Static and dynamic displacement near a fault with the discrete wavenumber method

When we represent seismic waves by standard wavenumber decomposition schemes such as the discrete wavenumber method, there is ambiguity in the choice of one parameter, the truncate number of the wavenumber integration. Although there is a conventional truncation number to obtain accurate seismograms, few studies have paid attention to seismograms including static displacement (i.e. w = 0). We estimate this parameter in order to accurately calculate not only body waves like P- and S-wave but also static displacement, by overcoming the poor convergence of integration over horizontal wavenumbers. In order to find a suitable value of the truncation number of wavenumbers, we utilize wavenumber-domain solutions from spatial-domain solutions for a homogeneous half space, using the analytic formulation of Okada [Bull. Seismol. Soc. Am. 75 (1985) 1135-1154] and estimate a region where spectral energy is concentrated. We shall identify the range in which the energy is reduced into 1/100 of the maximum amplitude in k(x) and k(y), and define the larger of k(x) and k(y) in this estimation as the truncation number of wavenumbers, k(max). For a finite rectangular fault buried in a half space, we find that a suitable value of the truncation number of wavenumber k(max) is 4 km(-1) by comparing the analytical solution of Okada for static deformation. If there is large moment release near the surface (e.g. artificial source such as explosion of dynamites), the convergence over horizontal wavenumbers becomes very poor not only for static but also for dynamic motions. Our results show that 4 km(-1) is a reasonable value when a fault extend to several kilometers under the surface even if the moment release at each, point on the fault surface is constant. In a realistic case, earthquake source has large moment release at several kilometers under the surface, so we can obtain surface displacement accurately, including static displacement in any realistic cases, with the above value of k(max). (C) 2003 Elsevier Science B.V All rights reserved.