Geoscience Reference
In-Depth Information
The second generalization is based on using distances
calculated with anisotropy, that is, preferred directions of
continuity. The anisotropy can be introduced by re-scaling
the directional distances appropriately, as has been used by
some major gold companies, also observed in the Western
Australia goldfields. Overall, the application of ID method
has been steadily decreasing through the years in favor of
geostatistical methods.
Kriging is a method that allows calculating weights that
are optimal according to the least-squares, or minimum ex-
pected error variance, criteria. Although estimation schemes
sometimes provide a measure of how good the estimates
are, there is no good measure of uncertainty attached to the
estimates. Probabilistic estimation (Chap. 9) or simulation
(Chap. 10) is needed for this purpose.
The choice of method depends on the geologic setting, the
amount of information available, and the characteristics of
the RF model envisioned. The most common estimation
method is OK, although the different variants that model
trends have become more popular in more recent years.
8.2.1
Simple Kriging
The purpose of kriging is to determine a set of optimal
weights that minimize the expected error variance. Consider
a linear estimator:
n
∑
Z
*
()
u
= ⋅ −+
λ
[( )
z
u
mm
]
i
i
i
=
1
n
n
∑
∑
= ⋅ +−
λ
z
() 1
u
λ
()
u·
m
i
i
i
i
i
=
1
i
=
1
8.2
Kriging Estimators
where
z
(
u
i
) are the data values and
Z*
(
u
) is the estimate. The
constant mean
m
is assumed known and stationary (location-
independent). In this case, the SK estimator is unbiased by
definition, and the estimation is performed in effect on the
residuals data values. The known mean
m
is subtracted from
the data values and then added back after the residuals have
been estimated. The estimation error is then expressed as a
linear combination of the residuals
The basis for the kriging framework is to calculate the
weights that minimize an expected error variance. There are
many flavors of kriging, but the basic forms differ mostly on
the assumptions they make regarding the local or stationary
domain mean. This is expressed as conditions on the set of
weights. Linear kriging has been presented in several clas-
sic references, such as Journel and Huijbregts (
1978
); Isaaks
and Srivastava (
1989
); Deustch and Journel (
1997
); and
Chilès and Delfiner (
2011
).
The more common types of kriging are:
• Simple kriging (SK): minimizes the error variance with
no constraints on the weights. The mean is a known con-
stant (inferred from the available samples) for the entire
domain.
• Ordinary kriging (OK): the local mean is implicitly re-
estimated as a constant within each search neighborhood.
OK is a common technique used to obtain interim esti-
mates.
• Kriging with a trend model or universal kriging (KT or
UK): this method estimates residuals from a specified
location-dependent mean
m
(
u
). The location-dependent
mean could be a specified constant (local-varying mean),
or a deterministic trend typically specified as a function
of the coordinates. This method is also called non-station-
ary kriging because of the location-dependent mean.
• Kriging with an external drift: in this variant, the trend
model is scaled from a secondary variable.
• Factorial kriging: the RF model
Z
(
u
) is split into indepen-
dent components (factors), which are then independently
estimated.
• Non-linear kriging, including Gaussian-based (disjunc-
tive kriging, uniform conditioning, multi-Gaussian),
indicator kriging (median, multiple, probability), and log-
normal kriging. These are discussed in Chap. 9.
*
Y
()
u
−
Y
()
u
.
SK
The error variance is defined as
*
2
*
2
*
EY
{[ ()
uu u uu
u
−
Y
()]}
=
EY
{[ ()]} 2 { () ()}
{[
−⋅
EY
⋅
Y
2
+
EY
( )] }
and can be expressed as a linear combination of covariance
values of the residuals:
nn
∑∑
λλ
EY
{
(
uu
)
⋅
Y
(
)}
i
j
i
j
i
==
11
j
n
nn
∑
∑∑
−⋅
2
λ
EY
{
( )
u u
⋅ + =
Y
(
)}
C
(0)
λλ
C
(
uu
,
)
i
i
i
j
i
j
i
=
1
i
==
11
j
n
∑
−⋅
2
λ
C
( ,
uu
)
+
C
(0)
i
i
i
=
1
It can be seen that the error variance is written in terms of the
(1) weights used for the estimate (the
values), (2) the vari-
ance (
C
(
0
)), (3) the covariance between the data locations
and the location,
(
C
uu
and (4) the covariance between
( ,
))
i
all pairs of data
(
C
uu
. The covariance is required be-
cause the estimate is linear and the estimation variance is a
quadratic form. The required covariance values are calculat-
ed from the variogram model.
The optimal weights
λ
i
, i = 1,…, n
are determined by tak-
ing partial derivatives of the error variance with respect to
the weights and setting them to zero:
(
,
))
i
j
