Physical geodesy applies potential theory to study the Earth’s gravitational field in space outside and up to a few km inside the Earth’s mass. Among various tools offered by this theory, boundary-value problems are particularly popular for the transformation or continuation of gravitational field parameters across space. Traditional problems, formulated and solved as early as in the nineteenth century, have been gradually supplemented with new problems, as new observational methods and data are available. In most cases, the emphasis is on formulating a functional relationship involving two functions in 3-D space; the values of one function are searched but unobservable; the values of the other function are observable but with errors. Such mathematical models (observation equations) are referred to as deterministic. Since observed data burdened with observational errors are used for their solutions, the relevant stochastic models must be formulated to provide uncertainties of the estimated parameters against which their quality can be evaluated. This article discusses the boundary-value problems of potential theory formulated for gravitational data currently or in the foreseeable future used by physical geodesy. Their solutions in the form of integral formulas and integral equations are reviewed, practical estimators applicable to numerical solutions of the deterministic models are formulated, and their related stochastic models are introduced. Deterministic and stochastic models represent a complete solution to problems in physical geodesy providing estimates of unknown parameters and their error variances (mean squared errors). On the other hand, analyses of error covariances can reveal problems related to the observed data and/or the design of the mathematical models. Numerical experiments demonstrate the applicability of stochastic models in practice.

物理大地测量学应用位势理论来研究地球质量外部和内部几公里范围内的地球引力场。在该理论提供的各种工具中，边值问题在跨空间引力场参数的变换或延续方面特别受欢迎。随着新的观测方法和数据的出现，早在十九世纪就提出和解决的传统问题逐渐被新问题所补充。在大多数情况下，重点是在 3D 空间中制定涉及两个函数的函数关系；一个函数的值被搜索但不可观察；另一个函数的值是可观察的，但有错误。这种数学模型（观测方程）被称为确定性的。由于带有观测误差的观测数据用于其解决方案，因此必须制定相关的随机模型以提供估计参数的不确定性，以便评估其质量。本文讨论了为物理大地测量学当前或可预见的将来使用的重力数据制定的势理论的边值问题。回顾了它们的积分公式和积分方程形式的解，制定了适用于确定性模型数值解的实用估计量，并介绍了它们相关的随机模型。确定性和随机模型代表了物理大地测量学问题的完整解决方案，提供了未知参数及其误差方差（均方误差）的估计。另一方面，误差协方差分析可以揭示与观测数据和/或数学模型设计相关的问题。 数值实验证明了随机模型在实践中的适用性。