Information Theory and the Analysis of Uncertainties in a Spatial Geological Context

Information Theory and the Analysis of Uncertainties in
a Spatial Geological Context
Florian Wellmann, Mark Lindsay and Mark Jessell
Centre for Exploration Targeting (CET)
PICO presentation — EGU 2014
May 9, 2014

Structural Geological Models and Uncertainties
Section view of a structural geological model
Model created during a mapping course by one team of students...
Yellow lines: surface contacts White lines: faults
(From: Courrioux et al., 34th IGC, Brisbane, 2012)

Structural Geological Models and Uncertainties
Section view of a structural geological model
Model created during a mapping course by one team of students...
...and results from multiple teams!

Stochastic Geological Modelling
Stochastic modelling approach
Primary Observations
Realisation 1
Realisation n
Realisation 3
Realisation 2
Model 1
Model n
Model 3
Model 2
c
(Jessell et al., submitted)
Generate multiple structural
geological models with a
stochastic approach

So how to analyse all those generated models?
Our approach taken here:
Calculate probabilities
for geological units in
discrete regions (cells) of
the model;

the model;
Determine information
entropy for each cell as
a measure of
uncertainty;

the model;
a measure of
uncertainty;
Evaluate conditional
entropy to determine
how knowledge at one
location could reduce
uncertainties elsewhere.

Application to Gippsland Basin model
We apply the concept here to a kinematic structural model of the
Gippsland Basin, SE Australia:
We assume that parameters related to the geological history are
uncertain and generate multiple realisations.

Analysis of a 2-D slice of the model
As an example, consider uncertainties in a E-W slice through the
model:
Information entropy shows high uncertainties in basin:
0 20 40 60 80
X
0
20
40
Y
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
entropy(bits)

Analysis of a 2-D slice of the model
As an example, consider uncertainties in a E-W slice through the
model:
Conditional entropy to determine potential uncertainty reduction (e.g.
during drilling):

Overview of Presentation
“PICO madness”
Geological
uncertainties
Brief overview:
important concepts of
Information Theory
Stochastic geo-
logical modelling
Application to
a kinematic struc-
tural model of the
Gippsland Basin

Back to overview .
Information theoretic concepts - intuitive introduction
Concepts from information theory used in this work
In the work presented here, we apply basic measures from information
theory to evaluate uncertainties in a spatial context:

Back to overview .
1 Information entropy as a measure of uncertainty at one spatial
location, and

Back to overview .
location, and
2 Conditional entropy to determine how knowledge at one location
could reduce uncertainties at another location. Finally,

Back to overview .
location, and
3 Multivariate conditional entropy is applied to evaluate how
gathering successive information, e.g. while drilling, reduces
uncertainty.

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location, and
3 Multivariate conditional entropy is applied to evaluate how
gathering successive information, e.g. while drilling, reduces
uncertainty.
In this subsection, we provide a brief and intuitive introduction into these
concepts.

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Information theory and the coin ﬂip
Coin ﬂip example
Simple example for the interpretation
of information entropy:
For a fair coin, p(head) =
p(tail) = 0.5: the uncertainty
is highest as no outcome is
preferred (•)

Back to overview .
Coin ﬂip example
preferred (•)
If the coin is unfair (and we
know it), uncertainty is
reduced (•)

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Coin ﬂip example
preferred (•)
If the coin is unfair (and we
know it), uncertainty is
reduced (•)
For a double-headed coin,
outcome is known, no
uncertainty remains (•)

Back to overview .
Conditional entropy and uncertainty reduction
Sharing information about a coin toss
Now we assume a related experiment: we ask someone who observed
the coin toss about the outcome. What is the remaining uncertainty about
the outcome?
Case 1: We ask a good friend
H(X) = 1 H(Y |X) = 0
Friend
100%
Always tells us the right result, no remaining uncertainty

Back to overview .
the outcome?
Case 2: We ask someone who might be a friend
H(X) = 1 H(Y |X) = 0.47
“Friend”
90%
Might tell us the outcome mostly correctly, but uncertainties remain...

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the outcome?
Case 3: We ask someone who may not be a friend at all...
H(X) = 1 H(Y |X) = 1
Friend
0%
We can not rely at all on the reply, the uncertainty is not reduced at all!

Back to overview .
Interpretation in a spatial context
Interpretation in a spatial
context:
the model;

Back to overview .
context:
the model;
a measure of
uncertainty;

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context:
the model;
a measure of
uncertainty;
Evaluate conditional
entropy to determine
how knowledge at one
location could reduce
uncertainties elsewhere.

Back to overview .
Conclusion
More information
For more information, see:
The landmark paper by Claude Shannon (1948);
As a good extended theoretic overview: Cover and Thomas:
Elements of Information Theory;
Our paper in Entropy (open access);
The wikipedia page for Information theory.

Back to overview .
Conclusion
More information
For more information, see:
The landmark paper by Claude Shannon (1948);
As a good extended theoretic overview: Cover and Thomas:
Elements of Information Theory;
Our paper in Entropy (open access);
The wikipedia page for Information theory.
Next ...
Continue with the next section: the overview of Geological
uncertainties
Or go back to the Overview

Back to overview .
Uncertainties in 3-D Geological Modelling
Types of uncertainty
Mann (1993):
Error, bias, imprecision
B´ardossy and Fodor (2001):
Sampling and
observation error

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Mann (1993):
Inherent randomness
Sampling and
observation error
Variability and
propagation error

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Mann (1993):
Inherent randomness
Incomplete knowledge
Sampling and
observation error
Variability and
propagation error
Conceptual and model
uncertainty

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Geological Uncertainties are real
Field example by Courrioux et al.: comparing multiple 3-D models,
created for same region, by diﬀerent teams of students

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Next...
Conclusion
Uncertainties in structural geological models can be signiﬁcant!
In practice, creating several models for the same region is not
feasible - we therefore attempt to simulate the eﬀect of
uncertainties with stochastic methods (see next section).

Back to overview .
Next...
Conclusion
Uncertainties in structural geological models can be signiﬁcant!
In practice, creating several models for the same region is not
feasible - we therefore attempt to simulate the eﬀect of
uncertainties with stochastic methods (see next section).
Next ...
Continue with the next section: Stochastic Modelling for
structural models

Back to overview .
Realisation 1
Realisation n
Realisation 3
Realisation 2
Model 1
Model n
Model 3
Model 2
c
ologies per voxel 6
Start with geological
parameters (observations or
aspects of geological history)

Back to overview .
Realisation 1
Realisation n
Realisation 3
Realisation 2
Model 1
Model n
Model 3
Model 2
c
ologies per voxel 6
Assign probability
distributions to observations

Back to overview .
Realisation 1
Realisation n
Realisation 3
Realisation 2
Model 1
Model n
Model 3
Model 2
c
ologies per voxel 6
Assign probability
Randomly generate new
parameter sets

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Realisation 1
Realisation n
Realisation 3
Realisation 2
Model 1
Model n
Model 3
Model 2
c
ologies per voxel 6
Assign probability
Randomly generate new
parameter sets
Create models for all sets

Back to overview .
3-D Modelling Methods
Diﬀerent methods to create 3-D models
Several methods exist to generate 3-D geological models. Most suitable
for stochastic structural modelling are:
Implicit modelling
method
SKUA%
Earthvision% Geomodeller%
Noddy%
Explicit(
Implicit(
Kinema/c/(
Mechanical(
Geophysical(
Inversion(
VPmg%
Kine3D%
Vulcan%(old)%

Back to overview .
Implicit modelling
method
Kinematic/ mechanical
modelling methods
SKUA%
Noddy%
Explicit(
Implicit(
Kinema/c/(
Mechanical(
Geophysical(
Inversion(
VPmg%
Kine3D%
Vulcan%(old)%

Back to overview .
Implicit modelling
method
Kinematic/ mechanical
modelling methods
We use in the application
in this presentation a
kinematic modelling
approach.
SKUA%
Noddy%
Explicit(
Implicit(
Kinema/c/(
Mechanical(
Geophysical(
Inversion(
VPmg%
Kine3D%
Vulcan%(old)%

Back to overview .
Next...
For more information, please see:
on stochastic structural geological modelling, e.g.:
Jessell et al., 2010
Lindsay et al., 2012
Wellmann et al., 2010
For implicit geological modelling, e.g. Calcagno et al., 2008
For kinematic modelling and Noddy: Jessell, 2001.

Back to overview .
Next...
For more information, please see:
on stochastic structural geological modelling, e.g.:
Jessell et al., 2010
Lindsay et al., 2012
Wellmann et al., 2010
For implicit geological modelling, e.g. Calcagno et al., 2008
For kinematic modelling and Noddy: Jessell, 2001.
Next ...
Continue with the next section: Application to a kinematic
structural model of the Gippsland Basin

Back to overview .
Example model: Gippsland Basin, SE Australia
The Gippsland Basin is a sedimentary basin, located in SE Australia:
( Lindsay et al., 2013)

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Example model: Gippsland Basin, SE Australia
Kinematic model reﬂects main geological events leading to the formation
of the basin:
6580
70
FoldUnconformity Unconformity Unconformity Fault Fault Fault Unconformity
Tectonic EvolutionTectonic Evolution
Kinematicmodel
Noddy
FINAL MODEL!F AL DEL!
90
Jessell(1981)
For more information, see also poster on Thursday, Session SSS11.1/ESSI3.6 B190, or the Abstract

Back to overview .
Kinematic block model
3-D view of the base model
E-WN-S
In a ﬁrst step, we evaluate uncertainties in an E-W slice through the
Graben structure.

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Model slice and uncertainties
Slice in E-W direction and considered uncertainties
The parameterisation of the geological events contains uncertainties,
and we consider here as uncertain:
0 20 40 60 80
X
0
20
40
Z
Parameters of geological
history:

Back to overview .
0 20 40 60 80
X
0
20
40
Z
history:
Fault positions and dip
angle (•)

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0 20 40 60 80
X
0
20
40
Z
1
2
3
history:
angle (•)
Age relationship
(order) of faults (•)

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0 20 40 60 80
X
0
20
40
Z
1
2
3
history:
angle (•)
Age relationship
Unit thickness (•)

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0 20 40 60 80
X
0
20
40
Z
1
2
3
history:
angle (•)
Age relationship
Unit thickness (•)
Position of
unconformity (•)

Back to overview .
Multiple model realisations
These are samples of the set of randomly generated models:
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X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z
0 20 40 60 80
X
0
20
40
Z

Back to overview .
Analysis of unit probabilities
Visualising probabilities for diﬀerent units provides an insight into
speciﬁc outcomes, but is not suitable to represent spatial uncertainty
for the entire model:
0 20 40 60 80
0
10
20
30
40
Probability of unit 15
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 20 40 60 80
0
10
20
30
40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 20 40 60 80
0
10
20
30
40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 20 40 60 80
0
10
20
30
40
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

Back to overview .
Analysis of information entropy
Visualisation of information entropy
0 20 40 60 80
X
0
20
40
Y
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
entropy(bits)
1
1 Uncertainties are highest in the
deep parts of the basin;
Entropy is calculated for each cell based on estimated unit probabilities
with Shannon’s equation:
H(X) = −
n
i=1
pi (X) log2 pi (X)

Back to overview .
0 20 40 60 80
X
0
20
40
Y
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
entropy(bits)
1
2
2 At shallow depth, only
uncertainty due to depth of
unconformity;
H(X) = −
n
i=1
pi (X) log2 pi (X)

Back to overview .
0 20 40 60 80
X
0
20
40
Y
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
entropy(bits)
1
2
3
3
2 At shallow depth, only
uncertainty due to depth of
unconformity;
3 In shoulders uncertainty due to
stratigraphic layer thickness.
H(X) = −
n
i=1
pi (X) log2 pi (X)

Back to overview .
Potential uncertainty reduction
Uncertainty reduction
After analysing uncertainties, the logical next question is how these
uncertainties can be reduced with additional information?

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Potential uncertainty reduction
Uncertainty reduction
After analysing uncertainties, the logical next question is how these
uncertainties can be reduced with additional information?
We use here (multivariate) conditional entropy to evaluate how
uncertainty at a position X2 is reduced when knowing the outcome at
another (or multiple other) position(s) X1:
H(X2|X1) =
n
i=1
pi (xi )H(X2|X1 = i)

Back to overview .
Uncertainty reduction with additional information
Gathering subsequent information at one location (“drilling”):
First approach: drilling into
area of highest uncertainty:
Conditional entropy of each
cell given information at
subsequent locations along a
line (“drillhole”):
uncertainty in the model is
reduced with new
knowledge.

Back to overview .
Gathering subsequent information at one location (“drilling”):
If we gather instead
information on the sholder,
something interesting
happens...

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Comparison of ”drillhole” positions
Comparison of remaining uncertainty for diﬀerent drillhole positions
The diﬀerence is clearly visible when we compare both results:
uncertainty in Graben reduced more when drilling on side!
This analysis can give us an insight ino where additional information
can be expected to reduce uncertainties.

Back to overview .
Kinematic block model
3-D view of the base model
We now brieﬂy evaluate uncertainties in a N-S slice that shows the
folding pattern. As additional parameters, fold wavelength and
amplitude are considered uncertain.
E-WN-S

Back to overview .
Information entropy in a N-S slice
0 20 40 60 80 100 120
X
0
20
40
Y
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
entropy(bits)
1
1 Uncertainties are highest at a
depth where several thin
stratigraphic units are possible;

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Information entropy in a N-S slice
0 20 40 60 80 100 120
X
0
20
40
Y
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
entropy(bits)
1
2
1 Uncertainties are highest at a
depth where several thin
stratigraphic units are possible;
2 Uncertainties generally increase
towards the right (South), as
folding patterns are anchored at
left where more data exists.

Back to overview .
Evaluation of uncertainty reduction with conditional entropy
A comparison of conditional entropies for gathering information at
diﬀerent locations shows again where we can expect to reduce the
uncertainty:

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Conclusion and Outlook
Conclusion
Measures from information theory provide a suitable framework to
visualise uncertainties and evaluate uncertainty reduction in a
geospatial setting.

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Conclusion
geospatial setting.
The analysis provides insights into the underlying model structure
that can lead to, sometimes counter-intuitive, insights.

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Conclusion
geospatial setting.
Outlook
Future work will focus on methods to determine the overall
reduction of uncertainty and more detailed analyses of uncertainty
correlations.

Back to overview .
Conclusion
geospatial setting.
Outlook
Future work will focus on methods to determine the overall
reduction of uncertainty and more detailed analyses of uncertainty
correlations.
In addition, we are working on algorithmic eﬃciency as computation
time becomes critical for large multivariate evaluations.

Back to overview .
More information
Thank you for your attention!

Back to overview .
More information
More information
If you are interested, please have a look at our publications on this topic:
Wellmann and Regenauer-Lieb, 2012 in Tectonophysics;
Wellmann, 2013 in Entropy (open access);

Back to overview .
More information
More information
The software to create the kinematic model realisations, pynoddy, is
available online on github!

Back to overview .
More information
More information
The software to create the kinematic model realisations, pynoddy, is
available online on github!
Also, come and visit us at our poster on Thursday B190 at Session
SSS11.1/ESSI3.6, or see the Abstract

Information Theory and the Analysis of Uncertainties in a Spatial Geological Context

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