Computational Geosciences with Mathematica

385 Pages·2004·13.372 MB·English

by  Dr. William C. Haneberg (auth.)

William C. Haneberg
Computational Geosciences with Mathematica
William C. Haneberg
Computational
Geosciences
with
Mathematica
With 297 Figures and a CD-ROM
123
Dr. William C. Haneberg
Haneberg Geoscience
10208 39th Avenue SW
Seattle WA 98146
USA
E-mail:[email protected]
Additional material to this book can be doenloaded from http://extra.springer.com.
Library of Congress Control Number: 2004106660
ISBN 978-3-642-62157-4 ISBN 978-3-642-18554-0 (eBook)
DOI 10.1007/978-3-642-18554-0
This work is subject to copyright.AII rights are reserved,whether the whole or part of the material is
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sp ri ngeronl i ne.com
© Springer-Verlag Berlin Heidelberg 2004
Originally published by Springer-Verlag Berlin Heidelberg New York in 2004
Softcover reprint of the hardcover 1st edition 2004
The use of general descriptive na mes, registered names, tradema rks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant
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Cover design: E. Kirchner
Production: A. Oelschlăger
Typesetting: F. Herweg, Germany
Printed on acid-free paper 32/3141/AO - 543210
Preface
Mathematica(cid:2)R is a comprehensive mathematics package that can be used to per-
form numerical calculations, manipulate symbolic expressions, develop compli-
cated computer programs, and create sophisticated scientific graphics. My objec-
tiveinwritingComputationalGeoscienceswithMathematicawastoshowhowthe
programcanbeappliedtosolveawiderangeofproblemsofinteresttogeologists,
geomorphologists,hydrologists,geophysicists,and othergeoscientists. Assuch, it
is partly a textbook on quantitative geoscience and partly a manual showing how
Mathematicacan beused to solvesomeproblemsof interestto geoscientists. Itis
written at a level appropriate for graduate students embarking on quantitative re-
searchprojects,professorswhoareinterestedinlearningaboutnewapproachesto
quantitativeproblemsolving,andpracticinggeoscientistswithaninterestinMathe-
matica.Whilesomeofthematerialismoreadvancedthanthattaughtintypicalun-
dergraduategeologyprogramsintheUnitedStates,muchofitwillbeaccessibleto
motivatedjuniorandseniorstudents.
Ithaslongpuzzledmethat,whileadvancedcomputationaltoolssuchasMathe-
maticahavebeenavailablefor15yearsormore,manygeoscientists(geologistsin
particular) seem to be stuck in a spreadsheet rut. While spreadsheets manipulate
rowsandcolumnsof numbersadequately,they arenotwellsuited formuchmore
thansimplearithmetic.Still,theirpopularitypersistsand,inmyopinion,continues
to make life more difficult for students or professionals trying to solve quantita-
tivegeoscientificproblems.IbeganusingMathematicain1989andithasbecome
an indispensablecomputationalandgraphicstoolin my researchandprofessional
practice.GerardMiddletonseemstoshareasimilarviewofspreadsheetstohisbook
DataAnalysisin the EarthSciencesUsingMatlab,andI am gladtosee thatIam
not alone in that regard. Inexpensivestudent versions make Mathematica particu-
larlywellsuitedforstudentsincomputermethodsorquantitativegeologyclasses.
The subject matter and examples in Computational Geoscience with Mathe-
matica were drawn largely from my experience as an applied researcher in engi-
neeringgeologyandhydrogeology,universityinstructor,andconsultinggeologist.
Ihavetriedtoincludeabroadrangeoftopics,buttherearemanygeoscientificand
mathematicaltopics that are not covered.Fractals, wavelets, and geostatistics, for
example,arealltopicsthatcanbe fruitfullyaddressed Mathematica,buttheseare
topicsthateitherfallwelloutsidemyrangeofexperienceorthespaceavailablein
VI Preface
thisbook.Ihopethat,ratherthansparkingcomplaints,theiromissionwillmotivate
specialistsinthosefieldstofillthevoid.
MathematicaispublishedbyWolframResearch,Inc.,andiscurrently(summer
2003)inversion5.0.Formoreinformation,contactthecompanyat:
WolframResearch,Inc.
100TradeCenterDrive
Champaign,IL61820-7237USA
[phone]
[email protected]
www.wolfram.com
Mathematica is a registered trademark of Wolfram Research, Inc. Matlab is a
registeredtrademarkofTheMathWorks,Inc.
Thisbookwouldnothavebeenwrittenwithoutthesupportandencouragement
of my wife Lisa. Others who deserve a measure of credit (but none of the blame
foranymistakes)includeArvidJohnsonandPaulPotterwho,respectively,taught
mehowtoformulategeologicalproblemsintermsofmechanicsandstatistics.John
HawleyandthelateFrankKottlowskihiredmeandprovidedafertileenvironment
forprofessionalgrowthattheNewMexicoBureauofMinesandMineralResources,
adivisionofNewMexicoTech.ThelateAllanGutjahr,alsoatNewMexicoTech,
wasafountainofstatisticalwisdomduringourcarpoolingyears.MikeWhitworth,
Marshall Reiter, Dave Love, Laurel Goodwin,Peter Mozley, and many other col-
leagues introduced me to a fascinating array of technical topics during my time
inNewMexico.Finally,WolframResearchgenerouslyprovidedcopiesof Mathe-
maticaandallowedmetouseapre-releaseversionofMathematica5.0duringthe
writingofthisbook.
WilliamC.Haneberg
PortOrchard,Washington
August2003
About theAuthor
William C. Haneberg is an engineering geologist, hydrogeologist, and accredited
Mathematica consultant living and working in the Seattle area. Before moving to
the Pacific Northwest to establish a consulting practice in 1999, he served as Se-
nior Engineering Geologist and Assistant Director of the New Mexico Bureau of
Mines and Mineral Resources in Socorro and Albuquerque,New Mexico. He has
alsotaughtundergraduateandgraduateclassesingeology,hydrology,geophysics,
andgeologicalengineeringatNewMexicoTechandPortlandStateUniversity.Dr.
Haneberg is an author or co-author of more than 25 papers and co-editor of two
Preface VII
multi-authormonographs(ClayandShaleSlopeInstability,publishedbytheGeo-
logicalSocietyofAmerica,andFaultsandFluidFlowintheShallowSubsurface,
publishedbytheAmericanGeophysicalUnion).HeearnedaPh.D.ingeologyfrom
the University of Cincinnati, where his dissertation research concerned precipita-
tion induced pore pressure increases in potentially unstable slopes. For additional
information,pleasevisitwww.haneberg.com.
Contents
1 IntroductiontoMathematica .................................... 1
1.1 WhatisMathematica? ...................................... 1
1.2 GettingHelp............................................... 2
1.3 InstallingandRunningMathematica........................... 2
1.4 HowtheBookisOrganized.................................. 3
1.5 ABriefTourofMathematica ................................ 4
1.5.1 SymbolicandNumericalOperations .................... 4
1.5.2 VectorandMatrixOperations.......................... 8
1.5.3 2-Dand3-DGraphing................................ 10
1.5.4 User-DefinedFunctions............................... 20
1.5.5 DataImportandExport............................... 21
1.5.6 MathematicaPackages ............................... 22
1.6 ReferencesandRecommendedReading........................ 23
2 SpecialPlotsforGeoscienceData ................................ 25
2.1 MathematicaPackagesYouWillNeed......................... 25
2.2 Overview ................................................. 25
2.3 StemPlots ................................................ 25
2.3.1 ImportingtheData ................................... 26
2.3.2 CreatingtheStemPlot................................ 26
2.4 RosePlots ................................................ 29
2.4.1 ImportingtheData ................................... 29
2.4.2 CreatingtheRosePlot................................ 30
2.5 TernaryPlots .............................................. 35
2.6 StereographicProjections.................................... 37
2.6.1 StereographicProjectionsofPlanes..................... 38
2.6.2 StereographicProjectionsofLines...................... 39
2.7 EqualAreaProjections...................................... 41
2.7.1 EqualAreaProjectionsofLines........................ 41
2.7.2 ContouringEqualAreaProjections ..................... 43
2.8 BoxandWhiskerPlots ...................................... 47
2.9 WellLogs................................................. 49
2.10 ReferencesandRecommendedReading........................ 55
X Contents
3 ManipulatingandSolvingEquations ............................. 57
3.1 MathematicaPackagesYouWillNeed......................... 57
3.2 BasicSymbolicManipulation ................................ 57
3.3 MatrixandVectorOperations ................................ 62
3.4 LinearEquationSolving..................................... 64
3.4.1 Solve,Roots,andReduce ............................. 64
3.4.2 NSolveandFindRoots................................ 66
3.4.3 GeoscienceExamples ................................ 68
3.5 OrdinaryDifferentialEquations .............................. 78
3.5.1 ManualManipulationandIntegration ................... 78
3.5.2 SolutionsUsingDSolveandNDSolve................... 80
3.5.3 GeoscienceExamples ................................ 81
3.6 PartialDifferentialEquations.................................103
3.6.1 HillslopeDiffusion...................................103
3.6.2 PeriodicHeatFlow...................................109
3.6.3 TopographicLoadingofEarth’sCrust...................113
3.6.4 TwoDimensionalSteadyGroundwaterFlow .............119
3.7 ReferencesandRecommendedReading........................129
4 RandomVariablesandUnivariateProbabilityDistributions ........131
4.1 MathematicaPackagesYouWillNeed.........................131
4.2 TheConceptofRandomVariables ............................131
4.3 SomeContinuousDistributions...............................133
4.3.1 NormalDistribution..................................133
4.3.2 Log-NormalDistribution..............................135
4.3.3 UniformDistribution .................................136
4.3.4 ExtremeValueDistribution............................137
4.3.5 BetaDistribution ....................................137
4.3.6 ParetoDistribution ...................................139
4.4 SomeDiscreteDistributions..................................140
4.4.1 PoissonDistribution..................................140
4.4.2 BinomialDistribution ................................142
4.5 RelatingDistributionstoData:MethodofMoments..............142
4.5.1 HowGoodAreThoseEstimates?.......................145
4.6 ParametricHypothesisTesting:tandFtests ....................147
4.6.1 ThetStatistic .......................................148
4.6.2 CriticaltValues .....................................149
4.6.3 ComparingTwoMeansorVariances ....................152
4.7 NonparametricHypothesisTesting:K-STests ..................154
4.8 GeneratingRandomNumbersfromProbabilityDistributions ......157
4.9 CareandFeedingoftheRandomNumberGenerator .............162
4.10 IllustratingtheCentralLimitTheorem.........................163
4.11 ThePitfallsofUndersampling................................166
4.12 ReferencesandRecommendedReading........................171
Contents XI
5 ProbabilisticSimulation ........................................173
5.1 MathematicaPackagesYouWillNeed.........................173
5.2 FloodFrequencyModeling ..................................173
5.2.1 PlottingtheData.....................................174
5.2.2 Log-NormalandExtremeValueDistributionFitting.......175
5.2.3 EmpiricalCumulativeDistribution......................177
5.2.4 ComparisonofResults................................178
5.2.5 ExceedanceProbabilityandRecurrenceIntervals .........179
5.3 Didn’tWeJustHavea100YearFlood? .......................181
5.4 MonteCarloSimulationofaWettingFront.....................183
5.5 MonteCarloAnalysisofInfiniteSlopeStability.................186
5.5.1 StaticFactorofSafety ................................186
5.5.2 EffectsofChangingIndependentVariableDistributions....189
5.5.3 ConditionalProbability:EarthquakesandSlopeStability...192
5.6 ApparentClastSizeDistributions:TheOutcropEffect ...........196
5.6.1 RandomlyRotatedEllipsoids ..........................202
5.7 ReferencesandRecommendedReading........................211
6 InterpolationandRegression ....................................213
6.1 MathematicaPackagesYouWillNeed.........................213
6.2 InterpolationorRegression:WhichisAppropriate?..............213
6.3 Interpolation...............................................214
6.3.1 FindingaSingleInterpolatingPolynomial ...............214
6.3.2 PiecewisePolynomialInterpolation.....................218
6.4 LinearRegression ..........................................221
6.4.1 DerivationofLinearLeastSquaresEquations ............222
6.4.2 Residuals...........................................226
6.4.3 Goodness-of-FitandtheCorrelationCoefficient ..........228
6.4.4 SignificanceofRegressionResults:ANOVA .............229
6.4.5 UsingFitandRegress ................................231
6.4.6 CanISolvefortheIndependentVariable? ...............233
6.4.7 ReducedMajorAxisRegression:TwoVariableswithError .240
6.5 NonlinearRegression .......................................244
6.5.1 NonlinearLeastSquares ..............................245
6.5.2 LogisticRegression ..................................247
6.6 ReferencesandRecommendedReading........................255
7 VisualizingandAnalyzingSurfaces ..............................257
7.1 MathematicaPackagesYouWillNeed.........................257
7.2 GriddedData ..............................................257
7.2.1 DigitalElevationModels..............................258
7.2.2 ImportingSDTSDEMFiles ...........................259
7.2.3 ContourPlots .......................................260
7.2.4 DensityPlots........................................263
7.2.5 ThreeDimensionalSurfacePlots .......................265
XII Contents
7.2.6 QuantitativeTerrainAnalysis ..........................268
7.2.7 CompositeGeomorphicMaps..........................281
7.3 IrregularlySpacedData .....................................284
7.3.1 ReciprocalDistanceGridding..........................290
7.3.2 ThinPlateSplineGridding ............................292
7.3.3 ANoteAboutKriging................................294
7.3.4 AddingWellLocationstoSurfacePlots .................294
7.3.5 ComparingResults...................................296
7.4 TrendSurfaceMapping .....................................298
7.5 ReferencesandRecommendedReading........................305
8 DigitalSignalandImageProcessing..............................307
8.1 MathematicaPackagesYouWillNeed.........................307
8.2 TheNatureofPeriodicWaveforms............................307
8.3 DiscreteFourierTranforms ..................................310
8.4 AutocovarianceandAutocorrelation...........................318
8.5 FiltersandConvolution......................................321
8.5.1 FirstDifferences.....................................322
8.5.2 MovingAveragesandSmoothing.......................325
8.5.3 High-PassFiltering...................................327
8.6 ImageProcessing...........................................328
8.6.1 ImportingDigitalImages..............................328
8.6.2 BasicMathematicalOperations ........................333
8.6.3 Thresholding........................................337
8.6.4 SmoothingorBlurring................................338
8.6.5 UnsharpMasking....................................340
8.6.6 EdgeDetection ......................................342
8.6.7 UsingListInterpolation ...............................345
8.7 RecommendedReading .....................................347
AppendixA MathematicaFunctionsintheComputationalGeoscience
Package.......................................................349
A.1 Introduction ...............................................349
A.2 PlottingandCalculations ....................................349
A.3 ColorFunctions............................................351
AppendixB WorkingwithColor.....................................353
B.4 MathematicaPackagesYouWillNeed.........................353
B.5 SpecifyingColorsinMathematica ............................353
B.5.1 Hue,Saturation,andBrightness ........................353
B.5.2 Red,Green,andBlue(RGB) ..........................355
B.5.3 Cyan,Magenta,Yellow,andBlack(CMYK) .............358
B.5.4 OtherColorSystems .................................358
B.6 UsingColorinPlotsandGraphics ............................359
B.6.1 PlotandListPlot.....................................359