To investigate deep Earth information, researchers often utilize geomagnetic observatories and satellite data to obtain the conversion function of geomagnetic sounding, C-response data, and employ traditional inversion techniques to reconstruct subsurface structures. However, the traditional gradient-based inversion produces geophysical models with artificial structure constraint enforced subjectively to guarantee a unique solution. This method typically requires the model parameterization knowledge a priori (e.g., based on personal preference) without uncertainty estimation. In this paper, we apply an efficient trans-dimensional (trans-D) Bayesian algorithm to invert C-response data from observatory and satellite geomagnetic data for the electrical conductivity structure of the Earth’s mantle, with the model parameterization treated as unknown and determined by the data. In trans-D Bayesian inversion, the posterior probability density (PPD) represents a complete inversion solution, based on which useful inversion inferences about the model can be made with the requirement of high-dimensional integration of PPD. This is realized by an efficient reversible-jump Markov-chain Monte Carlo (rjMcMC) sampling algorithm based on the birth/death scheme. Within the trans-D Bayesian algorithm, the model parameter is perturbated in the principal-component parameter space to minimize the effect of inter-parameter correlations and improve the sampling efficiency. A parallel tempering scheme is applied to guarantee the complete sampling of the multiple model space. Firstly, the trans-D Bayesian inversion is applied to invert C-response data from two synthetic models to examine the resolution of the model structure constrained by the data. Then, C-response data from geomagnetic satellites and observatories are inverted to recover the global averaged mantle conductivity structure and the local mantle structure with quantitative uncertainty estimation, which is consistent with the data.

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.