Geography Reference
In-Depth Information
where,
0
Z
at point x
0
and
= weighting
factors that minimize the variance of the estimation error (ordinary kriging
weights).
Now two conditions are imposed to Equation (3), i.e., the unbiased
condition and the condition of optimality. The unbiased condition means that
the expected value of estimation error or the mean difference between the
= estimation of function
Z
*
x
and the true (unknown)
xz
value of the concentration of
water quality variable should be zero. The condition of optimality means the
variance of the estimation error should be minimum.
The spatial structure defined by theoretical variogram, a kriging system of
linear equations combining neighbouring information can be defined as
estimated
0
*
z
x
0
n
C
x
,
x
C
x
,
x
j
i
j
i
0
, i = 1, 2, …n
(4)
j
1
subjected to the constraint on weights:
n
1
j
(5)
j
1
where,
= Lagrangian multiplier and
C
x
,
x
= value of covariance
i
j
between two points x
i
and x
j
.
When we deal with an intrinsic case, i.e., working with variogram, the
kriging Equation (4) and (5) are simply modified as follows (Marsily, 1986;
Ahmed, 2006):
C
x
,
x
C
0
x
,
x
(6)
i
j
i
j
C
x
,
x
C
0
x
,
x
(7)
i
0
i
0
Eqnuations (6) and (7) hold good only when both the covariance and the
variogram exist, i.e., variables are stationary.
(B) Geostatistical or Variogram Models
Experimental geostatistical or variogram model is the function of
separation vector between two points i and j.
