The inverse Laplace transform is one of the methods used to obtain time-domain electromagnetic (EM) responses in geophysics. The Gaver-Stehfest algorithm has so far been the most popular technique to compute the Laplace transform in the context of transient electromagnetics. However, the accuracy of the Gaver-Stehfest algorithm, even when using double-precision arithmetic, is relatively low at late times due to round-off errors. To overcome this issue, we have applied variable-precision arithmetic in the MATLAB computing environment to an implementation of the Gaver-Stehfest algorithm. This approach has proved to be effective in terms of improving accuracy, but it is computationally expensive. In addition, the Gaver-Stehfest algorithm is significantly problem dependent. Therefore, we have turned our attention to two other algorithms for computing inverse Laplace transforms, namely, the Euler and Talbot algorithms. Using as examples the responses for central-loop, fixed-loop, and horizontal electric dipole sources for homogeneous and layered mediums, these two algorithms, implemented using normal double-precision arithmetic, have been shown to provide more accurate results and to be less problem dependent than the standard Gaver-Stehfest algorithm. Furthermore, they have the capacity for yielding more accurate time-domain responses than the cosine and sine transforms for which the frequency-domain responses are obtained by interpolation between a limited number of explicitly computed frequency-domain responses. In addition, the Euler and Talbot algorithms have the potential of requiring fewer Laplace- or frequency-domain function evaluations than do the other transform methods commonly used to compute time-domain EM responses, and thus of providing a more efficient option.

A finite-element time-domain (FETD) electromagnetic forward solver for a complex-shaped transmitting loop is presented. Any complex-shaped source can be viewed as a combination of electric dipoles (EDs), each of which can be further decomposed into two horizontal EDs along the x- and y-directions, and one vertical ED along the z-direction. Using this method, a complex-shaped loop can be easily handled when implementing a FE method based on the total-field algorithm and an unstructured tetrahedral mesh. The FETD solver presented here used a vector FE method and the first-order backward Euler method to discretize in space and time, respectively. Unstructured tetrahedral girds combined with a local refinement technique was used to exactly delineate topography and a deformed loop. This FETD solver was tested by the five following scenarios: a rectangular loop on a flat-surface half-space, a circular loop on a stratified-medium, a rectangular loop laid on a slope-surface half-space, a rectangular loop laid on a slope with a conductive cubic body, and a complex-shaped loop on a real-life topography. The results of this FETD solver agreed well with the ones evaluated by the analytic methods for the first three examples, and with a frequency-domain FE solver combined with a cosine transform for the last two examples.

In controlled-source electromagnetic (CSEM) inversion with conventional regularization, the reconstructed conductivity image is usually blurry and only has limited resolution. To effectively obtain more compact conductivity models, we apply the concept of multinary transformation to CSEM inversion based on the finite-element method with an unstructured tetrahedral mesh. Within the framework of multinary inversion, the model conductivities are allowed to be clustered only within the designed values, which are usually obtained from other a priori information or from conventional inversion. Synthetic studies show that multinary inversion produces conductivity images with clearer model boundaries compared to the maximum smoothness inversion and the focusing inversion for realistic geoelectric models. We further apply the developed method to a land CSEM survey for mineral exploration. The multinary inversion results are closer to the ground truth compared to the conventional maximum smoothness inversion and the focusing inversion. The developed method and the numerical algorithm provide a new approach and workflow for CSEM inversion when the models need to have clear boundaries and clustering model values. Such geoelectric models could be very useful for geologic interpretation in oil and mineral exploration when a priori information (such as the estimated conductivity values) of the exploration targets is known.

The correct interpretation of time-domain or transient electromagnetic (TEM) data relies on effective inversion techniques. Herein, we have developed an effective 3D-TEM inversion method based on the vector finite-element method. The domain was discretized with unstructured tetrahedra to simulate complex geologic structures. We develop a second-order backward-Euler scheme, rigorously with nonuniform time-step sizes, to approximate the time derivative of the electric field. The waveform was directly simulated and considered in the forward modeling and inversion. In a 3D inversion, the most time-consuming element of the workflow is solving a large system of linear equations. To alleviate this problem, we adopt a parallel direct solver for solving a large system of equations involved in forward modeling and sensitivity calculations. We use the Gauss-Newton approach to solve the inverse problem and formulate the least-squares problem to obtain model updates. We avoided the explicit sensitivity calculation by solving the matrix-vector multiplication between the sensitivity matrix (or its transpose) and vector. The developed inversion algorithm is validated using several realistic synthetic models. Finally, we apply the method to practical TEM data with complex topography for coal mine goaf detection. We thus obtain a detailed 3D resistivity model that revealed the layering structure and some localized anomalies related to the coal mine goaf.

In transient electromagnetic (TEM) methods, the full transmitting-current waveform, not just the abrupt turn-off, can have effects on the measured responses. A 3D finite-element time-domain forward-modeling solver was used to investigate these effects. This was motivated by an attempt to match, via forward-modeling, real data from the Albany graphite deposit in northern Ontario, Canada. Initial modeling results for homogeneous half-spaces illustrate the effects that a full waveform can have on TEM responses, especially the durations of the steady stage and turn-off time. For the Albany data set, a geophysical conductivity model was developed from a geologic model that itself had been constructed predominantly from drillhole information. The conductivities of the various geologic units in the model were first estimated based on typical conductivity values for the respective rock types, then adjusted to fit the measured TEM data as closely as possible. We found that the TEM responses differed significantly from the pure step-off response and that incorporating the effects of the full waveform (particularly the linear ramp turn-off) greatly improved the match between observed and computed responses, especially for the early measurement times. In addition, this Albany example illustrates the presence of sign changes in TEM data caused primarily by localized conductivity targets.

Physics-informed neural networks (PINNs) have recently been utilized to tackle wave equation-based forward and inverse problems. However, they encounter challenges in accurately predicting the high-frequency wavefields, known as the spectral bias problem. Based on the previously used frequency upscaling (FU) and neuron splitting (NS) concepts to help with the high frequency wavefields, we present a sequence of strategies (i.e., multiscale Fourier feature mapping (MFFM), frequency transferring (FT), a revised NS (RNS), and denser sampling (DS)) to overcome the spectral bias challenge of PINN in solving the frequency-domain acoustic wave equation. MFFM projects PINN inputs onto sinusoids with off-axis frequency distributions, characterizing the PINN wavefield by a Fourier decomposition. FT initializes the NN for higher-frequency (e.g., 16 Hz) using the pre-trained NN for the nearest lower-frequency (e.g., 15 Hz), which is refined from FU (where the 16 Hz NN is initialized by that of 8 Hz). Thus, we show that FT is more effective than FU. We also introduce RNS, which is a modified strategy derived from NS by adding a small amount of random noise, drawn from a uniform distribution, to the replicated weights/biases to break the symmetry after replication, improving the NN’s representational capacity. Because MFFM, FT, and RNS alone are inadequate for generating high-frequency wavefields, we further develop DS to use a smaller spatial sampling interval within the same computational domain (the wavefield is more continuous and smoother). DS helps PINN simulate high-frequency components easier. The experiments on typical models validate the effectiveness of the introduced strategies. We share the codes and data through a public repository.