We first show a construction and make an analysis of a simulation algorithm using fast Fourier transform for a stationary Gaussian Random field. This kind of algorithm can be found in (Wood and Chan, 1994) (and later in (Chan and Wood, 1999)) using a circulant embedding approach) and a decade later in (Lang and Potthoff, 2011), with a Fourier transform on a generalized white noise. The main point behind those algorithms is essentially the spectral representation, in the stationary case, of the covariance, when the latter is continuous.
The first consequence of this common root is the error directly generated by the approximation used to compute the Fourier transform, here using a fast Fourier transform instead.
For the sake of applied matters, the quantification of such an approximation was indeed very important, especially if we want to use combinations of several simulated Gaussian Fields, for example, in order to simulate some non-stationary Gaussian Random fields.
It gives a very simple way to build non-stationary Gaussian Random fields as we will see in an application to a medical modelization of a layer of the human cornea; we will then discuss some ways to generalize this idea for the purpose of generating simulated non-stationary Gaussian Fields with this algorithm.