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Method of Soil Analysis 1.5 Geostatistics 1.5.1 Introduction 1.5.2 Using Geostatistical Methods. 1 Dec. 2004 D1 Takeshi TOKIDA. 1.5.1 Introduction. True understanding of the spatial variability in the soil map is very limited. Distinct boundary (too continuous or sudden change).

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method of soil analysis 1 5 geostatistics 1 5 1 introduction 1 5 2 using geostatistical methods

Method of Soil Analysis1.5 Geostatistics1.5.1 Introduction1.5.2 Using Geostatistical Methods

1 Dec. 2004

D1 Takeshi TOKIDA

1 5 1 introduction
1.5.1 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.
1 5 1 introduction3
  • 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.
1 5 1 1 geostatistical investigations 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
1 5 2 using geostaitstical methods 1 5 2 1 sampling


Appropriate data


Objective of the study

1.5.2 Using Geostaitstical Methods1.5.2.1 Sampling
  • Consider the appropriate sampling methodology (see Section 1.4)

The analysis of the data depend on the objective of the study and appropriate data collection.

table 1 5 1
Table 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.
table 1 5 2
Table 1.5-2


Na values can be used to estimates B content at lower cost.

randome function realization
Randome function & realization
  • Observed data are a single realization of the random field, Z(x).




Assumptions, i.e. stationarity

Random field

(Random function)

1 5 2 2 spatial autocorrelation 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

1 5 2 2 a variogram Variogram


Experimental variogram (Estimator)

How to create pairs?

variogram model
Variogram model

Variance is undefined



Practical range

important considerations when calculating the variogram 1
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.


compared 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 more pairs are sufficient to produce a reasonable sample variogram.

Fig. 1.5-3

important considerations when calculating the variogram 2
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

fig 1 5 1 1 5 5
Fig. 1.5-1 & 1.5-5
  • The variograms reproduce spatial structure of simulated random fields.

Fig. 1.5-1

example of variogram
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


1 5 2 2 d directional variograms 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 anisotropy

Fig. 1.5-7

Fig. 1.5-8

1 5 2 2 e stationarity Stationarity

A sample at a location

Impossible to determine the probability distribution at the point!

The joint distribution do not depend on the location.


A stationary Z(x) has the same joint probability distribution for all locations xi and xi+h.

second order stationarity


Second-Order Stationarity

1.5-3, 1.5-6





Nugget effect




intrinsic stationarity hypothesis


, the random field is stationary in terms of Intrinsic hypothesis.

Intrinsic Stationarity(Hypothesis)

No Drift

Theoretical Variogram


No Drift?

Fig. 1.5-9

1 5 2 2 c integral scale Integral Scale


A measure of the distance for which the attribute is spatially correlated.


Autocorrelation function:

normalized form of the autocovariance function

1 5 2 3 geostatistics and estimation Geostatistics and Estimation
  • Kriging produces a best linear unbiased estimate of an atribute together with estimation variance.
  • Multivariate or cokriging: Superior accuracy
  • Powerful tool, useful in a wide variety of investigations.
1 5 2 3 a ordinary kriging Ordinary Kriging

We wish to estimate a value at xo using the data values and combining them linearly with the weiths: λi



Z* should be unbiased:


derivation of equation 1 5 10
Derivation of equation 1.5-10

Z* should be best-linear, unbiased estimator.

Our goal is to reduce as much as possible the variance of the estimation error.

First, rewrite the estimation variance


derivation of equation 1 5 1024
Derivation of equation 1.5-10

Let’s rewrite the estimation variance in terms of the semivariogram.

We assume intrinsic hypothesis.

From the definition of the semivariogram we know:

derivation of equation 1 5 1026
Derivation of equation 1.5-10

We define an objective function φ containing a term with the Lagrange multiplier, 2β.

To solve the optimization problem we set the partial derivatives to zero:

kriging variance

Derivation of equation 1.5-10

Kriging Variance

equation 1.5-12

Block Kriging

Estimation of an average value of a spatial attribute over a region.

equation 1.5-13

Average variogram values


equation 1.5-14

equation 1.5-15

1 5 2 3 b validation Validation

Cross validation

Little bias

Estimated kriging variance is nearly equal to the actual estimation error.

1 5 2 3 c examples Examples

Isotropic Case, Kriging Matrix.

equation 1.5-18

equation 1.5-10


But we can’t find the values of a given attribute!

λ1=0.107, λ2=0.600, λ3=0.154, λ4=0.140

Note that the weight for point 1 is less than point 4, even though the distance from the estimation site is almost the same.

creating maps using kriging
Creating Maps Using Kriging

Directional variogram oriented in 0°& 90°

Length of each ray is equal to the range of the directional variogram.

Anisotropy ratio = major axes / minor axes

creating maps using kriging32
Creating Maps Using Kriging

Fig. 1.5-12 Based on Anisotropic variogram

Fig. 1.5-13 Based on isotropic variogram