HYDROLOGICAL PROCESSES
Hydrol. Process. (2014)
Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/hyp.10280
Advances in interpretation of subsurface processes with
time-lapse electrical imaging
K. Singha,1* F. D. Day-Lewis,2 T. Johnson3 and L. D. Slater4
1
2
Hydrologic Sciences and Engineering Program, Colorado School of Mines, Golden, CO 80401, USA
Office of Groundwater, Branch of Geophysics, U.S. Geological Survey, 11 Sherman Place Unit 5015, Storrs, CT 06269, USA
3
Pacific Northwest National Laboratory, 902 Battelle Blvd., Richland, WA 99352, USA
4
Department of Earth and Environmental Sciences, Rutgers University - Newark, 101 Warren St., Newark, NJ 07102, USA
Abstract:
Electrical geophysical methods, including electrical resistivity, time-domain induced polarization, and complex resistivity, have
become commonly used to image the near subsurface. Here, we outline their utility for time-lapse imaging of hydrological,
geochemical, and biogeochemical processes, focusing on new instrumentation, processing, and analysis techniques specific to
monitoring. We review data collection procedures, parameters measured, and petrophysical relationships and then outline the
state of the science with respect to inversion methodologies, including coupled inversion. We conclude by highlighting recent
research focused on innovative applications of time-lapse imaging in hydrology, biology, ecology, and geochemistry, among
other areas of interest. Copyright © 2014 John Wiley & Sons, Ltd.
KEY WORDS
electrical resistivity; induced polarization; complex resistivity; imaging; petrophysics; inversion
Received 18 November 2013; Accepted 30 June 2014
INTRODUCTION
Geophysical data have long been used to characterize
Earth systems for mineral and energy exploration,
geotechnical investigations, and water resources. The
development of time-lapse technologies, in particular
those with applications in near-surface processes,
however, is more recent. Electrical methods, defined here
as direct-current (DC) electrical resistivity imaging (ERI),
time-domain induced polarization (TDIP), and complex
resistivity imaging (CRI), have become increasingly
applied to monitor dynamic systems over the last decade.
These technologies, sensitive to pore fluid conductivity,
pore space, geometry and the electrochemical interaction
at grain interfaces, are important to hydrogeologists
attempting to characterize flow and transport processes.
Time-lapse electrical images and even simple time series
of electrical measurements readily provide qualitative
information to help infer changes associated
with unsaturated flow (e.g. Daily et al., 1992; Park,
1998; Binley et al., 2002), solute transport (Slater et al.,
2000; Kemna et al., 2002; Singha and Gorelick, 2005;
Muller et al., 2010), contaminant remediation operations
(e.g. Ramirez et al., 1993; Daily et al., 1995; Truex et al.,
*Correspondence to: Kamini Singha, Hydrologic Sciences and Engineering Program, Colorado School of Mines, Golden, CO 80401, USA.
E-mail: ksingha@mines.edu
Copyright © 2014 John Wiley & Sons, Ltd.
2013), aquifer/surface-water interaction (e.g. Henderson
et al., 2010; Coscia et al., 2012), biogeochemical processes
(Atekwana and Slater, 2009), and other hydrologic
processes of interest (e.g. Kemna et al., 2006). In most
work, the goal is to translate the time-lapse electrical results
into information on variations in moisture content or
salinity, although information on other time-varying
parameters (temperature, mineral exploration, or dissolution) also has been demonstrated. Early demonstrations of
time-lapse studies were of ERI in soil cores and experimental tanks (e.g. Binley et al., 1996; Slater et al., 2002) as
well as field-scale studies (Ramirez et al., 1993). Time-lapse
CR and TDIP may provide additional information beyond
what is measured with ERI (Slater and Binley, 2006; Flores
Orozco et al., 2011).
Extraction of quantitative hydrologic information from
geophysical data or images remains an important
challenge in hydrogeophysical research, as evidenced by
a burgeoning literature focused on translation of
geophysical information into hydrologic information. In
this paper, we discuss more recent developments in this
area and, in contrast to previous reviews (e.g. Rubin and
Hubbard, 2005; Linde et al., 2006; Singha et al., 2007;
Loke et al., 2013), we focus specifically on time-lapse
electrical imaging. We outline the state of the science in
time-lapse ERI, TDIP, and CR measurements, including
data collection and analysis, inversion, and petrophysical
K. SINGHA ET AL.
relations. We follow this section with an outline of capability
advances both in data collection and inversion as well as
emerging applications specific to time-lapse studies.
DATA COLLECTION AND PETROPHYSICS
BACKGROUND
ERI, TDIP, and CR measurements
The nomenclature describing electrical measurements
and electrical properties can be unclear in meaning to the
non-expert. Common terminology used to describe the
instrument or method includes DC resistivity, electrical
resistivity (ER), electrical impedance (EI), complex
resistivity (CR), induced polarization (IP), spectral
induced polarization (SIP), and time-domain induced
polarization (TDIP). ER is the same as DC resistivity,
where measurements are made using a DC (or lowfrequency alternating current) and the system is assumed
to be in steady state during the measurement. In this case,
only charge transport by electromigration is measured. In
all other methods (EI, CR, IP, and SIP), an additional
measurement is made to quantify temporary charge
storage (via electrochemical mechanisms) in addition to
electromigration. EI is commonly used in the literature for
process and medical tomography, where frequencies
higher than those in ER are used, and will not be
discussed here. In the geophysical literature, CR, IP, and
SIP are the most common terms used to describe
these frequency-dependent measurements. The terms IP,
SIP and TDIP originate from the development of
geophysical instruments in mineral exploration, whereas
the term CR has been adopted for environmental applications (e.g. Olhoeft, 1985). In most papers, CR, IP, and TDIP
mean probing frequency-independent impedance properties. SIP, on the other hand, includes measurements at
multiple frequencies to probe frequency-dependent
impedance properties, typically from a few mHz up to
1 kHz (e.g. Kemna et al., 2012). However, some authors
also refer to CR as meaning a frequency-dependent
measurement. Here, we use ER (or ERI, when discussing
imaging specifically) to describe all approximately DC
measurements, TDIP to describe polarization measurements
made in the time domain, and CR to describe polarization
measurements made in the frequency domain.
Measurements of ERI, TDIP, and CR are collected using
two current injection electrodes (source and sink), and two
potential measurement electrodes (positive and negative).
Both current and potential electrodes may be deployed on the
surface or beneath the surface (e.g. in boreholes), or some
combination of these. The key criterion is that there is an
electrical contact between the electrode and the earth. For a
given measurement, a current waveform is injected between
the current electrodes, and attributes describing the
corresponding potential waveform are measured across
the potential electrodes. Figure 1 shows a schematic of
frequency-domain measurements, which are collected during
CR experiments, and time-domain measurements, which are
collected during ERI and TDIP experiments. Figure 2 shows
a schematic for the workflow of time-lapse geophysical
experiments and subsequent translation to hydrologic
parameters and is referred to throughout this paper.
Figure 1. (A) Example electrode configuration for surface electrical measurement including current source (I+) and sink (I) electrodes, and positive (V+) and
negative (V) electrodes. (B) Diagram of frequency-domain measurement including transmitted current source waveform injected between current electrodes,
and corresponding potential waveform observed between potential electrodes. Attributes of the potential waveform measured include the amplitude |φ| and
phase ϕ. (C) Diagram of a time-domain measurement including transmitted current waveform and corresponding potential waveform. Attributes of the potential
waveform measured include the total potential ϕ T, and the apparent chargeability Ma (see Equations (1) and (2))
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
Figure 2. Schematic diagram depicting the workflow for electrical imaging of a subsurface manipulation (e.g. tracer experiment or biostimulation) and
translation to hydrologic parameters of interest
During ERI, CR, and TDIP experiments, an alternating
square current waveform (or in the case of CR, sometimes a
sinusoidal waveform) is injected across the current electrodes.
The resulting voltage difference developed across the
potential electrodes is measured in terms of the magnitude
φT for the ERI experiment, and both φT and apparent
chargeability (Ma) for the TDIP experiment, as described in
the succeeding section. The corresponding attributes measured in the frequency domain, for CR for example, are the
frequency-dependent magnitude |φ| and phase lag ϕ relative
to the current waveform, or the real or in-phase component of
the potential response φ ′ and the complex or quadrature
component of the potential response φ ″. Both the phase lag
and the apparent chargeability develop due to charge
polarization mechanisms in the subsurface, or the ability of
the subsurface to store charge, as reviewed in the next section.
The potential waveform can equivalently be represented
by the complex potential given by
′
φ ¼ φ þ iφ
″
(1)
where
φ′ ¼ jφj cosðϕ Þ
(2)
φ″ ¼ jφj sinðϕ Þ
(3)
and
Here, i ¼
pffiffiffiffiffiffiffi
1.
Copyright © 2014 John Wiley & Sons, Ltd.
An ERI, TDIP, or CR survey is acquired by collecting
many measurements on different current and potential
electrode pairs strategically chosen to provide adequate
imaging resolution. Time-domain electrical measurements
are attractive for electrical monitoring as they are generally
quicker to acquire than multi-frequency, frequency-domain
measurements. The time-domain measurement of the
charge storage is Ma, defined as (Siegel, 1959; Oldenburg
and Li, 1994)
Ma ¼
φM
φT
(4)
where φM is the part of the total potential (φT) arising due to
charge polarization mechanisms (Figure 1). In practice, φM
is difficult to measure precisely, and Ma is typically
approximated by
t2
Ma≈
1 1
∫ φðtÞdt
Δt φT t1
(5)
where Δt = t2 t1 is the length of the sampling window and
φ(t) is the voltage at time t (Figure 1). According to
Equations (4) and (5), φM is approximated as the average
potential recorded during the IP time window. Note that Ma
is partly a function of the settings of the TDIP instrument
and thus not an intrinsic property of the Earth, requiring
quantitative data-processing algorithms to decouple the
instrument response given some assumptions about the
phase angle (e.g. Kemna et al., 1997). Some IP instruments
measure a property known as percentage frequency effect
Hydrol. Process. (2014)
K. SINGHA ET AL.
(PFE) that quantifies the decrease of impedance magnitude
with frequency associated with polarization. PFE is not
typically used in time-lapse imaging but is a traditional IP
measure used in mining exploration.
Although Ma depends on the configuration of the TDIP
instrument settings, a linear proportionality between Ma and
ϕ can be both theoretically shown (Slater and Lesmes, 2002)
and experimentally demonstrated (Lesmes and Frye, 2001;
Slater and Lesmes, 2002; Mwakanyamale et al., 2012). A
linear proportionality between PFE, Ma, and ϕ can also be
expected (Slater and Lesmes, 2002). To be clear, a summary
of the measurement types provided by ERI, TDIP, and CR
measurements is given in Table I.
Low-frequency electrical properties
Electrical methods operate at low frequencies where
displacement currents, which are time-varying and
associated with magnetic fields, are insignificant in
comparison with galvanic, or direct, currents ( f < 1 kHz).
At these frequencies, the relationship between current,
potential, and electrical properties of the Earth is defined
by the Poisson equation. Here, we consider a complex
electrical conductivity (σ*) that contains information on
both electromigration and polarization, such that the
Poisson equation is given as
∇σ ðr; ωÞ∇φ ðr; ωÞ ¼ I ðr0 ; ωÞ
1
¼ iωε
ρ
Table I. Electrical technologies and measured properties
Method
ERI
TDIP
CR
Apparent
chargeability
Ma (V/V)
X
X
X
Frequencydependent
potential
magnitude
|φ|(ω) (V)
Frequencydependent
phase shift
ϕ(ω) (rad)
X
X
Copyright © 2014 John Wiley & Sons, Ltd.
(8)
At the low excitation frequencies employed for electrical
monitoring (typically less than 1000 Hz), electromigration
dominates over polarization, i.e. σ″ < < σ′. The phase angle,
ϕ ¼ tan1 σ″ =σ′ ≅ σ″ =σ′
(9)
determines the rotation of the resultant conductivity vector
away from the real conductivity axis and is small (typically,
ϕ < 100 mrad); the approximation shown in Equation (9) is
therefore often valid. While real and imaginary conductivities
and phase angles are measured with CR, ERI only estimates
the magnitude of σ*,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(10)
jσj ¼ ðσ′ Þ2 þ ðσ″ Þ2 ≅ σ′
where the approximation shown again holds for small phase
angles. In non-metallic soils, this low-phase assumption is
generally valid, but it can be violated over metal-rich soils
where phase angles of a few hundred mrad are possible due to
the strong polarization properties of disseminated metallic
minerals. As with the complex potential, the real and
imaginary conductivities are related to the conductivity
magnitude and phase by
σ′ ¼ jσj cos ϕ
(11)
σ″ ¼ jσj sin ϕ
(7)
as each of these parameters contains an energy storage
(polarization) part and an electromigration (conduction)
Total
potential
φT (V)
σ* ¼ σ′ þ iσ″
(6)
where I is the sinusoidal point current source injected at
position r0 and frequency ω, and φ* is the corresponding
complex potential field at position r and frequency ω.
Note that for ER, the equivalent conductivity and
potential terms appearing in Equation (6) are the real
terms |σ| and φT, respectively.
The measured complex conductivity describing the
properties of the Earth can equally be defined as a
complex electrical conductivity (σ*), a complex resistivity
(ρ*), or a complex permittivity (ε*),
σ ¼
part. Here, we use σ* for convenience when describing
measurements in terms of the parallel conduction model
commonly used to relate electrical properties to physicochemical properties of the subsurface, as described later. In
terms of σ*, the in-phase (real, σ′) conductivity component
is the property defining the ability of the Earth to transport
charge via electromigration, whereas the out-of-phase
(imaginary or quadrature) conductivity (σ″) is a property
defining the strength of the polarization associated with a
localized redistribution of charge,
Early formulations related the current and potential
to the electrical properties using the concept of an
intrinsic chargeability (M), which relates the conductivity (at DC) of a polarizable material (|σ|) to a
decreased conductivity of a non-polarizable material
(σnp) (Siegel, 1959),
σnp ¼ jσjð1 M Þ
(12)
based on measurement of Ma using TDIP instrumentation. The chargeability (M) is then related to the total
potential and the current by
∇jσjð1 M Þ∇φT ¼ I
(13)
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
Electrical petrophysics
The electrical properties sensed with ER, TDIP, and CR
measurements depend on the pore fluids and pore architecture.
The additional properties sensed with CR and TDIP are also
modified by mineral dissolution and precipitation, biomineralization, reactive contaminant transport, and biodegradation
of hydrocarbon contaminants. In other words, time-lapse
inverse estimates of polarization properties (i.e. ϕ or M in
addition to |σ|) provide an opportunity to monitor processes
that modify the physicochemical properties of the grain–fluid
interface. These processes may not be detectable using inverse
estimates of |σ| alone. We demonstrate the basic relations using
the real and imaginary parts of the complex conductivity
defined earlier (e.g. Lesmes and Friedman, 2005).
Most models for σ* of a porous material devoid of metals
at low frequencies (e.g. less than 100 Hz) are based on a
parallel addition of two conduction terms representing (1) an
electrolytic contribution via electromigration through the
interconnected pore space (σel) and (2) a complex mineral
surface conduction contribution (σ*surf) (e.g. Vinegar and
Waxman, 1984),
σ ¼ σel þ σsurf
(14)
Here, σ*surf arises due to both electromigration and
polarization of ions in the electrical double layer (EDL)
that forms at mineral–fluid interfaces. In aquifers containing
potable water, the EDL is most commonly considered a
layer of net negative ions adsorbed on colloidal particles that
attracts a layer of net positive ions in the surrounding
electrolytic solution (Grahame, 1947; Davis et al., 1978).
Other polarization mechanisms, such as the Maxwell–
Wagner mechanism that results from charge accumulation
at contrasts in electrical conductivity (dominant in the kHz
range) and molecular polarization (dominant in the GHz
range), are assumed to be insignificant at low frequencies.
For a fully saturated medium,
σ′ ¼
1
σw þ σ′surf
F
(15)
and
σ″ ¼ σ″surf
(16)
where F is the electrical formation factor and σw is the
fluid conductivity. The true formation factor F is related
to the interconnected porosity (n) of a fully saturated soil
via Archie’s law (Archie, 1942),
F ¼ nm
(17)
where m is the cementation exponent ranging from 1.3 for
clean sand to 4.4 for Mexican Altered Tuff (a clay-rich
mineral) (Lesmes and Friedman, 2005).
Copyright © 2014 John Wiley & Sons, Ltd.
Equation (16) states that the measured σ″ is exclusively
controlled by the surface conductivity, whereas the
measured σ′ is controlled by both electrolytic and surface
conductivity terms (Equation (15)). For simplicity,
surface conduction is often considered to be negligible
at high fluid salinities. Weller et al. (2013) show that this
assumption is typically only valid above 1000 mS/m and
that unreliable estimates of F and σw will result from
single salinity measurements if surface conduction is
ignored below these salinities (i.e. in the range of potable
aquifers). In such a situation, only an apparent formation
factor is determined; measurements at multiple high
salinities are needed to reliably compute the true
formation factor (e.g. Weller et al., 2013).
The effect of partial saturation on σel is most commonly
accounted for using a second power law such that
σel ¼ σw nm Sn
(18)
where S is the saturation (from 0 to 1) and n is the saturation
exponent that typically ranges between 1.3 to 2.7 (Schön,
1996) and is approximately 2 for unconsolidated sediments.
From consideration of the electrolytic conductivity alone, a
well-founded interpretational framework therefore exists to use
electrical monitoring to track changes in σw, n, and moisture
content, θ, (θ = nS) in the subsurface. This opportunity in large
part explains the growing popularity of electrical monitoring
within the hydrogeophysical community.
Empirical and mechanistic formulations for the surface
conductivity now exist (e.g. Revil et al., 2012) but are not
as well established as for σel via Archie’s classic law.
Such formulations describe the surface conductivity in
terms of (1) a volume-normalized surface area of the
material or the cation exchange capacity and (2) the
surface charge density and surface charge mobility within
the EDL (Waxman and Smits, 1968; Rink and Schopper,
1974; Vinegar and Waxman, 1984). In a study based on
114 samples from 10 independent datasets, Weller et al.
(2010) defined the following simple linear relation,
σ″ ¼ cp Spor
(19)
where Spor is the pore volume-normalized specific surface
area (typically determined from a Brunauer–Emmett–Teller
(BET) nitrogen adsorption and porosity measurement) and
cp is the specific polarizability invoked to describe the role
of EDL chemistry on surface polarization. Weller et al.
(2010) suggested that Equation (19) represents the equivalent of Archie’s law for the surface conductivity. The
concept of specific polarizability accounts for the fact that
some minerals are more polarizable than others. For
example, metallic minerals have a much higher polarizability per unit Spor than non-metallic minerals due to the
tendency of the metal to engage in redox reactions with ions
in the pore fluid. When the full complex conductivity is
Hydrol. Process. (2014)
K. SINGHA ET AL.
measured, the opportunity therefore exists to use electrical
monitoring to track changes in mineral surface area, as well
as changes in the chemistry of the EDL. Numerous
biogeochemical processes associated with contaminant
transformations, mineral–fluid chemistry, and microbial
activity have the potential to impact the electrical properties
of mineral–fluid interfaces. This fact in large part explains
the recent surge of interest in CR monitoring for
hydrogeophysical and biogeophysical applications
(Atekwana and Slater, 2009; Williams et al., 2009).
Given the opportunities to sense geochemical and
biogeochemical processes that cannot be captured with a
measurement of |σ| alone, interest in CR imaging is likely to
grow. When measured, the frequency dependence of the
complex surface conductivity can provide further information on the physical properties of the subsurface.
Polarization only occurs when charge cannot move freely
by electromigration such that a local accumulation of charge
results from application of an electrical field. For any
particular length scale where electromigration is discontinuous, a critical frequency will exist where this polarization
response is maximized. A relationship between this length
scale (dl) and the time constant (τ) describing this
polarization response is often formulated in terms of the
diffusion coefficient (D) for the ions in the EDL,
dl 2 ¼ τD
(20)
This length scale is usually equated to a grain diameter or
a pore length. The shape of the polarization spectrum can
therefore be related to the distribution of polarization length
scales within the porous medium, e.g. a distribution of grain
sizes (e.g. Lesmes and Morgan, 2001; Leroy et al., 2008) or
pore sizes (Titov et al., 2002; Tong et al., 2006). Assuming
that spectral measurements can be reliably obtained, CR
monitoring therefore also has the potential to track changes
in the distribution of grain or pore sizes (e.g. due to
cementation or dissolution) with time.
NUMERICAL MODELLING BACKGROUND
Analysis of electrical geophysical data commonly entails
solution of the forward problem (i.e. simulation of
measurements given physical parameters) and solution
of the inverse problem (i.e. estimation of physical
parameters given measurements). The forward problem
is commonly solved numerically using, for example the
finite-difference or finite-element method to identify
solutions to the governing partial differential equation
(e.g. Equation (6)). The inverse problem is commonly
solved using optimization methods to identify the 2-D or
3-D cross section or volume of electrical parameters that
provide the best fit to the measurements. In this section
Copyright © 2014 John Wiley & Sons, Ltd.
we review approaches to the forward and inverse
problems in time-lapse electrical geophysics.
ERI/TDIP/CR forward problem
The relationship between subsurface complex conductivity, potential, and current in the frequency domain is defined
by the Poisson equation given in Equation (6). The objective
of the forward solution is to simulate the potential field
generated by the current source and extract the simulated
voltages. The solution is typically produced by solving the set
of matrix equations arising from the discretization of
Equation (6) according to an appropriate numerical scheme
(e.g. Dey and Morrison, 1979a; Dey and Morrison, 1979b;
Rucker et al., 2006). The particular simulated measurement is
then produced by subtracting the simulated potential at the
negative potential electrode from the simulated potential at
the positive electrode. In cases where every electrode is used
as a source during a survey, it is more efficient to compute
the pole solution for each electrode than it is to compute
the dipole solution for each measurement. The pole solutions
(i.e. the solution for a current injection source with a current
sink at infinite distance) may be used to form the dipole
solutions arising from any source electrode pair by
superposition, thereby requiring only ne forward simulations
as opposed to nm, where ne is the number of electrodes and
nm is the number of measurements.
Although efficient algorithms for solving Equation (6)
have been available for some time (e.g. Saad and Schultz,
1986; Van der Vorst, 1992), recent developments in
meshing algorithms (Si, 2006) have facilitated greater
accuracy in forward solutions through explicit modelling
of known conductivity boundaries into the numerical
mesh. Accurate forward solutions are critical for
optimizing ERI/TDIP/CR resolution because they enable
misfits between observed and simulated data to be
reduced in the inversion process, and they enable accurate
Jacobian matrix calculations as discussed in the next
section, ultimately improving imaging resolution. For
example, Rucker et al. (2006) and Gunther et al. (2006)
presented a 3-D finite-element algorithm using meshes
generated by advanced unstructured tetrahedral mesh
generation software (Si, 2006) to accurately model
surface topography. To ease the computational demands
of 3-D inversion, they used the same meshing software to
produce a finely discretized forward mesh within a
concurrent coarsely discretized inversion mesh. Doetsch
et al. (2010) used the same algorithm to explicitly
incorporate fluid-filled borehole boundaries into the
forward and inversion mesh, which enabled smoothing
constraints, described in detail in the succeeding
paragraphs, to be relaxed between the inside and outside
of the borehole, which effectively reduced the borehole
artefacts that otherwise were evident in ERI images.
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
Using synthetic models, Robinson et al. (2013a) used a
similar approach to demonstrate how explicitly incorporating wells and dominant fractures into the mesh
substantially improved characterization and time-lapse
imaging of tracer movement within a fracture. They
investigated the potential of the approach on field data to
monitor groundwater movement within a dominant
fracture during pumping at a carboniferous limestone
quarry (Robinson et al., 2013a). Wallin et al. (2013) used
a mesh with finely divided unstructured triangular
elements to relax smoothing constraints across the
transient, but known, water-table boundary during an
experiment to image stage-driven river water intrusion
into a contaminated aquifer using 2-D time-lapse ERI.
The meshing that enabled the inversion to place a sharp
conductivity boundary across the moving water table
substantially improved the accuracy of the images.
ERI/TDIP/CR optimization problem
In the context of this paper, inversion is the numerical
process whereby a discretized estimate of subsurface
electrical conductivity (real or complex), called a
tomogram, is generated that (1) honours the data acquired
during a survey to a degree consistent with data noise and
(2) accurately represents the subsurface electrical conductivity at the spatial scale that the data can resolve
(Figure 2, step 2). This estimate is referred to herein as the
inverse solution. Because the inverse problem is generally
non-unique, there are an infinite number of solutions that
satisfy the first condition (Backus and Gilbert, 1968), and
the second condition is not automatically satisfied by the
first. To find a solution that satisfies both conditions, two
types of constraints are placed on the inversion: data
constraints that force condition 1, and solution constraints
that force condition 2. The solution constraints are often
referred to as regularization constraints (Tikhonov, 1963)
and, following Occam’s principle (Constable et al., 1987;
LaBrecque et al., 1996), enforce parsimonious solutions
that only resolve the conductivity structure that is
required to honour the data.
In time-lapse imaging, a chronological sequence of
inverse solutions is generated, with the objective that each
solution honour conditions 1, 2, and a third condition (3)
that the inverted sequence provides an accurate representation of subsurface conductivity at the temporal scale
resolvable by the data, which is determined primarily by
the time required to conduct a single survey. As with the
spatial dimensions, it is typically assumed that conductivity varies smoothly in the time dimension. Although
explicit solution constraints to enforce this condition are
not required, many practitioners have found it difficult to
produce smoothly varying conductivity in the time
dimension without some form of transient solution
Copyright © 2014 John Wiley & Sons, Ltd.
smoothing constraint (Cassiani et al., 2006). Hence,
many of the original and recent advancements in
time-lapse inversion have focused on methods of
imposing transient solution constraints through analysis
of difference data (e.g. Daily et al., 1992; LaBrecque and
Yang, 2001), differencing of multiple inversions using
images from previous time steps to define constraints or
prior information (e.g. Miller et al., 2008), or temporal
regularization constraints (e.g. Day-Lewis et al., 2003;
Kim et al., 2009; Karaoulis et al., 2011b).
The objective of the inversion is to minimize a function
of the general form (Farquharson and Oldenburg, 1998)
Φ ¼ Φd ðud Þ þ Φs ðus Þ
(21)
where Φd and Φs represent the data and solution norms of
the inverse solution(s). These norms are, respectively, a
scalar measure of the misfit between the observed and
predicted data, and a scalar measure of the misfit between
the inverse solution structure and the structure imposed by
the solution constraints in both space and time. The vectors
ud and us represent the data and solution constraint residual
(or misfit) vectors, respectively. The specific form of us
depends on how the spatial and temporal constraints are
implemented, and generally defines the time-lapse inversion
approach. The cascaded time-lapse inversion approach
(Miller et al., 2008), for example, which is a slight
modification of a common implementation for static
inversions, and the starting point for the succeeding
discussion, has the objective function
2 t1 2
(22)
Φ ¼ Wd dtobs dtsim þβWs mtest mest
Here Φ d = Φs = ‖‖2 represents the L2-norm operator,
ud ¼ Wd diobs dtsim is the data misfit term, and
i1
us ¼ βWs miest mest
is the model misfit term where
β is known as a trade-off parameter that determines the
relative weight on the misfit terms. Here, dtobs is the vector
of observed data at time t, dtsim is the corresponding
simulated data which are produced as described in the
forward modelling section, and Wd is the data error
covariance matrix or data weighting matrix, discussed in
the following paragraphs. The vectors mtest and mt1
est are
the inverse solutions at times t and t 1, respectively.
The regularization or solution constraint matrix Ws is
used to impose constraints on the inverse solution. For
example, Miller et al. (2008) used Ws to impose
smoothness constraints on mtest mt1
est , thereby imposing
t
a smooth temporal transition from mt1
est to mest . In this
t1
case, mest is provided from a previous time-lapse
inversion, or from a static inversion if mt1
est is the baseline
solution. Constraints also can be used to restrict timelapse changes to regions in space where data indicate that
Hydrol. Process. (2014)
K. SINGHA ET AL.
changes are occurring (e.g. Day-Lewis et al., 2003;
Karaoulis et al., 2011b).
Continuing with this example, the objective of the
inversion is to find a solution, mtest , that minimizes
Equation (22), subject to appropriately fitting the data (see
the Section on Data Weighting). The standard nonlinear
least squares solution for model updates is given by
T T
J Wd Wd J þ βWTs Ws δmtest
¼ JT WTd Wd dtobs dtsim
(23)
þβWTs Ws mtest mt1
est
where J is the Jacobian matrix with elements Jij being the
sensitivity of measurementt i with respect to the
∂d
t
discretized conductivity j ( ∂msim;i
t ). The vector δmest is a
j
solution update vector which, when added to mtest ,
decreases the value of the objective function. Equation
(23) forms a series of matrix equations that are typically
solved using a Gauss–Newton iteration algorithm.
Although the details vary for other time-lapse inversion
implementations, the general objective function formulation and solution strategy are similar to that presented
earlier. For example, Kim et al. (2009), Karaoulis et al.
(2011a), and Hayley et al. (2011) all presented different
variants of time-lapse inversion methods that simultat
neously invert for mt1
est and mest (or in theory for any
number of time steps) using augmented and slightly
modified versions of Equation (23). Each time-lapse
inversion approach presented in the literature has
advantages and disadvantages, such that the optimum
approach often depends on the application. For example,
the simultaneous inversion approaches mentioned earlier
are flexible and theoretically appealing but computationally demanding and impractical for large inverse
problems. The difference inversion method presented by
LaBrecque and Yang (2001) is computationally efficient
but requires that changes in conductivity with time are
small enough that the same Jacobian matrix can be used
for each time-lapse inversion.
As a note, inverse estimation of |σ| from ER measurements is relatively straightforward in comparison with CR
monitoring. The signal-to-noise ratio of the polarization
measurements (ϕ or Ma), made via CR, is typically 2–3
orders of magnitude smaller than for the corresponding
measurement of potential magnitude. Electromagnetic and
capacitive coupling between input and output channels, and
between these channels and the earth, can render the
measurements unusable. A rigorous assessment of measurement errors in CR is necessary to prevent noisy data
from generating image artefacts (Flores Orozco et al., 2012;
Mwakanyamale et al., 2012). Furthermore, the additional
information comes at the cost of longer data acquisition
times. Despite these limitations, inversion algorithms for
interpretation of time-lapse CR data have recently become
Copyright © 2014 John Wiley & Sons, Ltd.
available beyond the individual developer (Karaoulis et al.,
2011a; Karaoulis et al., 2013).
Data weighting
Data weighting refers to the scaling factors applied to
observed and simulated measurements in order to give
each the appropriate influence within the objective
function. For example, noisy measurements have greater
uncertainty and therefore should not have the requirement
of being closely fit by the simulated data. Data weights
are applied with the data covariance matrix Wd.
Measurement errors are typically assumed to be random
and uncorrelated such that Wd is a diagonal matrix with
each diagonal element being the reciprocal of the standard
deviation for the corresponding measurement. Errors are
generally quantified one of two ways: via stacking –
collecting the same measurement more than once, or by
reciprocal measurements – which are a measurement of
voltage with the current and voltage electrodes swapped.
In either case, the duplicate or reciprocal measurement
should provide the same data as the original in the
absence of noise, and any difference can be used as an
estimate of error. Given these errors, the first term of the
objective function in Equation (22) becomes the rootmean-square (RMS) error given by
RMS ¼ kWd ðdobs dsim Þk2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u nm uX dobs;i dsim;i 2
t
¼
sd2i
i¼1
(24)
where sdi is the standard deviation for measurement i and
nm is the number of measurements. Note that the quantity
within the radical is the χ 2 value. Both the RMS and χ 2
values are commonly used to determine convergence.
Assuming (1) data noise has zero mean and is normally
distributed, (2) measurement standard deviations
accurately represent the true error for each measurement,
and (3) the forward model simulation error is insignificant
with respect to the measurement error, the χ 2 value should
be equal to nm when the data are appropriately fit. With
normalization by nm, both the χ 2 and RMS values should
be equal to 1 when the data are appropriately fit. In
practice, estimates of measurement standard deviation
produced by repeat measurements and/or reciprocal
measurements often produce values which, when applied
to derive an appropriate convergence criteria by Equation
(24), are too small to be of use. Either the inversion
cannot fit the data to the degree suggested by the
estimated deviations, or if it can, the solution at RMS = nm
is obviously over fit. Assuming standard deviation
estimates are accurate and condition 1 above holds, this
suggests condition 3 is violated; the forward model is
unable to model the data to the precision consistent with
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
noise in the field. Indeed, forward modelling errors are
often assumed to be adequately small if they are less than
2% of transfer resistance magnitude. Data errors estimated through repeat and reciprocal measurements can be
smaller than the modelling errors. This suggests that, to
achieve optimal imaging resolution by adequately fitting
the data, forward modelling errors should be reduced as
much as possible. Forward model accuracy can be
improved by, for example, appropriately refining forward
meshes, accurately modelling surface topography and
other known conductivity interfaces, appropriately handling boundary conditions – for example, using the
known no-flux boundaries in tank experiments, and using
solution improvement techniques such as singularity
removal when possible (Lowry et al., 1998). In addition,
the conductivity discretization must be adequately fine for
the inversion to make smaller-scale changes that might be
required to honour the data. This has implications for
approaches where forward modelling is executed on a fine
mesh, and the inversion solution is done on a coarser
mesh to reduce computational demands. That is, if the
inversion mesh is too coarse, a conductivity distribution
that enables an adequate data fit is not possible, and the
resolving potential of the data is not fully realized.
TRANSLATING GEOPHYSICAL IMAGES INTO
HYDROLOGIC INFORMATION
Strategies for the translation from time-lapse electrical
tomograms to quantitative hydrologic information can be
divided into five general categories: (1) conversion by
petrophysical transformation, (2) correlation in time, (3)
condensing to summary information, (4) calibration of
process models, and (5) coupled inversion; these categories
are represented by step 3 in Figure 2. We note that other
categorizations are possible but suggest that this scheme
captures most of the ongoing work to capitalize on time-lapse
geophysical imaging. These five general strategies provide
different levels of information and require varying degrees of
different limiting assumptions and approximations.
Conversion by petrophysical transformation
The most straightforward approach to translate
geophysical information to hydrologic information is
through simple application of a petrophysical relation,
such as those listed previously (e.g. Equations 15–18), or
some empirical relation derived from laboratory or field
experimental data (e.g. by linear or nonlinear regression)
(e.g. Purvance and Andricevic, 2000; Bowling et al.,
2006). This approach falls under ‘direct mapping’ in the
classification proposed by Linde et al. (2006). The
petrophysical transformation is applied to convert each
time-lapse image in turn, producing a time-lapse 2-D
Copyright © 2014 John Wiley & Sons, Ltd.
cross section or 3-D volume of the hydrologic parameter
of interest. The strength of conversion-based approaches
lies in their simplicity, but the weaknesses of these
approaches are severe.
In past work monitoring flow and transport with ERI,
the limitations of conversion-based approaches manifested as poor recovery of spatial and temporal moments of
tracer plumes. For example, Binley et al. (2002) noted a
50% mass-balance error in their effort to monitor a fluid
tracer with ERI. Singha and Gorelick, (2005) and Muller
et al. (2010) observed similar or larger mass-balance
errors in their effort to monitor tracer experiments with
ERI. Singha and Gorelick (2005) offered three explanations for these problems. First, unresolved lithologic/soil
heterogeneity can result in a petrophysical relation with
parameters that vary spatially in a manner that is difficult
or impossible to map; consequently, the conversion is nonunique. Second, there commonly is a discrepancy between
the support volumes of hydrologic and geophysical
parameters; i.e. the two are averaged over different
volumes of rock or soil, and this averaging complicates
conversion. Third, geophysical image resolution can vary
in space and time, resulting in spatially and time-varying
loss in correlation between the estimated geophysical
parameter and hydrologic parameter of interest (e.g. Cassiani
et al., 1998; Day-Lewis and Lane, 2004; Day-Lewis et al.,
2005). The second and third issues are linked through the
physics underlying hydrologic and geophysical
measurements. The third issue, furthermore, is a function
of choices made in the inversion of both data types.
Assumptions that are necessary to constrain the inverse
solution (e.g. regularization, inversion constraints, and prior
information) can provide results that degrade the correlation
between estimated geophysical parameters and the hydrologic parameter of interest. In the last decade, considerable
effort has been devoted to understanding and addressing
these issues to develop more effective strategies to convert
tomograms to hydrologic parameter estimates.
Geophysical measurement support volume, sensitivity,
and tomographic resolution are longstanding topics of
research and well-established issues in the interpretation
of geophysical images (Menke, 1989; Ramirez et al.,
1993; Oldenburg and Li, 1999; Alumbaugh and Newman,
2000; Friedel, 2003; Dahlen, 2004). Geostatistical
approaches, such as co-kriging and conditional simulation
(e.g. McKenna and Poeter, 1995; Cassiani et al., 1998),
and Bayesian frameworks (Hubbard and Rubin, 2000)
can convert geophysical images to hydrologic estimates
while accounting for uncertainty in the parameters of
petrophysical relations, but the petrophysical relation may
not hold at the field scale or be appropriate for an inverted
image. For time-lapse imaging, identifying an effective
field-scale petrophysical relation is especially problematic, as measurement sensitivity and measurement error,
Hydrol. Process. (2014)
K. SINGHA ET AL.
and thus image resolution, may vary in time as well as
space; moreover, temporal smearing and aliasing can
occur as a result of the non-negligible time required to
collect a geophysical imaging dataset and the time
interval between datasets.
Several statistical approaches have been devised to
develop field-scale calibrations to convert geophysical
tomograms to estimates of hydrologic parameters. DayLewis and Lane (2004) used random field averaging
(RFA) to predict the field-scale correlation between two
properties (i.e. geophysical and hydrologic) correlated at
the point scale, when one property is found by inversion
and thus resolution-limited. Day-Lewis et al. (2005)
applied this approach to ERI, providing insight into the
pattern of correlation loss for ERI (Figure 3). Moysey
et al. (2005) developed an approach, Full Inverse
Statistical (FISt) calibration, to convert geophysical
tomograms to estimates of hydrologic parameters
accounting for the imperfect resolution of geophysical
tomograms. Their conversion strategy entails Monte
Carlo simulation of numerical analogues of the geophysical experiment, generating an ensemble of tomograms
from which a field-scale petrophysical relation is
developed. Singha and Gorelick (2006) applied the
approach to a field-scale ERI problem. The concept of
‘apparent petrophysical relations’, outlined in these
papers, provides a means to upscale point-scale relations,
as derived in the laboratory, to the field scale while
accounting for survey geometry, measurement physics,
regularization, and measurement error. Singha et al.
(2007) developed apparent petrophysical relations using
RFA and FISt and found that the two approaches
produced similar results for linear problems; however,
the RFA approach likely breaks down where assumptions
of second-order stationarity, Gaussian errors, and a linear
forward model are violated. Although FISt relaxes these
assumptions, the approach still requires some knowledge
of the spatial models of geophysical and hydrologic
variability, an underlying petrophysical model linking the
geophysical and hydrologic properties, and quantification
of measurement errors. Such information is commonly
not available a priori.
Correlation in time
Simple comparison of inverted images between time
steps is no longer the state of the science. Such qualitative
interpretation of time-lapse tomograms can provide
valuable insight into subsurface properties and changes
associated with diverse hydrologic processes and engineering practices; however, the objectives of time-lapse
geophysical surveys are increasingly quantitative in
nature. Time-lapse geophysical tomograms provide rich
datasets amenable to analysis using classical time series
Copyright © 2014 John Wiley & Sons, Ltd.
and spectral frameworks. However, such analysis has
only recently been reported. Johnson et al. (2012) and
Wallin et al. (2013) examined cross-correlation between
time series of 3-D and 2-D conductivity tomograms and
Columbia River stage proximal to the Hanford 300 Area
near Richland, Washington, to infer aquifer/stream
interaction. Correlation coefficient, time lag, and coefficient of variation between river stage and conductivity all
revealed preferential pathways (Figure 4) connecting the
aquifer and river. This work required no assumption of a
petrophysical model yet provided quantitative information for the timing of aquifer/river interaction. Timefrequency analysis using the S-transform provided
additional insight into non-stationary behaviour (Johnson
et al., 2012).
Condensing to summary statistics
Given direct sampling of tracer concentration or
moisture associated with an injection experiment or
infiltration experiment, it is common practice to condense
the sampled data to temporal or spatial moments, from
which plume morphology and evolution are inferred. In
the case of solute concentration, spatial and temporal
moments provide valuable insight into controlling
processes such as advection, dispersion, and rate-limited
mass transfer (e.g. Freyberg, 1986; Goltz and Roberts,
1987; Garabedian et al., 1991; Harvey and Gorelick,
1995b). The 3-D spatial moments at time t and the 1-D
temporal geometric moments at location (x,y,z) of
concentration C are given, respectively, by
i j k
mspatial
i; j;k ¼ ∫∫∫x y z C ðx; y; z; t Þdx dy dz; and
¼ ∫tl C ðx; y; z; tÞdt
mtemporal
l
(25)
(26)
where i, j, k, and l, are the moment orders for x, y, z, and t,
respectively.
Time-lapse images comprise valuable spatially distributed time series relevant to hydrologic processes and can
be summarized similarly to solute concentration or
moisture, as spatial or temporal moments of electrical
conductivity; these geophysical moments have been taken
as surrogates for moments of injected tracer (Singha and
Gorelick, 2005; Ward et al., 2010a) and moisture content
(Binley et al., 2002; Haarder et al., 2012). Time-lapse
ERI results for tracer injection experiments also have
been condensed to effective velocity and dispersivity
values, termed effective convection–dispersion equation
(CDE) parameters (Kemna et al., 2002; Vanderborght
et al., 2005; Koestel et al., 2008; Muller et al., 2010). In a
numerical study, Vanderborght et al. (2005) demonstrated
that with high-resolution ERI, the effective velocities
derived from ERI are strongly correlated with hydraulic
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
conductivity. In controlled laboratory experiments,
Koestel et al. (2008) found good agreement between
apparent CDE parameters from ERI and effective
parameters calculated from breakthrough curves.
The same issues impeding conversion between
geophysical and hydrologic estimates manifest to varying
degrees in calculation of moments, CDE parameters, or
other summary statistics. Day-Lewis et al. (2007)
evaluated the recovery of plume moments by tomograph-
ic imaging as a function of survey geometry, measurement error, and regularization. Although their examples
focused on linear ray-based tomography (i.e. not ERI,
TDIP, or CR), Day-Lewis et al. (2007) found that the
conventional pixel-based parameterization and Tikhonov
regularization (Tikhonov and Arsenin, 1977) commonly
used for electrical inversion can produce images from
which inferred moments only poorly approximate true
plume moments. In addition to issues of incomplete or
Figure 3. Synthetic models from Day-Lewis et al. (2005) demonstrating how spatially variable resolution in ERI images impact estimated hydrologic parameters (in
this case, water content based on cross-well data collection, hence the longer z-axis than x-axis). Cross sections of (a) true water content and (b) true resistivity, (c) the
log10 diagonal of the model resolution matrix, (d) the inverted tomogram of ln(resistivity), and (e) the predicted correlation coefficient between true and estimated ln
(resistivity). In the bottom are a series of assumed (white line) and estimated (surface plot) petrophysical relations from this example
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. (2014)
K. SINGHA ET AL.
limited survey coverage, regularization, and imperfect
resolution, another issue in time-lapse electrical data is
temporal smearing. Commonly, time-lapse electrical
data are inverted with the assumption that changes
in resistivity structure during data collection are
negligible, i.e. data acquisition for a single image is
substantially faster than any changes occurring as a
result of tracer migration, infiltration, or other processes
under study. Koestel et al. (2008) suggest that
calculation of effective CDE velocity and dispersivity
is more robust than calculation of moments from
images, but the resolving power of tomographic
imaging still limits the quality of results.
Recently, the ERI inverse problem has been reformulated
to invert directly for plume geometry (e.g. Miled and Miller,
2007). Pidlisecky et al. (2011) formulated the inverse
Figure 4. Summary statistics were extracted from time-lapse ERI by Wallin et al. (2013) to understand aquifer/river interaction along the Columbia River at the
Hanford 300 Area near Richland, Washington, USA. (Top) Time-lapse ERI images were collected on electrode lines 1–3. Statistics for line 1 are shown below.
(Bottom) (A) Normalized river stage and conductivity time series at the five example image pixel locations shown in (B). (B) Maximum correlation between
river stage and conductivity time series in each image pixel, indicating where river and aquifer between ERI lines are most hydraulically connected. (C) Lag time
to maximum correlation between river stage and conductivity in each pixel, which is indicative of flow velocity, (D) coefficient of variation of bulk conductivity
time series in each pixel, indicating most active zones of groundwater/river water interaction, (E) peak-to-peak time shift between stage and conductivity time
series, which is indicative of flow velocity. Red electrode points are located over former waste infiltration galleries
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
problem for distribution-based parameters, i.e. the parameters of a Gaussian and/or log-Gaussian plume. Laloy et al.
(2012) parameterized a tomographic inverse problem using
spatial orthogonal moments, which are linear combinations
of the conventional geometric moments (Equations (25) and
(26)). These alternative parameterizations effectively condense the information captured in the time-lapse geophysical datasets to a handful of summary statistics, each of
which provides direct insight into the transport process and
may be more interpretable than an over-parameterized
image. For example, the zeroth-order geometric moment
gives the total mass (controlled by decay processes), the first
and zeroth together give the centre of mass (controlled by
advection), and the second and zeroth together give the
spread (controlled by dispersion and heterogeneity). In
principle, it is possible to calculate parameters controlling
solute transport directly from temporal moments. For
example, for solute transport with dual-domain mass
transfer, the mass-transfer rate coefficient and the ratio of
immobile to mobile porosity can be calculated as simple
linear combinations of the temporal moments of electrical
and fluid data (Day-Lewis and Singha, 2008). Both
Pidlisecky et al. (2011) and Laloy et al. (2012) found
improved recovery of plume morphology and mass using
this new paradigm for inversion of time-lapse ERI data.
Knowledge of plume moments (or effective CDE
parameters) provides quantitative insight into hydrologic
processes and controlling parameters. Although valuable as
a final result of analysis, such condensed information may
be useful in further analysis involving model calibration. As
reviewed subsequently, calibration of process models to a
few inferred moments or CDE parameters is an efficient
alternative to calibration to time-lapse ERI results consisting
of time series for the hundreds or thousands of pixels or
voxels a tomogram comprises.
Calibration of hydrologic process models
It is reasonable to expect that changes in geophysically
imaged properties coincide, in space and time, with
changes in hydrologic parameters, such as moisture
content and salinity. Rather than seek to convert the
geophysical tomogram to a cross section (or volume) of
the hydrologic parameter, the calibration strategy seeks to
use the geophysical images as calibration data for the
hydrologic process model. In our definition and categorization of translation strategies, we draw an important
distinction between calibration and coupled inversion,
explained in more detail subsequently.
The hydrologic model may be calibrated to the
geophysical information in various ways, some involving
a petrophysical relation and conversion step and others
not. The level or rigour in calibration ranges from manual
calibration to formal parameter estimation using optimiCopyright © 2014 John Wiley & Sons, Ltd.
zation algorithms [e.g. PEST (Doherty and Hunt, 2010)
and UCODE (Poeter et al., 2005)]. In some cases,
calibration may be to summary statistics derived from
geophysical images. For example, Binley et al. (2002)
calibrated a vadose zone model to estimate saturated
hydraulic conductivity using spatial moments (i.e. centre
of mass) of cross-hole electrical resistivity images. Deiana
et al. (2008) converted resistivity tomograms to water
content and used moments from the resulting water
contents to calibrate an infiltration model and estimate
saturated hydraulic conductivity. Briggs et al. (2013) used
UCODE to calibrate a MT3D (Zheng and Wang, 1999)
solute-transport model with dual-domain mass transfer to
both concentration data and apparent conductivities.
Doetsch et al. (2013) calibrated a TOUGH2 (Pruess et al.,
1999) flow and transport model for CO2 migration to timelapse electrical resistivity tomograms using a petrophysical
relation to link gas saturation and resistivity.
Coupled inversion
Coupled inversion has the potential to directly translate
geophysical data into hydrologic information during a
hydrologic inversion procedure. There is no inversion for
geophysical parameters (e.g. conductivity), post-inversion
conversion of geophysical inversion results, or calibration
of a hydrologic process model to geophysical inversion
results. The term ‘coupled inversion’ has various
definitions in the literature, with synonyms sometimes
including joint inversion, data fusion, and data integration. We stress, however, that these terms have different
meanings in different publications. In the following
discussion, we attempt a review of definitions found in
the recent literature and clarify important differences.
The goal of coupled inversion, by any common
definition of the term, is to reduce uncertainty and better
constrain estimation of hydrologic properties or states
compared with inversion conditioned to a single data
type. The philosophy underlying coupled inversion
strategies is not new and goes back decades in the
hydrologic literature, with numerous papers combining,
for example, temperature and head data (e.g. Stallman,
1965; Woodbury and Smith, 1988; Lapham, 1989;
Woodbury, 2007) and tracer and head data (e.g. Graham
and McLaughlin, 1989a; Graham and McLaughlin,
1989b; Carrera, 1993; Harvey and Gorelick, 1995a).
Similarly, the geophysical literature is replete with articles
in which multiple data types are inverted simultaneously
to better constrain estimates of parameters common to the
multiple underlying forward models. We focus this
review on work in which ERI is used within a coupled
inversion framework, but emphasize that seminal work using
similar approaches with other data types (e.g. Kowalsky
et al., 2005) is outside the scope of this paper.
Hydrol. Process. (2014)
K. SINGHA ET AL.
In recent years, considerable effort within the
hydrogeophysics community has been dedicated to
development and application of techniques of coupled
inversion, called by other names depending on the author.
Yeh and Simunek (2002) proposed two levels of ‘data
fusion’ in the context of a study in which ERI is used to
inform estimation of hydrologic parameters governing
variably saturated flow. In Level 1 data fusion, the
hydrologic estimation problem is conditioned on results
from the geophysical image (i.e. calibration in our
terminology). In Level 2 data fusion (Figure 5), the
inverse problem is solved using both types of data, with a
strong coupling between the two forward models, in
which resistivity data are used alongside hydrologic data
to estimate hydrologic parameters, and the hydrologic
model output is used as input to the geophysical forward
model. Whereas calibration uses the geophysical
inversion results (i.e. the image) as additional data for
estimation of hydrologic parameters, Level 2 data fusion
(i.e. coupled inversion) uses geophysical data as
additional hydrologic data. The disadvantage of Level 1
data fusion is that geophysical images are resolutionlimited and bear the imprint of regularization, prior
information, temporal smearing, mapping of data errors
into the model, and limited survey geometry.
More recently, Hinnell et al. (2010) coined the term
‘hydrogeophysical coupled inversion’, for a workflow,
which functionally is identical to that of Yeh and Simunek’s
Figure 5. Workflow for hydrogeophysical inversion, adapted from Level 2
data fusion as outlined by Yeh and Simunek (2002) and functionally
identical to ‘coupled hydrogeophysical inversion’ as defined by
Hinnel et al. (2010)
Copyright © 2014 John Wiley & Sons, Ltd.
Level 2 data fusion. Coupled hydrogeophysical inversion
focuses on similar linkages between a hydrologic process
model (i.e. flow or transport) with geophysical data sensitive
to the hydrologic process (e.g. electrical conductivity).
Coupling as outlined by Hinnell et al. (2010) and in Level 2
data fusion as outlined by Yeh and Simunek (2002) is
achieved in the inversion of data by explicitly accounting
for the effect of the hydrologic state on the geophysical
property. Output from the hydrologic forward model (e.g. a
transport model to simulate a tracer test) is fed through a
petrophysical relation, and the result is used as input to the
geophysical forward model (e.g. electrical conduction to
simulate ERI data). The primary limitation of this approach
is that the petrophysical relationship is rarely known at
the field scale. The inverse solution will tend to produce
hydrological parameter estimates consistent with the
petrophysical model assumed, resulting in biased estimates
if the petrophysical model is inaccurate. Johnson et al.
(2009) proposed and synthetically demonstrated an
approach that bypasses the petrophysical relation by
maximizing the correlation between observed ERI data
and that produced by the coupled model. Irving and
Singha (2010) demonstrated a stochastic coupledinversion framework using Markov chain Monte Carlo
(MCMC) algorithm for ERI and tracer data. Such
stochastic approaches are amenable to modifications that
enable uncertainty in the petrophysical relationship to be
appropriately accounted for. Ultimately, petrophysical
uncertainty is one of the primary factors limiting the
utility of coupled inversion approaches. Resolving this
problem remains an area of active research.
Whereas most examples of coupled inversion to date
have focused on fusion of information considering
geophysical data and hydrologic data as contributing to
different terms in the inverse problem’s fitting criteria or
objective function, Pollock and Cirpka (2008; 2012)
presented a coupled inversion framework in which the
physics of the two data types are linked at a more basic
level. For the specific problem of monitoring ionic tracers
with ERI, Pollock and Cirpka (2008) derived temporal
moment generating equations for concentration and
electrical potential perturbations as a function of the
underlying hydraulic conductivity field.
At the time of this writing, coupled inversion remains
an active area of research in hydrogeophysics, with new
tools being developed to facilitate applications to real
datasets. For example, Lawrence Berkeley Laboratory’s
MPiTOUGH2 (Commer et al., 2013) now explicitly
supports coupled inversion. Recent developments in
parameter estimation codes such as PEST (Doherty and
Hunt, 2010) and UCODE_2005 (Poeter et al., 2005)
similarly allow for consideration of multiple data types.
With such advancements, we foresee proliferation of
coupled inversion studies in the literature.
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
TECHNOLOGICAL ADVANCES
Instrumentation
Advances in time-lapse electrical imaging capabilities
have been enabled in part by the development of autonomous
multi-electrode, multi-channel instrumentation. Modern day
survey instruments are capable of accommodating many tens
of electrodes using a single control unit, which can be
expanded with additional switching units to accommodate
many hundreds to thousands of electrodes, each capable of
either transmitting current or measuring potential. In
addition, multi-channel instruments enable the simultaneous
measurement of potential across many pairs of electrodes
for a given current injection, which has the potential to
decrease survey times by the reciprocal of the number of
channels, thereby substantially improving temporal
resolution. Most systems also allow fully customized
measurement sequences that are completed without
user intervention. System control and data transfer
through wireless internet connection facilitate long-term
field deployment and enable remote, continuous, and
autonomous time-lapse monitoring of subsurface processes
from data collection to offsite transmission for processing.
Coupled with equally autonomous data processing and
inversion on the software end, this provides the opportunity
to execute real-time autonomous monitoring of subsurface
process through time-lapse ERI (Versteeg et al., 2006;
Johnson et al., 2010a). In addition to commercially available
systems, the availability of highly customizable ‘smart’
electrical hardware components and advanced, high-level
control software have facilitated the development of capable
and inexpensive custom ‘home grown’ ERI survey
instruments (Sherrod et al., 2012).
High-performance computing
Advances in survey instrumentation have enabled
applications using large numbers of electrodes and have
reduced survey times between time-lapse surveys,
providing the opportunity to monitor over larger areas
and to improve spatial and temporal resolution. In many
cases, fully utilizing the monitoring capabilities afforded
by these systems is cumbersome or impossible with nonscalable or limited scalability computing systems due to
the computationally intensive nature of ERI, TDIP, or CR
inversion, particularly for 4-D imaging applications.
Several practitioners have addressed the computational
demands of large-scale applications by developing
inversion software capable of using parallel computing
hardware. For example, Loke et al. (2010) demonstrated a
reduction in computing time of more than two orders of
magnitude by optimizing memory usage and leveraging
the parallel computation capabilities of the graphical
processor unit when optimizing measurement sequences
Copyright © 2014 John Wiley & Sons, Ltd.
for 2-D ERI arrays on a commonly available microcomputer. In addition, most inversion codes are capable of
utilizing the shared-memory parallelism naturally provided by modern multi-core processors and/or shared
memory multi-CPU computing hardware.
Given the high cost of massively parallel shared-memory
systems, most high-performance parallel computing systems
are distributed memory, whereby many machines comprising
independent CPUs and memory are linked together by
high-speed networking hardware. This enables relatively
inexpensive access to large numbers of processors and large
volumes of memory, but requires parallelized inversion
software written specifically to run on distributed memory
systems. Commer et al. (2011) modified a parallel frequencydomain EM inversion code to invert time-lapse SIP data.
They used the code to image 4-D changes in conductivity
magnitude and phase during a bioremediation experiment
using 250 CPUs on a distributed memory parallel computing
system. In addition, Johnson et al. (2010b) described a
parallel inversion algorithm designed to invert 3-D and 4-D
ERI datasets, which has facilitated temporally dense 4-D
monitoring applications over extended periods requiring
hundreds to thousands of inversions (Johnson et al., 2012;
Truex et al., 2013; Wallin et al., 2013).
Survey design optimization
Survey design is the process whereby the number of
electrodes, electrode layout, and set of measurements
comprising a survey is selected. Many survey types have
been used, with practitioners often choosing from one
(or a combination) of traditional array types that were
defined in the early days of electrical prospection before
the advent of imaging. Five such arrays are the Wenner,
Schlumberger, dipole–dipole, pole–pole, and pole–dipole
configurations. Most of these arrays were designed to
address a particular issue. For example, the Wenner array
provides a high signal-to-noise ratio, whereas the
Schlumberger array is particularly sensitive to horizontal
contacts. The general advantages and disadvantages of
each array are well understood and have been published
in previous literature (Ward, 1990) and each continues to
be used today. However, it is also well recognized
that, for a given electrode layout and subsurface
conductivity, there are likely non-standard arrays that
can optimize imaging resolution for a given number of
measurements. This can be especially important in
time-lapse imaging applications because the maximum
possible temporal resolution is equal to the time required
to complete a survey, and therefore to the number of
measurements collected in a survey. Reducing the number
of measurements in a survey increases temporal resolution, but in general will decrease spatial resolution. It is
also important to consider efficient sequences for
Hydrol. Process. (2014)
K. SINGHA ET AL.
collecting error measurements, in particular reciprocal
measurements (Slater et al., 2000).
The number of four-electrode non-reciprocal
measurements nd to choose from given ne electrodes is
(Noel and Xu, 1991)
nd ¼ neðne 1Þðne 2Þðne 3Þ=8
(27)
which represents a remarkable number of possibilities, even
for a modest number of electrodes. For example, for 24
electrodes, nd is equal to almost 32 000. For a given
electrode layout, survey optimization methodologies in
general seek to identify sets of measurement sequences that
optimize spatial resolution with a minimum number of
measurements, thereby also optimizing temporal resolution.
Several approaches have been proposed for survey
optimization, with most research as of late focusing on
methods that increase the diagonal of the resolution matrix
(Menke, 1989; Lehmann, 1995; Stummer et al., 2004; Zhe
et al., 2007; Maurer et al., 2010; Blome et al., 2011;
Wilkinson et al., 2012). Wilkinson et al. (2006) also
proposed an optimization scheme that, in addition to
optimizing the diagonal of the resolution matrix, also seeks
to minimize electrode polarization effects by allowing
sufficient time between when a particular electrode is used
to inject current and subsequently measure potential. Such
an approach is potentially valuable when collecting TDIP or
CR data. It is worth noting, however, that the resolution
matrix is dependent upon the conductivity distribution,
so optimized surveys generated using an erroneous
conductivity distribution may not be optimal at all, although
the papers referenced earlier have demonstrated good
improvements in resolution, in synthetic and field studies,
when optimizing about a homogeneous conductivity field.
In many time-lapse studies, while the conductivity will
change during the monitoring period, it may not change
substantially, which allows this exercise to remain valuable.
Other possibilities for improved experimental design
include making sequential measurements that reduce the
size of the null space (e.g. Coles, 2008).
Although survey design methods proposed to date have
demonstrated impressive spatial resolution with a relatively small number of measurements, optimization
schemes that rely on calculating the resolution matrix
are computationally demanding, even with the more
efficient updating schemes (Wilkinson et al., 2006; Loke
et al., 2010), and have only been demonstrated on modest
2-D arrays to date. These can be used as a starting point
for 3-D survey optimization, but efficient methods for full
3-D optimization have not yet been demonstrated. Given
the computational demands of optimization, the value of
optimizing surveys for conductivity characterization (as
opposed to collecting and inverting a more comprehensive survey that provides similar resolution) is debatable.
However, optimized surveys may be needed in time-lapse
Copyright © 2014 John Wiley & Sons, Ltd.
imaging applications where subsurface conductivity is
evolving quickly and high temporal resolution is needed.
Regardless of the method used to design a time-lapse
survey, the resulting survey should be synthetically tested to
ensure the survey meets design objectives. Unfortunately,
this important step is often overlooked. Such a test
includes modelling subsurface conductivity distributions
representative of those anticipated during the field
experiment, generating the synthetic data resulting from
those distributions for the designed survey through forward
modelling and assessing forward model accuracy, adding
noise to the data consistent with that expected in the field,
inverting the data, and analysing the resulting images
against the design criteria. If possible, a pre-design survey
should be conducted to provide information concerning the
background conductivity distribution and noise conditions.
These can be used in the design and synthetic testing phase
as described earlier to provide an accurate assessment of
survey performance under field conditions. An assessment
of survey time should also be conducted to ensure the survey
design provides adequate time resolution. In this regard, one
sensible strategy is to first identify the maximum number of
measurements that can be collected and still satisfy time
resolution requirements, and then design the survey to that
number of measurements.
Emerging applications
Developments in measurement technology, inversion, and
joint inversion methods have lead to a tremendous growth in
application of electrical methods. Here, we highlight a few
major areas where process monitoring is being increasingly
performed with ERI, TDIP and/or CR: (1) groundwater
resources and exchange processes, (2) vadose zone processes,
(3) solute and contaminant transport, (4) ecological processes,
(5) biogeochemistry and remediation, and (6) carbon dioxide
storage. The multi-dimensionality of these methods and their
strength for long-term continuous monitoring, given
semi-permanently emplaced electrodes, can be highly
advantageous over traditional point-based methods for
studies of time-varying processes.
Groundwater resources and exchange
Electrical resistivity imaging has been particularly
successful in monitoring the movement of water at
interfaces, including exchange between surface water and
groundwater, or groundwater and the ocean or estuaries.
Recent work has used ERI methods to build relationships
between electrical conductivity, hyporheic exchange, and
alluvial deposits (Crook et al., 2008; Slater et al., 2010;
Bianchin et al., 2011; Doro et al., 2013) or to directly
monitor exchange between groundwater and surface
water (Nyquist et al., 2008; Coscia et al., 2011). Many
authors have used tracers to image these exchange
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
processes (Figure 6) (Ward et al., 2010b; Ward et al.,
2010a; Cardenas and Markowski, 2011; Doetsch et al.,
2012; Ward et al., 2012a; Ward et al., 2013). ERI and CR
have also been used to image natural exchange processes
(Slater and Sandberg, 2000; Breier et al., 2005), including
the exchange of seawater with a freshwater sand aquifer
in a tidewater stream (Acworth and Dasey, 2003) and
exchange associated with daily dam operations (Coscia
et al., 2012; Johnson et al., 2012; Wallin et al., 2013).
ERI has additionally been used to image groundwater
exchange with bays and oceans, and studies have shown
that ERI can be used to determine areas of discharge,
which can have important implications for nutrient
loading and water management (Bratton et al., 2003;
Day-Lewis et al., 2006b; Henderson et al., 2010).
Electrical resistivity imaging has also been used to image
transport between fast and slow paths in groundwater
systems, such as fractured rock or karst. Quantifying the
location of the fast flow paths in these heterogeneous
settings, how they may change in time, and the fluid
interactions between fast paths and a lower permeability
matrix that may include substantial storage capacity is
important to understanding the vulnerability of these
aquifers to contamination. In the case of karst systems,
targets may be large air-filled or fluid-filled conduits of
sufficient contrast to be easily mapped (e.g. Šumanovac
and Weisser, 2001; van Schoor, 2002; Deceuster et al.,
2006; Youssef et al., 2012). An overview of geophysical
techniques applicable to karst systems was recently
published by Chalikakis et al. (2011). Fractured rock
continues to be an area of characterization difficult for ERI
methods, which are often of poor enough resolution that
imaging specific features is difficult. Despite this issue,
recent contributions have highlighted possible approaches
to better imaging fractures and transport in fractures by
focusing on more appropriate inversion parameterization for
fractured rock settings given an understanding of fracture
locations from borehole logging data (Robinson et al.,
2013a; Robinson et al., 2013b).
It is important, especially in hydrologic systems, to account
for temperature effects on resistivity measurements. Electrical
data are often calibrated to remove changes associated with
temperature changes in the subsurface (e.g. Hayley et al.,
2007). In the absence of calibration, electrical measurements
may produce confounding signals in complicated settings
where temperature as well as other properties controlling
resistivity is changing (Rein et al., 2004; Revil et al., 2004).
That said, many have capitalized on this change to use ERI
to directly image changes in temperature in hydrogeologic
settings (Ramirez et al., 1993; Ramirez and Daily, 2001;
Musgrave and Binley, 2011). Temperature changes of a
few degrees have produced resistivity changes of 10–20%
Figure 6. Electrical resistivity imaging of solute transport in a stream and an abandoned stream channel during a 21-h tracer injection in Pennsylvania
from Ward et al. (2010b). Transects shown are perpendicular to the stream, with flow directed out of the page. (A) Pre-injection electrical resistivity
model. Areas of high resistivity at an elevation of 3-m local datum are suggested to be an abandoned cobble bed. (B) Model sensitivity as a function of
physical location. (C–G) Time-lapse ERI results, at time elapsed after beginning the conservative solute injection. Colour indicates the percent change in
bulk resistivity from background conditions. (H) Interpretation of resistivity images
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. (2014)
K. SINGHA ET AL.
in controlled studies (Revil et al., 1998). Additionally,
ERI associated with geothermal fields has occurred over
the past many years; only recently have these methods
been used to help calibrate fluid and heat flow models
(Hermans et al., 2012).
Vadose zone moisture dynamics
Electrical resistivity imaging has been frequently used
to monitor moisture dynamics in the vadose zone, going
back to early work by Daily et al. (1992). ERI is generally
used in the vadose zone to image infiltration, define
interface geometries, or map preferential flow paths.
Binley et al. (2002), for instance, monitored seasonal
dynamics of moisture within a sandstone aquifer, and
used ERI to map wetting and drying fronts. French et al.
(2002) used ERI to monitor tracer transport and
infiltration associated with snowmelt and quantified
movement with moment analysis. More recently,
Travelletti et al. (2012) used ERI in a simulated 67-h
rainfall experiment to monitor water infiltration and
subsurface flow to pinpoint when steady-state conditions
were reached, and determine bedrock depth and potential
preferential flow paths. Gasperikova et al. (2012) used
both cross-well and surface ERI to examine the impacts
of recharge on subsurface contamination, primarily
nitrate. Some recent work has used ERI to help look at
longstanding issues in hydrology where spatially exhaustive data may contribute, for example, to the idea that the
assumption of 1-D piston flow associated with recharge is
not valid (Nimmo et al., 2009; Arora and Ahmed, 2011).
Some authors have also explored new inversion techniques given a vadose zone motivation; Yeh et al. (2002)
developed geostatistically based inversion, and Mitchell
et al. (2011) explored different inversion techniques for
monitoring the decreasing effectiveness of a recharge
pond through time.
Recent work has also explored partially saturated flow
in alpine watersheds using ERI, including quantification
of storage and flow. Sass (2004) used ERI to image pore
water being pushed away from the freezing front
associated with increasing hydraulic pressure during
freezing in the Alps. Langston et al. (2011) imaged
perched water and focused infiltration, which could be
used to quantify the timing of alpine watershed discharge.
McClymont et al. (2010; 2011) used multiple geophysical
tools, including ERI, to image subsurface sediment
packages and bedrock topography, which controlled
available storage and the location of springs, respectively,
in the Canadian Rockies. Two recent studies outline
information provided from ERI about flow dynamics on
longer scales than are commonly considered. Hilbich et al.
(2008) describe continuous ERI monitoring of permafrost
Copyright © 2014 John Wiley & Sons, Ltd.
within the Swiss Alps for 7 years and found short-term,
seasonal, and interannual changes associated with
changes in melt and radiation. Both vertical and
horizontal infiltration of water affected these measurements, and the authors noted that moisture measurements
would be needed to separate moisture from temperature
changes. Scherler et al. (2010) explore infiltration along
heterogeneous flow paths to investigate energy and mass
transfer in Switzerland and found an interesting cycle of
decreased surface permeability from freezing, increased
pressure heads, and then infiltration and heat transfer, and
used ERI to constrain moisture content ignoring changes
in temperature.
Electrical resistivity imaging has also been successfully
used to explore vadose zone remediation, including
engineered vadose zone desiccation, which was recently
tested at the Hanford Site as a remediation technology for
slowing the migration of contaminants toward the water
table (Truex et al., 2013). Relying on the petrophysical
relationship between saturation and electrical conductivity, 4-D cross-hole ERI was used to monitor the
desiccation zone with time, providing detailed information concerning the subsurface structure and desiccation
performance (Figure 7).
Contaminant transport
Electrical resistivity imaging has been a successful
tool for monitoring contaminant transport in the case of
imaging electrically conductive contaminants like landfill leachates (e.g. Chambers et al., 2006; Mansoor and
Slater, 2007). Other contaminants are more difficult to
detect. For example, non-aqueous phase liquids
(NAPLs) are challenging targets for electrical methods
as the electrical properties of NAPLs change over time in
response to biodegradation (see, for review, Atekwana
and Atekwana, 2010). CR has been repeatedly explored
as a potential technology for characterization and
monitoring of NAPL contaminants in the subsurface;
early CR monitoring experiments focused on the
potential of CR to monitor the transport of dense nonaqueous phase liquid (DNAPL) (Daily et al., 1998;
Chambers et al., 2004). Such efforts were motivated by
laboratory experiments suggesting that reactions
between DNAPL and clay minerals (clay–organic
interactions) could be sensed with CR (e.g. Olhoeft,
1985). However, contrary to such laboratory studies, CR
monitoring experiments on recently spilled DNAPL
found the phase as measured by CR to be relatively
insensitive to the presence and/or transport of DNAPL,
with more information coming from |σ| as a result of the
resistive DNAPL replacing the electrolyte within the
pore space (Daily et al., 1998; Chambers et al., 2004).
More recent laboratory experiments indicate that CR
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
Figure 7. (Top) Pre-desiccation 3-D baseline ERI image showing bulk electrical conductivity along selected iso-surfaces modified from Truex et al.
(2013). Higher electrical conductivity lenses are diagnostic of fine-grained, elevated moisture units that tend to harbour contaminated pore water. Lower
electrical conductivity lenses represent coarser-grained, higher-permeability units that likely provide preferred flow pathways between gas injection and
extraction wells. (Bottom) Three-dimensional view of the change in volumetric water content (WC) over time derived from 3-D cross-hole ERI. Two
tomograms were produced per day over the 70-day monitoring period
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. (2014)
K. SINGHA ET AL.
signals of clay–organic interactions are uncertain and
sometimes small (Ustra et al., 2012). This in part
explains the limited success of these early CR imaging
experiments.
Ecological processes
Electrical resistivity imaging has been successfully
used in recent years to image processes driven by or
important to biology, from the microbiological scales up
to ecosystem dynamics. For example, ERI has been used
to image root uptake in variably vegetated systems
(Jayawickreme et al., 2008; Jayawickreme et al., 2010);
the authors found differences in soil moisture distribution
between forested and grassland ecosystems throughout
the growing season that are important for parameterizing
roots in global climate and hydrological models
(Figure 8). On shorter time scales, Robinson et al.
(2012) monitored ERI in an oak–pine forest, finding
results indicative of hydraulic lift, whereby trees pump
water from deep within the vadose zone and release it
near the surface. al Hagrey (2007) used ERI to
discriminate between types of roots (soft vs woody) as
well as imaging moisture content within the root zone and
in tree trunks (al Hagrey, 2006), inspiring similar work
differentiating sapwood from heartwood (Guyot et al.,
2013). Other authors have similarly used ERI to look at
moisture dynamics and its importance on vegetation, from
Mediterranean southern France (Nijland et al., 2010) to
the Mojave Desert (Nimmo et al., 2009).
Biogeochemistry and remediation
Significant research has recently explored the electrical
signatures resulting from microbial growth and interactions between microbes and mineral surfaces. The
monitoring of biogeochemical processes associated with
the natural or stimulated biodegradation of contaminants
is a promising application of CR monitoring. Numerous
recent laboratory experiments have indicated that CR is
sensitive to microbe-mineral interactions (see for review
Atekwana and Slater, 2009), particularly biomineralization associated with the sequestration of heavy metals
(Williams et al., 2005; Slater et al., 2007). Flores Orozco
et al. (2011) built on these laboratory findings and
successfully demonstrated the feasibility of monitoring
biogeochemical changes accompanying stimulation of
indigenous aquifer microorganisms known to sequester
uranium during and after acetate injection (Figure 9).
Spatiotemporal changes in the phase response of aquifer
sediments were shown to correlate with increases in Fe(II)
and precipitation of metal sulfides (e.g. FeS) following
the iterative stimulation of iron and sulfate-reducing
microorganisms. In contrast, only modest changes in |σ|
were observed over the monitoring period and could not
be clearly related to the biomineralization. Chen et al.
(2012) suggested a Bayesian framework for estimating
the geochemical parameters directly from the frequencydependent CR response observed by Flores Orozco et al.
(2011), although they recognized that the approach was
limited by the need for more robust petrophysical
Figure 8. Two-dimensional ERI across a sharp ecotone in Michigan from Jayawickreme et al. (2008) before the start of the growing season (May) and in
the peak of the growing season (August). ERI data show a significantly larger seasonal change in soil moisture for the forest and also that trees have
deeper effective roots
Copyright © 2014 John Wiley & Sons, Ltd.
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
Figure 9. (a) Schematic representation of the experimental setup for a biostimulation experiment using CR. Monitoring wells upgradient (open circles)
and downgradient (solid circles) of the injection gallery (solid black rectangle) are referred to the direction of the groundwater flow (solid arrow). The
dashed lines represent the location and extent of array A and B. All dimensions are given in metres. (b) CR images for the measurements collected along
array A (red line in part a) in 2007 after an injection of 5 mM acetate and 2 mM bromide. Elapsed time in days is referenced to the start of acetate
injection. (left) Resistivity images for data collected at 1 Hz. Phase images computed for data collected at (middle) 1 and (right) 4 Hz. Modified from
Flores Orozco et al. (2011)
relations for quantifying biomineral concentration from
complex resistivity data.
Electrical resistivity imaging has also been successful
in some situations for imaging solid phase transformations and the emplacement and migration of amendments
for remediation (e.g. Ramirez et al., 1993; Daily
and Ramirez, 1995). For example, ERI was recently
demonstrated as an effective bioremediation long-term
bioremediation monitoring tool at a site contaminated
with chlorinated solvents in Brandywine, Maryland, USA
(Johnson et al., 2010b). A lactate-based amendment and
pH buffer (sodium hydroxide) were injected into the
subsurface to moderate pH levels and stimulate the
bio-activity of reducing organisms. A 3-D cross-borehole
ERI array was installed at the site with the objective of
monitoring the injection and subsequent transport of
amendment to determine which regions had been treated.
The system operated autonomously for approximately
3 years, collecting one 3-D ERI dataset approximately
every 2 days. Extensive sampling was also conducted in
order to relate changes in conductivity with changes in
biogeochemical conditions. A set of selected ERI images
showing the change in conductivity with time from
baseline conditions is shown in Figure 10. The ERI
images show the amendment plume sinking and spreading from the injection zone over a lower confining unit
and slowly diluting from day 2 to day 371. Changes in
bulk conductivity were consistent with changes in fluid
conductivity during this period. From day 371 to 762, a
large secondary increase in conductivity was observed
Copyright © 2014 John Wiley & Sons, Ltd.
that was not associated with a corresponding increase in
fluid conductivity. Geochemical indicators sampled
during this period revealed enhanced bioactivity and
provided evidence for the formation of FeS precipitates.
Furthermore, fine FeS particulates were observed in
several wellbore samples during this period, suggesting
the large secondary increase in conductivity from day 371
to 762 was caused by FeS precipitation. Thus, in this case,
the ERI images provided detailed spatial and temporal
information concerning both the transport of injected
amendments, and concerning the distribution and timing
of corresponding biological activity, and stage of
bioremediation progress.
Carbon dioxide storage
ERI, TDIP, and CR can also be useful technologies for
monitoring processes associated with climate change. For
example, while capturing and storing CO2 has become a
popular idea for controlling greenhouse gas emissions to
the atmosphere, long-term storage of CO2 requires
monitoring for possible CO2 migration and leakage.
ERI is a promising method for these studies because CO2
displaces conductive brine in many storage areas. A
number of studies have used ERI to monitor CO2
propagation in storage sites (Ramirez et al., 2003; Giese
et al., 2009; Kiessling et al., 2010; Bergmann et al., 2012;
Doetsch et al., 2013), including a recent study that
imaged CO2 injection into an oil-and-gas field at 3000 m
depth (Carrigan et al., 2013).
Hydrol. Process. (2014)
K. SINGHA ET AL.
suggest that the ϕ recorded on sets of electrodes that
sample the subsurface properties close to the borehole
showed some evidence for additional information beyond
what was obtained from |σ| alone. They suggested that
this resulted from changes in the surface charge density,
along with possible mineral dissolution and ion exchange.
However, changes in ϕ largely tracked changes in |σ| in
this experiment, which can occur in the presence of only
changes in electrical conduction (i.e. σ′). Consequently,
this experiment was inconclusive in demonstrating the
additional information on geochemical transformations
that can potentially be extracted from CR measurements
relative to measuring |σ| alone.
DISCUSSION AND FUTURE DIRECTIONS
Figure 10. Four-dimensional ERI monitoring images during enhanced
bioremediation at a site in Brandywine, Maryland. The images show the
sodium enriched, lactate amendment plume sinking and spreading from
the injection zone over a lower confining unit, and slowly diluting from
day 2 to day 371. Changes in bulk conductivity were consistent with
changes in fluid conductivity during this period. From day 371 to 762, a
large secondary increase in conductivity was observed resulting from
increased bioactivity and subsequent FeS precipitation, which was
validated by sampling
Monitoring geochemical transformations associated
with CO2 storage and sequestration in aquifers represents
another promising application of CR monitoring. For
example, Dafflon et al. (2013) performed a CR
monitoring experiment on a test site where dissolved
CO2 was injected into the subsurface. The data were not
of sufficient quality to generate reliable time-lapse images
of phase, highlighting the challenges of acquiring reliable
CR data as stressed earlier. Dafflon et al. (2013) did
Copyright © 2014 John Wiley & Sons, Ltd.
In this paper, we outline the state of the science with
respect to time-lapse electrical data collection, inversion,
interpretation, and application. ERI methods have become
mainstream geophysical tools, with recent innovations
driven by the development of multichannel, autonomous
tools and improved inversion methodologies. These
developments have been applied to a wide range of
applications, from imaging changes in water quality or
quantity to mapping the moisture content of tree trunks to
mapping CO2 injection for sequestration. Despite all of
this work, there are a number of areas where these
technologies have not been used to their full potential.
SIP largely remains a research technology outside of its
long-standing use for mineral exploration. Application of
ERI/TDIP/CR methods to biological systems, in particular, is an area where a substantial amount of fundamental
research still remains to be conducted. Also, while these
methods are being more regularly applied for monitoring,
a largely missing step is to use these tools to develop
decision-support systems and improve environmental
stewardship; for example, these instruments could be
used to image changes in moisture content as a function
of climate change, or saline intrusion along coastal
aquifers as shown by the Automated Time-Lapse
Electrical Resistivity Tomography (ALERT) system
(Kuras et al., 2009; Ogilvy et al., 2009). In terms of
tools, the development of wireless receivers may change
the field; while recently on the market these remain rarely
used, but would be fundamental to mapping processes
over large scales, such as watersheds.
Major improvements have come with the development of
alternative parameterizations, for example to discretize
boreholes and regularization that allows for sharp boundaries, important where such contrasts are expected.
New inversion tools that move beyond over-parameterized
voxel models to solving for process-indicative parameters
directly also show promise. Parameterization based on the
Hydrol. Process. (2014)
TIME-LAPSE ELECTRICAL IMAGING
physics of plume shape evolution, for example, appears a
promising direction for future research to translate ERI results to
hydrologic information. Indeed, the medical imaging literature
offers many approaches to moment-based (e.g. Milanfar et al.,
1996) and shape-based parameterizations (e.g. Dorn et al., 2000;
Miller et al., 2000) that might prove useful in hydrogeophysics.
As a community, it is worth continuing to push toward
alternative inversion and regularization techniques; ideas such as
steering filters or prediction error filters (Schwab et al., 1996;
Clapp et al., 1998) from seismic processing may be used to
improve tomograms and thus geological understanding of Earth
systems. Another challenge is how to balance different types of
data within coupled inversion, where we may have hundreds or
thousands of geophysical measurements but only tens of
hydrologic observations. Future opportunities include further
refinement and continued adoption of joint inversion methods
for estimating parameters or processes of interest directly, and
development of rock physics relations for time-lapse inversions,
where sensitivity and measurement errors change with time.
ACKNOWLEDGEMENTS
This research was supported in part by National Science
Foundation Grant EAR-0747629 and Environmental Protection Agency Region 1 and the U.S. Geological Survey’s
Toxic Substances Hydrology Program and Groundwater
Resources Program. We thank two anonymous reviews and
Burke Minsley for thoughtful feedback. Any use of trade,
firm, or product names is for descriptive purposes only and
does not imply endorsement by the US Government.
Acronyms
CDE
CR
CRI
DC
EDL
EI
ER
ERI
FISt
IP
NAPL
PFE
RFA
RMS
SIP
TDIP
Convection–dispersion equation
Complex resistivity
Complex resistivity imaging
Direct current
Electrical double layer
Electrical impedance
Electrical resistivity
Electrical resistivity imaging
Full inverse statistical
Induced polarization
Non-aqueous phase liquid
Percentage frequency effect
Random field averaging
Root-mean-square
Spectral induced polarization
Time-domain induced polarization
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