Identifiability for non-stationary spatial structure

For modelling non-stationary spatial random fields Z = {Z(x) : x is an elem ent of R-n, n greater than or equal to 2} a recent method has been proposed to deform bijectively the index space so that the spatial dispersion D(x, y) = var[Z(x) - Z(y)], (x, y) is an element of R-n x R-n, depends only on t he Euclidean distance in the deformed space through an isotropic variogram gamma. We prove uniqueness of this model in two different cases: (i) gamma is strictly increasing; (ii) gamma(u) is differentiable for u > 0.