球面积分变换的远区效应 I:径向边界值问题及其衍生物的公式

IF 4.9 2区 地球科学 Q1 GEOCHEMISTRY & GEOPHYSICS Surveys in Geophysics Pub Date : 2024-05-03 DOI:10.1007/s10712-023-09818-4
Michal Šprlák, Martin Pitoňák
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引用次数: 0

摘要

积分变换是重力场建模的重要数学工具。积分变换的一个基本假设是数据覆盖全球,但高分辨率和精确引力数据的可用性可能受到限制。因此,我们将全局积分分解为两部分:(1) 通过对球冠内的数据进行数值积分计算得出的近区效应;(2) 通过谐波展开合成的球冠外数据得出的远区效应。对于球面上的各向同性积分变换,例如霍廷积分公式、泊松积分公式和斯托克斯积分公式,经常对这种分解的理论和数值方面进行研究。在本文中,我们系统地回顾了球面积分公式的远区效应数学理论,这些公式将扰动引力势或其纯径向导数转化为引力场的可观测量,即扰动引力势及其一阶、二阶或三阶径向、水平或混合导数。这些公式在 MATLAB 软件中实现,并在闭环模拟中得到验证。通过检查截断误差系数的行为,研究了谐波展开的某些特性。本文介绍的数学公式对于实际解决精确引力场建模中的直接或逆问题,或研究积分变换的统计特性都是不可或缺的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Far-Zone Effects for Spherical Integral Transformations I: Formulas for the Radial Boundary Value Problem and its Derivatives

Integral transformations represent an important mathematical tool for gravitational field modelling. A basic assumption of integral transformations is the global data coverage, but availability of high-resolution and accurate gravitational data may be restricted. Therefore, we decompose the global integration into two parts: (1) the effect of the near zone calculated by the numerical integration of data within a spherical cap and (2) the effect of the far zone due to data beyond the spherical cap synthesised by harmonic expansions. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transformations on the sphere, such as Hotine’s, Poisson’s, and Stokes’s integral formulas. In this article, we systematically review the mathematical theory of the far-zone effects for the spherical integral formulas, which transform the disturbing gravitational potential or its purely radial derivatives into observable quantities of the gravitational field, i.e. the disturbing gravitational potential and its radial, horizontal, or mixed derivatives of the first, second, or third order. These formulas are implemented in a MATLAB software and validated in a closed-loop simulation. Selected properties of the harmonic expansions are investigated by examining the behaviour of the truncation error coefficients. The mathematical formulations presented here are indispensable for practical solutions of direct or inverse problems in an accurate gravitational field modelling or when studying statistical properties of integral transformations.

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来源期刊
Surveys in Geophysics
Surveys in Geophysics 地学-地球化学与地球物理
CiteScore
10.00
自引率
10.90%
发文量
64
审稿时长
4.5 months
期刊介绍: Surveys in Geophysics publishes refereed review articles on the physical, chemical and biological processes occurring within the Earth, on its surface, in its atmosphere and in the near-Earth space environment, including relations with other bodies in the solar system. Observations, their interpretation, theory and modelling are covered in papers dealing with any of the Earth and space sciences.
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