Efficient Trans-Dimensional Bayesian Inversion of C-Response Data from Geomagnetic Observatory and Satellite Magnetic Data

Abstract

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.

1. Introduction

The mantle refers to the portion between the Moho discontinuity and the Gutenberg discontinuity of the Earth, constituting an important component of the Earth. Studying the structure of the mantle can be a means to explore the evolutionary history and dynamic characteristics of the Earth [1,2]. In particular, estimating the average water content within the mantle transition zone on a global scale is a crucial step in understanding global geodynamics. Conductivity is exceptionally sensitive to variations in temperature, water content, material composition, and so on within the mantle. Therefore, conductivity is a crucial physical parameter for revealing the mantle’s structure [3,4,5,6,7,8]. Geomagnetic data, either from geomagnetic observatories or satellites and typically covering a period range from several hours to several hundred days, are sensitive to the conductivity structure of the Earth’s mantle [9]. In practice, time-domain geomagnetic data are Fourier transformed to estimate geomagnetic responses, for instance, the commonly used C-responses [10]. Quantitative inversion in geomagnetic depth sounding (GDS) is routinely required to obtain the mantle conductivity from the response data [11].

Due to the non-uniqueness of inversion, the interpretation of the C-response data is commonly carried out based on gradient-based inversions, by subjectively enforcing a structure constraint (e.g., flattest structure) to yield a unique solution [12,13,14,15,16]. One obvious advantage of the over-parameterization method is that the best-fit solution can be obtained efficiently. However, the purpose of this approach is to achieve the optimal solution by incorporating regularization information based on gradient-based methods, and it cannot provide quantitative uncertainty estimation on the recovered model, indicating how well one specific structure of a model is constrained by the data. Due to the application of an artificial structure constraint, the recovered model typically manifests the important characteristics enforced artificially rather than from data itself. This complicates the interpretation process and reduces the credibility of the inference model [17,18,19,20,21].

Bayesian inversion produces models with parsimonious structure and quantitative uncertainty, determined from the actual data. Thus, it is constantly applied in geophysical data interpretation for subsurface structures [22,23,24,25,26]. It is attracting increasing attention in the interpretation of geomagnetic data from observatories and satellites [27,28,29,30]. In their work, the model parameterization is assumed a priori and considered fixed during inversions. This conflicts with the fact that the prior knowledge of model parameterization is typically not available in most geophysical applications.

In the most recent work [16], they developed a joint trans-D inversion based on an essentially single-chain sampling algorithm and physical parameter space for magnetotelluric data and local geomagnetic data from observatories. In their work, the model parameterization is treated as unknown and included in the inversion, determined by the actual data. The effect of unknown model parameterization is contained in the uncertainty estimation of the conductivity profile. To the best of our knowledge, trans-D Bayesian inversions for geomagnetic data both from observatories and satellites have not been reported in the literature.

Trans-D Bayesian inversion based on a single-chain sampling algorithm can potentially be trapped in local high-probability regions separated by low-probability areas [20]. This can be addressed by the application of a parallel tempering scheme to sample the model space more completely [31,32]. Another challenge is that if the parameters of a model are strongly correlated, the parameter perturbation can be inefficient. To minimize this effect, a parameter perturbation technique based on the unit-lag model covariance matrix can be applied [20,33].

In this paper, an efficient trans-D Bayesian inversion [20,33] is applied to invert both observatory and satellite geomagnetic data. In our algorithm, the parallel tempering scheme is utilized to guarantee a complete sampling over the model space for geomagnetic inversion. Parameter perturbation is carried out in the principal-component space, defined by the eigenvectors of the unit-lag model covariance. Two synthetic examples are designed to show the resolution of the model structure according to the data information. Finally, both real observatory and satellite data are inverted to obtain the local and global conductivity structure of the mantle.

2. Basics Theory

2.1. C-Responses

From the geomagnetic fields measured from ground geomagnetic observatories or satellites in the time domain, we can obtain frequency domain fields, and then transform these into C-responses. Assuming that we have the frequency domain magnetic data of the radial component $H_r\left(\omega \right)$ and the geomagnetic south component $H_{\theta }\left(\omega \right)$ at a particular site, $\theta$ is the geomagnetic colatitude. We can define the C-responses at $\omega$ as

$$C\left(\omega \right)=-{\displaystyle \frac{a_{0}tan\theta }{2}}{\displaystyle \frac{H_r\left(\omega \right)}{H_{\theta }\left(\omega \right)}}.$$

(1)

where $a_0=6371$ km, and C-responses (in km) is a complex variable (for the detailed derivation process, refer to Li et al., 2018 [34]). The data processing is generally similar for satellite and observatory data, used to estimate the spherical harmonic coefficients [35]. The real part of C-responses reflects the mean depth of magnetic fields penetrated into the Earth, and the imaginary part reflects the resistivity variation [11]. We can use real geomagnetic data to estimate the C-responses at different frequencies [34,36]. The forward modeling solutions for C-responses are provided by Semenov and Kuvshinov [36].

2.2. Model Parameterization

A spherical partition approach is applied to discretize the Earth into a set of spherical homogeneous layers (for instance k), with a half-space layer at the bottom as shown in Figure 1. Thus, the parameters of the model $\mathbf{m}(k,z,œ)$ consist of depths of the k − 1 interfaces $z=\left(z_1,z_2,\dots ,z_{k-1}\right)$ and the conductivities $œ=\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_k\right)$ of the k layers. Both the interface depths and the conductivities are represented in the log scale. However, for convenience, we will refer to them as interface depths and layer conductivities in this paper. In general, the layer number is unknown and can be determined by the data information. The priors applied are considered to be uniform on the bounded range $\left[z_{min},z_{max}\right]$ for interface depth (100 km is used for $z_{min}$ considering the frequencies used for GDS) and $\left[{\sigma }_{\min },{\sigma }_{\max }\right]$ for layer conductivities. The interface number is drawn from the uniform distribution over bounded range of $\left[k_{\min },k_{\max }\right]$. To avoid generating an extreme thin layer, the minimum thinness for the new generated layer is set to $h_{min}=50\mathrm{km}$.

2.3. Trans-Dimensional Bayesian Inversion Theory

In Trans-D Bayesian inversions, both data and model parameters as well as model parameterization are considered as random variables. Let K represent a countable number set indexed by k, for instance, the number of interfaces, which can also be considered as a random variable. For each k, there is an associated $M_k$-dimensional parameter space ${\mathcal{M}}_k$. For simplicity, we use m to represent a model of variable dimensions, with $M_k$ (or k) used to specify the specific choice of the dimension. In Bayesian inversion, N-dimensional data d and $M_k$-dimensional model m are related by Bayes’ rule, as defined by

$$P\left(\mathbf{m}|\mathbf{d}\right)={\displaystyle \frac{P\left(\mathbf{m}\right)P\left(\mathbf{d}|\mathbf{m}\right)}{P\left(\mathbf{d}\right)}}={\displaystyle \frac{P\left(\mathbf{m}\right)P\left(\mathbf{d}|\mathbf{m}\right)}{{\sum }_{k\in K}{\int }_{{\mathcal{M}}_k}P\left(\mathbf{m}\right)P\left(\mathbf{d}|\mathbf{m}\right)\mathrm{d}\mathbf{m}}},$$

(2)

where $p\left(\mathbf{m}\right)$ is the prior information on model m, and $p\left(\mathbf{d}\left|\mathbf{m}\right.\right)$ is the conditional data probability given model m. If the observation data d are given, $p\left(\mathbf{d}\left|\mathbf{m}\right.\right)$ can be interpreted as the likelihood function of m, $L\left(\mathbf{m}\right)$. $p\left(\mathbf{d}\right)$ is the total data evidence, a normalizing integral over the whole variable model space. $p\left(\mathbf{m}\left|\mathbf{d}\right.\right)$ is the conditional probability of m given observed or synthetic data, called the posterior probability density (PPD), defined over the trans-D model space, e.g., changes between different k. The complete solution to a Bayesian inverse problem is defined by the trans-D PPD, with multi-dimensional integration typically required for a useful interpretation, for instance, the average conductivity, conductivity variance, and 1D marginal PPD of conductivity at depth z (the calculation for a sequence of z leads to the probability profile for conductivity), as defined by

$$\overline{\sigma }\left(z\right)=\sum _{k\in K}{\int }_{{\mathcal{M}}_k}\sigma \left(\mathbf{m};z\right)P\left(\mathbf{m}|\mathbf{d}\right)\mathrm{d}\mathbf{m},$$

(3)

$$var\left(\sigma \left(z\right)\right)=\sum _{k\in K}{\int }_{{\mathcal{M}}_k}{(\sigma \left(\mathbf{m};z\right)-\overline{\sigma }\left(z\right))}^{2}P\left(\mathbf{m}|\mathbf{d}\right)\mathrm{d}\mathbf{m},$$

(4)

$$P\left(\sigma \left(z\right)|\mathbf{d}\right)=\sum _{k\in K}{\int }_{{\mathcal{M}}_k}\delta \left(\sigma \left(\mathbf{m};z\right)-\sigma \left(z\right)\right)P\left(\mathbf{m}|\mathbf{d}\right)\mathrm{d}\mathbf{m},$$

(5)

where $\sigma \left(z\right)$ is the conductivity parameter of interest at z depth, $\sigma \left(\mathbf{m};z\right)$ is the $\sigma \left(z\right)$ value in state m with k interfaces, and $\delta$ is the Dirac delta function.

For general inverse problems, an analytical solution to the multi-dimensional integration as in Equations (3)–(5) is impossible. Alternatively, it is commonly evaluated numerically by trans-D sampling over different subspace ${\mathcal{M}}_k$, such as the reversible-jump Markov-chain Monte Carlo sampling (rjMcMC; Green [37]) as used in this paper. To sample PPD in Equation (2), the Markov chain must keep the detailed balance requirement in order to transit in model space with variable dimensions [20].

An rjMcMC algorithm can be implemented by generating a model ${\mathbf{m}}^{\prime }$ with dimension of ${\mathit{k}}^{\prime }$ from the trans-D space based on the current model m with dimension of k for a Markov chain, with the acceptance probability determined by the Metropolis–Hastings–Green criterion [37], as defined by

$$\alpha ({\mathbf{m}}^{\prime },\mathbf{m})=min\left\{1,{\displaystyle \frac{p\left({\mathbf{m}}^{\prime }\right)}{p\left(\mathbf{m}\right)}}{\displaystyle \frac{L\left({\mathbf{m}}^{\prime }\right)}{L\left(\mathbf{m}\right)}}{\displaystyle \frac{q\left(\mathbf{m}\left|{\mathbf{m}}^{\prime }\right.\right)}{q\left({\mathbf{m}}^{\prime }\left|\mathbf{m}\right.\right)}}\left|\mathbf{J}\right|\right\},$$

(6)

where $\mathbf{J}$ is the Jacobian matrix for a diffeomorphism from state m to state ${\mathbf{m}}^{\prime }$, with values depending on the implementation of the rjMcMC algorithm. q is the proposal distribution from which the new state is proposed. In this paper, the birth-death scheme [22,37,38] is applied to implement the rjMcMC algorithm, leading to a unit Jacobian determinant, $\left|\mathbf{J}\right|=1$.

In the birth-death scheme, one of the perturbation, birth, and death steps is applied randomly at each iteration. At the perturbation step, the current model is transformed into the principal-component (PC) space defined by eigen vectors of the unit-lag model covariance matrix (see Xiang et al. [20] for more details), and each parameter in PC space is perturbated randomly from a Gaussian distribution with the corresponding variance defined by eigenvalues of the covariance matrix centered at the current PC space parameter without dimensional changes (i.e., ${\mathit{k}}^{\prime }$ = k). At the birth step, one interface is added into the current model with the interface depth generated randomly from the uniform distribution over $\left[z_{min},z_{max}\right]$. In this case, the new model has a dimension defined by ${\mathit{k}}^{\prime }$ = k + 1. The conductivity of the new generated layer is drawn from the prior, as in the work [20]. In the death step, one layer interface in the current model is chosen and deleted randomly, leading to ${\mathit{k}}^{\prime }$ = k− 1. The two original layers defined by the deleted interface are merged into one layer with the conductivity set to that of the original lower layer.

Based on the birth-death scheme, we need to formulate the model prior ratio, the model likelihood ratio, and the proposal density ratio from state m to state ${\mathbf{m}}^{\prime }$ to calculate Equation (6). This is addressed in the next section. Each of the three steps is accepted or rejected based on the acceptance criterion given by Equation (6). If the proposed ${\mathbf{m}}^{\prime }$ is rejected, an additional copy of m is retained in the Markov chain.

2.3.1. Prior Ratio

The prior for m is defined by

$$p\left(\mathbf{m}\right)=\underset{i=1}{\prod ^k}p\left({\sigma }_i\left|k\right.\right)p\left(z_i\left|k\right.\right)p\left(k\right).$$

(7)

The prior for k is uniform over $\left[k_{min},k_{max}\right]$, with a probability of ${\displaystyle \frac{1}{\mathsf{\Delta }k}}$($\mathsf{\Delta }k$ is the interval of bounded range of k). The possible arrangement of the k layers is k!. Since the interface depths and conductivity are physically bounded over $\left[z_{min},z_{max}\right]$ and $\left[{\sigma }_{min},{\sigma }_{max}\right]$, respectively, their respective priors are ${\displaystyle \frac{1}{\mathsf{\Delta }{\sigma }^k}}$ and ${\displaystyle \frac{1}{\mathsf{\Delta }{z}^k}}$ ($\mathsf{\Delta }\sigma$ and $\mathsf{\Delta }z$ are the bounded intervals for them). Then, Equation (7) can be given by

$$p\left(\mathbf{m}\right)={\displaystyle \frac{1}{\mathsf{\Delta }k}}{\displaystyle \frac{1}{\mathsf{\Delta }{\sigma }^k}}{\displaystyle \frac{k!}{\mathsf{\Delta }{z}^k}}.$$

(8)

In the birth step, $k^{\prime }=k+1$. Then, the prior ratio for the birth step is given by

$${\displaystyle \frac{p\left({\mathbf{m}}^{\prime }\right)}{p\left(\mathbf{m}\right)}}={\displaystyle \frac{\frac{1}{\mathsf{\Delta }k}\frac{1}{\mathsf{\Delta }{\sigma }^{k+1}}\frac{(k+1)!}{\mathsf{\Delta }{z}^{k+1}}}{\frac{1}{\mathsf{\Delta }k}\frac{1}{\mathsf{\Delta }{\sigma }^k}\frac{k!}{\mathsf{\Delta }{z}^k}}}={\displaystyle \frac{k+1}{\mathsf{\Delta }\sigma \mathsf{\Delta }z}}.$$

(9)

Similarly, for the death step $k^{\prime }=k-1$, the prior ratio for the death step can be calculated by

$${\displaystyle \frac{p\left({\mathbf{m}}^{\prime }\right)}{p\left(\mathbf{m}\right)}}={\displaystyle \frac{\frac{1}{\mathsf{\Delta }k}\frac{1}{\mathsf{\Delta }{\sigma }^{k-1}}\frac{(k-1)!}{\mathsf{\Delta }{z}^{k-1}}}{\frac{1}{\mathsf{\Delta }k}\frac{1}{\mathsf{\Delta }{\sigma }^k}\frac{k!}{\mathsf{\Delta }{z}^k}}}={\displaystyle \frac{\mathsf{\Delta }\sigma \mathsf{\Delta }z}{k}}.$$

(10)

In the perturbation step, there are no model dimension changes, $k^{\prime }=k$. Then, the prior ratio for the perturbation step is unity.

2.3.2. Likelihood Ratio

Assuming that data errors are Gaussian distributed, we have

$$L\left(\mathbf{m}\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^N\left|{\mathbf{C}}_e\right|}}exp\left[-\mathsf{\Phi }\left(\mathbf{m}\right)\right],$$

(11)

with

$$\mathsf{\Phi }\left(\mathbf{m}\right)={\left(\mathbf{f}\right(\mathbf{m})-\mathbf{d})}^T{\mathbf{C}}_d^{-1}(\mathbf{f}\left(\mathbf{m}\right)-\mathbf{d}).$$

(12)

where $\mathbf{f}$ is the forward operator and represented as the mapping from model space to data space. ${\mathbf{C}}_d$ represents the covariance matrix of the data error. In the Bayesian inversion, we assume that our error obeys a Gaussian distribution. Therefore, the likelihood ratio of our model’s birth, death, and perturbation steps can be written as follows:

$${\displaystyle \frac{L\left({\mathbf{m}}^{\prime }\right)}{L\left(\mathbf{m}\right)}}=exp\left\{\mathsf{\Phi }\left(\mathbf{m}\right)-\mathsf{\Phi }\left({\mathbf{m}}^{\prime }\right)\right\}.$$

(13)

$$\begin{array}{cc}\hfill {\displaystyle \frac{L\left({\mathbf{m}}^{\prime }\right)}{L\left(\mathbf{m}\right)}}& =exp\left\{\mathsf{\Phi }\left(\mathbf{m}\right)-\mathsf{\Phi }\left({\mathbf{m}}^{\prime }\right)\right\}\hfill \\ \hfill & =exp\left\{{\left(\mathbf{f}\right(\mathbf{m})-\mathbf{d})}^T{C}_d^{-1}(\mathbf{f}\left(\mathbf{m}\right)-\mathbf{d})-{(\mathbf{f}\left({\mathbf{m}}^{\prime }\right)-\mathbf{d})}^T{C}_d^{-1}(\mathbf{f}\left({\mathbf{m}}^{\prime }\right)-\mathbf{d})\right\}\hfill \end{array}$$

(14)

2.3.3. Proposal Probability Ratio

For the perturbation step, the new model is generated from a symmetrical distribution (e.g., Gaussian distribution used in this paper), centered at the current model. Therefore, the proposal probability ratio for the perturbation step is equal to one.

For the birth step, we randomly insert one layer over $\left[z_{min},z_{max}\right]$ and generate the conductivity for this layer from a uniform distribution over $\left[{\sigma }_{\min },{\sigma }_{\max }\right]$ (for the benefit of this choice see the discussion in Xiang et al. [20]). Thus, the proposal probability for the birth step is determined by

$$q\left({\mathbf{m}}^{\prime }\left|\mathbf{m}\right.\right)={\displaystyle \frac{1}{\mathsf{\Delta }\sigma \mathsf{\Delta }z}}.$$

(15)

The reverse step deletes one layer from the k+1 layers, which leads to the proposal probability for the reverse step, given by

$$q\left(\mathbf{m}\left|{\mathbf{m}}^{\prime }\right.\right)={\displaystyle \frac{1}{k+1}}.$$

(16)

Thus, the proposal probability ratio for the birth step can be obtained as

$${\displaystyle \frac{q\left(\mathbf{m}\left|{\mathbf{m}}^{\prime }\right.\right)}{q\left({\mathbf{m}}^{\prime }\left|\mathbf{m}\right.\right)}}={\displaystyle \frac{\mathsf{\Delta }\sigma \mathsf{\Delta }z}{k+1}}.$$

(17)

Similarly, we can derive the proposal probability ratio for the death step, given by

$${\displaystyle \frac{q\left(\mathbf{m}\left|{\mathbf{m}}^{\prime }\right.\right)}{q\left({\mathbf{m}}^{\prime }\left|\mathbf{m}\right.\right)}}={\displaystyle \frac{k}{\mathsf{\Delta }\sigma \mathsf{\Delta }z}}.$$

(18)

2.3.4. Acceptance Probability

Once we calculate the model prior ratio, the model likelihood ratio, and the proposal probability ratio, we can trivially calculate the acceptance probability for the perturbation, birth, and death steps as

$${\alpha }_{P,B,D}({\mathbf{m}}^{\prime },\mathbf{m})=min\left\{1,\left({\displaystyle \frac{L\left({\mathbf{m}}^{\prime }\right)}{L\left(\mathbf{m}\right)}}\right)\right\}.$$

(19)

For cases with strong nonlinearity and multiple physical properties, the rjMcMC algorithm mentioned above may fail to provide complete sampling of the PPD. In this case, a parallel tempering technique can be applied, which is not repeated again in this paper. Then, the acceptance at temperature T can be calculated by

$${\alpha }_{P,B,D}({\mathbf{m}}^{\prime },\mathbf{m})=min\left\{1,{\left({\displaystyle \frac{L\left({\mathbf{m}}^{\prime }\right)}{L\left(\mathbf{m}\right)}}\right)}^T\right\}.$$

(20)

3. Synthetic Data Test

In the section, we design two models to examine our trans-D Bayesian inversion algorithm. For C-response data, since the periods used are typically insensitive to structures shallower than 100 km, we fix the conductivity of the shallow region above 100 km during the inversion. The conductivity of the Earth core is set to ${10}^4\mathrm{S}/\mathrm{m}$. For both cases, the priors for the number of interfaces and the conductivity in log scale are bounded uniform distributions in the range of [1,30] and [−5,3], respectively. The maximum depth is set to ${10}^{3.5}\mathrm{km}$. Five Markov chains are applied in the parallel tempering technique, with $T_i={1.5}^{(i-1)},i=1,\dots ,5$. A half-space model with $0.1\mathrm{S}/\mathrm{m}$ is used as the initial model for inversion.

3.1. High-Conductivity Model

In this subsection, we design a high-conductivity model, as listed in

Table 1. Conductivity and interface depth information for the high-conductivity model.

Layer	1	2	3
$\sigma (\mathrm{S}/\mathrm{m})$	0.01	4.0	0.1
z (km)	300	300	2090

, to test our inversion algorithm for C-response data. The synthetic C-response data are generated at 27 logarithmically spaced periods from 1 day to 114 days, by adding Gaussian distributed random errors with standard deviation of $5\%$ of the corresponding C-responses magnitude.

In the inversion, one million samples are sampled for each chain. For T = 1, the first 100,000 samples are considered as burn-in and not used for the inference to guarantee good chain mixing and stationary sampling. The remaining 900,000 samples are further thinned by a factor of 10 to yield 90,000 samples. The convergence of the remaining samples is examined by visual inspection of the marginal profiles for samples at different sampling periods, which is considered sufficient in the literature [20].

The inversion result is shown in Figure 2. As shown in Figure 2a, the interface probability profile shows that high probabilities are centered at vicinities of the true interface depths. The first layer interface is well resolved by the data, with a sharp probability peak at the interface of the first layer. Relative to the first layer interface, the uncertainty of the second layer interface is much larger, which is physically reasonable due to the diffuse nature of magnetic fields. The conductivity marginal profile in Figure 2b shows that the three-layer model is generally well recovered with excellent constraint on the first two layers. The uncertainty of the conductivity in the second layer is even smaller than in the first layer due to the high sensitivity of geomagnetic fields to high-conductivity structures (even when buried deeper). As the geomagnetic fields go deeper, they have less resolution power, and the conductivity uncertainty becomes larger.

Figure 3a,b shows the sampling history and 1D marginal density of the sampled Markov chain at T = 1 for data misfit. The sampling history indicates that a well-mixed train is likely obtained. The misfit is bounded at a relative narrow range with peak of 126.8. Figure 3c shows 1D marginal densities at T = 1 for the interface number. The result indicates that the interface number is also bounded at a relative narrow range by the data information with a peak of 3, equal to the true interface number.

3.2. Low-Conductivity Model

In this part, we design a low-conductivity model, with the model information listed in

Table 2. Conductivity and interface depth information for the low-conductivity model.

Layer	1	2	3
$\sigma (\mathrm{S}/\mathrm{m})$	0.5	0.01	1
z (km)	300	400	2090

. C-response data are generated at 27 logarithmically spaced periods from 1 day to 114 days by adding Gaussian distributed errors with $5\%$ standard deviation.

Similar to the first case, we finally retain 90,000 samples for later inference. The inversion results are shown in Figure 4. As indicated in Figure 4, all the interfaces are well resolved by the data, with sharp peaks centered at the true interface depths. Again, the deeper interface manifests larger uncertainty than that of the shallower one. In general, the three-layer model is well recovered. However, the uncertainty of conductivity for the second layer is significantly larger than that of the rest of the layers. This physically makes sense, since the geomagnetic fields are insensitive to lower-conductivity structures. Due to the high conductivity, the deepest layer has been slightly better constrained than the first layer, indicating our algorithm can recover a physically meaningful model with information contained in the data.

Figure 5a,b shows the sampling history and 1D marginal density of the sampled Markov chain at T = 1 for data misfit, indicating good mixing of the sampled Markov chain and narrow constraint on the data misfit. Figure 5c shows 1D marginal density at T = 1 for the interface number. As in the previous case, the data information can constrain the interface number in a relative narrow range peaking at the true interface number.

4. Real Data

4.1. 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 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 $[100,2500]\mathrm{km}$, 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 Hitoshi and Fenglin [39], and Lev 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 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.

4.2. Field Data from the BJI Observatory

In this case, we consider inverting C-response data from BJI observatory, located in northern China, calculated by Zhang [41,42], using the BIRRP code [43]. The data, including 20 periods, cover a period range from 1.33 to 117.78 days. The same priors are used for the interface number, conductivity, and interface depth as in the previous case. A total of five tempering chains are used. The collected sample is thinned as in previous cases, and used for later inference. The inversion result is shown in Figure 9. The 1D marginal profile shows a four-layer model, with increasing conductivity as the depth increases. The second and third layers are generally constrained well with narrow uncertainty, indicating that the data are sensitive to the structure defined by the two layers. For the structure below 1250 km, the uncertainty increases with the depth, which is physically understood as the diffusion of geomagnetic fields decreasing the sensitivity to the deeper structure. The marginal profile for the interface number in Figure 9a indicates that the probability peaks at 550, 950, and 1250 km, supporting a four-layer model.

The fit of the data from inversion with the observation data is examined in Figure 10. As indicated in the figure, the predicted data both from the averaged model and the sampled ensemble are in good agreement with the measured C-response data. The sampling history of the Markov chain is shown Figure 11a, indicating a well-mixing chain. The 1D marginal distribution in Figure 11b for the misfit indicates a relatively narrow high-probability region with a low-probability area covering a large area. The 1D marginal distribution for the interface number shows that the probability peaks at 7, covering a large area due to the possible fact that the data are less sensitive to the interface number (see Figure 11c).

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 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.