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Contents
1 Introduction 9
1.1 The digital revolution 9
1.2 Digital Recording 11
1.3 Processing 13
1.4 Inversion 15
1.5 About this book 18
Part one: PROCESSING 23
2 Mathematical Preliminaries: the TVU and Discrete Fourier Transforms 25
2.1 The T -transform 25
2.2 The Discrete Fourier Transform 29
2.3 Properties of the discrete Fourier transform 34
2.4 DFT of random sequences 43
3 Practical estimation of spectra 47
3.1 Aliasing 47
3.2 Aliasing 51
3.3 Spectral leakage and tapering 51
3.4 Examples of Spectra 57
4 Processing of time sequences 65
4.1 Filtering 65
4.2 Correlation 71
4.3 Deconvolution 73
5 Processing two-dimensional data 82
5.1 The 2D Fourier Transform 82
5.2 2D Filtering 84
5.3 Travelling waves 87
Part two: INVERSION 93
6 Linear Parameter Estimation 95
6.1 The linear problem 95
6

Contents 7
6.2 Least squares solution of over-determined problems 99
6.3 Weighting the data 102
6.4 Model Covariance Matrix and the Error Ellipsoid 110
6.5 “Robust methods” 113
7 The Underdetermined Problem 120
7.1 The null space 120
7.2 The Minimum Norm Solution 122
7.3 Ranking and winnowing 123
7.4 Damping and the Trade-off Curve 125
7.5 Parameter covariance matrix 127
7.6 The Resolution Matrix 131
8 Nonlinear Inverse Problems 139
8.1 Methods available for nonlinear problems 139
8.2 Earthquake Location: an Example of Nonlinear Parameter Estimation 141
8.3 Quasi-linearisation and Iteration for the General Problem 144
8.4 Damping, Step-Length Damping, and Covariance and Resolution
Matrices 145
8.5 The Error Surface 146
9 Continuous Inverse Theory 154
9.1 A linear continuous inverse problem 154
9.2 The Dirichlet Condition 155
9.3 Spread, Error, and the Trade-off Curve 157
9.4 Designing the Averaging Function 159
9.5 Minimum Norm Solution 160
9.6 Discretising the Continuous Inverse Problem 163
9.7 Parameter Estimation: the Methods of Backus and Parker 165
Part three: APPLICATIONS 173
10 Fourier Analysis as an inverse problem 175
10.1 The Discrete Fourier Transform and Filtering 175
10.2 Wiener Filters 177
10.3 Multitaper Spectral Analysis 180
11 Seismic Travel Times and Tomography 185
11.1 Beamforming 185
11.2 Tomography 192
11.3 Simultaneous Inversion for Structure and Earthquake Location 198
12 Geomagnetism 203
12.1 Introduction 203
12.2 The Forward Problem 204
12.3 The Inverse Problem: Uniqueness 206
12.4 Damping 208

8 Contents
12.5 The Data 212
12.6 Solutions along the Trade-off Curve 216
12.7 Covariance, Resolution, and Averaging Functions 218
12.8 Finding Fluid Motion in the Core 221
Appendix 1 Fourier Series 228
Appendix 2 The Fourier Integral Transform 234
Appendix 3 Shannon’s Sampling Theorem 240
Appendix 4 Linear Algebra 242
Appendix 5 Vector Spaces and the Function Space 250
Appendix 6 Lagrange Multipliers and Penalty Parameters 257
Appendix 7 Files for the Computer Exercises 260
References 261
Index 265

List of Illustrations
1.1 Fig 1.1. BB seismogram Nov 1 page 13
1.2 Fig 1.2. First arrival 14
1.3 Gravity anomaly 16
2.1 The convolution process 27
2.2 In the Argand diagram, our choice for , , lies always on the unit
circle because . Discretisation places the points uniformly around
the unit circle. In this case and there are 12 points around the
unit circle. 30
2.3 Boxcar 32
2.4 Cyclic convolution 38
2.5 Stromboli velocity/displacement. 39
3.1 Aliasing in the time domain. 48
3.2 Aliasing in the frequency domain 49
3.3 Normal modes for different window lengths, boxcar window. 52
3.4 Tapers 55
3.5 Proton magnetometer data. Top panel shows the raw data, middle panel the
spectrum, lower panel the low frequency portion of the spectrum showing
diurnal and semi-diurnal peaks. 59
3.6 Airgun source depth 60
3.7 Microseismic noise 61
4.1 Aftershock for Nov 1 event 66
4.2 Butterworth filters, amplitude spectra 67
4.3 Low pass filtered seismograms: effect of . 68
4.4 Causal and zero-phase filters compared 69
4.5 Water level method 74
5.1 2D Filtering. 83
5.2 2D Aliasing. 84
5.3 Upward continuation 86
5.4 Separation by phase speed 87
5.5 Example of - -filtering of a seismic reflection walk-away noise test. See
text for details. 89
6.1 Two-layer density model 97
3

4 List of Illustrations
6.2 Picking errors 103
6.3 Dumont D’Urville jerk 105
6.4 Error ellipse 111
6.5 Moment of Inertia and Mass covariance 112
6.6 Histograms of residuals of geomagnetic measurements from a model of
the large scale core field. (a) residuals from an ! -norm model compared
with the double-exponential distribution (solid line) (b) residuals from a
conventional least squares model without data rejection compared with the
Gaussian distribution (c) least squares with data rejection. 116
7.1 Null space for mass/inertia problem 121
7.2 Trade-off curve 125
8.1 Hypocentre partial derivatives 142
8.2 Error Surfaces 147
9.1 Data kernels #
!$
#% for mass and moment of inertia 155
9.2 Averaging function '()+*-, designed to give an estimate of the mass of the
Earth’s core (solid line) and the desired averaging function '/.0)1*2, 161
10.1 The first 4 optimal tapers found by solving (10.29). Note that tapers 1 and 3
are symmetrical, 2 and 4 antisymmetrical. Taper 1 has least spectral leakage
but does not use the extremes of the time series; higher tapers make use of the
beginning and end of the record. The corresponding eigenvalues (bandwidth
retention factors) are (1) 1.00000 (2) 0.99976 (3) 0.99237 (4) 0.89584. 182
11.1 Top: array records from a borehole shot to 68 geophones on land with 10 m
spacing. Middle: results from the initial model. Trace 99 is the beam formed
by a simple stack with unit weighting. Bottom: results of the inversion. 189
11.2 A regional event (1 November 1991 436577879:;7 , 87=?7@;@7A S CB7B0 D;EFA W, Ms
6.7) at the broad-band, three-component array operated by the University of
Leeds, UK, in the Tararua Mountain area of North Island, New Zealand. The
time axes do not correspond to the origin time of the earthquake. 190
11.3 Results of the filtered P and S wavelets in the frequency band 1 to 2 Hz for
the regional event shown in Figure 5. Label G stands for data, H for initial
model and I for inverted model. The numbers at the left of each trace of
inverted model are stacking weights. A SP-converted phase is present in the
vertical and NS components of ltw1 and ltw3, close to the S wave. It is
suppressed in the S wave beams. 191
11.4 1D tomography 193
11.5 Geometry of rays and receivers for the ACH inversion. Only one ray path
passes through both blocks ' and J , so there will be a trade-off between
velocities in the two, creating an element of the null space. The same applies
to blocks K and L . Ray paths 1 and 2 both pass through block M but have
no others in common, which will allow separation of anomalies in block M
from those in any other sampled block. 196

List of Illustrations 5
12.1 Data kernels for the continuous inverse problem of finding the radial
component of magnetic field on the core surface from a measurement of
radial component (NPO4, and horizontal component (N
5
) at the Earth’s surface
as a function of angular distance between the two points. Note that radial
component measurements sample best immediately beneath the site, but
horizontal components sample best some 23A away. 210
12.2 Data distribution plots. From the top: epoch 1966, dominated by total
intensity from the POGO satellite. Epoch 1842. Epoch 1777.5, the time of
Cook’s voyages. Epoch 1715, containing Halley’s voyages and Feuilliée’s
measurements of inclination. 213
12.3 Trade-off curve 216
12.4 Solutions for epoch 1980 for 4 values of the damping constant marked on the
trade-off curve in Figure 12.3. J O is plotted at the core radius. They show
the effect of smoothing as the damping is increased. 218
12.5 Contour maps of the errors in the estimated radial field at the core-mantle
boundary for (a) 1715.0; (b) 1915.5; and (c) 1945.5. The contour interval is
5 QSR and the units are 10 Q T; the projection is Plate Carré. 225
12.6 Resolution matrices for three epochs with quite different datasets. (a) epoch
1980 (b) 1966 (c) 1715. 226
12.7 Averaging functions for the same three models as Figure 12.7 and two points
on the core mantle boundary. Good resolution is reflected in a sharp peak
centred on the chosen site. Resolution is very poor in the south Pacific for
epoch 1715.0. 227
A1.1 Square wave 232

1
Introduction
1.1 The digital revolution
Recording the output from geophysical instruments has undergone four stages of
development during this century: mechanical, optical, analogue magnetic, and dig-
ital. Take the seismometer as a typical example. The principle of the basic sensor
remains the same: the swing of a test mass in response to motion of its fixed pivot
is monitored and converted to an estimate of the velocity of the pivot. Inertia and
damping determine the response of the sensor to different frequencies of ground
motion; different mechanical devices measured different frequency ranges. Ocean
waves generate a great deal of noise in the range 0.1–0.5 Hz, the microseismic noise
band, and it became normal practice to install a short period instrument to record
frequencies above 0.5 Hz and a long period instrument to record frequencies below
0.1 Hz.
Early mechanical systems used levers to amplify the motion of the mass to drive
a pen. The classic short period, high-gain design used an inverted pendulum to
measure the horizontal component of motion. A large mass was required simply to
overcome friction in the pen and lever system.
An optical lever reduces the friction dramatically. A light beam is directed onto
a mirror, which is twisted by the response of the sensor. The reflected light beam
shines onto photographic film. The sensor response deflects the light beam and
the motion is recorded on film. The amplification is determined by the distance
between the mirror and film. Optical recording is also compact: the film may be
enlarged to a more readable size. Optical recording was in common use in the
1960’s and 1970’s.
Electromechanical devices allow motion of the mass to be converted to a volt-
age, which is easy to transmit, amplify, and record. Electromagnetic feedback
seismometers use a null method, in which an electromagnet maintains the mass in
9

10 Introduction
a constant position. The voltage required to drive the electromagnet is monitored
and forms the output of the sensor.
This voltage can be recorded on a simple tape recorder in analogue form. There
is a good analogy with tape recording sound, since seismic waves are physically
very similar to low frequency sound waves. The concept of fidelity of recording
carries straight across to seismic recording. A convenient way to search an ana-
logue tape for seismic sources is to simply play it back fast, thus increasing the fre-
quency into the audio range, and listen for bangs. Analogue magnetic records could
be converted to paper records simply by playing them through a chart recorder.
The digital revolution started in seismology in about 1975, notably when the
World Wide Standardised Seismograph Network (WWSSN) was replaced by Seis-
mological Research Observatories (SRO). These were very expensive installations
requiring a computer in a building on site. The voltage is sampled in time and
converted to a number for input to the computer. The tape recording systems were
not able to record the incoming data continuously so the instrument was triggered
and a short record retained for each event. Two channels (sometimes three) were
output: a finely-sampled short period record for the high frequency arrivals and a
coarsely-sampled channel (usually one sample each second) for the longer period
surface waves. Limitations of the recording system meant that SROs did not herald
the great revolution in seismology: that had to wait for better mass storage devices.
The great advantage of digital recording is that it allows replotting and process-
ing of the data after recording. If a feature is too small to be seen on the plot, you
simply plot it on a larger scale. More sophisticated methods of processing allow
us to remove all the energy in the microseismic noise band, obviating the need for
separate short and long period instruments. It is even possible to simulate an older
seismometer simply by processing, provided the sensor records all the information
that would have been captured by the simulated instrument. This is sometimes use-
ful when comparing seismograms from different instruments used to record similar
earthquakes. Current practice is therefore to record as much of the signal as pos-
sible and process after recording. This has one major drawback: storage of an
enormous volume of data.
The storage problem was essentially solved in about 1990 by the advent of cheap
hard disks and tapes with capacities of several gigabytes. Portable broadband seis-
mometers were developed at about the same time, creating a revolution in digital
seismology: prior to 1990 high-quality digital data was only available from a few
permanent, manned observatories. After 1990 it was possible to deploy arrays of
instruments in temporary sites to study specific problems, with only infrequent
visits to change disks or tapes.

1.2 Digital Recording 11
1.2 Digital Recording
The sensor is the electromechanical device that converts ground motion into volt-
age; the recorder converts the voltage into numbers and stores them. The ideal
sensor would produce a voltage that is proportional to the ground motion but such
a device is impossible to make (the instrument response would have to be constant
for all frequencies, which requires the instrument to respond instantaneously to any
input, see Section 2). The next best thing is a linear response, in which the output
is a convolution of the ground motion with the transfer function of the instrument.
Let the voltage output be WXZY4[ . The recorder samples this function regularly in
time, at a sampling interval ]Y , and creates a sequence of numbers:
^
W-_a`bW6Xdce]Yf[ghci`bjekml0kfn2kCoCoCoqpsr (1.1)
The recorder stores the number as a string of bits in the same way as any computer.
Three quantities describe the limitations of the sensor: The sensitivity is the
smallest signal that produces non-zero output; the resolution is the smallest change
in the signal that produces non-zero output; and the linearity determines the extent
to which the signal can be recovered from the output. For example, the ground
motion may be so large that the signal exceeds the maximum level; the record
is said to be “clipped”. The recorded motion is not linearly related to the actual
ground motion, which is lost.
The same three quantities can be defined for the recorder. A pen recorder’s
linearity is between the voltage and movement of the pen, which depends on the
electronic circuits and mechanical linkages; its resolution and accuracy are limited
by the thickness of the line the pen draws. For a digital recorder linearity requires
faithful conversion of the analogue voltage to a digital count, while resolution is
set by the voltage corresponding to one digital count.
The recorder suffers two further limitations: the dynamic range, t , the ratio
of maximum possible to minimum possible recorded signal, usually expressed in
deciBel: n0j uwvx2y{zX+t[ dB; and the maximum frequency that can be recorded. For a
pen recorder the dynamic range is set by the height of the paper, while the maxi-
mum frequency is set by the drum speed. For a digital recorder the dynamic range
is set by the number of bits available to store each member of the time sequence,
while the maximum frequency is set by the sampling interval ]Y or sampling fre-
quency |;}`~l70]Y (we shall see later that the maximum meaningful frequency is
in fact only half the sampling frequency).
A recorder employed for some of the examples in this book used 16 bits to record
the signal as an integer, making the maximum number it can record n
y{€
Ul and the
minimum is 1. The dynamic range is therefore n0j uwvx y{z Xƒ‚„„„-[:`‡†‚ dB. Another
popular recorder uses 16 bits but in a slightly more sophisticated way. One bit is

12 Introduction
used for the sign and 15 for the integer if the signal is in the range ân
y{‰
UŠl ; outside
this range it steals one bit, records an integer in the range ân
yŒ‹
Ul , and multiplies
by 20, giving a complete range ânFŽ0‚FŽj , or 116 dB. The stolen “gain” bit is used
to indicate the change in gain by a factor of 20; the increase in dynamic range has
been achieved at the expense of accuracy, but this is usually unimportant because
it only occurs when the signal is large. A typical “seismic word” for recorders in
exploration geophysics consists of 16 bits for the integer (“mantissa”), 1 for the
sign, and 4 bits for successive levels of gain. The technology is changing rapidly
and word lengths are increasing; most recorders now have being sold now have 24
bits as standard.
Storage capacity sets a limit to the dynamic range and sampling frequency. A
larger dynamic range requires a larger number and more bits to be stored; a higher
sampling frequency requires more numbers per second and therefore more numbers
to be stored. The following calculation gives an idea of the logistical considerations
involved in running an array of seismometers in the field. In 1990 Leeds Univer-
sity Earth Sciences Department deployed 9 3-component broadband instruments in
the Tararua Mountain region of North Island, New Zealand, to study body waves
travelling through the subducted Pacific Plate. The waves were known to be high
frequency, demanding a 50 Hz sampling rate. Array processing (see Chapter 11.1)
is only possible if the signal is coherent across the array, requiring a 10-km interval
between stations. The Reftek recorders used a 16 bit word and had 360 Mb disks
that were downloaded onto tape when full.
A single field seismologist was available for routine servicing of the array, which
meant he had to drive around all 9 instruments at regular intervals to download the
disks before they filled. How often would they need to be downloaded? Each
recorder was storing 16 bits for each of 3 components 50 times each second, or
2400 bits/second. Allowing 20% overhead for things like the time channel and
state-of-health messages gives 2880 bits/second. One byte is 8 bits, and dividing
2880 into the storage capacity i‘2o’‚“lCj•” bits gives about lCj
€
s, or 11.5 days. It
would be prudent to visit each instrument at least every 10 days, which is possible
for an array stretching up to 150 km from base over good roads. A later deployment
used Mars recorders with optical disks which could be changed by a local unskilled
operator, requiring only infrequent visits to collect data and replenish the stock of
blank disks. In this case, the limiting factor was the time required to back up the
optical disks onto tape, which can take almost as long as the original recording.
It is well worth making the effort to capture all the information available in a
broadband seismogram because the information content is so extraordinarily rich.
The seismogram in Figures 1.1 and 1.2 are good examples. P, S, and surface waves
are clearly seen. The surface waves are almost clipped (Figure 1.1), yet the onset of
the P wave has an amplitude of just one digital count (Figure 1.2). The frequency

1.3 Processing 13
Fig. 1.1. Seismogram of an event in the Kermadec Islands on 1 Nov 1991 recorded in
North Island, New Zealand. Note the P, S, and longer period surface waves. The dynamic
range meant maximum signals of –43;87E78 digital counts, and the scale shows that this was
nearly exceeded. The time scale is in minutes, and the surface waves have a dominant
period of about 20 s
of the surface waves is about 20 s, yet frequencies of 10 Hz and above may be seen
in the body waves. The full dynamic range and frequency bandwidth was therefore
needed to record all the ground motion.
1.3 Processing
Suppose now that we have collected some important data. What are we going to
do with it? Every geophysicist should know that the basic raw data, plus that field
note-book, constitute the maximum factual information he will ever have. This is
what we learn on field courses. Data processing is about extracting a few nuggets
from this dataset; it involves changing the original numbers, which always means
losing information. So we always keep the original data.
Processing involves operating on data in order to isolate a signal, the message
we are interested in, and to separate it from “noise” , which nobody is interested

14 Introduction
Fig. 1.2. Expanded plot of the first part of the seismogram in Figure 1.1. Note the very
small amplitude of the P arrival, which rides on microseisms of longer period. The arrival
is easy to see because its frequency is different from that of the microseisms.
in, and unwanted signals, which do not interest us at this particular time. For
example, we may wish to examine the P-wave on a seismogram but not the surface
waves, or we may wish to remove the Earth’s main magnetic field from a magnetic
measurement in order to determine the magnetisation of local rocks because it will
help us understand the regional geology. On another day we may wish to remove
the local magnetic anomalies in order to determine the main magnetic field because
we want to understand what is going on in the Earth’s core.
Often we do not have a firm idea of what we ought to find, or exactly where we
should find it. Under these circumstances it is desirable to keep the methods as
flexible as possible, and a very important part of modern processing is the interac-
tion between the interpreter and the computer. The graphical display is a vital aid
in interpretation, and often the first thing we do is plot the data in some way. For
example, we may process a seismogram to enhance a particular arrival by filtering
(Section 4.1). We know the arrival time roughly, but not its exact time. In fact,
our main aim is to measure the arrival time as precisely as possible; in practice,

1.4 Inversion 15
the arrival will be visible on some unprocessed seismograms, on others it will be
visible only after processing, while on yet more it may never be visible at all.
The first half of this book deals with the analysis of time series and sequences,
the commonest technique for processing data interactively without strict prior prej-
udice about the detailed cause of the signals or the noise. This book adopts Claer-
bout’s 1992 usage and restricts the word processing to distinguish this activity from
the more formal process of inversion explained below. Processing is interactive and
flexible; we are looking for something interesting in the data without being too sure
of what might be there.
Suppose again that we wish to separate P-waves from surface waves. We plot
the seismogram and find the P-waves arriving earlier because they travel faster
than surface waves, so we just cut out the later part of the seismogram. This does
not work when the P-waves have been delayed (by reflecting from some distant
interface for example) and arrive at the same time as the surface waves. The surface
waves are much bigger than body waves and the P-wave is probably completely
lost in the raw seismogram. P-waves have higher frequencies than surface waves
(period 1 second against 20 seconds for earthquake seismograms or 0.1 s against
1 s for a typical seismic exploration experiment), which gives us an alternative
means of separation. The Fourier transform, in its various forms (Appendix 2),
decomposes a time series into its component frequencies. We calculate and plot
the transform, identify the big contributions from the surface waves, zero them and
transform back to leave the higher frequency waves. This process is called filtering
(Section 4.1). In these examples we need only look at the seismogram or its Fourier
transform to see two separate signals; having identified them visually it is an easy
matter to separate them. This is processing.
1.4 Inversion
The processed data are rarely the final end product: some further interpretation
or calculation is needed. Usually we will need to convert the processed data into
other quantities more closely related to the physical properties of the target. We
might want to measure the arrival time of a P-wave to determine the depth of a
reflector, then interpret that reflector in the context of a hydrocarbon reservoir. We
might measure spatial variations in the Earth’s gravity, but we really want to find
the density anomalies that cause those gravity anomalies, and then understand the
hidden geological structure that caused the gravity variations.
Inversion is a way of transforming the data into more easily interpretable phys-
ical quantities: in the example above we want to invert the gravity variations for
density. Unlike processing, inversion is a formal, rigid procedure. We have already
decided what is causing the gravity variations, and probably have an of the depth

16 Introduction
and extent of the gravity anomalies, and even their shape. We have a mathematical
model set out that we wish to test and refine, using this new data. The inversion
excludes any radically different interpretation. For example, if we invert the seis-
mic arrival times for the depth of a pair of horizontal reflectors we would never
discover that they really come from a single, dipping reflector.
0
1.0
0.8
0.6
0.4
0.2
-3 -2 -1 0 1 2 3
g
x
D
R
ρ
X X
X
X
X X
X
X
X
X
X
Fig. 1.3. Gravity anomaly
Consider the gravity traverse illustrated in Figure 1.3, which we intend to invert
for density. The traditional method, before widespread use of computers, was
to compare the shape of the anomaly with theoretical curves computed from a
range of plausible density models simple enough for the gravity to be calculated
analytically. This is called forward modelling. It involves using the laws of physics
to predict the observations from a model. Computers can compute gravity signals
from very complicated density models. They can also be used to search a large
set of models to find those that fit the data. A strategy or algorithm is needed to
direct the search. Ideally we should start with the most plausible models and refine

1.4 Inversion 17
our ideas of plausibility as the search proceeds. These techniques are sometimes
referred to as Monte Carlo methods.
There are two separate problems, existence and uniqueness. The first require-
ment is to find one model, any model, that fits the data. If none are found the data
are incompatible with the model. Either there is something wrong with the model
or we have overestimated the accuracy of the data. This is rare. Having found one
model we search for others. If other models are found the solution is nonunique;
we must take account of the it in any further interpretation. This always happens.
The nonuniqueness is described by the subset of models that fit the data.
It is more efficient, when possible, to solve directly for the model from the data.
This is not just an exercise in solving the equations relating the data to the model
for a model solution, we must also characterise the nonuniqueness by finding the
complete set of compatible solutions and placing probabilities on the correctness
of each individual model.
Like most formal procedures, the mathematics of inverse theory is very attrac-
tive; it is easy to become seduced into thinking the process is more important than it
really is. Throughout this book I try to emphasise the importance of the measured
data and the desired end goal: these are much more important than the theory,
which is just the vehicle that allows you to proceed from the data to a meaningful
interpretation. It is like a car that lets you take luggage to a destination: you must
pack the right luggage and to get to the right place (and, incidentally, the place
should be interesting and worthy of a visit!). You need to know enough about the
car to drive it safely and get there in reasonable time, but detailed knowledge of its
workings can be left to the mechanic.
There are two sources of error that contribute to the final solution: one aris-
ing because the original measurements contain errors, and one arising because the
measurements failed to sample some part of the model. Suppose we can prove for a
particular problem that perfect, error-free data would invert to a single model: then
in the real case of imperfect data we need only worry about measurement errors
mapping into the model. In this book I call this parameter estimation to distin-
guish it from true inversion. Modern inversion deals with the more general case
when perfect data fail to provide a unique model. True inversion is often confused
with parameter estimation in the literature. The distinction is vital in geophysics,
because the largest source of error in the model usually comes from failure to ob-
tain enough of the right sort of data, rather than sufficiently accurate data.
The distinction between inversion and parameter estimation should not become
blurred. It is tempting to restate the inverse problem we should be solving by
restricting the model until the available data are capable of determining a unique
solution in the absence of errors, but this is philosophically wrong. The model is

18 Introduction
predetermined by our knowledge of the problem at the start of the experiment, not
by what we are about to do.
Some quotes illustrate the diversity of meaning attached to to the term ‘inver-
sion’. J. Claerbout, in his book “Earth Soundings Analysis—Processing versus In-
version”, calls it matrix inversion. This is only true if the forward problem is posed
as matrix multiplication. In exploration geophysics one sometimes hears inversion
called deconvolution (Section 4.3), an even more restrictive definition. Deconvo-
lution is treated as an inverse problem in Section 10.2. “Given the solution of a
differential equation, find its coefficients” was the definition used by Gel’fand and
Levitan (1955). More specifically “given the eigenvalues of a differential operator,
find the operator”, a problem that finds application in the use of normal mode fre-
quencies in determining average Earth structure. A more poetic statement of the
same problem is “can we hear the shape of a drum?”, which has received much
attention from pure mathematicians. None of these definitions cover the full extent
of inverse problems currently being studied in geophysics.
1.5 About this book
The book is divided into three parts: processing, inversion, and applications which
combine techniques introduced in both of the first parts. Emphasis is placed on
discrete, rather than continuous formulations, and deterministic, rather than ran-
dom, signals. This departure from most texts on signal processing and inversion
demands some preliminary explanation.
The digital revolution has made it easy to justify treating data as a discrete se-
quence of numbers. Analogue instruments that produce continuous output in the
form of a paper chart record might have the appearance of continuity, but they
never had perfect time resolution and are equivalent to discrete recordings with in-
terpolation imposed by the nature of the instrument itself—an interpolation that is
all too often beyond the control and sometimes even the knowledge of the operator.
The impression of continuity is an illusion. Part I therefore uses the discrete form
of the Fourier transform, which does not require prior knowledge of the integral
form.
It is much harder to justify discretising the model in an inversion. In this book
I try to incorporate the philosophy of continuous inversion within the discrete for-
mulation and clarify the distinction between true inversion, in which we can only
ever discover a part of the true solution, and parameter estimation, where the model
is adequately described by a few specially chosen numbers.
In the example of Figure 1.3 the model is defined mathematically by a set of
parameters—— , the radius of the cylinder, ˜ , the depth of burial, and ™ , the den-
sity or density difference with the surroundings. The corresponding gravity field

1.5 About this book 19
is computed using Newton’s inverse square law for each mass element šF™ and in-
tegrating over the volume. This gives a prediction for every measured value. We
can either select a curve and find the best fit by trial and error, then deduce the best
values of the three parameters, or do a full mathematical inversion using the meth-
ods described in the Part II. Either way should give the same answer, but both will
suffer from the same obvious drawback: the model is implausibly over-simplified.
It is a pity to use the power of computers to simply speed up old procedures. It
is better to prescribe a more general model and determine as much about it as the
data allow. Ultimately, for this problem, this might mean allowing the density to
vary with all three space coordinates, an infinite number of degrees of freedom.
For the most part, we shall content ourselves with a finite number of model
parameters, but still allow the model to be as complicated as we wish, for example
by dividing the half space into rectangular blocks and ascribing a different density
to each cell. Some such parameterisation will be needed ultimately to represent
the model numerically whichever inversion procedure is adopted. The methods
are sufficiently flexible allow for any previous knowledge we might have about the
structure without necessarily building it into the parameterisation.
In practice, virtually all real inverse problems end up on the computer, where any
continuous function is represented discretely. Continuous inversion has one great
pedagogical advantage: if the model is a continuous function of an independent
variable and therefore contains an infinite number of unknowns, it is difficult to
pretend that a finite number of data can provide all the answers. Given a discrete
formulation and a huge amount of (very expensive) data, it becomes all to easy to
exaggurate the results of an inversion. Chapter 9 gives a brief introduction to the
techniques of continuous inversion. Its main purpose is to explain how it fits with
discrete inversion, and to illustrate some potential pitfalls of discrete treatments,
rather than providing a practical approach to solving continuous inverse problems.
Deterministic signals are predictable. An error-free seismogram can be com-
puted from the properties of the source and the medium the waves pass through; a
gravity traverse can be computed from density; and a magnetometer survey can be
predicted from magnetisation. For random signals we can only predict statistical
properties: the mean, standard deviation, etc. Wind noise on a seismogram is one
example, its statistical properties depending on the weather, neighbouring trees,
etc. Electronic noise is another source of error in all modern digital instruments.
Many books use a statistical definition of the power spectrum (e.g. Percival and
Walden (1998); Priestley (1992) ) whereas this book uses the more familiar math-
ematical definition. The central problem in transferring the familiar mathematical
techniques of Fourier Series and Transforms to data analysis is that both assum-
ing the time function continues forever: Fourier Series require periodicity in time
while the Integral Fourier Transform (Appendix 2) requires knowledge for all time.

20 Introduction
Data series never provide this, so some assumptions have to be made about the sig-
nal outside the measurement range. The assumptions are different for random and
deterministic signals. For example, it would seem reasonable to assume a seismo-
gram is zero outside the range of measurement; the same could apply to a gravity
traverse if we had continued right across the anomaly. Random signals go on for
ever yet we only measure a finite portion of them. They are usually continued by
assuming stationarity, that the statistical properties do not vary with time.
The emphasis on deterministic signals in this book is prompted by the subject.
The maxim “one man’s signal is another man’s noise” seems to apply with more
force in geophysics than in other branches of physical science and engineering.
For example, our knowledge of the radius of the Earth’s core is limited not by
the accuracy with which we can read the arrival time of seismic reflections from
the boundary, but from the time delays the wave has acquired in passing through
poorly-known structures in the mantle. Again, the arrival time of shear waves are
made inaccurate because they appear within the P-wave code, rather than at a time
of greater background noisie. The main source of error in magnetic observations of
the Earth’s main field comes from magnetised rocks in the crust. not from inherent
measurement errors—even for measurements made as early as the late 18th cen-
tury. One has to go back before Captain Cook’s time, when navigation was poor,
to find measurement errors comparable with the crustal signal.
Something similar applies to errors in models obtained from an inversion. Grav-
ity data may be inverted for density, but an inherent ambiguity makes it impossible
to distinguish between small, shallow mass anomalies and large, deep ones. Seis-
mic tomography (Chapter 11), in which arrival times of seismic waves are inverted
for wave speeds within the Earth, is limited more by its frequent inability to distin-
guish anomalies in different parts of the Earth than by errors in reading the travel
times themselves.
Exercises
1.1 A geophone response is quoted as 3.28 volts/in/sec. Give the ground mo-
tion in metres/sec corresponding to an output of 1 millivolt.
1.2 A simple seismic acquisition system uses 14 bits to record the signal digi-
tally as an integer. Calculate the dynamic range. Convert to decibels (dB)
using the formula
›Vœ
`‡n0j uwvx y{z X
›V2žVŸ¢¡¤£¦¥fŸž
x§0[ (E1.1)
(The factor 10 converts “Bel” to “deciBel” and the factor 2 converts from
amplitude to energy.)

Exercises 21
1.3 The geophone in question 1 outputs linearly over the range 1 microvolt–0.1
volt. What is its dynamic range in dB?
1.4 You attach the geophone in question 1 to the recorder in question 2 and
set the recorder’s gain to 1 digital count/microvolt. What is the maximum
ground motion recorded?
1.5 A more sophisticated recording system has a 19-bit seismic word, 1 sign
bit (polarity), 4 gain bits (exponent plus its sign), and a 14-bit mantissa
(the digits after the decimal point). Calculate the dynamic range in dB.
1.6 What happens to the sensitivity of the more sophisticated recording system
for large signals?

2
Mathematical Preliminaries: the ¨¢© and Discrete
Fourier Transforms
Suppose we have some data digitized as a sequence of p numbers:
^
Wªra`‡W z kW y kW-«0kCoCoCo;kW-¬® y (2.1)
These numbers must be values of measurements made at regular intervals. The data
may be sampled evenly in time, such as a seismogram, or evenly in space, such as
an idealised gravity traverse. Whatever the nature of the independent variable, it
will be called time and the sequence a time sequence. The mathematics described
in this chapter requires only a sequence of numbers. The main tool for analysing
discrete data is the discrete Fourier transform or DFT. The DFT is closely related
to Fourier Series, with which the reader is probably already familiar. Some basic
properties of Fourier Series are reviewed in Appendix 1. This chapter begins not
with the DFT but with the TVU transform, a simpler construct that will allow us to
derive some fundamental formulae that apply also to the DFT.
2.1 The T -transform
The T -transform is made by simply forming a polynomial in the complex variable
T using the elements of the time sequence as coefficients:
¯
X°Te[ `bW z²± W y T ± W-«qT
«
± oCoCo ± W-¬® y T
¬® y
(2.2)
Many physical processes have the effect of multiplying the TªU transform of a time
sequence by another TVU transform. The best known of these is the effect of an
instrument: the output of a seismometer, for example, is different from the actual
ground motion it is supposed to measure. The relationship is one of multiplying
the TVU transform of the ground motion by another TVU transform that depends on
the instrument itself. For reasons that will become apparent later, it is physically
impossible to make a device that records the incoming time series precisely.
It is therefore often instructive to consider operations on the T -transform rather
25

26 Mathematical Preliminaries: the and Discrete Fourier Transforms
than on the original time series. Operations on the T -transform all have their coun-
terpart in the time domain. For example, multiplying by T gives
T
¯
X°Te[ `bW z T ± W y T
«
± W « TF³ ± oCoCo ± W ¬ y T
¬
(2.3)
This new T -transform corresponds to the time sequence:
^
We´:r`bjekW z kW y kW2«0kCoCoCo7kW2¬ y (2.4)
which has been shifted one space in time, or by an amount equal to the sampling
interval ]Y . Likewise, multiplying by a power of T , T
_
, shifts the time sequence by
ce]Y . T is called the unit delay operator.
In general, multiplication of two T -transforms is equivalent in the time domain
to a process called discrete convolution discrete convolution. This discrete convo-
lution theorem is the most important property of the TVU transform. Consider the
product of
¯
X°Te[ with µ¶X°Te[ , the T -transform of a second time sequence
^;·
r , whose
length ¸ is not necessarily the same as that of the original sequence
¹
X°Te[ `
¯
X°TS[dµŠX°TS[ `
¬® y
º
_q» z
W-_T
_½¼
y
º¾
» z
·
¾
T
¾
(2.5)
To find the time sequence corresponding to this TVU transform we must write it as
a polynomial and find the general term. Setting ¿`Àc ±Á for the new subscript ¿
and changing the order of the summation does exactly this:
¬® y
º
_C» z
¼
y
º¾
» z
W2_
·
¾
T
_ ´
¾
`
¼
´Â¬ª«
º
Ã » z
Ã
º
_C» z
W2_
·
Ã _;T
Ã
(2.6)
This is just the T -transform of the time sequence
^7Ä
r , where
Ä
Ã `
Ã
º
_C» z
W-_
·
Ã _ (2.7)
^7Ä
r is the discrete convolution of
^
Wªr and
^;·
r ; it has length p ± ¸ÅUÆl , one less
than the sum of the lengths of the two contributing sequences.
Henceforth the curly brackets on sequences will be omitted unless it leads to
ambiguity. The convolution is usually written as
Ä
`bWPÇ
·
(2.8)
and the discrete convolution theorem for T -transforms will be represented by the
notation:
W È
¯
X°Te[
·
È µ¶X°Te[
WPÇ
·
È
¯
X°Te[dµ¶X°Te[ (2.9)

2.1 The -transform 27
a0
a4 3
a a a1
2
0
b b b b
1 2 3
0
b b b b
1 2 3
0
b b b b
1 2 3
0
b b b b
1 2 3
0
b b b b
1 2 3
0
b b b b
1 2 3
0
b b b b
1 2 3
0
b b b b
1 2 3
c0
c1
c2
c3
c4
c5
c6
c7
Fig. 2.1. The convolution process. The first sequence H is plotted against subscript . The
second sequence I is reversed and shifted so that I . is at ]sÉ . Products of the adjacent
elements of the two sequences are then summed to give the convolution for that value of É .
As an example, consider the two sequences
WÊ`Ël0kfn2kf2kml0kfn pÌ`‡„
·
`Àl0kf2kf„2k4Ž ¸Í`ÏÎ (2.10)
The convolution c will have length p ± ¸ÐUbl]`Ñ and is given by the formula
(2.7). The formula must be applied for each of 8 values of ¿ :
Ä
z ` W z
·
z `Ël
Ä
y ` W z
·
yÒ± W y
·
z `‡„
Ä
«Í` W z
·
« ± W y
·
y½± W2«
·
z `ÀlqÎ
Ä
³
` W z
·
³
± W y
·
« ± W2«
·
y½± W
³
·
z `‡nFŽ
Ä
‹ ` W y
·
³
± W-«
·
« ± W
³
·
y½± W ‹
·
z `‡Î
Ä
‰ ` W «
·
³
± W
³
·
« ± W ‹
·
y `‡n
Ä
€ ` W
³
·
³
± W ‹
·
«`Àl7Ž
ÄCÓ
` W ‹
·
³
`ËlqÎ (2.11)
The procedure is visualised graphically in Figure 2.1. The elements of the first se-
quence (W ) are written down in order and those of the second sequence (
·
) written
down in reverse order (because of the Uc subscript appearing in (2.7)). To compute
the first element,
Ä
z , the start of the
·
sequence is aligned at cÊ`bj . The correspond-
ing elements of W and
·
are multiplied across and summed. Only
·
z lies below an
element of the W sequence in the top frame in Figure 2.1, so only the product W z
·
z

contributes to the first term of the convolution. In general,
Ä
Ã is found by aligning
·
z at ci`Ô¿ , multiplying corresponding elements and summing. It is often the case
that one sequence is much longer than the other, the long sequence representing
the seismogram and the shorter sequence some operation to be performed on the
seismogram. Think of the short sequence as a “train” running along the “rails” of
the long sequence as the subscript ¿ is increased.
If multiplication by a T -transform corresponds to convolution, division by a
T -transform corresponds to the opposite procedure, deconvolution since µ¶X°Te[Õ`
¹
X°Te[{
¯
X°Te[ . Inspection of the first equation in the example (2.11) shows we can
find
·
z if W zÊÖ`bj :
·
z `
Ä
z W z (2.12)
Knowing the first term
·
z we can use to it to find the next term of the
·
sequence
from the second of (2.11):
·
y `ËX
Ä
y UW y
·
z [{W z (2.13)
and so on. The general formula to invert (2.7) is obtained by moving the c`×j
term in the sum for
Ä
Ã to the left hand side and rearranging to give
·
Ã `ÙØ
Ä
Ã UÛÚ
Ã
_C» y W-_
·
Ã _7Ü
W z
(2.14)
This procedure is called recursion because the result of each equation is used in all
subsequent equations. Notice the division by W z at every stage: deconvolution is
impossible when W z `Ýj . Since µŠX°TS[²`
¹
X°TS[{
¯
X°Te[ , the recursion formula (2.14)
must be equivalent to division by the T -transform. Deconvolution is discussed in
detail in Section 4.3.
We may now define a generalised operation corresponding to multiplication of
the T -transform by a rational function (ratio of two polynomials):
Þ
X°TS[½`~ß
X°Te[
tàX°Te[
`
X°TPUsT z [ªX°TPUsT y [VoCoCo7XƒTUsT ¬ y [
X°T/U¶¿ z [ªXƒTU“¿ y [ªoCoCo7X°TPU“¿
¼
y [
In the time domain this corresponds to a combination of convolution and recursion.
The operation describes the relationship between the output of a seismometer
and the ground motion, or instrument response. Instrument responses of modern
seismometers are usually specified in terms of the roots of the two polynomials,
the poles and zeroes of the complex T -transform Þ
X°Te[ . An ideal seismometer has
Þ
X°Te[:`ál and records the ground motion exactly. Some modern seismometers ap-
proach this ideal and have rather simple instrument responses; older instruments
(and others not designed to record all aspects of the ground motion) have more
complicated responses. The SRO (which stands for seismological research obser-
vatory) was designed in the 1970s for the global network. Its full response was

properly described by no fewer than 14 zeroes and 28 poles. Both input and out-
put are made up of real numbers, so the TªU transforms have real coefficients and
therefore the roots (poles and zeroes) are either real or occur in complex conjugate
pairs.
2.2 The Discrete Fourier Transform
Setting T]`‡â
Vãåä-æÒç
into the formula for the T -transform (2.2) and normalising with
a factor p gives:
¯
XZè¦[½`
l
p
¬® y
º
_C» z
W-_â
Vãåä _ æÒç
(2.15)
This equation is a complex Fourier series (Appendix 1); it defines a continuous
function of angular frequency è which we discretise by choosing
èêéÊ`
n0ëì
í `
n0ëì
p‘]Y
`‡n0ëìîi| (2.16)
where i| is the sampling frequency. Substituting into (2.15) and using
í
`bp‘]Y
gives the Discrete Fourier Transform (DFT):
¯
é `
¯
XZèêéª[
`
l
p
¬ y
º
_C» z
W-_0â
ª«ðïãñé _Cò ¬
g
ìó`‡jekml0kfn2kCoCoCo;kpÌUl (2.17)
This equation transforms the p values of the time sequence
^
WŒ_Fr into another
sequence of numbers comprising the p Fourier coefficients
^ ¯
éªr . The values of
T that yield the DFT are uniformly distributed about the unit circle in the complex
plane (Figure 2.2).
An inverse formula exists to recover the original time sequence from the Fourier
coefficients:
W-_a`
¬® y
º
é » z
¯
éeâ
«ðïã _ é ò ¬
(2.18)
To verify this, substitute
¯
é from (2.17) into the right hand side of (2.18) to give
¬ y
º
é » z
l
p
¬® y
º¾
» z
W
¾
â
ª«ðïãåé
¾
ò ¬
â
«ðï;ã _ é ò ¬
`
¬® y
º¾
» z
W
¾
l
p
¬® y
º
é » z
â
ª«ðï;ãñé-ô
¾
_Cõ+ò ¬
(2.19)
The sum over ì is a geometric progression. Recall the formula for the sum of a

ο
30
z6
z7
z8
z9
z10
z11
z0
z1
z2
z3
z4
z5
Real
Imag
Fig. 2.2. In the Argand diagram, our choice for , ¦ , lies always on the unit circle
because Sö . Discretisation places the points uniformly around the unit circle. In
this case ÷ and there are 12 points around the unit circle.
geometric progression, which applies to both real and complex sequences:
¬® y
º
_C» z
Þ _
`
l®U
Þ ¬
lU
Þ (2.20)
In (2.19) the ratio is Þ
`‡â
ª«ðï;ã ô _
¾
õ1ò ¬
and Þ ¬
is â
ª«ðï;ã ô _
¾
õ
, which is unity provided
c Ö` Á , giving zero sum. When cÏ` Á every term is unity, giving a sum of p .
Nyquist’s theorem follows:
l
p
¬ y
º
é » z
â
ª«ðïãåé-ô _
¾
õ1ò ¬
`bøC_
¾
(2.21)
Substituting into (2.19) and using the substitution property of the Kronecker delta

(see box) leaves W;_ , which proves the result. Equations (2.17) and (2.18) are a
transform pair that allow us to pass between the time sequence W and its DFT
¯
and back without any loss of information.
THE KRONECKER DELTA
The Kronecker delta, also called the isotropic tensor of rank 2, is
simply an array taking the value zero or one:
øC_
¾
` jöc Ö` Á
` lÆcÊ` Á
It is useful in time sequence analysis in defining a spike, a se-
quence containing a single non-zero entry at time corresponding
to element ci` Á .
The substitution property of the Kronecker delta applies to the
subscripts. When multiplied by another time sequence and
summed over c the sum yields just one value of the sequence:
º
_
øC_
¾
W-_a`bW z¦ù j ± W y:ù j ± oCoCo ± W
¾
ù l ± oCoCo2`bW
¾
The subscript Á on the W on the right hand side has been substituted
for the subscript c on the left hand side.
In equations (2.17)–(2.18) subscript c measures time in units of the sampling
interval ]Y , so Y `Ýce]Y , up to a maximum time
í
`‡p]Y . ì measures frequency
in intervals of i|ú`Ål7
í
up to a maximum of the sampling frequency |°}`
p¢|û`ül70]Y . The complex Fourier coefficients
¯
é describe the contribution of
the particular frequency èÛ`‡n0ëìîi| to the original time sequence. Writing
¯
é in
terms of modulus and argument gives
¯
é `b— é â
ã¤ýSþ
(2.22)
A signal with just one frequency is a sine wave; — é is the maximum value of the
sine wave and ÿ é defines the initial point of the cycle – whether it is truly a sine,
or a cosine or some combination. — é is real and positive and gives a measure of
the amount that frequency contributes to the data; a graph of —Sé plotted against ì
is called the amplitude spectrum and its square, —
«
é , the power spectrum. ÿ é is an
angle and describes the phase of this frequency within the time sequence; a graph
of ÿ é plotted against ì is called the phase spectrum.
Consider first the simple example of the DFT of a spike. Spikes are often used in
processing because they represent an ideal impulse. The spike sequence of length
p has
š _a`bøC_ (2.23)

The spike is at time

]Y . Substituting into (2.17) gives
˜]éi`
l
p
â
ª«ðïã é ò ¬
(2.24)
When

` j , ˜Õé`
£
v
žŸž
; the spike is at the origin, the phase spectrum is
zero and the amplitude spectrum is flat. The spike is said to contain all frequencies
equally. Shifting the spike to some other time changes the phase spectrum but not
the amplitude spectrum.
Fig. 2.3. A boxcar sequence for NÑÆ=7= and Æ4= . Its amplitude spectrum is shown
in the lower trace. The sequence defined in (2.25) has been shifted to centre it on
@0 = s for clarity; the amplitude spectrum remains unchanged because of the shift theorem
(Section 2.3.2). Note the zeroes and slow decrease in maximum amplitude from ŠÔ= to
]Û@m= , caused by the denominator in equation (2.28).
An important example of a time sequence is the boxcar, a sequence that takes the
value 1 or 0 (Figure 2.3). It takes its name from the American term for a railway
carriage and its appearance in convolution, when it runs along the “rails” of the
seismogram. The time sequence is defined as
·
_ ` l j Ïc ¸

` j ¸ Æc p (2.25)
Substituting into (2.17) gives
µé]`
l
p
¼
y
º
_C» z
â
ª«ðï;ã _ é ò ¬
(2.26)
which is another geometric progression with factor §/U®n0ëdìêp and sum
µé `
lUâ
ª«ðï;ãñé
¼
ò ¬
p
Ø
lUâ
ª«ðïãñé ò ¬
Ü
` â
Vïãåé-ô
¼
y õ+ò ¬
¡1ž
ëìî¸bp
p
¡1ž
ëìêp
(2.27)
The amplitude and phase spectra are, from (2.22)
—é `

f¡¤ž
ëìî¸Ïp
p
¡¤ž
ëìêp
(2.28)
ÿ é ` U
ëìÊXƒ¸ Ul[
p
Uë (2.29)
where takes the values 0 or 1, depending on the sign of
¡1ž
ëìî¸Ïp .
The numerator in (2.27) oscillates as ì increases. Zeroes occur at ìó`bp¢q¸Ýkfn0p¢q¸ÝkCoCoCo ,
where ¸ is the length of the (non-zero part of the) boxcar. The denominator in-
creases from zero at ì` j to a maximum at ì` p¢0n then decreases to zero at
ìÛ`áp ; it modulates the oscillating numerator to give a maximum (height ¸ ) at
ìÏ` j to a minimum in the centre of the range ìb`×p¢0n . The amplitude spec-
trum is plotted as the lower trace in Figure 2.3. The larger ¸ the narrower the
central peak of the transform. This is a general property of Fourier transforms (see
Appendix 2 for a similar result): the spike has the narrowest time spread and the
broadest Fourier transform.
Inspection of (2.27) shows that the phase spectrum is a straight line decreasing
with increasing ì (frequency). The phase spectrum is ambiguous to an additive
integer multiple of n0ë ; it is sometimes plotted as a continuous straight line and
sometimes folded back in the range X°jekfn0ëÒ[ in a sawttooth pattern.
Cutting out a segment of a time sequence is equivalent to multiplying by a
boxcar—all values outside the boxcar are set to zero, while those inside are left
unchanged. This is why the boxcar arises so frequently in time series analysis, and
why it is important to understand its properties in the frequency domain.

2.3 Properties of the discrete Fourier transform
2.3.1 Relation to continuous Fourier Series and Integral Fourier Transform.

There exist analogies between the DFT on the one hand and Fourier series and the
Integral Fourier Transform on the other. Fourier Series and the FIT are summarised
in Appendices 1 and 2, but this material is not a prerequisite for this Chapter. The
forward transform, (2.17), has exactly the same form as the complex Fourier series
(1.10). The coefficients of the complex Fourier series are given by the integral
(1.12):
¯
é]`
l
í

z
W6XZY4[Œâ
Vãñé7ä-ç
šFY (2.30)
Setting èÌ` n0ëê
í
,
í
` p]Y , Yó` ce]Y , where ]Y is the sampling interval,
and using the trapezium rule to approximate the integral leads to (2.17), the DFT
equation for
¯
é . It is important to note that (2.17) is not an approximation, it is
exact. This follows from Nyquist’s theorem (2.21) and because we chose exactly
the right frequency interval for the given sampling interval.
Likewise, Fourier integrals may be approximated by the trapezium rule to give
sums similar to (2.17) and (2.18). Results for continuous functions from FITs and
Fourier series provide intuition for the DFT. For example, the FIT of a Gaussian
function â
ç is another Gaussian, X•l70n ë ²[Œâ
Sä! ò ‹ (Appendix 2). This useful
property of the Gaussian (the only function that transforms into itself) does not ap-
ply to the DFT because the Gaussian function never goes to zero, there are always
end effects, but Gaussian functions are still used in time series analysis. Note also
that when is large the Gaussian is very narrow and sharply peaked, but the trans-
form is very broad. This is another example of a narrow function transforming to a
broad one, a general property of the integral Fourier transform as well as the DFT.
The Fourier series of a monochromatic sine wave has just one non-zero coef-
ficient, corresponding to the frequency of the sine wave, and the integral Fourier
transform of a monochromatic function is a Dirac delta function (Appendix 2) cen-
tred on the frequency of the wave. The same is approximately true for the DFT.
It will contain a single spike provided the sequence contains an exact number of
wavelengths. Differences arise because of the discretisation in time and the finite
length of the record. Fourier Series are appropriate for analysing periodic functions
and FITs are appropriate for functions that continue indefinately in time. The DFT
is appropriate for sequences of finite length that are equally spaced in time, which
are our main concern in time sequence analysis.
This subsection may be omitted if the reader is unfamiliar with Fourier series or the integral transform

2.3.2 Shift Theorem.
The DFT is a special case of the T -transform, so multiplication by T Xd`úâ
Vã ä2æ ç
[
will delay the sequence by one sampling interval ]Y . Put the other way, shifting
the time sequence one space will multiply the DFT coefficient
¯
é by â
ª«ðïãñé ò ¬
.
The corresponding amplitude coefficient — é remains unchanged but the phase is
retarded by n0ëìêp . This is physically reasonable; delaying in time cannot change
the frequency content of the time sequence, only its phase. The spike provides a
simple illustration of the shift theorem. Equation (2.24) with

`Ëj gives ˜ éû`
p ; shifting in time by

]Y introduces the phase factor â
ª«ðïã ò ¬
by the shift
theorem, which agrees with (2.24).
2.3.3 Time Reversal
Reversing the order of a sequence is equivalent to reversing the time and shifting
all terms forward by one period (the full length of the sequence). Denote the time-
reversed sequence by W$# so that W%#_ `bW ¬ _ . The right hand side of equation (2.17)
then gives, with the substitution Á `‡p UÛc ,
l
p
¬® y
º
_q» z
W ¬® _0â
ª«ðïé _Cò ¬
`
l
p
¬ y
º¾
» z
W
¾
â
ª«ðïé-ôñ¬®
¾
õ+ò ¬
`
¯'
é (2.31)
Time reversal therefore complex conjugates the DFT.
Time reversal yields the T -transform of T
y
with a delay factor:
¯
# X°Te[½`bW2¬ y½± W2¬®ª«CT ± oCoCo ± W z T
¬® y
`bT
¬® y ¯)( l
T*
(2.32)
2.3.4 Periodic repetition.
Replace c with c ± p in (2.18):
W2_ ´Â¬ `
¬® y
º
é » z
¯
é2â
«ðïã ô _ ´Â¬ õ é ò ¬
`bW-_ (2.33)
Similar substitutions show that WC_ ª¬ `ÀW-_ , W2_ ´Â«•¬ `ÀW2_ , and so forth. The DFT
always “sees” the data as periodically repeated even though we have not specif-
ically required it; the inverse transform (2.18) is only correct if the original data
repeated with period p .

2.3.5 Convolution theorem.
The DFT is a special case of the T -transform and we would expect it to obey the
convolution theorem provided T has the same meaning in both series and we se-
lect the same frequencies èSé , which means the same sampling interval and number
of samples for each series. There is one important difference resulting from the
periodic repetition implicit in the DFT. In deriving (2.7) we omitted terms in the
sum involving elements of W or
·
with subscripts which lay outside the specified
ranges, j to p Ul for W and j to ¸ Ul for
·
. When the sequences repeat period-
ically this is no longer correct. We can make the convolution theorem work in its
original form by extending both sequences by adding zeroes (“padding”) to length
p ± ¸ ± l or more. There is then no overlap with the periodically repeated parts
of the sequences (Figure 2.4).
The effect of convolution is often difficult to understand intuitively, whereas
the equivalent process of multiplication in the frequency domain can be simpler.
Consider the example of convolution with a boxcar. This is a moving average (pro-
vided the boxcar has height l7q¸ ). A moving average might be thought suitable
for smoothing out high frequencies in the data, but this is not the case. Convolu-
tion with
·
in the time domain is equivalent to multiplication by µ in the frequency
domain. The amplitude spectrum in Figure 2.3 shows that the procedure will com-
pletely eliminate the frequencies p¢q¸Ýkfn0p¢q¸ kCoCoCo but will not do a very good job
of removing other high frequencies. The zeroed frequencies correspond to oscilla-
tions with an exact integer number of wavelengths within the length of the boxcar.
Thus a moving average with a length of exactly 12 hours will be very effective in
reducing a signal with exactly that period, (thermal effects from the Sun for exam-
ple), but will not do a good job for neighbouring frequencies (tides, for example,
which do not have periods that are exactly multiples of 12 hours). In general we
do not know the periods in advance, so we cannot design the right moving aver-
age, and few physical phenomena have a single precise frequency. The reduction
of other frequencies is determined mainly by the slowly varying denominator. In
Chapter 4.1 we shall explore more effective ways to remove the higher frequencies.
2.3.6 Cyclic Convolution.
If no padding is applied the product of two DFTs give the DFT of the cyclic convo-
lution. The same formula (2.7) applies but values of the sequence with subscripts
outside the range X°jekphU l[ are the periodically repeated values rather than ze-
roes. The cyclic convolution is periodic with period p , the same as the original
sequences.
The cyclic convolution for the two sequences W and
·
defined in (2.10) is quite

different from the ordinary convolution. First we make them the same length by
adding one zero to
·
, then apply (2.7) while retaining elements of
·
with subscripts
outside the range X°jek•Îe[ , which are filled in by periodic repetition (e.g.
·
y `
·
‹ `
j ,
·

³
`
·
³
`‡„ ). Equation (2.11) is replaced by
+ .Í H.I•.-,ûH
!
I/. !
,óH % I/. % ,ûH10qI/.!02,óH43I/.!3: 878
+
!
H.I
!
,ûH
!
Ið.5,ûH % I/. !
, H104I/. % ,ûH43qI/.!0²7
+ % H.I % ,ûH
!
I
!
,ûH % Ið.-,óH40I/. !
,óH43I/. % 7E
+ 0 H . I 0 ,ûH
!
I•%2,ûH0%I
!
,óH 0 I . ,óH 3 I . !
Û;B
+ 3Í H.I3-,ûH
!
I02,ûH % I % ,óH40I
!
,óH63qIð.:87
(2.34)
The convolution is of length 5, the same as both contributing series, and repeats
periodically with period 5. It is quite different from the non-cyclic convolution in
(2.11): X•l0kf„2kmlqÎSkfnFŽ-kfÎSkfnekml7Ž2kmlCÎe[ .
2.3.7 Differentiation and integration.
Differentiation and integration only really apply to continuous functions of time, so
first let us recover a continuous function of time from our time sequence by setting
Y `Ýce]Y and èîé]` n0ëìêp]Y in (2.18):
WXZY4[Ò`
¬® y
º
é » z
¯
é2â
ãåä þ ç
(2.35)
Differentiating with respect to time gives
š W
š Y
`
¬® y
º
é » z
°èêé
¯
éeâ
ãåä þ ç
(2.36)
Now discretise this equation by setting Y²` ce]Y and
8
W2_a`
š W
š Y
ç »Â_ æÒç
to give
8
W2_à`
¬® y
º
é » z
°è é
¯
é2â
«ðï;ã _ é ò ¬
(2.37)
This is exactly the same form as the inverse DFT (2.18), so
8
W_ and °èîé
¯
é must be
transforms of each other.
Differentiation with respect to time is therefore equivalent to multiplication by
frequency (and a phase factor ) in the frequency domain; integration with respect
to time is equivalent to division in the frequency domain. The same properties hold
for the FIT (Appendix 2).

a b
T
-T 2T
c
T 2T
a b c
-T
Fig. 2.4. Cyclic convolution of a long trace with a shorter one. The long trace is actually
a seismogram and the shorter one (9 ) a “filter” (Chapter 4.1) designed to enhance specific
reflections within the original seismogram. The sequences have been plotted as continuous
functions of time, as would normally be done in processing a seismogram. '
$
J
$
K show
the location of the shorter filter used for calculating different values of the convolution (É
in equation (2.7)). When one time sequence is short cyclic and ordinary convolutions only
differ near the ends. They can be made identical by adding zeroes to both sequences, as
shown in the lower figure. In this example, if the seismogram is outlasts the duration of
significant ground motion there is little difference between cyclic and standard convolution.
For example, we sometimes need to differentiate and integrate seismograms to
convert between displacement, velocity, and acceleration. The change in appear-
ance of the seismograms can sometimes be quite dramatic because multiplication
by frequency enhances higher frequencies relative to lower frequencies; thus the
velocity seismogram can appear noisier than the displacement seismogram [Fig-
ure 2.5].

TIME [min]
0.0 0.5 1.0 1.5 2.0 2.5
Velocity
[micron/s]
Displacement
[microns]
20
-20
10
-15
Fig. 2.5. Broadband velocity (upper trace) and displacement (lower trace) seismograms
obtained from the same record on Stromboli volcano. A depression is most noticeable in
the displacement; the displacement trace also has much less high frequency noise because
of the integration Neuberg (2000).
2.3.8 Parseval’s Theorem.
Substituting for
¯
é from (2.17) gives:
: ¯
é
: «
`
l
p
«
¬® y
º
_q» z
W-_0â
¬ y
º¾
» z
W
¾
â
ª«ðï;ãñé
¾
ò ¬
(2.38)
Summing over ì , using the Nyquist theorem, and the substitution property of the
Kronecker delta gives:
¬® y
º
é » z
: ¯
é
: «
`
l
p
«
º
_6;
¾
W2_7W
¾
¬® y
º
é » z
â
ª«ðïãåé-ô
¾
_Cõ¤ò ¬
`
l
p
º
_6;
¾
W2_;W
¾
øC_
¾

¬® y
º
é » z
: ¯
é
: «
`
l
p
¬® y
º
_C» z
:
W-_
: «
(2.39)
This is called Parseval’s theorem; it ensures equality of “energy” between the time
and frequency domains.
Parseval’s theorem applies generally to any orthogonal function expansion. It
equates the energy in the time sequence (sum of squares of elements) to that of
the function coefficients. Nyquist’s theorem (2.21) ensures that Parseval’s theorem
applies exactly to the DFT.
2.3.9 Fast Fourier Transform.
Equation (2.17) for the DFT requires a sum over p terms for each of p frequency
values ì ; the total computation required is therefore p
«
complex multiplications
and additions (operations), assuming all the exponential factors have been com-
puted in advance. A clever algorithm, called the fast Fourier transform or FFT,
accomplishes the same task with substantially less computation. To illustrate the
principle of the FFT suppose p is divisible by 2 and split the sum (2.17) into two
parts:
p
¯
éÊ`
¬® y
º
_q» z
W-_â
`
¬ ò « y
º
_C» z
W-_0â
±
¬ y
º
_C» ¬ ò «
W2_;â
ª«ðï;ãñé _Cò ¬
¬ ò « y
º
_C» z
X°W-_ ± W _ ´Â¬ ò « â
Vïãñé
[Œâ
ª«ðï;ãñé _qò ¬
(2.40)
This sum is achieved by first forming the combinations W•_ ± W _ ´Â¬ ò « â
Vïãñé
(N/2
operations) and then performing the sum of p¢0n terms. The total number of oper-
ations for all frequency terms is now p ù p¢0n ± p¢0n , a reduction of almost half for
large p when the additional p¢0n operations may be neglected. There is no need
to stop there: provided p can be divided by 2 again we can divide the sum in half
once more to obtain a further speed up of almost a factor of two. When p is a
power of 2 the sum may be divided uwvx « p times, with a final operation count for
large p of about ÎFpÀuwvx « p .
The modern version of the FFT algorithm dates from Cooley and Tukey (1965) It
was known much earlier but did not find widespread application until the advent of
digital computers led to large scale data processing. The power of 2 fast algorithm

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Arachnides
Solpugides.
Dimerosomata Phalangides.
Araneïdes.
Trombidides.
Gammasides.
Monomerosomata Acarides.
Cheyletides.
Eylaïdes.
Hydrachnides.
Ametabolia Thysanura.
Anoplura.
Coleoptera.
Insecta Dermaptera.
Orthoptera.
Dictyoptera.
Hemiptera.
Omoptera.
Metabolia Aptera.
Lepidoptera.
Trichoptera.
Neuroptera.
Hymenoptera.
Rhiphiptera.
Diptera.
Omaloptera.
I have before expressed my sentiments upon several of these Orders[1443]: I shall not here
repeat them, but shall merely observe, with respect to those I have not adopted, that, though
perhaps not entitled to rank as Orders, most of them form natural groups. His Orders, however,
of Arachnida must be excepted from this remark, since they are evidently artificial. His analyses
of his Orders, though in general they give natural groups, are usually not carried so far as those
of M. Latreille, so as seldom to indicate what may properly be denominated families. He has
made his nomenclature for his so-called families more uniform and satisfactory than that of the
French Entomologist: and we may say, with respect to the extent and effect of his zoological
labours,—Nihil non tetigit, et omnia quæ tetigit ornavit.
7. Era of MacLeay, or of the Quinary System. I have more than once stated to you in my former
letters the bases upon which the system which I am in the last place to explain to you is built.
You know the Sub-kingdoms and Classes into which its learned and ingenious author, upon a
novel and most remarkable plan, has divided the Animal Kingdom[1444]. I shall now copy for
you his diagram of the Annulosa.

ANNULOSA
I have before sufficiently noticed these Classes, or Orders as Mr. MacLeay terms them, of the
Sub-kingdom Annulosa: I shall here therefore only throw out a few remarks on their
composition. With regard to their circular distribution in the Crustacea, Mr. MacLeay thinks the
series runs from the Branchiopods or Monoculus L. to the Decapods or Cancer L.; and so on, till
by means perhaps of the genus Bopyrus, which Fabricius regards as a Monoculus, it returns to
the Branchiopods again. This circle, through Porcellio, a kind of wood-louse, c., which has only
a pair of antennæ and at first but six legs, is connected with the Ametabola Class, which
beginning with Glomeris goes by the other Chilognatha (Iulus L.), having also six legs at first,
and certain Vermes to the Anoplura, and terminates in the Chilopoda (Scolopendra L.) their
cognate tribe[1445]. From the Ametabola Mr. MacLeay proceeds to the Mandibulata, between
which two groups he has discovered no osculant one, but he takes the Anoplura of the former
as the transit to the Coleoptera in the latter; from whence passing to the Orthoptera, c., he
finally returns by the Hymenoptera. Between the Mandibulata likewise and Haustellata he finds
no osculant class: but as the affinity between the Trichoptera and Lepidoptera is evident,
proceeding by the Homoptera he returns to the Lepidoptera by certain Diptera, as Psychoda,
c. From the Aptera Lam. or Pulex L. he passes by the osculant class Nycteribida to the
Arachnida; and beginning with the Acaridea, he goes to the Scorpionidea, and so to the
Aranidea or spiders, which he connects with the Decapod Crustacea;—thus forming his great
circle of five smaller ones, each of which, as well as that which they form, returns into
itself[1446].
We next take his Circles of Mandibulata: thus—

MANDIBULATA
In this arrangement of the tribes, as he calls them, of Mandibulata,
Mr. MacLeay sets out from the Coleoptera, which he distributes,
according to the supposed typical forms of their larvæ, into five
minor groups, sufficiently noticed on a former occasion[1447]. From
this tribe or Order he proposes to pass by Atractocerus to the
osculant Order Strepsiptera, and from thence by Myrmecodes and
the Ants to the Hymenoptera. From hence he next proceeds to his

Trichoptera; in which, as we have seen[1447], he places not only
Phryganea L., but also Tenthredo L. and Perla Geoffr., making his
transit by Sirex L.; forming an osculant Order which he denominates
Bomboptera. From this his way to the Neuroptera is by the Perlides,
with Sialis as an osculant Order under the name of Megaloptera: he
enters by Chauliodes, and leaves it by Panorpa or Raphidia by means
of Boreus, forming also an osculant Order (Raphioptera) for the
Orthoptera; which he enters by Phasma, Mantis, c., and leaves by
Gryllus, entering the Coleoptera again by the osculant Order
Dermaptera formed of Forficula L.: and thus returning to the point
from which he set out[1448]. He has not, however, made this return
of the series into itself so clear in each order, excepting in the
Orthoptera, as he has done in the whole Class or Sub-class. Thus in
the Coleoptera there appears no particular affinity between the
Predaceous and Vesicant beetles, his first and fifth forms[1449], or
his Chilopodimorphous Coleoptera, and his Thysanurimorphous.
To enter fully into his doctrine of Analogies would lead us into a very
wide field, and occupy a larger space than I can afford; I must
therefore refer you to his work for more particular and detailed
information on that subject. With regard to the analogy between
opposite points of contiguous circles, you may get a very good idea
of it from his diagram of Saprophagous and Thalerophagous
Petalocerous beetles, which I here subjoin.

It is a very singular circumstance that in these two circles we have
two sets of insects,—one impure in its habits and feeding upon
putrescent food, and the other clean and nourished by food that has
suffered no decay,—set in contrast with each other, and that in each
of the opposite groups, the one has its counterpart in some respect
in the other. In none is this more striking than the Scarabæidæ and
Cetoniadæ, both remarkable for having soft membranous mandibles
unfit for mastication, and both living upon juices, the one in a
putrescent and the other in an undecayed state[1450]. Our learned
author in subsequent works has stated every circle to be resolvable
into two superior groups, which he denominates normal or typical,
and three inferior ones, which he calls aberrant or annectent[1451].
Before I conclude this account of the various general systems that
have distinguished the different entomological eras, i must say a few
words on those partial ones which have been founded on the
neuration of the wings of insects. Frisch, who died in 1743,
attempted something in this way[1452]: Harris, in his Exposition of
English Insects published in 1782, had arranged his Hymenoptera
and Diptera according to characters derived from this same
circumstance[1453]: Mr. Jones in the Linnean Transactions had made
good use of it in dividing the Diurnal Lepidoptera into groups[1454]:

and in the Monographia Apum Angliæ, the characters exhibited by
the various groups into which Linné's genus Apis was resolvable, as
to the neuration of their wings, were described[1455]. But M. Jurine
was the first Entomologist who made that circumstance the keystone
of a system; which indeed he restricted to Hymenopterous and
Dipterous insects, but which might be extended much further. As this
system has been before sufficiently enlarged upon[1456], I need here
only mention it.
To particularize the various entomological works in every department
of the science, that have appeared since the commencement of the
era of Fabricius, would require a volume. Such was its progress and
spread, that in every corner of Europe the pens and pencils of able
and eminent men, whose works have almost all been quoted in the
course of our correspondence, have been employed to illustrate
it[1457]. I may observe, however, that the Internal Anatomy of
Insects, a branch of Entomology which on account of its difficulty,
from the extreme nicety required in dissecting them, had before
been cultivated by scarcely more than a single student in an age,
has now attracted numerous votaries. In Germany—Carus, Gaede,
Herold, Posselt, Ramdohr, Rifferschweils, Sprengel, and others, have
distinguished themselves in this arena: and in France, besides the
illustrious Baron Cuvier (himself a host), Marcel de Serres, Leon
Dufour, and very recently, by his elaborate essays On the Flight of
Insects and its wonderful apparatus, one of the most acute of
anatomical physiologists, M. Chabrier,—have all contributed greatly
to the elucidation of this interesting part of the science. In our own
country very little has hitherto been effected in this line; but a
learned Oxford Professor (Kidd) has presented to the Royal Society
an account of the anatomy of the Mole-cricket, which entitles him to
an eminent station amongst the above worthies.

I may likewise further observe, that the pictorial department of
Entomology was, during the period I am speaking of, carried to its
greatest perfection. Painters of insects formerly were satisfied with
giving a representation generally correct, without attempting a
faithful delineation of all the minor parts, particularly as to number;
—for instance, the joints of the antennæ and tarsi, the areolets of
the wings, c.: but now no one gives satisfaction as an
entomological artist unless he is accurate in these respects.
I am, c.

LETTER XLIX.
GEOGRAPHICAL DISTRIBUTION OF INSECTS;
THEIR STATIONS AND HAUNTS; SEASONS;
TIMES OF ACTION AND REPOSE.
Though no subject is more worthy of the attention of the
Entomologist than the Geographical Distribution of insects, yet
perhaps there is none connected with the science, for the elucidation
of which he is furnished with fewer materials. The geographer of
these animals sitting by his fireside, even supposing his museum as
amply stored as that of Mr. MacLeay, and the habitats of its contents
as accurately indicated, still labours under difficulties that are almost
insuperable; so that it is next to impossible, with our present
knowledge of the subject, to give satisfactory information upon
every point which it includes. Had he the talents and opportunities of
a Humboldt, and could, like him, traverse a large portion of the
globe, he would endeavour to note the elevation, the soil and
aspect, the latitude and longitude, the mean temperature and
meteorological phænomena, the season of the year, the kind of
country, and other localities connected with the insects he captured,
and so might build his superstructure upon a sure basis. But these
are things seldom registered by travellers that take the trouble to
collect insects; who, if they specify generally the country in which
any individual was found, think they have done enough. But to say
that an insect was taken in India, China, New Holland, and North or
South America,—when we consider the vast extent of those regions,
—is saying little of what one wishes to know even with respect to its
habitat. You must regard therefore, after all, what I have been able
to collect,—and for which I am greatly indebted to the labours of my

few but able precursors in this walk,—as merely approximations to
an outline, rather than as a correct map of insect Geography.
Amongst the numerous obligations that he conferred upon Natural
History, Linné was the first Naturalist who turned his attention to the
Geographical Distribution of its objects, especially that of the
Vegetable Kingdom[1458]: and the accomplished traveller Baron
Humboldt, by the observations he made on this subject in the course
of his peregrinations in tropical America, has furnished the Botanist
with a clue which, duly followed, will enable him to perfect that part
of his science; an end to which the learned observations of Messrs.
R. Brown and Decandolle have greatly contributed[1459]. With regard
to animals, Mr. White, so long ago as 1773, had observed that they,
as well as plants, might with propriety be arranged
geographically[1460]: and in 1778 Fabricius in his Philosophia
Entomologica applied the principle to insects[1461]. Nearly forty years
elapsed before any improvement or enlargement of this last
department was attempted; when in 1815 M. Latreille, stimulated by
what had been effected in Botany, in a learned and admirable
memoir[1462] endeavoured to place Entomology in this respect by
the side of her more fortunate sister: and subsequently Mr. W. S.
MacLeay, in the memorable work so often quoted in our
correspondence, has viewed the subject in another light, and added
some important information to what had been before collected[1463].
The point now under consideration naturally divides itself into two
principal branches;—the numerical distribution of insects, and the
topographical.
I. By the numerical distribution of insects I mean not only the
number which Providence has employed to carry on its great plan on
this terraqueous globe, or any given portion of it; or of the species
of which each group or genus may be supposed to consist; or of the
comparative number of individuals furnished by each species,—
points of no easy solution: but more particularly their distribution

according to their functions, whether they prey upon animal or
vegetable matter, and in its living or decaying state.
We have no data enabling us to ascertain with any degree of
accuracy the actual number of species of insects and Arachnida
distributed over the surface of the globe; but it is doubtless
regulated in a great degree by that of plants. We should first then
endeavour to gain some just though general notion on that head.
Now Decandolle conjectures that the number of the species of
plants, 60,000 being already known, may be somewhere between
110,000 and 120,000[1464]. If we consider with reference to this
calculation, that though the great body of the mosses, lichens, and
sea-weeds are exempt from the attack of insects, yet as a vast
number of phanerogamous plants and fungi are inhabited by several
species, we may form some idea how immense must be the number
of existing insects; and how beggarly does Ray's conjecture of
20,000 species[1465], which in his time was reckoned a magnificent
idea, appear in comparison! Perhaps we may obtain some
approximation by comparing the number of the species of insects
already discovered in Britain with that of its phanerogamous plants.
The latter,—and it is not to be expected that any large number of
species have escaped the researches of our numerous Botanists,—
may be stated in round numbers at 1500, while the British insects,
(and thousands it is probable remain still undiscovered,) amount to
10,000; which is more than six insects to one plant. Now though this
proportion, it is probable, does not hold universally; yet if it be
considered how much more prolific in species tropical regions are
than our chilly climate, it may perhaps be regarded as not very wide
of a fair medium. If then we reckon the phanerogamous vegetables
of the globe in round numbers at 100,000 species, the number of
insects would amount to 600,000. If we say 400,000, we shall
perhaps not be very wide of the truth. When we reflect how much
greater attention has been paid to the collection of plants than to
that of insects, and that 100,000 species of the latter may be
supposed already to have a place in our cabinets[1466], we may very

reasonably infer that at least three fourths of the existing species
remain undiscovered.
Certain groups and genera are found to contain many more species
than others: for instance, the Coleoptera and Lepidoptera Orders
than the Orthoptera and Neuroptera; the Rhincophora than the
Xylophagi: the Dytiscidæ than the Gyrinidæ; Aphodius than
Geotrupes; Carabus than Calosoma. Again, some insects are much
more prolific than others. Thus the Diptera Order, though not half so
numerous with respect to species as the Coleoptera, exceeds it
greatly in the number of individuals, filling the air in every place and
almost at every season with its dancing myriads. We rarely meet
with a single individual of the most common species of Calosoma or
Buprestis; whilst the formicary, the termitary, the vespiary, and the
bee-hive send forth their thousands and tens of thousands; and
whole countries are covered and devastated by the Aphides and the
Locusts. An all-wise Providence has proportioned the numbers of each
group and species to the work assigned to them. And this is the view
in which the numerical distribution of insects is most interesting and
important: and we are indebted to Mr. W. S. MacLeay for calling the
attention of Entomologists more particularly to this part of our
present subject.
With regard to their functions, insects may be primarily divided into
those that feed upon animal matter and those that feed upon
vegetable. At first you would be inclined to suppose that the latter
must greatly exceed the former in number: but when you reflect that
not only a very large proportion of Vertebrate animals, and even
some Mollusca[1467], have more than one species that preys upon
them, but that probably the majority of insects, particularly the
almost innumerable species of Lepidoptera, are infested by parasites
of their own class, sometimes having a different one appropriated to
them in each of their preparatory states[1468], and moreover that a
large number of beetles and other insects devour both living and
dead animals,—you will begin to suspect that these two tribes may
be more near a counterpoise than at first seemed probable. In fact,

out of a list of more than 8000 British insects and Arachnida taken
several years ago, and furnished chiefly by Mr. Stephens, I found
that 3894 might be called carnivorous, and 3724 phytiphagous[1469];
so that, speaking roundly, they might be denominated
equiponderant.
Carnivorous and phytiphagous insects may be further subdivided
according to the state in which they take their food,—whether they
attack it while living, or not till after it is dead. To adopt Mr. W. S.
MacLeay's phraseology, the former may be denominated
thalerophagous, and the latter saprophagous. The British
saprophagous carnivorous insects, compared with those that are
thalerophagous, are about as 1:6; while the phytiphagous ones are
as 1:9. The thalerophaga in both tribes may be further subdivided as
they take their food by suction or mastication: in the carnivorous
ones, the suckers to the masticators in Britain are nearly as 1:6; but
with respect to the phytiphagous tribe you must take into
consideration that some insects imbibing their food by suction in
their perfect state (as the great body of the Lepidoptera), masticate
it when they are larvæ: deducting therefore from both sides the
insects thus circumstanced, the masticators will form about three
fourths of the remaining British thalerophagous insects. Another
circumstance belonging to this head must not be passed without
notice:—there are certain insects feeding upon liquid food that do
not suck, but lap it. This is the case with the Hymenoptera, who,
though they are mandibulate, generally lap their food (the nectar of
flowers) with their tongue, and may be called lambent insects: nor is
this practice confined to that order, but all the mandibulate insects
that feed on that substance merit the same appellation. The
absorption of this nectar is so important a point in the economy of
nature, that a very large proportion of the insect population of the
globe in their perfect state, are devoted to it. Considerably more
than half the species indigenous to Britain fulfill this function, and
probably in tropical countries the proportion may be still larger.

To push this analysis still further—Amongst our carnivorous
thalerophaga, aphidivorous insects are about as 1:14; and amongst
the phytiphagous, the fungivorous ones form about a twentieth; and
the granivorous about a twenty-fifth part of the whole. Again: in the
saprophaga the lignivorous tribes form more than half, and the
coprophagous ones more than a third.
If you wish to know further the relative proportions of the different
Orders to each other—The Coleoptera may be stated as forming at
least 1:2 of our intire insect population; the Orthoptera and
Dermaptera as about 1:160; the Hemiptera as 1:15; the Lepidoptera
as more than 1:4; the Neuroptera with the Trichoptera as 1:29; the
Hymenoptera as about 1:4; the Diptera as not 1:7; and the Aptera
and Arachnida as perhaps amounting to 1:19[1470].
To extend this inquiry to exotic and more particularly to extra-
European insects, in the present state of our knowledge, would lead
to no very satisfactory results. The lists we have are so imperfect,
that those which tell most in this country,—I mean the more minute
insects and the Brachyptera—have hitherto formed a very small, if
any part, of the collections made out of Europe. Mr. W. S. MacLeay
however, who, besides his father's (particularly rich in Petalocera),
has had an opportunity of examining the Parisian and other cabinets,
finds that the species of coprophagous insects within the tropics, to
those without, are nearly in the proportion of 4:3; and that the
coprophagous Petalocera, to the remainder of the saprophagous
ones, may be represented by 3:2[1471]. It may be inferred, from the
superabundance of plants and animals in equinoctial countries, that
the number of species of insects in general is greater within than
without the tropics: the additional momentum produced by the vast
size of many of the tropical species must also be taken into
consideration.
II. There are three principal points that call for attention under the
second branch of our present subject—the topographical distribution

of insects; namely, their Climates, their Range, and their
Representation.
i. Entomologists, taking heat for the principal regulator of the station
of insects, have divided the globe into entomological climates.
Fabricius considers it as divisible into eight such climates, which he
denominates the Indian, Egyptian, Southern, Mediterranean,
Northern, Oriental, Occidental, and Alpine. The first containing the
tropics; the second, the northern region immediately adjacent; the
third, the southern; the fourth, the countries bordering on the
Mediterranean sea, including also Armenia and Media; the fifth, the
northern part of Europe interjacent between Lapland and Paris; the
sixth, the northern parts of Asia where the cold in winter is intense;
the seventh, North America, Japan, and China; and the eighth, all
those mountains whose summits are covered with eternal
snow[1472]. M. Latreille objects to this division, as too vague and
arbitrary and not sufficiently correct as to temperature; and
observes, with great truth, that as places where the temperature is
the same, have different animals, it is impossible, in the actual state
of our knowledge, to fix these distinctions of climates upon a solid
basis. The different elevations of the soil above the level of the sea,
its mineralogical composition, the varying quantity of its waters, the
modifications which the mountains, by their extent, their height, and
their direction, produce upon its temperature; the forests, larger or
smaller, with which it may be covered; the effects of neighbouring
climates upon it,—are all elements that render calculations on this
subject very complicated, and throw a great degree of uncertainty
over them[1473]. This learned Entomologist would judiciously
consider entomological climates under another view,—that which the
genera of Arachnida and insects exclusively appropriated to
determinate spots or regions would supply[1474]. Linné's dictum with
regard to genera will here also apply; Let the insects point out the
climate, and not the climate the insects. If you expect invariably to
find the same insects within the same parallels of latitude, you will
be sadly disappointed; for, as our author further observes, The

totality or a very large number of Arachnida and insects, the
temperature and soil of whose country are the same, but widely
separated, is in general, even if the countries are in the same
parallel, composed of different species[1475]. The natural limits of a
country,—as mountainous ranges, rivers, vast deserts, c.,—often
also say to its insect population, No further shall ye come;
interposing a barrier that it never passes[1476]. Humboldt observes,
with respect to the Simulia and Culices of South America, that their
geographical distribution does not appear to depend solely on the
heat of the climate, the excess of humidity, or the thickness of
forests; but on local circumstances that are difficult to
characterize[1477]: and Mr. W. S. MacLeay makes a similar
observation upon that of Gymnopleurus[1478]. So that the real insect
climates, or those in which certain groups or species appear, may be
regarded as fixed by the will of the Creator, rather than as certainly
regulated by any isothermal lines. Still, however, under certain
limitations, it must be admitted that the temperature has much to do
with the station of insects. The increase of caloric is always attended
with a proportional increase in the number and kind of the groups
and species of these beings. If we begin within the polar regions of
ice and snow, the list is very meager. As we descend towards the
line, their numbers keep gradually increasing, till they absolutely
swarm within the tropics. Something like this takes place in
miniature upon mountains. Tournefort long since observed at the
summit of Mount Ararat the plants of Lapland; a little lower, those of
Sweden; next, as he descended, those of Germany, France, and
Italy; and at the foot of the mountain, such as were natural to the
soil of Armenia. And the same has been observed of insects. Those
that inhabit the plains of northern regions have been found on the
mountains of more southern ones; as the beautiful and common
Swedish butterfly Parnassius Apollo, on the mountains of France,
and Prionus depsarius on those of Switzerland[1479].
M. Latreille, having given a rapid survey of the peculiar insect-
productions of different countries, next attempts a division of the

globe into climates, which he thinks may be made to agree with the
present state of our knowledge, and be even applicable to future
discoveries. He proposes dividing it primarily into Arctic and Antarctic
climates, according as they are situated above or below the
equinoctial line; and taking twelve degrees of latitude for each
climate, he subdivides the whole into twelve climates. Beginning at
84° N. L. he has seven Arctic ones, which he names polar, subpolar,
superior, intermediate, supratropical, tropical, and equatorial: but his
antarctic climates, as no land has been discovered below 60° S. L.,
amount only to five, beginning with the equatorial and terminating
with the superior. He proposes further to divide his climates into
subclimates, by means of certain meridian lines; separating thus the
old world from the new, and subdividing the former into two great
portions,—an eastern, beginning with India, and a western,
terminating with Persia. He proposes further that each climate
should be considered as having 24° of longitude, as well as 12° of
latitude[1480]. In this chart of insect Geography he states that he has
endeavoured to make his climates agree with the actual distribution
of insects[1481]; and it should seem that in many cases such an
agreement actually does take place: yet the division of the globe into
climates by equivalent parallels and meridians, wears the
appearance of an artificial and arbitrary system, rather than of one
according with nature.
He has also pointed out another index to insect climates, borrowed
from the Flora of a country. Southern forms in Entomology, he
observes, commence where the vine begins to prosper by the sole
influence of the mean temperature; that they are dominant where
the olive is cultivated; that species still more southern are
compatriots of the orange and palmetto; and that some equatorial
genera accompany the date, the sugar-cane, the indigo and
banana[1482]. The idea is very ingenious, and, under certain
limitations, supplies a useful and certain criterion. For though none
of these plants are universal in isothermal parallels of latitude; yet,
as plants are more conspicuous than insects, the Entomologist,

furnished with an index of this kind, may by it be directed in his
researches for them; and in all countries in which there is a material
change of the climate, as in France, there will be a proportional
change in the vegetable accompanied by one in the insect
productions.
ii. In considering the range of insects I shall first advert to that of
individual species. At the extreme limits of phanerogamous
vegetation we find a species of humble-bee (Bombus arcticus),
which, though it is not known to leave the Arctic circle, has a very
extensive range to the westward of the meridian of Greenwich,
having been traced from Greenland to Melville Island; while to the
eastward of that meridian it has not been met with. In Lapland its
place appears to be occupied by B. alpinus and lapponicus, with the
former of which, though quite distinct, it was confounded by O.
Fabricius; but whether these range further eastward of that meridian
has not been ascertained. From its being found in the Lapland
Alps[1483], it may be conjectured that B. alpinus ranges as high on
this side as B. arcticus on the other, and may perhaps be found in
Nova Zembla. Some species that have been taken in Arctic regions
are not confined to them. Of this kind is Dytiscus marginalis, which
appears common in Greenland, abundant in Britain, and is dispersed
over all Europe; while D. latissimus is more confined, neither ranging
so far to the north or south; and though found in Germany, not yet
discovered in Britain. Other species have a still more extensive
range, and are common to the old world and the new. Thus
Dermestes murinus, Brachinus crepitans, Tetyra scarabæoides[1484],
Pentatoma juniperina, Cercopis spumaria, Vanessa Antiopa,
Polyommatus Argiolus, Hesperia Comma, Vespa vulgaris, Ophion
luteus, Helophilus pendulus, Oscinis Germinationis, and many
besides, though sometimes varying slightly[1485], inhabit both Britain
and Canada: and though vast continents and oceans intervene
between us, New Holland, and Japan; yet all have some insect
productions in common. With the former we possess the painted-
lady butterfly (Cinthia Cardui), with scarcely a varying streak: and

Thunberg, in his list of Japan insects, has mentioned more than forty
species that are found also in this country. Whether any species has
a universal range may be doubted, unless indeed the flea and the
louse may be excepted. On the other hand, some are confined
within very narrow limits. Apion Ulicis for instance, abundant upon
Ulex europæus in Britain, has not, I believe, been found upon that
plant on the continent.
The geographical distribution of groups, is, however, far more
interesting than that of individual species: for in considering this we
see more evidently how certain functions are devolved upon certain
forms, and can scan the great plan of Providence, in the creation of
insects, more satisfactorily than by confining our attention to the
latter. Groups, according to their range, may be denominated either
predominant, dominant, sub-dominant, or quiescent.
1. M. Latreille has observed, that where the empire of Flora ceases,
there also terminates that of Zoology[1486]. Phytiphagous animals
can only exist where there are plants; and those that are carnivorous
and feed upon the former, must of necessity stop where they stop.
Even the gnat, which extends its northern reign so high[1487], must
cease at this limit; while, where vegetation is the richest and most
abundant, there the animal productions, especially the insect, must
be equally abundant. I call that, therefore, a predominant group,
members of which are found in all the countries between these
points, or from the limits of animal-depasturing vegetation in the
polar regions to the line.
Generally speaking, the carnivorous insects, whether thalerophagous
or saprophagous, are of this description. Calosoma, which devours
Lepidopterous larvæ, though poor in species and individuals, is
widely scattered. Captain Frankland found C. calidum in his Arctic
journey; C. laterale and curvipes inhabit tropical America[1488]: C.
Chinense, as its name indicates, is Chinese[1489]; Mr. MacLeay has
an undescribed species from New Holland; and C. retusum was
taken in Terra del Fuego. Another genus, equally universal and richer

in numbers, is the lady-bird (Coccinella), which keeps within due
limits the Aphides of every climate from pole to pole. The Libellulina
pursue their prey both in Greenland and New Holland. The
saprophagous carnivora are also similarly predominant;—the
Silphidæ, the Dermestidæ, the Brachyptera, the Muscidæ, prey on
carcases wherever the action of the solar beam causes them to
become putrid. Many of the above insects have probably their capital
station, or that where the species are most numerous, in or near the
tropics; but the metropolis of the Brachyptera, at least as far as we
can judge from our present catalogues, is within the temperate
zone, particularly in Britain[1490]. The coprophagous Petalocera are
most abundant in the hottest climates; but the Aphodiadæ form a
predominant group: Professor Hooker took one species in
Iceland[1491], and it probably ascends higher; others are found in
India and China: but the metropolis of the group is within the
temperate zone. Perhaps no genus is more completely universal than
Bombus (Bremus Jur.), which, although its centre or metropolis is
likewise in the northern temperate zone, extends from Melville Island
to the line. It is remarkable that some of the tropical Bombi wear the
external aspect of Xylocopæ, the kindred genus most prevalent in
warm climates; and, vice versâ, some Xylocopæ resemble Bombi. I
have a Brazilian undescribed species of the latter genus, whose black
body and violet-coloured wings would almost cause it to be mistaken
for a variety of X. violacea; and B. antiguensis and caffrus F.,
(though their aspect belies it,) which misled Fabricius, are true
Xylocopæ. I shall mention only one other predominant group, but
that one of no common celebrity, formed of the gnats, or genus
Culex. These piping pests, with their quiver—venenatis gravida
sagittis—annoy man almost from the pole to the line. What
remarkably distinguishes them, (as was formerly observed[1492],)
and also the Simulium or true mosquito,—they appear to prevail
most in the coldest and the hottest climates, and the Laplander and
the tropical American are equally their prey; while the inhabitants of
the temperate zone, with some exceptions, suffer but little from

them: so that they may be stated to have both an arctic and a
tropical metropolis.
2. There are other groups which, though their empire extends to the
tropics, fall short of the polar circles:—these I call dominant groups.
Of this description are some of the Scarabæidæ. Onthophagus is
found both in the old world and in the new, and in the temperate
and torrid zones. Its principal seat appears to be within the tropics,
but it may almost be said to have also a northern metropolis. More
than one species have been taken in New Holland. In general,
tropical insects exceed those of colder climates in size; but in the
genus we are speaking of, the European species are usually larger
than the Indian. Copris seems more abhorrent of cold than its near
relation Onthophagus. C. lunaris, which ranges northward as far as
Sweden, is the only recorded species found in Europe out of Spain.
Latreille says, that all the large species of this genus are equinoctial:
but C. Tmolus, described and figured by Fischer[1493], found in Asia
near Orenburg, north of 50° N. L., is as big as C. Gigas or
bucephalus. Another dominant group of Petalocera, remarkable for
the bulk and arms of its tropical species, are the mighty Dynastidæ,
the giants and princes of the insect race. Though their metropolis is
strictly tropical, yet the scouts of their host have wandered even as
far as the south of Sweden, where one of them, Oryctes nasicornis,
is extremely common. O. Grypus[1494] and some other species are
found in South Europe; but though in a torpid state they can endure
unhurt the severity of a Scandinavian winter, they cannot when
revived stand the cold that often pinches Britons in the midst of
summer, and therefore are unknown in our islands[1495]. The
Sphæridiadæ, whose metropolis is within the northern temperate
zone, extend from thence beyond the line, since Dr. Horsfield found
two species in Java[1496]. It is probable, indeed, that this group is
predominant. Some dominant groups begin at a lower latitude. Of
this description are the carpenter-bees (Xylocopa), whose larvæ are
preyed upon by that of the Horiadæ[1497] under two forms, which
extend from the tropics to about 50° N. L. Others are not common

to both worlds. Thus, while Cantharis is the gift of Providence to
America as well as the old world, Mylabris is confined to the latter,
where its range is very extensive;—in Europe, from South Russia to
Italy and Spain; in Asia, from Siberia to India; and in Africa, from the
shores of the Mediterranean to the Cape of Good Hope; which last
continent, to judge from our present lists, especially the vicinity of
the Cape, may be called the metropolis of the group[1498]. On the
other hand, the Rutelidæ and Chlamys, which have a range from
Canada to the tropics, (within which is their metropolis,) are purely
American groups. Many more might be named under this head, but
these will suffice for examples.
3. I call those subdominant groups, which either never enter the
tropics, or those tropical ones whose range does not exceed 50° of
N. L. in the old world, or 43° in the new. I make this difference
because, as M. Latreille observes, the southern insects which in
Europe begin between 48° and 49° N. L., in America do not reach
43°.[1499] But though the winters in Canada, within the same
parallel as France, are longer and more severe than those even of
Great Britain or of Germany, yet the summers are intensely hot; so
that though tropical species do not range so high, those of a tropical
structure, as Mr. W. S. MacLeay has intimated[1500], may be found at
a higher latitude in the new world than in Europe.
The genus Melöe affords an instance of a subdominant group of the
first description. It ranges from Sweden to Spain and the shores of
the Mediterranean, and seems a tribe almost confined to Europe,
where it is not very unequally distributed. Of registered species
Britain possesses the largest proportion; but Mr. W. S. MacLeay is of
opinion that Spain is its true metropolis[1501]. I have a species of
this genus, taken in North America by Professor Peck. The splendid
genus Carabus ranges still further north than Melöe[1502]. A very fine
species (C. cribellatus) inhabits the polar regions of Siberia[1503]; but
the metropolis of the group appears to be the temperate zone:
some, however, have been found in northern Africa; and Sir Joseph

Banks captured one in Terra del Fuego. Of those whose range is
between the tropics and 50° N. L. we may begin with Cicada. One
species, indeed, has been found by Mr. Bydder and others, a little
higher, near the New Forest, Hampshire. We may take Scolia for an
example of a subdominant group beginning more southward. Its
species first appear about 43° N. L., and abound in warm climates.
In general most of those insects which M. Latreille denominates
meridional,—such as Scarabæus, Onitis, Brentus, Scarites, Mantis,
Fulgora, Termes, Scorpio, c.—come under the present head, and in
fact all tropical forms that wander to any distance within the above
limits from their metropolis.
4. By quiescent groups I mean those that have none, or no high
range as to latitude, from their centre or metropolis. I say as to
latitude, because these groups have often an extensive one as to
longitude. Thus, Mr. W. S. MacLeay has remarked to me, that
Goliathus appears to belt the globe, but not under one form. The
types of the genus are the vast African Goliaths (G. giganteus, c.),
which, as well as G. Polyphemus, and another brought from Java by
Dr. Horsfield, have, like Cetonia[1504], the scapulars interposed
between the posterior angles of the prothorax and the shoulders of
the elytra[1505]: while the South American species (G. micans, c.)
have not this projection of the scapulars; in this resembling Trichius.
Mr. MacLeay further observes, that the female of the Javanese
Goliathus is exactly a Cetonia, while that of the Brazilian is a
Trichius. But quiescent groups have not generally this ample
longitudinal range. Thus, Euglossa, in both its types,—one
represented by Eu. cordata, and the other by Eu. surinamensis,—is
confined to the tropical regions of America. Doryphora, likewise
American, seems equally confined. Asida, though a southern genus,
is not found to enter the tropics; and Manticora and Pneumora are in
nearly the same predicament.
Under the present head we may consider what may perhaps be
denominated without much impropriety endemial groups; by which I
mean those groups that are regulated, as to their limits, not so

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