## 4 Real Data  
### Field Data from the Satellites  
In this part, we consider applying the rjMcMC algorithm to invert C-response data derived from 10-year satellite magnetic data in which the night-time data and data from latitude \(>\)55\({}^{\circ}\) are excluded [14]. The data, including 27 periods, cover a period range  
Figure 4: Marginal posterior probability profiles of (**a**) interface depth and (**b**) conductivity for the low-conductivity model. Figure 5: Sampling results for nonphysical parameters at \(T=1\): (**a**) the sampling history of misfit, (**b**) 1D marginal density of misfit, and (**c**) 1D marginal distribution for the interface number. from 1.5 to 150 days. The ocean effect has been corrected from the data [14]. The prior for the interface number is the bounded uniform distribution on [1,30]. The priors for the conductivity and interface depth are bounded uniform distributions on [0.00001,1000] \(\sigma\) (\(\mathrm{S}/\mathrm{m}\)) and \([100,2500]\) km, respectively. A total of five tempering chains are used, with the tempering schedule defined by \(T=1.5^{(i-1)},i=1,\dots,5\). Similar preprocessing as used in the previous cases is applied to generate the sample for later inference. The inversion results are shown in Figure 6. The interface probability in Figure 6a indicates three interfaces, with two sharp peaks in the shallow region and relative large uncertainty for the deepest interface. The conductivity marginal profile shows a distinct four-layer Earth model, with a relative thin high-conductivity layer underlying a lower-conductivity top layer, followed by two layers with increasingly larger conductivity. This is similar to the inversion results from the work [14]. For the recovered structure with low conductivity above around 800 km, it is interesting to note that the uncertainty increases with depth, reflecting that the data generally decrease in resolution as the fields diffuse deeper. The recovered high-conductivity structure at a depth between 800 km and 900 km, consistent with the discontinuity at depth around 900 km inferred from [PERSON] and [PERSON] [39], and [PERSON] et al. [40], is resolved well with small uncertainty, indicating that the data are sensitive to the high-conductivity structure at this depth range. At a depth of [1000,1400] km, the conductivity increases slightly relative to the first layer. At depths deeper than 1400 km, the conductivity increases further due to the geodynamic state of deep Earth. At this depth, the recovered structure is determined with even less uncertainty than shallow ones, probably due to the deep large-scale high-conductivity structure. Compared with traditional gradient-based inversion, one impressive benefit of our inversion is that our result indicates how well each portion of the recovered structure is solved entirely by the actual data, rather than the subjective enforcement of an artificial structure (model smoothness). To consider the fit to the observed C-response data, we calculate the forward modeling response data as shown in Figure 7 for both the sampling ensemble and its ensemble average. Both the marginal probability density for the ensemble and the modeling data for the averaged model indicate that the predicted data are generally in good agreement with the corresponding observed data. Figure 6: Marginal posterior probability profiles of (**a**) interface depth and (**b**) conductivity for the 10-year satellite magnetic data [14]. Solid line in figure indicates the inversion result from the work [14]. Figure 8 show the sampling history and 1D marginal density of the sampled Markov chain at \(T=1\) for data misfit, indicating the well-mixing of the sampled Markov chain and relatively wide sampling range over the data misfit. Figure 8c show the 1D marginal density at \(T=1\) for the interface number, showing that the interface has maximum probability at 7 interfaces with a broad range.