Conventional 3D magnetotelluric (MT) inversion methods face substantial computational challenges when inverting large-scale data sets, demanding extensive processing time and computing resources. To address these limitations, we develop an advanced frequency-polarization joint parallel inversion framework for MT data that uses the nonlinear conjugate gradient (NLCG) method, integrating an adaptive algebraic multi-resolution sampling (AMRS) approach with a regularization technique. The AMRS approach adaptively discretizes modeling domains across different periods based on the normalized root-mean-square difference relative to the solution on the initial grid, which serves as a reference. The regularization technique ensures the conservation of current, which can significantly affect the efficiency of the forward-modeling problem and its adjoint when using iterative solvers, particularly at low frequencies. The reliability and efficiency of the regularization AMRS inversion are first examined by benchmarking it against the traditional NLCG inversion and the regularization inversion based on a two adjacent blocks model and a terrain model. Subsequently, the regularization AMRS inversion is applied to invert real MT data from the Cascadia region in the USA, demonstrating its capability to address real inverse problems.

Artificial-source frequency-domain electromagnetic methods are important tools for deep mineral exploration and concealed geological structure detection. Apparent resistivity is a key parameter linking measured electromagnetic fields to the interpretation of subsurface electrical structures, and its calculation method directly affects geological interpretation and engineering applicability. Although substantial efforts have been devoted to the theoretical development, data processing, and practical application of different apparent resistivity formulations, most previous studies have focused on the analysis and improvement of a single method. Systematic comparisons of the main apparent resistivity formulations under unified conditions remain limited, particularly in terms of deep basement characterization, anti-interference performance, and engineering applicability. To fill this gap, this study systematically compares the $E\text{–}{E}_x$ wide-field apparent resistivity, the $E\text{–}{E}_x$ far-zone apparent resistivity, and the $E\text{–}{Z}_{xy}$ Cagniard apparent resistivity. Through theoretical derivation, forward modeling of typical one-dimensional models, and field verification, the differences among these three formulations in geological characterization, anti-interference capability, and engineering applicability are analyzed, with the aim of clarifying their applicable boundaries and selection principles for artificial-source frequency-domain electromagnetic exploration.