As the structure of the underground space becomes increasingly complex, traditional 2-D seismoelectric methods are no longer adequate for comprehensive exploration. To achieve precise imaging of the underground space, there is an urgent need to develop 3-D full-waveform modeling techniques. In this article, we propose a 3-D time-domain finite-element method to solve the seismoelectric wavefield in saturated porous media. Since the electroosmotic feedback is very small, we can ignore the mechanical disturbance caused by the electromagnetic (EM) fields induced by seismic waves and, thereby, can decouple the electrokinetic coupling equations and separately solve the seismic and EM waves. For the simulation of seismic wavefield, we employ the explicit finite-element method and utilize a lumped mass matrix instead of a consistent mass matrix to facilitate explicit recursion. In addition, we apply the complex frequency-shifted unsplit perfectly matched layer technique to effectively handle seismic boundary conditions. Then, the velocity fields obtained by solving the poroelastic equations serve as the source term of the EM equations, and the finite-element method is used to solve the EM wavefield. Considering that the huge velocity difference exists between the EM and seismic waves, we adopt an unconditionally stable implicit method for the solution of the EM wavefield. By combining explicit and implicit recursion, the computational efficiency can be improved significantly. The accuracy of our time-domain finite-element algorithm is validated by checking our results against the analytical solutions for a half-space model. Furthermore, we conduct numerical simulations and analyses on a typical block model and a modified SEG/EAEG salt dome model.

We present an efficient method for uncertainty quantification in 3D magnetotelluric (MT) inversions based on variational inference principles and the variational autoencoder (VAE) framework. In this approach, we replace the VAE’s encoder and decoder respectively with a forward modeling and a 3D optimization module, where the latent variables represent updates in the constrained model space. Utilizing reparameterization, the gradient of objective function can propagate back to the mean and variance of the latent variables, and thus restores the Gaussian distribution of resistivity models. By setting appropriate prior and initial distributions, our method explores the solutions with high log-variance to fit the data. To ensure a stable convergence, we apply a smooth constraint on the log-variance of latent variables and employ a modified adaptive moment estimation (Adam) optimizer. Numerical experiments confirm that our method can accurately estimate both the resistivity model and standard deviation at a time cost only 2-3 times that of conventional inversion, which provides a highly efficient solution for uncertainty analysis. The tests with varying regularization parameters reveal that the standard deviation estimates are sensitive to both the Kullback-Leibler (KL) divergence and log-variance smoothness terms, while the adaptive smoothing strategy can well balance these effects. The application to a dataset from USArray survey in the Northwestern US demonstrates the method can achieve robust inversion and uncertainty quantification.

Magnetotelluric (MT) and teleseismic tomography are critical techniques in deep Earth exploration for geodynamic studies, internal energy circulation, plate tectonics, volcanic systems and so on. However, due to the uneven data distribution, the inherent non-uniqueness of inversion, and differing sensitivities to subsurface media, the velocity and resistivity structures derived from independent inversions often exhibit substantial discrepancies and contradict conventional geological understanding. To address these issues, we propose a novel three-dimensional (3D) joint inversion method for MT and teleseismic traveltime data, applicable to different types of grid discretization. The method first establishes a parameter mapping for different grid types. Then, a joint inversion framework based on the local Pearson correlation constraints (LPCC) is developed using a virtual grid technique. The inversion alternately updates the resistivity and velocity models by ensuring structural similarity between them. Numerical experiments demonstrate that the proposed method can effectively integrate the advantages of both techniques, enhance resolution and stability. Additionally, the tests on hyperparameter selection provides guidance for optimizing joint inversion parameters. Finally, the developed joint inversion method is applied to MT and teleseismic traveltime data in southeastern Australia and yields a constrained 3D resistivity and velocity model of the region. The results offer significant theoretical and practical insights into the lithospheric structure, the magma transport pathways, and the geodynamic processes in this area.

Localized water-rich areas in aquifers can cause severe accidents during coal mining, including property damage and casualties. Traditional ground-based geophysical methods often struggle in the mountainous terrains where coal mines are typically located. To address this, we applied a high-resolution exploration and interpretation strategy based on advancements in semi-airborne transient electromagnetic (SATEM) technology, integrating drilling and logging to detect water-rich areas in the Huangling coal mine. Our approach involved acquiring high signal-to-noise ratio electromagnetic (EM) data near the transmitting line source, using unstructured tetrahedral meshes to simulate complex topography, and employing a quasi-Newton optimization algorithm for detailed 3-D inversion. Synthetic tests demonstrate the accuracy of this method in resolving underground conductivity structures, even in challenging terrains. Field data inversion showed a good fit with observed data and good agreement with resistivity logging, confirming the reliability of the results. This study not only addresses the critical need for efficient water hazard detection in coal mining but also offers significant potential for mineral exploration, geological surveying, and disaster prevention.

The conventional magnetotelluric (MT) inversions rely primarily on L2norm to quantify the misfit between the observed data and the predicted ones. However, it often suffers from issues of unreasonable weighting of inversion data, sensitivity to outliers, and poor resolution to weak anomalies. To address these limitations and achieve better data fitting across varying scales, we propose to replace the L2norm with the Wasserstein distance to measure the distributional discrepancy between the predicted and observed data. This method adopts the Wasserstein Generative Adversarial Network (WGAN) framework and utilizes the Kolmogorov-Arnold Network (KAN) as the discriminator to implement the Wasserstein metric. By substituting the neural generator in WGAN with a physics-based forward modeling, we achieve a purely physics-driven iterative process that is similar to the conventional MT inversion workflows to ensure an adequate data fitting. Numerical experiments on synthetic models demonstrate that MTGAN-KAN can better fit data distributions and yield higher-resolution inversion results than the conventional methods. Compared to WGAN that employs a fully connected neural network (FCNN) as the discriminator, the incorporation of KAN can significantly improve the convergence and stability of MTGAN. An inversion test using MT data from Newer Volcanic Province in Australia further highlights the superior characteristics of data fitting and resolution of MTGAN-KAN in the inversion of MT data.

Induced polarization (IP) effects can distort late-time transient electromagnetic (TEM) signals, sometimes even leading to their sign reversal. These distortions are closely linked to subsurface IP effects. However, the significantly different sensitivities of the IP parameters often result in strong non-uniqueness in TEM data inversion. Here we present a three-dimensional (3-D) TEM-IP inversion method that incorporates local Pearson correlation constraints (LPCC). By establishing LPCC between the resistivity and the IP parameters, we achieve a joint inversion framework that stabilizes the estimation of these parameters. Numerical experiments on synthetic models demonstrate that, compared with unconstrained inversion, our method yields results that can more accurately recover the true distributions of chargeability, time constant, and frequency-dependent coefficient. We further investigate the impact of the size of correlation window in LPCC domain, and achieve practical guidance for optimal constraint selection. Finally, we apply the proposed method to a TEM dataset collected from a bauxite exploration site in Guangxi Province, Southern China, to validate our method for real-world geological settings.

Semi-airborne transient electromagnetic (EM) method is an effective exploration tool in complex environment due to its utilization of unmanned aerial platforms for data collection. With the escalating demands of detection accuracy, the conventional one-dimensional (1D) inversion is no longer sufficient. Over the last two decades, there has been rapid development in three-dimensional (3D) EM inversions. For semi-airborne transient EM method, it usually has a substantial number of receiver stations. To obtain accurate 3D solutions, one has to refine the grids near the receiver points, which causes huge number of grids and reduces computational efficiency. Thus, the computational complexity is one of primary factors restricting the practicality of 3D inversions. In this paper, we have developed an approach using decoupled meshes. This method uses a serial of meshes to parallelly calculate the forward modeling and Jacobian information, with one mesh containing only a subset of receiver points. This scheme aims to decrease the number of grids and improve the efficiency. After that, we map the results to an overarching inversion mesh using the node cloud technique and renew the model parameters. We check our algorithm via both synthetic and field data. Numerical experiments show that the decoupled mesh inversion method can effectively recover the location and resistivities of the anomalies under the complex topography. When compared with the conventional inversion method, the decoupled mesh inversion method can significantly reduce the calculation time.

At present, the mainstream technology for leachate detection in landfills is electrical resistivity tomography (ERT), known for its efficiency and nondestructive nature. However, the conventional ERT data interpretation primarily uses inversion based on structured grids, which cannot accurately simulate the complex and thin impermeable layers of landfills, leading to unreliable results. To address this issue, we propose a novel ERT observation system and a new 3-D inversion technology. In our observation system, all measuring electrodes are placed around the landfill at once, and only a limited number of transmitting sources are needed to sequentially inject current, which effectively reduces the time for data acquisition. For 3-D inversions, we employ an unstructured tetrahedral grid for fine discretization of structures at various scales. The node-based finite-element method is used for high-precision forward and adjoint forward calculations, while the gradient filtering method in combination with limited-memory quasi-Newtonian (L-BFGS) algorithm is used to update the inversion model. Numerical experiments indicate that the proposed method can mitigate the influence of thin impermeable landfill layers and provide more accurate imaging results compared to the conventional methods. In addition, we also test the effects of water-bearing layers, faults, and near-surface interferences on the leakage inversion results. The results show that the proposed method can achieve reliable high-resolution imaging under various conditions, making it an effective technique for landfill leachate detection.

The choice of data misfit measure has a great impact on the convergence of electromagnetic inversion. The conventional measure based onl2-norm tends to excessively amplify the weights of a larger misfit, inadvertently neglecting data with a smaller misfit during the inversion process, thereby diminishing the resolution to a certain degree. To solve this problem, we propose a robust inversion strategy based onl1-norm data misfit and adaptive moment estimation (Adam). In this scheme, we use the Ekblom-typel1-norm to simplify the derivative computation of the absolute value function. The Adam algorithm is further applied to optimize this type of non-smooth objective function, which incorporates momentum terms and adaptive steps, allowing it to better adapt to the irregularities in gradient changes. The inversion results obtained from both synthetic models and field measurements demonstrate that the Adam method performs considerably better than the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method for optimizing thel1-l2norm of objective function. Compared with the conventionall2-norm data misfit, thel1-norm data misfit can effectively avoid excessive optimization of data with large misfits and achieve high-resolution inversion results.

Airborne electromagnetic (AEM) technology is an efficient geophysical exploration tool for investigating subsurface electrical structures. In recent years, 3-D inversion of AEM data has been developed rapidly, but it still faces challenges such as low resolution and computational efficiency. To solve these problems, we propose a multiscale shearlet-based regularization inversion algorithm by establishing the relationship between spatial resolution and shearlet coefficients in the inversion process. In the initial stage of inversion, the coarse grids and sparse measurement points data are used to recover the main subsurface structure. When the data misfit reaches a certain level, the previous results are used as the coarse scale model in the shearlet domain to recover the model with fine grids and dense measurements. By building this coarse-to-fine inversion scheme, we can well utilize the multiscale information in AEM data and effectively achieve high-resolution inversions. We demonstrate the effectiveness and practicality of our 3-D MS inversion algorithm using two synthetic examples and a field dataset from Norway. The numerical experiments show that our inversion method can effectively reduce the computational time and improve inversion resolution.

Satellite magnetic data contain significant information about the Earth’s interior electrical structure. However, the altitudes of satellites vary over time and latitude. Theoretically, the signals from the external magnetosphere and ionosphere, along with the induced magnetic field from the Earth, exhibit considerable variation at different altitudes. The specific effects of altitude variations on the global electromagnetic induction signal, and their subsequent impact on the transfer function, are not yet fully understood. Given that the orbital altitude of the Challenging Minisatellite Payload (CHAMP) satellite changes continuously, we have calculated the Q-response and the C-response at different altitudes through numerical simulations and CHAMP satellite data, respectively. The Q-response represents the ratio of coefficients of the internal and external fields, while the C-response, derived from the Q-response, is altitude-dependent. The findings indicate that the amplitudes of the induced (internal) magnetic field decrease with increasing altitude, with discrepancies exceeding 10 nT, particularly in mid-latitude regions. Conversely, the inducing (external) fields increase with altitude. The Q-responses exhibit distinct behaviors across different degrees of Gaussian spherical harmonics (SHs). Specifically, the real part of the orbital C-response increases with satellite altitude, although higher degrees tend to yield lower magnitudes. This study offers insights into the design of satellite orbits in magnetic field measurements.

The conventional geo-electromagnetic data inversions are mostly based on gradient optimization methods. However, this type of method can only provide a single “optimal” inverse model under specific prior conditions, which cannot effectively evaluate the reliability and uncertainty of the inversion results. The widely used uncertainty quantification (UQ) methods are based on the theory of Bayesian inference. Although they have achieved success in many applications, they suffer from the curse of dimensionality and low efficiency. To overcome these problems, we propose a novel UQ strategy for geo-electromagnetic inversions based on Bayesian processes and surrogate modeling with adaptive deep neural network (DNN). In this method, an embedded DNN is used for forward modeling in the Bayesian inference to improve computational efficiency. The training of the DNN is divided into two stages. First, a predesigned small training set is used and the resulting DNN only gives a low-accuracy result. Second, this DNN is fine-tuned dynamically during the Metropolis-Hastings (M-H) sampling process, in which the training set is adaptively supplemented according to the modeling errors. Compared to the conventional data-driven approach, this dynamically adaptive constructing method of the training set can greatly reduce the training set and constantly maintain high accuracy in forward modeling. We demonstrate the effectiveness and practicality of our surrogate modeling Bayesian and analyze the effects of different sampling numbers, noise levels, prior distributions, and sampling radius. Compared with Occam’s inversion and conventional Bayesian inversions, our method shows good robustness and high accuracy, making it an effective Bayesian inversion technique.

Previous works have demonstrated that inverting magnetotelluric (MT) data jointly with gravity data can synergize the high lateral resolution of gravity and the vertical resolution of MT. However, the existing joint stabilizers usually work for structured grids instead of unstructured ones that are more powerful for characterizing complex geology. Here, we utilize the local Pearson correlation coefficient (LPCC) for the joint inversion of gravity and MT data based on unstructured grids. We first establish a background mesh by discretizing the research area into virtual rectangular grids and then enhance the structured similarity between density and resistivity via the LPCC. Compared to existing joint constraints, our method is more flexible in solving multiscale joint inversions thanks to the adjustable subdomain size. The synthetic experiments show that the joint stabilizer can recover the subsurface targets at a higher resolution, especially for gravity data, than the standalone inversions. This method is further applied to the joint inversion of gravity and MT data from the Yellowstone area and the inverted density and resistivity models are structurally consistent. Then, based on the inverted subsurface structures and incorporating the existing research, we infer that the two inverted zones with low density and low resistivity correspond to the partially molten rhyolitic and basaltic, respectively. The proposed joint constraint can be further extended to the inversion of other geophysical data.

In a heterogeneous medium (usually called a rough medium) with fractured formations, the propagation of an electromagnetic (EM) field is a type of subdiffusion. Current mainstream geophysical EM data processing methods cannot be applied to data acquired on heterogeneous Earth, as they are not governed by the classic diffusion theory. To evaluate the influence of roughness on the transient EM (TEM) signal for a complex model and contribute to data inversion, we proposed a novel three-dimensional (3-D) forward modeling scheme for TEM in rough media. First, we derived the governing equation with a fractional-order time derivative for the subdiffusion of EM waves in rough media. Then, we proposed a novel time discretization using an unequal step length for the Caputo operator, which significantly reduces the total number of time steps. Finally, an implicit time-domain finite-element method using unstructured tetrahedron discretization was adopted to solve the 3-D forward problem. Furthermore, an efficient time segmentation strategy combined with parallel RHS construction was proposed to accelerate modeling. The numerical results prove that the proposed method is accurate and efficient, and will be a powerful numerical method for analyzing TEM wave propagation and processing TEM data in areas with multiscale fractures or porosity.

The conventional magnetotelluric (MT) data inversion methods, such as the nonlinear conjugate gradient method, quasi-Newton method, and Gauss–Newton method and so on, can converge robustly, but their results are easily affected by the initial model and regularization term. Although supervised learning can break through the resolution limitation by directly learning the nonlinear relationship between the model and the data, it cannot guarantee the data fitting without considering physical constraints. Here, we propose a novel prior model generation method using deep learning for conventional inversion to jointly take advantages of the two techniques. We first combine Gaussian random rough surface scheme and random polygon generation algorithm to construct practical 2-D geoelectric models, in which the prior information on the geoelectric structure can be flexibly integrated. Then, a fast 2-D MT forward modeling method is applied to calculate the forward responses and establish the training set. Finally, we use the training set to complete the parameter optimization of U-shaped network (U-NET) and run the conventional inversion with prior model generated by the trained U-NET. Numerical experiments with synthetic data show that the proposed method can effectively integrate the advantages of conventional inversion and supervised learning, and remarkably improve the resolution in the inversion if proper training sets are used. The inversions of the USArray data also prove that our method can retain high-resolution structures predicted by the U-NET in the final inversion results with a data fitting as good as the traditional Gauss–Newton method.

The flight altitude has a large effect on the airborne electromagnetic (AEM) responses. Due to the dynamic environment of the aircraft, the recorded sensor altitudes may contain errors. Research demonstrates that the AEM responses caused by a several meters altitude errors can be larger than caused by some anomalous body. Ignoring these errors will create erroneous results in AEM data interpretation. Considering that there is not yet a published 3D AEM inversion method that takes into account the flight altitude, we develop in this paper a 3D inversion algorithm for AEM with the flight height treated as an inversion parameter. For the forward modeling we use the finite element method, while for the inversion we use the Gauss-Newton optimization method. To make our inversion works for variable flight altitudes, we propose a scheme of 3D Jacobean matrix calculation for both the resistivities and flight altitudes without much increasing the computational cost. The numerical simulation result confirms that the flight altitude really has a large effect on the AEM responses. The inversions of synthetic data show that our 3D inversion method can both recover the resistivity distribution in the underground and decrease the altitude errors recorded, while the field data inversion demonstrates that our method can deliver a better inversion model with a smaller data misfit.

The 3-D airborne electromagnetic (AEM) inversions have been restricted by the modeling efficiency resulting from the complex geology in exploration areas and massive amount of data collected by AEM systems. In order to improve the modeling efficiency, we develop an algorithm that combines the hexahedral vector finite element (FE) with octree meshes, in which the boundary conditions are imposed via an algebraic constraint to ensure the continuity of the FE solution. This makes the division with hexahedral meshes more flexible for complex geology such as rugged topography or underground structures so that we can reduce the number of elements while maintaining the accuracy. After formulating the forward problem, we check the accuracy of our algorithm by taking a homogeneous half-space model and comparing the results of our octree method with the semianalytical solutions. Furthermore, we demonstrate the efficiency of our octree method by comparing with the traditional FE method using tetrahedral meshes. Finally, we subdivide a complex topography constructed using the 2-D Gaussian rough surface and calculate the EM responses with and without anomaly embedded. The results show that the EM responses are overwhelmed by the Earth topography. We carry out the topographic correction by taking a method based on the ratio of EM responses with and without anomaly. The experiments show that after topographic correction to AEM data, the response of anomaly becomes more distinguishable so that the anomaly can be clearly identified. Furthermore, we also calculate the EM response for a realistic model—the Ovoid Zone ore body located at Voisey’s Bay, Labrador, Canada, to verify the flexibility and practicality of our algorithm.

Based on the spatial structure correlation in different geophysical parameters, we propose a new 3-D joint inversion method for frequency-domain airborne electromagnetic (AEM) and airborne magnetic (AirMag) data by incorporating a local Pearson correlation constraint (LPCC). For each iteration, the entire model is separated into multiple subdomains and the Pearson correlation coefficients of resistivity and magnetization in the subdomain are employed as the additional regularization term to do the joint constraint. This new regularization term is continuously updated in the inversion process to ensure that the resistivity and magnetization models in two separated inversions converge to a similar spatial structure. As a statistics technology, the LPCC-based joint inversion scheme not only has the advantages of the conventional joint inversions, but also can implement the structural constraints in different scales by selecting different sizes of the subdomain. This provides the flexibility for solving multiscale problems. Synthetic examples show that the joint inversion can improve the overall inversion resolution by combining the high vertical resolution of the EM method and large exploration depth and high horizontal resolution of the magnetic method. In the application to field survey datasets, the joint inversion delivers better results than those of separate inversions, which further verifies the effectiveness of our method.

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 conventional, L2-norm-based, regularization term in electromagnetic (EM) inversions implements smooth constraints on model complexity in the space domain, which can smoothen the boundaries of complex underground structures. To improve the resolution of 3-D frequency-domain airborne EM (AEM) inversions, we propose a new algorithm for sparse-regularized inversion based on the shearlet transform. Unlike traditional methods that invert the model parameters in the space domain, we first transform the 3-D resistivity model into the frequency domain and then invert the sparse coefficients using an L1-norm measure to ensure the sparseness of the solution. Finally, we transform the shearlet coefficients back to the space domain to update the model. The shearlet transform has inherent multiscale and multidirectional properties, making it capable of effectively extracting complex geometries such as curved boundaries. We adopt the finite-difference method and the iteratively reweighted least-squares scheme for our 3-D AEM modeling and inversions and apply the “moving footprint” technique to speed up the inversion. Tests using synthetic data show that sparse-regularized inversion based on the shearlet transform can obtain more-focused inversion results than conventional smoothness-constrained inversions based on the L2-norm. Tests using field survey data also reveal that the new method can achieve more realistic underground structures.

As an efficient geophysical exploration method, the time-domain airborne electromagnetic (AEM) data often show sign reversal in late-time channels due to induced polarization (IP) effect. The traditional imaging and inversion methods without considering the IP effect cannot recover the true electrical structure of the earth, so it is necessary to develop 3D EM forward modeling and inversion techniques with IP effect. In this paper, we propose a 3D forward modeling method for time-domain AEM with IP effect based on unstructured finite-element (FE) method. To describe the IP effect of a medium, we introduce the Cole-Cole model and transform it into fractional derivative form using frequency-time conversion. Then, we discretize it using the difference discretization format of Caputo fractional derivative. Finally, we use the vector FE method based on unstructured tetrahedral mesh and unconditionally stable second-order backward Euler’s scheme to discretize Maxwell’s equations in space and time. In this way, we can solve the 3D forward modeling problem for AEM with IP effect in time domain. We verify the accuracy of our algorithm by comparing it with the 1D semi-analytical solution for a half-space model, and then calculate EM responses for typical abnormal models and analyze the characteristics of IP effect.

In this article, to solve the frequency-domain airborne electromagnetic (EM) modeling problem, we express the electric and magnetic field by the vector magnetic potential and the scalar electric potential and derive the governing equation using the Galerkin weighted residual method under the Coulomb gauge. To avoid the strong singularity of the solution near the transmitting source, we directly solve the relatively slow-changing secondary potential. By introducing the spectral element method (SEM) based on unstructured tetrahedral grids, in which the EM field in each element is characterized by high-order Proriol–Koornwinder–Dubiner (PKD) orthogonal polynomials, we can obtain stable and accurate numerical solutions. By establishing the mapping relationship between the physical domain, the right-angled tetrahedral reference domain, and the orthogonal hexahedral domain, we can easily accomplish the element matrix analysis and calculation. The numerical examples show that, when the SEM is used to simulate the airborne EM response, a high accuracy can be obtained even if a very coarse grid is used. Furthermore, since the tetrahedral grids can easily model the complex boundaries, the SEM based on unstructured grids has significant advantages for simulating complex underground structures.