Geoscience Reference
In-Depth Information
n
n
n
∑
∑
*
z
()
u
=
λ
z
( )
u
+− ⋅
1
λ
m
= −−
∑
2
i
i
i
σ µ
CV V
(, )
λ
CV v
α
(, )
K
α
i
=
1
i
=
1
α
=
1
n
∑
λ
=
1
The condition
is the unbiasedness condition when
i
8.2.3
Kriging with a Trend
i
=
1
the mean
m
is not known. This is the essence of ordinary
kriging: the estimation variance is minimized under the con-
dition that the sum of the weights is 1.0. It can be shown that
ordinary kriging amounts to re-estimating, at each new loca-
tion
u
, the mean
m
as used in the SK expression. Since OK is
most often applied within moving search neighborhoods,
i.e., using different data sets for different locations
u
, the im-
plicit re-estimated mean denoted
m*
(
u
) depends on the loca-
tion
u
. Thus the OK estimator is a type of SK, where the
constant mean value
m
is replaced by a location-dependent
estimate
m*
(
u
).
Ordinary kriging is a non-stationary algorithm. It cor-
responds to a non-stationary RF model with varying mean
but stationary covariance. This ability to rescale locally the
RF model
Z
(
u
) to a different mean value
m*
(
u
) explains the
robustness of the OK algorithm. Ordinary kriging has been
and is likely to remain the anchor algorithm of geostatistics.
The OK system is also a system of normal equations, but
with an additional constraint: the sum of weights equal to
1. The Lagrange formalism is again used to obtain the opti-
mal weights and derive the OK system of equations. Using
the more general notation to take into account the different
support of the samples and the blocks being estimated, the
derivation of the OK system is
The term universal kriging has been traditionally used to de-
note what is, in fact, kriging with a prior trend model. The
terminology, kriging with a trend model (KT) is more appro-
priate since the underlying RF model is considered to be the
sum of a trend component plus a residual:
Z mR
()
u
=
()
u
+
()
u
The trend component defined as
m
(
u
)
= E
{
Z
(
u
)}
,
is usually
modeled as a smoothly varying deterministic function of the
coordinates vector
u
whose unknown parameters are fitted
from the data:
L
=
∑
m
()
u
af
()
u
ll
l
=
0
where
m
(
u
) is the local mean,
a
l
, l = 0…L
are unknown coef-
ficients of the trend model, and
f
l
(
u
) are low order monomi-
als of the coordinates. The trend value
m
(
u
) is itself unknown
since the parameters
a
l
are unknown.
The residual component
R
(
u
) is usually modeled as a sta-
tionary RF with zero mean and covariance
C
R
(
h
).
The Kriging with the trend model (KT) system is also a
system of constrained normal equations. The KT estimator
is written as
n
∑
2
Qi
λ µσµλ
=
(
,
= = +
1,...,
,
)
2
−→
1
minimum
i
E
j
j
1
n
=
∑
u uu
*
(
KT
)
Z
()
λ
() ( )
Z
KT
i
i
i
=
1
Taking the partial derivative with respect to the weights and
the Lagrange multiplier,
and the KT system is
∂
Q
n
�
∑
n
∑
K
λ
(
KT
)
( )
u u u
C
(
−+
)
µ
( )
u u uu
f
(
)
= − =
C
(
), i
1,...,
n
∑
=− ⋅
2 (, ) 2
CV v
+ ⋅
λ
Cv v
( , )
�
j
R j
i
k
k
i
R
i
j
=
1
k
=
0
α
β αβ
∂
λ
n
β
=
1
∑
α
�
(
KT
)
λ
( )
uu u
f
(
)
=
f
( ),
k
=
0,...,
K
j
k
j
k
j
=
1
+⋅= ∀=
2
µ
0,
α
1,...,
n
where the
µ
u
's are
the (
K + 1
) Lagrange parameters associated with the (
K + 1
)
constraints on the weights.
Ideally, the functions
f
k
(
u
) that define the trend should be
specified by the physics of the problem. For example, if a
periodic component is known to contribute to the spatial or
temporal variability of
z
(
u
), a sine function
f
k
(
u
) with specif-
ic period and phase could be considered; the amplitude of the
periodic component, i.e., the parameter
a
k
, would then im-
plicitly be estimated from the
z
data through the KT system.
In the absence of any information about the shape of the
trend, the split of the
z
data into trend and residual compo-
nents is somewhat arbitrary. What is regarded as stochastic
KT
λ
(
)
()
u
's are the KT weights and the
()
k
∂
= −=
∂
Q
n
∑
λ
10
j
µ
j
=
1
with
μ
being the Lagrange parameter introduced due to the
constraint that the weights sum up to 1. The resulting OK
system and the corresponding OK variance are
�
n
∑
λ
Cv v
(
,
)
+
µ
=
CV v
(
,
) ,
∀=
i
1,
…
,
n
š
�
š
j
i
j
i
j
=
1
n
∑
λ
=
1
š
j
