Leachate leakages from municipal landfills are significant environmental problems that threaten groundwater and soil resources. Geophysical techniques, such as the self-potential (SP) method, are commonly used to detect and delineate underground contaminated plumes. However, traditional inversion techniques for SP source information require precise knowledge of subsurface conductivity, which can be challenging to obtain. In this study, we proposed an inversion algorithm, called SP-Net, based on a convolutional neural network that can directly train the intrinsic relationship between SP signals and the location of SP sources, while being a great performance within a comparative heterogeneous resistivity setting. In this work, we used the U-shaped network as the structure of the SP-Net and treated the problem of locating SP sources as an image segmentation problem. We designed a sandbox experiment model by adding humus and the microorganism calledShewanella oneidensisMR-1 to simulate the scenario of microbial-mediated SP generation, which typically exists at organic-rich contaminated sites. We used this situation to generate numerous 3-D SP datasets for SP-Net training. We tested the SP-Net on both synthetic testing datasets and the measured laboratory case, and the results show the effectiveness of SP-Net. We also developed a field-scale synthetic model for landfills as a preliminary attempt to test the SP-Net in practical applications, and the results reveal that our study has a valuable reference for potential future applications. Our work provides a promising tool to locate SP sources from both laboratory and field-scale SP data.

The multigrid (MG) method has been widely used for large-scale electromagnetic (EM) forward modeling, primarily due to its linear dependence between computational time and grid size. However, in natural source EM forward modeling, the inclusion of air layers and the use of low frequencies can result in slow convergence or even divergence for MG solvers, due to the sharp change of conductivity and violation of the divergence-free condition for current. One typical solution is to develop an efficient and specialized smoother, for instance, with one property of satisfying the divergence-free condition in certain local areas. In this article, we modify the curl-curl differential equation by explicitly including the gradient of the divergence of current (CCGD). Hence, this ensures approximately the divergence-free condition at every iteration while solving EM fields. Based on the modified equation, an MG solver is then developed with traditional Krylov subspace solvers as the smoother, and no specialized smoothers are required. One benchmark model, one reservoir model with topography, and one realistic model from inversion are used to examine its numerical performance, in terms of accuracy, efficiency, and ability to handle grid stretching and size increase.

Multigrid (MG) solvers, typically increasing the computational time linearly with the grid size, are suitable for large-scale forward modeling problems. However, for electromagnetic (EM) problems as frequency decreases and grid is increasingly stretched, MG solvers for EM modeling can converge slowly or even diverge. We propose an efficient four-color Line Gauss-Seidel (GS) MG for finite difference (FD) frequency-domain EM solution. In this algorithm, the edge components attached to nodes on each line in one particular direction are updated simultaneously, leading to that the solution in each local region satisfies divergence free condition. Due to the fact that each local linear system of equations is completely uncoupled with that formed for its disjoint lines, we can group all lines of grid nodes into four colors with the requirement that all local systems formed for lines with the same color are disjoint. This can be utilized to parallelize or vectorize our algorithm. The correctness is verified by comparing with the analytical solution based on a three-layered model. The numerical performance is examined by comparing with other commonly used state-of-art solvers based on three increasingly more complex models, indicating the efficiency dominance, good parallelization and excellent ability on handling grid stretching for our algorithm.

Multigrid (MG) methods are among the best choice for stable and efficient three-dimensional (3D) forward modeling of electromagnetic (EM) fields over large area due to their linear dependence of computational time on grid size. However, as the frequency decreases to near zero and/or the grid is increasingly stretched, MG solvers for EM modeling based on the curl-curl equations converge slowly or even diverge. In this letter, we develop an efficient MG algorithm combined with a two-color plane Gauss-Seidel (GS) smoother for finite difference forward modeling of EM fields particularly at low frequencies. In this algorithm, we group different planes of the grid nodes into two colors. In each color, the components attached to different planes are totally decoupled and can be solved simultaneously, which can be distributed to different processors. The Dublin Test Model 1 is used to verify the accuracy of our algorithm and examine the numerical performance of our method against MG algorithms based on a four-color cell-block GS smoother and the Bi-Conjugate Gradient stabilized (BICGstab) smoother (as four-color cell-block GS MG and BICGstab-MG, respectively), and BICGstab and Quasi-Minimal Residual (QMR) both preconditioned with block incomplete lower-upper (blockILU) decomposition (as blockILU-BICGstab and blockILU-QMR, respectively). The numerical test based on OpenMP shows the good parallelization of our algorithm. Grids stretched to different degrees are designed to examine its ability to handle grid-stretching. The numerical performance comparison indicates its remarkable dominance in efficiency and stability.

We investigate an algorithm for the 3-D inversion of frequency-domain airborne electromagnetic (AEM) data based on the forward modeling and sensitivity calculation by footprint-guided compact finite element method (CFEM). Unlike the conventional approach, the modeling volume in our algorithm for each transmitter–receiver pair is a regular hexahedral that encloses the footprint, rather than a large mesh for the entire survey area or the local mesh with a number of grids extending from the footprint. After the electric fields in the modeling volume are solved by vector finite element method (FEM) with an integral equation boundary condition, the response and sensitivity are explicitly calculated by employing the product of the prepared Green’s functions and the vector of electric fields. The accuracy of this footprint-guided CFEM is validated by comparing it against conventional CFEM, and different synthetic models are tested by our inversion algorithm. The inversion tests of synthetic models show the feasibility of the combination of footprint-guided CFEM and Gauss–Newton optimization in recovering models within an acceptable error level, and the inversion results show a good agreement with the true models on both the model geometry and recovered conductivity.

Krylov subspace solvers for frequency-domain electromagnetic forward modeling problems converge remarkably more slowly as the period increases. In this article, we present an efficient four-color cellblock Gauss Seidel (GS) preconditioner for finite-difference (FD) electromagnetic modeling in geophysical applications. Rather than updating the FD electromagnetic (EM) equation edge by edge, as in a traditional GS scheme, we renew six edge components attached to one node simultaneously (i.e., in cellblock manner) effectively enforcing a local divergence free condition for currents. To improve implementation efficiency, we reorder the nodes on the FD grid into four colors so that nodes in each color are uncoupled, allowing the use of highly parallel vectorized algorithms. The four-color cellblock GS preconditioner is implemented in the MATLAB code, in conjunction with a BiCGstab solver. It is compared, in terms of iteration number and computing time, with other three commonly used preconditioners [GS, symmetric successive overrelaxation (SSOR) and incomplete lower and upper triangular matrix decomposition (ILU)] on three models-two synthetic and one modified from the version of real data inversion. The comparison indicates that the proposed algorithm is extremely stable and efficient compared with the other three preconditioners tested, over a range of periods (1-1000 s). Especially at long periods, the improvement of our proposed algorithm is substantial. In addition, a parallel implementation of the cellblock GS preconditioner is straightforward due to the independence of nodes in each color.

The airborne electromagnetic (AEM) method is an efficient tool for assessing conductivity structures near the earth's surface. The huge amounts of collected data over a survey area of tens to thousands of square kilometers result in an extremely high computational cost for rigorous modeling. Fortunately, for each transmitter and receiver (Tx-Rx) station, a volume of limited scale beneath the transmitter, called the footprint, contains the majority of the induced current and contributes most of the EM response at the receiver. In this letter, we develop a footprint-guided compact finite element method (CFEM), in which the inhomogeneous conductivity structure in the entire survey area is divided into small subareas based on the footprint so that the forward modeling for each subarea can be performed efficiently. The computational domain for every single Tx-Rx station consists of a small subarea and a surrounding layer. The accuracy of the algorithm is verified by comparing its solutions with semianalytical solutions on a layered earth model, and its applicability and efficiency are demonstrated with a more complex 3-D model consisting of a large inhomogeneous structure.

A label of an equation in the Four-Color Cellblock Gauss-Seidel Preconditioner section of the title article contains a writing mistake, so we are modifying it by: 1) changing “violation of (8)” to “violation of (6)” and 2) changing “free condition in (8)” to “free condition in (6).” This error does not affect the text or results presented in the article.