## 5 Conclusions  
In this paper, we applied an efficient trans-dimensional Bayesian inversion algorithm to invert the C-responses derived from both observatory and satellite geomagnetic data. In this algorithm, both the model parameter and the number of interfaces (or the number of layers) are considered as unknown and determined by the actual data, rather than enforced subjectively. For efficiency, the parameter is perturbated in the principal space, and the parallel tempering scheme is used to guarantee the completeness of the sampling. Our algorithm is firstly investigated by several synthetic models, and then applied to invert real C-response data from both observatory and satellite. The tests on two synthetic models (high- and low-conductivity three-layer models) show that our trans-D Bayesian inversion parsimoniously prefers simple models and can recover the model parameters with nonlinear uncertainty, and the model parameterization as well. The model uncertainty manifests the sensitivity of the data to the structure and provides information on how well the inference is determined by the data. Since magnetic fields are sensitive to high-conductivity structures, the uncertainty of the high-conductivity layer is significantly smaller than its surrounding structure for the high-conductivity model. For the low-conductivity model, the uncertainty of the low-conductivity layer is remarkably larger than its surrounding structure. The inversion for real satellite data indicates a similar four-layer structure as in the previous work [14]. However, our result clearly shows how well each part of the inverted structure is constrained by the data. The inversion of C-response data from the BJI geomagnetic observatory shows a four-layer model for this site, with  
Figure 11: Sampling results for nonphysical parameters at \(T=1\): (**a**) the sampling history of misfit, (**b**) and its 1D marginal density, and (**c**) 1D marginal distribution for the interface number.  
Figure 10: The marginal probability density for the sampling ensemble and the modeling data (real part (**a**), imaginary part (**b**)) for the ensemble averaged model in comparison to observation data from BJI Observatory.  
increasingly large conductivity as the depth increases. The uncertainty for this site generally increases with depth, except for the third layer, which has slightly smaller uncertainty. Through the inversion of the satellite and the observatory data, the one-dimensional models obtained both exhibit an increase in conductivity with depth. However, the model from the satellite observations suggests a sharp conductivity interface at around 900 km. In contrast, this is not shown in the model recovered for the northern region of China based on the BJI observatory. This indicates that the mantle structure in this region is continuous, with conductivity increasing with depth due to the fact that the temperature within the Earth increases with depth.  
Conceptualization, J.L.; funding acquisition, R.G., J.L. and Y.-a.C.; methodology, R.G., S.T. and C.C.; project administration, J.L. and Y.-a.C.; supervision, Y.-a.C.; writing--original draft, R.G. and S.T.; writing--review and editing, C.C. All authors have read and agreed to the published version of the manuscript.  
This work was financially supported by the National Natural Science Foundation of China (42130810, 42074165, 42174171, 42174170) and the Institute of Geological Survey of Hunan Province (NGSTP202201), Research Foundation of the Department of Natural Resources of Hunan Province (Grant No: 20230124 DZ), Hunan Provincial Natural Science Foundation (Grant No: 2023)J60164).  
Data from this research will be available upon request to the authors.  
The authors declare no conflicts of interest.