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INTRODUCTION TO INTRODUCTION TO GEOSTATISTICSGEOSTATISTICS

an ER&P program short course

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bybyAbaniAbani RanjanRanjan SamalSamal (Student, ER&P)(Student, ER&P)

Courses in Geostatistics:Courses in Geostatistics:1.1. MTechMTech (Mineral Exploration, I.S.M., India): (Mineral Exploration, I.S.M., India): nn Basic Geostatistics and Advanced GeostatisticsBasic Geostatistics and Advanced Geostatistics2.2. MScMSc (Mineral Exploration, Imperial College, (Mineral Exploration, Imperial College,

London)London)nn Geostatistics Geostatistics nn Short Course in MDE (Margaret Armstrong)Short Course in MDE (Margaret Armstrong)Work Experience:Work Experience:

Mineral Resource Evaluation using Micromine

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Outline:•Introduction:

•Classical statistics Vs Geostatistics, •Popular Estimation Methods Vs Kriging

•Semivariogram•Introduction, •Types and formulae

•Kriging•Formula(e), •Types, •B.L.U.E.

•ApplicationGeology/Mineral Exploration/Mining,

•GIS•Conclusion

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Conventional Statistics

11/12/2002 53.054877102Confidence

Level(95.0%)0.052222768

Confidence Level(95.0%)

546Count546Count36771.78Sum805.36Sum

225Maximum5.85Maximum2.33Minimum0.05Minimum

222.67Range5.8Range0.653851135Skewness

0.769522334Skewness

0.41431362Kurtosis4.369420709Kurtosis

1320.54349Sample Variance0.385909082Sample Variance

36.33928301Standard Deviation0.62121581Standard Deviation31Mode1.8Mode

64.65Median1.48Median67.34758242Mean

1.475018315Mean

AgAu

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Conventional Statistics

What is the value of the variable (say Au grade, Ag grade) at “xx”?

xx

Au: 1.475+0.621215

Ag:67.34758+36.33928

(from statistics)

Correct?

If Correct, Error?

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WhatsWhats Next?Next?INTERPOLATION

n Estimating values at un-sampled location using neighboring samples

Source:http://www.cee.vt.edu/program_areas/environmental/teach/smprimer/kriging/dt-arrow.gif

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INTERPOLATIONnWhat Interpolation Techniques we have? nDeterministicnExact Interpolator

§ Inverse Distance Weighted (IDW); popularly known as IDS and Radial Basis Functions

n Inexact interpolator§Global Polynomial and Local

Polynomial nGeoststistical: Incorporates

statistics of data in interpolation∑

=

=n

iii zw

1

0*Z

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HISTORY: n Started in 1960sn Began with D G Krige and Siecheln Developed by G Matheron with the concept of

Regionalized Variable (late 1960s)n Present Status: n Highly developedn Applied in almost all areas of regionalized variablen In almost all parts of the worldn Lots of software tools available

§ Free and Commercial

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HISTORY

Other Founders: D G Krige (South African Mining Engineer), Siechel (1960)

Other Famous Geostatisticians: Snowden (Australia), Goovaerts (Belgium), E H Isaaks (USA), R M Srivastav(USA) etc.

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IDS Vs KRIGINGn IDS:An Exact InterpolatornLeast Variance?

§Not as compared to Kriging

nUnbiased? §NO

nSum of all weight factors (wi)=1 ?§NO

nShould we discard IDS? nNO: Because where data is not good for

kriging, IDS is the best interpolation method.

∑

∑=

2i

i2i*

d1

Zd1

Z

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Though these two look different in their spatial distribution, but their mean and variances are identical.

IDS Vs Kriging:

Ref: Petroleum geostatistics for nongeostatisticians, THE LEADING EDGE MAY 2000

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IDS Vs Kriging:

IDS (left) and kriging (right) using a 3:1 anisotropic variogram model orientedN60E. The neighborhood search ellipse is identical for both.

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Why Why GeostatisticsGeostatistics??n Incorporates the position of sample in spacen Weights the sample size : support

n Theory of Regionalized Variable (Matheron)n Support: Shape and Volume of the Sample (David)

n Uses statistical parameters of the samples for estimation (kriging)

n Two important terms (Estimators ?)n VARIOGRAM: for structural analysis of data, ststistically

models the data in spacen KRIGING: for interpolationn Kriging uses the information from a variogram to find an

optimal set of weights (wi) that are used in estimating a surface at unsampled locations.n Variogram is not a part of Kriging rather a pre requirement

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Random Variable Vs

Regionalized Variable

n Random Variablen whose values are randomly generated

according to some probabilistic mechanismn Example: the result of casting a die can take one of

six equally probable values.

n Regionalized Variablen distributed in spacen Support: Shape and Volume

n Example: mostly natural phenomena. For instance, the heavy metal content of the top layer of a soil

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REGIONALIZED VARIABLEREGIONALIZED VARIABLETwo aspects: A structured and an erratic aspectn The structured aspect

ndistribution of the natural phenomenon. n The erratic aspect

nthe local behavior of the natural phenomenon. Distribution of the heavy metal content

seems to fluctuate randomly.

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Spatial Stationarity and Intrinsic Hypothesis

n Strict Stationarityn invariance of joint probability density function

under spatial shift (“translation”)n { z(x1), … , z(xk) } and { z(x1+h), … , z(xk+h) }n information about process the same no matter

where it is obtained

n Moment Stationarityn moments invariant under shift

n constant mean and constant variancen covariance only function of spatial separation h

nIntrinsic Hypothesisn The Mean, Variance of the increments z(xk+h)-z(xk) exists and independent of the point Xk.

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Spatial distribution of dataSpatial distribution of data

• Location (X,Y,Z)

• Support (Volume and Shape)

H Scatter plot:

• Describes “Spatial Continuity” (& Spatial dependence)

• Plot of value of one variable at position xagainst the value of the same variable at position x+h, h being a separation vector

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HH--Scatter plotScatter plot

• Tolerance

•Direction and magnitude

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H Scatter plotH Scatter plotnn Associated with each Associated with each γ(h) value of the experimental

semivariogramn Spatial Continuity

n Better continuity in H-Scatter plot = better meaning of directional variogram

n Pattern:n Butterfly pattern: Skewed distribution,

requires transformationn Groups of pairs: mixture of different populations

n A way to separate different populations

n Visual check of multigausian hypothesis(multigausian = limit distribution of central limit theorems.

elliptical cloud around the diagonal)

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Experimental Experimental SemivariogramSemivariogram

Lag (h) = 1unit

Lag (h) = 2unit

Lag (h) = 3unit

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Experimental Experimental SemivariogramSemivariogramnn Sample points: 3 4 6 5 7 7 6 4 3 5 5 6 5 7Sample points: 3 4 6 5 7 7 6 4 3 5 5 6 5 7nn N=14, 2N=14, 2nn Distance = hDistance = h 22??(h)= {(h)= {z(xz(x) ) -- z(x+h)}2z(x+h)}2nn 11 26/13 = 2.026/13 = 2.0nn 22 41/12 = 3.441/12 = 3.4nn 33 55/11 = 5.155/11 = 5.1nn 44 50/10 = 5.850/10 = 5.8nn 55 53/9 = 5.953/9 = 5.9

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Experimental Experimental SemivariogramSemivariogram

COMPONENTS

•Sill : Total Variance

•Range: The (max.) distance over which samples are correlated (zone of influence)•Nugget: γ(h) @ hà0 : magnitude of the discontinuity

• A result of human error• inability to sample at the very same sample location more than once• Laboratory error

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SemivariogramSemivariogram FormulaFormula

2γ(h) = E[{Z(x+h)-Z(x)}2]

= E[{(Z(x+h)-m)2 + (Z(x)-m)2} –

2(Z(x+h)-m)(Z(x)-m)}

=2C(0) – 2C(h)

èγ(h) = C(0) – C(h)

è γ(h) = Sill – C(h)

è γ(h) = σ2 – C(h)

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Directional Directional SemivariogramSemivariogram::Two Dimensional ExampleTwo Dimensional Example

Distance N Semivariance

1.000 24 1.5381.414 18 2.528 1.690 34 2.6622.000 16 2.813 2.384 32 3.1102.236 24 3.000 3.097 20 2.0002.828 8 3.438 3.733 10 3.9503.000 8 1.6253.162 12 2.250 3.309 30 2.6503.606 8 3.8754.243 2 4.250

55442222

66332244

44110022

33442211

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Directional Directional SemivariogramSemivariogram::Two Dimensional ExampleTwo Dimensional Example

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Directional Directional SemivariogramSemivariogram::

Variograms Variogram Surface

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Directional Directional SemivariogramSemivariogram::Another ExampleAnother Example

Anisotropic variogramshort-scale range of 800 m and long-scale range of 2200 m.

An anisotropic search ellipsewith eight sectors and a maximum of two data points per sector. The minor axis has a length of 1000 m.

Ref: Petroleum geostatistics for nongeostatisticians, THE LEADING EDGE MAY 2000

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AnisotropismAnisotropism: Directional : Directional SemivariogramSemivariogram

nn Spatial structure may not be the same in all Spatial structure may not be the same in all directions :”Anisotropic”directions :”Anisotropic”nn Spatial Structure same in all directions: IsotropicSpatial Structure same in all directions: Isotropic

nn Develop a semiDevelop a semi--variogram for each (major) variogram for each (major) directiondirection

nn Major Directions: 0Major Directions: 000,45,4500,90,9000,135,13500

nn Angle of tolerance (Angle of tolerance (<< 22.522.500))

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Semivariogram ModelingSemivariogram Modeling

1. Experimental Semivariogram2. Model fitting:

n Hand fittingn Weighted Least Square method (avoided always)

Why?1. Model must be positive definitive function

n Otherwise variance <0 (impossible)n Rarely satisfied by LS method

2. LS assumes : samples are independentn NOT true in case of spatially distributed data (natural

data)

3. Behavior @ origin at i.e., distance shorter than first lag (h)n IGNORED by LS method

∑=

+−=n

i

hxZxZhN

h1

2)]()([)(2

1)(γ

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SEMIVARIOGRAM FITTINGSEMIVARIOGRAM FITTINGRules of Thumb Rules of Thumb nn Fit the model well at the Fit the model well at the

originoriginnn Range = 3/2 of the distance Range = 3/2 of the distance

where the tangent to the where the tangent to the experimental experimental semivariogramsemivariogrammeets Sill.meets Sill.

nn Take 3Take 3--4 points with 4 points with comparatively more sample comparatively more sample pairs starting at the originpairs starting at the origin

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Semivariogram ModelsSemivariogram Models

• Spherical

• Linear

2)]()([21

)( hxZxZEh +−=γ

000)(

>==

hh

Chγ

ah

ah

Cah

ah

Ch

≥

<

−= 3

3

21

23

)(γ

BAhh +=)(γ

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Semivariogram ModelsSemivariogram Models

• Gausian

•

•Exponential

2)]()([21

)( hxZxZEh +−=γ

000)(

>==

hh

Chγ

))/exp(1()( ahCh −−=γ

−−= )exp(1)( 2

2

ah

Chγ

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Semivariogram ModelsSemivariogram Models

•Hole effect

•De Wijsian•(No model available in this slide)

•Power

•Nugget

2)]()([21

)( hxZxZEh +−=γ

000)(

>==

hh

Chγ

20,)( ≤<= αγ α withhCh

hh ln)(ln λγ =

−=

ahah

Chsin

1)(γ

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NESTED NESTED SemivariogramSemivariogram

• Sum of Exponential and Spherical Structures at Different Scales • does not resemble any single -semivariogram• sill value is the sum of the individual sills C0,1=1 and C0,2=2.

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Presenting and interpreting Presenting and interpreting Directional Directional SemivariogramSemivariogram::

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VALIDATION OF SEMIVARIOGRAM MODELVALIDATION OF SEMIVARIOGRAM MODEL

nn PKCVPKCVnn Goodness of fitGoodness of fitnn It gives a measure of how well the model It gives a measure of how well the model

adjusts the directional adjusts the directional variogramsvariogramsnn a number without units and a value close to a number without units and a value close to

zero indicates a good fitzero indicates a good fitnn a standardized measure of fita standardized measure of fit

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3D 3D variographyvariography

Source:http://www.erms.fr/english/applied_geostatistics.html#vario

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Goodness of Fit:Goodness of Fit:

n N = number of directional variogram measures used for the modeln n(k) = number of lag for the kth variogram measuren P(i) = number of pair for lag in h(i) = mean distance for lag in hmax(k)= maximum distance for the kth variogram measure

n γ(i) = experimental variogram measure for lag i

n γ *(i) = modeled variogram measure for the mean distance of lag in s2 = variance of the data for the semivariogram and the non

ergodic covariance, maximum experimental value of all variogram measures for the madogram, 1 for the correlogram and the standardized inverted covariance.

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Search CriteriaSearch Criteria

EllipsoidEllipseAnisotropicAnisotropicSphereCircleIsotropicIsotropic

3D3D2D2D

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Kriging:Kriging:nn Interpolation: Interpolation: what measured values tell us what measured values tell us

about the properties at unabout the properties at un--sampled locationsampled locations.s.nn a a geostatisticalgeostatistical interinter--polationpolation technique. technique. nn a linear weighteda linear weighted--averaging methodaveraging methodnn Kriging weights depend on a model of spatial Kriging weights depend on a model of spatial

correlation.correlation.

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KrigingKrigingnn Why Why KrigingKriging::

nn BBest : Minimum est : Minimum KrigingKriging Variance (Variance (σσ22kk))

nn LLinear: inear:

n : Weights, Z(x i ) are sample values

nn UUnbiased : Error nbiased : Error àà 0 0 i.ei.e, , nn EEstimatorstimator

0)]()([ * =− XZxZE

∑∑=

=1

)(*

i

iii XZZλ

λ

iλ

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Common Types of Common Types of KrigingKriging: : n Punctual Kriging (also termed Ordinary Kriging):

uses only the samples in the local neighborhood for the estimate. most common method used in environmental engineering.

n Block Kriging: Estimating the value of a block from a set of nearby sample values using kriging.

n Point Kriging: Estimating the value of a point from a set of nearby sample values using kriging.

When a kriged point happens to coincide with a sample location, the kriged estimate = the sample value.

n Universal Kriging : used when a trend, or slow change in average values, in the samples exists

n Co-kriging: for more than one variable

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KrigingKriging Formula:Formula:

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An Example:An Example:

Distance Matrix

3.61 0 --3

1.41 3.61 0 -2

2 5.39 3.16 0 1

p 3 2 1

Semivariogram Matrix

14.42 0 --3

5.66 14.42 0 -2

8 21.54 12.65 0 1

p 3 2 1

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The complete set of simultaneous equations are:

Wi are weights, and ΣWi = 1

λIs Lagrange multiplier ( to ensure minimum possible kriging variance

is obtained)

2kσ

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In matrix form:In matrix form:

[ ] [ ] [ ] 1−∗= ijipiW γγ

[ ] [ ] [ ]ipiij W γγ =∗

OR,

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SolutionSolution

Estimation of Unknown value:

YE,P = 0.3805(150) + 0.4964(110) + 0.1232(140) = 128.9 meters

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Semivariogram Matrix

14.42 0 --3

5.66 14.42 0 -2

8 21.54 12.65 0 1

p 3 2 1

Estimation of variance:

YE,P = 0.3805(8.0) + 0.4964(5.66) + 0.1232(14.42) - 0.9319(1.0) = 6.70 m2

Standard error : (6.70)1/2=2.59 meters

Assuming, errors of estimation are normally distributed:

YE,P = 128.9 +/- 5.18 meters, with 95% probability

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YE,Q = 138.6 +/- 6.45 meters, with 95% probability

Finally

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3D Data analysis3D Data analysis

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Free Software Systems:Free Software Systems:

n VARIOWIN: Software for Spatial Data Analysis in 2D Download from: http://www-sst.unil.ch/research/variowin/n Geo-EAS 1.2.1 n Geostatistical Toolbox 1.30 n GEOSTAT OFFICE Software n Geostat etcn Krigame: Software and data Download from

http://geoecosse.bizland.com/softwares/index.htm

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Commercial Packages:Commercial Packages:nn GS+ Version 5GS+ Version 5nn ISATISISATISnn Snowden ANALYSOR, Snowden VISOR etcSnowden ANALYSOR, Snowden VISOR etcApplications:

• Mining and Geology (Geochemistry, Geophysics, hydrology,Paleontology, Mine Planning, Production and development etc.)

• Mineral exploration

• GIS and Mapping

• Remote sensing

• Fishery

• Environment

• Forestry

• Public Administration (Crime analysis, Services etc.)

• Waste management

11/12/200211/12/2002 5555American Scientist (1998)