Geoscience Reference
In-Depth Information
7.2.1.3.
Stationarity of order 2
A random function is said to be stationary of order 2 if a correllogram exists and
if the averages and variances of the function are present for each point
x
G
:
(
m
G
= m (for each point
x
G
which is part of D),
)
(
x
σ =
σ
(for all points
x
G
which are part of D),
)
G
G
C
(
σ
x
i
x
,
)
i
x
G
and
x
G
which are part of D).
(ρ =
)
(for all points of
2
The assumption that was made at the end of the previous section rarely occurs in
practice, especially in the case of large drainage basins. A certain number of
geographical factors, such as relief, distance from the sea, latitude, and exposure,
highlight the fact that average rainfall is different in the area being investigated, i.e.
area D. Using geographical information reduces all of these different factors into
one pluviometric characteristic for which the hypothesis of stationarity is valid.
7.2.2.
Use with interpolation methods
7.2.2.1.
General principles
The problem with mathematical modeling comes to the fore when we want to
recreate at any given point
x
G
, during an eventω, and for which the value
R
G
as a
linear combination of observations that are recorded for the same period of rainfall
on the network of rain gauges. This network is made up of a different number of
points known as
R
G
∗
*
has not been recorded. The decision is then made to recreate
( ω
)
(
,
ω
)
x
G
:
∗
G
n
G
=
R
(
x
,
ω
)
=
a
+
λ
R
(
x
,
ω
)
[7.1]
o
o
i
i
i
1
The objective is to have an accurate estimation of the averages:
⎩
⎨
⎧
G
∗
G
⎭
⎬
⎫
⎩
⎨
⎧
G
n
G
⎭
⎬
⎫
E
R
(
x
,
ω
)
−
R
(
x
,
ω
)
=
0
⇒
E
R
(
x
,
ω
)
−
a
−
=
λ
R
(
x
,
ω
)
=
0
o
o
o
o
i
i
i
1
⎩
⎨
⎧
G
n
G
⎭
⎬
⎫
E
R
(
x
,
ω
)
−
=
λ
R
(
x
,
ω
)
=
a
o
i
i
o
i
1
G
n
G
a
=
m
(
x
)
−
=
λ
m
(
x
)
[7.2]
o
o
i
i
i
1
The final objective is to recreate a given point by providing the best estimation
of the averages by using the principle of the least squares:
