ABSTRACT

Analytical and numerical methods of solving inverse problems for logarithmic and Newtonian potentials are investigated. The following contact problem in the case of a Newtonian potential is considered: In a domain \(\Omega\{\Omega: -l\leq x,y \leq l, H - \varphi(x,y) \leq z \leq H\}\) there are sources with density \(\rho (x, y)\) that perturb the Earth’s gravitational field. Here \(\varphi (x, y)\) is a nonnegative compactly supported function with a support \(\Omega= [- l, l]^2\), \(0 \leq \varphi (x, y) \leq H\). It is required to simultaneously restore the depth \(H\) of the contact surface \(z = H\), the density \(\rho (x, y)\) of the sources, and the function \(\varphi (x, y)\). Methods of simultaneous determination based on nonlinear models of potential theory are developed in this paper. The following basic information is used in case of a Newtonian potential: (1) values of the gravity field and its first and second derivatives; (2) values of the gravity field at different heights. A method of simultaneous recovery of the functions \(\rho (x, y)\), \(\varphi (x, y)\) and the constant \(H\) in analytical form is demonstrated. Iterative methods for the simultaneous recovery are constructed. The efficiency of the numerical methods is demonstrated on model examples.

中文翻译：

---

接触表面反重力问题中密度和表面方程的同时恢复

摘要

研究了求解对数和牛顿势逆问题的解析和数值方法。考虑牛顿势的以下接触问题：在域\（\ Omega \ {\ Omega：-l \ leq x，y \ leq l，H-\ varphi（x，y）\ leq z \ leq H \} \） 有一些密度为\（\ rho（x，y）\）的源 会扰动地球的引力场。这里 \（\ varphi（x，y）\） 是具有支持 \（\ Omega = [-l，l] ^ 2 \）， \（0 \ leq \ varphi（x，y）\ leq H \）。需要同时恢复 接触面的深度 \（H \）\（z = H \）和密度\（\ rho（x，y）\） 源代码和函数\（\ varphi（x，y）\）。本文提出了一种基于势能理论非线性模型的同时确定方法。在牛顿势的情况下，使用以下基本信息：（1）重力场及其一阶和二阶导数的值；（2）不同高度处的重力场值。 演示了以解析形式同时恢复函数\（\ rho（x，y）\）， \（\ varphi（x，y）\） 和常数\（H \）的方法。构建了同时恢复的迭代方法。数值方法的有效性在模型示例中得到了证明。