on the value of information for spatial problems...
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ON THE VALUE OF INFORMATION FOR SPATIAL PROBLEMS IN THE EARTH SCIENCES
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF EARTH SCIENCES
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Whitney Jane Trainor-Guitton
May 2010
http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/gp948sj5713
© 2010 by Whitney Jane Trainor. All Rights Reserved.
Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Jef Caers, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Rosemary Knight
I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.
Tapan Mukerji
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.
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ABSTRACT
We propose two value of information (VOI) methodologies for spatial Earth problems.
VOI is a tool to determine whether purchasing a new information source would improve
a decision-maker’s chances of taking the optimal action. These actions represent
alternatives to spatial decisions regarding the sustainability of aquifers or enhanced
recovery of oil or minerals. These decisions are difficult due to uncertainty in the Earth
properties and the predictions made from Earth models. The risk of making poor
decisions due to this uncertainty may justify the collection of more information. Both the
prior geologic uncertainty and the information reliability must be quantified before the
data are collected to estimate the VOI, making it challenging to obtain. A flexible prior
geologic uncertainty modeling scheme is presented that allows for the inclusion of many
types of spatial parameters and used for both VOI methodologies.
In the first methodology, we describe how to obtain a physics-based reliability measure
by simulating the geophysical measurement on the generated prior models and
interpreting the simulated data. Repeating this simulation and interpretation for all
datasets, a table can be obtained that describes how many times a correct or false
interpretation was made by comparing them to their respective original model. This table
is the reliability measure, which is required for the VOI calculation. An example VOI
calculation is demonstrated for a spatial decision related to aquifer recharge where two
geophysical techniques are considered for their ability to resolve channel orientations. As
necessitated by spatial problems, this methodology preserves the structure, influence and
dependence of spatial variables through the prior geological modeling and the explicit
geophysical simulations and interpretations.
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The focus of the second methodology is to 1) represent the uncertainty of a dynamic
response of the unknown subsurface, 2) provide a quantitative data reliability and 3) use
both of these to propose a VOI workflow for spatial decisions and data. The dynamic
response of the subsurface is the result of some stress or perturbation (which is either
important for or representing the decision action) and the geologic spatial heterogeneity.
Prior models are used to capture the prior uncertainty of the dynamic response.
Geostatistical simulation and a dynamic simulation function are both used to actualize the
data reliability. The geostatistical simulation requires a likelihood function. Likelihood
functions describe a particular data attribute’s discrimination ability of the key geologic
indicators (deemed to influence the decision outcome). The geostatistical simulation
creates Earth models conditioned to the synthetic geophysical data, thereby representing
possible interpretations that could be made if the information was purchased. Using these
prior and conditioned models’ dynamic responses, a VOI workflow is proposed for
dynamic spatial problems. Specifically, two VOI calculations were determined using two
different likelihoods.
This thesis contributes VOI methodologies that incorporate the spatial element into Earth
science decisions. Previous VOI studies ignored the spatial dimension. Spatial modeling
is included in the prior model uncertainty and in both proposed information reliability
methodologies. In the first method, the spatial aspect is included by simulating the
interpretation of a geophysical image. The second method proposes geostatistical
simulation to represent the variability in the geophysical message and how geophysics
could resolve the static properties which influence a dynamic response. These methods
for obtaining spatial data reliability and consequently a VOI estimate will become
increasingly important with changing uncertainties and risks regarding decisions related
to the management of our natural resources.
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ACKNOWLEDGMENTS
I was inspired to take this path of study due to my upbringing in Idaho. I’m from Twin
Falls, Idaho, which lies on the south rim of the Snake River Canyon. Pictured above is
the Snake River aquifer discharging from the canyon walls at Thousand Springs. That
such a large volume of water could be flowing through the seemingly solid subsurface
always fascinated me growing up. The groundwater of Idaho is vital to both its people
and agricultural industry, yet relatively few resources are allocated to studying and
understanding how our actions affect the aquifer.
My PhD was possible because of the financial support from the Affiliates of Stanford
Center for Reservoir Forecasting and Schlumberger Water Services. I am very grateful
for their generosity in funding my six years at Stanford. My advisor Jef Caers has been
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extremely available over the last six years at Stanford. I admire his creative problem-
solving yet pragmatic approach to research. When I first arrived to Stanford, three years
out of undergrad and with no geostatistics background, Jef wasted no time in getting me
up to speed and producing results, thanks to both his teaching skills and regard for
learning by doing. The interdisciplinary nature of my work has involved my committee
members more than the traditional PhD thesis. Tapan Mukerji is a great professional
mentor who not only introduced me to the value of information concept but provided
great technical and professional advice throughout my research. Tapan is a true
interdisciplinary scientist whose breadth of knowledge in geophysics, rock physics,
statistics, decision analysis, etc, is a huge benefit for all the students that work with him.
Rosemary Knight is who initially inspired me to apply to Stanford. She has a six-sense
for why your code isn’t running or why your results are strange. Rosemary’s passion for
groundwater geophysics is contagious, and I’ve benefited greatly from participating in
her environmental geophysics seminars for the last six years.
Two mentors from my internships at Schlumberger and Chevron deserve special
recognition for their influence and support. Angeline Kneppers at Schlumberger Water
Services provided invaluable discussions on current groundwater challenges and issues
worldwide. Her knowledge and insights on current groundwater scientific and business
practices helped form the initial ideas about what I wanted my thesis to be. At Chevron, I
worked under Mike Hoversten for two internships. It’s no exaggeration to say that
working with Mike the summer of 2007 reinforced my resolve to stay in the PhD
program. His confidence in me, the freedom, and challenges of the project imparted the
kind of creative work I could expect if I finished the PhD. It was a delight to work with
such a smart, knowledgeable, modest, and great collaborator such as Mike.
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Esben Auken and Nikolaj Foged of the University of Århus, Denmark provided the
forward and inverse TEM code and helpful insights about the TEM measurement, which
were invaluable for the work in Chapter 2. Debarun Bhattacharjya (at the time a MS&E
Stanford PhD student) provided helpful conversations regarding the framing of our VOI
example problem of Chapter 2. Thomas Nyholm and Stine Rasmussen of the Danish
Ministry of the Environment provided useful information about aquifer vulnerability
issues used for the work in Chapter 3.
My fellow SCRF students have been some of my closest friends at Stanford. Our
research group is a fantastically diverse group both in research backgrounds and
nationalities. It’s been such a great experience to work with such brilliant and fun
students. I want to especially thank Yongshe Liu, Mehrdad Honarkhah, Celine Scheidt,
Amit Suman, Antoine Bertoncello and Kwangwon Park for their friendship and technical
advice. Alumni members Alex Boucher and Amisha Maharaja started out as my excellent
geostatistics TA’s and turned into great friends with whom I’ve enjoyed adventures of
snowshoeing, skiing and biking. Extra appreciation goes to alumna Scarlet Castro who
was my officemate my first year as a PhD student. She is one of my most highly
regarded mentors; I have so much respect for Scarlet as a scientist, a person and a great
friend.
One of the greatest benefits from this PhD has been the friendships I’ve made at Stanford,
especially the group of wonderful people I started at Stanford with: Hilary Schaffer
Boudet, Taku Ide, Egill Juliusson, Rob Stacey, and Lisa Stright. They have served as
bike, ski, running or travel adventure companions, a shoulder to cry on, and a joker when
a laugh (and some perspective) was badly needed. I would choose to do this all over
again just to make sure our paths crossed.
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Any success I have achieved is the direct result of the 31 years of love and support from
my family. My parents have always been actively involved in my life, either camping,
rafting, or skiing, or tirelessly driving around the state of Idaho (and even to Colorado,
Chicago, Spain, and Panama) to watch a basketball game, track meet, marathon, or
experience my latest life adventure. I’m so fortunate to have such an exceptional sister.
Although Meghan and I haven’t lived within 500 miles of each other for 13 years now,
she’s been one my best friends throughout all the stages of my life: always a phone call
away and making visits whenever either of us can. Most importantly, they’ve all
encouraged and supported my academic goals and always believed I could accomplish
whatever I set in my sights. I’d also like to acknowledge my grandparents Flora Jane and
Harvey Wirth and Martha and Francis Trainor. They all rose from very modest means
and made family top priority in life. Thanks for setting such great examples to live by.
The greatest and most unexpected benefit from Stanford is meeting my husband Antoine.
Before, I never understood how people could say at their PhD defense, that they “never
could have done it without ‘so and so’.” Now I understand: I definitely couldn’t have
finished without Antoine. He provided encouragement, love, countless healthy dinners,
feedback on rough drafts, and, most importantly, great perspective and advice since he
had been through the Stanford PhD process. Besides all that, he’s my best friend. He’s
the most loyal, thoughtful and fun life partner that I ever could have asked for. Thank
you for everything!
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TABLE OF CONTENTS
List of tables...................................................................................................................... xii
List of figures................................................................................................................... xiii
Chapter 1: Challenges in applying VOI methodologies to the Earth sciences ..............1
1.1 Introduction..............................................................................................................1
1.2 Decision analysis and the Value of Information (VOI) ...........................................2
1.2.1 Challenges of applying decision analysis and VOI to spatial
decision-making regarding the subsurface.........................................................3
1.3 Previous VOI work ..................................................................................................8
1.3.1 Petroleum engineering literature............................................................8
1.3.2 Hydrogeology literature .........................................................................9
1.3.3 Geophysics literature ...........................................................................10
1.4 A preview of the methodologies proposed in this thesis .......................................12
1.4.1 Chapter 2: Methodology for reliability assessment of spatial data......13
1.4.2 Chapter 3: Addressing dynamic responses in spatial decisions...........14
Chapter 2: Reliability through Interpretation ..............................................................15
2.1 Prior geological models .........................................................................................16
2.2 Geophysical data and reliability.............................................................................18
2.3 Value of information calculation ...........................................................................22
2.3.1 Generalization to several geologic input parameters ...........................30
2.4 VOI example..........................................................................................................32
Chapter 3: VOI for Dynamic Problems .......................................................................42
3.1 Demonstration case setting ....................................................................................42
3.1.1 Chapter 3 vs Chapter 2.........................................................................48
3.2 The proxy decision variable: motivation ...............................................................48
3.3 Estimating the proxy decision variable..................................................................50
3.4 Data reliability for measuring proxy decision variables ........................................52
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3.4.1 Reliability through geostatistical simulation .......................................52
3.4.2 Creating synthetic data sets with rock physics ....................................55
3.4.3 Deriving information content to create conditioned Earth models......60
3.4.4 Creating data-constrained proxy decision variables ............................62
3.5 VOI Methodology summary ..................................................................................63
3.5.1 Generalization to combinations of alternatives....................................70
3.6 Application of VOI calculation to the demonstration case....................................72
Chapter 4: Discussion and Future Work......................................................................76
4.1 Summary of the two methods ................................................................................76
4.2 Discussion of Chapter 2 method............................................................................77
4.3 Discussion of Chapter 3 method............................................................................80
4.4 Future work............................................................................................................82
APPENDIX A: Details of TEM measurement & modeling ..............................................87
Bibliography ......................................................................................................................90
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LIST OF TABLES
Number Page
Table 1: Airborne TEM Reliability....................................................................................40
Table 2: Land-based TEM Reliability ...............................................................................40
Table 3: VOIII Results for Aquifer Vulnerability Demonstration Case.............................73
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LIST OF FIGURES
Number Page
Figure 1: Illustration of a spatial decision. The decision is which possible
contamination sources should be removed. The uncertainty is whether
they are located on a significant surficial entry-point location. This is
determined by the regional heterogeneity of the buried valleys. .......................5
Figure 2: Schematic showing the two randomizations used to create the Earth models.
First, the geologic parameter(s) is (are) identified, then outcomes are
drawn and lastly these outcomes are used as input into stochastic
algorithms to create each model z(t)(θ).............................................................18
Figure 3: Example Decision Tree for a binary case demonstrating prior value (Vprior).
The tree represents chronology from left to right, but calculations are
made from right to left. (1) First calculations for each combination of
action and geologic parameter output (possibility node) must be made. (2)
The action values are calcualted as expected values of all the possibilities
with that action. (3) Finally, Vprior is the highest action value (the best
action considering our current uncertainty). ....................................................26
Figure 4: Decision Tree depicting the calculations for the Value with Imperfect
Information (VII). Chronologically, we can interpret the subsurface
possibility first (red node) and then chose the best action (blue node).
Again, calculations start from right and proceed left. (1) The possibility-
action combinations are already calculated from Vprior, now the same
possibilities iθ are grouped together. (2) The interpreted possibility
branches are calculated for each action. These weight the values of (1)
with the probability that the interpretation is correct or incorrect. (3)
Finally, the value with imperfect information (VII) is the expected value of
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the best action for each interpreted branch. This is weighted by the
probability of each interpretation being made ...........................29 )Pr( intjθ=Θ
Figure 5: Fluvial Training Image; Red = sand facies; Blue=non-sand facies ..................34
Figure 6: (A) Example rotation maps and (B) 2D Earth models from snesim for each
of the three channel scenarios (80 x 80 x 1 cells, 5.3km x 5.3km x 70m,
blue=non-sand facies and red=sand facies) .....................................................35
Figure 7: Decision tree schematic for the prior value (Vprior) of the aquifer recharge
example. The red nodes denote the 3 possible channel orientation
scenarios. The blue node denotes the 4 recharge options. ..............................36
Figure 8: Results of image analysis for the angle of maximum correlation. A)
Original and inversion results shown for a model (z(t)(θi)) from scenario
Northeast. Far right, the locations are shown in red where Northeast is
misclassified as Southeast and in blue where correctly classified. B)
Original and inversion results shown for a model (z(t)(θi)) from scenario
Mixed. Far right, the locations are shown in red where Northeast is
misclassified as Southeast and in blue where correctly classified. ..................39
Figure 9: Network of buried valleys; darker to lighter representing older to younger
buried valley generations (Jørgensen & Sanderson, 2006)..............................44
Figure 10: Schematic showing the two randomizations used to create the Earth
models of buried valleys. First, the geologic input parameters are
identified, then outcomes are drawn and lastly these outcomes are used as
input in to stochastic algorithms to create each model. ...................................45
Figure 11: The 4 steps of obtaining a data reliability for proxy decision variables
through geostatistical simulation. Step 1: Generate a synthetic dataset
from the likelihood and a prior model. Step 2: Generate a soft probability
cube for the valley lithofacies from a dataset and the information content.
Step 3: Generate a conditioned Earth model with the soft probability cube.
Step 4: Obtain the conditioned proxy variable by applying the dynamic
simulation function to the conditioned Earth model........................................55
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Figure 12: Synthetic data reliability describing a good contrast between the two
lithologies’ ranges of electrical resistivity .......................................................58
Figure 13: Synthetic data reliability describing a poor contrast between the two
lithologies’ ranges of electrical resistivity .......................................................59
Figure 14: Overall workflow of this VOI methodology. A) First the generation of the
prior models and calculation of Vprior . B) Second, the reliability measure
and how it is utilized with the prior models. C) The conditioned models
are generated and used to calculate VII. ...........................................................64
Figure 15: Tracer concentration thresholds are applied to flow simulation results such
that each continuous tracer body is identified. Topography is depicted by
the semi-transparent surface. ...........................................................................66
Figure 16: Conceptualization of how vulnerability maps (the proxy decision variable)
are made from the tracer concentration bodies. ...............................................67
Figure 17: Example vulnerability map which indicates locations that serve as entry
points into the aquifer. The magnitude reflects the volume of aquifer that
would be affected if a contaminant would enter at that particular location.....67
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CHAPTER 1: CHALLENGES IN APPLYING VOI METHODOLOGIES TO THE
EARTH SCIENCES
1.1 INTRODUCTION
Much uncertainty surrounds the future regarding the global climate’s impact on the water
cycle and the finite supply of natural resources. Both of these issues will affect actions
taken by the public and private sectors, as governmental organizations and stakeholders
try to meet water, energy and material demands. These actions are sometimes in the form
of spatial decisions made to protect groundwater resources or improve recovery of
resources such as oil and minerals. However, due to spatial uncertainty about the
subsurface properties or assets, the outcome of these protection and extraction endeavors
is unclear. Information from various measuring and monitoring techniques
(direct/indirect, qualitative/quantitative, etc) exists to decipher subsurface characteristics.
Assuming the information is relevant to the protection or extraction decision, two
important questions arise: 1) how accurate is the information at identifying the desired
subsurface properties and 2) is the information worth its cost/price? The cost is important
because currently there is little to no monetary funds for spatial data collection for
protecting groundwater resources, and there is a high data collection cost for oil
exploration and production.
To clearly illustrate the type of challenges and problems addressed in this thesis, a
synthetic demonstration case, which is modeled after a European groundwater problem
and decision, is described. In this example, we assume that a particular area relies solely
on its groundwater sources to supply drinking water. Over the past several decades, the
area aquifers have been compromised by surface-sourced contaminants due to urban
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growth and farming activities. Contamination will continue to be a threat until critical
surface locations that serve as entry points into the aquifer are identified. This can only
be successfully achieved if the hydraulically complex connections between the
contaminant sources at the surface and the underlying aquifers are understood. The
decision is to determine which contamination sources (e.g. farms or factories) need to be
removed to ensure a sustainable supply of drinking water. Decision making in this
context is difficult because of the uncertainty surrounding the properties of the unknown
Earth, which results in a difficult prediction of contaminant transport. We will first
describe the field of study known as decision analysis which is designed to aid in
decision processes beset with uncertainty and is critical to our methodology. This
description will be followed by the challenges of applying the decision analysis and value
of information (VOI) methods to subsurface problems, such as our demonstration case.
1.2 DECISION ANALYSIS AND THE VALUE OF INFORMATION (VOI)
The Value of Information (VOI) methodology is designed to determine if purchasing the
proposed information will assist in making better decisions. The VOI metric originates
from the discipline of decision analysis. The term decision analysis was coined by
Howard (1966). More recently Matheson (1990) and Clemen and Reilly (2001) give
overviews of decision analysis and VOI concepts. Decision analysis and VOI has been
widely applied to decisions involving engineering designs and tests, such as assessing the
risk of failure for buildings in earthquakes, components of the space shuttle and offshore
oil platforms [Cornell and Newmark, 1978, Paté-Cornell and Fischbeck, 1993, Paté-
Cornell, 1990]. As demonstrated in those example studies, the performance statistics of
these engineering apparatuses or components under certain conditions are readily
available; therefore, applying the decision analysis framework for decisions involving
engineering designs can be fairly straightforward and convenient. Additionally, the
statistics on the accuracy of the tests or information sources that attempt to estimate the
competency or condition of these designs or components are also available, as they are
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typically made repeatedly in controlled environments such as a laboratory or testing
facility. These statistics are required to complete a VOI calculation as they provide a
probabilistic relationship between the information message (the data) and the state
variables of the decision (the specifications of the engineering design or component).
This is often referred as the information’s reliability measurement. For engineering
decisions, the reliability is typically straightforward as the information or test we are
considering is more or less directly measuring the component on which we are trying to
make decisions. In other words, from repeated trials we can get the success rate of the
information in the form
( )component theof conditiontruethe|conditionscomponent' the is saystestthe whatPr
However as more complexity exists between the decision variables and what property the
information can actually test for, the reliability is more broadly described as
( )isworldrealthewhat|saysdatathewhatPr
The reliability must account for all the complexities that associate the decision variables
(which determine the outcome of the any taken decision) and the actual parameter
measured by the information.
1.2.1 CHALLENGES OF APPLYING DECISION ANALYSIS AND VOI TO SPATIAL DECISION-
MAKING REGARDING THE SUBSURFACE
Many challenges exist in applying this type of statistical analysis to an unknown Earth.
Instead of predicting the performance of an engineer’s design under different conditions,
the desired prediction is the response of a subsurface--which can have complex geologic
heterogeneity and be poorly understood-- to some external action representing the
decision action. Additionally obtaining a meaningful measure of how spatial data will
correctly estimate geologic variables of interest is not a trivial exercise.
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Some definitions
Before describing these challenges, it is necessary to first define some terms that are
important in communicating the contributions of this thesis. A decision has been defined
as “an irrevocable distribution of resources” [Howard, 1966]. We define spatial decisions
as decisions of which the outcomes are a result of the spatial distribution of some
property. In some cases, the decision-maker is faced with choosing between different
spatial alternatives. These spatial decisions are difficult because we are very uncertain
about the outcomes for different decision actions or alternatives. The outcomes are
uncertain because we are uncertain about the spatial distribution of the properties of the
Earth which will influence or determine the decision outcome. Therefore, the uncertainty
in space of the decision variables can be termed the spatial uncertainty. Figure 1
illustrates the spatial decision and uncertainty of the demonstration case: which of the
possible contamination sources needs to be removed? The heterogeneity of the buried
valley depositional system creates spatial uncertainty regarding the properties of the
aquifer system, thus making us uncertain about which locations may act as entry points
into the aquifer. The reliability of the considered information source would therefore
need to quantify how much the data reveal about the entry points of contamination into
the subsurface. This is not a trivial quantification. Data such as geophysical data would
only indirectly measure the subsurface properties, from which entry points (the hydraulic
structure) could be inferred.
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Figure 1: Illustration of a spatial decision. The decision is which possible contamination sources should be removed. The uncertainty is whether they are located on a significant
surficial entry-point location. This is determined by the regional heterogeneity of the buried valleys.
Uncertainty of subsurface heterogeneity
Representing the heterogeneity of the subsurface properties as accurately as possible is
important as it affects the decision actions that we take on the Earth, such as those related
to aquifer remediation [Wagner and Gorelick, 1987; Wagner and Gorelick, 1989; Freeze
and Gorelick, 1999] or the production of oil [Bickel and Bratvold, 2008]. Recent studies
have demonstrated how geologic facies heterogeneity can dominate the hydrogeologic
response of interest [Feyen and Caers, 2006; Ronayne et al, 2008] or the production of oil
[Hoffman and Caers, 2007].
The field of geostatistics has been developed to help represent our uncertainty of the
spatial distribution of the Earth’s properties. Geostatistics began by describing
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correlations in space through the variograms and solving the Kriging system [Isaaks and
Srivastava, 1989; Goovaerts, 1997]. The discipline has evolved to include more complex
spatial features which attempt to mimic the geologic process that create the
heterogeneity. Some of the existing geostatistical modeling techniques include the
traditional two-point statistics of the variogram, training images for multiple-point
statistics, object-based and lastly, process-based modeling (which is often a hybridization
of various physical and statistical techniques, e.g. Michaels et al, 2010). These are listed
from least to most geologically realistic and easiest to hardest to condition to data.
Respective examples or reviews for each of these can be seen in Deutsch and Journel,
(1997), Hu and Chugunova, (2008), Matheron et al (1987), and Koltermann and Gorelick
(1996). Hybrids of these groups have also being proposed [Deutsch and Wang, 1996;
Pyrcz and Strebelle, 2006; Michaels et al, 2010].
Yet geostatistics alone does not exhaustively capture all of our spatial uncertainty. All
geostatistical algorithms require some prior (and hence subjective) information on the
type of spatial continuity and its parameters (ranges, dimensions, choice of Boolean
model or training image). With this prior model information, many Earth models (or
realizations) can be created by changing a spatial random seed and thus represent the
possible spatial variability of the given prior model. However, these Earth models alone
can not represent our uncertainty of the initial model inputs given to the algorithm. The
uncertainty of this prior information (which can be termed as either the model uncertainty
or the geologic input parameter uncertainty) must also be represented to realistically
capture uncertainty [Caers, 2005; Scheidt and Caers, 2009]. Again, determining
reasonable geologic input parameters is quite challenging since the depositional history
of any subsurface location could be poorly understood or very complex. For the
demonstration case, the geologic depositional system is interpreted as glacial buried
valleys, which act as important subsurface groundwater resources. We may be able to
create one possible training image that represents a buried valley depositional system, but
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using only one training image would not realistically represent prior uncertainty of the
dimensions of the valleys themselves.
Determining a reliability measure of data
Obtaining an information reliability for the engineering applications listed previously can
be relatively straightforward since measurements can be repeated in a lab for many well-
defined designs or components. It is this reliability measure that allows for the
calculation of VOI, since it takes into account the success rate of the information.
Determining such a reliability measure for Earth science applications and data gathering
techniques is a considerable challenge, especially when considering spatial information
from a geophysical technique. The data’s ability to resolve the decision variables is
restricted by many different factors including: an indirect measurement of the decision
variable, non-unique solutions, complex physics, a complex Earth, and noise. All these
factors make it difficult to quantify the data reliability. We will now describe these
factors in detail, using the demonstration case as an example.
There are two facets of the indirect nature of the measurement. Most geophysical
techniques indirectly measure the property of interest. This requires transforms between
the geophysical observables and geologic indicators; the transforms themselves involve
uncertainty [Mavko and Mukerji, 1998]. For the demonstration case, a possible
geophysical observable is electrical resistivity, which can (imperfectly) distinguish
between different lithologies (the geologic indicators). Secondly, the decision outcome is
often determined by some cumulative response to a regional heterogeneity. This
describes the demonstration case situation: the decision outcome relies on preferential
flow paths from the surface to the aquifer which are influenced by the existence of valley
or non-valley facies. We have no instrument that directly detects flow paths; instead we
must attempt to discern this variable from properties that can be measured. Commonly
geophysical inversion is required to recover the geophysical attributes from the data.
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Geophysical inversion problems are notoriously ill-posed (usually requiring the recovery
of more Earth parameters than there are data measurements), and thus plagued with non-
unique solutions [Tarantola, 1986], which will also affect the success or accuracy of the
information message. Additionally, this inversion requires a forward model that
endeavors to simulate the complex physics of the measurement. Often the forward model
must make approximations in order for the computation to be feasible. Lastly, noise
(either from cultural or environmental sources) is an issue when collecting information
outside of a laboratory or controlled setting [Auken et al, 2008]. The reliability must
consider all of these challenges and estimate realistically how well the measurement can
resolve variables important to a decision for a particular situation and location. All of
these factors that may cause inaccuracies in the data must be estimated before data
collection, as the purpose of VOI is to determine if it should be collected or not. For
Earth science problems, this is especially difficult because each situation will have its
own unique challenges.
1.3 PREVIOUS VOI WORK
The Earth Science community is well equipped to develop tools to address these
challenges and achieve a VOI estimate. The VOI work in the literature demonstrates a
variety of ways to represent both the prior uncertainty and the reliability of the
information source. We now review how these were previously addressed and their
shortcomings.
1.3.1 PETROLEUM ENGINEERING LITERATURE
VOI has been applied to subsurface fluid flow decisions in the petroleum engineering
literature. Coopersmith et al. (2006) propose a qualitative approach to obtain the
information reliability measurement. Using a modified Sherman-Kent language
probability scale, they transfer language such as “certain,” “likely,” “unlikely,” and
“doubtful” from expert interviews into discrete intervals of likelihoods. Bratvold et al.
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(2009) give a thorough review of the 30 VOI papers in the petroleum engineering
literature from the past 44 years. Among other assessments and critiques, they note that
only 13 of the 30 papers published address the issue of reliability, and 11 of these 13 used
the subjective expert interview method. One of the two that use a quantitative approach
to reliability is Bickel et al. (2006), which utilizes forward modeling of seismic amplitude
to address the issue of seismic accuracy for reservoir development. The likelihood and
posterior are approximated by assuming that the seismic signal (data) relates to the
geologic observables (porosity and reservoir thickness) through a joint normal
distribution, thus only a covariance between the reservoir properties and seismic
attributes are needed. The examples include an analysis of how different error levels
affect the final VOI, but no spatial structure is included. Bickel et al (2008) is not a value
of information problem but uses decision analysis to tackle the problem of how to
determine the best drilling strategy: the option of drilling five wells for five determined
locations. They focus on the dependence of “geologic success” between the possible
drilling locations. The “geologic success” is described through marginal and conditional
probabilities of three geologic factors (charge, reservoir rock and seal) for and between
the 5 well locations. Even though the dependence of five wells is determined by geology,
this study does not consider geostatistics or spatial correlation methods as a way to model
these dependencies.
1.3.2 HYDROGEOLOGY LITERATURE
VOI examples also exist in the hydrogeologic literature. Reichard and Evans (1989)
present a framework for analyzing the role of groundwater monitoring in reducing
exposure to contamination. The reliability here is the efficiency of detecting arsenic in
the water (a scalar parameter), and no spatial dependence is included in the 1D
contaminant transport modeling. Wagner et al (1992) compare four different strategies
for representing uncertainty of hydraulic conductivities and solving for the optimal
solution for a groundwater contamination decision. Therefore, the “information” is
10
represented by the different deterministic and stochastic formulations and how their
respective uncertainty measures affect the decisions made and resulting outcomes. The
decision faced by groundwater managers in the example posed by Feyen and Gorelick
(2005) is how to maximize the extraction of groundwater (which is sold for profit)
without violating ecological constraints (specific groundwater table levels). They
consider how hydraulic conductivity information at certain locations may improve
hydrogeologic model predictions and allow for an increase in water production while still
observing the hydro-ecological balance. The aquifer is represented by many Earth
models with a fixed mean, variance and covariance. The models with the hydraulic
conductivity measurements are conditioned to these point location values. However, no
particular measurement technique is specified, and thus, no analysis is made on the
accuracy of a measurement, which would affect the value of information. The
hydrogeologic community adopted decision analysis and VOI early on, but none of the
examples offer a method for quantifying the reliability of a measurement’s accuracy.
1.3.3 GEOPHYSICS LITERATURE
The existing applications of the VOI method to geophysical data are quite recent.
Preceding these examples, however, are three particular studies that led towards a “data
worth” measure but aren’t strictly speaking VOI since they don’t use the decision
analysis framework. Lumley et al (1997) attempt at valuing 4D seismic data by using the
“scorecard” technique. The “scorecard” rates the 4D image quality, repeatability and
resolution for different reservoir conditions (depth, fluids, rock types, etc). Scorecards
are actually deterministic versions of influence diagrams, which are graphical tools that
demonstrate relationships between different random variables through conditional
probabilities. Bhattacharjya and Mukerji (2006) use the same 4D example from Lumley
et al but they make maximum expected value calculations using conditional probabilities
to describe the interaction between variables. The advantage of influence diagrams is
when situations entail “conflicting interrelated observables and the decision is not
11
straightforward.” George and Woodgate (2002) propose “A Framework for Considering
the Utility of Airborne Geophysics” in managing land salinity in Australia. They
recognize (using different terms) that geophysics will be “effective” when the prior
uncertainty is high and when information from geophysics can assist in choosing between
different management options. They even use the term “reliability” of airborne
geophysics to identify depth to salt layer. The vocabulary used is mostly in terms of cost-
benefit, where they identify that improved decisions will lead to benefits to private
(farmers) and public entities. However, the evaluation lacks an inclusion of how
uncertainty can be represented, but it ultimately identifies how useful a utility or value
calculation of information would be for gaining public confidence in decision-making
using data collection.
The first proper VOI example in the geophysics literature is Houck (2004). Houck
(2004) uses VOI to evaluate whether better 3D seismic coverage is worth the extra survey
costs. The seismic amplitude data are used to build 1D reservoir models that describe the
reservoir’s thickness, porosity and fluid type. The reliability of the amplitude data to
detect these reservoir properties is modeled through a simulated calibrated data set
(specifically linear regression is used to link the simulated data with the reservoir
parameters). Houck and Pavlov (2006) use detection theory for reconnaissance
controlled-source electromagnetics (CSEM) surveys for oil reservoir exploration. CSEM
surveys with different configurations (different numbers of sources and receivers) are
evaluated for their ability to detect economic and non-economic reservoirs (where these
terms refer to their absolute size). VOI is used to determine the survey design with the
optimal value considering the cost of the survey and CSEM's ability to resolve reservoir
size. However, full electromagnetic modeling is not performed; instead, sensitivity maps
are generated and used to determine the CSEM response to economic and uneconomic
reservoirs. For the sensitivity analysis, targets are limited to geometric shapes that are
not representative of geologic features. Houck (2007) examines the worth of 4D data for
12
two different reservoir conditions: one with and one without gas. The 4D “repeatability”
is modeled through conditional probabilities that are generated by two different sets of
triangle distributions of seismic attributes. Again, only 1D reservoir models are used.
Eidsvik et al. (2008) introduce statistical rock physics and spatial dependence within a
VOI methodology for the decision of whether or not to drill for oil. Spatial dependence
is included in the grids representing the porosity and saturation of the reservoir through a
covariance model. At each of the grid locations, CSEM and seismic amplitude-versus-
offset (AVO) data are drawn from likelihood models that represent the link between the
reservoir properties and the geophysical attributes. Many data sets are generated (drawn),
the posterior is approximated with a Gaussian, and then a posterior value is drawn from
it. By drawing the geophysical information from a spatially correlated porosity and
saturation field, this method attempts to preserve spatial dependence. However, while the
decision is explicitly modeled, the spatial structure’s influence on the flow of oil is not
taken into account. Finally, Bhattacharya et al (2010) propose a decision-analytic
approach to valuing experiments performed in situations that naturally exhibit spatial
dependence. They incorporate dependence by modeling the system as a Markov random
field, and also include the effects of constraints on the decisions. Their methodology is
illustrated with two examples related to conservation biology and reservoir exploration
geophysics. Bhattacharya et al (2010) additionally describe the measure of mutual
entropy which describes how new information may decrease the disorder of an observed
property of interest.
1.4 A PREVIEW OF THE METHODOLOGIES PROPOSED IN THIS THESIS
Even though studies have shown the importance of its impact, geologically realistic
heterogeneity has not been realistically and comprehensively included in the value of
information analysis for spatial decisions and information. Spatial data, such as from
geophysical measurements, may provide the kind of information needed to resolve the
heterogeneity that affects the outcome of spatial decisions. However, none of the
13
previous VOI work includes the spatial element in both the prior models and the
assessment of the information reliability. This shortcoming motivates this thesis. The
two methodologies developed for spatial decisions are described in Chapter 2 and
Chapter 3. It should be emphasized that the goal of this thesis is the development of
methodologies. Specific approaches regarding any of the geophysical, geostatistical or
fluid flow modeling can be replaced by others that are more relevant for particular
situations. The objective is to propose a framework of how to approach a spatial VOI
problem and for it to be transferable to many types of applications (climate, groundwater
or oil), data sources and physical modeling techniques.
1.4.1 CHAPTER 2: METHODOLOGY FOR RELIABILITY ASSESSMENT OF SPATIAL DATA
Much of the previous work fails to model the geologic prior uncertainty both realistically
and comprehensively. Much of the previous work has relied on either multi-Gaussian or
discrete conditional probabilities to describe the geophysical data’s reliability. Only a
few of the previous examples explicitly model the geophysical forward modeling but
none of them use interpretation to obtain the reliability measure. Therefore, the major
contribution of the work presented in Chapter 2 is a methodology that includes: 1)
geologic realism and flexibility to describe the prior model uncertainty, 2) a method to
obtain the reliability of information using both the entire prior model uncertainty and
rigorous geophysical forward modeling with interpretation to capture the spatial content,
and 3) the decision action is explicitly modeled to simulate the outcome of the spatial
decision options. The methodology of Chapter 2 therefore captures the spatial element
through the prior spatial uncertainty and the geophysical interpretation used for the
reliability measure. The VOIII (value with imperfect information) calculation uses both
results from the prior and reliability. A synthetic example built after a real-world
problem aims at demonstrating the various contributions of the proposed methodology,
which includes the different flow responses of the modeled spatial structure that represent
the different decision actions.
14
1.4.2 CHAPTER 3: ADDRESSING DYNAMIC RESPONSES IN SPATIAL DECISIONS
The content and contributions of Chapter 3 are also three-fold. The third chapter uses the
same framework of Chapter 2 to represent the prior uncertainty of spatial heterogeneity.
However, the uncertainty of the dynamic hydrogeologic response (deemed important or
directly related to the groundwater decision) is the challenge addressed in Chapter 3.
Uncertainty regarding this response can be captured using both these models and the
appropriate dynamic simulation function. Secondly, we propose a quantitative method of
achieving a data reliability through geostatistical and dynamic simulation. Using a
likelihood relationship between the observed geophysical attributes and the geologic
indicators, Earth models conditioned to geophysical information can be generated.
Performing the dynamic function on these conditioned Earth models represents how by
constraining the static properties, one can constrain the important dynamic response. In
short, our methodology captures geophysical information’s potential to resolve the spatial
heterogeneity and the subsequent hydrogeologic response to these spatial properties. The
last contribution is the VOI calculation for spatial data which uses the first two
contributions of the prior and posterior uncertainty analyses.
15
CHAPTER 2: RELIABILITY THROUGH INTERPRETATION
A catch-22 situation presents itself when determining the VOI of any measurement. VOI
is calculated before the proposed information is collected. However, the VOI calculation
cannot be completed without a measure that describes how well the proposed method
resolves the target parameters. In the VOI literature, this measure is known as the “data
reliability measure” [Bratvold et al, 2009] and is used to determine the discrimination
power of geophysical (or any other) measurements to resolve the target subsurface
characteristics (which have been deemed to influence the outcome of a decision).
Determining this measure becomes especially challenging when the subsurface
geological variability is highly uncertain, hence the motivation to resolve key geological
parameters by collecting data with the proposed geophysical technique.
According to decision analysis theory, there is no intrinsic value in collecting data unless
it can influence a specific decision goal [Bickel and Bratvold, 2008]. The aim of
collecting more data is to reduce uncertainty on those geologic parameters that are
influential to the decision making process. Therefore, there is no value (in a monetary
sense) in reducing uncertainty on the subsurface characteristic just for the sake of
reducing uncertainty; a particular decision goal needs to be formulated. Therefore, VOI
is dependent on the particular decision problem faced before taking any additional data.
As a result, three key components are involved in determining the VOI: 1) the prior
uncertainty on geological parameters or characteristics of the subsurface, 2) the data
reliability and 3) expressing the decision outcome in terms of value to achieve the VOI
calculation. These three components are described in the next sections.
16
2.1 PRIOR GEOLOGICAL MODELS
One of the most general ways of representing prior geological uncertainty is to generate a
set of models that include all sources of uncertainty prior to collecting the data in
question for the value of information assessment. Such a large set of model realizations
or “Earth models” may be created using a variety of techniques, including the variation of
subsurface structures, i.e. faults and/or layering, uncertainty in the depositional system
governing important properties (permeability, porosity, ore grade, saturations etc)
contained within these structures as well as the spatial variation of such properties
simulated with geostatistical techniques. This prior uncertainty may be elicited from
analog information and/or interpretation from other data gathered (drill-holes, wells,
cores, logs) in the field of study. Caers (2005) provides an overview of the various
sources of uncertainty as well as techniques to create “Earth models” that can readily be
used in mining, environmental, petroleum and hydrogeological applications.
In general, the creation of a large set of alternative Earth models can be seen through the
system view represented in Figure 2. First, the prior uncertainty on input geological
parameters needs to determined. These input parameters could be as simple as the range
or azimuth of a variogram model that is uncertain. Or the input geological parameters
could be a set of training images, each with a prior probability of occurrence. Next, a
particular outcome of the input parameter(s) is (are) drawn and a stochastic simulation
algorithm generates one or several Earth models. For example, in case of permeability
modeled with a variogram, this would involve the simulation of multiple permeability
models with a fixed range and azimuth angle using a multi-Gaussian based simulation
algorithm. Note that we distinguish two sources of randomization: randomization due to
uncertainty of the input geological parameters and randomization due to spatial
uncertainty for given fixed input. To explain clearly our method for value of information
calculation, we will assume that we have only one such input geological parameter,
17
denoted by the random variable Θ . For convenience, we will also assume that this
variable is discrete and has two possible outcomes θ1 and θ2, such that { 21 , }θθ∈Θ (we
will generalize to include more input geologic parameters and outcomes in a later
section). Some prior probability is stated on 1θ and 2θ . 1θ and could be two
variogram models, two training images, or two structural model concepts. Once such
outcome θ is known, then several alternative Earth models can be simulated. We will
denote such Earth models as
2θ
Equation 1
Tt(t) ,...,1)( =θz
where T is the total number of such models. Note that z is multi-variate since many
correlated (or uncorrelated) spatial variables can be generated (porosity, permeability,
soil-type, concentrations, fault position etc…) in a single Earth model.
18
Figure 2: Schematic showing the two randomizations used to create the Earth models. First, the geologic parameter(s) is (are) identified, then outcomes are drawn and lastly these outcomes are used as input into stochastic algorithms to create each model z(t)(θ).
2.2 GEOPHYSICAL DATA AND RELIABILITY
Geophysical techniques are used for several purposes, but in general they aim at
resolving key geological parameters. In this chapter, we assume geophysics is used to
provide more information in terms of the outcomes of the input geological parameters θ ,
thereby having the potential to improve decision making. We will not use the
geophysical data to constrain the spatial variation of properties for a given input
parameter as is done in many geostatistical algorithms (e.g. using co-kriging). Rather,
this chapter focuses on the uncertainty of the geological input parameters. For certain
19
problems, the uncertainty on the input parameters can be much larger and more
influential than the uncertainty in spatial variation for a given input parameter. This is
particularly pertinent in modeling oil fields and aquifers, or any other problem involving
modeling flow with very few data available. In such cases, the choice of a training image
is known to be much more influential than the uncertainty due to spatially varying
lithologies for a given fixed input training image [Suzuki and Caers, 2008; Scheidt and
Caers, 2009]. However, this does not mean that the randomization leading to spatial
uncertainty with fixed input parameters will be ignored in our methodology.
Most geophysical techniques indirectly measure geological parameters of interest. For
example, seismic data are used to make fluid, porosity, lithology, or horizon
interpretations; however seismic wave propagation is influenced to a first order by the
medium’s density and velocity. Interpretations of electrical and electromagnetic data are
used to decipher lithology or the saturating fluid, but the techniques are a measure of the
electrical resistivity of the rocks. In order to establish a measure of data reliability, one
first needs to figure out the relationship between variability in the subsurface and the
geophysical technique employed. This is possible through the field of rock physics,
which is an applied science that transforms geologic indicators (often the property of the
decision) to different geophysical observables. These transforms can come from either
theoretical models or statistical relationships based on observations [Mavko et al., 1998].
Next, we need to figure out how the results of rock physics are interpreted in terms of the
particular geological parameter of interest.
Since in the VOI calculation no data are yet available, we first need to simulate the data
collection technique on the Earth models, including the simulation of any measurement
error. The geophysical forward problem is described as
20
Equation 2
Ttf tt ,...,1))(()( )()( =+θ=θ εzd
where f is the forward model representing the physics of the geophysical technique (or
some simplification of it), is the vector of simulated geophysical data when
applied to each Earth model , and ε is some added error related to the
measurement. We assume the forward model is exact, in other words perfectly
simulating the physics of the particular geophysical technique. Note that the Earth model
needs to contain the simulated subsurface property relevant to the geophysical
data collection technique. For example, in the case of modeling seismic data, one will
need to know spatial variation of rock density and velocity.
)()( θtd
(tz )() θ
)()( θtz
Intrinsic to any geophysical data collection technique is some form of geophysical
interpretation. The resulting geophysical images need to be interpreted in terms of the
input geological parameter(s) θ at hand. For example, the θ that may need to be
determined could represent the azimuth of a variogram, the orientation of subsurface
channels or the appropriate depositional system depicted in a training image. Therefore,
in order to establish a data reliability measure we need to simulate the geophysical
interpretation process. Interpretation is a field of expertise on its own: either automatic
(using multi-variate statistical techniques or data mining techniques such as neural
network) or manual interpretation is performed on the geophysical data to extract
geological parameters of the subsurface formation. We assume that this interpretation
results in choosing one of the alternatives of θ . Even if a probabilistic technique were to
be used in determining the pre-posterior probability of each outcome of θ , we can
perform a classification by selecting the outcome with highest pre-posterior (symmetric-
loss Bayesian classification, see [Ripley, 1996]). Since in this case only two possible
21
outcomes are assumed, the interpretation process will result in classification of the
(simulated) data into one of the two possible outcomes 1θ and 2θ as follows:
Equation 3
Tth t ,...,1))(( )(int =θ=θ d
where h is the function representing the interpretation/classification process. By applying
such classification on the simulated datasets, we implicitly create a new random variable
Θint, namely the interpreted geological parameter. Note that this parameter depends on the
initial Θ, the spatial randomization, the geophysical forward model, the modeling of
measurement error as well as the interpretation process itself.
If the interpretation is performed on all simulated datasets, then we can construct a
frequency table called the Bayesian confusion matrix [Bishop, 1995], i.e. we can count
how many times a correct interpretation was obtained and how many times a false
interpretation occurred simply by comparing the outcome of Θ that was used to create the
Earth model with it geophysical interpretation Θ . This frequency is an estimate of the
conditional probabilities
int
2,1,)|Pr( int =θ=Θθ=Θ jiij
Equation 4
.
The conditional probabilities of Equation 4 forms a measure of data reliability, because if
the following probabilities were equal to unity,
22
Equation 5
ii
2,1,)|Pr( int =θ=Θθ=Θ jiji
2,1)Pr( int =θ=Θ jj
2,1)|Pr( int =θ=Θθ=Θ i
we would have perfect information, i.e. information that perfectly resolves the uncertain
geological parameter Θ. In the case where the conditional probabilities are equal to the
prior probabilities, the data are completely non-informative about Θ.
Note that from this analysis, we can also deduce by simple counting (or using Bayes’
rule) both the probability of the interpretation being correct or false (or posterior)
Equation 6
and the probability of a particular interpretation to occur (or pre-posterior)
Equation 7
.
2.3 VALUE OF INFORMATION CALCULATION
In this section, the determination of the value of information is described in three parts.
First, the outcome of a spatial decision will be expressed in terms of value as related to a
specific decision outcome (i.e. the result of choosing between several different and
possible locations for a well or mine extraction schemes). Expressing the outcome of a
decision in value allows us to evaluate and compare the decision outcomes of different
23
decision alternatives. Next, the definition of VOI will be explained, which essentially
reveals if the information has the capability to aid the decision-maker. Lastly, the
reliability measure of Equation 4 will be incorporated into the VOI definition to account
for the uncertainty of the information. These elements will now be described in detail.
Many types of spatial decisions exist in the Earth Sciences. In oil recovery, different
development schemes (where to drill wells and what kind of wells) represent different
possible actions or alternatives a to the decision of how to develop a particular field. In
mining, several alternative mine plans represent such actions. Therefore, the outcome
expressed in terms of value will be a combination of the action taken (the chosen
development scheme), the cost of taking that action, and the subsurface response to this
action (the amount of oil/ore recovered). The possible alternative actions are indexed by a
= 1,…,A, with A being the total number of possible alternatives that have been identified.
The action taken on the Earth is represented as function ga. Since the true subsurface
properties are unknown, the action ga is simulated on the generated models such
that
)()( θtz
Equation 8
( ) ( )( ) TtiAagv it
aita ,...,12,1,...,1)()( ===θ=θ z
)(ta
.
Note that the value v is a scalar. Value can be expressed in a variety of terms such as
ecological health [Polasky and Solow, 2001]; however, monetary units (usually expressed
in net-present-value, NPV) are conceptually the most straightforward and will be adopted
here. Again, expressing the decision outcome in value outcomes (Equation 8) allows for
comparison between the “successes” of different alternatives.
24
For any situation, the decision alternative that results in the best possible outcome should
be chosen. However, this is difficult to know in advance because of uncertainty
regarding how the subsurface will react to any proposed action. The values in Equation 8
could vary substantially due to such uncertainty. Assuming that the decision-maker is
risk-neutral (meaning we will use the expectation of the value of alternative actions), the
definition of VOI is the difference between the value with data (known in decision
analysis terminology as value with the free experiment VFE) and the value without the
data (known as prior value Vprior) [Raiffa, 1968; Bratvold et al, 2009]
Equation 9
priorFE VOI V −=
)()( θtz
V
Now, consider Va is the random variable describing the possible value outcomes
due to randomization of both input parameters and spatial variation of properties
of and for a given action a, then
)()( itav θ
Equation 10
[ ]aa
prior VEmax=
1θ 2θ
. V
In other words, the Vprior is defined as the highest expected value among all possible
actions given the average outcome for all of the possible input parameters. Note that
decision analysis and more specifically utility theory allows different criteria to be used
than the expectation, depending on the risk-attitude of the decision maker [Paté-Cornell,
2007; Bickel, 2008]. The following equation is an estimate of Vprior in the case where Θ
has two categories and
25
Equation 11
( ) ( ) AavViT
itai T
i ti
,...,11
Pr1
)(2
1
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛θθ=Θ ∑∑
θ
=θ=
)
. a
prior max=
( iθ=ΘPr represents the prior uncertainty for geologic input parameter iθ and are
the number of Earth models generated when
iTθ
iθ=Θ (therefore representing the spatial
uncertainty or variability within iθ ). We can generalize the computation of Vprior to Nθ
categories as follows:
Equation 12
( )
( ) AavT
Vi
i
T
ti
ta
N
ii
aprior ,...,1
1Prmax
1
)(
1
=⎟⎟⎟⎞
⎜⎜⎜
⎝
⎛θθ=Θ= ∑∑
θθ
=θ=.
⎠
Decision analysis uses so-called decision trees to represent clearly the decision, the
identified alternatives and the uncertain variables that will determine the decision
outcome. Figure 3 demonstrates the case of two possible assignments for Θ (where Θ is
the training image, deltaic and =θ1 θ =2
1
fluvial depositional systems) which are
represented by the red possibility nodes. Two possible actions are identified ( =a
2=a
:“to
drill at location X” or :“not drill at location X”) which are represented by the blue
square nodes. All decision trees go chronologically from left to right, yet calculations
proceed from right to left. Since Figure 3 represents the prior value, we are faced with
the decision (blue node) before we know what the possible subsurface is (red nodes). For
each of these combinations, the value is calculated as shown by the equation on the far
right and depicted with the green value node.
26
Figure 3: Example Decision Tree for a binary case demonstrating prior value (Vprior). The tree represents chronology from left to right, but calculations are made from right to left. (1) First calculations for each combination of action and geologic parameter output (possibility node) must be made. (2) The action values are calcualted as expected values of all the possibilities with that action. (3) Finally, Vprior is the highest action value (the
best action considering our current uncertainty).
Commonly, the value with data (VFE) is expressed as the expectation over all possible
datasets d of the highest expected value among all possible actions a for that dataset
[Raiffa, 1968]
27
Equation 13
⎥⎦⎢⎣= d|max a
aFE VEEV [ ] ⎤⎡ .
Recall that 1) randomization is over all possible (generated) datasets since no actual field
geophysical data have been collected and that 2) this methodology does not use the
geophysical data to condition properties in the Earth models. Instead, the reliability
measure in Equation 4 created from Θ and will be used to approximate these
expectations in terms of arithmetic averages. However, before this is done, we calculate
the value with perfect information.
intΘ
Perfect information would reveal, without fail, what the actual input parameter iθ is.
Although it is not realistic to assume that the information is perfect, it is useful to
calculate the value of hypothetical perfect information: if perfect information has no
value, then there is no reason to calculate the value of imperfect data (as it will always be
equal to or less than the value of perfect information). Given Equation 13, the value with
perfect information (VPI) for our binary example case is now
Equation 14
( ) ( ) AavT
Vi
T
ti
taPI
i
i
,...,11
maxPr2
1 1
)( =⎟⎟⎟
⎠
⎞⎜⎛
θθ=Θ= ∑ ∑= =θ
θ
. a
i ⎜⎜⎝
Equation 14 demonstrates how having d is equivalent to knowing iθ
⎟⎠⎞
⎝ amax
in that the best
decision action ⎜⎛ can be taken for each iθ . Again, if Θ would have Nθ categories,
VPI can be expressed as
28
Equation 15
( ) ( ) AavT
VN
i
T
ti
ta
aiPI
i
i
,...,11
maxPr1 1
)( =⎟⎟⎟⎞
⎜⎜⎜⎛
θθ=Θ= ∑ ∑θ θ
= =θ ⎠⎝
( ) ( )
.
In the case of imperfect information, we need to account for the information’s reliability
which is expressed through the conditional probabilities of Equation 4. This means that
the expectation of Equation 15 requires the integration of the posterior in Equation 6.
Hence, in terms of arithmetic averages, the value with imperfect information VII is
estimated as
Equation 16
( ) AavT
Vj t
ita
iji
ajII
i
,...,1|PrmaxPr1 1
)(
1
intint =⎟⎟⎠
⎜⎜⎝
⎟⎟⎠
⎜⎜⎝
θθ=Θθ=Θθ=Θ= ∑ ∑∑= =θ=
N TN i1 ⎟⎞
⎜⎛
⎟⎞
⎜⎛θ θθ
Notice that in the case of perfect information, Equation 16 will simplify to Equation 15,
as all values will now only be weighted by either 1 when i=j or 0 when i≠j and the pre-
posterior reverts to the prior )Pr( intjθ=Θ )Pr( iθ=Θ . Figure 4 demonstrates the
components of this equation and how chronologically the geologic parameter iθ is
interpreted before choosing the best decision action (with the maximum value outcome)
for that interpretation. The four values weighted by their posterior value (Equation 6) are
explicitly shown for the binary example. The highest weighted value for each
interpretation is then chosen for that branch. Finally, VII is the expected value of
these two values, weighted by their respective probability Pr( (
jθ
)intjθ=Θ Equation 7).
29
The value of imperfect information can then be calculated as the difference between VII
and Vprior
Equation 17
priorIIII VVVOI −= .
Figure 4: Decision Tree depicting the calculations for the Value with Imperfect Information (VII). Chronologically, we can interpret the subsurface possibility first (red
node) and then chose the best action (blue node). Again, calculations start from right and proceed left. (1) The possibility-action combinations are already calculated from Vprior,
now the same possibilities iθ are grouped together. (2) The interpreted possibility branches are calculated for each action. These weight the values of (1) with the
probability that the interpretation is correct or incorrect. (3) Finally, the value with imperfect information (VII) is the expected value of the best action for each interpreted
30
branch. This is weighted by the probability of each interpretation being made . )Pr( int
jθ=Θ
2.3.1 GENERALIZATION TO SEVERAL GEOLOGIC INPUT PARAMETERS
The VOI calculation has now been completed with a single input parameter Θ and two
possible outcome values for iθ . We now generalize the calculation of Vprior and VII to
more input geological parameters and their outcomes; we focus on VII as it utilizes the
reliability measure. Suppose that a total of M other geologic input parameters are
included, which all have outcomes iθ that are mutually exclusive from each other. These
additional input parameters can be represented by the vector Θ .
Equation 18
[ ]M1 ,...,ΘΘ=Θ
To distinguish the geologic input parameter and its outcome value from other input
parameter outcomes, an additional subscript m is introduced: where
, , andN is the number of possible outcomes (categories) for
input parameter mΘ
{ Mii ,...,1θ }
)(,...,1 mm Ni θ= Mm ,...,1=
.
) (mθ
For instance, let us assume that there are two possible additional input parameters: the
variogram model type and the paleo-direction of a lithofacies, represented by 2Θ and
respectively. Suppose that the variogram parameter 3Θ 2Θ models three ranges for the
continuity within the lithology classified as low, medium, and high range. On the other
hand, the rotation parameter 3Θ for the paleo-direction determines the orientation of
facies as either northeast or southwest. Therefore, the new possible categories will be
combinations of all the outcomes of all the included parameters [ ]321 ,, ΘΘΘ=Θ . For
31
this extended example, would denote the category where the training image is
fluvial, the variogram range is high, and the facies orientation deemed northeast.
}1,3,2{θ
With this additional index m denoting from which Θ the samples were drawn, all
previous definitions can be extended to include all M input parameters and their
respective outcomes. The generalized form of Vprior (Equation 12) is now
Equation 19
{ }( ){ }
{ }{ }
AavT
V
M
M
Mii
MMii
M
N
i
N
i
T
tii
taii
aprior ,,1)(
1Pr θ=Θ
{i ,,1 Kθ=
total
max
)( )1(
1
,,1
1,,1
11 1 1
,,)(
,, KKK
K
KK =⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛θ= ∑ ∑ ∑
θ θ θ
= = =θ
.
Most importantly, the total number of categories used for the reliability measure is now
Equation 20
∏=
θ=M
m
mtotal NC
1
)( .
Note that in all rigor, this may require M interpretation techniques (hm) for the M
geological input parameters. The geophysical method is expected to uniquely distinguish
between all these categories. Thus, the Bayesian confusion matrix, now expressed as
Equation 21
{ } } MmNi,j mijj MM
,,1,,1)|Pr( )(,,
int1
KKK ==Θθ=Θ θ ,
will be of dimension totalCx
C , such that all unique combinations of are
represented. Lastly, the calculation for the value with imperfect information of
{ }Mii ,,1 Kθ
Equation
32
16 is modified to include all the combinations of geological parameters and their values
as follows:
Equation 22
( )
{ }( )( )
{ } { }( ){ }
{ }( ){ }
vT
V
Mii
MMii
MM
M
M
M
M
M
T
tii
ta
N
ijjii
N
ia
N
jjj
N
jII
1|Pr...max
Pr
,,1
1,,1
)1(
1
11
)(
1
1
1,,
)(
1,,
int,,
1
1,,
int
1
K
LL
K
K
KKK
K
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛θθ=Θθ=Θ
θ=Θ=
∑∑∑
∑∑
θθθ
θθ
=θ==
==1
Aa ,,1K=
2.4 VOI EXAMPLE
To demonstrate the methodology of Chapter 2, we now present an example of a VOI
calculation for a synthetic aquifer recharge decision. VOI can be viewed as the interplay
of three components: the reliability of the information, the chance of making a poor or
suboptimal decision, and the magnitude of the decision. In this section, our results will
demonstrate the contribution of this work: a quantitative, repeatable and comprehensive
method to obtain VOI for spatial decision problems accounting for the reliability of
spatial data. The second and third VOI factors are outside of the scope of this work and
would need to be addressed by sensitivity studies on both the spatial uncertainty
modeling parameters as they relate to the decision and the VOI sensitivity to other
elements such as the costs and/or values assigned to different outcomes of the decision.
Nominal economic values have been chosen for both values and costs. Realistic values
could be obtained through experts (e.g. economists), but economic modeling is not the
purpose of this example.
33
Obtaining a meaningful data reliability will give more confidence in the calculated value
of information. The VOI metric is crucial for stakeholders faced with protection or
extraction decisions for assets such as groundwater resources, oil reservoirs, or mineral
deposits, where in all cases data collection represents a large expense. To demonstrate
this, a synthetic aquifer example, inspired by a real case on the California coast, will
demonstrate how this reliability measurement is used in a VOI calculation for a spatial
decision with subsurface uncertainty. In our case, artificial recharge is considered to
mitigate seawater intrusion. Intrusion leads to an increase in salinity in the groundwater
and hence a decrease in the amount of useable water in a coastal fluvial aquifer that is
critical for farming operations. The spatial decision concerns the location for performing
this recharge, if at all, given the uncertainty of the subsurface channel directions, how
they affect the success of the recharge actions, and the costs of the recharge operation.
First, the prior uncertainty model is described. Then, alternatives to this decision are
established, and the range of outcomes from the combination of models and alternatives
are generated. Finally, to illustrate a geophysical data measurement scenario, airborne
and land-based transient electromagnetic (TEM) techniques are evaluated for their ability
to resolve the channel orientation. We emphasize that this example serves to illustrate
the VOI calculation methodology under spatial uncertainty; we will not focus in detail on
the geophysics or any economics in the value calculation itself.
2.4.1 PRIOR GEOLOGIC UNCERTAINTY
Only one input geologic parameter (M=1) is used in the example: orientation of the
channels. The prior models are generated using three possible channel direction
scenarios ( =3): dominantly northeast, dominantly southeast and a mix of both. These
three scenarios are deemed equally probable (
)()( θtz
θN
( ) 3.01
Pr ==θ=ΘθN
i ); ideally, this prior
34
likelihood for each scenario would come from the expert geologist for any particular area.
For each of these three scenarios, 50 Earth models are generated using a channel training
image (Figure 5) and the algorithm snesim (Single Normal Equation Simulation) within
Stanford’s Geostatistical Earth Modeling Software (SGeMS) [Remy et al, 2009]. snesim
is a multiple-point geostatistical algorithm which has a rotation functionality [Strebelle,
2002]. snesim utilizes rotation maps to locally turn patterns of the training image. The
angle maps that represent the three channel scenarios and an example 2D realization
generated from each are illustrated in Figure 6.
Figure 5: Fluvial Training Image; Red = sand facies; Blue=non-sand facies
35
Figure 6: (A) Example rotation maps and (B) 2D Earth models from snesim for each of the three channel scenarios (80 x 80 x 1 cells, 5.3km x 5.3km x 70m, blue=non-sand
facies and red=sand facies)
2.4.2 DECISION ACTIONS
Four alternatives to the recharge decision are identified as 1) no recharge, 2) a central
recharge location, 3) a northern recharge location, and 4) a southern recharge location.
Flow simulation is performed for all combinations: the four recharge options on all the
three channel scenarios (specifically the 50 lithology Earth models within each), resulting
in a total of 50 x 3 x 4 = 600 flow simulations. The decision tree (Figure 7) visualizes
these decision options (blue nodes) and the possible states of the subsurface (red nodes).
36
Figure 7: Decision tree schematic for the prior value (Vprior) of the aquifer recharge example. The red nodes denote the 3 possible channel orientation scenarios. The blue
node denotes the 4 recharge options.
2.4.3 VALUE DEFINITION
For this example, we consider that groundwater is used for agriculture. Thus, the volume
of freshwater in the aquifer after 10 years, with or without recharge, is equated into
potential crop revenue in dollars for farmers. Essentially, if the water in any reservoir
cell is below a salinity threshold of 150 ppm (parts per million) of chloride at time=10
years, then the volume of water in that cell (m3) is converted to the production of crop
“X” (this is given by the required volume of water to produce 1 ton of crop “X”:
37
tons/m3). This can then be equated into dollars by the price of crop “X” ($/ton). For all
options where recharge is performed, initial and operational costs of the artificial
recharge are subtracted out of the final value (approximately 30% of the average profit
for the outcomes without recharge). The resulting value for each recharge-channel
direction scenario combination ( )( iav θ ) is seen on the far right of Figure 7. The prior
value Vprior (Equation 12) is $12.47x106 and the value with perfect information (Equation
15) is $12.8x106. Therefore the value of perfect information (VOIPI = VPI – Vprior)is
approximately $300,000.
2.4.4 TEM RELIABILITY
Transient or time-domain electromagnetic (TEM) data are considered to help determine
aquifer heterogeneity, specifically the channel orientation. TEM works with a transmitter
loop that turns on and off a direct current to induce currents and fields into the
subsurface; meanwhile, the larger receiver loop measures the changing magnetic field
response of induced currents in the subsurface [Christiansen, 2003]. The data (the
magnetic field response in time) must then be inverted to a layered model of electrical
resistivity and thickness values which is achieved through geophysical inversion [Auken
et al, 2008]. The recovered electrical resistivity can be an indication of the lithology type
as clay (non-sand) typically has an electrical resistivity less than 30 ohm-m, whereas sand
is usually greater than 80 ohm-m1. Therefore, from the TEM measurement, a lithological
image can be recovered.
In the prior models , the channels are represented as sand, and background facies
is non-sand. The orientation of these channels is here deemed, as an illustration, to be the
)()( θtz
1 See Appendix A for details on rock transforms used and the TEM measurement.
38
dominant geologic parameter that determines the success of the artificial recharge;
consequently, each of the inverted images must be evaluated (interpreted) in terms of the
channel directions. To achieve this by automatic interpretation (note this is needed
because we need to interpret 150 inversion results in terms of channel direction), we
calculate the directional variogram maps from the inversion image [Marcotte, 1996].
This analysis is also performed on the original models so that we can compare the “true”
channel-direction scenarios with the interpreted ones to obtain the probabilities of
Equation 4. Figure 8 shows two examples of the image analysis on both the lithology
models and its inversion result, followed by an illustration of the difference of these
specifically when a northeast channel orientation was misinterpreted as southeast.
39
Figure 8: Results of image analysis for the angle of maximum correlation. A) Original and inversion results shown for a model (z(t)(θi)) from scenario Northeast. Far right, the
locations are shown in red where Northeast is misclassified as Southeast and in blue where correctly classified. B) Original and inversion results shown for a model (z(t)(θi))
from scenario Mixed. Far right, the locations are shown in red where Northeast is misclassified as Southeast and in blue where correctly classified.
Recall the underlying goal of this analysis, is to determine how well TEM can identify
the channel orientation of the original model (i.e. the reliability measure), such that
ultimately the value with imperfect information (VII) can be calculated. With all the
original models and inversion images, the frequency of misclassification is obtained for
both airborne and land-based TEM. The conditional probabilities (Equation 21) for
airborne and land-based are represented in Table 1 and Table 2, where the rows and the
columns represent the true geologic scenario iθ=Θ and the interpreted scenario
respectively. In constructing jθ=Θint Table 1 and Table 2 two assumptions were made.
40
First, electrical resistivity was assigned deterministically to each lithology: sand was
assigned 80 ohm-m and non-sand 13 ohm-m. Hence, it was assumed that electrical
resistivity can discriminate between lithologies. Second, for the example problem and for
computational reasons, a 1D forward modeling code was used which does not capture the
effect of lateral heterogeneity on the TEM measurement.
Table 1: Airborne TEM Reliability
Interpreted Scenario
Mixed Northeast Southeast
Mixed 96% 0% 4%
Northeast 0% 98% 2%
True
Sce
nari
o
Southeast 0% 0% 100%
Table 2: Land-based TEM Reliability
Interpreted Scenario
Mixed Northeast Southeast
Mixed 64% 4% 32%
Northeast 24.4% 48.8% 26.8%
True
Sce
nari
o
Southeast 37.2% 25.6% 37.2%
41
The two reliabilities of airborne and land-based measurements (Table 1 and Table 2) are
quite different. This is due to the difference of in-line measurement separation. For land
TEM, in-line measurements are 250m apart as opposed to 50m for the airborne method.
Thus, the resolution of channel orientation is decreased for land TEM with fewer
locations of measurements. Utilizing the reliability measures of Table 1 and Table 2 in
Equation 22, the VOIII (VII – Vprior) for airborne and land-based TEM are ~$260,000 and
$0. Compared to VOIperfect ($300,000) the influence of the reliability measure is
apparent. Under our assumptions, if the price of acquiring airborne TEM is less than
$260,000, then it is deemed a sound decision to purchase this information.
42
CHAPTER 3: VOI FOR DYNAMIC PROBLEMS
As Keeney (1982) concisely describes, there are four characteristics of today’s decision
problems that makes them, and consequently VOI studies, challenging: complicated
structure, high stakes, no overall experts, and the need to justify decisions. Each of these
characteristics will be described in terms of certain Earth science decisions, while further
introducing the synthetic demonstration case modeled after a European groundwater
problem that was briefly described in the introduction chapter. This example will
motivate and illustrate new decision challenges which require the concept and
introduction of proxy decision variables. This case poses a significant problem for
achieving a VOI metric as no information source measures the proxy variable directly or
indirectly. For these proxy decision cases, a realistic VOI methodology for imperfect
geophysical data are proposed by conditioning models to generated synthetic data that are
derived from rock physics likelihoods.
3.1 DEMONSTRATION CASE SETTING
Complicated structure. Spatial decisions regarding the Earth vary from the frequently
asked “where to drill for oil,” to actions for protecting groundwater from contamination,
and increasingly, to decisions regarding the subsurface storage of greenhouse gases such
as CO2. To demonstrate these decision problem characteristics, we will use an example
which is inspired by a Northern European groundwater contamination decision case. The
reader should distinguish this “demonstration case,” serving as illustration of a general
methodology, from an actual case study since the main elements such as the data and the
actual decision have been altered. Therefore, no real-world implications should be drawn
from our analysis. In this example, we assume that a particular area relies solely on its
groundwater sources to supply drinking water. Over the past several decades, the area
aquifers have been compromised by surface-sourced contaminants due to farming
43
activities. Contamination will continue to be a threat until critical surface locations that
serve as entry points into the aquifer are identified. This can only be successfully
achieved if the hydraulically complex connections between the contaminant sources at
the surface and the underlying aquifers are understood. Thus, the principle uncertainty is
which surface locations act as entry points into the aquifer. The decision is to determine
which contamination sources (e.g. farms) need to be removed to ensure a sustainable
supply of drinking water.
Decision making in this context is difficult because of the uncertainty surrounding
properties of the unknown Earth. Thus, the uncertainty regarding the unknown
subsurface and how it will respond once the decision is made is the main facet of the
“complicated structure” that we consider in this chapter. As described in the Chapter 1,
decision analysis manages the unknown parameters through probabilistic representation.
As in Chapter 2, we represent uncertainty about the unknown subsurface by means of
generating many Earth models. Again, we differentiate two types of uncertainty: “spatial
uncertainty” (geostatistics) due to the limited amount of information to constrain such
models and “model uncertainty” because the input to the spatial stochastic simulation
algorithms is uncertain. The geological input parameters, represented by random variable
, are the dominant geologic features or characteristics of a particular locale that often
control the outcome of the decision.
Θ
For the demonstration case, the geologic depositional system is interpreted as glacial
buried valleys, which act as important groundwater resources in many countries in
Northern Europe. Buried valleys are the result of the “waxing and waning of Pleistocene
ice sheets” [BurVal Working Group, 2006]. These glacial valleys can be thought of as
the primary level of uncertainty in the aquifer system structure. If largely filled with
44
sand, the buried valley has potential for being a high volume aquifer (reservoir). The
superposition of three to five different generations of glaciation has been observed. Thus,
glacial valleys from multiple generations cross-cut each other and can also appear to
abruptly end as seen in Figure 9 [Jørgensen and Sanderson, 2006]. Figure 9 is not used in
this study as a deterministic aquifer model. Instead, it is used as a concept for generating
several training images which are necessary for multiple-point geostatistical modeling
and simulation [Strebelle, 2002; Caers, 2005].
Figure 9: Network of buried valleys; darker to lighter representing older to younger buried valley generations (Jørgensen & Sanderson, 2006)
Figure 10 depicts as the glacial buried valley training images with two possible
outcomes for the dimensions of the buried valley lithofacies. At the bottom
of
Θ
2θ{ 1 ,θ∈Θ }Figure 10, a few Earth models are shown that are generated from iθ and a stochastic
algorithm. Generally, any input geological parameter may have N number of outcomes:
45
Equation 23
N{ }i θθθ∈Θ ,...,,...,1
Figure 10: Schematic showing the two randomizations used to create the Earth models of buried valleys. First, the geologic input parameters are identified, then outcomes are drawn and lastly these outcomes are used as input in to stochastic algorithms to create
each model.
Ideally, experts (e.g. geologists) assign prior probabilities ( )iθ=ΘPr for these specific
geologic input possibilities to occur. The spatial stochastic variation accounts for the
spatial variability that may occur within any of the distinguishing qualities or features of
iθ . As seen in Chapter 2, many Earth models may be generated and represented by
46
Equation 24
i( ) T,1,t)( …=θtz
with T the total number of models. The ensemble of ( )it θ)(z Earth models captures the
prior uncertainty regarding the subsurface properties of interest, which are all captured in
the vector z.
High Stakes. Much effort is made to create realistic Earth models such that the
predictions of the decision alternatives (for example remove farm at location A or
location B or at both locations A and B) reasonably capture the range of possible
outcomes. The difference between these outcomes could represent severe environmental
damage or the loss of millions of dollars. Accordingly, the decision makers must identify
the possible alternatives to the decision, which are denoted by a=1,…,A, where A is total
number of alternatives identified. Each alternative may have different outcomes due to
the possible and unknown heterogeneity modeled with ( )it θ)(z . We can imagine how
removing contaminant sources at different locations with varying connectivity to the
underlying aquifers will result in different aquifer quality protection. The function ga
denotes the action taken on the Earth (such as the removal of contaminant sources). It
models the predicted outcome of alternative a with unknown subsurface captured by the
models ( )it θ)(z . Lastly, in order to evaluate and compare the different alternatives, the
outcome for each combination of subsurface model and alternative must be expressed in
terms of value
Equation 25
TtAagv it
aita ,...,1,...,1))(()( )()( ==θ=θ z
47
where value can be in terms of monetary units ($), ecological health [Polasky and Solow,
2001] or some other appropriate utility.
No overall experts. So far, several disciplines have been introduced and are necessary
for spatial Earth decisions: geology, geostatistics, modeling related to the physical,
chemical or others processes of the decision action, and economics. An effective
decision analysis requires the coordination of expertise from these fields. As identified
by Howard (1966), a sensitivity analysis should be completed by the geologist(s) and
modeler(s) to determine which uncertain parameters are most consequential and hence
control the outcome of a decision. For this example, we assume this sensitivity analysis
has identified the input geologic parameters iθ representing the different width, length
and thickness dimensions of the buried valley lithofacies. The sensitivity of other
parameters related to the economics or public policy constraints should be analyzed;
however, we deem these considerations outside the scope of this work. Therefore, the
sensitivity of values and costs of different decision alternative outcomes are ignored here.
Need to justify decisions. Both public and private decision makers will have to defend
their Earth Science decisions, such as their choices of their prior uncertainty ( )it θ)(z ,
their identification of the A alternatives, and their predicted outcomes ( )( )it θ)(za
ta gv =)(
iθ
.
Depending on the decision outcome, environmental regulatory authorities and/or
economic entities such as shareholders or supervisors will scrutinize the rigor of their
decision analysis. During this regulatory or economic scrutiny, the decision makers may
be asked why further information was not gathered to ensure a better result. Relevant
information sources may be able to reveal which input geologic parameters exist at a
certain locale, and subsequently, the best decision for iθ=Θ could be made.
48
3.1.1 CHAPTER 3 VS CHAPTER 2
Equation 13 of Chapter 2 is an expression for the value with the free experiment or
information:
[ ] AaVEEV aa
FE ,...,1|max =⎥⎦⎤
⎢⎣⎡= d
where the vector d represents the synthetic or forward simulated data related to the
proposed data source. All the defined uncertainties, alternatives, and value outcomes are
utilized, with an expectation over the data d. Recall that no data have been collected;
instead we must simulate the possible datasets d and evaluate how these data d would
influence the decision. If d includes realistic errors that may be in the data, then VFE
becomes the value of imperfect information VII. Attaining a realistic VII involves an
estimation of how accurate the data are in resolving certain Earth parameters that are
relevant to the decision. This is also known as the reliability of the information.
In defining the reliability using Equation 4 of Chapter 2 and using it for the VOI
calculation, it is implicitly assumed that knowledge of iθ solves the decision problem
deterministically; in other words, there is a direct link between the decision we make and
the uncertainty on the input geologic parameter iθ . This would mean in our
demonstration case that knowing the buried valley dimensions would determine which
farms should be removed.
3.2 THE PROXY DECISION VARIABLE: MOTIVATION
In many situations, however, knowing i
θ does not resolve the decision problem
deterministically. For our demonstration case, knowledge of the buried valley
dimensions will not tell us precisely which farms to remove. The network of connected
49
buried valleys is complex; “significant parts of the recharge area may therefore lie at
relatively large distances from the valley [which represents the deep aquifer]” [Sandersen
and Jørgensen, 2003]. Thus, contamination can be transported kilometers from its
surficial entry point into a deep aquifer. Knowledge that a valley or non-valley exists at
one particular location is not a thorough representation of possible risks of contamination
or aquifer vulnerability. This illustrates the complex relationship between the model
characteristics (the heterogeneity as represented by alternative iθ ’s) and the aquifer’s
vulnerability to surface-sourced contaminants. In this chapter, we introduce a new VOI
calculation that relies on a so-called “proxy decision variable.”
A “proxy decision variable” locally represents the geologic heterogeneity’s response to
some applied stress or process. This variable is required when facing a decision relying
on some dynamic response at a specific location within the Earth models. We will call
these responses the dynamic simulation function, which may model physical, chemical,
geomechanical, or any other processes. For our demonstration case, we will introduce a
so-called “aquifer vulnerability” as the proxy decision variable, and the simulation of
fluid flow in porous media is the dynamic simulation function. Defining this proxy
decision variable will provide an indication of the decision outcome for a particular
action at a particular location, hence removing the difficulty as to whether a polluting
farm “connects” with an aquifer several kilometers away.
The next three sections describe the three key parts of our VOI methodology for
dynamic, spatial decisions. First, we will show how the proxy variable can be estimated
for all the Earth models. This provides a range of proxy variable estimations that capture
the uncertainty regarding the outcomes of location-specific decisions. Then, we will
address how to generate datasets utilizing a likelihood of geophysical data to provide
50
information about lithology. Multiple Earth models conditioned to these synthetic
datasets are achieved through geostatistical simulation. These represent the possible
lithology interpretations that could be made from the data, and they are needed to
calculate the value of imperfect information (Equation 13). Next, the uncertainty of the
proxy decision variable and the multiple conditioned models will be integrated into the
VOI calculation framework. Explanation of details of the demonstration case will be
given for these two steps. Lastly, the methodology will be applied to our demonstration
case of aquifer vulnerability.
3.3 ESTIMATING THE PROXY DECISION VARIABLE
We illustrate the estimation of the proxy decision variable by means of our demonstration
case where the decision depends on aquifer vulnerability: the surficial locations where
contaminants can enter into the aquifer [Thomsen et al, 2004]. However, this workflow
can also be applied to other evaluations requiring proxy decision variables involving
decisions depending on some form of connectivity: well-to-surface or well-to-well due to
a fracture network [Kerrou et al, 2008; Karimi-Fard and Firoozabadi, 2001]. Similarly,
infiltration potential defines surficial locations’ ability to serve as artificial recharge
points, as seen in Stamos et al (2002).
The proxy decision variable depends on the response of the Earth models to some
dynamic simulation function which is denoted as f . In estimating the proxy decision
variable, the output of f may require some post-processing or interpretation.
Specifically, we assume a single dynamic simulation function exists to transform each
( )it θ)(z into the proxy variable s
51
Equation 26
( ) ( ) LTtNifs it
it ,,1,,1,1),()()( KlKKl ===θ=θ θz
il .
Here l denotes all the L locations within each model ( )it θ)(z where the decision must be
made. We must define because the goal is to determine the best decision alternative for
different locations within
l
l ( )it θ)(z . For our aquifer vulnerability study, will cover all
possible locations of farms. The function represents the A different actions or
alternatives that are deemed plausible. In our demonstration case, the two alternatives are
to either remove possible contaminant sources (e.g. farms) or do nothing. For now we
assume that action g can only be taken at one position at a time. This restriction will
be removed later.
l
ag
a
l
( ) ( )
l
The proxy variable has the information that will determine all the value outcomes at
each location
)(ts
Equation 27
( ) LTtNiAasgvii
tai
ta ,,1,,1,1,,1)()(, KlKKKll ====θ=θ θ .
With a proxy decision variable, we can determine the outcome of different decision
alternatives a at locations l . And since these are obtained for all Earth models ( )it θ)(z ,
we have a range of the possible value outcomes for these local decision actions,
representing the uncertainty on the local decision outcome. The location-specific values
of Equation 27 will eventually be used in Vprior (Equation 10). The details on how they
will be utilized will be explained later.
52
3.4 DATA RELIABILITY FOR MEASURING PROXY DECISION VARIABLES
3.4.1 RELIABILITY THROUGH GEOSTATISTICAL SIMULATION
In VOI methodologies, a data reliability measure is a conditional probability of the form
( ) isworldrealthewhat|saydatathewhatPr
However, since the “real world” includes both the complex subsurface variation as well
as the proxy decision variable, such a reliability measure is not trivial to explicitly
determine by means of forward model runs [Bickel et al, 2006; Houck and Pavlov, 2006;
Houck, 2007]. Since we cannot explicitly derive the reliability measure in terms of
conditional probabilities, we propose a geostatistical simulation approach for including
data reliability into the VOI calculation based on rock physics relationships.
Geostatistical simulation is used to represents the variability in the geophysical message
and how it could resolve the subsurface static properties which influence the dynamic
response. No forward modeling of geophysics is performed, but it could be included in
the workflow. In this proposed workflow, the intent is to capture the dynamic response
that is important to the decision.
In summary, our methodology is as follows:
1) Use a likelihood function to generate a synthetic data set from each prior Earth
model, where the likelihood describes the relationship between the geophysical
attribute and the key geologic indicator (which influences the outcome of a
decision).
2) From each of the synthetic data sets, derive a pre-posterior probability distribution
on how informative that synthetic data set is about the key geologic indicators.
53
3) Use that pre-posterior probability to generate multiple, new Earth models
constrained to each of the synthetic data sets. These conditioned Earth models
represent the variability in what we could interpret from the data.
4) Create realizations of the proxy decision variable by applying the dynamic
simulation function f to the conditioned (interpreted) Earth models.
These four steps are graphically shown in Figure 11. This is a valid approach to attain a
VOI measure considering the proxy decision variable. By using rock physics likelihood
relationships, synthetic data can be generated that contains local information. The local
information has the potential (if the data are sufficiently reliable) to constrain the
conditioned or interpreted Earth models, which in turn will influence the response to the
dynamic simulation function.
In our demonstration case, the key geologic indicators are lithology. Specifically for our
demonstration case, the geologic input parameters iθ represent different combinations of
buried valley dimensions. We assume exclusivity between lithology and facies ( iθ ):
sand is always interpreted as buried valley and non-sand (clay) as non-valley. This
assumption may be true for some geographic locations but not always [Sandersen and
Jørgensen, 2003]. However, this assumption could be relaxed or accounted for by
estimating a percentage of buried valleys that are filled with clay materials. But for this
demonstration, an exclusivity between lithology and iθ is represented in ( )it θ)(z .
To further demonstrate the validity of this four step approach to reliability, we can
imagine the two extreme cases: the geophysical attribute either perfectly informs
54
lithology or gives completely unreliable lithology information. With perfect information,
these four steps will allow us to retrieve the true proxy variable, as all the actual valley
and non-valley locations will be identified perfectly through the likelihood and soft
probability. With the true proxy variable we can than make the best decisions for each
location . Whereas completely unreliable data could generate conditioned lithology
models that are very different from their respective prior models and from each other (as
the pre-posterior is uninformative). Decisions made on proxy variables obtained through
these inaccurately interpreted models, will not represent the best decisions made for the
true proxy variable. This will be further validated later with the expression for VII. Next,
we will clarify each of the four steps in detail.
l
55
Figure 11: The 4 steps of obtaining a data reliability for proxy decision variables through geostatistical simulation. Step 1: Generate a synthetic dataset from the likelihood and a prior model. Step 2: Generate a soft probability cube for the valley lithofacies from a
dataset and the information content. Step 3: Generate a conditioned Earth model with the soft probability cube. Step 4: Obtain the conditioned proxy variable by applying the
dynamic simulation function to the conditioned Earth model.
3.4.2 CREATING SYNTHETIC DATA SETS WITH ROCK PHYSICS
Rock physics relationships associate geologic indicators with geophysical attributes
[Mavko et al, 1998]. Recall that all subsurface properties of interest can be captured in
the vector z of ( )it θ)(z . In our case, this includes the geological indicator of lithology,
56
therefore, we consider a link between the lithology and possible geophysical attributes.
For this synthetic example, the geophysical information source being considered is
transient or time-domain electromagnetic (TEM) data. As described in Chapter 2, TEM
works with a transmitter loop that turns on and off a direct current to induce currents and
fields into the subsurface; meanwhile, the larger receiver loop measures the changing
magnetic field response of induced currents in the subsurface [Fitterman and Stewart,
1986; Christiansen, 2003]. The magnetic field response in time is then inverted into a
layered model of electrical resistivity and thickness values [Auken et al, 2008]. The
recovered electrical resistivity can be an indication of the lithology type as clay typically
has an electrical resistivity less than 30 ohm-m, whereas sand is usually greater than 80
ohm-m.
The association between lithology (litho) and the electrical resistivity ( ) may be
described through an empirical relationship [Archie, 1942; de Lima and Sharma, 1990].
However, a probabilistic relationship is a more realistic description of what the indirect
geophysical data can resolve and is typically modeled as a conditional probability relating
the geologic indicators of the prior models (known) to the geophysical attributes
(unknown). This conditional probability (a likelihood) can be obtained through forward
models [Eidsvik et al, 2008], from some calibration dataset [Houck, 2004], from geologic
analogs, or could be synthetically created. Two likelihoods of the form
ρ
Equation 28
ρ = = −=Ρ 0}sandnon,sand{)|Pr( litholithoLitho < ρ < ∞
were synthetically created for our demonstration case to describe two possible
relationships between electrical resistivity and lithology. To utilize Archie’s relation,
information on porosity and saturation would be needed not just lithology, however we
57
base our likelihoods on a calibration of co-located inverted resistivity data and driller’s
logs, which respectively have electrical resistivity and lithology information. The first
data set (shown in Figure 12) demonstrates a decent electrical resistivity contrast between
sand (assumed here as valley) and clay (non-valley). Whereas the second data (Figure
13) has a less discriminating message about lithology as a greater overlap exists between
the electrical resistivity of the two lithologies.
58
Figure 12: Synthetic data reliability describing a good contrast between the two lithologies’ ranges of electrical resistivity
59
Figure 13: Synthetic data reliability describing a poor contrast between the two lithologies’ ranges of electrical resistivity
Most importantly, we can create many synthetic datasets d (which are in the form of the
geophysical observables) using Monte Carlo sampling of Equation 28 (represented in
Figure 12 and Figure 13). Creating many synthetic datasets d in this way allows us to
statistically represent the possible variation in the datasets. Namely, knowing the
occurrence of lithology at a certain location within ( )it θ)(z , an instance of electrical
resistivity (ρ ) can be drawn by Monte Carlo sampling of the cdf (cumulative distribution
function) form of either the top of Figure 12 (given that the considered location is
sand/valley) or the bottom (given the considered location is non-sand/non-valley).
Similarly, this can be performed with the cdf versions of Figure 13. This is repeated for
all locations within the model ( )it θ)(z to generate the dataset:
60
Equation 29
ii TtNi)(t
θ== ,,1,1 KKd .
Therefore, there is an electrical resistivity dataset ( )itd that corresponds to each of the
prior models ( )it θ)(z . Figure 11 demonstrates this Step 1 and displays an example
synthetic dataset of electrical resistivity that was generated using one prior model and the
cdf versions of the pdf’s in Figure 12.
3.4.3 DERIVING INFORMATION CONTENT TO CREATE CONDITIONED EARTH MODELS
While data reliability models a conditional distribution of the form:
( )isworldrealthewhat|saydatathewhatPr ,
“information content” of a data source about the unknown Earth is of the form:
( )saydatathewhat|isworldrealthewhatPr .
This conditional probability, also termed “pre-posterior” is important in generating new
Earth models constrained to the synthetic data sets as outlined in our workflow. More
specifically the pre-posterior probabilities we seek are of the form
Equation 30
. ρ = −=Ρ= 0}sandnonsand,{)|Pr( litholithoLitho ≤ ρ ≤ ∞
Through Bayes Law, Equation 30 can be obtained from Equation 28. Using Equation 28
and Equation 30, we notice that perfect information would imply that an exclusive
61
relationship exists between lithology and electrical resisitivity, such that any draw from
or )|Pr( lithoLitho =ρ=Ρ )|Pr( ρ=Ρ= lithoLitho would be 0 or 1.
We can create a lithology probability cube using each of the datasets ( )itd to obtain a
sand and non-sand probability from Equation 30
Equation 31
i TtNitθ== ,,1,...,1)( Ky . i
For our demonstration case, ( )ity contains a probability of sand and non-sand (valley or
non-valley) at each location within the dataset ( )itd , which is derived from the prior
model ( )it θ)(z (see Step 2 of Figure 11). This is known as the “soft probability” for
conditioning multiple-point realizations [Caers, 2005].
Step 3 involves creating multiple conditioned Earth models to this soft probability (see
Step 3 of Figure 11). We let the snesim algorithm (Single Normal Equation Simulation)
within Stanford’s Geostatistical Earth Modeling Software (SGeMS) generate Earth
models (realizations) of lithology
Equation 32
( )( )i
i TtNiWwitwt
θ===θ ,...,1,...,1,...,1,),( yz .
Here w represents the number of realizations generated from the same soft probability
cube. By generating several conditioned Earth models, we can capture the different
62
possible Earth model interpretations that could be made from the data which stems from
the overlap in the likelihood. In addition to being conditioned to the soft probability ( )ity , these Earth models reflect the prior through the training images ( ); furthermore,
they could be constrained to any available hard data (see Caers et al, 2001 and Strebelle
et al, 2003 for aplications of this methodology to modeling sand/shale sequence in oil
reservoirs). Depending on how discriminating the data are, these conditioned models
iθ
( )( )it i θ,ywt ),(z may be very different or very similar to the prior models ( )it θ)(z from
which they are derived.
3.4.4 CREATING DATA-CONSTRAINED PROXY DECISION VARIABLES
Finally, we arrive at Step 4: obtaining the proxy decision variables from the conditioned
Earth model (see Step 4 of Figure 11). As with the prior models, the dynamic simulation
function f must be applied to the new conditioned models ( )itwt i θ,)(),( yz to get the
conditioned or interpreted proxy variable
Equation 33
( ) ( ) ( ( ))( ) WwLTtNifsi
ii twti
twt ,...,1,,1,,1,1,,, )(),(, ====θ=θ θ KlKKll yzy . i
( )( ) ( )The conditioned proxy variable itwt is θ,, yl
( ) ( )
determines the outcome of the decision,
which for the demonstration case is whether a farm is removed or not. Again, as in the
prior proxy variable, we express the outcome of these decisions in terms of value:
Equation 34
(( ) ( ) ( )( )) WwLTtNiAasgvi
iii
twtai
twta ,...,1,,1,,1,1,,1,, ,,, =====θ=θ θ KlKKKll yy
.
63
These individual values from the conditioned models are used to calculate the value of
imperfect information VII since they are derived from proxy variables that have been
constrained by the data reliability measure. In the next section, we illustrate the
application of these four steps (1-4) to the aquifer vulnerability demonstration case as
well as present the complete VOI calculation methodology.
3.5 VOI METHODOLOGY SUMMARY
We now present the complete methodology, which allows us to assess the value of a
geophysical technique that does not directly measure the decision variable. In particular,
we need to compute the VOI components of Vprior and VII. We present a VOI workflow
consisting of three parts. First, the generation of the prior models and calculation of Vprior
is described. Second, we describe how the two synthetic likelihood measures are utilized
with the prior models to make many synthetic datasets. From these synthetic datasets,
soft probability cubes are created from the datasets and used to generate conditioned
Earth models that represent possible interpretations from the data. Third, the generated
conditioned models are used to calculate VII. Lastly in this section, the methodology will
be generalized to allow several local actions to be taken simultaneously. We would like
to emphasize that this thesis and specifically this chapter is about the overall
methodology and that some of the detailed components (e.g. tracer simulation,
vulnerability definition) can be re-defined based on specific needs.
Prior model generation. The details of Part A in the schematic of Figure 14, which
depicts the entire proposed workflow, are described. The generation of the prior models
( )it θ)(z utilizes eighteen training images (N=18) to represent the model uncertainty of
the buried valley length, width and thickness dimensions. All the training images are
64
deemed equally probable: ( ) 055.01
Pr ==θ=ΘNi . Within each of these , ten binary
facies Earth models are generated (examples shown at the bottom of
iθ
Figure 10) using the
snesim algorithm [Strebelle, 2002]. Each of the 180 models has 143 x 91 x 50 grid cells
with each cell dimension being 150m x 150m x 4m.
Figure 14: Overall workflow of this VOI methodology. A) First the generation of the prior models and calculation of Vprior . B) Second, the reliability measure and how it is utilized with the prior models. C) The conditioned models are generated and used to
calculate VII.
65
Dynamic simulation f . Once the prior Earth models ( )it θ)(z are established, the
dynamic simulation function f is applied to achieve prior proxy variable ( )its θ)(l . In
order to define aquifer vulnerability, flow simulation is performed with a tracer initially
placed at all L surface locations. The permeability of two facies is assigned
deterministically: valley (sand) is 1165 mD (2.8 ft/day or 9.8E-6 m/s) while non-valley
(non-sand) is 1.1 mD (2.6E-3 ft/day or 9.2E-9 m/s). The simulation is run for 20 years
with extraction and influx boundary conditions (representing pumping wells,
precipitation and regional recharge).
Proxy decision variable. We establish thresholds of the tracer concentration that will
allow us to delineate which surface locations are major entry points into the aquifer.
Thresholds are chosen to account for and remove situations where pooling occurs at the
surface or very insignificant amounts of tracer have reached into the aquifer. These
concentration thresholds define continuous concentration bodies; Figure 15 shows an
example of a 3D view of identified concentration bodies in unique colors and the
topography as a semi-transparent surface. Locations where these concentration bodies
intersect the surface are mapped as vulnerable. The volume of the concentration body
represents the potential damage a contaminant could do if released at that surface
location . Therefore, the magnitude of vulnerability
l
l ( )its θ)(l at these surface
intersections is equal to the volume of the concentration body. The conceptual 2D cross-
section of Figure 16 demonstrates the effective entry points at the surface and the volume
of concentration bodies. Figure 17 is an example vulnerability map. More refined
vulnerability maps could be constructed including chemical and biological processes, but
this is not the focus of this work; instead the overall methodology is our contribution.
66
Figure 15: Tracer concentration thresholds are applied to flow simulation results such that each continuous tracer body is identified. Topography is depicted by the semi-
transparent surface.
67
Figure 16: Conceptualization of how vulnerability maps (the proxy decision variable) are made from the tracer concentration bodies.
Figure 17: Example vulnerability map which indicates locations that serve as entry points into the aquifer. The magnitude reflects the volume of aquifer that would be affected if a
contaminant would enter at that particular location.
68
Decision action & Vprior. ( )its θ)(l directly determines the outcome of our decision ga.
The scalar value results ( )ia θ)t(,lv of Equation 27 can be used in the expression for Vprior
(Equation 12) resulting in:
Equation 35
AavVL T
ita
N
iprior
i
,...,1)(1
)Pr(max )(, =⎟
⎟⎞
⎜⎜⎛
θθ=Θ= ∑ ∑∑θ
T tiai1 11 ⎟
⎠⎜⎝= =θ=l
l .
This prior value includes a sum over the value outcomes of the action at each location l .
This summation is outside ( )La
max because each decision at can be made
independently from other locations; recall that we want to allow an independent action a
per each location . One could remove a farm (a=1) at location and not remove a
farm (a=2) at
l
l 2=l
1=l .
Reliability & synthetic data. The second step of the workflow is represented in part B of
Figure 14. The first of two tasks for Part B is to use the data likelihood (Equation 28) and
the prior models to generate synthetic datasets ( )itd since we are considering purchasing
information that we do not yet have. The synthetic data ( )itd is generated as described in
Section 2.3. For each of our buried valley prior models, we generate one electrical
resistivity dataset for a total of 180 datasets from the likelihood of Figure 12 and 180
datasets from the likelihood of Figure 13. These electrical resistivity datasets represent
the information we could expect to collect, given our uncertainty of both the subsurface
heterogeneity (represented with the prior models) and the ability of the TEM data in
resolving lithology (represented through the two respective data likelihoods).
69
The second task of Part B uses the conditional probability of Equation 30 and datasets ( )itd to generate a soft probability cube ( )ity . For each ( )ity , two (W=2) new
conditioned Earth models ( )itwt i θ,)(),( yz are generated. This may be considered the
minimum number of conditioned models that should be generated. More conditioned
models will capture the possible variability due to the imperfect geophysical information
message. However, again, our aim is to develop the complete methodology. Ultimately,
there are two sets of 360 conditioned models, each constrained to their respective
synthetic datasets (represented through the soft probabilities ) and the spatial
constraints of the original input geologic parameters
)( ity
iθ (through the training images).
Conditioned models to attain VII. Third, we arrive at part C of Figure 14 which
represents the part of the workflow that achieves the value with imperfect information
VII. The conditioned models of our demonstration case are put through the same
workflow as the prior models in order to achieve the aquifer vulnerability ( ) ( )( )itwt is θ,, yl
(Equation 33). These results are ultimately used to obtain all the individual value
outcomes ( ) ( )( )itwta
iv θ,,, yl (Equation 34). At last, all the individual conditioned values
are used to calculate the value with imperfect information
Equation 36
( ) ( )( ) AavVL N T W
itwt
iII
i
,,1,1
max1
)Pr( , K=⎟⎞
⎜⎛
⎟⎞
⎜⎛
θθ=Θ= ∑ ∑ ∑ ∑θ
yWTi t w
aai1 1 1 1
,l
l ⎟⎠
⎜⎝
⎟⎠
⎜⎝= = = =θ
( )
.
Chronologically, we first obtain the W possible interpretations of the proxy variable
( )( )i
twt is θ,, yl through the conditioned models before we make our decision. Therefore,
we can choose the best decision action on average for those interpretations, represented
70
by . Then the expected value from all the models for location is taken. Note that,
unlike
amax l
Equation 16, this VII is not weighted by probabilities derived from a reliability
measure. Instead, the reliability is captured in imperfect simulated data ( )itd . The
conditioned models ( ) ( )( )itwt i θ,, yz account for the possible inaccuracies of the
geophysical message to inform about lithology. The imperfect data will have value if
they can resolve the proxy variable and lead to decisions with a higher value outcome
than the Vprior.
Notice that in the case of perfect information, ( ) ( )( )itwt
aiv θ,,
, yl will be equal to
( ) (( ) )it iv θ,lwta,, y
( )
, as all prior models will be perfectly recovered through the data into
( )( it is θ,l )wt ,
y . Therefore, we are assured that the best possible decision is made given
our prior models. Whereas with data that has no information content, the interpreted or
conditioned model will create a proxy variable that poorly represents the true Earth
response and will be quite dissimilar from each other. Therefore decisions made on
“inaccurate interpretations” will lead to lower value outcomes on average (as the W very
different outcomes ( ) ( )( )itwta
iv θ,,, yl will be averaged) and will have a lower best
alternative compared to interpretations that are more similar to each other. Better quality
data will ultimately lead to higher valued decision outcomes and consequently, a higher
VOI.
3.5.1 GENERALIZATION TO COMBINATIONS OF ALTERNATIVES
Finally, the restriction of each location l being considered independently is eliminated,
such that several actions can be taken simultaneously at several locations. We
introduce variable c which represents different possible decision and spatial combinations
l
71
for the multiple l locations, with Κ total combinations deemed or identified as possible
and valuable. The notation of where acg { }Laa ,...,1=a denotes which action is taken at
which location, with including all the L possible locations at which a decision
action a may be made. Suppose that there are three locations where it is possible to take
a decision action (L=3), two unique combinations of the spatial and decision actions are
considered ( Κ =2) and there are two possible decision actions ( ) such that
L,,1Kl =
2A =
{ }removetdon,removea∈ ' . The two possible decision-spatial combinations are deemed
{ }2,1 a,1:1 === a =ac and { }2,2,1:2 ==== aaac . The notation g would indicate
that farms at and would be removed, and the farm at would not be
removed; whereas indicates that only the farm at
1=c
3=1=l 2=l
2=cg
l
1=l would be removed. With
the ability to apply combinations of different actions, Vprior now becomes
Equation 37
Aa ,...,1,1 =aaL }c ,,1 Κ=⎟⎟⎟
⎠
⎞Kv t
c)(aT
V ic
prior )max θ=Θ=a
ac
T
t
i
1∑
θ
=i
i
)(1
θθ
N
i⎜⎜⎜
⎝
⎛∑=
,K{=aPr(1
such that the best we can do with the present uncertainty is the highest valued spatial-
decision combination on average over all ∑=
θ
N
ii
T1
v Earth models. The value with
imperfect information becomes
Equation 38
( )W
( )( ) c Aaa aLvT
VN
i
T
tII
i
i
Pr(1 1
Θ= ∑ ∑= =
θ W
w
w1
1∑
=y i
t i ,...,1, =⎟⎟
⎠
⎞θt
c,ac
max⎜⎜
⎝
⎛
aiθ= ,,1{ 1 K== },K,
1)
θΚ a
)( itd
.
If the proposed information, represented synthetically with , can constrain the results
of the dynamic simulation function and subsequently the proxy decision variable, then
72
this imperfect information may have value. The degree of “constraining” is measured by
estimating VOIII = VII - Vprior.
3.6 APPLICATION OF VOI CALCULATION TO THE DEMONSTRATION CASE
To keep the example concise and clear, we assume the only possible decision alternatives
are concerned with which contaminant sources (e.g. farms) should be removed or not.
Thus there are only two alternatives (A=2) which are remove or don’t remove. With
these two alternatives, there are four possible outcomes at each surface location
depending if its vulnerability
l
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
==
=>
==
=>
=
)'(2,00
)'(2,02
)(1,01
)(1,01
cos
)(
)(
)(
)(
)(,
removetdonasif
removetdonasif
removeasif
removeasif
t
t
t
t
t
ta
l
l
l
l
l
where we express these outcomes in nominal, unitless costs. It is more straightforward to
use costs for these outcomes, but the value definition will be expressed in terms of
negative costs (value = -cost). The first two cost outcomes, when a farm is removed
regardless of its vulnerability, are deemed a unit cost (possibly representing the cost of
farmer compensation to stop using chemicals). The third cost outcome (when farm is not
removed on an effective entry point) is assigned a cost that is twice as much as
compensation, representing the environmental consequences. The last cost outcome has
no cost as no removal is taken at a location deemed not vulnerable. Table 3 contains the
VOIII results with these cost outcomes in column 1. The results for the two different
reliabilities are shown in the two rows. As expected, the data generated with the
73
reliability of Figure 13 has less value than that of Figure 12, which has a better electrical
resistivity contrast between the two lithologies. These VOIII results assume that the
remove or not decision can be made independently at each of the L= 7,776 surface
locations of the model. It is assumed that a farm exists at each one of these locations and
therefore, as seen in Equation 36, a sum is made over all the L locations, which account
for the VOIII results being on the order of 50 whereas the costs are 1 and 2. Again, if
only certain combinations of farm removal were deemed possible, this could be
accounted for using Equation 37 and Equation 38.
Table 3: VOIII Results for Aquifer Vulnerability Demonstration Case
VOIII = VII-Vprior
Costs = {1,1,2,0}
Costs scaled by vulnerability: b=0.5
Costs scaled by vulnerability: b=1
Data Reliability 1
(Figure 12) 57.56 4,259.4 4,522.0
Data Reliability 2
(Figure 13) 51.655 4,210.7 4,426.4
Now, we consider costs that utilize the magnitude of the proxy decision variable of
vulnerability. The vulnerability measure represents the magnitudes of the estimated
damage to the groundwater resources. We can reward decisions to remove farms when at
vulnerable locations by representing the costs savings with negative costs scaled to the
vulnerability.
74
⎪⎪⎪⎪
⎩
⎪⎪⎪⎪
⎨
⎧
==
=>
==
=>−
=
)'(2,00
)'(2,0
)(1,01
)(1,0
cos
)(
)()(
)(
)()(
)(,
removetdonasif
removetdonasifs
removeasif
removeasifsb
t
t
tt
t
tt
ta
l
ll
l
ll
l
The first cost outcome scales to the negative vulnerability through some scaling factor b.
The third cost outcome (when a farm is not removed on an effective entry point) is
equivalent to magnitude of damage done to the aquifer. The second and third columns of
Table 3 contain the VOIII results for this cost structure, where the second column scales
the negative vulnerability by 0.5 and the third by 1. Generally, by weighting more by the
vulnerability, the value of imperfect information increases. The VOIII scaled by
vulnerability is two orders of magnitude greater than the VOIII computed in the first
column of Table 3 because the aquifer vulnerability ranges from zero to 500. And again,
as expected, “good” data (Figure 12) have a higher VOI for all three cost outcome
structures (Table 3). We may expect to see this difference increase for W>2, as we
would be better representing the uncertainty in the data likelihood. In all cases, the VOIII
was positive. Therefore, under the assumptions made in the demonstration case, it would
be a sound decision to purchase the TEM data as long as the cost of acquisition was less
than the VOIII.
For this demonstration case, we make a significant simplifying assumption that all sand is
valley facies, and that vulnerability is driven only by the buried valley structure (i.e. no
sub-grid cell features exist that affect aquifer vulnerability). This lithology-facies
assumption is extended into the geophysical reliability measure, such that we assume
TEM’s ability to distinguish sand/clay lithologies means it also can exclusively identify
valley/non-valley facies. Therefore, the two generated likelihoods do not describe how
the geophysical data discerns the complex geologic input parameter iθ (describing the
75
dimensions of the buried valley) but the geologic indicator of lithology. This is because
of the nature of the decision itself. The demonstration case requires local flow response
information which is not deterministically captured in the iθ alone.
76
CHAPTER 4: DISCUSSION AND FUTURE WORK
4.1 SUMMARY OF THE TWO METHODS
The two VII (value with imperfect information) equations of Chapter 2 and Chapter 3
illustrate the common thread of this thesis as well as and the differences in how the two
methodologies handle the information reliability:
Equation 39: VII of Chapter 2
( ) ( ) ( ) AavVN T
ita
N
jijII
i
,...,11
|PrmaxPr )(intint =⎟⎟⎞
⎜⎜⎛
⎟⎟⎞
⎜⎜⎛
θθ=Θθ=Θθ=Θ= ∑ ∑∑θ
Tj tiai1 11 ⎟
⎠⎜⎝
⎟⎠
⎜⎝= =θ=
Equation 40: VII of Chapter 3
( ) ( )( ) AavWT
VL N
i
T
t
W
wi
twtaa
iII
i
i
,,1,1
max1
)Pr(1 1 1 1
,, K
ll =⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛θθ=Θ= ∑ ∑ ∑ ∑
= = = =θ
θ
y
amax
.
Within the innermost brackets of the two expressions is an average of the “interpreted”
results. In Equation 39, this is an average of value outcomes for the different alternatives,
weighted by the conditional probabilities that represent the information content of the
data on the geological input parameter θ. Whereas in Equation 40, this is an average of
the value outcomes for the different alternatives evaluated on models that were
conditioned to the data using the information content as pre-posterior probabilities in the
conditioning method (tau-model). From these interpreted results, the best alternative
( ) is taken from all interpretations. The averages outside these inner brackets
represent the additional averages for the geologic input parameter.
77
The reliability is more apparent in Chapter 2’s VII (Equation 39), as it is explicit in the
conditional probability and is used to weight the prior value outcomes. This tells us that
only the prior models were necessary for this work. The effort of Chapter 2 was making
geophysical interpretations on all the prior models. In contrast, Equation 40 of Chapter 3,
new posterior models were “drawn” to generate posterior value outcomes.
This thesis offers two quantitative ways to achieve the information reliability measure.
For some situations, this may be more appropriate than subjective expert interviews that
translate experience into probabilities. Forward modeling may avoid biases from some
experts. However, there may be times when these expert interviews are more suitable.
Perhaps an expert has worked with a lot of data in specific circumstances that are similar
to the proposed area and information. This kind of experience may be better translated
from the expert himself rather than in the forward models. Additionally, some decision
makers may not have the accurate codes that would capture the challenges. An expert
could have a better sense for these conditions that can’t be well emulated numerically.
4.2 DISCUSSION OF CHAPTER 2 METHOD
VOI quantifies how much a new information source would improve our chances of
making a better decision than made without it. VOI has the potential to be a powerful
tool for decision makers who are considering purchasing this new information. However,
any VOI calculation is only as realistic as the prior model of uncertainty on the studied
subsurface heterogeneity as well as the reliability measure of the measurement technique.
The methodology to create Earth models (utilized in both Chapters 2 and 3) is flexible to
many prior modeling approaches and is expandable to include many subsurface qualities
or features (i.e. variograms, histogram, training images, etc). Chapter 2 provides a
78
repeatable quantitative and Monte Carlo simulation-based methodology to achieve a
reliability measurement based on the resolving power of a particular geophysical
technique. Instead of generating many datasets to approximate a posterior distribution,
we use these datasets to make interpretations and assess the correctness of all the
interpretations, thereby including “geophysical interpretation” as part of VOI. If the
geophysical calculation is computationally less expensive than the decision action
calculation (which is true in the Chapter 2 example since the geophysical forward model
is 1D and the decision action is 2D flow simulation), then this technique is
computationally feasible. With the reliability measure in hand, the decision action
calculation is only performed for the prior models and the geophysical reliability measure
is used to calculate arithmetic averages for the value of imperfect information.
Otherwise, all decision actions A must be performed on all conditioned models (which
reflect the information gained from the simulated datasets) in addition to the prior
models.
Chapter 2 attempts to achieve a realistic estimation of VOI by proposing comprehensive
prior Earth modeling schemes along with a physics-based reliability measure. However,
limitations still exist. For any VOI calculation, the possibilities (state variables) affecting
the decision must be clearly defined and framed. Howard (1966) recommends a
sensitivity study to identify the essential state variables: the uncertain elements that have
the greatest influence on a decision outcome. For subsurface problems, this requires
coordination between the expert(s) on possible subsurface states and the modelers of the
sensitivity study. It is the responsibility of the local geologist to identify what is possible
and probable in the subsurface given the location’s depositional and structural history. It
is optimistic to assume (especially from the geologist’s point of view) that the subsurface
clearly classifies into different categories. However, this is the responsibility of a
sensitivity analysis: to identify which subsurface elements are most impacting the
decision. Therefore, the prior modeling requires close collaboration between the
79
geologist and the modeler to establish satisfactory geological categories .
Additionally, the sensitivity study would ideally identify the appropriate categories that
must be established for any continuous variable included in the prior modeling.
},...,{ 1 Miiθ
A significant limitation resides in the interpretation techniques’ ability to classify
complicated Earth models. Once rock physics relationships, physics of the measurement
and noise are included, it may be difficult or impossible to distinctly classify a
geophysical result into one of the initial },,{ 1 Mii Kθ . If the geologic input parameter of
orientation is established with two directions, it is possible that the interpretation could
result in a third direction that was not originally represented ( ). Sophisticated
interpretation techniques are required to analyze multi-dimensional subsurface features.
These problems could be alleviated by taking care to only include geologic input
parameters that impact the decision outcomes, thus reducing the complexity of the Earth
models.
Θ∉Θint
The premise of Chapter 2 is that the decision variable – the uncertain parameter deemed
to most influence the decision outcome – is identified as the geologic input parameter.
However, we can imagine how information about the property of interest at particular
locations would be important, i.e. resolving the spatial uncertainty. For the example
problem given, the property is lithology (channel or background facies) and this
information would likely influence the recharge actions. Therefore, another information
reliability may need to be described in terms of the information’s success at resolving this
property, and not just the ability of the geologic input scenarios to be interpreted
correctly.
80
eli
Finally, another significant assumption is that the physics of the geophysical technique is
simulated as accurately as possible for the forward mod ng )(⋅f . However 3D
modeling and inversion codes are often not available or too CPU demanding to be
repeated on a large set of models. This proposed methodology requires i
TCtotal θ×
generated forward responses (one for each of the Earth models). Efficient methods for
clustering the Earth models using distance and kernel methods (e.g. Scheidt and Caers,
2009) can help to reduce the computational demands in VOI calculations for spatial
decisions by eliminating geologic input parameters that do not impact the decision
outcome.
4.3 DISCUSSION OF CHAPTER 3 METHOD
The methodology of Chapter 3 does not assume that the decision outcome is
deterministically related to iθ . Instead, some decisions will involve the analysis of a
complex local response to some reaction of the Earth to a local or global “stress.”
Therefore, the information reliability can not be acquired just from forward models of
geophysics as seen in the previous work. Our methodology addresses the reliability of
the data by proposing comprehensive prior Earth modeling schemes along with rock-
physics likelihood relationships. Our focus is an estimation of VOIII for situations
necessitating proxy decision variables.
Although the methodology is presented using the aquifer vulnerability demonstration
case, it is flexible to other Earth science modeling schemes and decision problems. The
geologic parameter Θ could also describe the reservoir volume or fault structure type
which could affect oil production and development decisions. Other possible
applications that could apply this methodology are enhanced oil recovery scenarios with
81
mline simulation or a geomechanical response to
achieve location specific information.
geophysical data inversion to get earth resistivities from recorded elatctromagnetic fields.
ponses may not be
captured in the proxy variable and may need to be modeled explicitly.
water injection or a geomechanical model for subsidence problems or carbon storage at
different locations within the subsurface. We can imagine how the dynamic simulation
function could be modeled with strea
However, the methodology could be computationally expensive depending on the
dynamic simulation function. The methodology requires that the dynamic response be
simulated on all prior and conditioned models to capture the proxy decision variable. For
certain applications, more conditioned models may be needed to acheive a pragmatic
VOIII measure. For this demonstration case, it was already computationally expensive to
run 360 3D flow simulations for 20 years each. Additionally, this methodology does not
consider the errors that could result due to limitations related to the volume of support or
depth of investigation of the geophysical technique. The generation of synthetic datasets
only relies on the imperfect relationship between the geophysical attributes and geologic
indicators; it does not incorporate how the geometry or physics of the measurement may
distort the message as well. The synthetic data does not capture the actual process of
The last assumption that different combinations of actions can be evaluated on the same
proxy variable also depends on the dynamic simulation function and the proxy decision
variable. In the case of aquifer vulnerability, the proxy variable will allow us to
reasonably evaluate the outcome of these combinations (removal and non-removal). An
example where the proxy variable may not reasonably evaluate different combinations of
decision alternatives is when the alternatives represent different pumping or injection at
different locations and/or at varying rates. The interaction of these res
82
For some Earth science decisions, knowledge of the geologic input parameters or “model
uncertainty” will aid in decisions (e.g. don’t drill when iθ = small (uneconomic) oil
h
interpretation of the general heterogeneity
reservoir). Chapter 3 addresses situations w en we can not rely on knowledge or
iθ to make a sound decision. We have
defined a VOI methodology that accounts for such situations. Since the decision is based
on the simulated dynamic response on the Earth models, we introduced the proxy
decision variable which contains location-specific results of the dynamic response. In
conjuction, we proposed geostatistical simulation using synthetic datasets derived from
likelihoods to achieve conditioned proxy decision variables. Finally we described how
these two approaches together can achieve a value of imperfect information metric for
geophysical data. Specifically, we have shown how this methodology applies to our
demonstration case of aquifer vulnerability.
4.4 FUTURE WORK
In both Chapter 2 and 3 it was assumed that a sensitivity analysis had been performed to
identify the geologic input parameters that most mattered or influenced the decision
outcome. This work (done perhaps through collaboration of a geologist and a modeler)
establishes satisfactory prior models. Although no details were given on how this could
be achieved for Earth models made from multiple-point geostatistical methods, future
studies could design efficient methods for this work specifically for value of information
problems. Distance methods have proven to be effective tools in describing the
differences between models’ response (such as flow or seismic) [Ripley, 1996; Scheidt
and Caers, 2009; Suzuki and Caers, 2008]. Unlike experimental design schemes, this can
be used on categorical variables, such as choosing between different training images. In
the demonstration case of Chapter 3, the distance could represent how similar or
dissimilar the vulnerability maps are for each model. In Chapter 2, the response would
83
en allow for a focused information
reliability study: can the data source help us resolve that this feature is present or
inate between the different possible features?
mics along with thorough spatial
modeling would make VOI calculations incredibly useful for gaining public or private
aking using data collection.
situation, you may have the ability to adjust your situation by taking action, i.e. making
be the volume of fresh water in a model after either a particular artificial recharge scheme
or no recharge had been performed. Multidimensional scaling (MDS) and cluster
analysis offer methods to visualize which model features or characteristics are the drivers
of the response of interest. This knowledge would th
discrim
As stated at the end of Chapter 2, VOI can be viewed as the interaction of three
components: the reliability of the information, the chance of making a poor or suboptimal
decision, and the magnitude of the decision. The second and third VOI factors were
deemed outside of the scope of this thesis but the sensitivity of these to VOI could be
included in future work. Again, using sensitivity analysis techniques, the VOI sensitivity
to other elements such as the costs and/or values assigned to different outcomes of the
decision could be investigated. Nominal economic values have been chosen for values
and costs. Realistic values/costs could be obtained through experts (e.g. economists).
From all this analysis, the most impacting uncertainties could be ranked in their order of
degree of influencing the final VOI estimate. The most impacting uncertainty may be
either the geologic heterogeneity, the information reliability, the alternatives, or the
assigned outcome values or costs. Realistic econo
confidence in decision-m
Another important consideration is the inclusion of “flexibility” in the decision-making
process. Flexibility describes the ability (and necessity) to make sequential decisions.
For example, after making a decision and realizing its outcome, depending on the
84
Bratvold, 2004]. This must be included in the VOI calculation to achieve realistic results.
tion problem by maximizing
the data’s success rate at identifying the decision variable.
another decision. This flexibility or ability to react must be taken into account in the
original decision, as it will affect your preferences to certain outcomes [Begg and
In this thesis work, the role of optimization has not been utilized. There are several
opportunities where optimization could improve the decision process and the value of
information calculation. The opportunities to maximize VOI would come from the
aforementioned three inter-related factors that determine VOI: the information’s
reliability to detect or determine the decision variable, the chance of making the
suboptimal decision and the magnitude of the decision (the variability of costs and prices
that influence the value outcomes). To maximize reliability, the optimization would need
to consider which parameters of the data collection are adjustable and how they could
affect the resolution of the decision variable. For example, the flexibility to choose the
spatial and temporal sampling, or noise reduction through stacking or processing could be
parameters included in an information reliability optimiza
The chance of suboptimal decision is captured in the difference between the value
outcomes for different identified alternatives. One of the most useful characteristics of
decision analysis is its ability of finding creative solutions by discovering the best
decision alternatives (those with highest value outcome). Therefore, instead of assuming
a small set of alternatives are possible for any Earth model (as was the case for the
examples presented in this thesis), optimization could be performed on each Earth model
to discover the best alternatives for that model (maximizing the value outcomes). For the
some Earth science decision, this may be choosing the optimal locations for both
pumping and injection wells to maximize oil production; Onwunalu and Durlofsky
85
re, a VOI including
optimization on alternatives will be a more thorough assessment.
nd prices. These optimization results could then be used in the
final decision analysis.
considered reliable. Of course, if the model attribute is not the decision variable (as seen
(2010) explore the efficiency of two different optimization methods for this kind of
decision. Techniques such as these would fit well with a decision analysis study, as they
efficiently identify alternatives in a high dimensional search space which is likely in
spatial decisions. With the best alternatives identified with optimization, the VOI
estimation will better capture this risk (as opposed to situations where a small number of
alternatives are determined for all Earth models), and therefo
Finally, the magnitude of the decision entails the costs and prices which directly affect
the value outcomes of different alternatives. The costs and prices could also be free
parameters included in an optimization scheme, which would influence the difference
between alternatives (the chance at make a suboptimal decision). Depending on the
decision problem, it may be too complicated to include these as free parameters, and
instead separate optimizations could be performed on the perceived 10th, 50th and 90th
percentile of the costs a
Optimization may also play a role in the VOI methodology in either geophysical
inversion or history matching. Stochastic inversion of synthetic geophysical data using
Markov Chain Monte Carlo techniques draw many possible models from a prior and
evaluate the model’s likelihood of occurring. The resulting “chain” of accepted models
represents the posterior model space. The frequency of each model represents the
probability of occurring given the synthetic data. This method could give a reliability
measure by comparing the chain of models to original synthetic model. If the most likely
model in the chain is close to original model then the geophysical method could be
86
in Chapter 3), another connection would have to be taken into account in order to utilize
these inversion results to achieve an information reliability.
History-matching is also an inversion: it attempts to match petrophysical models to
observed production (fluid flow) data. History matching can also be performed non-
deterministically such that many models can be obtained that equally fit the observed
data. Like the stochastic geophysical inversion, the uncertainty measures on these
history-matched models can be used directly in the decision analysis framework. With
regard to VOI, one question to ask is: Does the set of models that do match historic data
really help our predictions (versus or compared to the entire prior model set)? Again,
using a synthetic, forward-modeling experiment, we could evaluate the reliability of the
history-match models to improve our predictions of future performance of the reservoir.
Specifically, what is the rate at which predictions improve by ensuring models match the
current information versus the rate of those that don’t match the data. Additionally, are
there certain situations (i.e. geologic heterogeneity, flow conditions, number or types of
wells) where the reliability of history matched-models is better? Ultimately, we could
understand which methods or conditions that may make history matching more useful in
terms of decision-making.
87
APPENDIX A: DETAILS OF TEM MEASUREMENT & MODELING
In Chapter 2, the models ( )it θ)(z must contain electrical resistivity R, the property
indirectly measured by TEM. Archie’s Law [Archie, 1942] is used to transform the sand
facies in resisitivity:
watermwater
sandbulkS
RR
φ=
where the sand is assumed to be fully saturated (Swater= 1), Rwater is the resistivity of
the pore water and m represents the cementation exponent for the model. For sand, this m
is typically 2. And a modified Archie’s law is used for the non-sand (clay) facies [de
Lima and Sharma, 1990]:
( ) ⎥⎦⎤
⎢⎣⎡ −
−+= swaterclaybulk Rfmm
Rf
R 11
1
where the cementation exponent m and the clay particle resistivity Rs are chosen
according to the clay type present and f is related to the usual formation factor 11−
−
φ= mf .
For the example demonstrated in Chapter 2, the following values were used for the two
forms of Archie’s Law:
146.0
07.1
5
=
=
Ω=
s
clay
water
R
m
mR
88
which results in sand and clay facies having 80 and 13 ohm-m respectively. These
lithological channel models represent the aquifer layer. An overburden layer of varying
thickness and resistivity is added above and basement layer below this layer.
How the TEM method works and measures the earth’s resistivity structure can be
summarized into six steps with the appropriate Maxwell equation2:
1) The constant current Iwire in the transmitter produces a primary magnetic field Hp:
wirep IH =×∇
2) Current suddenly terminates; the changing magnetic field induces a secondary
electric field Es in the earth that attempts to oppose the change (where μ is
magnetic permeability of the Earth): ( )t
HE ps ∂
μ∂−=×∇
3) The induced electric field produces an image current, which in turn produces a
secondary magnetic field Hs (where σ is electrical conductivity of the Earth):
ss
ss
IH
EI
=×∇
σ=
4) Current diffuses outward and downward over time (t). The diffusion rate:
a. is proportional to t
1
b. depends on earth conductivity ( σ )
2 modified from lecture notes of D. Alumbaugh, University of California, Berkeley
89
5) The diffusing, time varying current produces time varying secondary magnetic
field.( )
t
I
t
H ss
∂∂
=∂×∇∂
6) This decaying magnetic field Hs produces a time-varying secondary magnetic
field:( )
∫ ⋅∂μ∂
−=∇ dsnt
HV s ˆ
Noise modeling was performed according to Equations 1 and 2 of Auken et al, (2008).
The following values were used:
1ms at noise 8 9−=b
airborne for 400 2mAmpmoment −=
based‐land for 3 2mAmpmoment −=
03.0=unistd
5.0
310*
−
− ⎟⎠
⎞⎜⎝
⎛=
tbVnoise
( ) VV
VstdNVV noise
uni ⋅⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+⋅+=
5.0221,0~ where V is unperturbed synthetic data
90
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