Geology Reference
In-Depth Information
n
n
n
n
)
)
)
)
=
C ( x i , x j ) + C (O) - 2
C ( x i , x o ) - 2 1
+ 2
(6)
i
j
i
i
i
1
j
1
i
1
i
To minimize the above expression which is quadratic in
, we equate
- Q /- i , i = 1…… n and - Q /- to zero and obtain the following kriging
equations:
and
n
)
C ( x i , x j ) - = C ( x i , x o )
(7)
j
j
1
i = 1….. n
n
)
= 1
(8)
j
j
1
where C ( x i , x j ) is the value of covariance between two points x i and x j . Now
n
)
2
to express the value
k we multiply both sides of equation (7) by
and
i
i
1
n
n
)
)
put the value of
C ( x i , x j ) into equation (6) to get:
j
i
j
1
i
1
n
)
k 2 = C (O) -
C ( x i , x o ) +
(9)
i
i
1
which is the variance of the estimation error. Equations (7) and (8) are a set
of ( n + 1) linear simultaneous equations with ( n + 1) unknowns and on
solving them we get the values of
i , i = 1…… n and which are later used
to calculate the estimated value by equation (1) and the variance of the
estimation error by equation (9). Therefore, the square root of the expression
given by equation (9) gives the standard deviation
k , which means that with
the 95% confidence true value will be within z * ( x 0 )± 2 k .
When we deal with intrinsic case i.e. working with variogram, which is
common in hydrogeology the kriging equations (7) to (9) are simply modified
as follows:
( x i , x j ) (10)
C ( x i , x 0 ) = C (0) - ( x i , x 0 ) (11)
Equations (10) and (11) hold good only when both the covariance and the
variogram exist, i.e. in the case of stationary variables. In case the covariance
cannot be defined we can derive the following kriging equations:
C ( x i , x j ) = C (0) -
n
)
( x i , x j ) + = ( x i , x 0 )
(12)
j
j
1
i = 1….. n
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