ETL 1110-1-175
30 Jun 97
( (x , x ) = ( (h),
fitted by a smooth curve that represents a theoret-
1
2
ical variogram selected from a suite of possible
(2-20)
choices. Usually, the theoretical variogram is
h=
(u1 & u2)2 % (v1 & v2)2
monotonically increasing, signifying that the far-
ther two observations are apart, the more their
residuals tend to differ, on average, from one
another. Several properties common to many
or possibly, on the lag and angle between locations
theoretical variograms are shown in Figure 2-2. If
the variogram either reaches or becomes asymp-
( (x1, x2) = ( (h, a),
totic to a constant value as h increases, that value
is called the sill (Figure 2-2). The distance (value
of h) after which the variogram remains at or
h = (u1 & u2)2 % (v1 & v2)2,
(2-21)
close to the sill is called the range. Measurements
whose locations are farther apart than the range all
v2 & v1
have the same degree of association. Often, a
a = atan
u2 & u1
variogram will have a discontinuity at the origin,
signifying that even measurements very close
together are not identical. Such variation in the
(Figure 2-1). Equation 2-20 is called an isotropic
measurements at small scales is called the nugget
variogram and Equation 2-21 is a directional
effect. The size of the discontinuity is called the
variogram at angle a.
nugget. Although the nugget effect is sometimes
confused with measurement error, there is a subtle
d. For the isotropic case, the sample, or
difference between these two concepts that will be
empirical, variogram is obtained by averaging the
explained in section 2-4. A simple monotonic
square of all computed differences between resid-
function is usually selected to approximate the
uals separated by a given lag:
variogram. Four such functions that are often used
in practice are:
1
( (h) =
Z ( (x )
^
ave
1
2
the exponential variogram (parameters: sill, s >
0; nugget, 0 < g < s; range, r > 0)
(2-22)
(
2
&Z
(x )
:
2
h
h & )h < hij < h % )h
g % (s & g) 1 & exp &3
, h>0
r
( (h) =
(2-23)
where, as before, hij is the distance between xi and
xj. For a given h as more and more points sepa-
h=0
0,
rated by distance h )h are sampled and as )h
gets small, ( (h) should approach the theoretical
^
the spherical variogram (parameters: sill, s > 0;
variogram. More detail on variogram estimation
nugget, 0 < g < s; range, r > 0)
will be presented in Chapter 4, including the
directional case. In this section, it will be suffi-
cient to describe some general properties of iso-
s,
h> r
tropic variograms that will be referred to numerous
times in the application sections to follow.
(2-24)
3
( (h) ' g % (s & g) 1.5 h & 0.5 h
, 0< h#r
e. A plot of the sample variogram versus h
r
r
often has a considerable degree of scatter (Fig-
h'0
0,
ure 2-2), which is especially evident if the sample
size n is small. However, the points can usually be
2-7
Previous Page
