In a (stationary) Gaussian Process, values which are closeby are more similar than values far away from each other. The correlation function tends to zero as distance increases. Often, one models the decaying correlation functon $C$ as:
$Cor(x_i, x_j) = e^{-\theta||x_i - x_j||^2}$
I believe this model also underpins the Predict
function of Mathematica described here.
However, how does one generate a random field with such a property? You may, for simplicity, assume it's a one dimensional function $f(x)$ with mean $\mu = 0$ and standard deviation $\sigma = 1$.