Environmental Engineering Reference
In-Depth Information
The error variance,
σ
2
, is estimated by Equation
(10.182) as
mean can be assumed approximately constant (even
though the mean changes gradually in space). Interpola-
tion based on the intrinsic hypothesis is called
ordinary
kriging
, versus simple kriging, which requires second-
order stationarity and a known and constant mean.
Combining Equations (10.184) and (10.185) yields the
following expression for the semivariogram,
γ
(
h
),
4
∑
σ
2
=
σ
2
−
λ
C
e
i
iX
i
=
1
=
10 2
.
−
[( .
0 355 3 78
)( .
)
+
( .
0 134 1 69
)( .
)
+
( .
0 047
8 0 510
)( .
)
2
+
( .
0 0 841 1 04
.
)( .
)
=
8 51
.
cm
1
2
( )]
2
(10.186)
γ( )
h
=
[
Z
(
x h
+
)
−
Z
x
10.12.2 The Intrinsic Case
If the covariance exists, it can be shown that the
semivariogram and the covariance function are related
by
The hypothesis of second-order stationarity with a finite
variance is not satisfied in some cases of practical inter-
est. This is particularly true in cases where the variance
calculated using the measured data, called the
experi-
mental variance
, increases with the size of the area
under consideration (de Marsily, 1986), as is the usually
case for hydraulic conductivities in porous media. A
less stringent hypothesis, called the
intrinsic hypothesis
,
is invoked to make estimation by kriging possible.
The intrinsic hypothesis states that a random space
function (rSF)
Z
can be characterized by the following
statistics
γ( )
h
=
C
( )
0
−
C
( )
h
(10.187)
The covariance function at zero lag,
C
(0), is equal to
the variance of the random function, and the covariance
generally decreases as the separation between measure-
ments increases. Conversely, the semivariogram at zero
lag is equal to zero, and increases asymptotically to the
variance,
C
(0), with increasing separation. This asymp-
tote is called the
sill
of the semivariogram, and the sepa-
ration at which the semivariogram reaches its asymptotic
value is called the
range
of the semivariogram. The term
scale
is sometimes used to refer to the range (Piegorsch
and Bailer, 2005). The
nugget
is the limiting value of the
semivariogram as the separation approaches zero. A
typical semivariogram illustrating the nugget, sill, and
range is illustrated in Figure 10.8. Care should be taken
in defining the range of a semivariogram, since the sill
is frequently approached asymptotically. It is generally
recommended that the analyst define the criteria used
to identify the point where the sill begins. nugget effects
are associated with microscale variations and measure-
ment errors.
Z
(
x h
+
)
−
Z
( )
x
=
m
( )
h
(10.183)
and
var[
Z
(
x h
+
)
−
Z
( )]
x
= 2γ
( )
h
(10.184)
where the functions
m
and
γ
depend only on the separa-
tion,
h
, and the function
γ
(
h
) is called the
variogram
, or,
more commonly, the
semivariogram
, in which case 2
γ
(
h
)
is called the variogram. The term “variogram” was origi-
nally proposed by Matheron (1962). The basic assump-
tions of the intrinsic hypothesis, given in Equations
(10.183) and (10.184), require that the statistics of the
increments of
Z
(
x
) are homogeneous (in space), and
random space functions that obey the assumptions asso-
ciated with the intrinsic hypothesis (Eqs. 10.183 and
10.184) demonstrate
intrinsic stationarity
(Piegorsch
and Bailer, 2005). For the case in which the mean is
removed from the data or the mean is homogeneous,
then Equation (10.183) becomes
sill
g
(
h
)
Z
(
x h
+
)
−
Z
( ) ]
x
= 0
(10.185)
Equation (10.185) can also be applied in environ-
ments where the mean is unknown and changes gradu-
ally, provided that the mean can be assumed
approximately constant over a distance |
h
|. In such cases,
the intrinsic hypothesis, as given by Equations (10.184)
and (10.185), are applicable over distances where the
nugget
range
h
Figure 10.8.
Semivariogram properties.
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