An approach to characterise spatio-temporal drought dynamics

Delft University of Technology

An approach to characterise spatio-temporal drought dynamics

Diaz, Vitali; Corzo Perez, Gerald A.; Van Lanen, Henny A.J.; Solomatine, Dimitri; Varouchakis, Emmanouil

A.

DOI

10.1016/j.advwatres.2020.103512

Publication date

2020

Document Version

Final published version

Published in

Advances in Water Resources

Citation (APA)

Diaz, V., Corzo Perez, G. A., Van Lanen, H. A. J., Solomatine, D., & Varouchakis, E. A. (2020). An

approach to characterise spatio-temporal drought dynamics. Advances in Water Resources, 137, [103512].

https://doi.org/10.1016/j.advwatres.2020.103512

Important note

To cite this publication, please use the final published version (if applicable).

Please check the document version above.

Copyright

Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy

Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.

This work is downloaded from Delft University of Technology.

ContentslistsavailableatScienceDirect

Advances

in

Water

Resources

journalhomepage:www.elsevier.com/locate/advwatres

An

approach

to

characterise

spatio-temporal

drought

dynamics

Vitali

Diaz

_,

_Gerald

_A.

_Corzo

_Perez

_,

_Henny

_A.J.

_Van

_Lanen

_,

_Dimitri

_Solomatine

_,

Emmanouil

A.

Varouchakis

a IHE Delft Institute for Water Education, Hydroinformatics Chair group, Delft, 2601 DA, the Netherlands b Delft University of Technology, Water Resources Section, Delft, the Netherlands

c Hydrology and Quantitative Water Management Group, Wageningen University, Wageningen, the Netherlands d Water Problems Institute of the Russian Academy of Sciences, Moscow, Russia

e School of Environmental Engineering, Technical University of Crete, Chania, Greece

a

r

t

i

c

l

e

i

n

f

o

Keywords:

Spatio-temporal drought analysis Drought tracking

Drought dynamics Drought monitoring Drought characterisation

a

b

s

t

r

a

c

t

The spatiotemporal monitoring of droughts is a complex task. In the past decades, drought monitoring has been increasingly developed, while the consideration of its spatio-temporal dynamics is still a challenge. This study proposes a method to build the spatial tracks and paths of drought, which can enhance its monitoring. The steps for the drought tracks calculation are (1) identification of spatial units (areas), (2) centroids localisation, and (3) centroids linkage. The spatio-temporal analysis performed here to extract the areas and centroids builds upon the Contiguous Drought Area (CDA) analysis. The potential of the proposed methodology is illustrated using grid data from the Standardized Precipitation Evaporation Index (SPEI) Global Drought Monitor over India (1901-2013), as an example. The method to calculate the drought tracks allows for identification of drought paths delineated by an onset and an end in space and time. Tracks, severity and duration of the drought are identified, as well as localisation (onset and end position), and rotation. The response of the drought tracking method to different com- binations of parameters is also analysed. Further research is in progress to set up a model to predict the drought tracks for particular regions across the world, including India ( https://www.researchgate.net/project/STAND-Spatio-Temporal-ANalysis-of-Drought).

1. Introduction

Droughtisaregionalphenomenonthatoftencoverslargeterritorial extensions(WorldMeteorologicalOrganizationWMO,2006).Itcan oc-curanywhereintheworldwithsevereconsequences(impacts)inwater resourcesandsocioeconomicactivities(Belowetal.,2007;Sheffieldand Wood, 2011;Tallaksen andVanLanen, 2004; Wilhite,2000). WMO stressesthattoimprovedroughtimpactsmitigation,itisnecessaryto de-velopandimplementnationalpoliciesbasedonthebestdescriptionand characterisationofdrought(WorldMeteorologicalOrganizationWMO, 2006).

Thereisnouniquedefinitionofdrought.However,thereisan agree-mentthatitisananomalyinprecipitationandtemperaturethatwhen extendedoveraregioncausesalackofsoilmoisture,runoff and ground-water (Mishra andSingh, 2010; Van Loon, 2015). This lackof wa-teris expressedbyadroughtindicator,which transformsthe hydro-meteorologicalvariable into avalue thatis relatedtosuch a water anomaly(MishraandSingh,2011;Wandersetal.,2010).Indrought monitoring,thedroughtindicators aregenerallyusedtoidentifythe lackofwater.

∗_{Corresponding author at: IHE Delft Institute for Water Education, Westvest 7, 2601 DA, Delft, the Netherlands.} E-mailaddresses:v.diazmercado@tudelft.nl, vitalidime@gmail.com(V. Diaz).

Regarding drought monitoring, Hao et al. (2017) provide an overview of its statusfor regional andglobalapplications. They re-port as anessential advance the integration of more data resources tofeeddroughtindicators,allowingforabetterdescriptionof hydro-meteorologicalandvegetationcondition.Thisintegrationincludesthe useofhydrologicalsimulations,aswellasremotesensing,and forecast-ingdata.Forinstance,theEuropeanDroughtObservatory( Sepulcre-Cantó etal.,2012)providestheconditionofdroughtevolution (devel-opment)in Europebasedon satelliteobservationsandmodelledsoil moisture.Ontheonehand,currentdroughtmonitoringallowsfor fol-lowingdroughtdevelopmentforaspecificlocationoragivenregion, mainlythroughthevisualisationandanalysisoftimeseriesofdrought indicators.Ontheotherhand,thespatialconditionofdrought,including itsextent,ismonitoredwiththehelpoftimesnapshots,whichprovide qualitativeinformationonthespatialbehaviourofthephenomenon.

Intermsofthespatialdevelopmentofthedrought,nowadays,the availabledroughtmonitorsdeliverinformationaboutthespatialextent ofdroughts(i.e.snapshots).However,consistentproceduresfor track-ingofdroughtareasarelacking,not allowingforassessingtemporal variationsthatformitsspatio-temporaldynamics(Haoetal.,2017).In

https://doi.org/10.1016/j.advwatres.2020.103512

Received 26 July 2019; Received in revised form 8 January 2020; Accepted 15 January 2020 Available online 16 January 2020

addition,thespatialdistributionofdroughtataspecifictimedoesnot giveinformationaboutthespatialpathwayofthedroughts. Implement-ingthedataanalysisandhydroinformaticstechnologiestotracedrought inspaceandintimeondroughtmonitorscanenhanceitsspatial track-ingandprediction.

Thenecessitytoincreaseourunderstandingofthespatio-temporal developmentofdroughthasmotivatedthestudieswheredroughtis con-sidered asaphenomenonthat hasat leastthefollowing main char-acteristics:duration, intensity(magnitude),andspatial extent(area) (Andreadis et al., 2005; CorzoPerezet al., 2011;Diazet al., 2018;

Herrera-Estradaetal.,2017;Lloyd-Hughes,2012;Sheffieldetal.,2009;

Tallaksenetal.,2009;VanHuijgevoortetal.,2013;Vernieuweetal., 2019).Ageneralframeworkforcarryingoutspatio-temporalanalysisof droughtcanbeformulatedbasedonthesestudies,anditcanbebriefly describedasfollows.First,agivendroughtindicatorisusedtotransform thehydro-meteorologicalvariableintowateranomalies.Thedrought indicatoriscomputedin aspatialcontext,wherethestudyregionis embeddedinagrid.Then,byestablishingathresholdonthedrought indicator,theconditionofnon-drought/droughtisidentifiedineachof thecellsofthegrid.Thisconditioncanbeexpressedinabinaryway,i.e. using0sand1s.Finally,neighbouringcellsshowingthesamedrought conditionareaggregatedintoregions(clusters)byapplyingaclustering technique.Inthisway,droughtisdefinedinspaceandintime,witha spatialextentandduration.

The spatio-temporal analysis of drought that would also include the spatial drought tracking explicitly is however limited to a few studiessuchasDiazetal.(2018), Herrera-Estradaetal.(2017),and

Zhouetal.(2019).Thefirsttwoaddresstheanalysisforlarge-scale stud-iesandthelatterpresentsabasin-scaleapplication.Althoughthereare otherpublicationsthatconsiderthestudyofdroughtextentlocations, theymisstheexplicitcalculationofspatialdroughttracks.Followingthe frameworkmentionedinthepreviousparagraph,aftertheextractionof droughtextents(areas),itispossibletoidentifytheirlocationand fur-therconstructionofthespatialtracks(definedbythelinkagebetween consecutivecentroidsintime).Thecalculationandfurtheranalysisof thesetracks,alongwithoutcomeondroughtareas,mayhelptoanswer thefollowingquestionsregardingdroughtdynamics.Whatarethemain placeswheredroughtremains?Aretherepredominantroutesfollowed bydrought?Howfastdoes droughtchange (itsextentandlocation) alongitsspatialpath?Literaturereviewshowsthatthedevelopmentof methodologiestodescribedroughtdynamicsisstillinprogress, there-foremoreresearchisneededinthisregard(e.g.Herrera-Estradaetal., 2017;Vernieuweetal.,2019;Zhouetal.,2019).

Thisstudyaimstoexplainthemainprinciplesofanewmethodthat complementcurrentdroughtmonitoringbytrackingthespatialextent of drought(referredtoin thisdocument asarea, orcluster). Inthis study,thedescriptionandtheapplicationofthemethodology to cal-culatedroughttracksarepresentedindetail.Theproposedmethodis accompaniedbyanalgorithmtocalculatethedroughtcharacteristics. Bothmethodsaredescribedafterthisintroductionsection.The spatio-temporalContiguousDroughtArea(CDA)analysis(CorzoPerezetal., 2011)isusedasthebasisforthedevelopmentofthetrackingmethod. TheCDA isapplied toidentify theneighbouringcells thatform the droughtclusters.Adroughtisdefinedbyanonsetlocation,pathway overtime,andanendlocationbasedonthebuilttracks.Anewdrought characteristicisintroducedinthisstudy,namelyrotation(Sect.2.2),a featureoftenusedwhentrackingobjectsinspace(detailsinSect.2.2). Theapplicationofdroughttrackingmethodisperformedoverthe coun-tryofIndiafortheperiod1901-2013.

2. Methods

2.1. S-TRACK:spatialtrackingofdrought

Thespatialidentificationofdroughttracksisfirstlyintroducedby

Diazetal.(2018)andfurtherdevelopedinthisresearch.S-TRACK

con-Fig.1. Schematic overview of S-TRACK method for spatial drought tracking which involves: ( step1) spatial drought units (clusters) computation, ( step2) centroids localisation, and ( step3) centroids linkage (see Sect. 2.1). An example is presented for the case of three times steps: from t1to t3. Columns in the dia- gram show the sequence of the steps. Coloured cells in the first column indicate all cells in drought. Colours in the second column point out different clusters identified. In the third column, the largest contiguous area in drought is pre- sented with a different colour. Only the largest cluster is shown in the fourth column and its centroid ( p) is indicated by a point. Subscripts indicate time steps.

sistsofthethreemainsteps:(1)calculationofthespatialdroughtunits (referredtoherealsoasareasorclusters);(2)localisationofcentroids; and(3)linkageofcentroids(Fig.1).

Step1. Spatialdroughtunitscomputation

Inthespatialcontext,droughtunitsareidentifiedbymeansofthe ContiguousDroughtArea (CDA)analysis(CorzoPerezet al., 2011). A CDAis composedof neighbouringcells in drought.These cells in droughtareidentifiedineachtimestep.Whenthedroughtindicator is belowor equaltotheselected threshold,thevalueof1isusedto indicatethatthecellisindrought,otherwise,thevalueof 0isused, indicatingnon-drought.Droughtindicators(DIs)aremathematical rep-resentationsofawateranomaly(seeSect.2.3.1).Ingeneral,CDAcanbe appliedoveranyDIthatisinagridform.FollowingtheCDA method-ology,ineachtimestep,theCDAsarecomputed.

CDAanalysisfollowsaconnected-componentlabellingapproachto cluster thecells indrought(HaralickandShapiro,1992).Inthis ap-proach,atwo-scanalgorithmisapplied.Firstly,eachcellisnumbered forlocationissues.Then,thefirstrunis performedwherethebinary gridisexploredandconnected(contiguous)components(cells)are as-signedwithprovisionallabels.Theselabelspointouttheconnectionof everycellwithits8nearestneighbours.Withinthegrid,inasectionof 3×3cells,9cellsintotal,thecentralcellhas8surroundings.Inthis firstrun,thecell’slabeldoesnotrefertothenumberofclusteryetbutto thecellswithwhichthegivencellisconnected.Finally,asecondscanis carriedouttofindsimilarcellconnections,i.e.clusters,whicharegiven auniquelabel.Examinationofthegridcanbeperformedbycolumns orbyrows.CDAanalysisisconductedineachtimestepoverthewhole grid.FormoredetailsonCDAanalysisrefertoCorzoPerezetal.(2011). TheuseofCDAreliesontheassumptionthatthebinarydescription ofdroughtcondition(0sand1s)ishomogeneousoverthewholegrid. Thus,iftwoormorecellsdenotedroughtconditions(valueof1),andare contiguousinspace,itisassumedthatallofthemarepartofthesame droughtunit.Inthis respect,itisrecommendedtochooseadrought indicatorthatconsidersthenormalisationofthevaluesinthespatial domain. Inthis study,astandardiseddroughtindicatorisapplied as mentionedafterwards,whichallowstheclusteringofneighbouringcells indrought(cellswith1s).

Afterclusters(areasindrought)areidentified,themajor(largest) oneisidentifiedineachtimestept(Fig.1).Asthetrackingalgorithm

Fig.2. Flowchart showing the rules for linking drought areas (clusters) in time. Numbers in the boxes indicate the sequence of rules 1 to 4. The output of 1 is used to point out that the drought area Aat time tjoins its predecessor at time

t–1, otherwise 0 is retrieved. The distance between the centroids at times tand

t–1 is represented by Δl. The linking algorithm has the following parameters:

a,b,c and d. The first two used to control drought area A, and the last two, to check distance Δl.

focusesonthecalculationofthemajorspatialdroughtextentineach timestep,smallorone-cellunitsarediscriminatedwiththeselection ofthelargestone,allowingtheeliminationofpossibleartefactdrought areas.

Step2. Centroidslocalisation

Afteridentificationofthemajor(largest)droughtcluster,itscentroid (p)iscalculatedineachtimestep.Thisfeatureisusedasthelocation oftheclusterinasimilarwayasCorzoPerezetal.(2011)and Lloyd-Hughes(2012)present.Thewayinwhichtheclustersarejoinedintime isexplainedinthefollowingstep.Step2and3presentedinthis docu-ment,areanextensionoftheCDAanalysisofCorzoPerezetal.(2011). Anotherpossibility toindicate thelocationof agiven cluster is,for instance, to use the point with the lowest drought indicator value (Andreadisetal.,2005;Herrera-Estradaetal.,2017).Inthisresearch, wechosethecentroidsincewealreadyreducethespatial representa-tionofdroughtindicatorbyusingonly1sand0s,i.e.droughtand non-droughtcondition,respectively.

Step3. Centroidslinkage

Thealgorithmtolinkcentroidsofconsecutiveclustersintimeisa setofrulestoseparateorjointhesequenceintime(Fig.2).Therules considertwo typesof thresholdparameters:(1)twothatcontrolthe magnitude(size)ofthecluster(A,withdimensionsL2_),and(2)two

thatconstraintheEuclideandistancebetweenconsecutiveclusters(Δl, withdimensionsL)(Fig.2).Theparametersaredenotedasfollows:a, b,candd.ThefirsttwoareusedtothedroughtareaA,andthelasttwo tothedistanceΔl.Theoutputinthisstepisthetimeseriesof0sand1 s,denotedbyS(t).Here,thevalueof1indicatesthelinkageofclusters intime.Iftheclusterattimetisnotconnectedwiththeclusterattime

t–1,thevalueof0isusedinstead.Consecutivevaluesof1sinthetime seriesSshowtheoccurrenceofwhatisdefinedasadroughttrack.The flowchartoftherulesforlinkingthecentroidsispresentedinFig.2,and belowtheserulesareexplained.

CentroidslinkagestartsbyidentifyingiftheclusterareaAishigher thana(Fig.2,rule1).Thisfirstcomparisonhelpstodiscriminatesmall clusters.IfAisbelowa,thereis noconnectionbetweenconsecutive clustersandthisprocedurefinalises,retrieving0.Beforecomparingthe distancebetweenareas(Δl),thesecondcomparisonofAisappliedto identifyifitisa“verylarge” area(Fig.2,rule2).Parameterbis pro-posedtoconsidertheselargeareas.WhenAisbelowb,theparameterc

isusedtocomparedistancesbetweenclusters(Fig.2,rule3).Otherwise,

whenAisaboveb("verylarge"area),torestrictthedistances, param-eterdisconsideredinstead(Fig.2,rule4).Thereasonofthesecond comparisonofclusterareasandtheuseofparameterdisbecause cen-troidsofclusterswithaconsiderablesizemaybelocatedfartheraway fromeachotherandthenthedistanceΔlcouldfalloutsideofthelimit indicatedbyparameterc(Fig.3).

Anotherparameterthatcouldbeincludedinthislinkagealgorithm isthedegreeofoverlapbetweenconsecutiveclustersintime.Thisway ofintersectionisnotconsidereddirectlyinourlinkagealgorithmasa parameter(e.g.percentageofoverlapping).Theoverlapiscontemplated intheuseoftheparametersthatcontrolthedistancebetweenclusters. Anintersectionmayoccurwhenthedistancebetweencentroidsisshort (Fig.3).

2.2. Calculationofdroughtcharacteristics

Themethodologytobuilddroughttracksallowsfortheidentification ofpathswithanonsetandanendlocation.Theinformationcalculated alongthepathscanhelptodescribetheoccurrenceofdrought. Particu-larly,itispossibletoextractinformationregardingtheduration, sever-ity,aswellasrotation.Inthefollowinganalysisofthespatio-temporal droughtdynamics,severityhasadifferentmeaningcomparedtoon-site analysisorCDAstudies.Inthelatter,itexpressesacertaindegreeof wa-termissing,ananomalycomparedtonormalconditions.Herein, sever-ityhasaspatialmeaning,itisconnectedtothetemporalevolutionofthe sizeoftheareaindrought,irrespectiveofthestrengthofthedrought.In thefollowingparagraphs,theproceduretocalculatedrought character-isticsispresented.TheproposedapproachiscalledDDRASTIC-spatial (DroughtDuRAtion,SeveriTyandIntensityComputing-spatialevents). DDRASTIC-spatialisappliedafterdroughttracksareidentifiedbythe S-TRACK algorithm.Thisapproach hasasa predecessor(Diazetal., 2019),which howeverdoesnot considertheelementsrelatedtothe spatialdomain,suchasclusters,locationsandpaths.

Forthecalculationofthedroughtduration,firstlytheonsetandthe end areobtained:thetimeseriesS(t) of1sand0scalculatedwith S-TRACKmethodisanalysedtodoso.Asmentioned,theconsecutive sequenceof1sinthetimeseriesS,indicatestheoccurrenceofadrought track.Oneisolatedvalueof1showsthelinkingoftwoclustersintime. Twoconsecutivevaluesof1showthelinkageofthreeclustersintime, andsoon.Inasequenceof1s,thetimeofthefirstvalueof1(tfirst)is

thetimestepatwhichthesecondandfirstclusterareconnected.The timestepofthelastvalueof1(t_last)istheonewhenthelastandthe penultimateclustersarelinked.Theonsettiisdefinedasti=tfirst−1,

whiletheendtfastf=tlast.Theduration(dd)iscalculatedwithEq.(1).

𝑑𝑑= 𝑡𝑓 ∑ 𝑡 =𝑡𝑖

S(𝑡) (1)

Themagnitudesofareas ofthelargestclusterscalculatedineach timestepwithS-TRACKmethodaresavedinthetimeseriesDA(drought area).Thedroughtareaisusedasthemeasureofthedroughtseverity (ds),whichiscomputedasthesumofdroughtareasoftheperiod de-finedbytheonset(ti)andtheend(tf)(Eq.(2)).Droughtintensity(di)is definedastheratiobetweendroughtseverityandduration(Eq.(3)).

𝑑𝑠= 𝑡𝑓 ∑ 𝑡 =𝑡𝑖 DA(𝑡) (2) 𝑑𝑖=𝑑𝑠∕𝑑𝑑 (3)

Identificationoflocationswhereadroughtpathstartsandendscan provideitsmaindirection.Theinitialandfinallocationsareidentified usingthecentroidsofthefirstandlastcluster,respectively.Thelocation isarelativepositioninthespatialdomainofthestudyregion.Itrefers toapointintheaxessouth-north(S-N)andwest-east(W-E)(Fig.4).The originoftheaxesisassignedarbitrarily,hereitisproposedtoplacethis origininthecentroidofthestudyregion.Thecentroidofaparticular

Fig.3. Schematic overview of the four cases of linking clusters (drought areas) in time .Area at time tis indicated by At(bold circle) and its predecessor at time t–1 by At–1 (dashed circle). Centroids of areas Atand At–1are denoted by ptand pt–1(points), respectively. Distance between cen- troids is represented by Δl(arrow). An example of the size of parameters aand bis represented by the circles shown on the right. Centroids in both (i) and (ii) have the same location, in the same way, the centroids in both (iii) and (iv). Areas Atin (i) and (iii) are of similar size and between the parameters aand b. On the other hand, in (ii) and (iv), areas A_tare also equal but above those parameters (case of a “very large ” area). Only the parameters of drought area are represented in this figure. Schemes (i) to (iv) help to illustrate the relevance of using parameters that con- sider not only the magnitude of areas but also the distance between them within the linking algorithm. As a distance limit that helps in linking large areas may not be adequate in connecting smaller ones, as shown in (iv) and (iii), the two distances parameters are proposed in the linking algo- rithm (see Sect. 2.1 for details).

Fig.4. Schematic overview of the procedure to define centroid’s location of a cluster. A centroid can be located in one of nine positions: centre (C), east (E), northeast (NE), north (N), northwest (NW), west (W), southwest (SW), south (S) and southeast (SE). The symbol rstands for the distance between the cluster’s centroid and the one of the region. The angle between the W-E axis and the line defined by centroid’s cluster is indicated by 𝜃. The radius to define if a cluster is located in the centre (C) of the region is pointed out by rmin.

clustercanbelocatedinoneofthenineproposedpositions:centre(C), east(E),northeast(NE),north(N),northwest(NW),west(W),southwest (SW),south(S),andsoutheast(SE)(Fig.4).Centre(C)issituatedinthe centroidofthestudyregion(Fig.4).Apoint(centroid)isinthecentre ifthedistance(r)betweensuchpointandtheoriginiswithinthermin

radius(Fig.4).Ifdistance ris outof thermin radius,thelocationis

assignedbasedontheangle𝜃.ThisangleiscalculatedbetweentheW-E axisandthelinedefinedbetweenthecentroidandorigin(Fig.4).All therulestoidentifythecentroid’slocationarepresentedin Table1. Withinthealgorithm,insteadofletters,locationsaredenotedbymeans ofnumericalidentifiers(Ids)aspresentedinthefirstcolumnofTable1. Droughttracksprovidethevisualoverviewofhowdroughtmoves inthespatialdomain.Initialandendlocation(initialandendpointof thetrack)helptoidentifythedirectionfollowedbyagivendrought

Table1

Rules to define the location of a centroid’s cluster. Nine positions are pro- posed: centre (C), east (E), northeast (NE), north (N), northwest (NW), west (W), southwest (SW), south (S) and southeast (SE).

Id Rule Location 0 r ≤ r min C 1 r > r min and 0° ≤ 𝜃< 22.5° or 337.5° ≤ 𝜃< 360° E 2 r > r min and 22.5° ≤ 𝜃< 67.5° NE 3 r > r min and 67.5° ≤ 𝜃< 112.5° N 4 r > r min and 112.5° ≤ 𝜃< 157.5° NW 5 r > r min and 157.5° ≤ 𝜃< 202.5° W 6 r > r min and 202.5° ≤ 𝜃< 247.5° SW 7 r > r _{min and} 247.5° ≤ 𝜃< 292.5° S 8 r > r min and 292.5° ≤ 𝜃< 337.5° SE

r= distance between centroid’s cluster and the one of the region;

𝜃 =angle between W-E axis and the line defined by centroid’s cluster;

rmin= limit distance to consider the location in the centre (C) of the region.

cluster.Yetanothercharacteristicthatcomplementsthedescriptionof thedroughtdynamicsisitsrotation.Thischaracteristicisdefinedasthe circularorientationfollowedbythespatialextentofdrought.Rotation isafeaturecommonlyattributedtoobjectsthatexperiencechangesin space.Itisanessentialcharacteristicanalysedinotherweather-related phenomenasuchascyclones(e.g.Chavasetal.,2017;Rahmanetal., 2018) butthathasnotbeenusedandexploredmuchin droughtsso far.Thischaracteristicisincludedbecauseitisforeseenthatitcanhelp toanalysetheimpactofthedroughtdrivers,suchastheclimateand land surfacecontrolfactors,on thespatialdevelopment ofdroughts. Thedroughtrotationpatternsareexpectedtobedifferentforeach com-binationoftheaforementionedfactors.Weseethisstudyasaninitial steptowardsdevelopingatechnologicalframeworkforidentifyingand interpretingthedroughtrotation.

As thedroughttrackcan switch between clockwiseand counter-clockwisealongthepathway,weproposetoclassifytherotationina moregeneralwayas(1)mostlyclockwise(cw),or(2)mostly counter-clockwise(ccw)(Fig.5).Todeterminetherotation,aprocedureis sug-gestedwhichmakesuseofthecentroids’coordinates.Thealgorithmis basedoncomputingapolygon‘sarea(A)fromthevectorwiththe co-ordinatesxandyrepresentingthevertices(Eq.(4)).Inthisalgorithm, firstlythesumofproductsbetweenthecoordinatesxandy,denoted

Fig.5. Example of rotation calculation. Two types are considered: (1) mostly counter-clockwise when 𝜌<0 (left); and (2) mostly clockwise when 𝜌>0 (right). The number in each centroid (point) indicates the tracking sequence. Ar- rows show the track direction and the rotation. Rotation of each line segment is also pointed out by cw and ccw that stand for clockwise and counter-clockwise, respectively.

by𝜌 (Eq.(5)),iscalculated.Then,𝜌 isappliedtodefinetherotation direction(Eq.(6)).Thecoordinatesxandyaretakenfromtheonesof centroids’clusters.Whenthereareonlytwopoints (twoclusters),or whenthetrackishorizontalorvertical,therotationisnotdefined, be-cause𝜌 takesthevalueofzero.InFig.5,twoexamplesofthecalculation ofrotationareshownbywayofillustration.Oneexampleispresented formostlycounter-clockwise(Fig.5(left))andoneformostlyclockwise (Fig.5(right)).Wechosethisapproachtocomputerotationbecauseit distinguishesbetween “big” and“small” turnsin thecalculation(Eq. (5)).ThefourthcolumninbothtablespresentedinFig.5provides ex-amplesofhowthemagnitudeofeachturnisconsidereddifferentlyin therotationalgorithm.

𝐴=1 2|𝜌| (4) 𝜌 =(𝑥1−𝑥𝑛 )( 𝑦1+𝑦𝑛 ) + 𝑛 −1 ∑ 𝑖 =1 ( 𝑥𝑖 +1−𝑥𝑖 )( 𝑦𝑖 +1+𝑦𝑖 ) (5) 𝜔= ⎧ ⎪ ⎨ ⎪ ⎩ cw(mostlyclockwise)if𝜌 >0 ccw(mostlycouter−clockwise)if𝜌 <0 nan(notdef ined)if𝜌 =0

(6)

2.3. Experimentalsetup 2.3.1. Droughtindicatordata

DroughttrackswerecalculatedwithS-TRACKalgorithmforthe pe-riod1901to2013(113years).TheanalysiswasconductedforIndia, onamonthlybasis.DatafromtheStandardizedPrecipitation Evapora-tionIndex(SPEI)GlobalDroughtMonitor(http://spei.csic.es/)wasused (Beguería etal.,2014) totesttheproposedmethodologyfordrought trackingandcharacterisation.TheproceduretocalculateSPEI( Vicente-Serranoetal.,2010)issimilartotheoneusedtocomputethe Standard-izedPrecipitationIndex(SPI)(Mckeeetal.,1993),buttakinginto ac-countprecipitation(P)minuspotentialevaporation(E)insteadofonly

P.SPEIdatafrom thedroughtmonitorareina gridformfor differ-enttemporalaggregationperiods.Inthisstudy,weusedSPEI-6,which correspondstoanomaliesofthesix-monthaccumulationofP– E.This aggregationusuallyreferstoextendedperiodsoflackofwater availabil-ity,thereforeconsequencesofwhatiscommonlycalledmeteorological droughtareclosertothatcausedbytheso-calledhydrologicaldrought (WorldMeteorologicalOrganizationWMO,2012).

2.3.2. Droughtareasandcentroids

Beforetheapplicationofthedroughttrackingalgorithm,thesizeof thelargestclustersandthedistancesbetweenthecentroidsof consecu-tiveclustersintimewerecalculated.Thiscalculationwasperformed,on theonehand,tounderstandtheorderoftheirmagnitudeandfrequency,

andontheotherhand,tosetthevaluesofthetrackingalgorithm pa-rameters.

Forthedefinitionofdroughtareas,usually,thethresholdof -1is usedtoindicatedroughtconditioninthedroughtindicatorsthatfollow asimilarmethodologythanSPI,alsoreferredtoasstandardisedones. Inthisresearch,thesamethreshold(SPEI=-1)wasselectedtodefine droughtconditionineachcellofthegridineachtimestep.WhenSPEI wasbelow-1,with1sthedroughtconditionwasindicated,inanother case,with0sthenon-droughtstatuswaspointedout.Thisbinary rep-resentationallowedtheidentificationofspatialdroughtunits(clusters) throughtheapplicationofthespatio-temporalanalysisofContiguous DroughtArea(CDA)(Sect.2.1).

Thelargestclustersineachtimestepwerethenidentified.Thearea ofthelargestclusterwascomparedwiththetotalonetoidentifythe similarityinsizebetweenthem.Itisassumedthatthemoresimilarthe largerareatothetotalone,thebettertheidentificationofthedrought trackswillbe.This standsbecausethetrackingalgorithmfocuseson onlyoneareapertimestep.Forthecomparison,theareaofallclusters (DA_total)andtheareaofthelargestone(DA_largest)werecalculated. Bothareas wereexpressedaspercentages calculatedastheratio be-tweenthenumberofcellsindroughtandthetotalnumberofcells.The totalnumberofcellsconsideredforIndiawas1,173.

Oncethecentroidswereidentified,thedistancesbetween consec-utivecentroidswerecalculatedovertime(Sect2.1).Boththeclusters andthedistanceswerecalculatedfortheentireperiodofanalysisona monthlybasis.

2.3.3. Trackingalgorithmcalibrationandevaluation

S-TRACK usesfour parametersandtheyhave tobe user-defined, or,better,calibrated.Theproblemofcalibratingthisalgorithmisthat thereisnoground-truthdataonthedroughttracks,hence,somealiases shouldbeused.Afull-fledgedcalibrationprocedurecanbeapplied(e.g. one of therandomised searchalgorithms,like anevolutionary algo-rithm).Theoptimalparametersshouldbeselectedbasedoninformation ofreporteddrought.Intheabsenceofdroughttracks,itisnecessaryto havedataatleastontheonsetandendmonthofthereporteddroughts. Thenear-optimalparametersarethosethatprovidethebestmatch be-tweentheobservedandcalculatedonsets/ends.

However,inthispaper,weappliedasimplifiedprocedure. Consid-eringthatthereisnoavailableinformationtocomparethecalculated droughtpathsinthestudyarea,welimitedtheproceduretoa quali-tativeanalysisofthepathsofthemostseveredroughtsreportedinthe analysisperiod.Thedroughtsof1905,1942,1965,1972,1987,2000, and2002wereconsideredbecausetheirsevereimpactswerereferenced (Guha-Sapir,2018).Thequalitativeevaluationwasfocusedonthe anal-ysisoftheextremeincidencesusingacombinationofparameters.From thewholesetofcombinations,wehavechosenthree:theonethat pro-ducesthesmallestnumberofdroughtspaths(combination_1),theone that yieldsthelargest numberofdroughtspaths (combination_3),as

Fig.6. Percentage of drought area considering all clusters (DA_total, left), and considering only the largest one (DA_largest, centre). Right panel shows the difference between DA_total and DA_largest.

wellastheonethatproducesthenumberofdroughtpathssimilarto thenumberofyearsoftheanalysisperiod(combination_2).

Yetanotherimportantpartof thealgorithmevaluationisits sen-sitivityanalysis. Itallowsforassessingtherobustnessof themethod throughtheanalysisoftheoutputsunderthevariationofparameters (Pannell,1997).Sensitivityanalysisgenerallyallowsansweringthe fol-lowingquestionswhenevaluatinganalgorithm.Howparametersand outputarerelated?Whatlevelofaccuracyintheparametersisrequired? Whichparametersaremoresensitive,andwhatdroughtcharacteristics dotheyinfluencemost?Whataretheconsequencesofvaryingthe pa-rameters?

Inprinciple,calibrationandsensitivityanalysisstepshavetobe co-ordinated,e.g.allowingforremovaloflesssensitiveparametersfrom thesetoftheparameterstobecalibrated(e.g.tospeedupcalibration). Inthiswork,asthealgorithmisnotcomputationallycomplex,this ap-proachwasnotfollowed.Thesensitivityanalysiswasperformedto as-sesstheeffectofparametersovertheidentificationofdroughtstracks andcharacteristics.Thequestionsmentionedinthepreviousparagraph wereusedasaguidelinetoperformsuchananalysis.

3. Results

3.1. Droughtareasandcentroids

Droughtareasandcentroidswerecomputedfortheperiod1901to 2013.Withrespecttotheareas,firstlythecomparisonbetweenthearea ofallclusters(DA_total)andareaofthelargestone(DA_largest)was per-formed.Fig.6showsthemonthlyvaluesofbothDA_totalandDA_largest arrangedinmatrices.ColumnsindicatemonthsfromJanuary(J)to De-cember(D),whilerowspointouttheyearfrom1901to2013.Drought areamagnitudeisindicatedwithacolourscale,wherethedarkerthe colour,thehigherthedroughtareais.Thewhitecolourdenotesmonths withsmalldroughtareas(lessthan10%).Itisobservedthatforalmost allmonthsDA_totalandDA_largesthavesimilarvalues,andthis agree-mentisespeciallyhighforthelargestvalues.Thedroughtarea aver-agefortheperiodwas17.4%forDA_total,and11.5%forDA_largest.

Fig.6(right)presentsthedifferencebetweenDA_totalandDA_largest. Acrossthewholeperiod,theaverageofthedifferenceswas5.9%.As DA_largestandDA_totalwereverysimilar,itcanbeconsideredthatthe largestclusterisagoodproxytoanalysehowdroughtchangesinthe regionwithoutconsideringtheoccurrenceoftwoconsecutivedrought tracks.

Fig.7. Centroids of the largest clusters (DA_largest) identified on a monthly basis. Spatial drought extent is schematized by four symbols pointing out the drought area. The origin of the axes is placed in the centre of the country.

ThecentroidsofthelargestclustersarepresentedinFig.7.The spa-tialdroughtextentisshownschematicallywithsymbolsthatindicate fourintervalsofthepercentageofdroughtareawithrespecttothe coun-tryextent.Theoriginoftheaxesisplacedinthecentreofthecountry. Itisobservedthatthespatialdistributionofthecentroidsisalmost uni-formlydistributedoverIndia.However,ahigherdensityof theareas withaconsiderableextentcanbeseeninthecentralregion.

Thedistancesbetweenconsecutiveclustersintimewerecalculated alsoforthewholeperiod.Fig.A1(AppendixA)presentstheareaofthe largestcluster(DA_largest)andthedistance(Δl)betweenconsecutive clustersintime.ItcanbeobservedthattheoccurrenceofDA_largest is greaterthan25%duringalldecadesoftheanalysisperiod.A pat-ternisobservedbetweenDA_largestandΔl:whenDA_largestincreases, Δlusuallydecreases. Thisbehaviourwasexpected,becausethemore theareaincreases,thesmallerthedistancebetweencentroidsbecomes.

Fig.8. Relative frequency of the largest cluster area (DA_largest) and distances ( Δl) between consecutive clusters in time.

Thismeansthatthelocationoftheconsecutiveclustersisbecomingthe same.WhenΔldoesnotfollowthisbehaviour,itmightbebecausethe consecutiveareasintimeareveryfareachother,i.e.theyarepartof differentdroughtpaths.

Fig. 8 shows the frequency of both the largest cluster area (DA_largest)andthedistance(Δl)betweenconsecutiveclustersintime. Forbothvariables,resultsaredisplayed infourintervals.It was ob-servedthatastheareaincreases,thefrequencyof longdistances be-tweentheseareasdecreases,whilethefrequencyofsmalldistances in-creases.FortheDA_largesttheintervalsof25-50%and≥50%,the fre-quencyofthesmalldistances(Δl<250km)wasslightlygreaterthan halfofallthedistances.ThisresultsofDA_largestandthedistancesΔl

confirmquantitativelywhatisobservedinFig.A1:ingeneralwhenthe areagrows,thedistancesbetweenthecentroidstendtodecrease.Onthe otherhand,thesmallvalueofthefrequencyoflargedistancesinlarge areas(intervals25-50%and≥50%)indicatesthattherearelarge con-secutiveareasintimethatarenotnecessarilyconnectedtoeachother.

3.2. SensitivityofS-TRACKresultstothechoiceofparameters

S-TRACKalgorithmhasa numberof parameters.Forthereasons mentionedabove(Sect.2.3.3),itisusefultostudythesensitivityofits outputstotheseparameters.Basedontheresultsofareasanddistances betweenclusters(Sect.3.1),theS-TRACKalgorithmwassettotake pa-rametersvalueswithinthefollowingranges:a≤50,b≥50,c≥50,and

d≥50thpercentile(median).Asmentioned,aandbareparametersthat controlthesizeofclusters(areas),andcanddareparametersthat con-strainthedistancesbetweenconsecutiveclustersintime.Theaverage duration,averageseverity,onsetlocation,aswellasendlocation,were calculatedforthedifferentcombinationsofparameters.Resultsfora

(30,40,and50),b(50,70,and90),c(50,60,70,80,and90),andd

(50,60,70,80,and90)arepresentedinFigs.9andA2toA6(Appendix A).Theaandbparametersareexpressedaspercentageofdroughtarea andcanddaskm.Attheendofthissection,asummaryoftheresults ispresented.

Fig.9showsthenumberofdroughtpaths(combinationoftracks linkedintime).Itisobservedthatthenumberofdroughtpaths,in gen-eral,increasedwhenadecreased.Thisisexpectedsinceparameterais theonethatdeterminesifaclusterjoinstheconsecutiveclustersineach timestep.Whenaissmall,moreclustersareexpectedtobeconnectedin eachtimestepandthereforemoredroughtpathscanbeidentified.The valueofb(usedfor“verylarge” areas)influencedthenumberofpaths

lessthana,e.g.sothatwhenbincreased,therewasasmallproportional increaseinthenumberofpathsforallcombinationsofparameters.The combinedvariationofbandcinfluencedmorethenumberofpathsfor smallvaluesofd.Itwasobservedthatingeneral,thenumberofpaths dropswhenaincreasesandbothb,c,andddecrease.Ingeneral,the numberofdroughtpathswasmoresensitivetothechangesin parame-tera.

InFig.A2theaveragedurationofdroughtpaths ispresented. Al-thoughthevariationofaveragedurationwassmalltothechangesof parameters,aslightincreasewasobserved,asadecreasedandbothb,c

anddincreased.Theaveragedurationwasmoresensitivetotheincrease incanddthataretheparametersthatcontrolthedistancebetween con-secutiveclustersintime.

Regardingtheseverity,itwassmallerwhenaincreasedandbothb, c,andddecreased(Fig.A3).Severityiscalculatedastheratiobetween thetotalsumofdroughtareasandduration(numberofmonths),soitis gettinglowerasdurationincreases(seeEqs.(1),(2),and–(3)).Similarly tothenumberofdroughtpaths,theaverageseveritywasalsosensitive tochangesin parametera.Itwasobserved thatwhenthenumberof pathsdecreased,theaverageseverityincreased(Figs.A2andA3).This behaviourinseverityistheeffectoftheselectionofathatcontrolsthe sizeoftheareasthatarejoinedineachinstantoftime.Ifaissmall, moreareascanbejoinedandseveritymaydecreaseduetotheeffectit producesthepoolingofmoreareasofsmallsizesdividedbyalonger duration(seeEqs.(1),(2),and–(3)).

Figs.A4andA5showthemodeofonsetandendlocationofdrought paths, respectively. In Fig.A4, not many changes were observed in theonsetlocation.Eastwasthemostcommononsetlocationinmost combinations of parameters, followedby South.On theotherhand,

Fig.A5showstheendlocationsthatinmostcombinationstheSouth, followedbyEastwerethemostcommon.Whenbothadecreaseandb

increased,theSouthwasthemostcommonendlocation.

Fig.A6showsthemodeofrotation.Itwasobservedinmostcases that mostlyclockwise(cw)was thecommonrotationinthedrought paths.Whenadecreasedandbincreased,themostlyclockwiserotation wasthemostcommonrotation.Thiswasthecasewhenmoredrought pathswereobtained.Itwasobservedthatrotationwasmostsensitive tothevariationsofcanddthataretheparameterswhichcontrolthe distancebetweenconsecutiveclustersintime.

Summaryofresults

Table2showsasummaryofhowthetrackingalgorithmresponds todifferent combinationsofparameters. Inparticular,thebehaviour ofthenumberofpaths,duration,severity,onsetandendlocation,as wellasrotation,isindicated.Thecombinationswhereitwasobserved thatthevaluesofthesecharacteristicstendtoincreaseordecreaseis presented.Ingeneral,themostsensitiveparameter(important)isthe onethatcontrolstheminimumarea(parametera).Changesinthis pa-rameter havemoreinfluenceontheresultofthenumberofdrought pathsandduration.Regardingdurationandseverity,itisobservedthat asthepathslastlongertheseveritydecreases.Thismayapplybecause theseverityiscalculatedasthesumoftheareasofclustersthatbelong tothedroughtduration.Thus,whilethedurationincreases,theareas thatareaddedtendtobesmallerandthenthesumdoesnotincrease significantly.

Thecombination11(Table2)referstotheidentificationofpathsof “verylarge” areas.Inthiscombination,itisexpectedthattheinitialand finallocationswillbeinthecentre.Centroidsoftheseclusterareastend tobeidenticaltothatoftheregion.Forthesepaths,itisalsoobserved thattherotationtendstobeclockwise.

Incombinations6,7and14(Table2),bydecreasingtheparameter thatcontrolstheminimumarea(parametera),moredroughtpathsare identified,withthecharacteristicofbeinglongandwithasmallseverity (formedbyanumberofsmallerareas).Inthesecombinations,drought pathsusuallystartintheEastandendintheSouth,withaclockwise rotation.

Fig.9. Number of drought paths obtained with different combinations of parameters. Table2

Summary of drought characteristics obtained with different combinations of parameters. Numbers in parentheses indicate the location as presented in Figs.A4

and A5. Abbreviations ccw and cw stand for counter-clockwise and clockwise, respectively. # Parameters Number of paths Drought characteristics

a b c d Duration Severity Onset location End location Rotation

1 ↑ ↓ ↓ ↓ decreases tends to decrease

decreases tends to decrease

increases

2 ↑ ↓ ↑ ↑ decreases decreases increases

3 ↑ ↑ ↑ ↑ decreases decreases increases

4 ↑ ↑ ↓ ↓ decreases decreases increases

5 ↓ ↓ ↓ ↓ increases increases decreases

6 ↓ ↓ ↑ ↑ increases increases decreases tends to

the south (7)

tends to cw 7 ↓ ↑ ↑ ↑ increases increases tends to

increase decreases tends to the east (1) tends to cw

8 ↓ ↑ ↓ ↓ increases increases decreases

9 ↑ ↓ ↓ ↑ decreases decreases increases

10 ↑ ↓ ↑ ↓ decreases decreases increases

11 ↑ ↑ ↓ ↑ decreases decreases increases tends to

increase tends to the centre (0) tends to the centre (0) tends to cw

12 ↑ ↑ ↑ ↓ decreases decreases increases

13 ↓ ↓ ↓ ↑ increases increases decreases

14 ↓ ↓ ↑ ↓ increases increases decreases tends to

the south (7)

tends to the east (1)

tends to ccw

15 ↓ ↑ ↓ ↑ increases increases decreases

16 ↓ ↑ ↑ ↓ increases tends to increase

increases decreases tends to decrease

Fig.10. Occurrence of drought paths calculated with three combination of parameters: (left) combination_1 ( a= 50, b= 50, c= 50, d= 50), (centre) combination_2 ( a= 40, b= 50, c = 70, d= 80), and (right) combination_3 ( a= 30, b= 70, c = 90, d= 50). Consecutive cells in colour indicate the occurrence of a drought path (top). Frequency is calculated per column from January (J) to December (D) (bottom).

IfthedroughtpathstartsintheSouth,itusuallyendsintheEast,and inthiscase,therotationiscounter-clockwise,i.e.therotationfollows theminorturn(Table2(combination14)).Inotherwords,ifthepath startsintheSouthandendsintheEast,itismorelikelytobedirected towardstheEastshowingacounter-clockwiserotation,insteadofgoing firstlytotheWest,thenNorthandfinallyEast,showing aclockwise rotationinthiscase.

3.3. Qualitativeevaluationofdroughtpaths

Sevenof themost extremedroughtsreported duringtheanalysis periodwereselected fortesting S-TRACK.These droughts,as itwas mentionedearlier,correspondtothefollowingyears:1905,1942,1965, 1972,1987,2000,and2002.Intheabsenceofinformationregarding thedynamicsofthedroughts,suchastrajectories,ourvalidation fo-cusedontheanalysisofthecalculatedtracksintheperiodwhenthe droughtsoccurred.

Fromthesetofparametercombinationsshownintheprevious sec-tion,threewereselectedtoanalysethecalculateddroughttracks.For thefirstcombination(combination_1,a=50,b=50,c=50,d=50), thenumberofdroughtpathsobtainedwasthelowest.Forthesecond combination(combination_2,a=40,b=50,c=70,d=80),the num-berofdroughtpathswassimilartothenumberofyearsoftheanalysis period,i.e.therewasapproximatelyonedroughtpathperyear.Finally, inthethirdcombination(combination_3,a=30,b=70,c=90,d=50), thehighestnumberofdroughtpathswasidentified.

Fig.10presentstheoccurrenceofdroughtpathscalculatedforthe threecombinationsofparameters.Columnsindicatethemonthsfrom January(J)toDecember(D)andtherowsshowtheyears.Consecutive cellsincolourindicatetheoccurrenceofadroughtpath(Fig.10(top)). Thefrequencypermonthwascalculatedtoanalysethedistributionof thetracksoverthemonths(Fig.10 (bottom)).Ingeneral,themonth withthelessfrequencyofdroughttrackswasMarch.FromJanuaryto July,thefirstpartoftheyear,thefrequencywasfewerthanfromAugust

toDecember.Itwasobservedthatwhenthenumberofdroughtpaths increased(Fig.10(top,fromlefttoright)),thefrequencyofdrought tracksineachmonthincreasedaswell(Fig.10(bottom)).

Fig.11showstheresultsfromthecalculationofclustersand dis-tancesbetweencentroidstotheconstructionofdroughtpathsforthe droughtof1987-1988.InAppendixA,onecanseetheothersixdroughts (Figs.A7,A8,A9,A10,A11,andA12).InFig.11(top)clustersand cen-troidsarepresented.Areasoflargestcluster(DA_largest)anddistances betweenconsecutiveareasintime(Δl)areshownfortheperiodfrom 1987/1to1988/6(Fig.11(centre)).Durationofthedroughtpathsis in-dicatedinaschematicwaywithahorizontallineforeachcombination ofparameters.Droughttrackscalculatedwiththethreecombinationsof parametersarealsopresented(Fig.11(bottom)).Inmostoftheseven droughts,themaximumareasofthelargestclusterswereinthesecond halfoftheyearandthefirsthalfofthefollowingone(e.g.Fig.11 (cen-tre)).Itwasobservedthat,ingeneral,whenDA_largestincreased, Δl

usuallytendedtodecrease(e.g.Fig.11(centre)).Thisrelationshipcan beexploredinfurtherresearchtodefinequantitativelytheonsetand endofthedroughts.

Table3presentsasummaryofthedurationoftheselecteddroughts. Itwasobservedthatalthoughthenumberofdroughtpathsincreases fromthecombination_1(Fig.10(left))tothecombination_3(Fig.10

(right)),intermsofthemostseveredroughts,thedurationsremain al-most similar(Table 3(column2and4)andFig.11(bottom)).More droughttrackswereidentifiedinthefirstpartoftheyearin combina-tion_3.Iftheparameterscanddthatcontrolthedistancebetween cen-troidsaremoreflexible,i.e.considerlongerdistances,droughttracksof thesecondpartoftheyeararemorelikelytojointhoseofthefirstpart ofthenextyear,asoccursincombination_2.Inthecombination_2,the droughtpathsshowedthelongestdurations(Table3(column3)and

Fig.11(bottom,centre)).

Inalltheselecteddroughts(Figs.11andA7toA12),itwasobserved thatconsecutiveclustersintimeoverlapconsiderably,whichsuggests thatthespatialextentafterreachingaconsiderablesize,itremainsin

Fig.11. Results for the drought of 1987. Largest clusters and centroids are indicated from 1987/3 to 1988/6 (top). Area of largest cluster (DA_largest) and distance between consecutive clusters in time ( Δl) are displayed from 1987/1 to 1988/6 (centre). The drought duration is pointed out schematically with a horizontal line for each combination of parameters. Drought tracks calculated with the three combinations of parameters are also presented (bottom). Spatial drought extent is schematised by four symbols pointing out the size of area. The origin of the axes is placed in the centre of the country. Arrows point out the direction of each track segment. Insets show zoomed-in views.

Table3

Duration of selected droughts calculated with three combinations of parameters. In parentheses, the period is indicated.

Drought Duration [months]

Combination_1 Combination_2 Combination_3

a = 50, b = 50, c = 50, d = 50 a = 40, b = 50, c = 70, d = 80 a = 30, b = 70, c = 90, d = 50 1 6 (1905/7 to 1905/12) 12 (1905/6 to 1906/5) 6 (1905/7 to 1905/12) 2 5 (1942/10 to 1943/2) 6 (1942/10 to 1943/3) 5 (1942/10 to 1943/2) 3 6 (1965/7 to 1965/12) 22 (1965/5 to 1967/2) 6 (1965/7 to 1965/12) 4 3 (1972/8 to 1972/10) 16 (1972/4 to 1973/7) 3 (1972/8 to 1972/10) 5 6 (1987/9 to 1988/2) 8 (1987/7 to 1988/2) 6 (1987/9 to 1988/2) 6 5 (2000/8 to 2000/12) 11 (2000/8 to 2001/6) 6 (2000/7 to 2000/12) 7 5 (2002/8 to 2002/12) 12 (2002/4 to 2003/3) 6 (2002/8 to 2003/1) thesameregion.Thispresenceoflargedroughtareasinthesameregion

overtimemayexplaintheseverityofdroughteventsinthosedroughts. Therewasnopredominantpathwayfollowedbydroughtsinthoseyears. Intermsofspatialextent,2000and2002eventswerethelargestas showninFigs.A11andA12,respectively.Thedroughtwiththelongest durationwasthatof1965(Table3),whichisconsistentwiththe re-portedin(Guha-Sapir,2018).

4. Discussion

4.1. Droughtindicatorandareas

Inthepresentedversionofthetrackingmethod,weusedaunique threshold over the drought indicator to indicate drought and non-droughtconditionsineachgridcell(1sand0s).Thisthresholdisone ofthemost commonusedin droughtstudieswhenconsidering

stan-dardiseddroughtindicators.SPEIwasappliedinthisresearch,butitis possibletouseanyother,includingthresholdapproach(Wandersetal., 2010),withtheconditionofbeingspatiallydistributed.Theeffectsof other droughtindicatorthresholdsover theclustersizewerenot as-sessedbecausethescopeofthisstudywaslimitedtotestingthedrought trackingalgorithm.

Ontheotherhand,theclusteringalgorithmusedinthisstudy as-sumes thatallcellvaluesin thespacedomainarehomogeneous. To ensurethatthisassumptioniscorrect,itisrecommendedtheselection ofadroughtindicatorthatusesanormalizationprocedureintoits cal-culation.Inaddition,ourclusteringapproachisbasedonlyondrought indicatorvaluesanddoesnotconsiderothersaspectsthatcaninfluence thedelimitationofthespatialextentofdrought,suchastopography, landuse,andclimateregions.Infurtherstudies,itisrecommendedto incorporateotherelementstomaketheclusteringmethodmore gen-eral.Anotherwayofconsideringthefactorsmentionedabove,without

modifying/changingtheclusteringalgorithm,istheuseofadrought in-dicatorthattakesintoaccountvariablessuchassoilmoistureorrunoff.

4.2. Droughttrackingmethod

S-TRACKalgorithmisanextensionoftheContiguousDroughtArea (CDA)analysisofCorzoPerezetal.(2011).Thisdroughttracking algo-rithmwasfirstlyintroducedinDiazetal.(2018)andfurtherdeveloped inthisresearch.ThecurrentversionofS-TRACKfocusesonthelargest droughtareas.Inthisway,areaswithaconsiderableterritorialextent areidentified.Weareawarethatsmaller,intensedroughtswouldnot becapturedbythistrackingalgorithm.Also,thatmilddroughtsover largeareasobtainedbythealgorithmwouldovershadowsmaller, in-tensedroughts.

Although S-TRACK makes use of CDA analysis for the extrac-tion of drought clusters, other algorithms used for the same pur-posecanalsobeconsidered. Thesealgorithmsincludethe recursion-basedapproach(Andreadisetal.,2005;Herrera-Estradaetal.,2017;

Lloyd-Hughes, 2012; Sheffield et al., 2009), and variations of the connected-componentlabellingapproach(VanHuijgevoortetal.,2013;

Vernieuweetal.,2019).Thecompositionofdroughtclustersextracted withanyofthesealgorithmsshouldbesimilar.Themaindifference be-tweenthealgorithmsisinthecomputationalefficiencyandprocessing time,whichisanimportantelementtoconsiderwhenprocessingalarge amountofdata.Inthissense,algorithmsbasedonconnected-component labellingareconsideredtobemoreefficient(Heetal.,2009).

Toconnecttwoconsecutiveclustersintimeandensurethattheyare notfarinspace,thelengthbetweencentroidsoftheclustersistakeninto account,similartoHerrera-Estradaetal.(2017)andZhouetal.(2019). Thedegreeoftheoverlapbetweenthesetwoclusterscanbeanother waytohandletheconnectionbetween them.Yet another,andmore comprehensivewayof joiningclustersintime, isthroughtheuseof theCDAapproach butextendedtothetime domain,i.e. toconnect 26nearestneighbourcells,aformingacubeinspace-timedomain,as showninCorzoPerezetal.(2011),Lloyd-Hughes(2012),and Herrera-Estradaetal.(2017).

Incaseswhenmorethanonedroughttrackoccursatthesametime, thealgorithmwillaimtoidentifytheonethatiscomposedofthelargest areas.Initscurrentversion,thealgorithmneitherdetectssimultaneous droughttracksnormergestheareasofthesametimestepintoasingle one.

Inthisresearch,wecomparedtheareaofallclustersandthearea ofthelargestoneineachtimestep,toseeifthepresenceofmorethan onelargeareaispredominantornot.Wefoundthatdifferencebetween DA_total−DA_largestwas,inmostofthecases,closetozero(Fig.6). ThisdifferencebetweenDA_totalandDA_largestindicatesthatthesize oftheareaofthelargestclusterisverysimilartothetotalone.Basedon thelatter,itisassumedthatthepresenceofmorethanonelargecluster atthesametimestep,isnotdominant.Thentheresearchwasfocused ontestingthetrackingalgorithm,withoutconsideringtheeffectofthe presenceofmorethanonesimultaneousdroughttrack.

Ifthepresenceofmorethanoneconsecutivedroughttrackis sus-pected,anoptiontoperformthisalgorithmistocarryourtrackingfor differentsub-regionsofthestudyarea(analyseitbyparts)andthen su-perimposethedroughttracks.Inthisway,onewouldexpecttoidentify morethanonetrack,ifany.Infutureversionsofthetrackingalgorithm, itisrecommendedtoincludetheidentificationofmorethanone simul-taneousdroughttracks.

TheuseofCDAapproachcanretrieveareaswith“islands” of non-droughtcells(0s).Inthisresearch,wedo notconsiderthepossible effectsoftheseholesoverthedroughttracksconstruction(thecentroid couldbelocatedinoneoftheseholes).Weassumethatcentroidisa goodspottolocatethecontiguousdroughtarea.

Inlargestclusters,thecentroidapproachesthecentroidofthe anal-ysisregion.Thisisanexpectedoutcomebecauseiftheclustercovers theentireregionofanalysis, thecentroidwillbe similartothatone

oftheregion.Inourcase,themaximumDA_largestwas70.7%, there-foreintheperiodofanalysis,noclustercoveredtheentireterritory.In addition,twosimultaneouslargeclustersarenotexpected.

Althoughthedroughttracksthatoccurredneartheboundariesof thedomaincouldnotbeconsideredappropriately,i.e.thetrackscould bemiscalculated,itisassumedthattheseboundarytracksdonot signif-icantlyimpacttheregion.Toimprovethecalculationinsuchcases,itis recommendedtoincreasethesizeoftheanalysedregion.

5. Summaryandconclusions

Inthisstudy,amethodthatallowstheconstructionofdroughttracks inspaceisintroduced.Theonsetandendofdroughtpaths(combination oflinkeddroughttracks)areusedtocomputethedroughtduration.The informationobtainedduringthepathcalculationisemployedto com-putetheseverity,aswellastheonsetandendlocation,direction,and rotation.Allthesefeatureshavebeenidentifiedasdrought character-isticsandareframedwithintheDDRASTIC-spatialmethodology,also presentedinthispaper.OutputsofthetrackingalgorithmS-TRACKand themethodfordroughtcharacterisationDDRASTIC-spatialhelpto de-scribethedynamicsofdroughts.

S-TRACKhasfourparameters.Parametersaandbcontrolthesizeof thecluster(area)tobeincludedinthedroughttracks.Parameterscand

dlimitthedistancesbetweenconsecutiveclustersintime(Sect.2.1).In thispaper,S-TRACKisusedtoconstructthedroughttracksinspace.

FromtheapplicationofS-TRACK,somekeyfindingsarepresented:

• Thenumberofdroughtpaths,duration,andseverityaremore sensi-tivetothechangeoftheparameterthatlimitstheminimumdrought area(parametera)(Sect.2.1).

• Ifthedurationofthedroughtpathsincreases,severitydoesnot nec-essarilydoso,becausethelongertheduration,theareasthatmake upthepathtendtobesmaller(Sect.3.2).

• Toobtaindroughtpathswithlongerdurations,itisimportanttobe flexiblewiththeparametersthatcontrolthedistancebetweenareas (parameterscandd),i.e.toconsiderlargerdistances.

The outcome ofthe approachpresented in this paperis relevant for(i)droughtforecasting,i.e.droughttrackscanhelptopredicthow droughtmovesoveraparticularregion,and(ii)forimproving knowl-edgeondrought-generatingprocesses.Thefirstitemismorefor opera-tionalpurposes(shortterm)andtheseconditemforscientificresearch (longterm).

Regarding the improvementof knowledge on drought-generating processes,i.e.theinteractionbetweenclimateandlandsurface charac-teristics,anewdroughtcharacteristicisintroducedinthisresearch,the rotation(Sect.2.2).Thisfeatureisusedinthestudyofother weather-relatedphenomenasuch ascyclonesbecauseithelpsin the descrip-tion/identificationofforcingmechanismsbehindtheirspatial develop-ment(e.g. Chavasetal.,2017; Rahmanetal.,2018).Weareof the opinionthatthis droughtcharacteristiccan alsohelpinthe identifi-cationanddescriptionofclimateandlandsurfacecontrolfactorsthat drivethespatialbehaviourofdroughts.

FortheconsideredcasestudyinIndia,wefoundthatconsecutive clustersin timeoverlapconsiderablyin thedroughtsselected (1905, 1942,1965,1972,1987,2000,and2002),whichsuggeststhatthe spa-tialextentofdrought,afterreachingaconsiderablesize,remainsinthe sameregion.Thispresenceoflargedroughtareasinthesameregion overtimemayexplaintheseverityofdroughtsinthoseyears.Thereis nopredominantpathwayfollowedbydroughtsinthoseyears.Interms of spatialextent,2000and2002eventsarethelargest.Thedrought withthelongestdurationisthatof1965.Apaperwaspreparedwhere theparametersofthetrackingalgorithmwerecalibratedbasedonthe informationof droughtsreported. Inthatdocument,adescriptionof droughtsispresented basedonthedroughtpaths andcharacteristics (Diazetal.,2019).

Furtherresearchisaimedattryingtodevelopanapproachto pre-dictthesubsequentdevelopmentoftracksidentifiedbyS-TRACK.These progress of these developments and other aspects of the studycan be found at www.researchgate.net/project/STAND-Spatio-Temporal-ANalysis-of-Drought.

Authorcontributionstatement

VD,GACP,HvL,DS,andEAVcontributedtothedesignand imple-mentationoftheresearch,totheanalysisoftheresultsandtothewriting ofthemanuscript.

AppendixA

Fig.A3. Average severity of drought paths obtained with different combinations of parameters. Severity is expressed as the ratio between the total sum of areas (in percentage) and duration (months).

Fig.A6. Mode of rotation of drought paths obtained with different combinations of parameters. Rotation is indicated by ccw and cw that sand for mostly counter- clockwise, and mostly clockwise, respectively.

Fig.A7. Results of the drought of 1905. Largest clusters and centroids are indicated from 1905/3 to 1906/6 (top). Area of largest cluster (DA_largest) and distance between consecutive clusters in time ( Δl) are displayed from 1905/1 to 1906/6 (centre). The drought duration is pointed out schematically with a horizontal line for each combina- tion of parameters. Drought tracks calculated with the three combinations of parameters are also presented (bot- tom). Spatial drought extent is schematised by four symbols pointing out the size of area. The origin of the axes is placed in the centre of the country. Arrows point out the direction of each track segment. Insets show zoomed-in views.

Fig.A9. Same as Fig.A7but for the drought of 1965. ∗_In the figure only it is shown the tracks until 1966/6 but they end in 1967/2.

Fig.A10. Same as Fig.A7but for the drought of 1972. ∗_{In the figure only it is shown the tracks until 1973/6 but} they end in 1973/7.

Fig.A11. Same as Fig.A7but for the drought of 2000.

Fig.A12. Same as Fig.A7but for the drought of 2002.

DeclarationofCompetingInterest

Theauthorsdeclarethatthereisnoconflictofinterestregardingthe publicationofthispaper.

Acknowledgements

VDthankstheMexicanNationalCouncilforScienceandTechnology (CONACYT)andAlianzaFiiDEMforthestudygrand217776/382365. HvLissupportedbytheH2020ANYWHEREproject(GrantAgreement No.700099).DSacknowledgesthegrantNo.17-77-30006ofthe

Rus-sianScienceFoundation,andtheHydroinformaticsresearchfundofIHE Delftinwhoseframeworksomeresearchideasandcomponentswere developed.ThestudyisalsoacontributiontotheUNESCOIHP-VII pro-gramme(EuroFRIEND-Waterproject)andthePantaRheiInitiativeof theInternationalAssociationofHydrologicalSciences(IAHS).

Supplementarymaterials

Supplementarymaterialassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.advwatres.2020.103512.

References

Andreadis, K.M., Clark, E.A., Wood, A.W., Hamlet, A.F., Lettenmaier, D.P., 2005. Twentieth-century drought in the conterminous United States. J. Hydrometeorol. 6 (6), 985–1001. https://doi.org/10.1175/JHM450.1 .

Beguería, S., Vicente-Serrano, S.M., Reig, F., Latorre, B., 2014. Standardized precipitation evapotranspiration index (SPEI) revisited: Parameter fitting, evapotranspiration mod- els, tools, datasets and drought monitoring. International Journal of Climatology 34 (10), 3001–3023. https://doi.org/10.1002/joc.3887 .

Below, R., Grover-Kopec, E., Dilley, M., 2007. Documenting drought-related disasters: a global reassessment. J. Environ. Dev. 16 (3), 328–344. https://doi.org/10.1177/1070496507306222 .

Chavas, D.R., Reed, K.A., Knaff, J.A., 2017. Physical understanding of the tropical cyclone wind-pressure relationship. Nat. Commun. 8 (1), 1360. https://doi.org/10.1038/s41467-017-01546-9 .

Corzo Perez, G.A., Van Huijgevoort, M.H.J., Voß, F., Van Lanen, H.A.J., 2011. On the spatio-temporal analysis of hydrological droughts from global hydrological models. Hydrol. Earth Syst. Sci. 15 (9), 2963–2978. https://doi.org/10.5194/hess-15-2963-2011 .

Diaz, V., Corzo, G., Lanen, H.A.J.V., Solomatine, D.P., 2019a. 4 - Spatiotemporal drought analysis at country scale through the application of the STAND toolbox. In: Corzo, G., Varouchakis, E.A. (Eds.), Spatiotemporal Analysis of Extreme Hydrological Events. Elsevier, pp. 77–93. https://doi.org/10.1016/B978-0-12-811689-0.00004-5 . Diaz, V., Corzo Perez, G.A., Van Lanen, H.A.J., Solomatine, D., 2018. Intelligent drought

tracking for its use in Machine Learning: implementation and first results. In: La Log- gia, G., Freni, G., Puleo, V., De Marchis, M. (Eds.), HIC 2018. 13th International Con- ference on Hydroinformatics, 3 Palermo: EasyChair. https://doi.org/10.29007/klgg . Guha-Sapir, D. (2018). EM-DAT: the emergency events database - Université catholique

de Louvain (UCL) - CRED. Retrieved from emdat.be

Diaz, V., Corzo Perez, G.A., Van Lanen, H.A.J., Solomatine, D., Varouchakis, E.A., 2019b. Characterisation of the dynamics of past droughts. Sci. Total Environ., 134588 https://doi.org/10.1016/j.scitotenv.2019.134588 .

Hao, Z., Yuan, X., Xia, Y., Hao, F., Singh, V.P., 2017. An overview of drought monitoring and prediction systems at regional and global scales. Bull. Am. Meteorol. Soc. 98 (9), 1879–1896. https://doi.org/10.1175/BAMS-D-15-00149.1 .

Haralick, R.M. , Shapiro, L.G. , 1992. Computer and Robot Vision , Volume I. Addison-Wesley . He, L., Chao, Y., Suzuki, K., Wu, K., 2009. Fast Connected-component labeling. Pattern

Recogn. 42 (9), 1977–1987. https://doi.org/10.1016/j.patcog.2008.10.013 . Herrera-Estrada, J.E., Satoh, Y., Sheffield, J., 2017. Spatio-temporal dynamics of global

drought. Geophys. Res. Lett. 2254–2263. https://doi.org/10.1002/2016GL071768 . Lloyd-Hughes, B., 2012. A spatio-temporal structure-based approach to drought charac-

terisation. Int. J. Climatol. 32 (3), 406–418. https://doi.org/10.1002/joc.2280 . Mckee, T.B., Doesken, N.J., Kleist, J., 1993. The relationship of drought frequency and

duration to time scales. AMS 8th Conference on Applied Climatology January, 179– 184. https://doi.org/10.1002/10490403 .

Mishra, A.K., Singh, V.P., 2010. A review of drought concepts. J. Hydrol. 391 (1–2), 202– 216. https://doi.org/10.1016/j.jhydrol.2010.07.012 .

Mishra, A.K., Singh, V.P., 2011. Drought modeling - a review. J. Hydrol. 403 (1–2), 157– 175. https://doi.org/10.1016/j.jhydrol.2011.03.049 .

Pannell, D.J., 1997. Sensitivity analysis of normative economic models: theo- retical framework and practical strategies. Agric. Econ. 16 (2), 139–152. https://doi.org/10.1016/S0169-5150(96)01217-0 .

Rahman, S.M., Yang, R., Di, L., 2018. Clustering Indian ocean tropical cyclone tracks by the standard deviational ellipse. Climate 6 (2). https://doi.org/10.3390/cli6020039 . Sepulcre-Cantó, G., Horion, S., Singleton, A., Carrao, H., Vogt, J., 2012. Development of a combined drought indicator to detect agricultural drought in Europe. Nat. Hazard. Earth Syst. Sci. 12 (11), 3519–3531. https://doi.org/10.5194/nhess-12-3519-2012 . Sheffield, J. , Wood, E.F. , 2011. Drought: Past problems and future scenarios. P. Earthscan,

Ed. .

Sheffield, J., Andreadis, K.M., Wood, E.F., Lettenmaier, D.P., 2009. Global and conti- nental drought in the second half of the twentieth century: Severity-area-duration analysis and temporal variability of large-scale events. J. Climate 22 (8), 1962–1981. https://doi.org/10.1175/2008JCLI2722.1 .

Tallaksen, L.M. , Van Lanen, H.A.J. , 2004. Hydrological drought - processes and estima- tion methods for streamflow and groundwater. In: Tallaksen, L.M., Van Lanen, H.A.J. (Eds.). Developments in Water Sciences, 48. Elsevier B.V., The Netherlands . Tallaksen, L.M., Hisdal, H., Lanen, H.A.J.V., 2009. Space-time modelling of

catchment scale drought characteristics. J. Hydrol. 375 (3–4), 363–372. https://doi.org/10.1016/j.jhydrol.2009.06.032 .

van Huijgevoort, M.H.J., Hazenberg, P., van Lanen, H., Teuling, A.J., Clark, A.J., Fol- well, D.B., Uijlenhoet R, S., 2013. Global multimodel analysis of drought in runoff for the second half of the twentieth century. J. Hydrometeorol. 14 (5), 1535–1552. https://doi.org/10.1175/JHM-D-12-0186.1 .

Van Loon, A.F., 2015. Hydrological drought explained. Wiley Interdiscip. Rev. 2 (4), 359– 392. https://doi.org/10.1002/wat2.1085 .

Vernieuwe, H., De Baets, B., Verhoest, N.E.C., 2019. A mathematical morphology ap- proach for a qualitative exploration of drought events in space and time. Int. J. Cli- matol. 1–14. https://doi.org/10.1002/joc.6226 .

Vicente-Serrano, S.M., Beguería, S., López-Moreno, J.I., 2010. A multiscalar drought index sensitive to global warming: The standardized precipitation evapotranspiration index. Journal of Climate 23 (7), 1696–1718. https://doi.org/10.1175/2009JCLI2909.1 . Wanders, N., van Lanen, H.A.J., van Loon, A.F., 2010. Indicators for drought char-

acterization on a global scale. WATCH Technical Report No 24. Retrieved from http://www.eu-watch.org/media/default.aspx/emma/org/10646416/WATCH+ Technical+Report+Number+24+Indicators+For+Drought+Characterization+ on+a+Global+Scale.pdf .

Wilhite, D.A. , 2000. Drought as a natural hazard: concepts and definitions. DROUGHT, A Global Assessment, Vol I and II. Routledge Hazards and Disasters Series .

World Meteorological Organization (WMO), 2006. Drought monitoring and early warning: concepts, progress and future challenges. progress and future chal- lenges. WMO-No, WMO-No. 1006. WMO, Geneva, Switzerland Retrieved from http://www.droughtmanagement.info/literature/WMO_drought_monitoring_early_ warning_2006.pdf .

Zhou, H., Liu, Y., Liu, Y., 2019. An approach to tracking meteorological drought migration. Water Resour. Res. 55 (4), 3266–3284. https://doi.org/10.1029/2018WR023311 . World Meteorological Organization (WMO). 2012. Standardized Precipitation

Index user guide. WMO-No.1090. Geneva, Switzerland. Retrieved from http://library.wmo.int/pmb_ged/wmo_1090_en.pdf