Application of 1D inverse scattering and projection onto convex sets to inversion of seismic data

Abstract

The Born series provides a framework for constructing a connection between earth model parameters and a wavefield characterised by perturbation about a known reference wavefield. The inverse scattering series provides a linear and a nonlinear expression for approximating earth's model parameters. Based on our results, the linear expression provides an accurate approximation of the 1D depth varying velocity profile. However, with increasing contrast, one has to utilise the nonlinear expression for increased accuracy. In seismic inversion, a major problem relates to the bandlimited nature of recorded seismic data, limiting the inversion for absolute amplitudes. To reconstruct missing low frequencies, we employ a frequency extrapolation method, in this case, projection onto convex sets (POCS) to reconstruct missing low frequencies critical for accurate inversion and reconstruction of depth varying velocity profile. Numerical results on 1D synthetic data and succeeding results on 1D physical modelling data demonstrate the efficiency of POCS in reconstructing critical missing low frequencies and subsequent advantage of nonlinear inverse scattering in approximating the change in velocity profile. In comparison, the linear scattering expression fails to provide a reasonable estimate of reflectors beyond the first interface.