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A best linear unbiased estimator minimizes covariance among linear estimators unbiased for the same target. In $Y=X\beta +\epsilon$, assume $\mathbb{E}\epsilon =0$, $X$ is a full-column-rank design matrix and the known covariance matrix $\Sigma$ is a positive-definite matrix. The best linear unbiased estimator of the coefficient vector is $(X^T{\Sigma }^{-1}X{)}^{-1}{X}^T{\Sigma }^{-1}Y$. A spatial trend estimate excludes the correlated residual prediction included in universal kriging.

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.