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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.

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.

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.

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.

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.

Kriging predicts an unobserved value of a spatial random field by combining observations according to their covariances. Nearest-neighbor interpolation uses the closest sample; kriging also asks how strongly each sample is related to the target and how much information the samples share with one another. Two nearby, highly correlated samples need not provide twice the information of one. The different forms of kriging depend on whether the mean is known, an unknown constant, or an unknown regression drift.

Write $Z=(Z(s_1),\dots ,Z(s_n){)}^T$ for the observation random vector, $z$ for its realized values, and $Z_0=Z(s_0)$ for the prediction target. Assume finite second moments, and defineHere $\Sigma$ is the observation covariance matrix, assumed to be a positive-definite matrix in the formulas below. A covariance function supplies its entries and the target covariance vector $c$. For a second-order stationary random field, the covariogram $C(h)$ depends on displacement, and the semivariogram satisfies $\gamma (h)=C(0)-C(h)$. The intrinsically stationary random field formulation permits kriging using a semivariogram even when a stationary covariogram is unavailable; the finite covariance formulation below is a convenient important case.

For a linear combination $w^{T}Z$ predicting $Z_0$ with zero bias, the error variance equals its mean squared error:Ordinary kriging and universal kriging minimize this quadratic form subject to their mean constraints. Their optimality is among the specified linear predictors with zero bias under the chosen statistical model; covariance information alone does not establish optimality among arbitrary nonlinear predictors. The Gaussian random field interpretation supplies additional conclusions in simple kriging.

The reported error variance assumes a chosen covariance function, its statistical parameters, observation locations, and a mean structure. It omits uncertainty in fitted covariance parameters and does not validate the trend. A repeated-sampling mean squared error need not equal the error variance conditional on each realized data vector.^[ref]

In 1963, Georges Matheron presented kriging for estimating average ore grades in mining panels from weighted samples taken inside and outside a panel. This setting in geostatistics explains why prediction targets can be averages over regions as well as values at points.^[ref]

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 ${\tau }^2{1}_{\{h\ne 0\}}$.

Ordinary kriging assumes $\mathbb{E}[Z(s)]=\mu$ for an unknown constant $\mu$. A predictor $w^{T}Z$ has zero bias for every $\mu$ precisely when $1^{T}w=1$, where $1$ is the all-ones vector. Ordinary kriging minimizes the error variance $Q(w)$ subject to this constraint, giving the best linear unbiased predictor. Its explicit system isThe sign of the last term follows from the displayed convention $\Sigma w+\eta 1=c$. To derive the system, differentiate $Q(w)+2\eta (1^{T}w-1)$. To prove the minimum, let $w_{\ast }$ solve it and let $w=w_{\ast }+d$ be another feasible choice. Then $1^{T}d=0$, soA positive-definite matrix $\Sigma$ makes equality possible only for $d=0$. This argument uses second moments, not a normal distribution assumption. Ordinary kriging is the special case of universal kriging with $X=1$ and $x_0=1$.

For an original one-dimensional teaching example, observe $z=(1,3,2{)}^T$ at $s=(0,1,3)$, and predict at $s_0=2$. Choose unit marginal variance, exact observations without measurement noise, an unknown constant mean, and the exponential covariance functionThe exponential covariance function corresponds to a stationary Ornstein-Uhlenbeck process, so it gives a valid covariance model. Its semivariogram is $1-2^{-|h|}$. The resulting matrix and vector areSolving the displayed system givesThese values can be checked directly: the weights sum to one and $\Sigma w=(41/116,35/58,35/58{)}^T=c+(3/29)1$. Substitution into $Q(w)$ gives $18/29$. Nearest-neighbor interpolation must choose between the two equally close samples at one and three, returning either three or two; their error variances under this model are both $1+1-2(1/2)=1$. Thus the benefit is quantified under the chosen model, without knowing the unobserved target value.

The upper panel applies the same ordinary kriging calculation at each target location. The lower panel shows the square root of its error variance, in the units of the observations. It is zero at the exact observation locations and rises between and beyond them. No confidence interval is inferred from a variance alone; a probability distribution assumption is needed to attach a coverage level.

The weights are not required to be positive. Keep the same observations and target, but replace the covariogram by the smoother Gaussian kernel $C(h)=2^{-h^2}$. ThenThe negative coefficient compensates for dependence between samples; the constraint eliminating bias still holds. Kriging is not generally a convex combination, and its prediction need not stay between the observed values. A smaller model-based error variance does not establish that one covariance function is better supported by these three observations.

There is a literal domain error in the printed formula. A semivariogram indexed by a signed displacement must satisfy $\gamma (h)=\gamma (-h)$, since reversing the two observations leaves the squared increment unchanged. For positive ${\sigma }^2$ or ${\tau }^2$, the displayed positive-lag value is nonzero whereas the displayed negative-lag value is zero. Thus the claim on all of $\mathbb{R}$ is false as written.

Interpret the argument as a nonnegative distance, or use the even extension of the Gaussian semivariogram. For ${\sigma }^2,{\tau }^2\ge 0$ and $\varphi >0$, the intended model isIt admits the covariogramThe Gaussian kernel is a positive-definite kernel, as can be seen from the Fourier transform of a Gaussian: in one dimensionand the nonnegative spectral density makes every finite covariance quadratic form nonnegative. In higher dimensions use the product Gaussian density. Adding independent Gaussian white noise of variance ${\tau }^2$ adds the diagonal nugget covariance. Thus a zero-mean stationary Gaussian random field exists with this covariance, and $C(0)-C(h)$ gives the corrected semivariogram.

For nonzero structured variance, the nugget effect is the right limit ${\tau }^2$ at zero, the sill of a semivariogram is ${\tau }^2+{\sigma }^2$, and the range of a semivariogram is infinite if defined as the distance at which the sill is exactly reached. A 95-percent practical range solves $1-e^{-{\varphi }^2{h}^2}=0.95$:The scale $1/\varphi$ is sometimes called the range parameter; it is neither the exact range nor the 95-percent practical range. If ${\sigma }^2=0$, there is no nontrivial structured range.

The fact that a semivariogram does not determine stationarity is a further distinction: a semivariogram specifies increment variation, not a unique process or absolute covariance. Adding an independent random constant adds a constant to $C$ without changing $\gamma$. More strongly, subtracting $W(0)$ from a stationary field $W(s)$ preserves its semivariogram but generally makes its variance depend on $s$. The corrected statement is that this semivariogram admits a second-order stationary model, with the displayed covariance using the usual convention $C(h)\to 0$ at large distance; it does not force every compatible process to be stationary.

The first binned empirical semivariogram rises strongly through the displayed distances, approximately quadratically, with no visible sill. The spatial data plot shows higher depth at larger $y$. A stationary residual covariance should not be asked to explain that deterministic drift.

For a model $Z(s)=m(s)+W(s)$ with stationary zero-mean residual field, each raw squared difference has expectationWith linear drift $m(x,y)={\beta }_0+{\beta }_{1}y$, the added term is $\frac{1}{2}{\beta }_1^2(y_i-y_j{)}^2$. Its average within distance bins can grow with separation, producing the shape seen in the first plot. The empirical semivariogram uses half the average squared difference over the pairs in each bin, so removing only a common intercept does not remove this drift contribution.

The formula depth ~ y estimates and removes a linear drift before computing a residual empirical semivariogram. The second graph levels off near $0.47$ with a fitted zero nugget and a finite scale, making the Gaussian residual covariance a plausible working model. Estimate the drift and fit spatial dependence to residuals rather than the raw depth differences. Estimated residuals are not the true errors, so their semivariogram has estimation effects; sparse long-distance bins are also noisy. These graphs support the model but do not establish stationarity or isotropy.

The subpart refers to svgm and svgm2, while the code calls the objects smvg and smvg2. Also the printed code omits the line that fits gauss.model2 before plotting or using it. The supplied output must be interpreted as the result of that missing residual-variogram fit, rather than as an executable complete script.

The universal kriging model has a linear drift in $y$ and a zero-mean Gaussian residual field:The Gau range parameter in this software is $a=1/\varphi$, since its semivariogram uses $1-e^{-(h/a{)}^2}$. The output gives ${\hat{\tau }}^2=0$, ${\hat{\sigma }}^2=0.4726257$, and $\hat{a}=1.112235$, hence $\hat{\varphi }\simeq 0.899091$. The exact Gaussian range is infinite; the 95-percent practical range is approximately $1.92508$ in the coordinate distance units.

The BLUE=TRUE option asks for the best linear unbiased estimator of the drift, rather than the full field kriging prediction. Thus the two printed fitted trends give ${\hat{\beta }}_0=20.13793$ at $y=0$ and ${\hat{\beta }}_0+{\hat{\beta }}_1=22.08399$ at $y=1$. ThereforeThere is no $x$ coefficient in the drift. The reported variances $0.09456496$ and $0.06097459$ are the variances of the corresponding estimated trend values, not full-field prediction variances. They do not determine the slope variance without its covariance with the intercept. The interpolated map can still vary with $x$ through the correlated residual prediction, even though the fitted deterministic drift depends only on $y$.

Let $X$ have rows $(1,y_i)$ and let ${\Sigma }_{\vartheta }$ be the spatial covariance matrix with $\vartheta =({\tau }^2,{\sigma }^2,a)$. Conditional on chosen covariance parameters, the best linear unbiased estimator of the drift coefficients is the generalized least squares estimatorThe printed procedure first estimates the linear drift from depth ~ y, constructs a binned residual empirical semivariogram, fits a Gaussian semivariogram to it, and then uses the fitted covariance in generalized least squares trend estimation and universal kriging. The missing fit could be supplied aswhere those arguments are initial values, not the reported final estimates. Such semivariogram fitting typically minimizes a weighted sum of squared discrepancies between binned empirical values and model values; it is not automatically a joint Gaussian likelihood maximization. One can iterate: form residuals using the current drift, refit the covariance model, recompute the generalized least squares drift, and repeat until changes are small. The displayed commands themselves show a single sequence, not evidence that such iteration occurred.

A likelihood-based alternative estimates the two blocks coherently fromSubstitute $\hat{\beta }(\vartheta )$ to obtain a profile log-likelihood, maximize over ${\tau }^2,{\sigma }^2\ge 0$ and $a>0$ with a positive-definite matrix ${\Sigma }_{\vartheta }$, and recover the drift coefficients from the optimizing covariance. Restricted maximum likelihood adds the design determinant term $\log \det (X^T{\Sigma }_{\vartheta }^{-1}X)$ and uses $n-2$ in place of $n$ in the normalizing term, up to a fixed-design constant. It can reduce bias from estimating the drift before the covariance. A zero fitted nugget is a legitimate boundary estimate, not proof that measurement error is absent. Estimate drift using covariance-weighted regression, and spatial dependence from drift-adjusted variation, with the two stages or likelihood explicitly distinguished.

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.

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.

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.