Mainstream numerical methods for 3D time-domain airborne electromagnetic (AEM) modeling, such as the finite-difference (FDTD) or finite-element (FETD) methods, are quite mature. However, these methods have limitations in terms of their ability to handle complex geologic structures and their dependence on quality meshing of the earth model. We have developed a time-domain spectral-element (SETD) method based on the mixed-order spectral-element (SE) approach for space discretization and the backward Euler (BE) approach for time discretization. The mixed-order SE approach can contribute an accurate result by increasing the order of polynomials and suppress spurious solutions. The BE method is an unconditionally stable technique without limitations on time steps. To deal with the rapid variation of the fields close to the AEM transmitting loop, we separate a secondary field from the primary field and simulate the secondary field only, for which the primary field is calculated in advance. To obtain a block diagonal mass matrix and hence minimize the number of nonzero elements in the system of equations to be solved, we apply Gauss-Lobatto-Legendre integral techniques of reduced order. A direct solver is then adopted for the system of equations, which allows for efficient treatment of the multiple AEM sources. To check the accuracy of our SETD algorithm, we compare our results with the semianalytical solution for a layered earth model. Then, we analyze the modeling accuracy and efficiency for different 3D models using deformed physical meshes and compare them against results from 3D FETD codes, to further show the flexibility of SETD for AEM forward modeling.

The spectral-element (SE) method, which is based on the Galerkin technique, has been gradually implemented in geophysical electromagnetic (EM) 3D simulation. The accuracy and efficiency of this approach, implemented for deformed hexahedral and regular meshes, has been verified for airborne EM forward modeling. One advantage of the SE method over the conventional finite-element method is that it provides accurate results for earth models that can be adequately parameterized using a coarse mesh. However, realistic models can contain important small-scale conductivity variations or larger features with complicated boundaries. To overcome the limitations imposed by using the same mesh to parameterize the model and for implementing the forward-modeling approach, we have developed an adaptation of the conventional SE method. This is inspired by the ideas behind the element-free Galerkin (EFG) method, in which the conductivity is no longer assumed to be constant within a cell; instead, it is handled via the same kind of numerical integration as in the EFG method. This allows a coarse, regular hexahedral mesh to be used for the forward modeling for complex earth models. After presenting the theory for this new SE approach, we test it for airborne EM modeling of 1D and 3D models to verify its flexibility and accuracy. Finally, we model the Ovoid Zone massive sulfide ore body located at Voisey’s Bay, Labrador, Canada, to illustrate the flexibility and practicality of our approach.

The transient electromagnetic method (TEM) is widely used in the exploration of mineral, petroleum, and geothermal resources due to its sensitivity to low-resistivity bodies, limited site constraints, and strong resistance to interference. In practical applications, the TEM often uses a long wire source instead of an idealized horizontal electric dipole (HED) source as the excitation source. This is due to the complex external conditions and the relatively large distance between the receiving zone and the transmitter source. Compared to the HED, the long wire source can provide a larger excitation current, generating stronger signals to meet the requirements of a higher signal-to-noise ratio or deeper exploration. It also produces longer-duration signals, thereby providing better resolution. Additionally, for the interpretation of TEM data, three-dimensional forward modeling plays a crucial role. However, the mature traditional TEM forward method is based on a simple, sometimes inappropriate model, as it is well established that the induced polarization (IP) effect is widely present in the deep earth, especially in oil and gas reservoirs. The presence of the IP effect results in negative responses in field data that do not conform to the traditional theoretical decay law of TEM, which can significantly impact data processing and inversion results. To address this issue, a TEM forward modeling method considering the IP effect based on the spectral element method (SEM) has been developed in this study. Firstly, starting from the Helmholtz equation satisfied by the time domain electric field, we introduce the Debye model with polarization information into the forward modeling by utilizing the differential form of Ohm’s law. As a result, we derive the boundary value problem for the time domain electric field that considers the induced polarization effect. Using Gauss–Lobatto–Legendre (GLL) polynomials as the basis functions, the SEM is employed to discretize the governing equations at each time step and obtain spectral element discretization equations. Then, temporal discretization equations are derived using the second-order backward Euler formula, and the linear system of equations is solved using the Pardiso direct solver. Finally, the electromagnetic responses at any time channel are obtained via SEM interpolation and numerical integration, thereby achieving three-dimensional TEM forward modeling considering the IP effect. The results indicate that this method can effectively reflect the spatial distribution of polarizable subsurface media. It provides valuable references for studying the polarization parameters of subsurface media and performing a three-dimensional inversion of TEM data considering the induced polarization effect.

Airborne electromagnetic (AEM) surveys using airborne mobile platforms enable rapid and efficient exploration of areas where groundwork is difficult. They have been widely used in fields such as shallow resource exploration and environmental engineering. Three-dimensional AEM inversion is the main technique used in fine structural interpretation. However, most current methods focus on separate component data inversions, which limit the kinds of structures that can be recovered in the inversion results. To address this issue, a method for the robust 3D joint inversion of multicomponent frequency-domain AEM data was developed in this study. First, a finite element method based on unstructured tetrahedral grids was used to solve the forward problem of frequency-domain AEM data for both isotropic and anisotropic media. During inversion, a limited-memory quasi-Newton (L-BFGS) method was used to reduce the memory requirements and enable the joint inversion of large-scale multicomponent AEM data. The effectiveness of our algorithm was demonstrated using synthetic models for both isotropic and anisotropic cases, with 5% Gaussian noise added to the modeling data to simulate the measured data for separate and joint inversions. The results of the synthetic models show that joint inversion has advantages over separate inversion in that it enables the recovery of finer underground structures and provides a novel approach for the fine interpretation of frequency-domain AEM data.

The transient electromagnetic (TEM) method is a key detection and monitoring technology for safe coal-mine production. Surface TEM depth penetration is limited by real geological conditions and transmitter–receiver hardware performance. Compared with the surface TEM method, the tunnel TEM method can enhance the depth of exploration to some extent, but it is constrained by the limited working space of the roadway, which makes it difficult to perform the area-wide and multi-line data acquisition, and thus the lateral detection resolution is directly compromised. Consequently, either surface or tunnel TEM alone suffers inherent limitations. The multidimensional surface and surface-to-tunnel TEM method employs a single large-loop transmitter and records electromagnetic (EM) signals both on the surface and in the tunnel, enabling joint data interpretation. The joint TEM observation method effectively addresses the limitations by using a single observation mode, with the goal of achieving high-precision detection. To investigate the detection capabilities of the joint surface and surface-to-tunnel TEM method, we propose a three-dimensional (3D) joint surface and surface-to-tunnel TEM forward modeling method based on the spectral element method (SEM). The SEM, using high-order vector basis functions, enables high-precision modeling of TEM responses with complex geo-electric earth models. The accuracy of the SEM is validated through comparisons with one-dimensional (1D) TEM semi-analytical solutions. To further reveal TEM response characteristics and multi-dimensional resolution under joint surface and tunnel detection modes, we construct several typical 3D geo-electric earth models and apply the SEM algorithm to simulate the TEM responses. We systematically analyze the horizontal and vertical resolution of 3D earth model targets at different decay times. The numerical results demonstrate that surface multi-line TEM surveying can accurately delineate the lateral extent of the target body, while vertical in-tunnel measurements are crucial for identifying the top and bottom interfaces of geological targets adjacent to the tunnel. Finally, the theoretical modeling results demonstrate that compared to individual TEM methods, the multi-dimensional joint surface and tunnel TEM observation yields superior target spatial information and markedly improves TEM detection efficacy under complex conditions. The 3D TEM forward modeling based on the SEM provides the theoretical foundation for subsequent 3D inversion and interpretation of surface-to-surface and surface-to-tunnel joint TEM data.

Spectral-element (SE) method is a kind of higher-order finite-element method based on weighted residual technique; however, the basis functions for SE are polynomial, like Gauss–Lobatto–Legendre (GLL) or Gauss–Lobatto–Chebyshev (GLC) polynomials. Because of its high modeling accuracy and flexibility, it has been successfully used in computational electromagnetism. In this paper, we use the SE method for 3D frequency-domain airborne electromagnetic (AEM) modeling for an anisotropic earth and we take horizontal coplanar and vertical coaxial coil systems as example for the modeling. We first derive the discrete governing equation from Maxwell equations, in which the conductivity tensor is obtained by 3 Euler rotations of a principal conductivity tensor. GLL polynomial is selected as the vector SE basis functions, while GLL integration is applied for calculating matrix elements. A direct solver is used for the solution of the matrix equations system. The modeling accuracy is checked against a semi-analytical solution. Further, we calculate AEM responses for different anisotropic models and demonstrate that SE method canobtain high precision by either increasing SE order or refining meshes, so that it can save computation cost vastly. Numerical results further confirm that the anisotropy of both 3D body and host rock can be identified from the polar plots of ratio of magnetic field components.