Environmental Engineering Reference
In-Depth Information
is defined as a variable that is averaged over a large area
and is smooth enough to be differentiable (Kitanidis,
1993). The exponential model is popular in practice
because of its simple analytic form, and, along with the
spherical model, is applicable to random fields that are
are less smooth that those of regionalized variables.
random variables that are continuous but not differen-
tiable can be described by the exponential and spherical
models. The hole-effect model is used where the covari-
ance function of the random field does not decay mono-
tonically with distance, and is applicable to processes in
which fluctuations above the mean are compensated by
fluctuations below the mean. The nugget effect model is
used when all measurements are spatially uncorrelated,
but are locally variable.
In the above development, the spatial covariance
function,
C
(
h
), is expressed as a function of a separation
vector,
h
. In cases where the spatial covariance
function depends on both the magnitude and direction
of of the separation,
h
, the process is
anisotropic
, and
in cases where the covariance function depends only
on the magnitude of the separation, |
h
|, the process is
isotropic
. Most analyses assume that the process is
isotropic.
Solution
Comparing the covariance function of the rainfall with
the form of the exponential covariance function given
in Table 10.11 indicates that the rainfall variance,
σ
2
, is
10.2 cm
2
, and the length scale,
L
, is 3.2 km. The range
of the covariance function is equal to 3
L
= 3 ×
3.2 km = 9.6 km.
The station weights for each station,
λ
1
,
λ
2
,
λ
3
, and
λ
4
are obtained by solving Equation (10.175), which can be
written as
i
=
1
2
:
:
λ
C
+
λ
C
+
λ
C
+
λ
C
=
C
1
11
2
12
3
13
4
14
1
X
i
=
λ
C
+
λ
C
+
λ
C
+
λ
C
=
C
1
21
2
22
3
23
4
24
2
X
i
=
3
4
:
:
λ
C
+
λ
C
+
λ
C
+
λ
C
=
C
1
31
2
32
3
33
4
34
3
X
i
=
λ
C
+
λ
C
+
λ
C
+
λ
4
C
=
C
X
1
41
2
42
3
43
44
4
where
(
x
−
x
)
2
+
(
y
−
y
)
2
i
j
i
j
C
=
C x
(
−
x
)
=
10 2
. exp
−
ij
i
j
3 2
.
using the given station coordinates to calculate
C
ij
yields the following fourth-order system of equations
for the stations weights (noting that
C
ij
=
C
ji
):
EXAMPLE 10.31
i
i
=
1
:
λ
(
10 2
. )
+
λ
( .
0 893
)
+
λ
( .
0 00672
)
+
λ
( .
0 511
)
=
3 78
.
1
2
3
4
Synoptic analysis of rainfall measurements indicate that
the monthly rainfall in a region has a mean of 11.6 cm,
and the spatial correlation is described by the exponen-
tial covariance function
=
2
:
λ
( .
0
893
)
+
λ
(
10 2
. )
+
λ
( .
0 146
)
+
λ
( .
0 00330
)
=
1 69
.
1
2
3
4
i
=
3
:
λ
( .
0 00672
)
+
λ
( .
0 146
)
+
λ
(
10 2
. )
+
λ
( .
0 00874
)
=
0 510
.
1
2
3
4
i
=
4
:
λ
( .
0 511
)
+
λ
( .
0 003
30
)
+
λ
( .
0 00874
)
+
λ
(
10 2
. )
=
1 04
.
1
2
3
4
The solution of this system of equations is
h
C h
( )
=
10 2
. exp
−
3 2
.
λ
=
0 355
.
,
λ
=
0 134
.
,
λ
=
0 0478
.
,
λ
=
0 0841
.
1
2
3
4
According to Equation (10.176), the estimated rain-
fall at station
X
,
Z
X
, is given by
where
h
is in kilometers and
C
(
h
) is in cm
2
. What is the
range of the covariance function? Estimate the rainfall
at a location,
X
, based on the measured rainfall at four
nearby stations, where the coordinates and rainfall mea-
surements at the stations are given by
4
∑
Z
= +
µ
λ
(
Z
−
µ
)
X
i
i
i
=
1
where
μ
= 11.6 cm, and the rainfall measurements,
Z
i
,
are given as
Z
1
= 7.2 cm,
Z
2
= 6.5 cm,
Z
3
= 8.3 cm, and
Z
4
= 7.8 cm, hence
Station
Coordinate (km, km) Measured rainfall (cm)
X
(0, 0)
-
1
(1.27, 1.25)
7.2
Z
X
=
11 6
.
+
( .
0 355 7 2 11 6
)( .
−
. )
+
( .
0 134 6 5 11 6
)( .
−
. )
2
(−1.44, 1.92)
6.5
3
8.3
(−2.68, −1.55)
+
( .
0 0478 8 3 1
)( .
−
1 6
. )
+
( .
0 0
841 7 8 11 6
)( .
−
. )
4
(2.07, −1.74)
7.8
=
8 88 cm
.
The estimation rainfall at station
X
is therefore equal
to 8.88 cm.
What is the error variance of the estimated rainfall
at
X
?
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