The spherical radial basis function (SRBF) approach, widely used in gravity modeling, is theoretically surveyed from a viewpoint of random field theory. Let the gravity potential be a random field which is represented as an integral functional of another random field, namely an isotropic Gaussian random field (IGRF) on a sphere inside the Bjerhammar sphere with the SRBF as the integral kernel. When the integration is approximated by a discrete sum within a local region, one gets the widely applicable SRBF model. With this theoretical study, the following two findings are made. First, the IGRF implies a Gaussian prior on the spherical harmonic coefficients (SHCs) of the gravity potential; for this prior the SHCs are independent with each other and their variances are degree-only dependent. This should be reminiscent of two well-known priors, namely the power-law Kaula’s rule and the asymptotic power-law Tscherning-Rapp model. Second, the IGRF-SRBF representation is non-unique. Benefiting from this redundant representation, one can employ a simple IGRF, e.g., the simplest white field, and then design the SRBF accordingly to represent a potential with desired prior statistical properties. This can simplify the corresponding SRBF modeling significantly; to be more specific, the regularization matrix in parameter estimation of the SRBF modeling can be chosen to be a diagonal matrix, or even the naïve identity matrix.

广泛用于重力建模的球面径向基函数 （SRBF） 方法从随机场理论的角度进行理论研究。设重力势是一个随机场，它表示为另一个随机场的积分函数，即 Bjerhammar 球体内球体上的各向同性高斯随机场 （IGRF），SRBF 作为积分核。当积分由本地区域内的离散和近似时，可以得到广泛适用的 SRBF 模型。通过这项理论研究，得出以下两个发现。首先，IGRF 意味着重力势的球谐系数 （SHC） 存在高斯先验;对于这个先验，SHC 彼此独立，它们的方差仅与度数相关。这应该让人想起两个著名的先验，即幂律 Kaula 规则和渐近幂律 Tscherning-Rapp 模型。其次，IGRF-SRBF 表示是非唯一的。受益于这种冗余表示，人们可以使用一个简单的 IGRF，例如最简单的白色区域，然后相应地设计 SRBF 以表示具有所需先验统计特性的势能。这可以显著简化相应的 SRBF 建模;更具体地说，SRBF 建模参数估计中的正则化矩阵可以选择对角矩阵，甚至是朴素单位矩阵。