Introduction

Electrical resistivity surveys aim at producing an image of the sub-surface’s resistivity distribution from current injected directly into the ground. This direct injection makes the technique a galvanic source method; the strategy here is different from that of inductive source methods where electromagnetic coils are used to induce current flow. Researchers working with the galvanic source methods usually emphasize the measurement of resistivity; those working with inductive source methods emphasize the measurement of electrical conductivity. The two terms are transposable: conductivity is simply the inverse of resistivity.

In resistivity surveys, low-frequency alternating current is injected into the ground via electrodes arranged on the surface. The reason for the use of low-frequency alternating current in resistivity surveying is to avoid electrode polarization. Current I is injected via a pair of electrodes; the resulting potential difference V is measured between another pair of electrodes (Fig. 1). The transfer resistance R is then computed using Ohm’s law R = V/I. This transfer resistance (or impedance) is complex-valued, and comprises an amplitude and a phase. Its value is used to compute the apparent resistivity which is given by

Fig. 1
figure 1

General arrangement of electrodes in resistivity imaging. Current electrodes are denoted by A (or C1) and B (or C2); potential electrodes are denoted by M (or P1) and N (or P2). Current is injected into electrodes A and B; induced potential difference is measured between electrodes M and N

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$${\rho _a}=GR$$
(1)

where G denotes the geometric factor and R denotes the transfer resistance. The geometric factor relates the apparent resistivity measurements to the true subsurface resistances.

Geometric factors are dependent on the type of electrode array in use, hence computed apparent resistivities are also array dependent. The term “apparent resistivity” means that such measurements are not the true resistivities of the particular parts of the subsurface under the electrode dipoles. Rather, they represent the resistivity of an equivalent homogeneous half-space which gives the same data. Extracting true subsurface resistivities from apparent resistivity measurements requires inversion. In practice, a discretized model of the subsurface resistivities is obtained.

In resistivity surveying, the general formula for computing the geometric factor G is

$$\frac{1}{{2\pi }}\left[ {\left( {\frac{1}{{AM}} - \frac{1}{{BM}}} \right) - \left( {\frac{1}{{AN}} - \frac{1}{{BN}}} \right)} \right]$$
(2)

Here, AM, BM, AN, and BN denote the distances between electrodes. These distances depend on the electrode array used. An electrode array in this context means the particular arrangement of current and potential electrodes. Several types of standardized electrode arrays exist: Wenner, Schlumberger, dipole–dipole, pole-dipole, pole–pole, and square arrays. These arrays are based on a simplified geometry which allows the computation of the geometrical factors. As earlier mentioned, the geometric factor G is array-dependent. From Eq. (2), the Wenner array, for example, has the following values for electrode distances: AM = a, BM = 2a, AN = 2a, and BN = a. Its geometric factor G is therefore 2Пa, where a is the spacing between the dipoles. Similar computations are required to compute the geometric factors of other electrode arrays.

Research in resistivity surveying was pioneered by Schlumberger in the early 1900s. Soon the technique was found useful in differentiating the resistivities of different subsurface features, and routine use continued up till the late 1960s (Keller and Frischknecht 1966). Those early surveys were one-dimensional (1D) in nature, and were implemented either as profilings or vertical electrical soundings (VES). In profiling, equally spaced electrodes were moved along the survey profile. After the complete survey of the profile, computed apparent resistivity measurements were plotted against the distance from the center of the electrodes. These plots showed how apparent resistivity varied along the profile. In vertical electrical sounding, electrode spacings were increased on a logarithmic scaling about the array’s midpoint. These increments were undertaken to allow greater depths of subsurface investigation: the wider the electrode spacing, the greater the depth of investigation, and vice versa. Transfer resistances measured at each electrode spacing were used to compute apparent resistivity. Plots of apparent resistivity against the electrode spacing showed how the sub-surface’s apparent resistivity values varied with depth. Such plots were known as sounding curves.

The advent of computers and development of linear filter theory in the early 1970s impacted positively on electrical resistivity surveys. They made it possible to carry out computer-based interpretations of apparent resistivity measurements (Ghosh 1971). However, the initial computer-based interpretations were based on trial and error. They were undertaken by adjusting values of subsurface thicknesses and apparent resistivities on sounding curves to obtain a best fit between the resistivity model and the observed data. The adjustments continued until a best fit was obtained (Van Nostrand and Cook 1966). This best fit was chosen based on a specified stopping criterion. With the development of inversion schemes, this trial and error approach gave way to interpretations based on automatic inversion by software. The initial inversion algorithms worked only on one-dimensional (1D) models (Inman 1975). With further research, new algorithms were developed for two-dimensional (2D) models (Pelton et al. 1978; Smith and Vozoff 1984; Tripp et al. 1984).

Forward solvers were developed for these algorithms: some based on the finite difference scheme (Mufti 1976; Dey and Morrison 1979; Oldenburg and Li 1994; Loke and Barker 1996), others based on the finite element scheme (Coggon 1971; Pridmore et al. 1981; Loke and Barker 1996). These solvers generated synthetic resistivity models for the field data by dividing the subsurface model into cells. The cell size increased with depth. The inversion algorithms then searched for optimal resistivity models that matched the field data. When the synthetic model matched the field data with a defined error, the algorithm discontinued the search. When the synthetic model did not match, and disparity between the model and the data was large, the inversion algorithm iterated to adjust the synthetic model. These iterations continued until either a satisfactory fit between the synthetic model and field data was obtained or a solution could not be found. In cases where the field site had topography, topographic measurements were acquired on the field and incorporated into the forward solver’s grid (Tong and Yang 1990).

Yet another advancement was the development of electrical imaging concepts (Lytle and Dines 1978). These new concepts, adapted from principles developed for biomedical imaging (Plonsey and Fleming 1969; Gordon et al. 1975; Henderson et al. 1976; Swindell and Barrett 1977) in turn prompted the development of multi-electrode acquisition systems. The first prototypes of these systems, which appeared in the early 1980s, used manual switches (Barker 1981). Later versions used switches that were computer controlled. Prior to this time, resistivity survey kits comprised no more than a simple box of batteries, a voltmeter, four electrodes, and few cables. These automated systems allowed two-dimensional (2D) acquisition of resistivity data. This meant resistivity data could now be acquired both vertically and horizontally: a combination of profiling and vertical electrical sounding. While typical one-dimensional sounding surveys gave measurement data ranging from 10 to 20 readings, the two-dimensional imaging surveys gave measurement data ranging from 100 to 1000 readings.

Typical systems comprised an earth resistance meter, a central switching unit, electrodes, multi-core electrode cables, and an IBM compatible laptop computer (Griffiths and Turnbull 1985; Van Overmeeren and Ritsema 1988; Griffiths et al. 1990). Whereas some systems merged the resistance meter and computer in the same box, other systems used resistance meters that were external to the computers. Differences even existed in the nature of electrodes used: some systems used intelligent electrodes with switches at each takeout points, others used passive electrodes having a central switch. To acquire the two-dimensional dataset, field equipment were set up. Equally spaced electrodes were clipped to metallic takeouts along the multicore cable. The multicore cable connected the electrodes to the central switching unit, which then connected the resistance meter and computer. The function of the resistance meter was to generate the current signals and measure the resulting potential differences; the function of the computer was to control the acquisition process and store the measured data.

In cases where practitioners wished to extend the profile line, they used the roll-along technique (Dahlin 2001). Most surveys used a large number of electrodes, ranging from 25 electrodes and more, with varying electrode spacings. The electrode spacing chosen depended on the desired depths of subsurface investigation and the required image resolution. Small electrode spacings gave better resolution; however, they imaged less depth. Large electrode spacings, on the other hand, gave poor resolution but imaged greater depths.

Up till the late 1980s, profiling and vertical electrical sounding were the sole approaches for implementing resistivity surveys (Telford et al. 1990). Resistivity data was typically acquired using the traditional electrode arrays: Wenner, Schlumberger, dipole–dipole, pole-dipole, and pole–pole. With the development of automated multi-electrode electrical resistivity tomography systems (Griffiths and Turnbull 1985; Griffiths et al. 1990) and the innovation of two- and three-dimensional inversion routines (Loke and Barker 1995), resistivity surveying advanced. The new automated systems improved the quality of field data, reduced field survey time, and scaled down manpower requirements. Furthermore, they allowed flexibility within the measurement sequence.

The flexibility, unfortunately, did not initially change electrode arrangements used for resistivity data acquisition. Practitioners tended to keep using the traditional electrode arrays (Wenner, Schlumberger, dipole–dipole, pole-dipole, and pole–pole), which only allowed fixed sequences of measurements. This reluctance to change was possibly because the use of the traditional arrays had proved successful in the past. Besides, characteristics of the different traditional arrays were well-known (Wilkinson et al. 2006a) and researchers knew what array to use for a given target. Furthermore, the absence of efficient inversion software for inverting datasets obtained from complex electrode geometry probably contributed to continued use of traditional arrays. Nevertheless, the situation gradually changed as research efforts commenced on optimizing electrical resistivity tomography field surveys.

One of the early attempts to optimize electrical resistivity tomography in order to reduce field survey times and exploit the flexibility of automated multi-electrode resistivity systems was the development of the roll-along technique (Dahlin and Bernstone 1997). This technique involved adding measurements acquired on a given profile line to measurements acquired from the next profile line, and so on. The results were large datasets which were then inverted to obtain the resistivity distribution within the target area. Yet another effort at resistivity tomography optimization was the development of new electrode arrays as improvements over the conventional Wenner, Schlumberger, dipole–dipole and pole-dipole electrode arrays. Examples were the gradient array and the midpoint-potential-referred measurement array (Dahlin and Zhou 2002) developed to reduce field survey time whilst acquiring large datasets to improve image resolution.

The late 1990s saw the first attempts to optimize datasets for geophysical prospection, the aim being to obtain optimal suites of electrode configurations that would give good reconstructions. One of such attempts (Cherkaeva and Tripp 1996) was based on the principles developed in the late 1980s for biomedical imaging (Issacson 1986). Issacson’s work had involved making selections of optimal current patterns from targets of interest. Use of such optimal currents would generate best reconstructions of the targets’ conductivity distribution.

Adapting Issacson’s principles, Cherkaeva and Tripp (1996) attempted to optimise resistivity datasets for improved reconstructions of the subsurface resistivity. Their work focused on the selection of optimal injected currents that would give optimal geoelectrical datasets with few electrodes. Using optimal geoelectrical datasets in the inversion procedure would therefore give maximum information about the subsurface resistivity distribution.

Examples of other schemes developed to find optimal suites of electrode configurations were the sensitivity analysis scheme (Furman et al. 2003, 2004, 2007) and the object orientated focusing scheme (Hennig and Weller 2005; Hennig et al. 2008). The basic task of the sensitivity analysis strategy was to identify electrode arrays having greatest sensitivity to perturbations placed at particular locations within the subsurface. Arrays with highest sensitivities were considered optimal for resistivity imaging.

The object orientated focusing strategy, on the other hand, aimed at reducing the number of measurements required for a geoelectrical survey by optimizing the measurements’ sensitivity distribution in comparison to a given target’s sensitivity distribution. A superposition of all sensitivity distributions was carried out and a weight factor was applied to adjust its value to that obtained from computations of the target sensitivity distribution. The mathematical problem therefore was that of adjusting the weights to ensure an adequate fit between the target and the measurement sensitivities. To reduce the number of measurements required for the geo-electrical survey, weight factors obtained during object orientated focusing were analyzed. Measurements with low weights were ignored while those with high weights were archived. In all, only 30 percent of the measurements had high weights and a superposition of their sensitivities gave values comparable to that of the target. Hence the conclusion by the authors that the technique served as a useful option for reducing the number of measurements required for a geoelectrical survey by up to 70 percent, consequently saving valuable survey time.

Alternative strategies were developed to select optimal configurations based on the model resolution matrix. Stummer et al. (2004) developed the real-time experimental design scheme. This scheme was implemented by using datasets from a previous standard electrode array as its initial dataset. Arbitrary four-electrode configurations were then added to this initial dataset depending on their influence on the model resolution matrix. At every iteration of the real-time experimental design scheme, a goodness function was computed for every configuration within the comprehensive dataset. This function ranked configurations based on how well they improved the model resolution. At each iterative step, configurations that reduced the model resolution were excluded while those that improved model resolution were marked and used to constitute the optimal dataset on the condition that the configurations were linearly independent from the base set.

Development of the real-time experimental design scheme prompted further research on array optimization within the ERT community. New optimization schemes were developed.

Examples were the “modified goodness function” scheme (Wilkinson et al. 2006b) which amended the goodness function expression used in the experimental design scheme, and the “compare resolution” scheme (Wilkinson et al. 2006b) which computed directly the changes in the model resolution matrix for each new add-on configuration. The modified goodness function scheme and the compare resolution scheme have undergone more modifications (Loke et al. 2007; Wilkinson et al. 2007, 2009; Loke and Wilkinson 2009).

Further strategies were designed to amend the real-time experimental design scheme.

These were the use of a non-homogeneous initial model in place of the original homogeneous model (Athanasiou et al. 2006) and use of a finite element forward solver in place of the original Jacobian matrix inversion procedure (Athanasiou et al. 2009). Whereas use of the non-homogeneous model was to improve the scheme’s model resolution calculations, use of the finite element forward solver was to reduce the scheme’s computational time. It is expected that further selection scheme modifications will continue in the ERT research community.

The above optimization strategies are useful, however, no research has yet been undertaken to develop a general optimization approach in electrical resistivity tomography. There is yet no theoretical framework for finding optimal measurement locations for the technique. Besides, past schemes did not take into account the ill-posedness of the inverse problem of resistivity tomography and the influence of the inversion process on the optimization process. This research aims at such a framework that incorporates the important ingredients that define the inversion process: nature and size of error, choice of regularization scheme, and prior knowledge on solutions in the search for optimal measurement locations. Previous work (Udosen et al. 2018) described the forward and inverse solvers developed for carrying out simulations and reconstructions in electrical resistivity tomography. The inverse solver aimed to reconstruct the resistivity distribution from simulations modeled from the forward solver. This work describes the automated search for optimal electrode locations for a 2D resistivity tomography problem within a meta inverse framework. This meta inverse framework was incorporated into the forward solver and inverse solvers, the aim being to search for optimal electrode locations at which best reconstructions of the resistivity distribution could be obtained. The forward solver generated the simulations, the inverse solver generated the reconstructions, and the meta inverse framework solver aimed at finding the optimal electrode locations at which best reconstructions could be obtained.

Electrical resistivity tomography, just as other tomography inverse problems, is considerably ill-posed (Sasaki 1992; LaBrecque et al. 1996; Cardarelli and Fischanger 2006). This meta inverse framework therefore takes into consideration the influence of ill-posedness on the resistivity reconstruction problem. It also takes into account the influence of the components of an inversion process on optimization. For example, the choice of regularisation parameter, a component of the inversion process, has an influence on the reconstructions and the size of reconstruction errors (Colton and Kress 1992; Engl et al. 1996; Kirsch 1996; Kress 1999; Groetsch 2007). This framework therefore incorporates within its optimisation process important ingredients of an inversion process such as the choice of regularisation parameter, the nature and size of data error, and prior knowledge of solutions.

These are concepts that had not been considered in past schemes developed to explore optimal measurement configurations in resistivity tomography surveys. The meta inverse framework is developed for use by practitioners in applications where there is need to find optimal measurement setups. Use of the framework in these applications will enable the collection of optimal datasets, and consequently improve the resolution of reconstructed images. Further, the framework applies a systematic approach to optimization. Steps of the framework are applied in a methodological manner to find the optimal measurement setup. Use of the optimal measurement setup should result in improved reconstructions of the investigated quantity.

Theoretical foundations

In applications, a direct problem is given by simulations of a natural system and an inverse problem is given by reconstructions of the investigated quantity. We have named the task of finding an optimal measurement setup that improves reconstructions the meta inverse problem.

To solve the direct problem, a forward solver based on the finite integration technique (Weiland 1977, 1985) was developed in MATLAB and used to compute the simulated electric potentials. In electrical resistivity tomography, a volume \(\Omega\) with subsurface boundary \(\partial \Omega\) and unknown resistivity distribution \(\rho\)is imaged by injecting steady-state electric currents into electrodes placed on the boundary of \(\partial \Omega\)(Gautam and Biswas 2016, Olagunju, 2017). Given a source-free inhomogeneous conductor, any electric potential measurement \(V\)is governed by the partial differential equation with boundary conditions

$$V={V_0}~\quad {\text{on}}\quad \partial \Omega \quad {\text{and}}\quad {\rho ^{ - 1}}\frac{{\partial V}}{{\partial \eta }}={j_0}\quad {\text{on}}\quad ~\partial \Omega$$
(3)

Here \(\rho\) denotes resistivity (inverse of conductivity\(\sigma\)) within the investigated region, V denotes the electric potential, V0 denotes the potential at the boundary, j0 denotes the current density at the boundary, and \(\eta\) denotes the outward normal vector to the boundary. The electric potential V in Eq. (3) is typically solved for by using a numerical scheme such as the finite integration technique (FIT). FIT was used in this work to simulate the electric potentials.

FIT approximates values of electric potential on the problem domain by discretising Maxwell’s equations on grids. The finite integration technique was used as the forward solver because it was versatile, easy to implement, able to handle large computations, and capable of approximating arbitrary geometries.

To solve the inverse problem, a domain search algorithm was developed in MATLAB. The domain search algorithm takes the measured (or simulated) data and aims to reconstruct the parameters (i.e. the centre x, the radius r and resistance value R) of the cavity. The basic idea of the search for the function depending on one scalar variable is as follows:

When searching for a variable λ between a minimum value λmin and a maximum value λmax, a regular grid is set up in the form of equation

$${\lambda _{min}}+~\frac{k}{n}\left( {{\lambda _{max}} - ~{\lambda _{min}}} \right),~~~k=0, \ldots ,~n,$$
(4)

scanning the whole search space with particular discretization determined by \(n \in N\). The target function is calculated at all values of \({\lambda _k}\) for k = 0,…,n and the minimizer λ* is determined. The next level of search is set up by choosing an interval

$$[{\lambda _*} - ~\delta \lambda ,~{\lambda _*}+~\delta \lambda ]$$
(5)

with \(\delta \lambda =~\frac{1}{n}\left( {{\text{~}}{\lambda _{max}} - ~~{\text{~}}{\lambda _{min}}} \right)\) and setting up a grid similar to Eq. (4) with \(\left[ {{\lambda _{min}},~~{\text{~}}{\lambda _{max}}} \right]\) replaced by Eq. (5). In other words, an initial search involves scanning the entire search space for the minima λ*. The subsequent searches were undertaken around the region of the initial minima, the aim being to find a better minimum, and so on. This method was modelled using MATLAB in higher dimensions to search for the particular parameters which needed to be determined in the inverse search.

To simulate field conditions, synthetic noise was added to the simulated resistance measurement data using the MATLAB random function rand. To obtain the desired random number range of [a, b], the rand function was multiplied by (b − a) and the product was added to a, i.e., (b − a) \(\times\) rand + a. The range used in the numerical experiments was [− 1, 1]. Random entries generated by the random number function were then multiplied by varying percentages of noise.

To solve the meta inverse problem, that is, finding the optimal measurement locations at which best reconstructions can be obtained, the following steps were modeled in MATLAB.

Reconstruction error

As a first step, the reconstruction error was computed. This error is given by

$${E_1}(\alpha ,\varphi ,\xi ,q)=\left\| {\varphi _{\alpha }^{\xi } - \varphi } \right\|\quad {\text{where}}~\quad \varphi _{\alpha }^{\varepsilon }={R_\alpha }(A\left[ q \right]\varphi +\xi )$$
(6)

Here, φ ∈ U denotes the true solution, ε denotes the imposed random error within the range described by error level δ ≥ 0, α denotes the regularization parameter, and q ∈ Q denotes a measurement setup within a set of possible setups Q.

Regularization parameter

As second step, Eq. (6), i.e., the values of the obtained reconstruction errors were computed over all values of the regularization parameter. This computation was defined by

$${E_2}(\varphi ,\xi ,q)=\;_{{\alpha \varepsilon \left[ {{\alpha _1},{\alpha _2}} \right]}}^{{\hbox{min} }}{E_1}(\alpha ,\varphi ,\xi ,q)$$
(7)

over some interval [α1, α2] ⊂ R, which provides the error for the optimal α.

Error estimate

As third step, the supremum of the functional in Eq. (7) was computed over several estimates of the stochastic error \(\xi\) such that

$$E_{3}^{A} \left( {\delta ,\varphi ,q} \right) = \begin{array}{*{20}c} {\sup } \\ {\xi \in \chi ,\left\| \xi \right\| \le \delta } \\ \end{array} E_{2} \left( {\varphi ,\xi ,q} \right)$$
(8)

where stochastic error δ > 0.

Alternatively, one may assume that some appropriate distribution dpδ, depending on the error level δ is given. The error was then treated by estimating

$$E_{3}^{B}\left( {\delta ,~\varphi ,q} \right)=~{\mathbb{E}}\left[ {{E_2}\left( {\varphi ,.,~q} \right)} \right]=~\smallint {E_2}\left( {\varphi ,~\xi ,q} \right)~d{p_\delta }\left( \xi \right)~$$
(9)

Range of solutions

Our fourth step involved finding an optimal system setup for several φ ∈ U with data controlled by δ > 0. This step was undertaken by estimating the values of Eqs. (8) and (9) over all true solutions φ. This is given by the functional

$$E_{4}^{{AA}}\left( {\delta ,U,q} \right)\quad =\quad \sup E_{3}^{A}\left( {\delta ,\phi ,q} \right)$$
(10)
$$E_{4}^{{AB}}\left( {\delta ,U,q} \right)\quad =\quad \int\limits_{{\;\;\;U}}^{{\phi \Xi U}} {E_{3}^{A}\left( {\delta ,\phi ,q} \right)d\pi \left( \phi \right)}$$
(11)
$$E_{4}^{{BA}}\left( {\delta ,U,q} \right)\quad =\quad \mathop {\text{sup}}\limits_{{\varphi \text{EU}}} E_{3}^{B}\left( {\delta ,\varphi ,q} \right)$$
(12)
$$E_{4}^{{BB}}\left( {\delta ,U,q} \right)\quad =\quad \int_{U} {\text{E}_{3}^{\text{B}}\left( {\updelta ,\text{U},\text{q}} \right)\text{d}\uppi \left( \varphi \right)}$$
(13)

Optimal setup

The last step was that of determining the optimal measurement setup. This step minimizes \(\mathop E\nolimits_{4}^{{ZZ}}\)(with Z ∈ {AA, AB, BA, BB) over a range of measurement arrays q ∈ Q such that

$$\mathop q\nolimits_{{opt}}^{{ZZ}} \quad :=\;\arg {\hbox{min} _{q{\rm E}Q}}\mathop E\nolimits_{4}^{{ZZ}} \left( {\delta ,U,q} \right),$$
(14)

and

$$\mathop E\nolimits_{5}^{{ZZ}} \left( {\delta ,\varphi } \right)\quad :=\;\mathop {\hbox{min} }\limits_{{q\text{EQ}}} \mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right),$$
(15)

The functional \(\mathop E\nolimits_{5}^{{ZZ}} \left( {\delta ,\varphi } \right)\)in Eq. (15) describes the minimum reconstruction error within a system setup q ∈ Q for true solutions φ ∈ U. The minimization in Eq. (14) calculates an optimal measurement setup \(\mathop q\nolimits_{{opt}}^{{ZZ}} \in Q\) such that for measurements \(A\left[ q \right]\varphi +\xi\) the reconstruction value will be minimized. This final step solves the meta inverse problem, i.e., the task of finding an optimal measurement setup that will improve reconstructions of the resistivity distribution.

As mentioned earlier, the finite integration forward solver was used to simulate the data, the domain search algorithm was used to reconstruct the data, and the meta inverse framework solver was used to search for optimal measurement setups. Noise was added to the simulated data, and the domain search algorithm was then used to reconstruct the data. A Wenner configuration was used (Fig. 2). The spacing between the electrodes could either be small or large. See Figs. 3, 4, 5, 6, 7, 8, 9 and 10 for the case where a small electrode spacing was used, and see Figs. 11, 12, 13, 14, 15, 16, 17 and 18 for the case where a larger electrode spacing was used. The domain search algorithm was tested at different electrode setups, but the sake of space, only Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 are shown.

Fig. 2
figure 2

Measurement sequence for a Wenner array

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Fig. 3
figure 3

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 4
figure 4

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 5
figure 5

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 6
figure 6

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 7
figure 7

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 8
figure 8

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 9
figure 9

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 10
figure 10

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 11
figure 11

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 12
figure 12

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 13
figure 13

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 14
figure 14

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 15
figure 15

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 16
figure 16

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 17
figure 17

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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Fig. 18
figure 18

The electrode array is shown in image (a). Individual electrodes are denoted with circles. The reconstruction obtained from the domain search algorithm at 5% imposed data error is shown in image (b). The plot of error vector is shown in image (c). The small white circle on this plot denotes true location of anomaly

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The domain search could be visualized when the search space was set up as a two-dimensional set. An error vector which represents the target functional used for low-dimensional minimization was used to solve the inverse problem in the domain search algorithm. Given a point (x1, 0, x2) to be the center of the cavity on the search grid, the error vector was obtained by computing the difference between the simulated data and the given measurements for a given resistivity model. The error vector represented the objective function which calculated the error in the measurements. The error vectors were calculated over the search grid, and plots of the error vectors are shown in images (c) of Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. On the figures representing the electrode setup, i.e. images (a) of Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18, the circles aligned on the top of the grid denote individual electrodes. Our aim here was to observe the influence of each electrode setup on the nature of reconstructions and corresponding error vector setups.

Numerical results

Reconstructions generated from the different electrode arrays are shown in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. For all the reconstructions shown in Figs. 3, 4, 5, 6, 7, 8, 9 and 10 (where electrode spacing is small), and in Figs. 11, 12, 13, 14, 15, 16, 17 and 18 (where electrode spacing is larger), a noise level of 5% was added to the simulated measurement data. Work by Dahlin (1996), Dahlin and Loke (1998), and Loke et al. (2003) have shown that the data error ranges of < 1–3% is typical for many ERT field surveys. In most cases, the measurement error can be kept significantly small (i.e., < 1%), if a proper electrode-ground contact is ensured (Dahlin 1993).

When there exists geological inhomogeneities within the survey area that cannot be accounted for in the model, the inhomogeneities are considered as noise that may increase the overall noise level (Dahlin and Loke 1996). In the work by Loke et al. (2003), 3% random noise was added to the data. The authors stated that that value was similar to that typically observed in multi-electrode ERT field surveys. In this work, 5% random noise was added to the data.

The domain search algorithm aimed to reconstruct the simulated cavity. The electrode setups are shown in images (a), the resulting reconstructions are shown in images (b), and the error vector plots are shown in images (c) of Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. For ERT applications, the horizontal axis on the electrode setup image represents the length of the profile line, and the vertical axis represents the investigative depth, all in meters (m). The horizontal axis on the reconstruction represents the length of the profile line, and the vertical axis represents the investigative depth, all in meters (m). On the error plots, the horizontal and vertical axes represent the grid space over which the search for the anomaly was implemented.

Recall that the fifth step of the meta inverse framework (Eqs. 14 and 15) entailed computing the minimum of error \(\mathop E\nolimits_{4}^{{ZZ}}\)(with Z ∈ {AA,AB,BA,BB) over a range of measurement arrays q ∈ Q such that

$$\mathop E\nolimits_{5}^{{ZZ}} \left( {\delta ,\varphi } \right)\quad :=\;\mathop {\hbox{min} }\limits_{{q\text{EQ}}} \mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right),$$

The value of q ∈ Q that minimizes error \(\mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right)\!\!:\)

$$\mathop q\nolimits_{{opt}}^{{ZZ}} \quad:=\;\mathop {\arg \hbox{min} }\limits_{{q\text{EQ}}} \mathop E\nolimits_{4}^{{ZZ}} \left( {\delta ,U,q} \right),$$

denotes the optimal electrode locations at which best reconstructions of the resistivity distribution can be obtained. For an imposed noise level of 5%, a plot of error \(\mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right)\,\)obtained for the electrode arrays in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 is shown in Fig. 19. For an imposed noise level of 20%, a plot of error \(\mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right)\,\) obtained for the electrode arrays in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 is shown in Fig. 20.

Fig. 19
figure 19

Surface plot of fourth step of the meta inverse framework, error \(\mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right),\) for the various electrode arrays illustrated in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 at 5% error. The optimal electrode location as deduced from this figure is the location at which the plot has its minimum. The global minimum is denoted by the small white circle on the plot. The alternative local minima are those with the smallest values, with the deepest blue coloring. The alternative local minima illustrate alternative locations at which a solution could be found

Full size image
Fig. 20
figure 20

Surface plot of fourth step of the meta inverse framework, error \(\mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right),\) for the various electrode arrays illustrated in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 at 20% error. The optimal electrode location as deduced from this figure is the location at which the plot has its minimum. The global minimum is denoted by the small white circle on the plot. The alternative local minima are those with the smallest values, with the deepest blue coloring. The alternative local minima illustrate alternative locations at which a solution could be found

Full size image

The optimal measurement setup or electrode location q ∈ Q, i.e., qopt that minimises error \(\mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right)\) is illustrated in Figs. 19 and 20 as the location at which the plots have their minima. Taking measurements with an array having the interval length and distance scaling at which these minima are located will give best reconstructions of the resistivity distribution.

Discussion

Image (a) of Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 illustrates the variations in electrode setups. Figure 3 commenced with a small electrode spacing and the electrodes were moved gradually along the grid to the end of the grid space (Fig. 10), the aim being to overlap the entire grid. At subsequent iterations, the electrode spacings were gradually increased and moved along the entire grid. Further increments in electrode spacing continued until Figs. 11, 12, 13, 14, 15, 16, 17 and 18. The reason for progressively moving the electrodes across the grid was to enable coverage of the entire space. This enables one to observe the nature of reconstructions obtained as the electrodes are shifted along the length of the grid. This is equivalent to surveying every part of the subsurface area under investigation at a range of measurement scales. The subsurface anomaly over which these electrodes are deployed is denoted in image (a) of Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 as a small circle located between 1 and 2 metres on the horizontal axis of the plots (this circle is more clearly seen in Figs. 4, 12). Image (b) of Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 shows the reconstructions obtained at these varying electrode setups with use of the domain search algorithm. These figures demonstrate that the reconstructions depend on the electrode array used. For example, closely spaced electrodes that overlap the entire region comprising the anomaly are seen to give better reconstructions compared to electrodes deployed lateral to the subsurface spherical anomaly (contrast the reconstruction in Fig. 12 with the reconstruction in Fig. 16).

The surface plot shown on image (c) of Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 denotes the error curve that illustrates the error in the reconstruction. The small white circle on each of these plots denotes the true location of the cavity. Error curves with a minimal error at the true location of the cavity. e.g., Figs. 4 and 12, show that the corresponding electrode array used for the particular reconstruction was optimal for obtaining the best reconstruction of the resistivity distribution within the subsurface. In other words, experiments where there are strong minima of reconstruction error (e.g., Figs. 4, 12) denote that their corresponding electrode arrays are optimal for obtaining the best reconstruction. Contrast these error curves with those on Figs. 15, 16, 17 and 18 where there are no strong minima of reconstruction error on the error plot. There is a less clear optimum electrode setup in those figures, in that the electrode arrays do not overlap the cavity being imaged, hence their reconstructions are poorer compared to those obtained when the electrode setups are optimal.

For example, Fig. 10 shows the case where the electrodes do not overlap the investigated anomaly. A poor reconstruction is obtained, and the location of the reconstruction is different from the case where the electrodes overlap the anomaly (e.g., Fig. 4). Hence for the reconstruction in Fig. 10 where the electrodes do not overlap the anomaly, one end ups with a reconstruction shifted in location towards the upper right hand side of the grid, and this is different from where the true solution should have been. This reflects an instability that depends on electrode setup. Further, observe the contrast between the error vector plots of Figs. 4 and 10. Figure 4 has an error vector plot with a strong minima of reconstruction at the location of the anomaly, and its corresponding electrode setup is optimal (as it is aligned over the cavity) for finding the best reconstructions. The error plot in Fig. 10, on the other hand, does not have such strong minima, implying that its electrode setup is not optimal for obtaining the best reconstructions. Since the error curve searches for the location of the cavity given each electrode setup, it solves a form of the meta inverse problem, as it can be used to deduce optimal and non-optimal electrode setups. This implies that without actually seeing the electrode setup in use, one can observe the error vector plot and use the information obtained from the plot to determine if the electrode setup used was optimal or not. Error vectors with strong minima of reconstruction at the true location of anomaly denote optimal electrode setups, while error vectors without strong minima denote that the given electrode setup was non-optimal for obtaining best reconstructions of the resistivity distribution.

The reconstruction in Fig. 18 is different from Fig. 10, but again, it is a poor reconstruction, just as that of Fig. 10. This shows that just as in Fig. 10, the electrode spacing used in Fig. 18 was not optimal, in that did not overlap the anomaly in question, resulting in a poor reconstruction. Further, an observation of their error curves show that there are no strong minima of reconstruction, hence the corresponding electrode setups are non-optimal. This reflects one of the points of the meta inverse optimization procedure, that given non-optimal electrode configurations, the location and nature of reconstructions will differ from the simulated solutions and from reconstructions for which the electrode setups are optimal. Further, error curves can be used to find optimal setups, thereby solving a form of the meta inverse problem.

These examples also show that the electrode distance scaling and location of electrodes, relative to the anomaly to be investigated, influences reconstruction quality. In addition, closely spaced electrodes that overlap the entire region under investigation are seen to give better reconstructions; nevertheless, the spacing between electrodes should not be so small that the anomaly is not overlapped by the array (Fig. 3). This observation bears out previous research by Dahlin and Loke (1998) which linked the problem of poor spatial resolution in ERT surveys to the scale of electrode arrays used for subsurface investigation. Their work showed that the smaller the spacing between measurement electrodes, the better the reconstructions obtained, and vice versa. The quality of reconstructions is dependent on electrode spacing and location. For instance, both Fig. 7 (with very closely spaced electrodes) and Figure.16 (with widely spaced electrodes) that did not overlap the investigated region of the anomaly give worse reconstructions compared to Figs. 4 and 12 where the electrodes were spaced to overlap the investigated region.

Figures 19 and 20 denote plots of error \(\mathop {{E^{ZZ}}}\nolimits_{4} \left( {\delta ,U,q} \right),\)the fourth step of the meta inverse framework for the electrode arrays illustrated in Figs. 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18. In Figs. 19 and 20, there exists a global minimum, alongside other local minima that illustrate alternative locations at which a solution could be found. The global minimum is denoted by the small white circle on the plot, and in general, the alternative local minima considered are those with the smallest values. They reflect typical resonances resulting from the scaling and shift of the grid during the search for an optimal measurement location. In inversion schemes, similar scenarios are observed, where alongside a global minimum, other local minima are also obtained. Usually many local minima surround the global minimum (Fernandez-Martinez et al. 2008) and attempts are then made to choose those that will best minimize the error or find the solution.

The minima in Figs. 19 and 20 therefore denote the optimal electrode array locations at which best reconstructions of the resistivity distribution will be obtained. The significance of the numerical experiments is that they illustrate how one can quantitatively achieve optimization of electrode locations by applying a search within a meta inverse formulation of the optimization problem. The experiments show how sensitive the subsurface reconstruction is to electrode array position, for a given array type and level of noise. Further, they point the way to how optimization might be carried out, given more complex subsurface problems. Numerical results obtained show that the meta inverse problem framework can be implemented successfully to find optimal measurement locations from which one can obtain best reconstructions of the subsurface area under investigation.

Conclusion

The aim of this work was to develop a framework that incorporated the role of ill-posedness of an inverse problem and the components of an inversion process in the search for optimal measurement setups in electrical resistivity tomography surveys. The framework was developed within the context of inverse problems. The reason for the development of the framework within the context of inverse problem is because electrical resistivity tomography is an ill-posed inverse problem. Instability, an aspect of ill-posedness, means that the field data are not continuously dependent on the solution (i.e., the reconstructed image). Small errors within the data translate into large instabilities within the solution. Conversely, large improvements in data quality generate only small improvements in the reconstructed solution.

The problem that this research aimed to solve was to provide theorists and practitioners with a useful framework and inversion scheme for carrying out electrical resistivity tomography reconstructions and for selecting the optimal measurement locations at which best reconstructions of the resistivity distribution could be obtained. This work has demonstrated the integration of the meta inverse framework within  a forward solver and inverse solver, the aim being to search for optimal electrode locations at which best reconstructions of the resistivity distribution could be obtained. Numerical results obtained from incorporating the framework into the solvers have been presented: these results show that the framework is successful for determining optimal electrode locations for resistivity surveys.