Kriging with the intrinsic hypothesis

The hypothesis of second order stationarity of the RF is not always satisfied, for example $C(0)$ increases with the distance, violating hypothesis (2.2). In this case the ``Intrinsic hypothesis'' must be used, in which we assume that the first order increments

$$\delta = Y(x+h) - Y(x)$$

are second order RF:

$$E[Y(x+h) - Y(x)] = m(h) = 0$$ (2.6)

$$[Y(x+h) - Y(x)] = 2\gamma(h)$$ (2.7)

where the function $\gamma(h)$ is called the variogram. If the mean $m(h)$ is not zero an obvious change of variable is required. The variogram is defined as the mean quadratic increment of $Y(x)$ (divided by 2) for any two points $x_i$ and $x_j$ separated by a distance $h$:

$$\gamma(h) = \frac{1}{2} [Y(x+h) - Y(x)] = \frac{1}{2} E[(Y(x+h) - Y(x))^2]$$ (2.8)

and is related to the covariance function by:

$$\gamma(h) = \frac{1}{2} E[(Y(x+h) - Y(x))^2]$$

$$= \frac{1}{2} E[Y^2(x+h)] - E[Y(x+h)Y(x)] + \frac{1}{2} E[Y^2(x)]$$

$$= C(0) - C(h)$$

The intrinsic hypothesis requires a finite value for the mean of $Y(x)$ but not for its variance. In fact, hypothesis (2.2), as changed into (2.3), implies (2.8), but not viceversa.

Figure 2.1: Behavior of the covariance as a function of distance (left) and the corresponding variogram (right).

The covariance $C(h)$ has a decreasing behavior as shown in Fig. 2.1. When $C(h)$ is known then the variogram can be directly calculated. When $C(0)$ is finite, the variogram $\gamma(h)$ is bounded asymptotically by this value. The value of $h$ at which the asymptot can be considered achieved is called the ``range``, while $C(0)$ is called the ``sill'' (see Fig. 2.1).