Reducing non-stationary random fields to stationarity and isotropy using a space deformation

Second-order non-stationary modeling approaches for univariate geostatistical data

“…They show that if the deformation function f ðÁÞ is differentiable as well its inverse and the function c 0 ðÁÞ, then the space deformation model is identifiable up to a scaling for c 0 ðÁÞ and up to a homothetic Euclidean motion for f ðÁÞ. Perrin and Senoussi (2000) give a characterization of non-stationary correlation functions reducible to stationary or isotropic stationary under certain regularity conditions. Porcu et al (2010) address this problem of reducibility of a non-stationary random function in a wider framework.…”

Kriging groundwater solute concentrations using flow coordinates and nonstationary covariance functions

“…In geostatistics, coordinate transformations have been widely used to cope with spatially-varying anisotropy (Dagbert et al, 1984;Deutsch and Wang, 1996;Barabas et al, 2001;Legleiter and Kyriakidis, 2006;Boisvert and Deutsch, 2010) and for dealing with nonstationarity in general (Sampson and Guttorp, 1992;Perrin and Senoussi, 2000;Damian et al, 2001;Schmidt and O'Hagan, 2003;Anderes and Stein, 2008). Other type of kriging approaches (Soares, 1990; te Stroet and Snepvangers, 2005) deal with local anisotropy and curvilinear features, as well as some geostatistical and multiple-points simulation approaches (Xu, 1996;Strebelle, 2002).…”

“…When the location of points in the new space is not obtained directly from a geographical or geological model (Dagbert et al, 1984;Barabas et al, 2001;Legleiter and Kyriakidis, 2006), it is either estimated by multidimensional scaling (Sampson and Guttorp, 1992;Monestiez et al, 1993;Loland and Host, 2003;Almendral et al, 2008;Boisvert and Deutsch, 2010) or by semiparametric Bayesian inference (Schmidt, 2001;Damian et al, 2001;Schmidt and O'Hagan, 2003). However, according to Perrin and Senoussi (2000), stationary reducibility by spatial deformation requires the variance to be constant. Other approaches handle the problem differently by directly using nonstationary covariance functions.…”

On the non-reducibility of non-stationary correlation functions to stationary ones under a class of mean-operator transformations

“…Sampson and Guttorp (1992) refer only to stationarity and isotropic reducibility. Perrin and Senoussi (2000) study stationarity reducibility as well without restricting to isotropic conditions, thus analyzing (1). This is the approach we are taking here.…”