  Standard geostatistical models assume second order stationarity of the
underlying Random Function. In some instances, there is little reason to expect
the spatial dependence structure to be stationary over the whole region of
interest. In this paper, we introduce a new model for second order
non-stationary Random Functions as a convolution of an orthogonal random
measure with a spatially varying random weighting function. This new model is a
generalization of the common convolution model where a non-random weighting
function is used. The resulting class of non-stationary covariance functions is
very general, flexible and allows to retrieve classes of closed-form
non-stationary covariance functions known from the literature, for a suitable
choices of the random weighting functions family. Under the framework of a
single realization and local stationarity, we develop parameter inference
procedure of these explicit classes of non-stationary covariance functions.
From a local variogram non-parametric kernel estimator, a weighted local
least-squares approach in combination with kernel smoothing method is developed
to estimate the parameters. Performances are assessed on two real datasets:
soil and rainfall data. It is shown in particular that the proposed approach
outperforms the stationary one, according to several criteria. Beyond the
spatial predictions, we also show how conditional simulations can be carried
out in this non-stationary framework.
