# 3-D DC Resistivity Forward Modeling Using the Multi-resolution Grid

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## Abstract

We implemented a novel multi-resolution grid approach to direct current resistivity (DCR) modeling in 3-D. The multi-resolution grid was initially developed to solve the electromagnetic forward problem and helped to improve the modeling efficiency. In the DCR forward problem, the distribution of the electric potentials in the subsurface is estimated. We consider finite-difference staggered grid discretization, which requires fine grid resolution to accurately model electric potentials around the current electrodes and complex model geometries near the surface. Since the potential variations attenuate with depth, the grid resolution can be decreased correspondingly. The conventional staggered grid fixes the horizontal grid resolution that extends to all layers. This leads to over-discretization and therefore unnecessary high computational costs (time and memory). The non-conformal multi-resolution grid allows the refinement or roughening for the grid’s horizontal resolution with depth, resulting in a substantial reduction of the degrees of freedom, and subsequently, computational requirements. In our implementation, the coefficient matrix maintains its symmetry, which is beneficial for using the iterative solvers and solving the adjoint problem in inversion. Through comparison with the staggered grid, we have found that the multi-resolution grid can significantly improve the modeling efficiency without compromising the accuracy. Therefore, the multi-resolution grid allows modeling with finer horizontal resolutions at lower computational costs, which is essential for accurate representation of the complex structures. Consequently, the inversion based on our modeling approach will be more efficient and accurate.

## Keywords

Multi-resolution grid DCR 3-D forward modelling finite-difference## 1 Introduction

The direct current resistivity (DCR) method is one of the classical geophysical techniques, which is nowadays widely used in the mineral exploration (Oldenburg et al. 1997; Schoor 2005; Zhang et al. 2015), groundwater (Andrade 2011; Thompson et al. 2012), engineering (Chambers et al. 2014; Lysdahl et al. 2017), and environmental problems (LaBrecque et al. 1996; Rosales et al. 2012). The method is often used both for ground measurements and in the boreholes (Loke et al. 2013). Subsequently, the measured electric potential data are inverted to obtain a resistivity image of the subsurface to a depth depending on the separation distance between electrodes.

The forward modeling is an essential step of any inversion algorithm. Over the past few decades, the 1-D (O’Neil and Merrick 1984; Das and Verma 1980) and 2.5-D (Mundry 1984; Pidlisecky and Knight 2008) modeling of the DCR has been developed and used routinely. The 3-D DCR surveying became feasible with the development of the multi-electrode and multi-channel systems (Dahlin 2016; Loke et al. 2013). Therefore, the data in 3-D case can only be fully exploited using 3-D modeling and inversion algorithms. There is a number of algorithms developed so far for 3-D DCR modeling based on the integral equation method (Schulz 1985; Mendez-Delgado et al. 1999), finite-element method (Li and Spitzer 2002a; Rucker et al. 2006; Ren et al. 2018) and finite-difference method (Dey and Morrison 1979; Scriba 1981; Spitzer 1995; Zhang et al. 1995; Loke and Barker 1996; Zhao 1996; Wang et al. 2000; Wu et al. 2003a; Śebastien Penz et al. 2013). The integral equation approach is efficient to handle the models with a few anomaly bodies. On the contrary, the commonly used finite-element method (FEM) and finite-difference method (FDM) are more suitable to cope with models having an arbitrary number of structures (Li and Spitzer 2002b; Wu et al. 2003b). Despite the efforts to improve the efficiency, available 3-D modeling algorithms are still heavy in terms of computational requirements. Since the inversion requires numerous forward calculations, optimized modeling algorithm naturally leads to an efficient inversion.

The electrical field potentials induced by the current electrodes create a singularity effect around the source positions. In order to cope with it, the secondary potentials’ formulation is often used (Lowry 1989). However, for both of the total and secondary field approaches, the simulated electric field potentials generally attenuate with depth, meaning that no significant variations can be expected at larger depth. Therefore, responses can be modeled on a grid using coarser resolution (discretization) in the deeper regions, without loss of accuracy.

The conventional staggered (SG) grid employs rectangular cells to discretize the model, which simplifies the gridding process. However, the conformal SG grid fixes the horizontal resolution, which extends to all depths. This may cause over-discretization of the deeper regions, hence leading to redundant computational requirements. The unstructured grid, generally adopted by FEM, commonly employs the tetrahedral cells to discretize the model, which allows local refinement and roughening to avoid the over-discretization condition.

We present a new 3-D DCR forward modeling based on a finite-difference multi-resolution (MR) grid approach. The MR grid was initially developed for the electromagnetic forward problem and proved to be an efficient alternative to the conventional structured finite-difference approach (Cherevatova et al. 2018). The MR grid is a simplified implementation of the non-conformal grid, it resembles the approach suggested in the octree scheme of Haber and Heldmann (2007), but limited to only the horizontal resolution’s refinement. The MR grid can be derived from a fundamental SG grid by horizontally combining adjacent cells. Thus, MR grid represents a vertical stack of several SG grids (sub-grids) with different horizontal resolutions. In this way, the forward modeling operators developed for SG grid are readily applied for each sub-grid.

The main difficulty in MR modeling is the definition of the forward operators at common interfaces between the adjacent sub-grids. Several approaches to defining differential operators at the interfaces were considered and tested in Cherevatova et al. (2018). Generally, all grid nodes and edges are separated into two groups: ‘active’ and ‘inactive’. Inactive elements are not evaluated in the solution, but represented by their neighboring active elements through interpolation. This allows defining differential operators in the physically correct way at the common interfaces. The only difference between the approaches to handle operators on the interfaces lies in the definition of active and inactive grid elements and the interpolation scheme. As a result, different accuracy might be achieved, depending on the selected approach. Following the conclusion of Cherevatova et al. (2018), we selected the Coarse Active (CA) approach, which is shown to be the most efficient and accurate. Moreover, within the CA approach the discrete divergence operator is the adjoint of the discrete gradient operator, which leads to the symmetry of the coefficients matrix. In this case the linear equations set can be solved efficiently using the Preconditioned Conjugate Gradient method. As a result, the time required for solving the system of equations is nearly linear with respect to the degrees of freedom (*DoF*). The MR grid allows us to substantially reduce the amount of *DoF* leading to a significant speed-up of the forward modeling.

The paper is organized as follows. In the methods part, we briefly describe the standard staggered grid formulation of the DCR modeling problem, and its modification to define the problem on the multi-resolution grid. In order to verify our newly developed algorithm, in the following sections we present several synthetic examples to examine the speed and accuracy of the MR solution.

## 2 Methods

*I*is the current intensity, \((x_0,y_0,z_0)\) denotes the current electrode’s coordinate, and \(\delta (\cdot )\) is the delta function.

In order to solve the PDEs, boundary conditions are required on the boundaries confining \(\varOmega\) into a finite space. In our approach, we implemented Neumann boundary condition on the top boundary \(\varGamma _0\) to prevent the electrical current from flowing out. For the distant side boundaries \(\varGamma _{\infty }\), we simply employed the Dirichlet boundary condition (Mufti 1976; Tripp et al. 1984) in our tests.

Since the multi-resolution (MR) grid is based on the conventional staggered (SG) grid, first we briefly describe the DCR modeling on SG grid. Then we explain the implementation of MR grid approach and emphasize its differences.

### 2.1 Staggered Grid Approach

The modeling domain \(\varOmega\) on the SG is discretized into \(N_{x}\), \(N_{y}\) and \(N_{z}\) rectangular cells in the *x*, *y* and *z* directions, respectively. The entire modeling domain is divided into \(N_{c}\) cells (\(=N_{x}\times N_{y}\times N_{z}\)) in total, and each cell holds a constant conductivity value \(\sigma\).

A natural choice for solving a large symmetric system of linear equations would be one of the Krylov subspace iterative methods. We choose to solve Eq. (8) using the Preconditioned Conjugate Gradient (PCG) method, which takes advantage of the symmetry of \(\widetilde{\mathbf{A}}\) and is superior to the generalized solvers (Spitzer 1995). As a preconditioner, the Symmetric Successive Over-Relaxation (SSOR) method is employed. The combination (SSOR-PCG) shows a fast and stable convergence in our tests. Later we refer to the left and right-hand sides of Eq. (8) as \(\mathbf{A}\mathbf{\varphi }\) and \(\mathbf{b}\) correspondingly omitting the tilde.

### 2.2 Secondary Field Formulation

The most significant potential variations appear around the current electrodes, causing the singular solutions at those points. Local refinement of the grid around the source region is one of the options to alleviate the numerical errors, but it requires very fine discretization around the current electrodes positions, and therefore, inevitably increases the computational costs. Alternatively, secondary field approach is commonly applied (Lowry 1989) to tackle this problem.

### 2.3 Multi-resolution Grid Approach

*i*denotes the index of the sub-grid, \(N_{x}^{f} \times N_{y}^{f}\) is the finest horizontal resolution used in the whole MR grid, \(Cs^{(i)}\) is the Coarseness parameter (a non-negative integer) of the \(i\hbox {th}\) sub-grid. \(N_{x}^{(i)}\) and \(N_{y}^{(i)}\) between adjacent sub-grids can only change by a factor of two. It is noteworthy that \(Cs^{(i)}\) is not only limited to increase with depth but also allowed to decrease. Thus, \(N_{x}^{f}\times N_{y}^{f}\) is not fixed in the top sub-grid (although, we commonly implement it in this way), which could also be assigned to the lower sub-grids to describe complex geometries.

Based on the above rule, the MR grid can be derived from a fundamental SG grid with a discretization: \(N_{x}^{f} \times N_{y}^{f} \times N_{z}^{f}\). First, one split the SG grid into \(N_{sg}\) sub-grids, such as \(\sum _{i=1}^{N_{sg}}N_{z}^{(i)}=N_{z}^{f}\), where \(N_{z}^{(i)}\) is the vertical discretization of each sub-grid. Second, for a sub-grid with \(Cs^{(i)} > 0\), horizontally combine each set of \(2 ^ { Cs^{(i)} } \times 2 ^ { Cs^{(i)}}\) cells to form a coarser cell. Following the above rule, the discretization of the MR grid is only controlled by the fundamental SG grid and the parameters: \(\left. \left[ Cs^{(i)} , N_{z}^{(i)} ; \right] \right| _ { i = 1 , N_{ sg } }\), which leads to a very simple gridding process. For instance, the MR grid in Fig. 2a with parameters: \(\left. \left[ Cs^{(i)} , N_{z}^{(i)} ; \right] \right| _ {i = 1,2 } = ( 0,2 ; 1,2 )\) is derived from a \(4 \times 4 \times 4\) SG grid. Alternatively, the discretization can start from the coarsest SG grid by refining the sub-grids, however, we always describe sub-grids relative to the finest sub-grid.

### 2.4 Multi-resolution Grid Elements Classification

Since each sub-grid could be regarded as a standard SG grid, at the interior of each sub-grid, the spatial relations between the grid elements are the same as the SG grid, therefore the forward operators (\(\mathbf{G} ^ { \dagger }\), \(\mathbf{G}\) and \(\mathbf{W}\)) have the same structure as Eqs. (4), (5) and (6). However, the main challenge in the MR grid implementation is the definition of the forward operators on the overlapped interface between the adjacent sub-grids, e.g. the red interface in Fig. 2a.

On a common interface, part of the grid elements (nodes and edges) from the finer sub-grid overlap with the grid elements from the coarser sub-grid, which will lead to redundant elements. In addition, due to the changing of horizontal resolution, the remained grid elements from the finer sub-grid without the corresponding partner from the coarser sub-grid will be ‘hanging’ on the interface. See an illustration of the interface in Fig. 2b.

The nodes and edges in the MR grid are classified into two groups: active and inactive elements. The active elements are involved in the calculation explicitly, whilst the inactive elements are eliminated from the solution. All the nodes and edges at the interior of the sub-grids are classified as active, whilst the choice at the common interface will affect the overall solution accuracy. Three approaches were considered and tested in Cherevatova et al. (2018). (1) Face Active (FA) case classifies the nodes and edges from the finer sub-grid as active and the elements from the coarser sub-grid as inactive similar to Haber and Heldmann (2007) results and provides operators symmetry; (2) Ghost Faces (GF) case is similar to FA, but it extends one ‘ghost’ interpolated finer layer into the coarser sub-grid, resulting in a second-order accuracy but lacks symmetry (Horesh and Haber 2011); (3) Coarse Active case (CA) is a reverse to FA that takes the nodes and edges from the coarser sub-grid as active. It has accuracy similar to GF but also maintains symmetry. Therefore, CA case is a preferable approach, which we used to implement the MR DCR modeling. Figure 2b shows the classification of CA case. Later we use \(N_{n}^{active}\) and \(N_{e}^{active}\) to denote the number of the active nodes and edges on MR grid, respectively.

### 2.5 Multi-resolution Grid Forward Modeling Operators

- 1.
For an active node, \(\mathbf{A}_{coarse}\) acts as a self-mapping operator, therefore the corresponding interpolation coefficient in \(\mathbf{A}_{coarse}\) is 1.

- 2.
For an inactive node from the finer sub-grid overlapping with an active node from the coarser sub-grid, e.g. the inactive and active nodes with label

*i*1 and*a*1 respectively in Fig. 2b, the corresponding interpolation coefficient from*a*1 to*i*1 is 1 as well. - 3.
For an inactive node located on an active edge, the entries are defined by the linear interpolation. In Fig. 2b, the mapped inactive node

*i*2 is interpolated from the adjacent active nodes*a*2 and*a*3. - 4.
For a ‘hanging’ inactive node located on a face of a coarser cell, it is linearly interpolated from four surrounding active nodes. For example, the inactive node with label

*i*3 is interpolated by the active nodes with label*a*4 to*a*7 in Fig. 2b.

In general horizontally non-uniform case, the interpolation coefficients are derived by the corresponding metric elements.

Thus, \(\overline{\mathbf{G}}_{full}\) is the gradient topology matrix of the full MR grid that maps from the full node set to the full edge set, and the sparse selection matrix \(\mathbf{R}_{coarse}\) (\(N_{e}^{active} \times N_{ e }\)) picks out the rows corresponding to the active edges only.

It is also readily verified that the transpose of \(-\overline{\mathbf{G}}\) is a good approximation to the topology matrix of the discrete divergence operator \(\mathbf{G} ^ { \dagger }\) on MR grid, even around the common interface (Cherevatova et al. 2018), see the spatial illustrations and matrix forms of \(\overline{\mathbf{G}}\) and \(-\overline{\mathbf{G}}^{T}\) in Appendix. Thus, \(\mathbf{G}\) and \(\mathbf{G} ^ { \dagger }\) on MR grid can be derived in a similar manner as Eqs. (4) and (5) for SG grid. However, \(\tilde{\mathbf{V}}_{C}\) (\(N_{n}^{active} \times 1\)) includes only dual-cell-volumes around the active nodes, \(\mathbf{L}_{E}\) and \(\tilde{\mathbf{A}}_{F}\) vectors (\(N_{e}^{active} \times 1\)) include only edge-lengths and dual-face-areas elements of the active edges, respectively.

\(\overline{\sigma }\) (\(N_{e}^{active} \times 1\)) is only evaluated on the active edges and can be derived from Eq. (6) as well. The topology matrix \(\hat{\mathbf{W}}\) maps from the cells set to the active edges set. However, the rows of \(\hat{\mathbf{W}}\), which correspond to the active edges at the common interfaces, include more nonzero coefficients related to the extra cells from the adjacent finer sub-grid, see Appendix for further details.

We take the advantage of CA symmetry, i.e. \(\mathbf{G}^{ \dagger }\) is well approximated by the adjoint of \(\mathbf{G}\) similarly to SG case, therefore, the derived coefficients matrix \(\mathbf{A}\) is also symmetric. In the MR case, the system of equations can also be solved efficiently using SSOR-PCG iterative solver, but the degree of freedom (*DoF*) of the system can be significantly decreased compared to the original SG grid.

## 3 Results

In this section, several examples were presented to verify the accuracy and efficiency of our algorithm. All the modeling of the SG and MR grid approaches were computed on a PC with 2.60 GHz CPU and 16GB memory.

### 3.1 Example 1

The model was discretized by an SG grid with resolution \(120 \times 120 \times 40\) at first, the central region was divided by 5 m uniform-cells, see Fig. 3. An MR grid was derived from the SG grid and decomposed it into three sub-grids with parameters: \(\left. \left[ Cs^{(i) } , N_{z}^{(i)}; \right] \right| _ {i = 1,3 } = ( 0,9; 1,21; 2,10)\), i.e. the horizontal resolution was gradually roughened downwards as \(120 \times 120\), \(60 \times 60\) and \(30 \times 30\). The common interfaces between adjacent sub-grids were located inside layers, as shown in Fig. 3.

### 3.2 Example 2

Next, we considered a 3-D case to demonstrate the accuracy and speed of the MR grid approach against the SG grid approach. The model with \(100\; \varOmega \,{\text {m}}\) background included two anomalous blocks with the same resistivity of \(1 \varOmega m\) and size of \(60\,{\text {m}} \times 60\,{\text {m}} \times 40\,{\text {m}}\), where they were placed at different depths of \(20\,{\text {m}}\) and \(80\,{\text {m}}\), as shown in Fig. 5a. The data were calculated for one \(600\,{\text {m}}\) pole-dipole profile with 30 m electrode distance (*a*) and \(n = 1\) to 8 levels, which includes 21 electrodes in total.

An MR grid was designed subsequently based on the SG grid, it consists of three sub-grids with the parameters: \(\left. \left[ Cs^{(i) } , N_{z}^{(i)};\right] \right| _ {i = 1,3 } = ( 0,14; 1,26; 2,10 )\). The deeper block was placed in the middle sub-grid, which is one step coarser, see Fig. 5a.

The \(\mathbf{\varphi }_s\) on MR grid generated by the same current electrode at \((x,y,z) = 0\,{\text {m}}\) was calculated, and its distribution is shown in Fig. 7a. It is noteworthy that the contour map of each sub-grid was plotted separately without any interpolated connection between them. The contour lines of \(\mathbf{\varphi }_s\) demonstrate that the sub-grids are well-connected and there are no artifices or interruptions due to the horizontal resolution changing between sub-grids. It means that we have sufficient accuracy at the common interfaces.

Comparison of the modeling efficiency between SG and MR grid approaches from example 2

| \(\overline{ t } _ { F w d }\) | \({\overline{t}}_{eq}\) | RAM | |
---|---|---|---|---|

SG | 1,531,250 | 198.01 s | 9.3 s | 2348 Mb |

MR | 644,034 | 83.81 s | 4.03 s | 946 Mb |

\({\mathrm {MR}}/{\mathrm {SG}} \times 100\%\) | 42.06% | 42.33% | 43.33% | 40.29% |

The MR grid approach requires less time for solving the system of equations (43.33% of the SG grid), which is nearly linear with respect to the reduction in the amount of *DoF*. As solving the equations set is the most time-consuming part of the forward modeling, the MR approach allows us to greatly improve the efficiency of the forward modeling. The memory usage is also reduced accordingly.

### 3.3 Example 3

In order to avoid the influences from using an improper background model in the secondary field approach (e.g. using the flat half-space model for this topography case), the total field approach was employed. The \(\rho _ {a}\) pseudo-sections of the MR grid response are shown in Fig. 10a, b. As it can be seen from the figures, the high \(\rho _ {a}\) values caused by the blocks can be observed, where the anomaly responses of two shallow blocks are more obvious. The topography caused perturbations in \(\rho _ {a}\), which could heavily disturb the anomaly responses, especially for the response of block 3, which is hard to find out. Therefore, the topography effect should be taken into account when interpreting the data to avoid inadequate interpretations.

The MR and SG grid solutions were compared with the finite-element method (FEM) solution, which used unstructured grid (Ren et al. 2017). The relative differences between the SG and MR grid solutions against the FEM solution were calculated and presented in Fig. 10c–f. As one can see, the relative differences of the SG and MR grid solutions against the FEM solution are generally less than \(1\%\). The deviations could mainly come from the discretization approaches with respect to the topography. The topography was discretized by SG and MR grids using a mass of fine rectangular cells to fit the surface in a staircase approximation manner. Contrarily, the tetrahedral cells used by FEM can preferably fit the topography surface, which is more suitable for coping with the topography case. Even so, the overall solutions are comparable.

Comparison of the modeling efficiency between SG and MR grid approaches from example 3. The abbreviations denote: *DoF*, the degrees of freedom; \(\overline{t}_{F w d}\), averaged time of forward modeling; \(\overline{t} _ { e q }\), averaged time for solving the system of equations (one source); RAM, maximum memory usage. The last row compares the ratio of the above parameters in percent

| \(\overline{ t } _ { F w d }\) | \({\overline{t}}_{eq}\) | RAM | |
---|---|---|---|---|

SG | 4,270,630 | 671 s | 38.65 s | 5441 Mb |

MR | 1,638,451 | 223.88 s | 12.33 s | 2430 Mb |

\({\mathrm {MR}} /{\mathrm {SG}} \times 100 \%\) | 38.37% | 33.37% | 31.9% | 44.66% |

We present results of the speed-up test for example 3 in a similar manner as example 2, see Table 2. The fine horizontal resolution required to handle the topography leads to a higher number of *DoF* in the SG case and naturally caused higher computational demands (Table 2). The MR grid roughen horizontal resolution downwards to avoid the over-discretization in the deep. Consequently, it reduced the *DoF* significantly to only \(38.37\%\) of the SG grid, and resulted in a smaller forward problem. Comparing run-times and memory usage, the MR grid reduced the computational costs to nearly one third of SG grid case, but without loss of the modeling accuracy. It means that the MR grid is more suitable than the SG grid for solving large modeling problem.

## 4 Conclusions

We implemented the multi-resolution (MR) grid approach to 3-D Direct Current Resistivity (DCR) modeling. The MR grid allows to adjust the horizontal resolution with depth, which is more flexible than the conventional staggered (SG) grid with respect to description of the shallow geometries, topography, and measurements configuration.

We applied a Coarse Active (CA) approach, which was carefully studied in Cherevatova et al. (2018), to handle the common interfaces between the sub-grids. Beneficial from the definition of the differential operators on the interface layers, the derived coefficient matrix remains in symmetric form as in the SG grid approach. Preconditioned conjugate gradient method was used as the solver, which provides fast and stable convergence due to the symmetry of the system of linear equations. We presented three synthetic studies to verify the accuracy and efficiency of the MR DCR modeling. The SG grid solution with fine discretization guarantees the accurate response, but leads to higher computational demands. The MR grid allows us to reduce the computational costs (CPU time and RAM usage) substantially without loss of accuracy. This is specifically important for inversion algorithms, which require numerous modeling calculations. Taking into account, that MR code can be parallelized to further accelerate calculations, the MR solver opens perspectives for large inversions, which are in high demand for geophysical problems.

## Notes

### Acknowledgements

Open access funding provided by Lulea University of Technology. This research was supported by the ARN project, and the project is funded by the European Union, through the Swedish Agency for Economic and Regional Growth and the Norrbotten County Council, as research Project no. 20200552.

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