Statistical assumptions

Let $Z(\vec{x},\xi)$ be a random function (RF) simulating our hydrological quantity of interest. We recall that with this notation we imply that $\vec{x}$ denots a point in space (two or three dimensional space), while $\xi$ denotes the state variable in the space of the realizations. In other words:

• $Z(\vec{x}_0,\xi)$ is a random variable at point $\vec{x}_0$ representing the entire set of realizations of the RF at point $\vec{x}_0$;
• $Z(\vec{x},\xi_1)$ is a particular realization of the RF $Z$;
• $Z(\vec{x}_0,\xi_1)$ is a measurement point.

Given a set of sampled values $Z(\vec{x}_i,\xi_1),\;i=1,2,\ldots$ we want to reconstruct $Z(\vec{x},\xi_1)$, i.e. a possible realization of $Z(\vec{x},\xi)$.