Using ArcGIS Geostatistical Analyst · 2005-01-25 · 2 USING ARCGIS GEOSTATISTICAL ANALYST Exploratory spatial data analysis,

ArcGIS®

9Using ArcGIS® Geostatistical Analyst

Copyright © 2001, 2003 ESRIAll Rights Reserved.Printed in the United States of America.

The information contained in this document is the exclusive property of ESRI. This work is protected under United States copyright law and the copyrightlaws of the given countries of origin and applicable international laws, treaties, and/or conventions. No part of this work may be reproduced or transmittedin any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage or retrieval system, except asexpressly permitted in writing by ESRI. All requests should be sent to Attention: Contracts Manager, ESRI, 380 New York Street, Redlands, CA 92373-8100, USA.

The information contained in this document is subject to change without notice.

DATA CREDITSCarpathian Mountains data supplied by USDA Forest Service, Riverside, California, and is used here with permission.

Radioceasium data supplied by International Sakharov Environmental University, Minsk, Belarus, and is used here with permission. Copyright © 1996.

Air quality data for California supplied by California Environmental Protection Agency, Air Resource Board, and is used here with permission.Copyright © 1997.

Radioceasium contamination in forest berries data supplied by the Institute of Radiation Safety “BELRAD”, Minsk, Belarus, and is used here withpermission. Copyright © 1996.

CONTRIBUTING WRITERSKevin Johnston, Jay M. Ver Hoef, Konstantin Krivoruchko, and Neil Lucas

DATA DISCLAIMER

THE DATA VENDOR(S) INCLUDED IN THIS WORK IS AN INDEPENDENT COMPANY AND, AS SUCH, ESRI MAKES NO GUARANTEES AS TO THEQUALITY, COMPLETENESS, AND/OR ACCURACY OF THE DATA. EVERY EFFORT HAS BEEN MADE TO ENSURE THE ACCURACY OF THE DATAINCLUDED IN THIS WORK, BUT THE INFORMATION IS DYNAMIC IN NATURE AND IS SUBJECT TO CHANGE WITHOUT NOTICE. ESRI ANDTHE DATA VENDOR(S) ARE NOT INVITING RELIANCE ON THE DATA, AND ONE SHOULD ALWAYS VERIFY ACTUAL DATA AND INFORMATION.ESRI DISCLAIMS ALL OTHER WARRANTIES OR REPRESENTATIONS, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,THE IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. ESRI AND THE DATA VENDOR(S) SHALLASSUME NO LIABILITY FOR INDIRECT, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES, EVEN IF ADVISED OF THE POSSIBILITYTHEREOF.

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Attribution.pmd 11/25/2003, 4:46 PM1

iii

Contents 1 Welcome to ArcGIS Geostatistical Analyst 1 !"#

2 Quick-start tutorial 11$ %%&' %& %(&)&' #&**) (&+

3 The principles of geostatistical analysis 49, -, .*/* ( %0 " !" #

4 Exploratory Spatial Data Analysis 81.1"2#1"#1"# * (

TOC.p65 03/07/2001, 4:00 PM3

iv USING ARCGIS GEOSTATISTICAL ANALYST

12 * (#3* (($ * %-%3* %-3* %- %- %-#, %-(, %%%

5 Deterministic methods for spatial interpolation 11345$1. 5%%'$1.%%#45*5%-'*%455%'%45 *5%'6/7%(

6 Creating a surface with geostatistical techniques 131.82%, %, %' %, % , %#' %(, %'%, %-

TOC.p65 03/07/2001, 4:00 PM4

CONTENTS v

'%%, %, %' %, **%'**% , 9%(' 9%-, %'%

7 Using analytical tools when generating surfaces 167$&%# % 1* )%#%1* )%#+: %#(+: %(" %(' 5%( ' 5%(( * --,;</='< >-,-'* *-'* *-($ 9%%1 9%6 %6* & %#

TOC.p65 03/07/2001, 4:00 PM5

vi USING ARCGIS GEOSTATISTICAL ANALYST

8 Displaying and managing geostatistical layers 219.2-" .5?5"'6 '*(1-' 55* +

9 Additional geostatistical analysis tools 239'& -+ %+ *

Appendix A 247

Appendix B 275

Glossary 279

References 285

Index 287

TOC.p65 03/07/2001, 4:00 PM6

IN THIS CHAPTER

1

Welcome to ArcGIS Geostatistical Analyst 1• Exploratory spatial data analysis

• Semivariogram modeling

• Surface prediction and errormodeling

• Threshold mapping

• Model validation and diagnostics

• Surface prediction using cokriging

• Tips on learning GeostatisticalAnalyst

Welcome to the ESRI® ArcGIS® Geostatistical Analyst extension foradvanced surface modeling using deterministic and geostatistical methods.Geostatistical Analyst extends ArcMap™ by adding an advanced toolbarcontaining tools for exploratory spatial data analysis and a geostatisticalwizard to lead you through the process of creating a statistically validsurface. New surfaces generated with Geostatistical Analyst cansubsequently be used in geographic information system (GIS) models and invisualization using ArcGIS extensions such as ArcGIS Spatial Analyst andArcGIS 3D Analyst™.

Geostatistical Analyst is revolutionary because it bridges the gap betweengeostatistics and GIS. For some time, geostatistical tools have beenavailable, but never integrated tightly within GIS modeling environments.Integration is important because, for the first time, GIS professionals canbegin to quantify the quality of their surface models by measuring thestatistical error of predicted surfaces.

Surface fitting using Geostatistical Analyst involves three key steps(demonstrated on the following pages):

• Exploratory spatial data analysis

• Structural analysis (calculation and modeling of the surface properties ofnearby locations)

• Surface prediction and assessment of results

The software contains a series of easy-to-use tools and wizards that guideyou through each of these steps. It also includes a number of unique toolsfor statistical spatial data analysis.

ch01_Welcome.pmd 11/25/2003, 2:54 PM1

2 USING ARCGIS GEOSTATISTICAL ANALYST

Exploratory spatial data analysis, <!" 5@ 5!" *< < * @

A number of exploratory spatial data analysis tools are used to investigate the properties of ozone measurements takenat monitoring stations in the Carpathian Mountains.

Welcome.p65 03/05/2001, 2:27 PM2

WELCOME TO ARCGIS GEOSTATISTICAL ANALYST 3

Semivariogram modeling! 5&%> )< >@"* *!"< <<< <**< 9@

The two phases of geostatistical analysis of data are illustrated above. First, the semivariogram/covariance wizard was used tofit a model to winter temperature data for the USA. This model was then used to create the temperature distribution map.

Welcome.p65 03/05/2001, 2:27 PM3


Surface prediction and error modeling

Here, Geostatistical Analyst has been used to produce a prediction map of radioceasium soil contamination levels in the country ofBelarus after the Chernobyl nuclear power plant accident.

Welcome.p65 3/21/01, 7:58 AM4


Threshold mapping+*** 5 @

Locations shown in dark orange and red indicate a probability greater than 62.5% that radioceasium contamination exceeds the upperpermissible level (critical threshold) in forest berries.

Welcome.p65 03/05/2001, 2:27 PM5


Model validation and diagnostics$ *5*@** * @ 55? @

The validation wizard is used to assess a model developed to predict organic matter for a farm in Illinois.

Welcome.p65 03/05/2001, 2:27 PM6


Surface prediction using cokriging

In this example, exploratory spatial data analysis tools are used to explore spatial correlation betweenozone (primary variable) and nitrogen dioxide (secondary variable) in California. Because the variables arespatially correlated, cokriging can use the nitrogen dioxide data to improve predictions when mappingozone.

Welcome.p65 3/21/01, 8:01 AM7


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Contacting ESRI

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Welcome.p65 03/05/2001, 2:27 PM10

IN THIS CHAPTER

11

With Geostatistical Analyst, you can easily create a continuous surface,or map, from measured sample points stored in a point-feature layer, rasterlayer, or by using polygon centroids. The sample points may be measurementssuch as elevation, depth to the water table, or levels of pollution, as is the casein this tutorial. When used in conjunction with ArcMap, Geostatistical Analystprovides a comprehensive set of tools for creating surfaces that can be usedto visualize, analyze, and understand spatial phenomena.

Tutorial scenario

The U.S. Environmental Protection Agency is responsible for monitoringatmospheric ozone concentration in California. Ozone concentration is mea-

sured at monitoring stations throughout the state.The locations of the stations are shown here. Theconcentration levels of ozone are known forall of the stations, but we are also interested inknowing the level for every location in California.However, due to cost and practicality, monitoringstations cannot be everywhere. GeostatisticalAnalyst provides tools that make the best predic-tions possible by examining the relationshipsbetween all of the sample points and producing acontinuous surface of ozone concentration,standard errors (uncertainty) of predictions, andprobabilities that critical values are exceeded.

Quick-start tutorial 2• Exercise 1: Creating a surface

using default parameters

• Exercise 2: Exploring your data

• Exercise 3: Mapping ozone con-centration

• Exercise 4: Comparing models

• Exercise 5: Mapping the probabil-ity of ozone exceeding a criticalthreshold

• Exercise 6: Producing the finalmap

ch02_Tutorial.pmd 11/25/2003, 4:33 PM11


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Directional semivariograms

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Exercise 5: Mapping the probability of ozone exceeding a critical threshold

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77

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Tutorial.p65 03/07/2001, 2:39 PM48

IN THIS CHAPTER

49

The principles of geostatistical analysis 3• Understanding deterministic

methods

• Understanding geostatisticalmethods

• Working through a problem

• Basic principles behindgeostatistical methods

• Modeling a semivariogram

• Predicting unknown values withkriging

• The Geostatistical Analystextension

Geostatistical Analyst uses sample points taken at different locations in alandscape and creates (interpolates) a continuous surface. The samplepoints are measurements of some phenomenon such as radiation leakingfrom a nuclear power plant, an oil spill, or elevation heights. GeostatisticalAnalyst derives a surface using the values from the measured locations topredict values for each location in the landscape.

Geostatistical Analyst provides two groups of interpolation techniques:deterministic and geostatistical. All methods rely on the similarity of nearbysample points to create the surface. Deterministic techniques usemathematical functions for interpolation. Geostatistics relies on bothstatistical and mathematical methods, which can be used to create surfacesand assess the uncertainty of the predictions.

Geostatistical Analyst, in addition to providing various interpolationtechniques, also provides many supporting tools. These tools allow you toexplore and gain a better understanding of the data so that you create thebest surfaces based on the available information.

This chapter will provide an overview of the theory behind deterministic andgeostatistical interpolation techniques. The first part of the chapter willintroduce you to the deterministic interpolation methods. You will then beexposed to geostatistical methods through an example, and then you willread about the principles, concepts, and assumptions that provide thefoundation for geostatistics.

ch03_BasicConcepts.pmd 11/25/2003, 2:58 PM49


Understanding deterministic methods

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THE PRINCIPLES OF GEOSTATISTICAL ANALYSIS 51

Visualizing global polynomial interpolation

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Assess the quality of the output surface usingCross-Validation and Validation tools. This will helpyou understand how well the model predicts thevalues at unmeasured locations.

More than one surface can be produced. Thesurface can be compared using cross-validationstatistics.

Create subsets

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IN THIS CHAPTER

81

4• What is Exploratory Spatial Data

Analysis?

• The Exploratory Spatial DataAnalysis tools

• Examining the distribution of thedata

• Looking for global and localoutliers

• Looking for global trends

• Examining spatial autocorrelationand directional variation

• Understanding covariation amongmultiple datasets

Exploratory Spatial Data Analysis allows you to examine your data indifferent ways. Before creating a surface, ESDA enables you to gain adeeper understanding of the phenomena you are investigating so that youcan make better decisions on issues relating to your data. The ESDAenvironment is composed of a series of tools, each allowing a view into thedata. Each view can be manipulated and explored, allowing differentinsights about the data. Each view is interconnected with all other views aswell as with ArcMap. That is, if a bar is selected in the histogram, thepoints comprising the bar are also selected on the QQPlot (if opened), onany other open ESDA view, and on the ArcMap map.

The ESDA environment is designed to explore, as its name implies.However, there are certain tasks that are useful in most explorations.Exploring the distribution of the data, looking for global and local outliers,looking for global trends, examining spatial autocorrelation, andunderstanding the covariation among multiple datasets are all useful tasks toperform on your data. The ESDA tools can assist you with these tasks aswell as many others.

Exploratory Spatial Data Analysis

ch04_ESDA.pmd 11/25/2003, 3:02 PM81


What is Exploratory Spatial Data Analysis?

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ESDA.p65 3/21/01, 8:08 AM82

EXPLORATORY SPATIAL DATA ANALYSIS 83

Exploratory Spatial Data Analysis

Selection of Data Points

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Histogram tool

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ESDA.p65 3/21/01, 8:08 AM83


Exploratory Spatial Data Analysis tools

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ESDA.p65 03/07/2001, 12:44 PM84


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ESDA.p65 03/07/2001, 12:44 PM85


Tools (pan,zoom, etc.) Voronoi map Cell values

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Voronoi map

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ESDA.p65 03/07/2001, 12:44 PM86


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ESDA.p65 03/07/2001, 12:44 PM88


General QQPlotNormal QQPlot

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ESDA.p65 03/07/2001, 12:44 PM89


Trend analysis

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ESDA.p65 03/07/2001, 12:44 PM90


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ESDA.p65 03/07/2001, 12:44 PM91


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ESDA.p65 03/07/2001, 12:44 PM92


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ESDA.p65 03/07/2001, 12:45 PM93


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ESDA.p65 03/07/2001, 12:45 PM94


Examining the distribution of the data

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ESDA.p65 03/07/2001, 12:45 PM96


Understanding distributions with the QQPlot

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ESDA.p65 03/07/2001, 12:45 PM97


Examining thedistribution ofyour data

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1. Click on the point featurelayer in the ArcMap table ofcontents that you wish toexplore.

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Tip

The QQPlot

Exploring the distributionthrough the QQPlot

1. Click on the point featurelayer in the ArcMap table ofcontents that you wish toexplore.

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ESDA.p65 03/07/2001, 12:45 PM98


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ESDA.p65 03/07/2001, 12:45 PM99


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ESDA.p65 03/07/2001, 12:45 PM100

EXPLORING SPATIAL DATA ANALYSIS 101

Identifying globaland local outliers

Identifying global outliersusing the Histogram tool

1. Click on the point or polygonfeature layer in the ArcMaptable of contents that youwish to explore.

2. Click on the GeostatisticalAnalyst toolbar, click ExploreData, then click Histogram.

See Also

Looking for globaloutliers through theSemivariogram/Covariance Cloud


2. Click on the GeostatisticalAnalyst toolbar, click ExploreData, then click Semivari-ogram/Covariance Cloud.

ESDA.p65 3/21/01, 8:09 AM101


Finding local outliersusing Voronoi map


2. Click on the GeostatisticalAnalyst toolbar, click ExploreData, then click Voronoi Map.

ESDA.p65 03/07/2001, 12:45 PM102


Looking for global trends

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ESDA.p65 03/07/2001, 12:45 PM103


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ESDA.p65 3/21/01, 8:09 AM104


Looking forglobal trends # )

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Identifying global trendswith the Trend Analysistool


2. Click on the GeostatisticalAnalyst toolbar, click ExploreData, then click TrendAnalysis.

See Also

!

ESDA.p65 03/07/2001, 12:45 PM105


Examining spatial autocorrelation and directional variation

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ESDA.p65 03/07/2001, 12:45 PM106


Looking for directional influences with theSemivariogram/Covariance Cloud tool

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ESDA.p65 03/07/2001, 12:45 PM107


Examiningspatial structureand directionalvariation

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Understanding spatialstructure


2. Click on the GeostatisticalAnalyst toolbar, click ExploreData, then click Semivari-ogram/Covariance Cloud.

See Also

ESDA.p65 3/21/01, 8:10 AM108


Exploring covariation among multiple datasets

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ESDA.p65 03/07/2001, 12:45 PM109


Exploring the correlation between two datasets

Data pairing for the Los Angeles area with high cross-correlation between ozone and nitrogen dioxide

ESDA.p65 03/07/2001, 12:45 PM110


Understandingspatialcovariationamong multipledatasets

Understanding spatialcovariation using theCrosscovariance Cloud

1. Right-click on the pointfeature layer in the ArcMaptable of contents identifyingthe first layer in thecrosscovariance analysisand click Properties.

2. Click on Selection.

3. Click on the symbol radiobutton.

4. Click on the symbol.

5. Choose a color and size forthe selection.

Repeat steps 1–5 for thesecond layer to be used inthe crosscovariance analysis,but choose different selectionsizes and colors.

6. Highlight the layers in theArcMap table of contents byholding down the Ctrl keywhile left-clicking on the twolayers.

7. Click on GeostatisticalAnalyst, click Explore Data,and click CrosscovarianceCloud.

ESDA.p65 3/21/01, 8:15 AM111


8. Click on the appropriateattribute for each layer in theAttribute dropdown list.

9. Input the Lag Size andNumber of Lags.

10. Check Search Direction.

11. Click on the center blue linein the Covariance Surfaceand spin the search directionuntil it points to the anglewhere you believe there is ashift; in this example it is270 degress (given in theangle direction box).

12. Brush some points in thecovariance cloud by holdingdown the left mouse buttonand dragging it over some ofthe points. Examine where,on the ArcMap map, thepairs of points are that werebrushed.

ESDA.p65 3/21/01, 8:15 AM112

IN THIS CHAPTER

113

Deterministic methods for spatial interpolation 5• How Inverse Distance Weighted

interpolation works

• Creating a surface using InverseDistance Weighted interpolation

• How global polynomialinterpolation works

• Creating a map using globalpolynomial interpolation

• How local polynomialinterpolation works

• Creating a surface using localpolynomial interpolation

• How radial basis functionsinterpolation works

• Creating a surface using radialbasis functions interpolation

There are two main groupings of interpolation techniques: deterministic andgeostatistical. Deterministic interpolation techniques create surfaces frommeasured points, based on either the extent of similarity (e.g., InverseDistance Weighted) or the degree of smoothing (e.g., radial basis functions).Geostatistical interpolation techniques (e.g., kriging) utilize the statisticalproperties of the measured points. The geostatistical techniques quantify thespatial autocorrelation among measured points and account for the spatialconfiguration of the sample points around the prediction location.Geostatistical techniques will be discussed in Chapter 6, ‘Creating a surfacewith geostatistical techniques’.

Deterministic interpolation techniques can be divided into two groups: globaland local. Global techniques calculate predictions using the entire dataset.Local techniques calculate predictions from the measured points withinneighborhoods, which are smaller spatial areas within the larger study area.Geostatistical Analyst provides the global polynomial as a global interpolatorand the Inverse Distance Weighted, local polynomial, and radial basisfunctions as local interpolators.

A deterministic interpolation can either force the resulting surface to passthrough the data values or not. An interpolation technique that predicts avalue identical to the measured value at a sampled location is known as anexact interpolator. An inexact interpolator predicts a value that is differentfrom the measured value. The latter can be used to avoid sharp peaks ortroughs in the output surface. Inverse Distance Weighted and radial basisfunctions are exact interpolators, while global and local polynomial areinexact.

ch05_Deterministic.pmd 11/25/2003, 3:02 PM113


How Inverse Distance Weighted interpolation works

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Deterministic.p65 03/07/2001, 12:50 PM114

DETERMINISTIC METHODS FOR SPATIAL INTERPOLATION 115

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Creating a mapusing IDW'+ '" # " ,-# ( &# , -!

Creating a predictionmap

1. Click on the point layer onwhich to perform IDW in theArcMap table of contents.

2. Start the GeostatisticalAnalyst.

3. Click the Attribute dropdownmenu and click the attributeon which to perform IDW inthe Choose Input Data andMethod dialog box.

4. Click the Inverse DistanceWeighting method.

5. Click Next.

6. Specify the desired param-eters in the IDW Set Param-eters dialog box and clickNext.

7. Examine the results on theCross Validation dialog boxand click Finish.

8. Click on the Output LayerInformation dialog box andclick OK.

See Also

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Using a database fileinstead of a point layer' '( )* + ,( $

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DETERMINISTICAL METHODS OF SPATIAL DATA INTERPOLATION 119

Creating a predictionmap using validation

1. Click on the point layer onwhich to perform IDW in theArcMap table of contents.


3. Click the Attribute dropdownmenu and specify the field onwhich to perfom IDW in theChoose Input Data andMethod dialog box.

4. Check Validation.

5. Pick a point layer file in theInput Data dropdown menuor browse for the desiredlayer.

6. Click the Attribute dropdownmenu and specify the field onwhich to validate the IDWinterpolation within theChoose Input Data andMethod dialog box.

7. Click the Inverse DistanceWeighting method.

8. Click Next.

9. Specify the desired param-eters in the IDW Set Param-eters dialog box and clickNext.

10. Examine the results on theCross Validation andValidation dialog boxes andclick Finish.

11. Click on the Output LayerInformation dialog box andclick OK.

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Using validation(), +)



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1. Click on the point layer onwhich to perform GlobalPolynomial Interpolation inthe ArcMap table of contents.


3. Click the Attribute dropdownmenu and click the attributeon which to perform GlobalPolynomial Interpolation inthe Choose Input Data andMethod dialog box.

4. Click the Global PolynomialInterpolation method andclick Next.

5. Specify the desired order ofpolynomial in the GlobalPolynomial Interpolation SetParameters dialog box andclick Next.


7. On the Output Layer Informa-tion dialog box click OK.



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1. Click on the point layer onwhich to perform LocalPolynomial Interpolation inthe ArcMap table of contents.


3. Click the Attribute dropdownmenu and click the attributeon which to perform LocalPolynomial Interpolation inthe Choose Input Data andMethod dialog box.

4. Click the Local PolynomialInterpolation method.

5. Click Next.

6. Specify the desired param-eters in the LP InterpolationSet Parameters dialog boxand click Next.


8. On the Output Layer Informa-tion dialog box, click OK.

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How radial basis functions work

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Deterministic.p65 03/23/2001, 8:26 AM128


Creating a mapusing RBFs012 1 #012

Creating a predictionmap using RBFs

1. Click on the point layer onwhich to perform RadialBasis Functions in theArcMap table of contents.


3. Click the Attribute dropdownmenu and click the attributeon which to perfom RadialBasis Functions in theChoose Input Data andMethod dialog box.

4. Click the Radial BasisFunctions method.

5. Click Next.

6. Click on the Kernel Functionsdropdown menu and click thedesired Radial BasisFunction in the RBF Interpo-lation Set Parameters dialogbox.

7. Specify the desired param-eters in the RBF Set Param-eters dialog box and clickNext.



See Also

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131

IN THIS CHAPTER

Creating a surface with geostatistical techniques 6• What are geostatistical interpola-

tion techniques?

• Understanding kriging models

• Understanding output surfaces

• Creating a map using defaults

• Understanding transformationsand trends

• Understanding and mapping with:

• Ordinary kriging

• Simple kriging

• Universal kriging

• Indicator kriging

• Probability kriging

• Disjunctive kriging

• Cokriging

In the previous chapter, you learned about deterministic techniques forinterpolation. Deterministic techniques used the existing configuration of thesample points to create a surface (Inverse Distance Weighted) or fit amathematical function to the measured points (global and local polynomialand radial basis functions). In this chapter, you will get an overview of thedifferent geostatistical interpolation techniques. As their name implies,geostatistical techniques create surfaces incorporating the statisticalproperties of the measured data. Because geostatistics is based on statistics,these techniques produce not only prediction surfaces but also error oruncertainty surfaces, giving you an indication of how good the predictionsare.

Many methods are associated with geostatistics, but they are all in the krigingfamily. Ordinary, simple, universal, probability, indicator, and disjunctivekriging along with their counterparts in cokriging are available inGeostatistical Analyst. Not only do these kriging methods create predictionand error surfaces, but they can also produce probability and quantile outputmaps depending on your needs.

Kriging is divided into two distinct tasks: quantifying the spatial structure ofthe data and producing a prediction. Quantifying the structure, known asvariography, is where you fit a spatial-dependence model to your data. Tomake a prediction for an unknown value for a specific location, kriging willuse the fitted model from variography, the spatial data configuration, and thevalues of the measured sample points around the prediction location.Geostatistical Analyst provides many tools to help you determine whichparameters to use, but it also provides reliable defaults that you can use tomake a surface quickly.

ch06_GeoStatistics.pmd 11/25/2003, 3:03 PM131


What are geostatistical interpolation techniques?

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GeoStatistics.p65 03/07/2001, 2:46 PM132

CREATING A SURFACE WITH GEOSTATISTICAL TECHNIQUES 133

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1. Click the point layer on whichyou wish to perform kriging inthe ArcMap table of contents.

2. Start the GeostatisticalWizard.

3. Click the Attribute dropdownmenu, and click the attributeon which you wish to performkriging.

4. Click the Kriging method.

5. Click Next.

6. On all subsequent dialogboxes, click Next.

7. On the Cross Validationdialog box, click Finish.


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1. Click the point layer on whichyou wish to perform OrdinaryKriging in the ArcMap table ofcontents.


3. Click the Attribute dropdownlist, and click the attribute onwhich you wish to performordinary kriging.


5. Click Next.

6. Click Prediction underOrdinary Kriging.

7. Click Next.

8. Specify the desired param-eters in the Semivariogram/Covariance Modeling dialogbox and click Next.

9. Specify the desired param-eters in the SearchingNeighborhood dialog box andclick Next.


11. On the Output Layer Infor-mation dialog box, click OK.

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Important parameters' +8+



Creating a predictionstandard error map

1. Right-click on the predictionsurface in the ArcMap table ofcontents that was createdusing Ordinary Kriging andclick Create PredictionStandard Error Map.

Creating a prediction mapusing validation

1. Click the point layer on whichyou wish to perform ordinarykriging in the ArcMap table ofcontents.

2. Start Geostatistical Analyst.

3. Click the Attribute dropdownlist and specify the field onwhich you wish to performordinary kriging.


5. Check Validation and specifythe validation dataset andattribute.

6. Click Next.

7. Follow steps 6 through 10 in‘Creating a prediction map’ onthe previous page andexamine the results on theValidation dialog box andthen click Finish.

Tip

Creating training and testdatasets !"#$

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GeoStatistics.p65 3/21/01, 8:12 AM140


Creating a predictionmap while applying atransformation



3. Click the Attribute field onwhich you wish to performordinary kriging.


5. Click Next.

6. Expand the list underOrdinary Kriging and clickPrediction.

7. Click the desired transforma-tion from the Transformationdropdown menu.

8. Click Next.

9. Follow steps 9 through 12 in‘Creating a prediction mapusing detrending’ on thefollowing page.

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Creating a predictionmap using detrending



3. Click the Attribute dropdownlist and select the attribute onwhich you wish to performordinary kriging.


5. Click Next.

6. Click Prediction underOrdinary Kriging.

7. Click the Order of TrendRemoval dropdown menu andchoose an option.

8. Click Next.

9. Specify the desired param-eters in the Detrending dialogbox and click Next.


11. Specify the desired param-eters in the SearchingNeighborhood dialog boxand click Next.



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Understanding simple kriging

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1. Click the point layer on whichyou wish to perform simplekriging in the ArcMap table ofcontents.


3. Click the Attribute field onwhich you wish to performsimple kriging.


5. Click Next.

6. Expand the list under SimpleKriging and click Prediction.

7. Specify the Mean Value.

8. Click Next.





Creating a mapusing simplekriging. #4# # 5"

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Creating a quantile map

1. Follow the steps in ‘Creatinga prediction map’ on theprevious page, except clickQuantile Map in step 6,rather than Prediction.

2. Click the Quantile up anddown arrow buttons tospecify the quantile level.

3. Follow steps 7 through 12 in‘Creating a prediction map’on the previous page.

See Also

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Creating a probabilitymap

1. Follow the steps in ‘Creatinga prediction map’ on theprevious page, except clickProbability Map in step 6,rather than Prediction.

2. Type a value in the Thresholdinput or click the Set... buttonand set the threshold on thePrimary Threshold selectiondialog box.

3. Click either the Exceed orNOT Exceed radio buttons.

4. Click Next.

5. Follow steps 7 through 12 in‘Creating a prediction map’on the previous page.



Tip

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1. Right-click on the predictionsurface in the ArcMap tableof contents that was createdusing simple kriging andclick Create PredictionStandard Error Map.

Tip

Creating training and testdatasets-+. +/%09+ &1



Creating a predictionmap while applying atransformation





5. Click Next.

6. Expand the list under simplekriging and click Prediction.

7. Click the desired transforma-tion from the Transformationdropdown menu.

8. Click Next.

9. Follow steps 9 through 12 in‘Creating a prediction map’,earlier in the chapter.

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Creating a predictionmap while applying atransformation withdeclustering





5. Click Next.


7. Click Normal Score under theTransformation dropdownmenu.

8. Check Declustering beforeTransform.

9. Click Next.

10. Specify the desired param-eters in the Declusteringdialog box and click Next.

11. Specify the desired param-eters in the Normal ScoreTransformation dialog boxand click Next.

12. Follow steps 9 through 12 in‘Creating a prediction map’,earlier in the chapter.

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Examining the bivariatedistribution whencreating a prediction map





5. Click Next.


7. Check Examine BivariateDistribution.

8. Specify Mean Value and thenumber of Quantiles tocheck.

9. Click Next.


11. Explore the Semivariogram/Covariance Modeling(Examine Bivariate Distribu-tion) dialog box and clickNext.

12. Follow steps 10 through 12in ‘Creating a predictionmap’, earlier in the chapter.

Tip

Bivariate distribution

See Also

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Understanding universal kriging

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Creating a mapusing universalkriging. # 4# # '"


1. Click the point layer on whichyou wish to perform universalkriging in the ArcMap table ofcontents.


3. Click the Attribute field onwhich you wish to performuniversal kriging.

4. Click the Kriging method andclick Next.

5. Under Universal Kriging,expand the list and clickPrediction.

6. Click the Order of Trenddropdown menu and click thedesired order.

7. Click Next.

8. Specify the desired param-eters in the Detrendingdialog box and click Next.





See Also

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1. Right-click on the predictionsurface in the ArcMap table ofcontents that was createdusing universal kriging andclick Create PredictionStandard Error Map.

Tip

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Creating a mapusing indicatorkriging. '"


1. Click the point layer on whichyou wish to perform indicatorkriging in the ArcMap table ofcontents. Start the Geostatis-tical Analyst.

2. Click the Attribute field onwhich you wish to performindicator kriging.


4. Click Next.

5. Expand the list underIndicator Kriging and clickProbability Map.

6. Type the Threshold value orclick the Set... button and setthe threshold on the PrimaryThreshold selection dialogbox.


8. Click Next.

9. Set additional cutoffs on theAdditional Cutoffs selectiondialog box.

10. Specify the desired param-eters in the Semivariogram/Covariance Modeling andSearching Neighborhooddialog boxes and click Nextin each dialog box.



Tip

Important parameters3+;<=+;/=4+8+

See Also

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Selecting a threshold:+8+ + 9>+$>



Understanding probability kriging

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Creating a mapusing probabilitykriging.


1. Click the point layer on whichyou wish to perform probabil-ity kriging in the ArcMaptable of contents.


3. Click the Attribute field onwhich you wish to performprobability kriging.


5. Click Next.

6. Expand the list underProbability Kriging and clickProbability Map.

7. Type in the Threshold valueor click the Set... button andset the threshold on thePrimary Threshold selectiondialog box.


9. Click Next.

10. Specify the desired param-eters in the Semivariogram/Covariance Modeling andSearching Neighborhooddialog boxes and click Nextin each dialog box.

Tip

Important parameters3+;<=+;/=4+8+

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Selecting a threshold:+8+ + >, +$>

See Also

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Creating a mapusing disjunctivekriging


1. Click the point layer on whichyou wish to perform disjunc-tive kriging in the ArcMaptable of contents.


3. Click the Attribute field onwhich you wish to performdisjunctive kriging.


5. Click Next.

6. Expand the list underDisjunctive Kriging and clickPrediction. Specify MeanValue. Optionally, specifyNormal Score Transformation(NST) and click Next.

7. If an NST was indicated,specify the desired param-eters in the NST dialog boxand click Next.


9. Specify the desired param-eters in the SearchingNeighborhood dialog box andclick Next.



Tip

Using a database fileinstead of a point layer

Tip

Important parameters!"!#!!

See Also

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1. Follow the steps in ‘Creatinga prediction map’ on theprevious page, except clickon Probability Map in step 6rather than Prediction.

2. Type in the Threshold valueor click the Set... button andset the threshold on thePrimary Threshold selectiondialog box.


4. Specify Mean Value andNormal Score Transformation.Click Next.

5. Follow steps 7 through 11 in‘Creating a prediction map’earlier in the chapter.

See Also

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1. Right-click on the predictionsurface in the ArcMap table ofcontents that was createdusing disjunctive kriging andclick Create PredictionStandard Error Map.

Tip

A prediction standard errormap!, !!!-.!/0!



Creating a standard errorof indicators map

1. Follow the steps for ‘Creatinga probability map’ previouslyin this chapter, except clickStandard Error of Indicatorsin step 6 rather than Predic-tion.



Creating a predictionmap with declustering





5. Click Next.

6. Expand the list underDisjunctive Kriging and clickPrediction.

7. Click Normal Score under theTransformation dropdownmenu.


9. Click Next.

10. Specify the desired param-eters in the Declusteringdialog box and click Next.

11. Specify the desired param-eters in the Normal ScoreTransformation dialog boxand click Next.

12. Follow steps 8 through 11 in‘Creating a prediction map’,previously in this chapter.

Tip

Using declustering



Examining the bivariatedistribution whencreating a prediction map


2. Click the GeostatisticalAnalyst toolbar and clickGeostatistical Wizard.


4. Click the Kriging method.Click Next.

5. Expand the list underDisjunctive Kriging and clickPrediction.

6. Check Examine BivariateDistribution and specifyMean Value or NST.


8. Explore the Semivariogram/Covariance Modeling dialogbox (Examine BivariateDistribution) and click Next.

9. Follow steps 9 through 11 in‘Creating a prediction map’,previously in this chapter.

See Also

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Creating a mapusing cokriging


1. Click the point layers onwhich you wish to performcokriging in the ArcMap tableof contents.


3. Click the CoKriging method.

4. Click the Attribute field onwhich you wish to performcokriging for all datasets(switch between dataset tabsto specify the parameters).

5. Click Next.

6. Click the desired cokrigingmethod and output layer typein the Geostatistical Methodslist. Click Next.

7. Specify the parameters in theDetrending dialog box if theOrder of Trend Removal wasspecified for all datasets.Click Next.

8. Specify the parameters in theSemivariogram/CovarianceModeling dialog box for alldatasets. Click Next.

9. Specify the desired param-eters in the SearchingNeighborhood dialog box forall datasests and click Next.



See Also

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IN THIS CHAPTER

167

Using analytical tools when generating surfaces 7• Investigating spatial structure:

variography

• Determining the neighborhoodsearch size

• Performing cross-validation toassess parameter selections

• Assessing decision protocol usingvalidation

• Comparing one model withanother

• Modeling distributions and deter-mining transformations

• Checking for the bivariate normaldistribution

• Implementing declustering toadjust for preferential sampling

• Removing trends from the data

There are many steps that you go through when creating a surface. In eachof these steps, you specify a number of parameters. Geostatistical Analystprovides a series of dialog boxes containing analytical tools to assist you indetermining the values for the parameters. Some of these dialog boxes andtools are applicable to almost all interpolation methods such as specifyingthe search neighborhood, cross-validation, and validation. Others arespecific to the geostatistical methods (kriging and cokriging), such asmodeling semivariograms, transformations, detrending, declustering, andchecking for bivariate normal distributions.

Within each dialog box, there are a series of tasks that can be accomplishedusing the tools. In this chapter, the concepts for the most frequentlyperformed tasks are discussed and the steps identified. Depending on yourdata, none, some, or all of the tasks and their parameters might be explored.As with all parameters, Geostatistical Analyst provides reliable defaults,some of which have been calculated specifically for your data. However,you may have additional insight into your data from prior knowledge of thephenomena under study or that you gained through the exploratory toolsprovided with Geostatistical Analyst, which you can use to refine theparameters to create an even more accurate surface.

ch07_AnalyticalTools.pmd 11/25/2003, 3:04 PM167


Investigating spatial structure: variography

Semivariograms and covariance functions

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AnalyticalTools.p65 03/07/2001, 2:49 PM168

USING ANALYTICAL TOOLS WHEN GENERATING SURFACES 169

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The Semivariogram/Covariance Modeling dialog box

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Associatedparametervalues

Semivariogramcloud

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Modelingsemivariogramsand covariancefunctions6 "! "

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Selecting a model

1. On the Semivariogram/Covariance Modeling dialogbox, click on the desiredsemivariogram model.

You will notice that the yellowline modeling in theSemivariogram dialog boxwill change to reflect themodel that is selected.

See Also

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Selecting the model+))+!+9+



Exploring for directionalautocorrelation

1. Check Show Search Direc-tion.

2. Click the up or down arrowsof the Angle Direction toexplore for the desireddirectional angle.

Alternatively, in thesemivariogram map, clickand hold the middle blue lineand drag to the desireddirection.

3. Click the up or down arrowsof the Angle Tolerance toadjust the angle of tolerance.

Alternatively, in the semivari-ogram map, click and holdeither of the red directionalindicator lines and drag tothe desired angle.

4. Click the up or down arrowsof the Bandwidth.

Alternatively, in thesemivariogram map, clickand hold either of the purplelines located on the bound-ing square. Drag to thedesired width.

Tip

Autocorrelation in yourdata

Tip

This is a graphicalexploration only ! "#

AnalyticalTools.p65 3/21/01, 8:18 AM176


Modeling anisotropy

1. Check Anisotropy on theSemivariogram/CovarianceModeling dialog box.

Notice the single yellow linebecomes many lines whenanisotropy is checked.

Tip

Calculating the optimalparameters +A $+*)+

Altering the anisotropyparameters

1. Check Anisotropy on theSemivariogram/CovarianceModeling dialog box.

Note: The Minor Range andDirection check boxesbecome active.

2. To change the major range,click the pencil icon abovethe Major Range input(which makes the input boxactive) and type in thedesired range.

3. To change the minor range,click the pencil icon abovethe Minor Range input(which makes the input boxactive) and type in thedesired range.

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Checking the anisotropycheck box# )+34; =++B'3(AC(C'4



4. To change the angle direc-tion, click the pencil iconabove the Direction input(which makes the input boxactive) and type in thedesired angle.



Changing the lag sizeand number of lags

1. On the Semivariogram/Covariance Modeling dialogbox, type the desired LagSize.

2. Click the up or down arrowsor type in the desired valuein order to change theNumber of Lags.

Tip

Choosing a lag size' > 9

Tip

Calculating the optimalparameters + $+*)+

Changing the partial silland nugget

1. On the Semivariogram/Covariance Modeling dialogbox, click on the pencilabove the Partial Sill input(which makes the input boxactive) and type in thedesired value.

2. Check the Nugget check box,click on the pencil iconabove the Nugget input(which makes the input boxactive), and type in thedesired value.



Handling measurementerror

1. Check the Error Modelingcheck box.

2. Move the scroll bar todetermine the percentages ofMicroStructure and Measure-ment Error for the nugget.

Alternatively, type thepercentages or values intothe input boxes.

Tip

Zero measurement error,>;$=)

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Determining the neighborhood search size

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If not enough points areavailable within the searchneighborhood, the softwareselects the nearest availablepoints (this yellow pointbelongs to the upper-rightsector).

Only two points are availablein this sector.



The Searching Neighborhood dialog box

Click to previewthe surface.

Number of points usedto predict a value at a

test location.

The minimum number of pointsto be used (they may lie outside

the search ellipse).

Geometry andnumber of sectorsused in the search.

Crosshairsdefine the test

location.

Points used andassociated

weights.

Prediction fortest location.



Determining theneighborhoodsearch size $ " "

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Changing the number ofpoints to include in aneighborhood

1. On the Searching Neighbor-hood dialog box, click theNeighbors to Include up anddown arrows.

Alternatively, type in thedesired value.

2. To set a minimum of points toinclude in the neighborhood,check the Include at Leastcheck box and click the upand down arrows until thedesired value is reached.

Alternatively, type the valuein the input field.

Tip

Assessing a neighborhood 9 9+++ B9',



Altering the shape of theneighborhood

1. On the Searching Neighbor-hood dialog box, click thedesired ellipse icon tochange the default neighbor-hood shape type.

2. Check the Shape check box.

The controls in the shapeframe will become active.

3. Click the Angle up or downarrows or type in the desiredangle to alter the angle of theellipse.

4. Type the desired value in theMajor and Minor Semiaxisinput fields to alter the shapeof the ellipse.

In the display window of thedialog box, the ellipse willreflect the changes.

Tip

Weight values- +9DD;=

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Determining theprediction for a specificlocation

1. The number of neighborsused for prediction and theprediction are displayedbelow the input fields (in thebottom right of the dialogbox). To initiate a newprediction location, click thedesired location in thedisplay area of the dialogbox.

2. Alternatively, enter thelocation in the X and Y inputfields.

The prediction and number ofneighbors are updatedimmediately with the newlocation.

Tip

The predicted value7# $EF+G&@34



Altering the map view

1. On the Searching Neighbor-hood dialog box, to zoom inon the map display, click theZoom In button, then drag abox around the area of themap on which the zoom willoccur.

2. To zoom out on the mapdisplay, click the Zoom Outbutton.

3. To pan around in the mapdisplay, click the Pan buttonand move the mouse into themap display, hold down theleft mouse button, and movethe cursor.

The map will move incoordination with the cursor.

4. To display the map using thefull extent, click the FullExtent button.

5. To preview the outputsurface, click the Previewtype dropdown menu andclick Surface. To return to theprevious view, click Neigh-bors.



Performing cross-validation and validation

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The Cross Validation and Validation dialog box

Line of best fit 1:1 Line

Results fromcross-validationor validation

Summarystatistics

Cross-validationscatter plot

Save to a file



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Examining the predictedfit

1. On the Cross Validationdialog box, select either thePredicted, Error, Standard-ized Error, or QQPlot tabaccording to the desiredmethod in which you want toview the results.

Selecting a particularpoint

1. On the Cross Validationdialog box, in the table at thebottom right, click on the rowrepresenting the point ofinterest.

When a row is selected, thepoint is highlighted in thechart above.

Tip

Selecting points- +)

Tip

Viewing all rows andcolumns! +



Saving the cross-validation statistics to afile

1. Click Save Cross Validation.

2. Navigate to the location tosave the dataset.

3. Type the name of the dataset.

4. Click the type of dataset.

5. Click Save.

Tip

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Creating the subsets touse for validation

1. Add the dataset that you wishto subset to ArcMap.

2. Click Create Subsets.

3. Click the dropdown arrowand click on the dataset thatyou wish to subset.

4. Click Next.

5. Optionally, change thelocation and/or name for theoutput geodatabase file.

6. Optionally, change thedefault subset names.

7. Click and drag the slider tothe required position.

8. Click Finish.

Tip

Dividing the training andtest dataH

See Also

09++3)49++



Performing validation

1. Click the Add Data buttonand navigate to the test andtraining datasets (createdfollowing the steps on theprevious page). Click Add.

2. Start the GeostatisticalAnalyst Wizard.

3. Click the Input Datadropdown arrow and clickthe training layer (created bysubsetting the originaldataset on the previouspage).

4. Click the Attribute dropdownarrow and click the attributeyou want to use in theinterpolation.

5. Check the Validation checkbox.

6. Click the Input Datadropdown arrow and clickthe test dataset (created bysubsetting the originaldataset on the previouspage).

7. Click the Attribute dropdownarrow and click the sameattribute you chose for thetraining dataset.

8. Click the Method you wish touse.

9. Click Next on this and allsubsequent dialog boxesuntil you reach the Validationdialog box.

10. Optionally, click to save thevalidation table to a data-base.

Tip

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Tip

Highlighting values- +)

Viewing all rows andcolumns! +

Tip



Comparing one model with another

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Summarystatistics for

layer 1

Cross-validation

scatter plot forlayer 1

Summarystatistics forlayer 2

Fitted linefor layer 2

Cross-validationscatter plot for

layer 2

1:1 line forlayer 1

1:1 line forlayer 2



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Performing comparison

1. Right-click one of the layersin the ArcMap table ofcontents you wish to com-pare and click on Compare.

2. Click the second layer in thecomparison in the Todropdown menu.

3. Click the various tabs to seethe different results of thecomparison.

4. Click Close to close theCross Validation Comparisondialog box.

See Also

09++3)49++



Modeling distributions and determining transformations

Box–Cox, arcsine, and log transformations

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Choose number ofGaussian kernels

EmpiricalCumulativeDistributionfunction

FittedCumulativeDistributionfunction(red line)

Number ofhistogram bars

Approximation method:Direct (default), linear,Gaussian kernels

Fitted probability density(for Gaussian kernels only)

Histogram of datarepresenting probabilitydensity

Dataset



Comparing normal score transformation to othertransformations

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1. On the Geostatistical MethodSelection dialog box, clickthe desired transformation inthe Transformation dropdownmenu.

2. Click Next.

3. Follow the dialog boxes tocreate a surface.

Tip

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Using the normalscoretransformation " )

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Modeling distributions

1. Click the Simple or Disjunc-tive Kriging interpolationmethod to use for thedistribution.

2. On the Geostatistical MethodSelection dialog box, clickNormal Score in the Transfor-mation dropdown list.

3. Click Next.

4. Alternatively, click Cumula-tive Distribution to switch thedisplay of the graph.

5. Alternatively, type the numberof bars you wish to display inthe chart.

6. Click the Dataset Selectiondropdown arrow to switchbetween datasets (only forcokriging where you havetwo or more datasets).



Checking for the bivariate normal distribution

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The Examine Bivariate Distribution dialog box

Fitted covariancefunction toempirical

covariance onindicators (yellow

line)

Empiricalcovariance on

indicators

Theoreticalcurve of

indicatorcovariance,

assuming datacomes from

bivariate normaldistribution(green line)



Checking forbivariatedistribution

Checking for bivariatedistribution

1. Click the Add Data button onthe ArcMap toolbar and addthe layer you wish to checkfor bivariate distribution.

2. Start the GeostatisticalWizard.

3. Click either Kriging orCokriging.

4. Click Next on the ChooseInput Data and Methoddialog box.

5. Click the Simple or Disjunc-tive Kriging/Cokrigingmethod to use.

6. Check Examine BivariateDistribution and selectNormal Score Transformation.

7. Choose the Dataset combi-nation you wish to use(cokriging only).

8. Type the number of quantilesto check.

9. Click Next.

10. Click the ApproximationMethod in the dropdownmenu, set the parameters,and click either ProbabilityDensity or CumulativeDistribution radio buttons onthe Normal Score Transfor-

Tip

Methods that enable thebivariate distribution to bechecked




12. Explore the bivariatedistribution on the ExamineBivariate Distribution dialogbox. Click Next.



Implementing declustering to adjust for preferential sampling

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The Declustering dialog box

Griddisplayingcell size

Ratio of cell“height” to

“width”

Set cellsize

Color legend correspondingto the size of the polygons

Plot of weightedmean versus

grid size

Orientation of the grid



Declustering toadjust forpreferentialsampling

Performing celldeclustering

1. Click Kriging or Cokriging onthe Choose Input Data andMethod dialog box.

2. Click either Probability,Disjunctive, or SimpleKriging or Cokriging meth-ods.

3. Click the Normal Scoretransformation in the Trans-formation dropdown menu.


5. Click Next.

6. Click the Dataset Selectiondropdown arrow and clickthe dataset you wish todisplay (cokriging only).

7. Specify the desired param-eters.

8. Click the tabs to switchbetween the Cell Size,Anisotropy, and Angle charts.

9. Change cell size, anisotropy,shift, and angle to find theextremum in the graph.

10. Alternatively, click theDeclustering Methoddropdown arrow and clickPolgonal to switch to apolygon declusteringdisplay.

11. Click Next.

Tip

Using declustering



12. Click the ApproximationMethod in the dropdownmenu, set the parameters,and click either ProbabilityDensity or Cumulativedistribution radio buttons onthe Normal Score Transfor-mation dialog box. ClickNext.







Removing trends from the data

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The Detrending dialog box (Standard option)

The datasetbeing detrended

Controls for thetrend display

The neighborhoodsearch size

The power ofthe polynomial

Map legend

SearchingNeighborhood

settings

The estimatedtrend display



Removing globaland local trendsfrom the data:detrending. ( @#, -!

1# A #" 1 #

Using the slider toestimate the trend

1. On the Geostatistical MethodSelection dialog box, clickeither Ordinary Kriging,Universal Kriging, or Disjunc-tive Kriging and the desiredoutput surface type in theGeostatistical Method treeview.

2. Click the dropdown arrow onthe Order of Trend Removaland click the order for thetrend.

3. Click Next.

Note: If the order of trend isanything but None, then theDetrending dialog box willfollow when clicking Next.

4. On the Detrending dialogbox, by moving the sliderbetween the two extremes,you define the window sizefor fitting the polynomial fromGlobal to Local.

Alternatively, to set theneighborhood parametersclick Advanced Options>>.

5. Click Next.

Tip

The order of your trendsH9 +


IN THIS CHAPTER

219

Displaying and managing geostatistical layers 8• What is a geostatistical layer?

• Adding layers

• Working with layers in a map

• Managing layers

• Viewing geostatistical layers inArcCatalog

• Representing a geostatistical layer

• Changing the symbology of ageostatistical layer

• Data classification

• Setting the scales at which ageostatistical layer will bedisplayed

• Predicting values for locationsoutside the area of interest

• Saving and exporting geostatisti-cal layers

ArcMap and Geostatistical Analyst provide a wide variety of tools withwhich to display and manage your data. With the display tools, you cancreate fine cartographic output and explore or analyze your data to gaingreater insights in order to make more effective decisions. Exploration isparticularly important when using Geostatistical Analyst because throughthese insights you are able to build better models and create more accuratesurfaces.

Even though the management tools may not directly aid in the creation ofsurfaces with Geostatistical Analyst, these tools are indispensable fororganizing and ordering the map session as well as your own thinking.

Many of the display and management tools that are applicable to anyArcMap layer are also applicable to a geostatistical layer. In this chapter, wewill only discuss the general layer tools that are most often used whenworking with Geostatistical Analyst and the tools that are specific to ageostatistical layer. Please refer to Using ArcMap for additional tools thatcan be applied to all layer types, including geostatistical layers.

ch08_Display.pmd 11/25/2003, 3:04 PM219


! "

Filled contours

What is a geostatistical layer?

# $ % $ & ' %# ( ) )

$ " $ * $ $ + , - .

Display.p65 3/21/01, 8:21 AM220

DISPLAYING AND MANAGING GEOSTATISTICAL LAYERS 221

Contours

Grid

Hillshade

Combination of contours, filled contours, and hillshade

Display.p65 03/07/2001, 3:40 PM221


Adding layers'6 # # B # #

#"# B B ## 6 #

Adding layers

1. Click the Add Data button onthe ArcMap Standard toolbar.

2. Click the Look in dropdownarrow and navigate to thefolder that contains the layer.

3. Click the layer.

4. Click Add.

Adding a group layer

1. Right-click on Layers in theArcMap table of contents andclick New Group Layer.

2. Right-click the resulting NewGroup Layer in the table ofcontents and click Properties.

3. Click the General tab.

4. Optionally, name the grouplayer.

5. Optionally, check the Visiblecheck box to make the GroupLayer visible.

6. Optionally, set the ScaleRange.

7. Click the Group tab.

8. Click Add and navigate to adataset you wish to add.

9. Click Add.

10. Continue adding the desireddatasets to the group byrepeating steps 8 and 9.

11. Click OK.

Tip

Revealing hidden layersE ;=+9))2E

Display.p65 03/07/2001, 3:41 PM222


Working withlayers in a map

! "

# " "

Turning the display of alayer on or off

1. Check the check box to theleft of the layer name to turnthe layer on or off.

Zooming and panning alayer

1. Click View in the ArcMapmenu bar, click Toolbars, andcheck Tools.

2. Various tools, accessedthrough icons on the Toolstoolbar, can be used toexplore the map.

Moving a layer to changeits drawing order

1. In the table of contents, clickand drag the layer up ordown to the desired position.

A black line indicates wherethe layer will be placed.

2. Release the mouse pointer todrop the layer into the newposition.

See Also

Display.p65 3/21/01, 8:22 AM223


Managing layers6 # B ## #

"###"#"#A5CD!"" " "

( "" "

0 ""

Changing the name of alayer

1. Click the layer in the table ofcontents to select it.

2. Click again on the name.

This will highlight the nameand enable it to be changed.

3. Type the new name.

Copying a layer

1. Right-click the layer you wantto copy and click Copy.

2. Right-click the Layers dataframe.

3. Click Paste Layer(s).

Removing a layer

1. Right-click the layer you wantto remove and click Remove.

Display.p65 03/07/2001, 3:41 PM224


Viewinggeostatisticallayers inArcCatalogB ""((""4"

" # ""(

Starting ArcCatalog andenabling GeostatisticalAnalyst

1. Click the Start button on theWindows taskbar.

2. Point to Programs.

3. Point to ArcGIS.

4. Point to ArcCatalog.

The ArcCatalog windowappears. Now, click the Toolsmenu, click Extensions, andcheck Geostatistical Analyst.Click Close.

Tip

Metadata'+))

Tip

Accessing ArcCatalog' '( )'

Previewing data

1. Start ArcCatalog.

2. Navigate to the desiredgeostatistical layer in thetable of contents.

3. Click the Preview tab.

Display.p65 03/07/2001, 3:41 PM225


Viewing layer metadata

1. Start ArcCatalog.

2. Navigate to the desiredgeostatistical layer in thetable of contents.

3. Click the Metadata tab.

4. Click the Description tab toretrieve a general descriptionof the layer.

5. Click the Spatial tab toexplore the spatial character-istics of the layer such as itsbounding coordinates.

6. Click the Attributes tab toexamine other informationabout the layer.

7. Input or change any meta-data information.

See Also

Display.p65 03/07/2001, 3:41 PM226


Representing ageostatisticallayer "

2 '

B ' " !

( 4

E

(

Displaying ageostatistical layer asfilled contours

1. Right-click the desiredgeostatistical layer in theArcMap table of contents andclick Properties.

2. Click the Symbology tab.

3. Click Filled Contours in theShow list and check theaccompanying check box.

4. Set the desired parameters.

5. Click OK.

Displaying ageostatistical layer as agrid

1. Right-click the desiredgeostatistical layer in theArcMap table of contents andclick Properties.


3. Click Grid in the Show listand check the accompanyingcheck box.


5. Click OK.

Tip

Viewing multiple datasets++

Display.p65 03/07/2001, 3:41 PM227


Displaying thegeostatistical layer as ahillshade

1. Click the desired geostatisti-cal layer in the ArcMap tableof contents and click Proper-ties.


3. Click Hillshade in the Showlist and check the accompa-nying check box.


5. Click OK.

Tip

Shortcut to the layerproperties 9) '()E0 $

Tip

The Hillshade Z factor,2>+$,>++$+2

Tip

Viewing combinations ofrepresentations togetherI+ )+( : E

Displaying thegeostatistical layer ascontours

1. Click the desired geostatisti-cal layer in the ArcMap tableof contents and click Proper-ties.


3. Click Contours in the Showlist and check the accompa-nying check box.


5. Click OK.

Display.p65 03/07/2001, 3:41 PM228


Changing thesymbology of ageostatisticallayer6 . # #" $" "

6 E" #) #

Changing the colorscheme

1. Right-click the geostatisticallayer and click Properties.


3. Click on either Contours,Grid, or Filled Contours forthe layer from the Show list.

4. Click the Color Rampdropdown arrow and click acolor scheme.

5. Click OK.

Tip

Zooming'> >, C9))C

Changing the colorinteractively

1. Right-click a symbol from thelegend of a geostatisticallayer.

2. Click a color for the symbol.

All values in the displayrepresented with this symbolwill be displayed in thechosen color.

Display.p65 03/07/2001, 3:41 PM229


Data classification

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Why set your class ranges manually?

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Display.p65 03/07/2001, 3:41 PM230


Equal interval

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Display.p65 03/07/2001, 3:41 PM231


Smart quantiles

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Display.p65 03/07/2001, 3:41 PM232


Classifying data+#

'#

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Setting a predefinedclassification method

1. In the table of contents, right-click the geostatistical layeryou want to classify and clickProperties.


3. Click Classify.

4. Click the Method dropdownarrow and click a classifica-tion method.

5. Click the up/down arrows onthe Classes input box to setthe desired number ofclasses.

6. Click OK on the Classificationdialog box.

7. Click OK on the LayerProperties dialog box.

Display.p65 03/07/2001, 3:41 PM233


Manually altering theclass breaks

1. In the table of contents, right-click the geostatistical layeryou want to classify manuallyand click Properties.


3. Click Classify.

4. Click the Method dropdownarrow and click Manual.

5. Click the up/down arrow ofthe Classes input box untilthe desired number ofclasses is reached.

6. Click and drag the classbreaks to the desired posi-tion.

7. Alternatively, type in specificclass breaks.

8. Alternatively, check theCustom Min & Max checkbox, then type specificminimum and maximumvalues to include in theclassification.

9. Click OK on the Classificationdialog box.

10. Click OK on the Symbologytab.

Display.p65 03/07/2001, 3:41 PM234


Setting the scalesat which ageostatisticallayer will bedisplayed6 " $3# !

$ # ' " 2 #" " " "

" "=

Setting the scale range


2. Click the General tab.

3. Click Don’t show layer whenzoomed.

4. Set the scale range byinputting the out beyond andin beyond entries.

5. Click OK.

Display.p65 03/07/2001, 3:41 PM235


Predicting valuesfor locationsoutside the areaof interest1#B " E" # "

? ' #!#

Extrapolating values

1. Right-click the geostatisticallayer that you want toextrapolate values for in theArcMap table of contents andclick Properties.

2. Click the Extent tab.

3. Click a custom extent enteredbelow in the Set the extent todropdown list.

4. Type the new values into theVisible Extent.

Alternatively, use the extentof any other available layer.

5. Click OK.

Tip

$+ $):$

Display.p65 03/07/2001, 3:41 PM236


Saving andexportinggeostatisticallayers

! " #

Saving a mapcomposition

1. Click the File menu and clickSave As.

2. Navigate to the directory inwhich you want to save themap.

3. Change the map name ifdesired.

4. Click Save.

Tip

Distributing ageostatistical layer

Saving individual layers

1. Right-click the geostatisticallayer you wish to save in theArcMap table of contents andclick Save As Layer File.

2. Navigate to the directory inwhich you want to save thelayer.

3. Change the layer name ifdesired.

4. Click Save.

Display.p65 06/08/2001, 2:33 PM237


Exporting a geostatisticallayer to a raster

1. Right-click the geostatisticallayer in the ArcMap table ofcontents, click Data, and clickExport to Raster.

2. Set the desired properties forthe resulting raster such asthe number of rows andcolumns, cell size, number ofpredictions in each cell(Block Interpolations), andthe name and location for theraster.

3. Click OK.

Exporting a geostatisticallayer to a vector format

1. Right-click the geostatisticallayer in the ArcMap table ofcontents, click Data, and clickExport to Vector.

2. Specify the output format(e.g., shapefile, personalgeodatabase, or SDEdatabase).

3. Select Contours or FilledContours from the Exportdropdown menu.

4. Click OK.

Display.p65 3/21/01, 8:22 AM238

IN THIS CHAPTER

239

Additional geostatistical analysis tools 9• Changing the parameters of a

geostatistical layer: methodproperties

• Predicting values for specifiedlocations

• Performing validation on ageostatistical layer created from asubset

• Stratifying your data for betterpredictions

There are many supporting tools in ArcMap and Geostatistical Analyst toassist in geostatistical analysis. You can change the parameters in a model,retrieve predictions for specific locations, perform validation from subsets,spatially divide your data, interpolate each division, and then combine theresults. This chapter is not an exhaustive list of tools that will help withgeostatistical analysis. Instead, it discusses some of the more commonlyused tools that will aid in your analysis. But you should be aware thatbecause Geostatistical Analyst is integrated into ArcMap, there arecountless functions that you can and will use in your analysis. The morefamiliar you become with ArcMap and the supporting Geostatistical Analystextensions, the more tools you will find that meet your specific need tocreate even more accurate surfaces.

ch09_AdditionalTools.pmd 11/25/2003, 3:05 PM239


Changing theparameters of ageostatisticallayer: methodproperties " B# "6 1 # " " " 0 " # " # "$ "

Using method properties

1. Right-click the geostatisticallayer in the ArcMap table ofcontents and click MethodProperties.

The options availabledepend on the techniqueused to create the originalsurface. Follow these stepswhen the prediction surfaceis created by a krigingmethod.

2. Click on the desired krigingmethod.

3. Click Next on the Geostatisti-cal Method Selection dialogbox.

4. Optionally, change any of themodel parameters and clickNext on the Semivariance/Covariance Modeling dialogbox.

5. Optionally, define a newsearch neighborhood andclick Next on the SearchingNeighborhood dialog.

6. Assess the cross-validationresults. Has the outputimproved? If not, repeat steps2 to 5. If yes, Click Finish onthe Cross Validation dialogbox.

7. Click OK on the Output LayerInformation dialog box.

Tip

Understanding methodproperties(0 $ 9++

AdditionalTools.p65 03/08/2001, 3:49 PM240

ADDITIONAL GEOSTATISTICAL ANALYSIS TOOLS 241

Predicting valuesfor specifiedlocations 2 # "

' " # " E" # " # =

Using map tips

1. Create a geostatistical layerusing any of the methodsdiscussed in Chapters 5 or 6.


3. Click the Display tab.

4. Check Show MapTips.

5. Click OK.

6. Place the cursor over a pointof interest on the layer.

The value at that location isdisplayed.

Tip

Selecting points using theattribute table, 39))7' 4



Predicting at specificlocations from a pointlayer

1. Right-click in the ArcMaptable of contents thegeostatistical layer that wascreated using observed dataand click Prediction.

2. Click the Input Datadropdown arrow and clickthe layer containing thelocations for which you wantto obtain predictions.

3. Identify the directory to storethe output dataset in byclicking the browse buttonnext to Specify outputshapefile or feature class.

4. Browse or type the directoryand name for the output file.

5. Click Save and OK.

6. Add the prediction file to theArcMap table of contentswhen prompted.

7. Right-click the predictionlayer and click Open AttributeTable to display the results.

The predicted values at thespecified locations will bedisplayed in the table.

Tip

Other ways to view theresults

AdditionalTools.p65 3/21/01, 8:23 AM242


Performingvalidation on ageostatisticallayer createdfrom a subset " 4 " 2##!# A " A # # # #

Creating subsets

1. Click the GeostatisticalAnalyst toolbar and clickCreate Subsets.

2. Click the Input Layerdropdown arrow and clickthe layer to be divided.

3. Click Next.

4. Click and drag the slider barto an appropriate location toselect the relative percent-ages of training and testdata.

By default, the output datasetis named according to thefollowing convention: “inputfiledataset”+“_sets.mdb”, forexample,inputpoints_sets.mdb, whereinputpoints is the name of theinput dataset containing thepoints.

5. Click Finish.

The training and testdatasets form two tables inthe personal geodatabase.

Tip

Dividing the dataset+ + + @)+9 +



Performing validationusing subsets

1. Click the Add Data icon onthe Standard toolbar inArcMap. Navigate to thefolder in which the subsetdata was saved (if it is not inthe ArcMap table of con-tents).

2. Click both the training andtest layers (click one layer,then hold down the Shift keyand click the other).

3. Click Add.


5. Click the Input Datadropdown menu and click thetraining dataset.

6. Click an appropriate method.

7. Click Next and follow thedialog boxes to create asurface.

8. Right-click the newly createdgeostatistical layer and clickValidation....

9. Click the test dataset in theInput Data dropdown menu.

10. Click the same attribute inthe Attribute dropdown menuthat the surface was createdin.

11. Type a name and locationfor saving the output(validation) dataset.

12. Click OK.

Tip

Performing validation fromthe Geostatistical Wizard $B->,( $7+ > $+ $

Tip

Viewing the summarystatistics+$ +'('9

See Also

!9++



13. When queried, add thevalidation layer to theArcMap table of contents.

14. Right-click the new layerand click Open AttributeTable.

The results of the validationare displayed for assess-ment.

Tip

The resulting output files+.+ ++ $9+3& !4



IN THIS APPENDIX

247

Appendix A• Deterministic methods

• Inverse distance weightedinterpolation

• Global polynomial interpolation

• Local polynomial interpolation

• Radial basis function interpola-tion

• Geostatistical methods

• Declustering dialog box

• Distribution modeling dialog box

• Semivariogram/Covariancedialog box

• Bivariate distribution dialog box

• Kriging formulas

• Cross Validation dialog box

There are many formulas and unique implementation concepts underlyingGeostatistical Analyst. The formulas and technical concepts are presentedhere in this appendix. It is assumed that you have some training inmathematics or geostatistics before reading this appendix.

Following the appendix are references from commonly used textbooks andjournal articles for the mathematical details of the methods used inGeostatistical Analyst. In some cases, when the details are not easily foundin textbooks, we give greater detail.

AppendixA.pmd 11/25/2003, 3:17 PM247


Deterministic methods

Inverse distance weighted interpolation

Global polynomial interpolation

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AppendixA.p65 3/21/01, 8:24 AM248

APPENDIX A 249

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AppendixA.p65 3/21/01, 8:25 AM249


Geometric anisotropy

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AppendixA.p65 03/08/2001, 2:38 PM250

APPENDIX A 251

Declustering dialog box

Cell declustering

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Geostatistical methods

Z(s(1)) Z(s(2)) Z(s(3)) Z(s(4))

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Value

1/4

1/2

3/4

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NS /*2/1

Distribution modeling dialog box

Normal-score transformation

7 < * * @G5 * * @7<R 49*;%( #<@ #><$ ;%(#(<@(><';%((<@#%><6 ;%((<@P><!;%((#<@>< ' 1;%(((<@P#->@45< *5 *5 @$<@3 *;

><;

><X

; ><5;

>5 ;

>

@;O>E *55&

* 5@"5: *;%0$ @!" @

AppendixA.p65 03/08/2001, 2:38 PM251


∑=

=K

iiiii zpzp

1

2),;()( σµα

11

=∑=

K

iiα

( )

−−=2

22

2exp

2

1),;(

i

i

iiii

zzp

σµ

σπσµ

∫∞−

=z

dxxpzP )()(

Z(s(1)) Z(s(2)) Z(s(3)) Z(s(4)) 0-3 30

1

0

1

Original data Normal scores

Empirical cumulativedistribution

Normal-scoretransformation

Cumulativedistribution forstandard normal

Z(s(1)) Z(s(2)) Z(s(3)) Z(s(4)) 0-3 30

1

0

1

Original data Normal scores

Piece-wise linear interpolationsof empirical cumulativedistribution

Normal-scoretransformation

Cumulativedistribution forstandard normal

Direct

7 *<5HI@4 *5 @

Linear

:5 *@$ <

Gaussian mixture

"! ** ** @. **

5

<

<

* <

* @ * <

5-;!> * <95 @

AppendixA.p65 03/08/2001, 2:38 PM252

APPENDIX A 253

Semivariogram/Covariance dialog box

Variogram definitions

1 ;@@<R 49*<%( #<@%E'<%((<@#E!<%(( <@#E"<%((#<@%(E' 1<%(((<@%E<%(((<@(>@F ;> ;>@4<;> ;>< @

Covariance definitions

1;@@<R 49*<%( #<@%E$ <%(#(<@%E'<%((<@E!<%(( <@#E"<%((#<@%(E' 1<%(((<@-E<%(((<@%>@

Estimating the variogram

8* @;@@<R 49*<%( #<@%(E$ <%(#(<@-E'<%((<@(E!<%(( <@#E"<%((#<@ E' 1<%(((<@E<%(((<@(>@

Estimating the covariance

8* @;@@<R 49*<%( #<@%(E$ <%(#(<@(E'<%((< -E!<%(( <@#E' 1<%(((<@%E<%(((<@(>@

Binning the variogram and covariance estimates intolag classes

* * O;

<

>

< 5*@<* E55 @

AppendixA.p65 03/08/2001, 2:38 PM253


h y-direction

h x-direction

Lag bins

Lag vector

7< 5< !3$/;1 R<%((<@><;>< ";6>@$ <!" E55 (

h x-direction

h y-direction

Lag bins

Lag vector

45< <* **@$ * <!" 5 < 5@5 * 5<5@" @F5*5< 5@

AppendixA.p65 03/08/2001, 2:38 PM254

APPENDIX A 255

4<5 5**5 @*<5@ *@

Semivariogram and covariance models

Relationship between semivariograms and covariance

$5<5* @$< <*5 @7<

*;E>O;E>=;E><∞

∞0

1

0 1

Kernelfunctionprofile

h x-direction

h y-direction

;E>O*;&E>=*;E><

5;E> *;&E>@ < !"@

Geometric anisotropy

!*

;E'>O;[[[[E><

5 ;E> *5@4<O;

>

5 5@$!"<9< @9 58 ( @7 <R 49*;%( #<@% ><$ ;%(#(<@ ><';%((<@><!;%(( <@(-><";%((#<@#><' 1;%(((<(>@

Linear combinations of models

4* * *<

;E>O;E

>M

;E

>MX

!"5 * : @

AppendixA.p65 03/08/2001, 2:38 PM255


Nugget effect

5-@

Circular

<

5-

-

@

Spherical

5-

-

@

≠=

=0for

0for0);(

h

hh

sθγ

<

≤≤

+

−

=

h

hhhh

h

rs

rrrr

s

θθ

θθθθπ

θγ

for

0forarcsin12

);(

2

<

≤≤

−

=

h

hhh

h

rs

rrr

s

θθ

θθθ

θγ

for

0for2

1

2

3);(

3

Tetraspherical

5-

-

@

Pentaspherical

5-

-

@

<

≤≤

+

−

=

h

hhhh

h

rs

rrrr

s

θθ

θθθθ

θγ

for

0for8

3

4

5

8

15);(

53

( )

−+

−+

=

s

rrrrr

s

θ

θθθθθπθ

γ

2

322

13

21arcsin

2

;

hhhhh

h

<≤≤h

h

r

r

θθ

for

0for

AppendixA.p65 03/08/2001, 2:38 PM256

APPENDIX A 257

5-

-

@

Hole effect

5-@

K-Bessel

5-<

-<

-<

;>O-@(

<;

><

/

;"*5) <%(<@ >@

Exponential

5-

-

@

Gaussian

5-

-

@/ **5<* !" <8%Q%--- @

Rational quadratic

kθΩ

∫∞ − −=Γ

0

1 )exp()( dxxxy y

)(•k

Kθ

−−=

rs θ

θγh

h3

exp1);( '

2

2

191

19

);(

+

=

r

rs

θ

θθγ

h

h

h '

( )( )

≠−

== 0h

h

h0h

h for/2sin

/2sin1for0

);(

r

rs θπ

θπθγ

−−=

2

3exp1);(r

s θθγ

hh '

( ) ( )

Ω

ΓΩ

−= − r

k

r

s kkk

k

k K θθθ

θγ θθθ

θθ /

)(2

/1);(

1h

hh '

AppendixA.p65 03/08/2001, 2:38 PM257


J-Bessel

5-<

-<

-<<

;><

R:/;"*5) <%(<@#>@

Stable

5- -

@/ *

*5<* !" <8%Q%--- @

dθΩ

∫∞ − −=Γ

0

1 )exp()( dxxxy y

)(•d

Jθ

( ) ( )

Ω

Ω+Γ

−= r

r

ds ddd

d

d

J θθ

θθγ θθθθ

θ

//

)1(21);( h

hh

( ) ( ) rBBB

Bs

θγθγ

=<′=> 0,,0

min

−−=

e

rs

θ

θθγ

hh 3exp1);( '

Crosscovariance models

' !"H)I <5 *@' !"@ ;<%(>* ;R 49*<%( #<@E'<%((<@ E?4 '<%((>< 5@'5 ;R 49*<%( #<@%E?4 '<%((>< 5 @

') ;<* > * R 49*;%( #<@-><$ ;%(#(<@(-><!<;%(( <@%- >< ' 1;%(((<@(>@!" *5*55*;?4 '<%((>@ 5 *: : @

AppendixA.p65 03/08/2001, 2:38 PM258

APPENDIX A 259

! "#

# $ !

%&$ $

$"' " &

("

) * $ +!

) * $ & &#& * $ $ ) &" &'#& ,"'- .'

Fitting semivariogram and covariance models

)& """* & & / &$ "**

Stage 1

& 0

* $1""

* ** $ $ & 23$ $ $ &* ) & & $ 0' & ) &," 4"$ * $ * $ $ "* $ "* )& &* " "#

& $ &* & $

& $ *

" &* *

$ && $ "

)( ikjZ s

)(~

ikjZ s ki

kj sZ /)(s

)(~

ikjZ s

);(

)();(

)1(2)1(

iiii

kijiikij

N

h

hh

γϖ =

( ) ( )( )∑∑∑= = =

−T

i

T

j

n

kkijijkijkij

ij

CCw1 1 1

2ˆ;~

)( hhh

∑=

=ijn

mmijkijkij NNw

1

)(/)()( hhh '

∑=

=ijn

mijmijijkijkijw

1

)1()1( );(/);()( hhh ϖϖ (

( ) ( )( )

−∑

=

n

kkiiiikiikii

ij

ii

w1

2ˆ;~)(minarg hhh γγ

θ( )kij hC

);(~

);(~

);(~

)();(

)1(2)1()1(

)1(

ijkijjjjjiiii

kijijkij

CCC

N

h00

hh

+=ϖ

)[ 2/12/1 , +− kk dd

)2(ii

)2(ii

5

)logcosh(21 kk dd

AppendixA.p65 3/21/01, 8:25 AM259


Stage 2

1' *"* $ $ & 23$ $ &#" & & 1& * "&* $ $ &&&

6 * * *&

*$+ & " &

&$ "" & && $ " &

7

# $ 77

0

* * . *&$ "& &* ) &* " "#'* $ " "#& ( 5* $ )& & "$& $ & %", )& $

0

& $

0

& * # "'

TT ×

)(~

ikjZ s

);(~ )4(

ijijC h

);(~ )4(2iiiiis hγ

( ) ( ) ( ).;,;~

1∑

=

=S

uuuuij jiBC hh ρ

Bivariate distribution dialog box

89$, ! * & $"$ ")* &&* )$$*"$ * $&&" $* " $ $& $ "&8 : '*5 $ *';) $"* * $ " < &==> " ** $"$ " "&$ "& ),& "$ "" & *, ! * ""* " "*

AppendixA.p65 3/21/01, 8:25 AM260

APPENDIX A 261

Kriging formulas

& , "* ? $ !$" " * 3@/> * $ , "& A 5? $(A ( " ;' 6 ;51** $ " ;5 < ;';1: 5& , 1$(& * "", < B;& 9$, C$ &,

)* ) " ;C;5 ;5'BD5) $", & " EF* 3&* " * $ * # ) +,*" "E9*F* $ $ " * & E& F * * *< *"& * $ , " 3" ,"* * * & **&$ ", A &$ "**

)( ikjZ s

Estimating measurement error

& *" $* )& & *

& $ $" &" $0& " &*) * * $7CC* / $ & && " 7$ $ $ !* $

2ˆMEσ

D

iiD

n

jj

ME nN

ZZi

i

−

−=

∑∑∈ =

2

12

))()((

ˆ

sssσ

2ˆMEσ2ˆMEσ

2ˆMEσ2ˆMEσ

AppendixA.p65 3/21/01, 8:25 AM261


)<

;;>=()><

5)* < 5@"* <

;;>=()>O-<

(*+%<5 35)@<*8<

Ordinary kriging

';%((<@% =%> 5E< !" @"'< ) &

;>O;>M

;><

*5 <

;>O,;>M ;>M

;><

5;> )

<

*

*@GO%< S%< @5 &

W ;>O5< @

W ,;> : 5 *5@

W ;,;>>O-@

W ';,;><,;M>>O*;><

D;>@

W ;> : 5- *5 @

W ; ;>>O-@

W '; ;>< ;M>>O*;>5*

;\>O-@

W ;>5:

@

=

′ 1

c

01

1z

m

S ;>O

W ;;>>O-<

@

W ';;><

;M>>OO&

O<5-@

W ,;,>< ;,>< ;,> @

"<5 ) 5&@<)O*;&>M

@7 < <

$<55 ;>8;

>M,;

>M ;

>

E<

@$<;>O;>@G 5* <

==++≠=+

≠+=+

utCC

utCC

CC

ZZCov

y

y

y

ut

andif)()(

andif)()(

if)()(

))(),((2 0h00

0h00

0hhh

hss

ση

η

η

@

AppendixA.p65 03/08/2001, 2:39 PM262

APPENDIX A 263

53<

< ';)<;>>O';)<,;

>M ;

>>@

" ;,>-<';)< ;

>>O& <5

O<5

* E

';;>< ;

>>O';&><5 * @

* <* 5<)OM';&>@$<';&>O)=@8 E-%< O) ';&>O;%=>)* @$*<* 5@

G ';&> < 58@$ ;,>;@@<><8 @*<

+;:*>5O;*(

:%>Q*(

*<

@*<:8: <

;;>=()>

O*;&>M';&>-(:

O*;&>M;%=>)-(:

O

Predicting a new value for cross-validation andvalidation

1: < 5 ;><

<* ;><5

< ::8 : @+ H5I*

;>O

)<

;;>=()>@

"<<O

1<S

@+

*<8* <

53<

';)<;>>O';)<,;

>M ;

>

Mu;>>@<5*<

+;:*><5O;*(

-*>Q*(

*@

F5O1< ;

>

8* !;>E@45<

*:8 <

;;>=()>

O*;&>M';&>M

-(:<

O*;&>M)-(:

O@

=

′ 1

c

01

1z

m

Z

)(ˆ 0sSσ

uZ

’)1()(Cy mv −−−+ c0 π

<

)(ˆ 0sZσ ’)(Cy mv −−+ c0

AppendixA.p65 03/08/2001, 2:39 PM263


* @F5

O

* 1< 5*-@

Probability and quantile maps

$ * 5; : >< ;

>:

;> *5)

@F5**<8<8@

)(ˆ 0sSσ

S

)(ˆ 02 ssσ

AppendixA.p65 03/08/2001, 2:39 PM264

APPENDIX A 265

Simple kriging

4 ! / " ! $! 5

()

! (), (

)(

)6(

)6 (

)

Prediction with measurement error

!

!)&()6

%

2(()7()7)&89(

)6 (

)7():69(

)7(7:

! % &(

)7(

&

;()()6 (

)) !

!)&()6&(

()0)6(

)

,

2(()7()7)

&(&)6;(&)7(

&(&)6(<7)7(

Predicting a new value for cross-validation

=0 > 0, 0 ?

()&()6

%

2( ()7()7)

& @?

!

()()6&(

()0)6(

)

0,

2( ()7()7)

&(&)6;(&)6

(

&(&)66(

&

*

&

A

&

)(ˆ 0sZσ )(Cy c0 −+ v

)1()(Cy c0 −−+ vπ)(ˆ 0sSσ

Sˆ

S

Z

Z

AppendixA.p65 05/11/2001, 12:55 PM265



$ 9 *<5

+

z

y CCN

S

c’

c00s

z

s )()(,

)(~

)( 00 ηµ

S

S

;> <2;;

>[)><

5:5 * *;

><

;>[)>]%;(

;):>M;

><*

!&"M*!&":(

>@

/ *<5**<8<8@" 2;;

>[)>* <

:8 < <@

AppendixA.p65 03/08/2001, 2:39 PM266

APPENDIX A 267

=

)( 0sx

c

m0X’

Xz

53<

< ';)<;>>@<

* <

+;:.><5O;.(

.";.(

/!

">@

*8 <

;;>=()>

O*;&>M';&>-(;#.><

O*;&>M;%=>)-(;#.><

)(ˆ 0sSσ O@

Predicting a new value for cross-validation

1: < ;><

E< ;>5

::8 : @+ H5I* <

Z ;>()<

)<

;;>=()>@

"<<O1<S@+ *<8* <

@

)(’)1()(Cy Xmc0 −−−+ vπ

<* <

+;:.>5O;.(

.";.(

/;

>>@

Universal kriging

"5 <

;>OK/;>L(M;><

*5 <

;>O,;>M ;>M

;><

5. < 5 <5<

;>OK/;>L(<5/;>* +@

Prediction with measurement error

" < ;>8;

>K/;

>L(M,;

>M ;

>

@.

< @,5* <

;>O()<

)<

;;>=()><

5)* < 5@"* <

;;>=()>O-<

.(O/;>@+ 5 <

8* <

S

=

)( 0sx

c

m0X’

Xz

AppendixA.p65 03/08/2001, 2:40 PM267


F5O1<

;>

8* !;>E@*

:8 <

;;>=()>O

O*;&>M*;&>M

-(;M.><

O*;&>M)-(;M.><

<

5 * @F5

O

* 1<

5*-@


$ * 5; : >< ;

>

!> *5)

@F5**<8<8@

Z

)(ˆ 0sSσ

S

)(’)(C y Xmc0 −−+ v

AppendixA.p65 03/08/2001, 2:40 PM268

APPENDIX A 269

Lognormal linear kriging

!!!! ;(<BBC)$ 5

1D>, CEFA<C';(<BBC)

D>, <CG;(<BBC)

HD>;(<BBC), CEFA%

I !, /(

)

;(<BBC), CEF<5

1D0

D0!

HD0

0&(1) 0

Transgaussian kriging

#7; !!! ;(<BBC<C+)

Indicator kriging

! ( )! %

()&6()

(A<)

()&( ()@)

() < A % ( )

()&6()

!!( A ) () J()

()&( ()@)

()&6()

*! ()

K(<BLC)! (<BLB);(<BBCEL<)/ (<BB+EBC);=(<BBBCL<)

YX’X)(X’sxs 1Y

11Y

−−−= )]’([)(ˆ 00Yµ

)/(ˆ 11’Y1’ 1Y

1Y

−−=Yµ

M)]([’2/)(ˆ);(ˆexp);(ˆ 002

00 sxmssZsZ YYYz pp −+= σ

AppendixA.p65 03/08/2001, 2:49 PM269


Probability kriging

!"#$%&'()$$*&))*$ +)$,&)

(-.(

#() $$

(-(/(

*$(&$&#$*+)$,&*$ &&(

(-.(

0$ ( 1(2$)

$ 1(2$$ 3$ &&$

Disjunctive kriging

45 6$"7($&$&)

*$(()&&$(0$

!+$&*

,))(()(ˆ0 ∑

∈=

Ds ZgZ

s

ss

),)((ˆ)(ˆ0

000 ∑>

+=k

kk YHffZ ss

,))(())((ˆ1

0 ∑=

=n

iikkik YHYH ss λ

*$

&&

((8))

(

($)0$

(&)5 ) (*$9)$)&)0$$&5 )$!+&*$)$:(

AppendixA.p65 3/21/01, 8:23 AM270

APPENDIX A 271

Cokriging

*$$0$9&)& ;85 " (4##((!" ($%& (< ) *)))5$ ) 5 9))&(

Ordinary, simple, and universal cokriging

0$ )$) )

-

*$ )9

&)&

$$*$$&* )=

> ()$?*$ ) $$$*

> @ ((-#

> ( .((-*$-#$ &&$ ((

> ()$?*$ ) #$$$*

> @((-#

> (.((-*$3*$-#

> (.((-#*$3

>*$)&))

> @(-#&

> (-&-$*#

> .((-#&3

(((&$$&+)$$$ &&

)&*

=)))$-

(.

+$$5&)$

&(($#A)$)$

A)*$ ,&)&B$&(3

(C

(D

.

(.

(

$!+&$

-

$)),

@(?CD(

B *$ $&* 3

)(skjZ

===++≠==+

≠=+≠

=+

tjmkCC

tjmkCC

mkCC

mkC

ZZCov

kkkkk

y

kkkky

kkkky

kmy

mt

kj

andandif)()(

andandif)()(

andif)()(

if)(

))(),((

2 0h00

0h00

0hhh

h

hss

ση

η

η

)(ˆ0

1 sS 2211 zz ′+′

2211 zz ′+′

)(∞kmCη

)(skjδ

=

)( 01 sx

c

m0X’

Xz

)(skjδ

)(skjδ )( hs +k

jδ

)(skjδ

)(skjδ

)(hkmyC )(∞km

yC

)(hkmCη

AppendixA.p65 3/21/01, 8:23 AM271


= .

B *&?5*$)

)

$)9

) $

(( &

?(*$-

?

((

$)?3

@(E(

-.

-.E(.(

$

4&

10

01

$ & A)

)(11 0yC )(11 0ηC

)(11 0yC

)(ˆ01 sS

)(ˆ 01 sSσ )1()(C 1111y cc’0 z−−+ νπ

Indicator, probability, and disjunctive cokriging

45 )$$9&)))&(& *$$)$4 5 & (!" "($%&7(B &) * $ $ $ 3%5 ,&5 &!6 $%&(

Cross Validation dialog box

Cross-validation

?&) )$ F9$)$($**$* F$ $) )))$))A)&)?4'(#(!"#'(+) '($%&G( '(

)(’)1()(C 1111y Xmc0 +−−+ νπ)(ˆ 01 sSσ

-.-?(.

(

*$$

2221

1211

2

1

X0

0X

2

1

c

c

2

1

2

1

m

m-----

*$

AppendixA.p65 3/21/01, 8:24 AM272

APPENDIX A 273

Cross-validation summaries

* * : @3

* : <!;>*

* < *

@*

!"&

%@ <

@

@6::8 <

@

@" <

)(ˆiZ s

)(ˆ isσ

( )n

zZn

iii∑

=

−1

)()(ˆ ss

( )n

zZn

iii∑

=

−1

2)()(ˆ ss

n

n

ii∑

=1

)(ˆ sσ

@ ) <

@

@6::8 ) <

@

( )n

zZn

iiii∑

=

−1

)(ˆ/)()(ˆ sss σ

( )[ ]n

zZn

i

iii∑=

−1

2)(ˆ/)()(ˆ sss σ

@

AppendixA.p65 03/08/2001, 2:41 PM273

AppendixA.p65 03/08/2001, 2:41 PM274

IN THIS APPENDIX

275

Appendix BThis appendix provides an overview of the methods that are available withthe Geostatistical Analyst. The following chart describes thecharacteristics of each method, summarizes its advandages anddisadvantages, and shows the type of output it creates. By comparing thedifferences between these methods you can determine which ones youshould use in your application.

• A comparison of the GeostatisticalAnalyst methods

AppendixB.pmd 11/25/2003, 4:38 PM275


Method Deterministic/ Output Computing Exact Advantages Disadvantages Assumptions2

Stochastic Surface Time/ InterpolatorTypes Modeling

Time1

Inverse Deterministic Prediction Fast/Fast Yes Few parameter No assessment ofDistance decisions prediction errors;Weighted produces “bulls eyes”

around data locations None

Globalpolynomial Deterministic Prediction Fast/Fast No Few parameter No assessment of

decisions prediction errors;may be too smooth;edge points have largeinfluence None

Local Deterministic Prediction Moderatelypolynomial Fast/Moderate No More parameter No assessment of

decisions prediction errors;may be too automatic None

Radial Deterministic Prediction Moderately Yes Flexible and No assessment of Nonebasis Fast/Moderate automatic with prediction errors;functions some parameter may be too automatic

decisions

Kriging Stochastic Prediction; Moderately Yes without Very flexible; Need to make Data comes from aPrediction Fast/Slower measurement allows assessment many decisions on stationary stochasticStandard error; of spatial transformations, process, and someErrors; No with autocorrelation; trends, models, methods require thatProbability; measurement can obtain parameters, and the data comes from aQuantile error prediction standard neighborhoods normal distribution

errors; manyparameter decisions

Cokriging Stochastic Prediction; Moderate/ Yes without Very flexible; Need to make Data comes from aPrediction Slowest measurement can use information many decisions on stationary stochasticStandard error; in multiple datasets; transformations, process, and someErrors; No with allows assessment trends, models, methods require thatProbability; measurement of spatial cross- parameters, and the data comes fromQuantile error correlation; neighborhoods a normal distribution

many parameterdecisions

1. Computing time is computer-processing time to create a surface. Modeling time includes user-processing time to make decisions on model parameters and search neighborhoods.2. We assume that all methods are predicting a smooth surface from noisy data.

A comparison of the Geostatistical Analyst methods

AppendixB.p65 03/08/2001, 10:51 AM276

APPENDIX B 277

Descriptions

23=N,! ! ! ! 4 =N ,

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AppendixB.p65 03/08/2001, 10:51 AM277

AppendixB.p65 03/08/2001, 10:51 AM278

279

12Glossary anisotropy ( )

autocorrelation O ;

bandwidth O ()

bin #

cokriging ( ) ( ) ( );! !

correlation; 0< <

covariance ?

Glossary.p65 03/08/2001, 11:28 AM279


crosscorrelation

crosscovariance

cross-validation

crossvariogram !

deterministic "# #$"%&'( %')* &

%##&%&%#&

detrending %"&

directional influences+

dissimilarity,# #

empirical ! "$"

estimation

first-order polynomial%&-'

Glossary.p65 3/21/01, 8:26 AM280

GLOSSARY 281

"""./-%

.

".

&

geostatistics(0'( 01"

global polynomial interpolation2(

histogram

interpolate 3 "

intrinsic stationarity

inverse distance weighted interpolation2(

isotropy %& %&

kriging % &%&4#

lag%& %&%&

least-squares fit %& 5!

linear model of coregionalization # #

Glossary.p65 3/21/01, 8:26 AM281


local polynomial interpolation2( $

long-range variation6 7#

mean stationarity

nugget

ordinary kriging%&2## #

partial sill '

polynomial '"" #-""""

.

".

.8'

"

prediction

prediction standard error! ! 9: #;<

probability map % &

QQPlot !

quantile! =- !

Glossary.p65 3/21/01, 8:26 AM282

GLOSSARY 283

radial basis functions2(

regression %& " !

residuals6 %"&

range

searching neighborhood 2

second-order polynomial%"& '"""./<%

.

".

.

".

.

"&

second-order stationarity

semicrossvariogram

semivariogram

semivariogram values

short-range variation6

sill ! (

spatial dependence

Glossary.p65 3/21/01, 8:26 AM283


spherical model

spline interpolation2( $

stationarity

surface

trend ""

unimodal ""

univariate distribution #

validation

variogram !

variography' "

Glossary.p65 3/21/01, 8:26 AM284

285

References !

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286 USING THE GEOSTATISTICAL ANALYST

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INDEX 289

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index2.p65 03/09/2001, 11:13 AM300

Documents

Using ArcGIS Geostatistical Analyst · 2005-01-25 · 2 USING ARCGIS GEOSTATISTICAL ANALYST Exploratory spatial data analysis,