Kriging for Weak or Second Order stationary RF

An RF is said to be second order stationary if:

1. Constant mean:

   $$E[Z(\vec{x},\xi)] = m$$ (2.1)

   
   

2. the autocovariance (another name for the covariance) is a function of the distance between the reference points $\vec{x}_1$ and $\vec{x}_2$:

   $$[\vec{x}_1,\vec{x}_2] = E[(Z(\vec{x}_1,\xi) - m)(Z(\vec{x}_2,\xi) - m)] = C(\vec{h})$$ (2.2)

   
   

In practice second order stationarity implies that the first two statistical moments (expected value and covariance) be translation invariant. Note that by saying that $E[Z(\vec{x},\xi)] = m$ we imply that effectively the expected value taken on all the possible realizations $\xi$ does not vary with space. However, for a given realization $Z(\vec{x},\xi_1)$ is a function of $\vec{x}$.

Because of (2.1), the covariance (2.2) can be written as:

$$[\vec{x}_1,\vec{x}_2] = E[(Z(\vec{x}_1,\xi) - m)(Z(\vec{x}_2,\xi) - m)]$$

$$= E[Z(\vec{x}_1,\xi)Z(\vec{x}_2,\xi)] -mE[Z(\vec{x}_2,\xi)] - E[Z(\vec{x}_1,\xi)]m + m^2$$

$$= E[Z(\vec{x}_1,\xi)Z(\vec{x}_2,\xi)] - m^2$$ (2.3)

Obviously if $\vec{h}=0$ we have the definition of variance, also called the ``dispersion'' variance:

$$C(0) =$$   var$$[Z] = \sigma^2_Z$$

For simplicity of notation, from now on, we will denote $x=\vec{x}$, $h=\vec{h}$ and we will drop the variable $\xi$ in the RF.