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A methodology for quantifying the value of spatial information for dynamic Earth problems

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Abstract

We develop a methodology for assessing the value of information (VOI) from spatial data for groundwater decisions. Two sources of uncertainty are the focus of this VOI methodology: the spatial heterogeneity (how it influences the hydrogeologic response of interest) and the reliability of geophysical data (how they provide information about the spatial heterogeneity). An existing groundwater situation motivates and in turn determines the scope of this research. The objectives of this work are to (1) represent the uncertainty of the dynamic hydrogeologic response due to spatial heterogeneity, (2) provide a quantitative measure for how well a particular information reveals this heterogeneity (the uncertainty of the information) and (3) use both of these to propose a VOI workflow for spatial decisions and spatial data. The uncertainty of the hydraulic response is calculated using many Earth models that are consistent with the a priori geologic information. The information uncertainty is achieved quantitatively through Monte Carlo integration and geostatistical simulation. Two VOI results are calculated which demonstrate that a higher VOI occurs when the geophysical attribute (the data) better discriminates between geological indicators. Although geophysical data can only indirectly measure static properties that may influence the dynamic response, this transferable methodology provides a framework to estimate the value of spatial data given a particular decision scenario.
1 23
Stochastic Environmental Research
and Risk Assessment
ISSN 1436-3240
Stoch Environ Res Risk Assess
DOI 10.1007/s00477-012-0619-4
A methodology for quantifying the value
of spatial information for dynamic Earth
problems
Whitney J.Trainor-Guitton, Tapan
Mukerji & Rosemary Knight
1 23
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ORIGINAL PAPER
A methodology for quantifying the value of spatial information
for dynamic Earth problems
Whitney J. Trainor-Guitton
Tapan Mukerji
Rosemary Knight
Ó Springer-Verlag 2012
Abstract We develop a methodology for assessing the
value of information (VOI) from spatial data for ground-
water decisions. Two sources of uncertainty are the focus of
this VOI methodology: the spatial heterogeneity (how it
influences the hydrogeologic response of interest) and the
reliability of geophysical data (how they provide informa-
tion about the spatial heterogeneity). An existing ground-
water situation motivates and in turn determines the scope
of this research. The objectives of this work are to (1)
represent the uncertainty of the dynamic hydrogeologic
response due to spatial heterogeneity, (2) provide a quan-
titative measure for how well a particular information
reveals this heterogeneity (the uncertainty of the informa-
tion) and (3) use both of these to propose a VOI workflow
for spatial decisions and spatial data. The uncertainty of the
hydraulic response is calculated using many Earth models
that are consistent with the a priori geologic information.
The information uncertainty is achieved quantitatively
through Monte Carlo integration and geostatistical simula-
tion. Two VOI results are calculated which demonstrate that
a higher VOI occurs when the geophysical attribute
(the data) better discriminates between geological
indicators. Although geophysical data can only indirectly
measure static properties that may influence the dynamic
response, this transferable methodology provides a frame-
work to estimate the value of spatial data given a particular
decision scenario.
List of symbols
h Geologic Input Parameter (e.g. training image)
i Index of training images
N Total number of training images
z Vector of Earth parameters
t Index of realizations
T
hi
Total number of realizations for training image i
s Aquifer vulnerability
Surface location
a Decision alternative
g
a
Decision predictor (e.g. flow simulation)
v Value: metric to define outcome of decision
d Synthetic data
y Soft probability (pre-posterior)
Q, q Electrical resistivity
Litho Lithology
c Decision alternative combinations
1 Introduction
Groundwater managers are increasingly faced with deci-
sions that impact the sustainability of groundwater
resources. We refer to a class of these decisions as spatial
decisions, where a spatial decision is defined as any deci-
sion whose outcome is influenced by the spatial distribu-
tion of some property. A common theme in many of these
spatial decisions is the lack of adequate spatial information
W. J. Trainor-Guitton (&)
Lawrence Livermore National Laboratory, 7000 East Ave,
L-231, Livermore, CA 94550, USA
e-mail: trainorguitton@llnl.gov
T. Mukerji
Energy Resources Engineering, Stanford University,
367 Panama Mall, Room 65, Stanford, CA 94305, USA
e-mail: mukerji@stanford.edu
R. Knight
Department of Geophysics, Stanford University, 397 Panama
Mall, Room 65, Stanford, CA 94305, USA
e-mail: rknight@stanford.edu
123
Stoch Environ Res Risk Assess
DOI 10.1007/s00477-012-0619-4
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about the groundwater system, or hydrologic process, of
interest. With data typically acquired through the drilling
of boreholes, there is inevitably a limited understanding of
the spatial variability in subsurface properties and pro-
cesses. The value of information (VOI) metric from deci-
sion analysis can be useful for determining how and where
to acquire additional data that could provide support for the
spatial decision-making process.
Of interest in our research is the development of a VOI
framework that can be applied to spatial decisions related
to groundwater management where geophysical data are
used as the source of information about the subsurface
directly relevant to the spatial decision. The motivation for
this research is a groundwater contamination situation in
Northern Europe, impacting a particular area that relies
solely on its groundwater sources to supply drinking water.
Over the past several decades, the aquifers have been
compromised by surface-sourced contaminants due to
farming activities. Contamination will continue to be a
threat until the surface locations, i.e. the farms, that serve
as entry points into the aquifer are identified and the source
of contamination removed. The decision is to determine
which farms need to be compensated to cease their use of
pesticides and other chemicals. Efforts are currently
underway to use airborne electromagnetic data to map out
near-surface clay content, as a way of gauging the vul-
nerability of the aquifer. The geophysical data are thus the
key source of information in this decision.
In this situation, the lack of knowledge of the subsurface
directly impacts the land-use decision. Additionally, this
example only has two stakeholders: the farmers and the
groundwater regulatory organization which represents the
interests of those obtaining water from the aquifer. Both of
these considerations determine our focus and our main
research objective: the development of an application of
VOI that accounts for spatial uncertainty. Specifically, we
consider two main challenges (1) representing the uncer-
tainty in spatial heterogeneity which largely determines the
outcomes of groundwater decisions and (2) accounting for
the accuracy of the information that can be acquired to
characterize the relevant spatial heterogeneity.
The first challenge is to address the way in which
uncertainty about the spatial properties of the subsurface
can make it impossible to accurately predict the outcome of
many spatial decisions. For our example, the prediction is
whether or not the contaminants at the surface at the farms
can reach the groundwater. We define our uncertainty
about the spatial properties of the subsurface in terms of
aquifer vulnerability, where aquifer vulnerability indicates
whether a groundwater aquifer is vulnerable to being
contaminated by a surface source at a particular location
(Thomsen et al. 2004). The second challenge involves
obtaining the statistics on the accuracy of the considered
information, often referred to as the data likelihood or
reliability measure. The reliability provides a probabilistic
relationship between the information message (the data)
and the state variables of the decision (the subsurface
properties which control the hydrogeologic response).
Obtaining a meaningful reliability measure is not a trivial
exercise, especially since this must be obtained before the
proposed data are collected. The methodology developed
here will address both of these challenges, thus providing a
VOI approach that includes an assessment of spatial
uncertainty in both the heterogeneity modeling and in
the information reliability. Previous VOI studies lack the
evaluation of spatial uncertainty in the heterogeneity of the
subsurface properties or in the information reliability.
There are examples of VOI in the hydrogeologic liter-
ature which include spatial heterogeneity in their decision
analysis; however, none provide an approach for estimating
the reliability of a particular information source. Reichard
and Evans (1989) present a framework for assessing the
role of groundwater monitoring in reducing exposure to
contamination. The information reliability here is the
accuracy of detecting arsenic in groundwater (a scalar
parameter). No methodology for obtaining this accuracy
measure is presented, and no spatial dependence is inclu-
ded in the 1D contaminant transport modeling. Wagner
et al. (1992) and Feyen and Gorelick (2005) present VOI
studies that include uncertainty about spatial heterogeneity;
however, neither suggest a methodology for estimating
reliability of an information source. In Wagner et al.
(1992), the ‘information’ is represented by the different
deterministic and stochastic formulations of the hydraulic
conductivities, with an assessment of how their respective
uncertainty measures affect the decisions made and
resulting outcomes. Similarly Feyen and Gorelick (2005)
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. Neither Wagner
et al. (1992) nor Feyen and Gorelick (2005) specify a
measurement technique, and thus, there is no analysis of
how the accuracy of a measurement would affect the value
of information.
VOI has been used in decisions in petroleum engineer-
ing related to subsurface fluid flow. Bratvold et al. (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 reli-
ability, and 11 of these 13 used the subjective expert
interview method (Coopersmith et al. 2006).
Examples exist in the geophysics literature which provide
quantitative measures to estimate the information reliability.
However, spatial uncertainty has not been included explicitly
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in previous assessments of the reliability of geophysical data.
Houck (2004), Houck and Pavlov (2006), and Houck (2007)
all use 1D reservoir models to evaluate the value of seismic
amplitude data, controlled-source electromagnetics (CSEM)
and 4D seismic data respectively. Since these examples do not
provide a framework for including spatial heterogeneity, they
are not directly applicable to spatial decisions. Eidsvik et al.
(2008) introduce statistical rock physics and spatial depen-
dence within a VOI methodology for the decision of whether
or not to drill for oil. Spatial dependence of saturation and
porosity is included in the grids representing the reservoir
through a covariance model. Likelihood models link reservoir
properties and geophysical attributes and serve as the reli-
ability for both CSEM and seismic data. But these likeli-
hood models alone do not allow for the inclusion of
spatial uncertainty. Additionally, how the spatial structure of
porosity and saturation would influence the flow of oil is not
modeled. Bhattacharjya et al. (2010) present a VOI method-
ology for spatial decisions, where the spatial dependence of
reservoir sands and shales are modeled as a Markov random
field, and the value of seismic data is estimated for informing
drilling decisions. But again, the flow of oil is not modeled.
Our example problem of aquifer vulnerability requires
that the spatial heterogeneity be included both in predicting
the decision outcome and in determining the information
reliability. First, we present a flexible methodology for rep-
resenting the prior uncertainty of spatial heterogeneity,
which is used in the decision analysis framework. With this
methodology, the uncertainty of the dynamic hydrogeologic
response (deemed important or related to the groundwater
decision) can be captured. We then address the need for a
quantitative measure of information reliability that includes
the spatial geologic heterogeneity and the dynamic response.
The information reliability is determined quantitatively
through Monte Carlo integration and geostatistical simula-
tion. Finally, we provide a methodology to compute value of
imperfect information (VOI
II
) for spatial data by combining
the first two components of this study. Our complete meth-
odology is demonstrated using a synthetic case, based on the
true situation described above, where knowledge of aquifer
vulnerability is needed to make informed decisions. This
case poses a significant challenge for achieving a VOI metric
as no information source directly or indirectly measures
aquifer vulnerability. Our proposed VOI methodology can
be applicable to other similar scenarios.
2 The decision scenario and prior spatial uncertainty
2.1 Concepts of the value of information
In the founding work of Howard (1966a), a decision is
defined as an allocation of resources (e.g. some type of
financial commitment). Decisions with highly uncertain
outcomes (like most of the spatial decisions in the Earth
sciences) motivate the field of decision analysis. In deci-
sion analysis, a distinction is made between a good deci-
sion and a good outcome. A good outcome is an outcome
that is highly desired or valued by the decision makers,
whereas a good decision is one that identifies the decision
alternative deemed to have the highest expected value
(Clemen and Reilly 2001; Bratvold and Begg 2010).
Figure 1a demonstrates the basic elements of our exam-
ple decision in a decision tree: the decision alternatives, the
uncertainty, and the outcomes. There is a need to decide, for
each farm, between two decision alternatives: (1) compensate
the farm, and thus remove the surface contaminant source, or
(2) do nothing, i.e. do not compensate the farm. In the
decision tree, the blue square is the decision node with two
branches representing these two decision alternatives. The
parameter defined as our principle uncertainty, aquifer vul-
nerability, is also depicted as a binary variable; the aquifer is
either vulnerable due to having the farm at location X (S
X
)or
not (
S
X
). The uncertainty is depicted with the red possibility
nodes, with probabilities Pr S
X
ðÞand Pr
S
X
ðÞ¼1 Pr S
x
ðÞ
for each branch. The associated cost (which is the negate of
the value) for each combination of the decision alternative
and aquifer vulnerability represents the full range of possible
outcomes.
The prior value (V
prior
), depicted in the decision tree of
Fig. 1a, can be estimated with these three components,
along with the prior probabilities at the branches of the
possibility node Pr S
X
ðÞand Pr
S
X
ðÞ
.
V
prior
is also known as
the value without data, and it describes the best outcome
given the alternatives and present uncertainty (Raiffa
1968). Figure 1b demonstrates the same decision tree but
with the order of the possibility nodes and the decision
nodes swapped, indicating that we have information that
first gives us an assessment of aquifer vulnerability before
we make a decision to compensate a farmer or not. This is
a depiction of the value with perfect information V
PI
. Both
V
prior
and V
PI
are needed for the formal definition of the
value of information (VOI):
VOI ¼ V
PI
V
prior
ð1Þ
VOI gives a quantitative measure of how much an
information source can increase a decision-maker’s chance
of choosing an alternative with a higher-valued expected
outcome.
Figure 2 is a schematic of the entire proposed VOI
methodology. The variables and notations used in the figure
are described in the following sections. The remainder of
Section 2 will cover part A, which includes all the steps of
prior modeling that lead to the calculation of V
prior
. Sec-
tion 3 will develop a methodology to estimate the more
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useful metric of the value with imperfect information (V
II
)
by accounting for the information reliability. We will
eventually use this in place of V
PI
for our VOI calculations.
We will now describe how the principle uncertainty,
decision alternatives and outcomes can be represented with
an approach that includes the spatial uncertainty in all three
of these elements. Details of the geologic uncertainty, aqui-
fer vulnerability modeling, decision alternatives and values
will be described for the example. However, the formulation
is fairly general and therefore applicable to other decision
situations that are driven by spatial uncertainty.
2.2 Prior modeling of geologic uncertainty
Let us first consider how spatial uncertainty influences our
decision. For this study, we consider the uncertainty
regarding aquifer vulnerability to be exclusively due to the
unknown geologic heterogeneity. For our demonstration
example, the geologic depositional system is interpreted to
be buried glacial valleys. These buried glacial valleys act
as important groundwater resources in many countries in
Northern Europe. Glacial valleys are the result of the
‘waxing and waning of Pleistocene ice sheets’ (BurVal
Working Group 2006). 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
Fig. 3 (Jørgensen and Sandersen 2006). We assume for our
example that the buried valley facies are filled with sand
and represent high volume aquifers. Conversely, the non-
valley or background facies are assumed to be aquitards.
Decision analysis manages the unknown parameters
through probabilistic representation. We represent uncer-
tainty about the unknown subsurface by generating many
Earth models. While any technique can be used, we dif-
ferentiate two levels of uncertainty. We term the first level
Fig. 1 a Decision tree
exhibiting the decision node
(blue), the decisions
alternatives, the aquifer
vulnerability possibility nodes
(red), and the modeled outcome
(green) in terms of a user-
defined value (-cost). b Decision
tree demonstrating how perfect
information would reveal which
surface locations cause the
aquifer to be vulnerable. (Color
figure online)
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‘model uncertainty’ which represents the uncertainty
about the prior conceptualization of the statistics or large-
scale structure of the heterogeneity. The chosen statistical
form (e.g. a histogram, the variogram type or training
image) will be called the geological input parameters,
represented by random variable H. The geological input
parameters are used as input to the stochastic algorithms
which simulate many models for each particular geologic
input parameter. Hence, the ‘spatial uncertainty’ repre-
sented through geostatistical modeling is the second level
of uncertainty.
We treat the buried glacial valleys as the model uncer-
tainty in the aquifer system structure. Figure 3, along with
estimations of the height, width and length of the glacial
valleys, can be used to generate several training images
which are needed for multiple-point geostatistical model-
ing (Strebelle 2002; Hu and Chugunova 2008; Remy et al.
2009). Figure 4 depicts the geologic input parameter H as
training images with two possible outcomes H 2 h
1
; h
2
fg
for the dimensions of the buried valley lithofacies. Gen-
erally, any input geological parameter can have N number
of outcomes:
H 2 h
1
; ...; h
i
; ...; h
N
fg
ð2Þ
Ideally, experts (e.g. geologists) assign prior probabilities
PrðH ¼ h
i
Þ 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 h
i
. Many Earth models can be
generated and represented by
z
ðt
i
Þ
ðh
i
Þ t ¼ 1; ...; T
h
i
; i ¼ 1; ...; N ð3Þ
where T
h
i
is the total number of models for a particular
geologic input parameter h
i
. The ensemble of z
ðtÞ
ðh
i
Þ Earth
models captures the prior uncertainty regarding the sub-
surface properties of interest, which are all captured in the
vector z. At the bottom of Fig. 4, a few Earth models are
shown that are generated from a particular buried valley
training image h
i
and a stochastic algorithm.
The generation of the prior models z
ðtÞ
ðh
i
Þ for the dem-
onstration case utilizes 18 training images (N = 18) to
represent the model uncertainty of the buried valley length,
Fig. 2 Overall workflow of this
VOI methodology. a First the
generation of the prior models
and calculation of V
prior
.
b Second, the reliability
measure and how it is utilized
with the prior models. c The
conditioned models are
generated and used to calculate
V
II
Fig. 3 Network of buried valleys; darker to lighter representing older
to younger buried valley generations (Jørgensen and Sandersen 2006)
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width and thickness dimensions. The width and length of
the buried valleys are given 3 possible categories: high,
medium, and low dimensions. The thickness of the buried
valley in any given training image is described as either
high or low. All the training images are deemed equally
probable: PrðH ¼ h
i
Þ¼
1
N
¼ 0:055. The physical meaning
behind equally probable training images could represent
that the current state of information is very incomplete. The
methodology allows for the assigned weights to be unequal,
which may be based on some geologic information. Within
each of these h
i
, 10 facies Earth models are generated using
the snesim multipoint geostatistical simulation algorithm
(Strebelle 2002). These facies models are binary: each
location represented by either valley or non-valley facies.
Each of the 180 models has 143 9 91 9 50 grid cells with
each cell dimension being 150 m 9 150 m 9 4 m. The
reader should distinguish this demonstration case, serving
as an illustration of a general methodology, from an actual
case study since there is no local accuracy (i.e. no condi-
tioning to local data) within the models.
2.3 Prior modeling of aquifer vulnerability
In Trainor-Guitton et al. (2011), a general methodology
(for any Earth model and data source) was outlined to
attain a meaningful measure of the reliability of informa-
tion in terms of how often spatial data d would correctly
identify and distinguish the specified input geologic
parameters h
i
. This implicitly assumes that knowledge of h
i
solves the decision problem deterministically. In many
situations, however, knowing h
i
is not enough to resolve
the decision problem deterministically. For our demon-
stration case, knowledge of the buried valley dimensions
will not reveal precisely which farms should be compen-
sated to cease the use of chemicals. The network of con-
nected 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. This illustrates the complex relationship
between the model characteristics (the heterogeneity as
represented by alternative h
i
’s) and the aquifer’s vulnera-
bility to surface-sourced contaminants. Therefore, we must
explicitly include this dynamic response to account for
aquifer vulnerability in the decision framework.
Aquifer vulnerability is approximated through the sim-
ulation of fluid flow in porous media. Specifically, we
assume a single dynamic simulation function exists to
transform each z
ðtÞ
ðh
i
Þ into aquifer vulnerability s
s
ðtÞ
ðh
i
Þ¼f ðz
ðtÞ
ðh
i
Þ;‘Þ i ¼ 1; ...Nt¼ 1; ...; T
h
i
¼ 1; ...; L:
ð4Þ
Here denotes all the L locations within each simulated
model z
ðtÞ
ðh
i
Þ where the decision must be made. We define
because the goal is to determine the best decision
alternative for different locations within z
ðtÞ
ðh
i
Þ.
Fig. 4 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 into stochastic
algorithms to create each model
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For our aquifer vulnerability study, will cover all
possible locations of farms. Flow simulation is performed
with a tracer initially placed at all L surface locations. The
permeability of the valley facies is set equal to 1165 mD
(2.8 ft/day or 9.8 E-6 m/s) and 1.1 mD (2.6E-3 ft/day or
9.2E-9 m/s) for non-valley facies. These values are based
on observations from the area that is the motivation of our
study. The simulation is run for 20 years with extraction
and influx boundary conditions (representing pumping
wells, precipitation and regional recharge). We then pro-
cess the tracer concentrations obtained from flow simula-
tion to approximate a measure of aquifer vulnerability as
follows. 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 within the aquifer
at the end of the 20 year simulation. Locations where
these concentration bodies intersect the surface are mapped
as vulnerable. The volume of the concentration body rep-
resents the potential damage a contaminant could do if
released at that surface location . Therefore, the magni-
tude of vulnerability s
ðtÞ
ðh
i
Þ at these surface intersections is
equal to the volume of the concentration body. Figure 5 is
an example vulnerability map for one particular realization
of the Earth model. More refined vulnerability maps could
be constructed to account for chemical and biological
processes, but this is outside the scope of our study.
2.4 Alternatives and value outcomes
For the demonstration case, an estimate of aquifer vul-
nerability will provide an indication of the decision
outcome for a particular action at a particular location,
hence revealing whether a farm ‘connects’’ with an aquifer
several kilometers away. In general, decision makers must
identify all the possible alternatives to the decision; we will
denote these by a ¼ 1; ...; A, where A is total number of
alternatives identified. The function g
a
ðs
ðtÞ
ðh
i
ÞÞ predicts the
outcome of alternative on the unknown decision variable:
aquifer vulnerability. 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. The simulated aquifer vulnerability s
ðtÞ
has
the information that will determine all the value outcomes
due to different decision alternatives a at each location :
v
ðtÞ
‘;a
ðh
i
Þ¼g
a
ðs
ðtÞ
ðh
i
ÞÞ a ¼ 1; ...; Ai¼ 1; ...N
t ¼ 1; ...; T
h
i
¼ 1; ...; L:
ð5Þ
where value can be in terms of monetary units ($), eco-
logical health (Polasky and Solow 2001) or some other
appropriate utility. For the demonstration example, a sim-
ple cost structure is used and will be explained in Section 4.
For now we assume that action g
a
can only be taken at one
position at a time. This restriction will be removed in
Section 4.
The location-specific values of Eq. 5 are obtained for all
Earth models z
ðtÞ
ðh
i
Þ. Therefore, we have a range of pos-
sible value outcomes for these local decision actions, rep-
resenting the uncertainty on the local decision outcome.
These scalar value results v
ðtÞ
‘;a
ðh
i
Þ are used to define V
prior
:
V
prior
¼
X
L
¼1
max
a
X
N
i¼1
PrðH ¼ h
i
Þ
1
T
h
i
X
T
h
i
t¼1
v
ðtÞ
‘;a
ðh
i
Þ
!
a ¼ 1; ...; A:
ð6Þ
Recall that V
prior
is also known as the value without data
and is what is depicted in Fig. 1a. For our demonstration case,
A = 2 (compensate or do not compensate). This prior value
includes a sum over the value outcomes of the action at each
location . This summation is outside max
a
ðÞ 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 ¼ 2
and not remove a farm (a = 2) at ¼ 1. We now need an
expression for the value with imperfect information (V
II
)in
order to achieve a VOI methodology appropriate for spatial
decisions with spatial uncertainty.
3 Data uncertainty: information reliability
VOI is calculated before any data are collected: we are
trying to decide if it is worth acquiring or purchasing the
Fig. 5 Example vulnerability map which indicates locations that
serve as entry points into the aquifer. The vulnerability magnitude
reflects the volume of aquifer that would be affected if a contaminant
would enter at that location
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data given that we are faced with the decision. A generic
expression for the value with imperfect information is the
following
V
II
¼ E max
a
E½V
a
jd
hi
a ¼ 1; ...; A: ð7Þ
where V
a
is the random variable describing the possible
value outcomes due to randomization of both input
parameters and spatial variation of properties of z
ðtÞ
ðh
i
Þ and
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;
now there is one more expectation (the outer expectation)
over the data d. Since no data for the proposed location
have been collected, we must simulate the possible datasets
d and evaluate how these data d would influence the
decision. Since we are estimating the value with imperfect
information V
II
, realistic errors that may be in the data d
must be included. Obtaining a realistic V
II
involves an
estimation of how accurate the data are in resolving certain
Earth parameters that are relevant to the decision; thus it
utilizes the measure of information reliability. The reli-
ability measure can encompass errors from both technical
(e.g due to instrumentation or physical ambiguities) and
interpretational (e.g due to subjectivity) sources. This will
be demonstrated in our example reliabilities.
In this section, we will describe the second key com-
ponent of the proposed VOI methodology for dynamic,
spatial decisions: an information reliability that includes
spatial uncertainty. This represents the second phase of the
workflow (represented in part B of Fig. 2). An information
reliability measure can be described in the form of a con-
ditional probability. In its most general form, the reliability
describes
Pr(signal or message from the datajthe true complex reality):
Since the ‘complex reality’ includes both the possible
subsurface heterogeneity as well as the dynamic response
such as the aquifer vulnerability, a reliability measure in
terms of a conditional probability is not trivial to determine
explicitly by means of forward modeling (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 Monte Carlo
simulation approach based on rock physics relationships
for including information reliability into the VOI
calculation. To calculate the expected value in Eq. 7,we
use a Monte Carlo integration approach as follows
E max
a
E½V
a
jd
hi
¼
Z
all d
max
a
Z
all v
a
f ðv
a
jd Þv
a
dv
a
2
6
4
3
7
5
f ðdÞdd
where the integral is approximated using an arithmetic
average calculated through the Monte Carlo sampling, and
here f, unlike in Eq. 4, represents a generic probability
density function. The approximation can be described in 6
steps:
1. Generate Earth models z
ðtÞ
ðh
i
Þ; t ¼ 1; ...; T
h
; i ¼
1; ...; N
2. Use a likelihood function to generate a synthetic data
set d
ðt
i
Þ
from each prior Earth model z
ðtÞ
ðh
i
Þ, where the
likelihood describes the relationship between the
geophysical attribute and the key geologic indicator.
3. From each of the synthetic data sets d
ðt
i
Þ
, derive a
distribution on how informative that synthetic dataset
is about the key geologic indicators (an information
pre-posterior probability y
ðt
i
Þ
).
4. Use that information pre-posterior probability y
ðt
i
Þ
as
a soft probability in any geostatistical simulation
algorithm to generate multiple, new Earth models
constrained to each of the synthetic data sets.
5. Create realizations (w) of aquifer vulnerability
s
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ by applying the dynamic simulation
function f to the conditioned (interpreted) Earth models.
6. Calculate the value v
ðt;wÞ
‘;a
ðy
ðt
i
Þ
; h
i
Þ of the various
alternatives a based on the values of pre-posterior
aquifer vulnerability s
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ.
Repeating these steps provides multiple value outcomes
from which an estimate of the expectation in Eq. 7 can be
calculated. These steps are graphically shown in Fig. 6.
Note that specifically for our demonstration case, we
assume exclusivity between lithology and buried valley
facies (h
i
): sand is always interpreted as buried valley and
clay as non-valley. This assumption may be true for some
geographic locations (Sandersen and Jørgensen 2003) but
could be relaxed and accounted for by estimating a per-
centage of buried valleys that are filled with clay materials.
Next, we will clarify each of the six steps in detail.
3.1 Synthetic data sets accounting for data reliability
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 z
ðtÞ
ðh
i
Þ. In our case, this includes the geological
indicator of lithology, 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 electromag-
netic (TEM) data. TEM induces currents and fields in the
subsurface and measures the changing magnetic field
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response of induced currents in the subsurface (Fitterman
and Stewart 1986; Christiansen 2003). The magnetic field
response is inverted to obtain a layered model of electrical
resistivity and thickness values (Auken et al. 2008). For the
purposes of our study, we wish to differentiate between the
valley facies, which we presume to be composed of sand,
and the non-valley facies, which we presume to be com-
posed of clay. 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 the
resistivity of sand is usually greater than 80 ohm-m
(Jørgensen et al. 2003). However it is not a perfect indi-
cator as there is overlap between the values of resistivity
for clay and sand. In addition the inversion process is
imperfect and introduces additional spatial smoothing.
The association between lithology (litho) and the elec-
trical resistivity (q) 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 proba-
bility. 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
PrðP ¼ qjLitho ¼ lithoÞ litho ¼fsand, clayg
0\q\1
ð8Þ
were used for our demonstration case to describe two
possible relationships between electrical resistivity and
lithology (such two scenarios can represent the uncertainty
in terms of rock physics). Our first likelihood is based on
a calibration of co-located inverted resistivity data and
driller’s logs, which respectively have electrical resistivity
and lithology information. The driller’s logs include
interpretational errors since they are subjective assessments
of lithology, and the electrical resistivity will contain errors
Fig. 6 The steps of obtaining a data reliability through Monte Carlo
simulation. Step 1: generate prior models. Step 2: generate a synthetic
dataset from the likelihood and a prior model. Step 3: generate a soft
probability cube for the lithofacies valley from a dataset and the
information content. Step 4: generate conditioned Earth models with
the soft probability. Step 5: obtain the conditioned aquifer vulnera-
bility by applying the dynamic simulation function to the conditioned
Earth model
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due to the overlap between clay and sand. The first data set
(shown in Fig. 7) demonstrates a reasonable electrical
resistivity contrast between sand (assumed here as valley)
and clay (non-valley). We see that in general, clays have
lower electrical resistivity than sands. However, any
overlap of electrical resistivity between the two histograms
(the top belonging to sand and bottom to clay) demonstrate
the ambiguity of using the geophysical attribute (electrical
resistivity) to determine lithology. The second data set
(Fig. 8) was synthetically generated to be less discrimi-
nating in terms of lithology, so a greater overlap exists
between the electrical resistivity of the two lithologies. In
decision analysis terms, data likelihood is in the assessed
form: the variables influencing the decision outcome are
known and a probabilistic relationship is identified with the
variables from the information. This is depicted in Fig. 9a.
Most importantly, we can create many synthetic datasets
d (in the form of the geophysical observables) using Monte
Carlo sampling of Eq. 8 (represented in Figs. 7, 8).
Namely, knowing the occurrence of lithology at a certain
location within z
ðtÞ
ðh
i
Þ, an instance of electrical resistivity
(q) can be drawn by Monte Carlo sampling of the cdf
(cumulative distribution function) form of either the top of
Fig. 7 (given that the considered location is sand/valley) or
the bottom (given the considered location is clay/non-val-
ley). Similarly, this can be performed with the cdf forms of
Fig. 8. This is repeated for all locations within the model
z
ðtÞ
ðh
i
Þ to generate the dataset:
d
ðt
i
Þ
i ¼ 1; ...Nt¼ 1; ...; T
h
i
: ð9Þ
This Monte Carlo sampling is performed grid cell by
grid cell, relying only on the spatial correlation of the prior
models; the final synthetic data are achieved by smoothing
the Monte Carlo result with a moving average filter.
For each of our buried valley prior models z
ðtÞ
ðh
i
Þ,we
generate one electrical resistivity dataset d
ðt
i
Þ
, for a total of
180 datasets from the likelihood of Fig. 7 and 180 datasets
from the likelihood of Fig. 8. Figure 6 demonstrates this
Step 2 and displays an example synthetic dataset of elec-
trical resistivity that was generated using one prior model
and the cdf versions of the pdf’s in Fig. 7. These electrical
resistivity datasets represent the information we could
expect to collect, given our uncertainty of both the sub-
surface heterogeneity (represented with the prior models)
and the ability of the TEM data to resolve lithology (rep-
resented through the data likelihoods).
3.2 Conditioned Earth models from information
content
Converse to the data reliability, the ‘information content
or information pre-posterior of a data source about the
unknown Earth is of the form:
Pr(the true complex reality jsignal or message from the data):
More specifically the information pre-posterior probabilities
we seek are of the form
PrðLitho ¼ lithojP ¼ qÞ
litho ¼fsand,non-sandg 0 q 1:
ð10Þ
Figure 9b demonstrates that the information pre-
posterior is the inferred form: the information variables
are used to determine the probability of occurrence of the
lithology variables, which are more closely related to the
decision variable of aquifer vulnerability. This conditional
Fig. 7 Synthetic data reliability describing a good contrast between
the two lithologies’ electrical resistivities
Fig. 8 Synthetic data reliability describing a poor contrast between
the two lithologies’ electrical resistivities
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probability is important in generating new Earth models
constrained to the synthetic data sets as outlined in our
workflow. Through Bayes Law, Eq. 10 can be obtained
from Eq. 8. Using Eqs. 8 and 10, we note that perfect
information would imply that an exclusive relationship
exists between lithology and electrical resistivity, such that
any particular resistivity value would result in only one type
of lithology to be drawn from PrðLitho ¼ lithojP ¼ qÞ :
We can create a lithology probability cube using each of
the datasets d
ðt
i
Þ
to obtain a sand and clay probability from
Eq. 10
y
ðt
i
Þ
i ¼ 1; ...; Nt¼ 1; ...; T
h
i
: ð11Þ
For our demonstration case, y
ðt
i
Þ
contains a probability
of sand and clay (valley or non-valley) at each location
within the dataset d
ðt
i
Þ
, which is derived from the prior
model z
ðtÞ
ðh
i
Þ (see Step 3 of Fig. 6). This is known as the
‘soft probability’ for conditioning multiple-point
realizations (Caers 2005).
The next step involves creating multiple Earth models
conditioned to this soft probability (see Step 4 of Fig. 6).
We again use the snesim algorithm (Single Normal Equa-
tion Simulation) within Stanford’s Geostatistical Earth
Modeling Software (SGeMS) to generate Earth models
(realizations) of lithology (Remy et al. 2009)
z
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ w ¼ 1; ...; Wi¼ 1; ...; N
t ¼ 1; ...; T
h
i
:
ð12Þ
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
possible Earth model interpretations which result from the
overlap in the data likelihood, i.e. imperfect information.
For each y
ðt
i
Þ
, two (W = 2) new conditioned Earth models
z
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ are generated for the demonstration case.
This can be considered the minimum 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.
In addition to being conditioned to the soft probability
y
ðt
i
Þ
, these Earth models reflect the prior spatial constraints
through the training images (h
i
) (Caers et al. 2001;
Strebelle et al. 2003). Depending on how discriminating
the data are, these conditioned models z
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ may
be very different or very similar to the prior models z
ðtÞ
ðh
i
Þ
from which they are derived.
3.3 Value with imperfect information
Finally, we arrive at Step 5 of Fig. 6: obtaining aquifer
vulnerability from the conditioned Earth model. This also
represents the beginning of part C of the workflow in
Fig. 2. These conditioned Earth models represent the var-
iability in geologic ‘interpretations’ from the synthetic
data d
ðt
i
Þ
. As with the prior models (Fig. 2a), the dynamic
simulation function f must be applied to the new condi-
tioned models z
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ to get the conditioned or
interpreted aquifer vulnerability
s
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ¼f ð z
ðt;wÞ
ðy
ðt
i
Þ
; h
i
;‘ÞÞ i ¼ 1; ...N
t ¼ 1; ...; T
h
i
¼ 1; ...; Lw¼ 1; ...; W:
ð13Þ
The conditioned aquifer vulnerability s
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ
determines the outcome of the decision. Again, as in the
Fig. 9 a Assessed order of
information: decision variables
are known (geologic lithologies)
with a probabilistic relationship
to the data attribute (electrical
resistivities). b Inferred order of
information: the probability of
the message from the acquired
information accurately
identifying the decision variable
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prior aquifer vulnerability, we express the outcome of these
decisions in terms of value:
v
ðt;wÞ
‘;a
ðy
ðt
i
Þ
; h
i
Þ¼g
a
ðs
ðt;wÞ
ðy
ðt
i
Þ
; h
i
ÞÞ
a ¼ 1; ...; Ai¼ 1; ...Nt¼ 1; ...; T
h
i
¼ 1; ...; Lw¼ 1; ...; W: ð14Þ
These individual values from the conditioned models are
used to calculate the value with imperfect information V
II
since they are derived from aquifer vulnerability maps that
have been constrained by the data reliability measure.
V
II
¼
X
L
¼1
X
N
i¼1
PrðH ¼ h
i
Þ
1
T
h
i
X
T
hi
t¼1
max
a
1
W
X
W
w¼1
v
ðt;wÞ
‘;a
ðy
ðt
i
Þ
; h
i
Þ
!!
a ¼ 1; ...; A:
ð15Þ
Chronologically, we first obtain the W possible
interpretations of aquifer vulnerability s
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ
through the conditioned models before we make our
decision. Therefore, we can choose the best decision
alternative on average for those interpretations, represented
by max
a
. Then the expected value from all the models for
location is taken. The reliability is captured in the imperfect
simulated data d
ðt
i
Þ
and ultimately in the conditioned models
z
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ. These models account for the possible
inaccuracies of the geophysical information message to
inform about lithology. The imperfect data will have value if
they can resolve the aquifer vulnerability and lead to
decisions with a higher value outcome than the V
prior
.
4 VOI calculation results
This section shows how the uncertainty of the aquifer vul-
nerability and the multiple conditioned models can be inte-
grated into the VOI calculation framework and presents VOI
II
results for the demonstration case of aquifer vulnerability.
Also in this section, the methodology will be generalized to
allow several local actions to be taken simultaneously.
In the case of perfect information v
ðt;wÞ
‘;a
ðy
ðt
i
Þ
; h
i
Þ (Eq. 14)
will be equal to v
ðtÞ
‘;a
ðh
i
Þ (Eq. 5), as all prior models will be
perfectly recovered through the data into s
ðt;wÞ
ðy
ðt
i
Þ
; h
i
Þ.
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
models will poorly represent the prior Earth model they
originated from and will be quite dissimilar from each
other. Therefore decisions made on ‘inaccurate interpre-
tations’ will lead to lower value outcomes on average.
Higher quality data will ultimately lead to higher valued
decision outcomes and consequently, a higher VOI
II
. If the
proposed information, represented synthetically with d
ðt
i
Þ
,
can constrain the results of the dynamic simulation func-
tion and subsequently the decision variable, then this
imperfect information may have value. The degree of
‘constraining’ is measured by estimating the value of
imperfect information: VOI
II
= V
II
- V
prior
.
Assuming that the two decision alternatives are made
independently, there are four possible outcomes at each
surface location depending on that location’s vulnerability
cos t
ðtÞ
‘;a
¼
1ifs
ðtÞ
[ 0; a ¼ 1 ðcompensateÞ
1ifs
ðtÞ
¼ 0; a ¼ 1 ðcompensateÞ
2ifs
ðtÞ
[ 0; a ¼ 2 ðdon’t compensateÞ
0ifs
ðtÞ
¼ 0; a ¼ 2 ðdon’t compensateÞ
8
>
>
>
<
>
>
>
:
where we express the costs associated with the outcomes in
nominal, unitless costs. Since it is more straightforward to
use costs for these outcomes, the value definition can be
expressed in terms of negative costs (value =-cost).
The first two cost outcomes are deemed a unit cost, rep-
resenting the cost of compensating a farmer to stop using
chemicals and avoiding contamination. The third cost
outcome is assigned a cost that is twice as much as com-
pensation, representing the environmental consequences of
not compensating a farm located on an effective entry point
to the aquifer. The last cost outcome has a zero cost as no
compensation nor contamination occurs. Note that we have
chosen to threshold the aquifer vulnerability to two levels:
0 or positive. Other levels of vulnerability (i.e. s
ðtÞ
[ 100)
could be considered to be more appropriate for particular
situations.
Table 1 contains the VOI
II
results with these cost out-
comes. The results for the two different reliabilities are
shown in the columns. As expected, the data generated
with the reliability of Fig. 8 have less value than those
generated with a the reliability of Fig. 7, which has a better
electrical resistivity contrast between the two lithologies.
These VOI
II
results assume that the decision to compensate
or not 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 Eq. 15, a sum is made over all the L locations, which
account for the VOI
II
results being on the order of 50
whereas the costs are 0, 1 and 2.
We might expect to see the difference between the two
VOI
II
values become more accurate for W [ 2, as we
would be better representing the uncertainty in the data
likelihood with more conditioned Earth models. In both
cases, the VOI
II
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
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cost of acquisition was less than the VOI
II
. The VOI nat-
urally also depends on the ratio of the compensation cost
(here nominally 1) and the contamination cost (here 2).
4.1 Generalization of decision alternatives
As a final step, the restriction of each location being con-
sidered 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 for the multiple loca-
tions, with K total combinations deemed or identified as
possible and valuable. The notation of g
c
a
where a ¼
a
1
; ...; a
L
fg
denotes which action is taken at which location,
with ¼ 1; ...; L including all the L possible locations at
which a decision action a may be made. Suppose, as an
example, that there are three locations where it is possible to
take a decision action (L ¼ 3), only three unique combina-
tions of the spatial and decision actions are considered
possible (K ¼ 3) and there are two possible decision
actions (A ¼ 2 ) such that a 2 compensate farm,don’t
f
compensate farmg The three possible decision-spatial
combinations are c ¼ 1 : a ¼ 1; a ¼ 1; a ¼ 2
fg
, c ¼ 2 :
a ¼ 2; a ¼ 2; a ¼ 1
fg
and c ¼ 3 : a ¼ 1; a ¼ 2; a ¼ 1
fg
The notation g
c¼1
would indicate that farms at ¼ 1 and
¼ 2 would be compensated, and the farm at ¼ 3 would
not be; g
c¼2
indicates that only the farm at ¼ 3 would be
compensated and g
c¼3
has farm 1 and 3 compensated. With
the ability to apply combinations of different actions, V
prior
now becomes
V
prior
¼ max
c
a
X
N
i¼1
PrðH ¼ h
i
Þ
1
T
h
i
X
T
h
i
t¼1
v
ðtÞ
c
a
ðh
i
Þ
!
c ¼ 1; ...; K a ¼fa
1
; ...; a
L
ga ¼ 1; ...; A
ð16Þ
so that the best we can do with the present uncertainty is
the highest valued spatial-decision combination v
c
a
on
average over all
P
N
i¼1
T
h
i
Earth models. The value with
imperfect information becomes
V
II
¼
X
N
i¼1
PrðH ¼ h
i
Þ
1
T
h
i
X
T
hi
t¼1
max
c
a
1
W
X
W
w¼1
v
ðt;wÞ
c
a
ðy
ðt
i
Þ
; h
i
Þ
!
c ¼ 1; ...; K a ¼fa
1
; ...; a
L
g a ¼ 1; ...; A
ð17Þ
Equations 16 and 17 can be used to recalculate the VOI
II
of Table 1 if the removal of only certain combinations of
farms was deemed possible.
5 Discussion and conclusions
Our VOI methodology has been designed to capture both
the dynamic hydrogeologic response of a heterogeneous
system and the geophysical information’s potential to
resolve the spatial heterogeneity. Although the develop-
ment of this VOI methodology was motivated by a decision
dependent on aquifer vulnerability, other dynamic
responses could be used in its place in this workflow,
e.g. other physical, chemical, or geomechanical processes
or some combination of the three. However, this method-
ology has some inherent assumptions about the uncertain-
ties that most influence the outcomes of the decision;
assumptions have also affected the formulation of the
information reliability. Additionally, there are potential
limitations of the proposed workflow due to the simplified
approach used to represent the acquired geophysical data
method and the possible high computational costs of sim-
ulating fluid flow in numerous subsurface models. We will
briefly address these assumptions and limitations.
As identified by Howard (1966a), a sensitivity analysis
should be completed by the geologist(s), geostatistician(s)
and other relevant modeler(s) to establish satisfactory prior
models and to determine which uncertain parameters of the
subsurface are most consequential and most impact the
decision. For this example, we have assumed a sensitivity
analysis has previously been performed and has identified
the input geologic parameters h
i
representing the different
width, length and thickness dimensions of the buried valley
lithofacies as the most influential input that affects aquifer
vulnerability.
A second assumption affected how we formulated our
information reliability. Even though it is assumed that the
principal uncertainty is related to the buried valley
dimensions, our information reliability is not in terms of
how well the buried valley dimensions are identified. This
is possible because we make a significant simplifying
assumption that all sand is valley facies, and that vulner-
ability is driven only by the buried valley structure (i.e. no
sub-grid cell features exist that affect aquifer vulnerabil-
ity). With this lithology-facies assumption, we only are
concerned with the TEM’s ability to distinguish sand/clay
lithologies since this means it also can exclusively identify
valley/non-valley facies. Therefore, the two generated
likelihoods do not describe how the geophysical data dis-
cerns the complex geologic input parameter h
i
(describing
the dimensions of the buried valley) but the geologic
indicator of lithology.
A focus of our study is quantifying the reliability of the
geophysical information. We emphasize that what we have
done in demonstrating a methodology is simplified. We
have presumed that the geophysical data provide electrical
resistivity, which is related to lithology, at specific
Table 1 VOI
II
results for aquifer vulnerability demonstration case
VOI
II
= V
II
-
V
prior
Data reliability 1
(Fig. 7)
Data reliability 2
(Fig. 8)
Costs = {1,1,2,0} 57.56 51.655
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subsurface locations. Our approach of generating synthetic
datasets only relies on the imperfect relationship between
the geophysical attribute (electrical resistivity) and the
geologic indicator (lithology). We do not consider the
complex physics of the measurement or the effects of
processing and inversion of the data. In order to accurately
estimate the reliability of geophysical information, many
other factors such as these need to be accounted for.
As demonstrated with our results, VOI increases with
more reliable data. Data reliability depends not only on the
quality of acquisition, processing, etc. but also on the geo-
logic conditions and rock properties. For example, in some
geologic conditions and for some rock properties, even a
high-quality data acquisition scheme may not give very
reliable information, while in others even a fairly simple data
acquisition might give highly reliable information.
The methodology evaluates both the prior and pre-pos-
terior Earth models response to the dynamic function
(i.e. subsurface fluid flow). This approach has a significant
drawback; the methodology could be computationally
expensive depending on the dynamic simulation function,
the size of the models, etc. For certain applications, more
conditioned models may be needed to achieve a reliable
VOI
II
. For this demonstration case, it was already com-
putationally expensive to run 360 3D flow simulations for
20 years each. Running a parallel version of the Eclipse
flow simulator, each simulation took 40 minutes of run
time.
We have defined a VOI
II
methodology that accounts for
decision situations that depend on a dynamic response. We
demonstrated this methodology with aquifer vulnerability;
the prior uncertainty and modeling lead to a measure of the
value without data or V
prior
. Next, the reliability of the
information had to be addressed to quantify the value with
imperfect information (V
II
). First, we proposed generating
synthetic datasets derived from likelihoods, which describe
the variability between electrical resistivity and lithology.
Then we utilized the information content of these synthetic
datasets with geostatistical simulation to achieve condi-
tioned decision variables. Specifically, we have shown how
this methodology applies to our demonstration case of
aquifer vulnerability. As would be expected, the VOI
II
results demonstrate that a higher VOI
II
occurs when the
discriminating geophysical data are available.
Lastly, it should be noted that the true power of the
VOI
II
approach is not the final VOI
II
measure itself, but its
ability to act as a tool to organize and evaluate the
uncertainties that impact a decision outcome. Once the
general framework has been established for a particular
decision and information source, the sensitivities of the
VOI
II
outcome to different parameters (e.g. the alterna-
tives, probability of certain outcomes occurring, the data
reliability, costs and benefits, etc.) can be evaluated.
Knowing which of these parameters most affect the final
VOI
II
is likely to be more meaningful than the absolute
VOI
II
for particular decision and information parameters.
Acknowledgments This work was possible because of the support
from the Affiliates of Stanford Center for Reservoir Forecasting and
Schlumberger Water Services. We thank professor Jef Caers for his
early participation in this work. Esben Auken and Nikolaj Foged of
the University of A
˚
rhus, Denmark provided helpful insights about the
TEM measurement. Thomas Nyholm and Stine Rasmussen of the
Danish Ministry of the Environment provided useful information
about aquifer vulnerability issues. Prepared by LLNL under Contract
DE-AC52-158 07NA27344
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... Because geophysical field experiments are often complicated and expensive, an efficient experimental design often can save considerable data-collection effort and expense. The ability to quantify the value of new observations is key to designing an efficient experiment (De Gruttola et al. 1987;Trainor-Guitton et al. 2013). In the context of an inverse problem, value is understood as the ability of an observation to improve the quality of the solution, either by decreasing its variance or increasing its resolution, or both. ...
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Seismic data provide essential information for guiding reservoir development. Improvements in data quality hold the promise of improving performance even further, provided that the value of these data exceeds their cost. Previous work has demonstrated value-of-information (VOI) methods to quantify the value of seismic data. In these examples, seismic accuracy was obtained by means of expert assessment instead of being based on geophysical quantities. In addition, the modeled seismic information was not representative of any quantity that would be observed in a seismic image. Here we apply a more general VOI model that includes multiple targets, budgetary constraints, and quantitative models relating poststack seismic amplitudes and amplitude-variation-with-offset (AVO) parameters to the quantities of interest for reservoir characterization, such as porosity and reservoir thickness. Also, by including estimated changes in data accuracy based on signal-to-noise ratio, the decision model can provide objective estimates of the reliability of measurements derived from the seismic data. We demonstrate this methodology within the context of a west Texas 3D land survey. This example demonstrates that seismic information can improve reservoir economics and that improvements in seismic technology can create additional value. Introduction Reservoir characterization makes heavy use of seismic data both for selecting a target for drilling and, with time-lapse data, for monitoring the fluid movements in the reservoir to optimize production of hydrocarbons. Reservoir characterization requires good-quality seismic data for optimal results. Improvements in aspects of seismic acquisition, such as signal-to-noise ratio, bandwidth, receiver positioning, or maximum offset, may help improve images or AVO analyses, thereby increasing the level of knowledge about reservoir structure or properties. However, modifications to acquisition procedures to estimate rock properties better or to improve subsalt images, for example, may increase expense of data acquisition and possibly experiment duration. The improved data quality must always be weighed against the additional cost. Previous work has addressed valuing seismic data using the decision-analysis concept of VOI, including Stibolt and Lehman (1993), Waggoner (2000b, 2002), Begg et al. (2002), Pickering and Bickel (2006), and Bickel et al. (2006). Ballin et al. (2005) and Steagall et al. (2005) provide examples of actual seismic projects where VOI analyses shaped management decisions significantly. See Bratvold et al. (2007) for a review of VOI papers in the SPE literature. 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Many authors implicitly embed downstream decisions in the seismic-accuracy assessment by assuming the chance of geologic success can only go up after commissioning a seismic survey (Head 1999; Waggoner 2000b, 2002). This mixing of probability assessments and decision making makes it difficult to understand the value of seismic in a specific situation. Houck (2004) addressed some of these concerns by valuing seismic's ability to inform estimates of porosity in the context of a multiwell drilling program and tying the accuracy of seismic data to directly observable seismic signals. This paper also extends previous VOI studies by considering multiple targets and budgetary constraints. We extend Houck's results by investigating the accuracy and value of AVO and peak amplitude. Furthermore, we examine the ability of seismic information to inform estimates of multiple reservoir properties simultaneously (e.g., porosity, thickness, and water saturation). The resulting models allow quantification of the accuracy of information provided by seismic data and quantification of the information's economic value. The contributions of this paper are three-fold. First, we illustrate a VOI method that directly relates observable seismic signals to reservoir properties and reservoir-management decisions. Second, we develop a seismic model that allows us to quantify objectively the accuracy of seismic information across a range of acquisition and processing techniques. Third, we quantify both the absolute value of seismic information and the relative value of improved seismic information within the context of a 3D land example situated in a hypothetical carbonate reservoir modeled after the McElroy field in west Texas.
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Due to the nature of our industry, we understand the importance of being able to value information and continue to develop skill in that domain. Daily, we face decisions regarding buying 3D seismic, taking cores and designing well tests and pilots; all with the aim to determine if it is valuable to do so prior to taking large investment decisions on our assets. This is an analysis performed at "time zero??, before information is acquired. How we determine its worth post acquisition, is important, less common, and often performed incorrectly. It is also the focus of this work. This paper reviews how information is valued pre-acquisition and addresses how to take the post acquisition interpretation, and operational results, and translate them to correct lookback valuations. Many information lookbacks erroneously consider the value associated with acquiring information pre-acquisition as an integral part of the value of the information post acquisition, and then find that it does not "marry well?? with the actual results. The missing step in most post information acquisition lookbacks is the proper consideration of what was interpreted from the new information, applied to the proper comparison to the actual outcome of the variable measured. For illustrative purposes, this work considers a 3D seismic reprocessing program example. It notes the development drilling success rate immediately prior to reprocessing, the interpreted location results post reprocessing, and the associated drilling results. The paper describes proper pre-information asset characterization (i.e., drilling success rate), reliability of information assessment, post information interpretation statistics, and post information drilling results. Most importantly, the paper ties the pre-acquisition value-of-information (VOI) analysis, and the reliability estimate of that information, to the post information interpretation and outcome of the measured variable. Introduction New information allows us to "update?? our view of project risks and uncertainties. How well it does this depends on how reliable the information is at helping predict the true state of the variable to be measured. If information is reliable, it keeps us from making poor decisions (e.g., drilling a dry hole, or developing sub-economic fields) as often as we might have, without the new information. That in turn increases the value of our projects, given we acquire the information. Sometimes we "upgrade?? our view of a project uncertainty and sometimes we "downgrade?? our view. In either case, if information is reliable, we are less likely to make a poor decision (e.g., drill or develop sub-economic fields). In that respect, quality information helps clarify the risks in our projects and enables us to take appropriate action with more confidence. This increases project success by helping teams pursue more good projects and stay away from bad ones. How then do we estimate the value of information? As others have described before1, 2, in terms of asset management, we estimate the increase, or decrease, of an asset's value, after acquiring information; classically defining the value as: Value-of-Information (VOI) = (Asset Valuewith Information) - (Asset Valuewithout Information) The Information Timeline Information has different value at different points in time. Failing to recognize this leads to common errors encountered with many value-of-information lookbacks.
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Geological interpretation and seismic data analysis provide two complementary sources of information to model reservoir architecture. Seismic data affords the opportunity to identify geologic patterns and features at a resolution on the order of 10's of feet, while well logs and conceptual geologic models provide information at a resolution on the order of one foot. Both the large-scale distribution of geologic features and their internal fine-scale architecture influence reservoir performance. Development and application of modeling techniques that incorporate both large-scale information derived from seismic and fine-scale information derived from well logs, cores, and analog studies represents a significant opportunity to improve reservoir performance predictions. In this paper we present a practical new geostatistical approach for solving this difficult data integration problem and apply it to an actual, prominent reservoir. Traditional geostatistics relies upon a variogram to describe geologic continuity. However, a variogram, which is a two-point measure of spatial variability, cannot describe realistic, curvilinear or geometrically complex patterns. Multiple-point geostatistics uses a training image instead of a variogram to account for geological information. The training image provides a conceptual description of the subsurface geological heterogeneity, containing possibly complex multiple-point patterns of geological heterogeneity. Multiple-point statistics simulation then consists of anchoring these patterns to well data and seismic-derived information. This work introduces a novel alternative approach to traditional Bayesian modeling to incorporate seismic. The focus in this paper lies in demonstrating the practicality, flexibility and CPU-advantage of this new approach by applying it to an actual deep-water turbidite reservoir. Based on well log interpretation and a global geological understanding of the reservoir architecture, a training image depicting sinuous sand bodies is generated using a non-conditional object-based simulation algorithm. Disconnected sand bodies are interpreted from seismic amplitude data using a principal component cluster analysis technique. In addition, a map of local sand probabilities obtained from a principal component proximity transform of the same seismic is generated. Multiple-point geostatistics then simulates multiple realizations of channel bodies constrained to the local sand probabilities, partially interpreted sand bodies and well-log data. The CPU-time is comparable to traditional geostatistical methods. Introduction Geostatistics aims at building multiple alternative reservoir models thereby assessing uncertainty about the reservoir. One major challenge of geostatistical modeling is to integrate information from different sources obtained at different resolutions:well-data which is sparse but of high resolution, on the order of one foot,seismic data which is exhaustive but of much lower resolution, on the order of 10's of feet in thevertical direction,conceptual geological models, which could quantify reservoir heterogeneity from the layer scale to the basin scale. Variogram-based algorithms allow integrating well and seismic data using a pixel-based approach: First, the well data are assigned to the closest simulation grid nodes. Then, all unsampled nodes are simulated conditional to well and seismic data using some form of co-kriging1. Variogram-based geostatistics is inadequate in integrating geological concepts since the variogram is too limited in capturing complex geological heterogeneity. A variogram is a two-point statistics that poorly reflects a geologists' prior conceptual vision of the reservoir architecture.