ETL 1110-1-175
30 Jun 97
reader who may not be familiar with the area of
contribute to measurements taken at locations close
geostatistics.
together being more closely related than measure-
ments taken farther apart.
a. General considerations in spatial
prediction.
(4) The most obvious way one might proceed
for spatial prediction at unsampled locations is
(1) The principal technical issue considered in
simply to take an average of the sample values that
this ETL is spatial prediction or modeling values
one does have and assume that this value gives a
of a spatial process; in particular it is considered
reasonable prediction at all locations in the region
how best to make use of measurements of a vari-
of interest. This may work adequately in some
able (such as pollutant concentration) at sampled
cases, but one can also see the pitfalls in doing
locations to make inferences (or predictions) about
this. Using a single value for an entire region
that variable at unsampled locations or about
makes an implicit assumption of spatial homo-
values of the variable for the region as a whole.
geneity. It ignores any spatial trends that might
exist in the data and it also ignores spatial conti-
(2) A spatial process can be viewed as having
nuity. If it is known that the variable of interest
a large-scale or regional component and a smaller
does have the tendency to be spatially correlated,
scale or local component; both of these compo-
then it would make sense to use a weighted average
nents need to be accounted for when modeling a
rather than a simple average in making a spatial
spatial process. The large-scale component is
prediction, with measurements at sampled loca-
referred to as the mean field and is most often
tions that are nearer to the unsampled location
modeled by a spatial trend which may or may not
being given more weight. This then is the motiva-
be constant over the region. The smaller scale
tion for the geostatistical methods discussed in this
component is a random fluctuation which is mathe-
ETL. The method known as kriging, which is the
matically combined with the trend to make up the
principal subject to be considered here, is a tech-
sample at a point. The random component is
nique for determining in an optimal manner the
usually assumed to be zero on the average but can
weighting of measurements at sampled locations
be either positive or negative in individual samples.
for obtaining predictions at unsampled locations.
The separation of the trend from the random com-
These optimal weights depend on spatial trends
ponents is problem- and scale-dependent and
and correlations that may be present.
requires some judgment to determine. There can
be several "solutions" to the problem of separating
(5) There are a number of ways to go about
the trend and random components that may be
performing spatial prediction. The geostatistical
useful for various geostatistical purposes when
method of kriging covered in this ETL belongs to a
using a single set of data.
class of methods known as stochastic methods. In
these methods, it is assumed that the measure-
(3) Local-scale fluctuation of the variable of
ments, both actual and potential, constitute a single
interest (e.g. water levels or contaminant concen-
realization of a random (or stochastic) process.
trations) at a sample point, although random, can
One advantage of assuming the existence of such a
show some association (i.e. correlation) with the
random process is that measures of uncertainty,
random fluctuations at nearby points. This is
such as the variance used in kriging, can be
referred to as spatial correlation. Positive spatial
defined. These measures of uncertainty permit
correlation between measurements means that the
objective assessment of the performance of a
random components at both points tend to have the
spatial prediction technique on the basis of how
same sign, whereas negative correlation means the
small such measures are. Once a measure of
random components tend to have opposite signs.
uncertainty has been selected, the weights to be
Both the "large-scale" trend and the positive
used in spatial prediction may be determined so as
spatial correlation of the "local-scale" fluctuations
to explicitly minimize the measure of uncertainty.
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