Geoscience Reference
In-Depth Information
x G , the function
If we consider a particular fixed point, such as
G
R is used to
represent the probability distribution law of isolated rainfall that occurs at
R
(
ω
)
=
R
(
x
,
ω
)
is said to be a random variable. In our study
)
i
i
x G over a
period of time t.
x G , the
R G
is the random function average that occurs at the point
( ω
)
m G is expressed as follows:
{
function
(
)
} Ω
G
G
G
m
(
x
)
=
E
R
(
x
,
ω
)
=
R
(
x
,
ω
)
P
(
d
ω
)
i
i
i
m G represents the average of the total amount of rainfall
that has fallen over a period of time, at a particular rain gauge located at
Regarding rainfall,
(
)
x G . The
G
G
covariance function
C
(
x
i x
,
)
(whenever it exists) can be expressed as follows:
j
{
}
G
G
G
G
G
G
G
G
C
(
x
,
x
)
=
E
R
(
ω
)
R
(
ω
)
m
(
x
)
m
(
x
)
with
C
(
x
,
x
)
=
C
(
x
,
x
)
i
j
i
i
i
j
i
j
j
i
A random function is said to be of order two if both the average and the
covariance exist together. If a random function is of order two then a variance for
each point x G exists:
G
G
G
G
2
V
(
x
)
= σ
(
x
)
=
C
(
x
,
x
)
i
i
i
i
ρ
i x G
The correlation function
(
,
)
can be defined from the co-variance and
j
variance functions:
G
G
C
(
x
,
x
)
G
G
i
j
ρ
i x G
ρ
j x G
ρ
(
x
,
x
)
=
with
(
,
)
=
(
,
)
)
i
j
j
i
G
G
σ
(
x
)
σ
(
x
)
i
j
This formulation is quite similar to the representation that we have of
precipitation. However, in order to continue with our research some more
assumptions need to be made. Such assumptions include the fact that if it is possible
to record annual rainfall each year (climatologic data), but the estimation of average
rainfall has to be unique.
7.2.1.2. Homogenity and isotropy
x G and
x G and only depends
If the correlation function is independent of the points
j
on the vector h G =
x G , the area being studied is said to be homogenous, and an
anisotropic correlogram is present
x G -
j
ρ .
( h
)
x G and
x G and only
If the correlation function is independent of the points
j
G
x G -
x G , the area being
depends on the normal function
h
=
h
of the vector
j
studied is said to be isotropic and an isotropic correlogram is present
(ρ .
)
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