Abstract: A method for determining spatial distribution of properties of formations in a subsurface volume using geophysical sensor signals recorded proximate the volume includes inversion processing an initial model of the spatial distribution. The inversion processing comprises at least second order optimizing. The second order optimizing comprises calculating a scalar for the identity matrix in limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization, modifying the scalar using an adaptive gradient type scheme to estimate an inverse Hessian matrix, and using the modified plurality of scalars to optimize the inversion processing. The method of estimating the inverse Hessian matrix in L-BFGS can be further extended to include convolutional operators.

Abstract: A method for modelling and migrating seismic data, that includes using an acoustic wave equation and a spatial distribution of one or more earth-model parameters. The acoustic wave equation is modified by including at least one secondary source term, and based on a seismic acquisition configuration, either calculating the seismic signals that would be detected from the modelled wavefield or migrating observed seismic signals or migrating residual signals as part of an inversion.