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.

(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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