# Spherical Planting Inversion of GRAIL Data
## 2 Materials and Methods  
### Spherical Planting Inversion  
Following the ideas of [PERSON] and [PERSON] [36; 37], to avoid solving an extensive system of linear equations constructed by density imaging, the spherical planting inversion of GRAIL data is proposed using an L1-norm-based systematic search algorithm to reduce memory requirements and facilitate fast convergence by replacing iterative inversion with cumulative summation analysis. For this reason, a residual vector \(\mathbf{r}\) between the observation data \(\mathbf{d}\) and the prediction data \(\mathbf{g}\) is constructed by the L1-norm based on Equation (1). \[\mathbf{r}=\mathbf{d}-\mathbf{g} \tag{1}\]  
The data-misfit function is defined (based on the L2-norm) by least squares estimation, which can be expressed as  
\[\phi_{\mathbf{d}}=\left\|\mathbf{r}\right\|_{2}=\left(\sum_{i=1}^{N_{\mathbf{d}}}(g_{i}- \mathbf{d}_{i})^{2}\right)^{1/2} \tag{2}\]  
where the subscript \(i\) corresponds to the \(i\)-th observation point. \(N_{\mathbf{d}}=n_{x}\times n_{y}\). \(n_{x}\)\(n_{y}\), and \(n_{r}\) represent the section number of the interpretation model along the \(x\)-, \(y\)- and radial axes, respectively. Compared to the L2-norm, the L1-norm is not sensitive to data with significant errors so it can provide a robust estimation strategy. \[\phi_{\mathbf{d}}=\left\|\mathbf{r}\right\|_{1}=\sum_{i=1}^{N_{\mathbf{d}}}\lvert\mathbf{g}_{ i}-\mathbf{d}_{i}\rvert \tag{3}\]  
Conventional inversion applies regularization terms to reduce the instability and non-uniqueness of the solutions. Many inverse strategies have been proposed for recovering compact anomalous sources, such as maximizing their compactness [41], concentratingthem along one or more axes [42; 43], and minimizing their area/volume [44]. For example, [PERSON] [34] proposed a systematic search inversion of gravity data using the \"shape-of-anomaly\" function to reconstruct 2D compact bodies (i.e., with no hollows inside). However, this inversion algorithm only uses a single sign/density contrast to recover density distributions. Compared with [PERSON]'s (1986) method, [PERSON] and [PERSON] interpret multiple geologic sources with density contrasts of different signs [36]. Consider that recovering a density distribution from GRAIL data is an ill-conditioned problem, it can be converted into a well-conditioned problem by adopting the following constraints:  
(1) A compact scheme ensures that there are no voids in the recovered density distributions, which are obtained from spherical gravity inversion with a seed [36; 41; 42; 43]. (2) An optimal prism is selected one by one from a neighboring tesseroids list, which is derived from a seed with density contrast \(\rho_{s}\), and shares faces with the seed or its adjacent tesseroids. Then, the optimal prism's density value \(\rho_{j}\) is set to the same as the seed. (3) For other uninvolved tesseroids, their density value equals zero. Based on the above constraint strategies, the objective function of the spherical planting inversion of GRAIL data is given by  
\[P^{a}(\mathbf{m})=\phi_{d}+\mu\phi_{m}\rightarrow\min \tag{4}\]  
where, the regularization parameter \(\mu\) balances the trade-off between the data-misfit function \(\phi_{d}\) and the model-misfit function \(\phi_{m}\), and is defined in the model space that imposes geologic constraints on the recovered density distribution. \[\phi_{\mathbf{m}}=\sum_{k=1}^{N_{\rm s}}\left(\sum_{j=1}^{N_{\rm m}}\frac{\rho_{j} }{\rho_{j}+\varepsilon}\zeta(j,k)l_{j,k}^{\beta}\right) \tag{5}\]  
where, \(\rho_{j}\) is the \(j\)-th element of the density vector \(\mathbf{m}\), whose size is \(N_{\mathbf{m}}\), and \(N_{\rm s}\) is the number of seeds. \(\varepsilon\) is a small positive count and will avoid a singularity error when \(\rho_{j}=0\). The sign function \(\varsigma(j,k)\) returns 1 when the \(j\)-th tesseroid is derived from the \(k\)-th seed, as shown in Figure 1; otherwise, it returns 0. When \(N_{\rm s}=1\), Formula (5) is similar to Formula (10) in [PERSON] and [PERSON]'s paper [36].