Parsimonious slope tomography based on eikonal solvers and the adjoint-state method

Abstract

Velocity macromodel building is an essential step of the seismic imaging workflow. Indeed, obtaining acceptable results through migration or full waveform inversion is highly dependent on the kinematic accuracy of the background/initial velocity model. Two decades ago, stereotomography was proposed as an alternative to reflection traveltime tomography, the first relying on semi-automatic picking of locally coherent events associated with small reflection or diffraction segments tied to scatterers in depth by a pair of rays, while the latter on interpretive picking of laterally continuous reflections. The flexibility of stereotomography paved the way for many developments that have shown the efficiency of the method whilst emphasizing on the complementary information carried out by traveltimes and slopes of locally coherent events. A recent formulation recast stereotomography under a matrix-free formulation based on eikonal solvers and the adjoint-state method. In the latter, like in the previous works, the scatterer positions and the velocity field are updated jointly to tackle the ill-famed velocity–position coupling in reflection tomography. Following on from this adjoint-state formulation, we propose a new parsimonious formulation of slope tomography that offers the chance to restrain the problem to minimizing the residuals of a single data class being a slope, in search of a sole parameter class being the subsurface velocity field. This parsimonious formulation results from a variable projection, which is implemented by enforcing a consistency between the scatterer coordinates and the velocity macromodel through migration of kinematic attributes. We explain why the resulting reduced-parametrization inversion is more suitable for tomographic problems than the most common joint inversion strategy. We benchmark our method against the complex Marmousi model along with a validation through time domain full waveform inversion and then present the results of a field data case study.