Geostatistics

Geostatistics models spatial dependence for estimation and prediction. A mean or drift describes systematic variation, while a covariogram or semivariogram describes residual dependence. Kriging combines the two for spatial prediction.

Kriging predicts a spatial random field by a linear combination of observations chosen to minimize prediction variance subject to the appropriate mean constraints. Its different forms depend on whether the mean is known, an unknown constant, or an unknown regression drift.

For unknown drift $X\beta$, a full-column-rank design matrix $X$, a known positive-definite matrix observation covariance matrix $\Sigma$, and target drift row $x_0^T$, universal kriging minimizes prediction variance subject to $X^{T}w=x_0$. Equivalently, with $\hat{\beta }=(X^T{\Sigma }^{-1}X{)}^{-1}{X}^T{\Sigma }^{-1}z$, predict $x_0^T\hat{\beta }+c^T{\Sigma }^{-1}(z-X\hat{\beta })$. The first term is the estimated trend alone; the second is a correlated residual prediction.

For an unknown constant mean, ordinary kriging minimizes error variance subject to weights summing to one. This unbiasedness constraint distinguishes it from simple kriging.

For known zero mean, invertible observation covariance $\Sigma$, target covariance vector $c$ and target variance $v_0$, simple kriging predicts $c^T{\Sigma }^{-1}z$ with error variance $v_0-c^T{\Sigma }^{-1}c$. Completing the covariance quadratic form proves optimality. A multivariate normal distribution makes this the exact conditional mean and variance.

The covariogram is the displacement-dependent covariance $C(h)=\mathrm{Cov}[Z(s+h),Z(s)]$ of a second-order stationary field. It is an even positive-definite kernel. A semivariogram determines differences $C(0)-C(h)$, but not an arbitrary constant covariance component.

An intrinsically stationary random field has constant mean and finite increment variances depending only on displacement. Its semivariogram is $\gamma (h)=\frac{1}{2}\mathrm{Var}[Z(s+h)-Z(s)]$. Second-order stationarity implies this property, but the converse can fail: subtracting the value at a fixed origin from a stationary field preserves increments and makes marginal variance vary with location.

For an intrinsically stationary random field, the semivariogram is $\gamma (h)=\frac{1}{2}\mathrm{Var}[Z(s+h)-Z(s)]$. It is even and vanishes at zero. With constant mean, a second-order stationary field has $\gamma (h)=C(0)-C(h)$. An isotropic model writes it as a function of nonnegative distance.

For a distance bin $B$, the empirical semivariogram is $\hat{\gamma }(B)=(2|B|{)}^{-1}{\sum }_{(i,j)\in B}(z_i-z_j{)}^2$. A nonconstant drift adds half its squared pairwise differences to the expected raw semivariogram. Residual semivariograms estimate spatial dependence after adjusting for the drift.

The exact range is a separation beyond which the semivariogram reaches its sill. A Gaussian semivariogram with nonzero structured variance approaches its sill asymptotically and therefore has infinite exact range. Software range parameters may instead specify a scale.

A practical range specifies a distance at which a chosen fraction of the structured sill is reached. For the Gaussian semivariogram ${\sigma }^2(1-e^{-(h/a{)}^2})$, the 95-percent practical range is $a\sqrt{\log 20}$, obtained by solving $e^{-(h/a{)}^2}=0.05$.

The sill is the large-distance level of a semivariogram when that limit exists. For covariance tending to zero, the sill is the total marginal variance; a nugget plus a structured component has total sill ${\tau }^2+{\sigma }^2$. A nonzero constant covariance component is invisible to the semivariogram.

The nugget is the discontinuity of a distance semivariogram at zero. Independent location-specific noise of variance ${\tau }^2$ contributes covariance ${\tau }^2$ at zero displacement and zero covariance at nonzero displacement, so its semivariogram is ${\tau }^2{1}_{\{h\ne 0\}}$.

If $W(s)$ is stationary, then $Z(s)=W(s)-W(0)$ has exactly the same increments and semivariogram, but $\mathrm{Var}Z(s)=2{\gamma }_W(s)$ is generally nonconstant. Thus existence of a stationary covariance model for a semivariogram does not imply stationarity of every process with that semivariogram. Adding an independent random constant also leaves the semivariogram unchanged while changing the covariogram.

For $\varphi >0$ and nonnegative variances, $\gamma (h)={\tau }^2{1}_{\{h\ne 0\}}+{\sigma }^2(1-e^{-{\varphi }^2\Vert h{\Vert }^2})$ admits covariance $C(h)={\sigma }^2{e}^{-{\varphi }^2\Vert h{\Vert }^2}+{\tau }^2{1}_{\{h=0\}}$. The Gaussian kernel is positive definite because it is the Fourier transform of the density of a Gaussian distribution. Adding independent white noise supplies the nugget effect. Its scale is $a=1/\varphi$. For ${\sigma }^2>0$, its exact range of a semivariogram is infinite. If ${\sigma }^2=0$, it is a pure nugget effect with no nontrivial structured range.