# Spherical Planting Inversion of GRAIL Data
## 2 Materials and Methods
### Spherical Planting Inversion
\(l_{j,k}^{\beta}\) is the distance from the \(j\)-th tesseroid's centroid to the \(k\)-th seed's centroid. Regardless of how a tesseroid's length and width vary with its position, the modified version \(l_{j,k}^{\beta}\) is given by  
\[l_{j,k}^{\beta}=\sqrt{\left(j_{x}-k_{x}\right)^{2}+\left(j_{y}-k_{y}\right)^{ 2}+\left(j_{r}-k_{r}\right)^{2}} \tag{6}\]  
where \(\left(j_{x},j_{y},j_{r}\right)\) and \(\left(k_{x},k_{y},k_{r}\right)\) stand for the order of the \(j\)-th tesseroid and the \(k\)-th seed along the longitude, latitude, and radius, respectively. Here, \(j=j_{x}+\left(j_{y}-1\right)\times n_{x}+\left(y_{r}-1\right)\times n_{x} \times n_{y}\). The solution of the spherical planting inversion of GRAIL data is recovered through a growth procedure divided into the following four steps, as shown in Figure 1.  
1. Initially, this algorithm requires a set of \(N_{\rm s}\) seeds, each of which is a tesseroid of the geophysical model. Seed locations can be obtained by Euler deconvolution or by manual selection, and the user can specify the density contrast of seeds. Meanwhile, the density contrast of all other tesseroids equals zero. 2. Secondly, for each seed, a list is constructed of neighboring tesseroids that share a face with the seed or neighboring ones. In constraint 3, only seeds will have a specific density contrast, and all other tesseroids' density contrasts are set to zero. Then the initial total objective function \(P^{(0)}\) is given by  
\[P^{(0)}=\phi_{d}^{(0)}+\mu\phi_{m}^{(0)} \tag{7}\]  
where, the initial total data-misfit function is given by  
\[\phi_{d}^{(0)}=d-\sum_{i=1}^{N_{c}}\rho_{s}a_{s} \tag{8}\]  
Here, \(\rho_{s}\) is the density-contrast value of the \(s\)-th seed, and the vector \(a_{s}\) consists of kernel functions which are gravitational effects at all observation points due to the \(s\)-th seed. Due to there being no neighboring tesseroids in the initial step, \(\phi_{m}^{(0)}=0\). 3. An iteration involves \(N\)s accretions in a spherical planting inversion. This algorithm finds a tesseroid from its neighboring tesseroid list for every accretion by minimizing the objective function and guaranteeing convergence of the following two conditions:  
\[\left|\phi_{d}^{(new)}-\phi_{d}^{(old)}\right|/\phi_{d}^{(old)}\geq\delta \tag{9}\]  
where the total data-misfit function \(\phi_{d}^{(new)}\) is calculated for the chosen neighboring tesseroids in the current accretion and all tesseroids whose density contrast is not equal to zero, \(\phi_{d}^{(old)}\)  
\[\phi_{\mathbf{d}}=\left\|\mathbf{r}\right\|_{2}=\left(\sum_{i=1}^{N_{\mathbf{d}}}(g_{i}- \mathbf{d}_{i})^{2}\right)^{1/2} \tag{2}\]0 Four stages of the spherical planting algorithm: (**a**) Initialization of seeds (in cyan) at the first stage; (**b**) Initialization of neighboring tesseroids (in green) of seeds at the second stage; (**c**) Finding the optimal tesseroid (in red) at first accretion, and (**d**) updating neighboring tesseroids of the first seed at the third stage; (**e**) Finding the optimal tesseroid at second accretion, and (**f**) updating neighboring tesseroids of the second seed at the fourth stage. is the total data-misfit function at the previous accretion. The small positive scalar \(\delta\) ranges from \(10^{-2}\) to \(10^{-6}\). \[\phi_{\mathbf{d}}=\left\|\mathbf{r}\right\|_{2}=\left(\sum_{i=1}^{N_{\mathbf{d}}}(g_{i}- \mathbf{d}_{i})^{2}\right)^{1/2} \tag{2}\]1  
where \(P^{(new)}\) and \(P^{(old)}\) are total global functions of current accretion and precious accretion, respectively. Ensuring the chosen tesseroid satisfies Equations (9) and (10), and treating it as the optimal tesseroid in the current iteration, then, the algorithm changes its density contrast from zero to the seed's and removes it from the seed's neighboring tesseroids list to guarantee constraint 3; otherwise, the algorithm will traverse over other seeds. 4. In any accretion at the previous iteration, if the convergence conditions (Equations (9) and (10)) are met, the growth process continues.