We use the improved Parker−Oldenburg's formulas that include a reference depth into the exponential term and employ the Gauss‐FFT method to determine Moho depth beneath the Tibetan Plateau. The synthetic models demonstrate that the improved Parker's formula has high accuracy with the maximum absolute error less than 0.25 mGal compared to the analytical solution. Two inversion parameters, i.e., the reference depth and the density contrast are essential for the Moho estimation based on the gravity field, and they need to be determined in advance to obtain correct results. Therefore, the Moho estimates derived from existing seismic studies are used to reduce the non‐uniqueness of the gravity inversion and to determine these parameters by searching for the maximum correlation between the gravity‐inverted and seismic‐derived Moho depths. Another critical issue is to remove beforehand the gravity effects of other factors, which affect the observed gravity field besides Moho variations. In addition to the topography, the gravity effects of the sedimentary layer and crystalline crust are removed based on existing crustal models, while the upper mantle impact is determined based on the seismic tomography model. The inversion results show that the Moho structure under the Tibetan plateau is very complex with the depths varying from about 30 ~ 40 km in the surrounding basins (e.g., Ganges basin, Sichuan basin, and Tarim basin) to 60 ~ 80 km within the plateau. This considerable difference up to 40 km on the Moho depth reveals the substantial uplift and thickening of the crust in the Tibetan Plateau. Furthermore, two visible “Moho depression belts” are observed within the plateau with the maximum Moho deepening along the Indus‐Tsangpo Suture and along the northern margin of Tibet bounding the Tarim basin with the relatively shallow Moho in central Tibet between them. The southern “belt” is likely formed in compressional environment, where the Indian plate underthrusts northwards beneath the Tibetan Plateau, while the northern one could be formed by the southward underthrust of the Asian lithosphere beneath Tibet.

Seismic image interpolation is a currently popular research subject in modern reflection seismology. The interpolation problem is generally treated as a process of inversion. Under the compressed sensing framework, various sparse transformations and low-rank constraints based methods have great performances in recovering irregularly missing traces. However, in the case of regularly missing traces, their applications are limited because of the strong spatial aliasing energies. In addition, the erratic noise always poses a serious impact on the interpolation results obtained by the sparse transformations and low-rank constraints-based methods. This is because the erratic noise is far from satisfying the statistical assumption behind these methods. In this study, we propose a mathematical morphology-based interpolation technique, which constrains the morphological scale of the model in the inversion process. The inversion problem is solved by the shaping regularization approach. The mathematical morphological constraint (MMC)-based interpolation technique has a satisfactory robustness to the spatial aliasing and erratic energies. We provide a detailed algorithmic framework and discuss the extension from 2D to higher dimensional version and the back operator in the shaping inversion. A group of numerical examples demonstrates the successful performance of the proposed technique.

The characterization of fracture-induced tilted transverse isotropy (TTI) seems to be more suitable to actual scenarios of geophysical exploration for fractured reservoirs. Fracture weaknesses enable us to describe fracture-induced anisotropy. With the incident and reflected PP-wave in TTI media, we have adopted a robust method of azimuthal amplitude variation with offset (AVO) parameterization and inversion for fracture weaknesses in a fracture-induced reservoir with TTI symmetry. Combining the linear-slip model with the Bond transformation, we have derived the stiffness matrix of a dipping-fracture-induced TTI medium characterized by normal and tangential fracture weaknesses and a tilt angle. Integrating the first-order perturbations in the stiffness matrix of a TTI medium and scattering theory, we adopt a method of azimuthal AVO parameterization for PP-wave reflection coefficient for the case of a weak-contrast interface separating two homogeneous weakly anisotropic TTI layers. We then adopt an iterative inversion method by using the partially incidence-angle-stacked seismic data with different azimuths to estimate the fracture weaknesses of a TTI medium when the tilt angle is estimated based on the image well logs prior to the seismic inversion. Synthetic examples confirm that the fracture weaknesses of a TTI medium are stably estimated from the azimuthal seismic reflected amplitudes for the case of moderate noise. A field data example demonstrates that geologically reasonable results of fracture weaknesses can be determined when the tilt angle is treated as the prior information. We determine that the azimuthal AVO inversion approach provides an available tool for fracture prediction in a dipping-fracture-induced TTI reservoir.

For large-scale magnetotelluric (MT) forward modeling using vector finite-element methods, iterative solvers are commonly applied. However, as frequency decreases close to zero, the iterative solution process struggles to converge. This is mainly caused by the fact that the weak conductivity term in the curl-curl equation governing the electromagnetic (EM) diffusion in the earth fades during the iterative solution for EM fields. This can lead to violation of the divergence-free condition for current and false jumps in the calculated fields. We develop a regularization technique for vector finite-element MT forward modeling, in which a scaled grad-div operator for electrical fields is included in the curl-curl equation to enforce the divergence-free condition explicitly. Because of its use of basis functions, the direct edge element discretization of the scaled term can lead to zero coefficients. To address this, an equivalent substitute of the scaled grad-div operator based on potential representation of electrical fields is used, discretized with node elements. For one specific element, this is finally reduced to a scaled gradient of the surface integral of current across all the surfaces of the element, resulting in better connectivity of the linear system of equations. The correctness of our algorithm is verified with two synthetic models and an inversion model from real data. The numerical performance is compared to the traditional iterative divergence correction technique (and without the application of the divergence correction for one case) for each of the three models. The results indicate that our algorithm is generally more efficient and stable compared to the traditional technique for all models at all periods considered, with significant improvement at long periods.