Identifiability for non-stationary spatial structure

For modelling non-stationary spatial random fields Z = { Z ( x ) : x ∊ℝ n , n ≥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 )∊ℝ n xℝ n , depends only on the Euclidean distance in the deformed space through an isotropic variogram γ. We prove uniqueness of this model in two different cases: (i) γ is strictly increasing; (ii) γ( u ) is differentiable for u > 0.