ETL 1110-1-175
30 Jun 97
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
& w1D10 & w2D20 &
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 x0, x1, and
diction that hold in general. For example,
x2, were assumed to be distinct. When a prediction
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
x0 and x1 are the same. If there is only small-scale
because scaling each measurement by a constant c
variability, but no measurement error, then
has the effect of scaling s by c2. 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 w1 = 1, w2 = 0, and 8 = 0
more complicated.
and in a kriging variance of zero. That is, Z(x1) is
a perfect predictor of Z(x0). 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(x0) is more
predictor that does not exactly interpolate the data,
but instead smooths the measured data to account
w1 will be larger than w2, indicating that the mea-
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.
c. Universal kriging.
(f) Effect of data clumping. If Z(x1) 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).
w1 will be much larger [smaller] than w2. In either
2-12
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