Correction methods and applications of ERT in complex terrain

Review Highlights

• •This review discusses various terrain correction techniques for ERT, a geophysical exploration technique, to enhance the accuracy of subsurface imaging in complex terrains.
• •It covers historical developments and modern approaches, including the ratio method, numerical simulation methods, angular domain method, conformal transformation method, inversion method, and orthogonal projection method.
• •The paper also suggests potential future research directions, such as improving computational efficiency, reducing resource consumption, and integrating advanced technologies like deep learning for more precise and reliable corrections.

Abstract

Electrical Resistivity Tomography (ERT) is an efficient geophysical exploration technique widely used in the exploration of groundwater resources, environmental monitoring, engineering geological assessment, and archaeology. However, the undulation of the terrain significantly affects the accuracy of ERT data, potentially leading to false anomalies in the resistivity images and increasing the complexity of interpreting subsurface structures. This paper reviews the progress in the research on terrain correction for resistivity methods since the early 20th century. From the initial physical simulation methods to modern numerical simulation techniques, terrain correction technology has evolved to accommodate a variety of exploration site types. The paper provides a detailed introduction to various terrain correction techniques, including the ratio method, numerical simulation methods (including the finite element method and finite difference method), the angular domain method, conformal transformation method, inversion method, and orthogonal projection method. These methods correct the distortions caused by terrain using different mathematical and physical models, aiming to enhance the interpretative accuracy of ERT data. Although existing correction methods have made progress in mitigating the effects of terrain, they still have limitations such as high computational demands and poor alignment with actual geological conditions. Future research could explore the improvement of existing methods, the enhancement of computational efficiency, the reduction of resource consumption, and the use of advanced technologies like deep learning to improve the precision and reliability of corrections.

Specifications table

Subject area:	Earth Science; Environmental Science; Geophysics
More specific subject area:	Geophysical Exploration; Mathematics
Name of the reviewed methodology:	The Ratio Method; Numerical Simulation Method; Angular Domain Method; Conformal Mapping Method; Inversion Method; Orthogonal Projection Method
Keywords:	Electrical Resistivity Tomography (ERT); Terrain Correction Techniques; Geophysical exploration; Deep learning in data interpretation
Review question:	What are the main challenges faced by ERT in detecting underground structures under complex terrain conditions? What are the main topographic correction techniques proposed in existing literature, and how do they improve the interpretive accuracy of ERT data? What limitations do these topographic correction techniques have in practical applications, and how can these limitations be overcome? What role might advanced technologies such as deep learning play in future ERT topographic corrections? How should future research improve existing topographic correction methods to enhance the precision and reliability of corrections?

Background

Accurate and reliable geophysical exploration techniques are crucial in various fields, including groundwater exploration, environmental monitoring, engineering geological assessment, and archaeology, including my own upcoming work on the improvement and optimization of correction methods. Electrical Resistivity Tomography (ERT), a prominent geophysical method, is significantly impacted by the undulation of the terrain, which can introduce errors and complicate the interpretation of subsurface structures.

The context of this review is set against the backdrop of the continuous evolution of terrain correction techniques since the early 20th century. The progression from initial physical simulation methods to modern numerical simulation techniques has been pivotal in enhancing the adaptability of ERT to diverse exploration site types. However, existing correction methods, having made significant advancements, still face limitations such as high computational demands and challenges in aligning with actual geological conditions.

This review aims to consolidate and assess the current research status of terrain correction methods for ERT, offering geologists and geophysicists a comprehensive perspective on the subject. It delves into various correction techniques, evaluates their effectiveness under different topographical conditions, and discusses the potential for future improvements, including the integration of advanced technologies like deep learning to enhance the precision and reliability of corrections.

This review is particularly timely, given the increasing demand for precision in geophysical exploration and the advent of new computational and artificial intelligence techniques that may offer novel solutions to the longstanding problem of terrain-related distortions in ERT data. By examining both the historical development and the cutting-edge innovations in the field, this review seeks to contribute to the ongoing discourse on how to optimize ERT for complex terrains and improve its utility across various applications. Method details.

Current research status

The application of resistivity methods in geophysical exploration date back to the early 20th century, first introduced by the Schlumberger brothers. As the types of exploration sites diversified, the significant impact of terrain factors on the results of exploration was gradually recognized. ERT is based on the principles of traditional resistivity methods and, although the dense layout of measurement points has some suppressive effect on the impact of terrain, the influence of terrain cannot be completely eliminated. The “ISO/TR 21,414:2016(en) Hydrometry — Groundwater — Surface geophysical surveys for hydrogeological purposes” and the “Code for Geophysical Prospecting of Railway Engineering” issued by the National Railway Administration of the People's Republic of China explicitly state that for areas with slopes exceeding 15°, topographic corrections must be made to eliminate the interference of terrain on the results of resistivity measurements [64,65]. In the early stages of research, scholars primarily employed physical simulation methods to explore the effects of terrain. Takeshi (1953) was the first to use physical simulation techniques to study the impact of terrain on apparent resistivity curves [1]. Vaeshev (1959) followed, using physical simulation to investigate the apparent resistivity response patterns of V-shaped terrain. Chinese scholars such as XU ShiZhe and He Jishan (1975) used models like conductive paper, water tanks, and soil tanks to delve into the issues of two-dimensional and three-dimensional terrain effects [[2], [3], [4], [5], [6], [7],51,53]. Concurrently, Coggon (1971) calculated the resistivity response curves of simple terrains like ridges and valleys using the finite element method, laying the foundation for a qualitative understanding of terrain effects [8,9]. Subsequently, Papazian (1979) simulated terrain responses using Schwartz-Christoffel mapping techniques [10], and Spiegel (1980) performed simulation corrections for terrain effects under two-dimensional homogeneous medium conditions with Schwartz-Christoffel conformal transformations [11,52]. G. L. Oppliger (1984) achieved simulation of terrain effects in three-dimensional charge methods through integral equations [12]. In the same year, Xu S. Z. (1984) and Jiracek (1984) explored the application of the boundary element method in 2.5-dimensional undulating terrain electric field calculations and proposed a three-dimensional finite element algorithm [13,14]. Huang Lanzhen (1986) studied the apparent resistivity anomaly curves of point source resistivity methods in two-dimensional terrain using the boundary element method and successfully eliminated terrain effects with the ratio method [15]. Kostyanev (1985) studied terrain effects based on Green's theory using integral formulae [16]. XU ShiZhe (1987) calculated the apparent resistivity anomalies of low-resistivity spheres in two-dimensional ridge terrain using the boundary element method and corrected terrain effects with the ratio method [17,18,50]. Xu derived the Green's theory for the potential equation using the solid angle method, improving the efficiency of terrain simulation [19]. XU ShiZhe improved the boundary element algorithm, reduced computational load, and corrected the apparent resistivity anomalies of four-level profile curves with the ratio method [20,21]. Liu Yang (1996) introduced the simulated charge method to accurately calculate pure terrain anomaly responses [22,49]. Wu Xiaoping (2001) studied and discussed the forward and inverse techniques of the finite difference algorithm under three-dimensional undulating terrain [23]. Duan Changsheng derived the calculation formula for the grid node ω using the area calculation method of spherical triangles, effectively eliminating terrain effects [24]. Tang Jingtian used the combination superposition method for terrain correction model cases, highlighting the anomalies of local geological bodies [25]. Xiong Yong proposed the orthogonal projection method for high-density electrical method terrain correction, making up for the deficiencies of previous methods [26,47]. Han Xue and Zhu Weiguo implemented a three-dimensional resistivity inversion algorithm for undulating terrain using the weighted regularization conjugate gradient method [27,60]. In the same year, Zheng Zhijie used a high-density resistivity micro-measurement system to reveal the differences in the effectiveness of high-density electrical method detection of karst pipelines under different terrain conditions [28]. Meng Lin (2021) developed a 2.5D forward and inverse algorithm suitable for undulating terrain, studied the anomaly characteristics of target anomalies under undulating terrain, and laid the foundation for further research on three-dimensional well-to-ground resistivity inversion under undulating surfaces [29,30,56,62,63]. Wang Zhi used unstructured grid simulation for complex terrain, achieved forward simulation of three-dimensional well-to-ground resistivity method, and analyzed and summarized the anomaly morphological characteristics of typical geological bodies under undulating terrain [31,55,59]. Wu Rongxin constructed a geoelectric model of karst caves with different filling degrees and deeply analyzed the impact of terrain undulation, cave filling, and depth changes on numerical simulation results [32,54,57].

It can be observed that the main correction methods include the ratio method, physical simulation method, numerical simulation method, angular domain method, conformal transformation method, inversionmethod, and orthogonal projection method. Among these, the basic approach of the physical simulation method, numerical simulation method, and angular domain method is to first calculate the terrain response, and then correct the terrain influence using the ratio method(As shown in Table 1).

Table 1.

Summary of terrain correction methods for ERT.

Method Name	Basic Principle	Advantages	Disadvantages	References
Ratio Method	Adjusts discrepancies by comparing resistivity data affected and unaffected by terrain.	Simple to operate and implement.	May not completely eliminate the influence of terrain.	He Jishan & Zhou Zhengxiu [3,33]
Numerical Simulation Method	Includes the finite element method and finite difference method, simulating the influence of terrain on the electric field distribution through numerical computation.	High precision, suitable for complex terrains.	Computationally intensive and time-consuming.	Liang Fangmin et al. [35,36]
Angular Domain Method	Obtains potential formulas for simple angular domain terrain through mathematical means, then derives apparent resistivity distortion values.	Systematic calculation process, suitable for complex terrains.	Complex calculation process requiring specialized software support.	Xu Zhimin [39]
Conformal Mapping Method	Transforms boundary value problems with complex boundary shapes into simpler shapes through analytic functions (complex function theory).	Theoretically can precisely handle complex terrains.	Finding suitable analytic functions in practice may be challenging.	Xu Shizhe [44]
Inversion Method	Applies multiple corrections to raw data to mitigate the impact of terrain.	User-friendly interface, easy to operate.	May poorly match actual geological conditions.	Yan, J. Y [46].
Orthogonal Projection Method	Calculates the actual position of the sounding point relative to the surface line AB using trigonometric functions.	Takes into account the vertical influence of terrain.	May require additional computational steps.	Xiong Yong [26]

Terrain correction methods

The ratio method

In resistivity measurements, the undulation of the terrain can affect the distribution of the electric field, causing the measurement results to deviate from the actual subsurface resistivity. The ratio method of terrain correction adjusts for this discrepancy by comparing resistivity data that are influenced by terrain with those that are not. In practical application, the principle is to establish a standard model that represents the pure effect of terrain using terrain data, from which a terrain correction factor $\left({\rho }_{\mathrm{S}}^{\mathrm{D}}/{\rho }_0\right)$ can be obtained. The apparent resistivity measured in the field, when divided by this correction factor, yields the apparent resistivity after the terrain effect has been eliminated. The expression is given by [60]:

$${\rho }_S^G={\rho }_S^M/\left({\rho }_S^D/{\rho }_0\right)$$ (1–1)

In Equation (1–1), the term ${\rho }_{\mathrm{S}}^{\mathrm{G}}$ represents the apparent resistivity after terrain correction, the term ${\rho }_{\mathrm{S}}^{\mathrm{M}}$ represents the apparent resistivity actually measured by the instrument, the term ${\rho }_{\mathrm{S}}^{\mathrm{D}}$ represents the apparent resistivity under the pure influence of terrain, and the term ${\rho }_0$ represents the background resistivity.

The site is located in Hesi Gully, Jinheba Village, Chuanzhu Temple Town, Songpan County, Aba Tibetan and Qiang Autonomous Prefecture, China. The main rock types in the area consist of cobble soil with a resistivity of 820 Ω·m, carbonaceous slate with a resistivity of 41 Ω·m, sandy slate with a resistivity of 162 Ω·m, and limestone with a resistivity of 539 Ω·m. The survey line is 1700 m long with an elevation difference of 150 m. The measured apparent resistivity profile is shown in Fig. 1. It can be observed that there are high apparent resistivity anomalies at the terrain undulations at positions 105 m to 178 m and 286 m to 304 m, and there are distinctly low apparent resistivity areas between 178 m and 238 m as well as from 304 m to 380 m.

Fig. 1.

Field-measured apparent resistivity profile.

When inversion is performed without topographic correction, as shown in Fig. 2, it can be observed that there is a high-resistivity zone at the 152 m position (encircled in red in the figure), whereas in actual field work, a depression to the right of this location reveals a spring, surrounded by an aquifer soil layer. There exists a large low-resistivity body between the 300 m and 560 m positions. Boreholes were drilled at positions 456 m and 912 m at a later stage.

Fig. 2.

Direct inversion results of the site.

Using the ratio method, a geophysical model was established for this section using actual elevation data, and the topographic correction factor was calculated. The apparent resistivity was then back-corrected, and the inversion resulted in Fig. 3. It can be observed that the high-resistivity area within the red circle has disappeared, replaced by a low-resistivity anomaly, which is consistent with the actual situation of the area being an aquifer soil layer. The range of the low-resistivity body from 300 m to 560 m has decreased. According to the results from Brill hole 1, the surface is covered by a loess layer about 17 m thick with a resistivity value of 110 to 205 Ω·m, followed by a gravel and pebble layer about 26 m thick with a resistivity value of 310 to 1250 Ω·m, and below that is a limestone layer with a resistivity value of 220 to 1260 Ω·m. It is clear that it cannot be the low-resistivity area with a resistivity value of around 75 Ω·m.

Fig. 3.

Inversion results after terrain correction using the ratio method.

This method has been introduced in many works, such as in the second part of “The point and line source field problems in resistivity method terrain correction” by scholars He Jishan and Zhou Zhengxiu, where they provide a detailed introduction from the principle to the application [3,33].

Numerical simulation method

The numerical simulation method encompasses the finite element method and the finite difference method. These techniques simulate the influence of terrain on the distribution of the electric field through numerical computation, thereby facilitating correction. While these methods offer high precision, they require substantial computational resources, especially in the context of three-dimensional terrain correction [34,48].

Taking the finite element method as an example, during terrain correction, the patterns of terrain influence are derived from numerical simulation, followed by correction using the ratio method. As shown in Fig. 4, these are the pure terrain apparent resistivity curve characteristics obtained from the combined section of the 30° ridge and valley model tests. It can be observed that on the positive terrain, the top of the ridge exhibits a low-resistivity anomaly, with the two lines forming an anti-crossover point (at the ridge top and on both sides of the valley). The negative terrain anomaly is characterized by the curve at the valley bottom forming a high-resistivity crossover anomaly (at the valley bottom and on both sides of the ridge). On the slope, the two curves show separation, indicating the presence of a divergence zone [35].

Fig. 4.

Apparent resistivity curves for the combined ridge and valley cross-section [41].

When we have a set of terrain data that requires correction, the processing method is illustrated in Fig. 5, as exemplified in the article “Application of finite elements method for two-dimensional terrain correction in resistivity method” by scholars including Liang Fangmin [35,36].

Fig. 5.

Finite element simulation of local low-resistivity combined profile with ratio method terrain correction [35].

The geoelectric model represents a ridge, with a background resistivity of the terrain ρ_0 = 100Ω∙m. On the slope of the hill, there is a low-resistivity rectangular prism with a resistivity ρ_1 = 50Ω∙m, and the electrode distance $AB/2=40m$. When measuring the apparent resistivity of the model using a Combined Profiling array, the curves are mirror images of each other due to the opposite current directions from the ${\rho }_{s}A$ and ${\rho }_{s}B$ electrodes. As the device moves to the right (AMN), the low-resistivity body attracts the current, causing an increase in current density at the MN electrodes located to the right of the A electrode. Consequently, the ${\rho }_{s}A$ curve begins to rise and reaches a maximum value at a certain position. As the device continues to move to the right, the low-resistivity body continues to attract the current from the A electrode, causing the current density at the MN position to start decreasing, and thus the ${\rho }_{s}A$ curve begins to fall. The curve of the device (MNB) ${\rho }_{s}B$, due to the B electrode being located to the left of MN, follows the same logic, and the ${\rho }_{s}B$ curve is symmetrical to the ${\rho }_{s}A$ curve. Therefore, under normal circumstances, an intersection point is formed above the low-resistivity body, and to the left of the intersection point, ${\rho }_{s}A$ is greater than ${\rho }_{s}B$. Typically, such an intersection point with this property is called a “positive intersection point.” When the terrain exists, the joint profile curve of the measured data (see Fig. 5a) shows that, due to the influence of the terrain, the joint profile curve near the low-resistivity anomaly does not exhibit a positive intersection point. The apparent resistivity curve obtained from the finite element method simulation of pure terrain influence is shown in Fig. 5b. The joint profile curve after correction using the ratio method is shown in Fig. 5c, where a positive intersection point can be observed at the low-resistivity body.

Angular domain method

The so-called angular domain [37] refers to a two-dimensional region bounded by two semi-infinite planes, within which a uniform and isotropic conductive medium with a resistivity of $\rho$ is filled [41]. The key to terrain correction using the angular domain method lies in obtaining the potential formula for simple angular domain terrain through mathematical means, then deriving the apparent resistivity distortion values. By combining and superimposing these values, one can obtain the apparent resistivity distortions for complex angular domain terrains, thereby isolating the pure influence of the terrain. Finally, the ratio method is utilized to perform terrain correction [38,39].

Considering the angle between the two half-planes as the vertex angle of the angular domain, represented by $\varnothing$, and the angles between the sides and the horizontal plane represented by $\beta$ and $\alpha$, respectively, as the inclination angles for the left and right sides. It is specified that if the slope is below the horizontal axis, the slope angle takes a negative value, and if it is above, it takes a positive value [25]. Thus, we have:

$$\{\begin{array}{c}0<\varnothing <\pi \\ \varnothing +\alpha +\beta =\pi \end{array}$$ (2–1)

Therefore, when $\varnothing <\pi$, it represents a two-dimensional ridge with an infinitely long slope; when $\varnothing >\pi$, it represents a two-dimensional valley with an infinitely long slope.

To calculate the potential of a two-dimensional angular domain terrain, a coordinate system must be established, as depicted in Fig. 6. The X-axis is taken in the direction that passes through the power supply point A and is perpendicular to the intersection line of the two terrain slopes. The Y-axis is perpendicular to the horizontal plane of the terrain, and the Z-axis is the intersection line of the two side surfaces of the terrain [40]. The point of intersection of these three axes is the origin O. Point A is a point source with a current I, and its distance to the coordinate origin is r0 . The angular domain has an opening angle of $\varnothing$, below which is a uniform medium with a resistivity of $\rho$, and above which is air with an infinitely large resistivity [39,41,61].

Fig. 6.

Schematic diagram of the coordinate system.

The analytical expression for the potential at point M on the terrain is:

$$\cup \left(\mathrm{r},\theta ,\mathrm{z}\right)=\frac{\rho \mathrm{I}}{\pi \phi \sqrt{\mathrm{r}{\mathrm{r}}_0}}\left[\frac{1}{2}{\mathrm{Q}}_{-\frac{1}{2}}\left(\xi \right)+\sum _{m=1}^{\infty }{\left(-1\right)}^{\mathrm{m}}\text{cos}\frac{\mathrm{m}\pi \left(\phi -\theta \right)}{\phi }{\mathrm{Q}}_{\frac{\mathrm{m}\pi }{\phi }-\frac{1}{2}}\left(\xi \right)\right]$$ (2–2)

In Equation (2–2), $\xi =\frac{{\mathrm{r}}^2+{\mathrm{r}}_0^2+{\mathrm{z}}^2}{2\mathrm{r}{\mathrm{r}}_0}$; ${\mathrm{Q}}_{\mathrm{V}}\left(\xi \right)$ is the associated Legendre function of the second kind [25].

Equation (2–2) is incomplete, as ${\mathrm{Q}}_{\mathrm{V}}\left(1\right)$ does not exist; therefore, the solution at this point exhibits singularities and must be calculated using the subsequent Equation (2–3):

$$\cup =\frac{\rho \mathrm{I}}{\pi \phi {\mathrm{r}}_0}\left\{\frac{\pi }{4}+\sum _{m=1}^{\infty }{\left(-1\right)}^{\mathrm{m}}\left[\phi \left(m+\frac{1}{2}\right)-\phi \left(\frac{\mathrm{m}\pi }{\phi }+\frac{1}{2}\right)\right]\right\}$$ (2–3)

In Equation (2–3), $\phi \left(\mathrm{x}\right)=\underset{\mathrm{n}\to \infty }{lim}\left[\ln n-\frac{1}{\mathrm{x}}-\frac{1}{x+1}-\frac{1}{x+2}-\dots .-\frac{1}{x+n}\right]$.

When the electrode is at the angular domain vertex, the potential formula simplifies to Equation (2–4):

$$\cup =\frac{\pi }{\phi }{\cup }_0=\frac{\pi }{\phi }\frac{\rho \mathrm{I}}{2\pi \mathrm{R}}=\frac{\rho \mathrm{I}}{2\phi \sqrt{{\mathrm{r}}^2+{\mathrm{z}}^2}}$$ (2–4)

In Equation (2–4), ${\cup }_0$ represents the background field.

When the measuring electrode M is at the vertex, according to the principle of electrode reciprocity, it can be derived that [42]:

$$\cup =\frac{\rho \mathrm{I}}{2\phi \sqrt{{\mathrm{r}}^2+{\mathrm{z}}^2}}$$ (2–5)

Based on the aforementioned expressions (2–2), (2–3), (2–4), and (2–5), along with their associated formulas, one can calculate the electric field potential for an angular domain terrain under point source conditions.

To emphasize the impact of terrain using a ratio, the expression $\cup /{\cup }_0$ is used to denote the potential distortion value $\cup \left(\tau \right)$, where ${\cup }_0$ is the background value of the site. When z = 0, dividing Equation (2–2) by ${\cup }_0=\mathrm{I}\rho /\left(2\pi |r-{\mathrm{r}}_0|\right)$ results in the following expression [42]:

$$\cup \left(\tau \right)=\frac{2\left|1-\tau \right|}{\varnothing \sqrt{\mathrm{t}}}\left[\frac{1}{2}{\mathrm{Q}}_{-\frac{1}{2}}\left(\xi \right)+\sum _{m=1}^{\infty }{\left(\frac{\tau }{\mathrm{t}}\right)}^{\mathrm{m}}{\mathrm{Q}}_{\frac{\mathrm{m}\pi }{\phi }-\frac{1}{2}}\left(\xi \right)\right]$$ (2–6)

In Equation (2–6), $\tau =\frac{r_p}{r_c};t=\left|\tau \right|;\xi =\frac{1+t^2}{2t}$. According to the principle of electrode reciprocity, $\cup \left({\mathrm{r}}_{\mathrm{p}}/{\mathrm{r}}_{\mathrm{c}}\right)=\cup \left({\mathrm{r}}_{\mathrm{c}}/{\mathrm{r}}_{\mathrm{p}}\right)$, hence $\cup \left(\tau \right)=\cup \left(1/\tau \right)$. Therefore, when calculating $\cup \left(\tau \right)$, only the case where $-1\le \tau \le 1$ is considered [39].

Once the method for calculating the potential of angular domain terrain is known, it is possible to proceed with the calculation of potentials for complex terrains. Since the angular domain is the fundamental unit of two-dimensional terrain, complex angular domain terrains can be obtained by combining and superimposing angular domains [42]. As shown in Fig. 7, by using the superposition formula to multiply and sum the anomalies of each angular domain ${\phi }_1$,${\phi }_2$,${\phi }_3$,${\phi }_4$ an approximate value of the actual terrain anomaly can be obtained [39].

Fig. 7.

Schematic diagram of angular domain simplified superposition for terrain.

Once the actual horizontal positions of the power supply electrodes and measurement electrodes of several angular domains are aligned, the apparent resistivity distortion value of the point source field on complex terrain is determined by the following formula [25]:

$$\frac{{\rho }_s}{\rho }\approx {\left(\frac{{\rho }_s}{\rho }\right)}_1*{\left(\frac{{\rho }_s}{\rho }\right)}_2\frac{1}{\cos {\beta }_2}*{\left(\frac{{\rho }_s}{\rho }\right)}_3\frac{1}{\cos {\beta }_3}\cdots {\left(\frac{{\rho }_s}{\rho }\right)}_n\frac{1}{\cos {\beta }_n}$$ (2–7)

In the formula, $\frac{{\rho }_{\mathrm{s}}}{\rho }$ represents the terrain anomaly value at a certain point on a continuous finite terrain; ${\left(\frac{{\rho }_{\mathrm{s}}}{\rho }\right)}_1\cdots {\left(\frac{{\rho }_{\mathrm{s}}}{\rho }\right)}_{\mathrm{n}}$ represents the terrain anomaly values at corresponding points in each angular domain; ${\beta }_1\cdots {\beta }_{\mathrm{n}}$ represent the slopes corresponding to each angular domain [39].

Fig. 8 illustrates the high-density Schlumberger array's pure terrain anomaly resistivity profile for a specific terrain, with a surface resistivity of 100 Ω·m [39]. It can be observed that there are distinct electrical property changes at the locations where the terrain undulates.

Fig. 8.

Pure terrain anomaly resistivity profile.

When direct inversion is performed without terrain correction, the results, as shown in Fig. 9, reveal the presence of high-resistivity anomalies with resistivity ranging from 500 to 2000 Ω·m in certain areas of the surface. Given the physical property parameters of the survey area, the resistivity should typically range from 100 to 500 Ω·m, leading to the inference that the high-resistivity anomalies are likely false. Additionally, extensive low-resistivity anomalies are observed in multiple locations beneath the terrain slopes, which could be attributed to the influence of the terrain. After applying the terrain correction using the ratio method, the inversion results, as depicted in Fig. 10, demonstrate a significant improvement.

Fig. 9.

Resistivity inversion results before terrain correction.

Fig. 10.

Resistivity inversion results after terrain correction.

In Xu Zhimin's paper “The analytical calculation of the angle domain topographic point-source electric field and its application to high-density resistivity method for topographic correction,” there is a detailed discussion of this method [39].

Conformal mapping method

The Conformal Mapping Method involves transforming boundary value problems with complex boundary shapes in the Z-plane into boundary value problems with simpler shapes (typically circles, upper half-planes, or strip-like domains) in the W-plane through the transformation (or mapping) of analytic functions (complex function theory). The solutions to the latter problems are more easily obtained. The solution to the original problem can then be found through the inverse transformation [44]. The transformation from the Z-plane to the W-plane by an analytic function has the property of conformality at the point of application [18,43]. According to the theory of complex functions, for any given terrain line, there exists an analytic function that can transform the terrain line and the area below it into a horizontal line and the area below it, thereby solving the effects of numerous smooth terrain uniform electric fields and line source fields [18,44].

Let us assume a two-dimensional terrain with a homogeneous medium, and take the section perpendicular to the terrain's strike as the Z-plane, denoted by RS as the terrain line of the section. The current in the sub-terrain is a uniform field; hence, the terrain line can be considered as a current line. At this point, the current density at a certain point ${\mathrm{Z}}_0$ on the surface is defined as ${\mathrm{j}}_0$, which means that the electric field distribution and the current density at all points throughout the terrain are completely determined. The boundary conditions for the electric field are as follows: (1) The terrain line itself is a current line; (2) The current density at a certain point ${\mathrm{Z}}_0$ is ${\mathrm{j}}_0$ [18]. The functional expression is:

$$w=f\left(\mathrm{z}\right)$$ (3–1)

It must satisfy two conditions: a. Equation (3–1) must transform the entire terrain on the Z-plane into a horizontal line on the W-plane (see Fig. 11). b. The derivative of Equation (3–1) must meet the condition of Equation (3–2):

$$\underset{\mathrm{z}\to {\mathrm{z}}_0}{lim}\mathrm{m}\frac{\overline{\text{dw}}}{\text{dz}}={\mathrm{j}}_0$$ (3–2)

Here, m is an introduced undetermined constant, determined by the current density value at any point on the surfa ce[43]. At this time, the function:

$$\zeta =\text{mw}=\text{mf}\left(\mathrm{z}\right)$$ (3–3)

Fig. 11.

Subsurface current line distribution on the plane.

To reposition the sought electric field, the real part can form the point-to-line equation, and the imaginary part can form the current line equation, with the current density being:

$$j=\frac{\overline{\mathrm{d}\zeta }}{\text{dz}}=m\frac{\overline{\text{dw}}}{\text{dz}}$$ (3–4)

Set up a line current source parallel to the terrain strike as shown in Fig. 12(a), with a unit length current source carrying a current of I. The source is depicted as a point in the diagram with coordinates ${\mathrm{Z}}_{\mathrm{A}}$. The boundary conditions are: Curve AR is a current line, and curve AS is another current line. Since the current between the two lines is I, curve AR satisfies the current line equation: $\eta \left(\mathrm{x},\mathrm{y}\right)=c$, and curve AS satisfies: $\eta \left(\mathrm{x},\mathrm{y}\right)=c+I$, where c is any arbitrary real number [43].

Fig. 12.

Schematic diagram of conformal mapping of current lines for terrain.

Currently, an analytic function is selected: $w=f\left(\mathrm{z}\right)$, which transforms the terrain RS and the area below it on the Z-plane into a horizontal line on the W-plane (see Fig. 12(b)). The image coordinate of the power supply point A on the W-plane is ${\mathrm{w}}_{\mathrm{A}}=f\left({\mathrm{Z}}_{\mathrm{A}}\right)$. The coordinate origin of the W-plane is then moved to ${\mathrm{w}}_{\mathrm{A}}$, and a logarithmic function is used to transform the horizontal line on the W-plane into an infinite horizontal strip domain on the $\zeta$-plane (see Fig. 12(c)), ensuring that the distance between lines AR and AS is I. That is, on the $\zeta$-plane, curve AR is transformed into a horizontal line $\eta =-I$, and curve AS is transformed into a horizontal line $\eta =0$, thus satisfying the aforementioned boundary conditions [18]. Therefore, the complex expression for the electric field is:

$$\zeta =\frac{\mathrm{I}}{\pi }\ln \left(w-{\mathrm{w}}_{\mathrm{A}}\right)=\frac{\mathrm{I}}{\pi }\ln \left[\mathrm{f}\left(\mathrm{z}\right)-f\left({\mathrm{z}}_{\mathrm{A}}\right)\right]$$ (3–5)

The real part of Equation (3–5) represents the equipotential line equation, and the imaginary part represents the current line equation. The current density j is given by [45]:

$$j=\frac{\overline{\mathrm{d}\zeta }}{\text{dz}}=\frac{\overline{\mathrm{d}\zeta }}{\text{dw}}·\frac{\overline{\mathrm{d}\zeta }}{\text{dz}}=\frac{\mathrm{I}}{\pi }\frac{\overline{1}}{\left(w-{\mathrm{w}}_{\mathrm{A}}\right)}·\frac{\overline{\text{dw}}}{\text{dz}}=\frac{\mathrm{I}}{\pi }\frac{\overline{{\mathrm{f}}^{\prime }\left(\mathrm{z}\right)}}{\left[\mathrm{f}\left(\mathrm{z}\right)-f\left({\mathrm{z}}_{\mathrm{A}}\right)\right]}$$ (3–6)

From the aforementioned theory, it can be observed that addressing the impact of terrain merely requires finding the corresponding analytic function. According to the theory of complex functions, there exists a corresponding analytic function for any given terrain line.

By employing conformal mapping to transform the terrain's potential values onto a horizontal plane, the device coefficient K and apparent resistivity are recalculated, and the results are obtained through inversion using a horizontal model. Subsequently, the conformal mapping inverse transformation is applied back to the undulating terrain to form the profile [18]. The effects of terrain correction can be seen in Fig. 13, Fig. 14. It can be observed that the response results after conformal mapping correction correspond well with the geophysical model, essentially reflecting the model's resistivity, as well as the size, shape, and position of the anomalies.

Fig. 13.

Conformal mapping terrain correction results (below) and hill slope model (above).

Fig. 14.

Conformal mapping terrain correction results (below) and valley model (above).

Inversion method

At present, the most widely used commercial inversion software includes the Res2DINV and Res3DINV series from Geotomo Software, as well as the Earthimage2D and Earthimage3D series from AGI Company [46]. Inversion software significantly mitigates the impact of terrain by applying multiple corrections to the raw data. However, there are certain limitations [26]. It is necessary to combine these with other methods for processing.

Taking Res2DINV as an example, this software, developed by Dr. M. H. Loke, employs a forced smoothing least squares inversion technique. It uses data obtained from the ground to generate two-dimensional models of the subsurface. By applying multiple corrections to the raw data, it significantly mitigates the influence of topography. However, due to its use of a linear approach to correct the nonlinear resistivity inversion process, there are certain limitations. In practical applications, it has been found that when topography is present, the results obtained from inversion using this software poorly match the actual geological conditions, with significant discrepancies in the depth of anomalies. Often, there is an unexplained low-resistivity pseudo-anomaly at the bottom of the inversion profile (see Fig. 15) [26].

Fig. 15.

Direct inversion results from inversion software [26].

Another example, as shown in Fig. 16, the geoelectric model has a slope of 30°, with a background resistivity value of 100 Ω·m. At the position x = 16 m, there is a low-resistivity anomaly body with a depth of 1 m, dimensions of 1 m × 1 m, and a resistivity value of 10 Ω·m. Using Res2DINV with topographic inversion, the results are as shown in Fig. 17. It can be observed that although the response to the low-resistivity anomaly body is good in the results, the pseudo-anomalies caused by topography still persist. If not further processed, this may lead to misinterpretation and incorrect conclusions about the subsurface structure.

Fig. 16.

30° slope geoelectric model.

Fig. 17.

Topographic inversion results.

Orthogonal projection method

As shown in Fig. 18, when there is terrain present, the actual position of the sounding point is at the orthogonal projection location relative to the surface line AB. The position offset of the sounding point is calculated using trigonometric functions; this method is known as the “Orthogonal Projection Method.” The virtual position of the sounding point, as commonly interpreted in data, is indicated in the diagram. This is correct under horizontal conditions, but under undulating terrain conditions, the sounding data reflects the direction perpendicular to the surface line connecting electrodes A and B. The actual position should be the “actual position of the sounding point” as shown in the diagram [26].

Fig. 18.

Schematic diagram of orthogonal projection [26].

The example model, as shown in Fig. 19(a), features a steep slope with a 30° gradient and a scarp height of 4m. Within a homogeneous semi-space, there is a low-resistivity cube with dimensions of 4 m × 4 m × 4 m, a resistivity value of 1Ω·m, and a center coordinate of (0m, 0 m, −4 m). The background resistivity of the model is 100Ω·m. Three survey lines are laid out along the x-direction with a spacing of 2 m, and the central survey line is located at y = 0m. Each survey line is equipped with 25 electrodes spaced 1 m apart.

Fig. 19.

Geoelectric inversion results for the topographic model (a) before correction (b) and after correction (c).

We select the measurement data from the survey line at y = 0 m for inversion and orthogonal projection method correction processing. The results, as shown in Figs. 19(b) and (c), indicate that the inversion result without correction (Fig. 19(b)) demonstrates a good correspondence between the center of the anomaly and the actual model position, but the anomaly range is excessively broad. After the topographic correction is applied, as depicted in Fig. 19(c), it can be observed that the center of the anomaly has shifted upward relative to the actual model, and the size range has contracted, which is more consistent with the actual conditions. From the effectiveness perspective, it appears that this method should be used in conjunction with other methods to ensure the precision of the results.

The method is detailed in Xiong Yong's paper “Discussion on terrain correction methods in high-density resistivity exploration.” [26]

Summary and prospects

The issue of terrain correction in ERT is a challenging and crucial aspect of processing exploration data, especially with the increasing demand for precision across various fields. Particularly when the anomalous signals are weak, terrain can significantly affect the quality of resistivity measurement data. This paper reviews a variety of terrain correction methods, including the ratio method, numerical simulation method, angular domain method, conformal transformation method, inversion method, and orthogonal projection method, aiming to enhance the interpretative accuracy of ERT data and reduce errors caused by terrain factors. Among the existing correction methods, the ratio method corrects deviations by comparing resistivity data with and without the influence of terrain; the numerical simulation method simulates the impact of terrain using the finite element method and finite difference method; the angular domain method and conformal transformation method simplify the computation of terrain effects through mathematical transformations. Although existing correction methods can eliminate the influence of terrain to some extent, each method has its limitations. For instance, while the numerical simulation method is highly accurate, it requires substantial computational resources; inversion method can mitigate the impact of terrain but may poorly match the actual geological conditions.

The rapid development of artificial intelligence (AI) currently has numerous applications in the field of resistivity method exploration. Although there are no direct applications of deep learning for terrain correction in ERT methods at present, its use in remote sensing data processing and terrain correction suggests that this technology has the potential to enhance the accuracy and reliability of exploration data [58]. Future research could focus on improving existing terrain correction methods, increasing computational efficiency, reducing resource consumption, and leveraging methods such as deep learning to improve the precision and reliability of corrections.

CRediT authorship contribution statement

Mingdong Zhao: Writing – original draft, Visualization, Data curation. Menghan Jia: Resources. Luqi Yang: Software. Kui Suo: Writing – review & editing, Resources, Funding acquisition, Data curation. Yuanhang Lu: Resources. Qinghang Wei: Software, Formal analysis. Yifan Zhang: Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Method name: The Ratio Method; Numerical Simulation Method; Angular Domain Method; Conformal Mapping Method; Inversion Method; Orthogonal Projection Method