or dry soils where it is not
possible to plant the electrodes to a sufficient depth, and/or the soil is too dry such that it is
not possible to pass enough current into the ground. In the pseudosection, this is seen as an
inverted “V” shaped pattern of bad data points with the two legs originating from an
electrode. This problem is more severe when the electrode is used as a current electrode. The
potential electrode is less sensitive to poor ground contact, so this problem in certain
situations can be overcome by swapping the current and potential electrodes.
(c). Poor current penetration. The success of the resistivity method depends on a current
flowing through the areas to be mapped. If the top layer has a very high resistivity, it might
be very difficult to get enough current to flow through the ground at all. The opposite
problem occurs if the top layer has an extremely low resistivity. The current might be trapped
in the top layer, so not much information is expected from the lower layers.

Figure 4.21. Example of the use of narrower model cells with the Wenner-Schlumberger
array. (a) The apparent resistivity pseudosection for the PIPESCHL.DAT data set. The
inversion models using (b) cells with a width of 1.0 meter that is the same as the actual unit
electrode, and (c) using narrower cells with a width of 0.5 meter. The pipe is marked by the
low resistivity anomaly below the 12 meters mark. The Wenner-Schlumberger array was
used for this survey.
(d). Not letting the current charge decay. When an electrode is used as a current electrode,
charges tend to build up around the electrode. When the current is no longer flowing through


72

the electrode, it still takes a finite amount of time for the charges to disperse. If the same
electrode is used as a potential electrode immediately after it has been used as a current
electrode, this could result in an erroneous reading (Dahlin 2000).
(e). Mistakes in the field. This can arise from a variety of sources. It could be caused by
instrumentation errors during the field survey, poor electrode contact in dry, sandy or stony
ground, shorting of electrodes due to very wet conditions or metal objects (such as fences,
pipes etc.) or mistakes such as attaching electrodes to the wrong connectors. Fortunately, it is
usually very easy to pick out such bad data points by viewing the pseudosection or the data in
the form of profiles. Before inverting a data set, take a look at it!
(f). 3-D geology. It is assumed that the subsurface is 2-D when interpreting the data from a
single line. This assumption is valid if the survey is carried out across the strike of an
elongated structure. If there are significant variations in the subsurface resistivity in a
direction perpendicular to the survey line (i.e. the geology is 3-D), this could cause
distortions in the lower sections of the model obtained. Measurements made with the larger
electrode spacings are not only affected by the deeper sections of the subsurface, they are
also affected by structures at a larger horizontal distance from the survey line. This effect is
most pronounced when the survey line is placed near a steep contact with the line parallel to
the contact. Figure 4.22 shows the apparent resistivity pseudosection for the Wenner array
over a 3-D model that has two structures of 100 Ω⋅m that have the same widths within a 10
Ω.m medium. One structure is truly 2-D while the second is truncated near the middle of the
area at the y-axis coordinate of 3 meters. At a large distance from the edge of the second
structure, for example at the y-axis coordinate of 8 meters, the anomaly due to the right
structure is almost as high as that due to the left structure. However, close to the edge, for
example at y-axis coordinate of 5 meters, the anomaly due to the right structure is
significantly weaker. If a survey was carried out along this line, and the subsurface was
assumed to be 2-D, then the dimensions of the right structure obtained from an inversion of
the data set would be wrong. Dahlin and Loke (1997) did a comparison of the sensitivity of
different arrays to structures off the axis of a 2-D survey line. In general, it was found that the
dipole-dipole array was the most sensitive (i.e. suffered the greatest distortion) due to off-axis
structures. The best way to handle 3-D structures would be a full 3-D survey and data
inversion. We will look at 3-D surveys in the later part of these notes.
It must be emphasized that a 2-D survey with very good quality data and very dense
data coverage, and inverted with a good inversion algorithm, can still give the wrong results
if the assumption of a 2-D geology on which the model is based is seriously wrong. This is a
particularly a problem in mineral surveys (which commonly also involve IP measurements)
where very complex geological structures and mineralization patterns are usually
encountered. In such situations, the results from a 2-D resistivity and IP model should be
treated with some reservations unless it is confirmed by a 3-D survey and model. There have
been many cases where expensive drill-holes have passed through barren zones in areas
where the 2-D model shows a strong IP anomaly.
(g). Limits of the physics of the resistivity method. While 2-D and 3-D surveys and data
inversion has greatly extended the range of field problems that can be solved using the
resistivity method, there are still basic laws of physics that place certain limitations on these
techniques. The resolution of the resistivity method decreases exponentially with depth. We
see the subsurface “as through a glass darkly” and the image becomes increasingly fuzzier
with depth. It is unlikely to be able to map a structure with a size of 1 meter at a depth of 10
meters using the resistivity method. The resistivity phenomena is based on the diffusion
equations, so its resolution is inherently poorer than the seismic or ground radar method at
depths greater than one wavelength.


73

(a). 2-D apparent resistivity pseudosections

(b). The actual 3-D model

Figure 4.22. An example of 3-D effects on a 2-D survey. (a) Apparent resistivity
pseudosections (Wenner array) along lines at different y-locations over (b) a 3-D structure
shown in the form of horizontal slices.

(h). Non-uniqueness. It is well known that more than one model can produce the same
response that agrees with the observed data within the limits of the data accuracy. In 1-D
resistivity sounding modeling, the problems of equivalence and suppression are well known.
The problems, in different forms, also occur in 2-D and 3-D modeling. A good example was
shown in the paper by Oldenburg and Li (1999). In 2-D and 3-D modeling, constrains are
used so that a stable solution can be obtained. The use of a smooth or blocky constrain results
in the production of models that look more reasonable, but it is no guarantee that they are
indeed correct. The accuracy of the result is only as good as the accuracy of the assumptions
made. The resulting model thus depends to a significant extent on the constrain used, and will
closely approximate the true subsurface resistivity only if the constrains correspond to the
real situation.
(i). Optimization versus inversion. The RES2DINV program, like most non-linear
inversion programs, actually carries out an optimization (i.e. not a direct one-to-one inversion
in the sense it must have only one solution) in that it tries to reduce the difference between
the calculated and measured apparent resistivity values. If there is infinite data and a perfect
fit between the calculated and measured values, how the data is measured should not have an


74

effect on the results. However with real, noisy and limited data, how the data is measured
does have an effect. A model with 5% rms error in the fit between the measured and
calculated apparent resistivity values with one data set might not give the same model as a
5% rms error with another data set although both might be from the same place. For this
reason, the dipole-dipole array gives an inversion model with much better resolution than the
pole-pole array although in theory the dipole-dipole values can be extracted from the pole-
pole values.
(j). Increasing the electrode separation does not always increase the survey depth. It is
generally assumed that as the separation between the electrodes is increased, the region of the
subsurface that is ‘sensed’ by the array also increases. While this is true of most arrays, there
are certain important exceptions. In particular, this is not true of the pole-dipole and dipole-
dipole arrays under certain circumstances. In some surveys with the pole-dipole array, the
separation between the C1 current electrode and the P1-P2 dipole (Figure 2.16) is increased
in an effort to increase the depth of survey by the array. However, if this is done with the P1-
P2 dipole length (the ‘a’ factor in Figure 2.16) kept at a constant spacing, certain interesting
effects come into play. In section 2.5.8 this practice was strongly discouraged on the basis
that the potential will decrease with the square of the ‘n’ factor. This problem can be
overcome by a combination of using higher currents and more sensitive receivers. However,
the problem caused by the change in the array sensitivity pattern as the ‘n’ factor is
monotonically increased is usually not taken into account. The change in the sensitivity
pattern when the ‘n’ factor changes from 1 to 6 was shown earlier in Figure 2.17. Figure 4.23
shows what happens when the ‘n’ factor jumps form 6 to 12 to 18. Here, the dipole length is
kept constant at 1 meter. When ‘n’ is equals to 6 there are reasonably high sensitivity values
to a depth of about 3 to 4 meters between the C1 current and the P1 potential electrode. When
‘n’ is increased to 12, the zone of high sensitivity values becomes increasingly more
concentrated below the P1-P2 dipole in an even shallower region. This means that the array
with ‘n’ equals to 12 is in fact less sensitive to deeper structures than the array with ‘n’
equals to 6. This effect is even more pronounced when ‘n’ is increased to 18.
Thus increasing the separation between the current electrode and the potential dipole,
while keeping the dipole length fixed, does not increase the survey depth of the array. It, in
fact, effectively decreases the depth of the region sensed by the array!
Figure 4.24 shows the apparent resistivity anomaly due to a small near-surface high
resistivity block for the pole-dipole array for ‘n’ values of up to 28. Note that the amplitude
of the high resistivity anomaly due to the near-surface block increases with the ‘n’ value, i.e.
the array becomes increasingly more sensitive to the near-surface block as the separation
between the electrodes increase. In field surveys with the pole-dipole array where the ‘n’
factor is monotonically increased in the belief that this increases the survey depth, the
pseudosection is frequently dominated by a series of parallel slanting high-amplitude
anomalies due to near-surface inhomogeneities. The anomalies due to the near-surface
structures frequently mask the anomalies due to deeper structures that are of interest.


75

Figure 4.23. The 2-D sensitivity sections for the pole-dipole array with a dipole length of 1
meter and with (a) n=6, (b) n=12 and (c) n=18. Note that as the ‘n’ factor increases, the zone
of high positive sensitive values becomes increasingly concentrated in a shallower zone
below the P1-P2 dipole.


76

To increase the depth of penetration with the pole-dipole array, the ‘a’ dipole length
should be increased when the ‘n’ factor exceeds 6 to 8. This method was discussed in detail
in section 2.5.9. For example, instead of fixing the ‘a’ dipole length to 1 meter and increasing
the ‘n’ factor to 28 in Figure 4.24, a more prudent approach is increase the ‘a’ spacing from 1
to 4 meters while ensuring the ‘n’ factor does not exceed 8.

Figure 4.24. Example of apparent resistivity pseudosection with pole-dipole array with large
‘n’ values. Note that the anomaly due to a small near-surface high resistivity block becomes
greater as the ‘n’ factor increases. This means that the sensitivity of the array to the near-
surface region between the P1-P2 potential dipole becomes greater as the ‘n’ factor increases.


77

5 IP inversion
5.1 Introduction
One of the more recent developments in the instrumentation for electrical imaging
surveys has been the addition of Induced Polarization (IP) capability in the multi-electrode
resistivity meter system. Many of the early 2-D surveys were resistivity and IP surveys
carried out using conventional 4-electrode systems in the 1950's onwards for mineral
exploration, particularly for conductive sulfide ore bodies. Quantitative interpretation of such
historical data was rather limited due to the limited computing facilities available at that time.
Such historical data provides an interesting source of data for testing modern 2-D and 3-D
inversion software. Re-interpretation of such old data to produce quantitative models
sometimes has shed new light on the geological structures.
One of the distinctive characteristics of the IP method has been the different
parameters in the time and frequency domains used to represent the IP effect. The following
section briefly discusses the IP phenomena and the different IP parameters. This is followed
by a few exercises in the inversion of IP data with the RES2DINV program. The data format
used by the RES2DINV and RES3DINV programs is described in their respective manuals.
As such, we will not cover it here.

5.2 The IP effect
A very brief description of the IP effect will be given here. Further details can be
found in many fine textbooks, such as by Keller and Frischknecht (1966), Summer (1976),
Telford et al. (1990) and Zhdanov and Keller (1994).
The IP effect is caused by two main mechanisms, the membrane polarization and the
electrode polarization effects. The membrane polarization effect is largely caused by clay
minerals present in the rock or sediment. This is particularly relevant in engineering and
environmental surveys. The electrode polarization effect is caused by conductive minerals in
rocks such that the current flow is partly electrolytic (through groundwater) and partly
electronic (through the conductive mineral). This effect is of particular interest in surveys for
metallic minerals, such as disseminated sulfides.
IP measurements are made in the time-domain or frequency domain. In the time-
domain, the IP effect is measured by the residual decay voltage after the current is switched
off (Figure 5.2c). The time domain IP unit, the chargeability, is usually given in millivolt per
volt (mV/V) or in milliseconds. Figure 5.1 shows the IP values (in terms of mV/V) for
several mineralized rocks and common rocks. Note that the IP effect due to sulfide
mineralization (the electrode polarization effect) is much larger than that due to clay minerals
(membrane polarization) in sandstone and siltstones.
In the frequency domain (Figure 5.2b), it is measured by the change in apparent
resistivity value from low to high frequencies (typically 1 to 10 Hz) where the unit used is the
percent frequency effect. Another measure of the IP effect in the frequency domain is the
phase shift between the potential signal and the input current, where the unit used is in milli-
radians.
One mathematical model that attempts to explain variation of resistivity with
frequency observed in the IP method is the Cole-Cole mode (Pelton et al. 1978), which is
defined by
  1 
 1 + (iωτ )c 
ρ (ω ) = ρ 0 1 − m1 −  (5.1)

  


78

Figure 5.1. The IP values for some rocks and minerals.

Figure 5.2. The Cole-Cole model. (a) Simplified electrical analogue circuit model (after
Pelton et al. 1978). ρ = resistivity, m = chargeability, τ = time constant, c = relaxation
constant. (b) Amplitude and phase response to sine wave excitation (frequency domain). (c)
Transient response to square wave current pulse (time domain). Most IP receivers measure
the integral of the decay voltage signal over a fixed interval, mt, as a measure of the IP effect.


79

where ρ 0 is the DC resistivity, m is the chargeability, ω is the angular frequency (2πf), τ is a
time constant and c is the exponent or relaxation constant. While the DC resistivity and
chargeability determine the behavior of the material at very low and very high frequencies,
the variation of the amplitude and phase curves at intermediate frequencies are also affected
by the time and relaxation constants.
The time constant factor τ has a large range, from 0.01 second to several thousand
seconds. The relaxation constant factor c is bounded by 0.0 to 1.0, with values frequently
between 0.2 and 0.7. Much of the earlier work was on the use of spectral IP (SIP)
measurements to differentiate between different types of conductive minerals for mining
purposes (Van Voorhis et al.1973, Zonge and Wynn 1975, Pelton et al. 1978, Vanhalla and
Peltoniemi 1992). More recently, attempts have been made to use the SIP method for
environmental surveys, such as in the detection contaminants (Vanhala et al. 1992). Figure
5.2 shows a simplified electrical analogue circuit for the Cole-Cole model, together with
typical response curves in the frequency and time domain.

5.3 IP data types
Although the chargeability is defined as the ratio of amplitude of the residual voltage
to the DC potential (Figure 5.2c) immediately after the current cut-off, this is not used in
actual field measurements. Field measurements of the IP effect may be divided into two main
groups, the time-domain and frequency domain methods.
(a) Time domain IP measurements
In the time-domain method, the residual voltage after the current cut-off is measured.
Some instruments measure the amplitude of the residual voltage at discrete time intervals
after the current cut-off. A common method is to integrate the voltage electronically for a
standard time interval. In the Newmont M(331) standard (Van Voorhis et al. 1973), the
chargeability, mt, is defined as
1.1

mt = 1870
∫ 0.15
,
Vs dt
(5.2)
VDC
where the integration is carried out from 0.15 to 1.1 seconds after the current cut-off. The
chargeability value is given in milliseconds. The chargeability value obtained by this method
is calibrated (Summer 1976) so that the chargeability value in msec. has the same numerical
value as the chargeability given in mV/V. In theory, the chargeability in mV/V has a
maximum possible value of 1000.
IP surveys have traditionally been used in the mineral exploration industry,
particularly for metal sulfides, where heavy electrical generators producing high currents of
the order of 10 Amperes are used. The apparent IP values from such surveys are usually less
than 100 msec. (or mV/V). One recent development is the addition of IP capability to battery
based systems used in engineering and environmental surveys where currents of 1 Ampere or
less are normally used. An accompanying phenomenon is the observation of IP values of over
1000 msec. (or less than -1000 msec.) in some data sets. Such values are almost certainly
caused by noise due to a very weak IP signal. To check whether such high IP values are real,
first check the apparent resistivity pseudosection. If it shows unusually high and low values
that vary in an erratic manner, the data is noisy. If the apparent resistivity values are noisy,
then the apparent IP values are almost certainly unreliable. Next check the apparent IP
pseudosection. If the apparent IP values show an erratic pattern (frequently with anomalous
values lined up diagonally with an apex at a doubtful electrode), then the IP values are too
noisy to be interpretable.
(b) Frequency domain IP measurements


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In the frequency-domain methods, the apparent resistivity is measured at least two
frequencies, for example at 1 Hz. and 10 Hz. The higher frequency is usually set at 10 times
time the lower frequency. One commonly used frequency domain IP unit is the percent
frequency effect, PFE, which is defined by
ρ (ω L ) − ρ (ω H )
PFE = .100 (5.3)
ρ (ω H )
Another closely related unit that is also commonly used is the metal factor MF which is
defined by
ρ (ω L ) − ρ (ω H )
MF = .2π .10 5 (5.4)
ρ (ω L ).ρ (ω H )
Another common frequency domain IP unit is the phase angle, φ. It is the phase shift between
the transmitter current and the measured voltage, and the unit commonly used is milliradians.

(c) Relationship between the time and frequency domain IP units
From the measurement of the amplitude and phase spectrum of porphyry copper
mineralization, Van Voorhis et al. (1973) proposed the following equation to described the
observed spectra.
ρ (ω ) = K ( jω )− b (5.5)
K is constant and b is a measure of the IP effect. It is a positive number of more than 0 and
less than 0.1. This is also known as the constant phase model (Weller et al. 1996). By using
the above model, the following relationship between the different IP units and the b
parameter were derived.
(
PFE = 10 b − 1 .100 )
φ = 1571 b (5.6)
mt = 1320b
These relationships provide a numerical link between the different IP units (Van Voorhis et
al. 1973, Nelson and Van Voorhis 1973).

There have been a large number of 2-D IP surveys published over the years, so there is
certainly no lack of data to test the IP inversion with the RES2DINV program. Most of the
newer multi-electrode systems now come with an IP option. For environmental and
engineering surveys, the most useful application of the IP data is probably in differentiating
between sand and clay sediments.


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Exercise 5.1 : Tests with 2-D IP inversion
IPMODEL.DAT – A time- (1). After reading the data file, run the inversion with the
domain synthetic data set. default inversion options.
(2). Next try with the robust inversion option.
(3). The program also has an option to carry out the IP
inversion sequentially after the resistivity inversion (under
the “Type of IP inversion method” option in the
“Inversion” menu). Try this and see if there are any
differences in the results.
IPSHAN_PFE.DAT – A field (1). After reading the data file, carry out the inversion with
data set from Burma (Edwards the default inversion options. Try again using the option to
1977) with measurements in ‘Limit the range of model resistivity’ option.
PFE. (2). Try reducing the unit electrode spacing in the data file
with a text editor to half the given value. Does it improve
the inversion results?

IPMAGUSI_MF.DAT – A (1). After reading the data file, carry out the inversion with
field data set from Canada the default inversion options. Try again using the option to
(Edwards 1977) with ‘Limit the range of model resistivity’ option.
measurements in metal factor (2). Run the inversion again with the ‘Select robust
values. inversion’ option. This should make a significant difference
in the results.
(3). Try reducing the unit electrode spacing in the data file
with a text editor to half the given value. Does it improve
the inversion results?

IPKENN_PA.DAT – A field (1). After reading the data file, carry out the inversion with
data set with the IP values in the default inversion options.
phase angles (Hallof 1990).


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6 Cross-borehole imaging

6.1 Introduction
One of the most severe limitations of 2-D imaging surveys carried out along the
ground surface is the reduction in the resolution with depth. This is a fundamental physical
limitation that no amount of reconfiguration of the surface arrays or computer modeling can
overcome. In theory, the only way to improve the resolution at depth is to place the sensors
(i.e. the electrodes) closer to the structures of interest. This is not always possible, but when
such boreholes are present, cross-borehole surveys can give more accurate results than is
possible with surface surveys alone.
That has been many fine publications on such surveys; such as by Zhao et al. (1986),
Daily and Owen (1991), Sasaki (1992), LaBrecque et al. (1996), Slater et al. (1997, 2000)
and Zhou and Greenhalgh (1997, 2000). In the following section, I will attempt to summarize
the main results with regards to the choice of array configurations for cross-borehole surveys.
However, please refer to the original papers for the details.

6.2 Electrode configurations for cross-borehole surveys
In theory, any array that is used for normal surface surveys can be adapted for cross-
borehole surveys. Thus, we can have arrays with two electrodes, three electrodes and four
electrodes.
6.2.1 Two electrodes array – the pole-pole
There are two possible configurations, both electrodes can be in the same borehole
(Figure 6.1a), or the electrodes can be in different boreholes (Figure 6.1b). Figure 6.1c
considers a third possibility that is frequently not considered, with one electrode on the
surface. In all cases, the areas of highest sensitivities are concentrated near the electrodes,
particularly if the two electrodes are far apart in different boreholes as in Figure 6.1c. Note
that the electrodes do not actually scan the area between them, as would be expected for a
seismic survey with the source and receiver in different boreholes. Note that the region
between the two electrodes generally has negative sensitivity values.
If there are n electrodes (including surface electrodes, if any), there are a total of n(n-
1)/2 possible independent measurements. Most authors recommend measuring all the
possible readings at the other electrodes for a current electrode, i.e. measure all the possible
combinations. One problem with this array is the physical location of the two remote
electrodes, C2 and P2. They must be sufficiently far so that the pole-pole approximation is
sufficiently accurate. This means they must be located at a distance of at least 20 times the
maximum separation used by the active C1 and P1 electrodes in the boreholes. The large
distance between the P1 and P2 electrodes leads to the problem of contamination by telluric
noise. These problems are exactly the same as that faced with the pole-pole array for normal
surface arrays.
While many earlier researchers have used this array for cross-borehole surveys
(Dailey and Owen 1991, Shima 1992, Spies and Ellis 1995), more recent work tends to be
less enthusiastic about it. Sasaki (1992) and Zhou and Greenhalgh (2000) found that this
array has a significantly poorer resolution than the bipole-bipole and pole-bipole arrays.
The subsurface pole-pole array gives a useful illustration of the change in the
geometric factor for subsurface arrays. We have seen that for a pole-pole array with both
electrodes located on the surface of a half-space, the geometric factor k is given by
k = 2π a
where a is the spacing between the electrodes. For the case where both electrodes are within
an infinite medium, the geometric factor is given by


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k = 4π a
where r is the distance between the current and potential electrodes. For the case where both
electrodes are below the surface of a half-space, the geometric factor is given by
 rr ' 
k = 4π  
 r + r' 
where r’ is the distance of the reflected image of the current (Figure 6.2) from the potential
electrode.

Figure 6.1. The possible arrangements of the electrodes for the pole-pole array in the cross-
borehole survey and the 2-D sensitivity sections. The locations of the two boreholes are
shown by the vertical black lines.


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Figure 6.2. A schematic diagram of two electrodes below the surface. The potential measured
at P can be considered as the sum of the contribution from the current source C and its image
C’ above the ground surface.

6.2.2 Three electrodes array – the pole-bipole
According to Zhou and Greenhalgh (1997), there are six possible independent
configurations for the pole-bipole type of array (ignoring the surface electrodes). Of these six
configurations there are two basic combinations, with the current electrode and one potential
electrode in one borehole and the other potential electrode in the second borehole (Figure
6.3a), and with the current electrode in one borehole and both potential electrodes in the other
borehole (Figure 6.3b). Zhou and Greenhalgh (1997) recommend the configuration with one
current and potential electrode in the same borehole, and the second potential electrode in the
other borehole (the C1P1-P2 or AM-N configuration). In Figure 6.3a, it can be seen that his
configuration has high positive sensitivity values in between the two boreholes. This means
that it provides significant information about the resistivity of the material between the two
boreholes. The zones with the negative sensitivity values are confined to between the C1 and
P1 electrodes in the first borehole and to the left of the P2 electrode in the second borehole.
The second configuration, with the current electrode in one borehole and both
potential electrodes in the other borehole, has large negative sensitivity values between the
C1 and P1 electrode and large positive values between the C1 and P2 electrodes. In between
these two bands, there is a zone with small sensitivity values, i.e. the array does not give
significant information about the resistivity in this zone. Another possible disadvantage of
this array is that for some positions of the P1-P2 bipole, the potential value measured is very
small or zero. This causes the signal to noise ratio to be small.
Figure 6.3c shows the sensitivity pattern when all three electrodes are in the same
borehole. There are very high negative and positive sensitivity values in the vicinity of the
borehole. Sugimoto (1999) recommends that such measurements also be made as they give
valuable information about the dip of structures between the boreholes. Figure 6.3d shows the
sensitivity pattern when the current electrode is on the surface. The sensitivity values between
the C1 and P1 electrodes are relatively small (probably due to the large distance between
these two electrodes), while the sensitivity values between the P1 and P2 electrodes have
moderate positive values.


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Overall, many authors have positive remarks about this array. It provides better
resolution and is less sensitive to telluric noise (since the two potential electrodes are kept
within the survey area) compared to the pole-pole array. While in theory the resolution of the
array is slightly poorer than the bipole-bipole array, the potential values measured are
significantly higher.

Figure 6.3. The 2-D sensitivity patterns for various arrangements with the pole-bipole array.
The arrangement with (a) C1 and P1 in first borehole and P2 in second borehole, (b) C1 in
the first borehole and both P1 and P2 in the second borehole, (c) all three electrodes in the
first borehole and (d) the current electrode on the ground surface.


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6.2.3 Four electrodes array – the bipole-bipole
Zhou and Greenhalgh (1997) list four possible independent configurations for this
array, of which there two basic combinations. In the first arrangement, the positive current
and potential electrodes C1 and P1 are located in one borehole, while the negative current and
potential electrodes C2 and P2 are located in the second borehole (Figure 6.4a).

Figure 6.4. The 2-D sensitivity patterns for various arrangements of the bipole-bipole array.
(a) C1 and P1 are in the first borehole, and C2 and P2 are in second borehole. (b) C1 and C2
are in the first borehole, and P1 and P2 are in second borehole. In both cases, the distance
between the electrodes in the same borehole is equal to the separation between the boreholes.
The arrangements in (c) and (d) are similar to (a) and (b) except that the distance between the
electrodes in the same borehole is half the spacing between the boreholes.


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In this arrangement, the C1P1-C2P2 configuration (i.e. the AM-BN arrangement of
Zhou and Greenhalgh (1997), there are large positive sensitivity values in the area between
the two boreholes. This is a desirable property for a cross-borehole array since the intention is
to map the material between the two boreholes. The large negative sensitivity values are
confined to the region along the boreholes between the C1 and P1 (and C2 and P2
electrodes). Figure 6.4c shows a similar arrangement, but with the spacing between the
electrodes in the same borehole reduced to half the spacing between the boreholes. Not again
the large positive sensitivity values between the two boreholes.
In the second basic configuration (Figure 6.4b), the C1C2 current bipole is located in
one borehole while the P1P2 potential bipole is located in the other borehole. There is also a
large region with positive sensitivity values between the two boreholes. However it is flanked
by two zones with large negative sensitivity values. Thus the response of this C1C2-P1P2
arrangement to inhomogeneities between the boreholes is more complicated than the first
arrangement. When the bipole length is reduced, the positive region is significantly reduced.
These features make this arrangement less desirable for cross-borehole surveys. Another
disadvantage is that the potential signal strength is weaker in the C1C2-P1P2 arrangement
compared to C1P1-C2P2 configuration.
Overall, Zhou and Greenhalgh (1997,2000) recommend the C1P1-C2P2
configuration. Sasaki (1992) has found the bipole-bipole array to have better resolution
compared to the pole-pole and pole-dipole arrays. Figure 6.5 shows one possible field survey
measurement sequence. A series of measurements is first made with a short spacing (for
example half the distance between the boreholes) starting from the top. Next, measurements
are repeated with a larger spacing between the electrodes. Due to the symmetry in the
arrangements shown in Figures 6.5 and 6.5b, measurements should also be made with other
less symmetrical arrangements. Figures 6.5c and 6.5d show two other possible measurement
sequences.
Note that the length of the boreholes must be comparable to the distance between the
boreholes. Otherwise, if the spacing between the electrodes in the same borehole is much
smaller than the distance between the boreholes, the readings are likely to be more influenced
by the materials in the immediate vicinity of the boreholes, rather than the material in
between the boreholes. In such a situation, the only alternative is probably to include the
surface-surface and surface-borehole measurements.


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Exercise 6.1 : A few borehole inversion tests.
BOREHOLE.DAT – A (1). Read in the file, and try an inversion with the default
synthetic data set with the inversion parameters.
pole-bipole array. (2). Next you can try with the robust inversion option.

BORELUND.DAT – A pole- (1). Read in the file, and try an inversion with the default
pole array field data set from inversion parameters.
Lund University, Sweden. The (2). Next you can try different settings, such as the robust
survey was conducted to map inversion option.
fractures in a limestone-marl (3). The program also has an option to reduce the size of
formation (Dahlin pers. the model cells by half, which you might like to try.
comm.).
BORELANC.DAT – A bipole- (1). Read in the file, and run the inversion. The path of the
bipole array field data set from saline tracer is represented by regions with low resistivity
Lancaster University, U.K values. Can you identify the tracer in the model sections?
(Slater et al. 1997). The survey
was conducted to map the flow
of a saline tracer from the
ground surface through the
unsaturated zone.

Figure 6.5. Possible measurement sequences using the bipole-bipole array. Other possible
measurements sequences are described in the paper by Zhou and Greenhalgh (1997).


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6.3 Single borehole surveys
When only a single borehole is available, only the borehole to surface measurements
are possible. Figures 6.6a and 6.6b shows two possible arrangements with two electrodes in
the borehole and two electrodes on the surface. The first arrangement (Figure 6a) has the C1-
C2 current bipole in the borehole while the P1-P2 bipole is on the surface. This arrangement
has large positive sensitivity values between the C1-C2 bipole, and also between the P1-P2
bipole. The region of the subsurface between the two bipoles have moderate positive
sensitivity values, while an approximately parallel zone stretching from the C2 current
electrode (which is higher in the borehole) to the left of the P2 potential electrode has
moderate negative sensitivity values. Figure 6b shows the alternative arrangement with the
C1 and P1 electrodes in the borehole and the C2 and P2 pair on the surface. There is a small
region of large negative sensitivity values along the borehole between the C1 and P1
electrodes. There is also a small near-surface zone between the C2 and P2 electrodes with
large negative sensitivity values. However, note that most of the region between the two pairs
of electrodes has relatively high positive sensitivity values. Since the intention of the survey
is to map the subsurface between the two pairs of electrodes, this might be a better
arrangement than the one given in Figure 6.6a.
In some situations, it might only be possible to have only one subsurface electrode,
for example at the end of a drill bit or penetrometer (Sorensen 1994). Figures 6.6c and 6.6d
shows the arrangements where the upper electrode that was formerly in the borehole (in
Figures 6.6a and 6.6b) is now placed on the surface near the borehole. The arrangement with
the C1 electrode in the borehole and the C2 electrode on the surface (Figure 6.6c) has a near-
surface zone of large negative sensitive values between the C2 electrode and the P2 electrode.
However, there are moderately high positive sensitivity values in the region between the C1
electrode in the borehole and the P1-P2 bipole on the surface. The alternative arrangement
with the C1 electrode in the borehole and the P1 electrode near the top of the borehole has a
zone of large negative sensitivity values along the borehole. The region between the borehole
and the C2-P2 electrodes has generally high positive sensitivity values, so this arrangement
again might be better.
Figure 6.7 shows a pole-bipole configuration with the C1 current electrode in the
borehole and the P1-P2 potential bipole on the surface. The sensitivity pattern is fairly similar
to that obtained in Figure 6.6a. There is a region with moderate positive sensitivity values
between the C1 electrode and the P1-P2 bipole, together with an approximately parallel zone
of moderate negative sensitivity values between the C1 electrode and the region to the left of
the P1 electrode. Overall, the arrangement shown in Figure 6.6d appears to be best due to the
large zone of high positive sensitivity values between the borehole and the surface electrode
pair. For all the possible arrangements, it is assumed that the normal surface-to-surface
measurements are also made to fill in the gaps not covered by the borehole to surface
measurements. The surface-to-surface measurements are particularly important to accurately
map the resistivity distribution in the near-surface zone.


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Figure 6.6. Several possible bipole-bipole configurations with a single borehole. (a) The C1
and C2 electrodes at depths of 3 and 2 meters respectively below the 0 meter mark. (b) The
C1 and P1 electrodes at depths of 3 and 2 meters respectively below the 0 meter mark. (c).
The C1 electrode is at a depth of 3 meters below the 0 meter mark while the C2 electrode is
on the surface. (d). The C1 electrode is at a depth of 3 meters below the 0 meter mark while
the P1 electrode is on the surface.

Figure 6.7. A pole-bipole survey with a single borehole. The C1 electrode is at a depth of 3
meters below the 0 meter mark.


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7 2-D field examples

7.1 Introduction
Here we will look at a number of examples from various parts of the world to give
you an idea of the range of practical survey problems in which the electrical imaging method
has been successfully used.

7.2 Landslide - Cangkat Jering, Malaysia
A recent problem faced in Malaysia is landslides on hill slopes. The landslides are
often triggered by water accumulation within part of the slope that leads to weakening of a
section of the slope. Figure 7.1 shows the results from a survey conducted on the upper part
of a slope where a landslide had occurred in the lower section. Weathering of the granite
bedrock produces a clayey sandy soil mixed with core boulders and other partially weathered
material. The image obtained from this survey shows a prominent low resistivity zone below
the center of the survey line. This is probably caused by water accumulation in this region
that reduces the resistivity to less than 600 Ω⋅m. To stabilize the slope, it would be necessary
to pump the excess water from this zone. Thus, it is important to accurately map the zone of
ground water accumulation. This data set also shows an example with topography in the
model section.

Figure 7.1. Landslide field example, Malaysia. (a) The apparent resistivity pseudosection for
a survey across a landslide in Cangkat Jering and (b) the interpretation model for the
subsurface.

7.3 Old Tar Works - U.K.
A common environmental problem in industrial countries is derelict industrial land.
Before such land can be rehabilitated, it is necessary to map old industrial materials (such as
metals and concrete blocks) that are left buried in the ground. Another problem in such areas


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is chemical wastes that had been stored within the factory grounds. Due to the nature of such
sites, the subsurface is often very complex and is a challenging target for most geophysical
methods. The survey for this example was carried out on a derelict industrial site where
leachate was known, from a small number of exploratory wells, to be moving from a surface
waste lagoon into the underlying sandstones (Barker 1996). Eventually the leachate was seen
seeping into a nearby stream. However, the extent of the subsurface contamination was not
known.
An electrical imaging survey was carried out along an old railway bed between the
lagoon and the stream. The metal railway lines had been removed except for short lengths
embedded in asphalt below a large metal loading bay. In the apparent resistivity
pseudosection (Figure 7.2a), the area with contaminated ground water shows up as a low
resistivity zone to the right of the 140 metres mark. The metal loading bay causes a
prominent inverted V shaped low resistivity anomaly at about the 90 metres mark. In the
inversion model (Figure 7.2b), the computer program has managed to reconstruct the correct
shape of the metal loading bay near the ground surface. There is an area of low resistivity at
the right half of the section that agrees with what is known from wells about the occurrence
of the contaminated ground water. The plume is clearly defined with a sharp boundary at 140
metres along the profile. The contaminated zone appears to extend to a depth of about 30
metres.

Figure 7.2. Industrial pollution example, U.K. (a) The apparent resistivity pseudosection from
a survey over a derelict industrial site, and the (b) computer model for the subsurface.

7.4 Holes in clay layer - U.S.A.
This survey was carried out for the purpose of mapping holes in a clay layer that
underlies 8 to 20 feet of clean sand (Cromwell pers. comm.). The results from the electrical
imaging survey were subsequently confirmed by boreholes.
The pseudosection from one line from this survey is shown in Figure 7.3a. The data in
the pseudosection was built up using data from horizontally overlapping survey lines. One
interesting feature of this survey is that it demonstrates the misleading nature of the


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pseudosection, particularly for the dipole-dipole array. In the inversion model, a high
resistivity anomaly is detected below the 200 ft. mark, which is probably a hole in the lower
clay layer (Figure 7.3b). This feature falls in an area in the pseudosection where there is an
apparent gap in the data.
However, a plot of the sensitivity value of the cells used in the inversion model shows
that the model cells in the area of the high resistivity body have higher sensitivity values (i.e.
more reliable model resistivity values) than adjacent areas at the same depth with more data
points in the pseudosection plot (Figure 7.3c). This phenomenon is due to the shape of the
contours in the sensitivity function of the dipole-dipole array (Figure 2.11), where the areas
with the highest sensitivity values are beneath the C1C2 and P1P2 dipoles, and not at the
plotting point below the center of the array. This example illustrates the danger of only using
the distribution of the data points in the pseudosection to constrain the position of the model
cells (Barker 1992, Loke and Barker 1996a). If the model cells are placed only at the location
of the data points, the high resistivity body will be missing from the inversion model, and an
important subsurface feature would not be detected!

Figure 7.3. Mapping of holes in a clay layer, U.S.A. (a) Apparent resistivity pseudosection
for the survey to map holes in the lower clay layer. (b) Inversion model and (c) sensitivity
values of the model cells used by the inversion program.


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7.5 Time-lapse water infiltration survey - U.K.
Resistivity imaging surveys have not only been carried out in space, but also in time!
In some studies, the change of the subsurface resistivity with time has important applications.
Such studies include the flow of water through the vadose (unsaturated) zone, changes in the
water table due to water extraction (Barker and Moore 1998), flow of chemical pollutants and
leakage from dams (Johansson and Dahlin 1996).
A simple, but very interesting, experiment to map the flow of water from the ground
surface downwards through the unsaturated zone and into the water table was described by
Barker and Moore (1998). In this section, only some of the highlights of this experiment are
described as an illustration of a time-lapse survey. This experiment was carried out in the
Birmingham (England) area where forty thousand litres of water was poured on the ground
surface using a garden hose over a period of 10 hours. Measurements were made before and
during the irrigation of the ground surface, and after that for a period of about two weeks.
Figure 7.4 (a and b) shows the results of a survey carried out at the beginning of the
experiment before the irrigation started. The inversion model (Figure 7.4b) shows that the
subsurface, that consists of sand and gravel, is highly inhomogeneous. The water was poured
out near the 24 meters mark on this line, and Figure 7.4c shows the inversion model for the
data set collected after 10 hours of continuous irrigation. While the model resistivity values in
the vicinity of the 24 meters mark are generally lower than the initial data set model in Figure
7.4b, the subsurface distribution of the water is not very clear from a direct comparison of the
inversion models alone.

Figure 7.4. Water infiltration mapping, U.K. (a) The apparent resistivity and (b) inversion
model sections from the survey conducted at the beginning of the Birmingham infiltration
study. This shows the results from the initial data set that forms the base model in the joint
inversion with the later time data sets. As a comparison, the model obtained from the
inversion of the data set collected after 10 hours of irrigation is shown in (c).

The water distribution is more easily determined by plotting the percentage change in
the subsurface resistivity of the inversion models for the data sets taken at different times
(Figure 7.5) when compared with the initial data set model. The inversion of the data sets was


95

carried using a joint inversion technique where the model obtained from the initial data set
was used to constrain the inversion of the later time data sets (Loke 1999). The data set
collected at 5 hours after the pumping began shows a reduction in the resistivity (of up to
over 50 percent) near the ground surface in the vicinity of the 24 meters mark. The near-
surface low resistivity zone reaches its maximum amplitude after about 10 hours when the
pumping was stopped (Figure 7.5b). Twelve hours after the pumping was stopped, the low
resistivity plume has spread downwards and slightly outwards due to infiltration of the water
through the unsaturated zone. There is a decrease in the maximum percentage reduction in
the resistivity values near the surface due to migration of the water from the near surface
zone. This effect of the spreading of the plume becomes increasingly more pronounced after
24 hours (Figure 7.5d) and 36 hours (Figure 7.5e) due to further migration of the water. Note
that the bottom boundary of the zone with approximately 20 percent reduction in the
resistivity values tends to flatten out at a depth of about 3 meters (Figure 7.5e) where the
plume from the surface meets the water table.

Figure 7.5. Time-lapse sections from the infiltration study. The sections show the change in
the subsurface resistivity values with time obtained from the inversion of the data sets
collected during the irrigation and recovery phases of the study.


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7.6 Pumping test, U.K.
Resistivity imaging measurements were made during a pumping test in the Hoveringham area
of East Central England. The aquifer is a sand and gravel layer overlying mudstone. Figure
7.6 shows the initial apparent resistivity pseudosection, and model sections before pumping
and after 220 minutes of pumping. Figure 7.7 that shows the relative change in the resistivity
at at 40, 120 and 220 minutes after the start of the pumping test.

Figure 7.6. Hoveringham pumping test, U.K. (a) The apprent resistivity pseudosetcion at the
the beginning of the test. The inversion model sections at the (b) beginning and (v) after 220
minutes of pumping.

Figure 7.7. Percentage relative change in the subsurface resistivity values for the
Hoveringham pumping test. To highlight the changes in the subsurface resistivity, the
changes in the model resistivity are shown. Note the increase in the model resistivity below
the borehole with time.


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Figure 7.7 clearly shows the increase in the zone with higher resistivity values with time due
to the extraction of the water. By using Archie’s Law, and assuming the water resistivity does
not change with time, we can estimate the change in the water saturation values. The decrease
in the water saturation level within the aquifer, or desaturation values, is shown in Figure 7.8.
As Archie’s Law assumes that the conduction is due to the water content alone, the
desaturation values are likely to be lower than the true values if there is significant clay
content.

Figure 7.8. Use of Archie’s Law for the Hoveringham pumping test. Sections showing the
relative desaturation values obtained from the inversion models of the data sets collected
during the different stages of the Hoveringham pumping test. Archie’s Law probably gives a
lower limit for the actual change in the aquifer saturation.

7.7 Wenner Gamma array survey - Nigeria
The Wenner Gamma array (Figure 1.4c) has a relatively unusual arrangement where
the current and potential electrodes are interleaved. Compared to the Wenner Alpha and Beta
arrays, the Wenner Gamma array is much less frequently used in field surveys. However, in
some situations, there might be some advantage in using this array. The depth of investigation
is significantly deeper than the Wenner Alpha array (0.59a compared to 0.52a, see Table 2.1),
but the potential measured between the potential electrodes is only about 33% less than the
Alpha array. In comparison, the voltage that would be measured by the Wenner Beta array is
one-third that of the Alpha array which could be a serious disadvantage in noisy
environments.
Figure 7.9a shows the Wenner Gamma array pseudosection from a groundwater
survey in the Bauchi area of Nigeria (Acworth 1981). In this region, groundwater is
frequently found in the weathered layer above the crystalline bedrock. The weathered layer is
thicker in areas with fractures in the bedrock, and thus such fractures are good targets for
groundwater. In this area, the surveys were carried out with the Wenner Alpha, Beta and
Gamma arrays, together with electromagnetic profiling measurements using a Geonics


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EM34-3 system (Acworth 1987). Here, only the result from the Wenner Gamma array data
set is shown as an example.
To emphasize the boundary between the soil layer and the bedrock, the robust
inversion option was used (section 4.3). The inversion model in shown in Figure 7.9b. The
thickness of the lower resistivity weathered layer is generally about 10 to 20 meters. There is
a narrow vertical low resistivity zone with a width of less than 20 meters below the 190
meters mark that is probably a fracture zone in the bedrock. A borehole well that was placed
at the 175 meters mark that lies just at the edge of the fracture zone. It had yields that were
somewhat lower than expected (Acworth 1987). In such a situation, the 2D resistivity model
would be useful to pinpoint the exact location of the centre of the fracture zone to improve
the yield from the borehole. The placement of the well was largely based on resistivity and
EM profiling data, and many years before 2D resistivity inversion software and fast
microcomputers were widely available.

Figure 7.9. Groundwater survey, Nigeria. (a). Apparent resistivity pseudosection. (b) The
inversion model with topography. Note the location of the borehole at the 175 meters mark.

As a final note, it is possible to invert data collected with the Wenner Alpha, Beta and
Gamma arrays along the same line simultaneously with the RES2DINV program as a single
data set. This can be done by using the "non-conventional array" option in the program where
the positions of all the four electrodes in an array are explicitly specified. This might be an
interesting method to combine the advantages of the different variations of the Wenner array.

7.8 Mobile underwater survey - Belgium
Contrary to popular belief, it is actually possible to carry out resistivity surveys
underwater, even in marine environments. This example is one of the most unusual data sets
that I have come across, and a worthy challenge for any resistivity imaging inversion
software. It is not only the longest in physical length and number of electrode positions, but
also uses an unusual highly asymmetrical non-conventional electrode arrangement collected
by an underwater mobile surveying system. Mobile surveying systems have an advantage of
faster surveying speed, but on land they suffer from the problem of poor ground contact (for
the direct contact type) or low signal strength (for the electrostatic type). Please refer to
section 2.3 for the details. An underwater environment provides an almost ideal situation for
a direct contact type of mobile system since there is no problem in obtaining good electrode


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contact!
Figure 7.10a shows the data from an eight kilometers survey line along a river. This
survey was carried out by Sage Engineering of Belgium. The purpose of the survey was to
map the near surface lithology of the riverbed where there were plans to lay a cable. The data
set has a total of 7479 electrode positions and 6636 data points, whereas the inversion model
used has 19936 cells. On a 3.2 Ghz Pentium 4 computer, it took slightly less than 2 hours to
process this data set!

Figure 7.10. The inversion model after 4 iterations from an underwater riverbed survey by
Sage Engineering, Belgium.
In the inversion model (Figure 7.10b), most of the riverbed materials have a resistivity
of less than 120 Ω⋅m. There are several areas where the near-surface materials have
significantly higher resistivities of over 150 Ω⋅m. Unfortunately, geological information in
this area is rather limited. In the high resistivity areas, the divers faced problems in obtaining
sediment samples. The lower resistivity materials are possibly more coherent sediments
(possibly sand with silt/clay), whereas the higher resistivity areas might be coarser and less
coherent materials.

7.9 Floating electrodes survey – U.S.A.
This survey was carried out along the Thames River in Connecticut, USA using a streamer
floating on the water surface that was towed behind a boat. The steamer has 2 fixed current
electrodes and 9 potential electrodes. Measurements were made with an 8-channel resistivity
meter system. The water depth and conductivity were also measured during the survey. A
dipole-dipole type of array configuration was used but some of the measurements used a non-
symmetrical arrangement where the potential dipole length was different from the current
dipole length. The apparent resistivity pseudosection from a survey line is shown in Figure


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7.11a. The inversion model (Figure 7.11b) obtained when no special constraints were placed
on the part of the model that falls within the water layer shows artefacts near the surface due
to noise in the data. This model is fairly accurate when the material just below the river
bottom has a high resistivity, but poor in areas with low resistivities. These problems do not
occur in the model where the water layer resistivity was fixed during the inversion process
(Figure 7.11c).

Figure 7.11. Thames River (CT, USA) survey with floating electrodes. (a) The measured
apparent resistivity pseudosection. Inversion models obtained (b) without constraints on the
water layer, and (c) with a fixed water layer.

Besides these examples, 2-D imaging surveys have been carried for many other
purposes such as the detection of leakage of pollutants from landfill sites, areas with
undulating limestone bedrock, mapping of the overburden thickness over bedrock (Ritz et al.
1999), leakage of water from dams, saline water intrusion in coastal aquifers, freshwater
aquifers (Dahlin and Owen, 1998), monitoring of groundwater tracers (Nyquist et al., 1999)
and mapping of unconsolidated sediments (Christensen and Sorensen, 1994). The resistivity
imaging method has also been used in underwater surveys in lakes and dams.


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8 3-D Electrical Imaging Surveys
8.1 Introduction to 3-D surveys
Since all geological structures are 3-D in nature, a fully 3-D resistivity survey using a
3-D interpretation model (Figure 1.6c) should in theory give the most accurate results. At the
present time 3-D surveys is a subject of active research. However it has not reached the level
where, like 2-D surveys, it is routinely used. The main reason is that the survey cost is
comparatively higher for a 3-D survey of an area that is sufficiently large. There are two
current developments that should make 3-D surveys a more cost-effective option in the near
future. One is the development of multi-channel resistivity meters that enables more than one
reading to be taken at a single time. This is important to reduce the survey time. The second
development is faster microcomputers to enable the inversion of very large data sets (with
more than 8,000 data points and survey grids of greater than 30 by 30) to be completed within
a reasonable time.

8.2 Array types for 3-D surveys
The pole-pole, pole-dipole and dipole-dipole arrays are frequently used for 3-D
surveys. This is because other arrays have poorer data coverage near the edges of the survey
grid. The advantages and disadvantages of the pole-pole, pole-dipole and dipole-dipole arrays
that were discussed in section 2.5 with regards to 2-D surveys are also valid for 3-D surveys.

8.2.1 The pole-pole array
Figure 8.1 shows one possible arrangement of the electrodes for a 3-D survey using a
multi-electrode system with 25 nodes. For convenience the electrodes are usually arranged in
a square grid with the same unit electrode spacing in the x and y directions. To map slightly
elongated bodies, a rectangular grid with different numbers of electrodes and spacings in the
x and y directions could be used. The pole-pole electrode configuration is commonly used for
3-D surveys, such as the E-SCAN method (Li and Oldenburg 1992, Ellis and Oldenburg
1994b). The maximum number of independent measurements, nmax, that can be made with ne
electrodes is given by
nmax = ne (ne -1) / 2
In this case, each electrode is in turn used as a current electrode and the potential at all
the other electrodes are measured. Note that because of reciprocity, it is only necessary to
measure the potentials at the electrodes with a higher index number than the current electrode
in Figure 8.2a. For a 5 by 5 electrodes grid, there are 300 possible measurements. For 7 by 7
and 10 by 10 electrodes grids, a survey to measure the complete data set would have 1176
and 4500 data points respectively. For commercial surveys, grids of less than 10 by 10 are
probably not practical as the area covered would be too small.
It is can be very time-consuming (at least several hours) to make such a large number
of measurements, particularly with typical single-channel resistivity meters commonly used
for 2-D surveys. To reduce the number of measurements required without seriously
degrading the quality of the model obtained, an alternative measurement sequence is shown
in Figure 8.2b. In this proposed "cross-diagonal survey" method, the potential measurements
are only made at the electrodes along the x-direction, the y-direction and the 45 degrees
diagonal lines passing through the current electrode. The number of data points with this
arrangement for a 7 by 7 grid is reduced to 476 which is about one-third of that required by a
complete data set survey (Loke and Barker 1996b).


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Figure 8.1. The arrangement of the electrodes for a 3-D survey.

Figure 8.2. Two possible measurement sequences for a 3-D survey. The location of potential
electrodes corresponding to a single current electrode in the arrangement used by (a) a survey
to measure the complete data set and (b) a cross-diagonal survey.


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In some cases, 3-D data sets are constructed from a number of parallel 2-D survey
lines (section 8.3). Ideally there should be a set of survey lines with measurements in the x-
direction, followed by another series of lines in the y-direction. The use of measurements in
two perpendicular directions helps to reduce any directional bias in the data.
However, in some cases, only the data from a series of survey lines in one direction is
available. This is particularly common if the surveys were originally conducted to provide 2-
D images. Sometimes the spacing between the “in-line” electrodes is significantly smaller
than the spacing between the lines. One important question is the maximum spacing between
the lines that can be used for the data to be still considered “3-D”. A useful guide is the 3-D
sensitivity plot. Figure 8.3 shows the sensitivity values on horizontal slices through the earth.
The electrodes are arranged along the 0 and 1 meter marks along the x-axis. Near the surface,
there is an approximately circular region with negative sensitivity values in the top two slices
at depths of 0.07 and 0.25 meter. The zone with the largest sensitivity (using the 4 units
sensitivity contour line as a guide) extends in the y-direction to slightly over half the
electrode spacing. This means to get a complete 3-D coverage, if the measurements are only
made in the x-direction, the spacing between the lines should not be much more than the
smallest electrode spacing used.

Figure 8.3. 3-D sensitivity plots for the pole-pole array. The plots are in the form of
horizontal slices through the earth at different depths.

The pole-pole array has two main disadvantages. Firstly it has a much poorer


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resolution compared to other arrays. Subsurface structures tend to be smeared out in the final
inversion model. The second disadvantage, particularly for large electrode spacings, is that
the second current electrode and potential electrode must be placed at a sufficiently large
distance from the survey grid. Both disadvantages have been discussed in detail in Section
2.5.7. Park and Van (1991) who used this array for a field experiment found that about 15%
of the measurements did not satisfy reciprocity because the contribution for the remote
electrodes were significant. In general, this probably affects the readings with the larger
spacings.

8.2.2 The pole-dipole array
This array is an attractive alternative to the pole-pole array for surveys with medium
and large survey grids (12 by 12 and above). It has a better resolving power than the pole-
pole array (Sasaki 1992), and is less susceptible to telluric noise since both potential
electrodes are kept within the survey grid. Compared to the dipole-dipole array, it has a much
stronger signal strength. Although it has one “remote” electrode (the C2 electrode), the effect
of this electrode on the measurements is much smaller compared to the pole-pole array
(section 2.5.8). As the pole-dipole array is an asymmetrical array, measurements should be
made with the “forward” and “reverse” arrangements of the electrodes (Figure 2.16). To
overcome the problem of low signal strength for large values of the “n” factor (exceeding 8),
the “a” spacing between the P1-P2 dipole pair should be increased to get a deeper depth of
investigation with a smaller “n” factor. The use of redundant measurements with overlapping
data levels to increase the data density can in some cases help to improve the resolution of the
resulting inversion model (section 2.5.9).
Figures 8.4 and 8.5 show the sensitivity patterns for this array with the dipole
separation factor “n” equal to 1 and 4 respectively. There is prominent area with negative
sensitivity values between the C1 and P1 electrodes (located at 0.0 and 0.5 meter along the x-
axis). The plots are arranged such that the array length (in this case the distance between the
C1 and P2 electrodes) is set at 1.0 meter for both “n” factors. For a larger “n” factor, the area
of negative sensitivities between the C1 and P1 electrodes becomes larger and extends to a
greater depth. The array is more sensitive to structures off the array axis (i.e. in the y-
direction) when “n” is equals to 1. The area with the higher sensitivity values extends to
about 0.8 times the array length (Figure 8.4), or 1.6 times the unit electrode spacing. When
the “n” factor is larger (Figure 8.5), the array is more sensitive to off-axis structures near the
P1-P2 dipole. Note also the negative sensitivity values to the right of the P2 electrode.
If the 3-D survey is carried out with a series of parallel lines, and the cross-line
measurements are not made, the distance between the lines should preferably be within two
to three times the inline unit electrode spacing. This is to ensure that the subsurface material
between the lines is adequately mapped by the in-line measurements.

8.2.3 The dipole-dipole, Wenner and Schlumberger arrays
This array can be recommended only for grids that are larger than 12 by 12 due to the
poorer horizontal data coverage at the sides. The main problem that is likely to be faced with
this array is the comparatively low signal strength. Similar to 2-D surveys, this problem can
be overcome by increasing the “a” spacing between the P1-P2 dipole to get a deeper depth of
investigation as the distance between the C1-C2 and P1-P2 dipoles is increased. Also, the use
of overlapping data levels is recommended (section 2.5.9).


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Figure 8.4. 3-D sensitivity plots for the pole-dipole array with n=1 in the form of horizontal
slices through the earth at different depths. The C1 electrode is the leftmost white cross.

Figure 8.5. 3-D sensitivity plots for the pole-dipole array with n=4 in the form of horizontal
slices through the earth at different depths.


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Figure 8.6 and 8.7 shows the sensitivity patterns for the dipole-dipole array when the
“n” factor is equal to 1 and 4 respectively. The sensitivity values have small but negative
values outside the immediate vicinity of the array. Another interesting feature is that the
sensitivity contours tend to be elongated in the y-direction, particularly for the larger “n”
value. The 4 units sensitivity contour extends to about 0.6 times the array length in the y-
direction, or about 1.8 times the unit electrode spacing. This means that the array is more
sensitive to structures off the array axis compared to the pole-pole and pole-dipole arrays.
This feature is troublesome in 2-D surveys, but might be advantageous in 3-D surveys.
The off-axis elongation of the sensitivity contours agrees with the observation by
Dahlin and Loke (1997) that the dipole-dipole array is more sensitive to 3-D effects
compared to the other common arrays. This factor is important when the dipole-dipole array
is used in 2-D imaging surveys where it is assumed that the subsurface geology is 2-D.
In many cases, 3-D data sets for the dipole-dipole arrays are constructed from a
number of parallel 2-D survey lines, particularly from previous surveys. Due to the elongated
sensitivity pattern, the dipole-dipole array can probably tolerated a larger spacing between the
survey lines (to about three times the inline unit electrode spacing) and still contain
significant 3-D information.
In closing this section on four electrodes arrays, Figures 8.8, 8.9 and 8.10 show the
sensitivity patterns for the Wenner alpha, Schlumberger and Wenner gamma arrays (the
Wenner beta is the dipole-dipole with a “n” value of 1 as shown in Figure 8.6). The
sensitivity contours for the Wenner alpha array, outside of the immediate vicinity of the
electrodes, are elongated in the direction of the line of electrodes. This means that the
Wenner alpha array is less sensitive to off-line structures than the dipole-dipole array, i.e. it is
less sensitive to 3-D. This agrees with empirical observations by Dahlin and Loke (1997).
The sensitivity pattern for the Wenner-Schlumberger array (Figure 8.9) is similar to that for
the Wenner alpha array except for a slight bulge near the center of the array.
The sensitivity patterns for the Wenner gamma array (Figure 8.10) show characteristic
bulges near the C1 and C2 electrodes that were observed earlier in the 2-D sensitivity
sections (Figure 2.10). Thus it is expected to be more sensitive to 3-D structures near the C1
and C2 electrodes.

8.2.4 Summary of array types
For relatively small grids of less than 12 by 12 electrodes, the pole-pole array has a
substantially larger number of possible independent measurements compared to other arrays.
The loss of data points near the sides of the grid is kept to a minimum, and it provides better
horizontal data coverage compared to other arrays. This is an attractive array for small survey
grids with relatively small spacings (less than 5 meters) between the electrodes. However, it
has the disadvantage of requiring two “remote” electrodes that must be placed at a
sufficiently large distance from the survey grid. Due to the large distance between the two
potential electrodes, this array is more sensitive to telluric noise. The pole-dipole array is an
attractive option for medium size grids. It has a higher resolution than the pole-pole array, it
requires only one remote electrode and is much less sensitive to telluric noise. For surveys
with large grids, particularly when there is no convenient location for a remote electrode, the
dipole-dipole array can be used. For the pole-dipole and dipole-dipole arrays, measurements
with overlapping data levels using different “a” and “n” combinations should be used to
improve the quality of the results. The electrodes for 3-D surveys are normally arranged in a
rectangular grid with a constant spacing between the electrodes (Figure 8.1). However, the
RES3DINV resistivity and IP inversion program can also handle grids with a non-uniform
spacing between the rows or columns of electrodes.


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Figure 8.6. 3-D sensitivity plots for the dipole-dipole array with n=1 in the form of horizontal

Figure 8.7. 3-D sensitivity plots for the dipole-dipole array with n=4 in the form of horizontal


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Figure 8.8. The 3-D sensitivity plots for the Wenner alpha array at different depths.

Figure 8.9. The 3-D sensitivity plots for the Wenner-Schlumberger array with the n=4 at
different depths.


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Figure 8.10. The 3-D sensitivity plots for the Wenner gamma array at different depths.


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8.3 3-D roll-along techniques
Most commercial 3-D surveys will probably involve grids of at least 16 by 16 in order
to cover a reasonably large area. A 16 by 16 grid will require 256 electrodes which is more
than that available on many multi-electrode resistivity meter systems. One method to survey
such large grids with a limited number of electrodes is to extend the roll-along technique used
in 2-D surveys to 3-D surveys (Dahlin and Bernstone 1997). Figure 8.11 shows an example
of a survey using a multi-electrode resistivity-meter system with 50 electrodes to survey a 10
by 10 grid. Initially the electrodes are arranged in a 10 by 5 grid with the longer lines
orientated in the x-direction (Figure 8.10a). Measurements are made primary in the x-
direction, with some possible measurements in the diagonal directions. Next the entire grid is
moved in the y-direction so that the 10 by 5 grid now covers the second half of the 10 by 10
grid area. The 10 by 5 grid of electrodes is next orientated in the y-direction (Figure 8.11b).
The example data file PIPE3D.DAT was obtained from a survey using such a roll-
along technique. It was carried out with a resistivity-meter system with only 25 nodes with
the electrodes arranged in an 8 by 3 grid. The long axis of this grid was orientated
perpendicularly to two known subsurface pipes. The measurements were made using three
such 8 by 3 sub-grids so that the entire survey covers an 8 by 9 grid. For each 8 by 3 sub-grid,
all the possible measurements (including a limited number in the y-direction) for the pole-
pole array were made. In this survey, the second set of measurements in the y-direction (as in
Figure 8.11b) was not carried out to reduce the survey time, and also because the pipes have
an almost two-dimensional structure.
For practical reasons, the number of field measurements in some surveys might be
even less than the cross-diagonal technique. Another common approach is to just make the
measurements in the x- and y- directions only, without the diagonal measurements. This is
particularly common if the survey is made with a system with a limited number of
independent electrodes, but a relatively large grid is needed.
In some cases, measurements are made only in one direction. The 3-D data set
consists of a number of parallel 2-D lines. The data from each 2-D survey line is initially
inverted independently to give a series of 2-D cross-sections. The measured apparent
resistivity values from all the lines can also be combined into a 3-D data set and inverted with
RES3DINV to give a 3-D picture. While the quality of the 3-D model is expected to be
poorer than that produced with a complete 3-D survey, such a “poor man’s” 3-D data set
could reveal major resistivity variations across the survey lines (see also section 8.5). Until
multi-channel resistivity instruments are widely used, this might be the most cost-effective
solution to extract some 3-D information from 2-D surveys. This arrangement might be
particularly useful for surveys with the dipole-dipole array that can tolerate larger spacings
between the survey lines.

8.4 A 3-D forward modeling program
In the interpretation of data from 2-D resistivity imaging surveys, it is assumed that
the subsurface geology does not change significantly in the direction that is perpendicular to
the survey line. In areas with very complex geology, there are could be significant variations
in the subsurface resistivity in this direction (i.e. the geology is 3-D), which could cause
distortions in the lower sections of the 2-D model obtained (please refer to section 4.7).
The free 3-D resistivity forward modeling program, RES3DMOD.EXE, enables you
to calculate the apparent resistivity values for a survey with a rectangular grid of electrodes
over a 3-D structure. This is a Windows based program that can be used from within
Windows 95/98/Me/2000/NT. To take a look the operation of the program, use the “File”
option followed by “Read model data” to read in the file BLOCK11.MOD, which has a 11 by
11 survey grid. After that, click the “Edit/Display” option.


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Figure 8.11. Using the roll-along method to survey a 10 by 10 grid with a multi-electrode
system with 50 nodes. (a) Surveys using a 10 by 5 grid with the lines orientated in the x-
direction. (b) Surveys with the lines orientated in the y-direction.


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To modify the 3-D model, click the “Edit resistivity model” option. In this option,
you can change the resistivity of the 3-D cells in the mesh used by the finite-difference
method (Dey and Morrison 1979b) to calculate the apparent resistivity values. To quit from
the “Edit” mode, press the Q or the Esc key. To calculate the apparent resistivity values, click
the “Calculate” option. To take a look at the apparent resistivity pseudosections, click the
“Display apparent resistivity” option. You can choose to display the apparent resistivity
values in the form of horizontal pseudosections, or as vertical pseudosections as used in 2-D
surveys. Displaying the vertical pseudosections will give you an idea of the effect of a 3-D
structure on the measurements in a 2-D survey. In a study made by Dahlin and Loke (1997),
the dipole-dipole array was found to be the most sensitive to 3-D effects while the Wenner
array was the least sensitive.
The RES3DMOD program also has an option to save the apparent resistivity values
into a format that can be accepted by the RES3DINV inversion program. As an exercise, save
the apparent resistivity values as a RES3DINV data file for one of the models, and later carry
out an inversion of this synthetic data set.
Figure 8.12a shows an example of a 3-D model with a 15 by 15 survey grid (i.e. 255
electrodes). The model, which consists of four rectangular prisms embedded in a medium
with a resistivity of 50 Ω⋅m, is shown in the form of horizontal slices through the earth. The
apparent resistivity values for the pole-pole array (with the electrodes aligned in the x-
direction) are shown in the form of horizontal pseudosections in Figure 8.12b. Note the low
resistivity block with a resistivity of 10 Ω⋅m near the centre of the grid that extends from a
depth of 1.0 to 3.2 meters. For measurements with the shorter electrode spacings of less than
4 meters this block causes a low resistivity anomaly. However, for electrode spacings of
greater than 6 meters, this low resistivity prism causes a high resistivity anomaly! This is an
example of “anomaly inversion” which is caused by the near-surface zone of negative
sensitivity values between the C1 and P1 electrodes (Figure 8.3).
Exercise 8.1 : 3-D forward modeling examples
PRISM2.MOD – This is a 3-D (1). Read in the file, and then run the “Calculate” step to
model used to generate the calculate the potential values. This is a relatively large
pseudosections shown in model file, so you might to collect some coffee while the
Figure 8.12 to illustrate 3-D computer is running.
effects in 2-D surveys. (2). Next choose the “Display apparent resistivity” option
under the “Edit/Display” menu. Try first with pole-pole
array, and take a look at the pseudosections.
(3).Next try with other arrays.

BLOCK15.MOD – A 15 by 15 (1). Read in the file, and then run the “Calculate” step to
grid model with several calculate the potential values.
rectangular prisms. (2). Next choose the “Display apparent resistivity” option
under the “Edit/Display” menu. Take a look at the
pseudosections for a few arrays.
(3). Try using the “Edit model” option to change the model,
and then recalculate the potential values. Check out the
effect of your changes on the apparent resistivity
pseudosections.


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Figure 8.12. A 3-D model with 4 rectangular prisms in a 15 by 15 survey grid. (a) The finite-
difference grid. (b) Horizontal apparent resistivity pseudosections for the pole-pole array with
the electrodes aligned in the x-direction.


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8.5 3-D inversion algorithms and 3-D data sets
Two interesting questions that sometimes arise, particularly in non-academic circles,
is the definition of a 3-D inversion algorithm and what constitutes a 3-D data set. The first
question can be easily answered, but the second is less clear.
A defining feature of a 3-D algorithm inversion algorithm is that it allows the model
resistivity values to vary in all three directions, i.e. in the x-, y- and z-directions. This is in
contrast to 2-D inversion where the subsurface resistivity is assumed to vary only in the x-
and z-directions but constant in the y-direction; and in 1-D inversion where the resistivity is
only allowed to changed in the z-direction. The inversion model used by the RES3DINV
program consists of independent rectangular cells (Figure 8.13) where the model values are
allowed to vary in all three directions simultaneously, so it is uses a true 3-D inversion
algorithm. Note that a model constructed from a series of 2-D inversions along parallel lines
is not a true 3-D inversion model.
Another defining characteristic of a 3-D inversion algorithm is the use of a 3-D
forward modeling subroutine, such as the 3-D finite-difference and finite-element methods
(Dey and Morrison 1979b, Silvester and Ferrari 1990), to calculate the model apparent
resistivity and Jacobian matrix values.
A more difficult question is what constitutes a “3-D” data set. At present there is no
generally accepted definition of a 3-D data set. While the inversion algorithm used to invert
the data set is 3-D, whether the data set contains significant 3-D information is another
matter. It was stated in section 2.2 that to get a good 2-D model for the subsurface the data
coverage must be 2-D as well. For a 2-D survey, measurements are made with different
electrode spacings and at different horizontal locations to obtain such a 2-D coverage.
However, the degree of data coverage needed before a data set can be considered “3-D” is
less clear. Using the pole-pole survey as an example, the following classification system is
proposed for 3-D data sets. They are listed in decreasing order of 3-D information content.
Category 1 - An ideal 3-D survey with the electrodes arranged in a rectangular grid, and with
measurements in all possible directions (such as in Figure 8.2a), i.e. along the grid lines as
well as at different angles to the grid lines. The SEPTIC.DAT data file is such an example of
a “complete” 3-D data set. In this survey, 56 electrodes were arranged in an 8 by 7 gird and
all the possible 1540 (56x55/2) measurements were made.
Category 2 – The electrodes are arranged in a rectangular grid. All the measurements along
the grid lines (i.e. in the x- and y-directions) but only a limited number of measurements at an
angle to the grid lines( such as along the 45 degree diagonals for square grids as shown in
Figure 8.2b) are made. The ROOTS7.DAT data file is such an example of a survey with
limited measurements in the angular directions. In this case, the survey was carried out 49
electrodes in a 7 by 7 grid and 468 measurements were made. A “complete” 3-D data set
would have 1176 measurements.
Category 3 – Measurements are only made in the two directions along the grid lines, i.e. in
the x- and y-directions, and no measurements at an angle to the grid lines are made. This
measurement sequence is frequently used when there are insufficient nodes in the multi-
electrode system to cover the entire survey area at a single time. One possible measurement
sequence is shown in Figure 8.11. The data for the sludge deposit field example (Dahlin and
Bernstone 1997) in section 8.7.2 falls under this category.
Category 4 – Measurements in only one direction (for example the x-direction) along a series
of parallel 2-D survey lines. This situation is common for data from old surveys, particularly
IP surveys for mineral deposits. For this type of data set, a series of 2-D inversions is usually
first carried out. The 3-D inversion is then used on a combined data set with the data from all
the survey lines in an attempt to gleam new information out of old data, and to see whether 3-
D effects are significant (i.e. whether the results from the 2-D inversions are valid). The


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success of the 3-D inversion partly depends on the spacing between the lines and the type of
array used (section 8.2). As a general rule, the spacing between the lines should not be more
than twice the unit electrode spacing along the lines.
While ideally the data should be of Category 1 or at least Category 2 so that some
angular data is available, the RES3DINV program will also accept data that falls under
Categories 3 and 4. The accuracy of the 3-D models obtained from the Categories 3 and 4
data types will be lower than the Categories 1 and 2 data types, and will greatly depend on
the spacing between the lines and the type of array used. However, even for the Category 4
data type, the results from the 3-D inversion should provide a useful indicator on whether 3-D
effects are significant. This can provide a check on the validity of the results obtained from
independent 2-D inversions of the different survey lines.

8.6 A 3-D inversion program
3-D inversion of field data set can be carried out in a similar way using the
smoothness-constrained least-squares method used for the 2-D inversion. One model used to
interpret the 3-D data set is shown in Figure 8.13a. The subsurface is divided into several
layers and each layer is further subdivided into a number of rectangular cells. A 3-D
resistivity inversion program, RES3DINV, is used to interpret the data from 3-D surveys.
This program attempts to determine the resistivity of the cells in the inversion model that will
most closely reproduce the measured apparent resistivity values from the field survey. Within
the RES3DINV program, the thickness of the layers can be modified by the user. Two other
alternative models that can be used with the RES3DINV program are shown in Figures 8.13b
and 8.13c. The second inversion model subdivides the top few layers vertically as well as
horizontally by half. Another alternative is to subdivide the top few layers by half only in the
horizontal directions (Figure 8.13c). Since the resolution of the resistivity method decreases
rapidly with depth, it has been found that subdividing the blocks is only beneficial for the top
two layers only. In many cases, subdividing the top layer only is enough. By subdividing the
cells, the number of model parameters and thus the computer time required to invert the data
set can increase dramatically.

Figure 8.13. The models used in 3-D inversion. (a) Standard model where the widths of the
rectangular cells are equal to the unit electrode spacings in the x- and y-directions. (b) A
model where the top few layers are divided by half, both vertically and horizontally, to
provide better resolution. (c) A model where the model cells are divided in the horizontal
direction but not in the vertical direction.


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Please refer to the instruction manual (RES3DINV.PDF) for the RES3DINV program
for the data format. The set of files that comes with the RES3DINV program package has a
number of field and synthetic data files. You can carry out an inversion of some of these files
to get a feel of how the program works.

Exercise 8.2 : 3-D inversion examples
PIPE3D.DAT – A field survey (1). Read in the file, and then run the “Carry out inversion”
data file where a roll-along step.
technique was used. The total (2). After that, take a look at the results using the “Display”
grid size is 9 by 8. window.
ROOTS7.DAT – A “cross- (1). Read in the file, and then invert the data set.
diagonal survey” data set using (2). Try using the “Robust inversion” option, and see
a 7 by 7 survey. whether there is a significant change in the results.

ROOT7_REMOTE.DAT – Try inverting this data file and see whether the remote
The same data set with the electrodes have a significant effect of the results.
location of the C2 and P2
electrodes included.
BLOCKS11T.DAT – A (1). Read in the file, and then invert the data set.
synthetic data set with (2). Invert again using the “Robust inversion” option.
topography. It has a 11 by 11 The inversion of this data set could take a while for
grid. computers slower than 1 GHz.

8.7 Examples of 3-D field surveys
In this section, we will take a look at the results from a couple of 3-D field surveys
over areas with complex geology.

8.7.1 Birmingham field test survey - U.K.
This field test was carried out using a multi-electrode system with 50 electrodes
commonly used for 2-D resistivity surveys. The electrodes are arranged in a 7 by 7 grid with
a unit spacing of 0.5 meter between adjacent electrodes (Figure 8.14). The two remote
electrodes were placed at more than 25 meters from the grid to reduce their effects on the
measured apparent resistivity values. To reduce the survey time, the cross-diagonal survey
technique was used. The subsurface is known to be highly inhomogenous consisting of sands
and gravels. Figure 8.15a shows the horizontal sections of the model obtained at the 6th
iteration. The two high resistivity zones in the upper left quadrant and the lower right corner
of Layer 2 are probably gravel beds. The two low resistivity linear features at the lower edge
of Layer 1 are due to roots from a large sycamore tree just outside the survey area. The
vertical extent of the gravel bed is more clearly shown in the vertical cross-sections across the
model (Figure 8.15b). The inverse model shows that the subsurface resistivity distribution in
this area is highly inhomogenous and can change rapidly within a short distance. In such a
situation a simpler 2-D resistivity model (and certainly a 1-D model from conventional
sounding surveys) would probably not be sufficiently accurate.


117

Figure 8.14. Arrangement of electrodes in the Birmingham 3-D field survey.

8.7.2 Sludge deposit - Sweden
This survey covers a relatively large 21 by 17 grid by using a 3-D roll-along method
(Dahlin and Bernstone 1997). To reduce the survey time, a number of parallel multi-electrode
cables were used. This survey was carried out at Lernacken in Southern Sweden over a closed
sludge deposit. Seven parallel multi-electrode cables were used to cover a 21 by 17 grid with
a 5 meters spacing between adjacent electrodes. There were a total number of 3840 data
points in this data set.
In this survey, the cables were initially laid out in the x-direction, and measurements
were made in the x-direction. After each set of measurements, the cables were shifted step by
step in the y-direction until the end of the grid. In surveys with large grids, such as in this
example, it is common to limit the maximum spacing for the measurements. The maximum
spacing is chosen so that the survey will map structures to the maximum depth of interest
(section 2.5.2). In this case, the maximum spacing was 40 meters compared to the total length
of 100 meters along a line in the x-direction.
The model obtained from the inversion of this data set is shown in Figure 8.16. The
former sludge ponds containing highly contaminated ground water show up as low resistivity
zones in the top two layers (Dahlin and Bernstone 1997). This was confirmed by chemical
analysis of samples. The low resistivity areas in the bottom two layers are due to saline water
from a nearby sea. Figure 8.17 shows a 3-D view of the inversion model using the Slicer-
Dicer 3-D contouring program. On a 200 Mhz Pentium Pro computer, it took slightly over 4
hours to invert this data set. On newer computers with operating frequencies of 550 Mhz or
higher, the inversion time should be significantly shorter.


118

Figure 8.15. Horizontal and vertical cross-sections of the model obtained from the inversion
of the Birmingham field survey data set. The location of observed tree roots on the ground
surface are also shown.


119

Figure 8.16. The 3-D model obtained from the inversion of the Lernacken Sludge deposit
survey data set displayed as horizontal slices through the earth.


120

Figure 8.17. A 3-D view of the model obtained from the inversion of the Lernacken Sludge
deposit survey data set displayed with the Slicer/Dicer program. A vertical exaggeration
factor of 2 is used in the display to highlight the sludge ponds. Note that the color contour
intervals are arranged in a logarithmic manner with respect to the resistivity.
8.7.3 Copper Hill – Australia
This is an interesting example of a 3-D resistivity and IP survey provided by Arctan
Services Pty. and Golden Cross Resources, Australia. Copper Hill is the oldest copper mine
in NSW, Australia. An earlier survey was conducted in 1966 using mapping, rock chip
sampling, an IP survey and 7 drill-holes (White at al. 2001). Copper porphyry with minor
gold and palladium mineralization were found to occur in structurally controlled fractures and
quartz veins. However, due to the very complex geology (Figure 8.18), large differences in
ore grades were found in drill-holes that were less than 200 meters apart.
To map the ore deposit more accurately, a new 3-D resistivity and IP survey using the
pole-dipole array was used. The survey covered a large (1.6 x 1.1km) area using a series of
1.6 km lines with a spacing of 25m between adjacent electrodes. Figure 8.19 shows the
arrangement of the transmitter and receiver lines. Currents of up to 7 Amps were used. The
entire survey took 10 days giving a total of over 7000 measurements. Further details about
the survey layout and procedures used to improve the data quality as well as to reduce the
survey time are described in the paper by Denne at al. (2001). A copy of the paper can be
downloaded from the www.arctan.com.au web site. Other interesting information about the
mineralization at the Copper Hill area is available at the Golden Cross Resources Ltd.
www.reflections.com.au/GoldenCross/ web site.
The data was inverted with the RES3DINV program that produced a 3-D resistivity as
well as a 3-D IP model for the area. The 3-D IP model that shows the location of the
mineralized zones more clearly is shown in Figure 8.20. The inversion model output from the
RES3DINV program was rearranged into a VRML format that could be read by a 3-D
visualization program (please contact Arctan Services for the details) that enables the user to
display the model from any direction.


121

Figure 8.18. Geological map of the Copper Hill area (Chivas and Nutter 1975, White et al.
2001).

Figure 8.19. Electrodes layout used for the 3-D survey of the Copper Hill area (White et al.
2001).


122

The 3D IP model in Figure 8.20 shows two en-echelon north-south trends and two
approximately east-west trends forming an annular zone of high chargeability. The results
from existing drill-holes that had targeted the shallower part of the western zone agree well
with the resistivity and IP model. A drill-hole, CHRC58, intersected a 217m zone with 1.7 g/t
gold and 0.72% copper coincided well an IP zone of greater than 35mV/V. The lower
boundary of the western zone with high chargeability coincides well with low assay results
from existing drill-holes. The eastern zone with high chargeability and resistivity values do
not outcrop on the surface and very little drilling has penetrated it. Further surveys, including
drilling, is presently being carried out.

Figure 8.20. The IP model obtained from the inversion of the Copper Hill survey data set.
Yellow areas have chargeability values of greater than 35 mV/V, while red areas have
chargeability values of greater than 45 mV/V (White et al. 2001).

A characteristic feature of 3-D surveys is the large number of electrodes and measurements.
To carry out such surveys effectively, the multi-electrode system should have at least 64 (and
preferably 100 or more) electrodes. This is an area where a multi-channel resistivity meter
system would be useful. For fast computer inversion, the minimum requirement is a Pentium
4 system with at least 256 megabytes RAM and a 40 GB hard-disk.


123

Acknowledgments

Dr. Torleif Dahlin of Lund University in Sweden kindly provided the Odarslov Dyke
and Lernacken data sets. The Grundfor data set was provided by Dr. Niels B. Christensen of
the University of Aarhus in Denmark and Dr. Torleif Dahlin. The Rathcroghan data set was
kindly provided by Dr. Kevin Barton and Dr. Colin Brown from data collected by the
Applied Geophysics Unit of University College Galway, Ireland. Many thanks to Richard
Cromwell and Rory Retzlaff of Golder Assoc. (Seattle) for the survey example to map holes
in a clay layer. Dr. Andrew Binley of Lancaster University kindly provided the interesting
cross-borehole data set. The Bauchi data set was provided by Dr. Ian Acworth of the
University of New South Wales, Australia. The Redas river survey data set was kindly
provided by Jef Bucknix of Sage Engineering, Belgium. Mr. Julian Scott and Dr. R.D. Barker
of the School of Earth Sciences, University of Birmingham provided the Clifton, the Tar
Works and the 3-D Birmingham survey data sets. Slicer/Dicer is a registered trademark of
Visualogic Inc. Last, but not least, permission from Arctan Services Pty. and Gold Cross
Resources to use the Copper Hill 3-D example is gratefully acknowledged.


124

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