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.

The coupling of induced polarization (IP) effects and electromagnetic induction significantly complicates the transient electromagnetic (TEM) diffusion, leading to serious distortion of the observed responses. In this article, we develop a 3-D forward modeling algorithm based on a goal-oriented adaptive finite-element method for the semi-airborne TEM method with IP effects to obtain high-precision simulation results. First, the Caputo fractional derivative is discretized by piecewise linear interpolation, and the Cole–Cole model is wholly introduced into the time-domain finite-element governing equation. Then, we simulate the semi-airborne TEM responses with IP effect directly in time domain by solving this governing equation. Because the Caputo fractional derivative depends on all historical information, it leads to a large storage and calculation. In order to solve this problem, we adopt the piecewise local time step discretization strategy to reduce the storage occupation and improve the computational efficiency. Finally, combined with the goal-oriented adaptive grid refinement technology, the forward modeling mesh is automatically optimized based on the weighted hybrid posterior error estimations. The accuracy of our codes is verified by comparing them with 1-D analytical solutions. We further simulate the electromagnetic responses of the polarizable body model with complex shape and topography to analyze the influence of the posterior error estimation method and the IP effects on the adaptive mesh.

The Bayesian inversion of electromagnetic data can obtain key information on the uncertainty of subsurface resistivity. However, due to its high computational cost, Bayesian inversion is largely limited to 1-D resistivity models. In this study, a fast Bayesian inversion method is implemented by introducing the spatial correlation as prior information. The contributions of this article mainly include: 1) explicitly introduce the expression of spatial correlation prior information and provide a method to determine the parameters in the expression through the variogram theory. The influence of parameters in the spatial correlation prior information on the inversion results is systematically analyzed with the 1-D synthetic model. 2) The information entropy theory of continuous functions is introduced to quantify the degrees of freedom (DOF) of the parameters of the spatial correlation prior model. The analysis shows that the DOF of model parameters are significantly smaller than the number of model parameters when spatial correlation prior information is introduced, which is the main reason for the rapid Bayesian inversion. 3) Introducing the Sengpiel fast imaging algorithm, combined with the variogram theory, realized the direct acquisition of spatial correlation prior information from the observation data, minimizing the dependence on other information. The inversion results of 1-D and 2-D synthetic models and field datasets show that considering the spatial correlation prior information, hundreds of thousands of Markov chain Monte Carlo sampling steps are needed to enable the inversion of up to thousands of model parameters. This result provides a possible idea for future Bayesian inversion of complex 3-D models.

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.

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.

In this paper, we set up an efficient Fréchet derivative algorithm of inversion of multicomponent induction logging (MCIL) data in horizontal layered transversely isotropic (TI) formations based on transmission line method (TLM) in order to simultaneously reconstruct the model vector including both horizontal and vertical conductivities, horizontal interfaces, borehole dipping angle, and tool azimuth in deviated well from the MCIL data. First, MCIL responses in the TI model are efficiently determined by both the spectrum EM fields obtained by TLM and the semianalytical computation of Sommerfeld integrals based on the cubic spline interpolation. Then, the Born approximations are executed to derive the MCIL Fréchet derivatives as the sixfold integrals of the products of two spectrum EM fields in infinite domains. By using the integral characters of the 2-D Dirac function, the sixfold integrals of Fréchet derivatives are further simplified into twofold integrals: One is the Sommerfeld integral in the radial spectrum domain, and the other is the definite integral in the vertical spatial domain. They are efficiently computed by the semianalytical approach similar to the MCIL simulation. Therefore, we obtain the linear equations between changes in the MCIL responses and small perturbations in the model vector. After that, we iteratively modify all of the model parameters to realize the best fit between the input data and the synthetic data of the inverted model by the normalization of the Fréchet derivative and singular value decomposition technique. Finally, we apply numerical results to investigate the characteristics of the Fréchet derivatives and to validate the inversion method and its antinoise ability.

There are a lot of handovers failed in high-speed mobile environment because the handover can't be completed in time although the field strength values are sufficient availability along the tracks. A mobile relay based fast handover scheme is proposed which is suitable for high-speed mobile environment. Two reference points are introduced to ensure handover in time. Pre-preparation and packet bi-casting is introduced to reduce communication interruption time and realize seamless handover. The performance of the proposed scheme is analyzed in terms of handover delay, communication interruption time and bi-casting time.