Model re-parameterization via misfit-based coordinate transforms

Kristopher A. Innanen

Re-parameterizations of model space, which are widely applied in seismic inverse problems, are coordinate transforms between non-Cartesian systems. To develop this we identify objective functions as scalar functions of the model vectors, which themselves are contravariant vectors; gradients of the objective function are covariant vectors. A procedure for general transformation to a pre-defined coordinate system and optimization within that system is set out. We argue that a under a class of transformations constrained by the Hessian operator in the reference system, steepest-descent updates are precisely parallel to Gauss-Newton updates, and, provided the transform can be efficiently determined, optimization within the transformed system should have favourable convergence properties. This class of transforms includes an infinite number of variants, and seeking examples from within this class with other, additional, favourable features appears warranted.