Regularized wave equation migration for imaging and data reconstruction

The reflection seismic experiment results in a measurement (reflection seismic data) of the seismic wavefield. The linear Born approximation to the seismic wavefield leads to a forward modelling operator that we use to approximate reflection seismic data in terms of a scattering potential. We consider approximations to the scattering potential using two methods: the adjoint of the forward modelling operator (migration), and regularized numerical inversion using the forward and adjoint operators. We implement two parameterizations of the forward modelling and migration operators: source-receiver and shot-profile. For both parameterizations, we find requisite Green's function using the split-step approximation. We first develop the forward modelling operator, and then find the adjoint (migration) operator by recognizing a Fredholm integral equation of the first kind. The resulting numerical system is generally under-determined, requiring prior information to find a solution. In source-receiver migration, the parameterization of the scattering potential is understood using the migration imaging condition, and this encourages us to apply sparse prior models to the scattering potential. To that end, we use both a Cauchy prior and a mixed Cauchy-Gaussian prior, finding better resolved estimates of the scattering potential than are given by the adjoint. In shot-profile migration, the parameterization of the scattering potential has its redundancy in multiple active energy sources (i.e. shots). We find that a smallest model regularized inverse representation of the scattering potential gives a more resolved picture of the earth, as compared to the simpler adjoint representation. The shot-profile parameterization allows us to introduce a joint inversion to further improve the estimate of the scattering potential. Moreover, it allows us to introduce a novel data reconstruction algorithm so that limited data can be interpolated/extrapolated. The linearized operators are expensive, encouraging their parallel implementation. For the source-receiver parameterization of the scattering potential this parallelization is non-trivial. Seismic data is typically corrupted by various types of noise. Sparse coding can be used to suppress noise prior to migration. It is a method that stems from information theory and that we apply to noise suppression in seismic data.