SUMMARY: We propose a novel smoothing regularization scheme for 3-D magnetotelluric (MT) inversion based on unstructured tetrahedral discretization. Different from conventional methods that explicitly add smoothing constraints to model parameters, we choose to do the gradient filtering to smooth the model updates in an implicit way. By transforming the model into a constraint domain, the gradient of the objective function for the parameters in the new domain can be taken as a product of transpose of inverse transformation operator and the conventional gradient. Since the transpose of inverse transformation is designed to be an inverse distance interpolation operator for each tetrahedron, the data fitting term in the gradient can be smoothed in a filtering-like process. We compare our new strategy with the conventional explicit smoothing ones by testing on synthetic data for different noise levels, initial models and regularization factors. The numerical results show that suffering from the unequal volume and random centroid location of adjacent tetrahedrons, the inversion results of conventional methods often demonstrate scattered structures in slices. In contrast, our new method recovers the model in a smooth way and the convergence speed is largely improved. Finally, we adopt the USArray data for further testing and find that comparing to conventional inversion methods, our new strategy can provide more reliable underground structures with better data fitting.

For the mineral exploration in complex terrain areas, the semi-airborne transient electromagnetic (SATEM) technology is one of the most powerful methods due to its high efficiency and low cost. However, since the mainstream SATEM systems only observe the component dBz/dt and the data are usually processed by simple interpretation or one-dimensional (1D) inversion, their resolutions are too low to accurately decipher the fine underground structures. To overcome these problems, we proposed a novel 3D forward and inversion method for the multi-component SATEM system. We applied unstructured tetrahedron grids to finely discretize the model with complex terrain, subsequently we used the vector finite element method to calculate the SATEM responses and sensitivity information, and finally we used the quasi-Newton method to achieve high-resolution underground structures. Numerical experiments showed that the 3D inversion could accurately recover the location and resistivities of the underground anomalous bodies under the complex terrain. Compared to a single component data, the inversion of the multi-component data was more accurate in describing the vertical boundary of the electrical structures, and preferable for high-resolution imaging of underground minerals.

The choice of data misfit measure has a great impact on the convergence of electromagnetic (EM) inversion. The conventional measure based on the $l_{2}$ -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 on $l_{1}$ -norm data misfit and adaptive moment estimation (Adam). In this scheme, we use the Ekblom-type $l_{1}$ -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 the $l_{1}$ - $l_{2}$ norm of the objective function. Compared with the conventional ${l} _{2}$ -norm data misfit, the $l_{1}$ -norm data misfit can effectively avoid excessive optimization of data with large misfits and achieve high-resolution inversion results.

SUMMARY: We propose a novel method for 3-D magnetotelluric (MT) forward modelling based on hybrid meshless and finite-element (FE) methods. This method divides the earth model into a central computational region and an expansion one. For the central region, we adopt scatter points to discretize the model, which can flexibly and accurately characterize the complex structures without generating unstructured mesh. The meshless method using compact support radial basis function is applied to simulate this area's electromagnetic field. While in the expansion region, to avoid the heavy time consumption and numerical error of the meshless method caused by non-uniform nodes, we adopt a node-based finite-element method with regular hexahedral mesh for stability. Finally, the two discretized systems are coupled at the interface nodes according to the continuity conditions of vector and scalar potentials. Considering that the normal electric field is discontinuous at the interface with resistivity discontinuity, while the shape functions for the meshless method are continuous, we further adopt the visibility criterion in constructing the support region. Numerical experiments on typical models show that using the same degree of freedom, the hybrid meshless-finite element method (FEM) algorithm has higher accuracy than the node-based FEM and meshless method. In addition, the electric field discontinuity at interfaces is well preserved, which proves the effectiveness of the visibility criterion method. In general, compared to the conventional grid-based method, this new approach does not need the complex mesh generation for complex structures and can achieve high accuracy, thus it has the potential to become a powerful 3-D MT forward modelling technique.