3D models of slow motions in the Earth’s crust and upper mantle in the source zones of seismically active regions and their comparison with highly accurate observational data: I. Main relationships

Constructing detailed models for postseismic and coseismic deformations of the Earth’s surface has become particularly important because of the recently established possibility to continuously monitor the tectonic stresses in the source zones based on the data on the time variations in the tidal tilt amplitudes. Below, a new method is suggested for solving the inverse problem about the coseismic and postseismic deformations in the real non-ideally elastic, radially and horizontally heterogeneous, self-gravitating Earth with a hydrostatic distribution of the initial stresses from the satellite data on the ground surface displacements. The solution of this problem is based on decomposing the parameters determining the geometry of the fault surface and the distribution of the dislocation vector on this surface and elastic modules in the source in the orthogonal bases. The suggested approach includes four steps: 1. Calculating (by the perturbation method) the variations in Green’s function for the radial and tangential ground surface displacements with small 3D variations in the mechanical parameters and geometry of the source area (i.e., calculating the functional derivatives of the three components of Green’s function on the surface from the distributions of the elastic moduli and creep function within the volume of the source area and Burgers’ vector on the surface of the dislocations); 2. Successive orthogonalization of the functional derivatives; 3. Passing from the decompositions of the residuals between the observed and modeled surface displacements in the system of nonorthogonalized functional derivatives to their decomposition in the system of orthogonalized derivatives; finding the corrections to the distributions of the sought parameters from the coefficients of their decompositions in the orthogonalized basis; and 4. Analyzing the ambiguity of the inverse problem solution by constructing the orthogonal complement to the obtained basis. The described approach has the following advantages over the method of steepest descent which was used in our previous works: 1. Application of the perturbation method significantly reduces the volume of the computations in the real problems of coseismic and postseismic deformations (by three to four orders of magnitude when the data from a few dozens of observation points are used); 2. In contrast to the method of steepest descent, the suggested method always provides stable results. This means that adding the new satellite data does not alter the previously calculated coefficients in the low-order harmonics of the distributions of the sought parameters in the orthogonalized basis; this only changes the coefficients of the increasingly higher harmonics which determine the smallscale details in the sought distributions. 3. In contrast to the method of steepest descent, the suggested method is not only capable of constructing stable partial solutions of the inverse problem but also estimating the ambiguity of these solutions. The ambiguity is represented in terms of the superposition of the known functions contained in the orthogonal complement and, hence, with the growth of the amount of the analyzed data it is determined by the linear combination of the increasingly higher harmonics. In the second part of the paper, we present the results of the model numerical computations of Green’s function for the elastic displacements of the ground surface, which correspond to the case of the arbitrary geometry of the dislocation surface and arbitrary orientation of the dislocation vector for the real model of the radially heterogeneous gravitating Earth with the hydrostatic distribution of the initial stresses. The numerical calculations of the creep function in the upper mantle for the coseismic deformations and the ambiguity of the models of postseismic deformations in the vicinity of the source of the Great Tohoku earthquake (Japan) of March 11, 2011 are illustrated by the examples.