Application of misfit-based model space coordinate system design to seismic FWI
Kristopher A. H. Innanen
The re-parameterizations we encounter regularly in seismic full waveform inversion can all be formally identified as coordinate transforms between suitably general coordinate systems. For instance, changes-of-variable between elastic/acoustic parameter classes, and the variable changes underlying the model space pre-conditioning methods of Harlan, Claerbout and Fomel, can both be shown to be transforms between oblique-rectilinear coordinate systems when covariant notation is employed. With transformation rules in place for updates and displacements (contravariant vectors), and sensitivities and gradients (covariant vectors), as well as objective functions (scalars) and Hessian operators (2nd rank covariant tensors), we may then seek new and potentially useful re-parameterizations. Coordinate systems within which the Hessian is close to the identity have vastly improved optimization properties. Using a newly-derived numerical procedure for designing transformation matrices mapping to such systems, we provide an early examination of re-parameterized FWI.