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Spatio-temporal cokriging of temperature and precipitation in Africa

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Title:
Spatio-temporal cokriging of temperature and precipitation in Africa
Creator:
Ndiaye, Sokhna Mariama
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 Mathematical and Statistical Sciences, CU Denver
Degree Disciplines:
Applied mathematics
Committee Chair:
Cobb, Loren
Committee Members:
Bennethum, Lynn
Lodwick, Weldon A.

Notes

Abstract:
The research presented in this thesis focuses on two alternative statistical methods for spatial interpolation (universal kriging and cokriging), as applied to the problem of interpo- lating temperatures within the Sahara desert from weather station data at the periphery of the desert. Universal kriging is a direct interpolation method, while cokriging is an indirect method that uses covariables of the variable of interest to generate estimates. Kriging can be used to predict unknown temperatures in large geographic regions from measured tempera- tures in discrete locations that surround the unmonitored area. These kriging techniques are used to interpolate temperatures across the continent of Africa, with a focus on the Sahara Desert, for July of each year of data available from 2007 to 2012.

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University of Colorado Denver
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Auraria Library
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Copyright Sokhna Mariama Ndiaye. 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.

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SPATIO-TEMPORALCOKRIGINGOFTEMPERATUREANDPRECIPITATIONIN AFRICA by SOKHNAMARIAMANDIAYE BS,MetropolitanStateUniversityofDenver,2009 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof MasterofScience AppliedMathematics 2018

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ThisthesisfortheMasterofSciencedegreeby SOKHNAMARIAMANDIAYE hasbeenapproved fortheAppliedMathematicsProgram by LorenCobb LynnBennethum WeldonA.Lodwick ii

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NDIAYE,SOKHNAMARIAMAM.S.,AppliedMathematics Spatio-TemporalCokrigingofTemperatureandPrecipitationinAfrica ThesisdirectedbyResearchAssociateProfessorLorenCobb ABSTRACT Theresearchpresentedinthisthesisfocusesontwoalternativestatisticalmethodsfor spatialinterpolationuniversalkrigingandcokriging,asappliedtotheproblemofinterpolatingtemperatureswithintheSaharadesertfromweatherstationdataattheperipheryof thedesert.Universalkrigingisadirectinterpolationmethod,whilecokrigingisanindirect methodthatusescovariablesofthevariableofinteresttogenerateestimates.Krigingcanbe usedtopredictunknowntemperaturesinlargegeographicregionsfrommeasuredtemperaturesindiscretelocationsthatsurroundtheunmonitoredarea.Thesekrigingtechniquesare usedtointerpolatetemperaturesacrossthecontinentofAfrica,withafocusontheSahara Desert,forJulyofeachyearofdataavailablefrom2007to2012. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:LorenCobb iii

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DEDICATION ThisthesisisdedicatedtomysponsorCollegeofLiberalArtsandSciencesAdvisorJe Schweinfest.Ialsothankmyfamily,friends,JoeCahn,EvalynVanAllen-ShalashandSaadi Simaweforalltheirsupportduringmystudies.IthankmyadvisorLorenCobbforallhis supportandencouragements. iv

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ACKNOWLEDGMENTS IthankJeSchweinfestforallhissupportsincemyundergraduatestudies,andtomyacademicadvisorResearchAssociateProfessorLorenCobbforallhissupportaboutassisting myresearch.IacknowledgetheInternationalStudentandScholarServicesandtheExperimentalLearningCenter.IalsothanktheMathematicsdepartment,especiallyJanMandel andMikeFerrara,andmycommittee. v

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TABLEOFCONTENTS I.INTRODUCTION................................... II.KRIGING........................................ II.1OrdinaryandRegressionKriging.......................... II.2ParameterEstimation................................ II.2.1Assumptionsaboutthecovariancefunction................... II.2.2EmpiricalSemivariogram............................. 1 II.3DataAnalysiswithRegressionKriging....................... 1 II.3.1Regressionstep................................... 1 II.3.2Theinterpolationstep............................... 1 II.3.3RegressionKrigingresultsshowninMaps.................... III. COKRIGING OF TEMPERTURE ELEV ATION AND PRECIPIT ATION IN AFRICA 2 III.1Assumptionsaboutthedata............................. 2 III.2CokrigingModel................................... 2 III.3CrossVariogramModel............................... 3 III.4 Cokriging results shown in Maps . . . . . . . . . . . . . . . . . . . . . . . . . . IV. REGRESSION OF KRIGING AND COKRIGING EXPERIMENTAL RESULTS 4 IV.1Motivation....................................... 4 IV.2DiscussionoftheHighRoot-Mean-SquareError(RMSE)resultintheUniversal Krigingtechnique................................... 5 V.EFFECTOFDISTANCETOWEATHERSTATIONSONKRIGINGTECHNIQUES........................................ 5 vi

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V.1Motivation....................................... 5 V.2ResultsandDiscussionfromExperimentIIandExperimentIII......... 5 V.3PredictionoftheSaharaDesertAverageTemperatureoversixYearsofData usingCokriging.................................... VI. DISCUSSION OF ELEV ATION DA TA . . . . . . . . . . . . . . . . . . . . . . . . VII. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 vii

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TABLES Table IV.1ExperimentalresultsofCokrigingCK.......................47 IV.2ExperimentalresultsofCokrigingCK,Continued...............48 IV.3ExperimentalresultsofUniversalKrigingUK..................49 IV.4ExperimentalresultsofUniversalKrigingUK,Continued...........50 V.1ExperimentII:EstimatedMeanTemperatureJuly2008.............54 V.2ExperimentIII:EstimatedMeanTemperatureJuly2008.............54 V.3EstimatedMonthlyMeanTemperatureMNTM.pred..............55 VI.1SampleElevationDatabyYear...........................58 viii

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FIGURES Figure I.1Anomalydierencefrom1961-1990......................... II.1SemivariogramModels,July2007andJuly2008................. 1 II.2SemivariogramModels,July2009andJuly2010................. 1 II.3SemivariogramModels,July2011andJuly2012.................. 1 II.4AvailableWeatherStationsinourRegionofStudy................ 1 II.5RegressionKrigingResultJuly2007........................ II.6RegressionKrigingResultJuly2008........................ II.7RegressionKrigingResultforJuly2009...................... II.8RegressionKrigingResultJuly2010........................ 2 II.9RegressionKrigingResultJuly2011........................ 2 II.10RegressionKrigingResultJuly2012........................ 2 III.1CrossvariogramAnalysisJuly2007......................... III.2CrossvariogramAnalysisJuly2008......................... 3 III.3CrossVariogramAnalysisJuly2009........................ 3 III.4CrossVariogramAnalysisJuly2010........................ 3 III.5CrossVariogramAnalysisJuly2011........................ 3 III.6CrossVariogramAnalysisJuly2012........................ 3 III.7AvailableWeatherStationsinourRegionofStudy................ 3 III.8CokrigingResultJuly2007............................. III.9CokrigingResultJuly2008............................. III.10 Cokriging Result July 2009 ............................. III.11 Cokriging Result July 2010 ............................. 4 ix

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III.12 Cokriging Result July 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 III.13 Cokriging Result July 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 IV.1RemovedWeatherStationsfortheexperimentalAnalysis............. 4 IV.2RemovedWeatherStationsfortheexperimentalAnalysis............. 4 IV.3RemainingWeatherStationsfortheexperimentalAnalysis............ V.1PredictedJulyMeanTemperatureoftheSaharaDesert............. x

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I. INTRODUCTION Kriging, which is also known as Gaussian process regression, is comprised of at least four dierent statistical methods: ordinary kriging, universal kriging, regression kriging, and cokriging. Kriging is a spatial interpolation method that was frst proposed as a theory by Danie G. Krige, a South African mining engineer[22]. He developed the initial theory of kriging, and helped create the discipline of geostatistics for the purpose of evaluating mineral resources. Later on, Georges Matheron, in his 1969 paper, proposed a new mathematical approach to the theory of universal kriging by establishing a statistical model for universal kriging. In the universal kriging model, he considered the new estimated values as random functions in terms of linear combinations of available variables[25]. The kriging variance increases with distance, and Matheron proposed a universal kriging model for both discrete and continuous cases. There are many applications of the kriging methods in geosciences. For example, to cite a few, Jean-Paul Chiles used kriging to estimate the average of head surface in order to design a piezometric network for supervision of a waste storage site[1]. Kriging was also used to analyze heavy metal sources in soil. In this case, kriging helped to create spatial distribution maps for the interpolation of unknown sources of heavy metal contamination[13]. The literature contains very few spatial studies of temperature using universal kriging and/or cokriging. T. Ishida and S. Kawashima used the cokriging method to estimate surface air temperature using elevation data[15]. In 1997, Andrew E. Long and Donald E. Myers proposed a new cokriging model and showed a case where cokriging reduces to kriging[23] . In 1984, Donald E. Myers and James Carr concluded that cokriging which is a minimum variance(joint estimation) provides better estimation than separate estimation (universal kriging) when the variable of interest are spatially correlated and intercorrelated[27]. The Sahara Desert is the main focus for this study because it is important to be able to estimate the temperatures across its vast area with no available weather stations. That will help people living in the local villages to have a better understanding of the temperature changes in the whole region. The Sahara desert is one of the largest deserts in the world, and it constitutes 10 percent of the entire continent of Africa. Figure I.1 below shows that the Sahara desert is a very signifcant land area without weather stations; therefore, over large regions there are no recorded temperatures. The white areas in Figure I.1, including 1

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theSaharaDesert,aretheareastobeinterpolatedwithestimatedmeantemperatureusing universalandcokrigingstatisticalinterpolationmethods. Theprimaryworld-widesourceforhistoricaltemperaturereadingsacrosstheglobeis theGlobalHistoricalClimatologyNetwork(GHCN).Theocialsummaryofthisdataset specifcallyidentifestheSaharadesertasaregionwithevidentdatagaps: This data set contains monthly temperature, precipitation, sea-level pressure, and station-pressure data for thousands of meteorological stations worldwide. The database was compiled from pre-existing national, regional, and global collections of data as part of the Global Historical Climatology Network (GHCN) project, [. . . ]. It contains data from roughly 6000 temperature stations, 7500 precipitation stations, 1800 sea level pressure stations, and 1800 station pressure stations. [. . . ] Data gaps are evident over the Amazon rainforest, the Sahara desert , Greenland, and Antarctica."[7] FigureI.1: Anomalydierencefrom1961-1990 2

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ThetechniquesofuniversalkrigingandcokrigingareappliedoverthecontinentofAfrica anditssurroundingstopredicttheunknownmeantemperaturesfromtheavailabledata recordedfromover170weatherstationsprovidedbyNOAA.Datausedforthespatialanalysisinthispaperincludemonthlymeantemperatureandothervariablesrecordedatweather stationsprovidedbytheNOAANationalClimaticDataCenterfrom2007to2012,where themonthofJulyforeachyearisselectedtoapplyKrigingtechniques. First,theuniversalkrigingtechniqueisgoingtobeusedtointerpolatetheareaswithout temperaturemeasurements.Then,wewillusecokrigingtoestimatethemeantemperature overthecontinentofAfrica.Wewillperformanexperimenttovalidatebothmethodsby estimatingknowntemperaturesfromrandomlyselectedweatherstationsbyremovingthem fromthedatausedtoperformkriging.Finally,wewillperformanotherexperimentanalyzing theeectofthedistancebetweenweatherstationsandagivenunknownarea. ThemeantemperaturesoftheSaharaDesertisestimatedfrom2007to2012usingcokriging, andtheissuesinourrawdatathataecttheaccuracyofbothuniversalandcokriging methodsareaddressed. 3

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CHAPTERII KRIGING Theinterpolationmethodknownas\kriging"isusedinthisthesistoestimateaverage monthlytemperaturesoverregionswithintheSaharaDesertwheresystematicobservations ofgroundtemperaturearenotrecordedbecauseoflackofweatherstations.Mostofthose regionsareinaccessibleanduninhabited;evenamongtheverysmallnumberofvillages locatedwithintheSaharaDesertthereareveryfewweatherstations. Krigingisageostatisticalprocedureforoptimallyinterpolatingthevaluesofavariable ofinterestofaregionofspace,givenobservationsofthevariableatascatteredsetofpoints inthatregion.Regressionkriging,alsoknownasuniversalkriging,isanextensionofthe interpolationmethodwhichincludesthelineareectsofanumberofindependentvariables inordertoimprovetheaccuracyofinterpolation. Inthisapplication,thevariableofinterestisthegroundtemperaturethroughoutthe SaharaDesert.Availableindependentvariablesforregressionkrigingincludelatitude,longitude,elevation,andprecipitation. Aninterestingmultivariatevariantofregressionkrigingisalsoavailableforthisresearch, namedcokriging.Inthisvariant,thespatialcovariancebetweenthevariableofinterest(in thiscasetemperature)andotherspatialvariables(e.g.precipitation)isexploitedina multivariatecontexttogenerateresultsthatinsomecircumstancescanbeevenbetterthan canbeachievedwithregressionkriging.

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Thefollowingsubsectionsprovideabriefmathematicaldescriptionofallofthesevariants ofthekrigingmethod. II.1OrdinaryandRegressionKriging Regressionkrigingisdescribedhereasitwasoriginallydenedin1995byOdeh[14],and recentlysummarizedbyGaetanandGuyon[6],withslightlydierentnotation. Let Y s bea2Drandomeld,whichistosayareal-valuedrandomvariabledened onatwo-dimensionalregionofspace s 2 R 2 .Wearegivenasetofobservationsof Y at n dierentlocationsinspace, f Y s 1 ;Y s 2 ;:::;Y s n g .Thepurposeofkrigingisto providethebestlinearunbiasedpredictorBLUPof Y atanyarbitrarylocation s 0 ,based ontheobserveddataandcertainassumptionsabouttheformofthetwo-dimensionalspatial covariancefunction\050 u;v for Y .Ingeneralthelocation s 0 ofthepredictedvalueof Y doesnotcoincidewithanyoftheobservationlocations s 1 ;:::;s n ,sokrigingisaformof interpolation. Ifwealsohaveobservationsof,say, d spatially-distributedauxilliaryvariables linespce f X 1 s ;X 2 s ;:::;X d s g ,atspatiallocations f s 1 ;s 2 ;:::;s n g ,thentheparticular formofinterpolationisknownvariouslyas regression or universal or KED kriging.The threetermsareequivalent,inthesensethattheircalculationsyieldthesamenumerical predictions.Thestatisticalmodelforregressionkrigingis Y s = + d X i =1 i X i s + n X k =1 k s Y s k + ; II.1 5

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where israndomerrorusuallyassumedtobedistributedas N ; 2 ,thefunctions i : R 2 ! R areregressioncoecientsforthevariables X i , istheregressionconstant, andthefunctions k : R 2 ! R arethekrigingweightsforlocation s .Theinterpolation parametersatanylocation s mustsatisfythelinearrestriction P n k =1 k s =1.Thefamilyof interpolationmodelsdescribedbyII.1isstrictly univariate ,inthesensethatthedependent variablemustbescalar-valuedratherthanvector-valued. Iftherearenoauxiliaryvariables,i.e.if d =0,thenII.1reducestotheinterpolation modelknownas ordinary kriging,with settothemeanofallthespatialobservations, = =n P n k =1 Y s k .However,if isassumedtobepreciselyzero,thenthemodelis customarilyknownas simple kriging. Ontheotherhand,ifall n observationsareconcentratedonasinglepointinspacebut d> 0,thenthemodelreducesto multipleregression [6],and istheregressionconstant.In summary,theabovestatisticalmodelforregressionkrigingencompasses,asspecialcases, simplekriging,ordinarykriging,universalkriging,andmultipleregression. Insomearticlesonkriging,modelII.1iscalled krigingwithexternaldrift KED.In theterminologyofgeostatistics, externaldrift referstotheportionofthemodelII.1that describesthedependentvariable Y atanypointinspaceasafunctionoftheindependent variables,withouttheinterpolationcomponent: Drift[ Y s ]= + d X i =1 i X i s : II.2 Theconceptof`drift'canbeextendedtospatio-temporalmodelsinseveral 6

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ways.Time,oraperiodicfunctionoftimeformodelingseasonality,canbeaddedtothe listofindependentvariables.Alternatively,theentirekrigingmodelcanbeembeddedinan autoregressivetimeseriesframework. Intheapplicationofkrigingthatisthesubjectofthisthesis,theindependentvariable isthe meanmonthlytemperature recordedinvariousAfricanmeteorologicalstations,and theexternaldriftvariablesaretheprecipitation,elevation,latitude,andlongitudeateach station. Intheapplicationofregressionkrigingreportedinthisstudythedependentvariableis meanmonthlytemperatureateachweatherstation,whileprecipitationistreatedasoneof theexternaldriftvariables.Analternativestatisticalmodelcokriging,canbeconstructed fromthesamevariablesbyallowingthedependentvariabletobemultivariate.Inthisstudy thecokrigingdependentvariablewouldbeabivariatefunctionofspace,namely 2 6 4 Y 1 s Y 2 s 3 7 5 = 2 6 4 Temperature s Precipitation s 3 7 5 ; andtheexternaldriftvariableswouldbeelevation,latitude,andlongitude. CokrigingwillbedenedandcomparedtoregressionkriginginSectionII.3below. II.2ParameterEstimation GivenanestimateddriftfunctionDrift[ Y s ],wecandenetheresidualsurfaceafter removalofthedrift,as: Z s = Y s )]TJ/F15 11.9552 Tf 11.955 0 Td [(Drift[ Y s ] ; II.3 wherethedrifttermisgivenbyII.2. 7

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Theparameterestimationtaskforkrigingisthentondoptimalestimatesforthe interpolationparameters 1 ;:::; d ,usingthecalculatedresiduals, Z s 1 ;:::;Z s n .Itis easytondordinaryleastsquaresOLSestimatesfortheseparameters,buttheresulting estimatorsaretypicallybiasedanestimator ^ foraparameter is biased ifE[ ^ ] 6 = .A commonandeectivewaytopreventthisbiasistobuildaconstraintintotheestimation equations,usingaLagrangemultiplier.Theresulting d +1normalequationscanbewritten inmatrixformasfollows. Letthe covariancefunction )-326(fortheresidualrandomeld Z be \050 u;v =Cov[ Z u ;Z v ] : II.4 Usingthiscovariancefunction,thenormalequationsforndingtheinterpolatedvalueat anygivenlocation s are: 2 6 6 6 6 6 6 6 6 6 6 6 4 \050 s 1 ;s \050 s 2 ;s . . . \050 s n ;s 1 3 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 6 4 \050 s 1 ;s 1 \050 s 1 ;s 2 ::: \050 s 1 ;s n 1 \050 s 2 ;s 1 \050 s 2 ;s 2 ::: \050 s 2 ;s n 1 . . . . . . . . . . . . . . . \050 s n ;s 1 \050 s n ;s 2 ::: \050 s n ;s n 1 11 ::: 10 3 7 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 4 1 s 2 s ::: n s 0 3 7 7 7 7 7 7 7 7 7 7 7 5 : II.5 Thenalequationinthisarrayenforcestheconstraint n X i =1 i s =1,whichissucientto maketheestimatorsunbiased[5]. Variousmethodsandassumptionsforestimatingthevaluesofthecovariancefunction usedintheabovenormalequationsareoutlinedinthenext 8

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section. Once these estimates are in hand, the kriging weights 1 ( s) ; : : : ; d ( s) are then estimated by solving the normal equations ( II.5). In its bare essentials, this is the core idea behind all kriging procedures. The estimated kriging weights for location s, 1 ( s) ; : : : ; d ( s), are used to calculate the interpolated value of Z( s). For regression kriging, the calculated drift value at s is then added to the interpolated value of Z( s). The result is the fnal interpolated value for the random feld Y at location s. II.2.1 Assumptions about the co v ariance function All variants of kriging require the estimation of (or, in some cases, the exact knowledge of) the spatial covariance function of the dependent variable Y ( s), defned by C (u;v )=E[Z (u)Z (v )] ; (II.6) where Z (s)= Y (s) )Tj/T1_0 11.955 Tf11.955 0 Td[(Drift[ Y (s)] : (II.7) Notethatthecovariancefunctionissymmetric: C (u;v )= C (v;u). Estimationofthespatialcovariancefunction C fromdataisespeciallyeasyandconvenientif Z canbeassumedtobeboth stationary and isotropic.Stationarity,inthiscontext, meansthat C (u;v )= C (u + h;v + h); foranyspatialdisplacement h.Isotropy,ontheotherhand,meansthatthecovarianceis dependentonlyonthedistancebetweenlocations u and v : C (u;v )= C (0; ju )Tj/T1_1 11.955 Tf11.955 0 Td(v j);

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where jj istheordinaryEuclideandistancebetweenthevectors u and v . TheSaharaandArabiandesertsdisplaygoodevidenceforisotropyandstationarity oftemperature,quiteunlikelandscapeswithlinearmountainrangessuchasinColorado, whichishighlyanisotropic.Ifisotropyandstationarityapply,thenastatisticknownasthe semivariogram canbeusedtoecientlyestimatethecovariancefunction.Thesemivariogram r (h)ofarandomfeld Z (s)thatisbothstationaryandisotropicisdefnedforany non-negativedisplacement h as 2r (h)=Var[ Z (s) )Tj/T1_2 11.955 Tf11.955 0 Td(Z (s + h)] = C (s;s) )Tj/T1_2 11.955 Tf11.955 0 Td(C (s;s + h)+ C (s + h;s + h) )Tj/T1_2 11.955 Tf11.955 0 Td(C (s + h;s) =2 C (0; 0) )Tj/T1_2 11.955 Tf11.955 0 Td(C (s;s + h) )Tj/T1_2 11.955 Tf11.956 0 Td(C (s + h;s)(bystationarity) =2 C (0; 0) )Tj/T1_2 11.955 Tf11.955 0 Td(C (0;h) )Tj/T1_2 11.955 Tf11.956 0 Td(C (h; 0)(byisotropy) =2 C (0; 0) )Tj/T1_0 11.955 Tf11.955 0 Td(2C (0;h)(bysymmetry) ) r (h)= C (0; 0) )Tj/T1_2 11.955 Tf11.955 0 Td(C (0;h): .Thuswhenbothstationarityandisotropyapply,wehavetheidentity r (h) C (0; 0) )Tj/T1_2 11.955 Tf11.955 0 Td(C (0;h): (II.8) II.2.2EmpiricalSemivariogram Ifthemodelisstationaryandisotropic,wecanusethefollowingequationtocompute thesemivariogramvaluefromourreducedinputdataset. ^ r (h)= 1 2n (h) n(h) X i=1 (Z (s i ) )Tj/T1_2 11.955 Tf11.956 0 Td(Z (s i + h)) 2 ; (II.9) 1

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where ^ r (h)istheempiricalsemivarianceoftemperatureatdistance h, n(h)isthenumberofpairsofdatalocationsseparatedbydistance h,and Z (s i )istheobservedvalueof Y atlocation s i ,lesstheestimateddriftatthislocation. Givenanempiricalsemivariogram^ r andtheassumptionsofstationarityandisotropy, thefullcovariancefunction C canbereadilyestimatedfromthisidentity: ^ C (u;v )= ^ C (0; 0) )Tj/T1_0 11.955 Tf12.39 0 Td(^ r (ju )Tj/T1_1 11.955 Tf11.955 0 Td(v j); = ^ Var[ Z ] )Tj/T1_0 11.955 Tf12.39 0 Td(^ r (ju )Tj/T1_1 11.955 Tf11.955 0 Td(v j): 1

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II.3DataAnalysiswithRegressionKriging Regressionkriging,alsoknownasuniversalkriging,isatwo-stepprocedure.Thestatisticalmodelisstatedin(II.1),andrepeatedhereforconvenience: Y (s)= + d X i=1 f i X i (s)+ n X k =1 k (s)Y (s k )+ : (II.10) Inthefrststeptheparameters and ff i g d i=1 areestimatedbymultiplelinearregression. Thisistheregressionstep.Itisperformedonlyonce. Inthesecondstep,foranydesiredlocation s,theparameters f k (s)g n k =1 areestimated fromtheregressionresiduals,usingsimplekriging.Inotherwords,adistinctsetof n kriging weightsmustobtainedforeachdistinctlocationwhereaninterpolatedvalueof Y isdesired. Thisistheinterpolationstep. Thedetailsofourimplementationofthisalgorithmfollow. II.3.1Regressionstep Intheregressionstepofregressionkriging,weemploymultipleregressiontoestimate theparametersoftheexternaldriftfunction. Inthemonthlydataforthisstudy,providedbytheU.S.NationalClimaticDataCenter, thedependentvariable Y (s)isthereportedmeantemperaturefortheweatherstationat location s,andthedriftfunctionisgivenby m (s)= + d X k =1 f k X k (s): (II.11) Theexternaldriftvariablesavailableinthisdatasetare: Longitude, Latitude, Elevation, 1

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Precipitation, Extrememaximumdailytemperature, Extrememinimumdailytemperature. Fortheregressionstepoftheregressionkrigingprocedure,Iperformedastepwiselinear regressionwithbackwardselection,andselectedvariableswithp-valueslessthan5%.The stepwiseprocedureeliminatedseveraloftheabovedriftvariables,leavingLongitude,Latitude,Elevation,andPrecipitation.Thosearethevariablesfromwhichthedriftfunctionfor monthlymeantemperatureiscomputed. TheNCDC/NOAAdatasetforalargeareaencompassingboththeSaharaandneighboringregionswasusedtoestimatetheempiricalsemivariogram.Using(II.9),theestimated semivariogramfortemperaturewascomputedforeachyearofdata(FigureII.1,II.2and II.3).Theseannualestimatedsemivariogramsaredisplayedbelow.NotethatoursemivariogramgraphsareslightlydierentfromwhatMatheronsuggestedinhis1969paper(he suggestedgraphingtheresidualvariogram). 1

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FigureII.1: SemivariogramModels,July2007andJuly2008 1

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FigureII.2: SemivariogramModels,July2009andJuly2010

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FigureII.3: SemivariogramModels,July2011andJuly2012 1

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II.3.2Theinterpolationstep ThemapbelowshowstheweatherstationsinAfricaandsurroundingregionsthatwere availableinJuly2008,andtheareaswithoutweatherstationsthataregoingtobepredicted from2007to2012usingtheregressionkrigingtechnique.Afterfttingourdatatoour semivariogrammodelforJuly2008,newinterpolatedvaluesarecomputedtoestimatethe meantemperatureinemptyareasinJuly2008.Thesameprocessisrepeatedfvemore timestoestimatethemeantemperaturefrom2007to2012,choosingthemonthofJulyfor eachyear. FigureII.4: AvailableWeatherStationsinourRegionofStudy 1

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II.3.3RegressionKrigingresultsshowninMaps HerearethegraphsforregressionkrigingoverthecontinentofAfricafrom2007to2012 usingequations(II.8)and(II.10). FigureII.5: RegressionKrigingResultJuly2007 1

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FigureII.6: RegressionKrigingResultJuly2008

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FigureII.7: RegressionKrigingResultforJuly2009 2

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FigureII.8: RegressionKrigingResultJuly2010 2

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FigureII.9: RegressionKrigingResultJuly2011 2

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FigureII.10: RegressionKrigingResultJuly2012 2

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III COKRIGING OF TEMPERTURE ELEV A TION AND PRECIPIT A TION IN AFRICA Mean temperatures over six years was previously predicted by using universal kriging, which used the available mean temperature data. In this chapter, we are going to use other correlated variables longitude, latitude, elevation and total precipitation in order to predict the temperature. Cokriging method is explored using covariables (the predictands) of temperature, which are elevation and precipitation, to help calculate predictions for poorly sampled variables. III.1 Assumptions about the data Like universal kriging, in cokriging we also assume that the spatial covariance function is isotropic (directionally invariant). In this research, the semivariances are computed in terms of the distance h , which is called gamma in equation (III.1). It has been claimed that the anisotropy assumption is more accurate when estimating elevation data, but will produce more error . Cokriging is a spatial interpolation as well, and a multivariate variant of our previous spatial interpolation method. In this section, temperature is estimated by knowing its covariables. In cokriging, the variable of interest, can be less densely sampled than its covariables. Therefore, we assume that we have very little information about the state of the temperature in our region of study and well-sampled covariables (Precipitation and Elevation) that are relatively correlated to temperature. 2

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Cokrigingisapointinterpolationmethodwhichconvertsacollectionofpointsintoa continuousmap.Itestimatesthetemperatureusingtheavailablevariablessuchastotal precipitationandelevation.Thiscorrelationimprovesestimationoftheunder-sampledvariable,whichistemperatureinthisresearchpaper.Theobservedvaluesofelevationand totalprecipitation(TCP)areusedtoimproveestimates(predictions)ofthemonthlymean temperature(MNTM)[25] III.2CokrigingModel Cokrigingisarecenttheorywithoutawellestablishedequationmadetodescribeor applycokriging.However,manyequationsorcokrigingmodelswereproposed.Longand Myersproposedoneofnewestmodelforcokriging[23].In1996,AndrewE.LongandDonald E.Myersproposedamodeltocomputethecokriginginterpolationtechnique.Theystated thattherandomfunctionzcanbeexpressedinthissimplerequation, Z (x 0 )= N X i=1 )Tj/T1_5 7.97 Tf7.314 4.936 Td(T i Z (x i ): (III.1) )Tj/T1_5 7.97 Tf7.314 4.338 Td(T i aretheweightedmatricessatisfyingthefollowingconditions. F l (x 0 )= N X i=1 F l (x i ))Tj/T1_5 7.97 Tf11.866 4.936 Td(T i : (III.2) l =1 ;:::::;p . F l arecomposedof p matrices,andcanbeexpressedasafunctionofindependent functionsconstitutingabasisforthedriftsurface[23].Theweightmatricessatisfyingthe conditionsinequation(III.2)canbe 2

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expressedasafunctionoflinearequationsinvolvingthevariogrammatrixfunction V : V (x i )Tj/T1_1 11.955 Tf11.955 0 Td(x)= N X i=1 V (x i )Tj/T1_1 11.955 Tf11.955 0 Td(x j ))Tj/T1_2 7.97 Tf11.867 4.936 Td(T i + N X l =1 F l (x i ) i : (III.3) Theirmotivationofhavinganewformulaisbasedonthefactofreducingthesizeof thematricesinvolved.AccordingtoMyersandLong,thesizeofmatricesinvolvedinthe modelofcokrigingcreateissueswhenapplyingofcomputingcokrigingtechniques.Inlinear algebra,Gaussianeliminationmethodisoneofthebestmethodsforsolvingmatrix,andit wasusedfortheprogramcokrige.However,LongandMyersfoundittobeapoorselectionwhencomputingcokrigingmethodsbecauseitdoesnotfxthetheproblem,whichis reducingthesizeofmatricesinvolved.Asaresult,theycreateanewprogramthatwillhelp reducethesizeofmatricesinvolved[23]. Forthepurposeofmakingthemodelofcokrigingsimplerforapplicationpurposes,the weightedmatricesstatedinthemodelaboveequation(III.1)areexpressedintheformofa linearcombination.Themodelofcokrigingusedinthisresearchpaperstatesthattheunknownvalue Z 0 (meantemperatureinareawithnoweatherstation)isalinearcombination ofNvaluesofelevationandtotalprecipitation(spatialvariables).Notethatweneedatleast onecovariableoftheunknownvariableinordertoperformcokrigingtechniques.Besides longitudeandlatitudetodeterminethelocation,thecovariablesofmeantemperaturesare considered.Elevationandtotalprecipitationaretheindependentvariableinthisresearch toperformcokriging.Ingeneralthehighertheelevation,thelowerthetemperature,anda similarrelationshipholdsfortotalprecipitationandtemperature. 2

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Basedonthetheoryofcokriging,equation(III.4)isusedtocomputethecokrigingresults foreachmonthofJulyfrom2007to2012.Thesimplerthemodel,thebetterwecanapply thetheoryofcokriging,andthemodelproposedandusedinthisthesisasdescribedbelow. Consideringthatthereiscorrelationbetweentemperature,elevationandtotalprecipitation, wecanexpressourpredictedvariablemeantemperatureasfollows: Assumingthattheunknowntemperatureisalinearcombinationofelevationandtotal precipitation Z = a + bZ 1 + cZ 2 + dZ 3 ::: wherea,b,carecoecients,andtheycanbeexpressedina matrixformaswell. Let Z 1 betheestimatedvalueofthetotalprecipitation,expressedas Z 1 (x 0 )= n X k =1 k p k (x)+ e 0 (x): (III.4) Let Z 2 bethepredictedvalueofelevationandexpressedas Z 2 (x 0 )= n X k =1 f k p k (x)+ e 0 (x): (III.5) ThenthepredictionofZ(MeanTemperature)canbeobtainedbyusingthismodelof cokrigingequationbelow Z (x 0 )= 2 X k =1 r k Z k (x)+ s 0 (x) (III.6) Where x 0 isunknownarea x arelocationswherewehaveinputdata, r f istheweight assignedtothecovariable Z k ,and s 0 (x)iserror. 2

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Z (x 0 )isassumedtobearandomfunction,andistheestimatedvalueofmeanmonthly temperatures. Z k (x)isanavailablevariable,andisassumedtobearandomvariable. ThenewestimatedvalueoftemperaturesthroughoutthecontinentofAfricaiswritteninthe formofalinearcombinationofitscovariablesinequation(III.6).Eachcovariablesatisfes theconditionsinequation(III.4)and(III.5)Itmightbenecessarytoknowhowtocompute universalkrigingwhenperformingcokrigingeventhoughthetwomodelcanbeindependent becausetheprocessofbuildingcokrigingsystemissimilartothedevelopmentofuniversal krigingmodel.Tocomputethecokrigingequationforconcreteresults,wewillproceedon buildingacrossvariogrammodel. 2

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III.3CrossVariogramModel Unlikeuniversalkriging,incokrigingacrossvariogramisusedinsteadofsemivariogram. Variogramswereusedinuniversalkrigingtodescribethedependencyofcovariance(oftemperature)ondistance.Incokriging,however,weusevariogramstodescribethecovariance betweendierentspatialvariableselevation,totalprecipitationandmeantemperature;in thiscase,itiscalledacrossvariogram.Acrossvariogramcanbeexpressedasafunctionof thedistanceanddirectionseparatingtwolocations,usedtoquantifythecrosscorrelation.It defnesthecovariancebetweentwodierentspatialvariablesindierentlocations.Crosscovarianceisusedtocomputethecokrigingtechniquebyusingweightedleastsquaresmethod thatweusedinuniversalkriging.Fromtheindependentvariablesinourdataset,elevation andtotalprecipitationwerechosenbecausetheyarecorrelatedtothemeantemperature. Likethesemivariogramintheuniversalkrigingsection,wehaveempiricalcrossvariograms andacrossvariogrammodel. Theempiricalcrossvariogram,likeempiricalsemivariograms,areestimateddirectlyfrom thesampledatausingthecrossvariableequationbelow,representingourisotropiccross variogrammodel. Thecrossvariogramhasadierentmathematicalequationcomparetothesemivariogramequationbecausenow,unliketheuniversalkrigingmethodweareintroducingcovariablesofourdependentvariableinordertopredictthemeantemperature(dependent variable).

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EmpricalCrossVariogram: ^ r AB (h)= 1 2n (h) n X i=1 m X j =1 (Z A (x i ) )Tj/T1_2 11.955 Tf11.955 0 Td(Z A (x j + h))( Z B (x i ) )Tj/T1_2 11.955 Tf11.956 0 Td(Z B (x j + h))(III.7) where^ r AB (h)istheestimatedcrosscovariancesatdistance h. n and m arethepaired observationsseparatedbydistance h, Variable Z A andvariable Z B areobservedvaluesofelevationandtotalprecipitationat locations x i and x j ,and h isthedistancebetween x i and x j . Incokriging,thevariableshouldbecorrelatedandthecrossvariogramisamultivariate formofourspatialcorrelationoperation,unlikeuniversalkrigingwherewehaveaunivariateform.Inuniversalkriging,onlytheavailablevariables Z A ,meantemperatures,were considered.Crossvariogramincludesexperimentalsemivariogramvaluesthatwehaveseen fromtheuniversalkriging.Inthiscase,ithelpstovisualizethespatialcorrelationsbetween ourvariables.Wecouldusecrossvariogramtoindicatethecorrelationbetweentemperature elevation,andtotalprecipitationinsteadofthebackwardselectionswedidearliertochose thecovariablesoftemperatureforthecokrigingmethod,becauseonlythevariablesthat ftbestinourmodelarechosentoperformcokriging.Theinformationprovidedbyour crossvariogramaboutthespatialvariationofasampledvariable(thecovariable)isusedto interpolateasparselysampledvariable(thepredictands). Inotherwords,thecokrigingmethodisperformedusingthecrossvariogramresults.The empiricalcrossvariograminequation(III.7)determinetheexperimentalcrossvariograms, andwhenweftourmodelofcokrigingtotheexperimentalsemi-andcross-variograms, theCauchy-Schwartzrelationmustbecheckedtoguaranteeacorrectcokrigingestimation varianceinallcircumstances[19].FiguresIII.1toIII.6belowshowthecrossvariograms. MNTMismonthlymeantemperature,TPCPstandsfortotalprecipitation.

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FigureIII.1: CrossvariogramAnalysisJuly2007 3

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FigureIII.2: CrossvariogramAnalysisJuly2008

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FigureIII.3: CrossVariogramAnalysisJuly2009 3

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FigureIII.4: CrossVariogramAnalysisJuly2010

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FigureIII.5: CrossVariogramAnalysisJuly2011

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FigureIII.6: CrossVariogramAnalysisJuly2012

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III.4CokrigingresultsshowninMaps BelowisthegraphshowingtheNOAAWeatherstationsinAfrica.Theareawithoutweather stationsaregoingtobepredictedfrom2007to2012usingcokriging. FigureIII.7: AvailableWeatherStationsinourRegionofStudy 3

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BelowarethegraphsforcokrigingoverthecontinentofAfricafromJuly2007toJuly 2012usingequations(III.1)and(III.2).Casesareprovidedonlywhenelevationortotal precipitationisrecorded. . FigureIII.8: CokrigingResultJuly2007 . 3

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FigureIII.9: CokrigingResultJuly2008

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FigureIII.10: CokrigingResultJuly2009

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FigureIII.11: CokrigingResultJuly2010

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FigureIII.12: CokrigingResultJuly2011

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FigureIII.13: CokrigingResultJuly2012

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Cokrigingseemstobebetterthanuniversalkrigingforpredictingthetemperatureofan unreachablearea.Itproducedsmallererrors,asmeasuredbytherootmeansquarederror. Also,inourcaseitiseasiertocomputeanaccurateestimationforcokrigingbecauseelevation dataisconstantovertime.Atleastoneofthecovariablesfortemperatureisneededto generateanacceptableinterpolatedvaluefortemperature.Theaccuracyofthecovariables dataprovidebetterresultsfromcokrigingspatialinterpolationmethod.

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REGRESSION OF KRIGING AND COKRIGING EXPERIMENTAL RESULTS In this frst experiment (Experiment I), we will compare temperatures estimated using universal kriging and cokriging to known values. Estimates using universal kriging and cokriging will be generated using stations shown previously in maps. IV.1 Motivation This part will validate the universal and cokriging methods described previously as predictors of mean temperature in an unknown area. Twenty-eight weather stations from eight countries in the Sahara Desert were removed from the data, and the remaining data points { temperature, elevation, and precipitation { from others stations were used to predict mean temperatures (MNTM) at the removed stations in July of 2008 1 and compared results to known monthly mean temperature. Results are summarized in Tables IV.1 to IV.4, and are shown in the form of maps as well. The removed weather stations were chosen randomly. They are from: Algeria, Lybia, Niger, Mali, Ouarzazate, Mauritania, Egypt and Western Sahara. Three stations were removed from Libya, four weather stations from Algeria, three from Niger, fve from Mali, four from Ouarzazate, fve from Mauritania, three from Egypt and one from Western Sahara. Figure IV.1 and IV.2 show the weather stations used after removing the 28 stations for this experiment. 1 July2008waschosenbecauseitwasthemostcompletedatasetwhencomparedtootheryears2007to 2012 4 IV

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FigureIV.1: RemovedWeatherStationsfortheexperimentalAnalysis

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FigureIV.2: RemovedWeatherStationsfortheexperimentalAnalysis

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FigureIV.3: RemainingWeatherStationsfortheexperimentalAnalysis

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TableIV.1showsthepredictedMeanTemperature(MNTM)forcokrigingmethod(CK). TableIV.1: ExperimentalresultsofCokriging(CK). StationsbyCountryOriginalMNTMPredictedMNTMErrorSquaredError Algeria257.03258.24-1.211.46 Algeria279.02281.08-2.064.24 Algeria252.98254.67-1.692.85 Algeria267.03265.841.191.42 Egypt274.73271.772.968.76 Egypt272.82271.681.141.30 Egypt308.22304.214.0116.08 Lybia308.51297.411.11123.43 Lybia300.13299.260.870.75 Lybia309.47308.41.071.14 Mali214.69216.35-1.662.75 Mali205.88206.71-0.830.69 Mali242.91244.22-1.311.72 Mali227.44229.03-1.592.51 Mali243.3244.76-1.462.13 Mauritania213.69214.72-1.031.06 Mauritania201.17201.0720.10.01

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TableIV.2: ExperimentalresultsofCokriging(CK),Continued StationsbyCountryOriginalMNTMPredictedMNTMErrorSquaredError Mauritania182.04180.032.14.41 Mauritania195.26194.950.310.09 Mauritania200.54200.290.250.06 Niger280.71266.1614.55211.70 Niger248.19248.52-0.330.11 Niger261.59258.872.727.39 Ouarzazate219.28220.59-1.311.72 Ouarzazate218.48219.54-1.061.1236 Ouarzazate225.79227.65-1.863.46 Ouarzazate232.86233.01-0.150.02 WesternSahara292.34 285.37.0449.56

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TableIV.3showsthepredictedMeanTemperature(MNTM)forUniversalKriging method(UK). TableIV.3: ExperimentalresultsofUniversalKriging(UK). StationsbyCountryOriginalMNTMPredictedMNTMErrorSquaredError Algeria257.03239.9217.11292.75 Algeria279.02231.7647.262233.51 Algeria252.98252.670.310.09 Algeria267.03240.3226.71713.42 Egypt274.73249.6125.12631.01 Egypt272.82236.5936.231312.61 Egypt308.22245.6762.553912.50 Lybia308.51247.0761.443774.87 Lybia300.13223.0977.045935.16 Lybia309.47250.7458.733449.21 Mali214.69193.0521.64468.28 Mali205.88214.92-9.488.36 Mali242.91258.81-15.9252.81 Mali227.44274.66-47.222229.72 Mali243.3268.022-24.722611.17 Mauritania213.69221.74-8.0564.80 Mauritania201.17271.81-70.644,990.00

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TableIV.4: ExperimentalresultsofUniversalKriging(UK),Continued StationsbyCountryOriginalMNTMPredictedMNTMErrorSquaredError Mauritania182.04292.34-110.312166.09 Mauritania195.26269.35-74.095489.33 Mauritania200.54275.61-75.075635.50 Niger280.71287.66-6.9548.30 Niger248.19237.4810.71114.70 Niger261.59251.1310.46109.41 Ouarzazate219.28284.75-65.474286.32 Ouarzazate218.48236.69-18.21331.60 Ouarzazate225.79239.63-13.84191.54 Ouarzazate232.86238.46-5.631.36 WesternSahara292.34 270.2422.1488.41

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Bothmethodsworkedacceptably,butcokrigingismoreaccuratebasedon therootmeansquareerror.Theroot-mean-squareerror(RMSE)forthe 28observationsis4.01whenappliedtocokriging.Ontheotherhand,when appliedtoUniversalKrigingmethodforthesame28observations,theRMSEis 46.23.Precipitationandelevationarealsopredictedalongwiththeestimated temperatureasbythecokrigingmethodwhichispartoftheanalysis. Inourdatasets,someprecipitationandelevationdatapointswerenot recorded,andthesemethodsassignthemanestimatedvalue.Fewcountries suchasLybiahavelessnumberofweatherstations,andthataectstheaccuracyoftheestimatedmeantemperature.

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IV.2 Discussion of the High Root-Mean-Square Error (RMSE) result in the Universal Kriging technique The higher root mean square error in universal kriging is explained by the fact that only the ftted data in our semivariogram model are chosen to compute the kriging interpolation. Also wehave very sparse data that miss many data points compared to the cokriging model, which used more reliable variables such as elevation (which is constant over the six years). The semivariogram model works better in cokriging than in universal kriging because more available data points are used. Since recorded temperature is used to predict temperature, this method of universal kriging relies on the input data and its accuracy to minimize the sum of squared errors. In the universal kriging, there is less data because stations that did not record mean temperature were removed; on the other hand the same stations were kept in the cokriging model because they recorded elevation and total precipitation even though the mean temperature at the same location is missing. In July 2008, 173 weather stations were used, while only 150 weather stations were used in July 2007 to predict with the same area size, which is the continent of Africa and surrounding.

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V EFFECT OF DISTANCE TO WEATHER ST A TIONS ON KRIGING TECHNIQUES In order to analyze the eect of the distance between weather stations and areas without weather stations, two additional experiments were performed. In Experiment II, weather stations closest to the Sahara Desert were removed, and we used the remaining stations in Africa to interpolate the temperature in the Sahara Desert. In Experiment III, the stations that were not included in Experiment II were used exclusively to extrapolate the temperature in the rest of Africa. V.1 Motivation The purpose for these additional experiments is to evaluate kriging estimates with respect to distance from the area of estimation. Both universal kriging and cokriging are analyzed here in order to compare behavior with respect to distance. Experiments II and III begin by generating predicted temperatures in the Sahara Desert using both techniques { universal kriging and cokriging. This part of the study considers the portion of the Sahara Desert that does not have any weather stations. This section will give us a better understanding about the use and merit of these spatial interpolation methods. First, weather stations closest to the Sahara Desert were removed, and the remaining stations were used to estimate temperature of the desert and surroundings for the month of July 2008. Second, only the weather stations closest to the Sahara Desert are used to estimate the mean temperature of the desert and surroundings for the same month and year.

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V.2ResultsandDiscussionfromExperimentIIandExperimentIII ExperimentIIandExperimentIIIresultsshowthatthepredictionsaremorereasonable oraccuratewhenthestationsareclosertotheareaofinterest.Bothmethodswork,but cokriginghasmuchlowerrootmeansquareerror.Whenthestationsareclosertheprediction ofmonthlymeantemperatureismoreaccuratethanwhenthestationsarefurther.Because ofthenormalaveragetemperatureinthosearea,thetemperaturerecordedwhenstations areclosermakesmoresense.TheSaharaDesertisthehottestdesertintheworld,andwe expectitstemperaturetobearound300degreeKelvinonaverage,whichis80.33Fahrenheit and26.85degreeCelsius.Werecordedaround300degreekelvinwhenthestationswereclose totheunknownareasoftheSaharaDesert. TableV1andV.2showtheresultsfromExperimentII(noweatherstationsnearorinside SaharaDesert)andExperimentIII.

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TableV.1: ExperimentII:EstimatedMeanTemperatureJuly2008 LongitudeLatitudeMNTM.pred(U.K)MNTM.pred(C.K)MNTM.std(CK) 1123263.00279.82 30.42 12 23 263.44280.36 30.43 13 23 263.93280.99 30.43 14 23 264.49281.73 30.40 15 23 265.12282.59 30.35 16 23 265.83283.59 30.28 17 23 266.62284.74 30.19 TableV.2: ExperimentIII:EstimatedMeanTemperatureJuly2008 LongitudeLatitudeMNTM.pred(U.K)MNTM.pred(C.K)MNTM.std(CK) 1123313.01310.01 33.20 12 23 313.08309.34 33.48 13 23 312.77308.69 33.80 14 23 312.24308.07 34.14 15 23 311.63307.49 34.49 16 23 311.01306.94 34.86 17 23 310.43306.42 35.24

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V.3PredictionoftheSaharaDesertAverageTemperatureoversixYearsof DatausingCokriging ShownbelowintableV.1andfgureV.1arethecokrigingestimatesoftemperature intheSaharadesertforJuly2007toJuly2012.Alltheavailabletemperaturesavailable fromallweatherstationsinthecontinentofAfricaweretakenintoaccount.Thecokriging methodwaschosensinceitappearstobethebetterinterpolationmethodfromtheprevious section. TableV.3: EstimatedMonthlyMeanTemperature(MNTM.pred) MonthandYearMNTM.predMNTM.std July2007315.4126.17 July2008319.3228.40 July2009318.6128.05 July2010295.3048.11 July2011291.7833.27 July2012313.6453.46 Theoutputofthisspatialinterpolationincludesstandarddeviationofeachvariableand covariancebetweenvariables.Theprecipitationismeasuredinmillimeters,andstdstands forstandarddeviation.NotethatthemonthofJulywehavechosencanaecttheresultin temperatureandtotalprecipitationsinceitoftentherainyseasoninthatpartofAfrica. 5

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ShownbelowinFigureV.1isagraphoftheestimatedmonthlymeantemperaturein theSaharaDesertatLatitude23N,Longitude12E. FigureV.1: PredictedJulyMeanTemperatureoftheSaharaDesert Theanomaliesnoticedin2010and2011areduetotheinaccuracyofthedatain2010 and2011.Inaddition,wemisseddatasuchaselevationandprecipitationaswellasmean temperaturefromstations.Wealsolosesomestationsinourdataset.Therefore,elevation andortotalprecipitationvalueswerenotrecorded.Seesampledatatablefromnextsection. Inaddition,theelevationdataoverallisnotveryaccurateintherawdata,however,it producesacceptableresultsservingthepurposeofthisthesis.

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CHAPTER VI DISCUSSION OF ELEV A TION DA TA Cokriging method helps estimate the temperature over the inaccessible area, and it predicts their elevation and total precipitation. Knowing the high and low elevation to region uninhabitated, inaccessible or dicult to reach from graphs below, will provide a better understanding temperature and precipitation behavioral patterns. Even though precipitation have a stronger negative correlation to temperature than elevation, elevation as well have a negative correlation with temperature. In other words, the higher the temperature, the lower the elevation, and it holds a similar behavior with respect to precipitation. There are issues with elevation for these reasons given below. First, wedo not have the same number of stations for each year from 2007 to 2012. For example, the number of stations in 2008 is more than the in 2010, and wehave fewer stations in 2009 than in 2007 and 2008. Stations were removed or did not record any data for some given years. Second, few weather stations recorded mean temperature, but did not record their respective elevation and or precipitation, and this aect both the methods of universal kriging and cokriging. Below is a random selection of stations and their recorded elevation data from 2007 to 2012. In July 2007, we have 172 weather stations, in July 2008 173 stations, in July 2009 174 weather stations, in July 2010 163 weather stations , 136 weather stations in July 2011 and in July 2012 our data includes 159 weather stations. In conclusion , our data is less accurate in July 2010 and July 2011. The lack of data points does not aect our spatial interpolation methods, but it explains the lack of the accuracy of our data, and therefore the issues with high root square mean error in our analysis over the six years period.

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TableVI.1: SampleElevationDatabyYear StationsName200720082009201020112012 ABUHAMEDSU312312312312312312 ADIAKEIV393939393939 AGADEZNG505505505505505505 AGALEGA333333 ALGERDARELBEIDAAG242424242424 ALJOUFSA689689689689689689 ALMADINAHSA636636636636636636 ALMERIAAEROPUERTOSP2121NA212121 AMIANDOSCY136013601360NANANA AMTIMANCD436436436436436436 ANTANANARIVOIVATOMA127612761276127612761276 ANTSIRANANAMA105105105105105105

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CHAPTER VII CONCLUSION Cokriging and universal kriging work as methods of spatial interpolation. They can be applied to perform analysis of variables of interest in a given area. The closer the weather stations are to the unknown area the better the predictions. In this analysis, we found more error in universal kriging than in cokriging; therefore, we can assume that the cokriging method is more reliable and accurate for temperature interpolation. The research in this thesis can be expended by analyzing closely the eect of elevation and total precipitation and how wecan use it to model temperature and vice-versa. This will help predict important temperature changes in the near future; in addition, it provides estimated temperatures in areas where there is population but no weather stations such as villages in the Sahara Desert, and villages in the forest of Congo. Also, the use of better or improved statistical software can give an exact longitude and latitude of relatively small unknown areas such aslocal villages found in forest or desert areas. In experiment IIand III, the results could be improved if wecould identify each village belonging to the Sahara Desert by having exact longitude and latitude of each village.

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