ETL 1110-1-175
30 Jun 97
horizontal line and have no structure. If measured
nonstationarity occurs as a gradual change.
data are clustered and the lags have been mini-
HTRW site investigations may present cases,
mized to meet the required number of pairs of loca-
especially when dealing with water- quality data in
tions, the variogram may seem horizontal because
and around plumes, that have abrupt step-like
it is dominated by small-scale effects in the
changes at plume boundaries and do not appear as
clustered data. The investigator then has latitude
regional drift. In these cases the investigator needs
to adjust the lags and attempt to balance the lag
to be aware that without knowledge of the plume
spacing and required number of pairs of locations,
boundaries, points from within the plume will be
as described in section 4-4. However, the vario-
grouped with points from outside the plume in
gram could also seem horizontal because the actual
computing the sample variogram. The effect of this
sill is reached within a very small lag. If that lag is
problem is minimized as long as the investigator
smaller than the minimum spacing of measured
can define lags that allow data points within the
data, obtaining structure in the variogram would
plume to be grouped together.
not be possible. If the investigator has a vario-
c. The construction of the variogram needs to
gram with no structure, the measured data need to
be considered independent, and kriging techniques,
be described. The description needs to address the
at the lag of the measured data, would be ineffec-
number of pairs of locations in each variogram lag
tive or at least offer little advantage over other
and to demonstrate that the variogram has struc-
interpolation techniques.
ture. A plot of the variogram is helpful to demon-
strate the presence or absence of structure. The
variogram construction discussion also needs to
6-2. Important Elements of Kriging
establish the presence of or lack of isotropy. If
Applications
anisotropy is present, its nature needs to be estab-
lished, and it needs to be addressed by variogram
a. Many important elements of kriging appli-
adjustments similar to the adjustments presented in
section 4-5b.
cations have been discussed in this ETL. These
discussions have been presented as a systematic
d. The variogram cross-validation statistics
and sequential method designed to provide guid-
ance in kriging applications. Occasionally, an
described in section 4-9 are useful and, if avail-
investigator will be presented with the results of a
able, they can aid in the evaluation of a kriging
previous kriging application and will need to eval-
application; authoritative and definitive kriging
uate the application before deciding whether or not
applications should include cross-validation.
to use the results. This section presents a brief
Cross-validation statistics need to conform to the
guidelines discussed in section 4-9. Section 4-9b
review of some important elements of kriging
applications that such an investigator may use in
indicates that the cross-validation exercise needs to
that evaluation. For a more detailed discussion of
balance minimizing the kriging cross-validation
important elements of geostatistical applications,
errors with efforts to guard against bias. Also, as
discussed in section 4-9b, if probabilistic state-
the reader is referred to the American Society of
Testing and Materials (Standard D 5549-94) for
ments are part of the kriging application, there
content of geostatistical investigations.
needs to be some demonstration about the normal-
ity of the reduced kriging error such as the cross-
b. The presence of or lack of stationarity in
validation probability plots included with the
the spatial mean needs to be demonstrated defini-
examples in Chapter 5.
tively. If the spatial mean is nonstationary, then
e. Maps of the kriging estimates and standard
drift is indicated and appropriate measures to
establish stationarity, which are similar to the
deviations need to be presented or discussed. The
measures presented in section 4-3, need to be part
maps of kriging estimates need to conform to any
of the application. In ideal situations,
qualitative information about the information
6-2
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