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.