Environmental Monitoring Systems Laboratory
Las Vegas NV 89193-3478
EPA/600/8-91/008
April 1991
Anisotropy
In geostatistics, the situation where a variogram exhibits a longer range
(i.e., better correlation) in one direction than another.
Block Kriging
Estimating the value of a block from a set of nearby sample values using
kriging. In Geo-EAS, the block area is approximated by a 2x2, 3x3, or 4x4
array of points centered on each specified grid node.
Covariance
A statistical measure of the correlation between two variables. In geostatistics, covariance is
usually treated as the simple inverse of the variogram, computed as the overall sample variance
minus the variogram value. These covariance values, rather than variogram values, are actually
used in the Geo-EAS kriging matrix equations for greater computational efficiency.
Cross Validation
A technique for testing the validity of a variogram model by kriging each sampled location with
all of the other samples in the search neighborhood, and comparing the estimates with the true
sample values. Interpretation of results, however, can often be difficult. Unusually large
differences between estimated and true values may indicate the presence of “spatial outliers”, or
points which do not seem to belong with their surroundings.
Discretization
In kriging, the process of approximating the area of a block by a finite array of points.
Exponential Model
A function frequently used when fitting mathematical models to experimental variograms, often
in combination with a nugget model.
Gaussian Model
A function frequently used when fitting mathematical models to experimental variograms, often
in combination with a nugget model.
Geostatistics
A methodology for the analysis of spatially correlated data. The characteristic feature is the use
of variograms or related techniques to quantify and model the spatial correlation structure. Also
includes the various techniques such as kriging, which utilize spatial correlation models.
Inverted Covariance
(InvCov) Variogram
A variogram computed by subtracting lag covariances from the sample variance. This approach
compensates for cases where in directional variograms, the mean of the, say, west members of the
sample pairs is not the same as the mean of the east members. “InvCov” is a rather obscure term
referring to the fact that some probabilistic assumptions underlying the variogram computation
are no longer necessary. Inverted Covariance variograms may be modeled and used in kriging
in the same way as ordinary variograms.
Kriging
A weighted-moving-average interpolation method where the set of weights assigned to samples
minimizes the estimation variance, which is computed as a function of the variogram model and
locations of the samples relative to each other, and to the point or block being estimated.
Kriging Standard
Deviation
The standard error of estimation computed for a kriged estimate. By definition, kriging is the
weighted linear estimate with the particular set of weights which minimizes the computed
estimation variance (standard error squared). The relationship of the kriging standard deviation
to the actual error of estimation is very dependent on the variogram model used and the validity
of the underlying assumptions - therefore kriging standard deviations should be interpreted with
caution.
Lag
A distance class interval used for variogram computation.
Linear Model A function frequently used when fitting mathematical models to experimental variograms, often
in combination with a nugget model.
Madogram
A plot of mean absolute difference of paired sample measurements as a function of distance and
direction. Madograms are not true variograms, and generally should not be used in kriging. If
used, the kriged estimates might be “reasonable”, but the kriging standard deviations will be
meaningless.
Nested Variogram
Model
A model which is the sum of two or more component models such as nugget, spherical, etc.
Adding a nugget component to one of the other models is the most common nested model, but
more complex combinations are occasionally used.
Nugget Model
A constant variance model most often used in combination with one or more other functions when
fitting mathematical models to experimental variograms.
Octant Search
In Geo-EAS, the kriging search neighborhood ellipse may be divided into eight equal-angle
sectors, which may have minimum and maximum numbers of samples specified. A limit on the
number of consecutive empty sectors may also be specified. When the specified criteria are not
satisfied for a particular point or block then the kriged estimate is not produced.
Ordinary Kriging
A variety of kriging which assumes that local means are not necessarily closely related to the
population mean, and which therefore uses only the samples in the local neighborhood for the
estimate. Ordinary kriging is the most commonly used method for environmental situations.
Point Kriging
Estimating the value of a point from a set of nearby sample values using kriging. The kriged
estimate for a point will usually be quite similar to the kriged estimate for a relatively small block
centered on the point, but the computed kriging standard deviation will be higher. When a kriged
point happens to coincide with a sampled location, the kriged estimate will equal the sample value.
Quadrant Search
In Geo-EAS, the kriging search neighborhood ellipse may be divided into four equal-angle
sectors, which may have minimum and maximum numbers of samples specified. A limit on the
number of consecutive empty sectors may also be specified. When the specified criteria are not
satisfied for a particular point or block then the kriged estimate is not produced.
Range
For a spherical model, the distance at which the model reaches its maximum value, or sill. For
the exponential and gaussian models, which approach the sill asymptotically, Geo-EAS uses
range to mean the “practical”, or “effective” range, where the function reaches approximately
95% of the maximum. The nugget model effectively has a sill with a range of zero; the linear
model uses “sill/range” merely to define the slope.
Relative Variogram
A variogram in which the ordinary variogram value for each lag has been divided by the square
of the mean of the sample values used in computing the lag. This is sometimes useful when a
“proportional effect” is present; i.e., when areas of higher than average concentration also have
higher than average variance. When relative variogram models are used in kriging, the resulting
kriging standard deviations represent decimal fractions of the estimated values.
Search Neighborhood
In Geo-EAS, an elliptical area centered on a point or block being kriged. Only samples within
the ellipse will be used for kriging. When the next point is kriged, the ellipse will be re-centered,
and a different (perhaps) set of samples will be used.
Semi-variogram
Identical to the term “variogram” as defined in Geo-EAS. There is disagreement in the
geostatistical literature as to which term should be used. Geo-EAS uses “variogram” for
simplicity.
Sill
The upper limit of any variogram model which has such a limit, i.e., which tends to “level off”
at large distances. In Geo-EAS, the spherical, gaussian, exponential, and nugget models have
sills. For the linear model, “sill/range” is used merely to define the slope.
Simple Kriging
A variety of kriging which assumes that local means are relatively constant and equal to the
population mean, which is well-known. The population mean is used as a factor in each local
estimate, along with the samples in the local neighborhood. This is not usually the most
appropriate method for environmental situations.
Spherical Model
A function frequently used when fitting mathematical models to experimental variograms, often
in combination with a nugget model.
Variogram
A plot of the variance (one-half the mean squared difference) of paired
sample measurements as a function of the distance (and optionally of the
direction) between samples. Typically, all possible sample pairs are
examined, and grouped into classes (lags) of approximately equal distance
and direction. Variograms provide a means of quantifying the commonly
observed relationship that samples close together will tend to have more
similar values than samples far apart.
Geo-EAS 1.2.1 vi March 1991
Page by Andy Long.
Comments appreciated.
longa@nku.edu