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


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


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

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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


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:


equation 1.5-10

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Kriging VarianceKriging Variance

)()(1

oiji

n

jj xxxx


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

1 method of soil analysis 1.5 geostatistics 1.5.1 introduction 1.5.2 using geostatistical methods 1...

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