Consistent estimates of deformed isotropic Gaussian random fields on the plane

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$ when observing the deformed random field $Z\circ f$ on a dense grid in a bounded, simply connected domain $\mathrm{\Omega }$, where $Z$ is assumed to be an isotropic Gaussian random field on ${\mathbb{R}}^2$. The estimate $\widehat{f}$ is constructed on a simply connected domain $U$, such that $\bar{U}\subset \mathrm{\Omega }$ 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 $\widehat{f}\to R_{\theta }f+c$ uniformly on compact subsets of $U$ with probability one as the grid spacing goes to zero, where $R_{\theta }$ is an unidentifiable rotation and $c$ is an unidentifiable translation.