Optimized Arrays for 2-D Resistivity Survey Lines Using a Multi-Step Compare R Method

Abstract

The imaging quality of electrical resistivity tomography (ERT) crucially depends on the electrode array configuration. Although the symmetrical optimized ‘Compare R’ (CR) method improves computational efficiency, restricting the search to the symmetrical data set inherently limits the imaging accuracy. To address this limitation, this paper proposes a multi-step optimized CR method that progressively explores both symmetrical and asymmetrical arrays to extend the search space and further enhance imaging accuracy. Numerical experiments demonstrate that the multi-step optimized array yields the highest average relative model resolution (0.646) and structural similarity index measure (0.668), surpassing the symmetrical optimized array (0.615 and 0.630, respectively). Field experiments on pipeline detection confirm that the proposed array accurately identifies the location and geometry of underground anomalies and achieves superior imaging accuracy. Applications in karst cavity exploration further confirm that the proposed array effectively detects the deep karst caves and the bedrock interfaces, as validated by borehole drilling. Additionally, the detection performance of both optimized arrays is evaluated at different depths. The results indicate that the multi-step optimized array preserves anomaly geometry and resistivity more reliably at greater depths, attributed to the accumulation of asymmetrical data points in deep regions, which results in a significantly higher data density.

1. Introduction

Electrical resistivity tomography (ERT) is a widely used geophysical technique for imaging subsurface structures [1,2]. However, the imaging quality of the subsurface structures is strongly influenced by both complex structure’s conditions [3,4] and the selected electrode configuration. Standard arrays, such as the Wenner and dipole–dipole arrays, are constrained by fixed configurations and do not fully exploit the multichannel capabilities of instruments, particularly when imaging complex subsurface structures [5,6,7,8]. The comprehensive data set contains abundant subsurface information, and improves inversion results through extensively increased data density [9,10]. However, acquiring such a massive comprehensive data set is time-consuming and impractical in most field surveys. Therefore, it is critical to develop optimization algorithms that select the most informative electrode configurations, while using the same number of data points, thereby enhancing deep-imaging capability while maintaining survey efficiency.

Most existing array optimization algorithms either focus on maximizing the sum of sensitivity matrix elements [11,12,13] or on maximizing the sum of the diagonal elements of the model resolution matrix [14,15,16]. Stummer et al. [9] proposed the Goodness Function (GF) optimization algorithm, which uses a ranking function to evaluate array performance based on both the sensitivity matrix and model resolution matrix. Numerical and field tests have demonstrated that the optimized arrays provide significantly more subsurface information than standard arrays. To further enhance imaging quality, Wilkinson et al. [17] introduced the ‘Compare R’ (CR) method. Although the CR algorithm produces arrays with the highest average model resolution, it searches the full configuration space throughout the optimization process and therefore requires significantly more computational time than the GF algorithm.

To improve the computational efficiency of the CR method, previous research focused on four key directions: exploiting the symmetry of 2-D survey lines and grid models, leveraging GPU acceleration, optimizing the Sherman-Morrison update, and modifying the comprehensive data set. Loke et al. [16] reduced the size of the comprehensive data set by incorporating model symmetry constraints and derived a modified Sherman–Morrison Rank-1 update formula based on the model resolution increment matrix. They also utilized a GPU parallel architecture, significantly reducing computation time and memory consumption. Subsequently, Loke et al. [18] extended the Sherman–Morrison Rank-1 update from a matrix–vector form to a matrix–matrix form, enabling batch processing of multiple arrays. By combining GPU acceleration strategies, the computational efficiency of the array optimization algorithm is improved by approximately two orders of magnitude. More recently, Loke et al. [19] further improved the CR update process by replacing the four-electrode configuration form of the Sherman-Morrison Rank-1 update formula with a linear combination form of two-electrode pairs. By restricting the optimization to the symmetrical data set, the symmetrical optimized CR method achieved a further order of magnitude reduction in computational cost.

However, the symmetrical optimized CR method still suffers from an upper limit in attainable model resolution. The symmetrical data set accounts for only about 1% to 3% of the comprehensive data set, and its attainable model resolution is approximately 75% to 80% of that provided by the comprehensive data set, thus preventing the optimization from reaching a global optimum [19]. Additionally, in the presence of asymmetrical topography or complex geoelectrical structures with laterally uneven resistivity distributions, the images obtained from the symmetrical optimized array are prone to asymmetrical distortions [19,20]. Moreover, relatively few field studies have examined the practical performance of optimized arrays in complex environments [21]. For example, Abdullah et al. [22] utilized the Noise-Weighted ‘Compare R’ (NWCR) optimized arrays, demonstrating superior performance in delineating weak zone and bedrock interface compared to standard arrays.

To address these limitations, this paper proposes a multi-step optimized CR method, which extends the symmetrical optimized CR method by introducing a multi-step strategy. In the early stages of optimization, the method employs the symmetrical data set to maintain high computational efficiency and establish a stable data set, while in the later stages, a limited number of asymmetrical data sets are introduced to overcome the search space limitation, thereby further enhancing model resolution and imaging accuracy in deeper regions. Under the same number of data points, the proposed method redistributes the selected data points toward deeper regions, enabling better subsurface resolution and greater depth of investigation than standard arrays. The effectiveness and practical applicability of the proposed method are further validated through its application to karst cavity exploration.

The rest of this paper is organized as follows. Section 2 provides a detailed description of the proposed method. Section 3 presents comparative analyses based on numerical and field experiments. Section 4 demonstrates the application to karst cavity exploration. Section 5 discusses the parameter sensitivity and depth imaging performance. Finally, Section 6 summarizes the conclusions.

2. Methodology

2.1. Multi-Step Optimized CR Method

The model resolution matrix R was introduced by Menke [23] and its main diagonal elements provide an estimate of the model cells resolution. The model resolution matrix R is given by:

$$\begin{array}{l}R=BA\\ \mathbf{A}={\mathit{J}}^{\mathrm{T}}\mathit{J}\\ \mathbf{B}={({\mathit{J}}^{\mathrm{T}}\mathit{J}+\lambda \mathit{C})}^{-1,}\end{array}$$

(1)

where J is the Jacobian matrix, λ is the damping factor, and C is the roughness filter constraint matrix. The J and C are defined on the same model discretization, with horizontal mesh spacing equal to one electrode spacing.

The Sherman–Morrison Rank-1 update method [24] is used to calculate the change in the model resolution matrix ΔR_b when a new test configuration is added to the base data set:

$$\Delta R_{\mathrm{b}}={\displaystyle \frac{\mathbf{z}}{1+\mu }}(g^{\mathrm{T}}-{\mathbf{y}}^{\mathrm{T}}),$$

(2)

where g is the model sensitivity vector of the new array, z = B_bg, y = A_bz, and μ = g.z.

The ranking function F_CR is used to evaluate and rank the improvement in model resolution due to an add-on configuration:

$${\mathrm{F}}_{\mathrm{CR}}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m\frac{\Delta {\mathbf{R}}_{\mathrm{b}}(j, j)}{{\mathbf{R}}_{\mathrm{c}}(j,j)}},$$

(3)

where m is the number of the model cells, and R_c is the model resolution matrix of the comprehensive data set.

After computing the ranking function for all candidate configurations, the selection procedure described by Wilkinson et al. [17] is adopted. At each iteration, the configuration with highest ranking value (denoted by the sensitivity vector g₁) is added to the base data set. Subsequently, the next highest ranked configuration (g₂) is added only if it has a suitable degree of orthogonality to the first one.

$${\displaystyle \frac{\left|{\mathbf{g}}_1\cdot {\mathbf{g}}_2\right|}{\left|{\mathbf{g}}_1\right|\left|{\mathbf{g}}_2\right|}}<\theta ,$$

(4)

where θ is a specified threshold (0.97). The calculated dot product must remain below this value to ensure sufficient independence between configurations while avoiding excessive redundancy among selected configurations [16,17].

The symmetrical optimized CR method restricts the search to a symmetrical candidate set to reduce computational costs. In each iteration, the number of new configurations added is set to 0.05 n_b, where n_b denotes the size of the current base data set and 0.05 represents the step size. The optimization process terminates once the base data set reaches a predetermined total size. After adding the new configurations, the model resolution matrix of the expanded base set is recomputed using Equation (1).

Building upon the symmetrical optimized CR method, this paper proposes a multi-step optimized CR method. The workflow is illustrated in Figure 1. First, the comprehensive data set is partitioned into two subsets based on geometric symmetry: the symmetrical data set and the asymmetrical data set. The optimization starts from an initial data set composed of dipole–dipole array, for which the dipole length a is fixed at one electrode spacing and the separation factor n ranging from 1 to 10.

The proposed algorithm introduces a two-stage selection process. In the first stage, optimization is performed using the symmetrical candidate set. The iterative process continues until the iteration count exceeds the predefined threshold. Then, the algorithm switches to the asymmetrical candidate set, and the procedure is repeated until the base data set reaches the predetermined total size. The number of new configurations added in each iteration remains consistent with that of the symmetrical optimized CR method.

The average relative model resolution S_r [25,26] is defined as:

$$S_{\mathrm{r}}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{j=1}^m\frac{{\mathbf{R}}_{\mathrm{b}}(j, j)}{{\mathbf{R}}_{\mathrm{c}}(j,j)}},$$

(5)

where S_r is commonly used to quantify the performance of the optimized data set relative to the comprehensive data set. An S_r value closer to 1 indicates that the resolution performance of the optimized arrays more closely approximates that of the comprehensive arrays.

2.2. Inversion Method

Before the inversion, the apparent resistivity data are inspected and anomalous values are removed. The inversion is performed using an L1-norm inversion algorithm, which is well suited for recovering models with resistivity distributions that exhibit sharp contrasts [27]. The objective function of the L1-norm inversion algorithm is given by:

$$\mathbf{\Phi }={‖\mathit{J}\Delta \mathit{m}-\Delta \mathit{d}‖}_1+\lambda {‖\mathbf{C}(\mathit{m}+\Delta \mathit{m})‖}_1,$$

(6)

where Δd denotes the difference between the observed and calculated apparent resistivity values, Δm is the model update vector, m represents the model parameter vector, and λ is the regularization parameter.

This inverse problem is solved through a linearized iterative scheme. By differentiating the objective function with respect to Δm and setting the derivative equal to zero, the corresponding system of linear equations is obtained:

$$\left({\mathit{J}}^{\mathrm{T}}{\mathit{R}}_d\mathit{J}+\lambda {\mathbf{C}}^{\mathrm{T}}{\mathit{R}}_m\mathbf{C}\right)\Delta \mathit{m}={\mathit{J}}^{\mathrm{T}}{\mathit{R}}_d\Delta \mathit{d}-\lambda {\mathbf{C}}^{\mathrm{T}}{\mathit{R}}_m\mathbf{C}\mathit{m},$$

(7)

where ${\mathit{R}}_d={\left|\Delta \mathit{d}-\mathit{J}\Delta \mathit{m}\right|}^{-1}$ and ${\mathit{R}}_m={\left|\mathit{C}\left(\mathit{m}+\Delta \mathit{m}\right)\right|}^{-1}$ denote the data-weighting and model-weighting diagonal matrices, respectively.

A homogeneous half-space model defined by the mean apparent resistivity is adopted as the starting model. The L1-norm inversion is used to iteratively update the subsurface resistivity distribution and reduce the root mean square error (RMSE) between the observed and calculated apparent resistivity values. The inversion stops when the RMSE decreases below 5% or when the maximum number of iterations reaches 7.

2.3. Evaluation Metrics

To quantitatively evaluate the structural similarity between the inverted images and the true resistivity model, the structural similarity index measure (SSIM) is employed [28]. For the SSIM, the focus is placed only on the structural term s(x,y), while omitting the luminance l(x,y) and contrast terms c(x,y). Thus, with α = β = 0 and γ = 1, the equation simplifies as follows:

$$\mathrm{SSIM}(\mathit{x},\mathit{y})={\left[l(\mathit{x},\mathit{y})\right]}^{\alpha }\cdot {\left[c(\mathit{x},\mathit{y})\right]}^{\beta }\cdot {\left[s(\mathit{x},\mathit{y})\right]}^{\gamma }={\displaystyle \frac{{\sigma }_{\mathrm{xy}}+{\mathrm{C}}_1}{{\sigma }_{\mathrm{x}}{\sigma }_{\mathrm{y}}+{\mathrm{C}}_1}},$$

(8)

where x and y represent the normalized inverted image and the normalized true model, respectively. σ_x and σ_y are the standard deviations, and σ_xy is the covariance of the resistivity values. The constant C₁ is introduced to prevent numerical instability when the denominator approaches zero.

3. Comparative Study

3.1. Numerical Experiment

A synthetic two-layer earth model is constructed (Figure 2a). The upper layer is characterized by a resistivity of 500 Ω·m and a thickness of 40 m. Six low-resistivity anomalies (50 Ω·m) are embedded in the upper layer. The lower layer has a resistivity of 100 Ω·m and a thickness of 120 m, and contains three resistive anomalies with a resistivity of 1000 Ω·m. Forward modeling is performed using 85 electrodes with a unit electrode spacing of 5 m, corresponding to a total survey length of 420 m. Subsequently, synthetic apparent resistivity data are simulated using the finite element method (FEM), and 5% Gaussian noise is added to simulate realistic field conditions.

The comparative study includes the Wenner–Schlumberger array, the symmetrical optimized array, and the proposed multi-step optimized array. To maintain a consistent comparison, the number of data points for the optimized arrays is constrained to match that of the Wenner–Schlumberger array (4992 data points). Optimization results show that the average relative model resolution of the multi-step optimized array is 0.646, surpassing that of the symmetrical optimized array (0.615). These results demonstrate its superior capability in resolving subsurface structures.

As shown in Figure 2, all three arrays clearly delineate the two-layer structure and successfully identify the shallow low-resistivity anomalies. The multi-step optimized array is less affected by boundary effects and mitigates geometric distortion for shallow low-resistivity anomalies near the model edges. The Wenner–Schlumberger array exhibits difficulty in resolving the geometry and resistivity characteristics of deep high-resistivity anomalies, yielding an SSIM of only 0.469. In contrast, both optimized arrays significantly improve deep imaging performance, effectively recovering the geometry, location, and resistivity values of the deep anomalies, with SSIM values of 0.630 and 0.668, and comparable RMSE of 1.4% and 1.3%, respectively. Notably, the multi-step optimized array produces inversion results that more closely approximate the true resistivity values, and the reconstructed anomaly boundaries are more consistent with the true model geometry (Figure 2d).

In summary, the multi-step optimized array demonstrates superior imaging accuracy compared with the symmetrical optimized array. This advantage is primarily attributed to the multi-step optimization strategy. Unlike the symmetrical optimized CR method, which is restricted to the symmetrical data set, the proposed method expands the search space by incorporating selected asymmetrical data sets. These additional asymmetrical data sets provide crucial complementary sensitivity information.

3.2. Field Experiment of Pipeline

The field experiment investigated a drainage pipeline with a diameter of 0.65 m. The pipeline center is located at a horizontal distance of 5.6 m from the start of the survey line, and the pipeline is buried at a depth of 0.7 m. The ERT survey line was deployed perpendicular to the pipeline axis and consisted of 48 electrodes with a unit electrode spacing of 0.3 m, as shown in Figure 3.

The electrode array configurations were optimized prior to field data acquisition. The dipole–dipole array was selected as the baseline configuration, with a dipole length of a = 1 m and separation factors of n = 2, 4, 6. During optimization, a fixed number of new arrays (with an increment of 0.02n_b) were added to the baseline array in each iteration. The iterative process terminated once the total number of data points for both optimized arrays matched that of the standard dipole–dipole array (900 data points). To adapt to the field conditions, the optimized array configurations with an original electrode spacing of 1 m were rescaled to match the actual electrode spacing of 0.3 m. Optimization results demonstrate that the multi-step optimized array achieves an average relative model resolution of 0.660, exceeding that of the symmetrical optimized array (0.624).

The inversion results in Figure 4 obtained from the three arrays consistently indicate the presence of a pronounced high-resistivity anomaly within the horizontal distance range of 4.8 m–6.0 m, corresponding well with the known position of the hollow pipeline (black circle). However, the imaging accuracy of the pipeline differs considerably among these arrays. As shown in Figure 4a, the inversion result for the dipole–dipole array exhibits a relatively weak response of the pipeline, and the resulting anomalous zone appears blurred, with an internal maximum resistivity of only 600.3 Ω·m and an RMSE of 0.9%. In contrast, the inversion result for the symmetrical optimized array (Figure 4b) identifies a closed high-resistivity anomaly with an internal maximum resistivity of 735.4 Ω·m and a reduced RMSE of 0.5%. The inversion result for the multi-step optimized array (Figure 4c) further delineates a larger and more distinct high-resistivity anomaly, with the internal maximum resistivity rising to 767.8 Ω·m while maintaining a comparable low RMSE of 0.6%. The reconstructed anomaly shape closely matches the location and geometry of the hollow pipeline, which is consistent with the expected geoelectrical response of air-filled cavities. These field results confirm that the multi-step optimized array provides superior imaging accuracy compared to both the dipole–dipole and the symmetrical optimized array.

4. Application to Karst Cavity Exploration

The studied site is located in the Baiyun District of Guangzhou, China (Figure 5a). The area belongs to a humid subtropical monsoon climate zone with abundant precipitation. According to the meteorological records (http://data.cma.cn), the average and maximum annual rainfall are 1800 mm and 2200 mm, respectively. The principal rivers proximal to the study area are the Pearl River and the Shijing River. Hydrological processes play a key role in the development of karst environments [29]. The abundant surface water and active groundwater circulation create an environment suitable for the development of karst.

The study area is situated at the intersection of the Guangcong and Guangsan fault zones, accompanied by multiple secondary faults (Figure 5b). Available geological data indicate that the region is underlain by the Carboniferous Hutian Formation (C₂ht), with the lithology mainly composed of limestone, dolomitic limestone, and dolomite. The synergy between these active tectonic structures and highly soluble lithologies contributes to the development of karst features [30,31].

The ERT survey line layout is shown in Figure 5c, with the profile oriented from west to east. A total of 48 electrodes were deployed at 3.5 m intervals, resulting in a total profile length of approximately 164.5 m. To improve electrode–ground coupling and ensure stable current injection, saltwater was poured at the electrode positions. In this section, the multi-step optimized array was employed for subsurface exploration.

Figure 6 shows a shallow low-resistivity layer (<90 Ω·m) extending from the ground surface to approximately 15 m depth, which is interpreted as the Quaternary overburden. The underlying layer exhibits a relatively high resistivity, suggesting that it consists of fractured to intact limestone. In survey lines L3 and L4, the boundaries between the overburden and the limestone layer are comparatively smooth. In contrast, survey lines L1 and L2 exhibit a downward U-shaped variation in this interface. Notably, the L1 ERT profile delineates two vertically extended low-resistivity zones (A1 and A2): one at a depth of 15.0–23.8 m (distance 60.0–83.0 m), and another at a depth of 16.0–22.0 m (distance 92.0–120.0 m). In survey line L2, another downward U-shaped low-resistivity anomaly (A3) is delineated between 60.0 and 85.0 m and extends from 15.0 to 23.8 m depth. This U-shaped low-resistivity anomalies are interpreted as the presence of a water-bearing sand-filled karst cave.

To verify the geophysical interpretation, detailed borehole exploration was conducted along the survey line. Three boreholes, BH1, BH2, and BH3, were drilled at distances of 72.0 m, 79.0 m, and 90.5 m, with final depths of 40 m, 40 m, and 35 m, respectively. The borehole logs shown in Figure 7 provide direct evidence of karst development. In BH1, a sand-filled karst cave exists at depths from 29.5 to 30.7 m. In BH2, multiple sand-filled karst caves were identified at depth intervals of 19.7–24.0 m and 31.0–37.5 m. Therefore, the multi-step optimized array effectively detected the deep karst caves and the bedrock interfaces.

5. Discussions

5.1. Parameter Sensitivity Analysis

This section investigates the sensitivity of the optimization performance to key parameters, including step size, array type, and mesh size. All computations are performed on a workstation equipped with a 2.30 GHz Intel i7 14-core Intel CPU and an NVIDIA GeForce RTX 3060 GPU. A 48-electrode model with a prescribed total number of 4092 selected data points is used for the test. Five step sizes, namely 0.02, 0.03, 0.05, 0.07, and 0.09, are tested for the symmetrical optimized, multi-step optimized, and comprehensive array types. The iterations, computational time, and average relative model resolution are summarized in

Table 1. Comparison of obtained for different array types and step sizes using the 48-electrode model with 4092 selected data points.

Metric	Array Type	Step Size
		0.02	0.03	0.05	0.07	0.09
Iterations	Symmetrical optimized array	121	81	49	36	28
Computational time (s)		13.40	9.92	7.18	6.38	5.81
Average relative model resolution		0.702	0.702	0.702	0.702	0.702
Iterations	Multi-step optimized array	121	81	49	36	28
Computational time (s)		29.00	21.17	16.54	14.71	14.71
Average relative model resolution		0.769	0.768	0.768	0.765	0.764
Iterations	Comprehensive array	121	81	49	36	28
Computational time (s)		76.35	53.60	33.65	31.73	22.40
Average relative model resolution		0.785	0.785	0.785	0.784	0.784

.

For the symmetric optimized array, the average relative model resolution remains almost unchanged at 0.702 for all tested step sizes, whereas the computational time decreases from 13.40 s to 5.81 s as the step size increases from 0.02 to 0.09. A similar trend is observed for the multi-step optimized array and the comprehensive array. The smaller step sizes produce a slightly higher average relative model resolution, but require more iterations and longer computational time. This trend is consistent with previous optimized-array studies [18]. Therefore, the step size mainly affects the number of iterations and computational time.

The comparison among different array types indicates that, at a step size of 0.05, the comprehensive array requires the longest computational time of 33.65 s and achieves the highest average relative model resolution of 0.785, because it searches all possible symmetrical and asymmetrical configurations. In contrast, the multi-step optimized array requires 16.54 s, whereas the symmetrical optimized array requires only 7.18 s. Compared with the symmetrical optimized array, the multi-step optimized array improves the average relative model resolution from 0.702 to 0.768. Although the multi-step optimized array requires additional computational time by incorporating asymmetrical configurations in the later stage, it provides a practical compromise between the computational efficiency of the symmetrical optimized array and the higher resolution of the comprehensive array.

In addition to the step size and array type, the influence of mesh size is further examined while keeping the physical survey geometry, step size, array type, and number of selected data points unchanged. As shown in

Table 2. Influence of mesh size on the optimization performance of the multi-step optimized array.

Mesh Size	Iterations	Computational Time (s)	Average Relative Model Resolution
1.0 electrode spacing	49	16.54	0.768
0.5 electrode spacing	49	24.33	0.753
0.25 electrode spacing	49	46.53	0.735

, refining the horizontal mesh increases the computational time from 16.54 s to 24.33 s and 46.53 s, while the average relative model resolution decreases from 0.768 to 0.753 and 0.735, respectively. These changes are associated with the increased number of model cells caused by mesh refinement. Specifically, refining the mesh increases the dimensions of the Jacobian matrix J and the constraint matrix C, thereby increasing the computational cost of each iteration. At the same time, the fixed number of data points must constrain more model cells, which increases parameter correlation and consequently reduces the cell-wise model resolution. Therefore, for the multi-step optimized array considered in this study, the 1.0 electrode spacing mesh provides the most appropriate discretization scheme.

5.2. Depth Imaging Performance

This section investigates the impact of optimized arrays on imaging results using synthetic subsurface models with anomalies at different depths. All models consist of a two-layer structure: the upper layer has a resistivity of 100 Ω·m and a thickness of 10 m, while the lower layer has a resistivity of 900 Ω·m and a thickness of 35 m. A circular anomaly with a diameter of 15 m and a resistivity of 100 Ω·m is depicted as a black circle. This anomaly is centered at a horizontal distance of 115 m and is embedded at depths of 15 m, 20 m, and 25 m, respectively (Figure 8). Forward modeling utilizes 48 electrodes with a unit electrode spacing of 5 m, corresponding to a total survey length of 235 m. Subsequently, synthetic apparent resistivity data are simulated using the FEM. To simulate realistic measurement conditions, 5% Gaussian noise is added to the apparent resistivity data.

The inversion results clearly indicate that the imaging accuracy of both arrays deteriorates with increasing depth. This trend is consistent with the theoretical decrease in sensitivity with depth [32,33]. However, significant differences arise between the two arrays, underscoring the importance of array configuration. When the anomaly is located at a depth of 15 m (Figure 9a,d), the multi-step optimized array yields a more distinct U-shaped low-resistivity anomaly with an internal minimum resistivity (371.2 Ω·m), demonstrating its superior capability in resolving the subsurface target. In contrast, the symmetrical optimized array exhibits less clarity, particularly in delineating the lower boundary of the anomaly. With the anomaly at a depth of 25 m (Figure 9c,f), the multi-step optimized array still resolves a reasonably well-defined anomaly, whereas the target in the symmetrical optimized array result is indistinguishable from the background. Furthermore, the multi-step optimized array yields a lower RMSE (3.8%) than the symmetrical optimized array (3.9%). Therefore, the multi-step optimized array preserves anomaly geometry and resistivity more reliably at greater depths.

Figure 10 shows the distribution of data points for both arrays. The prescribed total number of data points is 4092. The symmetrical optimized array consists entirely of 4092 symmetric configurations, whereas the multi-step optimized array includes 2288 (55.9%) symmetric configurations and 1804 (44.1%) asymmetric configurations, respectively. The horizontal position of each datum was calculated based on the average horizontal location of the four electrodes involved in the measurement, while the vertical position was estimated using the median of the depth of investigation characteristic curve [34,35]. Statistical analysis of the data distribution reveals that for pseudo-depths below approximately 25 m, the symmetrical optimized array allocates 31.0% (1270 points) of the total data volume. In contrast, the multi-step optimized array increases this data proportion to 33.0% (1349 points). This quantitative evidence demonstrates that the multi-step optimized array contains more data points at deep levels, effectively capturing more deep-sensing electrical information and thereby providing stronger data constraints on deep structures. Therefore, considering both detection depth and imaging accuracy, the multi-step optimized array outperforms the symmetrical optimized array.

6. Conclusions and Future Work

6.1. Conclusions

This paper proposes a multi-step optimized CR method to address the imaging accuracy limitations associated with the symmetrical optimized CR method. Numerical simulations were performed for the Wenner–Schlumberger array, the symmetrical optimized array, and the multi-step optimized array. The results demonstrate that the multi-step optimized array yields the highest average relative model resolution (0.646) and SSIM (0.668), surpassing the symmetrical optimized array (0.615 and 0.630, respectively) and the Wenner–Schlumberger array (0.469 for SSIM). Pipeline experiments confirmed that the multi-step optimized array provides superior imaging accuracy compared to both the dipole–dipole and the symmetrical optimized arrays. Application to karst cavity exploration demonstrate that the proposed array reliably identifies the bedrock interfaces and the deep karst caves. Finally, by expanding the search space beyond the symmetrical data set through the inclusion of selected asymmetrical data sets, the multi-step optimized array contains substantially more asymmetrical data points at greater pseudo depths, thereby demonstrating superior performance in both depth of investigation and imaging accuracy.

6.2. Future Work

Future work will aim to extend the multi-step optimized CR method to 3D ERT surveys, leveraging its efficiency to mitigate the computational costs. Moreover, an adaptive switching mechanism based on the average relative model resolution increment or information gain will be developed to automatically determine the optimal transition point between the symmetrical and asymmetrical data sets, thereby ensuring a balance between computational efficiency and imaging accuracy.