Determination of the Regularization Parameter to Combine Heterogeneous Observations in Regional Gravity Field Modeling

Qing Liu,Roland Pail,Martin Willberg,Michael Schmidt

doi:10.3390/rs12101617

Abstract

Various types of heterogeneous observations can be combined within a parameter estimation process using spherical radial basis functions (SRBFs) for regional gravity field refinement. In this process, regularization is in most cases inevitable, and choosing an appropriate value for the regularization parameter is a crucial issue. This study discusses the drawbacks of two frequently used methods for choosing the regularization parameter, which are the L-curve method and the variance component estimation (VCE). To overcome their drawbacks, two approaches for the regularization parameter determination are proposed, which combine the L-curve method and VCE. The first approach, denoted as “VCE-Lc”, starts with the calculation of the relative weights between the observation techniques by means of VCE. Based on these weights, the L-curve method is applied to determine the regularization parameter. In the second approach, called “Lc-VCE”, the L-curve method determines first the regularization parameter, and it is set to be fixed during the calculation of the relative weights between the observation techniques from VCE. To evaluate and compare the performance of the two proposed methods with the L-curve method and VCE, all these four methods are applied in six study cases using four types of simulated observations in Europe, and their modeling results are compared with the validation data. The RMS errors (w.r.t the validation data) obtained by VCE-Lc and Lc-VCE are smaller than those obtained from the L-curve method and VCE in all the six cases. VCE-Lc performs the best among these four tested methods, no matter if using SRBFs with smoothing or non-smoothing features. These results prove the benefits of the two proposed methods for regularization parameter determination when different data sets are to be combined.

Highlights

Gravity field modeling is a major topic in geodesy, and it supports many applications, including physical height system realization, orbit determination, and solid earth geophysics
In study cases A and B, the differences between the results delivered by variance component estimation (VCE) and the ones from the proposed methods are large, i.e., the Root mean square error (RMS) errors obtained from the VCE-Lc or Lc-VCE are 41% and 61% smaller than the ones obtained by VCE in case A and B, respectively
VCE cannot provide sufficient regularization in this case. This result coincides with the conclusion drawn by Naeimi [60], who showed that VCE gives similar RMS errors as the L-curve method at the orbit level, but it is not able to provide sufficient regularization at the Earth surface for the regional solutions based on satellite data

Summary

Introduction

Gravity field modeling is a major topic in geodesy, and it supports many applications, including physical height system realization, orbit determination, and solid earth geophysics. To model the gravity field, approaches need to be set up to represent the input data as well as possible. The global gravity field is usually described by spherical harmonics (SH), due to the fact that they fulfill the Laplacian differential equation and are orthogonal basis functions on a sphere; see, e.g., [1,2] for more detailed explanations. The computation of the corresponding spherical harmonic coefficients requires a global homogeneous coverage of input data. As this requirement cannot be fulfilled, SHs cannot represent data of heterogeneous density and quality in a proper way [3,4]. The method based on SRBFs will be the focus of this work

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Conclusion