Geology Reference
In-Depth Information
precisely Intrinsic Random Function (IRF) of order zero because its first
order difference has become stationary. Similarly if (
k
+ 1)
th
order difference
of a variable becomes stationary, the variable is called IRF-
k
, i.e. Intrinsic
Random Function of order
k
. Most of the hydrogeological variables could be
categorized as IRF-0 from the practical point of view. We will show in the
subsequent text that the complete estimation theory can be developed using
variograms and thus there is no need to calculate the mean.
STRUCTURAL ANALYSIS AND COMPUTATION
OF COVARIANCE AND VARIOGRAM
As both the covariance and the variogram are the function of separation
vector between two points
i
and
j
, values of separation vectors e.g.,
h
1
,
h
2
etc. are decided first such that
h
=
xx
(19)
Depending upon the value of
h
, the data are grouped into pairs and some
function as defined below are averaged to obtain a covariance (
C
ij
) or a
variogram (
-
i
j
ij
).
N
1
h
)
C
z
(
h
)=
{ (
zx
)
m
}{ (
zx
h
)
m
}
(20)
i
z
i
z
N
h i
1
N
1
h
)
2
z
(
h
) =
{( )
zx
zx
(
h
)}
(21)
i
i
2
N
h
i
1
N
h
is the number of pairs for a given
h
. It is usual to write
C
(
h
) or
C
ij
or
C
(
x
i
,
x
j
) etc. and correspondingly for the variogram also. If we consider
equation (18), the expression in the equation (21) should be called semi-
variogram. However, in practice only semi-variograms are used; hence for
convenience we call it simply a variogram. It is now clear that in practice,
we cannot calculate a true mean. Thus we prefer to work on variogram
rather than covariance.
Properties of Covariance and Variogram
A relation between a variogram and a covariance can be established as
follows if both the functions exist (i.e., the mean exist or the variable is
stationary).
(22)
Here
C
(0) is equal to the variance of the variable. Thus in case of stationary
variables one function can be calculated from the other.
()
h
=
CCh
(0)
( )
Behaviour of the Variogram for Large Separation Distance,
h
As the distance
h
increases, (
h
) increases. However, after a large distance,
(
h
) stabilizes around a constant value except sometimes when it increases
