Case Study 4.1 (M)

GIS 585 GIS 585 LECTURE 3:LECTURE 3: GEOSTATISTICSGEOSTATISTICS

Paul A.Loglet, Michael F.Goodchild, David Paul A.Loglet, Michael F.Goodchild, David J.Maguire, David W.Rhind, Geographical J.Maguire, David W.Rhind, Geographical Information Systems and Science, chapter 14, Information Systems and Science, chapter 14, pp. 333-340pp. 333-340

Peter Burrough, Rachael McDonnell Peter Burrough, Rachael McDonnell “Principles of GIS”, chapter 5, pp. 133-161“Principles of GIS”, chapter 5, pp. 133-161

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Optimal InterpolationOptimal Interpolation using Geostatisticsusing Geostatistics

Lecture outline:Lecture outline:

 Regionalized variable theory.Regionalized variable theory.  Who is Krige ?Who is Krige ?  Variogram modelsVariogram models  Spatial analysisSpatial analysis  How to optimize sampling.How to optimize sampling.  Is it a “cool” tool ?Is it a “cool” tool ?

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INTERPOLATIONINTERPOLATION

 When data are abundant, most interpolation techniques give similar When data are abundant, most interpolation techniques give similar results.results.

 When data are sparse, the assumptions made about the underlying When data are sparse, the assumptions made about the underlying variation that has been sampled, and the choice of methods, and its variation that has been sampled, and the choice of methods, and its parameters can be critical if one is to avoid misleading results.parameters can be critical if one is to avoid misleading results.

 If we need to make a decision on the costs of cleaning up soil with If we need to make a decision on the costs of cleaning up soil with more than a certain level of chromium.more than a certain level of chromium.

 None of the methods can provide direct estimates of the QUALITY of None of the methods can provide direct estimates of the QUALITY of the predictions made in terms of an estimation variance for the the predictions made in terms of an estimation variance for the predicted value at unsampled location.predicted value at unsampled location.

 A further objection is- you don’t know whether the best values have A further objection is- you don’t know whether the best values have been chosen for the weighting parameters, and if the size, been chosen for the weighting parameters, and if the size, orientation, and shape of the search neighborhood is appropriate. orientation, and shape of the search neighborhood is appropriate.

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GEOSTATISTICS &GEOSTATISTICS & KRIGING. DefinitionsKRIGING. Definitions  French geomathematician Georges Matheron, and the South French geomathematician Georges Matheron, and the South

African mining engineer D.C.Krige tried to develop OPTIMAL African mining engineer D.C.Krige tried to develop OPTIMAL methods of interpolation for use in the mining industry. Now methods of interpolation for use in the mining industry. Now increasingly used. increasingly used.

 GEOSTATISTICAL METHODS for interpolation recognize that the GEOSTATISTICAL METHODS for interpolation recognize that the spatial variation of spatial variation of ANY CONTINUOUSANY CONTINUOUS attribute is TOO attribute is TOO IRREGULAR to be modeled by a smooth mathematical function.IRREGULAR to be modeled by a smooth mathematical function.

 It can be better described by a stochastic surface.It can be better described by a stochastic surface.  The attribute is then known as a REGIONALIZED VARIABLEThe attribute is then known as a REGIONALIZED VARIABLE  (atmospheric pressure, elevation above sea level, etc).(atmospheric pressure, elevation above sea level, etc).  Interpolation with geostatistics is known as KRIGING.Interpolation with geostatistics is known as KRIGING.

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Regionalized VariableRegionalized Variable TheoryTheory

 REGIONALIZED VARIABLE THEORYREGIONALIZED VARIABLE THEORY assumes that the spatial variation of assumes that the spatial variation of ANY variable can be expressed as ANY variable can be expressed as THE SUM OF 3 MAJOR COMPONENTS.THE SUM OF 3 MAJOR COMPONENTS.

 A. A STRUCTURAL component, having a constant mean or trend (A’);A. A STRUCTURAL component, having a constant mean or trend (A’);  B. A RANDOM, but spatially correlated component, known as the B. A RANDOM, but spatially correlated component, known as the

variation of the REGIONALIZED VARIABLE;variation of the REGIONALIZED VARIABLE;  C. Spatially uncorrelated random noiseC. Spatially uncorrelated random noise  The value of a RANDOM variable Z at X position is given byThe value of a RANDOM variable Z at X position is given by

Z(x) = m(x) + E(x) + E”Z(x) = m(x) + E(x) + E”

A

B C A’

Z

X

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Regionalized VariableRegionalized Variable TheoryTheory  The first step is to decide a suitable function for m(X).The first step is to decide a suitable function for m(X).  No trend or drift at present, and m(X) equals the MEAN value in No trend or drift at present, and m(X) equals the MEAN value in

the sampling area. The average or expected difference between the sampling area. The average or expected difference between any two places = 0.any two places = 0. E [Z(x) - Z(x+h)] = 0E [Z(x) - Z(x+h)] = 0

 Also, it is assumed that the variance of differences depends Also, it is assumed that the variance of differences depends ONLY on the DISTANCE between sites, h.ONLY on the DISTANCE between sites, h.

 This variance is known as the SEMIVARIANCE.This variance is known as the SEMIVARIANCE. (h) = 1/2n (h) = 1/2n  {Z(x) - Z(x + h)} {Z(x) - Z(x + h)}22, where n - the number of sample points of observations of the values of attribute Z separated by distance h.

 It is known as EXPERIMENTAL VARIOGRAM.

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Variogram ModelsVariogram Models  The experimental variogram is the first step towards a QUANTITATIVE The experimental variogram is the first step towards a QUANTITATIVE

description of the regionalized variation.description of the regionalized variation.

Nugget

C1

Sill

Range (A0)

C0 Lag (h)

(h)

1. At large values of the lag, h, it levels off. This horizontal part - THE SILL. There is NO spatial dependence. 2. Within the RANGE the closer sites are together the more similar they are. If the distance is greater than the RANGE, then the data point is no useful. 3.THE NUGGET - the residual spatially uncorrelated noise +errors.

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VariogramsVariograms  Samples separated by distances CLOSE than the RANGE are Samples separated by distances CLOSE than the RANGE are

Spatially Related.Spatially Related.  The Range also defines the MAXIMUM Radius from which The Range also defines the MAXIMUM Radius from which

neighboring samples are drawn for interpolation by kriging.neighboring samples are drawn for interpolation by kriging.  The Sum of the Nugget variance and the spatial variance The Sum of the Nugget variance and the spatial variance

approximately equals the sill or SAMPLE VARIANCE for approximately equals the sill or SAMPLE VARIANCE for stationary data.stationary data.

 A nugget variance of 0% of sill means that there is neither A nugget variance of 0% of sill means that there is neither measurement error nor significant short-range variation.measurement error nor significant short-range variation.

 A total absence of spatial correlation occurs when A total absence of spatial correlation occurs when (h) (h) equals the sill at all values of h.equals the sill at all values of h.

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VariogramsVariograms

 1. Nugget variance = 0 % ( A ).1. Nugget variance = 0 % ( A ).  2. Nugget variance = sill ( B ).2. Nugget variance = sill ( B ).

B

A

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Variograms forVariograms for Spatial AnalysisSpatial Analysis

 The form of the variogram can be quite revealing about the kind of spatial The form of the variogram can be quite revealing about the kind of spatial variation present in an area.variation present in an area.

 Models: 1. TRANSITIVE: a. Spherical; b. Exponential; c. Gaussian.Models: 1. TRANSITIVE: a. Spherical; b. Exponential; c. Gaussian.  2. NON-TRANSITIVE: Linear.2. NON-TRANSITIVE: Linear.  When the nugget variance dominates the local variation, the data are so noisy When the nugget variance dominates the local variation, the data are so noisy

that interpolation is not sensible.that interpolation is not sensible.  If semivariances are scattered, then too few samples have been taken to If semivariances are scattered, then too few samples have been taken to

compute. compute.  Generally, at least 50-100 data points are necessary to achieve a stable Generally, at least 50-100 data points are necessary to achieve a stable

variogram.variogram.  The range of the variogram provides information about the size of the search The range of the variogram provides information about the size of the search

window.window.  If the range is large then long-range variation dominates; if it is small, then the If the range is large then long-range variation dominates; if it is small, then the

major variation occurs over short distances.major variation occurs over short distances.  The distances can be modified by anisotropy.The distances can be modified by anisotropy.

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Kriging.Kriging.  The fitted variograms, that showed spatial dependence, can be used to The fitted variograms, that showed spatial dependence, can be used to

determine the WEIGHTS needed for local interpolation.determine the WEIGHTS needed for local interpolation.  The method is known as KRIGING. The procedure is similar to the weighted The method is known as KRIGING. The procedure is similar to the weighted

moving average interpolationmoving average interpolation  PUNCTUAL (point, simple) KRIGING gives an optimal interpolation estimate for PUNCTUAL (point, simple) KRIGING gives an optimal interpolation estimate for

a given point coordinate.a given point coordinate.  BLOCK KRIGING estimates a weighted average of values of the variable at BLOCK KRIGING estimates a weighted average of values of the variable at

points in the vicinity of the defined cell or block. points in the vicinity of the defined cell or block.  Kriging interpolation methods provide each cell with a local, optimal Kriging interpolation methods provide each cell with a local, optimal

prediction and a standard deviation that depends on the variogram and the prediction and a standard deviation that depends on the variogram and the spatial configuration of the data.spatial configuration of the data.

 After computing average and standard deviation surfaces using kriging, you After computing average and standard deviation surfaces using kriging, you can display results as grid cell maps.can display results as grid cell maps.

 Last step - to input the results to GIS and use them in conjunction with the Last step - to input the results to GIS and use them in conjunction with the other data.other data.

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KrigingKriging

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Types of KrigingTypes of Kriging  Simple Kriging assumes stationarity of the

first moment over the entire domain with a known mean

 Ordinary Kriging assumes constant unknown mean only over the search neighborhood

 Universal Kriging assumes a general polynomial trend model, such as linear trend model

 Disjunctive Kriging is a nonlinear generalization of Kriging

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KrigingKriging

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If I am hired…..If I am hired…..

 Explain to a decision-maker why it is worth Explain to a decision-maker why it is worth having interpolated data with a known level having interpolated data with a known level of accuracy and uncertainty.of accuracy and uncertainty.