On the anomalies in gravity associated with lateral inhomogeneities in temperature: 1. Main relationships

Abstract

Green’s functions determining the radial displacements of the geoid and the external surface caused by the action of the arbitrarily distributed sources are calculated for a real model of a radially heterogeneous viscoelastic gravitating Earth. The variations in the external potential are contributed by three factors: the reduction in density in the heated area; the increase in density in the external (not heated) area due to its elastic compression, and the attraction of the subsurface simple layer, which is formed due to the changes in the shape of the external surface under its elastic deformation. The total effect of these factors is represented in the form of a series expansion in the spherical functions. The asymptotic analysis of the solutions shows that in the case of the spherical functions of sufficiently high orders, the ratios between the radial displacements of the geoid and the radial displacements of the external surface tend to zero. Since at the high orders of spherical functions the effects of sphericity and radial heterogeneity of the Earth are negligible, this statement is reduced to the fact that at any thermoelastic deformation of the uniform viscoelastic halfspace with arbitrary rheology, the three mentioned effects compensate each other. This compensation results in an interesting effect: even the weak radial inhomogeneities of the medium (e.g., those associated with the depth variations of the rheology) are not only capable of significantly changing the magnitude of radial displacements of the geoid but can even change their sign. Thus, the cross-correlation analysis of the data on the topography and shape of the geoid opens the possibility in principle to obtain new and quite strict constraints on the depth dependence of the rheological parameters of the medium (e.g., effective viscosity). For practical application of this method, it is necessary to quite reliably separate the effects associated with the horizontal heterogeneity of temperature and chemical composition of the mantle and crust. In the first part of the paper, we present the key relationships that are used in the further analysis; in the second part, we will present the results of the numerical calculations for the real Earth model and, based on this, the first results of interpretation of the obtained proportions between the coefficients of spherical expansions of the geoid, topography, and seismic velocities.