A more technical presentation about Kriging by Alberto Bellin. It covers simple and ordinary kriging, and other topics

1. GEOSTATISTICS: Part II
geostatistics as spatial interpolation technique
Alberto Bellin
Department of Civil, Environmental and Mechanical Engineering
University of Trento
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 1 / 25

2. Outline
1 The kriging paradigm
2 Simple and Ordinary Kriging
3 Secondary Information
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 2 / 25

3. General structure of the model
Z⇤
(x) m(x) =
n(x)
X
i=1
i (x) [z(xi ) m(xi )]
Z⇤(x): best (in some sense) estimate of the SRF Z at the position x;
z(xi ): measurement of the SRF Z at the location xi ;
m(x): local mean (at the position x);
n(x): number of measurements used for the estimation (not necessarily all
measures available)
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 3 / 25

4. The Kriging paradigm (continued...)
SEVERAL FORMS OF THE GEOSTATISTICAL INTERPOLATION ARE
AVAILABLE DEPENDING ON
1 model of spatial variability for the mean;
2 number of points used in the interpolation
To compute the weights i we impose two conditions:
1 unbiased estimation: E [Z⇤(x) Z(x)] = 0
2 minimize the variance of the error, i.e.:
2
E = E
h
(Z⇤(x) Z(x))2
i
= min
Z(x): is the true (unknown) value of the SRF Z at the location x.
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 4 / 25

5. Simple Kriging
We assume: m(x) = m = const and known, therefore:
Z⇤
SK (x) =
n(x)
X
i=1
SK
i (x) [z(xi ) m] + m
=
n(x)
X
i=1
SK
i (x) z(xi ) +
2
41
n(x)
X
i=1
SK
i (x)
3
5 m
We can see immediately that: E [Z⇤
SK (x) Z(x)] = m m = 0
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 5 / 25

6. Simple Kriging (continued...)
By replacing the above expression of Z⇤
SK into the general expression of
the error variance and setting equal to zero the derivatives with respect to
i , we obtain after a few manipulations:
n(x)
X
i=1
SK
i (x) CR(xi , xj ) = CR(xj , x); j = 1, ..., n(x)
and
2
E = CR(0)
n(x)
X
i=1
SK
i (x) CR(xi , x)
where CR is the covariance function of the residual: R((x)) = Z((x)) m
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 6 / 25

7. Ordinary Kriging
We assume: m(x) = const, within a given searching area centered on x,
but unknown, therefore:
Z⇤
OK (x) =
n(x)
X
i=1
OK
i (x) z(xi ) +
2
41
n(x)
X
i=1
OK
i (x)
3
5 m(x)
By imposing the following non-bias condition:
n(x)
X
i=1
OK
i (x) = 1
We obtain:
Z⇤
OK (x) =
n(x)
X
i=1
OK
i (x) z(xi ) with
n(x)
X
i=1
OK
i (x) = 1
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 7 / 25

8. Ordinary Kriging (continued...)
The minimum of the error variance should be computed under the
condition that
Pn(x)
i=1
OK
i (x) = 1:
L = 2
E (x) + 2 µOK (x)
2
4
n(x)
X
i=1
OK
i (x) 1
3
5
The minimum of the above expression is obtained for:
( Pn(x)
i=1
OK
i (x) CR(xi , xj ) + µOK (x) = CR(xj , x); j = 1, ..., n(x)
Pn(x)
i=1
OK
i (x) = 1
while the error variance assumes the following expression:
2
E = CR(0)
n(x)
X
i=1
OK
i (x) CR(xi , x) µOK (x)
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 8 / 25

9. Ordinary kriging written with the semivariogram
( Pn(x)
i=1
OK
i (x) (xi , xj ) µOK (x) = (xj , x); j = 1, ..., n(x)
Pn(x)
i=1
OK
i (x) = 1
2
E =
n(x)
X
i=1
OK
i (x) (xi , x) µOK (x)
The semivariogram allows to filter out the local mean m(x) that is
constant but unknown, over the local neighborhood W (x).
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 9 / 25

10. Kriging the local mean
Estimation of the local mean
m⇤
OK (x) =
n(x)
X
i=1
OK
i,m z(xi ) (1)
with
( Pn(x)
i=1
OK
i,m (x) CR(xi , xj ) + µOK
m (x) = 0; j = 1, ..., n(x)
Pn(x)
i=1
OK
i,m (x) = 1
This system is similar to that of the OK except for the right-hand side of
the first n(x) equations, which is set to zero.
2
E = Var {m⇤
OK (x) m(x)} =
n(x)
X
i=1
n(x)
X
j=1
OK
i (x) CR(xi , xj )
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 10 / 25

11. Kriging the local mean: An example
●
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●
●
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●
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1 2 3 4 5 6
01234
Cd Estimates − SK vs OK
Distance [km]
CdConcentration[ppm]
SK
OK
Mean
●
●
●
●
●
●
●
●
●
●
1 2 3 4 5 6
01234
Trend Estimates − SK vs OK
Distance [km]
CdConcentration[ppm]
u1
u2
u3
u4
u5
u6
u7
u8
u9
u10
OK
SK
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 11 / 25

12. Relationship between Ordinary and Simple Kriging
Ordinary Kriging estimation is equivalent to:
1 Compute the local mean m⇤
OK (x) by using OK with the data within
the neighborhood of x;
2 apply the SK estimator to the residuals R(x) = Z(x) (m⇤
OK (x) m)
Therefore:
Z⇤
OK (x) =
n(x)
X
i=1
SK
i (x) [z(xi ) m⇤
OK (x)] + m⇤
OK (x)
n(x)
X
i=1
SK
i (x) z(x) + SK
m (x)m⇤
OK (x)
Z⇤
OK (x) = Z⇤
SK (x) + SK
m (x) [m⇤
OK (x) m]
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 12 / 25

13. Simple vs Ordinary Kriging: An example
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 13 / 25

14. How to infer the residual covariance
First the residual semivariogram R is inferred, then the pseudo residual
covariance is computed as follows: CR(r) = A R(r), where A is an
arbitrary constant
2 (r) = E
n
[Z(x) Z(x + r)]2
o
2 R(r) + [m(x) m(x + r)]2
This calls for selecting data pairs that are una↵ected (or slightly a↵ected)
by the trend: in this case m(xi ) ⇡ m(xi + r) for i = 1, ..., n(x), and
therefore R = . The residual semivariogram can be inferred directly
from the data
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 14 / 25

15. Cross validation
1 Remove a measurement and perform kriging with the remaining data
on the point removed. An estimation of the error is given by the
di↵erence between the estimate and the measurement;
2 re-insert the measurement into the dataset and repeat the procedure
with another measurement;
3 repeat the procedure with all the other measurements to obtain an
estimation of the error at each measurement’s location.
An example of error statistics estimated from cross-validation is shown in
the following slide
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 15 / 25

16. Cross validation (continued)
Statistics of true and estimated concentrations at 100 test locations
Cd Cu Pb Co
Mean
True value 1.23 23.2 56.5 9.8
OK estimates 1.36 24.0 55.4 9.4
Std deviation
True values 0.69 25.8 40.3 3.5
OK estimates 0.41 7.5 11.7 2.4
% contamination
True values 63.0 8.0 42.0 0.0
OK estimates 92.0 0.0 66.0 0.0
% misclassification
OK estimates 35.0 8.0 36.0 0.0
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 16 / 25

17. Cross validation: graphical representation of the errors
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 17 / 25

18. Use secondary information: Classification of the approaches
Exhaustive secondary information
!
8
<
:
Kriging within strata
Simple Kriging with varying local mean
Kriging with external drift
Primary data z(x), i = 1, ..., n are supplemented by secondary information
at all primary data locations xi plus the locations x where estimation is
performed
Type of secondary information:
- a categorical attribute s with K mutually exclusive states
- a smoothly varying continuous attribute y, for example the concentration
of another metal
Non-exhaustive secondary information ! Cokriging
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 18 / 25

19. Simple Kriging with varying local mean
The mean of the field, which is variable in space, is obtained from the
secondary information
1 if the SI is related to a categorical attribute s with K non-overlapping
states: m⇤
SK (x) = m|sk
with s(x) = sk. The conditional mean is given
by:
m|sk
=
1
nk
nX
i=1
i(xi ; sk) · z(xi )
nk =
Pn
i=1 i(xi ; sk) is the number of primary data locations belonging
to the category sk
2 if the SI is a continuous attribute y: m⇤
SK (x) = f (y(x)), where f is a
regression function.
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 19 / 25

20. Simple Kriging with varying local mean (continued...)
Simple Kriging is then applied to the residuals:
Z⇤
SKlm(x) = m⇤
SKlm(x) +
n(x)
X
i=1
SK
i (x) [z(xi ) m⇤
SKlm(xi )]
An alternative to regression:
The secondary attribute is discretized into K classes: (yk, yk+1] and the
local mean is computed as follows:
m⇤
SK (x) = m|sk
=
1
nk
nX
i=1
i(xi ; sk) · z(xi )
i(xi ; sk) =
⇢
1 if y(xi ) 2 (yk, yk+1]
0 otherwise
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 20 / 25

21. Simple Kriging with varying local mean: an example
1 2 3 4 5 6
-1.0-0.50.00.51.01.52.02.5
Simple Kriging of residuals and Trend component
Distance [Km]
Cdconcentration[ppm] Trend
SK
1 2 3 4 5 6
01234
SK varying mean
Distance [Km]
Cdconcentration[ppm]
distance
semivariance
0.2
0.4
0.6
0.8
0.5 1.0 1.5
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 21 / 25

22. Simple Kriging with varying local mean: an example
(continued)
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 22 / 25

23. Kriging with external drift
KED is a variant of the kriging with a trend model (KT). The trend m(x)
is modeled as a linear function of a smoothly varying secondary variable
y(x):
m(x) = a0(x) + a1(x) y(x)
Note that KT uses: m(x) = a0(x) +
KX
k=1
ak(x) fk(x).
The KED estimators is written as follows:
Z⇤
KED(x) =
n(x)
X
i=1
KED
i (x) z(xi )
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 23 / 25

24. Kriging with external drift (continued...)
The weight are computed by solving the following system of equations:
8
>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>:
n(x)
X
j=1
KED
j (x) CR(xi , xj ) + µ0(x) + µ1(x) y(xi ) = CR(xi , x),
i = 1, ..., n(x)
n(x)
X
j=1
KED
j (x) = 1
n(x)
X
j=1
KED
j (x) y(xj ) = y(x)
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 24 / 25

25. Kriging with external drift (continued...)
The conditional variance (the minimized error variance) becomes
2
E = CR(0)
n(x)
X
i=1
KED
i (x) CR(xi , x) µKED
0 (x)
µKED
1 (x)
n(x)
X
i=1
KED
i (x) y(xi ) =
CR(0)
n(x)
X
i=1
KED
i (x) CR(xi , x) µKED
0 (x) µKED
1 (x) y(x)
Alberto Bellin Department of Civil, Environmental and Mechanical Engineering University of TrentoGEOSTATISTICS: Part II geostatistics as spatial interpolation technique 25 / 25