On the Uniqueness of Solutions for Variational Problems in Mathematical Geophysics Within the Framework of the Method of Linear Integral Representations

Various versions (local and regional, as well as stationary and non-stationary) of the method of linear integral representations in solving inverse problems of geophysics are considered. Particular attention is paid to the issues of unique solvability of variational problems that arise in the interpretation of geophysical data. This paper emphasizes the application of the approximation approach in constructing analytical models of any physical fields, and highlights a class of problems that can be considered conditionally solvable in closed form. This article proves theorems on the linear independence of function systems, a finite linear combination of which represents the solution to the inverse problem. Theorems of this kind are of fundamental importance both for investigating the well-posedness of the inverse problem formulation and for practical application, as choosing an observation network with prescribed properties will reduce the costs of conducting geological and other surveys.