Forecast Error Correction using Dynamic Data Assimilation PDF
Preview Forecast Error Correction using Dynamic Data Assimilation
Springer Atmospheric Sciences Sivaramakrishnan Lakshmivarahan John M. Lewis Rafal Jabrzemski Forecast Error Correction using Dynamic Data Assimilation Springer Atmospheric Sciences Moreinformationaboutthisseriesathttp://www.springer.com/series/10176 Sivaramakrishnan Lakshmivarahan John M. Lewis • Rafal Jabrzemski Forecast Error Correction using Dynamic Data Assimilation 123 SivaramakrishnanLakshmivarahan JohnM.Lewis SchoolofComputerScience NationalSevereStormsLaboratory UniversityofOklahoma Norman,OK,USA Norman,OK,USA DesertResearchInstitute RafalJabrzemski Reno,NV,USA OklahomaClimatologicalSurvey UniversityofOklahoma Norman,OK,USA ISSN2194-5217 ISSN2194-5225 (electronic) SpringerAtmosphericSciences ISBN978-3-319-39995-9 ISBN978-3-319-39997-3 (eBook) DOI10.1007/978-3-319-39997-3 LibraryofCongressControlNumber:2016940961 ©SpringerInternationalPublishingSwitzerland2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland YoshikazuSasaki(1927–2015), FatherofVariationalData AssimilationinMeteorology Preface Inthisbook, wefocus ondeterministicforecastingbasedongoverning dynamical equations(typicallyintheformofdifferentialequations).Theseequationsrequire specification of a control vector for their solution (initial conditions, boundary conditions,physicaland/orempiricalparameters).Definingforecasterroriscentral toourstudyasthebooktitleimplies.Anall-inclusivedefinitionofforecasterroris difficulttoformulate.Therefore,wefinditbesttodefinethiserrorcategorically:(1) error due toincorrectly specified termsinthegoverning equations (ortheabsence ofimportanttermsintheseequations),(2)inexactnumericalapproximationstothe analyticformofthedynamicalequationsincludingartificialamplification/damping ofsolutionsinthenumericalintegrationprocess,and(3)uncertaintyintheelements of control. That is, error in prediction results from incorrect dynamical laws, numerical inexactitude, and uncertainty in control. And the true test of forecast goodnessrestsoncomparingtheforecastwithaccurateobservations.Ofttimes,we areunabletodefinitivelydeterminethesource(s)oferror.Butgiventhiserror,we ask the question: Can we improve the forecast by altering the control vector or empiricallycorrectingthedynamicallaw?Andjustasimportantaquestion,canwe determine the relative impact of the various elements of control on the forecast of interest? Immediately we see that there is a desire to determine sensitivity of forecast to elements of control—labeled forecast uncertainty. This uncertainty is then used to findoptimalelementsofcontrol—valuesoftheelementsthatminimizethesumof squared differences between the forecast and observations and ideal placement of observationsinspace-time.Clearly,thispathblendssensitivitywithleastsquaresfit ofmodeltoobservations.Amethodologythatconsidersallofthefactorsmentioned above is called the forecast sensitivity method (FSM), a relatively new form of dynamicdataassimilationthatisthecenterpieceofthisbook. There is a rich history of work in both sensitivity analysis (SA) and dynamic dataassimilation(DDA).Sensitivityanalysishasbeenamainstayofbothdynamical systemsandbiostatisticalsystems.TheworksofGregorMendelandRonaldFisher areexcellentexamplesofsensitivityinthefieldofpopulationgenetics—necessarily intheformofstatisticsforhybridizationofpeasinMendel’scaseandstatisticsof vii viii Preface cropproductioninFisher’scase.HenriPoincaréwasthepioneerintheuncertainty of dynamical forecasts with respect to elements of control. Even in the absence of computational power in the late nineteenth century, he clearly understood the extremesensitivityofthethree-bodyproblem’ssolutiontoslightchangesininitial conditions(presentedasanexerciseinthisbook).Oneofthemostfrequentlyquoted sentences in his studies was “La prediction deviant impossible” [The prediction becomes impossible]. Lorenz (1995) took this to mean that Poincaré was close to discoveringthetheoryofchaos.Theseissuesindeterministicforecastinghaveled to the investigation of predictability limits, that point in time when the forecast is no better than “climatology” (the average state of affairs in the system). In engineering and control theory, Hendrik Bode, a research mathematician at Bell Laboratories, developed sensitivity analysis in service to feedback control that he studiedduringWWIIwhenheaddressedproblemsofgunnerycontrol.Hisclassic workistitledNetworkAnalysisandFeedbackAmplifierDesign(Bodeetal.1945). Thetwo-volumetreatiseonSensitivityandUncertaintyAnalysisbyCacuci(2003) andCacucietal.(2005)dealsextensivelywiththediscussionofadjointsensitivity and its applications to geosciences. For other applications of sensitivity analysis to engineering systems, refer to Cruz (1982), Deif (2012), Eslami (2013), Fiacco (1984),Frank(1978),KokotovicandRutman(1965),Ronen(1988),Rozenwasser andYusupov(1999),Saltellietal.(2000,2004,2008),andSobral(1968). In regard to DDA, celebrated mathematician Carl Gauss did his fundamental work on least squares fitting of observations to models in the first decade of the nineteenthcentury(Gauss1809)(adiscussionofhisproblemisfoundinthisbook). Thismethodhasneverfallenoutofuse.Gauss’sworkwasexpandedfromparticle dynamicstocontinuousmediathroughtheworkofAlfredClebsch(Clebsch1857). In the 1950s, Japanese meteorologist Yoshi Sasaki used Gauss’ fundamental idea in combination with the Clebsch transformation to develop DDA for numerical weather prediction (NWP). This methodology has come to be called variational analysis in meteorology and in abbreviated form 4D-VAR (Sasaki 1958). Lewis etal.(2006)reviewvariationalanalysisinthecontextofNWP.AlsorefertoLewis and Lakshmivarahan (2008) for more details. A resource for numerous papers in sensitivityanalysiscanbefoundinRabitzetal.(1983),Nago(1971)andTomovic´ andVukobratovíc(1972). As might be expected, there is equivalence between classic Gaussian least squares methodology (variational analysis) and FSM. We carefully examine this connectioninthebookandofferseveralproblems(anddemonstrations)thatexplore thisconnectionalongwithadvantages/disadvantagesoftheseDDAmethods. The book is partitioned into two sections: Part I, a general theory of FSM with a variety of practical problems that give substance to the theory, and Part II, an in-depthanalysisofFSMappliedtowell-knowngeophysicaldynamicsproblems— thedynamicsofshallowwaterwavesandair-seainteractionundertheconditionof a convective boundary layer. Recently, FSM has been applied to solve estimation problems in ecology, hydrology, and interdependent security analysis. For lack of space,wecouldnotincludetheseinterestingapplications. Preface ix A good working knowledge at the BS level of the standard calculus sequence, differential equations, linear algebra, and a good facility with programming con- stitute adequate prerequisites for a course based on this book. We have used this book as a secondary text along with parts of our earlier book (Lewis et al. 2006) in a senior/first year-graduate-level course devoted to solving static and dynamic deterministicinverseproblemsattheUniversityofOklahoma,Norman,Oklahoma, andattheUniversityofNevada,Reno,Nevada,USA. We have strived to eliminate typographical errors, and we would very much appreciatehearingfromreaderswhoidentifyremainingerrors. Norman,OK,USA SivaramakrishnanLakshmivarahan Reno,NV,USA JohnM.Lewis Norman,OK,USA RafalJabrzemski Acknowledgments Our interest in forward sensitivity-based approach to forecast error correction began almost a decade ago. During the development of this methodology, we have interacted and gained from the discussions with several of our colleagues and students. We wish to thank Qin Xu, National Severe Storms Laboratory, Norman,Oklahoma,forhisconstantsupport;hehasbeenthesoundingboardwho steered us in the right direction with a multitude of questions and comments. We are deeply indebted to Bill Stockwell (formerly of Desert Research Institute) of HowardUniversityformanyhoursofdiscussionsrelatingtotheroleofparameter sensitivity analysis in reaction kinetics and atmospheric chemistry. We (SL) wish to record our sincere thanks to Elaine Spiller and Adam Mallen, Marquette University, and Amit Apte, International Center for Theoretical Studies (ICTS) Bangalore, for their interest and comments on the earlier versions of the results inChap.7.Wehave benefitted immenselyfromthequestions andcomments from theseminarparticipantsattheNationalCenterforAtmosphericResearch(NCAR), Marquette University; TIFR, the Center for Applicable Mathematics, Bangalore, India; the Center for Atmospheric and Ocean Studies (CAOS), Indian Institute of Science, Bangalore, India; the Meteorological Research Center (MRI), Tsukuba, Japan;theDesertResearch Institute(DRI),Reno,Nevada, USA;andTexas A&M University,CollegeStation,Texas,USA.WearegratefultoJeffAnderson(NCAR); Amit Apte (ICTS); Mythily Ramaswamy (TIFR); M. Vanninathan (TIFR); A.S. Vasudeva Murthy (TIFR); Ravi Nanjundiah, J. Srinivasan, S.K. Satheesh, and AshwinSeshadri(CAOSandDivechaCenterforClimateChange,IISc,Bangalore, India); Elaine Spiller (Marquette University); K. Saito (MRI); and I. Szunyogh (Texas A&M) for all their interest, help, and hospitality. We wish to record our sincere appreciation for Yiqi Luo and his team at the ECO Lab, University of Oklahoma (OU), for providing the data and the impetus to apply FSM-based methodtoassimilatedataintotheterrestrialcarbonecosystem(TECO)model.Our thanksaretoRandyKolar,KendraDersback,andBaxterVieux,CivilEngineering, OU,andTomLandersandKashBarker,IndustrialandSystemsEngineering,OU, for encouraging their PhD students to apply FSM-based approach to estimation problemsinhydrologyandinterdependentriskanalysis. xi xii Acknowledgments We are particularly grateful to several graduate students at the University of Oklahoma for using FSM in their MS thesis and PhD dissertations. Phan (2011) worked on applying FSM to assimilate data into a class of carbon sequestration modelscalledTECOmodelforhisMSthesis.AsapartoftheirPhDdissertations, Tromble(2011)usedFSMtoestimateparameterstoanalyzefoodinundationusing theADCIRChydrodynamicmodel,Looper(2013)usedFSMtocalibrateaclassof distributedhydrologicmodel,andPant(2012)usedFSMtoestimateparametersin aclassofdynamicriskinput-outputmodel. JunjunHu,HumbertoA.Vergara,andHongchengQiofferedseveralsuggestions to improve the presentations in Part I of this book. Parts of the book were used in the course Scientific Computing-I during the fall of 2015 at OU. We thank all the studentsinthisclassfortheirenthusiasm,especiallyBlakeJamesforhiscomments onPartI. TheauthorswishtorecordtheirthankstothemanagementteamintheNational Severe Storms Laboratory, Norman, Oklahoma, for all their support and encour- agement and at the National Oceanic and Atmospheric Administration (NOAA) management, in particular Patrina Gregory and Hector Benitez, for crafting a contractthatisfaithfultolegalrequirementsforcopyrightofworkbyaUSfederal governmentemployee(JL).Inaddition,authors(RJ)wouldalsoliketoexpresstheir thanksforsupportfromtheOklahomaMesonet. It has been a pleasure to work with the Springer team, especially to Ron Doering—thank you all for the interest, help, and understanding right from the inception.
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