Geoid’s potential $W_0$ from a weighted constrained optimization problem

Abstract

The aim of this study is to estimate the geoid’s potential value, $W_0$, based on a weighted constrained optimization problem (WCOP) utilizing the most recent mean sea surface (MSS) models and global geopotential models (GGMs). The weight factor introduced into this WCOP-based approach is defined as a function of three constituents, i.e., latitude, MSS height uncertainty, and GGM commission error. In the proposed method, according to the Gauss–Listing definition for the geoid, first, an equipotential surface is found that geometrically fits the global MSS in a least-squares sense. Next, the GGM-derived gravity potential value of this equipotential surface is regarded as $W_0$. In this work, an error analysis is also provided to estimate the uncertainty of $W_0$. Due to the importance of the input data in the computation of $W_0$, the sensitivity of $W_0$ to the input data is studied from various aspects. In this regard, we find that: (1) all the three constituents of the weight factor have a considerable effect on the $W_0$ value and therefore should duly be considered in the computational procedure; (2) the minimum degree of the GGM expansion to neglect the GGM omission error in the computation of $W_0$ is $n=300$; (3) the $W_0$ estimation is sensitive to the selected GGM through its commission error; and (4) the weight factor, which includes the inverse of the MSS height uncertainty, reduces the effect of the coastal regions on the estimated value of $W_0$. To demonstrate the impact of the WCOP-based approach on the $W_0$ value, this approach is compared with the approaches used in the International Association of Geodesy (IAG) JWG 0.1.1 based on the same input data. The results of this comparison show that the WCOP-based approach can make a considerable contribution to the estimation of $W_0$. Finally, using the DTU21 MSS model and the GO_CONS_GCF_2_DIR_R6 GGM, we obtain a new value of $62636853.745\pm 0.003\,\mathrm m^2\,s^{-2}$ for $W_0$ at the epoch 2010.0, which differs from the IAG conventional value (Drewes et al. in J Geod 90:907–1205, 2016. https://doi.org/10.1007/s00190-016-0948-z) by $0.345\pm 0.02\,\mathrm m^2\,s^{-2}$. We find that this discrepancy is mainly due to the use of different input data, although the contribution of the computational approach is also significant. The theoretical and numerical evaluations performed in this study show that the WCOP-based approach can properly realize the Gauss–Listing geoid definition, which is one of the conventions recommended by Sánchez et al. (J Geod 90:815–835, 2016. https://doi.org/10.1007/s00190-016-0913-x) to uniquely determine $W_0$. Therefore, this approach can be regarded as a candidate for updating the current conventional $W_0$ value, when it becomes obsolete.