Geology Reference
In-Depth Information
Therefore, before deciding on a new location for a measurement point (or
the deletion of an existing point) a variance can be calculated, and thus when
the variogram or covariance is known, a better network of data can be
designed based on the minimum variance, prior to any measurement.
UNIVERSAL KRIGING MOSTLY APPLIED TO
WATER LEVEL ESTIMATION
Water levels generally being a non-stationary variable, Ordinary Kriging
technique is not applicable and the technique of Universal Kriging is thus
applied. The water level estimate at any point
p
(say a well) using Universal
Kriging is written as follows:
N
)
h
*
p
=
D
p
+
(
h
i
-
D
i
)
(19)
i
i
1
where
h
*
p
and
D
p
are the estimated water level and the drift respectively at
any unmeasured well
p
;
h
i
and
D
i
are the values of water level and drift at
well
i
and
i
are kriging weights. The water levels
h
are split into a smoothly
varying deterministic part
D
and the residual (
h
-
D
). A bounded variogram
could be obtained for the residual, as this is a stationary random function.
Modelling the drift correctly is difficult as it is not possible to estimate
parameters of the drift and that of the variogram of residuals from a single
data set. Drift depends on the nature of water level variation, and could be
linear, quadratic or of higher order. Usually a drift is approximated by
polynomials of the space coordinates. For example, a linear drift can be
written as:
D
i
=
a
+
bx
i
+
cy
i
3
i
(20)
A quadratic drift can be written as:
D
i
=
a
+
bx
i
+
cy
i
+
dx
i
2
+
ey
i
2
+
fx
i
y
i
(21)
where
a
,
b
,
c
,
d
,
e
and
f
are the drift coefficients and are constants. It is
possible to estimate a drift by ordinary kriging but it has got the indeterminacy
problem, as that requires the knowledge of the variogram of residuals.
The kriging weights can be determined as follows:
N
)
+
0
+
1
x
j
+
2
y
j
=
jp
i j
i
1
j
= 1 --------
N
(22)
N
)
= 1
(23)
i
i
1
N
)
x
i
=
x
p
(24)
i
i
1
