One option of constructing three-dimensional distribution of the mass and its derivatives for a spherical planet Earth

: pp. 36 – 44
Received: September 01, 2016
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
Department of Cartography and Geospatial Modeling, Institute of Geodesy, Lviv Polytechnic National University
Department of Cartography and Geospatial Modeling of Lviv Polytechnic National University

Purpose. To build a three-dimensional function of the mass distribution of the Earth's interior according to the parameters (Stokes constant to the second order inclusive) of the external gravitational field of the Earth without considering the minimum deviation from its known density models in geophysics. Methodology. The classic methods of constructing mass distribution use only the Stoke’s constants zero and second orders. In iterative methods of determining the distribution models the reference model of density is taken for zero approximation which is agreed upon by Stoke’s constants up to the second order inclusive. Further, the coefficients of potential expansion to a certain order are taken into account, but their contribution to the function of mass density does not investigate. This research provides an attempt to obtain such an estimation. The proposed method is approximate, but in the iterative process a function of the density is not only used, but also its derivatives. Bringing the order moments of density toward the controlled values (values that are defined on the surface of a sphere) makes it possible to analyze the process of successive approximations. Results. In contrast to the second-order model, which describes the global gross irregularities, the obtained distribution function gives a detailed picture of the placement density anomalies (deviation of three-dimensional functions from the average on the sphere – “izoden”). Analysis of maps at different depths
(2891 km core-mantle, 5,150 km of the inner-outer core) allow making preliminary conclusions about global redistribution of mass due to the rotating component of gravity across the radius: its dilution along the axis of rotation and accumulation of rejecting it. This is particularly evident for the equatorial regions. On the contrary, there is minimum deviation in the polar  regions of the Earth, which also have their own justification since the value of the rotation force decreases when moving away from the equator. The function of mass distribution, which is constructed using the proposed method, describes the mass distribution better. Originality. This research is in contrast to the classical results which have been obtained from the Adams-William’s equations for the derivatives of the density of one variable (depth), and make attempted to obtain derivatives using Cartesian coordinates. Using the gravitational field parameters up to second order increases the order of approximation of the distribution function of the masses of three variables from two to four through the possibility of restoring the planet's mass distribution by its derivatives. At the same time, in contrast to previous research, geophysical information accumulated in the reference PREM model is used, therefore, features of the internal structure are taken into account. Practical significance. The received function of mass distribution of the Earth can be used as a zero-order approximation when used in the presented algorithm Stokes constant of higher order. Their applications give the possibility to interpretate of the global anomalies of the gravitational field, and explore the geodynamic processes deep inside the Earth.

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