ETL 1110-1-175
30 Jun 97
2-2. Regionalized Random Variables
Chapter 2
Technical Aspects of Geostatistics
a. General.
(1) Suppose the extent of groundwater con-
2-1. General
tamination of a particular pollutant over a given
study area is being determined. To simplify the
a. This chapter provides technical aspects or
presentation, all data are assumed to be distributed
the necessary theoretical background for under-
over a two-dimensional region. In three-
standing kriging applications. Emphasis will be
dimensional groundwater flow systems, one could
placed on presentation of the basic ideas; long
study the depth-averaged concentration of a pol-
formulas or derivations are kept to a minimum.
lutant or the concentration of the pollutant in a par-
Statistical terms that are commonly used in
ticular horizontal stratum of the flow system. Let
geostatistical applications will be highlighted with
a vector x=(u,v) denote an arbitrary spatial loca-
bold text and briefly defined as they are intro-
tion in the study area. Unless otherwise stated, it
duced; notation used in this ETL is also tabulated
will be assumed throughout the ETL that u is the
in Appendix B. The reader who wishes a more
east-west coordinate and v is the north-south
thorough discussion of these fundamental concepts
coordinate (Figure 2-1). Denote by z(x) a meas-
may consult the references cited in Chapter 3.
urement at location x, such as the concentration of
Previous exposure to engineering statistics at the
a pollutant. The ultimate goal of an investigator
level of Devore (1987) and Ross (1987) would be
would be to determine z(x) for all locations in the
helpful in understanding some parts of this chap-
study area. However, without explicit knowledge
ter. Readers with limited statistical experience
of the flow and transport field, this goal cannot be
may wish to briefly scan this chapter and refer
achieved. Therefore, suppose, instead, that the
back to it after reading the remaining chapters.
goal is to estimate the values of z(x) with a given
error tolerance. In other situations, small estima-
b. In section 2-2, regionalized random vari-
tion error over some parts of the study area (for
ables are discussed. Regionalized random varia-
instance, near a domestic water supply) may need
bles constitute the random process that is sampled
to be obtained, while allowing larger estimation
to obtain the observed data that are available for
errors in other parts of the study area. The theory
analysis. Basic ideas related to probability distri-
of regionalized random variables is designed to
butions, means, variances, and correlation are
accomplish these goals.
introduced. The variogram, which is the funda-
mental tool used in geostatistics to analyze spatial
(2) In the regionalized random variable theory,
correlation, is introduced in section 2-3. In sec-
the true measurement z(x) is assumed to be the
tion 2-4 how kriging is used to obtain the best
value of a random variable Z(x). Associating a
weights for spatial prediction is discussed, and
random variable Z(x) with a true measurement z(x)
how the mean squared prediction error for these
is done for the purpose of characterizing the degree
predictions is computed is also shown. Section 2-5
of uncertainty in the quantity of interest at point x.
deals briefly with co-kriging, which is prediction of
If there is no actual measurement taken at x, then
one variable based not only on measurements of
the values taken on by Z(x) represent "potential"
that variable but on measurements of other vari-
measurements at x; that is, Z(x) represents possible
ables as well. Finally, section 2-6 shows how
values that might be expected if a measurement
kriging may be applied to determine not just opti-
were taken at x. Because there is uncertainty asso-
mal spatial predictions but also probabilities
ciated with Z(x), it needs to be characterized by a
associated with various events, such as extreme
probability distribution, defined by P [Z (x) # c]
events that may be of importance in risk-based
where P denotes probability and c is any constant.
analyses.
2-1
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