Algorithm for constructing the subsoil distribution function considering its value on the surface

Abstract

The conventional approach to constructing a three-dimensional distribution of the Earth's masses involves using Stokes constants incrementally up to a certain order. However, this study proposes an algorithm that simultaneously considers all of these constants, which could potentially provide a more efficient method. The basis for this is a system of equations obtained by differentiating the Lagrange function, which takes into account the minimum deviation of the three-dimensional mass distribution of the planet's subsoil from one-dimensional referential one. An additional condition, apart from taking into account the Stokes constants, for an unambiguous solution to the problem is to specify the value of the function on the surface of the ellipsoidal planet. It is possible to simplify the calculation process by connecting the indices of summation values in a series of expansions to their one-dimensional analogues in the system of linear equations. The study presents a control example illustrating the application of the given algorithm. In its implementation, a simplified variant of setting the density on the surface of the ocean is taken. The preliminary results of calculations confirm the expediency of this approach and the need to expand such a technique with other conditions for unambiguously solving the inverse problem of potential theory. Objectives. To create and implement the algorithm that takes into account the density of the planet’s subsoil on its surface. Method. The mass distribution function of the planet's subsoil is represented by a decomposition into biorthogonal series, the coefficients of decomposition which are determined from a system of linear equations. The system of equations is obtained from the condition of minimizing the deviation function of the desired mass distribution from the initially determined two-dimensional density distribution (PREM reference model). Results. On the basis of the described algorithm, a three-dimensional model of the density distribution of subsoil masses in the middle of the Earth is obtained, which takes into account Stokes constants up to the eighth order inclusively and corresponds to the surface distribution of masses of the oceanic model of the Earth. Its concise interpretation is also presented.