Consistent estimates of deformed isotropic Gaussian random fields on the plane

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation f: .2..2 when observing the deformed random field Z.f on a dense grid in a bounded, simply connected domain ., where Z is assumed to be an isotropic Gaussian random field on .2. The estimate f. is constructed on a simply connected domain U, such that U... and is defined using kernel smoothed quadratic variations, Bergman projections and results from quasiconformal theory. We show, under mild assumptions on the random field Z and the deformation f, that f..R.f+c uniformly on compact subsets of U with probability one as the grid spacing goes to zero, where R. is an unidentifiable rotation and c is an unidentifiable translation.