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.

The induced polarization (IP) effect is an important phenomenon in geophysical exploration. Understanding the impact of the IP effect on transient electromagnetic (TEM), it is necessary to consider chargeability during the forward modeling process. Currently, most 3-D forward modeling for the IP effect relies on frequency-to-time-domain transformation techniques, using the Cole-Cole model for frequency-domain computations. Although this method is straightforward, it has low computational efficiency. Based on this, we propose a model order reduction method based on the rational Krylov subspace to achieve rapid 3-D TEM modeling for multiple sets of IP parameters. Specifically, using the finite volume method (FVM) with Octree mesh for spatial discretization of the governing equations, we transform the equations into a transfer function independent of frequency and Cole-Cole parameters. Then, the rational Krylov subspace projection method is used for model order reduction, where the large-dimensional coefficient matrix in the transfer function is reduced to a smaller dimensional matrix, allowing for fast solution of the transfer function. This transfer function is used for modeling at different frequencies and for different Cole-Cole parameters. The accuracy of our algorithm is validated by comparing the results with analytical solutions. In addition, forward modeling analysis of multiple sets of Cole-Cole parameters demonstrates the significant computational efficiency of our algorithm, indicating its suitability for forward modeling with various sets of Cole-Cole parameters.

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.

The topography has serious effects on time-domain airborne electromagnetic (AEM) signal, and the EM responses resulted from the topography frequently overwhelm those from the underground abnormal bodies. This brings big challenges to the traditional AEM interpretations based on a flat ground model. In this article, we develop a 3-D AEM inversion algorithm for a topographic earth model. The time-domain finite-element algorithm based on unstructured mesh is used to model the AEM responses. The tetrahedral grids provide the flexibility to fit the rugged topography. Furthermore, we adopt the Gauss–Newton method for our inversion of time-domain AEM data. In the forward modeling and the calculation of Jacobian matrix, we introduce an unstructured local mesh and decouple the meshes for forward modeling and inversion to improve the computational efficiency. For that purpose, we first set up an unstructured inversion mesh and then extract those cells corresponding to the sensitive area of AEM system for each survey station from the inversion mesh and construct a local forward mesh with the extracted cells as the core. After that, we take advantage of the spatial relationship between the local forward meshes and the inversion ones to set up the global sensitivity matrix for Gauss–Newton inversion. We test the effectiveness of our algorithm by applying our 3-D inversion code to both synthetic and survey data. The numerical experiments show that the Earth topography can have big influence on AEM inversions, and ignoring the topography can create serious distortion to AEM inversion results.

The transient electromagnetic (TEM) method is widely used in near-surface geophysical prospecting. The high-precision forward and inversion of large-scale complex models is a challenging problem. It is difficult to solve the large-scale problems using the direct solvers due to the large memory requirements. In this article, we present a restarting polynomial Krylov method for modeling 3-D large-scale TEM responses. The mimetic finite volume method is carried out for spatial discretization. The step-off TEM response then can be expressed as a matrix exponential function. The restarting polynomial Krylov method is used to solve the matrix exponential function. For a given restart subspace dimension, the residual is used to obtain the forward response at any time that meets the given accuracy. This method does not need to solve large-scale linear equations. Furthermore, the memory usage is mainly determined by the number of spatial discrete grids and the restart subspace order. The numerical experiments of large-scale model demonstrated that the method is accurate and uses limited memory.

Unexploded ordnance (UXO), which causes many civilian casualties every year, has become a serious environmental problem. To deal with the problem, we have developed a new drone-based transient electromagnetic (TEM) system, which was designed for the UXO detection through very low altitude measurements. Drone-based TEM system uses the rotorcraft equipped with central loop TEM device to realize the UXO detection. The system was more safe and efficient than a ground-based TEM system in UXO detection. Compared with the airborne magnetic method or helicopter-borne TEM system, it has the advantages of low cost, flight safety, and so on. In this letter, the system characteristics are introduced in detail, providing the theoretical analysis of the system in UXO detection and verification of model data. Finally, through the UXO detection in the former weapon test base, the results show that the system is effective and successful in UXO detection.

The 3-D time-domain electromagnetic (TEM) modeling is computationally expensive. Shift-and-invert (SAI) Krylov subspace technique has proved efficient for the modeling of OFF-time TEM data. However, the ON-time data, which also involves useful Earth conductivity information, have not been well treated. In this article, we extend the SAI Krylov subspace method to deal with the ON-time data and obtain a complete algorithm for modeling of the full-time data. To deal with the ON-time problem, while retaining the whole framework of an SAI-style method, we have developed a new time integration method, based on the exponential trapezoidal rule, to help discrete the integral terms into matrix exponential function. Next, the solution of this matrix exponential function is obtained by constructing a new type of SAI Krylov subspaces. Numerical results for typical transmitting waveforms, such as half-sine, versatile-time-domain-electromagnetic (VTEM), and MULTIPULSE, demonstrate that the novel algorithm is accuracy for full-time TEM modeling.