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.

积分变换是引力场建模的重要数学工具。积分变换的基本假设是全球数据覆盖，但高分辨率和准确的重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）通过球冠内数据的数值积分计算出的近区的影响和（2）由球冠之外的数据合成的远区的影响谐波展开。这种分解的理论和数值方面经常被研究用于球体上的各向同性积分变换，例如 Hotine 积分公式、Poisson 积分公式和 Stokes 积分公式。本文系统回顾了球积分公式远区效应的数学理论，将扰动引力势或其纯径向导数转化为引力场的可观测量，即扰动引力势及其径向、一阶、二阶或三阶的水平导数或混合导数。这些公式在 MATLAB 软件中实现，并在闭环仿真中进行验证。通过检查截断误差系数的行为来研究调和展开式的选定属性。这里提出的数学公式对于精确引力场建模中的正向或反演问题的实际解决或在研究积分变换的统计特性时是必不可少的。