Citation
Detecting shifts in temporal dependencies between rainfall and streamflow using information thoery : a Colorado headwaters case study

Material Information

Title:
Detecting shifts in temporal dependencies between rainfall and streamflow using information thoery : a Colorado headwaters case study
Creator:
Franzeb, Samuel E.
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English

Thesis/Dissertation Information

Degree:
Master's ( Master of science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Civil Engineering, CU Denver
Degree Disciplines:
Civil engineering
Committee Chair:
Mays, David
Committee Members:
Goodwell, Allison
Guo, James

Record Information

Source Institution:
University of Colorado Denver
Holding Location:
Auraria Library
Rights Management:
Copyright Samuel E. Franzen. Permission granted to University of Colorado Denver to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.

Downloads

This item is only available as the following downloads:


Full Text

PAGE 1

DETECTINGSHIFTSINTEMPORALDEPENDENCIESBETWEENRAINFALLAND STREAMFLOWUSINGINFORMATIONTHEORY:ACOLORADOHEADWATERSCASE STUDY by SAMUELE.FRANZEN B.S.,ColoradoStateUniversity,2011 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience CivilEngineeringProgram 2018

PAGE 2

ThisthesisfortheMasterofSciencedegreeby SamuelE.Franzen hasbeenapprovedforthe CivilEngineeringProgram by DavidMays,Chair AllisonGoodwell,Advisor JamesGuo DateDecember15,2018 ii

PAGE 3

Franzen,SamuelE.MS,CivilEngineering DetectingShiftsinTemporalDependenciesbetweenRainfallandStreamowUsingInformation Theory:AColoradoHeadwatersCaseStudy ThesisdirectedbyAssistantProfessorAllisonGoodwell ABSTRACT Achangingclimateandhumanimpactshaveaectedbothprecipitationandstreamowpatterns intheU.S.Moreover,theyhavelikelyaectedthedependency,ortemporalrelationship,between streamowandprecipitation.Precipitation,streamow,andtrendsinbothhavebeenstudiedinthe pastoverlargeareas,andinsomecases,thestrengthofthedependencybetweenthetwohasbeen analyzed.Operationalandresearch-basedhydrologicalstudiesoftenattempttopredictstreamow basedonprecipitationusingmodels.However,theutilityofthesemodelsandthebenetofthe resultsdependonthestrengthoftherelationshipbetweenstreamowandthevariablesaecting it.Inthispaperweuseinformationtheory,specicallymutualinformation,tocharacterizethe strengthofthedependencybetweenstreamowandprecipitation.Additionally,weexpandon previousstudiestoshowtherearesignicanttrendsinthisdependencyandthatthesetrends dierdependingonseason,rainfallthreshold,andwhetherweexaminedayswithprecipitation, withoutprecipitation,orboth.OuranalysisisfocusedontheColoradoHeadwatersBasinanduses UnitedStatesGeologicalSurveyUSGSstreamowdataandClimatePredictionCenterUSUnied Precipitationdata.Ourresultsindicatetheprecipitation-streamowdependencyisstrongestin fallandthatthedependencyisdecreasinginthefallandincreasingintheotherseasons.Many otherphysicalandclimaticfactorsaectthestrengthofthedependencybetweenstreamowand precipitation,e.g.temperature,soilmoisture,andbasintopographyandgeology,andshouldbe studiedalongsideprecipitationandstreamowinthefuturetobetterunderstandthedependency betweenthetwo.Thisstudyservesasastartingpointtofullyunderstandthecomplexdependency betweenthemultiplevariablesthataecttheprecipitation-streamowrelationship. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:AllisonGoodwell iii

PAGE 4

ACKNOWLEDGEMENTS ThankyoutomyadvisorDr.Goodwellforteachingmesomuchandguidingmethroughthis process. Thankyoutomythesiscommitteefortakingthetimetoassistmeinthecompletionofmy thesisanddegree. ThankyoutomyfamilyandfriendsforbeingthereeventhoughIhavenotbeenthereasoften whilecompletingmythesis. iv

PAGE 5

TABLEOFCONTENTS CHAPTER I.INTRODUCTION......................................1 II.BACKGROUND......................................5 II.1StudySite......................................5 II.2InformationTheoryBackground..........................6 III.DATAANDMETHODS.................................8 III.1Data.........................................8 III.2PreliminaryAnalysis................................9 III.3Methods.......................................9 IV.RESULTS.........................................14 IV.1AverageInformationValues............................14 IV.2TrendsinInformation................................20 V.DISCUSSION........................................25 REFERENCES............................................27 v

PAGE 6

LISTOFTABLES TABLE IV.1ComparisonofEntropy,MutualInformation,andTrendsforthe0.3mmThreshold...24 vi

PAGE 7

LISTOFFIGURES FIGURE I.1ColoradoHeadwatersMap...................................2 I.2ColoradoHeadwatersFacilitySiteMap............................3 II.1AverageYearlyandMonthlyFlows..............................5 III.1ColoradoHeadwatersMapwithPrecipitationGridCellsandStreamGauge.......9 III.2ProbabilityDistributionFunctionandInformationTheoryDiagrams...........12 IV.1AverageEntropyofPrecipitation...............................15 IV.2AverageMutualInformationValuesforeachPrecipitationGridwithPrecipitation greaterthan0.3mm......................................16 IV.3AverageMutualInformationValuesforeachPrecipitationGridwithPrecipitation greaterthanthe50thPercentileEvent............................16 IV.4AverageLagproducingtheMaximumMutualInformationValueforeachPrecipitation GridwithPrecipitationgreaterthan0.3mm........................19 IV.5AverageLagproducingtheMaximumMutualInformationValueforeachPrecipitation GridwithPrecipitationgreaterthanthe50thPercentileEvent..............19 IV.6TrendsinMutualInformationforDayswithPrecipitationgreaterthan0.3mm.....21 IV.7TrendsinMutualInformationforDayswithPrecipitationgreaterthanthe50th PercentileEvent........................................22 vii

PAGE 8

CHAPTERI INTRODUCTION ThesteadywarmingofEarth'sclimateoverthepast100yearsSchneider,1989hasintensied thegloballyaveragedwatercycle,precipitation,evaporation,andrunoClark etal. ,1999;Milly etal. ,2002.Additionally,thefrequencyandintensityofextremeprecipitationanddroughtevents havebeenincreasingoverthesametimeperiodKarlandKnight,1998;Easterling etal. ,2000.As theglobalwatercyclechanges,itisimportanttoevaluatehowtherelationshipsanddependencies amongstitscomponentshavechangedandmaycontinuetochangeinthefuture.Resultsfrom studiesontrendsinprecipitationacrosstheUnitedStatesKarlandKnight,1998;Kunkel etal. , 1999showanupwardtrendinprecipitationacrossthemajorityoftheUnitedStateswithsignicantincreasesintheSouthwest,CentralGreatPlains,MiddleMississippi,andSouthernGreat Lakesbasins.Thesestudiesalsoshowanincreaseintheportionoftotalprecipitationcontributed byextremeeventsforthemajorityofthecountryKarlandKnight,1998;Kunkel etal. ,1999. Inadditiontoanincreaseinextremeevents,otherstudiesevaluatingmorefrequentlyoccurring precipitationeventshavefoundanincreaseinthefrequencyofallrainfalleventsforamajorityof theUnitedStatesRoque-MaloandKumar,2017. Streamowisalaggedfunctionofprecipitation,butisalsoheavilyinuencedbyotherfactors andbasincharacteristics.Forexample,owsinmoststreamsaremodiedsignicantlyfromhistoric owpatternsbydamsandrunofromimperviousareascreatedbyhumans.Meanwhile,thesize, topography,soils,vegetation,andheterogeneityofabasindictatethetranslationofprecipitation tostreamow.InthispaperweanalyzeprecipitationandstreamowintheColoradoHeadwaters BasinFigureI.1.Thisbasinisrelativelysparselypopulatedwithacorrespondinglysmallamount ofdevelopmentinthearea.Themajorriverinthebasin,theColoradoRiver,isunobstructed byanysubstantialdams,althoughanumberofitstributariesinthebasinareimpactedbywater storageprojectsFigureII.1.Relativetodownstreamows,itislikelythattheColoradoRiver streamowintheregionofstudyisnotsubstantiallyaectedbyanthropogeniccausesbecauseof therelativelysparsepopulationinthebasinandthefewreservoirsordiversionsontheriver. Itisimportanttoanalyzeandunderstandtheeectsthatchangesinprecipitationhavealready hadandwilllikelycontinuetohaveonstreamowforseveralreasons.Mainly,manystudieshave 1

PAGE 9

FigureI.1:ColoradoHeadwatersDrainageBasinMap Source:uppercoloradoriver.org/co-river-headwaters usedcomputermodelstoreviewandpredictshiftsinprecipitationandstreamowinthewestern UnitedStatesFicklin etal. ,2013;Hidalgo etal. ,2009.Resultsfromthesemodelsareusedby dierententitiesformanydierentobjectives.Forexample,governmentocialsuseresultsfrom thesemodelstopredictrainfalleventsandstreamowswhichareusedinconjunctionwithother datatodecidehowmuchwatertostoreandreleasefromreservoirsHejazi etal. ,2008.These decisionshavefarreachingeectsonthepeoplewhodependonthereservoirsforoodcontrol, watersuppliesfordrinkingwater,powerproduction,irrigation,andindustry,andtheecosystems whichdependonthewaterforsurvivaldownstreamofthereservoirs.Ifthesemodelsarenot accuratelypredictingfuturerainfalleventsandstreamows,thereservoiroperatorsmayrelease toomuchwaterleavingfarmerswithoutwatertoirrigatetheircropsorpeoplewithoutadequate 2

PAGE 10

FigureI.2:Coloradoheadwatersfacilitysitemapshowingreservoirs,diversions,andpower plantsinandaroundtheColoradoHeadwatersBasin Source:www.usbr.gov/uc/index.html watersupplies.Alternatively,ifnotenoughwaterisreleased,runofromalargestormcouldover topadamorrequiremorewatertobereleasedfromthereservoirthanthemaximumdesignow ofthedownstreamwaterways.Anywatermanagementdecisionbasedonincorrectassumptions orinaccuratemodelscouldleadtodownstreamooding,infrastructuredamage,andpossiblyloss oflife.Alternately,toolittleowcouldleadtodecreasedagriculturalproduction,powerlossat hydroelectricfacilities,anddamagetoecosystems.Seekingtrendsinexistingdataisarststep towardsimprovingmodelsusedtomakestreamowpredictions. Itisalsoimportanttounderstandanyshiftintherelationshipbetweenrainfallandstreamow becausetheecosystemsthathaveadaptedtothegeneralpatternsofstreamowwilllikelybe aectedbyanyshiftinthesepatternsPalmer etal. ,2009.Theseecosystemsarevaluableas ltersfornutrients,foodproduction,habitat,andrecreationalareas.Aswehaveseenwhendams areconstructedonwaterways,modiedstreamowpatternsaectsurroundingecosystems,often 3

PAGE 11

indetrimentalways.Ifthequicklychangingclimateandprecipitationpatternscausesimilartypes ofchangestostreamows,theecosystemsdependentonthestreamswouldfacesimilarchallenges andcouldexperiencethesamedetrimentaleects. Climatechangehascausedallpartsofthewatercycletochangeincludingprecipitationand streamow.Itisexpectedthatshiftsinstreamowarepartiallyduetochangesinthefrequencyand intensityofprecipitation.Streamowsareintegraltotheoperationofsocietyprovidingwaterfor agriculture,consumption,industry,powergeneration,andtheenvironment.Thisthesisfocuseson evaluatingthedependencybetweenstreamowandprecipitation.Itisimportanttounderstandthe dependencybetweenstreamowandprecipitationandhowitischangingbecausethiswillincrease theaccuracyofmodelspredictingstreamowandallowbettermanagementofourrivers,reservoirs, andthewatertheyhold.Theorganizationofthisthesisisasfollows:InChapterIItheareaof studyisintroducedandpreviousstudiesonprecipitation,streamow,trendsintheirvalues,and therelationshipbetweenthetwoarediscussed.InChapterIIIthedataandinformationtheory methodsusedinthestudyarepresented.InChapterIVtheresultsofaveragemutualinformation valuesandtrendsinthevaluesareassessed.InChapterVwediscusstheroletheseresultshave andhowtheycouldbeexpandedinsubsequentstudies. 4

PAGE 12

a b FigureII.1:FlowsintheColoradoRivermeasuredatUSGSgauge09163500ColoradoRiverNear Colorado-UtahStateLineincludingaaverageyearlyowsfrom1952through2016andb averagemonthlyowsforthe1980s CHAPTERII BACKGROUND II.1StudySite OurareaofstudyistheColoradoHeadwatersBasin,HydrologicalUnitCodeHUC1401 locatedwestoftheContinentalDivideinthenorthcentralportionofColorado.Thebasinis approximately9,838squaremilesandmeasures175milesfromeasttowestby55milesfromnorth tosouth.Elevationsinthebasinvaryfrom4,400feetto14,000feet.Themajorwaterwayinthe basinistheColoradoRiverwhichowsforapproximately235milesfromeasttowestacrossthe basin.TheGunnisonRiverjoinstheColoradoRiver30milesbeforetheColoradoRiverexitsthe basin.Annualprecipitationvarieswidelyacrossthebasinfrom9to30inchesperyearbasedon datacollectedfrom1981to2010bytheNationalOceanicandAtmosphericAdministrationNOAA NationalClimateDataCenterNCDC. AverageyearlyowsintheColoradoRivermeasuredneartheoutletofthebasinareshownin FigureII.1aforthestudytimeperiod.Theplotshowshighvariabilityinowsfromyeartoyear. FigureII.1bshowstheaveragemonthlyowsmeasuredatthesamelocationforthe1980s.The monthlyowplotshowsthepeakowsaligningwithsnowmelteachspring,relativelystablelow 5

PAGE 13

owsfortheremainderoftheyear,andaslightincreasefollowedbyadecreaseinowsacrossthe decade. II.2InformationTheoryBackground Informationtheoryisarobustmethodforassessingpotentiallynonlinearrelationshipsbetween twovariables,i.e.arelationshipthatisnotdirectlyproportionalandthatcannotbedened byasingleequation.Incontrasttolinearcorrelationtechniques,informationtheorymetricsare basedonprobabilitydensityfunctions,andmeasuretheextenttowhichknowledgeofonevariable reducestheuncertaintyofanotherCoverandThomas,2006.Asdiscussedbelow,ourstudyuses measuresbasedoninformationtheorytoevaluatethespatialandtemporaldependencybetween streamowandprecipitation.Detailsonthesemeasuresandourspecicapplicationareprovided inthefollowingChapter. Toeectivelystudythechangesinprecipitationandstreamowalongwithhowtheyarerelated, atoolisneededthatisabletondrelationshipsbetweeninterconnecteddatasets.Recently,informationtheory,andmorespecicallyShannonEntropy,hasbeenusedinmanyhydrologyrelated studiestodeterminethespatialandtemporalvariabilityofmanyhydroclimaticvariablesPechlivanidis etal. ,2016;Sang etal. ,2018.Entropyisausefultoolforevaluatingclimaticcharacteristics becauseitmeasurestheuncertaintyofavariableCoverandThomas,2006.Knowledgeoftheentropyofavariableisusefulinquantifyingandpredictingthespatialandtemporaldistributionofan eventandisthereforeavaluablemeasureinthestudyofprecipitationandstreamow.Entropywas originallyusedinthecommunicationseld.Itwasusedtodescribetherandomnessinamessage, whichdenedtherequiredchannelsizetotransmitandreceivemessagesShannon,1948. Manypreviousstudieshaveinvestigatedthespatialandtemporalvariabilityofprecipitation Mishra etal. ,2009;deP.RodriguesdaSilva etal. ,2016;Brunsell,2010;Kunkel etal. ,1999; GoodwellandKumar,2018,streamowdeP.RodriguesdaSilva etal. ,2016;Lettenmaier etal. , 1994;AmorochoandEspildora,1973;LinsandMichaels,1994,andprecipitationdecitSang etal. ,2018usingentropyasameasureofvariability.Someofthesestudiesalsoevaluatedlong termtrendsinthevariabilityofprecipitationusingtheHurstexponentandtheMann-Kendalltest Mishra etal. ,2009;Brunsell,2010orasimplelineartrendanalysisKunkel etal. ,1999,streamow variabilityusingtheMann-KendalltestLinsandMichaels,1994;Lettenmaier etal. ,1994,andthe 6

PAGE 14

variabilityofprecipitationandstreamowusingtheMann-KendalltestdeP.RodriguesdaSilva etal. ,2016.Thesestudiesusedentropyasameasureofuncertaintytodetermineiftherewere anynoticeabletrendsinthevariabilityofthefrequencyorintensityofprecipitationeventsand/or streamowswiththegoalofassessingwhataectsthesetrendsarelikelytohaveonhumanityin thefuture. Inadditiontothestudyofthevariabilityofprecipitationandstreamowseparately,therelationshipbetweenprecipitationandstreamowhasbeenevaluated.Forexample,deP.Rodriguesda Silva etal. 2016evaluatedtherelationshipbetweenrainfallandstreamowbyanalyzingtrends intherelativeentropy,ameasureofthedierencebetweentwodistributionsCoverandThomas, 2006.TheresultsofthestudydidnotshowanystatisticallysignicanttrendsovertimeintherelativeentropybetweenthevariabilitydistributionsofprecipitationandstreamowdeP.Rodriguesda Silva etal. ,2016.Lettenmaier etal. 1994analyzedrelativechangesbetweenstreamowandprecipitationoverthecontiguousUnitedStatesusingabivariatetest.Hefoundthatattheannual timescaletherearerelativelyfewsignicantchangesinstreamowrelativetoprecipitation,but atthemonthlytimescaletherearesignicantchangesinstreamowinrelationtoprecipitation duringthemonthsofJanuaryandFebruaryintheGreatLakesRegion.AmorochoandEspildora 1973usedtransinformation,ormutualinformation,toevaluatetheutilityofamodelpredicting streamowusingprecipitationastheprimaryinputbycomputingtransinformationbetweenthe measuredstreamowvaluesandthestreamowvaluesproducedbythemodel.AmorochoandEspildora1973foundtheuncertaintyofthestreamowvalueswerereducedsignicantlyformuch oftheyearwhenpredictedbythemodelusingprecipitationvalues. Previousresearchhasnotdirectlyevaluatedtheprecipitation-streamowrelationshipusinginformationtheory.Inthispaperweexpandonthestudiesthatusedinformationtheoryandother methodstoevaluatethevariabilityinandrelationshipsbetweenprecipitationandstreamow.We furtherexploretheserelationshipstodenethestrengthandtimescalesofthesedependenciesand towhatextentthesedependencieshavechangedovertime. 7

PAGE 15

CHAPTERIII DATAANDMETHODS III.1Data TheareaofstudyistheColoradoHeadwatersBasin.Weselectedasmallerareaofstudyas comparedtootherstudies,whichoftenfocusedonanentirecountryorcountries,tobeableto utilizemorepreciserainfalldatainsteadofrainfalldatainterpolatedoveralargearea.Wechose thisspatialscaletostudyprecipitation-streamowdependenciesatahighresolutionthatcaptures dierencesinsoil,topology,andlanduse,butoveralargeenoughareatoberelevanttobasin scalemodelingandoperations.Weuseddailyprecipitationdataoverthewatershed.Initially,we experimentedwithdatafromdierentgaugesthroughoutthebasin.However,duetodicultiesin theanalysiscausedbydierencesintherecordingperiodforeachgauge,datagapsintherecords fornumerousgauges,andconcernswiththesparsecoverageofsomeareasofthebasinbythegauge network,wedecidedtouseagriddedprecipitationdatasetprovidedat0.25 resolutionfrom1948 to2018providedbytheClimatePredictionCenterCPCUSUniedPrecipitationdataNOAA etal. ,2018.Thisdatasetusesavailableinformationfromanumberofgaugeswithinthebasin andinterpolatesvaluesforgridswithoutgaugesasdiscussedbyXie etal. 2007.Onlygridcells withacentroidwithintheboundariesoftheColoradoHeadwatersBasinwereused,whichresulted in43dierentgridcellsandcorrespondingrainfalldatasets.Thedatasetsforeachgridcellwere convertedtobinarydatasetsbasedonathresholdprecipitationvalue,zerofordrydaysandonefor rainydays,toevaluatetheeectofrainfalloccurrenceonstreamowratherthanrainfallmagnitude. Thebinarydatasetalsoreducestheimpactofthemajorityofprecipitationvaluesbeingzerodue tothedryclimateinColorado,whichwouldcausetheprobabilitydistributionfunctionpdfto besparseifrainfallmagnitudeswerediscretizedintomanybins.Thethresholdvaluesusedinthis studywere0.3millimetersmm,thesmallestamountofmeasurableprecipitation,andthe50th percentileprecipitationeventsforeachprecipitationgridcell. Streamowdatacollectedfrom1952to2018bytheUnitedStatesGeologicalSurveyColorado WaterServiceCenteratgauge09163500ColoradoRiverNearColorado-UtahStateLine,located nearwheretheriverexitstheColoradoHeadwatersBasin,wasusedintheanalysis.Precipitation 8

PAGE 16

gridsandthestreamgaugelocationareshowninFigureIII.1.Thegridcellswithpointsinthem arethegridcellsusedinthisstudy. FigureIII.1:ColoradoheadwatersdrainagebasinmapwithprecipitationgridcellsandColorado RivernearColorado-Utahstatelinestreamgauge III.2PreliminaryAnalysis Initially,wesearchedforastatisticallysignicantrelationshipbetweenlaggedprecipitationand currentstreamowbycalculatingthePearsoncorrelationcoecientwhichmeasuresthelinearrelationshipbetweentwodatasetsPearson,1900.Fromthisanalysis,wefoundveryfewstatistically signicantcorrelationvalues.WebelievethismethodwasineectivebecausegenerallythePearson correlationworksbestwithnormallydistributeddatasets,whichneitherprecipitationorstreamoware,andcanbesignicantlyaectedbyalargenumberofzerovalues,whichtheprecipitation datasethas.Afterndinglinearcorrelationineectiveforanalyzingtherelationshipbetween precipitationandstreamow,wedecidedtouseanalternatemethod. III.3Methods ShannonEntropy,denedas H X t = )]TJ/F25 10.9091 Tf 10.303 8.182 Td [(P p x log 2 p x ,foratime-seriesvariable X t witha probabilitydensityfunction p X t isameasureofvariabilityorrandomnessofavariable. H X t hasunitsofbitswhenthelogisbase2Shannon,1948;CoverandThomas,2006.Bythisdenition H X t onlyappliestodatasetswithdiscretedistributionsMays etal. ,2002;CoverandThomas, 2006.Otherwise,therewouldbeinnitepossiblevaluesandaninnitenumberofquestionswould 9

PAGE 17

needtobeaskedtogaincertaintyAmorochoandEspildora,1973.Weconsiderprecipitationand streamowdatasetsusedinthisstudyasdiscretedatasetswithnitenumbersofvalues,making H X t anacceptablemeasureforboth. Wemeasuredthedependencyofstreamowvaluesonprecipitationusingmutualinformation. Mutualinformationiscalculatedusingentropy, H X t ,andconditionalentropy.Conditionalentropyaccountsforreductionsinthevariabilityofonevariableduetohavingknowledgeabout anothervariable.Conditionalentropyisdenedas H X j Y = )]TJ/F25 10.9091 Tf 10.303 8.182 Td [(PP p x;y log 2 p x j y andisa measureoftheentropyofavariable, X ,ifthevalueofanothervariable, Y ,isknownCoverand Thomas,2006.Forexample,inthisstudyweexpectthattheuncertaintyofstreamowwillbe reducedgivenknowledgeofpreviousrainfallevents.Mutualinformationuses H X t and H X j Y tomeasurethereductionofuncertaintyofXwithknowledgeofY,andviceversaFigureIII.2b. MutualinformationiscalculatedasfollowsCoverandThomas,2006: I X t ;Y t = H X t )]TJ/F19 10.9091 Tf 10.909 0 Td [(H X t j Y t III.1 Acommonlyusedexampleofinformationtheorycalculationsisacointosswhichhastwoequally likelyoutcomes,headsortails,eachwiththeprobabilityofoccurrenceof0.5.Asdenedabovethe entropyofacointossisequalto log 2 =1 becauseoftheequallikelihoodofeitheroutcome.If thecoinisbiasedsothatoneresultbecomesmorelikelythananother,theentropy,oruncertainty oftheoutcomewillbereduced.Ifweconsiderthecointosstobearandomvariable,wenotethat onecointosshasnoinuenceonthenext.Inotherwords,givensomedistributionofheadsortails, knowledgeofpastcointossesprovidesnoinformationaboutfutureresults.However,real-world processes,suchasprecipitation,havetimedependentfeatures,wheretheuncertaintyofthefuture statecanbereducedbytheknowledgeofthepast.Forexample,rainfalloftenhappensinmultiple daysequences,suchthattheknowledgeofrainfallyesterdayhelpstopredictthestateofrainfall today.Thisexempliesareductioninentropyduetotheknowledgeofanothervariableorapast state.Thisreductioninuncertainty,ormutualinformation,iscalculatedasthedierencebetween entropyofavariableandtheconditionalentropyofthesamevariablegivenknowledgeofsomething else. 10

PAGE 18

Hereweassessdependenceofstreamow, Q t ,onprecipitation, P t ,bycalculating I Q t ;P t whichmeasuresthereductioninuncertaintyofstreamowvalues, H Q t ,duetohavingknowledge of P t .Tocalculate I Q t ;P t wecreatedatwo-dimensionalprobabilitydistributionfunction.First, wetransformedourprecipitationdataintoabinarydatasetwithavalueof1assignedtodays withrainfalldepthaboveathresholdvalueandavalueof0assignedtodayswitharainfalldepth belowathresholdvalue.Thethresholdvaluesusedinthisstudywere0.3millimetersmm,the smallestamountofmeasurableprecipitation,andthe50thpercentileprecipitationeventsforeach precipitationgridcell.Sinceeachdayhasarainfallvaluedenedas P t asetofrainfalldatafor acertaintimeperiodhasattributes p P t = p P t =1 and p P t =0=1 )]TJ/F19 10.9091 Tf 11.41 0 Td [(p P t whichequate totheprobabilityofrainyordrydays,respectivelyGoodwellandKumar,2018.Similarly,we assigned Q t valuestooneofseventeenlinearlyspacedbinsofequalwidth.Ournaltwo-dimensional probabilitydistributionfunctionpdfissize,17andisshownschematicallyinFigureIII.2a Ourperiodofstudyis1956through2015duetodataavailability.Forthisperiod,wecalculated H Q t , H Q t j P t ,and I Q t ;P t valuesforeachseasonandyear.Wechosetoconductouranalysis seasonallyasopposedtoannuallysothatparticularseasonaltrends,whichmaybeobscuredwithin ayearlydataset,wouldbeapparent.Weuseda5-yearmovingwindowforourcalculationsto evaluatewhetherdependencieschangedovertime. Theeectof P t on Q t isexpectedtobedelayed,particularlyforlocationsfartherfromtheoutlet andstreamgauge.ThelengthoftheColoradoRiverinourbasinisapproximately235miles.If weassumeanaverageowvelocityof5milesperhourintheriver,precipitationfallingdirectly intotheriverinthegridcellfurthestfromtheoutletwilltakealmosttwodaystoreachtheoutlet. Thedelaywouldincreaseforprecipitationnotfallingdirectlyintotheriverduetoinltrationand overlandow.Toallowforthisdelay P t datawaslagged daysascomparedto Q t data,where isfrom0to7days.Sevendayswasselectedasthemaximumtimelagbecauseaweexpect mostrainfalltoexitthebasinatthistimescaleorshorterandbsincewearelookingatdaily dataoveraseasone.g.ourdataforoneseasonis90days,wewanttoretainasmuchdataas possible.Calculationsfor I Q t ;P t werecompletedusingeach P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( datasetandthe Q t dataset. 11

PAGE 19

a b FigureIII.2:Diagramsshowingatheprobabilitydistributionfunctionwith2precipitation binsand17linearlyspacedstreamowbinsandbaninformationtheorydiagramshowingthe relationshipsbetweenentropy,conditionalentropy,andmutualinformation. Thisproducedone I Q t ;P t )]TJ/F19 10.9091 Tf 11.138 0 Td [( valueforeach ineachgridcellforeachseasonandyearofour study. Toverifythestatisticalsignicanceofthe I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( valueforeachdelayateachprecipitation gridcellineachyearacritical I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( value,denedas I crit ,wascalculatedusingthefollowing procedure.Oneofthetwodatasetswasshuedtodestroyanycorrelationbetweenthedatasetsand theanalysisdescribedabovewasrunusingtheshueddatasettoproducearandom I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [( valueforeach ateachprecipitationgaugeineach5-yearwindow.Thiswasdoneasetnumber oftimestoproducemultiplerandom I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [( values.Then,themean, I mean ,andstandard deviation, I std ,ofthesetofrandom I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [( valueswerefoundandacriticalvaluewascomputed by I crit = I mean +3 I std .Thisequatestotestingforstatisticalsignicanceata99.7%condence levelassumingaGaussiandistribution.Ifthe I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [( calculatedfromtherealdatasetswas greaterthanthe I crit value,thecalculated I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [( wasconsideredtobestatisticallysignicant. Ifitwasnot,the I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( valuewasnotusedandwassettozero.Theresultwasadataset foreachgridcellcontainingeightvalues,oneforeach ,foreachseasonandyear.Themaximum I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [( valueineachdatasetwasselectedproducingfourdatasetsforeachgridcell,oneeach containingthemaximum I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( valuesforeachseasonofeachyear. Eachofthefourseasonalinformationdatasetsforeachgridcellwasthenexaminedfortrends usingtheTheil-SenestimatorwhichcomputestheTheilslopeforadatasetalongwithalowerand uppercondenceintervalTheil,1992;Sen,1968. Asdiscussedpreviouslyweconvertedourprecipitationdatasetintoabinarydatasetbasedon athresholdvalue.Thisallowedustopartitionthemutualinformationintotwoparts:information 12

PAGE 20

thatthepresenceofrainfallprovidestostreamowandinformationthatanabsenceofrainfall providestostreamow,asshownintheEquationIII.2. I Q t ;P t = I Q t ;P t =1+ I Q t ;P t =0 III.2 Bydoingthiswewereabletoassesstheimpactofthepresenceorabsenceofprecipitation onstreamowseparately.Therstterm, I Q t ;P t =1 ,representstheprecipitation-streamow dependencyfordayswithprecipitation.Statisticallysignicantmutualinformationvaluesfromthis termsignifyareductionintheuncertaintyofstreamowswithknowledgethatitrained.Thesecond term, I Q t ;P t =0 ,representstheprecipitation-streamowdependecyfordayswithoutrainfall. Statisticallysignicantvaluesfromthistermindicateareductioninuncertaintyofstreamowswith knowledgethatitdidnotrain.Anytrendintheoverall I Q t ;P t couldbeattributabletoeitheror bothofthetermsdiscussedabove. 13

PAGE 21

CHAPTERIV RESULTS IV.1AverageInformationValues Astheclimateevolvesandpatternsinprecipitation, P t ,andstreamow, Q t ,changeitwillbeimportanttounderstandhowthestatisticallysignicantmutualinformation, I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( ,valuesshow adependencybetween Q t and P t .Thevariabilityofprecipitation, H P t ,aectsthisdependency considerably.Forourbinaryprecipitationdataset, H P t hasamaximumvalueof1bitoccurring whenthereisanequalchanceofitbeingarainyordryday,when p P t =1= P t =0=0 : 5 . Asthecertaintyofwhetheritisrainingornotrainingincreasesand p P t increasesordecreases from0.5, H P t decreasesfrom1bit.Thisisarelativelysmallmaximumvariabilitycomparedto themaximumvariabilityofstreamow, H Q t = log 2 =4 : 09 ,sinceweconsider17possible categories.Duetothisdiscretizationof P t and Q t , H P t 1 bit denesthemaximumboundfor mutualinformation,sinceinformationcontentmustbelessthanorequaltothesmallestentropy ofeithervariableFigureIII.2. Theaverage H P t overthestudyareaandtimeperiodisshowninFigureIV.1. H P t across themajorityofthebasinisbetween0.75and0.98bits.Smaller H P t valuesoccuralongthe westernedgeofthebasinwithvaluesincreasingacrossthebasintotheeastforallseasons.Since thebasinisinadryclimatetheentropyvaluesbeinglessthanonereectthattherearemoredry daysthanrainydays.Thelowervaluesalongthewesternedgemeanthereislessuncertaintyin precipitationpredictionsinthisarea,becauseitismoreconsistentlydry.Theincreaseinvalues acrossthebasinfromwesttoeastislikelybecauseoftheorthographiceectsofthemountains creatingmoredayswithprecipitationathigherelevationsintheeasternportionofthebasin.This wouldleadtoamoreequalnumberofdayswithandwithoutrainfallandcauseuncertaintyand entropytoincreaseintheseareas. Statisticallysignicant I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( valueswerefoundforatleastonelagtimeforeachgridin each5-yearwindow,andinmostcasesmultiplelagtimesproducedsignicantvalues.Themaximum statisticallysignicant I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [( value,whichwedesignateas I max ,foreachgridcellandeach year,representingthegreatestdependencybetweenstreamowandprecipitation,wasusedinthe analysis.Themagnitudeoftheaverage I max valuesforeachprecipitationgridandseasonoverthe 14

PAGE 22

FigureIV.1:Average H P t ofprecipitationforaWinterDJF,bSpringMAM,cSummer JJA,anddFallSON.Circlesizesindicatethemagnitudeof H P t foreachgridcell.Color istheinterpolationof H P t valuesforeachgridcelloverthebasin. 60-yearperiodofstudy,withrainydaysdenedasdayswithrainfalleventsproducingmorethan 0.3mmisshowninFigureIV.2.Wealsoperformedtheanalysisusingthe50thpercentilerainfall eventastherainydaythreshold.Themagnitudesoftheaverage I max valuesforeventsgreaterthan the50thpercentileareshowninFigureIV.3. Theaverage I max valuesforeachseasonaresimilarforboththresholdvaluesindicatinglarger eventsinuenceinformationcontentmorestronglythansmallereventsFiguresIV.2andIV.3. Removingsmalleventsthatdonotproducerunocouldbesimilartoremoving"noise"fromthe precipitationdata.Thelackofchangein I max valueswhenthe"noise"isremovedindicatesthe precipitation-streamowrelationshipismaintainedwhensmalleventsareexcluded.Thismakes sensebecauseeventswithgreaterprecipitationvalueswouldbeexpectedtocreatemorenoticeable changesinstreamows. Whilemutualinformationvaluesdierdependingontheseasonandthedemarcationbetween "rain"and"norain"states,itisexpectedthatlarger I max valuesindicatingagreaterdependency betweenstreamowandprecipitationwouldbenearthesouthwestcornerofthebasinwherethe basinoutletandstreamgaugearelocated.ThistrendisnoticeableonthefallSONmap,Figures IV.2andIV.3,paneld,andlessnoticeableornonexistentonthewinterDJF,springMAM, 15

PAGE 23

FigureIV.2:Average I max valuesforeachprecipitationgridwithrainydaysdenedasdayswith precipitationgreaterthan0.3mmforaWinterDJF,bSpringMAM,cSummerJJA,and dFallSON.Circlesizesindicatethemagnitudeofaverage I max valuesforeachgridcell.Color istheinterpolationofaverage I max valuesoverthebasin. FigureIV.3:Average I max valuesforeachprecipitationgridwithrainydaysdenedasdayswith precipitationgreaterthanthe50thpercentileeventforaWinterDJF,bSpringMAM,c SummerJJA,anddFallSON.Circlesizesindicatethemagnitudeofaverage I max valuesfor eachgridcell.Coloristheinterpolationofaverage I max valuesoverthebasin. 16

PAGE 24

andsummerJJAmapsFiguresIV.2andIV.3,panelsa,b,andc.Thisrelationshipmaybe shownmoststronglyinthefalldueprecipitationfallingasrainatlowerelevationinthewestern halfofthebasinandassnowathigherelevationsinthemajorityoftheeasternhalfofthebasin. PanelsbandcinFiguresIV.2andIV.3indicatetherelationshipmaybeskewedinthespringand summerbyrainonsnoweventsathigherelevationswhichwouldcreatemorerunothanonlya precipitationeventandlarger I max valuesfortheseevents.ThewintermapsFiguresIV.2and IV.3,panela,indicatelarger I max valuesneartheoutletofthebasinaswellasinisolatedareas athigherelevations.Winterresultsarelikelyimpactedbythemajorityoftheprecipitationinthe basinfallingassnow.Winter I max valuesaregreatestatlowerelevationsinthewesternportion ofthebasinbecausethisiswhereprecipitationmayfallasrain,aswellasatseeminglyrandom higherelevationsthroughoutthebasinlikelybecausesnowonslopeswithsouthernexposureswill meltonsunnydays.Winter I max valuesaresmallestatlowerelevationsontheeasternsideofthe basinlikelybecausetheseareasarelikelyinvalleyswhichseeminimalsunlightduringthewinter monthsandsnowwithinthemdoesnotmelt. Themagnitudeofthelaggeddependencybetweenrainfallandstreamow, I max ,variesgreatly betweenseasons.Thisislikelyduetootherfactorswhichplayaroleinthehydrologiccycle,such asevapotranspiration,soilmoisture,andbasingeology,averagestreamowduringeachseason,and precipitationpatternswithineachseason.Duringthespringandsummer,theevapotranspiration withinthebasinwillbegreatestreducingtheamountofrunoreachingthebasinoutlet.Also, theaveragestreamowsduringthespringandsummerwouldbeexpectedtobegreaterthanthose duringfallandwinterduetothetimingofrunofromsnowmelt.Asaresult,precipitationevents inthewinterandfallwouldlikelyproduceamorenoticeablechangeinstreamowwhereasthe changeinstreamowduetoaprecipitationeventduringthespringorsummermaybemaskedby thealreadyhighowsbeingproducedbysnowmelt.Precipitationpatternswithineachseason wouldalsoaectthemagnitudeof I max .Seasonswithmoreprecipitationeventswouldbeexpected tohavehigher I max valuesbecause H P t ,whichistheupperboundfor I Q t ;P t ,willbegreater forseasonswithhigherfrequenciesofrainfall. Theseasonwiththelargestaverage I max valuesisfall.Thisislikelycausedbyacombinationof factorsincludingalowstreamowsinthefallthatmakerunofromprecipitationmorenoticeable, bthetypeofprecipitationinthefallmonthsisstillpredominantlyraininthewesternportion 17

PAGE 25

ofthebasin,orclossesduetotranspirationarereducedtowardstheendofthegrowingseason inthebasin,allowingmoreoftheprecipitationtoreachtheriver.Springistheseasonwith thelowestmutualinformationvaluespossiblybecause,asdiscussedabove,thisisthemainsnow meltseasonandstreamowchangescausedbyprecipitationeventsmaybemaskedbythealready highstreamows.Also,springisthebeginningofthegrowingseasonandalargerpercentageof precipitationmaybetakenupbytranspiration. Asmentionedpreviouslyprecipitationwaslaggedfrom0to7dayswithstreamowbecauseitwas expectedthatprecipitationwouldnotaectstreamowimmediately,especiallyiftheprecipitation gridcellwasfurtherawayfromthestreamgaugeand/orthemainriverchannel.Herewediscuss theaveragedominantlagvalueforeachprecipitationgridcellwithineachseasonforrainfalldened asgreaterthan0.3mmFigureIV.4orgreaterthanthe50thpercentilerainfallFigureIV.5for thegridcell.Inthemaps,largerdotscorrespondtolongerlagtimes.Thewinterandspringseasons panelsaandbshowtheexpectedtrendofshorterdominanttimelagsneartheoutletatthewest endofthebasinandalongtheriverandlargerdominanttimelagsontheupstreamportionsof thebasinandinareasfurtherfromtheriver.Thesummerandfallmapspanelscanddshow thistrendtoalesserextentwhenarainydayisdenedasgreaterthanthe50thpercentileFigure IV.3,andshownotrend,withlargerandsmallerlagtimesinterspersedthroughoutthebasin,when arainydayisdenedasgreaterthan0.3mmFigureIV.2.Thisisasurprisingresultthatmay beexplainedbystormsduringtheseseasonstypicallybeinghighintensityshortdurationstorms. Thistypeofstormandthetopologyandgeologyofthebasinmaycreateeithershortburstsof increasedowswhichmaynotbecapturedadequatelybythedailystreamowdataorrunofrom theseeventsmaybecompletelyinterceptedorabsorbedbydepressions,ora,andsoilsleavingno runo.Therearealsoisolatedpocketsoffasterorslowertimelaggeddependenciesinsomeseasons; thesearelikelyexplainedbythetopographyinthebasinwhichcreateslongerorshorterowpaths totheriverduetotributaryriversorstreamschoosingthepathofleastresistance,whichisnot alwaysthemostdirectpath,totheColoradoRiver. Themagnitudeofthelagtimesalsodiersbetweenseasons.Averagelagtimesforallseasons werebetween2.3and4.7meaningthatonaverageprecipitationtakesatleast2daystomaximally aectstreamow,regardlessofwheretheprecipitationoccurs.Thismaynotseemrealisticassome gridcellsarelocatedveryclosetotheoutletofthebasin,butcanbeexplainedbytheotherfactors 18

PAGE 26

FigureIV.4:Averagelagvaluesproducingthemaximummutualinformationvalueforeachprecipitationgridcellineachseasonforprecipitationdenedasgreaterthan0.3mm. FigureIV.5:Averagelagvaluesproducingthemaximummutualinformationvalueforeachprecipitationgridcellineachseasonforprecipitationdenedasgreaterthanthe50thpercentile precipitationeventforthatgridcell. 19

PAGE 27

whichinuencehowquicklyprecipitationreachesthewaterwaysuchasinterception,absorption intothesoil,andevapotranspirationwhichwouldalldelaytheeectofprecipitationonstreamow especiallyforsmallerstormevents.Lagtimesalsoreectrainfallpersistence,whichmayexplain averagelagtimesbeinggreaterthan2days.Forexample,ifrainfalloccursin3-dayintervals,the maximumdependencymayappearafewmoredaysthanexpectedaftertherstdayofrainfall.The largestaveragelagtimesoccurinthesummer.Evapotranspirationinthesummerisatitshighest andwilltakeupalargerportionoftheprecipitationfromeachstorm,therebyreducingthespeed withwhichtherunoreachesthebasinoutlet.Springhasthewidestrangeoflagtimeslikelydue toprecipitationfallingasrainatthewestendofthebasinneartheoutlet,creatingrelativelyquick lagtimesandsnowtowardstheeastendofthebasinwhichhastomelt,creatinglongerlagtimes. Lagtimesgenerallybecomefasterorremainthesameforeventsgreaterthanthe50thpercentile ascomparedtoalleventsgreaterthan0.3mm.Thisisexpectedsincelargereventssaturatethesoil andcreateoverlandowmorequicklythansmallerevents.Overlandowreachestheriverchannel andincreasesstreamowmorequicklythanrunotravelingthroughthesoil.However,thereare areasofthebasinwherelagtimeincreasesforlargerevents,likethesoutheastcorner.Runofrom theseareasmaybeslowedornegatedbydiversionsordamsontributariestotheColoradoRiver. Watermaybestoredordivertedduringlargeeventswhenjuniorwaterrightsareenactedordam operatorsmayholdbackwaterfromlargeeventstopreventdownstreamoodingorsothewater canbereleasedlaterwhenitisneeded. IV.2TrendsinInformation Trendsinmaximummutualinformation, I max ,foreachgridcellrevealthatthedependency betweenstreamowandprecipitationhaschangedoverthepast60years.FiguresIV.6andIV.7 showstatisticallysignicanttrendsin I max foreachgrid,ineachseason.TheFiguresshowtrends foralldays,dayswithprecipitation, P t =1 ,anddayswithoutprecipitation, P t =0 forthe twodierentthresholdvalues.Gridswithoutdotsdidnothavestatisticallysignicanttrends. FiguresIV.6andIV.7showonlypositivetrendsforalldaysinthewinter,spring,andsummerand negativetrendsforalldaysinfall.Thisindicatesanincreaseinmutualinformation, I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [( , betweenstreamowandprecipitationinthewinter,spring,andsummerandadecreaseinthefall. Thisrelatestotheaverage I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [(tau valuesshowninFiguresIV.2andIV.3.Thedecreasein 20

PAGE 28

FigureIV.6:Seasonaltrendsinmutualinformationforalldays,dayswithprecipitationgreater than0.3mm,anddayswithprecipitationlessthan0.3mm. thelargeraverage I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [(tau valuesinfallandtheincreaseinthesmallervaluesintheother seasonsmaypointtowardsthedierenceininformationbetweenseasonsmovingtoanequilibrium. Additionally,fallaverage I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [(tau valuesreactedoppositelytotheotherseasonswhenthe precipitationdemarcationwaschangedfrom0.3mmtothe50thpercentileevent.Overalltrends inlaggeddependencies, I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [(tau ,inthefallaresimilarlyoppositetheoveralltrendstheother seasons. Overallmutualinformation, I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [(tau ,isthesumofthethemutualinformationforrainy days, I Q t ;P t =1 ,anddrydays, I Q t ;P t =0 ,asshowninEquationIII.2.Weexpectthereto beatendencyfor I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [(tau tobedominatedby I Q t ;P t =0 ,ordayswithoutprecipitation, becausetheremanymoredayswithoutprecipitationthandayswithprecipitation.Anytrendin 21

PAGE 29

FigureIV.7:Seasonaltrendsinmutualinformationforalldays,dayswithprecipitationgreater thanthe50thpercentilerainfall,anddayswithprecipitationlessthanthe50thpercentilerainfall. overallmutualinformationcouldbearesultoftrendsinthetwopartsorcouldbedominated byatrendinoneoftheparts.FiguresIV.6andIV.7showthetrendsin I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [(tau valuesare producedprimarilybypositivetrendsfordayswithoutprecipitationforwinter,spring,andsummer andnegativetrendsfordayswithprecipitationforfall.Thisdemonstratesthatforwinter,spring, andsummerthereisanincreasingpredictabilityof Q t associatedwithknowledgethatitdidnot rain daysagoandforfallthereisadecreasingpredictabilityassociatedwithknowledgethatitdid rain daysago.Similarly,becauseFigureIV.6showsnegativetrendsfor P t =1 forwinter,spring, andsummerwhileFigureIV.7showsprimarilypositivetrendsforthesameseasonsweconcludethe predictabilityofstreamowisdecreasingwiththeknowledgethattherewasasmallprecipitation event daysagoandincreasingwiththeknowledgethattherewasalargeprecipitationevent 22

PAGE 30

daysago.Astherearealmostnotrendsin I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [(tau =0 inthefall,havingknowledgethatit didnotrain daysagointhefalldoesnotreduceanyuncertaintyin Q t .Thesetrendsmaybe duetolessprecipitationfromsmallereventsreachingthebasinoutletduetoevapotranspirationor inltration.Thetrendsmayalsobecontributedtobyapossibleincreaseinintensityoflargeevents leadingtoacorrespondingincreaseintheinuenceoflargeeventson Q t .Anotherpossiblecause forthesetrendsisashiftintheclimatewhichisincreasingordecreasingthenumberofdayswith precipitationineachseasonand/orthenumberofconsecutivedayswithorwithoutprecipitation. Bothofthesemodicationstotheprecipitationregimewouldaectthedependencybetween Q t and P t directlybychangingthetypicalpatternsofotherbasincharacteristicssuchassoilmoistureand evapotranspiration.Forexample,iftherearelongerperiodswithoutprecipitationwithinthebasin streamowmayreturntoitsbaseowlevelforalongerperiodoftime,coincidingwith P t =0 . Thiswouldleadtolessvariabilityin Q t onthesedaysandlarger I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [(tau fordayswithout precipitation.Also,aspreviouslymentioned P t fromsmalleventsmaybecompletelyconsumed byevapotranspirationandabsorptionintothesoilcausingthebaseowconditiontopersistduring andaftersmallevents.Inthisrainfall-streamowcontext,smallrainfalleventsconstitute"noise" intherelationshipwhichdecreasesinformationcontent.Thiswouldreducemutualinformationfor dayswithrainfallwhenthesmallereventsareincludedinthisgroupwhichwouldcreateanegative trendin I Q t ;P t )]TJ/F20 7.9701 Tf 6.587 0 Td [(tau for P t =1 ifthefrequencyofsmallereventsisincreasing. Asdiscussed,theresultsshowincreasingtrendsintheoverallmutualinformationoverthe periodofstudyforwinter,spring,andsummeranddecreasingtrendsforfall.Thepositivetrend forwinter,spring,andsummerindicatesstreamowisbecomingmoredependentonprecipitation fortheseseasonsandthatusinglagged P t topredict Q t duringtheseseasonscanproducemore accuratepredictionsthanpreviously.Dependingontheclimatetrajectory,thepotentialaccuracy ofthesepredictionsmayormaynotcontinuetoincreaseinthefuture.However,becausedays withrainfallshowedanegativeornotrendwhenalleventsproducinggreaterthan0.3mmof precipitationwereincluded,itmaybeprudenttoonlyuseprecipitationeventsaboveacertain thresholdwhenpredictingstreamow.Thisisfurthersupportedbythecomparingpanelsi,j,k, andlFiguresIV.6andIV.7tooneanother.Whilethereareminordierencesbetweenthemaps theyaresimilar,leadingtotheconclusionthatincreasingthe P t thresholdfrom0.3mmtothe50th percentilerainfall,whichmovesdatapointsfrom P t =1 forthe0.3mmthreshold,to P t =0 for 23

PAGE 31

TableIV.1:Comparisonofentropyofprecipitation,mutualinformation,andtrendsinmutual informationforthe0.3mmthreshold SeasonAverage H P t Average I Q t ;P t I/H j I j over60-years j I j /I bitsbits%bits% Winter0.920.0667.2+0.01827 Spring0.910.0535.8+0.01630 Summer0.910.0636.9+0.01930 Fall0.920.0737.9-0.02636 the50thpercentilethresholddoesnotsubstantiallyshifttheresultsforthatcategory.Similarly,for falltherearerelativelyminorchangestotrendsin I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [(tau whenthe P t thresholdisincreased from0.3mmtothe50thpercentilerainfall.However,becausefalltrendresultswerenegativeand dominatedby P t =1 ,orthepresenceofrainfallratherthanitsabsence,weshouldusecarewhen using P t valuesinthefalltopredict Q t asthedependencybetweenthetwoisdecreasing. Theincreasinganddecreasingtrendsin I Q t ;P t )]TJ/F20 7.9701 Tf 6.586 0 Td [(tau FiguresIV.6andIV.7appeartobesmall withaverageabsolutemagnitudesbetweenbetween0.00027and0.00043bitsperyear.However, overour60yearperiodofstudythisamountstoatotalchangeof0.016to0.026bitsTableIV.1. Thisequatestoa27to36%changeintheaveragemutualinformationvaluesTableIV.1.Inother words,wehaveonaveragea30%increaseordecreaseinthepredictabilityofstreamowsbasedon pastprecipitationeventsoverourperiodandareaofstudy. Themagnitudeofmutualinformationvalueswasalsocomparedtothemagnitudeoftheentropiesofprecipitation,whichrepresentstheupperboundofmutualinformation.Theaverage mutualinformationvalueforeachseasonisbetween5.8and7.9%oftheaverageentropyforeach seasonTableIV.1.Thesepercentagesdemonstratethattheprecipitation-streamowdependency isrelativelyweakwithonlyasmallamountoftheuncertaintyinstreamowvaluesbeingremoved withknowledgeofpastprecipitationevents. 24

PAGE 32

CHAPTERV DISCUSSION Thechangingclimateandcorrespondingshiftsinprecipitationpatternswillaectallpartsofthe hydrologiccycleincludingthedependencybetweenstreamowandprecipitation.Thisdependency andanychangesinitwilllikelychangeourcondenceinthestreamowpredictionsproducedby models.Ourabilitytopredictstreamowsbasedonprecipitationeventshaswidereachingeects especiallyasprecipitationeventsbecomemorefrequentandintenseRoque-MaloandKumar,2017; Kunkel etal. ,1999;KarlandKnight,1998.Forexample,ifdamoperatorshavemorecondencein streamowpredictions,theymayleavemorewaterinreservoirstobeusedforagriculture,industry, andmunicipaldrinkingwatersupplies.Further,ifweareabletobetterpredictthetimingand intensityofooding,reservoirscouldbeemptiedpriortolargerunoeventsallowingmorewaterto becaptured,reducingthedestructionofoodevents.Understandingthedependencywillalsobe importantforprotectingecosystemswhichhaveadaptedtocertainpatternsofstreamows.Ifwe understandthedependencybetweenstreamowandprecipitationandhowitischanging,wewill beabletomoreaccuratelypredicthowstreamowpatternswillchangeandattempttolimitthe eectsofthischangeonriverineandoodplainecosystems. Thespatialandtemporaltrendsdetectedinthisstudyprovideastartingpointforunderstandingthedependencybetweenstreamowandprecipitationandhowitmaybeexpectedtochange. However,manyotherwatershedcharacteristicsaecttherelationshipbetweenstreamowandprecipitation.Landuse,soilmoisture,vegetation,andtopologyareafewofthecharacteristicsthat playaroleinhowprecipitationaectsstreamow.Theseothercharacteristicsshouldbestudied alongsideprecipitationandstreamowtofullyunderstandhowtheentiresystemmightrespondto changesinoneormultiplecharacteristics. Itisdiculttostudytemporaltrendsandpatternsusingdatasetsthatareonly60yearsin length.Futurestudiesmaywanttoselectbasinswithlongeravailableprecipitationandstreamow recordstobeabletobetterassesslongtermtrends.Resultsofthisstudymayalsohavebeen inuencedbytheGunnisonRiverjoiningtheColoradoRiverupstreamofthestreamgaugeused inthestudy.TheGunnisonRiverowsarecontrolledmorebydamsanddiversionsincludinga diversionjustbeforeitjoinstheColoradoRiver,butrainfalleventsintheGunnisonRiverbasin, 25

PAGE 33

whichwerenotaccountedforinthisstudy,likelyaectowsintheColoradoRiver.Expanding thisstudytoincludetheGunnisonRiverbasinandcomparingtheresultsmayproduceinteresting ndings. Aspopulationsincreasesthroughouttheworldandclimatechangealtersthehydrologiccyclein unanticipatedways,waterresourcesmanagementwillneedtobeoptimizedtomeetincreasingdemandandchangingconditions.Modelspredictingstreamowwillplayalargeroleinwaterresource management.Usingtoolslikemutualinformationtounderstandthedependencybetweenstreamowandprecipitationcanhelpusbenchmarktheaccuracyofourmodelsandidentifypotentially importanttrends. 26

PAGE 34

REFERENCES AmorochoJ,EspildoraB.Entropyintheassessmentofuncertaintyinhydrologicsystems andmodels. WaterResourcesResearch , 9 ,1511. BrunsellN.Amultiscaleinformationtheoryapproachtoassessspatialtemporalvariability ofdailyprecipitation. JournalofHydrology , 385 ,165172. ChapmanTG.Entropyasameasureofhydrologicdatauncertaintyandmodelperformance. JournalofHydrology , 85 ,111126. ChenM,ShiW,XieP,SilvaVBS,KouskyVE,WayneHigginsR,JanowiakJE.Assessing objectivetechniquesforgauge-basedanalysesofglobaldailyprecipitation. JournalofGeophysical Research:Atmospheres , 113 D4. ClarkPU,AlleyRB,PollardD.NorthernHemisphereice-sheetinuencesonglobalclimate change. Science , 286 ,1104. CoverTA,ThomasJA. Elementsofinformationtheory ,volume2.2ndedition.Wiley. dePRodriguesdaSilvaV,FilhoAFB,AlmeidaRSR,deHolandaRM,daCunhaCamposJHB .Shannoninformationentropyforassessingspacetimevariabilityofrainfallandstreamowinsemiaridregion. ScienceofTheTotalEnvironment , 544 ,330338. EasterlingDR,MeehlGA,ParmesanC,ChangnonSA,KarlTR,MearnsLO.Climate extremes:observations,modeling,andimpacts. Science , 289 ,2068. FicklinDL,StewartIT,MaurerEP.ClimatechangeimpactsonstreamowandsubbasinscalehydrologyintheUpperColoradoRiverBasin. PLOSONE , 8 ,1. GoodwellAE,KumarP.Achangingclimatologyofrainfallpersistenceacrossthecontinental U.S.usinginformation-basedmeasures. SubmittaltoPNAS,October2018 . HejaziMI,CaiX,RuddellBL.Theroleofhydrologicinformationinreservoiroperation learningfromhistoricalreleases. AdvancesinWaterResources , 31 ,16361650. HidalgoHG,DasT,DettingerMD,CayanDR,PierceDW,BarnettTP,BalaG,MirinA,Wood AW,BonlsC,SanterBD,NozawaT.Detectionandattributionofstreamowtiming changestoclimatechangeintheWesternUnitedStates. JournalofClimate , 22 ,3838. KarlTR,KnightRW.Seculartrendsofprecipitationamount,frequency,andintensityin theUnitedStates. BulletinoftheAmericanMeteorologicalSociety , 79 ,231. KunkelKE,AndsagerK,EasterlingDR1999.Long-termtrendsinextremeprecipitationevents overtheconterminousUnitedStatesandCanada. JournalofClimate , 12 ,2515. LettenmaierDP,WoodEF,WallisJR94.Hydro-climatologicaltrendsinthecontinentalUnited States,1948-88. JournalofClimate , 7 ,586. LinsHF,MichaelsPJ4.IncreasingU.S.streamowlinkedtogreenhouseforcing. Eos, TransactionsAmericanGeophysicalUnion , 75 ,281. 27

PAGE 35

MaysDC,FaybishenkoBA,FinsterleS.Informationentropytomeasuretemporalandspatialcomplexityofunsaturatedowinheterogeneousmedia. WaterResourcesResearch , 38 , 49. MillyPCD,WetheraldRT,DunneKA,DelworthTL.Increasingriskofgreatoodsina changingclimate. Nature , 415 ,514. MishraAK,zgerM,SinghVP.Anentropy-basedinvestigationintothevariabilityof precipitation. JournalofHydrology , 370 ,139154. NOAA,OAR,ESRLPSD.CPCuniedgauge-basedanalysisofdailyprecipitationover CONUS. PalmerMA,LettenmaierDP,PoNL,PostelSL,RichterB,WarnerR.Climatechange andriverecosystems:protectionandadaptationoptions. EnvironmentalManagement , 44 , 1053. PearsonK.Mathematicalcontributionstothetheoryofevolution.III.Regression,heredity, andpanmixia. PhilosophicalTransactionsoftheRoyalSocietyofLondon.SeriesA,Containing PapersofaMathematicalorPhysicalCharacter , 187 ,253. PearsonK.Onthecriterionthatagivensystemofdeviationsfromtheprobableinthecase ofacorrelatedsystemofvariablesissuchthatitcanbereasonablysupposedtohavearisenfrom randomsampling. TheLondon,Edinburgh,andDublinPhilosophicalMagazineandJournalof Science , 50 ,157. PechlivanidisIG,JacksonB,McmillanH,GuptaHV.Robustinformationalentropy-based descriptorsofowincatchmenthydrology. HydrologicalSciencesJournal , 61 ,1. Roque-MaloS,KumarP.Patternsofchangeinhighfrequencyprecipitationvariabilityover NorthAmerica. ScienticReports , 7 . SangYF,SinghVP,HuZ,XieP,LiX.Entropy-aidedevaluationofmeteorologicaldroughts OverChina. JournalofGeophysicalResearch:Atmospheres , 123 ,740. SchneiderSH.Thechangingclimate. ScienticAmerican,adivisionofNatureAmerica, Inc. , 261 ,70. SenPK.EstimatesoftheregressioncoecientbasedonKendall'sTau. Journalofthe AmericanStatisticalAssociation , 63 ,1379. ShannonCE.Amathematicaltheoryofcommunication. BellSyst.Tech.J. , 27 ,3793. TheilH. Arank-invariantmethodoflinearandpolynomialregressionanalysis.In:RajB., KoertsJ.edsHenriTheil'scontributionstoeconomicsandeconometrics.Advancedstudiesin theoreticalandappliedeconometrics ,volume23.Springer,Dordrecht. USGS.USGSSurface-waterdailydataforthenation. XieP,ChenM,YangS,YatagaiA,HayasakaT,FukushimaY,LiuC.Agauge-based analysisofdailyprecipitationoverEastAsia. JournalofHydrometeorology , 8 ,607. 28