# Three-Dimensional Inversion of Borehole-Surface Resistivity Method Based on the Unstructured Finite Element

Authors: Zhi Wang; Sinan Fang
Journal: International Journal of Antennas and Propagation
 (2021)

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## Abstract

The electromagnetic wave signal from the electromagnetic field source generates induction signals after reaching the target geological body through the underground medium. The time and spatial distribution rules of the artificial or the natural electromagnetic fields are obtained for the exploration of mineral resources of the subsurface and determining the geological structure of the subsurface to solve the geological problems. The goal of electromagnetic data processing is to suppress the noise and improve the signal-to-noise ratio and the inversion of resistivity data. Inversion has always been the focus of research in the field of electromagnetic methods. In this paper, the three-dimensional borehole-surface resistivity method is explored based on the principle of geometric sounding, and the three-dimensional inversion algorithm of the borehole-surface resistivity method in arbitrary surface topography is proposed. The forward simulation and calculation start from the partial differential equation and the boundary conditions of the total potential of the three-dimensional point current source field are satisfied. Then the unstructured tetrahedral grids are used to discretely subdivide the calculation area that can well fit the complex structure of subsurface and undulating surface topography. The accuracy of the numerical solution is low due to the rapid attenuation of the electric field at the point current source and the nearby positions and sharply varying potential gradients. Therefore, the mesh density is defined at the local area, that is, the vicinity of the source electrode and the measuring electrode. The mesh refinement can effectively reduce the influence of the source point and its vicinity and improve the accuracy of the numerical solution. The stiffness matrix is stored with Compressed Row Storage (CSR) format, and the final large linear equations are solved using the Super Symmetric Over Relaxation Preconditioned Conjugate Gradient (SSOR-PCG) method. The quasi-Newton method with limited memory (L_BFGS) is used to optimize the objective function in the inversion calculation, and a double-loop recursive method is used to solve the normal equation obtained at each iteration in order to avoid computing and storing the sensitivity matrix explicitly and reduce the amount of calculation. The comprehensive application of the above methods makes the 3D inversion algorithm efficient, accurate, and stable. The three-dimensional inversion test is performed on the synthetic data of multiple theoretical geoelectric models with topography (a single anomaly model under valley and a single anomaly model under mountain) to verify the effectiveness of the proposed algorithm.


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## Body

## 1. Introduction

Owing to the need for energy development and the development of remote sensing technology, the research of electromagnetic wave propagation and its application in communication and detection has made significant progress. It has been used in radio communications in mine tunnels, railway tunnels, and military tunnels; communications with submarines, command, and navigation; and the electromagnetic wave detection of mineral resources and crustal structures [1, 2] (including faults, glaciers, caves, pipelines, water sources, and objects in the ocean). The wave-field structure is an important problem for communication and detection systems [3–9]. Electromagnetic exploration is based on the electrical differences between different rocks in the Earth’s crust (such as differences in electrical conductivity, magnetic permeability, dielectric, and electrochemical properties). The electromagnetic field signal sent by the field source passes through the underground medium to reach the target geological body and then generates an induction signal. These electromagnetic waves containing the induction signal of the target geological body are received by the receiver arranged in the well or on the ground. In this paper, the influence of geometric and electromagnetic characteristics of various geological structures (including man-made structures) on electromagnetic wave propagation is studied. The time and space distributions of artificial or natural electromagnetic fields are observed and analyzed to determine useful underground mineral resources, identify underground geological structures, and solve the geological problems [1]. Electromagnetic exploration methods are mainly divided into direct current method based on the principle of geometric sounding, magnetotelluric method (MT/AMT/CSAMT) based on the principle of frequency domain sounding, and transient electromagnetic method (TEM) based on the principle of time-domain sounding [2]. The direct current (DC) resistivity method is one of the classic methods of geoelectric exploration. It has been widely and effectively applied in mineral resources (metallic and nonmetallic minerals, coal fields, oil, and gas), environmental engineering (groundwater, geological landslides, and environmental monitoring), geotechnical engineering (tunnel construction and mine water inrush), and other fields. Moreover, the DC resistivity method has been expanded to hydrology, archaeology, and other fields that are closely related to national economic construction and human social life. The DC resistivity method has a variety of flexible observation methods such as the electrical profile method and electrical sounding method with different electrode arrays, pole-pole, dipole-dipole, and multipole. The high-density electrical method has been introduced, which can efficiently obtain large observation data, making it possible for the three-dimensional resistivity inversion of underground fine structures [10]. The borehole-surface resistivity method is a type of electrical method in which electrodes are placed in the well and on the ground. The electrode in the well is the source electrode and the electrode on the ground receiving the electromagnetic field is measured. The source electrode is always placed in the deep part of the borehole to make it close to the object to be detected, thereby increasing the current intensity or the received abnormal response [11]. The borehole-surface resistivity method is mainly used for secondary resource exploration in metal mines and the prediction of oil reservoir boundaries [12]. Data are collected on a grid along parallel lines with different electrode arrays, and a 3D inversion algorithm is used. With the development of computer and numerical computing technology, the three-dimensional electromagnetic forward and inversion algorithms have made significant progress in the mesh design (structured [13–16] and unstructured [17–25]), as well as the numerical method in the forward solution (finite difference [26–30] and finite element [31, 32]), solving the objective function (Gauss–Newton (GN) method [33–37], quasi-Newton (QN) method [38–50], nonlinear conjugate gradient (NLCG) method [51–53], etc.).The unstructured finite element method (FEM) has achieved promising success in the three-dimensional numerical simulation of resistivity in complex topography. The unstructured grid allows local densification and can simulate complex geometric models. The structure also has controllable element quality and its solution efficiency significantly increases in the three-dimensional unstructured FEM. The calculation time and the storage capacity of the unstructured grid can be reduced by about an order of magnitude retaining the same calculation accuracy as the structured grid [10]. Owing to a large number of inversion parameters and a vast amount of data in the three-dimensional resistivity inversion, the Jacobian matrix (partial derivative matrix) has huge calculation and storage requirements. Many inversion algorithms have been proposed which can avoid the calculation of the Jacobian matrix. Zhang et al. [26] and Wu and Xu [54] introduced the conjugate gradient method to achieve fast and effective three-dimensional resistivity inversion and resolve the issues of solution and storage of the Jacobian matrix in the three-dimensional inversion to improve the efficiency. The optimization methods used in 3D data inversion mainly include the nonlinear conjugate gradient (NLCG) method, the Gauss-Newton (GN) method, and the quasi-Newton (QN) method. Both the NLCG and the QN only need the gradient information of the objective function, and no explicit sensitivity matrix is needed. The GN method has second-order sensitivity information, and the inversion convergence speed is better but the calculation speed is lower than that of the QN and the NLCG methods. The QN method approximately computes the inverse Hessian matrix in the iterative process and is more efficient than the NLCG method in the step size search. In the large-scale 3D data inversion, the QN method still has the problem of occupying memory. Therefore, the limited memory quasi-Newton method (L_BFGS) has been developed. The L_BFGS method only needs to store the last m iterations information to generate the inverse Hessian matrix, which significantly reduces the required memory. With the expansion of the application range of the DC resistivity method, the study of inversion accuracy and inversion speed in the 3D resistivity inversion method has important practical significance and theoretical value.In the above-mentioned studies, the L_BFGS method has the advantages of fast convergence, less memory, and better inversion efficiency than the other inversion algorithms. The L_BFGS is also more suitable for solving large-scale 3D electromagnetic inversion problems. Therefore, an inversion algorithm for the three-dimensional resistivity method with the undulating terrain is developed in this paper by combining the L_BFGS method, the borehole-to-surface observation method, and the FEM with unstructured tetrahedrons. Numerical results of the theoretical model inversion validate the effectiveness of the proposed method.


## 2. Forward Modeling Theory

### 2.1. Base Equation

The partial differential equation and its boundary-value problem satisfied by the total potential of the three-dimensional point current source field are given by the following equation with mixed boundary conditions [14, 55]:(1)∇⋅σx,y,z∇ux,y,z=−4πωAIδrA,inΩ,∂u∂n=0,onΓs,∂u∂n+cosθru=0,onΓ∞,where σ is the conductivity distribution on the surface, u is the electrical potential, I is the strength of the source, δ is the Dirac delta function, rA is the coordinate of the source electrode A, ωA is the opening angle of the source to the underground Earth by Ω, n is the outward normal to the boundary surface Γ∞ of the modal domain, r is the location of an arbitrary potential electrode from the source point, Γs and Γ∞ are the natural boundary condition (surface-air interface) and the infinite boundary condition (artificially cut off the interface), respectively, and θ is the angle between the radial distance r from the source point and the outward normal spatial coordinate n on the boundary. If the source point is on the ground, then ωA=2π, while if the source point is underground, then ωA=4π. The weighted residual method can be used to derive the integral equation of the variational problem corresponding to equation (1) [15, 55]:(2)Fu=∫Ω12σ∇u2−4πωAδAudΩ+∫Γ∞12σu2cosr,nrdΓ,δFu=0.The calculation area adopts tetrahedral division and linear difference and finally forms a large sparse symmetric linear equation system. The matrix expression is as follows:(3)Ku=P,where K is an n×n symmetric matrix, u is an n×1 column vector representing the potential vector on the three-dimensional grid node, and P is a column vector containing field source information. In order to save memory, the Compressed Sparse Row or Compressed Row Storage is used to store the coefficient matrix K, and the Super Symmetric Over Relaxation Preconditioned Conjugate Gradient (SSOR-PCG) algorithm [16, 25] is used to solve equation (3).


### 2.2. Algorithm Verification

In order to verify the correctness of the proposed algorithm, a buried spherical model in a uniform half-space is selected. All calculations in this article are done on a computer consisting of an Intel i7-4712 MQ CPU with a frequency of 2.3 GHz and 16 G memory. Both forward and inversion programs are compiled and run by Intel Fortran. Gmsh 4.8.4 [56] and ParaView 5.6.0 [57] are used for generating and visualizing the unstructured tetrahedral meshes, respectively.Figure1 shows the model for the spherical anomaly embedded in a uniform half-space (Ren and Tang [20]). The radius, the center coordinates, and the resistivity of the sphere are R = 2.25 m, (0, 0, −4.5), and ρ0=1Ωm, respectively, the resistivity of the half-space is ρ1=10Ωm, the point current source electrode A is at (−5, 0, 0), and a current source of strength 1A is injected into the Earth. The measuring electrodes M are along the X direction with spacing of 0.25 m. The spatiality of the entire calculation area is 500m×500m×500m.Figure 1
The geometric structure of the sphere model.The target area size is100m×100m×100m. The meshes at the source and the measurement points are refined, the total number of grid nodes is 56737, and the total number of grid cells is 277375. Figure 2 shows the partial magnification effect of the mesh division. The source point, the measuring point, and the anomalous body were refined meshes. The potential value at the measuring point is calculated by the finite element forward modeling program in this article with the pole-pole device and compared with the analytical solution given by Cook and Van Nostrand [58]. Figure 3 compares the analytical and numerical solutions of apparent resistivity of an underground spheroid. Figure 4 presents the relative error of the analytical and numerical solutions of apparent resistivity in the sphere model. It can be seen from Figure 4 that the maximum error is less than 1.4%.Figure 2
Partial magnification effect of the mesh division.Figure 3
Comparison of analytical and numerical solutions of apparent resistivity of the underground spheroid.Figure 4
Relative error.


## 2.1. Base Equation

The partial differential equation and its boundary-value problem satisfied by the total potential of the three-dimensional point current source field are given by the following equation with mixed boundary conditions [14, 55]:(1)∇⋅σx,y,z∇ux,y,z=−4πωAIδrA,inΩ,∂u∂n=0,onΓs,∂u∂n+cosθru=0,onΓ∞,where σ is the conductivity distribution on the surface, u is the electrical potential, I is the strength of the source, δ is the Dirac delta function, rA is the coordinate of the source electrode A, ωA is the opening angle of the source to the underground Earth by Ω, n is the outward normal to the boundary surface Γ∞ of the modal domain, r is the location of an arbitrary potential electrode from the source point, Γs and Γ∞ are the natural boundary condition (surface-air interface) and the infinite boundary condition (artificially cut off the interface), respectively, and θ is the angle between the radial distance r from the source point and the outward normal spatial coordinate n on the boundary. If the source point is on the ground, then ωA=2π, while if the source point is underground, then ωA=4π. The weighted residual method can be used to derive the integral equation of the variational problem corresponding to equation (1) [15, 55]:(2)Fu=∫Ω12σ∇u2−4πωAδAudΩ+∫Γ∞12σu2cosr,nrdΓ,δFu=0.The calculation area adopts tetrahedral division and linear difference and finally forms a large sparse symmetric linear equation system. The matrix expression is as follows:(3)Ku=P,where K is an n×n symmetric matrix, u is an n×1 column vector representing the potential vector on the three-dimensional grid node, and P is a column vector containing field source information. In order to save memory, the Compressed Sparse Row or Compressed Row Storage is used to store the coefficient matrix K, and the Super Symmetric Over Relaxation Preconditioned Conjugate Gradient (SSOR-PCG) algorithm [16, 25] is used to solve equation (3).


## 2.2. Algorithm Verification

In order to verify the correctness of the proposed algorithm, a buried spherical model in a uniform half-space is selected. All calculations in this article are done on a computer consisting of an Intel i7-4712 MQ CPU with a frequency of 2.3 GHz and 16 G memory. Both forward and inversion programs are compiled and run by Intel Fortran. Gmsh 4.8.4 [56] and ParaView 5.6.0 [57] are used for generating and visualizing the unstructured tetrahedral meshes, respectively.Figure1 shows the model for the spherical anomaly embedded in a uniform half-space (Ren and Tang [20]). The radius, the center coordinates, and the resistivity of the sphere are R = 2.25 m, (0, 0, −4.5), and ρ0=1Ωm, respectively, the resistivity of the half-space is ρ1=10Ωm, the point current source electrode A is at (−5, 0, 0), and a current source of strength 1A is injected into the Earth. The measuring electrodes M are along the X direction with spacing of 0.25 m. The spatiality of the entire calculation area is 500m×500m×500m.Figure 1
The geometric structure of the sphere model.The target area size is100m×100m×100m. The meshes at the source and the measurement points are refined, the total number of grid nodes is 56737, and the total number of grid cells is 277375. Figure 2 shows the partial magnification effect of the mesh division. The source point, the measuring point, and the anomalous body were refined meshes. The potential value at the measuring point is calculated by the finite element forward modeling program in this article with the pole-pole device and compared with the analytical solution given by Cook and Van Nostrand [58]. Figure 3 compares the analytical and numerical solutions of apparent resistivity of an underground spheroid. Figure 4 presents the relative error of the analytical and numerical solutions of apparent resistivity in the sphere model. It can be seen from Figure 4 that the maximum error is less than 1.4%.Figure 2
Partial magnification effect of the mesh division.Figure 3
Comparison of analytical and numerical solutions of apparent resistivity of the underground spheroid.Figure 4
Relative error.


## 3. Inversion Theory

### 3.1. Objective Function

According to Tikhonov’s regularization theory, the inversion objective function in the sense of least squares is used. The objective function is described as [10, 18](4)Φm=Φdm+λΦmm=Wddobs−Fm22+λWmm−mref22,where Fm is the forward response function, m is the model parameter mi,i=1,2,…,M, dobs is the observation data, Wd is an N×N data weighting matrix (N is the number of data) in which the diagonal elements are the measured data and the remaining elements are zero, σi is the standard deviation of the i-th measured data, Wm is the model-weighted matrix usually defined by the discrete difference operator of the model unit and generally takes the first-order regularization constraint, λ is the regularization parameter used to balance the weight of data fitting and model smoothness, and mref is a reference model containing prior information about the model parameters. The inverted measured data dobs are the pole-pole potential value and the model parameter is the conductivity value of the element. Usually, the logarithm is used to calibrate the measured data and the model parameters mainly due to the large variation range and to invert the stability. The logarithm is defined as d=lnϕobs,m=lnσ. The resistivity inversion problem is generally a mixed problem, which often leads to equation (4) as an ill-conditioned equation. To solve this problem, smooth constraints are introduced into the inversion equation. Unstructured grid with disorderly arrangement was used in the forward modeling, so we adopt smooth constraint method, that is, judge the adjacency of the grid according to whether the grid unit has contact surface to determine the adjacency of the grid unit. For adjacency to generate matrix Wm, Wm(i, j) represents the contribution of the j-th unit to the smoothness of the i-th unit, which is generated according to the following formula:(5)Wm=−1/xij∑j=1ki1/xij,iandjare adjacent units,1,i=j,0,other,where xij=xi−xj, and it is the distance in the X direction between the center of the i-th unit and the j-th unit, and ki is the number of adjacent units to the i-th unit.


### 3.2. Inversion Framework

For the large-scale inversion problem, the traditional BFGS method requires a large amount of memory. Nocedal [50] improved the BFGS method and proposed a limited memory BFGS (L_BFGS) method to solve the nonlinear optimization problem. In the L_BFGS method, the inverse Hessian matrix approximation formula is defined as(6)Hk+1=Vk−1T,…,Vk−mTHk0Vk−m,…,Vk−1+ρk−mVk−1T,…,Vk−m+1Tsk−msk−mTVk−m+1,…,Vk−1+ρk−m+1Vk−1T,…,Vk−m+2Tsk−m+1sk−m+1TVk−m+2,…,Vk−1+⋯+ρk−1sk−1sk−1T,ρk=1ykTsk,Vk=I−ρkykskT,sk=mk+1−mk,yk=gk+1−gk,where m is the number of previous iterations with a value between 3 and 20. The previous iteration information of gradient and the model modification is used to modify the inverse Hessian matrix. Hk0 defined by Nocedal and Wright [59] gives an update method as(7)Hk0=γkI.In this paper, the identity matrixI is selected to initialize matrix Hk0 and γk can be defined as(8)γk=sk−1Tyk−1yk−1Tyk−1.The inversion steps of the L_BFGS method are defined in Algorithm1 as follows [49, 59].In the algorithm,gk is the gradient of the objective function (4), and it is expressed as(9)gk=∇Φmk=−2JmTWdTWddobs−Fmk+2λWmTWmmk,where J represents the Jacobian matrix. It can be seen from formula (9) that the calculation of the gradient lies in the calculation of the Jacobian matrix. The explicit calculation requires massive computation and memory storage. Therefore, it is necessary to avoid directly calculating the Jacobian matrix and calculate the product of the transpose of the Jacobian matrix and any one-dimensional vector. Thus, there is no need for storing the Jacobian matrix, and the calculation can be obtained together after the forwarding in each inversion, which significantly speeds up the inversion calculation. The calculation details are provided by Zhanget al. [26].Nocedal and Wright [59] presented a double-loop recursive method to update Step 2 in Algorithm 1. The detailed calculation process can be found in the literature [59]. In the process of minimizing the objective function, in contrast to the CG’s method in which the step size is obtained using an analytical method, both the NLCG and the L_BFGS methods need to obtain the iterative step size through an inexact one-dimensional linear search method. In this article, the iterative step size is required to meet the sufficient descent condition and the curvature condition. The Wolfe-Powell criterion can be obtained as(10)Φmk+αkpk≤Φmk+c1αk∇ΦmkTpk,∇Φmk+αkpkTpk≥c2∇ΦmkTpk,where Φ is the forward operator, mk is the model parameter of the k-th inversion iteration, αk is the iteration step length, c1 and c2 are constants satisfying 0<c1<c2<1, and pk is the search direction. The search methods product can be calculated using the process described in the work of Nocedal and Wright [59]. In general, the smaller c2 is, the more accurate the linear search will be. If c2=0.1, a fairly accurate linear search will be obtained, while c2=0.9 will result in a relatively weak linear search. The smaller c2, the longer the search time. According to the literature, the commonly used values c1 and c2 in the L_BFGS method are c1=10−4andc2=0.9.Algorithm 1: Limited Memory BFGS.
(1)
Setk = 1, choose initial model m0, integer m > 0 and initial matrix Hk0 (the identity matrix);(2)
Computepk=−Hkgk,mk+1=mk+αkpk, where αk is selected to satisfy the Wolfe-Powell conditions;(3)
Ifk> m thenDiscard the vector pairsk−m,yk−m from the storageCompute and savesk=mk+1−mk,yk=gk+1−gk;end(4)
UpdateHk0 using formula (7) for m times to obtain Hk+1 from formula (6);(5)
Setk = k + 1, go to step 2.


## 3.1. Objective Function

According to Tikhonov’s regularization theory, the inversion objective function in the sense of least squares is used. The objective function is described as [10, 18](4)Φm=Φdm+λΦmm=Wddobs−Fm22+λWmm−mref22,where Fm is the forward response function, m is the model parameter mi,i=1,2,…,M, dobs is the observation data, Wd is an N×N data weighting matrix (N is the number of data) in which the diagonal elements are the measured data and the remaining elements are zero, σi is the standard deviation of the i-th measured data, Wm is the model-weighted matrix usually defined by the discrete difference operator of the model unit and generally takes the first-order regularization constraint, λ is the regularization parameter used to balance the weight of data fitting and model smoothness, and mref is a reference model containing prior information about the model parameters. The inverted measured data dobs are the pole-pole potential value and the model parameter is the conductivity value of the element. Usually, the logarithm is used to calibrate the measured data and the model parameters mainly due to the large variation range and to invert the stability. The logarithm is defined as d=lnϕobs,m=lnσ. The resistivity inversion problem is generally a mixed problem, which often leads to equation (4) as an ill-conditioned equation. To solve this problem, smooth constraints are introduced into the inversion equation. Unstructured grid with disorderly arrangement was used in the forward modeling, so we adopt smooth constraint method, that is, judge the adjacency of the grid according to whether the grid unit has contact surface to determine the adjacency of the grid unit. For adjacency to generate matrix Wm, Wm(i, j) represents the contribution of the j-th unit to the smoothness of the i-th unit, which is generated according to the following formula:(5)Wm=−1/xij∑j=1ki1/xij,iandjare adjacent units,1,i=j,0,other,where xij=xi−xj, and it is the distance in the X direction between the center of the i-th unit and the j-th unit, and ki is the number of adjacent units to the i-th unit.


## 3.2. Inversion Framework

For the large-scale inversion problem, the traditional BFGS method requires a large amount of memory. Nocedal [50] improved the BFGS method and proposed a limited memory BFGS (L_BFGS) method to solve the nonlinear optimization problem. In the L_BFGS method, the inverse Hessian matrix approximation formula is defined as(6)Hk+1=Vk−1T,…,Vk−mTHk0Vk−m,…,Vk−1+ρk−mVk−1T,…,Vk−m+1Tsk−msk−mTVk−m+1,…,Vk−1+ρk−m+1Vk−1T,…,Vk−m+2Tsk−m+1sk−m+1TVk−m+2,…,Vk−1+⋯+ρk−1sk−1sk−1T,ρk=1ykTsk,Vk=I−ρkykskT,sk=mk+1−mk,yk=gk+1−gk,where m is the number of previous iterations with a value between 3 and 20. The previous iteration information of gradient and the model modification is used to modify the inverse Hessian matrix. Hk0 defined by Nocedal and Wright [59] gives an update method as(7)Hk0=γkI.In this paper, the identity matrixI is selected to initialize matrix Hk0 and γk can be defined as(8)γk=sk−1Tyk−1yk−1Tyk−1.The inversion steps of the L_BFGS method are defined in Algorithm1 as follows [49, 59].In the algorithm,gk is the gradient of the objective function (4), and it is expressed as(9)gk=∇Φmk=−2JmTWdTWddobs−Fmk+2λWmTWmmk,where J represents the Jacobian matrix. It can be seen from formula (9) that the calculation of the gradient lies in the calculation of the Jacobian matrix. The explicit calculation requires massive computation and memory storage. Therefore, it is necessary to avoid directly calculating the Jacobian matrix and calculate the product of the transpose of the Jacobian matrix and any one-dimensional vector. Thus, there is no need for storing the Jacobian matrix, and the calculation can be obtained together after the forwarding in each inversion, which significantly speeds up the inversion calculation. The calculation details are provided by Zhanget al. [26].Nocedal and Wright [59] presented a double-loop recursive method to update Step 2 in Algorithm 1. The detailed calculation process can be found in the literature [59]. In the process of minimizing the objective function, in contrast to the CG’s method in which the step size is obtained using an analytical method, both the NLCG and the L_BFGS methods need to obtain the iterative step size through an inexact one-dimensional linear search method. In this article, the iterative step size is required to meet the sufficient descent condition and the curvature condition. The Wolfe-Powell criterion can be obtained as(10)Φmk+αkpk≤Φmk+c1αk∇ΦmkTpk,∇Φmk+αkpkTpk≥c2∇ΦmkTpk,where Φ is the forward operator, mk is the model parameter of the k-th inversion iteration, αk is the iteration step length, c1 and c2 are constants satisfying 0<c1<c2<1, and pk is the search direction. The search methods product can be calculated using the process described in the work of Nocedal and Wright [59]. In general, the smaller c2 is, the more accurate the linear search will be. If c2=0.1, a fairly accurate linear search will be obtained, while c2=0.9 will result in a relatively weak linear search. The smaller c2, the longer the search time. According to the literature, the commonly used values c1 and c2 in the L_BFGS method are c1=10−4andc2=0.9.Algorithm 1: Limited Memory BFGS.
(1)
Setk = 1, choose initial model m0, integer m > 0 and initial matrix Hk0 (the identity matrix);(2)
Computepk=−Hkgk,mk+1=mk+αkpk, where αk is selected to satisfy the Wolfe-Powell conditions;(3)
Ifk> m thenDiscard the vector pairsk−m,yk−m from the storageCompute and savesk=mk+1−mk,yk=gk+1−gk;end(4)
UpdateHk0 using formula (7) for m times to obtain Hk+1 from formula (6);(5)
Setk = k + 1, go to step 2.


## 4. Synthetic Data Inversion

In actual exploration, the influence of topography is unavoidable and will cause deviations in the inversion results. Topography correction is usually used to eliminate the impact. However, since the underground structure is complex, the topography correction can only be approximated and will still have large errors. Therefore, regardless of the topography correction in the data or the model space, the resistivity inversion under undulating topography conditions cannot eliminate the influence of the terrain. The influence of the topography can be accurately eliminated only by incorporating the topography into the inversion algorithm [10]. In this paper, the topography information is directly introduced in the inversion and the three-dimensional resistivity inversion with topography is conducted. Numerical examples over different scenarios are provided to illustrate the validity of the inversion algorithm proposed in this paper. A single low-resistance anomaly model is embedded with measured data for three-dimensional inversion under the flat, valley, and mountain topographies.


### 4.1. Inversion of Buried Cuboid Model under Flat Topography

The rectangular model is shown in Figure5. The resistivities of the background half-space and the low-resistance body are ρ0=100Ωm and ρ1=10Ωm, respectively. The size of the low-resistance cuboid is 10m×10m×5m. The buried depths of the cuboid from the top to the ground and from the bottom to the ground are h = 5 m and 10 m, respectively. The blue triangles represent the locations of the two wellheads (40, 50, 0) and (60, 50, 0). The red five-pointed star represents the origin of the Cartesian coordinate system (50, 50, 0), and the horizontal distance between the anomalous body and the drilling on both sides is d = 5 m. The survey line range used in inversion is 1∼99 m and the spacing between survey lines is 1 m. The number of selected measuring points in each survey line is 99, and the position coordinates of measuring point M (measuring electrode) along the X direction are x = 1∼99 m and y = 1∼99 m, while the spacing is 1 m. The range of the source point A (source electrode) is from −5 m to −25 m downhole, with an interval of 5 m. Figures 6(a) and 6(b) are the unstructured tetrahedral grids used in forward modeling and inversion, respectively. In order to improve the accuracy of forward modeling and reduce the numerical simulation errors near the source point, the forward modeling grids are measured, the source point and the vicinity of the anomalous body are refined meshes, and the total number of tetrahedra grids is 1706788. The inversion grid is refined at the source and the measurement points, and the total number of tetrahedra grids is 1512967. A total of 98,010 pieces of “potential measured data” of the primary field were obtained by using a pole-pole array and the three-dimensional finite element forward modeling program. In order to verify the stability of the proposed inversion algorithm, 3% Gaussian noise was added to the theoretical measured data. The selection of the inversion parameters is as follows: the regularization parameterλ = 0.05, which remains unchanged during the inversion process, the convergence coefficient of the inversion termination, and the number of inversion iterations is 12 times. Figures 7(a) and 7(b) show the objective function fitting and the root mean square (RMS) error during the inversion process, respectively. It can be seen from the figures that the data fitting is poor and the objective function decreases steadily, indicating that the L_BFGS in this paper has good convergence for the three-dimensional borehole-surface resistivity method. Figures 8(a) and 8(b) show the inversion result profiles of XOZ and YOZ, respectively. It can be seen that the inversion result of the synthetic model data is still in good agreement with the real model in the presence of noise, and the location of the underground low-resistance anomaly is the same as the resistivity value, which verifies the effectiveness of the inversion method proposed in this paper.Figure 5
Geometrical model of a flat surface with a cube. (a) XOY plane view of the model; (b) XOZ cross-sectional view of the model.
(a)(b)Figure 6
Forward and inversion grid of the buried rectangular model under the flat surface. (a) Forward mesh and (b) inversion mesh.
(a)(b)Figure 7
The relationship between inversion parameters and the number of iterations. (a) Data relative fit difference RMS; (b) fitting of objective function.
(a)(b)Figure 8
Inversion results of anomalous bodies under the flat surface. (a) Slice of resistivity inversion result at XOZ profile; (b) slice of resistivity inversion result at YOZ profile. The outline of the true model is indicated by the white rectangles.
(a)(b)


### 4.2. Inversion of Buried Cuboid Model under Valley

The model of the buried anomaly under the valley is shown in Figure9. The lowest part of the valley is 10 m above the ground, and the entire depression is symmetrical about the minimum. The horizontal span is 45 m. The resistivities of the background half-space and the low-resistance body are ρ0=100Ωm and ρ1=10Ωm, respectively. The size of the low-resistance cuboid is 10m×10m×5m. The buried depths of the cuboid from the top to the ground and from the bottom to the ground are h = 5 m and 10 m, and the horizontal distance between the anomalous body and the drilling on both sides is d = 5 m. The blue triangles represent the locations of the two wellheads (40, 50, −8) and (60, 50, −8). The red five-pointed star represents the origin of the Cartesian coordinate system (50, 50, 0). There are a total of 5 survey lines in the survey area in inversion. The distance between survey lines is 5 m. The number of measuring points selected in each survey line is 79. The position coordinates of measuring point M (measuring electrode) along the X direction are x = 11∼89 m and y = 40∼60 m with an interval of 1 m. The total measuring points are 395. The range of the source point A (source electrode) is from −5 m to −25 m downhole, with an interval of 5 m. Figure 10 shows the unstructured tetrahedral meshes used in the forwarding and inversion of the model. In order to improve the accuracy of the forward modeling and reduce the numerical simulation error near the source point, the forward meshes are refined in the vicinity of the survey point, the source point, and the anomalous body. The inversion mesh is at the source point. The total number of tetrahedral cells is 1247660. Encrypted with the measuring point, the total number of inversion grid tetrahedral cells is 1169531. The selection of the inversion parameters is consistent with the inversion model under a flat surface. The number of inversion iterations is 17 times. The relative fitting error RMS of the data during the inversion process is shown in Figure 11(a). It can be seen from the figure that the data fitting is poor. The steady decline indicates that the L_BFGS inversion method has good convergence in this paper. Figure 11(b) shows the slice of resistivity 3D inversion result, while Figures 11(c) and 11(d) show the results of slices at XOZ and YOZ profiles, respectively, under valley topography. The topography effects can be seen in the figures. Under the circumstances, the inversion result of the synthetic model data is in good agreement with the real model, and the location of the underground low-resistance anomaly is the same as the resistivity value, which effectively eliminates the influence of topography and verifies the effectiveness of the proposed inversion method.Figure 9
The geometry of valley topographical model with a cube. (a) XOY plane view; (b) XOZ cross-sectional view.
(a)(b)Figure 10
Forward and inversion grid of buried rectangular model under valley topography. (a) Forward mesh; (b) inversion mesh.
(a)(b)Figure 11
Inversion results of anomalous bodies under valley topographical model. (a) The relative fitting error RMS during the inversion process; (b) slice of resistivity 3D inversion result; (c) slice of resistivity inversion result at XOZ profile; (d) slice of resistivity inversion result at YOZ profile. The outline of the true model is indicated by the white boxes.
(a)(b)(c)(d)


### 4.3. Inversion of the Buried Cuboid Model under Mountain

The buried rectangular model under the mountain is shown in Figure12. The highest part of the mountain relative to the ground is 10 m high, the entire uplift is symmetrical about the maximum, and the horizontal span is 45 m. The resistivities of the background half-space and the low-resistance body are ρ0=100Ωm and ρ1=10Ωm, respectively. The size of the low-resistance rectangular block is 10m×10m×5m. The buried depths of the cuboid from the top to the ground and from the bottom to the ground are h = 5 m and 10 m, respectively. The horizontal distance between the abnormal body and the drilling on both sides is d = 5 m. The blue triangles represent the locations of the two wellheads (40, 50, 8) and (60, 50, 8). The red five-pointed star represents the origin of the Cartesian coordinate system (50, 50, 0). There are a total of 5 survey lines in the survey area of inversion. The survey lines are laid on the ground. The measuring device is placed along the positive direction of the x-axis with 99 measuring electrodes separated by 1 m to pass through the top of the mountain. The distance between the survey lines is 5 m. The position coordinates of measuring point M (measuring electrode) are x = 1∼99 m and y = 40∼60 m, and the total number of measuring points is 495. The range of the source point A (source electrode) is from −5 m to −25 m downhole, with an interval of 5 m. Figure 13 shows the unstructured tetrahedral meshes used in the forwarding and inversion of the model. In order to improve the accuracy of the forward modeling and reduce the numerical simulation error near the source point, the forward meshes are refined in the vicinity of the measuring point, the source point, and the anomalous body. The total number of tetrahedral cells is 1105427. The inversion mesh is refined at the source and the measuring points. The total number of tetrahedral cells is 732012. The selection of inversion parameters is consistent with the inversion model under a flat surface. The number of inversion iterations is 9 times. The RMS change during the inversion process is shown in Figure 14(a). It can be seen from the figure that the relative RMS of the data is poorly fitted. The steady decline of RMS indicates that the L_BFGS inversion method in this paper has good convergence. Figure 14(b) shows the slice of resistivity 3D inversion result, while Figures 14(c) and 14(d) show the results slices at XOZ and YOZ profiles, respectively, under mountain topography. It can be seen from the figures that the inversion result of the synthetic model data is in good agreement with the real model. The location of the underground low-resistance anomaly is the same as the resistivity value, which effectively eliminates the influence of topography and verifies the effectiveness of the inversion method proposed in this paper.Figure 12
The geometry of mountain topographical model with a cube. (a) XOY plane view; (b) XOZ cross-sectional view.
(a)(b)Figure 13
Forward and inversion grid of buried rectangular model under mountain topography. (a) Forward mesh; (b) inversion mesh.
(a)(b)Figure 14
Inversion results of anomalous bodies under mountain topographical model. (a) The change in the relative RMS during the inversion process; (b) slice of resistivity 3D inversion result; (c) slice of resistivity inversion result at XOZ profile; (d) slice of resistivity inversion result at YOZ profile. The outline of the true model is indicated by the white rectangles.
(a)(b)(c)(d)


## 4.1. Inversion of Buried Cuboid Model under Flat Topography

The rectangular model is shown in Figure5. The resistivities of the background half-space and the low-resistance body are ρ0=100Ωm and ρ1=10Ωm, respectively. The size of the low-resistance cuboid is 10m×10m×5m. The buried depths of the cuboid from the top to the ground and from the bottom to the ground are h = 5 m and 10 m, respectively. The blue triangles represent the locations of the two wellheads (40, 50, 0) and (60, 50, 0). The red five-pointed star represents the origin of the Cartesian coordinate system (50, 50, 0), and the horizontal distance between the anomalous body and the drilling on both sides is d = 5 m. The survey line range used in inversion is 1∼99 m and the spacing between survey lines is 1 m. The number of selected measuring points in each survey line is 99, and the position coordinates of measuring point M (measuring electrode) along the X direction are x = 1∼99 m and y = 1∼99 m, while the spacing is 1 m. The range of the source point A (source electrode) is from −5 m to −25 m downhole, with an interval of 5 m. Figures 6(a) and 6(b) are the unstructured tetrahedral grids used in forward modeling and inversion, respectively. In order to improve the accuracy of forward modeling and reduce the numerical simulation errors near the source point, the forward modeling grids are measured, the source point and the vicinity of the anomalous body are refined meshes, and the total number of tetrahedra grids is 1706788. The inversion grid is refined at the source and the measurement points, and the total number of tetrahedra grids is 1512967. A total of 98,010 pieces of “potential measured data” of the primary field were obtained by using a pole-pole array and the three-dimensional finite element forward modeling program. In order to verify the stability of the proposed inversion algorithm, 3% Gaussian noise was added to the theoretical measured data. The selection of the inversion parameters is as follows: the regularization parameterλ = 0.05, which remains unchanged during the inversion process, the convergence coefficient of the inversion termination, and the number of inversion iterations is 12 times. Figures 7(a) and 7(b) show the objective function fitting and the root mean square (RMS) error during the inversion process, respectively. It can be seen from the figures that the data fitting is poor and the objective function decreases steadily, indicating that the L_BFGS in this paper has good convergence for the three-dimensional borehole-surface resistivity method. Figures 8(a) and 8(b) show the inversion result profiles of XOZ and YOZ, respectively. It can be seen that the inversion result of the synthetic model data is still in good agreement with the real model in the presence of noise, and the location of the underground low-resistance anomaly is the same as the resistivity value, which verifies the effectiveness of the inversion method proposed in this paper.Figure 5
Geometrical model of a flat surface with a cube. (a) XOY plane view of the model; (b) XOZ cross-sectional view of the model.
(a)(b)Figure 6
Forward and inversion grid of the buried rectangular model under the flat surface. (a) Forward mesh and (b) inversion mesh.
(a)(b)Figure 7
The relationship between inversion parameters and the number of iterations. (a) Data relative fit difference RMS; (b) fitting of objective function.
(a)(b)Figure 8
Inversion results of anomalous bodies under the flat surface. (a) Slice of resistivity inversion result at XOZ profile; (b) slice of resistivity inversion result at YOZ profile. The outline of the true model is indicated by the white rectangles.
(a)(b)


## 4.2. Inversion of Buried Cuboid Model under Valley

The model of the buried anomaly under the valley is shown in Figure9. The lowest part of the valley is 10 m above the ground, and the entire depression is symmetrical about the minimum. The horizontal span is 45 m. The resistivities of the background half-space and the low-resistance body are ρ0=100Ωm and ρ1=10Ωm, respectively. The size of the low-resistance cuboid is 10m×10m×5m. The buried depths of the cuboid from the top to the ground and from the bottom to the ground are h = 5 m and 10 m, and the horizontal distance between the anomalous body and the drilling on both sides is d = 5 m. The blue triangles represent the locations of the two wellheads (40, 50, −8) and (60, 50, −8). The red five-pointed star represents the origin of the Cartesian coordinate system (50, 50, 0). There are a total of 5 survey lines in the survey area in inversion. The distance between survey lines is 5 m. The number of measuring points selected in each survey line is 79. The position coordinates of measuring point M (measuring electrode) along the X direction are x = 11∼89 m and y = 40∼60 m with an interval of 1 m. The total measuring points are 395. The range of the source point A (source electrode) is from −5 m to −25 m downhole, with an interval of 5 m. Figure 10 shows the unstructured tetrahedral meshes used in the forwarding and inversion of the model. In order to improve the accuracy of the forward modeling and reduce the numerical simulation error near the source point, the forward meshes are refined in the vicinity of the survey point, the source point, and the anomalous body. The inversion mesh is at the source point. The total number of tetrahedral cells is 1247660. Encrypted with the measuring point, the total number of inversion grid tetrahedral cells is 1169531. The selection of the inversion parameters is consistent with the inversion model under a flat surface. The number of inversion iterations is 17 times. The relative fitting error RMS of the data during the inversion process is shown in Figure 11(a). It can be seen from the figure that the data fitting is poor. The steady decline indicates that the L_BFGS inversion method has good convergence in this paper. Figure 11(b) shows the slice of resistivity 3D inversion result, while Figures 11(c) and 11(d) show the results of slices at XOZ and YOZ profiles, respectively, under valley topography. The topography effects can be seen in the figures. Under the circumstances, the inversion result of the synthetic model data is in good agreement with the real model, and the location of the underground low-resistance anomaly is the same as the resistivity value, which effectively eliminates the influence of topography and verifies the effectiveness of the proposed inversion method.Figure 9
The geometry of valley topographical model with a cube. (a) XOY plane view; (b) XOZ cross-sectional view.
(a)(b)Figure 10
Forward and inversion grid of buried rectangular model under valley topography. (a) Forward mesh; (b) inversion mesh.
(a)(b)Figure 11
Inversion results of anomalous bodies under valley topographical model. (a) The relative fitting error RMS during the inversion process; (b) slice of resistivity 3D inversion result; (c) slice of resistivity inversion result at XOZ profile; (d) slice of resistivity inversion result at YOZ profile. The outline of the true model is indicated by the white boxes.
(a)(b)(c)(d)


## 4.3. Inversion of the Buried Cuboid Model under Mountain

The buried rectangular model under the mountain is shown in Figure12. The highest part of the mountain relative to the ground is 10 m high, the entire uplift is symmetrical about the maximum, and the horizontal span is 45 m. The resistivities of the background half-space and the low-resistance body are ρ0=100Ωm and ρ1=10Ωm, respectively. The size of the low-resistance rectangular block is 10m×10m×5m. The buried depths of the cuboid from the top to the ground and from the bottom to the ground are h = 5 m and 10 m, respectively. The horizontal distance between the abnormal body and the drilling on both sides is d = 5 m. The blue triangles represent the locations of the two wellheads (40, 50, 8) and (60, 50, 8). The red five-pointed star represents the origin of the Cartesian coordinate system (50, 50, 0). There are a total of 5 survey lines in the survey area of inversion. The survey lines are laid on the ground. The measuring device is placed along the positive direction of the x-axis with 99 measuring electrodes separated by 1 m to pass through the top of the mountain. The distance between the survey lines is 5 m. The position coordinates of measuring point M (measuring electrode) are x = 1∼99 m and y = 40∼60 m, and the total number of measuring points is 495. The range of the source point A (source electrode) is from −5 m to −25 m downhole, with an interval of 5 m. Figure 13 shows the unstructured tetrahedral meshes used in the forwarding and inversion of the model. In order to improve the accuracy of the forward modeling and reduce the numerical simulation error near the source point, the forward meshes are refined in the vicinity of the measuring point, the source point, and the anomalous body. The total number of tetrahedral cells is 1105427. The inversion mesh is refined at the source and the measuring points. The total number of tetrahedral cells is 732012. The selection of inversion parameters is consistent with the inversion model under a flat surface. The number of inversion iterations is 9 times. The RMS change during the inversion process is shown in Figure 14(a). It can be seen from the figure that the relative RMS of the data is poorly fitted. The steady decline of RMS indicates that the L_BFGS inversion method in this paper has good convergence. Figure 14(b) shows the slice of resistivity 3D inversion result, while Figures 14(c) and 14(d) show the results slices at XOZ and YOZ profiles, respectively, under mountain topography. It can be seen from the figures that the inversion result of the synthetic model data is in good agreement with the real model. The location of the underground low-resistance anomaly is the same as the resistivity value, which effectively eliminates the influence of topography and verifies the effectiveness of the inversion method proposed in this paper.Figure 12
The geometry of mountain topographical model with a cube. (a) XOY plane view; (b) XOZ cross-sectional view.
(a)(b)Figure 13
Forward and inversion grid of buried rectangular model under mountain topography. (a) Forward mesh; (b) inversion mesh.
(a)(b)Figure 14
Inversion results of anomalous bodies under mountain topographical model. (a) The change in the relative RMS during the inversion process; (b) slice of resistivity 3D inversion result; (c) slice of resistivity inversion result at XOZ profile; (d) slice of resistivity inversion result at YOZ profile. The outline of the true model is indicated by the white rectangles.
(a)(b)(c)(d)


## 5. Conclusion

This paper successfully realizes and develops the three-dimensional inversion of the borehole-to-surface resistivity method in nonflat surface topography based on the unstructured finite element method and the limited memory L_BFGS method. In order to verify the effectiveness of the proposed algorithm, numerical simulations of inversion of synthetic data for flat, valley, and mountain topographies are conducted. The obtained results validate that the proposed algorithm has high stability, and the inversion results better restore the distribution characteristics of low-resistance anomalies under various topography conditions. Future research will focus on utilizing the algorithm proposed in this paper to carry out the inversion of field-data applications resistivity. A reasonable application of a variety of information constraints can improve the accuracy of the position and shape of anomalies in the imaging results. Future research plan also includes the use of regularization and related techniques to achieve high-precision underground imaging with prior information.



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*Source: 5154985-2021-09-24.xml*