Multiscale induced polarization tomography in hydrogeophysics: A new approach

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Multiscale induced polarization tomography in hydrogeophysics: A new approach

A Ahmed, A Revil, L Gross

To cite this version:

A Ahmed, A Revil, L Gross. Multiscale induced polarization tomography in hydrogeophysics: A new approach. Advances in Water Resources, Elsevier, 2019, 134, �10.1016/j.advwatres.2019.103451�.

�hal-03005866�

Contents lists available at ScienceDirect

Advances in Water Resources

journal homepage: www.elsevier.com/locate/advwatres

Multiscale induced polarization tomography in hydrogeophysics: A new approach

A. Soueid Ahmed ^{a , b} , A. Revil ^{a , b , ∗} , L. Gross ^c

a

Université Grenoble Alpes, USMB, CNRS, EDYTEM, 73000 Chambéry, France

b

Université Grenoble Alpes, USMB, CNRS, IRD, IFSTTAR, ISTerre, 73000 Chambéry, France

c

School of Earth and Environmental Sciences, The University of Queensland, Brisbane, Australia

a r t i c l e i n f o

Keywords:

Hydrogeophysics

Electrical resistivity tomography Inverse theory

Numerical modelling Tomography Electrical properties

a b s t r a c t

Induced polarization is a geophysical method that has gained ground in the last decade in hydrogeophysics.

Yet the acquisition of high quality induced polarization data may be challenging using the current technologies mostly because of capacitive coupling effects especially for short durations in the injected current. In addition, making a true 3D induced polarization survey is very tedious and time consuming in field conditions. We discuss the advantages of a new generation of induced polarization equipment composed of individual stations able to measure the two components of the electric field along the ground surface. We show how this approach allows for integrating the whole data avoiding negative apparent resistivity/chargeability data. The use of decentral- ized recording stations solves the issue of capacitive coupling effects. We present a completely novel induced polarization data inversion methodology based on the measurement of the electric field components. In addition, a robust geostatistical inversion approach is discussed for recovering the conductivity and chargeability fields using the electric field components measured on the ground surface. We also treat the case of time lapse mon- itoring by using a low-rank Kalman filter approximation, which is computationally very appealing in terms of computational time and storage savings. The effectiveness of the new methodology is demonstrated on several realistic synthetic case studies. These numerical tests show that this electric field-based approach is robust and very promising for applications in hydrogeophysics.

1. Introduction

Geoelectrical methods such as the electrical resistivity tomography and induced Polarization methods are widely used to image the struc- tures and dynamic processes of the subsurface. In practice, performing an electrical resistivity tomography or induced polarization survey is relatively easy. It mainly consists in laying out a series of electrodes and switching on the equipment to perform the measurements. The same equipment can be used for acquiring both resistivity and induced polarization data. Electrical resistivity tomography has been adopted for various applications in the field of hydrogeology (e.g., Daily et al., 1992 ; Descloitres et al., 2008 ; Coscia et al., 2011 ), contaminants tracking (e.g., Abu-Zeid et al., 2004 ; Doetsch et al., 2012 ), imaging of volcanoes (e.g., Revil et al., 2010 ; Soueid Ahmed et al., 2018 ), geological engineering, e.g., to study and locate leaks in dams and enbamkments (e.g., Chambers et al., 2006 ; Lucas et al., 2017 ), and for agriculture (e.g., Samouëlian et al., 2005 ).

∗

Corresponding author at: Université Savoie Mont-Blanc EDYTEM CNRS UMR 5204 5 Boulevard de la Mer Caspienne, 73370 Le Bourget-du-Lac, France E-mail addresses: [email protected] (A.S. Ahmed), [email protected] (A. Revil), [email protected] (L. Gross).

While electrical resistivity tomography is restricted to mapping the electrical resistivity of the medium, the induced polarization method goes beyond this scope by offering the possibility of characterizing an additional physical property of the subsurface namely the chargeability ( Schlumberger, 1920 ). Chargeability describes the ability of subsurface porous materials to reversibly store electrical charges under the applica- tion of an external electrical field. Historically, induced polarization was first developed for ore body exploration (e.g., Bleil, 1953 ; Van Voorhis et al., 1973 ; Zonge and Wynn, 1975 ; Pelton, 1978 ; Telford et al., 1990 ; Oldenburg et al., 1997 ) and later for archeological purpose e.g., for slag heaps (e.g., Florsch et al., 2017 ). Today, induced polarization is applied to contaminants mapping (e.g., Barker, 1990 ; Kemna et al., 2000 , 2004 ; Chambers et al., 2004 ), landfills mapping and monitoring (e.g., Leroux et al., 2007 ; Auken et al., 2011 ; Gazoty et al., 2012 ; Dahlin and Leroux, 2012 ), permeability imaging (e.g., Hördt et al., 2007 ; Attwa and Günther, 2013 ; Revil and Florsch, 2010 ; Revil et al., 2015 ; Weller et al., 2015 ), coal seam fires prospection and delineation (e.g., Shao et al.,

https://doi.org/10.1016/j.advwatres.2019.103451

Received 23 June 2019; Received in revised form 20 October 2019; Accepted 23 October 2019 Available online 28 October 2019

0309-1708/© 2019 Elsevier Ltd. All rights reserved.

2017 ; Soueid Ahmed et al., 2018 ), and permafrost characterization (e.g., Doetsch et al., 2015 ; Wu et al., 2017 ; Duvillard et al., 2018 ).

The success of an induced polarization survey basically relies on two aspects: (i) the use of an equipment able of recording high quality data, (ii) the robustness of the numerical model used to invert these data including for time-lapse survey. Currently available acquisition systems use cables to connect the different injecting and measuring electrodes and offer the possibility of reaching a depth of few hundreds of meters (generally ~ 500 m, e.g., Revil et al., 2018b ). This classical approach has several drawbacks: (i) Performing true 3D surveys with cables can be a complicated task. (ii) Very long cables are needed for deep surveys. (iii) Data quality may be an issue because of capacitive and inductive coupling effects associated with the long wires.

Recently, a new generation of induced polarization instrumentation has been developed based on remote recording stations. An example of such instrumentation corresponds for instance to the FullWaver (e.g., Truffert et al., 2017 ). It gives the possibility of measuring the components of the electric field tangential to the ground surface (there are no components normal to the ground surface). This system uses a set of autonomous stations synchronized by the Global Positioning System (GPS). As no wires are used between the transmitter and the recording stations, this overcomes the issue mentioned above associated with electromagnetic coupling effects. This configuration is also suitable for sites characterized by severe topography.

In the past, the interpretation of electrical resistivity and induced polarization data through data inversion has been the subject of intensive research. Oldenburg and Li (1994) presented three meth- ods for time domain induced polarization data. They show that the chargeability can be modeled as a perturbation of the conductivity of the medium. Loke and Barker (1996) introduced a fast inversion technique of 2D resistivity data, based on the Quasi-Newton approach.

Kim et al. (2009) presented a 4D resistivity inversion approach that considers that the resistivity of the medium can change in time in a continuous way. Nenna et al. (2011) used an extended Kalman filter to invert time-lapse electrical resistivity data to monitor temporal changes of electrical resistivity of a recharge pond.

Although the Kalman filter is very efficient for predicting the evo- lution of parameters in time for instance in geoelectrical monitoring, it has never been used for induced polarization data. The Kalman filter approach is also difficult to apply for large scale investigations as they require assembling and storing large dense matrices. To alleviate the computational burden involved in the implementation of the Kalman filter, one can compress the large Kalman filter covariance matrix, using low-rank approximation techniques (e.g., Kitanidis, 2015 ). Li et al . (2015) extended this technique to the nonlinear state estimation and applied it to synthetic CO

₂

monitoring examples. In addition, matrix free approaches (e.g., Kitanidis and Lee, 2014 ) can be used to avoid the assembly of the Jacobian matrix at each iteration of the Kalman filter.

We refer to this large-scale adapted Kalman filter as the Compressed State Kalman Filter (CSKF).

In this paper, we propose a new method able to tackle three major points: (1) We avoid the use of apparent resistivity data, and we based our approach on directly using the two components of the electric field. This gives the possibility of avoiding the presence of negative data, which are often considered as biased data by practitioners and systematically excluded from the inversion process at the cost of potentially reducing the resolution of the inverted conductivity and chargeability tomograms. (2) Solving the induced polarization inverse problem by using a large-scale geostatistical inversion approach as mentioned above. (3) We develop a time lapse inversion approach for induced polarization by using a novel approach based on a low rank approximation of the Kalman filter namely the CSFK mentioned above.

In order to show the advantages of our approach, we performed four numerical case studies. In the first case, we show the robustness of the geostatistical inversion approach in imaging the electrical conductivity and chargeability fields based on the two components of the electric

field. The second case presents the time-lapse inversion of conductivity and chargeability data. In the third case, we discuss a time lapse electrical conductivity tomography monitoring of a saline intrusion in an aquifer. The last case study involved the monitoring of water infiltration in a large portion of an unconfined aquifer. Considerable attention is paid to making the study cases physically meaningful, especially in terms of the petrophysical relationships used to relate all the physical parameters with each other in a consistent way.

2. Measuring the components of the electric field

When the current is injected into the ground, the response of the medium can be recorded in terms of the electric field. The development of such instruments makes it possible to record the electric field without the need for connecting the receivers through cables. In practice, this system is composed of two kind of units: (1) A current recording unit that is connected to a transmitter. It records the time stamped samples of the injected current during the survey. (2) A set of receivers measuring the two components of the electric field. These receivers are GPS synchronized.

The current is injected between two electrodes A and B connected by a long cable. The A electrode can be placed far from the area of interest, then the second electrode B is placed at different positions that cover the region of interest. This current injection line (dipole) can be set in any direction to increase the sensitivity with respect to the target.

This approach is also suitable for 3D surveys.

Each recording station has two channels and is connected to three potential electrodes: P

₁

, P

₂

and P

₃

. These electrodes are placed so that the two orthogonal components of the electric field are recorded

Fig. 1. Fullwaver survey design . The Fullwaver system contains two kind of units: (i) the I-Fullwaver unit (red box) which is used to record time series of the current injected by the means of a transmitter (ii) the V-Fullwaver (black boxes) which is the recording station that is used to measure the electric field. Several autonomous stations can be used in a survey. Each station has three electrodes P

₁

, P

₂

and P

₃

. The electric field tangential component E

₁

is measured on the first channel between P

₁

and P

₂

, while the normal component of the electric field E

₂

is measured on the second channel between P

₂

and P

₃

. Two coordinates systems can be employed: a global one denoted by the (x,y,z) axes on Fig. 1 and a local one defined on each station in which E

₁

denotes the component of 𝐸 ⃗ along the horizontal direction and E

₂

is the component of 𝐸 ⃗ along the vertical direction.

P

_F

denotes a fictitious electrode that can be used to derive the formulation of

the apparent resistivity and chargeability. The current is injected between the

electrodes A and B which are connected by a cable. The electrode A remains

fixed while the position of B can be switched for each current injection. (For

interpretation of the references to colour in this figure legend, the reader is

referred to the web version of this article.)

for each station ( Fig. 1 ). These two components can be seen as the horizontal components of the electric field E . Note that in the presence of topographic effects, it will be more appropriate to designate them as the tangential components of the electric field. The normal component of the electric field (with respect to the ground surface) is null since air is insulating. Using the relationship between the electric field and the electric potential, the horizontal and vertical components of the electric field are:

𝐸

1

= − Δ 𝜑

1

|| 𝑃

1

− 𝑃

2

|| (1)

𝐸

2

= − Δ 𝜑

2

|| 𝑃

3

− 𝑃

2

|| (2)

where Δ 𝜙

1

denotes the potential difference along the horizontal direction (channel 1), Δ 𝜙

2

denotes the potential difference along the perpendicular direction (channel 2), | P

₁

− P

₂

| and | P

₃

− P

₂

| are the distances between P

₁

and P

₂

and P

₂

and P

₃

, respectively. One can use a fictitious electrode P

_F

that can be placed anywhere along the direction of the electric field E so that this electric field can be written

𝐸 = − Δ 𝜑

_𝐹

|| 𝑃

_𝐹

− 𝑃

2

|| ̂𝑖 (3)

where Δ 𝜙

F

is the potential difference between P

_F

and P

₂

, ̂𝑖 denotes the unit vector defined in the E -direction ( Fig. 1 ). | P

_F

− P

₂

| denotes the distance between the fictitious electrode P

_F

and P

₂

. Using the intensity of the electric field E

Δ 𝜑

_𝐹

= || 𝑃

_𝐹

− 𝑃

2

|| E (4)

Δ 𝜑

𝐹

= || 𝑃

𝐹

− 𝑃

2

|| √

𝐸

₁²

+ 𝐸

₂²

(5)

Δ 𝜑

_𝐹

= || 𝑃

_𝐹

− 𝑃

2

||

√ ( Δ 𝜑

1

|| 𝑃

1

− 𝑃

2

||

)

2

+ ( Δ 𝜑

2

|| 𝑃

3

− 𝑃

2

||

)

2

. (6)

The apparent resistivity on the station can then be obtained by 𝜌

𝑎

= 𝐾 Δ 𝜑

_𝐹

𝐼 , (7)

where K denotes the geometrical factor and I is the intensity of the injected current.

A similar reasoning applies for the apparent chargeability deriva- tion. In fact, the apparent chargeabilities in the two perpendicular horizontal directions can be written as

𝜂

1

= Δ 𝜑

^𝑆₁

Δ 𝜑

1

, (8)

𝜂

2

= Δ 𝜑

^𝑆₂

Δ 𝜑

2

, (9)

respectively, and where Δ 𝜑

^𝑆₁

and Δ 𝜑

^𝑆₂

denote the secondary potentials computed on channels 1 and 2, respectively.

Similarly to Eq. (6) , the secondary voltage obtained along the direction of E is

Δ 𝜑

^𝑆_𝐹

= || 𝑃

𝐹

− 𝑃

2

||

√ √

√ √

√ ( Δ 𝜑

^𝑆₁

|| 𝑃

1

− 𝑃

2

||

)

2

+ ( Δ 𝜑

^𝑆₂

|| 𝑃

3

− 𝑃

2

||

)

2

. (10)

Given that 𝜂

_𝑎

= Δ 𝜑

^𝑆_𝐹

∕Δ 𝜑

_𝐹

, we obtain after some algebraic manipu- lations (see Appendix A ):

𝜂

_𝑎

=

√ √

√ √

√ ( 𝜂

1

Δ𝜑

₁

)

2

+ ( 𝜂

2

Δ𝜑

₂

)

2

( Δ 𝜑

₁

)

2

+ (

Δ 𝜑

₂

)

2

. (11)

Alternatively one can decide to express all the data in terms of the electric field, which is numerically more convenient. In other words, we

solve the electrical potential equation to obtain the electric potential 𝜙:

−∇ ⋅ ( 𝜎 ∇ 𝜑 ) = 𝐼𝛿 ( 𝑥 − 𝑥

_𝑠

)

𝛿 ( 𝑦 − 𝑦

_𝑠

)

𝛿 ( 𝑧 − 𝑧

_𝑠

)

, (12)

where 𝛿 denotes the delta Dirac function, x

_s

, y

_s

and z

_s

are the spatial coordinates of the current injecting electrode. Once Eq. (12) is solved, we compute the electric field as,

𝐸 = −∇ 𝜑. (13)

The intensity of the electric field is given by 𝐸 =

√ 𝐸

²₁

+ 𝐸

²₂

. (14)

It is measured on each station and can be used as input data for inverting for the electrical conductivity.

As stated by Siegel ( 1959 ), the effect of the chargeability 𝜂 can be considered as a perturbation of the electrical conductivity field of the medium as 𝜎(1 − 𝜂). Eq. (12) can be formulated in terms of the electric field. The apparent chargeability 𝜂

a

at each station is given in the local system coordinate by (see Appendix A ):

𝜂

𝑎

=

√

1 + 𝐸

^𝑇

( 𝐸 − 2 𝐸

^𝜂

)

𝐸

^𝜂𝑇

𝐸

^𝜂

, (15)

where E

^𝜂

= − 𝜎(1 − 𝜂) ∇ 𝜙, E

^𝜂T

and E

^T

denotes the transposes of E

^𝜂

and E , respectively. In other words, we need to solve the electrical potential equation twice: one to compute the electrical potential associated to 𝜎 and one with the electrical potential associated to 𝜎(1 − 𝜂), then the electric fields E and E

^𝜂

are used to calculate the apparent chargeability following Eq. (15) . Note that Eq. (15) can be rewritten as:

𝜂

_𝑎

= ||

|| 𝐸

^𝜂

− 𝐸 𝐸

^𝜂

||

|| , (16)

where |.| denotes the Euclidean norm i.e. |𝑥 | = √

𝑥

^𝑇

𝑥 . Eq. (16) imlies that 𝜂

a

is a real quantity. Using Eqs. (6) and (11) or Eqs. (14) and (16) ensure that we are working with data sets that are made positive and therefore their logarithms can be used as input data for the inversion, which generally reduces the artefacts in the tomograms.

3. Inverse scheme

In this section, we present an algorithm to solve the inverse problem.

It consists in retrieving the distributions of the intrinsic electrical con- ductivity and chargeability fields computed from the two components of the electric field measured on the ground surface including some error or noise in the measurements. We propose to use a geostatistical inversion approach. When working on time-lapse data, the parameters of interest i.e. the electrical conductivity and the intrinsic chargeability can also evolve in time, therefore the inverse problem needs to be solved at each time step to update the fields and to compute the new changes in these distributions. We propose to use a Kalman filter approach for this purpose.

3.1. Geostatistical inversion

The inverse problem for conductivity and chargeability is un-

determined and is ill posed. On way to address these issues is by

imposing constraints on the desired solution. Usually, in geophysics

these constraints are imposed using horizontal or vertical smoothing of

the model parameter distributions. That said, this strategy may lead to

oversmooth tomograms. Alternative approaches exist, for instance by

using a covariance matrix based on a variogram to guide the inversion

to converge towards the best solution, whose spatial distributions are

adequately being described by a given kernel such as Gaussian or

exponential one for instance. One major limitation to this approach is

the computation and storage of the covariance matrix in large-scales

applications where the number of unknowns can be very large. A

principal component analysis shows that, actually the information contained in the covariance matrix can be accurately captured by its leading values, which means that this covariance matrix can be compressed resulting in significant storage and computational savings (e.g., Kitanidis and Lee, 2014 ). This means that major features of the field can be captured by a low rank truncated covariance matrix. In this regard, Kitanidis and Lee (2014) adapted the geostatistical inversion to large-scale problems by introducing the so-called Principal Component Geostatistical Approach (PCGA). Using this approach, the covariance matrix is never computed or stored. Products of the Jacobian matrix with given vectors or matrices are evaluated without explicit storing of it in memory which allows for significant computational savings. In the PCGA framework, we consider that the optimal parameter set is obtained by minimizing the following objective function:

𝑚𝑖𝑛

_𝑠

{ 1

2 ( 𝑑 − Φ( 𝑠 ) )

^𝑇

𝑅

⁻¹

( 𝑑 − Φ( 𝑠 ) ) + 1

2 ( 𝑠 − 𝑋𝛽 )

^𝑇

𝑄

⁻¹

( 𝑠 − 𝑋𝛽 ) }

, (17) where s denotes the M × 1 parameter vector to be estimated, d is the N × 1 data vector, Φ(.) is the forward operator, R is the N × N data covariance matrix, X is a M ×1 vector of ones, 𝛽 denotes the scalar constant denoting the mean of the parameter field, and Q denotes the M × M prior spatial covariance matrix. The matrix Q can be compressed for instance by factorizing it in a product of low-rank matrices by using its eigen-decomposition:

𝑄 ≈ 𝑄

_𝐾

= 𝑃

_𝐾

Λ

_𝐾

𝑃

_𝐾^𝑇

=

∑

𝐾 𝑖=1

𝑣

_𝑖

𝑣

^𝑇_𝑖

, (18)

𝑣

_𝑖

=

√ || 𝜆

_𝑖

|| 𝑃

_𝑖

, (19)

where K is the truncation order, Q

_K

is the M ×M truncated covariance matrix, Λ

_K

is the K × K diagonal matrix of the first K largest eigenvalues of Q, P

_K

is the M × K matrix of the corresponding eigenvectors v

_i

, where 𝜆

i

is the i

^th

eigenvalue of Q and P

_i

is the i

^th

column of P . The choice of K is important, it should not be too small to not loose information and too large to not loose computational efficiency. A compromise can be obtained by making sure that K makes the truncation error small enough, i.e ‖Q − Q

_K

‖/ ‖Q ‖ ≪ 1 which means that the ratio between the K + 1 eigenvalue of Q and the largest one is much smaller than 1 (e.g., Kitanidis and Lee, 2014 ). The norm ‖. ‖ is a given norm operator. Given that generally M ≫ N > K ( M is the number of unknowns, N is the number of measurements and K is the truncation order), the low rank approximation of Q yields significant computational savings.

The solution of the minimization problem (17) is now given by:

𝑠

_𝑘+1

= ̃𝜉 ( 𝑑 − Φ (

𝑠

_𝑘

) + 𝐻

_𝑘

𝑠

_𝑘

)

, (20)

where k is the current iteration number of the inversion process, H

_k

is the N × M Jacobian matrix of the measurements with respect to the parameter model s

_k

.

The vector ̃𝜉 is found by solving the following linear system:

( 𝐻

_𝑘

𝑄𝐻

_𝑘^𝑇

+ 𝑅 𝐻

_𝑘

𝑋 ( 𝐻

_𝑘

𝑋 )

𝑇

0

) ( ̃𝜉

̃𝛽 )

= ( 𝐻

_𝑘

𝑄

𝑋 )

. (21)

The Jacobian products that appear in (21) can be computed as:

𝐻

_𝑘

𝑋 ≈ 1

𝛿 [ Φ( 𝑠 + 𝛿𝑋 ) − Φ( 𝑠 ) ] , (22)

where 𝛿 is a finite difference step.

𝐻

_𝑘

𝑄 ≈ 𝐻

_𝑘

𝑄

_𝐾

=

∑

𝐾 𝑖=1

𝜁

_𝑖

𝑣

^𝑇_𝑖

, (23)

where 𝜁

_𝑖

= 𝐻

_𝑘

𝑣

_𝑖

= 1

𝛿 [ Φ (

𝑠 + 𝛿𝑣

_𝑖

)

− Φ( 𝑠 ) ]

, (24)

Fig. 2. Geometry of the simulation domain . The dots denotes the locations of the receivers. A denotes the location of the injecting current electrode. The electrode B is supposed to move around the stations. 144 stations are used and 17 current injections are performed.

and

𝐻

_𝑘

𝑄𝐻

_𝑘^𝑇

≈ 𝐻

_𝑘

𝑄

_𝐾

𝐻

_𝑘^𝑇

=

∑

𝐾 𝑖=1

𝜁

_𝑖

𝜁

_𝑖^𝑇

. (25)

Computing all the Jacobian products (i.e., 𝐻

_𝑘

𝑋, 𝐻

_𝑘

𝑄, 𝐻

_𝑘

𝑄𝐻

_𝑘^𝑇

) only require K + 3 forward computations, which can lead to drastic com- putational savings. For more details on the computational performance of the PCGA method over traditional inversion methods, the reader can refer to Kitanidis and Lee (2014) and Soueid Ahmed et al. (2018) .

3.2. Compressed state Kalman filter

Retrieving the conductivity and chargeability fields is an estimation problem. If we are considering time-lapse monitoring, these fields are expected to evolve with time and therefore we seek to monitor the evo- lution of their states. In other words, in addition to a spatial regular- ization, a temporal one will be needed as well. Kalman (1960) intro- duced an iterative filter that used a series of data measured in time to predict the unknown parameters (conductivity and chargeability in our case). An advantage of the Kalman filter is that it incorporates all the knowledge taken from the previous time data to estimate the current pa- rameter field. The Kalman filter was extended to the nonlinear case by Anderson and Moore (1979) , the so-called Extented Kalman filter (EKF).

In essence, the EKF follows the same spirit as the Kalman filter. We con- sider a nonlinear system where the state variables are described as:

𝑠

_𝑘+1

= 𝜓 ( 𝑠

_𝑘

)

+ 𝜔

_𝑘

, (26)

where s

_k

denotes the estimation of the state variable at time k , and 𝜓 is the state transition model. The quantity 𝜔

k

represents any model errors such as poor knowledge of initial and boundary conditions or some other parameters related to the forward modeling. The quantity s

_k

and 𝜔

k

are M × 1 vectors, and m denotes the number of unknowns.

Similarly, the observations can be related to the state variables as 𝑑

_𝑘+1

= Φ (

𝑠

_𝑘

)

+ ̃𝑣

_𝑘

, (27)

where Φ(.) is the discretized forward operator, in our case it is given by the electrical potential equation, ̃𝑣

_𝑘

is the measurements error vector. d

_k+1

and ̃𝑣

_𝑘

are N × 1 vectors, with N the number of observations.

The EKF involves two steps: a forecast step and an analysis one. The

forecast step consists in predicting the state variable at time k + 1 while

the analysis step can be seen as an update which seeks to improve the

state variable using the observations at time k + 1. In fact, updating

the state variables in the analysis step can be seen as a minimization

Fig. 3. Electrical conductivity inversion of case study 1. a. True conductivity field. b. Estimated conductiv- ity field. 17 current injection resulting in 2248 electric fields measurements were used for this inversion. The inverse algorithm converged after 5 iterations. One can notice that despite the small number of current injec- tions, the main features of the conductivity are well recovered both in terms of shapes and magnitudes.

problem, in which the update state minimizes the objection function:

𝐿 = (

𝑑

_𝑘+1

− Φ( 𝑠

_𝑘

) )

_𝑇

𝑅

⁻¹

(

𝑑

_𝑘+1

− Φ( 𝑠

_𝑘

) ) + (

𝑠

_𝑘+1

− 𝜓 (

𝑠

_𝑘|𝑘

))

𝑇

𝐶

_𝑘⁻¹_+1|𝑘

( 𝑠

_𝑘+1

− 𝜓 (

𝑠

_𝑘|𝑘

))

, (28)

R is N × N the measurement covariance matrix, s

_k|k

is the parameter estimate at time k based on the measurements up to time k , likewise, 𝐶

_𝑘+1|𝑘

is the K × K compressed covariance matrix at time k + 1 based on the observation up to time k .

A major limitation of the conventional EKF appears in Eq. (28) . Indeed, we need to compute, store and invert the covariance matrix 𝐶

_𝑘+1|𝑘

. This is difficult for large-scale applications. A solution resides in reducing the dimensionality of the problem by approximating the covariance matrix by low-rank matrices (e.g., Kitanidis, 2015 ).

Following the same idea as the one presented in the previous section for the geostatictal inversion, the covariance matrix C is approximated by a low rank approximation in the form:

𝐶 ≃ ̄𝐶 = 𝐴 𝑉

_𝐶

𝐴

^𝑇

, (29)

where A is a M ×K matrix and V

_C

is a K ×K symmetric positive definite matrix. The idea is to fix A during all the KF iterations and to only update the compressed covariance matrix V

_C

. This algorithm will be referred to as the Compressed Extended Kalm Filter (CSFK). In this work, A is cho- sen to compose the K leading eigenvalues of a given covariance matrix that can be assembled using a variogram. The initial small covariance matrix C

₀

is given by a kernel matrix for instance a Gaussian one.

The forecast step allows for computing the optimal state variable and its associated compressed covariance matrix

𝑠

_𝑘+1|𝑘

= 𝜓 ( 𝑠

_𝑘|𝑘

)

, (30)

𝐶

_𝑘+1|𝑘

= ( 𝐴

^𝑇

𝐴

_𝜓

)

𝐶

_𝑘|𝑘

( 𝐴

^𝑇

𝐴

_𝜓

)

𝑇

+ 𝑉

_𝑄

, (31)

where V

_Q

( K ×K ) comes from the factorization of the covariance matrix

Q = AV

_Q

A

^T

. A

_𝜓

= H

_𝜓

A ( M × K ) (with 𝐻

_𝜓

= 𝜕 𝜓 ∕ 𝜕 𝑠 |

_{𝑠𝑘|𝑘}

) denotes the prod-

uct of the Jacobian matrix associated to 𝜓 and the matrix A . The analysis

step updates the state and compressed covariance. At step at time k + 1,

Fig. 4. Chargeability inversion of case study 1. a. True chargeability field. b. Estimated chargeability field.

17 current injection and 2248 electric fields measure- ments were used. The inverse algorithm converged after 7 iterations. This chargeability inversion was carried out using the conductivity field illustrated in Fig. 3 . Note that during the inversion process, we in- vert for the logarithm of the chargeability to enforce positivity as the intrinsic chargeability is, by definition, a positive quantity.

the state is updated using observations at the same time step:

𝑠

_𝑘+1|𝑘+1

= 𝑠

_𝑘+1|𝑘

+ K ( 𝑑

_𝑘+1

− Φ (

𝑠

_𝑘+1|𝑘

))

. (32)

The matrix K ( M × N ) is known as the Kalman gain. It can be obtained from the solution of the following linear system:

( 𝐴

Φ

𝐶

_𝑘+1|𝑘

𝐴

^𝑇_Φ

+ 𝑅 )

Σ = 𝐴

Φ

𝐶

_𝑘+1|𝑘

, (33)

where 𝐴

Φ

( 𝑁 × 𝐾 ) is the Jacobian product associated to forward opera- tor F , 𝐴

Φ

= 𝐻

Φ

𝐴 with 𝐻

Φ

= 𝜕 Φ∕ 𝜕 𝑠 |

_{𝑠𝑘|𝑘}

. Then the Kalman gain is given by

K = A Σ

^𝑇

. (34)

The Kalman gain plays a major role in the filtering process. It acts as a weight for adjusting the state variables to better fit the observations.

Inaccurate values of the Kalman gain will lead to unrealistic estimates values and artifacts in the tomograms and may lead to divergence of the filtering process.

Once the Kalman gain is computed, it can be used to update the compressed covariance matrix at time k + 1 as:

𝐶

_𝑘+1|𝑘+1

= (

𝐼 − Σ

^𝑇

𝐴

Φ

) 𝐶

_𝑘+1|𝑘

(35)

where I denotes the identity matrix. At the light of the presented algo- rithm, the computational advantages of the CSKF can be highlighted: (1) The full-covariance matrix is never computed explicitly, we only with deal with the small compressed covariance matrix, which is updated at each time step. (2) The Jacobian matrices (i.e., 𝜕 𝜓 ∕ 𝜕 𝑠 |

_{𝑠𝑘|𝑘}

and 𝜕 Φ∕ 𝜕 𝑠 |

_{𝑠𝑘|𝑘}

) are never computed explicitly, only their products are calculated. Such computation can be done efficiently using the finite differences scheme previously presented below (CKSF algorithm):

Initialization step: Decompose covariance matrices, as following:

C = AV CA ^T, and Q = AV QA ^T. Forecast step: m k+1|k= 𝜓( m k|k)

Compute A _𝜓= H _𝜓A C k+1|k= ( A ^TA _𝜓) C k|k( A ^TA _𝜓) ^T+ V Q

Analysis step: Compute 𝐴 Φ= 𝐻 Φ𝐴 Compute Kalman gain K by :

• Solving for Σ, ( 𝐴 Φ𝐶 _𝑘+1|𝑘𝐴 ^𝑇_Φ+ 𝑅 )Σ = 𝐴 Φ𝐶 _𝑘+1|𝑘

• Compute K = A Σ^T Update state variable

s k+1|k+1= s k+1|k+ K( d k+1− Φ( s k+1|k)) Update compressed covariance matrix

𝐶 _𝑘+1|𝑘+1= ( 𝐼 − Σ^𝑇𝐴 Φ) 𝐶 _𝑘+1|𝑘

4. Numerical investigations

We investigate now the relevance of this new method of acquiring induced polarization data and the algorithms presented in the previous sections to interpret these data. Four synthetic case studies are pre- sented. For all the cases, the data are acquired using the electric field components-based approach.

4.1. Case study 1

The goal of this test is to benchmark the geostatistical inversion of the electric field components. The domain of simulation is a volume of 1km ×1km × 100m (see Fig. 2 ). We use the SGEMS software (e.g., Deutch and Journel, 1992 ) to generate hypothetical true electrical conductivity and intrinsic chargeability fields (see Figs. 3 a and 4 a). The electrical conductivity values vary between 10

^−3.6

S m

^-1

(2.5 ×10

⁻⁴

S m

⁻¹

) and 10

^−1.8

S m

^-1

(1.6 × 10

⁻²

S m

⁻¹

). These true fields are assumed to be unknown and the inverse scheme is used to estimate their spatial distributions. They were generated using Gaussian var- iograms: 𝛾 ( h ) = 0.5[1 − exp (( − h /100)

²

)] for the conductivity and 𝛾 ( h ) = 1 − exp (( − h /135)

²

) for the chargeability, h is the distance expressed in m. The variogram depicts the spatial autocorrelation of the measured sample points. A variogram is characterized by a sill and a range. The physical meaning of the sill is the total variance contribution. The range is that pairs of parameter values at locations that are this distance or greater apart are not spatially correlated. A total of 144 receivers are placed on the ground surface. Two current injection electrodes A and B are used. The position of the electrode A is fixed and does not change during the acquisition process while the position of the electrode B is switched for each current injection at a set of predetermined locations. This makes sure that we stimulate different regions of the domain to obtain complete information in the electric response. In fact, when the current is injected, it propagates through the domain, even in the deeper areas of the subsurface. Nevertheless, it remains stronger at the vicinity of the injecting electrodes. This suggests that the current injections should be performed in several locations across the simulation domain to avoid recording redundant data that do not improve the resolution of the final tomogram.

A total of 17 current injections are performed, for each injection the electric field norm is computed on each of the 144 stations, this means that the total number of measurements is 2248. As shown in Section 2 , the apparent resistivities and chargeabilities data acquired using this methodology are always expected to be positive. An advantage of hav- ing positive data entries is that we can use their logarithms to invert the fields, which is generally more stable. This is especially true for datasets where there may be variations over several order of magnitude. We first estimate the electrical conductivity field, which is then used to estimate the chargeability field. Both inversions are launched with homogenous starting models that correspond to the geometric mean of the apparent electrical conductivity and apparent chargeability values. To be repre- sentative of the field conditions, the data have been corrupted with a 5% Gaussian noise. Although true 𝜎 and 𝜂 distributions have been gen- erated with known variograms, this is generally not the case on the field and the properties of the variograms need to be evaluated to construct the elements of the covariance matrix. Several approaches exist for estimating the variogram parameters (e.g., Soueid Ahmed et al., 2018 ).

One can estimate them together with inverse problem unknowns, or they can be deduced from our prior knowledge of the field, for instance, based on geological cross-sections. Another approach consists in using experimental variograms to fit theoretical varigorams (e.g., Deutsch and Journel, 1992 ; Kitanidis, 1997 ; Minsasny and McBratney, 2005 ). The idea is to use a theoretical variogram (for instance, exponential or Gaus- sian) and to solve a least square minimization problem to fit the experi- mental data. The unknowns are the sill and the correlation length. Fig. 5 shows the experimental 𝜎 and 𝜂 variograms and the theoretical ones.

Fig. 5. Plot of the experimental variogram against the theoretical one. a . Elec- trical conductivity . b. Chargeability. The variograms are those of the case study 1. The red squares denote the experimental variogram values while the blue line denotes the theoretical varigoram. The recovered sill and range are 0.57 and 110 for the conductivity and 0.73 and 140 for the chargeability. (For inter- pretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

One last issue needs to be addressed before being able to start the inversion process, that is the truncation of the covariance matrix.

As stated in the previous section, for computational efficiency, the covariance matrix is factorized into a product of low rank matrices. This approximation accuracy Is the related to choice of the truncation order K . In other words, K needs to be chosen in a way that will guarantee that the approximated covariance matrix still captures the major information about the field. As explained in Section 3 , the relative error between the covariance matrix and its squeezed form should be ≪ 1.

We chose K = 125 which leads to an error of 0.001. The eigenvalues decay of the covariance matrix is shown in Fig. 6 for the 𝜎 and 𝜂 co- variance matrices. This fast decay indeed suggest that the most relevant information for the fields can be described by the leading eigenvalues.

The domain of interest is discretized into regular rectangular cells.

The PCGA method is used to invert the data. The convergence is con-

sidered to be reached if after two consecutive iterations of the inverse

problem, the objective value function decrease is smaller than 0.001.

Fig. 6. Covariance matrix singular values decay for case study 1. a. Electrical conductivity . b. Chargeability. One can observe the fast decay of the singular values.

This suggests that the main information about the conductivity and chargeability structures can be captured by the leading eigenvalues, and thus the dimensionality of the inverse problem can be significantly reduced.

After 5 iterations the convergence criteria were met for the inversion of the electrical conductivity. Fig. 3 b shows the inverted conductivity 𝜎 , The major features are well reconstructed and the scatterplot of the true field against the estimated one shows a linear trend with a high coefficient of correlation, R

²

= 0.80. This estimated electrical conductivity field is used in the estimation of the chargeability field.

After 7 iterations, the algorithm has converged giving the chargeability estimate that is shown on Fig. 4 b. The inversion results are satisfactory and show high resemblance between the true and estimated chargeabil- ity fields (see Fig. 4 a and b). The chargeability coefficient of correlation R

²

= 0.78. The magnitudes of both the chargeability and electrical conductivity agree with the magnitudes of the true fields as illustrated by the residuals fields and scatter plots shown in Figs. 7 and 8 .

4.2. Case study 2

This case study is an extension of the previous case study but introducing now some time dependence. The same geometry as well as the electrodes configuration of the previous case are used. The numbers of measurements and unknowns remain unchanged. That said, in this case, the electrical conductivity and the chargeability fields are not stationary. To account for this, we computed several realizations of the conductivity field and chargeability fields using the SGEMS software.

These true fields are represented on Figs. 9 and 11 . They are generated with Gaussian variograms. Each snapshot illustrates the spatial distri- butions of the fields at a given time. The purpose of this case study is to see to what extent the CSKF will be able to monitor the temporal evolution of the parameter fields. All the input data are contaminated with a 5% Gaussian noise. The measurement covariance matrix is set to be diagonal with its entries given by (e.g., Lehikoinenen et al., 2009 ):

𝑑 𝑖𝑎𝑔 ( 𝑅

^𝑘

)

= 𝜔𝐼 ( 𝑎𝑏𝑠 (

𝑑

_𝑜𝑏𝑠^𝑘

) + 𝑚𝑎𝑥 (

𝑑

_𝑜𝑏𝑠^𝑘

))

(36) where 𝑅

^𝑘

is the measurements covariance matrix at time step k , 𝜔 = 5%

in our case, I is the identity matrix and 𝑑

_𝑜𝑏𝑠^𝑘

is the observed data vector at time step k , abs (.) denotes the absolute value operator.

The initial covariance matrices for both the conductivity and charge- ability are constructed using the properties of the variogram of the first time step. These properties are assumed to be unknown and are esti- mated using the approach discussed in the previous synthetic case. Sim- ilarly to the previous case, the truncation order was set to be K = 125. As the true fields do not significantly change from an iteration to another one, it is reasonable to choose the random walk model to update the state variables, which means that in Eq. (30) , 𝜓 = I . The conductivities are first estimated. Fig. 9 shows the predicted conductivities. They are

in general well estimated and the heterogeneities of the field are well re- covered. The magnitudes of the estimates are in the same range as those of the true parameters as shown by the residuals plots (see Fig. 10 ). The conductivity at each step is used in the estimation of the chargeability at the corresponding time step using the CSKF. Fig. 11 shows the charge- ability tomograms. The anomalies of the medium are well reconstructed even if some smoothness is observed due to the effect of compressing the covariance matrices. The uncertainties observed both in the conduc- tivity and chargeability tomograms around the top right corner of the simulation domain are due to a lack of observation stations in this area.

Indeed, Fig. 2 shows that there is no station coverage in this region of the simulation domain. The estimation seems to be less reliable at the begin- ning of the process i.e. the first two time steps. In fact, the CSFK, updates the current field based on the previous measurements, which means that the more we iterate the more the information we accumulate for pre- dicting the next state. The residual fields ( Fig. 12 ) show that the uncer- tainties are larger at the first two time steps and they become smaller.

One last issue that deserves to be discussed is the Kalman gain ma- trix. As stated before, the Kalman gain allows for adequately adjusting the update of the state variables and the compressed covariance matrix.

Each column of the Kalman gain is related to a given observation. In other words, each column of the Kalman gain matrix gives the sensitiv- ity of the field to an observation. Fig. 13 shows the two columns of the Kalman gain matrix for the conductivity and the chargeability at the last time step. As expected, the Kalman gain reaches its maximum values in the vicinity of the receiver at which the measurements are performed, i.e. the corresponding column. This means that the sensitivity to the corresponding measurements (the 60

^th

and the 1800

^th

measurements) is at its maximum around the location of these measurements.

This case study validates the CSFK approach and the electric field components acquisition approaches for monitoring transient processes.

In the next case study, we move to the induced polarization monitoring of a more complicated dynamic process.

4.3. Case study 3

This case deals with the modeling of a saline intrusion in a hetero-

geneous aquifer. Electrical conductivity imagining will be performed to

delineate in space and time the evolution of the contamination. As this

phenomenon is governed by several aspects of physics, we first would

like to present these governing equations, that is, the groundwater flow

and transport equations as well as the petrophysical relationships that

connect the concentration of the salt to the electrical conductivity of

the medium.

Fig. 7. Case study 1 Electrical conductivity uncertainties. a. Conductivity residual field. b. Scatter plot of true field versus estimated one.

Fig. 8. Case study 1 Chargeability uncertainties. a. Chargeability residual field. b . Scatter plot of true field versus estimated one.

Under steady state conditions and in a homogenous isotropic medium, the groundwater flow equation reads:

∇ . ( 𝐾 ∇ ℎ ) = 𝑄. (37)

We need to add some boundary conditions:

ℎ = ℎ

𝐷

on Γ

_𝐷

, (38)

− ̂𝑛 .𝐾 ∇ ℎ = 𝑞

_𝑁

on Γ

_𝑁

, (39)

where h denotes the hydraulic head (in m), K denotes the hydraulic conductivity field (in m.s

⁻¹

), Q (in m

³

. s

⁻¹

) is a flow source term, h

_D

denotes the hydraulic head value imposed at the Dirichlet’s boundary, while q

_N

is the hydraulic flux (in m

³

. s

⁻¹

) imposed on Γ

_N

. The unit vector ̂𝑛 is the unit outward vector normal to the Neumann’s boundary Γ

_N

. The transport equations are obtained by coupling two constitutive equations, namely Darcy’s law and Fick’s law:

𝑢 = − 𝐾 ∇ ℎ, (40)

𝑗

_𝑑

= − 𝜌

_𝑓

𝜙𝐷 ∇ 𝑐 + 𝜌

_𝑓

𝑐𝑢, (41)

where u (in m s

⁻¹

) denotes the Darcy velocity (hydraulic volumetric flux density), j

_d

is the salt flux (in kg m

⁻²

s

⁻¹

), D (in m

³

s

⁻¹

) is the hy-

drodynamic dispersion tensor, 𝜌

f

(in kg m

⁻³

) is the solute bulk density, 𝜑 (dimensionless) is the porosity, c (dimensionless) is the solute mass fraction. The hydrodynamic dispersion tensor D is expressed as 𝐷 =

( 𝐷

_𝑚

𝛼 + 𝛼

𝑇

𝑣

)

𝐼 + 𝛼

_𝐿

− 𝛼

_𝑇

𝑣 𝑣 ⊗ 𝑣, (42)

where D

_m

(in m

²

s

⁻¹

) is the molecular diffusion coefficient of the salt (at 25°C and at high salinities, D

_m

= 1.44 × 10

⁻⁹

m

²

s

⁻¹

), 𝛼 denotes the tortu- osity of the pore space (product of the formation factor by the porosity), 𝛼

L

is the longitudinal dispersitivity, 𝛼

T

is the transverse dispersitivity and v is the average velocity, and ⊗ denotes the tensorial product.

The continuity equation (conservation equation for the mass of solute) states that:

𝜕 ( 𝜌

_𝑓

𝜙𝑐 )

𝜕𝑡 + ∇ ⋅ 𝑗

_𝑑

= 0 . (43)

Inserting Eq. (41) into Eq. (43) yields the advection-dispersion equation:

𝜙 𝜕𝑐 𝜕𝑡 + ∇ ⋅ ( − 𝜙𝐷 ∇ 𝑐 ) + 𝑢 ∇ 𝑐 = 0 . (44) Eq. (44) can be completed by the following boundary conditions

𝑐 ( 𝑥, 𝑡 ) = 𝑐

0

on Γ

_𝑢

, (45)

Fig. 9. Case study 2 electrical conductivity.

The conductivity fields are represented for

each time step. The first column illustrates

the true fields and the second columns the

represents the estimated fields. We note that

the inversion is doing a good job in retriev-

ing the fields.

Fig. 10. Case study 2 electrical conductivity residual fields at various time steps. a. Time step1 . b. Time step 2 c. Time step 3. d. Time step 4. e. Time step 5.

− ̂𝑛 ⋅ ( 𝐷 ∇ 𝑐 ) = 0 on Γ

_𝑑

, (46)

where Γ

_u

corresponds to the upstream boundary (where the salt injection takes place) and Γ

_d

denotes the downstream boundary. At the initial time t = 0, we assume that c ( x, t ) = 0, where x denotes any spatial location of the 3D domain. Solving Eq. (44) allows for having the salt salinity for each time at any region of the aquifer.

On the other hand, according to Waxman and Smits (1968) , the electrical conductivity of the medium can be seen as the combination of a bulk conductivity related to the pore water conductivity 𝜎

f

(in S m

⁻¹

) and a surface conductivity 𝜎

s

(in S m

⁻¹

) related to conduction in

the double layer coating the surface of the grains:

𝜎 = 𝜎

_𝑓

𝐹 + 𝜎

_𝑠

, (47)

where F (dimensionless) is the formation factor, F = 𝜑

^−m

and where m (dimensionless) is the cementation or porosity exponent, ( m > 1). In this case study, we assume that 𝜎

s

(S m

⁻¹

) is constant in the aquifer which means that the electrical conductivity changes in the aquifer are induced by the changes in pore water conductivity, which are related to the solute concentration. In fact, the pore water conductivity is related to solute concentration though:

𝜎

_𝑓

= 𝛼𝑐 ( 𝛽

(+)

+ 𝛽

(−)

) 𝑒, (48)

Fig. 11. Case study 2 chargeability. The

true chargeabilities are represented at left

and the estimated ones at the right. The re-

sults are shown for each time step.

Fig. 12. Case study 2 chargeability residual fields at different time steps a. Time step 1 . b. Time step 2 . c. Time step 3. d. Time step 4. e. Time step 5.

where 𝛼 is a constant used to account for the conversion of the unit of concentration from kg m

⁻³

to molecules per m

³

, 𝛽

(+)

and 𝛽

(−)

are the mobility of the cations and anions in the pore water, at 25

^◦

𝐶 𝛽

(+)

= 5 × 10

⁻⁸

( m

²

S

⁻¹

V

⁻¹

) , and 𝛽

(−)

= 7 × 10

⁻⁸

( m

²

S

⁻¹

V

⁻¹

) , and 𝑒 denotes the elementary charge of the electron ( e = 1.6 × 10

⁻¹⁹

C).

Combining Eqs. (47) and (48) allows for explicitly writing the de- pendence of the electrical conductivity on the solute concentration as 𝜎 = 𝛼𝑐𝑒 𝜙

^𝑚

(

𝛽

(+)

+ 𝛽

(−)

) + 𝜎

_𝑠

. (49)

Now that the governing equations have been presented, we describe the time-lapse monitoring survey. In this numerical case study, we sim- ulate the contamination of an aquifer by a saline intrusion. The solute is supposed to be non-reactive NaCl salt. The synthetic aquifer covers a volume of 1km ×1km ×100m. A hydraulic gradient is created through the aquifer by imposing a 10 m hydraulic head at one vertical boundary and a 5 m hydraulic head at the opposite boundary. The porosity of the aquifer is constant 𝜑 = 0.30. The hydraulic conductivity of the aquifer is heterogeneous and varies between 10

⁻⁷

m s

⁻¹

and 10

⁻³

m s

⁻¹

.

This hydraulic conductivity field was generated using an exponential variogram. The migration of the contaminant occurs due to the natural hydraulic gradient through the diffusion and advection processes.

The contamination is assumed to take place at the left vertical boundary at a rate of 2 kg m

⁻³

. The transport equation is solved for a period of 10 days, using a time step of 1h. This allows for mapping the NaCl plume evolution throughout the aquifer for different times.

The contamination starts from the left region of the aquifer where the time dependent boundary condition is applied and progressively moves across preferential pathways to reach the right boundary after approximately 10 days, while augmenting the solute concentration in the aquifer. Since the electrical conductivity of the medium is related to the solute concentration, electrical conductivity imaging is expected to reveals some information about the spatial distribution and temporal evolution of the solute plume. We assume that the surface conductivity is constant in the aquifer and equals to 10

⁻²

S m

⁻¹

.

An electrical resistivity tomography survey using decentralized

stations is performed using a grid of 144 stations placed as in case study

Fig. 13. Kalman gain for case study 2. a. 60

^th

measure- ment. b. 1800

^th

measurement. The Kalman gain is a ma- trix, each of its columns is associated with a measurement.

This figure shows the 60

^th

and the 1800

^th

columns.

1. The injecting electrode A is placed at the point of coordinate: (10 m, 10 m, 0 m) and the second injecting electrode is moving between the stations. 17 current injections are performed, resulting in a 2448 electric field intensity measurements. The injected current has an intensity of 1 A. The domain of simulation is discretized into 8,000 regular cells. The initial covariance matrix is constructed based on an exponential kernel whose parameters are estimated using the least mean square approach presented in the first study case. The truncation order is K = 125, this is a reasonable choice given the fast decay of the eigenvalues of the covariance matrix. The electric field data were intentionally contaminated with a 5% Gaussian noise.

The random walk model is used to update the state variable, this means that we do not expect the conductivity to abruptly or quickly change within two consecutive time steps. Eq. (49) is used to derive the electrical conductivity from the solute concentration, these conductivity fields are considered to be the true conductivity fields and are used to generate to the electric field data. The CSKF is then used to predict the evolution of the electrical conductivity for 6 observation times: 1 day, 2 days, 3 days, 4 days, 5 days, 6 days.

Fig. 14 shows snapshots of the simulated and estimated electrical

conductivities. We can notice that the main features of the electric con-

ductivity front resulting from the pollutant transport are well captured

and they follow as well the structures of the solute concentration. The

estimated parameters have a high resolution even for the first time

steps where the diffusion process just start to emerge. Table 1 lists the

RMS and R

²

values of the true and estimated parameters. The low RMS

and large R

²

values indeed suggest that at least the major features

are well resolved. The residuals field are plotted on Fig. 15 . One can

see that the uncertainties are relatively low even in the vicinity of

the bottom of the domain. This shows that the contaminant can be

tracked even in the deeper areas of the aquifer without the need of

installing electrodes in boreholes, which is both expensive (because of

the high cost of drilling) and tedious. Finally, we would like to stress

that, although, saline intrusions generally take more time to occur over

a kilometric distance, we believe that is case study remains interesting

for proving the ability of the new system that we present to capture the

evolution of large scale physical processes that are of interest for the

hydrogeology community such as saline intrusion.

Fig. 14. Case study 3 electrical conductivities. The true conduc-

tivities are shown on the left while the true ones are shown on

the right side.

Table 1

Root mean square errors and coefficients of correlation . The values are listed for each time step.

Case study 1 Case study 2 Case study 3 Case study 4

RMSE 𝜎a 0.009 0.07, 0.19, 0.29, 0.17, 0.18 0.015, 0.014, 0.013, 0.015, 0.015, 0.018 0.018, 0.014, 0.015, 0.015, 0.015, 0.014 𝜎 0.15 0.26, 0.39, 0.46, 0.42, 0.36 0.041, 0.043, 0.046, 0.068, 0.074, 0.092 0.05, 0.047, 0.049, 0.05, 0.06, 0.06

𝜂a 0.004 0.01, 0.024, 0.026, 0.019, 0.026 – –

𝜂 0.02 0.018, 0.04, 0.041, 0.03, 0.05 – –

R ² 𝜎a 0.99 0.99, 0.98, 0.96, 0.98, 0.98 1, 1, 1, 1, 1, 1 0.99, 1, 0.99, 0.99, 0.99, 1 𝜎 0.8 0.8, 0.8, 0.8, 0.7, 0.9 0.9, 0.94, 0.95, 0.91, 0.88, 0.78 0.57, 0.73, 0.8, 0.81, 0.8, 0.84

𝜂a 0.8 0.87, 0.79, 0.76, 0.87, 0.91 – –

𝜂 0.78 0.72, 0.61, 0.45, 0.74, 0.81 – –

Fig. 15. Case study 3 residual fields at differ- ent observation times. a. 1 day. b. 2 days. c. 3 days . d. 4 days. e. 5 days. f. 6 days.

4.4. Case study 4

In this case study, we simulate a recharge experiment in a 1km × 1km × 200m region. This domain is considered to be variably saturated. Geophysical methods such as the induced polarization method can be used to map the infiltration of water in the vadoze zone and monitor the recharge of the saturated zone of the aquifer. From a numerical point of view, this requires modelling the groundwater flow and the electric mechanisms associated with it. To simulate the groundwater flow in a variably saturated medium, the Richard’s equation needs to solved. This will give the saturation of the water in the medium which in turns can be used to compute the electrical con- ductivity and the chargeability. We first briefly recall the constitutive equations and petrophysical relations used to describe the saturation

processes. Richard’s equation (e.g., Richards, 1931 ) is written as:

( 𝐶 𝜌𝑔 + 𝑆

𝑤

𝑆

) 𝜕ℎ

𝜕𝑡 − ∇ ⋅ (

𝐾

𝑟

( 𝑆

𝑤

) 𝐾

𝜌𝑔 ∇ ℎ )

= 𝑄

𝑠

, (50)

where C is the specific moisture capacity (m

⁻¹

), S

_w

(dimensionless) is the water saturation, S is the storage coefficient (m

⁻¹

), h (m) is the hy- draulic head, t is the time (s

⁻¹

), K

_r

is the relative hydraulic conductivity, K (m s

⁻¹

) is the hydraulic conductivity of saturated zone, and Q

_s

(s

⁻¹

) denotes a source term that represents any external fluid flow sources.

The van Genuchten model is used to describe the effect of saturation S

_w

on the capillary pressure and the relative hydraulic conductivity K

_r

: 𝑆

_𝑒

= 1

( 1 + |𝛼𝜓 |

^𝑁

)

𝑀

, (51)