1 method of soil analysis 1.5 geostatistics 1.5.1 introduction 1.5.2 using geostatistical methods 1...
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Method of Soil Analysis
1.5 Geostatistics 1.5.1 Introduction1.5.2 Using Geostatistical Methods
1 Dec. 2004
D1 Takeshi TOKIDA
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1.5.1 Introduction1.5.1 Introduction True understanding of the spatial variability in
the soil map is very limited. Distinct boundary (too continuous or sudden change). Assumption of uniformity within a mapping unit is not
necessarily valid.
Spatial and temporal variability diversify our environment. It’s Benefit!
However Soil variation can be problematic for landscape management.
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1.5.11.5.1 IntroductionIntroduction
There is a need to study surface variations in a systematic manner.
Geostatistical methods are used in a variety of disciplines.e.g. mining, geology, and recently biological sciences also.
Numerous books have been published.
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1.5.1.1 Geostatistical Investigations1.5.1.1 Geostatistical Investigations
Geostatistics is used to… map and identify the spatial patterns of
given attributes across a landscape. improve the efficiency of sampling
networks. identify locations in need of remediation.
Disjunctive kriging→Probability map
predict future effects in the landscape. Random field generation→Conditioned→Predict
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1.5.2 Using Geostaitstical Methods1.5.2 Using Geostaitstical Methods1.5.2.1 Sampling1.5.2.1 Sampling Consider the appropriate sampling methodology (see
Section 1.4)
Analysis
Appropriate data collection
Objective of the study
The analysis of the data depend on the objective of the study and appropriate data collection.
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Table 1.5-1Table 1.5-1
If the Kolmogrov-Smirnov statistic is greater than the critical value, the hypothesis of “not being normal” is adopted.
If the distribution is completely normal, skew and kurtosis values are 0.
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Table 1.5-2
θ ρb Na ln(Na) B ln(B)
θ 1 -0.49 0.06 0.14 0.09 0.19
ρb 1 -0.14 -0.25 -0.02 -0.16
Na 1 0.67 0.58 0.53
ln(Na) 1 0.57 0.83
B 1 0.76
ln(B) 1
?
Na values can be used to estimates B content at lower cost.
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Randome function & realizationRandome function & realization Observed data are a single realizationrealization of
the random fieldrandom field, Z(x).
Random field(Random function)
Realization
+Assumptions, i.e. stationarity
Z(xα)
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1.5.2.2 Spatial Autocorrelation1.5.2.2 Spatial Autocorrelation
Only if a spatial correlation exists, geostatistical analysis can be used.
Fig. 1.5-1 A: No spatial correlation
Fig.1.5-1 B: spatially correlated
Fig. 1.5-1
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1.5.2.2.a VariogramVariogram
Experimental variogram (Estimator)
How to create pairs?
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Variogram model
95%
Practical range
Var(Z) Variance is undefined
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Important considerations when calculating the variogram 1
Between a lag interval, in this case 1.5 to 4.5, a wide range of actual separation distance occurs.
ImprecisionImprecisioncompared with a situation where every sampling pair has the same distance
A large number of pairs are used to calculate a variogram value.
It is generally accepted that
30 or more30 or more pairs are sufficient to produce a reasonable sample variogram.
Fig. 1.5-3
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Important considerations when calculating the variogram 2
Width of the lag interval can affect the variance. This is not the case. The value for h (actual separation distance) is
affected by the lag width.
Fig. 1.5-4
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Fig. 1.5-1 & 1.5-5
The variograms reproduce spatial structure of simulated random fields.
Fig. 1.5-1
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Example of variogram
Some information at the smaller scales (less than 48 m) has been lost.
For both attribute, the range is about 900 m.
Nugget effect
Sill
Sill
Range
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1.5.2.2.d Directional Variograms
Often there is a preferred orientation with higher spatial correlation in a certain direction.
For many situations, the anisotropic variogram can be transformed into an isotropic variogram by a linear transformation.
Geometric anisotropyGeometric anisotropy
Fig. 1.5-7 Fig. 1.5-8
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1.5.2.2.e Stationarity1.5.2.2.e Stationarity
A sample at a location
Impossible to determine the probability distribution at the point!
A stationary Z(x)stationary Z(x) has the same joint probability distribution for all locations xi and xi+h.
The joint distribution do not depend on the The joint distribution do not depend on the location.location.
Assumption:
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Second-Order StationaritySecond-Order Stationarity
)(
)]()([
)])()()([(
)])]([)()])(([)([()](),(cov[
)]([)]([
2
h
xhx
xhx
xxhxhxxhx
xhx
C
ZZE
ZZE
ZEZZEZEZZ
ZEZE
Autocovariance
1.5-3, 1.5-6
*
)()0(
)0(2
1)()0(
2
1
]))([(2
1)]()([]))([(
2
1
]))()([(2
1)(
222
22
2
h
h
xxhxhx
xhxh
CC
CCC
ZEZZEZE
ZZE
)()0()( hh CC
h
C(0)
C(h)
(h)
Nugget effect
Range
Sill
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Intrinsic StationarityIntrinsic Stationarity (Hypothesis)(Hypothesis)
)(2)]()(var[
0)()]()([
hxhx
hxhx
ZZ
mZZE
Theoretical Variogram
]))()([(2
1)( 2xhxh ZZE
No Drift
Fig. 1.5-9
0)(
lim 2 h
hh
If , the random field is stationary in terms of Intrinsic hypothesis.
Drift? No Drift?
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1.5.2.2.c Integral Scale1.5.2.2.c Integral Scale
A measure of the distance for which the attribute is spatially correlated.
Autocorrelation function:normalized form of the autocovariance function
1.5-4
1.5-5
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1.5.2.3 Geostatistics and Estimation1.5.2.3 Geostatistics and Estimation
Kriging produces a best linear unbiased best linear unbiased estimateestimate of an atribute together with estimation varianceestimation variance.
Multivariate or cokriging: Superior accuracy
Powerful tool, useful in a wide variety of investigations.
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1.5.2.3.a Ordinary Kriging1.5.2.3.a Ordinary Kriging
Z* should be unbiased:
We wish to estimate a value at xo using the data values and combining them linearly with the weiths: λi
xo
101
)]()([
11
100
*
n
ii
n
ii
n
iii ZEZZE
xxx
1.5-7
1.5-9
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Derivation of equation 1.5-10
200
111
200
111
2
01
2
00*
2
00*
00*
00*
)()()(2)()(
)()()(2)()(
)()(
)()(
)()())()(()()(var
xxxxx
xxxxx
xx
xx
xxxxxx
ZEZZEZZE
ZEZZEZZE
ZZE
ZZE
ZZEZZEZZ
i
n
iiji
n
jji
n
i
i
n
iii
n
iii
n
ii
i
n
ii
0
First, rewrite the estimation variance
Z* should be best-linear, unbiased estimator.
Our goal is to reduce as much as possible the variance of the estimation error.
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22
2
)(2
1)()()(
2
1
)()(2
1)(
jjii
jiji
ZEZZEZE
ZZE
xxxx
xxxx
22 )(2
1)(
2
1)()()( jijiji ZEZEZZE xxxxxx
220 )(
2
1)(
2
1)()()( oioii ZEZEZZE xxxxxx
2000
20 )()()( xxxx ZEZE
Derivation of equation 1.5-10
Let’s rewrite the estimation variance in terms of the semivariogram.
We assume intrinsic hypothesisintrinsic hypothesis.
From the definition of the semivariogram we know:
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2
1
22
11
00111
2000
22
1
22
11
200
11100
*
)()()(2
1
)()(2)(
)()(
)(2
1)(
2
1)(2
)(2
1)(
2
1)(
)()()(2)()()()(var
i
n
iiji
n
jji
n
i
oi
n
iiji
n
jji
n
i
oioi
n
ii
jiji
n
jji
n
i
i
n
iiji
n
jji
n
i
ZEZEZE
ZE
ZEZE
ZEZE
ZEZZEZZEZZ
xxx
xxxxxx
xxx
xxxx
xxxx
xxxxxxx
0
Derivation of equation 1.5-10
Just substitute:
2
11
2
1
2
11
2
1
22
11
)(21)()()()()( j
n
jj
n
iij
n
jjj
n
jj
n
jij
n
iiji
n
jji
n
i
ZEZEZEZEZEZE xxxxxx
1 1
26
We define an objective function φ containing a term with the Lagrange multiplier, 2β.
To solve the optimization To solve the optimization problem we set the partial problem we set the partial derivatives to zero:derivatives to zero:
0
,
,...1for0,
i
i
i ni
Derivation of equation 1.5-10
12)()(2)(
12)()(var,
100
111
100
*
n
iioi
n
iiji
n
jji
n
i
n
iii ZZ
xxxxxx
xx
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Ordinary Kriging systemOrdinary Kriging system
nioiji
n
jj
oiji
n
jj
,...,2,1for)()(
2)(2)(2
1
1
xxxx
xxxx
11
n
jj
Example:
Derivation of equation 1.5-10
equation 1.5-10
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Kriging VarianceKriging Variance
)()(1
oiji
n
jj xxxx
Derivation of equation 1.5-10
equation 1.5-12
)()(
)()(2)(
)()(2)()()(var
001
0011
00111
00*
xxxx
xxxxxx
xxxxxxxx
oi
n
ii
oi
n
ii
n
ioii
oi
n
iiji
n
jji
n
i
ZZ
Block KrigingBlock KrigingEstimation of an average value of a spatial attribute over a region.
Average variogram values equation 1.5-13
VarianceVariance
equation 1.5-15
equation 1.5-14
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1.5.2.3.b Validation1.5.2.3.b Validation
Cross validation
Little bias
Estimated kriging variance is nearly equal to the actual estimation error.
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1.5.2.3.c Examples1.5.2.3.c Examples Isotropic Case, Kriging Matrix.
equation 1.5-18 equation 1.5-10
1.5-11
λ1=0.107, λ2=0.600, λ3=0.154, λ4=0.140But we can’t find the values of a given attribute!
Note that the weight for point 1 is less than point 4, even though the distance from the estimation site is almost the same.
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Creating Maps Using KrigingCreating Maps Using Kriging
Directional variogram oriented in 0°& 90°
Anisotropy ratio = major axes / minor axes
Length of each ray is equal to the range of the directional variogram.
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Creating Maps Using KrigingCreating Maps Using Kriging
Fig. 1.5-12 Based on Anisotropic variogram
Fig. 1.5-13 Based on isotropic variogram