It is very useful for the exploration of metallic sulfide deposits to simulate the induced polarization (IP) effect in semi-airborne transient electromagnetic (TEM) data and analyze its response characteristics. To improve the computational efficiency and the ability to handle the complex models, we develop a fast 3D forward modeling algorithm for semi-airborne TEM method with IP effect directly in time domain based on the sum-of-exponentials (SOE) approximation and the time-domain unstructured vector finite-element method. By adopting the SOE method to approximate the kernel function of Caputo fractional derivative, the convolution operation in the time direction is transformed into a piecewise analytical solution and a recursive relationship. This approach resolves the problem of huge storage and calculation caused by the traditional L1 approximation relying on all historical information, thereby improving the efficiency of 3D forward modeling. Meanwhile, the flexibility of the unstructured finite-element method provides our algorithm with the ability to deal with the models with undulating terrain and complex-shaped bodies. The numerical accuracy and efficiency of our method are verified by comparing it with the 1D analytical solution on a polarized half space and the L1 approximation respectively. On this basis, we analyze the influence of transmitting-waveform parameters on the IP responses on a polarizable ellipsoid model. Finally, our algorithm is applied to a polarizable model with topography to check its ability to handle the complex model. We further analyze the influence of the observation mode and the direction of the transmitter sources on the IP responses.

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.

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.