Sill (geostatistics)

In spatial statistics the theoretical variogram $2\gamma(\mathbf_1,\mathbf_2)$ is a function describing the degree of spatial dependence of a spatial random field or stochastic process $Z(\mathbf)$. The semivariogram $\gamma(\mathbf_1,\mathbf_2)$ is half the variogram.

In the case of a concrete example from the field of gold mining, a variogram will give a measure of how much two samples taken from the mining area will vary in gold percentage depending on the distance between those samples. Samples taken far apart will vary more than samples taken close to each other.

Definition

The semivariogram $\gamma(h)$ was first defined by Matheron (1963) as half the average squared difference between the values at points ($\mathbf_1$ and $\mathbf_2$) separated at distance $h$. Formally

:\gamma(h)=\frac\iiint_V \left (M+h) - f(M) \right2dV,

where $M$ is a point in the geometric field $V$, and $f(M)$ is the value at that point. The triple integral is over 3 dimensions. $h$ is the separation distance (e.g., in meters or km) of interest.
For example, the value $f(M)$ could represent the iron content in soil, at some location $M$ (with geographic coordinates of latitude, longitude, and elevation) over some region $V$ with element of volume $dV$.
To obtain the semivariogram for a given $\gamma(h)$, all pairs of points at that exact distance would be sampled. In practice it is impossible to sample everywhere, so the empirical variogram is used instead.

The variogram is twice the semivariogram and can be defined, equivalently, as the variance of the difference between field values at two locations ($\mathbf_1$ and $\mathbf_2$, note change of notation from $M$ to $\mathbf$ and $f$ to $Z$) across realizations of the field (Cressie 1993):

:2\gamma(\mathbf_1,\mathbf_2)=\text\left(Z(\mathbf_1) - Z(\mathbf_2)\right) = E\left (Z(\mathbf_1)-\mu(\mathbf_1))-(Z(\mathbf_2) - \mu(\mathbf_2)))^2\right

If the spatial random field has constant mean $\mu$, this is equivalent to the expectation for the squared increment of the values between locations $\mathbf_1$ and $s_2$ (Wackernagel 2003) (where $\mathbf_1$ and $\mathbf_2$ are points in space and possibly time):

:2\gamma(\mathbf_1,\mathbf_2)=E\left left(Z(\mathbf_1)-Z(\mathbf_2)\right)^2\right.

In the case of a stationary process, the variogram and semivariogram can be represented as a function $\gamma_s(h)=\gamma(0,0+h)$ of the difference $h=\mathbf_2-\mathbf_1$ between locations only, by the following relation (Cressie 1993):

:$\gamma(\mathbf_1,\mathbf_2)=\gamma_s(\mathbf_2-\mathbf_1).$

If the process is furthermore isotropic, then the variogram and semivariogram can be represented by a function $\gamma_i(h):=\gamma_s(h e_1)$ of the distance $h=\, \mathbf_2-\mathbf_1\,$ only (Cressie 1993):

:$\gamma(\mathbf_1,\mathbf_2)=\gamma_i(h).$

The indexes $i$ or $s$ are typically not written. The terms are used for all three forms of the function. Moreover, the term "variogram" is sometimes used to denote the semivariogram, and the symbol $\gamma$ is sometimes used for the variogram, which brings some confusion.

Properties

According to (Cressie 1993, Chiles and Delfiner 1999, Wackernagel 2003) the theoretical variogram has the following properties:
* The semivariogram is nonnegative $\gamma(\mathbf_1,\mathbf_2)\geq 0$, since it is the expectation of a square.
* The semivariogram $\gamma(\mathbf_1,\mathbf_1)=\gamma_i(0)=E\left((Z(\mathbf_1)-Z(\mathbf_1))^2\right)=0$ at distance 0 is always 0, since $Z(\mathbf_1)-Z(\mathbf_1)=0$.
* A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights $w_1,\ldots,w_N$ subject to $\sum_^N w_i=0$ and locations $s_1,\ldots,s_N$ it holds:
::$\sum_^N\sum_^N w_\gamma(\mathbf_i,\mathbf_j)w_j \leq 0$
: which corresponds to the fact that the variance $var(X)$ of $X=\sum_^N w_i Z(x_i)$ is given by the negative of this double sum and must be nonnegative.
* If the covariance function of a stationary process exists it is related to variogram by

$2\gamma(\mathbf_1,\mathbf_2)=C(\mathbf_1,\mathbf_1)+C(\mathbf_2,\mathbf_2)-2C(\mathbf_1,\mathbf_2)$

* If a stationary random field has no spatial dependence (i.e. $C(h)=0$ if $h\not= 0$), the semivariogram is the constant $var(Z(\mathbf))$ everywhere except at the origin, where it is zero.
* \gamma(\mathbf_1,\mathbf_2)=E\left even function