Plane-wave decomposition and reconstruction of spherical-wave seismograms as a linear inverse problem

The plane-wave decomposition of the vertical displacement component of the spherical-wave field corresponding to a compressional point source is solved as a set of inverse problems. The solution utilizes the power and stability of the Backus & Gilbert (smallest and flattest) model-construction techniques, and achieves computational efficiency through the use of analytical solutions of the integrals which are involved. The theory and algorithms developed in this work allow stable and efficient reconstruction of the spherical-wave field from a relatively sparse set of their plane-wave components. However, the algorithms do not formally conserve the correct amplitudes of the seismograms. Comparison of the algorithms with direct integration of the Hankel transform shows very little or no advantage for the transformation from the time-distance (t-x) domain to the delay time - angle of emergence (τ-γ) domain if the seismograms are equi-sampled spatially. However, for cases where the observed seismograms are not equally spaced and for the transformation τ-γto t-x, the proposed schemes are superior to the direct integration of the Hankel transform. Applicability of the algorithms to reflection seismology is demonstrated via the solution to the problem of trace interpolation and that of separation of converted S modes from other modes presented in common-source gathers. In both cases the application of the algorithms to a set of synthetic reflection seismograms yields satisfactory results.Science, Faculty ofEarth, Ocean and Atmospheric Sciences, Department ofGraduat