Spherical approximating and interpolating moving least squares in geodesy and geophysics: a case study for deriving gravity acceleration at sea surface in the Persian Gulf

M Kiani

doi:10.1515/jogs-2020-0112

Abstract

Abstract This paper is aimed at introducing the concept of Spherical Interpolating Moving Least Squares to the problems in geodesy and geophysics. Based on two previously known methods, namely Spherical Moving Least Squares and Interpolating Moving Least Squares, a simple theory is formulated for using Spherical Moving Least Squares as an interpolant. As an application, a case study is presented in which gravity accelerations at sea surface in the Persian Gulf are derived, using both the approximation and interpolation mode of the Spherical Moving Least Squares. The roles of the various elements in the methods-weight function, scaling parameter, and the degree of spherical harmonics as the basis functions-are investigated. Then, the results of approximation and interpolation are compared with the field data at sea surface, collected by shipborne gravimetry approach. Finally, the results are compared with another independent interpolation method-spline interpolation. It is shown that in this particular problem, SMLS approximation and SIMLS interpolation present a better accuracy than spherical splines.

Highlights

IntroductionThe interpolation and approximation problems that frequently appear in geodesy and geophysics are of key importance for geodetic community and as the result, geodesists and geophysicists have investigated many different approximants and interpolants
With the technological advances in geodesy and geophysics, there has always been the need to devise new mathematical methods.The interpolation and approximation problems that frequently appear in geodesy and geophysics are of key importance for geodetic community and as the result, geodesists and geophysicists have investigated many different approximants and interpolants
A case study is presented in which gravity accelerations at sea surface in the Persian Gulf are derived, using both the approximation and interpolation mode of the Spherical Moving Least Squares

Summary

Introduction

The interpolation and approximation problems that frequently appear in geodesy and geophysics are of key importance for geodetic community and as the result, geodesists and geophysicists have investigated many different approximants and interpolants. The observations in geodesy and geophysics are almost never continuously sampled, i.e. they are not known in every point and do not have an analytical formula. One of the most widely used assumptions is smoothness. This is because provably most of the functions in geodesy and geophysics are smooth, especially when one considers the gravity eld functions-potential and acceleration. This assumption leads to the concept of spline interpolation and approximation. There have been many works on the theoretical and applied aspects of the spline functions, including (Freeden, et al, 2018a), (Freeden, et al, 1998), (Freeden, 2009), (Freeden, 1981), (Kiani , 2019b), (Kiani , 2020), (Wahba, 1981), and (Wahba, 1990)

Objectives

Conclusion