Forward scattering series and seismic events: Far field approximations, critical and postcritical events

Inverse scattering series is the only nonlinear, direct inversion method for the multi-dimensional, acoustic or elastic equation. Recently developed techniques for inverse problems based on the inverse scattering series [Weglein et al., Geophys., 62 (1997), pp. 1975-1989; Top. Rev. Inverse Problems, 19 (2003), pp. R27-R83] were shown to require two mappings, one associating nonperturbative description of seismic events with their forward scattering series description and a second relating the construction of events in the forward to their treatment in the inverse scattering series. This paper extends and further analyzes the first of these two mappings, introduced, for 1D normal incidence, in Matson [J. Seismic Exploration, 5 (1996), pp. 63-78] and later extended to two dimensions in Matson [An Inverse Scattering Series for Attenuating Elastic Multiples from Multicomponent Land and Ocean Bottom Seismic Data, Ph.D. thesis, Department of Earth and Ocean Sciences, University of British Columbia, Vancouver, BC, Canada, 1997]. It brings a new and more rigorous understanding of the mathematics and physics underlying the calculation of terms in the forward scattering series and the events in the seismic model. The convergence of the series for 1D acoustic models is examined, and the earlier precritical analysis is extended to critical and postcritical reflections. An explanation is proposed for the divergence of the series for postcritical incident planewaves.