Construction of analytical approximations of geopotential fields with the consideration of their fractal structure

Abstract

Principal trends in the development of mathematical geophysics at the modern stage include the comprehensive development of the approximation approach [1]. Researchers widely use approximations of external elements of gravitational and magnetic fields by equivalent source assemblages ( source-like approximations [2]) and numerical field simulations [3]. We propose a new method for the approximation of geopotential fields by grid distribution of sources. This method takes into account the fractal structure of the fields and is based on the quadrotree technique used for the compression of digital graphic images [4]. The essence of the considered approximation transformations is as follows: all information related to the observed geophysical field U ( x , y , z ) is stored as a certain number k of vectors of parameter P = { p 1 , p 2 , …, p n } of sources creating the simulation field U mod ( x , y , z ) , which is practically equivalent to the field U ( x , y , z ) . The unknown parameters of sources are found by solving an inverse problem (IP) usually consisting in minimization of the functional within the ensemble µ of the field specification points. In the process of minimization, we solve a system of linear algebraic equations (SLAE) for the linear IP formulation and a sequence of SLAE for the nonlinear IP formulation. At F ( k , P ) ≤ e , where e is a sufficiently Fk P