For model reduction techniques, there have been relatively few studies performed regarding the forward modeling of anisotropic media in comparison to transient electromagnetic (TEM) forward modeling of isotropic media. The transient electromagnetic method (TEM) responses after the current has been turned off can be represented as a homogeneous ordinary differential equation (ODE) with an initial value, and the ODE can be solved using a matrix exponential function. However, the order of the matrix exponential function is large and solving it directly is challenging, thus this study employs the Shift-and-Invert (SAI-Krylov) subspace algorithm. The SAI-Krylov subspace technique is classified as a single-pole approach compared to the multi-pole rational Krylov subspace approach. It only takes one LU factorization of the coefficient matrix, along with hundreds of backward substitutions. The research in this paper shows that the anisotropic medium has little effect on the optimal shift |${\gamma _{opt}}$| and subspace order m. Furthermore, as compared to the mimetic finite volume method (SAI-MFV) of the SAI-Krylov subspace technique, the method proposed in this paper (SAI-FEM) can further improve the computing efficiency by roughly 13%. In contrast to the standard implicit time step iterative technique, the SAI-FEM method does not require discretization in time, and the TEM response at any moment within the off-time period can be easily computed. Next, the accuracy of the SAI-FEM algorithm was verified by 1D solutions for an anisotropic layer model and a 3D anisotropic model. Finally, the electromagnetic characteristics of the anisotropic anomalous body of the center loop device and separated device of the airborne transient electromagnetic method were analyzed, and it was found that horizontal conductivity has a considerable influence on the TEM response of the anisotropic medium.

Krylov subspace projection model order reduction technique for matrix function has proved efficient for the modeling of three-dimensional (3D) off-time transient electromagnetic (TEM) data. However, this method is time-consuming to compute full-time TEM response due to the time-dependent source terms. In this paper, we propose a new model reduction algorithm for fast computation of full-time TEM responses. First, the governing equations of full-time TEM are spatially discretized using a finite volume with octree meshes. The transmitter source is decoupled as the multiplication of a time-varying current term and a spatially distributed constant term. Then, a shift-and-inverse (SAI) Krylov subspace based on the governing equation coefficient matrix and the spatially distributed constant term of source is constructed. A reduced-order governing equations are obtained by projecting the governing equations into the SAI Krylov subspace. Finally, the full-time TEM response is achieved by solving the reduced-order governing equations. This algorithm only needs to construct a SAI Krylov subspace once, results in significantly faster solution times than previously proposed schemes. Numerical results for typical transmitting waveforms demonstrate that the novel algorithm is more than 100 times faster than previously proposed schemes without reducing the accuracy for full-time TEM modeling.

Transient electromagnetic (TEM) data are affected by resistivity anisotropy, which should be considered in 3D modelling. The influence of anisotropy on full-time response is the main focus of this research. For spatial discretisation of an anisotropic model, the mimetic finite volume approach was applied. The accuracy of the shift-and-invert (SAI) Krylov subspace approach and the two-step backward differentiation formula (BDF2) for modelling 3D full-time electromagnetic data has been demonstrated. However, both algorithms require time-consuming calculations. The SAI technique requires a number of projection subspace constructions, whereas the BDF2 algorithm necessitates numerous coefficient matrix decompositions. We proposed a novel mixed BDF2/SAI algorithm in this paper, which combines the advantages of the two algorithms. The on-time response is computed using BDF2, while the off-time response is computed using the SAI-Krylov subspace method. The forward results of a 1D model with a half-sine waveform demonstrated that the new algorithm is accurate and faster than both the BDF2 algorithm and the SAI algorithm. During the full-time period, the forward results of a 3D anisotropic model with half-sine waveform show that abnormal responses can be induced. It was shown that the relative abnormal of |${{{\bf b}}_{\boldsymbol{z}}}$| is higher during the on-time period, while the relative abnormal of |$\partial {{{\bf b}}_{\boldsymbol{z}}}/\partial t$| is higher during the off-time period. Furthermore, the change in relative anomaly is more obvious as the anisotropic block rotates around the x-axis. And the larger the rotation angle, the larger the relative anomaly.

We present a self-adaptive algorithm of inverse wavefield transform for the vertically magnetic components of the loop-source transient electromagnetic method (TEM) data to highlight the subsurface interfaces of electric resistivity. Based on the theory of wavefield transform between the electromagnetic (EM) diffusion and the EM wave, the transform formula of the TEM field and its time derivative are presented. The velocity of the pseudo-wavefield is given. To numerically accomplish the inverse wavefield transform in a finite interval, the shortest time window for the pseudo-wavefield is determined by a newly proposed formula. Then the pseudo-time window can be directly figured out according to the time wind of TEM measuring time. To solve the inverse wavefield transform equation accurately, the precise integral method (PIM) is introduced with an automatic iteration control scheme. In the numerical results, the pseudo-wavefield traces transformed from the modeled TEM responses are physically rational as the pseudo-wavefield reflection arrives as expected. The data misfit between the fit and the modeled data is smaller than 10% and is smaller than 1% for the TEM responses earlier than 1 ms. The results illustrate that the transformed pseudo-wavefield can highlight the location and features of the electric interfaces through the reflection arrival and waveform.

In this study, to obtain higher-resolution geoinformation in the electrical source transient electromagnetic method (TEM) interpretation, the pseudo-seismic Born approximation imaging method is applied to the vertical magnetic induction intensity ${B_z}$ and the horizontal electric field ${E_y}$, which is paralleled to the source. First, the differences between the two components are compared, and the results reveal that different components can reflect varied geological information. ${E_y}$ has a stronger signal and is more sensitive to the high-resistivity layer at a long offset, whereas ${B_z}$ has a greater relative difference in the low-resistivity layer at a short offset. The TEM field is then transformed into the pseudo-wavefield using the wavefield transformation method and the Born approximation pseudo-seismic imaging method is adopted to locate the electrical interface. After calculating the 3D model, the results indicate that the multi-component Born approximate imaging can intuitively and effectively locate the depth of the geological interface. Furthermore, the imaging results of a coal mine located in Gansu Province, China, proves that the proposed method can pinpoint the buried location of the coal goaf as well as the stratum trend. In general, the multi-component pseudo-seismic imaging technique plays a significant role in interpreting the electrical source TEM data.

For large-scale geophysical models, the order of the coefficient matrix in 3-D transient electromagnetics (TEMs) forward modeling can reach millions or even tens of millions. Balancing computational efficiency and memory usage presents a challenge worthy of in-depth exploration. In this letter, we utilize an integral representation of the iterative error in the Arnoldi method to construct an efficient quadrature-based restarted forward algorithm. First, the mimetic finite volume (MFV) method on a staggered hexahedral grid is employed to discretize the time-domain Maxwell’s equations, expressing the TEM response after the step-off waveform shutoff as the product of the matrix exponential function $f({\text {A}})$ and vector b. Then, using Cauchy’s integral formula, the expression of ${f}({\text {A}}){b}$ is transformed into an integral form and approximated using the restarted Arnoldi (RA) algorithm. Our method does not require solving linear systems and can leverage GPU parallel technology and optimize the RA algorithm parameters to enhance computational efficiency. Comparative studies with other numerical methods validate the advantages and accuracy of our approach, which numerical example demonstrates can fully achieve large-scale fast 3-D TEM forward modeling.