Geoscience Reference
In-Depth Information
G
G
G
G
G
G
G
G
(
G
2
2
2
m
x
)
=
m
V
(
x
)
=
σ
(
x
)
=
C
(
x
,
x
)
=
σ
=
V
C
(
x
,
x
)
=
σ −
γ
(
x
,
x
)
i
i
i
i
i
i
j
i
j
The spatial average of the rainfall in a drainage basin with area
S
is a random
variable
R
(ω
)
so that:
1
G
)
G
R
(
ω
)
=
R
(
ω
,
x
d
∫
S
S
S
It is possible to show that the average m
s
of
R
(ω
)
is the same as the average of
[
]
m
the rainfall m:
(ω . The variance σ
s
² of average rainfall in a
drainage basin can be expressed as follows:
m
=
E
R
)
=
S
S
1
2
G
G
G
G
2
σ
=
σ
−
∫∫
γ
(
x
−
x
)
d
x
d
x
2
i
j
i
j
S
S
SS
If isolated rainfall is distributed according to the same law that governs average
rainfall across a drainage basin, then it is possible to calculate average rainfall and
variance of average rainfall occurring in a drainage basin. Average rainfall and the
variance of average rainfall can be calculated from the variogram of the isolated
rainfall and from the domain S (the drainage basin which is used to calculate the
reduction coefficient). Variograms can often be modeled by using an exponential
equation:
G
G
G
h
γ
(
x
−
x
)
γ
(
h
)
−
i
j
p
=
=
1
−
e
2
2
σ
σ
The variance
σ of average rainfall in a drainage basin can be expressed as:
2
2
σ =
σ
g
(
S
,
p
)
S
where
Sg
is only a function of the area and shape of the drainage basin and the
parameter
p
describes the spatial structure of rainfall. If the drainage basin is
considered as being a square, the following equation is true:
(
,
p
)
0
245
S
g
(
S
,
p
)
=
1
−
p
Measuring the reduction coefficient involves calculating the average and
variance of isolated rainfall (depending on the station that has been in use for the
longest period of time). It also involves determining which parameter p should be
used to represent the spatial structure of rainfall (depending on the correlogram or
variogram that has been created from different series of data of varying length; these
series of data are recorded by the rain gauges). From this information, it is possible
to work out the average, and variance of the distribution of average rainfall for a
