# Spherical Planting Inversion of GRAIL Data
## 1 Introduction  
Kernel matrices are usually generated using partial differential and integral equation methods in gravity field inversion. Partial differential equations, such as the finite-difference [1, 2], finite-element [3], and finite-volume methods [4, 5], are used to discretize Poisson's equation to obtain potential fields. However, these methods are difficult to apply to gravity inversion because higher-order derivatives involve higher-order differences or the fast Fourier transform. Comparatively, integral equations utilize the superposition principle to calculate the gravity contribution of complex geological bodies. The observation surface of gravity data in small areas can be approximated as a plane; in the Cartesian coordinate system, the potential field calculation in the spatial domain is usually performed using prisms or general polyhedral cells [6, 7, 8, 9]. In large regional and even global gravity problems, the observation surface is generally expressed and processed in spherical coordinates because of the need to consider the observation surface's curvature [10]. As a result, interpretation models are usually divided into tesseroids in a spherical coordinate system using the parameters radius, latitude, and longitude. The gravitational effect generated by each unit is calculated and then summed to obtain global gravity/tensor gravity data. As a result of the CHAMP (Challenging Mini-Satellite Payload), GRACE (Gravity Recovery and Climate Experiment), GOCE (Gravity Field and Steady-State Ocean Circulation Explorer), and GRACE Follow-On satellite launches [11], it is now possible to obtain large-scale high-precision gravity data economically. Inversion of the gravity field is a powerful tool for understanding the density distribution inside planets. Many gravity inversions have been proposed in recent years. [PERSON] et al. applied tesseroid units to the lunar GRAIL gravity inversion and obtained the 3D density structure of the Moon [12]. [PERSON] et al. [13] analyzed the Moho surface depth of the South American continent using tesseroid grids for a fast nonlinear inversion of GOCOSS model data in a spherical coordinate system. [PERSON] et al. [14] developed an efficient algorithm for gravity gradient full-tensor data, which was used to study the density distribution of the Mare Smythii mascon on the Moon. [PERSON] et al. [15] performed a 3D spherical gravity inversion with tesseroid units for the GL1500E model data to obtain the 3D density distribution of the Imbrium and Serenitatis mascon basins. ETOPO1 is a one-arc-minute global relief model of the Earth's surface that integrates land topography and ocean bathymetry provided by the National Oceanic and Atmospheric Administration [16]. Take a spherical gravity inversion with the ETOPO1 model as an example. When a 1- by 1- minute latitude-longitude grid is used to cover the globe, the number of observation points is 180 \(\times\) 60 \(\times\) 360 \(\times\) 60 = 2.3328 \(\times\) 10\({}^{8}\). The number of kernel matrix elements for potential inversion is the product of the number of tesseroid units and observation points. The memory requirements of the kernel matrix, whether storing double-precision or single-precision data, are far more than the physical memory size of existing mainstream workstations. Constructing kernel matrices is a time-consuming problem with high memory requirements for potential inversion, especially for large-potential data [17]. Fast and accurate three-dimensional gravity inversion of large-scale, multi-scale and massive data is currently performed by improving the computational performance of equipment, reducing data dimensionality, and speeding up the convergence of the inversion iterations. First, high-performance computation relies on multi-core CPUs, clustered computer systems, and GPUs. Then, massively parallel acceleration studies are performed to reduce time-consuming forward modeling and inversion [18]. For example, [PERSON] et al. [19] and [PERSON] et al.