Consistent estimates of deformed isotropic Gaussian random fields on the plane

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation $f:\Bbb{R}^2\to\Bbb{R}^2$ when observing the deformed random field $Z\circ f$ on a dense grid in a bounded, simply connected domain $\Omega$, where $Z$ is assumed to be an isotropic Gaussian random field on $\Bbb{R}^2$. The estimate $\hat{f}$ is constructed on a simply connected domain $U$, such that $\overline{U}\subset\Omega$ 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 $\hat{f}\to R_{\theta}f+c$ 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.