Algorithm for calculating synthetic seismograms in a layered half-space with application of matrix impedance

A new algorithm is proposed for calculating the complete synthetic seismograms from a point source in the form of the sum of a single force and a dipole with an arbitrary seismic moment tensor in a plane layered medium composed of homogenous elastic isotropic layers. Following the idea of (Alekseev and Mikhailenko, 1978), an artificial cylindrical boundary is introduced, on which the boundary conditions are specified. For this modified problem, the exact solution (in terms of the displacements and stresses on the horizontal plane areal element) in the frequency domain is derived and substantiated. The unknown depth-dependent coefficients form the motion-stress vector, whose components satisfy the known system of ordinary differential equations. This system is solved by the method that involves the matrix impedance and propagator for the vector of motion, as previously suggested by the author in (Pavlov, 2009). In relation to the initial problem, the reflections from the artificial boundary are noise, which, to a certain degree, can be suppressed by selecting a long enough distance to this boundary and owing to the presence of a purely imaginary addition to the frequency. The algorithm is not constrained by the thickness of the layers, is applicable for any frequency range, and is suitable for computing the static offset.