# The methodology of approximate construction of the three-dimensional mass distribution function and its gradient for the ellipsoidal planet subsidies

2020;
: 21-32

Authors:
1
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
2
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
3
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
4
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University

Purpose. To create an algorithm for constructing a three-dimensional masses distribution function of the planet and its derivatives taking into account the Stokes constants of arbitrary orders. Being based on this method, the task is to perform the research on the internal structure of the Earth. Methodology. The derivatives of the inhomogeneous mass distribution are presented by linear combinations of biorthogonal polynomials which coefficients are obtained from the system of equations. These equations follow from integral transformations of the Stokes constants, the calculation process is carried out by a sequential approximation and for the initial approximation we take a one-dimensional density model that is consistent with Stokes constants up to the second inclusive order. Next, the coefficients of expansion of the potential of higher orders are determined up to a predetermined order. In this case, the information on the power moments of the density of surface integrals makes it possible to analyze and control the iterative process. Results. The results of calculations using the software according to the described algorithm are obtained. A fairly high degree of approximation (sixth order) of three-dimensional mass distributions function is achieved. Carto diagrams were created being based on the values of deviations of the three-dimensional average distributions (“isodens”), which give a fairly detailed picture of the Earth’s internal structure. The presented maps of “inhomogeneity’s” at characteristic depths (2891 km core – mantle, 5150 km internal – external core) allow us to draw preliminary conclusions about global mass movements. At the same time, the information on derivatives is significant for interpretation. First of all, it should be noted that the gradient of “inhomogeneity’s” is directed toward the center of mass. The presented projections of this gradient on a plane perpendicular to the rotation axis (horizontal plane) show the tendency of spatial displacements. Scientific novelty. Vector diagrams of the gradient, in combination with carto diagrams, give a broad picture of the dynamics and possible mechanisms of mass movement within the planet. To a certain extent, these studies confirm the phenomenon of gravitational convection of masses. Practical significance. The proposed algorithm can be used in order to build regional models of the planet, and numerical results can be used to interpret global and local geodynamic processes inside and on the Earth’s surface.

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