Stable plane‐wave decomposition and spherical‐wave reconstruction: Applications to converted S-mode separation and trace interpolation

Plane‐wave decomposition of the vertical displacement component of a spherical‐wave field corresponding to a compressional point source is solved as a set of inverse problems. The solution method utilizes the power and stability of Backus and Gilbert (smallest and flattest) model‐construction techniques, and achieves computational efficiency through the use of analytical solutions to the involved integrals. The theory and algorithms developed in this work allow stable and efficient reconstruction of spherical‐wave fields from a relatively sparse set of their plane‐wave components. Comparison of the algorithms with discrete integration of the Hankel transform shows very little or no advantage for the transformation from the time‐distance (t-x) domain to the intercept time‐angle of emergence (τ-γ) domain if the seismograms are equisampled spatially. However, when the observed seismograms are not equally spaced or the transformation τ-γ to t-x is performed, the proposed schemes are superior to the discrete integration of the Hankel transform. Applicability of the algorithms to reflection seismology is demonstrated by means of the solution of the problem of trace interpolation, and also that of the 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.