Environmental Engineering Reference
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where
T
i
are the measured transmissivities given as
T
1
= 2150 m/d,
T
2
= 2390 m/d,
T
3
= 2280 m/d, and
T
4
= 2060 m/d. Hence,
Transmissivity (m
2
/d)
Station
Coordinates (km, km)
X
(0, 0)
-
1
(173, 100)
2150
2
(134, −120)
2390
T
X
=
( .
0 284 2150
)(
)
+
( .
0 330 2390
)(
)
+
( .
0 081 2280
)(
)
3
2280
(−163, −281)
0 305 2060
= 2210
+
( .
)(
)
4
(−235, 85)
2060
m /d
What is the error variance of the estimated transmis-
sivity at
X
?
The error variance,
σ
2
, is estimated by Equation
(10.198) as
4
Solution
∑
2
σ
=
λγ
+
α
e
i
iX
i
=
1
The station weights for each station,
λ
1
,
λ
2
,
λ
3
, and
λ
4
are
derived from Equation (10.197), which can be written
as
=
( .
0 284 32 600
)(
)
+
( .
0 330 28 700
)(
)
+
( .
0 081 58 4
)(
00
)
+
( .
0 305 42 600
)(
)
−
1 12 10
.
×
4
=
2 53 10
.
×
4
m /d
4
2
i
i
=
1
2
:
:
λγ
+
λγ
+
λγ
+
λγ
+
α γ
=
1 11
2
12
3
13
4
14
1
X
=
λγ
+
λγ
+
λγ
+
λγ
24
+
α γ
=
which gives
1
21
2
22
3
23
4
2
X
i
i
=
3
4
:
:
λγ
+
λγ
+
λγ
+
λγ
+
α γ
=
1
31
2
32
3
33
4
34
3
X
4
2
σ
e
=
2 53 10
.
×
=
159
m /d
=
λγ
+
λγ
+
λγ
+
λγ
+
α γ
=
1
41
2
42
3
43
4
44
4
X
:
λ
+
λ
+
λ
+
λ
+
0
=
1
1
2
3
4
It has been shown that for relatively uniform and
dense sampling grids, ordinary kriging (described
above and based on the intrinsic hypothesis) provides
an effective interpolation scheme (Ellsworth et al.,
1999). For irregular and sparse spatial sampling
grids, nonlinear kriging techniques, such as
rank order
kriging
(Journel and Deutsch, 1996) might work better.
Kriging is referred to as nonlinear when the weights
assigned to neighboring points depend on the actual
data values.
where the semivariogram,
γ
ij
, can be written as
(
)
1 2
.
γ
=
γ
(
x
−
x
)
=
56 5
.
(
x
−
x
)
2
+
(
y
−
y
)
2
ij
i
j
i
j
i
j
using the given station coordinates to calculate
γ
ij
yields the following system of equations for the station
weights
PROBLEMS
i
i
=
1
:
λ
( )
0
+
λ
(
37 200
,
)
+
λ
(
99 800
,
)
+
λ
(
76 800
,
)
+ =
α
32 600
,
1
2
3
4
=
2
:
λ 37 200
(
,
)
+
λ
( )
0
+
λ
(
61 200
,
)
+
λ
(
79 900
,
)
+ =
α
28 700
,
1
2
3
4
10.1.
A variable,
W
, is calculated using the formula
i
=
3
:
λ
(
99 800
,
)
+
λ
(
61 200
,
)
+
λ
( )
0
+
λ
(
68 900
,
)
+ =
α
58 400
,
1
2
3
4
i
=
4
:
λ
(
76 800
,
)
+
λ
(
7
9 900
,
)
+
λ
(
68 900
,
)
+
λ
( )
0
+ =
α
42 600
,
1
2
3
4
Q
R
:
λ
+
λ
+
λ
+
λ
+ =
0
1
W
=
1
2
3
4
/ ν
The solution of this system of equations is
where
Q
is a n(0, 1) variate and
R
is a chi-square
variate with
ν
degrees of freedom. If
ν
= 28 and
the calculated value of
W
is 1.80, estimate the
probability that
W
is greater than 1.80. What is
the expected value of
W
?
λ
=
0 284
.
,
λ
=
0 330
.
,
λ
=
0 081
.
,
λ
=
0 305
.
,
1
2
3
4
α
= −
1 12 10
.
×
4
10.2.
A random variable,
Z
, is defined by
According to Equation (10.188), the estimated trans-
missivity,
T
X
, at Station
X
is given by
X
Y
/
/
ν
ν
X
Z
=
Y
4
∑
λ
1
where
X
and
Y
are chi-squared variates with
ν
X
and
ν
Y
degrees of freedom, respectively. If
ν
X
= 15
T
=
T
X
i
i
i
=
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