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Stochastic Environmental Research
and Risk Assessment
ISSN 14363240
Stoch Environ Res Risk Assess
DOI 10.1007/s0047701206194
A methodology for quantifying the value
of spatial information for dynamic Earth
problems
Whitney J.TrainorGuitton, 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. TrainorGuitton
•
Tapan Mukerji
•
Rosemary Knight
Ó SpringerVerlag 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 (preposterior)
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. TrainorGuitton (&)
Lawrence Livermore National Laboratory, 7000 East Ave,
L231, Livermore, CA 94550, USA
email: trainorguitton@llnl.gov
T. Mukerji
Energy Resources Engineering, Stanford University,
367 Panama Mall, Room 65, Stanford, CA 94305, USA
email: mukerji@stanford.edu
R. Knight
Department of Geophysics, Stanford University, 397 Panama
Mall, Room 65, Stanford, CA 94305, USA
email: rknight@stanford.edu
123
Stoch Environ Res Risk Assess
DOI 10.1007/s0047701206194
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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 decisionmaking 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 surfacesourced 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
nearsurface 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 landuse 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 hydroecological 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
Stoch Environ Res Risk Assess
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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, controlledsource 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 decisionmaker’s chance
of choosing an alternative with a highervalued 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 crosscut
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 multiplepoint 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 nonvalley 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 TrainorGuitton 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 surfacesourced 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 E6 m/s) and 1.1 mD (2.6E3 ft/day or
9.2E9 m/s) for nonvalley 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 locationspeciﬁc 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
ðÞ 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
preposterior probability y
ðt
i
Þ
).
4. Use that information preposterior 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 preposterior
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 nonvalley. 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 timedomain 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 nonvalley 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 ohmm, whereas the
resistivity of sand is usually greater than 80 ohmm
(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 colocated 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 (nonvalley). 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/nonval
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 preposterior 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 preposterior probabilities
we seek are of the form
PrðLitho ¼ lithojP ¼ qÞ
litho ¼fsand,nonsandg 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 nonvalley) 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 multiplepoint
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 decisionspatial
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 spatialdecision 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
subgrid cell features exist that affect aquifer vulnerabil
ity). With this lithologyfacies assumption, we only are
concerned with the TEM’s ability to distinguish sand/clay
lithologies since this means it also can exclusively identify
valley/nonvalley 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
highquality 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 prepos
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
DEAC52158 07NA27344
References
Archie GE (1942) The electrical resistivity log as an aid in
determining some reservoir characteristics. Trans Inst Min
Metall Eng 146:54–62
Auken E, Christiansen A, Jacobsen L, Sørensen K (2008) A resolution
study of buried valleys using laterally constrained inversion of
TEM data. J Appl Geophys 65:10–20
Bhattacharjya D, Eidsvik J, Mukerji T (2010) The value of
information in spatial decision making. Math Geosci 42:141–163
Bickel JE, Gibson RL, McVay DA, Pickering S, Waggoner J (2006)
Quantifying 3D land seismic reliability and value. Soc Petrol
Eng J 102340
Bratvold RB, Bickel JE, Lohne HP (2009) Value of information in the
oil and gas industry: past, present, and future. Soc Petrol Eng J
12(4):630–638. SPE110378PA.
Bratvold RB, Begg SH (2010) Making good decisions. Dallas, SPE
BurVal Working Group (2006) Groundwater resources in buried
valleys: a challenge for geosciences. Leibniz Institute for
Applied Geosciences. Hannover, Germany
Caers J, Avseth P, Mukerji T (2001) Geostatistical integration of rock
physics, seismic amplitudes and geological models in NorthSea
turbidite systems. Lead Edge 20(3):308–312
Caers JK (2005) Petroleum geostatistics. Society of Petroleum
Engineers, Richardson
Christiansen AV (2003) Application of airborne TEM methods in
Denmark and layered 2D inversion of resistivity data. PhD
dissertation, University of A
˚
rhus, Denmark, 136 p
Clemen RT, Reilly T (2001) Making hard decisions. Duxbury, Paciﬁc
Grove
Coopersmith E, Burkholder M, Schluze J (2006) Valueofinforma
tion lookbacks—was the information you gathered really worth
getting?. In: SPE annual technical conference and exhibition,
24–27 September 2006, San Antonio, Texas, USA. Soc Petrol
Eng J SPE Paper 101540
de Lima O, Sharma M (1990) A grain conductivity approach to shaly
sandstones. Geophysics 55(10):1347–1356
Eidsvik J, Bhattacharjya D, Mukerji T (2008) Value of information of
seismic amplitude and CSEM resistivity. Geophysics 70(4):
R59–R69
Feyen L, Gorelick S (2005) Framework to evaluate the worth of
hydraulic conductivity data for optimal groundwater resources
management in ecologically sensitive areas. Water Resour Res
41(3):13
Fitterman D, Stewart M (1986) Transient electromagnetic sounding
for groundwater. Geophysics 51(4):995–1005
Howard RA (1966) Decision analysis: applied decision theory,
proceedings of the fourth international conference on operational
research. WileyInterscience, New York, pp 55–71
Stoch Environ Res Risk Assess
123
Author's personal copy
Howard RA (1966) Information value theory. IEEE Trans Syst Sci
Cybernet SSC2(1):22–26
Houck RT (2004) Predicting the economic impact of acquisition
artifacts and noise. Lead Edge 23(10):1024–1031
Houck RT, Pavlov DA (2006) Evaluating reconnaissance CSEM
survey designs using detection theory. Lead Edge 25:994–1004
Houck RT (2007) Timelapse seismic repeatability—how much is
enough? Lead Edge 26(7):828–834
Hu LY, Chugunova T (2008) Multiplepoint geostatistics for mod
eling subsurface heterogeneity: a comprehensive review. Water
Resour Res 44:W11413. doi:10.1029/2008WR006993
Jørgensen F, Sandersen PBE, Auken E (2003) Imaging buried
Quaternary valleys using the transient electromagnetic method.
J Appl Geophys 53:199–213
Jørgensen F, Sandersen PBE (2006) Buried and open tunnel valleys in
Denmark—erosion beneath multiple ice sheets. Quatern Sci Rev
25:1339–1363
KarimiFard M, Firoozabadi X (2001) Effect of Pc on water injection
in discrete fractured media. Soc Petrol Eng J Paper 71615
Kerrou J, Renard P, HendricksFranssen H, Lunati I (2008) Issues in
characterizing heterogeneity and connectivity in nonmulti
Gaussian media. Adv Water Res 31(2008):147–159
Kirsch R, Rumpel HM, Scheer W, Wiederhold H (2006) Ground
water resources in buried valleys: a challenge for geosciences.
Leibniz Institute for Applied Geosciences (GGAInstitut),
Hannover
Mavko G, Mukerji T, Dvorkin J (1998) The rock physics handbook.
Cambridge University Press, Cambridge
Polasky S, Solow AR (2001) The value of information in reserve site
selection. Biodivers Conserv 10:1051–1058
Raiffa H (1968) Decision analysis. AddisonWesley, Reading
Reichard EG, Evans JS (1989) Assessing the value of hydrogeologic
information for riskbased remedial action decisions. Water
Resour Res 25(7):1451–1460
Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGEMS:
a user’s guide. Cambridge University Press, New York
Sandersen P, Jørgensen F (2003) Buried Quaternary valleys in
western Denmark—occurrence and inferred implications for
groundwater resources and vulnerability. J Appl Geophys
53(2003):229–248
Stamos C, Martin P, Predmore S (2012) Simulation of water
management alternatives in the Mojave river groundwater
basin, California, USGS OpenFile Report 02430
Strebelle S (2002) Conditional simulation of complex geological
structures using multiplepoint statistics. Math Geol 34(1):1–26
Strebelle S, Payrazyan K, Caers J (2003) Modeling of a deepwater
turbidite reservoir conditional to seismic data using multiple
point geostatistics. SPE J 8:227–235
Thomsen R, Søndergaard V, Sørensen K (2004) Hydrogeological
mapping as a basis for establishing sitespeciﬁc groundwater
protection zones in Denmark. Hydrogeol J 12:550–562
TrainorGuitton W, Caers J, Mukerji T (2011) A methodology for
establishing a data reliability measure for value of spatial
information problems. Math Geosci 43(8):929–949
Wagner J, Shamir U, Nemati H (1992) Groundwater quality
management under uncertainty: stochastic programming
approaches and the value of information. Water Resour Res
28(5):1233–1246
Stoch Environ Res Risk Assess
123
Author's personal copy