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 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.

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 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 paper, we develop a 3D 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. Firstly, 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 1D 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.

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.