Algorithm for calculation of the static displacements in the layered spherical earth based using the matrix impedance technique

Abstract

The author proposes a semianalytic algorithm for calculating the static displacements from a point source with a symmetrical seismic moment tensor in a layered ball. The algorithm is based on the solution representation using vector surface harmonics. Unknown functions of the radial variable ― radial functions ― form a motion−stress vector, which satisfy a system of ordinary differential equations (SODEs). The algorithm of calculation of the radial function is similar to that proposed by the author (Pavlov, 2009, 2013) for calculation of depth functions in layered half-space with replacement of the exponential function of the depth for power functions of the radial variable. The problem is reduced to the calculation of the impedance ― the matrix that converts the motion vector into the stress vector ― and the propagator of the motion vector. To obtain the solution of SODEs, an analytical solution for a spherical layer is used. In contrast to the case of infinite space for the solution construction, it is necessary to eliminate the translation and rotation of the ball as a rigid body. These conditions are formulated as the immobility of the center of mass and the equality to zero of angular momentum with respect to the ball center.