In short, the use of stochastic techniques provides

and makes explicit the background assumptions

the investigator with a way of objectively quanti-

that are being made.

fying errors and determining weights. In practice,

spatial predictions obtained using kriging are

almost always accompanied by a measure of the

are some of the key ideas in geostatistics that will

associated error. Most kriging practitioners

be given detailed attention in this ETL. They are

consider such an error evaluation to be an integral

introduced in much the same order that they are

part of the analysis, and point to error analysis as

discussed in Chapter 2, where more detail is

one of the principal advantages of using kriging (or

presented.

stochastic techniques in general) over other

procedures.

(1) Variograms.

(6) Nonstochastic techniques, on the other

(a) A central idea in geostatistics is the use of

hand, are typically applied strictly empirically,

spatial correlation to improve spatial predictions,

with no assumptions concerning the existence of an

or interpolations. The variogram is the principal

underlying random process and with no theoretical

tool used to characterize the degree of spatial

framework with which to evaluate statistically the

correlation present in the data and is fundamental

performance or optimality of the techniques.

to kriging. The correlation between measurements

When they are applied in such a manner, it is not

at two points is usually assumed, as described

possible to evaluate in advance whether such a

above, to depend on the separation between the two

procedure would be expected to yield results that

points. Values for all possible pairings of sample

are satisfactory. Two techniques that are com-

points can be examined by squaring the difference

monly applied in a nonstochastic setting are simple

between the values in each pair. The squared

averaging, mentioned above, and trend analysis,

differences are then categorized according to the

which is a least- squares method for fitting a

distance separating the pair. For small separa-

smooth surface to the data. Even though these

tions, or lags, the squared differences are usually

techniques are usually applied nonstochastically, it

small and increase as the lag increases. A plot of

is still possible to assess their performance if a

the squared differences per sample pair as a func-

stochastic setting is assumed. Loosely speaking

tion of lag is referred to as the sample variogram.

(these ideas are discussed more precisely in Chap-

ter 7), simple averaging would perform well if

(b) The general behavior of the sample vari-

there is no trend and no spatial correlation, and

ogram points relates to the spatial correlation

trend analysis would perform well if there is a

between sample sites and can provide investigators

trend that can be modeled, but no spatial correla-

with qualitative information about the spatial pro-

tion. Lack of correlation in the observations is one

cess, but in order to use this information in a math-

assumption that is made in ordinary statistical

ematically rigorous manner as a basis for inter-

regression analysis, and in fact trend analysis, if it

polation, a function with specific properties must

is placed in a stochastic setting, is actually one

be fit to the sample variogram points. The fit, as

special type of regression. The stochastic method

with all curve-fitting procedures, takes the scat-

of kriging explicitly incorporates the spatial corre-

tered points and passes a smooth curve through the

lations which are ignored in trend analysis. In

points. The curve, which can be represented by a

Chapter 7, a few other common techniques that are

mathematical expression or function, is called a

usually applied in a nonstochastic setting will be

model. Several named models with characteristic

discussed briefly. Most of these techniques are

features introduced in Chapter 2 are commonly

designed to incorporate the notion of spatial con-

used in geostatistics. The resultant variogram

tinuity, but the way it is incorporated may be

model is used to determine kriging weights for use

subjective. Kriging provides an objective means of

in interpolation.

incorporating the presence of spatial correlation

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