Numerically stable computation of complete synthetic seismograms including the static displacement in plane layered media

We present a new method for calculating dynamic and static displacements arising from a buried point source in a plane layered, isotropic, elastic half-space. We avoid the problem of the loss of precision in solutions for P-SV wave motion by integrating a system of minor vector equations with solutions constructed from pairs of solutions to the equations of motion in the frequency-wavenumber domain. The resulting algorithm is efficient, and numerically stable at zero frequency and at high frequencies, and thus can be used to compute complete synthetic seismograms that include the static offset. For the special case of the static deformation of a homogeneous half-space, we show that our results are equivalent to formulae that are often currently used in the geodetic community. An advantage of our method is that a single algorithm can be used to model both seismological and geodetic measurements of ground motion. We illustrate one seismological application of our method using continuous Global Positioning System (GPS) data obtained for a recent earthquake; seismograms calculated in a realistic layered crustal model emulate the observed dynamic and static displacements well. If the calculation is only carried out at 0 Hz, we obtain an image of the static surface deformation, and thus our method also has potential applications in the modelling of coseismic surface displacements measured with, for example, Synthetic Aperture Radar Interferometry.