Consistent Estimates of Deformed Isotropic Gaussian Random Fields on the Plane

This paper proves fixed domain asymptotic results for estimating a smooth invertible transformation $f\colon \Bbb R^2\to \Bbb R^2$ 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 $\bar U\subset\Omega$ 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.