The resulting kriging variance is

case, the kriging variance will increase to reflect

the redundant information in the two measure-

2

FK (*x*)0

ments. Automatic adjustment of the kriging

weights and kriging variance to account for data

(2-42)

clumping is an important property of the kriging

3

1

& *w*1D10 & *w*2D20 &

D % D20 & D12

=*s*

predictor.

2 10

2

(3) Example 2 (Nugget effect versus measure-

Although there are only three sample locations in

ment error).

this example (two actual and one potential), it indi-

cates several properties of best linear unbiased pre-

(a) In example 1, all three locations *x*0, *x*1, and

diction that hold in general. For example,

location happens to coincide with a measurement

(c) ** Effect of sill**. The kriging weights

location, there is an important distinction that

depend on *s *only through the relative nugget *p*.

needs to be made between a true nugget effect and

However, the kriging variance is directly propor-

a measurement error. Suppose that in example 1,

tional to *s*. The sill is called a scaling parameter

because scaling each measurement by a constant *c*

variability, but no measurement error, then

has the effect of scaling *s *by *c*2. When the relative

repeated measurements at the same location should

nugget is allowed to vary so that *s *and *g *can

be identical, that is, D10 = 1. In this case, the krig-

change independently, the effect of *s *is somewhat

ing equations result in *w*1 = 1, *w*2 = 0, and 8 = 0

more complicated.

and in a kriging variance of zero. That is, *Z*(*x*1) is

a perfect predictor of *Z*(*x*0). This property, called

(d) **Effect of nugget**. Increasing *p *has the

effect of drawing each of the weights closer to 1/2.

the data are assumed to contain no measurement

In fact, as *p *approaches 1, both weights will equal

errors. However, suppose instead that the nugget

1/2. The larger *g *is in relation to *s*, the more

is interpreted as measurement error rather than

small-scale variability there is in the data and the

small-scale variability. In that case, repeated

less important the correlation between neighboring

measurements at the same location would not be

locations becomes. The increased small-scale

perfectly correlated, but rather, D10 = 1-*g/s*.

variability also causes an increase in the kriging

variance.

(b) Substituting this correlation into the krig-

ing equations and solving the equations results in a

(e) **Effect of correlations**. If *Z*(*x*0) is more

predictor that does not exactly interpolate the data,

but instead smooths the measured data to account

for the measurement error. In this ETL, prediction

surement at the first location has more predictive

locations are assumed not to coincide with mea-

information than the measurement at the second

surement locations, in which case no distinction

location. Also, correlation in the data always

needs to be made between nugget and measurement

decreases the kriging variance compared to the

error.

variance with uncorrelated data.

(f) **Effect of data clumping**. If *Z*(*x*1) and

(1) Universal kriging is an extension of ordi-

being close to 1, then the two measurements con-

nary kriging, that, due to the fact that environ-

tain much of the same information. Two situations

mental data often contain drift, can be important in

can occur: D10 = D20, in which case the weights are

HTRW site investigations. Universal kriging

both equal, or D10 > D20 [D10 < D20], in which case

addresses the case of a nonconstant mean ( *x*).

2-12

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