Geology Reference
In-Depth Information
Expectation
If we can calculate the expected value of a distribution, it is known as
Expectation and is denoted by
E
. It is also known as the first moment of the
variable. Practically, one calculates the expected value of a variable using
arithmetic mean keeping in mind that the number of values available can be
approximated by
2
. However, we write an expected value of
Z
as follows:
m
z
=
(13)
Ez
[ ]
3
z
Variance
The variance of the variable
Z
can be written in the following form:
z
=
E
[(
z
i
-
m
z
)
2
] (14)
It is important to note that the variance in the general case is the variance
of many realization available at a single location. However, the variance
given by the equation (14) is the spatial variance of the variable and it does
not depend on the location
i
.
2
Covariance
Similarly an expression for covariance of a variable
Z
between two points
i
and
j
in space can be given as:
C
ij
=
E
[(
z
i
-
m
x
)(
z
j
-
m
x
)] (15)
By definition the expected value
m
z
should not change by adding or
removing one or a few values of the variable from the data. However, since
we simply calculate an arithmetic mean, it will not be possible to ensure this.
This means that a true mean or a true expected value can never be calculated
from the data. Thus we can neither calculate a correct variance nor a correct
covariance. We have to then introduce another hypothesis known as Intrinsic.
If we work on another variable
y
defined as:
y
=
z
i
-
z
j
(16)
i.e. the first order difference of the primary variable
Z
, then the expected
value of the variable
y
can be shown to be 0 provided the variable
Z
has an
unknown but constant mean
m
z
. Since the expected value of
Y
is 0, we need
not calculate it from the data. If we calculate the variance of this new
variable
Y
, we find the following expression.
Var (
y
)=
E
[(
z
i
-
z
j
)
2
] (17)
Though it is the variance of the new variable
Y
, it is a new function for the
variable
Z
that does not depend on the location of points nor is affected by
the value of
m
z
. This new function is called variogram of the variable
Z
between two points in space
i
and
j
and is denoted by
2
ij
=
E
[(
z
i
-
z
j
)
2
] (18)
The variogram depends only on the separation vector between the two points
i
and
j
and not on the location of
i
and
j
. Here
Z
is called Intrinsic and more
