GLOSSARY

Anisotropic: This term is applied both to a random function and to it's variogram (or covariance) when the values of the variogram depend on both the distance and the direction. Also see Isotropic.

Cross-validation: A method for comparing two or more conjectured variogram (or covariance) models. The technique depends on Jackknifing the data and on the exactness of the kriging estimator.

Drift: The expected value of a random function, it may be constant or it may depend on the coordinates of the location. In order for a random function to be stationary, second-order stationary or to satisfy the Intrinsic Hypothesis; the drift must be a constant. The drift is a characteristic of a random function and not of data.

Exact (ness): A property of an estimator/interpolator, namely that if estimating a value at a data location and if that data value is used in the estimation; then the estimated value will coincide with the data value. In some literature this is called Perfect.

Intrinsic Hypothesis: A weak form of stationarity for a random function sufficient for deriving the kriging equations corresponding to the (Ordinary) kriging estimator. See (i) and (ii).

Isotropic: A term applied both to a random function and to its variogram. See anisotropic which is the complementary property.

Kriging Equations: A set of linear equations whose solution includes the values of the weights in the kriging estimator.

Kriging Estimator: While the estimator may be a linear or a non-linear function of the data, in both instances the weights in the estimator are determined by requiring the estimator to be unbiased and have minimum error variance.

Kriging Variance: The minimized value of the estimation variance, i.e., the variance of the error of estimation. This variance is not data dependent but rather is determined by the variogram and the sample location pattern as well as the location of the point to be estimated relative to the sample locations.

Nugget: The variogram may exhibit an apparent discontinuity at the origin. The magnitude of the discontinuity is called the nugget.

Positive Definite: A term applied both to matrices and to functions, (Auto) covariance functions must be positive definite whereas the negative of variograms must be conditionally positive definite. Conditional positive definiteness is a weaker condition.

Random Function: A random function may be seen in two different forms; it may be thought of as a collection of dependent random variables with one for each possible sample location. Alternatively it may be thought of as a "random variable" whose values are functions rather than numbers.

Range (of a variogram): The distance at which the variogram becomes a constant. The Power model does not have a (finite) range. The Exponential and Gaussian models have only an apparent range.

Sill (of a variogram): The value of the variogram for distances beyond the range of the variogram. The Power model does not have a sill.

Spatial Correlation Used both as a generic term to denote that data at two locations is correlated in some sense as a function of their locations and also to denote the value of a spatial structure function such as a variogram or (auto)covariance for a pair of location.s

Stationarity (of a random function) Several different forms of stationarity are utilized in geostatistics. Stationarity, in one of its forms, is a property of a random function rather than of a data set. It expresses the property that certain joint distributions are translation invariant or that certain moments of the random function are translation invariant. See second order stationarity and the Intrinsic hypothesis.

Support The term is used in both a mathematical and in a physical sense. Many, if not most variables of interest in geostatistics, such as the concentrations of chemical elements or compounds only have values at "points" in an idealized sense although the random function treats them in this manner. The data values are usually associated with a physical sample having a length, area or volume; the concentration then represents an average concentration over this length, area or volume. This length, area or volume is called the support. Although it is common to report laboratory analyses in such a way as to not reflect the original support, non-point support has a significant effect on the variogram modeling process and there is a significant difference in estimating the average value over a large volume and in estimating the average value over a small volume. The kriging estimator and equations allow this to be incorporated.

Trend While sometimes used interchangeably with the term "drift", in geostatistics the two are considered separate. The term is usually reserved to denote the deterministic representation obtained by the use of Trend Surface Analysis, i.e, a functional representation for spatially located data (usually taken as a polynomial in the position coordinates). The "trend" is obtained by a least squares fit to the data. As an estimator to the mean of a random function it is sub-optimal. If the residuals from trend surface analysis are used to model the variogram, a biased variogram estimator results.

Variogram (originally called semi-variogram) This function quantifies the spatial correlation and in the case of second order stationarity it is expressible in terms of the (auto)covariance function. See part (ii) in the Intrinsic Hypothesis and equation (2). In order to apply kriging to a data set it is necessary to model the variogram. The variogram must satisfy certain positive definiteness conditions.