Assessment of the direct inversion scheme for the quasigeoid modeling based on applying the Levenberg–Marquardt algorithm

Abstract

Methods for a spherical harmonic analysis and synthesis are often used in describing the global gravity field by means of spherical harmonics to a certain degree of spectral resolution. In local gravity field modeling, the radial basis functions (RBFs) can be used to parameterize the gravity field with a high spatial/spectral resolution because of their local support. Since the global integration is required in both cases, these two methods are somehow complementary. In physical geodesy, this is done practically by adopting a remove-compute-restore numerical scheme. According to this scheme, a long-wavelength part of the gravity spectrum is defined by means of the spherical harmonics while a residual (high-frequency) gravity contribution is treated based on using the RBFs. To find an optimal parameterization of the gravity field, the number of RBFs and their spatial configuration are optional. Moreover, the regularization is applied to stabilize the (ill-posed) gravity inversion. In this study, we utilize the Levenberg–Marquardt (LM) algorithm for finding the optimal number of RBFs and their 3-D spatial configuration (i.e., horizontal location and depth) simultaneously with a regularization parameter. The optimal choice of these parameters is based on minimizing the least-squares residuals between the predicted and observed values of the potential and gravity field. In numerical experiment, we apply the LM algorithm in two inverse schemes of solving the Molodensky’s problem. The results reveal that a direct inversion of the observed gravity data to the potential field (i.e., quasigeoid heights) yields a systematic bias. Moreover, this gravimetric solution has a low approximation quality in terms of the potential field. The systematic discrepancies are thus modeled and corrected for by combining the gravity and GPS-leveling data. We propose and apply the RBF-parameterization, which again utilizes the LM optimization algorithm in combining these data. We demonstrate that only a two-step approach provides a satisfactory approximation to both the gravity and potential field.