Jos_von_Asmuth_Thesis.pdf - repository

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fåíêçÇìÅíáçå=Chapter==1 General introduction= NR==Based in part on:sçå=^ëãìíÜI=gK=oKI=~åÇ=hK=j~~ë=EOMMNFI=qÜÉ=ãÉíÜçÇ=çÑ=áãéìäëÉ=êÉëéçåëÉ=ãçãÉåíëW=~=åÉï=ãÉíÜçÇ=áåíÉÖê~íáåÖ=íáãÉ=ëÉêáÉëJI=ÖêçìåÇï~íÉêJ=~åÇ=ÉÅçJÜóÇêçäçÖáÅ~ä=ãçÇÉääáåÖI=áå=fãé~Åí=çÑ=eìã~å=^Åíáîáíó=çå=dêçìåÇï~íÉê=aóå~ãáÅëI=ÉÇáíÉÇ=Äó=dÉÜêÉäëI=gK`KI=mÉíÉêëI=kKbKI=eçÉÜåI=bKI=gÉåëÉåI=hKI=iÉáÄìåÇÖìíI=`KI=dêáÑÑáçÉåI=gKI=tÉÄÄI=_KI=~åÇ=w~~ÇåççêÇáàâI=General introductiontKgKI=fêp=mêÉëëI=`ÉåíêÉ=Ñçê=bÅçäçÖó=~åÇ=eóÇêçäçÖóI=t~ääáåÖÑçêÇI=RNJRUK=1Based partly on:Von Asmuth, J. R., and K. Maas (2001)The method of impulse response moments: a new methodintegrating time series-, groundwater- and eco-hydrologicalmodelling.in: Impact of Human Activity on Groundwater Dynamics, editedby Gehrels, J.C., Peters, N.E., Hoehn, E., Jensen, K., Leibundgut,C., Griffioen, J., Webb, B., and Zaadnoordijk, W.J., IAHS Press,Centre for Ecology and Hydrology, Wallingford, 51-58.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=N=NS=1.1 Initial objectives and scope^ë=éÉêÜ~éë=áë=åçí=ìåÅçããçå=áå=mÜa=êÉëÉ~êÅÜI=íÜÉ=áåáíá~ä=çÄàÉÅíáîÉë=ÇáÑÑÉê=ê~íÜÉê=ëíêçåÖäó=Ñêçã=íÜÉ=Ñáå~ä=êÉëìäíë=éêÉëÉåíÉÇ=áå=ÜÉêÉK=få=íÜÉ=çêáÖáå~ä=éêçéçë~äë=xj~~ëI=NVVRX=sçå=^ëãìíÜI=OMMMzI=íÜÉ=éêáã~êó=çÄàÉÅíáîÉ=ï~ë=íç=áãéêçîÉ=ÉÅçJÜóÇêçäçÖáÅ=ãçÇÉäáåÖ=ãÉíÜçÇë=ïáíÜ=êÉëéÉÅí=íç=íÜÉ=ï~ó=áå=ïÜáÅÜ=íÜÉ=êÉä~íáçåëÜáé=ÄÉíïÉÉå=ÖêçìåÇï~íÉê=Çóå~ãáÅë=~åÇ=íÜÉ=îÉÖÉí~íáçå=Åçãéçëáíáçå=áå=ÖêçìåÇï~íÉê=ÇÉéÉåÇÉåí=ÉÅçëóëíÉãë=ï~ë=ãçÇÉäÉÇK=^í=íÜ~í=íáãÉ=Äìí=~äëç=íç=Ç~íÉI=Åçããçå=TimeseriesmodellingGroundwatermodellingMomentsof theIR-functionEco-hydrologicalmodellingfigure 1.1: Overview of the method of impulse responsemoments, showing the three fields of modeling from whichmoments can be derived and be mutually exchanged [VonAsmuth and Maas, 2001].ãÉíÜçÇë=íç=éêÉëÉåí=~åÇ=ÅÜ~ê~ÅíÉêáòÉ=ÖêçìåÇï~íÉê=äÉîÉä=ÑäìÅíì~íáçåë=~êÉ=íÜÉ=ëçJÅ~ääÉÇ=Çìê~íáçå=äáåÉë=xÉKÖKIqΩñÉåI=NVRQX=dêççíà~åëI=NVURX=aÉ=e~~åI=NVVOzI=êÉÖáãÉ=ÅìêîÉë=~åÇ=çîÉê~ää=ÅÜ~ê~ÅíÉêáëíáÅë=äáâÉ=jñdi=ëí~íáëíáÅë=xs~å=ÇÉê=päìáàë=~åÇ=aÉ=dêìáàíÉêI=NVURzK=pìÅÜ=ãÉíÜçÇëI=ÜçïÉîÉêI=Ü~îÉ=íïç=áãéçêí~åí=Çê~ïÄ~ÅâëW==• pí~íáëíáÅë=Ä~ëÉÇ=çå=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=çîÉê=äáãáíÉÇ=éÉêáçÇë=~êÉ=ëÉåëáíáîÉ=íç=äçåÖ=íÉêã=î~êá~íáçå=áå=íÜÉ=ÇêáîáåÖ=ÑçêÅÉë=xhåçííÉêë=~åÇ=s~å=t~äëìãI=NVVTX=_~êíÜçäçãÉìë=Éí=~äKI=OMMUzK=• dê~éÜë=çê=äáåÉë=~êÉ=ÉëëÉåíá~ääó=îÉÅíçêëI=~åÇ=äÉëë=ÅçåîÉåáÉåí=Ñçê=Ç~í~=ëíçê~ÖÉI=Eëé~íá~äF=éêÉëÉåí~íáçåI=~å~äóëáë=~åÇ=ãçÇÉäáåÖ=éìêéçëÉë=íÜ~å=ëÅ~ä~êëK=lîÉê~ää=ëí~íáëíáÅëI=çå=íÜÉ=çíÜÉê=Ü~åÇI=çåäó=Å~éíìêÉ=ÅÉêí~áå=~ëéÉÅíë=çÑ=íÜÉ=Çóå~ãáÅë=xsçå=^ëãìíÜ=~åÇ=håçííÉêëI=OMMQzK===_ÉÅ~ìëÉ=çÑ=íÜÉëÉ=Çê~ïÄ~ÅâëI=áí=ï~ë=ÜóéçíÜÉëáòÉÇ=íÜ~í=~å=~äíÉêå~íáîÉ=êçìíÉ=Ä~ëÉÇ=çå=íÜÉ=ëçJÅ~ääÉÇ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=EëÉÉ=ëÉÅíáçå=OKPKQF=ÅçìäÇ=áãéêçîÉ=ÉÅçÜóÇêçäçÖáÅ=ãçÇÉäáåÖ=ãÉíÜçÇëK=få=íÜáë=~äíÉêå~íáîÉI=~=âÉó=åçíáçå=áë=íÜ~í=ëé~íá~ä=ÇáÑÑÉêÉåÅÉë=áå=ÖêçìåÇï~íÉê=äÉîÉä=Çóå~ãáÅë=~êÉ=ã~áåäó=ÇÉíÉêãáåÉÇ=Äó=ëé~íá~ääó=î~êá~ÄäÉ=ëóëíÉã=éêçéÉêíáÉëI=ïÜáäÉ=íÉãéçê~ä=Çóå~ãáÅë=~êÉ=ã~áåäó=ÇêáîÉå=Äó=ëé~íá~ääó=äÉëë=î~êá~ÄäÉ=ãÉíÉçêçäçÖáÅ=Çóå~ãáÅëK=`çåëÉèìÉåíäóI=áí=ï~ë=ÜóéçíÜÉëáòÉÇ=íÜ~í=ëé~íá~ä=ÇáÑÑÉêÉåÅÉë=áå=îÉÖÉí~íáçå=ÅçìäÇ=ÄÉ=ãçÇÉäÉÇ=ãçêÉ=~ÅÅìê~íÉäó=ìëáåÖ=ëóëíÉã=éêçéÉêíáÉë=~äçåÉI=çê=áå=çíÜÉê=ïçêÇë=Äó=ÚÑáäíÉêáåÖ=çìíÛ=íÉãéçê~äI=ãÉíÉçêçäçÖáÅ=Çóå~ãáÅëK=qç=ÄÉ=ãçêÉ=ëéÉÅáÑáÅI=íÜÉ=ìëÉ=çÑ=íáãÉ=ëÉêáÉë=ãçÇÉäë=ï~ë=éêçéçëÉÇ=Ñçê=áåÑÉêêáåÖ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåë=Ñêçã=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåëI=~åÇ=áå=íìêå=íç=ÅÜ~ê~ÅíÉêáòÉ=íÜÉëÉ=Äó=íÜÉáê=ãçãÉåíë=EëÉÉ=ëÉÅíáçå=OKPKQFK=jçãÉåíë=~êÉ=ëÅ~ä~êë=~åÇ=Åçåëí~åíë=áå=äáåÉ~êI=íáãÉJáåî~êá~åí=ëóëíÉãëI=~åÇ=íçÖÉíÜÉê=ïáíÜ=íÜÉ=ëé~íá~ääó=äÉëë=î~êá~ÄäÉ=ÇêáîáåÖ=ÑçêÅÉëI=íÜÉó=ÅçãéäÉíÉäó=ÅÜ~ê~ÅíÉêáòÉ=íÜÉ=EÇÉíÉêãáåáëíáÅ=é~êí=çÑ=íÜÉF=Çóå~ãáÅë=~í=~=ÅÉêí~áå=äçÅ~íáçåK=^ë=~å=~ÇÇáíáçå~ä=~Çî~åí~ÖÉI=ãçãÉåíë=Å~å=~äëç=ÄÉ=ëáãìä~íÉÇ=ÇáêÉÅíäó=~åÇ=ëé~íá~ääó=ìëáåÖ=ÇáëíêáÄìíÉÇ=ÖêçìåÇï~íÉê=ãçÇÉäë=EëÉÉ=ëÉÅíáçå=OKQKPFK===..…………………………………………………………………………………………….….

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fåíêçÇìÅíáçå=çÄí~áåÉÇ=áë=ÖáîÉåI=ïÜáäÉ=ëéÉÅáÑáÅ~ääó=ÑçÅìëáåÖ=çå=íÜÉ=áåíÉêÑ~ÅÉ=ÄÉíïÉÉå=Ç~í~JÄ~ëÉÇ=~åÇ==éÜóëáÅ~ääóJÄ~ëÉÇ=ãÉíÜçÇëK=cáå~ääóI=ÅÜ~éíÉê=U=Åçåí~áåë=íÜÉ=ÖÉåÉê~ä=ÅçåÅäìëáçåë=çå=~åÇ=ëìãã~êó=çÑ=íÜÉ=ãÉíÜçÇë=~åÇ=êÉëìäíë=éêÉëÉåíÉÇ=áå=íÜáë=íÜÉëáëK=== NV=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=N=OM=Abstract:Appreciating the contents of this thesis requires some knowledge of its background, i.e.the methods and theory on (geohydrologic) time series analysis and systemidentification. In this chapter, a basic introduction is given, and different views on thesubject are presented. First, some thought is given to the systems approach in general,as to date, most (geo)hydrologist are less familiar with system identification methodsthan with e.g., spatially-distributed groundwater models. In short, the system viewpointcan be characterized by the fact that in essence it is top-down. The system viewpoint treatsa groundwater system first as a 'whole' and not bottom-up, as an aggregate of cells, layersand/or elements, which is still the mainstream viewpoint. Having said this, also from thesystems perspective, there are different ways in which a groundwater system, or anysystem for that matter, can be perceived, modeled and/or analyzed.Time series analysis is a method that originates from the statistical sciences. In principle, itdoes not require any knowledge of the physical functioning of the system underconsideration. In its basics, it can be seen as a variant of simple, linear regression, and thecoefficients in the regression equation, either autoregressive or moving averageparameters, do not have a physical meaning a priori. From a physical point of view, on theother hand, a central concept is the so-called impulse response function, as it completelycharacterizes the functioning of a linear time-invariant system at a certain point in space.Impulse response functions can be inferred from a data set through time series analysis,but also using 'purely' physically-based methods, either analytic or numeric (in case oflinear systems). This means that impulse response functions can also be derived from thedifferential equation and boundary conditions that belong to a certain geohydrologicsystem and its schematization.In this thesis, a 'mix' between both worlds is developed and presented. In this approach,the time series analysis problem is formulated in a continuous time domain. It allows forthe use of (continuous) distribution functions that have a statistical origin, as well asphysically-based analytic response functions. Distribution functions of skew-Gaussiannature, like the scaled gamma distribution, prove to fit the behavior of a wide range ofsystems quite well. Next to that, a further link between the physically-based world ofgroundwater modeling and time series analysis is established using moments of impulseresponse functions, as these can also be generated directly and spatially using momentgeneratingdifferential equations, implementable in any standard groundwater model...…………………………………………………………………………………………….….

2_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=Chapter2 General background, methods andtheory=Based in part on:General sçå=^ëãìíÜI=gK=oKI=jK=håçííÉêëI=~åÇ=`K=j~~ë=EOMMSFI=qáãÉ=ëÉêáÉë=~å~äóëáë=Ñçê=background, methodsEÉÅçFÜóÇêçäçÖáëíëI=Ä~ÅâÖêçìåÇ=ÇçÅìãÉåí~íáçå=~åÇ=ÅçìêëÉ=ã~åì~ä=Eáå=aìíÅÜFI=háï~= =t~íÉê=oÉëÉ~êÅÜL^äíÉêê~I=káÉìïÉÖÉáåLt~ÖÉåáåÖÉåK=== ON=and theoryAbstract^ééêÉÅá~íáåÖ=íÜÉ=ÅçåíÉåíë=çÑ=íÜáë=íÜÉëáë=êÉèìáêÉë=ëçãÉ=âåçïäÉÇÖÉ=çÑ=áíë=Ä~ÅâÖêçìåÇI=áK=áë=Éëí~ÄäáëÜÉÇ=ìëáåÖ=ãçãÉåíë=çÑ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåëI=~ë=íÜÉëÉ=Å~å=~äëç=ÄÉ=ÖÉåÉê~íÉÇ=ÇáêÉÅíäó=~åÇ=ëé~íá~ääó=ìëáåÖ=ãçãÉåíJÖÉåÉê~íáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåëI=áãéäÉãÉåí~ÄäÉ=áå=~åó=ëí~åÇ~êÇ=ÖêçìåÇï~íÉê=ãçÇÉäK= Based partly on:Von Asmuth, J. R., M. Knotters, and C. Maas (2006)Time series analysis for (eco)hydrologists, backgrounddocumentation and course manual (in Dutch).Kiwa Water Research / Alterra, Nieuwegein / Wageningen.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=2.1 Facets of the system approachOO=2.1.1 On the system concept^ë=áí=áë=~=ïáÇÉäó=ìëÉÇ=íÉêã=áå=~åÇ=çìíëáÇÉ=íÜÉ=ëÅáÉåíáÑáÅ=ÅçããìåáíóI=ÚëóëíÉãÛ=Å~å=Ü~îÉ=ã~åó=ÇáÑÑÉêÉåí=ãÉ~åáåÖëK=lêáÖáå~ääóI=ëóëíÉã=ëíÉãë=Ñêçã=íÜÉ=dêÉÉâ=ïçêÇ=σύστηµαI=ïÜáÅÜ=íê~åëä~íÉë=~ë=Ú~=ïÜçäÉ=ÅçãéçìåÇÉÇ=çÑ=ëÉîÉê~ä=é~êíë=çê=ãÉãÄÉêëÛ=çê=ÚÅçãéçëáíáçåÛ=xmÉêëÉìë=aáÖáí~ä=iáÄê~êózK=_ÉáåÖ=~=ÅÉåíê~ä=ÅçåÅÉéí=áå=ëçãÉ=ÇáëÅáéäáåÉëI=íÜÉêÉ=Ü~îÉ=ÄÉÉå=ëÉîÉê~ä=~ííÉãéíë=íç=ÖáîÉ=~=Ñçêã~ä=ÇÉÑáåáíáçå=çÑ=ëóëíÉã=~åÇ=êÉä~íÉÇ=íÉêãë=xëÉÉ=ÉKÖKI=j~êÅÜ~äI=NVTRX=táääÉãëI=NVVNX=_~ÅâäìåÇI=OMMMX=häáêI=OMMNzK=cêçã=Üáë=êÉîáÉïI=_~ÅâäìåÇ=ÅçåÅäìÇÉë=íÜ~í=éêÉîáçìë=InputSystemStructureBehaviorboundaryEnvironmentOutput=figure 2.1: Schematic representation of an open, physicalsystem. The boundary separates the system from theenvironment, with which it exchanges matter, energyand/or information.ÇÉÑáåáíáçåë=çÑ=ëóëíÉã=~êÉ=áãéêÉÅáëÉK=eÉ=~êÖìÉë=íÜ~í=çåäó=îá~=~=ÇÉÑáåáíáçå=Ä~ëÉÇ=çå=íÜÉ=ÅçåëíáíìíáåÖ=é~êíë=~åÇ=íÜÉáê=Ñçêã~ä=áåíÉêêÉä~íáçåëÜáéëI=áí=áë=éçëëáÄäÉ=íç=ìå~ãÄáÖìçìëäó=ÇÉÑáåÉ=ïÜ~í=áë=é~êí=çÑ=~=ëóëíÉã=~åÇ=ïÜ~í=áë=åçí=EÉKÖKI=ïÜ~í=~êÉ=ÉñíÉêå~ä=ëíáãìäáFI=çê=ïÜ~í=~êÉ=~Åíì~ääó=íïç=ëóëíÉãë=áå=ëíÉ~Ç=çÑ=çåÉK=få=ëÜçêíI=çåäó=ëìÅÜ=~=ÇÉÑáåáíáçå=ïçìäÇ=ÇÉÑáåÉ=ïÜ~í=áë=~=ëóëíÉãI=~åÇ=ïÜ~í=áë=åçíK=cçê=çìê=éìêéçëÉëI=ÜçïÉîÉêI=ïÉ=éêÉÑÉê=~=ãçêÉ=ÖÉåÉê~ä=ìë~ÖÉ=çÑ=íÜÉ=íÉêãI=~ë=ÇçÉë=xiàìåÖI=NVVVz=ïÜç=ÇÉÑáåÉë=~=ëóëíÉã=áå=äççëÉ=íÉêãë=~ë=ÚÁ~å=çÄàÉÅí=áå=ïÜáÅÜ=î~êá~ÄäÉë=çÑ=ÇáÑÑÉêÉåí=âáåÇë=áåíÉê~Åí=~åÇ=éêçÇìÅÉ=çÄëÉêî~ÄäÉ=ëáÖå~äëÛK=få=~=éÜóëáÅ~ä=çê=íÜÉêãçÇóå~ãáÅ=ÅçåíÉñíI=ëóëíÉã=Ü~ë=~=íÉÅÜåáÅ~ä=ãÉ~åáåÖ=~åÇ=ëáãéäó=êÉÑÉêë=íç=íÜÉ=éçêíáçå=çÑ=íÜÉ=éÜóëáÅ~ä=ìåáîÉêëÉ=ÅÜçëÉå=Ñçê=íÜÉ=~å~äóëáëK=tÜ~í=áë=áåëáÇÉ=~åÇ=çìíëáÇÉ=~=ëóëíÉã=áë=åçí=ÑáñÉÇ=Äó=ëçãÉ=çÄàÉÅíáîÉI=Ñçêã~ä=ÇÉÑáåáíáçåI=~ë=~ííÉãéíÉÇ=Äó=_~ÅâäìåÇI=Äìí=áë=áå=Åçåíê~ëí=~=ÑêÉÉ=ÅÜçáÅÉI=ÖÉåÉê~ääó=ã~ÇÉ=íç=ëáãéäáÑó=íÜÉ=~å~äóëáëK=bîÉêóíÜáåÖ=çìíëáÇÉ=íÜÉ=ëóëíÉã=áë=íÜÉå=êÉÑÉêêÉÇ=íç=~ë=íÜÉ=ÉåîáêçåãÉåíI=ïÜáÅÜ=áë=áÖåçêÉÇ=áå=íÜÉ=~å~äóëáë=ÉñÅÉéí=Ñçê=áíë=ÉÑÑÉÅíë=çå=íÜÉ=ëóëíÉãK=^äÄÉáí=íÜÉ=Ñ~Åí=íÜ~í=áí=áë=~=îÉêó=ÖÉåÉê~ä=ÅçåÅÉéíI=éÜóëáÅ~ä=ëóëíÉãë=ëÜ~êÉ=Åçããçå=ÅÜ~ê~ÅíÉêáëíáÅëK=få=ÖÉåÉê~äI=ëóëíÉãë=Ñçê=áåëí~åÅÉ=Ü~îÉ=EÑáÖìêÉ=OKNFW==• ~=ÄçìåÇ~êó==EÇÉÑáåáåÖ=ïÜ~í=áë=é~êí=çÑ=~=ëóëíÉã=~åÇ=ïÜ~í=áë=ÉåîáêçåãÉåíF=• ëíêìÅíìêÉ======EÇÉÑáåÉÇ=Äó=íÜÉ=áåíÉêå~ä=ÅçãéçåÉåíë=~åÇ=íÜÉáê=ÅçãéçëáíáçåI== ÇÉíÉêãáåáåÖ=íÜÉ=çéÉê~íáçåF=• ÄÉÜ~îáçê======EáåîçäîáåÖ=íÜÉ=EäáåÉ~ê=çê=åçåJäáåÉ~êF=êÉëéçåëÉ=íç=áåéìíë=çÑ=ã~ííÉêI= ÉåÉêÖó=~åÇLçê=áåÑçêã~íáçåF==få=~=ëóëíÉã=áÇÉåíáÑáÅ~íáçå=çê=íáãÉ=ëÉêáÉë=~å~äóëáë=ÅçåíÉñíI=áí=áë=Åçããçå=éê~ÅíáÅÉ=íç=ÇÉåçíÉ=ÑçêÅáåÖ=î~êá~ÄäÉë=~ë=ÚáåéìíÛ=~åÇ=ÑçêÅÉÇ=î~êá~ÄäÉë=~ë=ÚçìíéìíÛK=s~êá~ÄäÉë=êÉä~íÉÇ=íç=ëóëíÉãëI=ÜçïÉîÉêI=Çç=åçí=åÉÅÉëë~êáäó=êÉéêÉëÉåí=ëçãÉíÜáåÖ=éÜóëáÅ~ääó=ÉåíÉêáåÖ=áí=çê=ÅçãáåÖ=çìíK=qÜÉ=ÚçìíéìíÛ=ã~ó=Ñçê=áåëí~åÅÉ=ïÉää=ÄÉ=íÜÉ=Éîçäìíáçå=çÑ=~=ÅÉêí~áå=..…………………………………………………………………………………………….….

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`Ü~éíÉê=O=OQ====HYDROLOGIC SIMULATIONSYSTEM INVESTIGATION SYSTEM INVESTIGATIONPHYSICAL HYDROLOGYPHYSICAL MODELS PARAMETRIC METHODS STOCHASTIC METHODSPHYSICALLY-BASEDMATHEMATICAL METHODSStatiscal treatment of systemson the basis of the variables,without reference to the systemon which they operate.Statiscal treatment of systemswith deterministic inputs andoutputsBoundary-value problems usingpartial differential equationsand potential theory.MATHEMATICAL MODELSSpatially and SequentiallyLUMPEDDISTRIBUTEDANALOG COMPUTERSOLUTIONSDIGITAL COMPUTERSOLUTIONSPHYSICALLY-BASEDDIGITALLY - SIMULATEDHYDROLOGIC RESPONSE MODEL==figure 2.2: Blueprint for a physically-based, digitally-simulated hydrologic response model [Blue line, Freeze and Harlan, 1969, in part after Amorochoand Hart, 1964]. ‘System investigation’ is holded to be the opposite of ‘physical hydrology’.=..…………………………………………………………………………………………….….

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`Ü~éíÉê=O=PO=• ^ë=~=íÜáêÇ=~ééêç~ÅÜI=áå=íÜáë=íÜÉëáë=íÜÉ=ëí~íáëíáÅ~ääó=çêáÉåíÉÇ=ôj^=ãçÇÉä=ëíêìÅíìêÉ=áë=~Ä~åÇçåÉÇ=~åÇ=êÉéä~ÅÉÇ=Äó=~=ãçÇÉä=ëíêìÅíìêÉ=áå=ïÜáÅÜ=Åçåíáåìçìë=íáãÉ=êÉëéçåëÉ=ÑìåÅíáçåë=~êÉ=ìëÉÇ=EëÉÉ=ÅÜ~éíÉê=PFK=_ó=ÇçáåÖ=ëçI=ãçêÉ=ÅçãéäÉñI=~å~äóíáÅ=ëçäìíáçåë=íç=ÖÉçÜóÇêçäçÖáÅ=éêçÄäÉãë=Å~å=ÄÉ=ìëÉÇ=~ë=êÉëéçåëÉ=ÑìåÅíáçåK=qÜÉ=éÜóëáÅ~ä=ÑìåÅíáçåáåÖ=çÑ=~=ëóëíÉãI=áå=êÉëéçåëÉ=íç=~=ëéÉÅáÑáÅ=ÉñÅáí~íáçåI=Å~å=áå=éêáåÅáéäÉ=íÜìë=ÄÉ=áãéçëÉÇ=çå=íÜÉ=ãçÇÉäK=få=íÜÉ=ÅçãéìíÉê=éêçÖê~ã=jÉåó~åíÜÉë=EëÉÉ=ÅÜ~éíÉê=TFI=e~åíìëÜÛ=ïÉää=Ñçêãìä~=áë=Ñçê=áåëí~åÅÉ=ìëÉÇ=Ñçê=ãçÇÉäáåÖ=íÜÉ=ÉÑÑÉÅíë=çÑ=éìãéáåÖ=EëÉÉ=ëÉÅíáçå=OKPKRKRFI=Äìí=ãçêÉ=ÖÉåÉê~ä=ÇáëíêáÄìíáçå=ÑìåÅíáçåë=ã~ó=ÄÉ=ìëÉÇ=~äëç=EëÉÉ=ëÉÅíáçå=OKQKNFK===_ÉÅ~ìëÉ=çÑ=íÜÉ=áåÅêÉ~ëáåÖ=éÜóëáÅ~ä=çêáÉåí~íáçåI=~äëç=áå=íÜÉ=ÜóÇêçäçÖáÅ=ëÅáÉåÅÉë=íÜÉ=ÑçÅìë=íÜìë=ëÜáÑíë=Ñêçã=íáãÉ=ëÉêáÉë=~å~äóëáë=íç=ëóëíÉã=áÇÉåíáÑáÅ~íáçåI=çê=Ñêçã=~å~äóòáåÖ=íÜÉ=ÅçêêÉä~íáçåë=áå=~åÇ=ÄÉíïÉÉå=íáãÉ=ëÉêáÉë=íç=áÇÉåíáÑóáåÖ=íÜÉ=âÉó=é~ê~ãÉíÉêë=áå=~=ëóëíÉã=~åÇ=íÜÉ=ÉÑÑÉÅíë=çÑ=ÉñÅáí~íáçåë=áãéçëÉÇ=íç=áíK=qáãÉ=ëÉêáÉë=~å~äóëáëI=~ë=ëìÅÜI=íÜÉå=ÄÉÅçãÉë=~=íççä=áå=íÜÉ=ãçêÉ=ÖÉåÉê~ä=~êÉ~=çÑ=ëóëíÉã=áÇÉåíáÑáÅ~íáçå=E~å~äçÖçìë=íç=íÜÉ=ï~ó=áí=áë=éêÉëÉåíÉÇ=áå=ÉKÖKI=xaççÖÉI=NVTPzFK=få=íÜÉ=åÉñí=ëÉÅíáçåëI=Ñáêëí=íÜÉ=ëí~íáëíáÅ~ä=çê=Ää~ÅâJÄçñ=îáÉïéçáåí=çå=íáãÉ=ëÉêáÉë=ãçÇÉäë=áë=íêÉ~íÉÇK=pÉÅçåÇI=íÜÉ=éÜóëáÅ~ä=îáÉïéçáåí=~åÇ=êÉëéçåëÉ=çÑ=ëÉîÉê~ä=ÉäÉãÉåí~êó=ÖêçìåÇï~íÉê=ëóëíÉãë=~êÉ=ÇáëÅìëëÉÇK=qÜáêÇ=~=ÚãáñÛ=ÄÉíïÉÉå=íÜÉëÉ=ïçêäÇë=áë=éêÉëÉåíÉÇI=ïÜáÅÜ=áåîçäîÉë=íÜÉ=ìëÉ=çÑ=EÇáëíêáÄìíáçåF=ÑìåÅíáçåë=íÜ~í=Çç=åçí=Ü~îÉ=~å=~éé~êÉåí=éÜóëáÅ~ä=ãÉ~åáåÖ=çê=ïÜçëÉ=éÜóëáÅ~ä=ãÉ~åáåÖ=áë=êÉä~ñÉÇK===..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=2.2 Statistical, ‘black box‘ viewpoint2.2.1 ARMA models as linear regression equationcêçã=~=ëí~íáëíáÅ~ä=îáÉïéçáåíI=~=íáãÉ=ëÉêáÉë=ãçÇÉä=Å~å=ÄÉ=êÉÖ~êÇÉÇ=~ë=~=î~êá~åí=çÑ=äáåÉ~ê=êÉÖêÉëëáçåK=qÜÉ=Éèì~íáçå=ÇÉÑáåáåÖ=~=ëáãéäÉ=êÉÖêÉëëáçå=äáåÉ=Å~å=ÄÉ=ïêáííÉå=~ëW====y = ω x + µ + n(2.1)=ïÜÉêÉ=y = W=çÄëÉêî~íáçåë=çÑ=íÜÉ=Éñéä~áåÉÇ=î~êá~ÄäÉ=EÉKÖKI=ÖêçìåÇï~íÉê=ÜÉ~ÇF=x = W=çÄëÉêî~íáçåë=çÑ=íÜÉ=Éñéä~å~íçêó=î~êá~ÄäÉ=EÉKÖKI=êÉÅÜ~êÖÉF=ω = W=é~ê~ãÉíÉê=ÇÉÑáåáåÖ=íÜÉ=ëäçéÉ=çÑ=íÜÉ=êÉÖêÉëëáçå=äáåÉ=µ = W=é~ê~ãÉíÉê=ÇÉÑáåáåÖ=íÜÉ=áåíÉêÅÉéí=ïáíÜ=íÜÉ=óJ~ñáë=n = W=Éêêçê=çê=êÉëáÇì~ä=íÉêã==cçê=åçïI=ïÉ=ïáää=áÖåçêÉ=íÜÉ=êÉëáÇì~ä=íÉêãI=çå=ïÜáÅÜ=ïÉ=ïáää=Éä~Äçê~íÉ=áå=íÜÉ=åÉñí=ëÉÅíáçåK=lÑ=ÅçìêëÉI=äáåÉ~ê=êÉÖêÉëëáçå=Å~å=ÄÉ=~ééäáÉÇ=íç=~ää=ëçêíë=çÑ=î~êá~ÄäÉëI=áåÅäìÇáåÖ=íáãÉ=ëÉêáÉëK=få=íÜÉ=ä~ííÉê=Å~ëÉI=íÜÉ=~î~áä~ÄäÉ=çÄëÉêî~íáçåë=~êÉ=áåÇÉñÉÇ=ïáíÜ=íáãÉ= t xJzI=ïÜáÅÜ=ÖáîÉë=íÜÉ=ÑçääçïáåÖ=Çóå~ãáÅ=êÉÖêÉëëáçå=Éèì~íáçåW==y = ω x + µ(2.2)tt=få=ã~åó=Çóå~ãáÅ=ëóëíÉãëI=ÜçïÉîÉêI=íÜÉ=ÉÑÑÉÅí=çÑ=çåÉ=î~êá~ÄäÉ=çåíç=íÜÉ=çíÜÉê=áë=åçí=Eçê=åçí=çåäóF=áåëí~åí~åÉçìëK=få=ëìÅÜ=Å~ëÉëI=íÜÉ=ëóëíÉã=áë=ë~áÇ=íç=Ü~îÉ=~=ãÉãçêóK=qÜÉ=êáëÉ=áå=ï~íÉê=äÉîÉä=áå=~å=~êÄáíê~êó=ÜóÇêçäçÖáÅ=ëóëíÉã=Å~ìëÉÇ=Äó=~=ê~áå=ëÜçïÉêI=Ñçê=áåëí~åÅÉI=ïáää=åçí=Çáë~ééÉ~ê=áããÉÇá~íÉäó=ïÜÉå=íÜÉ=ê~áå=ëíçéëK=`çåëÉèìÉåíäó=~åÇ=îáÅÉ=îÉêë~I=íÜÉ=ÅìêêÉåí=ï~íÉê=äÉîÉä=áå=ÜóÇêçäçÖáÅ=ëóëíÉãë=áë=~=ÑìåÅíáçå=çÑ=éêÉëÉåí=•åÇ=éêÉîáçìë=ê~áåÑ~ää=ÉîÉåíëK=j~íÜÉã~íáÅ~ääóI=íÜÉ=ÉÑÑÉÅíë=çÑ=éêÉîáçìë=ëí~íÉë=çÑ=íÜÉ=Éñéä~å~íçêó=î~êá~ÄäÉ=Å~å=ÄÉ=ëáãéäó=~ÇÇÉÇ=íç=íÜÉ=êÉÖêÉëëáçå=Éèì~íáçå=áå=íÜÉ=ÑçääçïáåÖ=ã~ååÉêW==yt ω0xt ω1xt −1 ..... ωnsxt −nsµ=qÜÉ=é~ê~ãÉíÉêë== + + + (2.3)ωns~êÉ=âåçïå=~ë=ãçîáåÖ=~îÉê~ÖÉ=Ej^F=é~ê~ãÉíÉêë=E~ë=EOKPF=êÉëÉãÄäÉë=íÜÉ=Å~äÅìä~íáçå=çÑ=~=ïÉáÖÜíÉÇ=ãçîáåÖ=~îÉê~ÖÉ=çÑ= x FI=~åÇ=íÜÉ=êÉëìäíáåÖ=ãçÇÉä=áë=ë~áÇ=íç=Ü~îÉ=çêÇÉê=jÊ ns FK=fÑ=ïÉ=éäçí=ω =îÉêëìë= ns =EÑáÖìêÉ=OKTFI=íÜÉ=êÉëìäí= t0 0 1 2 31 2 3 4 45 6tfigure 2.7: Transfer function of a hypotheticalMA(4) system.=== PP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=PQ=ëÜçïë=ïÜ~í=áë=âåçïå=~ë=íÜÉ=íê~åëÑÉê=çê=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=çÑ=íÜÉ=ëóëíÉã=ìåÇÉê=ÅçåëáÇÉê~íáçåI=ïÜáÅÜ=ïÉ=ÇÉåçíÉ=Äó= Θt=K=qÜÉ=å~ãÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=ëíÉãë=Ñêçã=íÜÉ=Ñ~Åí=íÜ~íI=~ÑíÉê=~=ëçäÉ=ÚáãéìäëÉÛ=çÑ=ìåáí=î~äìÉ=çÑ= x ~í=íáãÉ= t I= y ïáää=êÉëéçåÇ=ïáíÜ=íÜÉ=ëìÄëÉèìÉåí=î~äìÉë=çÑ=ω =áÑ=ïÉ=ãçîÉ=ÑçêíÜ=áå=íáãÉK==mäÉ~ëÉ=åçíÉI=ÜçïÉîÉêI=íÜ~í=ïÜ~í=áë=í~âÉå=íç=ÄÉ=~å=áãéìäëÉ=çÑ= xtáå=ÇáëÅêÉíÉJíáãÉ=åçí~íáçåI=çÑíÉå=áë=~=ã~íÜÉã~íáÅ~ä=~Äëíê~Åíáçå=çÑ=Åçåíáåìçìë=íáãÉ=êÉ~äáíó=áå=ïÜáÅÜ= xtáë=éÉêÜ~éë=~=ë~ãéäÉI=íáãÉ=~îÉê~ÖÉ=çê=ëìã=çÑ= x( t)K=_ÉÅ~ìëÉ=çÑ=íÜ~íI=ïÉ=ïáää=~îçáÇ=íÜÉ=íÉêã=ÚáãéìäëÉ=êÉëéçåëÉÛ=ïÜÉå=ÇáëÅìëëáåÖ=ÇáëÅêÉíÉJíáãÉ=ã~íÜÉã~íáÅë=~åÇ=ìëÉ=Úíê~åëÑÉê=ÑìåÅíáçåÛ=áå=ëíÉ~ÇK=få=íÜÉçêóI=íÜÉ=åìãÄÉê=çÑ=ëíÉéë= ns =ïáíÜ=ïÜáÅÜ=ïÉ=Å~å=äççâ=Ä~Åâ=áå=íáãÉ=áå=íÜáë=ï~ó=ã~ó=ÄÉ=áåÑáåáíÉK=få=éê~ÅíáÅÉI=ÜçïÉîÉêI=ïÉ=Å~ååçí=ÇÉÇìÅÉ=~å=áåÑáåáíÉ=åìãÄÉê=çÑ=é~ê~ãÉíÉêë=Ñêçã=~=ÑáåáíÉ=Ç~í~=ëÉíK=qÜÉêÉÑçêÉI=ÉëéÉÅá~ääó=ïÜÉå=íÜÉ=íáãÉ=ëíÉéë=~êÉ=ëã~ää=~ë=Åçãé~êÉÇ=íç=íÜÉ=êÉëéçåëÉ=íáãÉ=çÑ=íÜÉ=ëóëíÉãI=íÜÉ=ÇáêÉÅí=ìëÉ=çÑ=Éèì~íáçå=EOKPF=~ë=~=êÉÖêÉëëáçå=ãçÇÉä=Ü~ë=éê~ÅíáÅ~ä=äáãáí~íáçåëK===^=ëÉÅçåÇ=çéíáçå=Ñçê=ÇÉ~äáåÖ=ïáíÜ=ãÉãçêó=áë=íç=ãçÇÉä=~=î~êá~ÄäÉ=~ë=~=ÑìåÅíáçå=çÑ=áíë=î~äìÉ=~í=~=éêÉîáçìë=íáãÉ=ëíÉéI=áå=íÜÉ=ÑçääçïáåÖ=ã~ååÉêW==yt − µ = δ ( yt −1− µ ) + ωxt(2.4)=eÉêÉI=δ áë=âåçïå=~ë=~å=~ìíçêÉÖêÉëëáîÉ=EôF=é~ê~ãÉíÉêI=~åÇ=íÜÉ=ãçÇÉä=áë=êÉÑÉêêÉÇ=íç=~ë=Ü~îáåÖ=çêÇÉê=ôENFK=fÑ=ïÉ=äççâ=~í=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=çÑ=ëìÅÜ=~=ëóëíÉã=EÑáÖìêÉ=OKUFI=ïÉ=ÑáåÇ=íÜ~í=áí=ëÜçïë=ÉñéçåÉåíá~ä=ÇÉÅ~óI=ÖáîÉå=ÄóW==tΘt= ωδ(2.5)=^äëç=ÜÉêÉI=ïÉ=Å~å=ÅÜççëÉ=íç=äççâ=ÑìêíÜÉê=Ä~Åâ=áå=íáãÉ=~åÇ=~ÇÇ=ãçêÉ=ô=íÉêãë=íç=Éèì~íáçå=EOKQFK=_ó=ÇçáåÖ=ëçI=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=ïáää=í~âÉ=íÜÉ=ëÜ~éÉ=çÑ=~=ãáñíìêÉ= t 1çÑ=ÉñéçåÉåíá~äë=EëÉÉ=ëÉÅíáçå=OKPKNFK=pìãã~êáòáåÖI=~=äáãáí~íáçå=çÑ=ìëáåÖ=j^= 2íÉêãë=áå=Çóå~ãáÅ=êÉÖêÉëëáçå=Éèì~íáçåë=áë=íÜÉ=Ñ~Åí=íÜ~í=íÜÉó=Å~å=çåäó=ãçÇÉä=íÜÉ=3êÉëéçåëÉ=çÑ=~=ëóëíÉã=Ñçê=~=äáãáíÉÇ=éÉêáçÇ= 4Ä~Åâ=áå=íáãÉK=^å=~Çî~åí~ÖÉ=çÑ=j^=5 6é~ê~ãÉíÉêë=áë=íÜ~í=íÜÉó=Å~å=í~âÉ=çå=~åó=0 1 2 3 4î~äìÉI=ëç=íÜÉ=ëÜ~éÉ=çÑ=íÜÉ=íê~åëÑÉê=5 6t =ÑìåÅíáçå=áë=ÑêÉÉ=áå=íÜÉ=j^=é~êíK=^=äáãáí~íáçå=çÑ=ìëáåÖ=ô=íÉêãëI=çå=íÜÉ= figure 2.8: Transfer function of a hypotheticalçíÜÉê=Ü~åÇI=äáÉë=áå=íÜÉ=Ñ~Åí=íÜ~í=íÜÉó=äáãáí=AR(1) system.íÜÉ=ëÜ~éÉ=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=íç=çåÉ=çê=ãçêÉ=ÉñéçåÉåíá~äëK=^=äçÖáÅ~ä=ëçäìíáçå=íç=íÜáë=éêçÄäÉã=áë=íç=ìëÉ=ÄçíÜ=ô=~åÇ=j^=íÉêãë=áå=çåÉ=ãçÇÉäI=ïÜáÅÜ=êÉëìäíë=áå=íÜÉ=ÖÉåÉê~ä=Éèì~íáçå=çÑ=~=íê~åëÑÉê=ÑìåÅíáçå=ãçÇÉä=çÑ=çêÇÉê= ( nr, ns)W===..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=y − µ − δ ( y − µ ) − ... − δ ( y − µ ) = ω x + ω x + ... + ω x(2.6)t 1 t−1 nr t−nr 0 t 1 t−1ns t−ns=tÜÉå=ïÉ=áåíêçÇìÅÉ=íÜÉ=Ä~Åâï~êÇ=ëÜáÑí=çéÉê~íçê= B I=ÇÉÑáåÉÇ=ÄóW==B b yt yt − b= (2.7)=~åÇ=~ìíçêÉÖêÉëëáîÉ=~åÇ=ãçîáåÖ=~îÉê~ÖÉ=çéÉê~íçêë=çÑ=çêÇÉê= nr ~åÇ= ns I=ÇÉÑáåÉÇ=êÉëéÉÅíáîÉäó=ÄóW==2⎧ ⎪ = − 1 − 2 − −δ (B) 1 δ B δ B ... δnrB⎨2ns⎪⎩ ω(B) = 1+ ω1 B+ ω2B + ... + ωnsB=Éèì~íáçå=EOKSF=Å~å=ÄÉ=ïêáííÉå=ãçêÉ=ÉÅçåçãáÅ~ääó=~ëW==nr(2.8)δ (B) ỹ = ω(B)x(2.9)tt=çêW===ỹ −= δ1 (B) ω(B)x(2.10)t=ïÜÉêÉ=t~åÇ===(B) =ỹ =~êÉ=ÇÉîá~íÉë=çÑ=−1tyt=Ñêçã µ I=Θ δ (B) ω(B)(2.11)=áë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçåK=tÜÉå=~=ëóëíÉã=Ü~ë=~=ÇÉä~óÉÇ=êÉëéçåëÉI=çê=ÇÉ~Ç=íáãÉ=çÑ=Çìê~íáçå= b I=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=ÄÉÅçãÉëW==−1Θ (B) = δ (B) ω(B)B b (2.12)=^äíÉêå~íáîÉäóI=EOKNMF=ã~ó=ÄÉ=ïêáííÉå=~ë=~=ëçJÅ~ääÉÇ=ÇáëÅêÉíÉ=Åçåîçäìíáçå=éêçÇìÅí=EëÉÉ=~äëç=ëÉÅíáçå=OKPKOFW==t∑∞∑ỹ = Θ x ≡ Θ x ≡ Θ(B) x ≡ ( Θ∗x)(2.13)t t−i i i t−i t ti=−∞ i=0=ïÜÉêÉ= Θt=áë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=EÑáÖìêÉ=OKVFK=ôj^=íê~åëÑÉê=ÑìåÅíáçå=ãçÇÉäë=Å~å=ëÉêîÉ=~ë=ÅäçëÉ=~ééêçñáã~íáçåë=Ñçê=ã~åó=èìáíÉ=ÅçãéäáÅ~íÉÇ=Çóå~ãáÅ=ëóëíÉãëK=^é~êí=Ñêçã=íÜÉ=ôj^=ëíêìÅíìêÉI=ÜçïÉîÉêI=íïç=Ä~ëáÅ=~ëëìãéíáçåë=~êÉ=ã~ÇÉK=qÜÉ=Ñáêëí=áë=íÜÉ= tdelay free (MA) part exponential (AR) part01 2 3 45 6tfigure 2.9: Transfer function of a hypotheticalARMA(1,2) system with delay 1.==== PR=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=Ñ~Åí=íÜ~í=íê~åëÑÉê=ÑìåÅíáçå=ãçÇÉäë=~ëëìãÉ=íÜ~í=íÜÉ=ëóëíÉã=áë=äáåÉ~êI=çê=Å~å=ÄÉ=äáåÉ~êáòÉÇK=få=ëÜçêíI=íÜáë=áãéäáÉë=íÜ~í=íÜÉ=ÉÑÑÉÅíë=çÑ=~ää=éìäëÉë=~åÇ=Éñéä~å~íçêó=î~êá~ÄäÉë=Å~å=ÄÉ=~ÇÇÉÇI=êÉÖ~êÇäÉëë=çÑ=íÜÉ=ëí~íÉ=çÑ=íÜÉ=Éñéä~áåÉÇ=î~êá~ÄäÉK=pÉÅçåÇI=íÜÉ=ãçÇÉä=~ëëìãÉë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=íç=ÄÉ=íáãÉJáåî~êá~åí=çê=êÉã~áå=Åçåëí~åí=çîÉê=íáãÉK==PS=2.2.2 Effects of noise and noise modelfÑ=ïÉ=äççâ=~í=íÜÉ=êÉ~ä=ïçêäÇI=íÜÉêÉ=~êÉ=ëÉîÉê~ä=êÉ~ëçåë=ïÜó=ã~íÜÉã~íáÅ=Éèì~íáçåë=ïáää=åÉîÉê=ÄÉ=~ÄäÉ=íç=ã~íÅÜ=çÄëÉêî~íáçåë=Ñêçã=Çóå~ãáÅ=ëóëíÉãë=Éñ~ÅíäóK=lÄëÉêî~íáçåë=~êÉ=ÄçìåÇ=íç=ÄÉ=~ÑÑÉÅíÉÇ=Äó=åçáëÉI=áÑ=çåäó=ÄÉÅ~ìëÉ=çÑ=Éêêçêë=~åÇ=ìåÅÉêí~áåíáÉë=áå=íÜÉ=ãÉ~ëìêÉãÉåí=éêçÅÉëëK=låÉ=çÑ=íÜÉ=ã~áå=éêçÄäÉãë=áå=ëáÖå~ä=éêçÅÉëëáåÖ=çê=ëóëíÉã=áÇÉåíáÑáÅ~íáçå=áë=íç=ëÉé~ê~íÉ=íÜÉ=ëáÖå~ä=çê=ìëÉÑìä=áåÑçêã~íáçå=Ñêçã=íÜÉ=åçáëÉ=çê=ÇáëíìêÄ~åÅÉë=áå=~=ÖáîÉå=íáãÉ=ëÉêáÉëK=tÜÉå=ÅêÉ~íáåÖ=~=ãçÇÉä=çÑ=~=Çóå~ãáÅ=ëóëíÉãI=éçëëáÄäÉ=ëçìêÅÉë=çÑ=åçáëÉI=É~ÅÜ=ïáíÜ=~=ÇáÑÑÉêÉåí=ÅÜ~ê~ÅíÉê=~åÇ=ÉÑÑÉÅíI=~êÉ=áå=ÖÉåÉê~ä=EÑáÖìêÉ=OKNMFW==• bêêçêë=áå=áåéìí=ãÉ~ëìêÉãÉåíë=• bêêçêë=áå=ãçÇÉä=ÅçåÅÉéí=çê=é~ê~ãÉíÉêë=• råâåçïå=ëóëíÉã=ÇáëíìêÄ~åÅÉë=• bêêçêë=áå=çìíéìí=ãÉ~ëìêÉãÉåíë==få=ãçÇÉä=íÉêãëI=íÜÉ=çìíéìí=çÑ=íÜÉ=EÇÉíÉêãáåáëíáÅ=é~êí=çÑ=íÜÉF=Éèì~íáçåë=áë=Å~ääÉÇ=íÜÉ=ãçÇÉä=éêÉÇáÅíáçåK=qÜÉ=ÇáÑÑÉêÉåÅÉë=ÄÉíïÉÉå=ãçÇÉä=éêÉÇáÅíáçåë=~åÇ=íÜÉ=çÄëÉêîÉÇ=çìíéìí=Ñçêã=~=íáãÉ=ëÉêáÉë=çÑ=íÜÉáê=çïåI=Å~ääÉÇ=íÜÉ=ÚêÉëáÇì~äëÛK=få=ÖÉåÉê~äI=~=êÉëáÇì~ä=ëÉêáÉë=Å~ååçí=ëáãéäó=ÄÉ=ãçÇÉäÉÇ=çê=í~âÉå=íç=ÄÉ=~=ëÉí=çÑ=áåÇÉéÉåÇÉåí=d~ìëëá~å=ÇÉîá~íÉëK=póëíÉã=ÇáëíìêÄ~åÅÉëI=ãçÇÉä=Éêêçêë=~åÇ=Éêêçêë=áå=íÜÉ=áåéìí=ïáääI=àìëí=~ë=íÜÉ=áåéìí=áíëÉäÑI=Ü~îÉ=~å=ÉÑÑÉÅí=íÜ~í=áë=åçí=çê=åçí=çåäó=áåëí~åí~åÉçìëK=_ÉÅ~ìëÉ=ÉÑÑÉÅíë=äáåÖÉê=çåI=íÜÉ=î~äìÉ=çÑ=~=ãçÇÉä=êÉëáÇì~ä=~í=~=ÅÉêí~áå=éçáåí=áå=íáãÉ=ïáää=ÄÉ=ÅçêêÉä~íÉÇ=ïáíÜ=áíë=î~äìÉ=~í=éêÉîáçìë=íáãÉëK=qÜáë=éÜÉåçãÉåçå=áë=Å~ääÉÇ=~ìíçÅçêêÉä~íáçåK=^ìíçÅçêêÉä~íáçå=InputSystemOutputNoise SignalMeasurementnoiseDeterministicmodel(empirical orphysically based)ModelerrorsResponsePredictionObservationsDisturbanceSystem noiseMeasurement noiseResiduals=figure 2.10: Sources and effects of noise in models of (linear, open) systems.=..…………………………………………………………………………………………….….

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`Ü~éíÉê=O=InputSystemOutputTransferfunctionNoiseResidualsObservationsNoisemodelPU=(white)(colored)=figure 2.11: Scheme of a combined single input, transfer function-noise model.=ÇÉä~óI=ïÉ=~êêáîÉ=~í=íÜÉ=ÖÉåÉê~ä=Éèì~íáçå=çÑ=~=ÅçãÄáåÉÇ=íê~åëÑÉê=ÑìåÅíáçå=~åÇ=åçáëÉ=ãçÇÉäW==−1 b−1ỹ t= δ (B) ω(B)B xt + ϕ(B) θ (B) at(2.15)=ïÜÉêÉ= ϕ (B) =~åÇ= θ ( B)=~êÉ=íÜÉ=~ìíçêÉÖêÉëëáîÉ=~åÇ=ãçîáåÖ=~îÉê~ÖÉ=çéÉê~íçêë=Ñçê=íÜÉ=åçáëÉ=éêçÅÉëëI=ïáíÜ=çêÇÉêë= np =~åÇ= nq =êÉëéÉÅíáîÉäóK=^=ãçêÉ=ÖÉåÉê~ä=ëçäìíáçå=íç=íÜáë=éêçÄäÉã=áë=éêçîáÇÉÇ=Äó=íÜÉ=ëç=Å~ääÉÇ=h~äã~å=ÑáäíÉê=xh~äã~åI=NVSMzK=qÜÉ=h~äã~å=ÑáäíÉê=~ääçïë=Ñçê=íÜÉ=ÇáëíáåÅíáçå=ÄÉíïÉÉå=ëóëíÉã=çê=ÅçêêÉä~íÉÇ=~åÇ=ãÉ~ëìêÉãÉåí=çê=ìåÅçêêÉä~íÉÇ=åçáëÉI=~åÇ=Å~å=ÉîÉå=ÄÉ=ìëÉÇ=Ñçê=ïÜ~í=áë=Å~ääÉÇ=~Ç~éíáîÉ=ÑáäíÉêáåÖI=ïÜÉå=íÜÉ=ëóëíÉã=é~ê~ãÉíÉêë=~êÉ=~ääçïÉÇ=íç=ÅÜ~åÖÉ=çîÉê=íáãÉK=ôj^JíóéÉ=íê~åëÑÉê=ÑìåÅíáçåë=~åÇ=åçáëÉ=ãçÇÉäë=Å~å=ÄÉ=ÉãÄÉÇÇÉÇ=áå=~=h~äã~åJÑáäíÉê=xÉKÖKI==_áÉêâÉåë=Éí=~äKI=NVVVX=_ÉêÉåÇêÉÅÜíI=OMMQzI=ïÜáÅÜ=ëÉêîÉë=~ë=~=ëçäìíáçå=íç=íÜÉ=äáãáí~íáçå=çÑ=ôj^JãçÇÉäë=çÑ=Ü~åÇäáåÖ=áêêÉÖìä~ê=Ç~í~=EëÉÉ=~äëç=åÉñí=ëÉÅíáçåFK=2.2.3 Limitations of ARMA modelsfå=ëéáíÉ=çÑ=íÜÉ=ÖÉåÉê~ä=~ééäáÅ~ÄáäáíóI=ïáÇÉ=ëéêÉ~Ç=ìëÉ=~åÇ=çÑíÉå=~ÅÅìê~íÉ=éêÉÇáÅíáçåë=çÑ=ôj^JíóéÉ=íáãÉ=ëÉêáÉë=ãçÇÉäëI=íÜÉêÉ=~êÉ=~äëç=ëÉîÉê~ä=äáãáí~íáçåë=íç=íÜÉáê=ìëÉK=mÉêÜ~éë=ÑçêÉãçëí=áãéçêí~åíI=íÜÉ=äáíÉê~íìêÉ=~åÇ=íÜÉçêó=çå=íáãÉ=ëÉêáÉë=~å~äóëáë=Ü~îÉ=~=ëíêçåÖ=ëí~íáëíáÅ~ä=ÑçÅìë=~åÇ=ìëÉ=~=ëéÉÅáÑáÅ=ã~íÜÉã~íáÅ~ä=åçí~íáçåI=à~êÖçå=~åÇ=îáÉïéçáåíK=`çåëÉèìÉåíäóI=áí=í~âÉë=~=äçåÖ=íáãÉ=Ñçê=ëÅáÉåíáëíëI=ÉåÖáåÉÉêë=~åÇ=çíÜÉêë=ïÜç=ãçëíäó=êÉÅÉáîÉÇ=áåíÉåëáîÉ=íê~áåáåÖ=áå=íÜÉáê=çïå=ÇáëÅáéäáåÉI=Äìí=åçí=áå=ëí~íáëíáÅë=äÉí=~äçåÉ=íáãÉ=ëÉêáÉë=~å~äóëáëI=íç=ìåÇÉêëí~åÇ=íÜÉ=áåë=~åÇ=çìíëK=låÅÉ=íÜÉ=íÜÉçêó=áë=~ÇçéíÉÇI=ÜçïÉîÉêI=íÜÉêÉ=~êÉ=ëíáää=ëÉîÉê~ä=éê~ÅíáÅ~ä=äáãáí~íáçåë=~åÇ=ÇáÑÑáÅìäíáÉë=íç=çîÉêÅçãÉW===..…………………………………………………………………………………………….….

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_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=çêW==1 −1 1 −1y − A x = exp( −At){ y(0) − A x }(2.28)c1 A0 c1 A0=rëáåÖ=EOKOQFI=íÜáë=êÉÇìÅÉë=íçW==1 { exp( t)}−1y = I − −A A x (2.29)c1 A0=ïÜÉêÉ= I =áë=íÜÉ=áÇÉåíáíó=ã~íêáñK=_ó=ÇÉÑáåáíáçåI=íÜÉ=ÇÉêáî~íáîÉ=çÑ=íÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçåë= y =íç= t =áë=~=îÉÅíçê=çÑ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåë= θ I=ïÜáÅÜ=Éèì~äëIìëáåÖ=EOKOPFW==d y 1 ⎡1⎤θ = = exp( −A t ) d t c 01A⎢ ⎥(2.30)0 ⎣ ⎦=qÜÉ=ÉñéçåÉåíá~ä=áå=EOKPMF=áë=~=ëçJÅ~ääÉÇ=ã~íêáñ=ÉñéçåÉåíá~ä=íÜ~í=Å~å=ÄÉ=Éî~äì~íÉÇ=ìëáåÖ=~=éêçÖê~ã=äáâÉ=j~íä~ÄK=få=Å~ëÉ=çÑ=áÇÉåíáÅ~ä=êÉëÉêîçáêëI=íÜÉ=ëçäìíáçå=çê=áãéìäëÉ=êÉëéçåëÉ=çÑ=íÜÉ=áåÇáîáÇì~ä=êÉëÉêîçáêë=áë=EëÉÉ=~ééÉåÇáñ=^FW==nr1θi = ∑ eije1jexp( −λjt)(2.31)c A1 0 j=1=ïÜÉêÉ=i =áë=íÜÉ=éçëáíáçå=çÑ=íÜÉ=êÉëÉêîçáêI= nr =áë=íÜÉ=åìãÄÉê=çÑ=êÉëÉêîçáêë=~åÇ= ei=íÜÉ= i th =î~äìÉ=áå=ÉáÖÉåîÉÅíçê= j ïáíÜ=ÉáÖÉåî~äìÉ= λ K=få=Å~ëÉ=çÑ=åçåJáÇÉåíáÅ~ä=êÉëÉêîçáêëI=íÜÉ=ëçäìíáçå=êÉ~Çë=EëÉÉ=~ééÉåÇáñ=^FW==nr1θi = ∑ αij exp( −λjt)(2.32)c A1 0 j=1=ïÜÉêÉ=íÜÉ=α Ûë=ïÉáÖÜ=íÜÉ=ÉñéçåÉåíá~ä=ÑìåÅíáçåëK=få=xp~ÜìèìáääçI=NVUPzI=~=ãÉíÜçÇ=áë=éêÉëÉåíÉÇ=Ñçê=ëçäîáåÖ=íÜÉ=ÖêçìåÇï~íÉê=Ñäçï=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=ìëáåÖ=~=êÉä~íÉÇ=~ééêç~ÅÜ=xëÉÉ=~äëç=päç~åI=OMMMX=mìäáÇçJsÉä~òèìÉò=Éí=~äKI=OMMRX=_áÇïÉääI=OMMRzK=få=Å~ëÉ=çÑ=äáåÉ~êáíóI=íÜÉ=ÅÉääë=áå=~=ÑáåáíÉ=ÇáÑÑÉêÉåÅÉ=çê=ÑáåáíÉ=ÉäÉãÉåí=ãçÇÉä=Å~å=ÄÉ=ÅçåëáÇÉêÉÇ=íç=ÄÉ=äáåÉ~ê=êÉëÉêîçáêëK=_~ëÉÇ=çå=íÜáëI=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåë=Eçê=áåÑäìÉåÅÉ=ÑìåÅíáçåë=áå=p~ÜìèìáääçÛë=íÉêãëF=Å~å=ÄÉ=ÇÉêáîÉÇ=ÇáêÉÅíäó=Ñêçã=íÜÉ=ã~íêáñ=çÑ=ÅÉääë=áå=~=ÖêçìåÇï~íÉê=ãçÇÉä=ENI=O=çê=PaFI=~åÇ=ÄÉ=ëíçêÉÇ=Ñçê=ëáãìä~íáçå=éìêéçëÉëK=få=íÜáë=ï~óI=íÜÉ=ëçäìíáçå=íç=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=áë=Éñ~Åí=áå=íÜÉ=íáãÉ=Ççã~áåI=~åÇ=áí=áë=åçí=åÉÅÉëë~êó=íç=ëçäîÉ=íÜÉ=ÉåíáêÉ=ã~íêáñ=Ñçê=ÉîÉêó=íáãÉ=ëíÉéK=kÉñí=íç=íÜÉ=Ñ~Åí=íÜ~í=íÜáë=ãÉíÜçÇ=áë=ìëÉÑìä=áå=áíëÉäÑI=ÉëéÉÅá~ääó=ïÜÉå=êÉëìäíë=Ñêçã=çåäó=~=ÑÉï=äçÅ~íáçåë=~êÉ=åÉÉÇÉÇI=áí=~äëç=åáÅÉäó==== QP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=θ ( t)=Å~å=ÄÉ=íÜçìÖÜí=çÑ=~ë=íÜÉ=êÉëéçåëÉ=íç=~=îÉêó=ëÜçêí=ëÜçïÉê=çÑ=ìåáí=ÜÉáÖÜíI=ïÜÉå=íÜÉ=ÜÉ~Ç=áë=çíÜÉêïáëÉ=Åçåëí~åí=çê=Éèì~äë=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉK=få=ã~íÜÉã~íáÅ~ä=íÉêãëI= θ ( t)=áë=ÇÉÑáåÉÇ=Äó=íÜÉ=ÑçääçïáåÖ=ÅçåÇáíáçåëW==⎧θ ( t) = h( t)− d⎪⎨h( t) = d, t < 0(2.34)⎪⎩ p( t) = δ( t)=fÑ= θ ( t)=~åÇ= d =~êÉ=âåçïåI= h( t)Å~å=ÄÉ=çÄí~áåÉÇ=Ñçê=~å=áåéìí p( t)íÜ~í=î~êáÉë=~êÄáíê~êáäó=áå=íáãÉ=íÜêçìÖÜ=Åçåîçäìíáçå=EaìÜ~ãÉäÛë=éêáåÅáéäÉ=xaìÜ~ãÉäI=NUPPzFW==t∞h( t) − d = θ ( t −τ ) p( τ )d τ ≡ θ( t) p( t −τ )d τ ≡ ( θ ∗ p)( t)∫ ∫(2.35)−∞0=bèì~íáçå=EOKPRF=áãéäáÉë=íÜ~í=íÜÉ=Çóå~ãáÅë=çÑ=~å=~êÄáíê~êóI=äáåÉ~ê=Çóå~ãáÅ=ëóëíÉã=~í=~=ÅÉêí~áå=äçÅ~íáçå=~êÉ=ÅçãéäÉíÉäó=ÖçîÉêåÉÇ=Äó=íÜÉ=Çóå~ãáÅë=çÑ=íÜÉ=áåéìí=~åÇ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåK=qÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=ÅçãéäÉíÉäó=ÅÜ~ê~ÅíÉêáòÉë=íÜÉ=Çóå~ãáÅ~ääó=êÉäÉî~åíI=éÜóëáÅ~ä=éêçéÉêíáÉë=çÑ=íÜÉ=ëóëíÉã=~í=íÜ~í=äçÅ~íáçåI=~åÇ=áë=~ë=ëìÅÜ=~å=áåíÉÖê~ä=éêçéÉêíó=íÜÉêÉçÑK===== QR=2.3.3 Evaluation of a convolution integrallÑ=ÅçìêëÉI=íÜÉ=Ñáêëí=ëíÉé=áå=~ééäóáåÖ=~=ãçÇÉä=áë=Éî~äì~íáçå=çÑ=íÜÉ=Éèì~íáçåëK=cçê=Éî~äì~íáåÖ=íÜÉ=Åçåîçäìíáçå=áåíÉÖê~ä=áå=EOKPRFI=Ñáêëí=íÜÉ=áåéìí=Ç~í~=Ü~ë=íç=ÄÉ=íê~åëÑçêãÉÇ=íç=Åçåíáåìçìë=íáãÉI=~ë=ãçëí=Ç~í~=~êÉ=ÇáëÅêÉíÉ=ÇìÉ=íç=íÜÉ=çÄëÉêî~íáçå=~åÇ=ÇáÖáí~ä=ëíçê~ÖÉ=éêçÅÉëëK=få=Å~ëÉ=çÑ=ÖêçìåÇï~íÉê=ëóëíÉãëI=Åçããçå=ÉñÅáí~íáçåë=~êÉ=ÉáíÜÉê=äÉîÉäë=EëìêÑ~ÅÉ=ï~íÉê=çê=çíÜÉê=ÖêçìåÇï~íÉê=äÉîÉäëF=çê=ÑäìñÉë=EéêÉÅáéáí~íáçåI=Éî~éçê~íáçåI=éìãéáåÖFK=cçê=ÑäìñÉëI=íÜÉ=ÇáëÅêÉíÉ=Ç~í~=ã~ó=çÑíÉå=ÄÉ=êÉÖ~êÇÉÇ=~ë=ÅÜ~åÖÉë=áå=íÜÉ=éêáãáíáîÉ=ÑìåÅíáçå=çÑ=íÜÉ=ìåÇÉêäóáåÖ=Åçåíáåìçìë=Ñäìñ=çê=áåíÉåëáíóI=~ë=áë=íÜÉ=Å~ëÉ=ïáíÜ=éêÉÅáéáí~íáçå=~ãçìåí=ëÉêáÉë P txizW===P = P( t ) − P( t − ∆t ) = ∫ p( τ )dτ(2.36)tii i ititi−∆ti=lÑ=ÅçìêëÉI=íÜÉ=Åçåíáåìçìë=éêÉÅáéáí~íáçå=áåíÉåëáíó=ëÉêáÉë= p( t)xiq JN z=Å~ååçí=ÄÉ=êÉÅçåëíêìÅíÉÇ=Éñ~ÅíäóI=Äìí=áí=Å~å=ÄÉ=~ééêçñáã~íÉÇ=Äó=~ëëìãáåÖ=íÜ~í=íÜÉ=Ñäìñ=áë=Åçåëí~åí=çîÉê=íÜÉ=íáãÉ=ëíÉé ∆ t ~åÇ=Éèì~äë=íÜÉ=~îÉê~ÖÉK=bèì~íáçå=EOKPSF=ã~ó=íÜÉå=ÄÉ=ïêáííÉå=~ëW==Pt p( τ ) =i, ti − ∆ ti < τ < ti(2.37)∆tiKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=çê=Ç~í~=ÑêÉèìÉåÅóK=mêçÄäÉãë=êÉä~íÉÇ=íç=íÜÉ=ÑêÉèìÉåÅóI=ëìÅÜ=~ë=Ü~åÇäáåÖ=áêêÉÖìä~ê=çê=ÜáÖÜJÑêÉèìÉåÅó=Ç~í~I=~êÉ=ÅçåëÉèìÉåíäó=ÅáêÅìãîÉåíÉÇK=få=Åçåíáåìçìë=íáãÉI=íÜÉêÉ=áë=åç=ëìÅÜ=íÜáåÖ=~ë=~=Ö~é=çê=ÚãáëëáåÖÛ=çÄëÉêî~íáçå=çÑ= h( t)K==få=íÜÉ=Åçåîçäìíáçå=áåíÉÖê~ä=çÑ=Éèì~íáçå=EOKPRFI=íáãÉ=ëí~êíë=~í=ãáåìë=áåÑáåáíóK=^ë=~=ÅçåëÉèìÉåÅÉI=ïÉ=Ü~îÉ=íç=ÇÉÑáåÉ=~å=áåáíá~ä=î~äìÉ=Ñçê=íÜÉ=áåéìí=ëÉêáÉë=Ñêçã= t = −∞ =íç=t = 0 I=íÜÉ=íáãÉ=çÑ=íÜÉ=Ñáêëí=~î~áä~ÄäÉ=çÄëÉêî~íáçåI=áå=çêÇÉê=íç=ÄÉ=~ÄäÉ=íç=Éî~äì~íÉ=áíK=cçê=ëíêÉëëÉë=íÜ~í=~êÉ=ëí~íáçå~êó=çê=áå=ÖÉåÉê~ä=ÑäìÅíì~íÉ=~êçìåÇ=ëçãÉ=~îÉê~ÖÉ=äÉîÉäI=äáâÉ=éêÉÅáéáí~íáçåI=ïÉ=Å~å=ÇÉÑáåÉ=íÜÉ=áåáíá~ä=î~äìÉ= p =íç=ÄÉ=íÜÉ=íÉãéçê~ä=~îÉê~ÖÉ=çÑ=íÜÉ=~î~áä~ÄäÉ=çÄëÉêî~íáçåë=E~=ÜáëíçêáÅ~ä=~îÉê~ÖÉI=áÑ=~î~áä~ÄäÉI=ã~ó=~äëç=ÄÉ=ìëÉÇFK=cçê=åçåJëí~íáçå~êó=ëíêÉëëÉëI=ÉKÖKI=~=éìãéáåÖ=ïÉää=íÜ~í=ï~ë=ÅçåëíêìÅíÉÇ=áå=~=ÅÉêí~áå=óÉ~ê=~åÇ=Öê~Çì~ääó=áåÅêÉ~ëÉÇ=áå=ê~íÉI=íÜÉ=íÉãéçê~ä=~îÉê~ÖÉ=áë=áå~ééêçéêá~íÉ=~ë=áåáíá~ä=î~äìÉK=få=íÜ~í=Å~ëÉI=íÜÉ=áåáíá~ä=î~äìÉ=Å~å=ÉáíÜÉê=ÄÉ=í~âÉå=íç=ÄÉ=òÉêç=ÄÉÑçêÉ=éìãéáåÖ=ëí~êíÉÇI=çê=íç=ÄÉ=ëçãÉ=~îÉê~ÖÉ=ÜáëíçêáÅ=ê~íÉ=éêáçê=íç=íÜÉ=Ñáêëí=~î~áä~ÄäÉ=êÉÅçêÇÉÇ=ÇáëÅÜ~êÖÉK=^äëçI=~=êìåJáå=éÉêáçÇ=áë=åÉÉÇÉÇ=Ñêçã= t = 0 =íç= t = thI=íÜÉ=ëí~êí=çÑ=íÜÉ=Å~äáÄê~íáçå=éÉêáçÇ=çê=íáãÉ=çÑ=íÜÉ=Ñáêëí=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåI=íç=çÄí~áå=~=êÉ~ëçå~ÄäÉ=Éëíáã~íÉë=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉäë=~í=É~êäó=íáãÉK==qÜÉ=~Åíì~ä=çìíÅçãÉ=çÑ=íÜÉ=Åçåîçäìíáçå=áåíÉÖê~ä=áë=íÜÉ=ëìã=çÑ=íÜÉ=ÉÑÑÉÅí=çÑ=íÜÉ=áåáíá~ä=î~äìÉ=~åÇ=íÜÉ=~î~áä~ÄäÉ=çÄëÉêî~íáçåë=çÑ=íÜÉ=áåéìí=ëÉêáÉëI=áå=ÄçíÜ=íÜÉ=êìåJáå=~åÇ=Å~äáÄê~íáçå=éÉêáçÇK=qçÖÉíÜÉê=ïáíÜ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉ= d I=íÜáë=óáÉäÇë=~=éêÉÇáÅíáçå=Ñçê= h( t)íÜ~í=Å~å=ÄÉ=Åçãé~êÉÇ=ïáíÜ=íÜÉ=~î~áä~ÄäÉ=çÄëÉêî~íáçåë=EÑáÖìêÉ=OKNRFK==`çåîçäìíáçå=áë=~=îÉêó=éê~ÅíáÅ~ä=ãÉíÜçÇ=íÜ~í=ìåÑçêíìå~íÉäó=áë=ãáëëáåÖ=áå=ã~åó=íÉñíÄççâë=çå=ÖêçìåÇï~íÉê=ÜóÇêçäçÖó=xläëíÜççêåI=OMMUzK=fí=Å~å=åçí=çåäó=ÄÉ=ìëÉÇ=Ñçê=íáãÉ=ëÉêáÉë=~å~äóëáëI=Äìí=~äëç=Ñçê=ÉÑÑáÅáÉåíäó=ÖÉåÉê~íáåÖ=ÜáÖÜJêÉëçäìíáçå=íáãÉ=ëÉêáÉë=ìëáåÖ=êÉëéçåëÉ==== QT=Initial valuehRun-in periodh p tCalibration periodh( t)( )td-p0p( t)figure 2.15: Role of the initial value, run-in and calibration period in the evaluation of a convolutionintegral.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁKt h=

`Ü~éíÉê=O=ÑìåÅíáçåë=çÄí~áåÉÇ=Ñêçã=ãçêÉ=Åçãéìí~íáçå~ääó=ÇÉã~åÇáåÖ=ÇÉíÉêãáåáëíáÅ=ãçÇÉäë=xp~ÜìèìáääçI=NVUPzK==QU=2.3.4 Impulse responses and their characteristicsfå=ÖêçìåÇï~íÉê=ÜóÇêçäçÖóI=~=íóéáÅ~ä=áãéìäëÉ=êÉëéçåëÉ=í~âÉë=íÜÉ=ëÜ~éÉ=çÑ=~=ëâÉïÉÇ=ÇáëíêáÄìíáçå=ÑìåÅíáçå=EÑáÖìêÉ=OKNSFK=få=íÜÉ=ÑáÉäÇ=çÑ=ëí~íáëíáÅëI=ÇáëíêáÄìíáçå=ÑìåÅíáçåë=~êÉ=Åçããçåäó=ÅÜ~ê~ÅíÉêáòÉÇ=Äó=íÜÉáê=ãçãÉåíëK=eÉêÉI=ïÉ=~äëç=ìëÉ=íÜÉ=ÅçåÅÉéí=çÑ=ãçãÉåíë=íç=ÅÜ~ê~ÅíÉêáòÉ=íÜÉ=éêçéÉêíáÉë=çÑ=áãéìäëÉ=thêÉëéçåëÉëK=qÜÉ= n JçêÇÉê=íÉãéçê~ä=ãçãÉåí Mn=çÑ=~å=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=áë=ÇÉÑáåÉÇ=ÄóW==nM = ∫ t θ ( t)dt(2.39)n∞−∞with mean µ and variance σ .=eáÖÜÉê=çêÇÉê=ãçãÉåíë=~êÉ=ÖÉåÉê~ääó=ëÅ~äÉÇ=Äó=íÜÉ=òÉêçíÜ=ãçãÉåí=E~äíÜçìÖÜ=Ñçê=ÑêÉèìÉåÅó=çê=éêçÄ~Äáäáíó=ÇáëíêáÄìíáçåë=Äó=ÇÉÑáåáíáçå= M0= 1I=ëç=íÜÉêÉ=áí=ÇçÉë=åçí=ã~âÉ=~=ÇáÑÑÉêÉåÅÉF=~åÇ=ÅÉåíê~äáòÉÇ=~Äçìí=íÜÉáê=ãÉ~å µ K=`çåëÉèìÉåíäóI=ÅÉåíê~ä=ãçãÉåíë=~êÉ=ÖáîÉå=ÄóW==cMn=∞∫−∞nt θ ( t − µ )dt∞∫−∞θ ( t)dt(2.40)=qÜÉ=òÉêçíÜ=íÜêçìÖÜ=ëÉÅçåÇ=ÅÉåíê~ä=ãçãÉåíë=~êÉ=~äëç=Åçããçåäó=âåçïå=Äó=íÜÉ=íÉêãë=2~êÉ~I=ãÉ~å= µ =~åÇ=î~êá~åÅÉ= σ K=líÜÉê=ÅÜ~ê~ÅíÉêáëíáÅë=çÑ= θ ( t)=~êÉ=áíë=ëâÉïåÉëëI=ÇÉÑáåÉÇ=ÄóW==c1 3response factor (−)Mγ = (2.41)3σ==~åÇ=áíë=âìêíçëáë=çê=ÉñÅÉëë=âìêíçëáëW==µσtime (days)figure 2.16: Skewed impulse response function2..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=c2 M 434γ = − (2.42)σ=qÜÉ=ÉÑÑÉÅí=çÑ=ÇáÑÑÉêÉåÅÉë=áå=âìêíçëáë=áë=γ = 3áåëí~åíäó=ÅäÉ~ê=áÑ=çåÉ=Åçãé~êÉë=ïÉääJγ = 2âåçïå=ÇáëíêáÄìíáçåë=ïáíÜ=íÜÉ=ë~ãÉ=ãÉ~åI=γ = 0γ =−1î~êá~åÅÉ=~åÇ=ëâÉïåÉëëI=Äìí=~=ÇáÑÑÉêÉåí=γ =−1.2âìêíçëáë=EÑáÖìêÉ=OKNTFK=qÜÉëÉ=ÇáëíêáÄìíáçåë=Å~å=ÄÉ=ïêáííÉå=~ë=ëéÉÅá~ä=Å~ëÉë=çÑ=íÜÉ=pâÉï=bñéçåÉåíá~ä=mçïÉê=ÇáëíêáÄìíáçå=xcÉêå~åÇÉò=Éí=~äKI=NVVRX=pÅÜçìéë=~åÇ=sêìÖíI=OMNMzI=ïÜáÅÜ=Ü~ë=íÜÉ=~ÇÇÉÇ=~Çî~åí~ÖÉ=íÜ~í=áíë=ëâÉïåÉëë=~åÇ=âìêíçëáë=Å~å=ÄÉ=î~êáÉÇ=Öê~Çì~ääóK=^å=~äíÉêå~íáîÉI=~åÇ=áå=ëçãÉ=Å~ëÉë=ãçêÉ=ÅçåîÉåáÉåí=ï~ó=çÑ=ÅÜ~ê~ÅíÉêáòáåÖ=ÇáëíêáÄìíáçå=ÑìåÅíáçåë=áë=Äó=íÜÉáê=Åìãìä~åíëK=cêçã=~=ÜóÇêçäçÖáÅ=éÉêëéÉÅíáîÉI= M0ÇÉÑáåÉë=íÜÉ=ÉÑÑÉÅí=çÑ=~=ëí~íáçå~êó=ÉñÅáí~íáçå=çÑ=ìåáí=ëíêÉåÖíÜ=EÉKÖKI=ëí~íáçå~êó=éêÉÅáéáí~íáçåFK=få=íáãÉ=ëÉêáÉë=~å~äóëáë=äáíÉê~íìêÉI= M0=áë=Å~ääÉÇ=íÜÉ=Ö~áåK= µ I=çå=íÜÉ=çíÜÉê=Ü~åÇI=áë=íÜÉ=ãÉ~å=ÇÉä~ó=çê=êÉëéçåëÉ=íáãÉ=íç=~å=áåëí~åí~åÉçìë=áåéìí=EÉKÖKI=~=ëìÇÇÉå=ëÜçïÉê=çÑ=éêÉÅáéáí~íáçåFK=qÜÉ=ëí~åÇ~êÇ=ÇÉîá~íáçå=σ =áë=~=ãÉ~ëìêÉ=Ñçê=íÜÉ=íÉãéçê~ä=ÚÇáëéÉêëáçåÛ=çÑ=~å=áãéìäëÉK===response factor (−)time (days)figure 2.17: Well-known distributions withdifferent kurtosis (resp. uniform, Wignersemicircle, normal, hyperbolic secant, Laplacedouble exponential (source: Wikipedia)).==== QV=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=2.3.5 Responses of elementary groundwater systemsRM=2.3.5.1 Recharge on a system with a uniform internal head^=ÅçåëÉèìÉåÅÉ=çÑ=íÜÉ=ä~ï=çÑ=ã~ëë=ÅçåëÉêî~íáçå=áë=íÜ~í=íÜÉ=~ãçìåí=çÑ=ï~íÉê=áå=~=îçäìãÉ=ãìëí=áåÅêÉ~ëÉ=ïÜÉå=íÜÉ=áåÑäçï=~í=~=ÅÉêí~áå=íáãÉ=ëíÉé=ÉñÅÉÉÇë=íÜÉ=çìíÑäçïK=tÜÉå=ïÉ=ëáãéäáÑó=íÜÉ=ÑìåÅíáçåáåÖ=çÑ=~=ëçáä=Åçäìãå=íç=~=ëáãéäÉ=äáåÉ~ê=êÉëÉêîçáê=ïáíÜ=áåÑäçï=~í=íÜÉ=íçé=~åÇ=çìíÑäçï=~í=íÜÉ=ÄçííçãI=~å~äçÖçìë=íç=xhåçííÉêë=~åÇ=_áÉêâÉåëI=OMMMzI=íÜÉ=ï~íÉê=Ä~ä~åÅÉ=áë=ÖáîÉå=Äó=EÑáÖìêÉ=OKNUFW==r d h− q = S (2.43)d t=ïÜÉêÉ=q = W=ÇáëÅÜ~êÖÉ=éÉê=ìåáí=~êÉ~=~åÇ=íáãÉ=== = xiq JN z==h = W=ÖêçìåÇï~íÉê=ÜÉ~Ç=~í=äçÅ~íáçå= z =xiz=S = W=ëíçê~íáîáíó=xJz=r = W=êÉÅÜ~êÖÉ=éÉê=ìåáí=~êÉ~=~åÇ=íáãÉ=xiq JN z==tÜÉå=ÅçåëáÇÉêáåÖ=ëìêÑ~ÅÉ=ï~íÉêI=íÜÉ=ëíçê~íáîáíó= S =Éèì~äë=N=~ë=çåÉ=ãáääáãÉíÉê=çÑ=éêÉÅáéáí~íáçå=Å~ìëÉë=~=êáëÉ=áå=ï~íÉê=äÉîÉä=çÑ=çåÉ=ãáääáãÉíÉêK=få=ÖêçìåÇï~íÉê=ëóëíÉãëI=ÜçïÉîÉêI= S =áë=ëã~ääÉê=ÄÉÅ~ìëÉ=çåäó=íÜÉ=ÑêÉÉ=éçêÉ=ëé~ÅÉ=áë=~î~áä~ÄäÉ=Ñçê=~ÇÇáíáçå~ä=ëíçê~ÖÉ=çÑ=ï~íÉê=Eáå=éÜêÉ~íáÅ=ëóëíÉãëFI=çê=ÄÉÅ~ìëÉ=íÜÉ=éçêÉ=ëé~ÅÉ=Å~å=çåäó=ÅÜ~åÖÉ=~=äáííäÉ=Äó=ÅçãéêÉëëáçå=çê=Éñé~åëáçå=çÑ=íÜÉ=Öê~áå=ëâÉäÉíçåI=ï~íÉêI==çê=áåÅäìÇÉÇ=Ö~ë=ÄìÄÄäÉë=Eáå=ÅçåÑáåÉÇ=~èìáÑÉêëFK=bèì~íáçå=EOKQPF=Åçåí~áåë=íïç=ÇáÑÑÉêÉåí=î~êá~ÄäÉëI=q ~åÇ= h I=íÜ~í=~êÉ=åçêã~ääó=ÄçíÜ=ìåâåçïåI=ïÜáäÉ=íÜÉ=íÜáêÇ=E r F=áë=åçêã~ääó=ÖáîÉåK=få=çêÇÉê=íç=ëçäîÉ=íÜÉ=Éèì~íáçåI=ïÉ=åÉÉÇ=~=ëÉÅçåÇI=áåÇÉéÉåÇÉåí=êÉä~íáçåëÜáé=ÄÉíïÉÉå= q ~åÇ= h I=ïÜáÅÜ=ïÉ=ÑáåÇ=Ñêçã=a~êÅóÛë=ä~ïW===h − dq = (2.44)c=ïÜÉêÉ= c =áë=íÜÉ=Çê~áå~ÖÉ=êÉëáëí~åÅÉ=xqz=~åÇ d íÜÉ=Çê~áå~ÖÉ=äÉîÉä=xizK=bäáãáå~íáçå=çÑ= q =Ñêçã=EOKQPF=óáÉäÇë=íÜÉ=ÑçääçïáåÖ=Ñáêëí=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåW==drdhh( t+ d t)h( t)figure 2.18: Water balance of a linear reservoirsystem.cq..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=d h d − hS = + r(2.45)dtc=fí=áë=åçíÉÇ=íÜ~í=Éèì~íáçå=EOKQRF=ÇçÉë=åçí=Åçåí~áå=~åó=êÉÑÉêÉåÅÉ=íç=~=ëé~íá~ä=ÇáãÉåëáçåK=qÜÉ=áåíÉêå~ä=ÜÉ~Ç=çÑ=íÜÉ=ëóëíÉã=áë=ìåáÑçêãI=~åÇ=çåäó=~=ÇáÑÑÉêÉåÅÉ=áå=ÜÉ~Ç=ïáíÜ=íÜÉ=çìíëáÇÉ=ïçêäÇ=áë=ëéÉÅáÑáÉÇI=ïáíÜçìí=~=ÇáêÉÅíáçåK=dê~éÜáÅ~ääó=E~ë=áå=ÑáÖìêÉ=OKNUF=~åÇ=áåíìáíáîÉäóI=ÜçïÉîÉêI=áí=ã~ó=ÄÉ=ÅçåîÉåáÉåí=íç=éáÅíìêÉ=íÜÉ=ÜÉ~Ç=ÇáÑÑÉêÉåÅÉ=~ë=~=îÉêíáÅ~ä=Åçäìãå=çÑ=ï~íÉêI=ïÜáÅÜ=çÑ=ÅçìêëÉ=êÉèìáêÉë=íÜÉ=îÉêíáÅ~ä=ÇáãÉåëáçåK=fÑ=ïÉ=ëçäîÉ=EOKQRF=Ñçê=~å=áãéìäëÉ=çÑ=êÉÅÜ~êÖÉ=~åÇ=ìëÉ=EOKPQF=I=ïÉ=ÑáåÇ=íÜ~í=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=xJz=áë=~å=ÉñéçåÉåíá~ä=~åÇ=Éèì~äë=íÜ~í=çÑ=~=ëáåÖäÉI=äáåÉ~ê=êÉëÉêîçáê=EëÉÉ=~äëç=ëÉÅíáçå=OKPKNFW==θ 1 t( t) = exp( )S− cS(2.46)=^ë=Ñçê=~åó=äáåÉ~ê=ëóëíÉãI=~äëç=áå=íÜáë=Å~ëÉ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ÑäìÅíì~íáçåë=Å~å=åçï=ÄÉ=ÑçìåÇ=íÜêçìÖÜ=ÅçåîçäìíáçåK=qÜÉ=äáåÉ~ê=êÉëÉêîçáêI=ÜçïÉîÉêI=Ü~ë=íÜÉ=êÉã~êâ~ÄäÉ=~åÇ=ìëÉÑìä=éêçéÉêíó=íÜ~í=íÜÉ=ÑäìÅíì~íáçåë=áå=äÉîÉä=ã~ó=~äëç=ÄÉ=ÅçãéìíÉÇ=êÉÅìêëáîÉäóK=få=çíÜÉê=ïçêÇëI=íÜÉ=î~äìÉ=çÑ= h( t ) =~í=~åó=íáãÉ=ëíÉé=ã~ó=ÄÉ=ÅçãéìíÉÇ=Ñêçã= h( t − ∆ t)=~åÇ=íÜÉ=íçí~ä=áåéìí=çîÉê=íÜÉ=íáãÉ=ëíÉé R tI=~ë=xaÉ=wÉÉìï=~åÇ=eÉääáåÖ~I=NVRUzW==∆t∆th( t) = { h( t − ∆t) − d}exp( − ) + Rtc{1 − exp( − )} + d(2.47)cScS=bèì~íáçå=EOKQTF=áë=Éèìáî~äÉåí=íç=~å=ôuENF=ãçÇÉä=EÉèì~íáçå=EOKQF=F=áå=ÇáëÅêÉíÉ=íáãÉ=xhåçííÉêë=~åÇ=_áÉêâÉåëI=OMMMzK=qÜÉ=Ñ~Åí=íÜ~í= h Å~å=ÄÉ=ÅçãéìíÉÇ=êÉÅìêëáîÉäóI=~ääçïë=íÜÉ=êÉëéçåëÉ=é~ê~ãÉíÉêë=x c,S z=íç=ÅÜ~åÖÉ=éáÉÅÉïáëÉ=çîÉê=íáãÉK=`çåëÉèìÉåíäóI=Éèì~íáçå=EOKQTF=Å~å=~äëç=ÄÉ=~ééäáÉÇ=íç=åçåJëí~ÄäÉ=çê=åçåJäáåÉ~ê=ëóëíÉãë=EïÜÉêÉ= c =~åÇLçê= S ~êÉ=~=ÑìåÅíáçå=çÑ= h FK==1/Sresponse factor (−)cS = 0.05cS = 0.1cS = 0.2cS = 0.40 50 100 150 200time (days)figure 2.19: Example IR functions of the reservoirsystem of figure 2.18 for different values of cS .==== RN=2.3.5.2 Recharge on a system with a one-dimensional head gradientfå=~=ëóëíÉã=ïáíÜ=~=çåÉJÇáãÉåëáçå~äI=Üçêáòçåí~ä=ÜÉ~Ç=Öê~ÇáÉåíI=íÜÉ=ï~íÉê=Ä~ä~åÅÉ=Éèì~íáçå=áë=ÖáîÉå=ÄóW==∂qx∂h− = S − r(2.48)∂x∂tKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=rRO=dhdxh( t+ d t)h( t)q outq indHxLfigure 2.20: Water balance of a system with a one-dimensional, horizontal head gradient and parallelsurface waters.råÇÉê=éÜêÉ~íáÅ=ÅçåÇáíáçåëI=íÜÉ=ë~íìê~íÉÇ=íÜáÅâåÉëë H xiz=çÑ=~å=~èìáÑÉê=ÇÉéÉåÇë=çå h E~åÇ=íÜÉêÉÑçêÉ=çå= t ~åÇ= x FK=fÑ= H − h áë=ëã~ää=êÉä~íáîÉ=íç H I=ÜçïÉîÉêI=ïÉ=Å~å=~ééêçñáã~íÉ H ~ë=Åçåëí~åíK=få=Å~ëÉ=çÑ=~å=áãéÉêãÉ~ÄäÉ=Ä~ëÉI=Ñìääó=éÉåÉíê~íáåÖ=ÇáíÅÜÉë=~åÇ=~=Çáëí~åÅÉ=ÄÉíïÉÉå=íÜÉ=ëìêÑ~ÅÉ=ï~íÉêë L xiz=íÜ~í=áë=ãìÅÜ=ä~êÖÉê=íÜ~å= H I=ïÉ=Å~å=~äëç=~ééêçñáã~íÉ=íÜÉ=Ñäçï=~ë=Üçêáòçåí~ä=xaìéìáíJcçêÅÜÜÉáãÉê=~ééêçñáã~íáçåI=aìéìáíI=NUSPX=cçêÅÜÜÉáãÉêI=NVMNzK=a~êÅóÛë=ä~ï=ã~ó=áå=ëìÅÜ=~=Å~ëÉ=ÄÉ=ïêáííÉå=~ëW===d hqx= − KH (2.49)d x=ïÜÉêÉ==KH =W=íê~åëãáëëáîáíó=xi O q JN z=K = W=éÉêãÉ~Äáäáíó=xiq JN z==bäáãáå~íáçå=çÑ= q =Ñêçã=EOKQUF=~åÇ=EOKQVF=óáÉäÇë=íÜÉ=ÑçääçïáåÖ=ëÉÅçåÇ=çêÇÉê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåW===..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=2∂ h ∂hKH = S − r(2.50)2∂x∂t=eÉêÉI=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=Åçåí~áåë=çåÉ=ëé~íá~ä=ÇáãÉåëáçåI=~ääçïáåÖ=íÜÉ=ÜÉ~Çë=áå=íÜÉ=ëóëíÉã=íç=ÇÉéÉåÇ=çå=íÜÉ=äçÅ~íáçåK=^Ö~áåI=ïÉ=ìëÉ=~=ëÉÅçåÇI=îÉêíáÅ~ä=ÇáãÉåëáçå=íç=Öê~éÜáÅ~ääó=êÉéêÉëÉåí=íÜÉ=ëóëíÉã=EÑáÖìêÉ=OKOMFK=bèì~íáçå=EOKRMF=Å~å=ÄÉ=ëçäîÉÇ=~å~äóíáÅ~ääó=Ñçê=íÜÉ=ÉÑÑÉÅíë=çÑ=êÉÅÜ~êÖÉK=cçê=ÖêçìåÇï~íÉê=éêçÄäÉãëI=íÜÉ=ëçäìíáçå=Ü~ë=ÄÉÉå=ÑçìåÇ=Ñáêëí=Äó=däçîÉê=~åÇ=ï~ë=éìÄäáëÜÉÇ=áå=xaìããI=NVRQz=~åÇ=~äëç=ÇáëÅìëëÉÇ=áå=xhê~áàÉåÜçÑÑ=î~å=ÇÉ=iÉìêI=NVRUzK=tÜÉå=íÜÉ=ï~íÉê=äÉîÉä=~í=íÜÉ=ÄçìåÇ~êáÉë=áë=Åçåëí~åí= ( hx= 0= hx=L= d)I=íÜÉ=êÉÅÜ~êÖÉ=áë=~å=áãéìäëÉ=~åÇ=EOKPQF=áë=ìëÉÇI=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=xJz=ã~ó=ÄÉ=ÑçìåÇ=~åÇ=ÄÉ=ïêáííÉå=~ëW==2 2∞1 4 1 −n π KHt nπxθ ( t) = ∑ exp( )sin(2.51)π2S n=1,3,5,.... n SL L=ïÜÉêÉ x xiz=ÇÉÑáåÉë=íÜÉ=éçëáíáçå=çÑ=íÜÉ=äçÅ~íáçå=ìåÇÉê=ÅçåëáÇÉê~íáçåK=qÜÉ=êÉëéçåëÉ=çÑ=ëìÅÜ=~=ëóëíÉã=áë=åçí=çåäó=~=ÑìåÅíáçå=çÑ=íÜÉ=ëíçê~íáîáíó=~åÇ=êÉëáëí~åÅÉI=~ë=áå=íÜÉ=äáåÉ~ê=êÉëÉêîçáê=Å~ëÉI=Äìí=~äëç=çÑ=íÜÉ=êÉä~íáîÉ=éçëáíáçå=çê=ÅÉåíê~äáíó= C xJz=çÑ=íÜÉ=äçÅ~íáçå=ìåÇÉê=ÅçåëáÇÉê~íáçåI=ïÜáÅÜ=ïÉ=ÇÉÑáåÉ=ÄóW==⎧2xL, x ≤⎪ L2C = ⎨(2.52)⎪ 2( L − x) L , x >⎪⎩ L2=få=íÜáë=ï~óI=íÜÉ=ÅÉåíê~äáíó=áë=N=áÑ=íÜÉ=äçÅ~íáçå=áë=áå=íÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ëóëíÉã=~åÇ=M=áÑ=áí=äáÉë=çå=áíë=ÄçìåÇ~êáÉëK=få=ÑáÖìêÉ=OKONI=Éñ~ãéäÉ=fo=ÑìåÅíáçåë=~êÉ=éäçííÉÇ=Ñçê=ÇáÑÑÉêÉåí=î~äìÉë=çÑ= C I=ïÜáÅÜ=ëÜçïë=íÜÉ=ÇáÑÑÉêÉåÅÉ=áå=ÄÉÜ~îáçê=ÇÉéÉåÇáåÖ=çå=íÜÉ=äçÅ~íáçåK=få=ëÜçêíW=áå=íÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ëóëíÉãI=íÜÉ=äÉîÉä=Å~å=çåäó=Çêçé=~ÑíÉê=íÜÉ=äÉîÉäë=åÉ~ê=íÜÉ=ÄçìåÇ~êáÉë=Ü~îÉ=ÇêçééÉÇ=ÑáêëíK=kÉ~ê=íÜÉ=ÄçìåÇ~êáÉëI=íÜÉ=ï~íÉê=äÉîÉä=Çêçéë=èìáÅâäó=~í=Ñáêëí=Äìí=ëí~ÄáäáòÉë=ä~íÉê=çå=ÄÉÅ~ìëÉ=çÑ=ï~íÉê=ÑäçïáåÖ=áå=Ñêçã=íÜÉ=ÅÉåíÉêK=1/Sresponse factor (−)SL^2/KHtime (days)C = 1C = 1/2C = 1/4C = 1/8figure 2.21: Example IR functions of the system offigure 2.20 for different values of C .==== RP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=RQ=2.3.5.3 From infiltration to recharge in the unsaturated zoneré=ìåíáä=íÜáë=éçáåíI=ïÉ=Ü~îÉ=åÉÖäÉÅíÉÇ=íÜÉ=Ñ~Åí=íÜ~í=áå=êÉ~äáíó=êÉÅÜ~êÖÉ=áë=ÇáÑÑáÅìäí=íç=ãÉ~ëìêÉ=~åÇ=ÖÉåÉê~ääó=ìåâåçïåK=oÉÅÜ~êÖÉI=ÜçïÉîÉêI=Å~å=ÄÉ=ëÉÉå=~ë=íÜÉ=çìíÅçãÉ=çÑ=íÜÉ=éêçÅÉëëÉë=áå=íÜÉ=ìåë~íìê~íÉÇ=òçåÉI=çÑ=ïÜáÅÜ=íÜÉ=ÇêáîáåÖ=ÑçêÅÉë=~êÉ=éêÉÅáéáí~íáçå= p =Root zone~åÇ=EéçíÉåíá~äF=Éî~éçê~íáçå= e K=qÜÉ=ä~ííÉê=~êÉ=ÖÉåÉê~ääó=~î~áä~ÄäÉ=Ñêçã=ãÉíÉçêçäçÖáÅ=ëí~íáçåëI=~åÇ= PercolationÅ~å=íÜÉêÉÑçêÉ=ÄÉ=ìëÉÇ=~ë=áåéìí=áÑ= zoneïÉ=~êÉ=~ÄäÉ=íç=ãçÇÉä=çê=ãáãáÅ=íÜÉ=ìåë~íìê~íÉÇ=òçåÉ=éêçÅÉëëÉëK=få=Å~ëÉ=çÑ=ìåë~íìê~íÉÇ=ÑäçïI=íÜÉ= Saturatedï~íÉê=Ä~ä~åÅÉ=Éèì~íáçå=~åÇ= zonea~êÅóÛë=ä~ï=~Ö~áå=í~âÉ=ÇáÑÑÉêÉåí=ëÜ~éÉëI=ÄÉÅ~ìëÉ=çÑ=íÜÉ=î~êóáåÖ=ï~íÉê=ÅçåíÉåí=~åÇ=éÉêãÉ~ÄáäáíóK=fÑ=ïÉ=ÅçåëáÇÉê=çåÉJÇáãÉåëáçå~äI=îÉêíáÅ~ä=Ñäçï=çåäóI=íÜÉ=ï~íÉê=Ä~ä~åÅÉ=Éèì~íáçå=áë=ÖáîÉå=ÄóW==∂q∂ϑ=(2.53)∂z∂t=~åÇ=a~êÅóÛë=ä~ï=ÄóW==d hq = K( ϑ) (2.54)d z=ïÜÉêÉ=z = W=ÇÉéíÜ=xiz==ϑ = W=îçäìãÉíêáÅ=ï~íÉê=ÅçåíÉåí=xJz=K( ϑ)W=ìåë~íìê~íÉÇ=éÉêãÉ~Äáäáíó=xiq JN z==_ó=ÅçãÄáåáåÖ=íÜÉ=Éèì~íáçåë=~ÄçîÉI=ïÉ=ÑáåÇ=íÜÉ=ÑçääçïáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=Ñçê=Ñäçï=áå=íÜÉ=ìåë~íìê~íÉÇ=òçåÉW==∂ ⎡ ∂K( ) h ∂ϑϑ⎤ =(2.55)=∂z ⎢ z ⎥⎣ ∂ ⎦ ∂tConv.-Disp.Sat.ZoneField capacityWilting pointfigure 2.22: Schematic representation of the unsaturatedzone.prhe..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=ïÜáÅÜ=áë=~=ãçÇáÑáÉÇ=Ñçêã=çÑ=oáÅÜ~êÇëÛ=Éèì~íáçå=xoáÅÜ~êÇëI=NVPNzK=qÜÉ=ìåë~íìê~íÉÇ=òçåÉ=áë=ÖÉåÉê~ääó=ÇáîáÇÉÇ=áåíç=~F=íÜÉ=êççí=òçåÉI=ïÜÉêÉ=íÜÉêÉ=áë=ÇáêÉÅí=áåéìí=~åÇ=ìéí~âÉ=çÑ=ï~íÉêI=ÄF=íÜÉ=éÉêÅçä~íáçå=òçåÉI=ïÜÉêÉ=ï~íÉê=éêáã~êáäó=éÉêÅçä~íÉë=Ççïåï~êÇë=ÇìÉ=íç=Öê~îáí~íáçå~ä=ÑçêÅÉ=~åÇ=ÅF=íÜÉ=Å~éáää~êó=òçåÉI=áå=ïÜáÅÜ=íÜÉêÉ=áë=Å~éáää~êó=Åçåí~Åí=~åÇ=éçëëáÄäÉ=ÑÉÉÇáåÖ=Ñêçã=íÜÉ=ë~íìê~íÉÇ=òçåÉK=^ë=áí=áë=åÉ~êÉëí=íç=íÜÉ=ÇêáîáåÖ=ÑçêÅÉëI=íÜÉ=ï~íÉê=ÅçåíÉåí=áå=ÉëéÉÅá~ääó=íÜÉ=êççí=òçåÉ=áë=ÜáÖÜäó=Çóå~ãáÅ=~åÇ=ÅçåëÉèìÉåíäóI=ÄÉÅ~ìëÉ=çÑ=íÜÉ=î~êóáåÖ=éÉêãÉ~Äáäáíó K( ϑ)I=áíë=ÑìåÅíáçåáåÖ=áë=ÜáÖÜäó=åçåJäáåÉ~êK=få=~=ïÉí=ëí~íÉI=ïÜÉå=íÜÉ=ëçáä=áë=~í=ÑáÉäÇ=Å~é~ÅáíóI=éÉêãÉ~Äáäáíó=áë=ÜáÖÜI=Éî~éçê~íáçå=áë=~í=áíë=éçíÉåíá~ä=ê~íÉ=~åÇ=éêÉÅáéáí~íáçå=Å~å=áåÑáäíê~íÉ=~åÇ=éÉêÅçä~íÉ=ÑêÉÉäóK=få=~=Çêó=ëí~íÉI=Å~éáää~êó=éêÉëëìêÉ=áë=ÜáÖÜI=íÜÉ=éÉêãÉ~Äáäáíó=çÑ=íÜÉ=ëçáä=áë=ÖêÉ~íäó=êÉÇìÅÉÇ=~åÇ=Éî~éçê~íáçå=~åÇ=éÉêÅçä~íáçå=~êÉ=ëíêçåÖäó=äáãáíÉÇK=qÜÉ=ÉÑÑÉÅí=çÑ=î~êá~ÄäÉ=ëíçê~ÖÉ=çÑ=ï~íÉê=áå=íÜÉ=êççí=òçåÉ=áë=ëçãÉíáãÉë=ãçÇÉäÉÇ=ÅçåÅÉéíì~ääó=~ë=~=êÉëÉêîçáê=áå=ïÜáÅÜ=ìéí~âÉ=çê=Éî~éçê~íáçå=ëíçéë=~í=íÜÉ=ëçJÅ~ääÉÇ=ïáäíáåÖ=éçáåíI=~åÇ=Ñêçã=ïÜáÅÜ=ï~íÉê=çåäó=éÉêÅçä~íÉë=~í=ÑáÉäÇ=Å~é~Åáíó=EÑáÖìêÉ=OKOOFK=qÜÉ=Ñ~Åí=íÜ~í=ëìÅÜ=åçåJäáåÉ~ê=ÄÉÜ~îáçê=Å~ååçí=ÄÉ=ÇÉëÅêáÄÉÇ=ïáíÜ=~=äáåÉ~ê=ãçÇÉä=Å~å=~ÅÅçìåí=Ñçê=íÜÉ=ìÄáèìáíçìë=ëÉ~ëçå~ä=é~ííÉêåë=áå=íÜÉ=êÉëáÇì~ä=ëÉêáÉë=çÑ=äáåÉ~ê=íáãÉ=ëÉêáÉë=ãçÇÉäëK=fí=Å~å=~äëç=ÄÉ=~=êÉ~ëçå=Ñçê=ÅçãÄáåáåÖ=íáãÉ=ëÉêáÉë=ãçÇÉäë=ïáíÜ=ãçêÉ=çê=äÉëë=ÅçãéäÉñ=ãçÇÉäë=çÑ=ìåë~íìê~íÉÇ=òçåÉ=éêçÅÉëëÉë=xÉKÖKI=aÉ=hÉáòÉêI=OMMPX=_ÉêÉåÇêÉÅÜí=Éí=~äKI=OMMSz=K===få=íÜÉ=éÉêÅçä~íáçå=òçåÉI=áå=ëáíì~íáçåë=ïáíÜ=ÇÉÉéÉê=ï~íÉê=í~ÄäÉëI=íÜÉ=ï~íÉê=ÅçåíÉåí=áë=äÉëë=î~êá~ÄäÉ=~åÇ=áíë=ã~áå=ÉÑÑÉÅí=áë=êÉí~êÇ~íáçå=~åÇ=ÇáëéÉêëáçå=çÑ=áåÑáäíê~íáåÖ=ï~íÉê=x_ÉëÄÉë=~åÇ=ÇÉ=j~êëáäóI=NVUQX=m~êä~åÖÉ=Éí=~äKI=NVVOzK=tÜÉå=oáÅÜ~êÇëÛ=Éèì~íáçå=áë=äáåÉ~êáòÉÇ=~êçìåÇ=ëçãÉ=Åçåëí~åíI=Ú~îÉê~ÖÉÛ=ï~íÉê=ÅçåíÉåí= ϑc=ìëáåÖ=q~óäçê=ëÉêáÉë=Éñé~åëáçåI=áí=íê~åëÑçêãë=áåíç=íÜÉ=ÅçåîÉÅíáçåJÇáëéÉêëáçå=Éèì~íáçå=xgìêó=~åÇ=oçíÜI=NVVMX=wï~ãÄçêåI=NVVRzK=qÜÉ=ÅçåîÉÅíáçåJÇáëéÉêëáçå=Éèì~íáçå=Å~å=ÅçåëÉèìÉåíäó=ÄÉ=ìëÉÇ=íç=ÇÉëÅêáÄÉ=íÜÉ=Ú~îÉê~ÖÉÛ=ÉÑÑÉÅí=çÑ=íÜÉ=ìåë~íìê~íÉÇ=òçåÉ=çå=íÜÉ=Ç~ãéáåÖ=~åÇ=êÉí~êÇ~íáçå=çÑ=éêÉÅáéáí~íáçå=~åÇ=Éî~éçê~íáçå=éìäëÉëK=eÉêÉI=ïÉ=ÇÉÑáåÉ= z íç=ÄÉ=òÉêç=~í=íÜÉ=íçé=çÑ=íÜÉ=éÉêÅçä~íáçå=òçåÉK=fÑ=ïÉ=~ééäó=íÜÉ=ÑçääçïáåÖ=áåáíá~ä=ÅçåÇáíáçåëW==⎧ϑ( z < 0, t = 0) − ϑ = 1⎨⎩ϑ( z > 0, t = 0) − ϑc= 0=~åÇ=ÄçìåÇ~êó=ÅçåÇáíáçåëW==cflux [LT^−1]z/vtime (days)σ = µ/16σ = µ/8σ = µ/4σ = µ/2figure 2.23: Example IR functions of a linearizedunsaturated zone (convection-dispersion equation)for different values of σ .(2.56)=== RR=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=RS=⎧ϑ( z = −∞, t) − ϑc= 1⎨(2.57)⎩ϑ( z = ∞, t) − ϑc= 0=~åÇ=íÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçå Ω =xJz=íç=ÄÉW==Ω ( z, t) = ϑ( z, t) − ϑc(2.58)=íÜÉ=ÑçääçïáåÖ=ëçäìíáçå=íç=íÜÉ=ÅçåîÉÅíáçåJÇáëéÉêëáçå=Éèì~íáçå=ã~ó=ÄÉ=ÑçìåÇ=x`ê~åâI=NVRTzW==1 z − vtΩ ( z, t) = erfc{ }(2.59)2 2Dt=ïÜÉêÉ=D = W=EÉÑÑÉÅíáîÉF=ÇáÑÑìëáçåJÇáëéÉêëáçå=ÅçÉÑÑáÅáÉåí=xi O =q JN z==v = W=EÉÑÑÉÅíáîÉF=éêçé~Ö~íáçå=îÉäçÅáíó=xiq JN z===qÜÉ=áãéìäëÉ=êÉëéçåëÉ=θ =xq JN z=áë=íÜÉ=ÇÉêáî~íáîÉ=çÑ=EOKRVF=çê=xÉKÖKI=gìêó=~åÇ=péçëáíçI=NVURX=j~~ëI=NVVQzW==d Ω( z, t) z ( z − vt)θ ( z, t) = = exp{ − }(2.60)dt34 π Dt 4Dt=^=ëÜçêíÅçãáåÖ=çÑ=EOKRVF==áë=íÜ~í=áí=~äëç=éêÉÇáÅíë=ìéï~êÇ=ÇáëéÉêëáçå=ÄÉóçåÇ z = 0 K=tÜÉå=íÜÉ=ÄçìåÇ~êó=ÅçåÇáíáçåë=~êÉ=ÅÜçëÉå=ëìÅÜ=íÜ~í=ìéï~êÇ=ÇáëéÉêëáçå=áë=éêÉîÉåíÉÇ=( ϑ( z = 0, t) − ϑc= 1) I=~=ëÉÅçåÇ=íÉêã=áë=áåíêçÇìÅÉÇ=áå=íÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçå=xoáÑ~á=Éí=~äKI=NVRSzK=qÜÉ=ÅçåíêáÄìíáçå=çÑ=íÜáë=ëÉÅçåÇ=íÉêãI=ÜçïÉîÉêI=ê~éáÇäó=ÇÉÅ~óë=ïáíÜ=Çáëí~åÅÉI=ëç=íÜ~í=çåäó=EOKSMF=êÉã~áåë=~í=ëçãÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=ÄçìåÇ~êó=xëÉÉ=j~~ëI=NVVQI=é~ÖÉ=SNzKθ =áå=íÜáë=Å~ëÉ=áë=åçí=~=êÉëéçåëÉ=áå=äÉîÉä=Äìí=áå=ï~íÉê=ÅçåíÉåíI=~åÇ=áíë=~êÉ~=çê=òÉêçíÜ=ãçãÉåí=Éèì~äë=çåÉ=ÄÉÅ~ìëÉ=çÑ=íÜÉ=éêáåÅáéäÉ=çÑ=ã~ëë=ÅçåëÉêî~íáçåK==mäÉ~ëÉ=åçíÉ=~äëç=íÜ~í=íÜÉ=éêçé~Ö~íáçå=îÉäçÅáíó= v áë=íÜÉ=îÉäçÅáíó=~í=ïÜáÅÜ=íÜÉ=éìäëÉ=çê=éêÉëëìêÉ=ï~îÉ=éêçé~Ö~íÉëK=fí=áë=åçí=íÜÉ=EÉÑÑÉÅíáîÉF=îÉäçÅáíó=çÑ=ï~íÉê=áíëÉäÑ=áå=íÜÉ=ìåë~íìê~íÉÇ=òçåÉK=qÜÉ=îÉäçÅáíó=çÑ=ï~íÉê=áë=ãìÅÜ=ëã~ääÉê=íÜ~å=íÜ~í=íÜÉ=éêÉëëìêÉ=éêçé~Ö~íáçå=îÉäçÅáíóI=~åÇ=Éèì~äë=íÜÉ=êÉÅÜ~êÖÉ=ê~íÉ=áå=~=äáåÉ~êáòÉÇ=ëáíì~íáçåK=qÜÉ=Ñáêëí=E µ x=q JN 2zF=~åÇ=ëÉÅçåÇ=Eσ xq JO zF=ÅÉåíê~ä=ãçãÉåíë=çÑ=θ =~êÉ=ÖáîÉå=ÄóW==⎧ zµ =⎪ v⎨(2.61)⎪ 2 2zDσ =3⎪⎩ v=2..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=få=ÑáÖìêÉ=OKOPI=ëçãÉ=Éñ~ãéäÉ=fo=ÑìåÅíáçåë=~êÉ=éäçííÉÇ=íÜ~í=Ü~îÉ=íÜÉ=ë~ãÉI=~îÉê~ÖÉ=~êêáî~ä=íáãÉ µ I=Äìí=~=ÇáÑÑÉêÉåí=ëí~åÇ~êÇ=ÇÉîá~íáçå=σ K=^å=çìíÅçãÉ=çÑ=íÜÉ=ÅçåîÉÅíáçåJÇáëéÉêëáçå=Éèì~íáçå=áë=íÜ~í=íÜÉ=ëé~íá~ä=EîÉêíáÅ~äF=ÇáëíêáÄìíáçå=çÑ=ï~íÉê=éìäëÉë=áë=ëóããÉíêáÅI=ïÜáÅÜ=áë=~=ÅçåëÉèìÉåÅÉ=çÑ=íÜÉ=äáåÉ~êáò~íáçå=çÑ=oáÅÜ~êÇëÛ=Éèì~íáçå=~êçìåÇ=íÜÉ=~îÉê~ÖÉ=ï~íÉê=ÅçåíÉåíK==få=Çóå~ãáÅI=åçåJäáåÉ~ê=êÉ~äáíóI=íÜÉ=ÚïÉííáåÖÛ=Ñêçåí=ïáää=ÄÉ=ëÜ~êéÉê=ÄÉÅ~ìëÉ=çÑ=íÜÉ=áåÅêÉ~ëÉÇ=éÉêãÉ~Äáäáíó=ïÜÉå=íÜÉ=ï~íÉê=ÅçåíÉåí=áë=ÜáÖÜK=lå=íÜÉ=çíÜÉê=Ü~åÇI=ã~Åêç=éçêÉë=~åÇ=éêÉÑÉêÉåíá~ä=Ñäçï=ïáää=áåÅêÉ~ëÉ=íÜÉ=Ñäìñ=~í=êÉä~íáîÉäó=É~êäó=~êêáî~ä=íáãÉë=~åÇ=ÅçåëÉèìÉåíäó=ÅçìåíÉê~Åí=íÜÉ=ëÜ~êéåÉëëK===2.3.5.4 Convolution of response functionsfå=ëÉÅíáçå=OKPKRKPI=ïÉ=Ü~îÉ=ÇáëÅìëëÉÇ=íÜÉ=ÉÑÑÉÅí=çÑ=íÜÉ=ìåë~íìê~íÉÇ=òçåÉ=áå=ÇÉä~óáåÖ=~åÇ=Ç~ãéáåÖ=éêÉÅáéáí~íáçå=éìäëÉëK=^ë=êÉÅÜ~êÖÉ=áë=ÖÉåÉê~ääó=ìåâåçïåI=ÜçïÉîÉêI=ïÉ=Å~å=åçí=çÄí~áå=~=ëÉé~ê~íÉ=Éëíáã~íÉ=çÑ=íÜÉ=éêçéÉêíáÉë=çê=êÉëéçåëÉ=çÑ=íÜÉ=ìåë~íìê~íÉÇ=òçåÉ=íÜêçìÖÜ=íáãÉ=ëÉêáÉë=~å~äóëáëK=fåëíÉ~ÇI=áÑ=ïÉ=ìëÉ=éêÉÅáéáí~íáçå=~ë=áåéìí=~åÇ=Éëíáã~íÉ=áíë=ÉÑÑÉÅí=çå=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉäI=ïÉ=çÄí~áå=~å=Éëíáã~íÉ=çÑ=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=ÅçãÄáåÉÇ=ë~íìê~íÉÇ=~åÇ=ìåë~íìê~íÉÇ=òçåÉK=fÑ=äáåÉ~ê=ëóëíÉãë=~êÉ=éä~ÅÉÇ=áå=ëÉêáÉëI=íÜÉ=çìíéìí=çÑ=çåÉ=áë=áåéìí=íç=íÜÉ=çíÜÉêI=ëç=íÜÉáê=ÅçãÄáåÉÇ=ÉÑÑÉÅí=Éèì~äëW==tt∫ sz 2 ∫ uz 1 1 1 2(2.62)−∞−∞h( t) = θ ( t −τ ) θ ( t −τ ) p( τ )dτ dτ=ïÜÉêÉ= θsz~åÇ= θuz~êÉ=íÜÉ=êÉëéçåëÉ=ÑìåÅíáçåë=çÑ=íÜÉ=ë~íìê~íÉÇ=~åÇ=ìåë~íìê~íÉÇ=òçåÉI=êÉëéÉÅíáîÉäóK=j~íÜÉã~íáÅ~ääóI=Éèì~íáçå=EOKSOF=Éèì~äë=~=ëáåÖäÉ=Åçåîçäìíáçå=áåíÉÖê~ä=ïáíÜ==== RT=Convection-Dispersion eq.Glover &DummUnsaturatedZoneSaturatedZoneInputp( t)Outputh( t)CombinedResponse =figure 2.24: Combination of the responses of the unsaturated and saturated zones.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=RU=~=ÅçãÄáåÉÇ=êÉëéçåëÉ=íÜ~í=áë=ÑçìåÇ=Äó=ÅçåîçäìíáåÖ=íÜÉ=ëÉé~ê~íÉ=êÉëéçåëÉëW==tθc ( t) = ∫ θsz ( t −τ ) θuz( τ )dτ(2.63)−∞=få=ÑáÖìêÉ=OKOQ=íÜÉ=éêçÅÉëë=~åÇ=Éñ~ãéäÉ=êÉëéçåëÉë=~êÉ=áääìëíê~íÉÇK=aÉéÉåÇáåÖ=çå=íÜÉ=ëáíì~íáçåI=íÜÉ=ÅçãÄáåÉÇ=êÉëéçåëÉ=Å~å=í~âÉ=íÜÉ=ëÜ~éÉ=çÑ=~=ÑìåÅíáçå=íÜ~í=êáëÉë=Öê~Çì~ääó=~í=ÑáêëíI=ÇìÉ=íç=íÜÉ=ÉÑÑÉÅí=çÑ=íÜÉ=ìåë~íìê~íÉÇ=òçåÉI=ìåíáä=íÜÉ=éÉ~â=áë=êÉ~ÅÜÉÇK=^ÑíÉê=íÜ~íI=íÜÉ=êÉëéçåëÉ=ëäçïäó=ÇÉÅ~óë=~ë=êÉÅÜ~êÖÉ=ÇáãáåáëÜÉë=~åÇ=ÖêçìåÇï~íÉê=Çê~áåë=íçï~êÇë=íÜÉ=ëìêÑ~ÅÉ=ï~íÉêë=çê=Çê~áå~ÖÉ=ãÉ~åëK=få=ëÜçêíI=íÜÉ=ÅçãÄáåÉÇ=êÉëéçåëÉ=í~âÉë=íÜÉ=ëÜ~éÉ=çÑ=~=ëâÉïÉÇ=ÇáëíêáÄìíáçå=ÑìåÅíáçåK=tÜÉå=~=ëáÖå~ä=áë=íê~åëÑÉêêÉÇ=íÜêçìÖÜ= n äáåÉ~ê=thëóëíÉãë=éä~ÅÉÇ=áå=ëÉêáÉëI=íÜÉáê= k JçêÇÉê=áãéìäëÉ=êÉëéçåëÉ=ãçãÉåíë=ÅçãÄáåÉ=áå=íÜÉ=ÑçääçïáåÖ=ï~ó=xÉKÖKI=j~~ëI=NVVQzW==n⎧M0, total= ∏M k,i, k = 0;⎪i=1⎨(2.64)cn c⎪M k, total = Mk k,i , k > 0.⎪ ∑⎩i=1=^äíÜçìÖÜ=íÜÉêÉ=áë=åç=ÅäçëÉÇ=ã~íÜÉã~íáÅ~ä=ÉñéêÉëëáçåI=Ñçê=íáãÉ=ëÉêáÉë=~å~äóëáë=éìêéçëÉë=áí=áë=ïÉää=éçëëáÄäÉ=íç=ìëÉ=Ñçê=áåëí~åÅÉ=íÜÉ=ÅçåîÉÅíáçåJÇáëéÉêëáçå=~åÇ=däçîÉêÛë=Éèì~íáçå=~ë=ëÉé~ê~íÉ=ëçäìíáçåë=Ñçê=íÜÉ=ìåë~íìê~íÉÇ=~åÇ=ë~íìê~íÉÇ=òçåÉ=~åÇ=ÅçãÄáåÉ=íÜÉã=åìãÉêáÅ~ääó=îá~=Åçåîçäìíáçå=xëÉÉ=~äëç=_áÉêâÉåë=~åÇ=t~äîççêíI=NVVUX=_áÉêâÉåë=~åÇ=_êçåI=OMMMzK=tÜÉå=ÇçáåÖ=ëçI=ÜçïÉîÉêI=íÜÉ=é~ê~ãÉíÉêë=çÑ=ÄçíÜ=Éèì~íáçåë=éêçîÉ=íç=ÄÉ=ÜáÖÜäó=ÅçêêÉä~íÉÇ=~åÇ=ÄçíÜ=êÉëéçåëÉë=~êÉ=ÅçåëÉèìÉåíäó=ÇáÑÑáÅìäí=íç=ëÉé~ê~íÉ=xhêìáíÜçÑI=OMMNzK=2.3.5.5 Other situations and excitationsqÜÉ=éêÉîáçìë=ëÉÅíáçåë=ëÉêîÉÇ=íç=áääìëíê~íÉ=íÜÉ=ëÜ~éÉë=íÜ~í=êÉëéçåëÉë=çÑ=ÖêçìåÇï~íÉê=ëóëíÉãë=í~âÉ=Ñêçã=~=éÜóëáÅ~ä=éçáåí=çÑ=îáÉïK=^ë=ïÉ=çåäó=ÇáëÅìëëÉÇ=ÉäÉãÉåí~êó=ÖêçìåÇï~íÉê=ëóëíÉãëI=íÜÉ=íÉñí=Å~å=Ü~êÇäó=ëÉêîÉ=~ë=~å=çîÉêîáÉï=çÑ=íÜÉ=éçëëáÄäÉ=î~êá~Äáäáíó=áå=êÉëéçåëÉë=çÑ=ÖêçìåÇï~íÉê=ëóëíÉãëK=cçê=~=íÜçêçìÖÜ=íêÉ~íãÉåí=çå=íÜÉ=ÉñáëíáåÖ=~å~äóíáÅ=ëçäìíáçåë=áå=î~êáçìë=ÖÉçÜóÇêçäçÖáÅ=ëáíì~íáçåëI=ïÉ=êÉÑÉê=íç=x_êìÖÖÉã~åI=NVVVzK=få=íÜÉ=ÅçãáåÖ=ëÉÅíáçåëI=ÜçïÉîÉêI=ïÉ=ïáää=ëíÉé=~ï~ó=Ñêçã=íÜÉ=íê~Çáíáçå~äI=éÜóëáÅ~ä=îáÉïéçáåí=~åÇ=áåíêçÇìÅÉ=íÜÉ=ìëÉ=çÑ=ÄÉÜ~îáçê~ä=êÉëéçåëÉ=ÑìåÅíáçåë=íÜ~í=Ü~îÉ=ÖÉåÉê~ä=ìë~Äáäáíó=Äìí=åç=E~éé~êÉåíF=éÜóëáÅ~ä=ãÉ~åáåÖI=~åÇ=ÇáëÅìëë=áíë=ÅçåëÉèìÉåÅÉëK=^=éçáåí=íÜ~í=êÉã~áåë=ÜÉêÉ=áë=íÜÉ=Ñ~Åí=íÜ~í=ïÉ=çåäó=ÇáëÅìëëÉÇ=íÜÉ=ÉÑÑÉÅíë=çÑ=éêÉÅáéáí~íáçå=~åÇ=Éî~éçê~íáçåK=^ë=íÜÉ=ÇáãÉåëáçå~äáíó=çÑ=~=ëóëíÉã=Å~å=Ü~îÉ=~=éêçåçìåÅÉÇ=ÉÑÑÉÅí=çå=áíë=êÉëéçåëÉI=ëç=ÇçÉë=íÜÉ=ÇáãÉåëáçå~äáíó=çê=ëé~íá~ä=ÇáëíêáÄìíáçå=çÑ=íÜÉ=ÉñÅáí~íáçåëK=få=Åçåíê~ëí=íç=éêÉÅáéáí~íáçå=íÜ~í=Ü~ë=~êÉ~ä=ÅçîÉê~ÖÉI=êáîÉêëI=ÇáíÅÜÉë=~åÇ=çíÜÉê=ëìêÑ~ÅÉ=ï~íÉêë=Å~å=ÄÉ=ëÉÉå=~ë=äáåÉ=ÑÉ~íìêÉë=áå=íÜÉ=Üçêáòçåí~ä=éä~åÉI=~åÇ=éìãéáåÖ=ïÉääë=~ë=éçáåíëK=få=ÑáÖìêÉ=OKORI=ëÅÜÉã~íáò~íáçåë=~êÉ=ÖáîÉå=çÑ=~=éìãéáåÖ=ïÉää=~ÅÅçêÇáåÖ=íç=e~åíìëÜÛ=ïÉää=ÑìåÅíáçå=xe~åíìëÜI=NVRSX=sÉäáåÖ=~åÇ=j~~ëI=OMNMz=~åÇ=~=êáîÉê=çê=Å~å~ä=~ÅÅçêÇáåÖ=íç=íÜÉ=éçäÇÉê=ÑìåÅíáçå=x_êìÖÖÉã~åI=NVVVzK=få=ÄçíÜ=ëÅÜÉã~íáò~íáçåëI=íÜÉêÉ=áë=..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=q( t)ccrKHxxKHfigure 2.25: Schematization of a pumping well according to Hantush (left), and a river according tothe Polder function of Bruggeman (right).=== RV=~å=~èìáÑÉê=ïáíÜ=íê~åëãáëëáîáíó KH ~åÇ=ëíçê~íáîáíó= S =~åÇ=~å=~èìáí~êÇ=ïáíÜ=êÉëáëí~åÅÉ= c I=ïáíÜçìí=ëíçê~ÖÉK=cìêíÜÉêãçêÉI=ÄçíÜ=íÜÉ=éìãéáåÖ=ïÉää=~åÇ=íÜÉ=êáîÉê=Ñìääó=éÉåÉíê~íÉ=íÜÉ=~èìáÑÉêK=^=ÇáÑÑÉêÉåÅÉI=ÜçïÉîÉêI=áë=íÜ~í=áå=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=Ñçê=ïÜáÅÜ=e~åíìëÜÛ=ïÉää=ÑìåÅíáçå=áë=ÇÉêáîÉÇ=~=ê~Çá~ä=ëóããÉíêáÅ=ÜÉ~Ç=Öê~ÇáÉåí=áë=~ëëìãÉÇI=ïÜÉêÉ~ë=áå=íÜÉ=ëáíì~íáçå=çÑ=íÜÉ=éçäÇÉê=ÑìåÅíáçå=íÜÉ=ÜÉ~Ç=Öê~ÇáÉåí=Ü~ë=Äáä~íÉê~ä=ëóããÉíêóK=^ë=~=ëí~åÇ~êÇ=ïÉää=íÉëí=óáÉäÇë=~=ëíÉé=êÉëéçåëÉI=ÜÉêÉ=ïÉ=ëÉÉâ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=çê=ÇÉêáî~íáîÉ=çÑ=e~åíìëÜÛ=ïÉää=ÑìåÅíáçå=ïáíÜ=êÉëéÉÅí=íç=íáãÉK=qÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=ÇÉëÅêáÄÉë=íÜÉ=êÉëéçåëÉ=íç=~å=áåëí~åí~åÉçìë=Éñíê~Åíáçå=çÑ=~=ìåáí=îçäìãÉ=çÑ=ï~íÉê=~åÇ=Éèì~äëW==21 r S tθ ( t) = − exp( − − )(2.65)4πKHt 4KHt cS==få=íÜÉ=éçäÇÉê=ÑìåÅíáçåI=íÜÉ=ÉñÅáí~íáçå=áë=åçí=~=éìãéÉÇ=~ãçìåí=Äìí=~=êáëÉ=áå=ëìêÑ~ÅÉ=ï~íÉê=äÉîÉäK=^äëç=ÜÉêÉI=íÜÉ=éçäÇÉê=ÑìåÅíáçå=áë=~=ëíÉé=êÉëéçåëÉI=ïÜçëÉ=ÇÉêáî~íáîÉ=Éèì~äëW==2θ ( t) = −1x S texp( − − )34πKHt 4KHtcS(2.66)2x S=2.4 Distribution functions as response models2.4.1 The skew-Gaussian convolutional limitsfå=ëí~íáëíáÅë=~åÇ=éêçÄ~Äáäáíó=íÜÉçêóI=~ëëìãéíáçåë=~Äçìí=íÜÉ=éêçÄ~Äáäáíó=ÇáëíêáÄìíáçå=çê=ÚÄÉÜ~îáçêÛ=çÑ=î~êá~ÄäÉë=~êÉ=Åçããçåäó=ã~ÇÉ=ïáíÜçìí=éÜóëáÅ~ä=àìëíáÑáÅ~íáçå=çê=éÜóëáÅ~ä=~å~äóëáë=çÑ=íÜÉ=ëóëíÉã=íç=ïÜáÅÜ=íÜÉó=~êÉ=êÉä~íÉÇK=^å=~ééêçéêá~íÉ=ÇáëíêáÄìíáçå=áë=ëáãéäó=ëÉäÉÅíÉÇ=Ñêçã=~=äáãáíÉÇ=ëÉí=çÑ=Å~åÇáÇ~íÉëI=Ä~ëÉÇ=çå=íÜÉáê=ã~íÅÜ=ïáíÜ=íÜÉ=~î~áä~ÄäÉ=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=EOKSQFFK=j~~ë=~äëç=ëÜçïë=íÜ~í=ÄÉÑçêÉ=êÉ~ÅÜáåÖ=íÜáë=äáãáíI=áãéìäëÉ=êÉëéçåëÉë=ïáää=íÉåÇ=íç=~=ëâÉïJd~ìëëá~å=ëÜ~éÉK=cìêíÜÉêãçêÉI=ÜÉ=ëÜçïë=íÜ~í=íÜÉëÉ=Åçåîçäìíáçå~ä=äáãáíë=~äëç=ÜçäÇ=Ñçê=é~ê~ääÉä=ëÉêáÉë=çÑ=ëóëíÉãë=ïáíÜ=ä~íÉê~ä=áåíÉê~ÅíáçåëK===qÜÉ=ÅÉåíê~ä=äáãáí=íÜÉçêÉã=~åÇ=ëâÉïJd~ìëëá~å=Åçåîçäìíáçå~ä=äáãáí=çÑ=xj~~ëI=NVVQzI=ÜçïÉîÉêI=~êÉ=åçí=ÇáêÉÅíäó=~ééäáÅ~ÄäÉ=íç=íÜÉ=Å~ëÉ=çÑ=ÖêçìåÇï~íÉê=ÜÉ~ÇëK=fÑ=ïÉI=~å~äçÖçìë=íç=j~~ëI=ÅçåëáÇÉê=~=éçêçìë=ãÉÇáìã=~ë=~=ëÉêáÉë=çÑ=áÇÉåíáÅ~ä=ëÉÅíáçåë=çê=ëìÄëóëíÉãëI=ïÉ=Å~å=ÇáëíáåÖìáëÜ=íÜêÉÉ=íóéÉë=çÑ=áåéìíJçìíéìí=êÉä~íáçåëÜáéë=EÑáÖìêÉ=OKOSFK=få=íÜÉ=Ñáêëí=Å~ëÉI=íÜÉ=êÉëéçåëÉ=çÑ=áåÇáîáÇì~ä=ëÉÅíáçåë=áë=áåÇÉéÉåÇÉåí=çÑ=íÜÉ=çíÜÉêëK=qÜáë=áë=~ééêçñáã~íÉäó=íÜÉ=Å~ëÉ=Ñçê=íê~åëéçêí=çÑ=ëçäìíÉë=~åÇ=éÉêÅçä~íáçå=áå=íÜÉ=ìåë~íìê~íÉÇ=òçåÉK=låäó=áå=íÜáë=Å~ëÉI=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=ÅçãÄáåÉÇ=ëóëíÉã=áë=ëáãéäó=~=Åçåîçäìíáçå=çÑ=íÜÉ=ëÉé~ê~íÉ=êÉëéçåëÉë=~åÇ=íÜÉ=Åçåîçäìíáçå~ä=äáãáíë=ÇáêÉÅíäó=~ééäóK=få=íÜÉ=ëÉÅçåÇ=Å~ëÉI=íÜÉ=êÉëéçåëÉ=çÑ=áåÇáîáÇì~ä=ëÉÅíáçåë=ÇÉéÉåÇë=çå=íÜÉ=éêçéÉêíáÉë=çÑ=íÜÉ=ëóëíÉã=áå=íçí~ä=Eçê= n =áå=Å~ëÉ=çÑ=áÇÉåíáÅ~ä=ëÉÅíáçåë=çÑ=ÑáñÉÇ=ëáòÉFI=ïÜáÅÜ=áë=ÖÉåÉê~ääó=íÜÉ=Å~ëÉ=áå=ë~íìê~íÉÇ=Ñäçï=éêçÄäÉãëK=tÜÉåI=Ñçê=Éñ~ãéäÉI=ÜóéçíÜÉíáÅ~ääó=Ú~ÇÇáåÖÛ=ëÉÅíáçåë=íç=~=ëóëíÉã=~í=ïÜçëÉ=ÄçìåÇ~êáÉë=~=Åçåëí~åí=ÜÉ~Ç=ÇáÑÑÉêÉåÅÉ= ∆h =áë=áãéçëÉÇI=íÜÉ=äÉåÖíÜ=∆x =çÑ=íÜÉ=ëóëíÉã=ÅÜ~åÖÉë=~åÇ=íÜÉêÉÑçêÉ=ëç=Çç= ∆ h∆x çê=íÜÉ= M 0,i =çÑ=áåÇáîáÇì~ä=ëÉÅíáçåëK=== SN=h in ( t) ( t) ( t ) ... n ( t)hout( t)h in( t) ( t,n) ( t,n ) ... n ( t,n)hout( t)qin( t)qin( t)hout( t)qin( t)water divide ( t,n) ( )...t,n n( t,n)qout( t)figure 2.26: Three types of input-output relationships, when considering a porous medium as a seriesof identical sections: a) signal h( t ) travels through sections with independent ( t), b) as a), butwith dependent θ ( t, n), c) as b) but with h ( t ) and q ( t ) (partly) at the same location.outinθ=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=SO=^äëçI=éÜóëáÅ~ääó=áÇÉåíáÅ~ä=ëÉÅíáçåë=Çç=åçí=åÉÅÉëë~êáäó=Ü~îÉ=íÜÉ=ë~ãÉ= M 0,i =áå=ëìÅÜ=Å~ëÉëK=^ë=íÜÉ=áåÇáîáÇì~ä=êÉëéçåëÉë=~êÉ=åçí=áåÇÉéÉåÇÉåí=~åÇ=áÇÉåíáÅ~äI=íÜÉ=ÅÉåíê~ä=äáãáí=íÜÉçêÉã=~åÇ=ëâÉïJd~ìëëá~å=äáãáí=~êÉ=åçí=ÇáêÉÅíäó=~ééäáÅ~ÄäÉK=oÉã~êâ~ÄäóI=ÜçïÉîÉêI=áãéìäëÉ=êÉëéçåëÉë=çÑ=ëÉîÉê~ä=ëóëíÉãë=íÜ~í=Ñ~ää=áå=íÜáë=Å~íÉÖçêóI=äáâÉ=íÜÉ=mçäÇÉê=ÑìåÅíáçåI=e~åíìëÜÛ=tÉää=ÑìåÅíáçå=~åÇ=ÅçåîÉÅíáçåJÇáëéÉêëáçå=Éèì~íáçåI=~êÉ=~äëç=çÑ=ëâÉïJd~ìëëá~å=å~íìêÉ=~åÇ=~êÉ=ëéÉÅá~ä=Å~ëÉë=çÑ=çåÉ=ÇáëíêáÄìíáçå=ÑìåÅíáçå=EëÉÉ=ä~íÉê=çåFK=få=íÜÉ=íÜáêÇ=Å~ëÉI= θi( t, n)=ÇÉéÉåÇë=çå=íÜÉ=éêçéÉêíáÉë=çÑ=íÜÉ=ëóëíÉã=áå=íçí~ä=~äëçK=eÉêÉI=ÜçïÉîÉêI=íÜÉ=çìíéìí=ëáÖå~ä=h ( t ) =áë=çÄí~áåÉÇ=Ñêçã=íÜÉ=ë~ãÉ=äçÅ~íáçå=~ë=outïÜÉêÉ=íÜÉ=áåéìí= qin( t)E~í=äÉ~ëí=é~êíäóF=ÉåíÉêë=íÜÉ=ëóëíÉãK=qÜáë=áë=Ñçê=Éñ~ãéäÉ=íÜÉ=Å~ëÉ=ïáíÜ=áåéìíë=íÜ~í=Ü~îÉ=~êÉ~ä=ÅçîÉê~ÖÉI=äáâÉ=êÉÅÜ~êÖÉ= figure 2.27: Cascade of reservoirs asçê=ëÉÉé~ÖÉI=~åÇ=íÜÉáê=ÉÑÑÉÅí=çå=íÜÉ=éÜêÉ~íáÅ= physical analog of the gamma distribution.ÖêçìåÇï~íÉê=äÉîÉäK=_ÉÅ~ìëÉ=áå=íÜáë=Å~ëÉI=íÜÉ=ëáÖå~ä=ÇçÉë=åçí=Ü~îÉ=íç=íê~îÉä=ÑáêëíI=íÜÉêÉ=áë=åç=ÇÉä~ó=áå=íÜÉ=áåáíá~ä=êÉëéçåëÉ= θ (0) =~åÇ=áå=éêáåÅáéäÉ= θ (0) =áë=~äëç=áåÇÉéÉåÇÉåí=çÑ=íÜÉ=ëìêêçìåÇáåÖ=ëóëíÉãK=få=Å~ëÉ=çÑ=êÉÅÜ~êÖÉI=θ (0) =Éèì~äë= 1 S I=áå=äáåÉ=ïáíÜ=Éèì~íáçåë=EOKQSF=~åÇ=EOKRNFK=`çåëÉèìÉåíäóI=áå=ëìÅÜ=Å~ëÉëI=íÜÉ=êÉëéçåëÉ=ÇçÉë=åçí=~íí~áå=~=EëâÉïJFd~ìëëá~å=ëÜ~éÉ=ïáíÜ=áåÅêÉ~ëáåÖ=ëóëíÉã=ëáòÉI=~ë=θ (0) =ÇçÉë=åçí=~ééêç~ÅÜ=òÉêçK==2.4.2 The Pearson type III, scaled gamma and generalized moving Gaussiandistribution^å=Éñ~ãéäÉ=çÑ=~=ÇáëíêáÄìíáçå=ÑìåÅíáçå=íÜ~í=ã~íÅÜÉë=íÜÉ=ëâÉïJd~ìëëá~å=Åçåîçäìíáçå~ä=äáãáíë=çÑ=xj~~ëI=NVVQz=EáKÉK=áí=Å~å=ÄÉ=ëÉÉå=~ë=íÜÉ=çìíéìí=çÑ=~=ëÉêáÉë=çÑ=áåÇÉéÉåÇÉåíI=áÇÉåíáÅ~ä=ëóëíÉãëF=áë=íÜÉ=mÉ~êëçå=íóéÉ=fff=ÇáëíêáÄìíáçå=ÑìåÅíáçåK=qÜáë=ÑìåÅíáçå=áë=~äëç=ìëÉÇ=Äó=gìêó=EïáíÜ= b = 0 F==~åÇ=j~~ë=Ñçê=ãçÇÉäáåÖ=ëçäìíÉ=íê~åëéçêíI=~åÇ=áë=ÖáîÉå=ÄóW=n n−1⎧ − − −a ( t b) exp{ a( t b)}⎪θ( t) = , t ≥ b;⎨Γ( n)⎪⎩θ( t) = 0 , t < b.(2.67)ïÜÉêÉ= a I b =~åÇ= n =~êÉ=é~ê~ãÉíÉêëK=tÜÉå= b =Éèì~äë=òÉêç=Eåç=ÇÉä~óFI=Éèì~íáçå=EOKSTF=êÉÇìÅÉë=íç=ïÜ~í=áë=âåçïå=~ë=íÜÉ=Ö~ãã~=ÇáëíêáÄìíáçåK=qÜÉ=Ö~ãã~=ÇáëíêáÄìíáçå=ÇçÉë=Ü~îÉ=~=éÜóëáÅ~ä=~å~äçÖI=~ë=áí=ÇÉëÅêáÄÉë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=çÑ=~=ëÉêáÉë=çÑ=ÅçìéäÉÇ=äáåÉ~ê=êÉëÉêîçáêë=EÑáÖìêÉ=OKOTFI=~äëç=âåçïå=~ë=íÜÉ=k~ëÜ=Å~ëÅ~ÇÉ=áå=ëìêÑ~ÅÉ=ï~íÉê=..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=Aaresponse factor (−)A = 80A = 40A = 20A = 10Aa = 1Aa = 0.5Aa = 0.25Aa = 0.125n = 2n = 1.5n = 1n = 0.50 50 100 150 200time (days)0 50 100 150 200time (days)0 50 100 150 200time (days)=figure 2.28: Example curves of the scaled gamma distribution, illustrating its variability. In a) thedrainage resistance ( Aa = 1, n = 1), in b) the ‘storage coefficient’ ( a = 0.01, n = 1) and in c) the‘centrality’ of the location and/or ‘influence’ of the unsaturated zone ( A = 400 ⋅ n, a = 0.01 ) werevaried.=ÜóÇêçäçÖó=xk~ëÜI=NVRUzK=få=íÜ~í=êÉëéÉÅíI=íÜÉ=é~ê~ãÉíÉê= n ÇÉåçíÉë=íÜÉáê=Eåçí=åÉÅÉëë~êáäó=áåíÉÖÉêF=åìãÄÉê=~åÇ= a =Éèì~äë=íÜÉ=áåîÉêëÉ=çÑ=íÜÉ=êÉëÉêîçáê=ÅçÉÑÑáÅáÉåíK==få=Å~ëÉ=çÑ=ÖêçìåÇï~íÉê=äÉîÉä=êÉëéçåëÉëI=íÜÉ=êÉëéçåëÉ=ÇçÉë=åçí=ÇÉëÅêáÄÉ=íÜÉ=íê~åëÑçêã~íáçå=çÑ=çåÉ=Ñäìñ=áåíç=~åçíÜÉêI=Äìí=íÜÉ=íê~åëÑçêã~íáçå=çÑ=~=äÉîÉä=çê=Ñäìñ=áåíç=~=äÉîÉäK=få=íÜ~í=Å~ëÉI=~=ÅçåëÉêî~íáçå=ä~ï=ÇçÉë=åçí=åÉÅÉëë~êáäó=~ééäó=~åÇ=EOKSTF=Ü~ë=íç=ÄÉ=ãìäíáéäáÉÇ=ïáíÜ=~=Ñ~Åíçê A =áå=çêÇÉê=íç=~ääçï=íÜÉ=~êÉ~=íç=ÇáÑÑÉê=Ñêçã=çåÉK=få=íÜÉ=~ÄëÉåÅÉ=çÑ=~=ÇÉä~ó= b I=íÜÉ=êÉëìäí=áë=ÖáîÉå=ÄóW==n n−⎧ a t1 exp( −at)⎪θ( t) = A, t ≥ 0;⎨Γ( n)(2.68)⎪⎩θ( t) = 0 , t < 0.=få=íÜÉ=ÑçääçïáåÖI=ïÉ=ëÜ~ää=êÉÑÉê=íç=EOKSUF=~ë=íÜÉ=ëÅ~äÉÇ=Ö~ãã~=ÇáëíêáÄìíáçå=ÑìåÅíáçå=Epd=ÇÑFK=få=ÑáÖìêÉ=OKOUI=Éñ~ãéäÉ=ÅìêîÉë=çÑ=íÜÉ=pd=ÇÑ=~êÉ=éäçííÉÇ=Ñçê=ÇáÑÑÉêÉåí=î~äìÉë=çÑ=A,a =~åÇ= n I=íç=áääìëíê~íÉ=áíë=ÑäÉñáÄáäáíó=~åÇ=íÜÉ=ÉÑÑÉÅíë=çÑ=íÜÉ=ÇáÑÑÉêÉåí=é~ê~ãÉíÉêëK=tÜÉå=ìëÉÇ=Ñçê=ãçÇÉäáåÖ=ÖêçìåÇï~íÉê=ÜÉ~Ç=ëÉêáÉëI=íÜÉ=é~ê~ãÉíÉêë=áå=éêáåÅáéäÉ=ãÉêÉäó=ÇÉÑáåÉ=íÜÉ=ëÜ~éÉ=çÑ=íÜÉ=pd=ÇÑ=~åÇ=Ü~îÉ=åç=ÇáêÉÅí=éÜóëáÅ~ä=ãÉ~åáåÖK==få=Å~ëÉ=çÑ=íÜÉ=êÉëéçåëÉ=íç=éêÉÅáéáí~íáçå=~åÇ=Éî~éçê~íáçåI=ÜçïÉîÉêI=íÜÉ=é~ê~ãÉíÉêë=Å~å=ÄÉ=êÉä~íÉÇ=íç=ÖÉçÜóÇêçäçÖáÅ=é~ê~ãÉíÉêë=áå=ÖÉåÉê~ä=íÉêãëI=áå=íÜÉ=ÑçääçïáåÖ=ï~óI=íç=áääìëíê~íÉ=íÜÉ=ê~åÖÉ=çÑ=éÜóëáÅ~ä=î~êá~Äáäáíó=íÜÉ=pd=ÇÑ=Å~å=ÅçîÉêW==A =Ó= =Éèì~äë=íÜÉ=~êÉ~=çÑ=íÜÉ=pd=ÇÑK=fí=~äëç=áë=íÜÉ=ê~íáç=çÑ=íÜÉ=ãÉ~å=ÅçåîÉñáíó=çÑ=íÜÉ=ÖêçìåÇï~íÉê=ÜÉ~Ç=~ÄçîÉ=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉ=íç=íÜÉ=ãÉ~å=ÖêçìåÇï~íÉê=êÉÅÜ~êÖÉK= A =íÜÉêÉÑçêÉ=Äó=ÇÉÑáåáíáçå=Éèì~äë=íÜÉ=EäçÅ~äF=Çê~áå~ÖÉ=êÉëáëí~åÅÉK=táíÜ=áåÅêÉ~ëáåÖ=Çê~áå~ÖÉ=êÉëáëí~åÅÉI=íÜÉ=ÇÉÅ~ó=ê~íÉ= a =çÑ=íÜÉ=êÉëéçåëÉ=ÇÉÅêÉ~ëÉë=E a ≈1/A FK=få=ëáãéäÉ=ïçêÇëI=íÜÉ=ï~íÉê=í~ÄäÉ=ïáää=Çêçé=ëäçïäó=ïÜÉå=íÜÉ=Çê~áå~ÖÉ=êÉëáëí~åÅÉ=áë=ÜáÖÜK==== SP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó=ÅìêîÉI=áë=ÇáëÅìëëÉÇ=~åÇ=Åçãé~êÉÇ=íç=Éñé~åëáçåë=Ä~ëÉÇ=çå=íÜÉ=éìêÉ=d~ìëëá~å=ÇáëíêáÄìíáçåK=däçîÉêÛë=Éèì~íáçå=EOKRNFI=ÜçïÉîÉêI=áë=åçí=~å=Éñ~ÅíI=ëéÉÅá~ä=Å~ëÉ=çÑ=EOKSVFI=áå=äáåÉ=ïáíÜ=íÜÉ=Ñ~Åí=íÜ~í=~ë=ÇáëÅìëëÉÇ=ÄÉÑçêÉI=ÜÉêÉ=íÜÉ=áåáíá~ä=êÉëéçåëÉ=áë=ÑáñÉÇ=íç=íÜÉ=áåîÉêëÉ=çÑ=íÜÉ=ëíçê~íáîáíóK=mÉêÜ~éëI=áå=Å~ëÉ=çÑ=ëÜ~ääçï=ï~íÉê=í~ÄäÉëI=EOKSUF=~åÇ=EOKSVF=ã~ó=íÜÉêÉÑçêÉ=åçí=ÄÉ=çéíáã~ä=Ñçê=ãçÇÉäáåÖ=íÜÉ=É~êäó=íáãÉ=ÉÑÑÉÅíë=çÑ=ÉKÖKI=éêÉÅáéáí~íáçå=ìëáåÖ=ÜáÖÜ=ÑêÉèìÉåÅó=Ç~í~K=få=xsÉäáåÖI=OMNMz=~åÇ=ÉäëÉïÜÉêÉI=ÇáëíêáÄìíáçå=ÑìåÅíáçåë=äáâÉ=EOKSUF=çê=EOKSVF=~êÉ=êÉÑÉêêÉÇ=íç=~ë=~ééêçñáã~íÉK=qÜÉ=íÉêã=~ééêçñáã~íÉI=ÜçïÉîÉêI=ëÉÉãë=åçí=ÅçãéäÉíÉäó=~ÇÉèì~íÉI=~ë=EOKSVF=Éñ~Åíäó=ã~íÅÜÉë=íÜÉ=éÜóëáÅ~ääóJÄ~ëÉÇ=ëçäìíáçå=Ñçê=ëçãÉ=ëÅÜÉã~íáò~íáçåëI=ïÜÉêÉ~ë=Ñçê=çíÜÉêëI=áí=áë=~ééêçñáã~íÉK=kÉñí=íç=íÜ~íI=çìê=êÉÑÉêÉåÅÉ=áë=åçí=íÜÉ=Éñ~Åí=ëçäìíáçå=íç=ëçãÉ=~ééêçñáã~íÉ=ëÅÜÉã~íáò~íáçåI=Äìí=êÉ~äáíó=áíëÉäÑI=ïÜáÅÜ=~åó=ãçÇÉä=ïáää=~äï~óë=çåäó=~ééêçñáã~íÉK=få=éêáåÅáéäÉI=~å=Ú~ééêçñáã~íÉÛ=êÉëéçåëÉ=ÑìåÅíáçå=ã~ó=ÄÉííÉê=Ñáí=íÜÉ=êÉ~ä=ïçêäÇ=ÄÉÜ~îáçê=íÜ~å=~å=ÚÉñ~ÅíÛI=éÜóëáÅ~ääóJÄ~ëÉÇ=ëçäìíáçåK=rëÉ=çÑ=íÜÉ=íÉêã=ÄÉÜ~îáçêI=çå=íÜÉ=çíÜÉê=Ü~åÇI=ëíêÉëëÉë=íÜ~í=íÜÉ=ëé~íá~ä=ëíêìÅíìêÉ=çÑ=íÜÉ=ëóëíÉã=ìåÇÉê=ÅçåëáÇÉê~íáçå=ÇçÉë=åçí=Ü~îÉ=íç=ÄÉ=ÉñéäáÅáíäó=ÇÉÑáåÉÇ=íç=ãçÇÉä=áíë=Çóå~ãáÅëK=qÜÉ=íÉêã=ÄÉÜ~îáçê=áë=~äëç=ìëÉÇ=áå=ëìÅÜ=~=ëÉåëÉ=áå=íÜÉ=ÅçåíÉñí=çÑ=ëóëíÉãë=íÜÉçêóI=ëóëíÉã=áÇÉåíáÑáÅ~íáçå=~åÇ=ÉèìáÑáå~äáíó=xÉKÖKI=tÜáíÉÜÉ~Ç=~åÇ=vçìåÖI=NVTVX=mçäÇÉêã~å=~åÇ=táääÉãëI=NVVUX=_ÉîÉåI=OMMSzK==== SR=2.4.3 Matching temporal moments with spatial models^=ÇáêÉÅí=ÅçåëÉèìÉåÅÉ=çÑ=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=ëé~íá~ä=ëíêìÅíìêÉ=çÑ=~=ëóëíÉã=ÇçÉë=åçí=Ü~îÉ=íç=ÄÉ=ÇÉÑáåÉÇ=áå=~=íáãÉ=ëÉêáÉë=ãçÇÉä=EïÜáÅÜ=áë=Åçãé~ê~ÄäÉ=íç=íÜÉ=ëí~íÉãÉåí=ã~ÇÉ=Äó=xgìêó=~åÇ=oçíÜI=NVVMzI=ÇáëÅìëëÉÇ=áå=ëÉÅíáçå=OKNKPFI=áë=íÜ~í=íáãÉ=ëÉêáÉë=ãçÇÉäë=Å~å=åçí=ÄÉ=ìëÉÇ=íç=ã~âÉ=éêÉÇáÅíáçåë=~Äçìí=äçÅ~íáçåë=çíÜÉê=íÜ~å=íÜçëÉ=çÄëÉêîÉÇK=qÜáë=çÑ=ÅçìêëÉ=áë=~å=áãéçêí~åí=äáãáí~íáçåK=eçïÉîÉêI=áå=íÜÉ=ÑçääçïáåÖ=ïÉ=ëÜ~ää=ëÉÉ=íÜ~í=íÜÉ=çìíÅçãÉë=çÑ=íáãÉ=ëÉêáÉë=ãçÇÉäë=Å~å=ÄÉ=ã~íÅÜÉÇ=ïáíÜ=íÜçëÉ=çÑ=ëé~íá~ä=ãçÇÉäë=Äó=íê~åëÑçêãáåÖ=çêÇáå~êó=ëé~íáçíÉãéçê~ä=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåë=áåíç=ãçãÉåíJÖÉåÉê~íáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåëK=qÜÉ=ÅçåÅÉéí=çÑ=ãçãÉåíJÖÉåÉê~íáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåë=Ü~ë=ÄÉÉå=áåíêçÇìÅÉÇ=íç=ëìÄëìêÑ~ÅÉ=ÜóÇêçäçÖó=Ñáêëí=Äó=xe~êîÉó=~åÇ=dçêÉäáÅâI=NVVRzI=ïÜç=ìëÉÇ=áí=Ñçê=~å~äóòáåÖ=ëçäìíÉ=íê~åëéçêíI=~åÇ=áåÇÉéÉåÇÉåíäó=ÑçìåÇ=Äó=xj~~ëI=NVVRzK=^å=Éä~Äçê~íÉ=íêÉ~íãÉåí=çÑ=ìëáåÖ=ãçãÉåíë=áå=íÜÉ=ÑáÉäÇ=çÑ=ëçäìíÉ=íê~åëéçêí=Å~å=ÄÉ=ÑçìåÇ=áå=xdçîáåÇ~ê~àì=~åÇ=a~ëI=OMMTzK=xiá=Éí=~äKI=OMMRz=ìëÉÇ=ãçãÉåíë=íç=ÅÜ~ê~ÅíÉêáòÉ=íÜÉ=Çê~ïÇçïå=áå=éìãéáåÖ=íÉëíëK=qÜÉ=ìëÉ=çÑ=ãçãÉåíJÖÉåÉê~íáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåë=~åÇ=ãçãÉåí=ã~íÅÜáåÖ=Ñçê=ÖêçìåÇï~íÉê=ÜÉ~ÇëI=çê=ëóëíÉãë=ïáíÜ=ãìäíáéäÉ=ÉñÅáí~íáçåë=áå=ÖÉåÉê~äI=êÉèìáêÉë=íÜÉ=ìëÉ=çÑ=~=íáãÉ=ëÉêáÉë=ãçÇÉä=íç=Éëíáã~íÉ=~åÇ=ëÉé~ê~íÉ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåëK=qÜáë=ï~ë=éêçéçëÉÇ=Äó=xi~åâÉëíÉê=~åÇ=j~~ëI=NVVSz=~åÇ=xsçå=^ëãìíÜ=~åÇ=j~~ëI=OMMNzK=^Åíì~ä=éê~ÅíáÅ~ä=íÉëíáåÖ=çÑ=íÜÉ=ãÉíÜçÇ=ï~ë=éÉêÑçêãÉÇ=áå=x_~ââÉê=Éí=~äKI=OMMUzI=ïÜÉêÉ~ë=áå=x_~ââÉê=Éí=~äKI=OMMTz=ãçãÉåí=ã~íÅÜáåÖ=ï~ë=ìëÉÇ=Ñçê=íê~åëáÉåí=ãçÇÉäáåÖ=çÑ=ÖêçìåÇï~íÉê=ÜÉ~Çë=ìëáåÖ=~å~äóíáÅ=ÉäÉãÉåíëK===få=ëÉÅíáçå=OKPKRKOI=ïÉ=Ü~îÉ=ÇÉêáîÉÇ=~=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=Ñçê=íÜÉ=Å~ëÉ=çÑ=çåÉ=ÇáãÉåëáçå~äI=Üçêáòçåí~ä=ÑäçïK=^ééäáÅ~íáçå=çÑ=íÜÉ=ë~ãÉ=éêçÅÉÇìêÉ=íç=~=ëóëíÉã=ïáíÜ=íïç=ÇáãÉåëáçå~ä=Ñäçï=~åÇ=~=ÜçãçÖÉåÉçìëI=áëçíêçéáÅ=~èìáÑÉê=óáÉäÇë=íÜÉ=ÑçääçïáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåW=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=O=SS==figure 2.29: East to west cross-section of the dune area near Wijk aan Zee (The Netherlands),showing the surface level and the course of M0 [Van de Vliet & Boekelman, 1998].=2 ∂hKH∇ h = S − r(2.70)∂t=2 22∂ ∂ïÜÉêÉ= ∇ áë=íÜÉ=i~éä~Åá~å= + K=fÑ=ïÉ=êÉéä~ÅÉ=íÜÉ=êÉÅÜ~êÖÉ=áå=EOKTMF=Äó=~=aáê~Å=2 2∂x∂yÇÉäí~=ÑìåÅíáçåI=íÜÉ=ÜÉ~Ç=Äó=ÇÉÑáåáíáçå=Éèì~äë=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=θ W==2 ∂θ( t)KH∇ θ ( t) = S − δ( t)(2.71)∂t=thfå=çêÇÉê=íç=çÄí~áå=íÜÉ= n J=íÉãéçê~ä=ãçãÉåí=çÑ=θ I=ÄçíÜ=ëáÇÉë=çÑ=Éèì~íáçå=EOKTNF=~êÉ=nãìäíáéäáÉÇ=ïáíÜ= t ~åÇ=áåíÉÖê~íÉÇ=ïáíÜ=êÉëéÉÅí=íç=íáãÉ=Ñêçã= −∞ =íç= +∞ I=ïÜáÅÜ=ÖáîÉëW==∞ ∞ 2 n n S ∂θ( t)P∞0 n∫ ∫ ∫(2.72)∇ t θ ( t)dt = t dt − t δ( t)dtKH ∂t KH−∞ −∞ −∞=qÜáë=óáÉäÇëI=ìëáåÖ=íÜÉ=ÇÉÑáåáíáçå=çÑ=íÉãéçê~ä=ãçãÉåíë=ÖáîÉå=áå=EOKPVFW=..…………………………………………………………………………………………….….

_~ÅâÖêçìåÇ=~åÇ=íÜÉçêó==⎧ 2 1∇ M0 = − , n = 0⎪ KH⎨(2.73)⎪ 2 S∇ Mn = − Mn−1, n > 0⎪⎩ KH=kçíÉ=íÜ~í=íÜÉ=Ñ~Åíçê=íáãÉ=áë=åçí=éêÉëÉåí=áå=Éèì~íáçå=EOKTPFK=cçê= n = 0 I=Éèì~íáçå=EOKTPF=áë=ã~íÜÉã~íáÅ~ääó=áÇÉåíáÅ~ä=íç=íÜÉ=Éèì~íáçå=çÑ=ëí~íáçå~êó=ÖêçìåÇï~íÉê=Ñäçï=~åÇ=Å~å=ÄÉ=ëçäîÉÇ=ìëáåÖ=~å=çêÇáå~êó=ÖêçìåÇï~íÉê=ãçÇÉä=ìëáåÖ=íÜÉ=ÑáåáíÉ=ÇáÑÑÉêÉåÅÉI=ÑáåáíÉ=ÉäÉãÉåí=çê=~å~äóíáÅ=ÉäÉãÉåí=ãÉíÜçÇK=cçê=ÜáÖÜÉê= n I= Mn=Å~å=ÄÉ=ëçäîÉÇ=ìëáåÖ=íÜÉ=ãçãÉåíë=çÑ=çêÇÉê= n − 1=~ë=áåéìíK=xs~å=ÇÉ=säáÉí=~åÇ=_çÉâÉäã~åI=NVVUz=ìëÉÇ=íÜÉ=ÑáåáíÉ=ÉäÉãÉåí=ÅçÇÉ=qoft^`l=íç=çÄí~áå= M0=~åÇ= M1=Ñêçã=~=ÇìåÉ=~êÉ~=áå=qÜÉ=kÉíÜÉêä~åÇëK=få==ÑáÖìêÉ=OKOVI=~å=É~ëí=íç=ïÉëí=ÅêçëëJëÉÅíáçå=çÑ=íÜÉ=ãçÇÉäÉÇ=ÇìåÉ=~êÉ~=áë=ÖáîÉåK=qÜÉ=î~äìÉë=Ñçê= M0=çÄí~áåÉÇ=Ñêçã=qoft^`l=éêçîÉÇ=íç=ÄÉ=ÜáÖÜäó=Åçãé~ê~ÄäÉ=Eãçëí=ÇáÑÑÉêÉåÅÉë=ïÉêÉ=äÉëë=íÜ~å=RBF=íç=íÜÉ=î~äìÉë=çÄí~áåÉÇ=Ñêçã=~å~äóëáë=çÑ=íáãÉ=ëÉêáÉë=ÖÉåÉê~íÉÇ=Äó=íÜÉ=ë~ãÉ=ÖêçìåÇï~íÉê=ãçÇÉäK=qÜÉ=êÉëìäíë=Ñçê= M1=ïÉêÉ=åçí=Åçãé~ê~ÄäÉI=éÉêÜ~éë=ÄÉÅ~ìëÉ=íÜÉ=êÉí~êÇ~íáçå=áå=íÜÉ=ìåë~íìê~íÉÇ=òçåÉ=ï~ë=åçí=~ÇÉèì~íÉäó=íêÉ~íÉÇK=^å=~Çî~åí~ÖÉ=çÑ=íÜÉ=~å~äóíáÅ=ÉäÉãÉåí=ãÉíÜçÇ=áë=íÜ~í=áíë=Åçåíáåìçìë=ëé~íá~ä=ã~íÜÉã~íáÅë=ÅçãÄáåÉ=ÉäÉÖ~åíäó=ïáíÜ=íÜÉ=Åçåíáåìçìë=íáãÉ=mfocf`q=ãÉíÜçÇ=çÑ=íáãÉ==== ST=figure 2.30: Color image of 0M created with the analytic element method [Bakker et al., 2007] in anarea with shallow water tables and a detailed drainage pattern (created with Tim ML by Frans Schaars,Artesia (www.artesia-water.nl)KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK=

`Ü~éíÉê=O=ëÉêáÉë=~å~äóëáë=ÇÉîÉäçéÉÇ=áå=íÜáë=íÜÉëáëK=^ë=áí=áë=~=ÖêáÇäÉëë=ãÉíÜçÇI=íÜÉ=~å~äóíáÅ=ÉäÉãÉåí=ãÉíÜçÇ=~ääçïë=Ñçê=íÜÉ=éêÉÅáëÉ=éä~ÅÉãÉåí=çÑ=ÇáíÅÜÉëI=ëíêÉ~ãë=~åÇ=éìãéáåÖ=ïÉääëK=cìêíÜÉêãçêÉI=ãçãÉåíë=Å~å=ÄÉ=ÅçãéìíÉÇ=~å~äóíáÅ~ääó=~í=íÜÉ=Éñ~Åí=äçÅ~íáçå=çÑ=~å=çÄëÉêî~íáçå=ïÉääI=ïÜáÅÜ=ÅáêÅìãîÉåíë=ëÅ~äÉ=~åÇ=ÇáëÅêÉíáò~íáçå=éêçÄäÉãëK=qÜÉ=éçïÉê=çÑ=íÜÉ=~å~äóíáÅ=ÉäÉãÉåí=ãÉíÜçÇ=ÄÉÅçãÉë=ÉëéÉÅá~ääó=ÅäÉ~ê=áå=~êÉ~ë=ïáíÜ=ëÜ~ääçï=ï~íÉê=í~ÄäÉë=~åÇ=~=ÇÉí~áäÉÇ=Çê~áå~ÖÉ=é~ííÉêåI=~ë=áë=íÜÉ=ëáíì~íáçå=áå=ä~êÖÉ=é~êíë=çÑ=íÜÉ=kÉíÜÉêä~åÇë=EÑáÖìêÉ=OKPMFKSU=..…………………………………………………………………………………………….….

3rëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=Chapter=== SV=Adopted from:Von Asmuth, J.R., M.F.P. Bierkens and K. Maas (2002)Transfer function noise modeling in continuous time usingpredefined impulse response functions.Water Resources Research, 38(12), 23_1-23_12.3 Time series modeling using=================^ÇçéíÉÇ=ÑêçãW=continuous response functionsTime series modeling usingcontinuous response functionsReproduced by permission of American Geophysical Unioncopyright 2002 American Geophysical Union.Von Asmuth, J. R., M. F. P. Bierkens, and K. Maas (2002), Transfer function noise modelingin continuous time using predefined impulse response functions, t~íÉê=oÉëçìêÅÉë=oÉëÉ~êÅÜ, 38(12), 23_1-23_12. Reproduced by permission of American Geophysical Union,copyright © 2002 American Geophysical Union.=Abstract: Abstract=få=íÜáë=ÅÜ~éíÉêI=~=ãÉíÜçÇ=çÑ=íê~åëÑÉê=ÑìåÅíáçåJåçáëÉ=EqckF=ãçÇÉäáåÖ=áë=éêÉëÉåíÉÇ=íÜ~í=çéÉê~íÉë=áå=Åçåíáåìçìë=íáãÉ=~åÇ=ìëÉë=éêÉÇÉÑáåÉÇ=áãéìäëÉ=êÉëéçåëÉ=EfoF=In this chapter, a method of transfer function-noise (TFN) modeling is presented thatÑìåÅíáçåëK=qÜÉ=êÉëìäíáåÖ=Åä~ëë=çÑ=ãçÇÉäë=áë=êÉÑÉêêÉÇ=íç=~ë=mfocf`q=EmêÉÇÉÑáåÉÇ=fo=operates in continuous time and uses predefined impulse response (IR) functions. ThecìåÅíáçå=få=`çåíáåìçìë=qáãÉFK=fí=éêçîáÇÉë=~=ìëÉÑìä=íççä=Ñçê=ëí~åÇ~êÇáòÉÇ=~å~äóëáë=çÑ=resulting class of models is referred to as PIRFICT (Predefined IR Function In ContinuousíáãÉ=ëÉêáÉëI=~ë=áí=Å~å=ÄÉ=Å~äáÄê~íÉÇ=ìëáåÖ=áêêÉÖìä~êäó=ëé~ÅÉÇ=Ç~í~=~åÇ=ÇçÉë=åçí=êÉèìáêÉ=~=Time). It provides a useful tool for standardized analysis of time series, as it can beãçÇÉä=áÇÉåíáÑáÅ~íáçå=éÜ~ëÉ=éêáçê=íç=Å~äáÄê~íáçåK=få=íÜÉ=ãÉíÜçÇçäçÖáÅ~ä=ëÉÅíáçåI=íÜÉ=calibrated using irregularly spaced data and does not require a model identification phaseÇáëÅêÉíÉ=ôj^=E^ìíçoÉÖêÉëëáîÉJjçîáåÖ=^îÉê~ÖÉFJíóéÉ=qck=ãçÇÉä=çÑ=_çñ=~åÇ=gÉåâáåë=prior to calibration. In the methodological section, the discrete ARMA (AutoRegressivexNVTMz=áë=éêÉëÉåíÉÇ=~åÇ=íê~åëÑçêãÉÇ=áåíç=Åçåíáåìçìë=íáãÉ=íç=çÄí~áå=íÜÉ=mfocf`q=ãçÇÉäK=Moving Average)-type TFN model of Box and Jenkins [1970] is presented and transformedqÜÉ=ÇáëÅêÉíÉ=ôj^=íê~åëÑÉê=ÑìåÅíáçåI=ïÜáÅÜ=áë=ã~ÇÉ=ìé=çÑ=~=î~êá~ÄäÉ=åìãÄÉê=çÑ=into continuous time to obtain the PIRFICT model. The discrete ARMA transfer function,é~ê~ãÉíÉêëI=áë=êÉéä~ÅÉÇ=Äó=~=ëáãéäÉ=~å~äóíáÅ=ÉñéêÉëëáçå=íÜ~í=ÇÉÑáåÉë=íÜÉ=fo=ÑìåÅíáçåK=which is made up of a variable number of parameters, is replaced by a simple analyticcêçã=íÜÉ=fo=ÑìåÅíáçåI=ÄäçÅâ=êÉëéçåëÉ=ÑìåÅíáçåë=~êÉ=ÇÉêáîÉÇ=íÜ~í=Éå~ÄäÉ=íÜÉ=ãçÇÉä=íç=expression that defines the IR function. From the IR function, block response functions areÜ~åÇäÉ=áêêÉÖìä~êäó=ëé~ÅÉÇ=Ç~í~K=få=íÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçåI=íÜÉ=é~ê~ãÉíÉê=Éëíáã~íÉë=derived that enable the model to handle irregularly spaced data. In the example~åÇ=éÉêÑçêã~åÅÉ=çÑ=íÜÉ=ôj^=~åÇ=mfocf`q=ãçÇÉä=~êÉ=Åçãé~êÉÇ=ìëáåÖ=~=Ç~í~=ëÉí=çÑ=application, the parameter estimates and performance of the ARMA and PIRFICT modelNR=éáÉòçãÉíÉêë=~åÇ=~=ëáãìä~íÉÇ=ëÉêáÉëK=fí=ï~ë=ÑçìåÇ=íÜ~í=íÜÉ=Éëíáã~íÉÇ=íê~åëÑÉê=~åÇ=are compared using a data set of 15 piezometers and a simulated series. It was found that_o=ÑìåÅíáçåë=çÑ=ÄçíÜ=ãçÇÉäë=Ñçääçï=íÜÉ=ë~ãÉ=ÖÉåÉê~ä=é~ííÉêåI=~äíÜçìÖÜ=íÜÉ=ôj^=the estimated transfer and BR functions of both models follow the same general pattern,íê~åëÑÉê=ÑìåÅíáçåë=~êÉ=é~êíäó=áêêÉÖìä~êK=qÜÉ=éÉêÑçêã~åÅÉ=çÑ=ÄçíÜ=ãçÇÉäë=éêçîÉë=íç=ÄÉ=although the ARMA transfer functions are partly irregular. The performance of bothÜáÖÜäó=Åçãé~ê~ÄäÉ=Ñçê=~ää=éáÉòçãÉíÉêëK=models proves to be highly comparable for all piezometers.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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`Ü~éíÉê=P=TO==få=íÜáë=é~éÉê=ïÉ=éêÉëÉåí=~=ãÉíÜçÇ=çÑ=íê~åëÑÉê=ÑìåÅíáçåJåçáëÉ=ãçÇÉäáåÖ=áå=Åçåíáåìçìë=íáãÉI=ïÜáÅÜ=ìëÉë=éêÉÇÉÑáåÉÇ=áãéìäëÉ=êÉëéçåëÉ=EfoF=ÑìåÅíáçåëK=qÜÉ=êÉëìäíáåÖ=Åä~ëë=çÑ=qck=ãçÇÉäë=áë=êÉÑÉêêÉÇ=íç=~ë=mfocf`q=ãçÇÉäë=Emfocf`q=ëí~åÇë=Ñçê=mêÉÇÉÑáåÉÇ=fo=cìåÅíáçå=få=`çåíáåìçìë=qáãÉFI=~åÇ=ÅáêÅìãîÉåíë=~=åìãÄÉê=çÑ=äáãáí~íáçåë=çÑ=ÇáëÅêÉíÉ=qck=ãçÇÉäë=äáåâÉÇ=íç=íáãÉ=ÇáëÅêÉíáò~íáçå=~åÇ=ãçÇÉä=áÇÉåíáÑáÅ~íáçåK=qÜÉ=é~éÉê=áë=çêÖ~åáòÉÇ=~ë=ÑçääçïëK=cáêëíI=íÜÉ=íÜÉçêó=~åÇ=Ä~ëáÅ=Éèì~íáçåë=çÑ=ÇáëÅêÉíÉ=qck=ãçÇÉäë=~êÉ=ÖáîÉå=~åÇ=ëìÄëÉèìÉåíäó=íê~åëÑçêãÉÇ=áåíç=Åçåíáåìçìë=íáãÉ=íç=çÄí~áå=íÜÉ=mfocf`q=ãçÇÉäK=kÉñíI=íÜÉ=áãéäÉãÉåí~íáçå=çÑ=íÜÉ=mfocf`q=ãçÇÉä=áë=ÉäìÅáÇ~íÉÇ=ëíÉéJÄóJëíÉéI=~åÇ=áë=ëìãã~êáòÉÇ=~í=íÜÉ=ÉåÇ=çÑ=íÜÉ=ãÉíÜçÇçäçÖáÅ~ä=ëÉÅíáçåK=få=íÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçåI=íÜÉ=éÉêÑçêã~åÅÉ=çÑ=íÜÉ=mfocf`q=ãçÇÉä=áë=Åçãé~êÉÇ=íç=íÜ~í=çÑ=íê~Çáíáçå~ä=ôj^JíóéÉ=qck=ãçÇÉäë=~åÇ=Ñáå~ääóI=ÇáëÅìëëáçå=~åÇ=ÅçåÅäìëáçåë=~êÉ=ÖáîÉåK=cçê=êÉ~ëçåë=çÑ=ëáãéäáÅáíóI=íÜÉ=íÜÉçêó=áë=ÇÉîÉäçéÉÇ=~åÇ=áääìëíê~íÉÇ=ìëáåÖ=~=ëáåÖäÉ=áåéìí=qck=ãçÇÉä=ïáíÜ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=~ë=çìíéìí=ëÉêáÉë=~åÇ=éêÉÅáéáí~íáçå=ëìêéäìë=~ë=áåéìí=ëÉêáÉëI=Äìí=íÜÉ=Éèì~íáçåë=Å~å=ÄÉ=êÉ~Çáäó=ÉñíÉåÇÉÇ=íç=áåÅäìÇÉ=ãìäíáéäÉ=áåéìí=ëÉêáÉëK==3.2 Methods and theory3.2.1 The discrete ARMA TFN modelcçê=äáåÉ~êI=ìåÇáëíìêÄÉÇ=éÜêÉ~íáÅ=ëóëíÉãë=íÜ~í=~êÉ=áåÑäìÉåÅÉÇ=Äó=éêÉÅáéáí~íáçå=ëìêéäìë=çåäóI=íÜÉ=ÑçääçïáåÖ=ëáåÖäÉ=áåéìí=ÇáëÅêÉíÉ=qck=ãçÇÉä=Å~å=ÄÉ=ìëÉÇ=íç=ãçÇÉä=ÖêçìåÇï~íÉê=äÉîÉä=ÑäìÅíì~íáçåëW==h = hˆ+ n + d(3.1)t t t∞hˆ= Θ(B)p = ∑Θ p(3.2)t t i t−ii=0∞n = Φ(B)a = ∑Φ a(3.3)t t i t−ii=0=ïÜÉêÉ=t = W=ÇáëÅêÉíÉ=íáãÉ=ëíÉé=E t ∈ N I=xJzF=h = W=çÄëÉêîÉÇ=ÖêçìåÇï~íÉê=äÉîÉäI=êÉä~íáîÉ=íç=ëçãÉ=êÉÑÉêÉåÅÉ=äÉîÉä=xiz=ĥ = W=éêÉÇáÅíÉÇ=ÖêçìåÇï~íÉê=äÉîÉäI=~ííêáÄìí~ÄäÉ=íç= p ~åÇ=êÉä~íáîÉ=íç= d xiz=p = W=éêÉÅáéáí~íáçå=ëìêéäìë=xiz=d = W=äÉîÉä=çÑ= h ïáíÜçìí=éêÉÅáéáí~íáçåI=çê=áå=çíÜÉê=ïçêÇë=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=äÉîÉä=xiz=n = W=êÉëáÇì~ä=ëÉêáÉë=xiz=Θ = W=ÇÉíÉêãáåáëíáÅ=íê~åëÑÉê=ÑìåÅíáçå=xJz=B = W=Ä~Åâï~êÇ=ëÜáÑí=çéÉê~íçê=xJzI=ÇÉÑáåÉÇ=~ë= B i pt= pt− i=Φ = W=åçáëÉ=íê~åëÑÉê=ÑìåÅíáçå=xJz=a = W=ÇáëÅêÉíÉ=ïÜáíÉ=åçáëÉ=éêçÅÉëë=ïáíÜ=òÉêç=ãÉ~å=xiz===..…………………………………………………………………………………………….….

ëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=få=ôj^=qck=ãçÇÉäë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå= Θ (B) =áë=ÇÉÑáåÉÇ=~ë=~=Ñê~ÅíáçåI=ïÜÉêÉ=íÜÉ=åìãÉê~íçê=áë=~=ëçJÅ~ääÉÇ=ãçîáåÖ=~îÉê~ÖÉ=Ej^F=ÑìåÅíáçå= ω(B)I=~åÇ=íÜÉ=ÇÉåçãáå~íçê=~å=~ìíçêÉÖêÉëëáîÉ=EôF=ÑìåÅíáçå= δ (B) I=ëç=íÜ~í= Θ(B) = ω(B) / δ (B)N K==qÜÉ=ïÉáÖÜíë=Θ0I1Θ I=Á Θ∞=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=~êÉ=åçêã~ääó=êÉÑÉêêÉÇ=íç=~ë=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=EfoF=ÑìåÅíáçåI=Äìí=Ñêçã=~=Åçåíáåìçìë=éçáåí=çÑ=îáÉï=íÜáë=êÉëéçåëÉ=áë=~Åíì~ääó=íÜÉ=êÉëéçåëÉ=íç=~å=áåéìí=ëÉêáÉë=ïáíÜ=íÜÉ=ëÜ~éÉ=çÑ=~=ÄäçÅâ=EëÉÉ=ëÉÅíáçå=OKPFK=qç=~îçáÇ=ÅçåÑìëáçåI=ïÉ=ïáää=ìëÉ=íÜÉ=íÉêã=íê~åëÑÉê=ÑìåÅíáçå=áå=íÜÉ=ÇáëÅêÉíÉ=Å~ëÉ=~åÇ=êÉëÉêîÉ=íÜÉ=íÉêã=fo=ÑìåÅíáçå=Ñçê=íÜÉ=êÉëéçåëÉ=íç=~å=~Åíì~ä=áåëí~åí~åÉçìë=áãéìäëÉK=få=íÜÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éÜ~ëÉ=çÑ=ÇáëÅêÉíÉ=qck=ãçÇÉäëI=íÜÉ=ãçÇÉä=çêÇÉê=áë=ÇÉÑáåÉÇ=Äó=íÜÉ=ÅÜçáÅÉ=çÑ=~=ÇÉä~ó=íáãÉ=~åÇ=íÜÉ=åìãÄÉê=çÑ=j^=~åÇLçê=ô=é~ê~ãÉíÉêëI=ïÜáÅÜ=íçÖÉíÜÉê=Ñçêã=íÜÉ=åçÇÉë=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçåë=áå=Éèì~íáçåë=EPKOF=~åÇ=EPKPFI=~åÇ=ÇÉÑáåÉ=íÜÉáê=ÖÉåÉê~ä=ëíêìÅíìêÉW===⎧ + + +⎪Θ(B) = B = B⎪⎨2np⎪ θ (B) 1+ θ1B + θ2B + .. θnpB⎪Φ(B)= =2nq⎩ϕ(B) 1+ ϕ1B + ϕ2B + .. ϕnqB=ïÜÉêÉ=nr = W=åìãÄÉê=çÑ=j^=é~ê~ãÉíÉêë=çÑ=íÜÉ=íê~åëÑÉê=ãçÇÉä=ns = W=åìãÄÉê=çÑ=ô=é~ê~ãÉíÉêë=çÑ=íÜÉ=íê~åëÑÉê=ãçÇÉä=np = W=åìãÄÉê=çÑ=j^=é~ê~ãÉíÉêë=çÑ=íÜÉ=åçáëÉ=ãçÇÉä=nq = W=åìãÄÉê=çÑ=ô=é~ê~ãÉíÉêë=çÑ=íÜÉ=åçáëÉ=ãçÇÉä=b = W=ÇÉä~ó=íáãÉ=2 nr−1b ω(B)b ω0 ω1B ω2B .. ωnr−1B2nsδ (B) 1+ δ1B + δ2B + .. δnsB(3.4)=qÜìëI=íÜÉ=çêÇÉê=çÑ=~å=ôj^=qck=ãçÇÉä=áë=ìëì~ääó=ëéÉÅáÑáÉÇ=~ë=[ nr, ns, np, nq, b]K=^äíÜçìÖÜ=áí=áë=çÑíÉå=åçí=ÉñéäáÅáíäó=ãÉåíáçåÉÇI=íÜÉ=ãçÇÉäÉê=~äëç=Ü~ë=íç=ëéÉÅáÑó=íÜÉ=íáãÉ=ÇáëÅêÉíáò~íáçå= ∆thçÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=~åÇ= ∆tp çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëÉêáÉëI=óÉí=íÜÉ=çéíáçåë=~êÉ=åçêã~ääó=êÉëíêáÅíÉÇ=áå=éê~ÅíáÅÉI=ÄÉÅ~ìëÉ=çÑ=Ç~í~=~î~áä~Äáäáíó=~åÇ=ÄÉÅ~ìëÉ= ∆ th = ∆tp Ñçê=ôj^=ãçÇÉäëK=tÉ=íÜÉêÉÑçêÉ=ëéÉÅáÑó=íÜÉ=çêÇÉê=çÑ=~å=ôj^=qck=ãçÇÉä=~ë=[ nr, ns, np, nq, b][ ∆ t = ∆t] K=qÜÉ=çîÉê~ää=ÉÑÑÉÅí=çÑ=ãçîáåÖJ~îÉê~ÖÉ=hpé~ê~ãÉíÉêë=çå=íÜÉ=ãçÇÉä=ëíêìÅíìêÉ=áë=íÜ~í=íÜÉó=éêçîáÇÉ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=ÑêÉÉÇçã=çÑ=ëÜ~éÉ=ïáíÜ=êÉëéÉÅí=íç=êÉ~ä=íáãÉ=Ñêçã= b ⋅ ∆tp íç= ( b + nr) ⋅ ∆tp I=ïÜáäÉ=~å=~ÇÇáíáçå~ä=ô=é~ê~ãÉíÉê=Å~ìëÉë=íÜÉ=í~áä=çÑ=íÜÉ=fo=ÑìåÅíáçå=íç=ÄÉ=ÉñéçåÉåíá~ä=EëÉÉ=ÑáÖìêÉ=PKNFK==N=^äíÜçìÖÜ=çíÜÉê=~ìíÜçêë=~äëç=ìëÉ=íÜáë=áåíÉêéêÉí~íáçå=~åÇ=åçí~íáçåI=áí=áë=áåÅçêêÉÅíK=qÜÉ==== TP=−1ÅçêêÉÅí=åçí~íáçå=áë= Θ(B)= δ (B) ω(B)EëÉÉ=~äëç=é~ê~Öê~éÜ=OKOFI=ïÜáÅÜ=áë=åçí=êÉ~ääó=~=Ñê~Åíáçå=~åÇ=çåäó=Ü~ë=~=ëóãÄçäáÅ=ãÉ~åáåÖK=qÜáë=éçáåíI=ÜçïÉîÉêI=çåäó=ÅçåÅÉêåë=íÜÉ=åçí~íáçå=~åÇ=åçí=íÜÉ=ÅçåíÉåíë=çÑ=íÜáë=ÅÜ~éíÉêKKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=P=TQ==få=íÜÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉÇìêÉ=~ë=éêçéçëÉÇ=Äó=_çñ=~åÇ=gÉåâáåëI=íÜÉ=~å~äóëí=áÇÉåíáÑáÉëI=Äó=áíÉê~íáîÉäó=~ÇÇáåÖ=çê=êÉãçîáåÖ=j^=é~ê~ãÉíÉêëI=íÜÉ=éçáåí=Ñêçã=ïÜáÅÜ=íÜÉ=êÉã~áåÇÉê=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=Å~å=ÄÉ=~ÇÉèì~íÉäó=~ééêçñáã~íÉÇ=Äó=~å=ÉñéçåÉåíá~ä=ÑìåÅíáçåI=ïÜáÅÜ=çÑíÉå=äáÉë=àìëí=ÄÉóçåÇ=íÜÉ=éÉ~â=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçåK=qÜÉ=åìãÄÉê=çÑ=é~ê~ãÉíÉêë=åÉÅÉëë~êó=íç=êÉ~ÅÜ=íÜáë=éçáåíI=ÜçïÉîÉêI=áë=ÇÉéÉåÇÉåí=çå=íÜÉ=çÄëÉêî~íáçå=ÑêÉèìÉåÅóI=~ë=~å=j^=é~ê~ãÉíÉê=áë=åÉÅÉëë~êó=Ñçê=ÉîÉêó=íáãÉ=ëíÉé=áå=ÄÉíïÉÉåK=qÜÉêÉÑçêÉI=íÜÉ=ãçÇÉäÉê=~äëç=Ü~ë=íç=Ä~ä~åÅÉ=íÜÉ=çÄëÉêî~íáçå=ÑêÉèìÉåÅó=response (-)2.51.50.5321moving-average partdelay time00 200 400 600 800 1000 1200time (in days)auto-regressive tail=figure 3.1: Example ARMA transfer function of a modelwith order [10 1 0 1 3][30.4], illustrating the portions ofthe transfer function that are determined by the MA andAR parameters.~Ö~áåëí=íÜÉ=åìãÄÉê=çÑ=é~ê~ãÉíÉêëI=áå=çêÇÉê=íç=ÖÉí=íÜÉ=ÄÉëí=ãçÇÉä=êÉëìäíK===fí=Å~å=ÄÉ=êÉ~Çáäó=ëÉÉå=íÜ~í=íÜÉ=äáãáí~íáçåë=çÑ=ÇáëÅêÉíÉ=qck=ãçÇÉäë=ïáíÜ=êÉÖ~êÇ=íç=áêêÉÖìä~êäó=çÄëÉêîÉÇ=íáãÉ=ëÉêáÉë=Ñçääçï=ÇáêÉÅíäó=Ñêçã=bèëK=ENJPFK=få=íÜÉëÉ=Éèì~íáçåëI=íáãÉ=áë=ÅçåëáÇÉêÉÇ=íç=ÄÉ=~=ÇáãÉåëáçåäÉëë=áåÇÉñ= t ∈ N I=ëìÅÜ=íÜ~í=É~ÅÜ=íáãÉ=ëíÉé=áë=Éèì~ä=íç=çåÉI=êÉÖ~êÇäÉëë=çÑ=íÜÉ=ÇáëÅêÉíáò~íáçå=áå=êÉ~ä=íáãÉK=cìêíÜÉêãçêÉI=íÜÉ=ë~ãÉ=áåÇÉñ=áë=ìëÉÇ=Ñçê= h =~åÇ= p I=ëç=íÜÉ=ë~ãéäÉ=áåíÉêî~äë=çÑ=íÜÉ=áåéìí=~åÇ=çìíéìí=ëÉêáÉë=Ü~îÉ=íç=ÄÉ=áÇÉåíáÅ~äK=`çåëÉèìÉåíäóI=íÜÉ=~å~äóëí=áë=ÑçêÅÉÇ=íç=äçïÉê=íÜÉ=ÑêÉèìÉåÅó=çÑ=~ää=íáãÉ=ëÉêáÉë=íç=íÜÉ=äçïÉëí=çåÉ=~î~áä~ÄäÉ=~åÇ=íÜìë=ÇáëêÉÖ~êÇ=êÉäÉî~åí=áåÑçêã~íáçå=~Äçìí=íÜÉ=ÇáëíêáÄìíáçå=çÑ=íÜÉ=áåéìí=ëÉêáÉë=áå=ÄÉíïÉÉå=íÜÉ=íáãÉ=ëíÉéëK===3.2.2 The continuous time PIRFICT modeljçëí=éêçÅÉëëÉë=ãçÇÉäÉÇ=ïáíÜ=qck=ãçÇÉäëI=ëìÅÜ=~ë=éêÉÅáéáí~íáçå=~åÇ=ÖêçìåÇï~íÉê=äÉîÉä=ÑäìÅíì~íáçåëI=~êÉ=áå=êÉ~äáíó=åçí=ÇáëÅêÉíÉ=Äìí=ÅçåíáåìçìëK=få=Åçåíáåìçìë=íáãÉI=íÜÉ=íê~åëÑçêã~íáçå=çÑ=~=íáãÉ=ëÉêáÉë=ïáíÜ=~=äáåÉ~êI=íáãÉJáåî~êá~åí=íê~åëÑÉê=ÑìåÅíáçå=áë=ÖáîÉå=Äó=~=Åçåîçäìíáçå=áåíÉÖê~ä=xnìáãéçI=NVTNzK=qÜìëI=Éèì~íáçåë=ENJOF=Å~å=ÄÉ=ïêáííÉå=áå=Åçåíáåìçìë=íáãÉ=~ëW==h( t) = hˆ( t) + n( t)+ d(3.5)thˆ( t ) = ∫ θ ( t −τ ) p ( τ )d τ(3.6)−∞=råäáâÉ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=çÑ=ÇáëÅêÉíÉ=qck=ãçÇÉäëI=íÜÉ=fo=ÑìåÅíáçå= θ ( t)çÑ=~=Åçåîçäìíáçå=áåíÉÖê~ä=ÇçÉë=åçí=ÇÉéÉåÇ=çå=íÜÉ=çÄëÉêî~íáçå=ÑêÉèìÉåÅó=çÑ=íÜÉ=áåéìí=ëÉêáÉëK=fí=ÇÉëÅêáÄÉë=íÜÉ=Çóå~ãáÅ=êÉëéçåëÉ=çÑ=~=ëóëíÉã=íç=~å=áåëí~åí~åÉçìë=áãéìäëÉ=~åÇ=áë=íáãÉJáåî~êá~åí=~åÇ=~å=áåíÉÖê~ä=éêçéÉêíó=çÑ=~=ëéÉÅáÑáÅ=ëóëíÉã=xgìêó=~åÇ=oçíÜI=NVVMX=..…………………………………………………………………………………………….….

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`Ü~éíÉê=P=TS=Å~å=Ü~îÉ=~=ÑäÉñáÄäÉ=ëÜ~éÉ=~åÇ=ÄÉ=Éèìáî~äÉåí=íç=~=ëÉêáÉë=çÑ=ôj^=íê~åëÑÉê=ÑìåÅíáçåëK=pÉÅçåÇI=íÜÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉÇìêÉ=áë=ëáãéäáÑáÉÇI=ÄÉÅ~ìëÉ=íÜÉ=ãçÇÉä=ÑêÉèìÉåÅó=ÇçÉë=åçí=áåíÉêÑÉêÉ=ïáíÜ=íÜÉ=ãçÇÉä=çêÇÉêK=qÜáêÇI=~=Åçåíáåìçìë=fo=ÑìåÅíáçå=Å~å=ÄÉ=çÄàÉÅíáîÉäó=ÅÜçëÉå=~ë=íÜÉ=ÑìåÅíáçå=íÜ~í=êÉéêÉëÉåíë=íÜÉ=éÜóëáÅë=çÑ=íÜÉ=~å~äóòÉÇ=ëóëíÉã=ÄÉëíK=^=éÜóëáÅ~ääóJÄ~ëÉÇ=fo=ÑìåÅíáçå=çå=íÜÉ=çåÉ=Ü~åÇ=êÉÇìÅÉë=íÜÉ=ëÉåëáíáîáíó=çÑ=íÜÉ=ãçÇÉä=íç=ÅçáåÅáÇÉåí~ä=ÅçêêÉä~íáçåë=áå=íÜÉ=Ç~í~I=Äìí=çå=íÜÉ=çíÜÉê=Ü~åÇ=áí=Å~å=êÉÇìÅÉ=íÜÉ=Ñáí=áÑ=Ñçê=ëçãÉ=êÉ~ëçå=íÜÉ=éÜóëáÅ~ä=~ëëìãéíáçåë=éêçîÉ=áåÅçêêÉÅíK=få=ôj^=qck=ãçÇÉäë=íÜÉ=ãçÇÉä=çêÇÉê=Å~å=ÄÉ=ÅÜçëÉå=çå=éÜóëáÅ~ä=ÖêçìåÇë=xvçìåÖ=~åÇ=_ÉîÉåI=NVVQzI=Äìí=ïáíÜ=ãçÇÉäë=íÜ~í=Åçåí~áå=j^=é~ê~ãÉíÉêë=íÜÉ=Éñ~Åí=ëÜ~éÉ=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=Å~ååçíK===eÉêÉ=ïÉ=ÅÜççëÉ=íÜÉ=mÉ~êëçå=íóéÉ=fff=ÇáëíêáÄìíáçå=ÑìåÅíáçå=Emfff=ÇÑFI=ïáíÜ=~å=Éñíê~=é~ê~ãÉíÉê=^=íÜ~í=~Çàìëíë=íÜÉ=~êÉ~I=íç=ÇÉëÅêáÄÉ=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=ï~íÉê=í~ÄäÉ=íç=éêÉÅáéáí~íáçå=ëìêéäìëW===n n−1a ( t − b) exp{ −a( t − b)}θ ( t)= A(3.11)Γ( n)=ïáíÜ=[ a, n, b, A]ÄÉáåÖ=é~ê~ãÉíÉêëK=qÜÉ=éÜóëáÅ~ä=Ä~ëáë=çÑ=íÜÉ=mfff=ÇÑ=äáÉë=áå=íÜÉ=Ñ~Åí=íÜ~í=áí=ÇÉëÅêáÄÉë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=çÑ=~=ëÉêáÉë=çÑ=ÅçìéäÉÇ=äáåÉ~ê=êÉëÉêîçáêë=xk~ëÜI=NVRUzI=íÜÉ=é~ê~ãÉíÉê= n =ÇÉåçíáåÖ=íÜÉáê=åìãÄÉêI= a =Éèì~äáåÖ=íÜÉ=áåîÉêëÉ=çÑ=íÜÉ=êÉëÉêîçáê=ÅçÉÑÑáÅáÉåí=åçêã~ääó=ìëÉÇI=~åÇ= b =ÄÉáåÖ=íÜÉ=ÇÉä~ó=íáãÉK=qÜÉ=Éñíê~=é~ê~ãÉíÉê A ==áë=åÉÅÉëë~êó=ÄÉÅ~ìëÉ=áå=íÜÉ=Å~ëÉ=çÑ=Éèì~íáçå=EPKSFI=ïÜÉêÉ=~=éêÉÅáéáí~íáçå=ëìêéäìë=ëÉêáÉë=áë=íê~åëÑçêãÉÇ=áåíç=~=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉëI=íÜÉ=ä~ï=çÑ=ÅçåëÉêî~íáçå=çÑ=ã~ëë=ÇçÉë=åçí=~ééäóK=j~íÜÉã~íáÅ~ääóI= n =áë=åçí=êÉëíêáÅíÉÇ=íç=áåíÉÖÉê=î~äìÉëI=ïÜáÅÜ=ÑìêíÜÉê=áåÅêÉ~ëÉë=íÜÉ=ÑäÉñáÄáäáíó=çÑ=íÜÉ=mfff=ÇÑ=fo=ÑìåÅíáçåK=qÜÉ=mfff=ÇÑ=Å~å=í~âÉ=ëÜ~éÉë=Öê~Çì~ääó=ê~åÖáåÖ=Ñêçã=ëíÉÉéÉê=íÜ~å=ÉñéçåÉåíá~äI=îá~=ÉñéçåÉåíá~ä=íç=d~ìëëá~å=EëÉÉ=ÑáÖìêÉ=PKOFK=få=ÇáëÅêÉíÉ=íÉêãëI=áí=Å~å=ÄÉ=Éèìáî~äÉåí=íç=~å=ôENF=ãçÇÉäI=Äìí=~äëç=íç=~ää=ôj^=ãçÇÉäë=çÑ=ïÜáÅÜ=íÜÉ=ïÉáÖÜíë=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=íçÖÉíÜÉê=Ñçêã=çåÉ=çÑ=íÜÉ=ÅìêîÉë=áå=íÜÉ=mfff=ÇÑ=Ñ~ãáäóK=håçííÉêë=~åÇ=_áÉêâÉåë=xOMMMz=ëÜçïÉÇ=íÜ~í=~å=ôuENIMF=ãçÇÉä=áë=~=ÇáëÅêÉíáòÉÇ=îÉêëáçå=çÑ=~=ëáåÖäÉ=äáåÉ~ê=êÉëÉêîçáêI=~åÇ=ëìÄëÉèìÉåíäó=áÇÉåíáÑáÉÇ=áí=~ë=íÜÉ=ãçëí=~ééêçéêá~íÉ=äáåÉ~ê=íáãÉ=ëÉêáÉë=ãçÇÉä=Ñçê=íáãÉ=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=äÉîÉäK=qÜÉáê=ÅÜçáÅÉ=çÑ=íÜÉ=ôu=ãçÇÉäI=ÜçïÉîÉêI=ï~ë=response (−)10.90.80.70.60.50.40.30.20.1n = 0.5n = 1n = 1.3n = 1.7n = 2.300 50 100 150 200 250 300time (in days)=figure 3.2: A selection of curves of the Pearson type III df( n = [0.5 1 1.3 1.7 2.3], A = n ⋅ 100 , a = 0.01 ,b = 0 ), giving examples of the range of shapes the PIII dfcan take.=..…………………………………………………………………………………………….….

ëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=Ä~ëÉÇ=çå=~=ëáãéäÉ=éÜóëáÅ~ä=ãçÇÉä=çÑ=~=çåÉ=ÇáãÉåëáçå~ä=ëçáä=ÅçäìãåI=ÇáëÅ~êÇáåÖ=ä~íÉê~ä=Ñäçï=~åÇ=íÜÉ=ÑìåÅíáçåáåÖ=çÑ=íÜÉ=ìåë~íìê~íÉÇ=òçåÉK=få=íÜáë=ëÉåëÉ=íÜÉ=mfff=ÇÑ=Ñçêãë=~å=ÉñíÉåëáçå=íç=íÜÉáê=ãÉíÜçÇI=~ë=áí=áåÅäìÇÉë=íÜÉ=ôu=ãçÇÉä=Äìí=Å~å=Ñçê=Éñ~ãéäÉ=~äëç=ÇÉëÅêáÄÉ=íÜÉ=ÅçãÄáåÉÇ=êÉëéçåëÉ=çÑ=~=ë~íìê~íÉÇ=~åÇ=ä~óÉêÉÇ=ìåë~íìê~íÉÇ=òçåÉK====^ë=íÜÉ=êÉëáÇì~äëI=ïÜáÅÜ=~êÉ=íÜçìÖÜí=íç=ÄÉ=íÜÉ=çìíéìí=çÑ=íÜÉ=åçáëÉ=ãçÇÉäI=~êÉ=íÜÉ=êÉëìäí=çÑ=~=î~êáÉíó=çÑ=Å~ìëÉë=EÉKÖKI=Éêêçêë=áå=íÜÉ=çÄëÉêî~íáçåë=çÑ=íÜÉ=áåéìí=~åÇ=çìíéìí=ëÉêáÉëI=Éêêçêë=áå=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêëI=ëáãéäáÑáÅ~íáçåë=çê=Éêêçêë=áå=íÜÉ=ãçÇÉä=ÅçåÅÉéíFI=áí=áë=ÇáÑÑáÅìäí=íç=ã~âÉ=~=ÅäÉ~ê=ÅÜçáÅÉ=çÑ=íÜÉ=åçáëÉ=fo=ÑìåÅíáçå=çå=éÜóëáÅ~ä=ÖêçìåÇëK=tÉ=íÜÉêÉÑçêÉ=ÅÜççëÉ=~=ëáãéäÉ=ôENF=åçáëÉ=ãçÇÉäI=~åÇ=êÉäó=çå=Çá~ÖåçëíáÅ=ÅÜÉÅâë=íç=íÉëí=áíë=~ÇÉèì~ÅóK=qÜÉ=ÅÜçáÅÉ=çÑ=~å=ôENF=ãçÇÉä=Éèì~äë=íÜÉ=ÅÜçáÅÉ=çÑ=~å=ÉñéçåÉåíá~ä=fo=ÑìåÅíáçå=áå=Åçåíáåìçìë=íáãÉ=xÉKÖKI=_çñ=~åÇ=gÉåâáåëI=NVTMX=`Ü~íÑáÉäÇI=NVUVzK=få=çêÇÉê=íç=ÖÉí=~å=ÉñéçåÉåíá~ä=åçáëÉ=ãçÇÉä=ïáíÜ=~å=~ééêçéêá~íÉ=áååçî~íáçå=î~êá~åÅÉ=ÑìåÅíáçåI=ÜÉêÉ=ïÉ=ìëÉ=~å=fo=ÑìåÅíáçå=çÑ=íÜÉ=ÑçääçïáåÖ=Ñçêã=xsçå=^ëãìíÜ=~åÇ=_áÉêâÉåëI=OMMRzW===2nφ( t) = 2ασ exp( − αt)(3.12)=2ïáíÜ=íÜÉ=é~ê~ãÉíÉê=α =ÇÉíÉêãáåáåÖ=íÜÉ=ÇÉÅ~ó=ê~íÉ=çÑ=φ =~åÇ= σn=ÇÉåçíáåÖ=íÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=êÉëáÇì~äëK===== TT=3.2.3 Evaluation, parameter estimation and diagnostic checkingtÜÉå=íÜÉ=fo=ÑìåÅíáçåë=Ü~îÉ=ÄÉÉå=ÅÜçëÉåI=íÜÉ=mfocf`q=ãçÇÉä=áë=áÇÉåíáÑáÉÇ=~åÇ=Å~å=ÄÉ=Éî~äì~íÉÇK=få=íÜÉ=ÑçääçïáåÖI=ïÉ=ïáää=ÇÉëÅêáÄÉ=íÜÉ=ÇáÑÑÉêÉåí=ëíÉéë=áå=íÜÉ=áãéäÉãÉåí~íáçå=çÑ=íÜÉ=mfocf`q=ãçÇÉäK=cáêëíI=íÜÉ=~î~áä~ÄäÉ=íáãÉ=ëÉêáÉë=Ü~îÉ=íç=ÄÉ=íê~åëÑçêãÉÇ=íç=Åçåíáåìçìë=ëÉêáÉëI=~ë=ãçëí=íáãÉ=ëÉêáÉë=~êÉ=åçí=~î~áä~ÄäÉ=~ë=Åçåíáåìçìë=ëÉêáÉë=ÇìÉ=íç=íÜÉ=ÇáëÅêÉíÉ=çÄëÉêî~íáçå=éêçÅÉëëK=j~åó=ÅçääÉÅíÉÇ=íáãÉ=ëÉêáÉëI=ÜçïÉîÉêI=Å~å=ÄÉ=êÉÖ~êÇÉÇ=~ë=íÜÉ=ÅÜ~åÖÉ=áå=íÜÉ=éêáãáíáîÉ=ÑìåÅíáçå=çÑ=ëçãÉ=ìåÇÉêäóáåÖ=Åçåíáåìçìë=éêçÅÉëëI=~ë=áë=íÜÉ=Å~ëÉ=ïáíÜ=éêÉÅáéáí~íáçå=ëìêéäìëW==p = P( t ) − P( t ) = ∫ p( τ )dτ(3.13)t i i−1titi−1=tÜÉå=éêÉÅáéáí~íáçå=ëìêéäìë=Ç~í~=~êÉ=çåäó=~î~áä~ÄäÉ=~í=ÇáëÅêÉíÉ=áåíÉêî~äëI=íÜÉ=Åçåíáåìçìë=ëÉêáÉë= p( τ ) =Å~ååçí=ÄÉ=êÉÅçåëíêìÅíÉÇ=Éñ~ÅíäóI=Äìí=áí=Å~å=ÄÉ=~ééêçñáã~íÉÇ=Äó=~ëëìãáåÖ=íÜ~í=íÜÉ=ÇáëíêáÄìíáçå=çÑ= p =áë=ìåáÑçêã=ÇìêáåÖ=íÜÉ=éÉêáçÇ= ti− 1íç= t i=xwáÉãÉê=Éí=~äKI=NVVUzK=bèì~íáçå=EOKPSF=Å~å=íÜÉå=ÄÉ=ïêáííÉå=~ëW==ptp( τ ) = , ti−1≥ τ > ti(3.14)ti− ti−1=få=íÜáë=ï~óI=íÜÉ=ÜáÖÜÉê=íÜÉ=ÑêÉèìÉåÅó=çÑ=çÄëÉêî~íáçåI=íÜÉ=ãçêÉ= ti− ti− 1=~ééêç~ÅÜÉë=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=P=TU=òÉêç=~åÇ=íÜÉ=ÄÉííÉê=íÜÉ=~ééêçñáã~íáçå=çÑ= p( τ ) =EïáíÜ=10.9dt = 0dt = 50dt = 100ìåáíë=xiq JN dt = 150zF=ïáää=ÄÉK=kçíÉ=íÜ~í=íÜÉ= 0.8dt = 200Éêêçê=ã~ÇÉ=Äó=íÜáë=~ëëìãéíáçå=ïáää=0.7î~êó=áå=íáãÉ=ïÜÉå=åçåJÉèìáÇáëí~åí=éêÉÅáéáí~íáçå=ëìêéäìë=çÄëÉêî~íáçåë=~êÉ=ìëÉÇI=ïÜáÅÜ=ïáää=áåíêçÇìÅÉ=åçåJëí~íáçå~êáíó=áå=íÜÉ=ãçÇÉä=êÉëáÇì~äëK=eçïÉîÉêI=ïÉ=~ëëìãÉ=íÜ~íI=ïÜÉå=íÜÉ=íáãÉ=ëíÉéë=~êÉ=åçí=íçç=ä~êÖÉ=~åÇ=áêêÉÖìä~êI=íÜáë=ÉÑÑÉÅí=áë=ëã~ää=ïÜÉå=Åçãé~êÉÇ=íç=íÜÉ=0.60.50.40.30.20.1çíÜÉê=ëçìêÅÉë=çÑ=ãçÇÉä=Éêêçê=~åÇ=0Å~å=íÜÉêÉÑçêÉ=ÄÉ=åÉÖäÉÅíÉÇK==0 50 100 150 200 250 300time (in days)===pÉÅçåÇI=~ë=Éèì~íáçå=EPKSF=~åÇ= figure 3.3: Example BR functions of a single linearEPKTF=Åçåí~áå=íáãÉ=Ñêçã= −∞ I=ïÉ= reservoir ( ∆ t = [0,50,100,150,200] , n = 1 ,Ü~îÉ=íç=ÇÉÑáåÉ=~å=áåáíá~ä=î~äìÉ=íç=A = 100 , a = 0.01 ), illustrating the effect of differentÄÉ=~ÄäÉ=íç=Éî~äì~íÉ=íÜÉ=Éèì~íáçåë=time steps on the shape of the BR function.ïáíÜ=çÄëÉêî~íáçåë=ëí~êíáåÖ=Ñêçã=t = 0 K=eÉêÉ=ïÉ=ÇÉÑáåÉ=íÜÉ=áåáíá~ä=î~äìÉ=çÑ= p =Ñêçã= t = −∞ íç= t = 0 íç=ÄÉ= p I=íÜÉ=íÉãéçê~ä=~îÉê~ÖÉ=çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=ëÉêáÉë=ìëÉÇ=Ñçê=íÜÉ=ëáãìä~íáçåë=EÄìí=ïÜÉå=~î~áä~ÄäÉ=~=ÜáëíçêáÅ~ä=~îÉê~ÖÉ=ÅçìäÇ=~äëç=ÄÉ=ìëÉÇFK=táíÜ=íÜÉ=~áÇ=çÑ=Éèì~íáçå=EPKNQFI=íÜÉ=íê~åëÑÉê=ãçÇÉä=EÉèì~íáçå=EPKSFF=Å~å=åçï=ÄÉ=Éî~äì~íÉÇ=ìëáåÖ=íÜÉ=ÄäçÅâ=êÉëéçåëÉ=E_oF=ÑìåÅíáçåK=qÜÉ=_o=ÑìåÅíáçå= Θ( t)Å~å=ÄÉ=çÄí~áåÉÇ=Äó=ÅçåîçäìíáåÖ=íÜÉ=fo=ÑìåÅíáçå=ïáíÜ=~=ÚÄäçÅâÛ=çÑ=éêÉÅáéáí~íáçå=ëìêéäìë=ïáíÜ=ìåáí=áåíÉåëáíó=çîÉê=~=éÉêáçÇ= ∆tI=ïÜáÅÜ=Éèì~äëW==tΘ ( t) = ∫ θ ( τ )dτ(3.15)t−∆tresponse (−)=qç=ã~âÉ=íÜÉ=_o=ÑìåÅíáçå=Éèìáî~äÉåí=íç=íÜÉ=ÇáëÅêÉíÉ=íê~åëÑÉê=ÑìåÅíáçåI=áí=Ü~ë=íç=ÄÉ=ÇáîáÇÉÇ=Äó= ∆ t =áå=çêÇÉê=íç=ëÅ~äÉ=áí=íç=~=ÄäçÅâ=áåéìí=ïáíÜ=ìåáí=~êÉ~K=få=ÑáÖìêÉ=PKP=ëçãÉ=Éñ~ãéäÉ=_o=ÑìåÅíáçåë=çÑ=~=ëáåÖäÉ=äáåÉ~ê=êÉëÉêîçáê=~êÉ=éäçííÉÇ=Ñçê=ÇáÑÑÉêÉåí=íáãÉ=ëíÉéëK=*qÜÉ=éêÉÇáÅíÉÇ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë= h Å~å=ÄÉ=çÄí~áåÉÇ=Äó=~ÇÇáåÖ=íÜÉ=êÉëéçåëÉë=çÑ=~ää=ÚÄäçÅâëÛ=çÑ=éêÉÅáéáí~íáçåK=_ÉÅ~ìëÉ= Θ =áë=~=Åçåíáåìçìë=ÑìåÅíáçåI= h =áë=~äëç=Åçåíáåìçìë=~åÇ=Ñçê=ÉîÉêó=çÄëÉêî~íáçå=çÑ= h =~=ë~ãéäÉ=çÑ=íÜÉ=êÉëáÇì~ä=ëÉêáÉë= n =áë=çÄí~áåÉÇK===kÉñíI=íÜÉ=åçáëÉ=ãçÇÉä=EÉèì~íáçå=EPKTFF=áë=Éî~äì~íÉÇ=áå=çêÇÉê=íç=çÄí~áå=~=ëÉêáÉë=çÑ=áååçî~íáçåë=ν K=qç=Éî~äì~íÉ=íÜÉ=åçáëÉ=ãçÇÉä=ïáíÜçìí=Ü~îáåÖ=íç=ìëÉ=~=h~äã~å=cáäíÉê=EïÜáÅÜ=áë=Åçãéìí~íáçå~ääó=ÉñéÉåëáîÉF=ïÉ=ïáää=ÇÉêáîÉ=~=ÇáêÉÅí=êÉä~íáçå=ÄÉíïÉÉå=íÜÉ=êÉëáÇì~äë= n ~åÇ=íÜÉ=áååçî~íáçåë=ν K=`çåëáÇÉê=íÜÉ=áååçî~íáçå=ëÉêáÉë=ν =~ë=íÜÉ=åçåJÉèìáÇáëí~åíäó=ë~ãéäÉÇ=ÅÜ~åÖÉ=áå=íÜÉ=ëçäìíáçå=íç=íÜÉ=ëíçÅÜ~ëíáÅ=áåíÉÖê~ä=ÇÉëÅêáÄáåÖ=íÜÉ=*..…………………………………………………………………………………………….….

ëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=êÉëáÇì~ä=ëÉêáÉëW==tν ( t) = ∫ φ( t −τ )d W ( τ )(3.16)t−∆t=tÜÉå=íÜÉ=åçáëÉ=fo=ÑìåÅíáçå=φ =Éèì~äë=Éèì~íáçå=EPKNOF=I=Éèì~íáçå=EPKTF=Å~å=ÄÉ=ïêáííÉå=~ëW==t∫2n(3.17)t−∆tn( t) = exp( −α∆t) n( t − ∆t)+ 2ασ exp{ −α ( t −τ )}dW( τ )=ïÜáÅÜ=áë=âåçïå=~ë=~å=lêåëíÉáåJrÜäÉåÄÉÅâ=éêçÅÉëë=xrÜäÉåÄÉÅâ=~åÇ=lêåëíÉáåI=NVPMzK=_ó=ÅçãÄáåáåÖ=Éèì~íáçåë=EPKNSF=~åÇ=EPKNTFI=ν =Å~å=ÄÉ=Å~äÅìä~íÉÇ=Ñêçã=íÜÉ=~î~áä~ÄäÉ=Ç~í~=~ëW===v( t) = n( t) − exp( −α∆t) n( t − ∆ t)(3.18)=pìÄëÉèìÉåíäóI=íÜÉ=é~ê~ãÉíÉê=ëÉí= Ψ = [ A, a, n, b, α]=Åçåí~áåÉÇ=áå=íÜÉ=fo=ÑìåÅíáçåë=Ü~ë=íç=ÄÉ=Éëíáã~íÉÇ=Ñêçã=íÜÉ=Ç~í~K=_ó=~ÇàìëíáåÖ=íÜÉ=î~äìÉ=çÑ=íÜÉ=é~ê~ãÉíÉêë=íÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉä=Å~å=ÄÉ=Å~äáÄê~íÉÇ=çå=~=ëÉí=çÑ=çÄëÉêî~íáçåë=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=h( t i) Äó=ãáåáãáòáåÖ=~=ÅÉêí~áå=çÄàÉÅíáîÉ=ÑìåÅíáçåK=_áÉêâÉåë=Éí=~äK=xNVVVz=ìëÉ=~=äçÖJäáâÉäáÜççÇ=ÑìåÅíáçå=xpÅÜïÉééÉI=NVTPz=~ë=çÄàÉÅíáîÉ=ÑìåÅíáçå=Ñçê=íÜÉ=áååçî~íáçåë=çÑ=~=h~äã~å=ÑáäíÉê=ïáíÜ=~å=çÄëÉêî~íáçå=Éêêçê=î~êá~åÅÉ=çÑ=MI=íç=ÖÉí=~=ã~ñáãìã=äáâÉäáÜççÇ=Éëíáã~íÉ=çÑ=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêë=EìåÇÉê=íÜÉ=~ëëìãéíáçå=íÜ~í=íÜÉ=áååçî~íáçåë=~êÉ=d~ìëëá~åFK=eçïÉîÉêI=Ñçê=êÉ~ëçåë=çÑ=ÉÑÑáÅáÉåÅóI=ïÉ=ëÉÉâ=~å=çÄàÉÅíáîÉ=ÑìåÅíáçå=íÜ~í=Å~å=ÄÉ=ÉñéêÉëëÉÇ=áå=íÉêãë=çÑ=áåÇáîáÇì~ä=áååçî~íáçåëK=cêçã=EPKNOF=~åÇ=EPKNSF=ïÉ=Ü~îÉW==2 2ν ( ∆ t, ) = {1 − exp( −2 ∆t)} nσ Ψ α σ(3.19)=táíÜ=EPKNVF=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=Å~å=ÄÉ=~ééêçñáã~íÉÇ=Äó=íÜÉ=ÑçääçïáåÖ=ïÉáÖÜíÉÇ=äÉ~ëí=ëèì~êÉë=ÅêáíÉêáçå=xsçå=^ëãìíÜ=~åÇ=_áÉêâÉåëI=OMMRzW==j=1N∏{1 − exp( −2 α∆t)}NNii=1 2∑S{ Ψ | h( ti)} =ν ( tj, Ψ )(3.20)1− exp( −2 α∆t)j=ïáíÜ= ν2 ( , Ψ ) =Å~äÅìä~íÉÇ=Ñêçã=íÜÉ=êÉëáÇì~ä=ëÉêáÉë=ìëáåÖ=Éèì~íáçå=EPKNUFK=cêçã=t jÉèì~íáçå=EPKOMFI=~=g~ÅçÄá~å=ã~íêáñ=Å~å=ÄÉ=É~ëáäó=çÄí~áåÉÇI=ëç=áí=Å~å=ÄÉ=ãáåáãáòÉÇ=ïáíÜ=êÉëéÉÅí=íç= Ψ =ìëáåÖ=~=iÉîÉåÄÉêÖJj~êèì~êÇí=ãÉíÜçÇK=qÜáë=ã~âÉë=íÜÉ=é~ê~ãÉíÉê==== TV=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=P=UM=Éëíáã~íáçå=éêçÄäÉã=ãìÅÜ=ãçêÉ=ÉÑÑáÅáÉåí=íÜ~å=ìëáåÖ=~=h~äã~å=ÑáäíÉê=áå=ÅçåàìåÅíáçå=ïáíÜ=~=äçÖJäáâÉäáÜççÇ=ÑìåÅíáçå=~åÇ=ëçãÉ=ÖäçÄ~ä=çéíáãáò~íáçå=~äÖçêáíÜãK==cáå~ääóI=íÜÉ=~ÅÅìê~Åó=~åÇ=î~äáÇáíó=çÑ=íÜÉ=ãçÇÉä=êÉëìäíë=áë=ÅÜÉÅâÉÇ=Äó=Éñ~ãáåáåÖ=íÜÉ=~ìíçJ=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåë=çÑ=íÜÉ=áååçî~íáçåëI=íÜÉ=Åçî~êá~åÅÉ=ã~íêáñ=çÑ=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêë=~åÇ=íÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=fo=ÑìåÅíáçåëK=qÜÉ=Åçî~êá~åÅÉ=ã~íêáñ=çÑ=íÜÉ=é~ê~ãÉíÉêë= C( Ψ)áë=Éëíáã~íÉÇ=ìëáåÖ=íÜÉ=g~ÅçÄá~å=ã~íêáñ=çÄí~áåÉÇ=Ñêçã=íÜÉ=Å~äáÄê~íáçå=2νêçìíáåÉ=~åÇ= σ K=cêçã=íÜÉ=Åçî~êá~åÅÉ=ã~íêáñI=~äëç=íÜÉ=ÅçêêÉä~íáçå=ÄÉíïÉÉå=íÜÉ=é~ê~ãÉíÉêë=áë=Å~äÅìä~íÉÇK=qÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=fo=ÑìåÅíáçå=θ =Å~å=ÄÉ=çÄí~áåÉÇ=ÄóW==2 ∂θ ( t) 2 2 ∂θ ( t) 2 2 ∂θ ( t) 2 2 ∂θ ( t) ∂θ( t)σθ( t)= ( ) σA+ ( ) σa+ ( ) σn+ 2( )( ) C( A, a) + ....∂A ∂a ∂n ∂A ∂a(3.21)∂θ ( t) ∂θ ( t) ∂θ ( t) ∂θ( t)2( )( ) C( A, n) + 2( )( ) C( a, n)∂A ∂n ∂a ∂n=^ëëìãáåÖ=~=åçêã~ä=ÇáëíêáÄìíáçåI=~=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=Ñçê=θ =Å~å=ÄÉ=éäçííÉÇ=~ë=HLJ=Oσ K=^ë=áå=ÇáëÅêÉíÉ=qck=ãçÇÉäëI=ëÉêáçìë=ãçÇÉä=áå~ÇÉèì~Åó=Å~å=ÄÉ=ÇÉíÉÅíÉÇ=Äó=Éñ~ãáåáåÖ=íÜÉ=~ìíçÅçêêÉä~íáçå=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçå=ëÉêáÉë=ν =EïÜáÅÜ=áåÇáÅ~íÉë=ïÜÉíÜÉê=íÜÉ=ïÜáíÉ=åçáëÉ=~ëëìãéíáçå=ÜçäÇëF=~åÇ=íÜÉ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçå=çÑ=ν =~åÇ=íÜÉ=áåéìí=ëÉêáÉë= p =EïÜáÅÜ=áåÇáÅ~íÉë=ïÜÉíÜÉê=íÜÉêÉ=~êÉ=ëíáää=é~ííÉêåë=äÉÑí=áå=íÜÉ=áååçî~íáçå=ëÉêáÉë=íÜ~í=ÅçìäÇ=ÄÉ=Éñéä~áåÉÇ=Äó=íÜÉ=áåéìí=ëÉêáÉëFK=qÜÉ=~ìíçÅçêêÉä~íáçå=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåë=~í=ä~Ö= k =~êÉ=ÇÉÑáåÉÇ=áå=íÜÉ=ë~ãÉ=ï~ó=~ë=áå=ÇáëÅêÉíÉ=qck=ãçÇÉäëI=Äìí=ÄÉÅ~ìëÉ=çÑ=íÜÉ=åçåJÉèìáÇáëí~åí=ë~ãéäáåÖ=~=íçäÉê~åÅÉ=~êçìåÇ=ä~Ö= k =çÑ=±=MKR k =áë=áãéäáÉÇK=^=ëáãáä~ê=~ééêç~ÅÜ=áë=ìëÉÇ=áå=íÜÉ=ÑáÉäÇ=çÑ=ÖÉçëí~íáëíáÅë=íç=çÄí~áå=ëé~íá~ä=î~êáçÖê~ãë=xgçìêåÉä=~åÇ=eìáàÄêÉÖíëI=NVTUzK=^å=Éñ~ãéäÉ=çÑ=íÜÉ=~ìíçÅçêêÉä~íáçå=ÑìåÅíáçåë=çÑ=~å=ôj^=~åÇ=~=åçåJÉèìáÇáëí~åí=mfocf`q=ãçÇÉä=áë=ÖáîÉå=áå=ÑáÖìêÉ=PKQK=1correlation coefficient(−)0.50−0.50 5 10 15 20 25time lag (−)==figure 3.4: Autocorrelation functions of an [10 0 1 1 0][30.4] ARMA TFN model and a [30.4 30.4]PIRFICT model. The dotted lines denote the 95% confidence interval. From the figure it can be seenthat for both models, the autocorrelation functions are very similar and the white noise assumption isvalid...…………………………………………………………………………………………….….

ëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=3.2.4 Summary of methodfå=ëìãã~êóI=íÜÉ=ãÉíÜçÇ=ÇÉëÅêáÄÉÇ=~ÄçîÉ=Åçåëáëíë=çÑ=íÜÉ=ÑçääçïáåÖ=ëíÉéëK=cáêëíI=Ñçê=ÉîÉêó=áåéìí=ëÉêáÉë=~å=fo=ÑìåÅíáçå=áë=ÅÜçëÉåI=ïÜáÅÜ=áå=éêáåÅáéäÉ=Å~å=ÄÉ=~åó=Åçåíáåìçìë=ÑìåÅíáçå=çê=ÅçãÄáå~íáçå=çÑ=ÑìåÅíáçåëI=Äìí=áå=éê~ÅíáÅÉ=ïáää=çÑíÉå=ÄÉ=Ä~ëÉÇ=çå=íÜÉ=éÜóëáÅ~ä=ä~ïë=çÑ=íÜÉ=~å~äóòÉÇ=ëóëíÉãK=qÜÉ=áåéìí=ëÉêáÉë=~êÉ=~ëëìãÉÇ=íç=ÄÉ=ìåáÑçêãäó=ÇáëíêáÄìíÉÇ=áå=ÄÉíïÉÉå=íÜÉ=íáãÉ=ëíÉéë=~åÇ=íê~åëÑçêãÉÇ=áåíç=Åçåíáåìçìë=ëÉêáÉë=ìëáåÖ=EPKNQFK=qÜÉ=íê~åëÑÉê=Åçåîçäìíáçå=áåíÉÖê~ä=EÉèì~íáçå=EPKSFF=Å~å=åçï=ÄÉ=Éî~äì~íÉÇ=ìëáåÖ=ÄäçÅâ=êÉëéçåëÉ=ÑìåÅíáçåë=Ñçê=ÉîÉêó=ÄäçÅâ=éìäëÉI=íç=çÄí~áå=~=Åçåíáåìçìë=éêÉÇáÅíáçå=çÑ=íÜÉ=çìíéìí=ëÉêáÉëK=rëáåÖ=EPKNUFI=~=ë~ãéäÉ=çÑ=íÜÉ=áååçî~íáçå=ëÉêáÉë=áë=çÄí~áåÉÇ=Ñçê=ÉîÉêó=çÄëÉêî~íáçå=çÑ=íÜÉ=çìíéìí=ëÉêáÉëI=ïÜÉíÜÉê=çê=åçí=ÉèìáÇáëí~åíK=^å=Éëíáã~íÉ=çÑ=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêë=áë=ã~ÇÉ=ïáíÜ=íÜÉ=~áÇ=çÑ=~=iÉîÉåÄÉêÖJj~êèì~êÇí=~äÖçêáíÜãI=ïÜáÅÜ=åìãÉêáÅ~ääó=ãáåáãáòÉë=ïÉáÖÜíÉÇ=äÉ~ëí=ëèì~êÉë=ÅêáíÉêáçå=EPKOMF=íÜ~í=áë=Ä~ëÉÇ=çå=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçåëK=cáå~ääó=íÜÉ=~ÅÅìê~Åó=~åÇ=î~äáÇáíó=çÑ=íÜÉ=ãçÇÉä=áë=ÅÜÉÅâÉÇ=ìëáåÖ=íÜÉ=~ìíçJ=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåë=çÑ=íÜÉ=áååçî~íáçåëI=íÜÉ=Åçî~êá~åÅÉ=ã~íêáñ=çÑ=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêë=~åÇ=íÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=fo=ÑìåÅíáçåëK==3.3 Example application=== UN=3.3.1 Set-up and data setqÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçå=áë=ÇÉîáëÉÇ=íç=áääìëíê~íÉ=íÜÉ=éÉêÑçêã~åÅÉ=çÑ=íÜÉ=mfocf`q=ãçÇÉä=áå=éê~ÅíáÅÉ=Äó=Åçãé~êáåÖ=áíë=Å~äáÄê~íáçå=~åÇ=î~äáÇ~íáçå=êÉëìäíë=~åÇ=é~ê~ãÉíÉê=Éëíáã~íÉë=ïáíÜ=íÜçëÉ=çÑ=ôj^=qck=ãçÇÉäë=ÑáííÉÇ=çå=íÜÉ=ë~ãÉ=Ç~í~K=cçê=íÜáë=éìêéçëÉI=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=Ñêçã=NR=éáÉòçãÉíÉêë=~êÉ=ëÉäÉÅíÉÇI=~ää=äóáåÖ=áå=~=ÇìåÉ=êÉëÉêîÉ=áå=íÜÉ=éêçîáåÅÉ=kçêíÜJeçää~åÇI=qÜÉ=kÉíÜÉêä~åÇëI=åÉ~ê=íÜÉ=íçïå=çÑ=bÖãçåÇK=máÉòçãÉíÉêë=~êÉ=ëÉäÉÅíÉÇ=~í=äçÅ~íáçåë=íÜ~í=~êÉ=äáííäÉ=ÇáëíìêÄÉÇ=Äó=íÜÉ=ÖêçìåÇï~íÉê=~Äëíê~Åíáçå=áå=íÜÉ=~êÉ~=xoçäÑ=~åÇ=iÉÄÄáåâI=NVVUzI=íÜìë=~ääçïáåÖ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=íç=ÄÉ=ãçÇÉäÉÇ=ïáíÜ=éêÉÅáéáí~íáçå=ëìêéäìë=~ë=~=ëáåÖäÉ=áåéìí=ëÉêáÉëK=qÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=áå=~ää=éáÉòçãÉíÉêë=áë=ïÉää=çÄëÉêîÉÇ=áå=íÜÉ=ë~ãÉ=éÉêáçÇI=ïáíÜ=çÄëÉêî~íáçåë=í~âÉå=ã~åì~ääó=~Äçìí=íÜÉ=NQíÜ=~åÇ=OUíÜ=çÑ=ÉîÉêó=ãçåíÜI=ëç=íÜ~í=íÜÉ=ãçÇÉä=êÉëìäíë=~êÉ=äáííäÉ=áåÑäìÉåÅÉÇ=Äó=~=ÇáÑÑÉêÉåÅÉ=áå=äÉåÖíÜ=çÑ=íÜÉ=Å~äáÄê~íáçå=éÉêáçÇ=çê=íÜÉ=åìãÄÉê=çÑ=çÄëÉêî~íáçåë=áå=íÜ~í=éÉêáçÇK=cçê=íÜÉ=Å~äáÄê~íáçå=éêçÅÉëëI=çÄëÉêî~íáçåë=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=Ñêçã=NJNJNVVM=ìåíáä=NJNJOMMN=~êÉ=ìëÉÇ=~åÇ=çÄëÉêî~íáçåë=çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=EéêÉÅáéáí~íáçå=ãáåìë=éçíÉåíá~ä=Éî~éçê~íáçåF=ëí~êíáåÖ=Ñêçã=NJNJNVUTK=qÜÉ=éêÉÅáéáí~íáçå=ëÉêáÉë=áë=~î~áä~ÄäÉ=çå=~=Ç~áäó=Ä~ëáë=~åÇ=áë=çÄëÉêîÉÇ=Äó=íÜÉ=mêçîáåÅá~ä=t~íÉê=`çãé~åó=çÑ=kçêíÜJeçää~åÇ=áå=íÜÉ=ÇìåÉë=åÉ~ê=íÜÉ=íçïå=çÑ=`~ëíêáÅìãI=ïÜÉêÉ~ë=íÜÉ=Ç~áäó=éçíÉåíá~ä=Éî~éçê~íáçå=ëÉêáÉë=çêáÖáå~íÉë=Ñêçã=~=ëí~íáçå=çÑ=íÜÉ=oçó~ä=aìíÅÜ=jÉíÉçêçäçÖáÅ=fåëíáíìíÉ=åÉ~ê=íÜÉ=íçïå=çÑ=aÉ=hççóK===^ë=íÜÉ=éÉêÑçêã~åÅÉ=çÑ=ãçÇÉäë=ã~ó=ÄÉ=ÇÉéÉåÇÉåí=çå=íÜÉ=éêçéÉêíáÉë=çÑ=~=Ç~í~=ëÉíI=íÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçå=áë=åçí=ëçäÉäó=ÑçÅìëÉÇ=çå=~ëëÉëëáåÖ=íÜÉ=éÉêÑçêã~åÅÉ=çÑ=ÄçíÜ=ãçÇÉäë=Ñçê=íÜáë=ëéÉÅáÑáÅ=Ç~í~=ëÉíI=Äìí=~äëç=çå=Åä~êáÑóáåÖ=íÜÉ=ãÉÅÜ~åáëãë=ïÜáÅÜ=áåÑäìÉåÅÉ=ãçÇÉä=éÉêÑçêã~åÅÉK=cáêëí=çÑ=~ääI=ïÉ=ïáää=áääìëíê~íÉ=íÜÉ=éÉêÑçêã~åÅÉ=çÑ=~=ê~åÖÉ=çÑ=ôj^=qck=ãçÇÉäëI=íÜÉáê=ÇÉéÉåÇÉåÅÉ=çå=ìëÉê=ÇÉÑáåÉÇ=ÅÜçáÅÉë=ëìÅÜ=~ë=ãçÇÉä=çêÇÉê=~åÇ=íáãÉ=ÇáëÅêÉíáò~íáçåI=~åÇ=íÜÉ=ÇÉéÉåÇÉåÅÉ=çÑ=íÜÉ=êÉëìäíë=çÑ=mfocf`q=ãçÇÉäë=ÑáííÉÇ=çå=íÜÉ=ë~ãÉ=Ç~í~=Ñêçã=~å=Éñ~ãéäÉ=ëÉêáÉëK=cçê=íÜáë=éìêéçëÉI=éáÉòçãÉíÉê=NV^wtOQS|N=áë=ëÉäÉÅíÉÇK=^í=íÜáë=äçÅ~íáçå=íÜÉ=ï~íÉê=í~ÄäÉ=ëÜçïÉÇ=íÜÉ=ëäçïÉëí=êÉëéçåëÉ=íç=éêÉÅáéáí~íáçå=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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ëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=table 3.1: Calibration results for ARMA TFN models of different order and PIRFICT modelscalibrated on the same data (piezometer 19AZW246_1).= ôj^=qck=ãçÇÉäë== =[ nr, ns, np, nq, b][ ∆ t = ∆ t ] =hp=xR=N=M=N=Mz= =xNM=N=M=N=Mz=xPMKQz= xPMKQz==xOM=N=M=N=Mz=xPMKQz=xOM=N=M=N=Mz=xNRKOz==xR=N=M=N=Mz=xSMKUz=ojpb=EÅãF= NTKUO= NSKPV= NTKMU= NSKRU= NTKPP=ojpf=EÅãF= UKSN= UKQO= UKMS= TKMQ= NMKQN=bsm=EBF= UOKPR= URKMU= UPKTV= UQKUP= UPKRV== mfocf`q=ãçÇÉä= ==== UP=ojpb=EÅãF= = NSKOR= = NSKQN= NTKQP=ojpf=EÅãF= = VKMP= = TKOT= NMKRO=bsm=EBF= = URKPO= = URKNQ= UPKPV==íÜÉ=êÉëáÇì~ä=ëÉêáÉë=Å~å=ÄÉ=áåÑäìÉåÅÉÇ=Äó=áåíÉêéçä~íáçå=~åÇ=êÉë~ãéäáåÖ=çéÉê~íáçåëI=~äëç=íÜÉ=Éñéä~áåÉÇ=î~êá~åÅÉ=éÉêÅÉåí~ÖÉ=EbsmF=áë=ÖáîÉåK=qÜÉ=bsm=áë=ÇÉÑáåÉÇ=~ëW==2 2σh( t) −σn( t)EVP = *100%(3.22)2σh( t)=^=äçÖáÅ~ä=ï~ó=çÑ=Åçãé~êáåÖ=ãçÇÉä=êÉëìäíë=ëÉÉãë=íç=ÄÉ=íÜÉ=ìëÉ=çÑ=~ìíçã~íáÅ=ãçÇÉä=çêÇÉê=ëÉäÉÅíáçå=ÅêáíÉêá~=ëìÅÜ=~ë=^f`=~åÇ=cmbK=eçïÉîÉêI=ÄçíÜ=ÅêáíÉêá~=ìëÉ=íÜÉ=áååçî~íáçå=î~êá~åÅÉ=çê=íÜÉáê=äáâÉäáÜççÇ=Ñçê=ÇÉíÉêãáåáåÖ=íÜÉ=ÄÉëí=ãçÇÉä=çêÇÉêI=ïÜáÅÜ=~êÉ=áåÑäìÉåÅÉÇ=Äó=íÜÉ=ë~ãéäÉ=ÑêÉèìÉåÅóI=ëç=íÜÉëÉ=ÅêáíÉêá~=Å~ååçí=ÄÉ=ìëÉÇ=íç=Åçãé~êÉ=ãçÇÉäë=ïáíÜ=ÇáÑÑÉêÉåí=ë~ãéäÉ=ÑêÉèìÉåÅáÉë=çê=Ç~í~=ëÉíëK=cêçã=íÜÉ=êÉëìäíë=áå=í~ÄäÉ=PKN=áí=Å~å=Ñáêëí=çÑ=~ää=ÄÉ=ëÉÉå=íÜ~í=íÜÉ=ÇáÑÑÉêÉåÅÉë=ÄÉíïÉÉå=íÜÉ=Ñáí=çÑ=ÄçíÜ=ãçÇÉäë=~êÉ=ëã~ääK=^äíÜçìÖÜ=íÜÉ=mfocf`q=ãçÇÉä=ÇçÉë=ëÜçï=íÜÉ=äçïÉëí=ojpb=~åÇ=ÜáÖÜÉëí=bsmI=íÜÉ=î~êá~íáçå=Å~ìëÉÇ=Äó=ÇáÑÑÉêÉåÅÉë=áå=çÄëÉêî~íáçå=ÑêÉèìÉåÅó=~åÇ=ãçÇÉä=çêÇÉê=áë=ä~êÖÉê==íÜ~å=íÜÉ=ÇáÑÑÉêÉåÅÉë=ÄÉíïÉÉå=ÄçíÜ=ãçÇÉä=íóéÉëK=qÜÉ=çéíáã~ä=êÉëìäí=Ñçê=íÜÉ=ôj^=ãçÇÉä=~ééÉ~êë=íç=ÄÉ=ÖáîÉå=Äó=íÜÉ=xNM=N=M=N=Mz=xPMKQz=ãçÇÉäI=ëç=ïÉ=Å~å=áÇÉåíáÑó=íÜáë=Ñçê=íÜÉ=ãçãÉåí=~ë=íÜÉ=çéíáã~ä=ãçÇÉä=çêÇÉêK=^ë=ÉñéÉÅíÉÇI=íÜÉ=ojpf=çÑ=ÄçíÜ=ãçÇÉäë=î~êáÉë=ïáíÜ=íÜÉ=íáãÉ=ä~Ö=ÄÉíïÉÉå=íÜÉ=çÄëÉêî~íáçåë=çÑ=íÜÉ=çìíéìí=ëÉêáÉëK=tÜÉå= ∆ t = =NRKOI=íÜÉ=ojpf=äáÉë=áå=íÜÉ=çêÇÉê=çÑ=T=íç=U=ÅãI=ïÜÉêÉ~ë=íÜÉ=ojpf=áë=V=íç=NM=Åã=ïÜÉå= ∆ t h= =SMKUK=cìêíÜÉêãçêÉI=íÜÉ=êÉëìäíë=ëÜçï=íÜ~í=íÜÉ=ojpf=çÑ=íÜÉ=ôj^=qck=ãçÇÉä=ÇÉÅêÉ~ëÉë=ïÜÉå=íÜÉ=åìãÄÉê=çÑ=é~ê~ãÉíÉêë=áåÅêÉ~ëÉëI=ïÜáäÉ=íÜÉ=ojpb=ëÜçïë=~å=çéíáãìã=Ñçê=íÜÉ=xNM=N=M=N=Mz=ãçÇÉä=Ñçê= ∆ t h= =PMKQK=qÜáë=éÜÉåçãÉåçå=ÅçìäÇ=ïÉää=ÄÉ=~ííêáÄìíÉÇ=íç=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=ãçÇÉäë=~êÉ=Å~äáÄê~íÉÇ=Äó=ãáåáãáòáåÖ=íÜÉ=ojpf=EïÜáÅÜ=áë=äáåâÉÇ=íç=íÜÉ=åçáëÉ=é~êí=çÑ=íÜÉ=ãçÇÉäF=ê~íÜÉê=íÜ~å=íÜÉ=ojpb=EïÜáÅÜ=áë=äáåâÉÇ=íç=íÜÉ=íê~åëÑÉê=ãçÇÉäFK=_ÉÅ~ìëÉ=çÑ=íÜáëI=~ÇÇáåÖ=Éñíê~=é~ê~ãÉíÉêë=~åÇ=íÜÉêÉÄó=çîÉêÑáííáåÖ=íÜÉ=Ç~í~=ïáää=êÉëìäí=áå=~=Öê~Çì~ääó=áãéêçîáåÖ=Ñáí=çÑ=íÜÉ=åçáëÉ=ãçÇÉäI=Äìí=Å~å=~í=íÜÉ=ë~ãÉ=Ü~îÉ=~=åÉÖ~íáîÉ=ÉÑÑÉÅí=çå=íÜÉ=Ñáí=çÑ=íÜÉ=íê~åëÑÉê=ãçÇÉäK=lîÉêÑáííáåÖ=ÄÉÜ~îáçêI=çê=íÜÉ=ÖÉåÉê~ääó=ÜáÖÜÉê=hKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=P=2.52figure a)nb=5nb=10nb=202.52figure b)Box−JenkinsPIRFICTresponse (−)1.51response (−)1.510.50.5UQ=00 200 400 600 800 1000time (in days)00 200 400 600 800 1000time (in days)2.52figure c)th=tp=15.2th=tp=30.4th=tp=60.82.52figure d)th=tp=15.2th=tp=30.4th=tp=60.8response (−)1.51response (−)1.510.50.500 200 400 600 800 1000time (in days)00 200 400 600 800 1000time (in days)=figure 3.5: BR functions of several TFN models of piezometer 19AZW246_1. a) Three [nr 1 0 1 0][30.4] ARMA models with nr = 5, 10 and 20. b) An [10 1 0 1 0][30.4] ARMA model and a PIRFICTmodel using the same data. c) three [10 1 0 1 0][ ∆ t ] ARMA models with ∆ t = 15.2, 30.4 and 60.8.d) Three PIRFICT models with h p∆ t = ∆ t = 15.2, 30.4 and 60.8. The dotted lines denote the 95%confidence interval of the nb = 5 model in a), both models in b), and the∆ t = 30.4 model in c) and d).åìãÄÉê=çÑ=é~ê~ãÉíÉêë=çÑ=~å=ôj^=ãçÇÉäI=ÅçìäÇ=~äëç=Éñéä~áå=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=ojpf=çÑ=íÜÉ=ôj^=qck=ãçÇÉä=áë=äçïÉê=íÜ~å=íÜ~í=çÑ=íÜÉ=mfocf`q=ãçÇÉäI=ïÜáäÉ=áíë=ojpb=áë=ÜáÖÜÉêK==qÜÉ=é~ê~ãÉíÉê=Éëíáã~íÉë=çÑ=íÜÉ=ÇáÑÑÉêÉåí=ãçÇÉäë=Å~å=ÄÉëí=ÄÉ=Åçãé~êÉÇ=Äó=éäçííáåÖ=íÜÉ=Éëíáã~íÉÇ=_o=ÑìåÅíáçåëK=få=ÑáÖìêÉ=PKRI=íÜÉ=íê~åëÑÉê=ÑìåÅíáçåë=çÑ=íÜêÉÉ=x nr =N=M=N=MzxPMKQz=ôj^=ãçÇÉäë=~êÉ=éäçííÉÇI=ïáíÜ= nr = 5 I=NM=~åÇ=OMK=qÜÉ=êÉëìäíë=~ééÉ~ê=íç=ÄÉ=ëáÖåáÑáÅ~åíäó=áåÑäìÉåÅÉÇ=Äó=íÜÉ=ãçÇÉä=çêÇÉêI=~ë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçåë=çÑ=íÜÉ= nr = 10 =~åÇ=OM=ãçÇÉä=äáÉ=é~êíäó=çìíëáÇÉ=íÜÉ=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=çÑ=íÜ~í=çÑ=íÜÉ= nr = 5 =ãçÇÉäK=qÜÉ=åìãÄÉê=çÑ=j^=é~ê~ãÉíÉêë=Ñçê=íÜÉ=Å~ëÉ= nr = 5 =áë=~éé~êÉåíäó=íçç=äçï=íç=ãçÇÉä=íÜÉ=ëäçï=êÉëéçåëÉ=çÑ=íÜÉ=ëóëíÉã=ïÉääK=^ÅÅçêÇáåÖ=íç=íÜáë=ÑáÖìêÉ=~åÇ=íÜÉ=ojpbI=áå=íÜáë=Å~ëÉ=NM=j^=é~ê~ãÉíÉêë=àìëí=~Äçìí=ëìÑÑáÅÉ=íç=ãçÇÉä=íÜÉ=Ñáêëí=é~êí=çÑ=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=ëóëíÉãI=ïÜáäÉ=íÜÉ=êÉã~áåÇÉê=çÑ=íÜÉ=êÉëéçåëÉ=ÑìåÅíáçå=Å~å=ÄÉ=ÇÉëÅêáÄÉÇ=~ÇÉèì~íÉäó=Äó=~=ëáåÖäÉ=ô=é~ê~ãÉíÉêK=cêçã=ÑáÖìêÉ=PKRÄ=áí=Å~å=ÄÉ=ëÉÉå=íÜ~í=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=..…………………………………………………………………………………………….….

ëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=mfocf`q=ãçÇÉä=Ñçääçïë=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=xNM=N=M=N=MzxPMKQz=ôj^=ãçÇÉä=ê~íÜÉê=ÅäçëÉäóK=qÜÉ=êÉëìäíë=çÑ=ÄçíÜ=ãçÇÉäë=ëÜçìäÇ=íÜÉêÉÑçêÉ=ÄÉ=ÜáÖÜäó=Åçãé~ê~ÄäÉI=~ë=íÜÉ=çåäó=ÇáÑÑÉêÉåÅÉ=áå=íÜÉ=íê~åëÑÉê=ÑìåÅíáçåë=áë=íÜÉ=áêêÉÖìä~ê=é~ííÉêå=çÑ=íÜÉ=ôj^=ãçÇÉä=~êçìåÇ=íÜÉ=ëãççíÜ=ÅìêîÉ=çÑ=íÜÉ=mfocf`q=ãçÇÉäK=få=ÑáÖìêÉ=PKRÅ=íÜÉ=çêÇÉê=çÑ=íÜÉ=ôj^=ãçÇÉäI=áå=íÜÉ=íê~Çáíáçå~ä=ëÉåëÉI=áë=âÉéí=Åçåëí~åí=ïÜáäÉ=íÜÉ=ë~ãéäÉ=ÑêÉèìÉåÅó=áë=î~êáÉÇI=êÉëìäíáåÖ=áå=íÜêÉÉ=[10 1 0 1 0][ ∆t]ôj^=ãçÇÉäë=ïáíÜ= ∆ t = =NRKOI=PMKQ=~åÇ=SMKUK=^ë=ÉñéÉÅíÉÇI=íÜÉ=é~ê~ãÉíÉê=Éëíáã~íÉë=çÑ=íÜÉ=ôj^=ãçÇÉä=éêçîÉ=íç=ÄÉ=~äëç=ëáÖåáÑáÅ~åíäó=áåÑäìÉåÅÉÇ=Äó=íÜÉ=ë~ãéäÉ=ÑêÉèìÉåÅóI=ïÜáÅÜ=áåíÉêÑÉêÉë=ïáíÜ=íÜÉ=ãçÇÉä=çêÇÉê=ïÜÉå=íáãÉ=áë=ìëÉÇ=~ë=~=ÇáãÉåëáçåäÉëë=áåÇÉñK=cáå~ääóI=áå=ÑáÖìêÉ=PKRÇI=íÜÉ=_o=ÑìåÅíáçåë=çÑ=íÜêÉÉ=mfocf`q=ãçÇÉäë=~êÉ=éäçííÉÇI=Å~äáÄê~íÉÇ=çå=íÜÉ=ë~ãÉ=Ç~í~=~ë=íÜÉ=ôj^=ãçÇÉä=EáKÉK=∆ t = NRKOI=PMKQ=~åÇ=SMKUFK=cêçã=íÜÉ=êÉëìäíëI=íÜÉ=Ñáêëí=íïç=_o=ÑìåÅíáçåë=éêçîÉ=íç=ÄÉ=~äãçëí=áÇÉåíáÅ~äI=ïÜáäÉ=íÜÉ=ãçÇÉä=~éé~êÉåíäó=Ü~ë=ÇáÑÑáÅìäíáÉë=Éëíáã~íáåÖ=íÜÉ=Ñáêëí=é~êí=çÑ=íÜÉ=_o=ÑìåÅíáçå=ÅçêêÉÅíäó=Ñçê= ∆ t = =SMKUI=~äíÜçìÖÜ=áí=ÇçÉë=åçí=ÇáÑÑÉê=ëáÖåáÑáÅ~åíäóK=qÜáë=ÉÑÑÉÅí=áë=éêçÄ~Ääó=Å~ìëÉÇ=Äó=íÜÉ=áåÅêÉ~ëáåÖ=íáãÉ=áåíÉêî~ä=~åÇLçê=ÇÉÅêÉ~ëáåÖ=åìãÄÉê=çÑ=çÄëÉêî~íáçåëI=~ë=íÜÉ=ôj^=xSMKUz=ãçÇÉä=Éëíáã~íÉë=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=áå=~Äçìí=íÜÉ=ë~ãÉ=ï~óK==== UR=3.3.3 Multiple series and validation studycçê=~=Äêç~ÇÉê=Åçãé~êáëçå=ÄÉíïÉÉå=íÜÉ=ôj^=~åÇ=mfocf`q=qck=ãçÇÉäI=ÄçíÜ=ãçÇÉäë=~êÉ=Å~äáÄê~íÉÇ=çå=íáãÉ=ëÉêáÉë=Ñêçã=NR=éáÉòçãÉíÉêë=ïáíÜ=çÄëÉêî~íáçåë=ê~åÖáåÖ=Ñêçã=NJNJNVVM=ìåíáä=NJNJOMMNK=cçê=êÉ~ëçåë=çÑ=Åçãé~ê~Äáäáíó=~åÇ=çÄàÉÅíáîáíóI=åç=_çñ=~åÇ=gÉåâáåë=ëíóäÉ=áíÉê~íáîÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉÇìêÉ=áë=éÉêÑçêãÉÇ=Ñçê=É~ÅÜ=ëÉé~ê~íÉ=5.85.65.4PIRFICT modelBJ modelObservationsgroundwater level (m)5.254.84.64.44.241976 1979 1982 1984 1987 1990 1993 1995 1998 2001time (date)=figure 3.6: Simulations from a [10 1 0 1 0][30.4] ARMA model and a [30.4 30.4] PIRFICT modelcalibrated on observations of the water table depth of piezometer 19AZW246_1 between 1990 and2001. From the figure it can be seen that the simulation results in both the calibration and validationperiod are similar.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=P=US=éáÉòçãÉíÉê=íç=çÄí~áå=íÜÉ=ãçëí=~ééêçéêá~íÉ=ãçÇÉä=çêÇÉê=Eáå=çìê=Å~ëÉ=ÇÉÑáåÉÇ=Äó= nr,b =~åÇ= ∆thFK=fåëíÉ~ÇI=~=xNM=M=N=N= b zxPMKQz=ôj^=ãçÇÉä=áë=Å~äáÄê~íÉÇ=çå=~ää=éáÉòçãÉíÉêëI=ïÜáÅÜ=éêçîÉë=íç=ÖáîÉ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=àìëí=~Äçìí=ÉåçìÖÜ=j^=é~ê~ãÉíÉêë=íç=ãçÇÉä=íÜÉ=ëäçïÉëí=êÉëéçåëÉ=çÑ=íÜÉ=NR=éáÉòçãÉíÉêë=ïÉää=EëÉÉ=ëÉÅíáçå=PKOFK=cçê=~ää=ãçÇÉäëI=~=ÇÉä~ó=íáãÉ=áë=~ééäáÉÇI=ïÜáÅÜ=Ü~ë=ëÜçïå=íç=áãéêçîÉ=ÄçíÜ=íÜÉ=Å~äáÄê~íáçå=~åÇ=î~äáÇ~íáçå=êÉëìäíëK=^ë=ÉñéÉÅíÉÇI=íÜÉ=áãéêçîÉãÉåíë=~êÉ=ÖêÉ~íÉëí=Ñçê=íÜÉ=mfocf`q=ãçÇÉä=ÄÉÅ~ìëÉ=çÑ=áíë=éêÉÇÉÑáåÉÇ=ëÜ~éÉK=lå=íÜÉ=Ä~ëáë=çÑ=~=ã~åì~ä=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉÇìêÉI=íÜÉ=ÇÉä~ó=íáãÉ=Ñçê=íÜÉ=NR=éáÉòçãÉíÉêë=áë=ÅÜçëÉå=íç=ÄÉ= b = xNO=S=M=M=M=P=N=NP=NP=S=S=U=N=R=Mz=Ç~óëK=fí=áë=~ééäáÉÇ=Äó=ëÜáÑíáåÖ=íÜÉ=éêÉÅáéáí~íáçå=ëÉêáÉë=~äçåÖ=íÜÉ=íáãÉ=~ñáëK=pÜáÑíáåÖ=íÜÉ=áåéìí=ëÉêáÉë=ê~íÜÉê=íÜ~å=íÜÉ=êÉëéçåëÉ=ÑìåÅíáçå=ã~âÉë=áí=éçëëáÄäÉ=íç=~ééäó=ÇÉä~ó=íáãÉë=ïÜáÅÜ=~êÉ=ëã~ääÉê=íÜ~å=íÜÉ=ÇáëÅêÉíÉJíáãÉ=áåíÉêî~äI=~åÇ=Äó=ÇçáåÖ=ëç=íÜÉ=ÇÉä~ó=íáãÉ=ÅçìäÇ=ÄÉ=âÉéí=Éèì~ä=Ñçê=íÜÉ=ôj^=~åÇ=mfocf`q=ãçÇÉäK=qÜÉ=mfocf`q=ãçÇÉä=áë=Ñáêëí=Å~äáÄê~íÉÇ=ìëáåÖ=íÜÉ=ë~ãÉ=Ç~í~=~ë=íÜÉ=ôj^=ãçÇÉäI=ïÜáÅÜ=áë=íÜÉêÉÑçêÉ=ÇÉåçíÉÇ=~ë=ÉèìáÇáëí~åí=xPMKQ=PMKQzI=~åÇ=ëÉÅçåÇ=ìëáåÖ=åçåJÉèìáÇáëí~åí=xPMKQ=Nz=Ç~í~K=qÜÉ=êÉëìäíë=ëÜçìäÇ=íÜÉêÉÑçêÉ=ëÜçï=íÜÉ=ÅçãÄáåÉÇ=ÉÑÑÉÅí=çÑ=íÜÉ=áåíÉêéçä~íáçå=çéÉê~íáçåë=åÉÉÇÉÇ=íç=ã~âÉ=íÜÉ=Ç~í~=ÉèìáÇáëí~åí=~åÇ=íÜÉ=ÉÑÑÉÅí=çÑ=ëìããáåÖ=íÜÉ=Ç~áäó=éêÉÅáéáí~íáçå=ëìêéäìë=ëÉêáÉë=áåíç=PMKQ=Ç~ó=íçí~äëK=få=~ÇÇáíáçåI=Ñçê=ëáñ=çÑ=íÜÉ=éáÉòçãÉíÉêëI=~=î~äáÇ~íáçå=áë=éÉêÑçêãÉÇ=çå=íÜÉ=çÄëÉêî~íáçåë=í~âÉå=ÄÉÑçêÉ=NJNJNVVMK=^=íáãÉ=éäçí=çÑ=íÜÉ=~î~áä~ÄäÉ=çÄëÉêî~íáçåë=áå=ÄçíÜ=íÜÉ=Å~äáÄê~íáçå=~åÇ=î~äáÇ~íáçå=éÉêáçÇ=Ñçê=éáÉòçãÉíÉê=NV~òïOQS|N=áë=ëÜçïå=áå=ÑáÖìêÉ=PKSI=~äçåÖ=ïáíÜ=éêÉÇáÅíáçåë=çÑ=~=xNM=N=M=N=MzxPMKQz=ôj^=ãçÇÉä=~åÇ=~=mfocf`q=ãçÇÉä=ìëáåÖ=íÜÉ=ë~ãÉ=Ç~í~K=qÜÉ=ÑáÖìêÉ=ÅäÉ~êäó=ëÜçïë=íÜ~í=íÜÉ=çÄëÉêî~íáçå=ÑêÉèìÉåÅó=áå=íÜáë=éÉêáçÇ=áë=åçí=Éèì~ä=íç=íÜ~í=çÑ=íÜÉ=Å~äáÄê~íáçå=éÉêáçÇI=Äìí=Ü~ë=ÄÉÉå=ÅÜ~åÖÉÇ=~í=íÜÉ=ÉåÇ=çÑ=NVUV=Ñêçã=Q=íáãÉë=~=óÉ~ê=áåíç=OQ=íáãÉë=~=óÉ~êI=çå=~îÉê~ÖÉK=_ÉÅ~ìëÉ=íÜÉ=ÑêÉèìÉåÅó=çÑ=íÜÉ=çÄëÉêî~íáçåë=áë=ãìÅÜ=äçïÉê=íÜ~å=íÜÉ=ÑêÉèìÉåÅó=çÑ=íÜÉ=éêÉÇáÅíáçåëI=íÜÉ=éêÉÇáÅíÉÇ=î~äìÉë=~êÉ=äáåÉ~êäó=áåíÉêéçä~íÉÇ=íç=ã~íÅÜ=íÜÉ=Ç~íÉë=çÑ=íÜÉ=çÄëÉêî~íáçåëI=~åÇ=åçí=îáÅÉ=îÉêë~=~ë=áå=íÜÉ=Å~äáÄê~íáçå=êçìíáåÉK=cçê=íÜÉ=mfocf`q=ãçÇÉäI=íÜÉ=ëáãìä~íÉÇ=î~äìÉë=Çç=åçí=Ü~îÉ=íç=ÄÉ=áåíÉêéçä~íÉÇI=~ë=íÜÉ=ëáãìä~íÉÇ=ï~íÉê=í~ÄäÉ=ÇÉéíÜ=áë=Åçåíáåìçìëäó=ÇÉÑáåÉÇK===qÜÉ=~îÉê~ÖÉ=êÉëìäíë=çÑ=~ää=NR=éáÉòçãÉíÉêë=~êÉ=ëìãã~êáòÉÇ=áå=í~ÄäÉ=PKOK=^ë=áå=í~ÄäÉ=PKNI=íÜÉ=ojpbI=ojpfI=~åÇ=bsm=~êÉ=ÖáîÉåI=~äçåÖ=ïáíÜ=íÜÉ=î~äáÇ~íáçå=ojpbK=^Ö~áåI=íÜÉ=ÇáÑÑÉêÉåÅÉë=~êÉ=ëã~ää=~ë=íÜÉ=~îÉê~ÖÉ=ojpb=çÑ=íÜÉ=xNM=N=M=N=ÄzxPMKQ=PMKQz=ôj^=ãçÇÉäI=ïáíÜ=~=ÇáÑÑÉêÉåÅÉ=çÑ=çåäó=MKU=ãáääáãÉíÉêI=~äãçëí=Éèì~äë=íÜÉ=ojpb=çÑ=íÜÉ=åçåJÉèìáÇáëí~åí=xPMKQ=Nz=mfocf`q=ãçÇÉäK=få=T=çìí=çÑ=íÜÉ=NR=éáÉòçãÉíÉêë=íÜÉ=Ñáí=áë=table 3.2: Average calibration and validation results ofthe ARMA and PIRFICT model for 15 and 6 piezometersrespectively.= ôj^=xNM=N=M=N=Mz=xPMKQz=mfocf`q=xPMKQ=PMKQz=ÉèìáK=mfocf`q=xPMKQ=Nz=åçåJÉèìáK=ojpb=EÅãF= NOKQO= NOKRS= NOKRM=ojpf=EÅãF= TKOM= TKSR= TKUV=bsm=EBF= UUKSS= UUKPT= UUKPS=sJojpb=EÅãF= NRKTS= NRKSQ= NRKPU==ëäáÖÜíäó=ÄÉííÉê=ïÜÉå=íÜÉ=mfocf`q=ãçÇÉä=áë=ìëÉÇI=ïÜÉêÉ~ë=U=éáÉòçãÉíÉêë=ëÜçï=~=ÄÉííÉê=êÉëìäí=ïáíÜ=íÜÉ=ôj^=ãçÇÉäK=^í=íÜÉ=ë~ãÉ=íáãÉI=ÜçïÉîÉêI=íÜÉ=ojpf=çÑ=íÜÉ=ôj^=ãçÇÉä=áë=äçïÉê=Ñçê=~ää=éáÉòçãÉíÉêëI=ïÜÉêÉ~ë=íÜÉ=çééçëáíÉ=áë=íêìÉ=Ñçê=íÜÉ=î~äáÇ~íáçå=ojpbK=qÜÉ=äçïÉê=ojpf=áë=éêçÄ~Ääó=íç=~=ÅÉêí~áå=ÉñíÉåí=Å~ìëÉÇ=Äó=íÜÉ=áåíÉêéçä~íáçå=..…………………………………………………………………………………………….….

ëáåÖ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçåë=2.5 figure a)Box−JenkinsPIRFICT22.5 figure b)Box−JenkinsPIRFICT2response (−)1.51response (−)1.510.500 100 200 300 400 500time (in days)0.500 200 400 600 800 1000time (in days)=figure 3.7: Transfer functions of a [10 1 0 1 b][30.4] ARMA TFN model and a PIRFICT modelcalibrated on the same data from piezometers 19AZL5038_1, shown in a), and 19AZW195_1 shownin b). The dotted lines denote the 95% confidence intervals.==== UT=çéÉê~íáçåë=íÜ~í=ïÉêÉ=Å~êêáÉÇ=çìí=çå=íÜÉ=Ç~í~=íç=ã~âÉ=áí=ÉèìáÇáëí~åíI=ïÜáÅÜ=íÉåÇë=íç=ëãççíÜ=ÜáÖÜJÑêÉèìÉåí=î~êá~íáçåë=çÑ=~=ëáÖå~äK=qÜáë=áë=ÅçêêçÄçê~íÉÇ=Äó=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=mfocf`q=ãçÇÉä=óáÉäÇë=~=ojpf=íÜ~í=áë=OKQ=ãáääáãÉíÉê=ëã~ääÉê=Ñçê=íÜÉ=ÉèìáÇáëí~åí=Ç~í~=íÜ~å=Ñçê=íÜÉ=åçåJÉèìáÇáëí~åí=Ç~í~K=eçïÉîÉêI=íÜÉ=ojpf=çÑ=íÜÉ=ôj^=ãçÇÉä=áë=çå=~îÉê~ÖÉ=QKR=ãáääáãÉíÉê=ëã~ääÉê=íÜ~å=íÜ~í=çÑ=íÜÉ=xPMKQ=PMKQz=mfocf`q=ãçÇÉä=ïÜáäÉ=íÜÉ=sojpb=çÑ=íÜÉ=ôj^=ãçÇÉä=áë=ÜáÖÜÉêK=qÜáëI=~Ö~áåI=éçáåíë=íç=çîÉêÑáííáåÖ=ÄÉÜ~îáçê=çÑ=ôj^=ãçÇÉäëK=qÜÉ=ôj^=ãçÇÉä=áë=~ÄäÉ=íç=éêçÇìÅÉ=~=äçïÉê=î~äìÉ=çÑ=íÜÉ=çÄàÉÅíáîÉ=ÑìåÅíáçå=ìëáåÖ=íÜÉ=ë~ãÉ=Ç~í~I=Äìí=~éé~êÉåíäó=íÜÉ=ÅçêêÉä~íáçåë=ÑáííÉÇ=~êÉ=íç=ëçãÉ=ÉñíÉåí=ÅçáåÅáÇÉåí~ä=~ë=íÜÉó=~êÉ=~ÅÅçãé~åáÉÇ=Äó=~=ÜáÖÜÉê=sojpbK=tÜÉå=íÜÉ=êÉëéçåëÉ=ÑìåÅíáçåë=çÑ=ÄçíÜ=ãçÇÉäë=~êÉ=Åçãé~êÉÇI=íÜÉ=ëãççíÜ=Åçåíáåìçìë=_o=ÑìåÅíáçåë=éêçîÉ=íç=Ñáí=íÜÉ=ÖÉåÉê~ä=ëÜ~éÉ=çÑ=íÜÉ=ãçêÉ=áêêÉÖìä~ê=ÇáëÅêÉíÉ=íê~åëÑÉê=ÑìåÅíáçåë=ïÉää=Ñçê=~ää=éáÉòçãÉíÉêëI=áåÇáÅ~íáåÖ=íÜ~í=íÜÉ=mfocf`q=ãçÇÉä=Å~å=~ÇÉèì~íÉäó=ãçÇÉä=éÜêÉ~íáÅ=ëóëíÉãë=ïáíÜ=~=êÉä~íáîÉäó=Ñ~ëíI=áåíÉêãÉÇá~íÉ=~åÇ=ëäçï=êÉëéçåëÉ=EÑáÖìêÉ=PKT~=~åÇ=Ä=~åÇ=ÑáÖìêÉ=PKRÄFK==3.3.4 A small simulation study_ÉÅ~ìëÉ=ïÉ=ïÉêÉ=åçí=~ÄäÉ=íç=éÉêÑçêã=~=ÅêçëëJî~äáÇ~íáçå=~åÇ=Å~ååçí=ÇÉíÉêãáåÉ=íÜÉ=ÚíêìÉÛ=êÉëéçåëÉ=çÑ=íÜÉ=êÉ~ä=ïçêäÇ=éÜêÉ~íáÅ=ëóëíÉãëI=íÜÉêÉ=áë=çåäó=áåÇáêÉÅí=ÉîáÇÉåÅÉ=íÜ~í=íÜÉ=áêêÉÖìä~ê=ëÜ~éÉ=çÑ=íÜÉ=ôj^=íê~åëÑÉê=ÑìåÅíáçå=áë=íç=ëçãÉ=ÉñíÉåí=ÅçáåÅáÇÉåí~ä=çê=ÇìÉ=íç=çîÉêÑáííáåÖ=ÄÉÜ~îáçêK=få=çêÇÉê=íç=Ü~îÉ=ãçêÉ=ÇáêÉÅí=ÉîáÇÉåÅÉI=ïÉ=éÉêÑçêãÉÇ=~=ëÜçêí=ëíìÇó=áå=ïÜáÅÜ=ïÉ=ëáãìä~íÉÇ=~=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=Äó=íê~åëÑçêãáåÖ=~=Ç~áäó=ÑêÉèìÉåÅó=éêÉÅáéáí~íáçå=ëìêéäìë=ëÉêáÉë=áå=~=ëóåíÜÉíáÅ=ëóëíÉã=ïáíÜ=~=âåçïå=êÉëéçåëÉ=E~=mfff=fo=ÑìåÅíáçå=ïáíÜ= A = NRMMI= a = MKMMOI= n = NKRI= b = 0 FK=qç=íÜÉ=ëÉêáÉë=íÜìë=çÄí~áåÉÇI=ïÉ=~ÇÇÉÇ=~=êÉëáÇì~ä=ëÉêáÉë=íÜ~í=ï~ë=ÖÉåÉê~íÉÇ=Äó=ÅçåîçäìíáåÖ=~=ÚÇáëÅêÉíÉ=ïÜáíÉ=åçáëÉÛ=ëÉêáÉë=çÑ=åçêã~ääó=ÇáëíêáÄìíÉÇ=ê~åÇçã=åìãÄÉêë=ïáíÜ=~=âåçïåI=ÉñéçåÉåíá~ä=åçáëÉ=fo=ÑìåÅíáçå=Eα = =ORI= σ ν = 2.5 =Åã=çå=~=Ç~áäó=ÑêÉèìÉåÅóFK=cêçã=íÜÉ=ëÉêáÉë=íÜìë=çÄí~áåÉÇ=íÜÉ=é~ê~ãÉíÉêë=íÜ~í=ïÉ=ëÜçìäÇ=Ü~îÉ=êÉíêáÉîÉÇ=ìëáåÖ=íÜÉ=qck=ãçÇÉäë=~êÉ=âåçïå=Éñ~ÅíäóK=qÜÉ=é~ê~ãÉíÉê=Éëíáã~íÉë=çÑ=~=xNM=N=M=N=MzxPMKQz=ôj^=ãçÇÉä=~åÇ=~=mfocf`q=ãçÇÉä=~êÉ=ÇÉéáÅíÉÇ=áå=ÑáÖìêÉ=PKU=~ë=_o=ÑìåÅíáçåëI=~äçåÖ=ïáíÜ=íÜÉ=íêìÉ=_o=ÑìåÅíáçå=çÑ=íÜÉ=ëóåíÜÉíáÅ=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=P=UU=ëóëíÉãK=qÜÉ=Éëíáã~íÉÇ=é~ê~ãÉíÉêë=çÑ=íÜÉ=åçáëÉ=ãçÇÉä=~êÉ=xα = =OTKVI=σ = =OKQP=Åãz=~åÇ=α = =OVKSI=νσ ν = OKPV=Åãz=Ñçê=íÜÉ=mfocf`q=~åÇ=ôj^=ãçÇÉä=êÉëéÉÅíáîÉäóK=^ë=ï~ë=íÜÉ=Å~ëÉ=ïáíÜ=íÜÉ=êÉ~ä=ïçêäÇ=Ç~í~I=íÜÉ=ôj^=íê~åëÑÉê=ÑìåÅíáçå=ëÜçïë=~=é~êíäó=áêêÉÖìä~ê=é~ííÉêå=~åÇ=áå=íÜáë=Å~ëÉ=ÉîÉå=Ñ~ääë=é~êíäó=çìíëáÇÉ=íÜÉ=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=çÑ=íÜÉ=mfocf`q=êÉëéçåëÉK=tÜÉå=Åçãé~êáåÖ=íÜÉ=Éëíáã~íÉë=çÑ=ÄçíÜ=ãçÇÉäë=ïáíÜ=íÜÉ=íêìÉ=êÉëéçåëÉ=ÑìåÅíáçå=íÜÉ=ÇÉîá~íáçå=çÑ=íÜÉ=Åçåíáåìçìë=_o=ÑìåÅíáçå=áë=äÉëë=íÜ~å=íÜ~í=çÑ=íÜÉ=ÇáëÅêÉíÉ=íê~åëÑÉê=ÑìåÅíáçåI=~ë=áë=íÜÉ=Å~ëÉ=Ñçê=íÜÉ=é~ê~ãÉíÉêë=çÑ=íÜÉ=åçáëÉ=ãçÇÉäK===3.4 Discussion andconclusionsresponse (−)1.81.61.41.210.80.60.40.2Box−JenkinsPIRFICTTrue response00 200 400 600 800 1000time (in days)=figure 3.8: Transfer functions of a [10 0 1 1 0] [30.4]ARMA model and a PIRFICT model calibrated on thesame simulated series. The dotted lines denote the 95%confidence intervals. The figure illustrates that the partlyirregular shape of the ARMA transfer function is due tooverfitting the data with too high a model order and notconform the ‘true’ response of the synthetic system.qÜÉ=ãÉíÜçÇ=éêÉëÉåíÉÇ=Ü~ë=ëÜçïå=íç=ÅáêÅìãîÉåí=~=åìãÄÉê=çÑ=íÜÉ=äáãáí~íáçåë=çÑ=ÇáëÅêÉíÉ=ôj^=qê~åëÑÉê=cìåÅíáçåJkçáëÉ=ãçÇÉäëK=cáêëíI=íÜÉ=mfocf`q=ãçÇÉä=Å~å=ÄÉ=Å~äáÄê~íÉÇ=çå=Ç~í~=~í=~åó=ÑêÉèìÉåÅó=~î~áä~ÄäÉI=ÄÉÅ~ìëÉ=áí=çéÉê~íÉë=áå=~=Åçåíáåìçìë=íáãÉ=Ççã~áå=~åÇ=íÜÉ=íáãÉ=ëíÉéë=çÑ=íÜÉ=çìíéìí=î~êá~ÄäÉ=~êÉ=åçí=ÅçìéäÉÇ=íç=íÜÉ=íáãÉ=ëíÉéë=çÑ=íÜÉ=áåéìí=î~êá~ÄäÉëK=qÜìëI=~äëç=íÜÉ=ÑêÉèìÉåÅó=çÑ=íÜÉ=áåéìí=ëÉêáÉë=Å~å=ÄÉ=áêêÉÖìä~êK=pÉÅçåÇI=Åçãé~êÉÇ=íç=íÜÉ=ÅçãÄáåÉÇ=ôu=ãçÇÉä=~åÇ=h~äã~å=ÑáäíÉêI=íÜÉ=mfocf`q=ãçÇÉä=çÑÑÉêë=~=ÑìêíÜÉê=ÉñíÉåëáçå=çÑ=íÜÉ=éçëëáÄáäáíáÉë=çÑ=Å~äáÄê~íáåÖ=qck=ãçÇÉäë=çå=áêêÉÖìä~êäó=ëé~ÅÉÇ=íáãÉ=ëÉêáÉëI=ÄÉÅ~ìëÉ=íÜÉ=ëÜ~éÉ=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=áë=åçí=êÉëíêáÅíÉÇ=íç=~å=ÉñéçåÉåíá~äK=qÜáêÇI=ìëáåÖ=íÜÉ=mfocf`q=ãçÇÉäI=íÜÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉëë=áë=ëáãéäáÑáÉÇ=ÄÉÅ~ìëÉ=íÜÉ=ãçÇÉä=ÑêÉèìÉåÅó=ÇçÉë=åçí=áåíÉêÑÉêÉ=ïáíÜ=íÜÉ=ãçÇÉä=çêÇÉê=~åÇ=é~ê~ãÉíÉê=î~äìÉëI=~åÇ=íÜÉ=ÑäÉñáÄáäáíó=çÑ=~=ëáåÖäÉ=Åçåíáåìçìë=fo=ÑìåÅíáçå=Å~å=ÄÉ=ëìÅÜ=íÜ~í=áí=ÅçãéêáëÉë=~=ê~åÖÉ=çÑ=ôj^=íê~åëÑÉê=ÑìåÅíáçåëK=cìêíÜÉêãçêÉI=íÜÉ=ãçÇÉä=Å~å=ÄÉ=êÉ~Çáäó=áÇÉåíáÑáÉÇ=ìëáåÖ=éÜóëáÅ~ä=áåëáÖÜíK=^äíÜçìÖÜ=íÜÉ=ãÉíÜçÇ=áë=éêÉëÉåíÉÇ=áå=íÜÉ=Ñçêã=çÑ=~=ëáåÖäÉ=áåéìí=qck=ãçÇÉä=Ñçê=íáãÉ=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=ÜÉ~ÇI=áí=áë=áå=Ñ~Åí=èìáíÉ=ÖÉåÉê~äI=~åÇ=ÅçìäÇ=ÄÉ=ìëÉÇ=Ñçê=~=î~êáÉíó=çÑ=ëáåÖäÉ=çê=ãìäíá=áåéìíI=EåçåF=ÜóÇêçäçÖáÅ=éêçÄäÉãë=EáKÉK=Ñçê=íÜÉ=ë~ãÉ=éìêéçëÉë=íÜ~í=ôj^=qck=ãçÇÉäë=~êÉ=ìëÉÇI=ëÉÉ=áåíêçÇìÅíáçåFK=qÜÉ=Åçåíáåìçìë=íáãÉ=~ééêç~ÅÜ=éêçÄ~Ääó=çÑÑÉêë=ãçëí=~Çî~åí~ÖÉë=áå=Å~ëÉë=ïÜÉêÉ=íÜÉ=~ÄçîÉ=ãÉåíáçåÉÇ=äáãáí~íáçåë=çÅÅìêI=ïÜÉêÉ=íÜÉ=~å~äóëí=áë=áåíÉêÉëíÉÇ=áå=íÜÉ=ÑìåÅíáçåáåÖ=çÑ=~=ëóëíÉã=çå=ÇáÑÑÉêÉåí=çê=ëã~ää=íáãÉ=ëÅ~äÉëI=áå=íÜÉ=~ìíçã~íÉÇ=~å~äóëáë=çÑ=ä~êÖÉ=èì~åíáíáÉë=çÑ=íáãÉ=ëÉêáÉëI=çê=áå=~ëëÉëëáåÖ=íáãÉ=áåî~êá~åí=êÉëéçåëÉ=ÅÜ~ê~ÅíÉêáëíáÅë=çÑ=ëóëíÉãëK=tÜáäÉ=íÜÉ=mfocf`q=ãçÇÉä=éêçÄ~Ääó=éÉêÑçêãë=ÄÉëí=ïÜÉå=íÜÉ=Çóå~ãáÅ=ÄÉÜ~îáçê=çÑ=~=ëóëíÉã=Å~å=ÄÉ=ÉñéêÉëëÉÇ=áå=íÜÉ=Ñçêã=çÑ=~=ëáãéäÉ=~å~äóíáÅ=Ñçêãìä~I=íÜáë=ëÜçìäÇ=åçí=éçëÉ=~=êÉëíêáÅíáçå=íç=íÜÉ=ìëÉ=çÑ=íÜÉ=ãçÇÉäI=~ë=íÜÉ=ëÜ~éÉ=çÑ=íÜÉ=fo=ÑìåÅíáçå=Å~å=~äëç=ÄÉ=ÖÉåÉê~äáòÉÇ=Äó=ìëáåÖ=ëìãë=çÑ==mfff=ÇÑ=Eçê=çíÜÉêF=ÑìåÅíáçåëK===..…………………………………………………………………………………………….….

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4^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=Chapter4 A continuous noise model forautocorrelated data of irregularA continuous noise model forfrequencyautocorrelated data of irregular==frequency======= VN===Adopted from:=Von Asmuth, J.R., and M.F.P. Bierkens (2005)=Modeling irregularly spaced residual series as a continuous=stochastic process.^ÇçéíÉÇ=ÑêçãW=Water Resources Research, 41(12), W12404, doi: 10.1029 /Von Asmuth, J. R., and M. F. P. Bierkens (2005), Modeling irregularly spaced residual series2004WR003726.as a continuous stochastic process, t~íÉê=oÉëçìêÅÉë=oÉëÉ~êÅÜ, 41(12), W12404,doi:10.1029/2004WR003726. Reproduced Reproduced by permission by permission of American of American Geophysical Geophysical Union Union,copyright © 2005 AmericancopyrightGeophysical2002 AmericanUnion.Geophysical Union.==Abstract: Abstract=få=íÜáë=ÅÜ~éíÉêI=íÜÉ=Ä~ÅâÖêçìåÇ=~åÇ=ÑìåÅíáçåáåÖ=çÑ=~=ëáãéäÉ=Äìí=ÉÑÑÉÅíáîÉ=Åçåíáåìçìë=íáãÉ=~ééêç~ÅÜ=Ñçê=ãçÇÉäáåÖ=áêêÉÖìä~êäó=ëé~ÅÉÇ=êÉëáÇì~ä=ëÉêáÉë=~êÉ=In this chapter, the background and functioning of a simple but effective continuous timeéêÉëÉåíÉÇK=qÜÉ=Ä~ëáÅ=Éèì~íáçåë=ïÉêÉ=éìÄäáëÜÉÇ=É~êäáÉê=áå=xsçå=^ëãìíÜ=Éí=~äKI=OMMOz=approach for modeling irregularly spaced residual series are presented. The basicïÜÉêÉ=íÜÉó=ïÉêÉ=ìëÉÇ=~ë=é~êí=çÑ=~=Åçåíáåìçìë=íáãÉ=íê~åëÑÉê=ÑìåÅíáçå=åçáëÉ=EqckF=equations were published earlier in [ Von Asmuth et al., 2002] where they were used asãçÇÉäK=fí=áë=ëÜçïå=íÜ~í=íÜÉ=ãÉíÜçÇë=ÄÉÜáåÇ=íÜÉ=ãçÇÉä=~êÉ=ÄìáäÇ=çå=íïç=éêáåÅáéäÉëW=part of a continuous time transfer function noise (TFN) model. It is shown that the methodsÑáêëíI=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=Éèì~íáçåë=çÑ=~=h~äã~å=ÑáäíÉê=ÇÉÖÉåÉê~íÉ=íç=~=Ñçêã=íÜ~í=áë=behind the model are build on two principles: first, the fact that the equations of a KalmanÉèìáî~äÉåí=íç=ÚÅçåîÉåíáçå~äÛ=~ìíçêÉÖêÉëëáîÉ=ãçîáåÖ=~îÉê~ÖÉ=Eôj^F=ãçÇÉäë=ïÜÉå=íÜÉ=filter degenerate to a form that is equivalent to 'conventional' autoregressive movingãçÇÉäÉÇ=Ç~í~=~êÉ=ÅçåëáÇÉêÉÇ=íç=ÄÉ=ÑêÉÉ=çÑ=ãÉ~ëìêÉãÉåí=ÉêêçêëK=qÜáë=~ëëìãéíáçåI=áå=average (ARMA) models when the modeled data are considered to be free ofÅçãé~êáëçå=íç=íÜÉ=ÚÑìääÛ=h~äã~å=ÑáäíÉê=~äëç=óáÉäÇë=~=ÄÉííÉê=éêÉÇáÅíáçå=ÉÑÑáÅáÉåÅó=x^Üë~å=measurement errors. This assumption, in comparison to the 'full' Kalman filter also yields a~åÇ=lD`çååçêI=NVVQzX=ëÉÅçåÇI=íÜÉ=ã~íÜÉã~íáÅ~ä=Éèìáî~äÉåÅÉ=ÄÉíïÉÉå=ÇáëÅêÉíÉJíáãÉ=better prediction efficiency [ Ahsan and O'Connor, 1994]; second, the mathematicalô=é~ê~ãÉíÉêë=~åÇ=Åçåíáåìçìë=ÉñéçåÉåíá~äë=~åÇ=íÜÉ=éçáåí=íÜ~í=Åçåíáåìçìë=íáãÉ=equivalence between discrete-time AR parameters and continuous exponentials and theãçÇÉäë=éêçîáÇÉ=~å=ÉäÉÖ~åí=ëçäìíáçå=Ñçê=ãçÇÉäáåÖ=áêêÉÖìä~êäó=ëé~ÅÉÇ=çÄëÉêî~íáçåë=xÉKÖKI=point that continuous time models provide an elegant solution for modeling irregularlye~êîÉóI=NVUVzK=_ÉÅ~ìëÉ=ëáãéäÉ=äÉ~ëíJëèì~êÉë=ãÉíÜçÇë=Çç=åçí=~ééäó=áå=Å~ëÉ=çÑ=spaced observations [e.g., Harvey, 1989]. Because simple least-squares methods do notãçÇÉäáåÖ=áêêÉÖìä~ê=Ç~í~I=~=ëìã=çÑ=ïÉáÖÜíÉÇ=ëèì~êÉÇ=áååçî~íáçåë=EptpfF=ÅêáíÉêáçå=áë=apply in case of modeling irregular data, a sum of weighted squared innovations (SWSI)áåíêçÇìÅÉÇ=~åÇ=ÇÉêáîÉÇ=Ñêçã=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçåëK=få=~å=Éñ~ãéäÉ=criterion is introduced and derived from the likelihood function of the innovations. In an~ééäáÅ~íáçåI=áí=áë=ëÜçïå=íÜ~í=íÜÉ=Éëíáã~íÉë=çÑ=íÜÉ=ptpf=ÅêáíÉêáçå=ÅçåîÉêÖÉ=íç=ã~ñáãìã=example application, it is shown that the estimates of the SWSI criterion converge toäáâÉäáÜççÇ=Éëíáã~íÉë=Ñçê=ä~êÖÉê=ë~ãéäÉ=ëáòÉëK=cáå~ääóI=ïÉ=éêçéçëÉ=íç=ìëÉ=íÜÉ=ëçJÅ~ääÉÇ=maximum likelihood estimates for larger sample sizes. Finally, we propose to use the socalledinnovation variance function as an additional diagnostic check, next to the well-áååçî~íáçå=î~êá~åÅÉ=ÑìåÅíáçå=~ë=~å=~ÇÇáíáçå~ä=Çá~ÖåçëíáÅ=ÅÜÉÅâI=åÉñí=íç=íÜÉ=ïÉääJâåçïå=~ìíç=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåëK==known auto and crosscorrelation functions.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=ñ = W== ÇÉîá~íáçåë=çÑ=íÜÉ=êÉëáÇì~ä=ëÉêáÉë= n Ñêçã=áíë=ãÉ~å= E( n t) =ϕ = W== ~ìíçêÉÖêÉëëáîÉ=é~ê~ãÉíÉê=a = W== ÇáëÅêÉíÉ=ïÜáíÉ=åçáëÉ=éêçÅÉëë=ïáíÜ=éêçéÉêíáÉë= E{ at} = 0 =Eëç=íÜ~í= at = ã tFI==2 2= = E{ at} = σaI= E[ ata t + i] = 0 Ñçê i ≠ 0 ==få=ôj^=ãçÇÉäë=áå=ÖÉåÉê~äI=~=ïÜáíÉ=åçáëÉ=ëÉêáÉë=~í=Éèì~äë=íÜÉ=ëÉêáÉë=çÑ=çåÉJëíÉéJ~ÜÉ~Ç=éêÉÇáÅíáçå=ÉêêçêëI=çê=áååçî~íáçåëK=cçê=íÜÉ=ôENF=ãçÇÉäI=~å=Éëíáã~íÉ=çÑ=~í==áë=çÄí~áåÉÇ=Äó=ëìÄíê~ÅíáåÖ=íÜÉ=çåÉ=ëíÉé=~ÜÉ~Ç=éêÉÇáÅíáçå= ñ ˆ = ϕñ Ñêçã=íÜÉ=̃−t| nt−1t 1çÄëÉêîÉÇ=î~äìÉ=çÑ= ñtK=_ÉÅ~ìëÉ=çÑ=íÜáëI=~=ãáëëáåÖ=çÄëÉêî~íáçå=çÑ= ñ =~í=íáãÉ= t =áãéäáÉë=íÜ~í= at~åÇ== at + 1=Å~å=åçí=ÄÉ=Å~äÅìä~íÉÇK=mêÉÇáÅíáçå=çÑ= ñt=ìëáåÖ= ñt − 2çê=ãçêÉ=íáãÉ=ëíÉéë=Ä~Åâ=ÇçÉë=åçí=çÑÑÉê=~=ëíê~áÖÜíÑçêï~êÇ=ëçäìíáçåI=ÄÉÅ~ìëÉ=íÜÉ=äÉ~Ç=íáãÉ=áåÑäìÉåÅÉë=íÜÉ=éêÉÇáÅíáçå=Éêêçê=çê=î~êá~åÅÉ=çÑ= atI=ïÜáÅÜ=ëÜçìäÇ=ÄÉ=ëí~íáçå~êóK=qÜÉêÉÑçêÉI=~=íáãÉ=ëÉêáÉë=ãìëí=ÄÉ=ÅçãéäÉíÉ=~åÇ=ÖáîÉå=~í=êÉÖìä~ê=áåíÉêî~äë=áå=çêÇÉê=íç=ÄÉ=~ÄäÉ=íç=Ñáí=~å=ôj^=ãçÇÉäë=çå=íÜÉ=Ç~í~K=qÜÉ=ÑçêÉÅ~ëíáåÖ=ãçÇÉ=çÑ=ôj^=ãçÇÉäëI=ÜçïÉîÉêI=ÇçÉë=éêçîáÇÉ=íÜÉ=åÉÅÉëë~êó=Éèì~íáçåë=íç=éêÉÇáÅí=çîÉê=~=î~êá~ÄäÉ=äÉ~Ç=íáãÉ=~åÇ=íç=èì~åíáÑó=íÜÉ=~ÅÅçãé~åóáåÖ=éêÉÇáÅíáçå=Éêêçê=xëÉÉ=_çñ=~åÇ=gÉåâáåëI=NVTMzK=cçê=íÜÉ=ôENF=ãçÇÉäI=íÜÉ=éêÉÇáÅíáçå=~åÇ=áíë=î~êá~åÅÉ=~êÉ=~=ÑìåÅíáçå=çÑ=íÜÉ=íáãÉ=ä~Ö=ä=~åÇ=ÖáîÉå=ÄóW==⎧ ˆ lñt+l= φ ñt⎪2l⎨ 2 2 (1 −ϕ)(4.2)⎪σe= σt+l a 2⎩ (1 −ϕ)=t 2=== VR=n tnt-3*a t 2n t-2v*t?*+time indexKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK+2 vt1 n 2t-3t-2t-2==figure 4.2: Schematic representation of the functioning of an AR(1) model combined with a‘degenerate’ Kalman filter. The (*) denotes a residual, whereas the (+) denotes a predicted residual.The model predicts missing values of the residual series at every time step, along with the predictionerror varianceK==nnt+n2 vt

`Ü~éíÉê=Q=ïÜÉêÉ=̃ = = W=ìåÄá~ëÉÇ=éêÉÇáÅíáçå=çÑ= ̃t lÖáîÉå=íÜÉ=~î~áä~ÄäÉ=çÄëÉêî~íáçåë=ìé=íç=íáãÉ=ëíÉé= t =nˆt+ln +2σ e t+ l== W=î~êá~åÅÉ=çÑ=éêÉÇáÅíáçå=Éêêçê= et + l==få=çíÜÉê=ïçêÇëI=íÜÉ=ã~áå=êÉ~ëçå=íÜ~í=ôj^=ãçÇÉäë=Å~å=åçí=ÅçéÉ=ïáíÜ=áêêÉÖìä~ê=Ç~í~=~êÉ=íÜ~í=é~ê~ãÉíÉê=Éëíáã~íáçå=~åÇ=Éëíáã~íáçå=çÑ=ãáëëáåÖ=î~äìÉë=Å~å=åçí=ÄÉ=ÇçåÉ=ëáãìäí~åÉçìëäóK=få=x_êìÄ~ÅÜÉê=~åÇ=táäëçåI=NVTSzI=íÜáë=ï~ë=~äêÉ~Çó=êÉÅçÖåáòÉÇ=~åÇ=íÜÉó=ÇÉîáëÉÇ=~=íÉÅÜåáèìÉ=íÜ~í=êÉÖÉåÉê~íÉë=íÜÉ=êÉëáÇì~äë=ìëáåÖ=íÜÉ=ÑçêÉÅ~ëíáåÖ=~åÇ=Ä~ÅâÅ~ëíáåÖ=ãçÇÉ=çÑ=ôj^=ãçÇÉäëI=Äìí=~ÅÅçêÇáåÖ=íç=xeáéÉä=~åÇ=jÅiÉçÇI=NVVQI=éK=SVRz=áå=éê~ÅíáÅÉ=áí=ï~ë=åçí=îÉêó=ÅçåîÉåáÉåíK=VS=4.2.2 The combined AR(1) model and ‘degenerate’ Kalman filtertÉ=êÉÑÉê=íç=^Üë~å=~åÇ=lD`çååçê=xNVVQz=Ñçê=~å=ÉñíÉåëáîÉ=íêÉ~íãÉåí=çå=Üçï=íÜÉ=h~äã~å=Å~å=ÄÉ=ÚÇÉÖÉåÉê~íÉÇÛ=íç=~=ëáãéäÉê=Ñçêã=áå=íÜÉ=éìêÉ=éêÉÇáÅíáçå=ëÅÉå~êáçI=~åÇ=Üçï=íÜÉ=ëí~íÉ=ëé~ÅÉ=êÉéêÉëÉåí~íáçå=áå=ÖÉåÉê~ä=êÉä~íÉë=íç=ôj^=ãçÇÉäë=áå=ëí~åÇ~êÇ=åçí~íáçåK=qç=Ñ~Åáäáí~íÉ=Åçãé~êáëçåI=ÜÉêÉ=ïÉ=ïáää=ìëÉ=ëí~åÇ~êÇ=åçí~íáçåK=få=ÑáÖìêÉ=QKOI=~=ëÅÜÉã~íáÅ=êÉéêÉëÉåí~íáçå=çÑ=~=ÅçãÄáåÉÇ=ôENF=ãçÇÉä=~åÇ=íÜÉ=ÚÇÉÖÉåÉê~íÉÛ=h~äã~å=ÑáäíÉê=áë=ÖáîÉåK=tÜÉå=çÄëÉêî~íáçåë=~êÉ=ãáëëáåÖ=çê=ëÅ~êÅÉI=íÜÉ=î~äìÉ=çÑ= a =Å~ååçí=ÄÉ=ÇÉíÉêãáåÉÇ=Ñçê=ÉîÉêó=íáãÉ=ëíÉéK=fåëíÉ~ÇI=~å=áêêÉÖìä~êäó=ëé~ÅÉÇ=áååçî~íáçå=ëÉêáÉë=ν =áë=2a2v tÉëíáã~íÉÇK=tÜáäÉ= σ =áë=~=Åçåëí~åíI= σ ÇÉéÉåÇë=çå=íÜÉ=íáãÉ=ä~Ö=ÄÉíïÉÉå=íïç=çÄëÉêî~íáçåëK=få=íÜáë=~ééêç~ÅÜI=ÑáííáåÖ=íÜÉ=ãçÇÉä=íç=~å=áêêÉÖìä~ê=ëÉêáÉë=óáÉäÇë=~å=Éëíáã~íÉ=çÑ=ϕ I= σ I=ν =~åÇ=~=ëéÉÅáÑáÅ=î~êá~åÅÉσÑçê=ÉîÉêó=ν K=qÜÉ=é~ê~ãÉíÉêë=~êÉ=2aÉëíáã~íÉÇ=Äó=çéíáãáòáåÖ=~=ji=ÑìåÅíáçåI=ã~ÇÉ=ìé=çÑ=íÜÉ=áååçî~íáçåë=~åÇ=íÜÉáê=î~êá~åÅÉëK=qÜÉ=Éèì~íáçåë=çÑ=íÜÉ=h~äã~å=ÑáäíÉê=~êÉ=Éî~äì~íÉÇ=êÉÅìêëáîÉäóI=ïÜáäÉ=ÇáÑÑÉêÉåí=~Åíáçåë=~êÉ=í~âÉå=ÇÉéÉåÇáåÖ=çå=ïÜÉíÜÉê=çê=åçí= n =áë=~î~áä~ÄäÉ=~í=~=íáãÉ=ëíÉéK=pí~êíáåÖ=2ïáíÜ=áåáíá~ä=ÅçåÇáíáçåë= n ⌣̃ 0=~åÇ= σ e I=íÜÉ=ÑçääçïáåÖ=Éèì~íáçåë=~êÉ=Éî~äì~íÉÇ=áå=íÜÉ=ëçJ0Å~ääÉÇ=íáãÉ=ìéÇ~íÉ=x~ÑíÉê=_áÉêâÉåë=Éí=~äKI=NVVVzW==⌣⎧ ˆ⎪ ñt= ϕñt−1⎨(4.3)2 2 2 2⎪⎩σe= σt a+ ϕ σet− 1=ïÜÉêÉ=ñˆt=W=éêÉÇáÅíáçå=çÑ= ñt=áå=íÜÉ=íáãÉ=ìéÇ~íÉ=n ⌣̃ t= W=éêÉÇáÅíáçå=çÑ= ñt=áå=íÜÉ=ãÉ~ëìêÉãÉåí=ìéÇ~íÉ=2σ e t= W=î~êá~åÅÉ=çÑ=íÜÉ=Éêêçê=áå=íÜÉ=íáãÉ=ìéÇ~íÉ===tÜÉå=~å=çÄëÉêî~íáçå=áë=~î~áä~ÄäÉI=íÜÉ=ãÉ~ëìêÉãÉåí=ìéÇ~íÉ=áë=Éî~äì~íÉÇW==2 vt..…………………………………………………………………………………………….….

^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=⎧ ν ˆt= ñt− ñt⎪2 2⎪σv = σt et⎨(4.4)⎪ n⌣̃t= n ̃t⎪ 2⎩σe= 0t=fÑI=ÜçïÉîÉêI=åç=çÄëÉêî~íáçå=áë=~î~áä~ÄäÉW==n⌣̃ ˆt= n ̃t(4.5)=qÜÉ=êÉÅìêëáîÉ=~ééäáÅ~íáçå=çÑ=Éèì~íáçåë=EQKPFIEQKQF=~åÇ=EQKRF=ã~íÜÉã~íáÅ~ääó=Éèì~äë=íÜÉ=ÑçêÉÅ~ëíáåÖ=ãçÇÉ=çÑ=~å=ôENF=ãçÇÉä=~ë=ÇÉëÅêáÄÉÇ=Äó=EQKOFI=ÄÉÅ~ìëÉ=~ÑíÉê=~=íáãÉ=ä~Ö= l =íÜÉ=Éêêçê=áå=íÜÉ=íáãÉ=ìéÇ~íÉ=áëW==2l2 2 2 2l−2 2 (1 −ϕ)σ e = σ (1 ... )t l a + ϕ + + ϕ = σ+a(4.6)2(1 −ϕ)=eÉåÅÉI=áå=íÜÉ=ppo=ÚíÜÉ=ÑçêÉÅ~ëíáåÖ=ãçÇÉÛ=çÑ=íÜÉ=ôENF=ãçÇÉä=íÜ~í=Ü~åÇäÉë=éêÉÇáÅíáçåë=çîÉê=î~êá~ÄäÉ=äÉ~Ç=íáãÉë=áë=é~êí=çÑ=íÜÉ=Ä~ëáÅ=ãçÇÉä=Éèì~íáçåëK=eçïÉîÉêI=ÉîÉå=áå=íÜÉ=ÚÇÉÖÉåÉê~íÉÛ=ÑçêãI=íÜÉ=êÉÅìêëáîÉ=Éî~äì~íáçå=çÑ=íÜÉ=Éèì~íáçåë=áå=ppo=ÄÉÅçãÉë=Åçãéìí~íáçå~ääó=áåÅêÉ~ëáåÖäó=áåÉÑÑáÅáÉåí=ïáíÜ=~å=áåÅêÉ~ëáåÖ=ãçÇÉä=ÑêÉèìÉåÅóK=bîÉå=ãçêÉ=ëç=ïÜÉå=áí=áë=ìëÉÇ=áå=ÅçãÄáå~íáçå=ïáíÜ=~=ÇÉíÉêãáåáëíáÅ=ãçÇÉäI=ÄÉÅ~ìëÉ=~äëç=íÜ~í=Ü~ë=íç=çéÉê~íÉ=çå=íÜÉ=ë~ãÉ=ÑêÉèìÉåÅóK=kÉñí=íç=íÜ~íI=íÜÉ=~ìíçêÉÖêÉëëáîÉ=é~ê~ãÉíÉê=ϕ =ÄÉÅçãÉë=Ä~Çäó=ëÅ~äÉÇ=~ë=áí=~ëóãéíçíáÅ~ääó=~ééêç~ÅÜÉë=íÜÉ=î~äìÉ=çÑ=N=ïÜÉå=íÜÉ=íáãÉ=ëíÉé= ∆ t =Eáå=êÉ~ä=íáãÉ=ìåáíëF=~ééêç~ÅÜÉë=MK=cáå~ääóI=çéÉê~íáåÖ=çå=~=ÜáÖÜ=ÑêÉèìÉåÅó=~äëç===== VT=n ( t )n( t-t)*t* *n( t-t)en( t)+real timetv( t) ( t )d W ( )==figure 4.3: Schematic representation of the functioning of the Ornstein-Uhlenbeck based model. The(*) denotes a residual, whereas the (+) denotes a predicted residual. The model directly predicts thevalue of the residual series for the next observation available.*tt tKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=Q=éçëÉë=~=éêçÄäÉã=Ñçê=ôj^=íóéÉ=íê~åëÑÉê=ãçÇÉäëI=ÄÉÅ~ìëÉ=íÜÉ=åìãÄÉê=çÑ=j^=é~ê~ãÉíÉêë=áåÅêÉ~ëÉë=äáåÉ~êäó=ïáíÜ=íÜÉ=ÑêÉèìÉåÅó=xsçå=^ëãìíÜ=Éí=~äKI=OMMOzK=VU=4.2.3 The Ornstein-Uhlenbeck based modelfå=ÑáÖìêÉ=QKPI=~=ëÅÜÉã~íáÅ=êÉéêÉëÉåí~íáçå=çÑ=íÜÉ=ÑìåÅíáçåáåÖ=çÑ=íÜÉ=lr_=åçáëÉ=ãçÇÉä=áë=ÖáîÉåK=få=íÜáë=~ééêç~ÅÜ=~ää=íáãÉ=ëÉêáÉë=~åÇ=ÑìåÅíáçåëI=áåÅäìÇáåÖ=íÜÉ=åçáëÉ=éêçÅÉëëI=~êÉ=êÉÖ~êÇÉÇ=~ë=ÄÉáåÖ=ÅçåíáåìçìëK=bèì~íáçå=EQKNF=Å~å=ÄÉ=íê~åëÑçêãÉÇ=íç=Åçåíáåìçìë=íáãÉ=Äó=ïêáíáåÖ=áí=áå=ãçîáåÖJ~îÉê~ÖÉ=ÑçêãW==iñ = ϕ a(4.7)t∞∑i=0t−i=~åÇ=êÉéä~ÅáåÖ=~í=Äó=~=Åçåíáåìçìë=ïÜáíÉ=åçáëÉ=éêçÅÉëë= d W ( t)K=qÜÉ=êÉëáÇì~ä=ëÉêáÉë= n =Å~å=íÜÉå=ÄÉ=ãçÇÉäÉÇ=~ë=~=Åçåíáåìçìë=ëíçÅÜ~ëíáÅ=éêçÅÉëëI=ÖáîÉå=ÄóW==tñ ( t) = ∫φ( t - τ )d W ( τ )(4.8)−∞=ïÜÉêÉ=φ = W== åçáëÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=W ( t)W==táÉåÉê=éêçÅÉëë=xizI=ïáíÜ=éêçéÉêíáÉë= E{d W ( t )} = 0 I= E[{d W ( t)} ] = βdtI== = E[d W ( t1)d W ( t2)] = 0 Ñçê= t 1≠ t 2K=β = W== ~=é~ê~ãÉíÉê=xi O q JN z==^ë=áå=íÜÉ=h~äã~åJÑáäíÉê=~ééêç~ÅÜI=íÜÉ=ãçÇÉä=ÇçÉë=åçí=óáÉäÇ=~å=Éëíáã~íÉ=çÑ=íÜÉ=ïÜáíÉ=åçáëÉ=ëÉêáÉë= d W ( t)I=Äìí=áåëíÉ~Ç=ÖáîÉë=~å=Éëíáã~íÉ=çÑ=íÜÉ=áååçî~íáçå=ëÉêáÉë=ν K=ν =áë=ãçÇÉäÉÇ=~ë=íÜÉ=áêêÉÖìä~êäó=ë~ãéäÉÇ=ÉÑÑÉÅí=çÑ=íÜÉ=åçáëÉ=éêçÅÉëë=çå=íÜÉ=êÉëáÇì~ä=ëÉêáÉë=ÄÉíïÉÉå=íáãÉ=ëíÉéë= t − ∆ t =~åÇ= t I=ïÜáÅÜ=áë=ÖáîÉå=ÄóW==tν ( t) = ∫ φ( t −τ )d W ( τ )(4.9)t−∆t=cçê=íÜÉ=åçáëÉ=áãéìäëÉ=êÉëéçåëÉ=EfoF=ÑìåÅíáçå=íÜÉ=ÑçääçïáåÖ=ÉñéçåÉåíá~ä=áë=ÅÜçëÉåI=ëç=íÜ~í=EQKUF=êÉÇìÅÉë=íç=~å=ôENF=ãçÇÉä=ïÜÉå=áí=ìëÉÇ=çå=Ç~í~=ïáíÜ=êÉÖìä~ê=íáãÉ=ëíÉéëW===2n2ασφ( t) = exp( − αt)(4.10)β=2..…………………………………………………………………………………………….….

^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=ïáíÜ=α =ÇÉÑáåáåÖ=íÜÉ=ÇÉÅ~ó=ê~íÉ=çÑ=íÜÉ=åçáëÉI=~åÇ= σ =ÇÉåçíáåÖ=íÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=êÉëáÇì~äëK=táíÜ=íÜÉ=ÅÜçáÅÉ=çÑ=~å=ÉñéçåÉåíá~ä=foJÑìåÅíáçåI=Éèì~íáçå=EQKUF=Å~å=ÄÉ=ïêáííÉå=~ë=xëÉÉ=ÉKÖKI=d~êÇáåÉêI=NVVQzW==t2n2ñ ( t) = exp( −α∆t ) ñ ( t − ∆t)+ ∫ασ exp{ −α ( t −τ )}d W ( τ )(4.11)βt−∆t=ïÜáÅÜ=áë=âåçïå=~ë=~å=lêåëíÉáåJrÜäÉåÄÉÅâ=éêçÅÉëë=xrÜäÉåÄÉÅâ=~åÇ=lêåëíÉáåI=NVPMX=d~êÇáåÉêI=NVVQzK=_ó=ÅçãÄáåáåÖ=Éèì~íáçå=EQKVF=~åÇ=EQKNNFI=íÜÉ=áååçî~íáçå=ëÉêáÉë=ν =Å~å=ÄÉ=Å~äÅìä~íÉÇ=Ñêçã=íÜÉ=~î~áä~ÄäÉ=êÉëáÇì~äë=ìëáåÖ=ëáãéäóW===v( t) = ñ ( t) − exp( −α∆t) ñ ( t − ∆t)(4.12)=ïÜÉêÉ=íÜÉ=ä~ëí=íÉêã=çå=íÜÉ=êáÖÜí=ëáÇÉ=Éèì~äë= ñˆt|ñt−∆tEáå=ÑáÖìêÉ=QKP=áääìëíê~íÉÇ=Äó=íÜÉ=ÇçííÉÇ=ÉñéçåÉåíá~äFK=_ÉÅ~ìëÉ=çÑ=áíë=Åçåíáåìçìë=Ñçêãìä~íáçåI=EQKNOF=ÖáîÉë=~å=Éñ~Åí=ëçäìíáçå=~åÇ=çåäó=åÉÉÇë=íç=ÄÉ=Éî~äì~íÉÇ=çåÅÉ=Ñçê=ÉîÉêó=çÄëÉêî~íáçå=çÑ= n K=qÜáë=Å~å=êÉÇìÅÉ=Åçãéìí~íáçå=íáãÉë=ëìÄëí~åíá~ääóI=Åçãé~êÉÇ=íç=~=êÉÅìêëáîÉ=~äÖçêáíÜã=ïÜáÅÜ=ÇáëÅêÉíáòÉë= ∆ t =áå=ëã~ää=íáãÉ=ëíÉéëK===2n=== VV=4.2.4 Parameter estimation and SWSI criterionkÉñí=íç=íÜÉ=Éèì~íáçåë=ÇáëÅìëëÉÇ=áå=íÜÉ=éêÉîáçìë=ëÉÅíáçåI=~å=áãéçêí~åí=~ëéÉÅí=çÑ=ÇÉ~äáåÖ=ïáíÜ=áêêÉÖìä~ê=Ç~í~=äáÉë=áå=íÜÉ=çéíáãáò~íáçå=ãÉíÜçÇë=ìëÉÇK=cçê=ÑáñÉÇ=íáãÉ=ëíÉéëI=2σa=áë=Åçåëí~åí=~åÇ=ãáåáãáòáåÖ=íÜÉ=ëìã=çÑ=ëèì~êÉë=çÑ= a =óáÉäÇë=Éñ~Åí=ji=Éëíáã~íÉëI=ìåÇÉê=íÜÉ=~ëëìãéíáçå=çÑ=~=d~ìëëá~å=ÇáëíêáÄìíáçåK=qÜáë=áëI=ÜçïÉîÉêI=~äëç=ÅçåÇáíáçå~ä=çå=íÜÉ=ÅÜçáÅÉ=çÑ=íÜÉ=ëí~êíáåÖ=î~äìÉë= a0=~åÇ= n0I=~åÇ=áë=íÜÉêÉÑçêÉ=Å~ääÉÇ=íÜÉ=ÅçåÇáíáçå~ä=ëìã=çÑ=ëèì~êÉë=x_çñ=~åÇ=gÉåâáåëI=NVTMzK=táíÜ=~=î~êá~ÄäÉ= σ2v( t)I=ëáãéäÉ=äÉ~ëíJëèì~êÉë=ãÉíÜçÇë=~êÉ=åç=äçåÖÉê=ëíê~áÖÜíÑçêï~êÇäó=~ééäáÅ~ÄäÉK=fåëíÉ~ÇI=~=äáâÉäáÜççÇ=ÑìåÅíáçå=áë=ÅçåëíêìÅíÉÇ=~åÇ=ã~ñáãáòÉÇ=ìëáåÖ=íÜÉ=h~äã~å=ÑáäíÉê=Éëíáã~íÉë=çÑ= v =~åÇ= σ2v( t)=~í=ÉîÉêó=íáãÉ=ëíÉé=xëÉÉ=ÉKÖKI=pÅÜïÉééÉI=NVTPX=j¨ä~êÇI=NVUQzK=qÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçåI=ÜçïÉîÉêI=áë=~=ê~íÜÉê=ÅçãéäáÅ~íÉÇ=ÉñéêÉëëáçå=~åÇ=ÑäÉñáÄäÉ=~äÖçêáíÜãë=~êÉ=åÉÉÇÉÇ=áå=çêÇÉê=íç=ã~ñáãáòÉ=áí=xeáéÉä=~åÇ=jÅiÉçÇI=NVVQz=K=få=íÜÉ=ÑçääçïáåÖI=ïÉ=ïáää=ëÜçï=íÜ~í=~äëç=Ñçê=áêêÉÖìä~ê=Ç~í~I=~=ëáãéäÉ=äÉ~ëí=ëèì~êÉë=ÅêáíÉêáçå=Å~å=ÄÉ=ÇÉêáîÉÇ=ìëáåÖ=íÜÉ=áååçî~íáçå=2î~êá~åÅÉ=ÑìåÅíáçå=EfscF=íÜ~í=ÇÉÑáåÉë=íÜÉ=ÖÉåÉê~ä=êÉä~íáçåëÜáé=ÄÉíïÉÉå= σ ~åÇ= ∆tK=cçê=ôfj^=ãçÇÉäëI=ïÜÉå=íÜÉ=ëÉí=çÑ=é~ê~ãÉíÉêë=íÜ~í=áë=Éëíáã~íÉÇ=áë= Ψ = [ ψ1, ψ 2,.., ψ p ] I=~åÇ=~ëëìãáåÖ=íÜ~í= a =áë=~=ëÉêáÉë=çÑ= N =ê~åÇçã=î~êá~ÄäÉë=ïÜáÅÜ=~êÉ=åçêã~ääó=2aáåÇÉéÉåÇÉåíäó=ÇáëíêáÄìíÉÇ=EkfaEMI σ FFI=íÜÉå=íÜÉáê=àçáåí=éÇÑ=Å~å=ÄÉ=ïêáííÉå=~ëW==v( t)KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=Q=NMM=N−N 22 −a2t0 0=a ∑ 2i=1 2σaP( a | Ψ , a , n ) (2 πσ ) exp( )(4.13)=áå=ïÜáÅÜ=0ta =~åÇ= n0ÇÉåçíÉ=íÜÉ=áåáíá~ä=ÅçåÇáíáçåë=çÑ= a =~åÇ= n K=rëáåÖ=EQKNPF=íÜÉ=äçÖ=äáâÉäáÜççÇ=ÑìåÅíáçå=çÑ= Ψ =áë=ÖáîÉå=ÄóW==N 22 at0 0 = − π − σ a − ∑ 2i=1 σ aΛ( Ψ | a , a , n ) 0.5N ln(2 ) 0.5Nln( ) 0.5( )(4.14)t=cçê=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=çÑ=Åçåíáåìçìë=íáãÉ=áååçî~íáçåëI=ïÉ=Ñáêëí=~Ç~éí=íÜÉ=ÇáëÅêÉíÉ=åçí~íáçå=íç=~ääçï=Ñçê=ÇáëÅêÉíÉäó=ë~ãéäÉÇI=áêêÉÖìä~êäó=ëé~ÅÉÇ=çÄëÉêî~íáçåëK=cçê=íÜáë=êÉ~ëçå=ïÉ=ìëÉ= t =áå=êÉ~ä=íáãÉI=~åÇ=áåÇÉñ=íÜ~í=ïáíÜ= i =áåëíÉ~ÇK=^=Ç~í~=ëÉí= O =çÑ= N =çÄëÉêî~íáçåë=çÑ=~=Åçåíáåìçìë=éêçÅÉëë=äáâÉ= n =áë=íÜÉå=ÖáîÉå=ÄóW==O = [ n( t1 ), n( t2),........, n( t N)](4.15)=2^ë=ëí~íÉÇ=É~êäáÉêI=íÜÉ=áååçî~íáçå=î~êá~åÅÉë= σ =íÜ~í=~êÉ=çíÜÉêïáëÉ=áåÇáîáÇì~ääó=Éëíáã~íÉÇ=~åÇ=ëíçêÉÇ=Ñçê=ÉîÉêó=íáãÉ=ëíÉéI=Å~å=ÄÉ=àçáåíäó=ÇÉëÅêáÄÉÇ=Äó=íÜÉ=fsc=~ë=~=ÑìåÅíáçå=çÑ=íÜÉ=íáãÉ=ëíÉé= ∆ t =E~åÇ=íÜÉ=é~ê~ãÉíÉê=ëÉí= Ψ FK=få=íÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçåI=ïÉ=ïáää=é~ó=ãçêÉ=~ííÉåíáçå=íç=íÜÉ=fsc=~ë=áí=éä~óë=~=ÅêìÅá~ä=ÑìåÅíáçå=áå=Ü~åÇäáåÖ=áêêÉÖìä~ê=Ç~í~K=qÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=Å~å=åçï=ÄÉ=ïêáííÉå=~ë=xpÅÜïÉééÉI=NVTPzW==NN 22ν ( ti, Ψ)Λ( Ψ | O) = −0.5N ln(2 π ) − 0.5∑ln{ σν( ∆ti, Ψ)} − 0.5∑(4.16)2i= 1 i=1 σν( ∆ti, Ψ)=få=çêÇÉê=íç=êÉÇìÅÉ=íÜÉ=~ãçìåí=çÑ=é~ê~ãÉíÉêë=íÜ~í=Ü~ë=íç=ÄÉ=åìãÉêáÅ~ääó=çéíáãáòÉÇI=ïÉ=ïáää=ëí~êí=ïáíÜ=Éäáãáå~íáåÖ E{ n( t)}K=cçê=ë~ãéäÉ=ëáòÉë=ìëì~ääó=ÅçåëáÇÉêÉÇI=íÜÉ=ãÉ~å=çÑ=~=íáãÉ=ëÉêáÉë=Å~å=ÄÉ=~ÇÉèì~íÉäó=Éëíáã~íÉÇ=~ë=x_çñ=~åÇ=gÉåâáåëI=NVTMzW===N∑n( ti)i=1E{ n( t)}=(4.17)N=eÉåÅÉI=ïÉ=Å~å=Éëíáã~íÉ= ñ =ÇáêÉÅíäó=Ñêçã=Ç~í~=ëÉí= O K=_ÉÅ~ìëÉ= σ2 v( ∆t, Ψ ) =áë=~=ÑìåÅíáçå=çÑ=íÜÉ=íáãÉ=ëíÉéI=áí=Å~ååçí=ÄÉ=ëíê~áÖÜíÑçêï~êÇäó=Éëíáã~íÉÇ=Ñêçã=íÜÉ=~î~áä~ÄäÉ=áååçî~íáçåëK=eçïÉîÉêI=ìëáåÖ=EQKVF=~åÇ=EQKNMFI= σ2 v( ∆t, Ψ)Å~å=ÄÉ=ïêáííÉå=~ë=~=ÑìåÅíáçå=çÑ=íÜÉ=êÉëáÇì~ä=î~êá~åÅÉ=~ë=xd~êÇáåÉêI=NVVQzW==2 2σ ( ∆ t, Ψ) = {1 − exp( −2 α∆t)} σ ( Ψ )(4.18)vnv( t)..…………………………………………………………………………………………….….

^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä==ïÜáÅÜ=áå=íìêå=Å~å=óáÉäÇ=~å=Éëíáã~íçê=Ñçê= σn2 ( Ψ ) =ìëáåÖ=íÜÉ=áåÇáîáÇì~ä=áååçî~íáçåë=EëÉÉ=~ééÉåÇáñFW==N⎧ 1 ⎫ 2∑⎨⎬ν( ti, Ψ)2 i=11− exp( −2 α∆t)σn( Ψ ) =⎩⎭(4.19)N=táíÜ=EQKNUF=~åÇ=EQKNVF=áå=EQKNSFI=ïÉ=Å~å=åçï=Éäáãáå~íÉ= σ2 v( ∆t, Ψ ) =Ñçêã=íÜÉ=Éèì~íáçåëI=~åÇ=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=Å~å=ÄÉ=ïêáííÉå=~ë=EëÉÉ=~ééÉåÇáñFW==N∏{1 − exp( −2α∆t)}NNii=1 2∑ν ( t j , Ψ)j=11−exp( −2α∆t)(4.20)jΛ( Ψ | O) = −0.5N ln(2 π ) − 0.5Nln[ ] − ...N0.5N=qÜìëI=íÜÉ=çåäó=é~ê~ãÉíÉê=íÜ~í=Ü~ë=íç=ÄÉ=åìãÉêáÅ~ääó=çéíáãáòÉÇ=Ñçê=íÜÉ=åçáëÉ=ãçÇÉä=áë=α I=ÇÉÑáåáåÖ=íÜÉ=ÇÉÅ~ó=ê~íÉ=çÑ=íÜÉ=åçáëÉK=_ÉÅ~ìëÉ=íÜÉ=Ñáêëí=~åÇ=ä~ëí=íÉêãë=áå=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=~êÉ=Åçåëí~åíI=~å=Éëíáã~íÉ=çÑ= Ψ =Å~å=ÄÉ=çÄí~áåÉÇ=Äó=ãáåáãáòáåÖ=íÜÉ=ÑçääçïáåÖ=ÅêáíÉêáçåW==NNi2 i=12∑j=1N∏{1 − exp( −2α∆t)}S ( Ψ | O) =ν ( tj, Ψ )(4.21)1−exp( −2α∆t)j=ïÜáÅÜ=ã~ó=ÄÉ=êÉÑÉêêÉÇ=íç=~ë=íÜÉ=pìã=çÑ=tÉáÖÜíÉÇ=pèì~êÉÇ=fååçî~íáçåë=EptpfF=ÅêáíÉêáçåK=qÜÉ=ptpf=ÅêáíÉêáçå=áë=ëáãáä~ê=íç=çíÜÉê=ïÉáÖÜíÉÇ=äÉ~ëí=ëèì~êÉë=ÅêáíÉêá~I=~ë=~äëç=ÜÉêÉ=íÜÉ=ïÉáÖÜíë=êÉÑäÉÅí=íÜÉ=î~êá~åÅÉë=çÑ=íÜÉ=áååçî~íáçåëK=cçê=çéíáãáò~íáçå=~åÇ=é~ê~ãÉíÉê=Éëíáã~íáçåI=ïÉ=Å~å=åçï=ìëÉ=ëí~åÇ~êÇ=åçåäáåÉ~ê=äÉ~ëí=ëèì~êÉë=êÉÖêÉëëáçå=ãÉíÜçÇë=xëÉÉ=ÉKÖKI=påÉÇÉÅçê=~åÇ=`çÅÜê~åI=NVSTzK=cáêëíI=íÜÉ=é~êíá~ä=ÇÉêáî~íáîÉë=çÑ=EQKONF=~êÉ=çÄí~áåÉÇI=ÉáíÜÉê=åìãÉêáÅ~ääó=çê=~å~äóíáÅ~ääóI=~åÇ=ìëÉÇ=íç=ÅçåëíêìÅí=~=g~ÅçÄá~å=ã~íêáñW=====NMN=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=Q=NMO=⎡ ∂S1 ∂S1⎤⎢ … ⎥⎢∂β1 ∂β1⎥J = ⎢ ⋮ ⋱ ⋮ ⎥(4.22)⎢ ⎥⎢∂SN∂SN⋯ ⎥⎢ ∂β1∂β⎥⎣p ⎦=rëáåÖ= J =íÜÉ=ãçÇÉä=áë=Å~äáÄê~íÉÇ=çå=~=ëÉí=çÑ=çÄëÉêî~íáçåë=ïáíÜ=~=iÉîÉåÄÉêÖJj~êèì~êÇí=çéíáãáò~íáçå=~äÖçêáíÜã=xj~êèì~êÇíI=NVSPz=íÜ~í=~Çàìëíë Ψ I=ïÜáäÉ=ãáåáãáòáåÖ= S{ Ψ | O} K=qÜÉåI=íÜÉ=Åçî~êá~åÅÉ=ã~íêáñ=çÑ=íÜÉ=é~ê~ãÉíÉêë=Å~å=ÄÉ=Éëíáã~íÉÇ=ïáíÜW==22 min( S ) ( ' ) −1σ β = J J(4.23)N − p=2kÉñíI= σnáë=Éëíáã~íÉÇ=ìëáåÖ=EQKNVFK==qÜÉ=ãçÇÉä=êÉëìäíë=Å~å=ÄÉ=ÅÜÉÅâÉÇ=Äó=Éñ~ãáåáåÖ=íÜÉ=Åçî~êá~åÅÉ=ã~íêáñ=~åÇ=íÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=foJÑìåÅíáçåK=qÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=foJÑìåÅíáçå=φ =Éèì~äëW==2n2σ2 2var{ φ( t)} ={ exp( −α t) − t 2ασ n exp( − αt)} var( α)(4.24)2 α=^=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=Ñçê=φ =Å~å=ÄÉ=éäçííÉÇ=~ë=HLJ=Oσ I=ïÜÉå=~ëëìãáåÖ=~=åçêã~ä=ÇáëíêáÄìíáçå=Ñçê=α K=^ë=áå=íÜÉ=ôfj^=~ééêç~ÅÜI=ëÉêáçìë=ãçÇÉä=áå~ÇÉèì~Åó=Å~å=ÄÉ=ÇÉíÉÅíÉÇ=Äó=Éñ~ãáåáåÖ=íÜÉ=~ìíçÅçêêÉä~íáçå=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçå=ëÉêáÉë=ν =EïÜáÅÜ=ÖáîÉë=~å=áåÇáÅ~íáçå=çå=ïÜÉíÜÉê=íÜÉ=ïÜáíÉ=åçáëÉ=~ëëìãéíáçå=ÜçäÇëF=~åÇ=íÜÉ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçå=ÄÉíïÉÉå=ν =~åÇ=íÜÉ=íê~åëÑÉê=ãçÇÉä=áåéìí=ëÉêáÉë= p =EïÜáÅÜ=áåÇáÅ~íÉë=ïÜÉíÜÉê=íÜÉêÉ=~êÉ=ëíáää=é~ííÉêåë=äÉÑí=áå=íÜÉ=áååçî~íáçå=ëÉêáÉë=íÜ~í=ÅçìäÇ=ÄÉ=Éñéä~áåÉÇ=Äó=íÜÉ=áåéìí=ëÉêáÉëFK=qÜÉ=~ìíçÅçêêÉä~íáçå=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåë=~í=ä~Ö= k =~êÉ=ÇÉÑáåÉÇ=áå=íÜÉ=ë~ãÉ=ï~ó=~ë=áå=ÇáëÅêÉíÉ=qck=ãçÇÉäëI=Äìí=ÄÉÅ~ìëÉ=çÑ=íÜÉ=áêêÉÖìä~êáíó=çÑ=íÜÉ=íáãÉ=ëíÉéëI=~=íçäÉê~åÅÉ=~êçìåÇ=ä~Ö= k =çÑ=±=MKR k =áë=áãéäáÉÇK=^=éêççÑ=Ñçê=íÜÉ=Ñ~Åí=íÜ~íI=äáâÉ=ôENF=ãçÇÉäëI=íÜÉ=áååçî~íáçåë=çÑ=íÜÉ=Åçåíáåìçìë=ãçÇÉä=~êÉ=åçí=~ìíçÅçêêÉä~íÉÇI=åçê=ÅêçëëÅçêêÉä~íÉÇ=ïáíÜ=çÄëÉêî~íáçåë=çÑ=íÜÉ=éêçÅÉëë= n =áíëÉäÑI=áë=ÖáîÉå=áå=íÜÉ=~ééÉåÇáñK=qÜáë=ã~âÉë=íÜÉ=ãçÇÉä=ëìáí~ÄäÉ=Ñçê=ëáãìä~íáçå=éìêéçëÉëI=~åÇ=~äëç=áë=~=éêÉêÉèìáëáíÉ=Ñçê=íÜÉ=é~ê~ãÉíÉê=Éëíáã~íáçå=éêçÅÉëëK=4.2.5 Summary of methodtÜÉå=íÜÉ=é~ê~ãÉíÉêë=çÑ=~=ÅçãÄáåÉÇ=ÇÉíÉêãáåáëíáÅ=çê=íê~åëÑÉê=ãçÇÉä=~åÇ=åçáëÉ=ãçÇÉä=~êÉ=Éëíáã~íÉÇ=ëáãìäí~åÉçìëäóI=íÜÉ=éêçÅÉÇìêÉ=áë=~ë=Ñçääçïë=EëÉÉ=~äëç=ÑáÖìêÉ=QKQFK=cáêëíI=ìëáåÖ=íÜÉ=áåáíá~ä=î~äìÉë=çÑ= Ψ I=íÜÉ=ÇÉíÉêãáåáëíáÅ=ãçÇÉä=áë=Éî~äì~íÉÇ=áå=çêÇÉê=íç=ÖÉí=~=íáãÉ=ëÉêáÉë=çÑ=êÉëáÇì~äëK=råÇÉê=íÜÉ=~ëëìãéíáçå=çÑ=ÉñéçåÉåíá~ä=åçáëÉ=ÇÉÅ~ó=EÉèì~íáçå=EQKNMFFI=íÜÉ=áååçî~íáçå=ëÉêáÉë=áë=çÄí~áåÉÇ=Ñêçã=íÜÉ=êÉëáÇì~ä=ëÉêáÉë=ìëáåÖ=Éèì~íáçå=EQKNOFK==..…………………………………………………………………………………………….….

^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=kÉñíI=íÜÉ=ptpf=ÅêáíÉêáçå=áë=Å~äÅìä~íÉÇ=Äó=ïÉáÖÜíáåÖ=íÜÉ=ëèì~êÉÇ=áååçî~íáçåë=~ÅÅçêÇáåÖ=íç=íÜÉáê=êÉëéÉÅíáîÉ=î~êá~åÅÉë=~åÇ=êÉä~íáîÉ=íç=íÜÉ=ÖÉçãÉíêáÅ~ä=~îÉê~ÖÉ=çÑ=íÜÉ=áååçî~íáçå=î~êá~åÅÉë=Ñçê=~ää=íáãÉ=ëíÉéë=EÉèì~íáçå=EQKONFFK=pìÄëÉèìÉåíäóI=íÜÉ=é~ê~ãÉíÉê=ëÉí= Ψ =áë=Éëíáã~íÉÇ=Äó=ãáåáãáòáåÖ=íÜÉ=ÅêáíÉêáçå=ìëáåÖ=~=iÉîÉåÄÉêÖJj~êèì~êÇí=~äÖçêáíÜãK=cáå~ääó=íÜÉ=î~äáÇáíó=~åÇ=éÉêÑçêã~åÅÉ=çÑ=íÜÉ=ãçÇÉä=~êÉ=ÅÜÉÅâÉÇ=ìëáåÖ=íÜÉ=~ìíçJ=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåë=çÑ=íÜÉ=áååçî~íáçåëI=íÜÉ=ÅçêêÉä~íáçå=ã~íêáñ=çÑ=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêë=~åÇ=íÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=fo=ÑìåÅíáçåK===4.3 Exampleapplication4.3.1 Set-up and data setqÜÉ=ÉÑÑÉÅíáîÉåÉëë=çÑ=~=ÅçãÄáåÉÇ=ôENF=ãçÇÉä=~åÇ=h~äã~å=ÑáäíÉê=áå=ãçÇÉäáåÖ=áêêÉÖìä~êäó=çÄëÉêîÉÇ=ÜóÇêçäçÖáÅ=Ç~í~=Ü~ë=~äêÉ~Çó=ÄÉÉå=ëÜçïå=ê~íÜÉê=Éä~Äçê~íÉäó=x_áÉêâÉåë=Éí=~äKI=NVVVX=_ÉêÉåÇêÉÅÜí=Éí=~äKI=OMMPX=vá=~åÇ=iÉÉI=OMMQzK=qÜÉêÉÑçêÉI=ïÉ=ïáää=åçí=ÑçÅìë=çå=íÜ~í=ÜÉêÉK=qÜÉ=ã~íÜÉã~íáÅ~ä=Éèìáî~äÉåÅó=çÑ=íÜÉ=lr_=ãçÇÉä=~åÇ=íÜÉ=ôENF=ãçÇÉä=~êÉ=ëíê~áÖÜíÑçêï~êÇI=ëç=äçÖáÅ~ääó=áíë=êÉëìäíë=ïáää=~äëç=ÄÉ=Éèìáî~äÉåíK=fåëíÉ~ÇI=ïÉ=ïáää=Ñáêëí=ÑçÅìë=çå=Åçãé~êáåÖ=íÜÉ=ptpf=ÅêáíÉêáçå=ïáíÜ=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçåK=içÖáÅ~ääóI=áÑ=íÜÉ=ãáåáã~=çÑ=ÄçíÜ=ÅêáíÉêá~=~êÉ=áÇÉåíáÅ~äI=íÜÉó=ïáää=óáÉäÇ=áÇÉåíáÅ~ä=é~ê~ãÉíÉê=Éëíáã~íÉë=~åÇ=íÜÉêÉÑçêÉ=~äëç=áÇÉåíáÅ~ä=ãçÇÉä=éêÉÇáÅíáçåëK=pÉÅçåÇI=ïÉ=ïáää=Éñ~ãáåÉ=íÜÉ=áååçî~íáçå=î~êá~åÅÉ=ÑìåÅíáçåI=ïÜáÅÜ=áë=ÅêìÅá~ä=Ñçê=Ü~åÇäáåÖ=áêêÉÖìä~ê=Ç~í~=áå=ÄçíÜ=íÜÉ=lr_=~åÇ=íÜÉ=h~äã~å=ÑáäíÉê=~ééêç~ÅÜK=tÉ=ïáää=áääìëíê~íÉ=íÜ~í=~äëç=áå=íÜáë=Å~ëÉI=íÜÉ=ãçÇÉä=ÉÑÑÉÅíáîÉäó=ïÜáíÉåë=íÜÉ=êÉëáÇì~äë=~åÇ=íÜÉ=~ëëìãéíáçå=çÑ=ÉñéçåÉåíá~ä=åçáëÉ=ÇÉÅ~ó=áë=íÜÉêÉÑçêÉ=î~äáÇK=^ë=~=íÉëí=Å~ëÉI=ïÉ=ïáää=ìëÉ=íÜÉ=åçáëÉ=ãçÇÉä=áå=ÅçåàìåÅíáçå=ïáíÜ=~=Åçåíáåìçìë=íê~åëÑÉê=ÑìåÅíáçå=ãçÇÉä=áå=íÜÉ=ÅçåíÉñí=Ñçê=ïÜáÅÜ=áí=ï~ë=ÇÉîÉäçéÉÇI=áKÉK=ãçÇÉäáåÖ=íÜÉ=êÉëáÇì~äë=çÑ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåëK=få=íÜÉ=ëáãéäÉ=Å~ëÉ=ïÜÉêÉ=ÖêçìåÇï~íÉê=äÉîÉä=ÑäìÅíì~íáçåë=~êÉ=áåÑäìÉåÅÉÇ=Äó=éêÉÅáéáí~íáçå=ëìêéäìë=çåäóI=íÜÉ=ÅçãÄáåÉÇ=íê~åëÑÉê=ÑìåÅíáçå=åçáëÉ=ãçÇÉä=áë=ÖáîÉå=ÄóW==th( t) = ∫ θ ( t − τ ) p( τ )d τ + n( t)+ d(4.25)−∞=ïÜÉêÉ=t = W=íáãÉ=xqz=Optimisation method(Levenberg-Marquardt)AdjustparametersnoConvergence?yesDiagnosticCheckInitial estimateΨ = [ ψ1, ψ2,.., ψ p]EvaluatedeterministicmodelResidualsEvaluatenoise modelInnovationsComputeSWSI-criterionTObservations=figure 4.4: Procedure for applying the noise model incombination with a deterministic model.===NMP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=Q=NMQ=h = W=çÄëÉêîÉÇ=ÖêçìåÇï~íÉê=äÉîÉäI=êÉä~íáîÉ=íç=ëçãÉ=êÉÑÉêÉåÅÉ=äÉîÉä=xiz=p = W=éêÉÅáéáí~íáçå=ëìêéäìë=xiq JN z=θ = W=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=xJzI=Ñçê=ïÜáÅÜ=~=ëÅ~äÉÇ=Ö~ãã~=ÇáëíêáÄìíáçå=áë=ÅÜçëÉåI==n n−b t1 exp( −bt)= ==áKÉK= θ ( t)= A=Γ( n)n = W=êÉëáÇì~ä=ëÉêáÉë=xizI=ãçÇÉäÉÇ=ÅçåÑçêã=Éèì~íáçå=EQKNNF=d = W=äçÅ~ä=Çê~áå~ÖÉ=äÉîÉäI=êÉä~íáîÉ=íç=ëçãÉ=êÉÑÉêÉåÅÉ=äÉîÉä=xiz==jçêÉ=ÇÉí~áäë=çå=íÜÉ=Ä~ÅâÖêçìåÇ=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçå=ãçÇÉä=Å~å=ÄÉ=ÑçìåÇ=áå=xsçå=^ëãìíÜ=Éí=~äKI=OMMOzK=qÜÉ=qck=ãçÇÉä=áë=Å~äáÄê~íÉÇ=çå=~=NRJóÉ~ê=ENVUNJNVVSF=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=çÄëÉêîÉÇ=ïáíÜ=~=Ç~áäó=ÑêÉèìÉåÅóK=eçïÉîÉêI=íÜáë=ëÉêáÉë=áë=åçí=íçí~ääó=ÅçãéäÉíÉ=Äìí=UKTRB=çÑ=íÜÉ=çÄëÉêî~íáçåë=~êÉ=ãáëëáåÖK=qÜÉ=ëÉêáÉë=çêáÖáå~íÉë=Ñêçã=~=éáÉòçãÉíÉê=äçÅ~íÉÇ=çå=íÜÉ=ã~áå=ãÉíÉçêçäçÖáÅ=ÑáÉäÇ=çÑ=íÜÉ=oçó~ä=aìíÅÜ=jÉíÉçêçäçÖáÅ=fåëíáíìíÉ=~í=íÜÉ=íçïå=çÑ=aÉ=_áäí=áå=íÜÉ=ÅÉåíÉê=çÑ=íÜÉ=kÉíÜÉêä~åÇë=xëÉÉ=~äëç=_áÉêâÉåë=Éí=~äKI=NVVVzK=qÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=áë=çÄí~áåÉÇ=Ñêçã=Ç~áäó=~îÉê~ÖÉÇ=çÄëÉêî~íáçåë=çÑ=éêÉÅáéáí~íáçå=~åÇ=éçíÉåíá~ä=Éî~éçíê~åëéáê~íáçå=~í=íÜÉ=ãÉíÉçêçäçÖáÅ=ÑáÉäÇK=^=íáãÉ=éäçí=çÑ=íÜÉ=~î~áä~ÄäÉ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=áë=ÖáîÉå=áå=ÑáÖìêÉ=QKRK=2h (m +NAP)1.510.511982 1984 1986 1988 1990 1992 1994 1996date (years)n (m)0.50−0.50.51982 1984 1986 1988 1990 1992 1994 1996date (years)ν (m)0−0.51982 1984 1986 1988 1990 1992 1994 1996date (years)=figure 4.5: Time plot of the available groundwater level observations (dots) from piezometer 32cl0034relative to the national reference level (h+NAP), the predictions of the transfer model (full line), themodel residuals (n) and the innovation series (ν).=..…………………………………………………………………………………………….….=

^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=qÜÉ=é~ê~ãÉíÉê=ëÉí= Ψ =çÑ=íÜÉ=ÅçãÄáåÉÇ=ãçÇÉä=áë=áåíÉÖê~ääó=Éëíáã~íÉÇ=ìëáåÖ=~=åìãÉêáÅ~ääó=ÇÉêáîÉÇ=g~ÅçÄá~å=ã~íêáñ=~åÇ=íÜÉ=ãÉíÜçÇë=ÇÉëÅêáÄÉÇ=áå=íÜÉ=éêÉîáçìë=ëÉÅíáçåK=^ë= d =áë=Éäáãáå~íÉÇ=Ñêçã=íÜÉ=Éèì~íáçåëI=íÜÉ=é~ê~ãÉíÉêë=íÜ~í=Ü~îÉ=íç=ÄÉ=Éëíáã~íÉÇ=~êÉ=table 4.1: Calibration results, estimated parameters andcharacteristics for piezometer 32cl0034.máÉòçãÉíÉê=2R =ojpb=ojpf=A ====EHLJ=OσF=POÅäMMPQ=MKSSR=NNIM=Åã=PIS=Åã=SNKS==EHLJ=QKOF=Ç~ó=A, b,n =Ñêçã=íÜÉ=íê~åëÑÉê=ãçÇÉäI= b =====EHLJ=OσF=MKOU==EHLJ=MKMOF=Ç~ó JN=~äçåÖ=ïáíÜ=α =Ñêçã=íÜÉ=åçáëÉ= n =====EHLJ=OσF=NKUU==EHLJ=MKMSF=ãçÇÉäK=cçê=íÜáë=éìêéçëÉI=~ää= α =====EHLJ=OσF= NUKO==EHLJ=PKOF=Ç~ó JN =~î~áä~ÄäÉ=çÄëÉêî~íáçåë=áå=íÜÉ= =éÉêáçÇ=ïÉêÉ=ìëÉÇK=qÜÉ=êÉëìäíë=çÑ=íÜÉ=qck=ãçÇÉä=~êÉ=ÖáîÉå=áå=ÑáÖìêÉ=QKRI=~ë=íáãÉ=éäçíë=çÑ=íÜÉ=çÄëÉêî~íáçåë=~åÇ=éêÉÇáÅíáçåëI=~åÇ=çÑ=íÜÉ=êÉëáÇì~äë=~åÇ=áååçî~íáçåëK=qÜÉ=2é~ê~ãÉíÉê=Éëíáã~íÉë=~åÇ=Å~äáÄê~íáçå=êÉëìäíë=EÅçÉÑÑáÅáÉåí=çÑ=ÇÉíÉêãáå~íáçå=E R FI=êççí=ãÉ~å=ëèì~êÉÇ=Éêêçê=EojpbFI=êççí=ãÉ~å=ëèì~êÉÇ=áååçî~íáçå=EojpfFF=~êÉ=äáëíÉÇ=áå=í~ÄäÉ=QKNK=qÜÉ=~ìíçÅçêêÉä~íáçå=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçåëI=Ñçê=ïÜáÅÜ=~=íáãÉ=ä~Ö=áåÅêÉãÉåí=çÑ=çåÉ=ãçåíÜ=áë=ÅÜçëÉå=áå=çêÇÉê=íç=êÉîÉ~ä=ëÉ~ëçå~ä=é~ííÉêåë=áå=íÜÉ=~ìíçÅçêêÉä~íáçåI=áåÇáÅ~íÉë=íÜ~í=íÜÉ=ïÜáíÉ=åçáëÉ=~ëëìãéíáçå=ÜçäÇë=EÑáÖìêÉ=QKSFK=_ÉÅ~ìëÉ=çÑ=íÜÉ=ä~êÖÉ=åìãÄÉê=çÑ=~î~áä~ÄäÉ=çÄëÉêî~íáçåëI=íÜÉ=~ìíçÅçêêÉä~íáçå=ÑìåÅíáçå=áë=ê~íÜÉê=ëãççíÜ=~åÇ=íÜÉ=~ÅÅçãé~åóáåÖ=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=å~êêçïK====NMR=4.3.2 Comparison of the likelihood function and the SWSI criterionmäçíë=çÑ=ÄçíÜ=ÑìåÅíáçåë=ïÉêÉ=ã~ÇÉ=Äó=î~êóáåÖ=íïç=çÑ=íÜÉ=é~ê~ãÉíÉêë=~êçìåÇ=íÜÉáê=Éëíáã~íÉÇ=î~äìÉI=ïÜáäÉ=âÉÉéáåÖ=íÜÉ=çíÜÉêë=Åçåëí~åí=EÅÉíÉêáë=é~êáÄìëFK=qÜÉ=é~ê~ãÉíÉêë=ïÉêÉ=î~êáÉÇ=HLJ=MKR=Ü~äÑ=íÜÉáê=Éëíáã~íÉÇ=î~äìÉK=få=çêÇÉê=íç=ã~âÉ=ÄçíÜ=ÑìåÅíáçåë=Åçãé~ê~ÄäÉI=íÜÉó=ïÉêÉ=åçêã~äáòÉÇ=Äó=êÉëéÉÅíáîÉäó=ëÉííáåÖ=íÜÉáê=ã~ñáãìã=~åÇ=ãáåáãìã=íç=òÉêçI=~åÇ=ÇáîáÇáåÖ=íÜÉ=êÉëìäí=Äó=íÜÉ=ê~åÖÉK=få=ÑáÖìêÉ=QKTI=~=Åçåíçìê=éäçí=çÑ=íÜÉ=åçêã~äáòÉÇ=äçÖ=äáâÉäáÜççÇ=ÑìåÅíáçå=~åÇ=ptpf=ÅêáíÉêáçå=Ñçê=íÜÉ=Ñáêëí=íïç=é~ê~ãÉíÉêë=çÑ=íÜÉ=íê~åëÑÉê=ãçÇÉä=E A =~åÇ= b F=áë=ÖáîÉåK=qÜÉ=ÑáÖìêÉ=ëÜçïë=íÜ~í=íÜÉ=Åçåíçìêë=çÑ=ÄçíÜ=çÄàÉÅíáîÉ=ÑìåÅíáçåë=~êÉ=~äãçëí=áÇÉåíáÅ~äI=ïÜáÅÜ=ÅçåÑáêãë=íÜ~í=íÜÉ=ptpf=ÅêáíÉêáçå=óáÉäÇë=~äãçëí=Éñ~Åí=ji=Éëíáã~íÉëK=qÜáë=Å~å=ÄÉ=ÉñéÉÅíÉÇI=~ë=íÜÉ=çåäó=ÇáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=íÜÉ==0.2correlation coefficient (−)0.150.10.050−0.05−0.10 5 10 15 20 25time step (months)=figure 4.6: Autocorrelation function of the innovations of the OUB model. The dotted lines denote the95% confidence interval. The autocorrelation function indicates that the white noise assumptionholds.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=Q=NMS=ji=ÑìåÅíáçå=~åÇ=ptpf=ÅêáíÉêáçå=áë=íÜ~í=áå=Ñáêëí=Å~ëÉ= σ EÑçê=íÜÉ=ãáåáã~ä=íáãÉ=ëíÉéF=áë=é~êí=çÑ=íÜÉ=é~ê~ãÉíÉê=ëÉí=íç=ÄÉ=Éëíáã~íÉÇI=ïÜáäÉ=áå=íÜÉ=ptpf=ÅêáíÉêáçå= σ =áë=~ëëìãÉÇ=âåçïå=~åÇ=áãéäáÅáíäó=~ééêçñáã~íÉÇ=ïáíÜ=bèì~íáçå=EQKNVFK=qÜáë=~ééêçñáã~íáçå=ÅçåîÉêÖÉë=íç=íÜÉ=íêìÉ=î~äìÉ=çÑ= σ =Ñçê=ä~êÖÉ= N K=`çåëÉèìÉåíäóI=~=Åçãé~êáëçå=çÑ=2nσ2 a( Ψ ) =Éëíáã~íÉÇ=~ë=~=é~ê~ãÉíÉê=ïáíÜ=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=çê=Éëíáã~íÉÇ=~ÑíÉêï~êÇë=ìëáåÖ=íÜÉ=ptpf=~ééêç~ÅÜ=ïáää=ÉñÉãéäáÑó=íÜÉ=äáâÉåÉëë=çÑ=ÄçíÜ=ÅêáíÉêá~=Ñçê=~ää=é~ê~ãÉíÉêëK=rëáåÖ=íÜÉ=ptpf=ÅêáíÉêáçå= σ2 a( Ψ ) =áë=Éëíáã~íÉÇ=Ñêçã=íÜÉ=Ç~í~=áå=íÜÉ=ÑçääçïáåÖ=ï~óW==n⎧ 1−exp( −2α) ⎫ 2∑⎨⎬ν( ti, Ψ)2 i=11−exp( −2α∆ti)σa( Ψ ) =⎩⎭(4.26)nν ( ti, Ψ)i=1ïÜáÅÜ=áë=Éèìáî~äÉåí=íç=Éèì~íáçå=EQKSF=Ñçê=ÑáñÉÇ= ∆ t =ïÜÉå=náë=êÉéä~ÅÉÇ=Äóσ v( Ψ)I=~åÇ= φ = exp( −α) K=få=ÑáÖìêÉ=QKUI=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçåë==áë=éäçííÉÇ=~åÇ=Åçãé~êÉÇ=íç= σ2 ( Ψ ) =çÄí~áåÉÇ=ïáíÜ=Éèì~íáçå=EQKOSF=EáKÉK=MKMMNP=ãOFK=a2an∑22n1normalized ML and SSWI criteria0.80.60.40.20405060708090 0.15..…………………………………………………………………………………………….….*0.20.250.3parameter bparameter A==figure 4.7: Contour plot of the normalized log likelihood function of the innovations (dotted lines) andthe SSWI criterion (full lines) as a function of the parameters A and b (ceteris paribus). The marker(* ) denotes the estimated values. From the figure it can be seen that both functions are almostidentical.0.350.4

^=Åçåíáåìçìë=åçáëÉ=ãçÇÉä=950094009300log likelihood (−)92009100900089008800===NMT==87000.6 0.8 1 1.2 1.4 1.6 1.8 2noise variance (m)x 10 −3figure 4.8: Plot of the log likelihood function of the innovations as a function of the parameterThe dotted line denotes the value of σ2 a( Ψ ) obtained with the SWSI criterion. The minimum of thelog likelihood function proves to coincide well with the value obtained from the SWSI criterion.=cêçã=íÜáë=ÑáÖìêÉI=íÜÉ=Éëíáã~íÉ=çÑ= σ2 a( Ψ ) =éêçîÉë=íç=ÅçáåÅáÇÉ=ïÉää=ïáíÜ=íÜÉ=ã~ñáãìã=çÑ=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=Ñçê=íÜáë=é~ê~ãÉíÉêK==4.3.3 A check on the innovation variance function_ÉÅ~ìëÉ=íÜÉ=fsc=áë=~=âÉó=Ñ~Åíçê=áå=ÄçíÜ=íÜÉ=ÇÉêáî~íáçå=çÑ=íÜÉ=ptpf=ÅêáíÉêáçå=~åÇ=áå=ÖÉåÉê~ä=Ñçê=ëíçÅÜ~ëíáÅ=ãçÇÉäáåÖ=~í=ãáñÉÇ=ÑêÉèìÉåÅáÉëI=áå=íÜÉ=ÑçääçïáåÖ=ïÉ=ïáää=Éñ~ãáåÉ=áí=ãçêÉ=ÅäçëÉäó=ìëáåÖ=íÜÉ=Ç~í~=çÑ=ëÉÅíáçå=QKPKNK=cçê=íÜáë=éìêéçëÉI=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=ï~ë=êÉë~ãéäÉÇ=~í=ÑêÉèìÉåÅáÉë= ∆ t = =xNIOIPÁKKTRz=Ç~óëK=qÜÉ=íê~åëÑÉê=~åÇ=åçáëÉ=ãçÇÉä=ïÉêÉ=Éî~äì~íÉÇ=ìëáåÖ=íÜÉ=Éëíáã~íÉ=çÑ= Ψ =çÄí~áåÉÇ=ïáíÜ=íÜÉ=Ç~áäó=ëÉêáÉëI=ïÜáÅÜ=äçÖáÅ~ääó=ëÜçìäÇ=ÄÉ=çéíáã~äK=qÜìëI=~=êÉëáÇì~ä=~åÇ=áååçî~íáçå=ëÉêáÉë=ïÉêÉ=çÄí~áåÉÇ=Ñçê=ÉîÉêó=ë~ãéäÉ=ÑêÉèìÉåÅóK=^ë=ëí~íÉÇ=É~êäáÉêI=íÜÉ=Ç~áäó=ëÉêáÉë=ï~ë=åçí=ÅçãéäÉíÉ=~åÇ=íÜÉêÉÑçêÉ=~äëç=íÜÉ=êÉë~ãéäÉÇ=ëÉêáÉë=ïÉêÉ=åçíK=_ÉÅ~ìëÉ=çÑ=íÜáëI=çåäó=íÜçëÉ=áååçî~íáçåë=~åÇ=êÉëáÇì~äë=ïÉêÉ=ëÉäÉÅíÉÇ=Ñêçã=íÜÉ=êÉë~ãéäÉÇ=ëÉêáÉë=Ñçê=ïÜáÅÜ=∆ t i=Éèì~äÉÇ=íÜÉ=ë~ãéäáåÖ=ÑêÉèìÉåÅó=Éñ~ÅíäóK=_çíÜ=íÜÉ=áååçî~íáçå=~åÇ=êÉëáÇì~ä=î~êá~åÅÉ=çÑ=íÜÉ=êÉëìäíáåÖ=ëÉêáÉë=ïÉêÉ=Éëíáã~íÉÇ=~åÇ=éäçííÉÇ=áå=ÑáÖìêÉ=QKVK=få=íÜÉ=ë~ãÉ=ÑáÖìêÉI=Éèì~íáçå=EQKNUF=ÄÉáåÖ=íÜÉ=íÜÉçêÉíáÅ~ä=fscI=áë=éäçííÉÇ=íçÖÉíÜÉê=ïáíÜ=íÜÉ=VRB=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=çÑ=íÜÉ=Éëíáã~íÉëK=qÜÉ=ä~ííÉê=áë=ÖáîÉå=Äó=xpåÉÇÉÅçê=~åÇ=`çÅÜê~åI=NVSTzW===2σa.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=Q=0.0180.0160.014innovation variance (m)0.0120.010.0080.006NMU=0.0040.002estimated innov.var.theoretical innov.var.95% conf. intervalresidual var.00 10 20 30 40 50 60 70time step (days)==figure 4.9: Plot of the estimated and theoretical IVF as a function of the time step, and of theestimated residual variance. In general, the estimated innovation variances do not differ significantlyfrom the theoretical IVF, although the agreement is better for smaller time steps.2 2v ∆ i Ψ2 σ v ∆ i Ψ≤ σ ( , )2 v ∆ti≤2χ0.975 χ0.025σ ( t , ) *( N -1) ( t , ) *( N -1)Ψ (4.27)=cáÖìêÉ=QKV=íÜìë=~ääçïë=Ñçê=~=Åçãé~êáëçå=ÄÉíïÉÉå=íÜÉ=Éëíáã~íÉÇ=~åÇ=íÜÉçêÉíáÅ~ä=fscK=^=ÅÜÉÅâ=çå=ïÜÉíÜÉê=íÜÉ=Éëíáã~íÉÇ=î~êá~åÅÉë=ÇáÑÑÉê=ëáÖåáÑáÅ~åíäó=Ñêçã=íÜÉ=íÜÉçêÉíáÅ~ä=fsc=ÅçìäÇ=ïÉää=ÄÉ=ìëÉÇ=~ë=~=ÑìêíÜÉê=Çá~ÖåçëíáÅ=ÅÜÉÅâ=çå=íÜÉ=î~äáÇáíó=çÑ=íÜÉ=ãçÇÉäI=åÉñí=íç=ÅÜÉÅâáåÖ=íÜÉ=~ìíçJ=~åÇ=ÅêçëëÅçêêÉä~íáçå=ÑìåÅíáçåëK=cêçã=ÑáÖìêÉ=QKV=áí=áë=ÅçåÅäìÇÉÇ=íÜ~í=íÜÉ=Éëíáã~íÉë=çÑ=íÜÉ=áååçî~íáçå=î~êá~åÅÉ= σ2 ( ∆t, Ψ ) =Çç=åçí=ëáÖåáÑáÅ~åíäó=ÇáÑÑÉê=Ñêçã=víÜÉ=íÜÉçêÉíáÅ~ä=fscI=~äíÜçìÖÜ=íÜÉ=~ÖêÉÉãÉåí=áë=äÉëë=Ñçê=íÜÉ=áåíÉêãÉÇá~íÉ=íáãÉ=ëíÉéëK=qÜáë=éçáåíë=íç=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=é~ê~ãÉíÉê=α I=ÇÉÑáåáåÖ=íÜÉ=ÇÉÅ~ó=ê~íÉ=çÑ=íÜÉ=åçáëÉ=ãçÇÉäI=ÅçìäÇ=ÄÉ=ëÉåëáíáîÉ=íç=íÜÉ=ë~ãéäáåÖ=ÑêÉèìÉåÅóK=qÜÉêÉÑçêÉI=ëéÉÅá~ä=Å~êÉ=Ü~ë=íç=ÄÉ=í~âÉå=ïÜÉå=íÜÉ=~ëëìãéíáçå=çÑ=ÉñéçåÉåíá~ä=ÇÉÅ~ó=Ñçê=íÜÉ=åçáëÉ=ãçÇÉä=ÇçÉë=åçíI=çê=åçí=ïÉääI=ÜçäÇ=~åÇLçê=ïÜÉå=íÜÉ=ãçÇÉä=áë=ìëÉÇ=Ñçê=ëáãìä~íáçåë=~í=~=ÇáÑÑÉêÉåí=ÑêÉèìÉåÅó=íÜ~å=íÜ~í=çÑ=íÜÉ=Ç~í~=ïáíÜ=ïÜáÅÜ=áí=ï~ë=Å~äáÄê~íÉÇK=^=ëí~åÇ~êÇ=Çá~ÖåçëíáÅ=ÅÜÉÅâ=äáâÉ=íÜÉ=~ìíçÅçêêÉä~íáçå=ÑìåÅíáçå=çÑ=íÜÉ=áååçî~íáçåë=ïáää=åçí=éêçîáÇÉ=ÉåçìÖÜ=áåÑçêã~íáçå=çå=íÜáë=éçáåíK==4.4 Discussion and conclusions^=Åçãé~êáëçå=çÑ=íÜÉ=ôENF=ãçÇÉäI=ÚÅçåîÉåíáçå~äÛ=çê=ÉãÄÉÇÇÉÇ=áå=~=h~äã~å=ÑáäíÉêI=~åÇ=íÜÉ=lêåëíÉáåJrÜäÉåÄÉÅâ=Ä~ëÉÇ=åçáëÉ=ãçÇÉä=ëÜçïë=íÜ~í=íÜÉáê=Éèì~íáçåë=Ñçê=êÉëéÉÅíáîÉäó=íÜÉ=ÑçêÉÅ~ëíáåÖ=ãçÇÉ=áå=íÜÉ=ôENF=ãçÇÉäI=íÜçëÉ=Ü~åÇäáåÖ=íÜÉ=éêÉÇáÅíáçåë=~åÇ=íÜÉáê==..…………………………………………………………………………………………….….

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5jçÇÉäáåÖ=ãìäíáéäÉ=ëíêÉëëÉë=Chapter5 Modeling groundwater head seriessubjected to multiple stressesModeling groundwater head==series = subjected to multiple^ÇçéíÉÇ=ÑêçãW=sçå=^ëãìíÜI=gK=oKI=hK=j~~ëI=jK=_~ââÉêI=~åÇ=gK=mÉíÉêëÉå=EOMMUFI=jçÇÉäáåÖ=íáãÉ=ëÉêáÉë=stressesçÑ=ÖêçìåÇï~íÉê=ÜÉ~Ç=ÑäìÅíì~íáçåë=ëìÄàÉÅíÉÇ=íç=ãìäíáéäÉ=ëíêÉëëÉëI=dêçìåÇ=t~íÉêI=QSI=ÇçáW=NMKNNNNLàKNTQRJSRUQKOMMTKMMPUOKñENFI=PMJQMK=oÉéêçÇìÅÉÇ=ïáíÜ=éÉêãáëëáçå=Ñêçã=_ä~ÅâïÉää=mìÄäáëÜáåÖ=fåÅKI=ÅçéóêáÖÜí=©=OMMU=_ä~ÅâïÉää=mìÄäáëÜáåÖ=fåÅK===Adopted from:Abstract=få=íÜáë=ÅÜ~éíÉêI=íÜÉ=ãÉíÜçÇë=ÄÉÜáåÇ=íÜÉ=mfocf`q=EmêÉÇÉÑáåÉÇ=fãéìäëÉ=Von Asmuth, J. R., K. Maas, M. Bakker, and J. PetersenoÉëéçåëÉ=cìåÅíáçå=få=`çåíáåìçìë=qáãÉF=íáãÉ=ëÉêáÉë=ãçÇÉä=~êÉ=ÉñíÉåÇÉÇ=íç=ÅçîÉê=ãçêÉ=(2008)ÅçãéäÉñ=ëáíì~íáçåë=ïÜÉêÉ=ãìäíáéäÉ=ëíêÉëëÉë=áåÑäìÉåÅÉ=ÖêçìåÇï~íÉê=ÜÉ~Ç=ÑäìÅíì~íáçåë=Modeling time series of groundwater head fluctuationsëáãìäí~åÉçìëäóK=få=Åçãé~êáëçå=íç=ôj^=E^ìíçJoÉÖêÉëëáîÉ=jçîáåÖ=^îÉê~ÖÉF=íáãÉ=ëÉêáÉë=subjected to multiple stresses.ãçÇÉäëI=íÜÉ=mfocf`q=ãçÇÉä=áë=çéíáãáòÉÇ=Ñçê=ìëÉ=çå=ÜóÇêçäçÖáÅ=éêçÄäÉãëK=qÜÉ=çÄàÉÅíáîÉ=Ground Water, 46, doi:10.1111/j.1745-6584.2007.00382.x(1)çÑ=íÜÉ=é~éÉê=áë=íïçÑçäÇK=cáêëíI=~å=~ééêç~ÅÜ=áë=éêÉëÉåíÉÇ=Ñçê=Ü~åÇäáåÖ=ãìäíáéäÉ=ëíêÉëëÉë=30-4.áå=íÜÉ=ãçÇÉäK=b~ÅÜ=ëíêÉëë=Ü~ë=~=ëéÉÅáÑáÅ=é~ê~ãÉíêáÅ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåK=Reproduced with permission from Blackwell Publishing Inc.^ééêçéêá~íÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåë=Ñçê=çíÜÉê=ëíêÉëëÉë=íÜ~å=éêÉÅáéáí~íáçå=~êÉ=copyright 2008 Blackwell Publishing Inc.ÇÉêáîÉÇ=Ñêçã=~å~äóíáÅ=ëçäìíáçåë=çÑ=ÉäÉãÉåí~êó=ÜóÇêçÖÉçäçÖáÅ=éêçÄäÉãëK=cìêíÜÉêãçêÉI=Abstract: ÇáÑÑÉêÉåí=ëíêÉëëÉë=Çç=åçí=åÉÉÇ=íç=ÄÉ=ÅçååÉÅíÉÇ=áå=é~ê~ääÉä=áå=íÜÉ=ãçÇÉäI=~ë=áë=íÜÉ=ëí~åÇ~êÇ=éêçÅÉÇìêÉ=áå=ôj^=ãçÇÉäëK=pÉÅçåÇI=ÖÉåÉê~ä=éêçÅÉÇìêÉë=~êÉ=éêÉëÉåíÉÇ=Ñçê=In this chapter, the methods behind the PIRFICT (Predefined Impulse Response FunctionãçÇÉäáåÖ=~åÇ=áåíÉêéêÉí~íáçå=çÑ=íÜÉ=êÉëìäíëK=qÜÉ=ãìäíáéäÉJáåéìí=mfocf`q=ãçÇÉä=áë=~ééäáÉÇ=In Continuous Time) time series model are extended to cover more complex situationsíç=íïç=êÉ~ä=Å~ëÉëK=få=íÜÉ=Ñáêëí=çåÉI=áí=áë=ëÜçïå=íÜ~í=íÜáë=ãçÇÉä=Å~å=ÉÑÑÉÅíáîÉäó=where multiple stresses influence groundwater head fluctuations simultaneously. InÇÉÅçãéçëÉ=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=ÜÉ~Ç=ÑäìÅíì~íáçåë=áåíç=é~êíá~ä=ëÉêáÉëI=É~ÅÜ=comparison to ARMA (Auto-Regressive Moving Average) time series models, the PIRFICTêÉéêÉëÉåíáåÖ=íÜÉ=áåÑäìÉåÅÉ=çÑ=~å=áåÇáîáÇì~ä=ëíêÉëëK=qÜÉ=ëÉÅçåÇ=~ééäáÅ~íáçå=Ü~åÇäÉë=model is optimized for use on hydrologic problems. The objective of the paper is twofold.ãìäíáéäÉ=çÄëÉêî~íáçå=ïÉääëK=fí=áë=ëÜçïå=íÜ~í=ÉäÉãÉåí~êó=éÜóëáÅ~ä=âåçïäÉÇÖÉ=~åÇ=íÜÉ=First, an approach is presented for handling multiple stresses in the model. Each stress hasëé~íá~ä=ÅçÜÉêÉåÅÉ=áå=íÜÉ=êÉëìäíë=çÑ=ãìäíáéäÉ=ïÉääë=áå=~å=~êÉ~=ã~ó=ÄÉ=ìëÉÇ=íç=áåíÉêéêÉí=a specific parametric impulse response function. Appropriate impulse response functions~åÇ=ÅÜÉÅâ=íÜÉ=éä~ìëáÄáäáíó=çÑ=íÜÉ=êÉëìäíëK=qÜÉ=ãÉíÜçÇë=éêÉëÉåíÉÇ=Å~å=ÄÉ=ìëÉÇ=for other stresses than precipitation are derived from analytic solutions of elementaryêÉÖ~êÇäÉëë=çÑ=íÜÉ=ÜóÇêçÖÉçäçÖáÅ=ëÉííáåÖK=qÜÉó=~êÉ=áãéäÉãÉåíÉÇ=áå=~=ÅçãéìíÉê=é~Åâ~ÖÉ=hydrogeologic problems. Furthermore, different stresses do not need to be connected inå~ãÉÇ=jÉåó~åíÜÉë=EïïïK=ãÉåó~åíÜÉëKåäFK=parallel in the model, as is the standard procedure in ARMA models. Second, generalprocedures are presented for modeling and interpretation of the results. The multipleinputPIRFICT model is applied to two real cases. In the first one, it is shown that thismodel can effectively decompose series of groundwater head fluctuations into partialseries, each representing the influence of an individual stress. The second applicationhandles multiple observation wells. It is shown that elementary physical knowledge andthe spatial coherence in the results of multiple wells in an area may be used to interpretand check the plausibility of the results. The methods presented can be used regardless ofthe hydrogeologic setting. They are implemented in a computer package namedMenyanthes (www. menyanthes.nl).===NNN=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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`Ü~éíÉê=R=table 5.1: Model results and parameter estimates for the groundwater head series observed in well46DP0032NOM=bsm= VOKM=B= = = =ojpb=EãF= MKMSU= = = =aê~áå~ÖÉ=Ä~ëÉ=EãF= NNKRP= = = == = = = =fåéìí=ëÉêáÉë= píêÉëë=íóéÉ= m~ê~ãÉíÉê= s~äìÉ= σ =sbko^v=mêÉÅáéáí~íáçå= A( M 0, p ) =OVVKO=OP== = b = MKMMPNNT= MKMMMPP== = n = MKUVU= MKMOR=bfkaelsbk= bî~éçê~íáçå= f = NKNS= MKMTS=mKpK=_bodbk= mìãéáåÖ=ïÉää= α OKRU= OKQ== = β MKMROO= MKMOT== = γ MKMMRUR= MKMP== =M 0,w JPKSPbJMR= RKSUbJMS=p~ãÄÉÉâ_çîÉå= oáîÉê= α′ MKMOROT= MKMVQ== = β′ MKMPSU= MKNP== = γ ′ MKTSRS= MKMRT== ==5.3 Example application5.3.1 Single seriesM 0,sMKTOTV=MKMVPV=qÜÉ=ÑìåÅíáçåáåÖ=~åÇ=êÉëìäíë=çÑ=~=ãìäíáéäÉ=áåéìí=mfocf`q=ãçÇÉä=çå=~=ëáåÖäÉ=ïÉää=äçÅ~íÉÇ=áå=íÜÉ=åçêíÜÉêå=é~êí=çÑ=íÜÉ=éêçîáåÅÉ=çÑ=iáãÄìêÖ=EíÜÉ=kÉíÜÉêä~åÇëF=~êÉ=éêÉëÉåíÉÇ=áå=íÜáë=ëìÄëÉÅíáçåK=qÜÉ=ïÉääI=ïáíÜ=íÜÉ=å~íáçå~ä=ÅçÇÉ=QSamMMPOI=Ü~ë=íïç=ëÅêÉÉåëK=tÉ=ÅçåëáÇÉê=çåäó=íÜÉ=íçé=çåÉI=ïÜáÅÜ=áë=äçÅ~íÉÇ=NP=ãÉíÉêë=ÄÉäçï=íÜÉ=ëìêÑ~ÅÉK=qÜÉ=ïÉää=áë=äçÅ~íÉÇ=çå=íÜÉ=ÉÇÖÉ=çÑ=íÜÉ=ÑäççÇéä~áå=çÑ=íÜÉ=êáîÉê=jÉìëÉI=áå=~å=~èìáÑÉê=íÜ~í=Åçåëáëíë=çÑ=Åç~êëÉI=Öê~îÉääó=ë~åÇë=çîÉêä~áå=Äó=ÑáåÉê=ë~åÇë=ïÜáÅÜ=çêáÖáå~íÉ=Ñêçã=êáîÉê=ÇÉéçëáíëK=få=~ÇÇáíáçå=íç=íÜÉ=êáîÉê=äÉîÉä=ÑäìÅíì~íáçåëI=íÜÉ=ÜÉ~Ç=áë=áåÑäìÉåÅÉÇ=Äó=éêÉÅáéáí~íáçå=~åÇ=Éî~éçê~íáçå=~åÇ=Äó=~=éìãéáåÖ=ëí~íáçå=åÉ~ê=íÜÉ=íçïå=çÑ=_ÉêÖÉå=ïÜÉêÉ=ÖêçìåÇï~íÉê=áë=ïáíÜÇê~ïå=Ñçê=ÇêáåâáåÖ=ï~íÉê=éêçÇìÅíáçåK===qáãÉ=ëÉêáÉë=Ç~í~=çÑ=~ää=ëíêÉëëÉë=~êÉ=~î~áä~ÄäÉK=qÜÉ=éêÉÅáéáí~íáçå=~åÇ=éçíÉåíá~ä=Éî~éçê~íáçå=ëÉêáÉë=çêáÖáå~íÉ=Ñêçã=ëí~íáçåë=çÑ=íÜÉ=oçó~ä=aìíÅÜ=jÉíÉçêçäçÖáÅ=fåëíáíìíÉ=áå=sÉåê~ó=~åÇ=báåÇÜçîÉåI=êÉëéÉÅíáîÉäóK=qÜÉ=êáîÉê=äÉîÉäë=ïÉêÉ=ãçåáíçêÉÇ=~í=~=Ç~ã=áå=p~ãÄÉÉâI=ÇçïåëíêÉ~ãë=çÑ=ïÉää=QSamMMPOX=íÜÉ=éìãéáåÖ=ê~íÉë=ïÉêÉ=çÄí~áåÉÇ=Ñêçã=íÜÉ=ÇêáåâáåÖ=ï~íÉê=Åçãé~åó=çÑ=iáãÄìêÖK=qÜÉ=é~ê~ãÉíÉêë=çÑ=~ää=íÜêÉÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåë=~êÉ=çéíáãáòÉÇ=ìëáåÖ=íÜÉ=ãÉíÜçÇë=ÇÉëÅêáÄÉÇ=áå=sçå=^ëãìíÜ=~åÇ=_áÉêâÉåë=xOMMRzK=eÉêÉI=ïÉ=ïáää=ÅçåëáÇÉê=íÜÉ=êÉëìäíë=çÑ=íÜÉ=íê~åëÑÉê=é~êí=çÑ=íÜÉ=ãçÇÉä=~åÇ=çÑ=íÜáë=..…………………………………………………………………………………………….….

jçÇÉäáåÖ=ãìäíáéäÉ=ëíêÉëëÉë=áåÇáîáÇì~ä=ëÉêáÉë=çåäóI=~ÑíÉê=íÜÉ=Ñáêëí=êìå=çÑ=íÜÉ=ãçÇÉäK=qÜÉ=ãçÇÉä=êÉëìäíë=~åÇ=é~ê~ãÉíÉê=Éëíáã~íÉë=~êÉ=ëìãã~êáòÉÇ=áå=í~ÄäÉ=RKNK=få=íÜÉ=í~ÄäÉI=íïç=é~ê~ãÉíÉêë=~êÉ=ÖáîÉå=íÜ~í=ÇÉÑáåÉ=íÜÉ=ÖççÇåÉëë=çÑ=ÑáíW=íÜÉ=Éñéä~áåÉÇ=î~êá~åÅÉ=éÉêÅÉåí~ÖÉ=EbsmF=~åÇ=íÜÉ=oççí=jÉ~å=pèì~êÉÇ=bêêçê=EojpbFK=få=íÜÉ=ÇÉÑáåáíáçå=çÑ=bsmI=íÜÉ=êÉëáÇì~ä=î~êá~åÅÉ= σ =áë=ïÉáÖÜÉÇ=~ÅÅçêÇáåÖ=íç=íÜÉ=î~êá~åÅÉ=ã~ååÉêW==2 2h( t) −σn( t)2σh( t)σ2h(t)çÑ=íÜÉ=çêáÖáå~ä=ëáÖå~ä=áå=íÜÉ=ÑçääçïáåÖ=σEVP = *100%(5.15)=qÜÉ=êÉëìäíë=çÑ=íÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉä=~êÉ=ëÜçïå=áå=ÑáÖìêÉ=RKOI=ïÜÉêÉ=íÜÉ=ãÉ~ëìêÉÇ=ÜÉ~Çë= h =EÇçíëF=~åÇ=éêÉÇáÅíÉÇ=ÜÉ~Çë=N∑i=1hi2n(t)+ d EëçäáÇF=~êÉ=éäçííÉÇ=íçÖÉíÜÉê=áå=íÜÉ=ìééÉê=Öê~éÜI=ïÜáäÉ= hi=áë=éäçííÉÇ=Ñçê=ÉîÉêó=ëíêÉëë=áå=íÜÉ=Öê~éÜë=ÄÉåÉ~íÜK13PredictionObservationsResults of series 46DP0032===NON=Head − refl (m)12.51211.5Rise (m) Rise (m) Rise (m) Rise (m)0.80.60.4−0.4−0.6−0.810.500.10−0.1VENRAYEINDHOVENSambeekBovenP.S. BERGEN1980 1985 1990 1995 2000Datefigure 5.2: The ground water head fluctuations in well 46DP0032 (top) decomposed in four partialseries due to (from top to bottom) rainfall, evaporation, river level fluctuations and a pumpingstation. Adding the partial series and the estimated drainage base ( d , see table 5.1) results in thepredicted series.=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=R=NOO=qÜìëI=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=áë=ÇÉÅçãéçëÉÇ=áåíç=Ñçìê=é~êíá~ä=ëÉêáÉëI=ïÜáÅÜ=ëÜçï=íÜÉ=ÉÑÑÉÅíë=çÑ=íÜÉ=áåÇáîáÇì~ä=ëíêÉëëÉëK=cçê=Éñ~ãéäÉI=áí=ã~ó=ÄÉ=çÄëÉêîÉÇ=íÜ~í=íÜÉ=êÉÅÉåí=ëÉêáÉë=çÑ=ïÉí=óÉ~êë=áå=íÜÉ=kÉíÜÉêä~åÇë=ENVVVJOMMOF=Ü~ë=äÉÇ=íç=~å=çîÉê~ää=áåÅêÉ~ëÉ=áå=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉäëK=cìêíÜÉêãçêÉI=íÜÉ=ÉÑÑÉÅí=çÑ=Éî~éçê~íáçå=ÇçÉëåÛí=î~êó=ãìÅÜ=Ñêçã=óÉ~ê=íç=óÉ~êI=Äìí=ëÜçïë=~=ÇáëíáåÅí=ëÉ~ëçå~ä=é~ííÉêå=~ë=áí=áë=ã~áåäó=áåÑäìÉåÅÉÇ=Äó=íÉãéÉê~íìêÉ=~åÇ=ëçä~ê=ê~Çá~íáçåK=qÜÉ=êáîÉê=äÉîÉä=Ü~ë=äáííäÉ=ÉÑÑÉÅí=~ë=áí=áë=ã~áåí~áåÉÇ=Äó=Ç~ãëI=ÉñÅÉéí=Ñçê=ÜáÖÜJï~íÉê=ÉîÉåíë=ïÜÉå=íÜÉ=êáîÉê=äÉ~îÉë=áíë=ÅÜ~ååÉä=~åÇ=ÉåíÉêë=íÜÉ=ÑäççÇéä~áåK=få=íÜçëÉ=Å~ëÉëI=íÜÉ=ÜÉ~Ç=êÉëéçåÇë=îÉêó=èìáÅâäó=~åÇ=ëÜçïë=ÇáëíáåÅí=éÉ~âëK=cáå~ääóI=ÜÉ~Çë=ëÜçï=~=Öê~Çì~ä=ÇÉÅäáåÉ=ëáåÅÉ=NVVQ=ÇìÉ=íç=íÜÉ=ÖêçìåÇï~íÉê=ïáíÜÇê~ï~ä=íÜ~í=ëí~êíÉÇ=~êçìåÇ=íÜ~í=íáãÉK=kçíÉ=íÜ~í=íÜÉëÉ=ÇáëíáåÅí=ÇáÑÑÉêÉåÅÉë=áå=Çóå~ãáÅ=ÄÉÜ~îáçê=~ääçï=íÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉä=íç=ÇáëíáåÖìáëÜ=íÜÉ=ÉÑÑÉÅíë=çÑ=íÜÉ=áåÇáîáÇì~ä=ëíêÉëëÉëK=^äëç=åçíÉ=íÜ~í=íÜÉ=ÑêÉèìÉåÅó=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=ÅÜ~åÖÉë=~êçìåÇ=NVVO=Ñêçã=Ñçìê=íáãÉë=~=óÉ~ê=íç=çåÅÉ=~=ïÉÉâK=qÜáë=ÇçÉë=åçí=éçëÉ=~=éêçÄäÉã=Ñçê=íÜÉ=ãçÇÉäI=~ë=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=Éèì~íáçåë=~êÉ=Åçåíáåìçìë=áå=íáãÉ=~åÇ=éêÉÇáÅíáçåë=~êÉ=åçí=ÑáñÉÇ=íç=~=ÅÉêí~áå=íáãÉ=ÇáëÅêÉíáò~íáçåK==cçê=É~ÅÜ=çÑ=íÜÉ=ÇáÑÑÉêÉåí=ëíêÉëëÉëI=~å=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=áë=Éëíáã~íÉÇ=E~äíÜçìÖÜ=åçí=~äï~óë=áåÇÉéÉåÇÉåí=Ñêçã=íÜÉ=çíÜÉê=ëíêÉëëÉëI=ëÉÉ=Éèì~íáçåë=ERKPF==íç=ERKRFFK=qÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=Ñçêãë=íÜÉ=ÜÉ~êí=çÑ=íÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉä=~åÇ=êÉéêÉëÉåíë=íÜÉ=êÉëéçåëÉ=çÑ=ÜÉ~Çë=íç=~=ìåáí=áãéìäëÉ=çÑ=íÜÉ=ëíêÉëëK=qÜÉ=ëÜ~éÉ=çÑ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=ÇÉéÉåÇë=çå=íÜÉ=éçëáíáçå=çÑ=íÜÉ=çÄëÉêî~íáçå=ïÉää=~åÇ=íÜÉ=éêçéÉêíáÉë=çÑ=íÜÉ=ëóëíÉãK=^ë=~å=Éñ~ãéäÉI=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=çÑ=íÜÉ=äÉîÉä=çÑ=íÜÉ=êáîÉê=jÉìëÉ=áë=éäçííÉÇ=áå=ÑáÖìêÉ=RKPK=cêçã=íÜÉ=Öê~éÜI=çåÉ=Å~å=áåÑÉê=íÜ~í=íÜÉ=ÜÉ~Çë=áåÇÉÉÇ=êÉëéçåÇ=èìáÅâäó=íç=~=êáëÉ=~åÇ=Ñ~ää=áå=êáîÉê=äÉîÉäI=~ë=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=EëçäáÇ=äáåÉF=éÉ~âë=áå=äÉëë=íÜ~å=Ü~äÑ=~=Ç~óK=^é~êí=Ñêçã=áíë=Çóå~ãáÅ=êÉëéçåëÉI=çåÉ=áë=çÑíÉå=~äëç=0.70.6Response Factor (−)0.50.40.30.20.1Step ResponseImpulse Response95% Conf. Int.00 2 4 6 8 10Time (days)=figure 5.3: Estimated impulse and step response functions of observation well 46DP0032 for thelevel of the river Meuse (measured at dam Sambeek-Boven)...…………………………………………………………………………………………….….

jçÇÉäáåÖ=ãìäíáéäÉ=ëíêÉëëÉë=5.946y−coordinate (m)5.9455.9445.9435.9425.9415.945.93990%80%70%60%5.9385.93750%5.936x 10 6==figure 5.4: Spatial distribution of the explained variance percentage (EVP), in plan view (left figure).Model performance in general is lowest near the 15 pumping well screens, which are clustered in tworows (red dots in right figure).=áåíÉêÉëíÉÇ=áå=íÜÉ=ëí~íáçå~êó=áåÑäìÉåÅÉ=çÑ=~=ÅÉêí~áå=ëíêÉëëK=qÜáë=áë=êÉéêÉëÉåíÉÇ=Äó=íÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçå=EÇ~ëÜÉÇ=äáåÉFI=ïÜáÅÜ=áë=íÜÉ=áåíÉÖê~ä=çÑ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=ïáíÜ=êÉëéÉÅí=íç=íáãÉX=íÜÉ=ëíÉé=êÉëéçåëÉ=êÉéêÉëÉåíë=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=ëóëíÉã=íç=~=ëìÇÇÉå=~åÇ=íÜÉå=çåÖçáåÖ=êáëÉ=çÑ=íÜÉ=êáîÉê=äÉîÉäK=qÜÉ=äÉîÉä=íÜ~í=íÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçå=~ééêç~ÅÜÉë=íçï~êÇë=áåÑáåáíó=áë=íÜÉ=òÉêçíÜ=ãçãÉåí=E M0FI=~äëç=âåçïå=~ë=íÜÉ=Ö~áå=áå=íÜÉ=íáãÉ=ëÉêáÉë=~å~äóëáë=äáíÉê~íìêÉK=rëáåÖ=ERKNNFI=ïÉ=Å~å=çÄí~áå= M0=Ñçê=ëíêÉëëÉë=çÑ=íóéÉ= s =~ëW==∞M0, s= ∫ θs( t)dt= γ ′ exp(-2 α′)(5.16)0=få=íÜÉ=Å~ëÉ=éêÉëÉåíÉÇI= M 0,s =Ñçê=íÜÉ=êáîÉê=jÉìëÉ=áë=MKTP=ãÉíÉêI=ïÜáÅÜ=ãÉ~åë=íÜ~í=~=êáîÉê=äÉîÉä=êáëÉ=çÑ=N=ãÉíÉê=ïáää=ÉîÉåíì~ääó=äÉ~Ç=íç=~=ÖêçìåÇï~íÉê=äÉîÉä=êáëÉ=çÑ=TP=ÅÉåíáãÉíÉêë=çå=íÜáë=äçÅ~íáçåK=cçê=~ë=äçåÖ=~ë=íÜÉ=~ëëìãéíáçå=çÑ=äáåÉ~êáíó=ÜçäÇëI=íÜÉ=òÉêçíÜ=ãçãÉåí=Å~å=ÄÉ=ìëÉÇ=Ñçê=èìáÅâ=ëÅÉå~êáç=Å~äÅìä~íáçåëI=~ë=íÜÉ=ãìäíáéäáÅ~íáçå=çÑ=M =~åÇ=~=éä~ååÉÇ=êáëÉ=áå=íÜÉ=êáîÉê=äÉîÉä=óáÉäÇë=~=éêÉÇáÅíáçå=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=0,s3.402 3.404 3.406 3.408 3.41 3.412x−coordinate (m)x 10 63.402 3.404 3.406 3.408 3.41 3.412x−coordinate (m)êáëÉK=kçíÉ=íÜ~í=íÜÉ=ëí~åÇ~êÇ=ÇÉîá~íáçå=çÑ= M 0,s =áë=êÉ~ëçå~ÄäÉI=ïÜÉêÉ~ë=íÜÉ=ëí~åÇ~êÇ=ÇÉîá~íáçå=çÑ=é~ê~ãÉíÉê=α =áë=ä~êÖÉ=E~=ëáãáä~ê=Å~ëÉ=áë=íêìÉ=Ñçê= M 0,w =çÑ=íÜÉ=ÖêçìåÇï~íÉê=ïáíÜÇê~ï~äFK=qÜáë=áë=Å~ìëÉÇ=Äó=íÜÉ=Åçî~êá~åÅÉ=ÄÉíïÉÉå=íÜÉ=é~ê~ãÉíÉêëK=40%===NOP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=R=x 10 44.54WithdrawalFlow rate (m^3/day)3.532.521.51NOQ=1965 1972 1979 1986 1993 2000Date=figure 5.5: Groundwater withdrawal history at location ‘Harlingerland’. The flow rate is obtained bysumming the rates of the individual pumping wells.=5.3.2 Multiple seriestÉ=ïáää=áääìëíê~íÉ=íÜÉ=éêçÅÉÇìêÉ=áå=ïÜáÅÜ=íÜÉ=êÉëìäíë=Ñêçã=ãìäíáéäÉ=çÄëÉêî~íáçå=ïÉääë=ã~ó=ÄÉ=áåíÉêéêÉíÉÇ=~åÇ=ÅÜÉÅâÉÇ=áå=~=ëÉÅçåÇ=~ééäáÅ~íáçåK=qÜáë=~ééäáÅ~íáçå=ÅçåÅÉêåë=~=ÇêáåâáåÖ=ï~íÉê=éêçÇìÅíáçå=~êÉ~=ïáíÜ=ãìäíáéäÉ=çÄëÉêî~íáçå=ïÉääë=äçÅ~íÉÇ=áå=íÜÉ=åçêíÜJïÉëíÉêå=é~êí=çÑ=dÉêã~åóI=áå=~å=~êÉ~=Å~ääÉÇ=Úe~êäáåÖÉêä~åÇÛ=åÉ~ê=íÜÉ=qçïå=çÑ=bëÉåëK=dêçìåÇï~íÉê=áë=ïáíÜÇê~ïå=Äó=íÜÉ=ÇêáåâáåÖ=ï~íÉê=Åçãé~åó=läÇÉåÄìêÖáëÅÜJlëíÑêáÉëáëÅÜÉê=t~ëëÉêîÉêÄ~åÇ=~í=~=ÅìêêÉåí=ê~íÉ=çÑ=V=ãáääáçå=ÅìÄáÅ=ãÉíÉêë=~=óÉ~êK=qÜÉ=ÑáÑíÉÉå=éìãéáåÖ=ïÉääë=~êÉ=ÅäìëíÉêÉÇ=áå=íïç=êçïëI=ïÜáÅÜ=~êÉ=ãçêÉ=çê=äÉëë=éä~ÅÉÇ=áå=ëÉêáÉë=EÑáÖìêÉ=RKQFK=qÜÉ=ïÉää=ëÅêÉÉåë=~êÉ=äçÅ~íÉÇ=~í=~=ÇÉéíÜ=çÑ=OR=íç=QM=ãÉíÉêë=Ñêçã=íÜÉ=ëçáä=ëìêÑ~ÅÉK=qÜÉ=ïáíÜÇê~ï~ä=ê~íÉ=Üáëíçêó=EÑáÖìêÉ=RKRF=ëÜçïë=~=ã~êâÉÇ=áåÅêÉ~ëÉ=áå=íÜÉ=éÉêáçÇ=Ñêçã=NVTO=ìåíáä=NVTSK=pìÅÜ=~=ê~íÉ=ÅÜ~åÖÉ=áë=îÉêó=áãéçêí~åí=Ñçê=~=êÉäá~ÄäÉ=Éëíáã~íÉ=çÑ=íÜÉ=áåÑäìÉåÅÉ=çÑ=~=éìãéáåÖ=ïÉääI=~ë=áí=ïáää=Ü~îÉ=Å~ìëÉÇ=~=ã~êâÉÇ=Çê~ïÇçïåK=cìêíÜÉêãçêÉI=íÜÉ=ïáíÜÇê~ï~ä=ê~íÉë=ëÜçï=~=ÇáëíáåÅí=ëÉ~ëçå~ä=ÅóÅäÉI=Å~ìëÉÇ=Äó=Ñ~Åíçêë=äáâÉ=íÜÉ=áåÅêÉ~ëÉÇ=ï~íÉêáåÖ=çÑ=ó~êÇë=áå=Çêó=éÉêáçÇëK=qÜÉ=ïáíÜÇê~ï~ä=ê~íÉ=áë=çÄí~áåÉÇ=Äó=ëìããáåÖ=íÜÉ=éìãéáåÖ=ê~íÉë=çÑ=íÜÉ=áåÇáîáÇì~ä=ïÉääëK=a~í~=çå=íÜÉ=áåÇáîáÇì~ä=ïÉääëI=ïÜáÅÜ=~êÉ=ÅçåíêçääÉÇ=ëÉé~ê~íÉäóI=áë=çåäó=~î~áä~ÄäÉ=ëáåÅÉ=NVVOK=^åóï~óI=ìëÉ=çÑ=íÜÉ=áåÇáîáÇì~ä=éìãéáåÖ=ê~íÉë=áë=åçí=ëíê~áÖÜíÑçêï~êÇI=~ë=íÜÉ=åìãÄÉê=çÑ=NR=ïÉääë=áë=íçç=ÜáÖÜ=íç=Éëíáã~íÉ=íÜÉáê=áåÑäìÉåÅÉ=áåÇÉéÉåÇÉåíäóK=dêçìåÇï~íÉê=Çóå~ãáÅë=~êÉ=êÉÅçêÇÉÇ=~í=NNS=çÄëÉêî~íáçå=ïÉääë=ïáíÜ=~=íçí~ä=çÑ=NPV=ëÅêÉÉåë=áå=~=ÅáêÅäÉ=ïáíÜ=~=ê~Çáìë=çÑ=R=âáäçãÉíÉêë=~êçìåÇ=íÜÉ=éìãéáåÖ=ïÉääëK=qÜÉ=ëÅêÉÉåë=ÇÉéíÜë=ê~åÖÉ=Ñêçã=äÉëë=íÜ~å=N=íç=ãçêÉ=íÜ~å=VM=ãÉíÉêë=Ñêçã=íÜÉ=ëìêÑ~ÅÉK=m~êí=çÑ=íÜÉ=ïÉääë=~êÉ=ãçåáíçêÉÇ=ÉîÉêó=íïç=ïÉÉâëI=~åÇ=íÜÉ=çíÜÉê=é~êí=ãçåíÜäóK=qÜÉ=éÉêáçÇ=áå=ïÜáÅÜ=íÜÉ=ïÉääë=ïÉêÉ=ãçåáíçêÉÇ=ÇáÑÑÉêë=Ñêçã=ïÉää=íç=ïÉääI=ïáíÜ=íÜÉ=É~êäáÉëí=ãÉ~ëìêÉãÉåíë=Ç~íáåÖ=Ä~Åâ=íç=NVSQI=ïÜáäÉ=çíÜÉê=ïÉääë=ïÉêÉ=áåëí~ääÉÇ=~ë=êÉÅÉåíäó=~ë=NVVVK=få=ëçãÉ=çÑ=íÜÉ=ïÉääëI=ãçåáíçêáåÖ=..…………………………………………………………………………………………….….

jçÇÉäáåÖ=ãìäíáéäÉ=ëíêÉëëÉë=table 5.2: Range of model results and parameter estimates for all 139 groundwater head series.m~ê~ãÉíÉê= jáåáãìã=EHLJ=Oσ )= jÉÇá~å= j~ñáãìã=EHLJ=Oσ )=bsm= PUKN= URKV= VTKR=j MIé= RV=EHLJ=NKPÉP)= USQ= SRUM=EHLJ=PKRÉRF=j MIÉ= NTO=EHLJNOOF= TVM= TVTS=EHLJRVNPF=j MIï= NKVPÉJU=EHLJ=QKNÉJR)= QKRMÉJR= NKOR=ÉJO=EHLJ=MKOM)=Ñ= MKMV=EHLJ=MKPN)= NKOM= QKOV=EHLJ=UT)=Ç= JPKON= NKOS= TKMN==ëíçééÉÇ=áå=NVTRI=ïÜáäÉ=áå=ëçãÉ=çíÜÉêë=íÜÉ=éÉêáçÇ=ÄÉíïÉÉå=NVTR=~åÇ=NVVU=áë=ãáëëáåÖK=qÜÉ=~èìáÑÉê=áå=íÜÉ=~êÉ~=Åçåëáëíë=ã~áåäó=çÑ=ë~åÇëX=áå=ëçãÉ=é~êíë=~=Åä~ó=ä~óÉê=ëÉé~ê~íÉë=íÜÉ=éÜêÉ~íáÅ=Ñêçã=íÜÉ=ÇÉÉéÉê=~èìáÑÉêK==qÜÉ=ÜÉ~Ç=ÑäìÅíì~íáçåë=~êÉ=ãçÇÉäÉÇ=ïáíÜ=éêÉÅáéáí~íáçåI=êÉÑÉêÉåÅÉ=Éî~éçíê~åëéáê~íáçå=~åÇ=ÖêçìåÇï~íÉê=ïáíÜÇê~ï~äëX=íÜÉêÉ=~êÉ=åç=êáîÉêë=çê=çíÜÉê=áãéçêí~åí=ÑäìÅíì~íáåÖ=ëìêÑ~ÅÉ=ï~íÉêë=áå=íÜÉ=~êÉ~K=oÉëìäíë=çÑ=íÜÉ=NPV=íáãÉ=ëÉêáÉë=ãçÇÉäë=~êÉ=ëìãã~êáòÉÇ=áå=í~ÄäÉ=RKOI=ïÜÉêÉ=íÜÉ=ãáåáãìãI=ãÉÇá~å=~åÇ=ã~ñáãìã=î~äìÉ=çÑ=íÜÉ=é~ê~ãÉíÉêë=áå=~ää=NPV=ãçÇÉäë=~êÉ=ÖáîÉåI=~äçåÖ=ïáíÜ=íÜÉáê=VRB=ÅçåÑáÇÉåÅÉ=áåíÉêî~äK=qÜÉ=ãÉÇá~å=î~äìÉ=çÑ=bsm=éçáåíë=çìí=íÜ~í=íÜÉ=Ñáí=çÑ=ãçëí=ãçÇÉäë=áë=ÖççÇI=~äíÜçìÖÜ=íÜÉêÉ=~êÉ=ÅäÉ~êäó=çìíäáÉêë=áå=íÜÉ=êÉëìäíëI=~ë=íÜÉ=ÉñíêÉãÉë=çÑ=ãçëí=é~ê~ãÉíÉê=Éëíáã~íÉë=~êÉ=åçí=ïáíÜáå=~=ê~åÖÉ=íÜ~í=áë=éÜóëáÅ~ääó=éä~ìëáÄäÉK=qÜáë=ÇçÉë=åçí=ãÉ~å=íÜ~í=íÜÉ=Éëíáã~íÉë=~êÉ=åÉÅÉëë~êáäó=Äá~ëÉÇI=~ë=ãçëí=ÉñíêÉãÉë=~êÉ=~ÅÅçãé~åáÉÇ=Äó=ä~êÖÉ=ëí~åÇ~êÇ=ÇÉîá~íáçåëK=cçê=íÜÉëÉ=Å~ëÉëI=~=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=çÑ=HLJ=Oσ áåÅäìÇÉë=íÜÉ=î~äìÉ=òÉêçI=Äìí=~äëç=~=êÉ~äáëíáÅ=î~äìÉI=áåÇáÅ~íáåÖ=íÜ~í=íÜÉ=èì~äáíó=çê=èì~åíáíó=çÑ=íÜÉ=Ç~í~=áë=åçí=ëìÑÑáÅáÉåí=íç=ÇÉíÉêãáåÉ=íÜÉ=î~äìÉ=çÑ=íÜ~í=é~ê~ãÉíÉêK===qÜÉ=çåäó=Éëíáã~íÉ=áå=í~ÄäÉ=RKO=íÜ~í=ÇçÉë=ëÉÉã=Äá~ëÉÇ=áë=íÜÉ=ãáåáãìã=î~äìÉ=Ñçê=íÜÉ=Éî~éçê~íáçå=Ñ~Åíçê= f =I=ïÜáÅÜ=áë=îÉêó=äçï=EìééÉê=VRB=äáãáí=áë=MKQFK=få=íÜáë=Å~ëÉI=ÜçïÉîÉêI=íÜÉ=òÉêçíÜ=ãçãÉåí=çÑ=íÜÉ=Éî~éçê~íáçå= M 0,e ïÜáÅÜ=áë=ÇÉÑáåÉÇ=~ëW===∞M0, e= ∫ f θp ( t )d t = fA(5.17)0=ÇçÉë=Ü~îÉ=~=ä~êÖÉ=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=EHLJ=VQNFI=~ë=íÜÉ=áåÑäìÉåÅÉ=çÑ=éêÉÅáéáí~íáçå=áíëÉäÑ=E A F=Å~ååçí=ÄÉ=ÇÉíÉêãáåÉÇ=Ñêçã=íÜÉ=Ç~í~=~ÅÅìê~íÉäó=EíÜÉ=éêçÄ~ÄäÉ=Å~ìëÉ=áë=íÜ~í=íÜÉ=éÉêáçÇ=áå=ïÜáÅÜ=Ç~í~=~êÉ=~î~áä~ÄäÉ=Ñçê=íÜáë=é~êíáÅìä~ê=ëÉêáÉë=çåäó=ëé~åë=NS=ãçåíÜëI=ïáíÜ=çåÉ=çÄëÉêî~íáçå=éÉê=ãçåíÜFK======NOR=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=R=3Results of series F2320151_12Head − refl (m)10−1−2−3Results of series B2311630_1Head − refl (m)2.521.510.50NOS=1969 1975 1981 1987 1993 1999=figure 5.6: Head observations and predictions of wells F230151(near the pumping station) andB2311630 (at some distance), giving an indication of good and bad model fits.qç=ÇÉíÉêãáåÉ=íÜÉ=êÉ~ëçå=Ñçê=íÜÉ=äçïÉê=bsm=çÑ=ëçãÉ=ãçÇÉäëI=áíë=ëé~íá~ä=ÇáëíêáÄìíáçå=áë=éäçííÉÇ=EÑáÖìêÉ=RKQFK=cêçã=íÜÉ=ÑáÖìêÉI=áí=áë=ÅäÉ~ê=íÜ~í=íÜÉ=ãçÇÉä=Ñáíë=~êÉ=äçïÉëí=áå=íÜÉ=áããÉÇá~íÉ=îáÅáåáíó=çÑ=íÜÉ=éìãéáåÖ=ïÉääëK=qÜáë=ëìÖÖÉëíë=íÜ~í=íÜÉ=áåÑäìÉåÅÉ=çÑ=íÜÉ=áåÇáîáÇì~ä=ïÉääë=áë=åçí=ãçÇÉäÉÇ=ÅçêêÉÅíäóI=ïÜáÅÜ=áë=áåÇÉÉÇ=íÜÉ=Å~ëÉI=~ë=íÜÉ=ïáíÜÇê~ï~ä=ê~íÉ=çÑ=íÜÉ=íçí~ä=ïÉää=ÑáÉäÇ=áë=áåÅçêéçê~íÉÇ=áå=íÜÉ=ãçÇÉäI=åçí=íÜÉ=ê~íÉë=çÑ=íÜÉ=áåÇáîáÇì~ä=ïÉääëK=qÜÉ=áåÇáîáÇì~ä=ãçÇÉä=Ñáíë=EÑáÖìêÉ=RKSF=ÖáîÉ=íÜÉ=ë~ãÉ=áåÇáÅ~íáçåK=kÉ~ê=íÜÉ=ïÉää=ÑáÉäÇI=íÜÉ=ÜÉ~Ç=ÑäìÅíì~íÉë=ïáäÇäó=ïÜÉå=íÜÉ=áåÇáîáÇì~ä=éìãéáåÖ=ïÉääë=~êÉ=ëÜìí=çå=~åÇ=çÑÑK==qÜÉ=éêÉÇáÅíáçåëI=ÜçïÉîÉêI=çåäó=Ñçääçï=íÜÉ=ÖÉåÉê~ä=Çê~ïÇçïå=é~ííÉêåK=^í=ëçãÉ=Çáëí~åÅÉI=íÜÉ=éêÉÇáÅíáçåë=Ñáí=íÜÉ=ÜÉ~Ç=ÑäìÅíì~íáçåë=ãìÅÜ=ÄÉííÉêK=qÜÉ=ãçÇÉä=Ñáí=áë=~äëç=äçïÉê=áå=ëçãÉ=çÑ=íÜÉ=ëÜ~ääçïÉê=ïÉääëK=qÜÉ=ÜÉ~Ç=ëÉêáÉë=áå=íÜçëÉ=Å~ëÉë=ëÜçï=ÇáëíáåÅí=ëáÖåë=çÑ=EíÜêÉëÜçäÇF=åçåJäáåÉ~êáíó=xÉKÖKI=håçííÉêë=~åÇ=aÉ=dççáàÉêI=NVVVzK=qÜÉ=mfocf`q=ãÉíÜçÇ=Å~å=ÅìêêÉåíäó=Ü~åÇäÉ=íÜêÉëÜçäÇ=åçåJäáåÉ~êáíó=Ñçê=éêÉÅáéáí~íáçå=~åÇ=Éî~éçê~íáçåI=Äìí=åçí=óÉí=Ñçê=çíÜÉê=ëíêÉëëÉëI=ëç=íÜáë=ï~ë=åçí=ÅçåëáÇÉêÉÇ=ÑìêíÜÉê=~í=íÜáë=íáãÉK===kÉñíI=ïÉ=ÑçÅìë=çå=íÜÉ=Éëíáã~íÉÇ=áåÑäìÉåÅÉ=çÑ=íÜÉ=ïÉää=ÑáÉäÇI=ïÜáÅÜ=ï~ë=~å=áãéçêí~åí=çÄàÉÅíáîÉ=Ñçê=ÇçáåÖ=íÜÉ=íáãÉ=ëÉêáÉë=~å~äóëáë=çÑ=íÜáë=ëáíÉK=rëáåÖ=ERKVFI=ïÉ=ÑáåÇ=íÜ~í=íÜÉ=òÉêçíÜ=ãçãÉåí=çÑ=~=ïÉää=áë=ÖáîÉå=Äó=xe~åíìëÜI=NVRSzW===∞M = ∫ θ ( t)dt= −2γ K (2 α )(5.18)0, w w 00..…………………………………………………………………………………………….….

jçÇÉäáåÖ=ãìäíáéäÉ=ëíêÉëëÉë=x 10 −5x 10 −52200−2−2Gain of well field−4−6−8Transect−4−6−8Transect−10−10−12−123000 4000 5000 6000 7000 8000 9000Distance on transect (m)0 1000 2000 3000 4000 5000 6000 7000 8000 9000Distance on transect (m)=figure 5.7: Estimated gains (dots) for the well field in the different observation wells, presented inwest-east (left) and north-south (right) cross sections. The error bars indicate the 95% confidenceinterval of the estimates.áå=ïÜáÅÜ= K0=áë=íÜÉ=ãçÇáÑáÉÇ=_ÉëëÉä=ÑìåÅíáçå=çÑ=íÜÉ=ëÉÅçåÇ=âáåÇ=~åÇ=òÉêçíÜ=çêÇÉê=x^Äê~ãçïáíò=~åÇ=píÉÖìåI=NVSQzK=cáêëíI=ïÉ=êÉãçîÉ=~ää=ïÉääë=ïáíÜ=Éëíáã~íÉë=çÑ= M 0,w =ïáíÜ=ä~êÖÉ=ÅçåÑáÇÉåÅÉ=áåíÉêî~äëI=~ë=íÜÉó=~êÉ=çÑ=äáííäÉ=î~äìÉ=~åÇ=çåäó=Ääìê=çìê=îáÉï=çå=íÜÉ=çíÜÉê=êÉëìäíëX=ïÉ=ÅÜççëÉ=~=ÅìíJçÑÑ=î~äìÉ=çÑ=UÉJMRK=qÜÉ=PQ=ëÉêáÉë=íÜ~í=ïÉêÉ=ÇáëêÉÖ~êÇÉÇ=~ää=ä~Åâ=íÜÉ=éÉêáçÇ=Ñêçã=NVTO=íç=NVTS=áå=ïÜáÅÜ=íÜÉ=ïáíÜÇê~ï~ä=ê~íÉ=ï~ë=áåÅêÉ~ëÉÇ=ëáÖåáÑáÅ~åíäóK=få=ÑáÖìêÉ=RKTI=íÜÉ=ëé~íá~ä=ÇáëíêáÄìíáçå=çÑ=íÜÉ=NMR=êÉã~áåáåÖ= M 0,w =Éëíáã~íÉë=~åÇ=íÜÉáê=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=áë=éäçííÉÇ=áå=íïç=Åêçëë=ëÉÅíáçåëI=çåÉ=é~ê~ääÉä=~åÇ=çåÉ=éÉêéÉåÇáÅìä~ê=íç=íÜÉ=ëÉêáÉë=çÑ=éìãéáåÖ=ïÉääëK=^ë=ÉñéÉÅíÉÇI=íÜÉ= M 0,w =Éëíáã~íÉë=~êÉ=ÜáÖÜÉëí=åÉ~ê=íÜÉ=éìãéáåÖ=ïÉääëX=íÜÉ=äÉÑí=êçï=çÑ=ïÉääëI=ïÜáÅÜ=Ü~îÉ=ÇÉÉéÉê=ëÅêÉÉåëI=Å~ìëÉ=ëíêçåÖÉê=Çê~ïÇçïåë=íÜ~å=íÜÉ=êáÖÜí=êçïK=få=íÜÉ=éÉêéÉåÇáÅìä~ê=Åêçëë=ëÉÅíáçåI=íÜÉ=é~ííÉêå=çÑ= M 0,w =~ééêçñáã~íÉë=íÜÉ=ëÜ~éÉ=çÑ=~=Çê~ïÇçïå=ÅçåÉK=qÜÉ=M 0,w =Éëíáã~íÉë=~í=ëÜ~ääçï=ïÉää=ëÅêÉÉåë=~êÉ=ÅäÉ~êäó=äçïÉê=íÜ~å=~í=ÇÉÉéÉê=ïÉää=ëÅêÉÉåëX=íÜáë=ã~ó=ÄÉ=Å~ìëÉÇ=Äó=~=êÉëáëí~åÅÉ=ä~óÉê=çê=Äó=êÉÅÜ~êÖÉ=Ñêçã=ÇáíÅÜÉë=çê=ÅêÉÉâëK=qÜáë=áë=áå=äáåÉ=ïáíÜ=íÜÉ=ÜÉ~Ç=ëÉêáÉë=çÑ=íÜÉëÉ=çÄëÉêî~íáçå=ïÉääë=Eåçí=ëÜçïå=ÜÉêÉFI=ïÜÉêÉ=ÜÉ~Ç=ÇáÑÑÉêÉåÅÉë=ÄÉíïÉÉå=ÜáÖÜÉê=~åÇ=äçïÉê=ëÅêÉÉåë=ê~åÖÉ=ìé=íç=Q=ãÉíÉêëK=^äíÜçìÖÜ=íÜÉ=ãçÇÉä=Ñáí=áë=äçï=áå=íÜÉ=îáÅáåáíó=çÑ=íÜÉ=éìãéáåÖ=ïÉää=ëÅêÉÉåë=EïáíÜáå=~=ê~Çáìë=çÑ=~Äçìí=OMM=íç=PMM=ãÉíÉêëFI=~ää=áå=~ääI=íÜÉ= M 0,w Éëíáã~íÉë=ëÉÉã=åÉîÉêíÜÉäÉëë=êÉ~ëçå~ÄäÉK=qÜÉ=Éëíáã~íÉë=çÑ=çíÜÉê=é~ê~ãÉíÉêë=åÉ~ê=íÜÉ=ïÉääëI=ÜçïÉîÉêI=~êÉ=ÅäÉ~êäó=Äá~ëÉÇK=få=ÑáÖìêÉ=RKU=Åêçëë=ëÉÅíáçåë=~êÉ=ëÜçïå=çÑ=íÜÉ=bsm=~åÇ= M 0,e =Ñçê=íÜÉ=çÄëÉêî~íáçå=ïÉääë=áå=íÜÉ=îáÅáåáíó=çÑ=íÜÉ=éìãéáåÖ=ïÉääëK=qÜÉ=äçï=î~äìÉë=çÑ=bsm=åÉ~ê=íÜÉ=éìãéáåÖ=ïÉää=ëÅêÉÉåë=ÅçêêÉä~íÉ=====NOT=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=R=1050−580%70%60%50%−500−1000Depth−refl.(m)−10−15−20−2540%90%80%70%−1500−2000−3060%−3550%−400 500 1000 1500 2000 2500 3000Distance on transect (m)40%0 500 1000 1500 2000 2500 3000Distance on transect (m)NOU=figure 5.8: West-east cross sections near the pumping wells, showing the percentage of varianceaccounted for (left) and the estimated gain of the evapotranspiration (right) in the differentobservation wells. The values of both are lowest near the pumping well screens (the red markers in thefigure bottom right).ïáíÜ=íÜÉ=äçï=î~äìÉë=çÑ= M 0,e K=^ë=áí=áë=åçí=äçÖáÅ~ä=Ñçê=íÜÉ=áåÑäìÉåÅÉ=çÑ=Éî~éçíê~åëéáê~íáçå=íç=ÄÉ=íÜ~í=ÜÉíÉêçÖÉåÉçìë=áå=ÇÉÉéÉê=ëçáä=ä~óÉêëI=íÜáë=áåÇáÅ~íÉë=íÜ~í=íÜÉ=áåÑäìÉåÅÉ=çÑ=Éî~éçíê~åëéáê~íáçå=áë=çîÉêÉëíáã~íÉÇ=~åÇ=é~êíäó=ÅçêêÉä~íÉë=ïáíÜ=íÜÉ=ÉÑÑÉÅí=çÑ=íÜÉ=áåÇáîáÇì~ä=éìãéáåÖ=ïÉääëK===5.4 Discussion and conclusionsfå=íÜáë=é~éÉêI=íÜÉ=mfocf`q=ãçÇÉä=Ñçê=íáãÉ=ëÉêáÉë=~å~äóëáë=ï~ë=ÉñíÉåÇÉÇ=íç=Ü~åÇäÉ=ãìäíáéäÉ=áåéìíëK=cçê=ëíêÉëëÉë=çíÜÉê=íÜ~å=~êÉ~ä=êÉÅÜ~êÖÉI=~å~äóíáÅ=ëçäìíáçåë=çÑ=ëáãéäÉ=ÜóÇêçÖÉçäçÖáÅ=ëÅÜÉã~íáò~íáçåë=ïÉêÉ=ìëÉÇ=~ë=~=ÖìáÇÉ=íç=ÇÉîÉäçé=~ééêçéêá~íÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåëK=aáÑÑÉêÉåí=ëíêÉëëÉë=~êÉ=åçí=åÉÅÉëë~êáäó=ÅçååÉÅíÉÇ=áå=é~ê~ääÉä=áå=íÜÉ=ãçÇÉäI=~ë=áë=íÜÉ=ëí~åÇ~êÇ=éêçÅÉÇìêÉ=áå=ôj^=ãçÇÉäëK=^=Å~ëÉ=ïáíÜ=~=ëáåÖäÉ=çÄëÉêî~íáçå=ïÉää=ï~ë=ìëÉÇ=íç=áääìëíê~íÉ=Üçï=íÜÉ=ãçÇÉä=Å~å=ÉÑÑÉÅíáîÉäó=ÇÉÅçãéçëÉ=ÜÉ~Ç=ëÉêáÉë=áåíç=é~êíá~ä=ëÉêáÉë=íÜ~í=É~ÅÜ=ëÜçï=íÜÉ=ÉÑÑÉÅí=çÑ=~å=áåÇáîáÇì~ä=ëíêÉëëK=få=íÜÉ=Éñ~ãéäÉ=ïáíÜ=ãìäíáéäÉ=çÄëÉêî~íáçå=ïÉääëI=áí=ï~ë=ëÜçïå=íÜ~íI=åÉñí=íç=áíë=~=éêáçêá=ìëÉ=áå=ÇÉÑáåáåÖ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåëI=éÜóëáÅ~ä=âåçïäÉÇÖÉ=áë=~äëç=î~äì~ÄäÉ=áå=ÅÜÉÅâáåÖ=íÜÉ=ÅçåëáëíÉåÅó=~åÇ=éä~ìëáÄáäáíó=çÑ=íÜÉ=ãçÇÉä=êÉëìäíë=~=éçëíÉêáçêáK=qÜÉ=é~ê~ãÉíÉê=î~äìÉë=ëÜçìäÇ=Ñ~ää=ïáíÜáå=~=ê~åÖÉ=íÜ~í=áë=éÜóëáÅ~ääó=éä~ìëáÄäÉK=qÜÉ=ëé~íáçíÉãéçê~ä=é~ííÉêåë=çÄëÉêîÉÇ=áå=íÜÉ=î~êá~ÄäÉë=ëìééäó=áãéçêí~åí=~åÇ=áåÇÉéÉåÇÉåí=ÑÉÉÇÄ~Åâ=çå=íÜÉ=êÉëìäíëI=~ë=íÜÉêÉ=áë=åç=ëé~íá~ä=ÇÉéÉåÇÉåÅó=áãéçëÉÇ=çå=íÜÉ=ãçÇÉäëK=_ó=ÑçÅìëáåÖ=çå=íÜÉ=ãçÇÉä=êÉëáÇì~äëI=ãáëëáåÖ=ëíêÉëëÉëI=éêçÅÉëëÉë=çê=çíÜÉê=ëçìêÅÉë=çÑ=Éêêçê=ã~ó=ÄÉ=êÉ~Çáäó=áÇÉåíáÑáÉÇK=eáÖÜ=Éêêçê=äÉîÉäë=~êÉ=~=éçëëáÄäÉ=ëçìêÅÉ=çÑ=Äá~ë=áå=íÜÉ=Éëíáã~íÉëI=~ë=çíÜÉê=ëíêÉëëÉë=ã~ó=é~êíäó=ÅçãéÉåë~íÉ=ãáëëáåÖ=ëíêÉëëÉë=ïÜÉå=íÜÉáê=áåÑäìÉåÅÉ=áë=ÅçêêÉä~íÉÇK=få=íÜáë=Å~ëÉI=íÜÉ=ã~áå=ëçìêÅÉ=çÑ=Éêêçê=ï~ë=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=ÄÉÜ~îáçê=çÑ=íÜÉ=áåÇáîáÇì~ä=éìãéáåÖ=ïÉääë=ï~ë=åçí=~ÅÅçìåíÉÇ=ÑçêK=få=ëéáíÉ=çÑ=íÜáëI=íÜÉ=..…………………………………………………………………………………………….….

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`Ü~éíÉê=S=6NPM=Abstract:For e.g., visual interpretation, mapping or empirical modeling purposes, the amount ofinformation contained in a full spatiotemporal description of the groundwater tabledynamics is simply too large. For such purposes, the data has to be compressed withoutloosing too much information. Methods have been developed to visualize thegroundwater regime in overall graphs, or statistically characterize the dynamics with alimited set of parameters. More recently, methods have been sought to identify theproperties that determine the dynamics of a groundwater system. In such approaches, itis believed that the spatial differences in the groundwater dynamics are determined bythe system properties, while its temporal variation is driven by the dynamics of the inputinto the system. In this chapter, a method is presented that links the dynamics of the inputto the spatially variable system properties, and results in a new set of parameters thatcharacterize the groundwater dynamics (GD). While the dynamics of the input arecharacterized by its mean level and annual amplitude, the functioning of thegroundwater system is characterized by its impulse response (IR) function. The IRfunction can for instance be estimated empirically using a time series model.Subsequently, the input and system characteristics are combined into a set of parametersthat describe the output, or groundwater dynamics, using simple analytic expressions. Itis shown that these so-called GD characteristics (the mean depth, convexity, annualamplitude and phase shift), can describe the groundwater dynamics in detail (for as far asthe time series model can). In the example application, the GD characteristics arecompared to other methods for characterizing the groundwater regime, using twoexample series of groundwater level observations. It is shown that the so-called MxGLstatistics (Mean Highest, Lowest or Spring Groundwater Level) that are often used havesome important drawbacks, as they filter out the low-frequency dynamics of a systemand mix-up annual with higher frequencies. Consequently, it is concluded that thecapability of MxGL statistics in characterizing the groundwater dynamics at differentlocations is less than that of GD characteristics...…………………………………………………………………………………………….….

6`Ü~ê~ÅíÉêáòáåÖ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=ChapterCharacterizing groundwaterdynamics based on responseCharacterizing groundwatercharacteristicsdynamics based on response==characteristicsAdopted from:Von Asmuth, J. R., and M. Knotters (2004), Characterising spatial differences ingroundwater dynamics based on a system identification approach, gçìêå~ä=çÑ=eóÇêçäçÖó,296(1-4), 118-134. Reproduced with permission from Elsevier B.V., copyright © 2004 ElsevierB.V.=Adopted from:=Abstract:== Von Asmuth, J. R., and M. Knotters (2004)cçê=ÉKÖKI=îáëì~ä=áåíÉêéêÉí~íáçåI=ã~ééáåÖ=çê=ÉãéáêáÅ~ä=ãçÇÉäáåÖ=éìêéçëÉëI=íÜÉ=~ãçìåí=Characterising spatial differences in groundwater dynamicsçÑ=áåÑçêã~íáçå=Åçåí~áåÉÇ=áå=~=Ñìää=ëé~íáçíÉãéçê~ä=ÇÉëÅêáéíáçå=çÑ=íÜÉ=ÖêçìåÇï~íÉê=í~ÄäÉ=based on a system identification approach.Çóå~ãáÅë=áë=ëáãéäó=íçç=ä~êÖÉK=cçê=ëìÅÜ=éìêéçëÉëI=íÜÉ=Ç~í~=Ü~ë=íç=ÄÉ=ÅçãéêÉëëÉÇ=Journal of Hydrology, 296(1-4), 118-134.ïáíÜçìí=äççëáåÖ=íçç=ãìÅÜ=áåÑçêã~íáçåK=jÉíÜçÇë=Ü~îÉ=ÄÉÉå=ÇÉîÉäçéÉÇ=íç=îáëì~äáòÉ=íÜÉ=Reproduced with permission from Elsevier B.V.ÖêçìåÇï~íÉê=êÉÖáãÉ=áå=çîÉê~ää=Öê~éÜëI=çê=ëí~íáëíáÅ~ääó=ÅÜ~ê~ÅíÉêáòÉ=íÜÉ=Çóå~ãáÅë=ïáíÜ=~==copyright 2004 Elsevier B.V.äáãáíÉÇ=ëÉí=çÑ=é~ê~ãÉíÉêëK=jçêÉ=êÉÅÉåíäóI=ãÉíÜçÇë=Ü~îÉ=ÄÉÉå=ëçìÖÜí=íç=áÇÉåíáÑó=íÜÉ= ==NPN=éêçéÉêíáÉë=íÜ~í=ÇÉíÉêãáåÉ=íÜÉ=Çóå~ãáÅë=çÑ=~=ÖêçìåÇï~íÉê=ëóëíÉãK=få=ëìÅÜ=~ééêç~ÅÜÉëI=áí=áë=ÄÉäáÉîÉÇ=íÜ~í=íÜÉ=ëé~íá~ä=ÇáÑÑÉêÉåÅÉë=áå=íÜÉ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=~êÉ=ÇÉíÉêãáåÉÇ=Äó=íÜÉ=ëóëíÉã=éêçéÉêíáÉëI=ïÜáäÉ=áíë=íÉãéçê~ä=î~êá~íáçå=áë=ÇêáîÉå=Äó=íÜÉ=Çóå~ãáÅë=çÑ=íÜÉ=áåéìí=áåíç=íÜÉ=ëóëíÉãK=få=íÜáë=ÅÜ~éíÉêI=~=ãÉíÜçÇ=áë=éêÉëÉåíÉÇ=íÜ~í=äáåâë=íÜÉ=Çóå~ãáÅë=çÑ=íÜÉ=áåéìí=íç=íÜÉ=ëé~íá~ääó=î~êá~ÄäÉ=ëóëíÉã=éêçéÉêíáÉëI=~åÇ=êÉëìäíë=áå=~=åÉï=ëÉí=çÑ=é~ê~ãÉíÉêë=íÜ~í=ÅÜ~ê~ÅíÉêáòÉ=íÜÉ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=EdaFK=tÜáäÉ=íÜÉ=Çóå~ãáÅë=çÑ=íÜÉ=áåéìí=~êÉ=ÅÜ~ê~ÅíÉêáòÉÇ=Äó=áíë=ãÉ~å=äÉîÉä=~åÇ=~ååì~ä=~ãéäáíìÇÉI=íÜÉ=ÑìåÅíáçåáåÖ=çÑ=íÜÉ=ÖêçìåÇï~íÉê=ëóëíÉã=áë=ÅÜ~ê~ÅíÉêáòÉÇ=Äó=áíë=áãéìäëÉ=êÉëéçåëÉ=EfoF=ÑìåÅíáçåK=qÜÉ=fo=ÑìåÅíáçå=Å~å=Ñçê=áåëí~åÅÉ=ÄÉ=Éëíáã~íÉÇ=ÉãéáêáÅ~ääó=ìëáåÖ=~=íáãÉ=ëÉêáÉë=ãçÇÉäK=pìÄëÉèìÉåíäóI=íÜÉ=áåéìí=~åÇ=ëóëíÉã=ÅÜ~ê~ÅíÉêáëíáÅë=~êÉ=ÅçãÄáåÉÇ=áåíç=~=ëÉí=çÑ=é~ê~ãÉíÉêë=íÜ~í=ÇÉëÅêáÄÉ=íÜÉ=çìíéìíI=çê=ÖêçìåÇï~íÉê=Çóå~ãáÅëI=ìëáåÖ=ëáãéäÉ=~å~äóíáÅ=ÉñéêÉëëáçåëK=fí=áë=ëÜçïå=íÜ~í=íÜÉëÉ=ëçJÅ~ääÉÇ=da=ÅÜ~ê~ÅíÉêáëíáÅë=EíÜÉ=ãÉ~å=ÇÉéíÜI=ÅçåîÉñáíóI=~ååì~ä=~ãéäáíìÇÉ=~åÇ=éÜ~ëÉ=ëÜáÑíFI=Å~å=ÇÉëÅêáÄÉ=íÜÉ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=áå=ÇÉí~áä=EÑçê=~ë=Ñ~ê=~ë=íÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉä=Å~åFK=få=íÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçåI=íÜÉ=da=ÅÜ~ê~ÅíÉêáëíáÅë=~êÉ=Åçãé~êÉÇ=íç=çíÜÉê=ãÉíÜçÇë=Ñçê=ÅÜ~ê~ÅíÉêáòáåÖ=íÜÉ=ÖêçìåÇï~íÉê=êÉÖáãÉI=ìëáåÖ=íïç=Éñ~ãéäÉ=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåëK=fí=áë=ëÜçïå=íÜ~í=íÜÉ=ëçJÅ~ääÉÇ=jñdi=ëí~íáëíáÅë=EjÉ~å=eáÖÜÉëíI=içïÉëí=çê=péêáåÖ=dêçìåÇï~íÉê=iÉîÉäF=íÜ~í=~êÉ=çÑíÉå=ìëÉÇ=Ü~îÉ=ëçãÉ=áãéçêí~åí=Çê~ïÄ~ÅâëI=~ë=íÜÉó=ÑáäíÉê=çìí=íÜÉ=äçïJÑêÉèìÉåÅó=Çóå~ãáÅë=çÑ=~=ëóëíÉã=~åÇ=ãáñJìé=~ååì~ä=ïáíÜ=ÜáÖÜÉê=ÑêÉèìÉåÅáÉëK=`çåëÉèìÉåíäóI=áí=áë=ÅçåÅäìÇÉÇ=íÜ~í=íÜÉ=Å~é~Äáäáíó=çÑ=jñdi=ëí~íáëíáÅë=áå=ÅÜ~ê~ÅíÉêáòáåÖ=íÜÉ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=~í=ÇáÑÑÉêÉåí=äçÅ~íáçåë=áë=äÉëë=íÜ~å=íÜ~í=çÑ=da=ÅÜ~ê~ÅíÉêáëíáÅëK==KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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`Ü~ê~ÅíÉêáòáåÖ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=0.6a) b)2.5yearly coursesine20.5Fouriercoefficient (−)0.40.30.2precipitation surplus (mm/d)1.510.500.1−0.5010 −1 10 0 10 1Period (years)−1Jan Jan Jantime (month)=figure 6.1: a) Amplitude spectrum of the precipitation surplus series at the town of De Bilt (period1983-2001), showing an isolated peak at annual frequency. b) Average annual course of theprecipitation surplus and a sine with the same frequency, phase and amplitude.p =t p e∫t pbtp( τ ) dτpe− tpb(6.1)=ïáíÜ= t pb =~åÇ= t pe=ÇÉåçíáåÖ=íÜÉ=ëí~êí=~åÇ=ÉåÇ=çÑ=íÜÉ=éÉêáçÇ=çîÉê=ïÜáÅÜ=íÜÉ=ãÉíÉçêçäçÖáÅ=ÅÜ~ê~ÅíÉêáëíáÅë=~êÉ=Å~äÅìä~íÉÇK=kÉñíI=íáãÉ=áë=ëéäáí=áåíç=óÉ~ê=Y =~åÇ=gìäá~å=Ç~ó= D I=~åÇ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=áë=~îÉê~ÖÉÇ=çîÉê=Y I=ïÜáÅÜ=ÉÑÑÉÅíáîÉäó=ÑáäíÉêë=çìí=áíë=óÉ~êäó=ÅçìêëÉ= p̃ W==Ype∑Yp( Y, D)pbp̃ ( D) = , 1 D 365Y − Y≤ ≤(6.2)pepb=_ÉÅ~ìëÉ=íÜÉ=íÉãéÉê~íìêÉ=ä~êÖÉäó=ÇÉíÉêãáåÉë=íÜÉ=~ååì~ä=Éî~éçê~íáçå=ÅóÅäÉ=~åÇ=áë=ãçêÉ=çê=äÉëë=Ü~êãçåáÅI=ëç=áë=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìëI=~åÇ=íÜÉ=~ååì~ä=~ãéäáíìÇÉ=Å~å=ÄÉ=çÄí~áåÉÇ=Äó=ã~íÅÜáåÖ=~=ëáåÉ=íç=íÜÉ=óÉ~êäó=ÅçìêëÉ=EÑáÖìêÉ=SKNÄFK====NPR=6.2.2 Response characteristicsqÜÉ=ÑìåÅíáçåáåÖ=çÑ=äáåÉ~ê=ëóëíÉãë=Å~å=ÄÉ=ÅçãéäÉíÉäó=ÅÜ~ê~ÅíÉêáòÉÇ=Äó=íÜÉáê=áãéìäëÉ=êÉëéçåëÉ=EfoF=ÑìåÅíáçå=xwáÉãÉê=Éí=~äKI=NVVUX=sçå=^ëãìíÜ=~åÇ=j~~ëI=OMMNzK=få=íÜáë=ëÉÅíáçåI=ïÉ=ïáää=ÇÉëÅêáÄÉ=íÜÉ=ãÉíÜçÇë=ïáíÜ=ïÜáÅÜ=íÜÉ=fo=ÑìåÅíáçå=áë=Éëíáã~íÉÇI=~åÇ=ìëÉ=áíë=ãçãÉåíë=íç=~îçáÇ=êÉëíêáÅíáåÖ=çìêëÉäîÉë=íç=íÜÉ=ëéÉÅáÑáÅ=íóéÉ=çÑ=foJÑìåÅíáçå=ìëÉÇ=áå=çìê=ãçÇÉäK=^å=ÉäÉÖ~åí=ï~ó=íç=çÄí~áå=Éëíáã~íÉë=çÑ=íÜÉ=êÉëéçåëÉ=çÑ=ÜóÇêçäçÖáÅ=ëóëíÉãë=áë=íÜÉ=ìëÉ=çÑ=~=íê~åëÑÉê=ÑìåÅíáçå=åçáëÉ=EqckF=íáãÉ=ëÉêáÉë=ãçÇÉä=x_çñ=~åÇ=gÉåâáåëI=NVTMzK=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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`Ü~éíÉê=S==ïáíÜ= i =ÄÉáåÖ=íÜÉ=ãçãÉåí=çêÇÉêK=NPU=6.2.3 Combination into GD-characteristicslÑíÉå=åçí=íÜÉ=ëóëíÉã=éêçéÉêíáÉëI=Äìí=íÜÉ=ÄÉÜ~îáçê=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=áíëÉäÑ=áë=çÑ=ÇáêÉÅí=áåíÉêÉëí=Ñçê=ï~íÉê=ã~å~ÖÉãÉåí=çê=çíÜÉê=éìêéçëÉëK=få=íÜÉ=ÑçääçïáåÖ=ïÉ=ïáää=ÇÉêáîÉ=~=ëÉí=çÑ=da=ÅÜ~ê~ÅíÉêáëíáÅë=Ñêçã=íÜÉ=ãçãÉåíë=çÑ=íÜÉ=Éëíáã~íÉÇ=fo=ÑìåÅíáçå=~åÇ=íÜÉ=Çóå~ãáÅ=ÅÜ~ê~ÅíÉêáëíáÅë=çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=ëÉêáÉëK=tÜÉå=~=qck=ãçÇÉä=áë=Å~äáÄê~íÉÇ=çå=~=äáãáíÉÇ=ëÉí=çÑ=çÄëÉêî~íáçåëI=~å=Éëíáã~íÉ=çÑ=íÜÉ=ãÉ~å=ÖêçìåÇï~íÉê=äÉîÉä=h ~åÇ=~ååì~ä=~ãéäáíìÇÉ= h̃ =Å~å=ÄÉ=çÄí~áåÉÇ=ìëáåÖ=íÜÉ=ëíÉé=~åÇ=ÑêÉèìÉåÅó=êÉëéçåëÉ=ÑìåÅíáçåëK=qÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçå=áë=íÜÉ=êÉëéçåëÉ=çÑ=~=ëóëíÉã=íç=~å=áåéìí=ëÉêáÉë=ïáíÜ=ìåáí=áåíÉåëáíó=~åÇ=íÜÉ=ëÜ~éÉ=çÑ=~=ëíÉéI=~åÇ=Éèì~äëW==tΘ ( t) θ ( τ )dτ(6.8)=∫0=qÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçå=~ëóãéíçíáÅ~ääó=êÉ~ÅÜÉë=~=Åçåëí~åí=äÉîÉä=íÜ~í=áë=âåçïå=~ë=íÜÉ=Ö~áå=x_çñ=~åÇ=gÉåâáåëI=NVTMz=~åÇ=Éèì~äë=íÜÉ=òÉêçíÜ=ãçãÉåí= M0=çÑ=θ K=få=ÜóÇêçäçÖáÅ=íÉêãëI=áí=áë=íÜÉ=äÉîÉä=íç=ïÜáÅÜ=íÜÉ=ï~íÉê=í~ÄäÉ=êáëÉë=ïÜÉå=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=áåíÉåëáíó=ïçìäÇ=ÄÉ=Åçåëí~åí=~åÇ=çÑ=ìåáí=áåíÉåëáíóK=cêçã=íÜáëI= h =Å~å=ÄÉ=Éëíáã~íÉÇ=áå=íÜÉ=ÑçääçïáåÖ=ï~óI=í~âáåÖ=áå=ãáåÇ=íÜ~í=íÜÉ=êáëÉ=ëí~êíë=Ñêçã=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉ= d W==h = d + pM 0(6.9)=ïáíÜ= M0Éèì~ääáåÖ= A ïÜÉå=θ =Éèì~äë=~=pd=ÇÑ=EëÉÉ=ESKTFFK=_ÉÅ~ìëÉ=ãçëí=ÉÅçåçãáÅ~ä=~åÇ=ÉÅçäçÖáÅ~ä=~Åíáîáíó=í~âÉë=éä~ÅÉ=~í=ëìêÑ~ÅÉ=äÉîÉä=E s FI=çÑíÉå=íÜÉ=ÖêçìåÇï~íÉê=ÇÉéíÜ=EëJ h F=áë=çÑ=ãçêÉ=áåíÉêÉëí=íÜ~å=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=E h FK=eçïÉîÉêI=íÜÉ=ìëÉ=çÑ=~=êÉÑÉêÉåÅÉ=äÉîÉä= s =çê= d =íÜ~í=î~êáÉë=áå=ëé~ÅÉ=áåíêçÇìÅÉë=~åçíÜÉê=ÇÉÖêÉÉ=çÑ=ÑêÉÉÇçãI=íÜ~í=áë=ÅçãéÉåë~íÉÇ=Äó=ëéÉÅáÑóáåÖ=íÜÉ=ÅçåîÉñáíó=çê=ãÉ~å=Çáëí~åÅÉ=çÑ=íÜÉ=ÖêçìåÇï~íÉê=í~ÄäÉ=Ñêçã=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉ=E h − d FK===qÜÉ=ÑêÉèìÉåÅó=êÉëéçåëÉ=áë=íÜÉ=cçìêáÉê=íê~åëÑçêã=çÑ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåI=çê=íÜÉ=êÉëéçåëÉ=çÑ=~=ëóëíÉã=íç=~å=áåéìí=ëÉêáÉë=áå=íÜÉ=ëÜ~éÉ=çÑ=~=ëáåÉ=ïáíÜ=ìåáí=~ãéäáíìÇÉ=~åÇ=ÑêÉèìÉåÅó=ξ =xwáÉãÉê=Éí=~äKI=NVVUzW====tΞ ( t) = ∫ sin( ξt) θ ( t −τ )dτ(6.10)−∞=tÜÉå=ïÉ=ëçäîÉ=Éèì~íáçå=ESKNMF=Ñçê=~=ëóëíÉã=ïáíÜ=~=pd=ÇÑ=ëÜ~éÉÇ=fo=ÑìåÅíáçåI=ïÉ=ÑáåÇ=EëÉÉ=~ééÉåÇáñFW=..…………………………………………………………………………………………….….

`Ü~ê~ÅíÉêáòáåÖ=ÖêçìåÇï~íÉê=Çóå~ãáÅë==An ξΞ ( t) = sin{ ξt− arctan( )}(6.11)2 nξξ a(1 + )22a=qÜÉ=íÉêã=ÄÉÑçêÉ=íÜÉ=ëáåÉ=ÑìåÅíáçå=áë=íÜÉ=~ãéäáíìÇÉ=êÉëéçåëÉ=ÑìåÅíáçå=~åÇ=Å~å=ÄÉ=ìëÉÇ=2πíç=Éëíáã~íÉ=íÜÉ=~ååì~ä=~ãéäáíìÇÉ=EáKÉK=ÑêÉèìÉåÅó= ξ = = 0.0172 F=çÑ=íÜÉ=365.24ÖêçìåÇï~íÉê=äÉîÉä=ÑäìÅíì~íáçåëW===Ah̃ λ=(6.12)2 n0.0172(1 + )22a=ïáíÜ= λ =ÄÉáåÖ=íÜÉ=~îÉê~ÖÉ=~ãéäáíìÇÉ=çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=çîÉê=~=ÅÉêí~áå=éÉêáçÇ=EÑáÖìêÉ=SKNÄFK=qÜÉ=ä~ëí=íÉêã=çÑ=ESKNNF=áë=íÜÉ=éÜ~ëÉ=êÉëéçåëÉ=ÑìåÅíáçåK=cêçã=íÜáë=~=Ñ~Åíçê=η =Å~å=ÄÉ=çÄí~áåÉÇ=íÜ~í=Éèì~äë=íÜÉ=éÜ~ëÉ=ëÜáÑí=ÄÉíïÉÉå=íÜÉ=~ååì~ä=ÖêçìåÇï~íÉê=~åÇ=éêÉÅáéáí~íáçå=ëìêéäìë=ÅóÅäÉW==n 0.0172Input System Outputη = arctan( ) (6.13)0.0172 a====NPV=6.2.4 Summary of methodfå=ëìãã~êóI=íÜÉ=ãÉíÜçÇ=éêçéçëÉÇ=Åçåëáëíë=çÑ=íÜÉ=ÑçääçïáåÖ=ëíÉéë=EëÉÉ=ÑáÖìêÉ=SKOFK=cáêëíI=íÜÉ=âÉó=Çóå~ãáÅ=ÅÜ~ê~ÅíÉêáëíáÅë=çÑ=íÜÉ=áåéìí=íç=íÜÉ=ëóëíÉã=~êÉ=Å~äÅìä~íÉÇI=Ñçê=ìåáî~êá~íÉ=íáãÉ=ëÉêáÉë=ãçÇÉäë=ÄÉáåÖ=íÜÉ=ãÉ~å=~åÇ=~ååì~ä=~ãéäáíìÇÉ=çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìëK=pÉÅçåÇI=~å=Éëíáã~íÉ=çÑ=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=ÖêçìåÇï~íÉê=ëóëíÉã=áë=ã~ÇÉI=íÜ~í=áë=ÅÜ~ê~ÅíÉêáëíáÅ=çÑ=áíë=ÑìåÅíáçåáåÖK=cçê=íÜáë=éìêéçëÉI=~=Åçåíáåìçìë=íáãÉ=ëÉêáÉë=ãçÇÉä=áë=ìëÉÇ=~åÇ=Å~äáÄê~íÉÇ=çå=íáãÉ=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=~åÇ=éêÉÅáéáí~íáçå=ëìêéäìëK=cêçã=íÜÉ=fo=ÑìåÅíáçåI=ãçãÉåíë=Å~å=ÄÉ=ÇÉêáîÉÇ=~åÇ=ìëÉÇ=~ë=ëóëíÉã=ÅÜ~ê~ÅíÉêáëíáÅëK=qÜáêÇI=ÄçíÜ=ëóëíÉã=~åÇ=áåéìí=ÅÜ~ê~ÅíÉêáëíáÅë=~êÉ=ÅçãÄáåÉÇ=áåíç=ÅÜ~ê~ÅíÉêáëíáÅë=çÑ=íÜÉ=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁKpp~sdMh~hnpps - dMc~ ~==McM012s - hh - dh-1average precipitation surplus intensity [LT ]-1annual amplitude of ( ) [LT ]= surface level [L]p t= local drainage level [L]th= n order moment of the IR function [-]= average groundwater level [L]= annual amplitude of h( t) [L]= time shift [T]=figure 6.2: Relationship between input, systemand output characteristics.=

`Ü~éíÉê=S=çìíéìí=çê=ÖêçìåÇï~íÉê=í~ÄäÉ=Çóå~ãáÅëI=ìëáåÖ=ëáãéäÉ=~å~äóíáÅ=ÉñéêÉëëáçåëK=qÜÉ=~êêçïë=áå=ÑáÖìêÉ=SKO=áääìëíê~íÉ=íÜÉ=ï~ó=áå=ïÜáÅÜ=íÜÉ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=~êÉ=ÇÉíÉêãáåÉÇ=Äó=íÜÉ=éêçéÉêíáÉë=çÑ=íÜÉ=ëóëíÉã=~åÇ=íÜÉ=áåéìíI=Äìí=íÜÉ=äáåâë=Å~å=~äëç=ÄÉ=ìëÉÇ=îáÅÉ=îÉêë~K=qÜÉ=~îÉê~ÖÉ=ÖêçìåÇï~íÉê=ÇÉéíÜ= s − h I=Ñçê=Éñ~ãéäÉI=áë=ÇÉíÉêãáåÉÇ=Äó=íÜÉ=Çáëí~åÅÉ=çÑ=íÜÉ=ëçáä=ëìêÑ~ÅÉ=Ñêçã=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=äÉîÉä= s − d I=~åÇ=Äó= M0=~åÇ= p I=Äìí==îáÅÉ=îÉêë~I= p I=çê=íÜÉ=~îÉê~ÖÉ=ÖêçìåÇï~íÉê=êÉÅÜ~êÖÉI=Å~å=~äëç=ÄÉ=Éëíáã~íÉÇ=ïÜÉå= M0=~åÇ= h − d =~êÉ=âåçïåK=qÜÉ=ÇÉíÉêãáåáëíáÅ=é~êí=çÑ=íÜÉ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=áë=ÅçãéäÉíÉäó=ÇÉíÉêãáåÉÇ=Äó=íÜÉ=Ñçìê=é~ê~ãÉíÉêë=íÜ~í=ÇÉëÅêáÄÉ=íÜÉ=çìíéìí=EÑçê=~ë=Ñ~ê=~ë=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=êÉ~ä=ïçêäÇ=ëóëíÉãë=Ñáíë=íÜÉ=pd=ÇÑ=êÉëéçåëÉFI=ëáåÅÉ=Ñçê=äáåÉ~ê=ëóëíÉãëI=íÜÉ=íê~åëÑçêã~íáçå=çÑ=áåéìí=íç=çìíéìí=áë=ÅçãéäÉíÉäó=ÇÉíÉêãáåÉÇ=Äó=íÜÉ=fo=ÑìåÅíáçåK===6.3 Example applicationNQM=6.3.1 Set-up and data setfå=íÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçåI=íÜÉ=éÉêÑçêã~åÅÉ=çÑ=íÜÉ=daJÅÜ~ê~ÅíÉêáëíáÅë=áå=ÅÜ~ê~ÅíÉêáòáåÖ=íÜÉ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=áë=Åçãé~êÉÇ=íç=íÜ~í=çÑ=jñdi=ëí~íáëíáÅëI=êÉÖáãÉ=ÅìêîÉë=~åÇ=ÑêÉèìÉåÅó=çÑ=ÉñÅÉÉÇÉåÅÉ=Öê~éÜëK=cçê=íÜáë=éìêéçëÉI=íïç=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=~êÉ=ëÉäÉÅíÉÇ=íÜ~í=çêáÖáå~íÉÇ=Ñêçã=ê~íÜÉê=ÇáÑÑÉêÉåí=ÜóÇêçäçÖáÅ=ëóëíÉãëI=~åÇ=~äëç=ëÜçï=~=ÇáÑÑÉêÉåí=íóéÉ=çÑ=Çóå~ãáÅ=ÄÉÜ~îáçêK=cìêíÜÉêãçêÉI=ÄçíÜ=éáÉòçãÉíÉêë=~êÉ=ìåÇáëíìêÄÉÇ=Äó=ÖêçìåÇï~íÉê=~Äëíê~ÅíáçåI=ÜóÇêçäçÖáÅ=áåíÉêîÉåíáçåë=çê=çíÜÉê=áåÑäìÉåÅÉëI=íÜìë=~ääçïáåÖ=íÜÉã=íç=ÄÉ=ãçÇÉäÉÇ=ïáíÜ=éêÉÅáéáí~íáçå=ëìêéäìë=~ë=íÜÉ=çåäó=áåéìí=ëÉêáÉëK=qÜÉ=Ñáêëí=ëÉêáÉë=çêáÖáå~íÉë=Ñêçã=~=ÇìåÉ=êÉëÉêîÉ=áå=íÜÉ=éêçîáåÅÉ=çÑ=kçêíÜJeçää~åÇI=qÜÉ=kÉíÜÉêä~åÇëI=åÉ~ê=íÜÉ=íçïå=çÑ=bÖãçåÇK=qÜÉ=ÇìåÉë=áå=íÜ~í=~êÉ~=Ñçêã=~å=~ééêçñáã~íÉäó=NJR=âáäçãÉíÉê=ïáÇÉ=êáÇÖÉ=ÄÉíïÉÉå=íÜÉ=ëÉ~=~åÇ=íÜÉ=~Çà~ÅÉåí=éçäÇÉê=~êÉ~I=ëç=íÜÉ=ëóëíÉã=áë=êÉä~íáîÉäó=ä~êÖÉ=~åÇ=áëçä~íÉÇK=qÜÉ=íÉêê~áå=áë=ìåÇìä~íáåÖ=~åÇ=íÜÉ=ëçáä=Åçåëáëíë=çÑ=eçäçÅÉåáÅ=~Éçäá~å=ë~åÇëK=qÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=áå=íÜáë=éáÉòçãÉíÉê=ENV`kiROUNF=áë=ïÉää=çÄëÉêîÉÇI=ïáíÜ=çÄëÉêî~íáçåë=í~âÉå=ã~åì~ääó=~Äçìí=íÜÉ=NQ íÜ =~åÇ=OU íÜ =çÑ=ÉîÉêó=ãçåíÜ=áå=íÜÉ=éÉêáçÇ=Ñêçã=RJNOJNVUO=ìåíáä=OVJNJOMMNK=^ë=áåéìí=Ñçê=íÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉäI=çÄëÉêî~íáçåë=çÑ=íÜÉ=éêÉÅáéáí~íáçå=~åÇ=éçíÉåíá~ä=Éî~éçê~íáçå=~êÉ=ìëÉÇ=ëí~êíáåÖ=Ñêçã=NJNJNVTPK=qÜÉ=éêÉÅáéáí~íáçå=ëÉêáÉë=áë=~î~áä~ÄäÉ=çå=~=table 6.1: Calibration results, estimated parameters and characteristics for piezometers 19CNL5281and 32cl0034.máÉòçãÉíÉê= NV`kiROUN= POÅäMMPQ=2R = =MKVNV= MKUNR=ojpb= NNKO=Åã= UKQ=Åã=A = =EHLJ= 2σ F= NPVS= EHLJ=PPSF= NTO= =EHLJ=NSF=a = =EHLJ 2σ F= MKMMNV= EHLJ=MKMMMSF= MKMNRN= =EHLJ=MKMMQMF=n = =EHLJ 2σ F= NKMSUT= EHLJ=MKMQQOF= MKSTOP= =EHLJ=MKMRRTF=xjidi=jdi=jpdi=jediz= xPKTT===QKMO===QKNU===QKPz= xMKTR===MKVS===N===NKOSz=x h− d h = h̃ ==η z= xOKMU===QKMP===MKON=VMKUz= xMKOQ===MKVS==MKNQ=PPKPz=..…………………………………………………………………………………………….….

`Ü~ê~ÅíÉêáòáåÖ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=Ç~áäó=Ä~ëáë=~åÇ=áë=çÄëÉêîÉÇ=Äó=íÜÉ=mêçîáåÅá~ä=t~íÉê=`çãé~åó=çÑ=kçêíÜJeçää~åÇ=áå=íÜÉ=ÇìåÉë=åÉ~ê=íÜÉ=íçïå=çÑ=`~ëíêáÅìãI=ïÜÉêÉ~ë=íÜÉ=Ç~áäó=éçíÉåíá~ä=Éî~éçê~íáçå=ëÉêáÉë=çêáÖáå~íÉë=Ñêçã=~=ëí~íáçå=çÑ=íÜÉ=oçó~ä=aìíÅÜ=jÉíÉçêçäçÖáÅ=fåëíáíìíÉ=åÉ~ê=íÜÉ=íçïå=çÑ=aÉ=hççóK=qÜÉ=ëÉÅçåÇ=ëÉêáÉë=çêáÖáå~íÉë=Ñêçã=~=éáÉòçãÉíÉê=EPOÅäMMPQF=äçÅ~íÉÇ=çå=íÜÉ=ã~áå=ãÉíÉçêçäçÖáÅ=ÑáÉäÇ=çÑ=íÜÉ=oçó~ä=aìíÅÜ=jÉíÉçêçäçÖáÅ=fåëíáíìíÉ=~í=íÜÉ=íçïå=çÑ=aÉ=_áäí=áå=íÜÉ=ÅÉåíêÉ=çÑ=íÜÉ=kÉíÜÉêä~åÇë=xëÉÉ=~äëç=_áÉêâÉåë=Éí=~äKI=NVVVzK=qÜÉ=íÉêê~áå=áë=Ñä~íI=äáÉë=~í=íÜÉ=ÉÇÖÉ=çÑ=~å=áÅÉJéìëÜÉÇ=êáÇÖÉ=íÜ~í=áë=~=êÉãå~åí=çÑ=íÜÉ=Öä~ÅáÉêë=íÜ~í=ÅçîÉêÉÇ=íÜÉ=åçêíÜ=çÑ=íÜÉ=kÉíÜÉêä~åÇë=ÇìêáåÖ=íÜÉ=p~~äáÉå=áÅÉ=~ÖÉI=~åÇ=ÄçêÇÉêë=~=ëã~ää=êáîÉê=å~ãÉÇ=íÜÉ=Ú_áäíëÉ=dêáÑíÛK=qÜÉ=ëÉêáÉë=áë=çÄëÉêîÉÇ=ïáíÜ=~=Ç~áäó=ÑêÉèìÉåÅó=ÇìêáåÖ=~=NSJóÉ~ê=éÉêáçÇ=ERJNJNVUN=ìåíáä=NJNJNVVTFK=eçïÉîÉêI=íÜÉ=ëÉêáÉë=áë=åçí=íçí~ääó=ÅçãéäÉíÉ=Äìí=UKTRB=çÑ=íÜÉ=çÄëÉêî~íáçåë=~êÉ=ãáëëáåÖK=qÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=áë=çÄí~áåÉÇ=Ñêçã=a)1.8observedpredictedgroundwater level (m)1.61.41.210.80.6===NQN=0.45.254.8observedpredicted1982 1985 1987 1990 1992 1995 1997time(date)b)=groundwater level (m)4.64.44.243.83.63.43.21982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002time(date)==figure 6.3: Time plot of the available groundwater level observations for piezometers 19CNL5281 and32cl0034, together with the predictions (transfer part, at daily frequency) from the time series model.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=S=Ç~áäó=~îÉê~ÖÉÇ=çÄëÉêî~íáçåë=çÑ=éêÉÅáéáí~íáçå=~åÇ=éçíÉåíá~ä=Éî~éçíê~åëéáê~íáçå=~í=íÜÉ=ë~ãÉ=ãÉíÉçêçäçÖáÅ=ÑáÉäÇI=ëí~êíáåÖ=Ñêçã=NJNJNVTNK==NQO=6.3.2 Comparison of MxGL statistics and GD-characteristicsfå=çêÇÉê=íç=ÖÉí=~å=Éëíáã~íÉ=çÑ=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉëÉ=ëóëíÉãëI=~=ëáåÖäÉ=áåéìí=Åçåíáåìçìë=2íáãÉJqck=ãçÇÉä=EÉèì~íáçå=ESKPFF=ï~ë=Å~äáÄê~íÉÇ=çå=íÜÉ=Ç~í~=çÑ=ÄçíÜ=ëÉêáÉëI=ïáíÜ=~= R =EÅçÉÑÑáÅáÉåí=çÑ=ÇÉíÉêãáå~íáçåF=çÑ=êÉëéÉÅíáîÉäó=MKVNV=~åÇ=MKUNRK=qÜÉ=êÉëìäíë=~êÉ=ëìãã~êáòÉÇ=áå=í~ÄäÉ=SKNK=qáãÉ=éäçíë=çÑ=íÜÉ=~î~áä~ÄäÉ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=Ñçê=ÄçíÜ=ëÉêáÉëI=íçÖÉíÜÉê=ïáíÜ=çìíéìí=Ñêçã=íÜÉ=qck=ãçÇÉäëI=~êÉ=ÖáîÉå=áå=ÑáÖìêÉ=SKPK=pìÄëÉèìÉåíäóI=jñdi=ëí~íáëíáÅë=~åÇ=da=ÅÜ~ê~ÅíÉêáëíáÅë=ïÉêÉ=Å~äÅìä~íÉÇ=ìëáåÖ=íÜÉ=éáÉòçãÉíÉê=Ç~í~=~åÇ=íÜÉ=Éëíáã~íÉÇ=fo=ÑìåÅíáçåëK=_ó=ÇÉÑáåáíáçåI=jñdi=ëí~íáëíáÅë=~êÉ=Å~äÅìä~íÉÇ=Ñêçã=ëÉêáÉë=ïáíÜ=~å=áåíÉêî~ä=çÑ=NQ=Ç~óëI=ëç=ïÜÉêÉ=åÉÅÉëë~êóI=íÜÉ=éáÉòçãÉíÉê=Ç~í~=ï~ë=êÉë~ãéäÉÇ=íç=ã~íÅÜ=íÜáë=ÑêÉèìÉåÅóK=qÜÉ=jedi=~åÇ=jidi=~êÉ=Å~äÅìä~íÉÇ=~ë=íÜÉ=~îÉê~ÖÉ=çÑ=íÜÉ=edP=çê=idP=EáKÉK=íÜÉ=~îÉê~ÖÉ=çÑ=íÜÉ=íÜêÉÉ=ÜáÖÜÉëíI=êÉëéÉÅíáîÉäó=äçïÉëí=î~äìÉë=áå=~=ÅÉêí~áå=óÉ~êF=çîÉê=~=åìãÄÉê=çÑ=óÉ~êëK=qÜÉ=jpdiI=çå=íÜÉ=çíÜÉê=Ü~åÇI=áë=íÜÉ=~îÉê~ÖÉJçÑJíÜÉJ~îÉê~ÖÉ=çÑ=íÜÉ=íÜêÉÉ=î~äìÉë=åÉáÖÜÄçêáåÖ=íÜÉ=Ñáêëí=çÑ=^éêáä=Eçê=lÅíçÄÉê=çå=íÜÉ=ëçìíÜÉêå=ÜÉãáëéÜÉêÉF=çÑ=ÉîÉêó=óÉ~êK=qÜÉ=jñdi=ëí~íáëíáÅë=~åÇ=da=ÅÜ~ê~ÅíÉêáëíáÅë=Ü~îÉ=íç=ÄÉ=Å~äÅìä~íÉÇ=çîÉê=íÜÉ=ÅçêêÉÅí=éÉêáçÇ=áå=çêÇÉê=íç=ã~âÉ=íÜÉ=êÉëìäíë=Åçãé~ê~ÄäÉK=kçíÉ=íÜ~í=íÜÉ=éÉêáçÇ=ìëÉÇ=Ñçê=Å~äÅìä~íáåÖ=íÜÉ=ãÉíÉçêçäçÖáÅ=ÅÜ~ê~ÅíÉêáëíáÅë p ~åÇ p̃ ÇçÉë=åçí=Éèì~ä=íÜÉ=éÉêáçÇ=ìëÉÇ=Ñçê=Å~äÅìä~íáåÖ=íÜÉ=jñdi=ëí~íáëíáÅëK=qÜÉ=êÉ~ëçå=Ñçê=íÜáë=áë=íÜ~í=íÜÉ=ÖêçìåÇï~íÉê=ëóëíÉã=Ü~ë=~=ãÉãçêóI=~åÇ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë= p =çìíëáÇÉ=íÜÉ=éÉêáçÇ=çÑ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=~äëç=Ü~ë=~=ÅÉêí~áå=áåÑäìÉåÅÉK=qÜÉêÉÑçêÉI= p =Ü~ë=íç=ÄÉ=ïÉáÖÜíÉÇ=~ÅÅçêÇáåÖ=íç=áíë=áåÑäìÉåÅÉ=çå=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉëI=çê=áå=çíÜÉê=ïçêÇë=íÜÉ=ÑçääçïáåÖ=ÄäçÅâ=êÉëéçåëÉ=ÑìåÅíáçåW==tΘ ( t) = ∫ θ ( τ )dτ(6.14)−( − )t the thb=ïáíÜ= t hb=~åÇ= t he=ÇÉåçíáåÖ=íÜÉ=ëí~êí=~åÇ=ÉåÇ=çÑ=íÜÉ=éÉêáçÇ=Ñêçã=ïÜáÅÜ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=~êÉ=~î~áä~ÄäÉK=rëáåÖ=ESKNQFI=íÜÉ=ïÉáÖÜíÉÇ=~îÉê~ÖÉ=~åÇ=ïÉáÖÜíÉÇ=~ãéäáíìÇÉ=çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=ïÉêÉ=Å~äÅìä~íÉÇ=~ëW=p =the∫t pbYhe∑Yp( τ ) Θ( t −τ ) dτthe−∫0t pbYhe0heΘ( t)dτp( Y, D) Θ( Y − Y )hepbp̃( D) = , 1≤ D ≤ 365−Ypb∑Θ( Y )(6.15)..…………………………………………………………………………………………….….

`Ü~ê~ÅíÉêáòáåÖ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=qÜÉ=~ééäáÅ~íáçå=çÑ=~=ïÉáÖÜíÉÇ=~îÉê~ÖÉ=~åÇ=~ãéäáíìÇÉ=áãéäáÉë=íÜ~í=da=ÅÜ~ê~ÅíÉêáëíáÅë=Å~å=ÄÉ=êÉ~Çáäó=Å~äÅìä~íÉÇ=Ñçê=~åó=éÉêáçÇ=çê=Åäáã~íçäçÖáÅ=ëÅÉå~êáçI=çê=ìëáåÖ=~åó=ïÉáÖÜíáåÖ=EïÜÉå=Ñçê=áåëí~åÅÉ=íÜÉ=êÉÅÉåí=Åäáã~íçäçÖáÅ=ÅáêÅìãëí~åÅÉë=~êÉ=ÇÉÉãÉÇ=ãçêÉ=áãéçêí~åí=íÜ~å=íÜçëÉ=çÑ=PM=óÉ~êë=~ÖçFK=eçïÉîÉêI=íÜÉ=ìëÉ=çÑ=~=PMJóÉ~ê=éÉêáçÇ=êÉéêÉëÉåíáåÖ=íÜÉ=éêÉëÉåí=Ç~ó=Åäáã~íÉ=E~ë=ëìÖÖÉëíÉÇ=áå=xhåçííÉêë=~åÇ=s~å=t~äëìãI=NVVTz=~åÇ=~äëç=ìëÉÇ=áå=íÜÉ=ãÉíÉçêçäçÖáÅ=ëÅáÉåÅÉë=xpäìáàíÉê=~åÇ=kÉääÉëíáàåI=OMMOzFI=êÉã~áåë=~=äçÖáÅ~ä=ÅÜçáÅÉ=Ñçê=ã~åó=~ééäáÅ~íáçåëK=qÜÉ=jñdi=ëí~íáëíáÅë=~åÇ=da=ÅÜ~ê~ÅíÉêáëíáÅë=çÑ=ÄçíÜ=éáÉòçãÉíÉêë=~êÉ=éäçííÉÇ=áå=ÑáÖìêÉ=SKQI=íçÖÉíÜÉê=ïáíÜ=~=êÉÖáãÉ=ÅìêîÉ=çÄí~áåÉÇ=~å~äçÖçìë=íç=Éèì~íáçå=ESKOFK=qÜÉ=ÑáÖìêÉ=ÅäÉ~êäó=ëÜçïë=íÜ~í=jdi=~åÇ=h ~êÉ=~äãçëí=Éèì~äI=ïÜáäÉ=jediJjidi=áë=ÖêÉ~íÉê=íÜ~å=íÜÉ=~ååì~ä=~ãéäáíìÇÉ= h̃ =Ñçê=ÄçíÜ=éáÉòçãÉíÉêëK=oÉã~êâ~ÄäóI=jediJjidi=Ñçê=éáÉòçãÉíÉê=NV`kiROUN=~äãçëí=Éèì~äë=jediJjidi=Ñçê=éáÉòçãÉíÉê=POÅäMMPQK=`çåëÉèìÉåíäóI=íÜÉ=jedi=~åÇ=jidi=Çç=åçí=ÇáÑÑÉêÉåíá~íÉ=îÉêó=ïÉää=ÄÉíïÉÉå=íÜÉ=Çóå~ãáÅë=çÑ=íÜÉëÉ=íïçI=ÜóÇêçäçÖáÅ~ääó=ÇáÑÑÉêÉåíI=ëóëíÉãëK=qÜÉêÉ=áëI=ÜçïÉîÉêI=~=ÇáÑÑÉêÉåÅÉ=ÄÉíïÉÉå=íÜÉáê=êÉëéçåëÉ=~ãéäáíìÇÉëK=cêçã=íÜÉáê=~ãéäáíìÇÉ=ëéÉÅíê~=EÑáÖìêÉ=SKSFI=Äìí=~äëç=ÇáêÉÅíäó=Ñêçã=íÜÉ=ÖêçìåÇï~íÉê=ÜóÇêçÖê~éÜ=EÑáÖìêÉ=SKPFI=áí=Å~å=ÄÉ=ëÉÉå=íÜ~í=íÜÉ=ÜáÖÜ=ÑêÉèìÉåÅó=ÑäìÅíì~íáçåë=çÑ=íÜÉ=ëã~ää=~åÇ=Ñ~ëí=ãÉ~Ççï=ëóëíÉã=~êÉ=ãçêÉ=éêçåçìåÅÉÇ=íÜ~å=íÜçëÉ=çÑ=íÜÉ=ÄáÖÖÉê=~åÇ=ëäçïÉê=ÇìåÉ=ëóëíÉãI=ïÜáäÉ=íÜÉ=çééçëáíÉ=áë=íêìÉ=Ñçê=íÜÉ=~ååì~ä=~ãéäáíìÇÉK=içÖáÅ~ääóI=íÜÉ=ÜáÖÜ=ÑêÉèìÉåÅó=ÑäìÅíì~íáçåë=áåÑäìÉåÅÉ=íÜÉ=ÉñíêÉãÉë=EedP=~åÇ=idPFK=^éé~êÉåíäóI=íÜÉ=ÇáÑÑÉêÉåí=ÄÉÜ~îáçê=áå=íÜÉ=ÜáÖÜ=ÑêÉèìÉåÅáÉë=ÅçãéÉåë~íÉë=íÜÉ=ÇáÑÑÉêÉåÅÉ=áå=íÜÉ=~ååì~ä=~ãéäáíìÇÉëI=~åÇ=êÉëìäíë=áå=Åçãé~ê~ÄäÉ=jediJjidiëK=^=Ñ~Åíçê=íÜ~í=ÇçÉë=ÇáÑÑÉêÉåíá~íÉ=ÄÉíïÉÉå=ÄçíÜ=éáÉòçãÉíÉêë=áë=íÜÉ=jÉ~å=péêáåÖ=dêçìåÇï~íÉê=iÉîÉä=Eçê=ãçêÉ=Éñ~Åíäó=jpdiJjdiF=ïÜáÅÜ=áëI=áå=Åçåíê~ëí=íç=jedi=~åÇ=jidiI=åçí=~=êÉëìäí=çÑ=ÑäìÅíì~íáçåë=áå=ëÉîÉê~ä=ÑêÉèìÉåÅáÉëI=Äìí=áë=áå=Ñ~Åí=~å=çêÇáå~íÉ=çÑ=íÜÉ=~ååì~ä=ÖêçìåÇï~íÉê=ÅóÅäÉK=qÜáë=Ñ~Åíçê=áë=ÅäçëÉäó=êÉä~íÉÇ=íç=íÜÉ=Ñ~Åíçê=η =EëÉÉ=í~ÄäÉ=SKNFI=ïÜáÅÜ=Éèì~äë=íÜÉ=éÜ~ëÉ=ëÜáÑí=ÄÉíïÉÉå=íÜÉ=~ååì~ä=ÖêçìåÇï~íÉê=~åÇ=éêÉÅáéáí~íáçå=ëìêéäìë=ÅóÅäÉK=pÜáÑíáåÖ=íÜÉ=ëáåìëçáÇ~ä=ÖêçìåÇï~íÉê=ÅóÅäÉ=áå=íáãÉ=ïáää=~äíÉê=íÜÉ=î~äìÉ=çå=íÜÉ=Ñáêëí=çÑ=^éêáä=~åÇ=ÜÉåÅÉ=íÜÉ=jpdiK=eçïÉîÉêI=äçÖáÅ~ääó=~äëç=íÜÉ=~ãéäáíìÇÉ=çÑ=íÜÉ=~ååì~ä=ÅóÅäÉ=áë=çÑ=áåÑäìÉåÅÉ=íç====NQP=4.4a)1.3b)4.3MHGLMHGL1.2groundwater level (m)4.24.14+AmpMSGLMGL / hgroundwater level (m)1.110.9+AmpMSGLMGL / h3.9−Amp0.83.8−AmpMLGLMLGL0.7MLGLMLGLJan Apr Jul Oct Jan Apr Jul Oct JanmonthJan Apr Jul Oct Jan Apr Jul Oct Janmonth=figure 6.4: Comparison of MxGL statistics and GD characteristics for piezometers 19CNL5281 and32cl0034, respectively. From the figure it can be seen that MHGL - MLGL is larger than two timesthe annual amplitude, and does not differentiate much between these systems. MGL and h arealmost equal and their lines are hard to distinguish.=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=S=groundwater level (m)54.543.5a) b)2Regime CurveRegime Curve90% Percentiles90% Percentilesgroundwater level (m)1.510.5groundwater level (m)Jan Feb Mar Apr May Jun Jul Sep Oct Nov Dectime (month)c)54.543.5Annual F.O.E. graphMean F.O.E. graphTotal F.O.E. graphgroundwater level (m)0Jan Feb Mar Apr May Jun Jul Sep Oct Nov Dectime (month)d)2Annual F.O.E. graphMean F.O.E. graphTotal F.O.E. graph1.510.5NQQ=0 10 20 30 40 50 60 70 80 90 100frequency of exceedence (%)00 10 20 30 40 50 60 70 80 90 100frequency of exceedence (%)=figure 6.5: Plots of the regime curves h̃ ( D)and the 90% confidence interval of . h( Y, D ) . ofpiezometers 19CNL5281 (a) and 32cl0034 (b), and respective annual, mean and total frequency ofexceedence graphs (c and d).=íÜÉ=jpdiK===6.3.3 GD-characteristics and fluctuations of non-annual frequency^=Åçããçå=ãÉíÜçÇ=íç=áääìëíê~íÉ=~åÇ=èì~åíáÑó=íÜÉ=äçåÖJíÉêã=î~êá~Äáäáíó=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=~í=~=ÅÉêí~áå=äçÅ~íáçå=áë=íç=éäçí=ÄçíÜ=íÜÉ=ãÉ~å=EáKÉK=íÜÉ=êÉÖáãÉ=ÅìêîÉ=h̃ ( D)F=~åÇ=íïç=çê=ãçêÉ=éÉêÅÉåíáäÉë=çÑ=íÜÉ=ÇáëíêáÄìíáçå=ÑìåÅíáçå=çÑ= h( Y | D)EÑáÖìêÉ=SKR~=~åÇ=ÄFK=^å=~äíÉêå~íáîÉ=áë=íÜÉ=ìëÉ=çÑ=ÑêÉèìÉåÅó=çÑ=ÉñÅÉÉÇÉåÅÉ=Öê~éÜë=EáKÉK=íÜÉ=Åìãìä~íáîÉ=êÉä~íáîÉ=ÇáëíêáÄìíáçå=çÑ= h( Y, D)FI=ïÜáÅÜ=ÜçïÉîÉê=Çç=åçí=Åçåí~áå=áåÑçêã~íáçå=~Äçìí=íÜÉ=éÉêáçÇ=çÑ=çÅÅìêêÉåÅÉ=çÑ=íÜÉ=ÉñíêÉãÉë=EÑáÖìêÉ=SKRÅ=~åÇ=ÇFK=eÉêÉ=ïÉ=Ü~îÉ=ìëÉÇ=~=VMB=ÅçåÑáÇÉåÅÉ=áåíÉêî~äI=~åÇ=~äëç=éäçííÉÇ=íÜÉ=jedi=~åÇ=jidi=íç=ÄÉ=~ÄäÉ=íç=Åçãé~êÉ=íÜÉ=~ååì~äI=ÜáÖÜÉê=~åÇ=äçïÉê=ÑêÉèìÉåÅó=ÑäìÅíì~íáçåëK=cêçã=íÜÉ=êÉëìäíë=áí=Å~å=ÄÉ=ëÉÉå=íÜ~í=ÄçíÜ=ëóëíÉãë=êÉ~Åí=ê~íÜÉê=ÇáÑÑÉêÉåíäó=íç=ÇáÑÑÉêÉåí=ÑêÉèìÉåÅáÉë=áå=íÜÉ=áåéìí=ëáÖå~äK=cçê=íÜÉ=ëäçïÉê=ëóëíÉãI=áí=áë=ÅäÉ~ê=íÜ~í=íÜÉ=äçï=ÑêÉèìÉåÅó=ÑäìÅíì~íáçåë=~êÉ=ä~êÖÉI=~åÇ=jedi=~åÇ=jidi=~êÉ=Äó=Ñ~ê=êÉëéÉÅíáîÉäó=äçïÉê=~åÇ=ÜáÖÜÉê=íÜ~å=íÜÉ=ÉñíêÉãÉë=íÜ~í=Å~å=çÅÅìê=áå=Åäáã~íçäçÖáÅ~ääó=åçåJ~îÉê~ÖÉ=óÉ~êëK=cçê=íÜÉ=Ñ~ëí=ëóëíÉã=íÜÉ=ÜáÖÜ=ÑêÉèìÉåÅó=ÑäìÅíì~íáçåë=~êÉ=ãçêÉ=éêçåçìåÅÉÇI=~åÇ=jediJjidi=ÉñÅÉÉÇë=íÜÉ=~îÉê~ÖÉ=~ååì~ä=ÅóÅäÉK=qÜÉ=ÑáåÇáåÖ=íÜ~í=ÇáÑÑÉêÉåí=ÜóÇêçäçÖáÅ=ëóëíÉãë=êÉ~Åí=ÇáÑÑÉêÉåíäó=íç=ÑäìÅíì~íáçåë=áå=ÇáÑÑÉêÉåí=ÑêÉèìÉåÅáÉë=áë=áå=Ñ~Åí=êÉéçêíÉÇ=áå=ã~åó=çíÜÉê=éìÄäáÅ~íáçåë=xëÉÉ=~äëç=dÉÜêÉäë=Éí=~äKI=NVVQX=iÉÇìÅ=Éí=~äKI=NVVTX=jçäÉå~í=Éí=~äKI=NVVVX=`ÜÉå=Éí=~äKI=OMMOzK==qÜÉ=éêçéçêíáçå~äáíó=çÑ=ÑäìÅíì~íáçåë=çÑ=ÇáÑÑÉêÉåí=ÑêÉèìÉåÅó=ïáíÜáå=~=ëáÖå~ä=Å~å=ÄÉ=Éñ~ãáåÉÇ=áå=ãçêÉ=ÇÉí~áä=Äó=äççâáåÖ=~í=áíë=~ãéäáíìÇÉ=ëéÉÅíêìãI=çÄí~áåÉÇ=ïáíÜ=~=Ñ~ëí=cçìêáÉê=íê~åëÑçêã=EÑáÖìêÉ=SKS~=~åÇ=ÄFK=cêçã=íÜÉ=~ãéäáíìÇÉ=ëéÉÅíêìãI=áí=Å~å=~Ö~áå=ÄÉ==..…………………………………………………………………………………………….….

`Ü~ê~ÅíÉêáòáåÖ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=5Ampl. spectruma)5Ampl. spectrumb)44Fouriercoefficient (−)32Fouriercoefficient (−)3211Amplitude response (−)12001000010 −1 10 0 10 1800600400200Ampl. resp.95% Conf. int.Period (years)010 −1 10 0 10 1Period (years)c)Amplitude response (−)010 −1 10 0 10 1Period (years)010 −1 10 0 10 1Period (years)=figure 6.6: a) and b) Amplitude spectra of the groundwater level observations for piezometers19CNL5281 and 32cl0034. c) Amplitude response of piezometers 19CNL5281 and 32cl0034 d)Amplitude response of a variety of synthetic systems with fixed annual Amplitude and M 0 butvariable η .= ëÉÉå=íÜ~í=íÜÉ=~ååì~ä=~ãéäáíìÇÉ=çÑ=íÜÉ=ëäçï=ëóëíÉã=áë=ÜáÖÜÉê=íÜ~å=íÜ~í=çÑ=íÜÉ=Ñ~ëíÉê=ëóëíÉãK=_çíÜ=ëóëíÉãëI=ÜçïÉîÉêI=~äëç=ëÜçï=~=îÉêó=ÇáëíáåÅí=çîÉê~ää=é~ííÉêå=áå=íÜÉáê=~ãéäáíìÇÉ=ëéÉÅíêìãK=_ÉÅ~ìëÉ=áå=íÜÉ=ÑêÉèìÉåÅó=Ççã~áåI=íÜÉ=~ãéäáíìÇÉ=ëéÉÅíêìã=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=áë=íÜÉ=êÉëìäí=çÑ=~=ãìäíáéäáÅ~íáçå=çÑ=íÜÉ=~ãéäáíìÇÉ=êÉëéçåëÉ=çÑ=íÜÉ=ëóëíÉã=~åÇ=íÜÉ=~ãéäáíìÇÉ=ëéÉÅíêìã=çÑ=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìëI=áí=ã~ó=ÄÉ=ïçêíÜïÜáäÉ=íç=Éñ~ãáåÉ=íÜÉ=~ãéäáíìÇÉ=êÉëéçåëÉë=çÑ=ÄçíÜ=ëóëíÉãëK=rëáåÖ=Éèì~íáçå=ESKNNF=~åÇ=íÜÉ=é~ê~ãÉíÉêë=çÄí~áåÉÇ=ïáíÜ=íÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉäI=ïÉ=Ü~îÉ=éäçííÉÇ=ÄçíÜ=~ãéäáíìÇÉ=êÉëéçåëÉë=áå=ÑáÖìêÉ=SKSÅI=íçÖÉíÜÉê=ïáíÜ=íÜÉáê=VRB=ÅçåÑáÇÉåÅÉ=áåíÉêî~äI=íÜ~í=ï~ë=Å~äÅìä~íÉÇ=ïáíÜW==2 ∂Ξ( t) 2 2 ∂Ξ( t) 2 2 ∂Ξ( t) 2 2 ∂Ξ( t) ∂Ξ( t)2σ Ξ( t) = ( ) σ A + ( ) σ a + ( ) σ n + 2( )( ) σ A,a +∂A ∂a ∂n ∂A ∂a(6.16)∂Ξ( t) ∂Ξ( t) 2 ∂Ξ( t) ∂Ξ( t)2+ 2( )( ) σA, n+ 2( )( ) σa,n∂A ∂n ∂a ∂n1000800600400200Ampl. resp.d)====NQR=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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`Ü~éíÉê=S=NRM=Abstract:The program Menyanthes combines a variety of functions for managing, editing,visualizing, analyzing and modeling hydrogeologic time series. Menyanthes was initiallydeveloped within the scope of the PhD research of the first author, whose primary aimwas the integration of data and physically-based methods for modeling time series ofgroundwater heads. As such, time series analysis forms the heart of Menyanthes. WithinMenyanthes, time series can be modeled using both the ARMA and PIRFICT methods. ThePIRFICT method is a new method of time series analysis that has practical advantages andfacilitates physical interpretation and implementation of knowledge on physical behavior.Analytic solutions to specific hydrogeologic problems may be used as response function,along with their physically-based parameters. A more general approach is possible usingSkew-Gaussian distribution functions, which prove to fit the behavior of hydrogeologic(and other) systems well. Use of such functions within the PIRFICT method substantiallysimplifies the model identification procedure, as compared to the traditional Box-Jenkinsprocedure. PIRFICT models may be fitted to a large number of time series in batch. Spatialpatterns that emerge in the results provide useful, additional, and independentinformation, which adds another dimension to time series analysis. Their interpretation issupported by the spatial visualization and analysis tools of Menyanthes. The PIRFICTmethod also facilitates the integration of time series and spatially-distributed models via,e.g., moment-generating differential equations. The PIRFICT method may prove to be ofuse for other types of time series as well, both within and outside the realm ofenvironmental sciences...…………………………………………………………………………………………….….

7jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=Chapter=7 Menyanthes: software forgroundwater head data=Adopted Menyanthes: from: software forVon Asmuth, J. R., C. Maas, M. Knotters, M. F. P. Bierkens, M. Bakker, T. N. Olsthoorn, D. G.Cirkel, I. Leunk, F. Schaars, and D. C. Von Asmuth (submitted), Menyanthes: software forhydrogeologic groundwater time series analysis, interfacing data head with physical insight, data båîáêçåãÉåí~ä=jçÇÉääáåÖ=C=pçÑíï~êÉ.=^Äëíê~Åí=qÜÉ=éêçÖê~ã=jÉåó~åíÜÉë=ÅçãÄáåÉë=~=î~êáÉíó=çÑ=ÑìåÅíáçåë=Ñçê=ã~å~ÖáåÖI=ÉÇáíáåÖI=îáëì~äáòáåÖI=~å~äóòáåÖ=~åÇ=ãçÇÉäáåÖ=ÜóÇêçÖÉçäçÖáÅ=íáãÉ=ëÉêáÉëK=jÉåó~åíÜÉë=ï~ë=áåáíá~ääó=ÇÉîÉäçéÉÇ=ïáíÜáå=Adopted from:íÜÉ=ëÅçéÉ=çÑ=íÜÉ=mÜa=êÉëÉ~êÅÜ=çÑ=íÜÉ=Ñáêëí=~ìíÜçêI=ïÜçëÉ=éêáã~êó=~áã=ï~ë=íÜÉ=áåíÉÖê~íáçå=çÑ=Ç~í~=~åÇ=éÜóëáÅ~ääóJÄ~ëÉÇ=ãÉíÜçÇë=Ñçê=ãçÇÉäáåÖ=íáãÉ=ëÉêáÉë=çÑ=ÖêçìåÇï~íÉê=ÜÉ~ÇëK=^ë=ëìÅÜI=Von Asmuth, J.R., C. Maas, M. Knotters, M.F.P. Bierkens, M.íáãÉ=ëÉêáÉë=~å~äóëáë=Ñçêãë=íÜÉ=ÜÉ~êí=çÑ=jÉåó~åíÜÉëK=táíÜáå=jÉåó~åíÜÉëI=íáãÉ=ëÉêáÉë=Å~å=ÄÉ=Bakker, T.N. Olsthoorn, D.G. Cirkel, I. Leunk, F. Schaars, andãçÇÉäÉÇ=ìëáåÖ=ÄçíÜ=íÜÉ=ôj^=~åÇ=mfocf`q=ãÉíÜçÇëK=qÜÉ=mfocf`q=ãÉíÜçÇ=áë=~=åÉï=ãÉíÜçÇ=çÑ=D.C. Von Asmuth (submitted)íáãÉ=ëÉêáÉë=~å~äóëáë=íÜ~í=Ü~ë=éê~ÅíáÅ~ä=~Çî~åí~ÖÉë=~åÇ=Ñ~Åáäáí~íÉë=éÜóëáÅ~ä=áåíÉêéêÉí~íáçå=~åÇ=Menyanthes: software for hydrogeologic time series analysis,áãéäÉãÉåí~íáçå=çÑ=âåçïäÉÇÖÉ=çå=éÜóëáÅ~ä=ÄÉÜ~îáçêK=^å~äóíáÅ=ëçäìíáçåë=íç=ëéÉÅáÑáÅ=ÜóÇêçÖÉçäçÖáÅ=éêçÄäÉãë=ã~ó=ÄÉ=ìëÉÇ=~ë=êÉëéçåëÉ=ÑìåÅíáçåI=~äçåÖ=ïáíÜ=íÜÉáê=éÜóëáÅ~ääóJÄ~ëÉÇ=é~ê~ãÉíÉêëK=^=interfacing data with physical insight.ãçêÉ=ÖÉåÉê~ä=~ééêç~ÅÜ=áë=éçëëáÄäÉ=ìëáåÖ=pâÉïJd~ìëëá~å=ÇáëíêáÄìíáçå=ÑìåÅíáçåëI=ïÜáÅÜ=éêçîÉ=íç=Environmental Modelling & Software.Ñáí=íÜÉ=ÄÉÜ~îáçê=çÑ=ÜóÇêçÖÉçäçÖáÅ=E~åÇ=çíÜÉêF=ëóëíÉãë=ïÉääK=rëÉ=çÑ=ëìÅÜ=ÑìåÅíáçåë=ïáíÜáå=íÜÉ=Reproduced with permission from Elsevier B.V.mfocf`q=ãÉíÜçÇ=ëìÄëí~åíá~ääó=ëáãéäáÑáÉë=íÜÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉÇìêÉI=~ë=Åçãé~êÉÇ=íç=íÜÉ=copyright 2012 Elsevier B.V.íê~Çáíáçå~ä=_çñJgÉåâáåë=éêçÅÉÇìêÉK=mfocf`q=ãçÇÉäë=ã~ó=ÄÉ=ÑáííÉÇ=íç=~=ä~êÖÉ=åìãÄÉê=çÑ=íáãÉ= =ëÉêáÉë=áå=Ä~íÅÜK=pé~íá~ä=é~ííÉêåë=íÜ~í=ÉãÉêÖÉ=áå=íÜÉ=êÉëìäíë=éêçîáÇÉ=ìëÉÑìäI=~ÇÇáíáçå~äI=~åÇ=áåÇÉéÉåÇÉåí=áåÑçêã~íáçåI=ïÜáÅÜ=~ÇÇë=~åçíÜÉê=ÇáãÉåëáçå=íç=íáãÉ=ëÉêáÉë=~å~äóëáëK=qÜÉáê===NRN=áåíÉêéêÉí~íáçå=áë=ëìééçêíÉÇ=Äó=íÜÉ=ëé~íá~ä=îáëì~äáò~íáçå=~åÇ=~å~äóëáë=íççäë=çÑ=jÉåó~åíÜÉëK=qÜÉ=mfocf`q=ãÉíÜçÇ=~äëç=Ñ~Åáäáí~íÉë=íÜÉ=áåíÉÖê~íáçå=çÑ=íáãÉ=ëÉêáÉë=~åÇ=ëé~íá~ääóJÇáëíêáÄìíÉÇ=ãçÇÉäë=îá~I=ÉKÖKI=ãçãÉåíJÖÉåÉê~íáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåëK=qÜÉ=mfocf`q=ãÉíÜçÇ=ã~ó=éêçîÉ=íç=ÄÉ=çÑ=ìëÉ=Ñçê=çíÜÉê=íóéÉë=çÑ=íáãÉ=ëÉêáÉë=~ë=ïÉääI=ÄçíÜ=ïáíÜáå=~åÇ=çìíëáÇÉ=íÜÉ=êÉ~äã=çÑ=ÉåîáêçåãÉåí~ä=ëÅáÉåÅÉëK===pçÑíï~êÉ=~î~áä~Äáäáíó=aÉîÉäçéÉêëW==hto=t~íÉêÅóÅäÉ=oÉëÉ~êÅÜ=fåëíáíìíÉ=EìëÉê=áåíÉêÑ~ÅÉI=~ééäáÅ~íáçåëFI=aÉäÑí=råáîÉêëáíó=çÑ=qÉÅÜåçäçÖó=EãÉíÜçÇë=~åÇ=~äÖçêáíÜãëFI=^äíÉêê~=t~ÖÉåáåÖÉå=ro=EåçåJäáåÉ~êáíóI=ÅçìêëÉF=Software availabilitycáêëí=~î~áä~ÄäÉ=óÉ~êW=OMMP=Developers:KWR Watercycle Research InstitutepçÑíï~êÉ=êÉèìáêÉãÉåíëW=táåÇçïë∆=umI=OMMPI=sáëí~I=T=Delft University of TechnologymêçÖê~ã=ä~åÖì~ÖÉW=j~íä~ÄI=`=Alterra Wageningen UR^î~áä~Äáäáíó=~åÇ=ÅçëíW=ëÉÉ=ïïïKãÉåó~åíÜÉëKåä=Artesiamêáã~êó=Åçåí~ÅíW=gçë=îçå=^ëãìíÜ=First available year: 2003bJã~áäW=àçëKîçåK~ëãìíÜ]Öã~áäKÅçã=Software requirements: Windows® XP, 2003, Vista, 7Program language: Matlab, CAvailability and cost: see www.menyanthes.nlPrimary contact:Jos von AsmuthKKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=T==7.1 IntroductionNRO=7.1.1 Time series models: their strong points and limitationsaóå~ãáÅ=ëóëíÉãë=~êÉ=ìÄáèìáíçìë=áå=Systemçìê=ÉåîáêçåãÉåíI=ÄçíÜ=å~íìê~ä=~åÇ=ã~åJã~ÇÉI=~åÇ=Ñçêã=íÜÉ=StructureëìÄàÉÅí=çÑ=ã~åó=ÇáÑÑÉêÉåí=ëÅáÉåíáÑáÅ=ÇáëÅáéäáåÉëK=`çåëÉèìÉåíäóI=íÜÉ=InputíÜÉçêó=~åÇ=íÉÅÜåáèìÉë=Ñçê=~å~äóòáåÖ=~åÇ=ãçÇÉäáåÖ=ÅçääÉÅíÉÇ=BehavioríáãÉ=ëÉêáÉë=Ç~í~=Ü~îÉ=~=î~ëí=~ééäáÅ~íáçå=~êÉ~K=låÉ=ã~ó=ÇáëÅÉêå=ÇáÑÑÉêÉåí=ÇáëÅáéäáåÉë=ëìÅÜ=~ë=íáãÉ=ëÉêáÉë=~å~äóëáëI=ëóëíÉã=áÇÉåíáÑáÅ~íáçåI=Åçåíêçä=ÉåÖáåÉÉêáåÖI= =ëáÖå~ä=éêçÅÉëëáåÖ=~åÇ=ÑáäíÉêáåÖI=ÇÉéÉåÇáåÖ=çå=íÜÉ=íóéÉ=çÑ=ëóëíÉã=~ÇÇêÉëëÉÇI=íÜÉáê=éÜóëáÅ~ä=ÑìåÅíáçåáåÖI=íÜÉ=~î~áä~ÄäÉ=Ç~í~=~åÇ=íÜÉ=éêçÄäÉã=íç=ÄÉ=ëçäîÉÇK=qÜÉ=íÜÉçêó=~åÇ=íÉÅÜåáèìÉë=ìëÉÇ=áå=boundaryEnvironmentOutputfigure 7.1: Schematic representation of an open physicalsystem. The spatial structure of environmental systems isgenerally far more complex than their temporal behavior,and largely unknown. The structure does not have to beexplicitly defined for modeling the behavior.íÜÉëÉ=ÇáëÅáéäáåÉëI=ÜçïÉîÉêI=~êÉ=çÑíÉå=êÉã~êâ~Ääó=ëáãáä~êK=qÜÉ=ôfj^=E^ìíçoÉÖêÉëëáîÉ=fåíÉÖê~íÉÇ=jçîáåÖ=^îÉê~ÖÉF=íáãÉ=ëÉêáÉë=ãçÇÉäë=~êÉ=ãçëí=ïáÇÉäó=ìëÉÇI=~äíÜçìÖÜ=íÜÉêÉ=Ü~ë=ÄÉÉå=~=ÇáîÉêëáÑáÅ~íáçå=çÑ=ãÉíÜçÇëK=qÜÉ=~ééäáÅ~íáçå=çÑ=ôfj^=ãçÇÉäë=ÇÉîÉäçéÉÇ=ê~éáÇäó=~ÑíÉê=íÜÉ=éìÄäáÅ~íáçå=çÑ=íÜÉ=íÉñí=Äççâ=Äó=_çñ=~åÇ=gÉåâáåë=xNVTMzK=^äíÜçìÖÜ=áí=íççâ=ëçãÉ=íáãÉ=ÄÉÑçêÉ=íÜÉ=ãÉíÜçÇë=~åÇ=éêáåÅáéäÉë=ïÉêÉ=éáÅâÉÇ=ìé=áå=íÜÉ=ÜóÇêçäçÖáÅ=ëÅáÉåÅÉëI=íÜáë=ï~ë=~ãéäó=ÅçãéÉåë~íÉÇ=Äó=íÜÉ=éìÄäáÅ~íáçå=çÑ=íÜÉ=îçäìãáåçìë=Äççâ=Äó=eáéÉä=~åÇ=jÅiÉçÇ=xNVVQzK=få=íÜÉáê=ÄççâI=eáéÉä=~åÇ=jÅiÉçÇ=éêÉëÉåí=ã~åó=ÇáÑÑÉêÉåí=~ééäáÅ~íáçåëI=~ë=ïÉää=~ë=ëÉîÉê~ä=ÉñíÉåëáçåë=íç=íÜÉ=ôfj^=ãçÇÉäë=~ë=éìÄäáëÜÉÇ=Äó=_çñ=~åÇ=gÉåâáåëK=oÉä~íáîÉäó=ëáãéäÉ=Ç~í~JÄ~ëÉÇ=ãçÇÉäë=äáâÉ=íÜÉ=ôfj^=íáãÉ=ëÉêáÉë=ãçÇÉä=~êÉ=ëíáää=ïáÇÉäó=~ééäáÉÇ=áå=íÜÉ=ÉåîáêçåãÉåí~ä=ëÅáÉåÅÉëI=ÇÉëéáíÉ=íÜÉ=êáëÉ=çÑ=éÜóëáÅ~ääóJÄ~ëÉÇI=ëé~íá~ääóJÇáëíêáÄìíÉÇ=ãçÇÉäëK=^Çî~åí~ÖÉë=çÑ=íáãÉ=ëÉêáÉë=ãçÇÉäë=çîÉê=éÜóëáÅ~ääóJÄ~ëÉÇI=ÇáëíêáÄìíÉÇ=ãçÇÉäë=~êÉ=íÜÉáê=êÉä~íáîÉ=~ÅÅìê~ÅóI=É~ëÉ=çÑ=ÅçåëíêìÅíáçåI=~åÇ=ïÉääJÇÉîÉäçéÉÇ=ëí~íáëíáÅ~ä=Ä~ÅâÖêçìåÇK=qÜÉ=Ñáêëí=íïç=éçáåíë=Ñçääçï=ÇáêÉÅíäó=Ñêçã=íÜÉ=Ñ~Åí=íÜ~í=~=íáãÉ=ëÉêáÉë=ãçÇÉä=íêÉ~íë=~=éÜóëáÅ~ä=ëóëíÉã=Ñáêëíäó=~ë=~=ÚïÜçäÉÛK=qÜÉ=ÉÑÑÉÅíáîÉI=çîÉê~ää=ÄÉÜ~îáçê=çÑ=~=ëóëíÉã=~í=~=ÅÉêí~áå=äçÅ~íáçå=áë=ãçÇÉäÉÇ=ïáíÜçìí=êÉèìáêáåÖ=íÜÉ=ÉñéäáÅáí=ÇÉÑáåáíáçå=çÑ=áíë=ÉåíáêÉ=ëé~íá~ä=ëíêìÅíìêÉ=EÑáÖìêÉ=TKNFK=få=Åçåíê~ëíI=éÜóëáÅ~ääóJÄ~ëÉÇI=ÇáëíêáÄìíÉÇ=ãçÇÉäë=ëí~êí=ÄçííçãJìé=Ñêçã=íÜÉ=ëçJÅ~ääÉÇ=êÉéêÉëÉåí~íáîÉ=ÉäÉãÉåí~êó=îçäìãÉ=EobsFI=~åÇ=~äãçëí=áåÜÉêÉåíäó=áåîáíÉ=íÜÉ=áåÅäìëáçå=çÑ=~ää=ÇÉí~áä=~ÄçîÉ=íÜ~í=ëÅ~äÉK=^ë=~=ÅçåëÉèìÉåÅÉI=ëìÅÜ=ãçÇÉäë=ÖÉåÉê~ääó=êÉèìáêÉ=~=ÖêÉ~í=~ãçìåí=çÑ=Ç~í~I=ïÜáäÉ=íÜÉ=ä~êÖÉê=é~êí=çÑ=íÜÉ=ãçÇÉä=ëíáää=áë=íÜÉ=êÉëìäí=çÑ=ÅçêêÉä~íáçåëI=Éñíê~éçä~íáçåë=~åÇ=~ëëìãéíáçåëI=ïÜáÅÜ=Å~ååçí=ÄÉ=îÉêáÑáÉÇ=çê=î~äáÇ~íÉÇ=xëÉÉ=ÉKÖKI=hçåáâçï=~åÇ=_êÉÇÉÜçÉÑíI=NVVOX=lêÉëâÉë=Éí=~äKI=NVVQzK=pìÅÜ=éêçÄäÉãëI=ïÜáÅÜ=~êÉ=ãçêÉ=çê=äÉëë=áåíêáåëáÅ=íç=íÜÉ=obs=ëÅ~äÉ=~ééêç~ÅÜI=Ü~îÉ=äÉ~Ç=ëçãÉ=íç=éêçÅä~áã=íÜ~í=áí=ïáää=éêçîÉ=íç=ÄÉ=~=ÇÉ~Ç=ÉåÇ=Ñçê=ÅçãéäÉñI=ÉåîáêçåãÉåí~ä=ëóëíÉãëI=ïÜáäÉ=çíÜÉêë=ÇÉëáÖåÉÇ=~äíÉêå~íáîÉ=~ééêç~ÅÜÉë=çå=..…………………………………………………………………………………………….….

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jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=7.2 Methods and theory7.2.1 Differential equations, impulse responses and convolutiontÜáäÉ=ã~åó=é~éÉêë=ëí~êí=Ñêçã=íÜÉ=íáãÉ=ëÉêáÉë=~å~äóëáë=îáÉïéçáåíI=ïÉ=ÅÜççëÉ=íç=ëí~êí=Ñêçã=~=éÜóëáÅ~ä=îáÉïéçáåí=ÜÉêÉ=íç=ÄÉííÉê=ëÉêîÉ=íÜÉ=åçåJëéÉÅá~äáëí=ÅçããìåáíóK=qÜÉ=ÅçåëíêìÅíáçå=çÑ=~=éÜóëáÅ~ääóJÄ~ëÉÇ=ãçÇÉä=Ñçê=íÜÉ=Çóå~ãáÅë=çÑ=~å=~êÄáíê~êó=î~êá~ÄäÉ=ëí~êíë=ïáíÜ=íÜÉ=ÇÉêáî~íáçå=çÑ=~å=~ééêçéêá~íÉ=ã~íÜÉã~íáÅ~ä=ÉñéêÉëëáçåI=Åçããçåäó=~=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåI=Ñçê=áíK=aáÑÑÉêÉåíá~ä=Éèì~íáçåë=~êÉ=ÇÉêáîÉÇ=Ñêçã=íïç=íóéÉë=çÑ=Éèì~íáçåë=xÉKÖKI=_É~êI=NVTOzW==• qÜÉ=Åçåíáåìáíó=Éèì~íáçåEëF=• qÜÉ=ÅçåëíáíìíáîÉ=Éèì~íáçåEëF==cçê=ÖêçìåÇï~íÉêI=íÜÉ=Åçåíáåìáíó=Éèì~íáçå=áë=ÑçìåÇ=Äó=~ééäóáåÖ=íÜÉ=éêáåÅáéäÉ=çÑ=ã~ëë=ÅçåëÉêî~íáçå=EïÜáäÉ=áå=çíÜÉê=Å~ëÉëI=~äëç=çíÜÉê=ÅçåëÉêî~íáçå=ä~ïë=ã~ó=~ééäóI=äáâÉ=íÜçëÉ=Ñçê=ÉåÉêÖó=çê=ãçãÉåíìãFK=få=ëáãéäÉ=ïçêÇëI=íÜáë=éêáåÅáéäÉ=ëí~íÉë=íÜ~í=åç=ã~ëë=Å~å=Çáë~ééÉ~ê=ïáíÜçìí=êÉ~ëçå=~åÇ=áí=áë=î~äáÇ=Ñçê=~ää=íóéÉë=çÑ=ã~ëëÉëK=få=Å~ëÉ=çÑ=ï~íÉêI=áí=áë=âåçïå=~ë=íÜÉ=ï~íÉê=Ä~ä~åÅÉ=Éèì~íáçåK=qÜÉ=ÅçåëíáíìíáîÉ=Éèì~íáçå=ëéÉÅáÑáÉë=~=éêçéÉêíó=çÑ=íÜÉ=ëéÉÅáÑáÅ=ãÉÇáìã=ìåÇÉê=ÅçåëáÇÉê~íáçåI=ïÜáÅÜ=áå=Å~ëÉ=çÑ=ÖêçìåÇï~íÉê=áë=a~êÅóÛë=ä~ïK=qÜáë=Éèì~íáçå=íêÉ~íë=~=êÉéêÉëÉåí~íáîÉ=ÉäÉãÉåí~êó=îçäìãÉ=EobsF=çÑ=~=éçêçìë=ãÉÇáìã=~ë=~=ÅçåíáåììãI=íÜÉêÉÄó=ÇáëêÉÖ~êÇáåÖ=íÜÉ=ëé~íá~ä=î~êá~Äáäáíó=ÄÉäçï=íÜáë=ëÅ~äÉ=EÅçåëáëíáåÖ=çÑ=Öê~áåë=~åÇ=éçêÉëI=çê=ãçäÉÅìäÉë=çå=~å=ÉîÉå=ëã~ääÉê=ëÅ~äÉFK==^=ëí~åÇ~êÇ=íÉÅÜåáèìÉ=Ñçê=ëçäîáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåë=çÑ=äáåÉ~ê=Çóå~ãáÅ=ëóëíÉãë=Éñ~Åíäó=áë=íç=ÇÉíÉêãáåÉ=íÜÉáê=ëçäìíáçå=Ñçê=~=aáê~Å=ÇÉäí~=ÑìåÅíáçå δ xaáê~ÅI=NVQTz=~ë=áåéìí p K=qÜÉ=ÇÉäí~=ÑìåÅíáçå=Ü~ë=íÜÉ=ÑçääçïáåÖ=éêçéÉêíáÉëW===⎧ δ( t) = 0, t ≠ 0⎪ ∞⎨⎪ δ( t)dt = 1∫ (7.1)⎩−∞=~åÇ=Å~å=ÄÉ=íÜçìÖÜí=çÑ=~ë=íÜÉ=äáãáí=çÑ=EÑçê=áåëí~åÅÉF=~=d~ìëëá~å=ÇáëíêáÄìíáçå=ÑìåÅíáçå=ïáíÜ=ãÉ~å=òÉêç=~åÇ=~=î~êá~åÅÉ=íÜ~í=~äëç=~ééêç~ÅÜÉë=òÉêçK=få=ïçêÇëI=íÜÉ=ÇÉäí~=ÑìåÅíáçå=áë=~å=áåëí~åí~åÉçìë=éìäëÉ=çê=áãéìäëÉ=çÑ=ìåáí=~êÉ~K=qÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=θ =áë=ÇÉÑáåÉÇ=~ë=íÜÉ=ÉÑÑÉÅí=çÑ= δ =~ë=áåéìí= p =çå=íÜÉ=ëí~íÉ= h =çÑ=~=ëóëíÉãI=áKÉK=íÜÉ=ÇÉîá~íáçå=áå=íáãÉ=Ñêçã=~å=çíÜÉêïáëÉ=ëíÉ~Çó=ëí~íÉ= d K=få=íÉêãë=çÑ=ÖêçìåÇï~íÉê=ÜÉ~Ç=~åÇ=éêÉÅáéáí~íáçåI==θ =Å~å=ÄÉ=íÜçìÖÜí=çÑ=~ë=íÜÉ=êÉëéçåëÉ=íç=~=îÉêó=ëÜçêí=ëÜçïÉê=çÑ=ìåáí=ÜÉáÖÜíI=ïÜÉå=íÜÉ=ÜÉ~Ç=áë=çíÜÉêïáëÉ=Åçåëí~åí=çê=Éèì~äë=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉK=få=ã~íÜÉã~íáÅ~ä=íÉêãëI=θ =áë=ÇÉÑáåÉÇ=Äó=íÜÉ=ÑçääçïáåÖ=ÅçåÇáíáçåëW=====NRR=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=T=⎧θ ( t) = h( t)− d⎪⎨h( t) = d, t < 0(7.2)⎪⎩ p( t) = δ( t)=fÑ=θ =~åÇ= d =~êÉ=âåçïåI= h =Å~å=ÄÉ=çÄí~áåÉÇ=Ñçê=~å=áåéìí p =íÜ~í=î~êáÉë=~êÄáíê~êáäó=áå=íáãÉ=íÜêçìÖÜ=Åçåîçäìíáçå=EaìÜ~ãÉäÛë=éêáåÅáéäÉ=xaìÜ~ãÉäI=NUPPzFW==t∞h( t) − d = θ ( t −τ ) p( τ )d τ ≡ θ ( t) p( t −τ )d τ ≡ ( θ ∗ p)( t)∫ ∫(7.3)−∞0=ïÜÉêÉ=íÜÉ=ä~ëí=Ñçêã=áë=àìëí=~=Åçãé~Åí=ï~ó=çÑ=ëóãÄçäáòáåÖ=~=Åçåîçäìíáçå=éêçÇìÅíK=bèì~íáçå=EOKPRF=áãéäáÉë=íÜ~í=íÜÉ=Çóå~ãáÅë=çÑ=~å=~êÄáíê~êóI=äáåÉ~êI=Çóå~ãáÅ=ëóëíÉã=~í=~=ÅÉêí~áå=äçÅ~íáçå=~êÉ=ÅçãéäÉíÉäó=ÖçîÉêåÉÇ=Äó=íÜÉ=Çóå~ãáÅë=çÑ=íÜÉ=áåéìí=~åÇ=íÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåK=qÜÉ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=θ =ÅçãéäÉíÉäó=ÅÜ~ê~ÅíÉêáòÉë=íÜÉ=Çóå~ãáÅ~ääó=êÉäÉî~åíI=éÜóëáÅ~ä=éêçéÉêíáÉë=çÑ=~=ëóëíÉãI=~åÇ=áë=~ë=ëìÅÜ=~å=áåíÉÖê~ä=éêçéÉêíó=íÜÉêÉçÑK=θ I=ÜçïÉîÉêI=áë=~=ÑìåÅíáçå=çÑ=íÜÉ=äçÅ~íáçå=áå=íÜÉ=ëóëíÉãI=~åÇ=çÑ=íÜÉ=ëéÉÅáÑáÅ=ÉñÅáí~íáçå=ìåÇÉê=ÅçåëáÇÉê~íáçåK===NRS=7.2.2 Methods of constructing response functionsqÜÉ=ëçäìíáçå=íç=EOKPRF=çåäó=Ñáíë=íÜÉ=Çóå~ãáÅë=çÑ=~=ëóëíÉã=Éñ~Åíäó=áÑ=θ =E~åÇ= p F=~êÉ=~äëç=Éñ~Åíäó=âåçïå=E~åÇ=íÜÉ=ëóëíÉã=áë=äáåÉ~ê=~åÇ=íáãÉJáåî~êá~åíFK=lÑ=ÅçìêëÉI=θ =áë=åÉîÉê=âåçïå=Éñ~Åíäó=áå=éê~ÅíáÅÉI=Äìí=íÜÉêÉ=~êÉ=ëÉîÉê~ä=ãÉíÜçÇë=íç=çÄí~áå=~å=~ééêçñáã~íáçå=Ñçê=áíK=qÜÉ=ÑáêëíI=ÄçííçãJìé=~åÇ=åçï~Ç~óë=ã~áåëíêÉ~ã=~ééêç~ÅÜI=áë=íç=ÅçåëíêìÅí=~=ëé~íá~ääóJÇáëíêáÄìíÉÇ=ãçÇÉä=íç=ëçäîÉ=íÜÉ=ÖçîÉêåáåÖ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK=aáëíêáÄìíÉÇ=ãçÇÉäë=ã~ó=ÄÉ=íê~åëÑçêãÉÇ=áåíç=ëÉíë=çÑ=áãéìäëÉ=êÉëéçåëÉëI=~ë=çìíäáåÉÇ=áå=xp~ÜìèìáääçI=NVUPzK=^äíÜçìÖÜ=íÜáë=éêçÅÉÇìêÉ=áë=åçí=ÅçããçåI=áíë=~Çî~åí~ÖÉë=çîÉê=íÜÉ=ÇáêÉÅí=ìëÉ=çÑ=íÜÉ=ÇáëíêáÄìíÉÇ=ãçÇÉä=~êÉ=áíë=Åçãéìí~íáçå~ä=ÉÑÑáÅáÉåÅó=~åÇ=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=Å~å=ÄÉ=ëçäîÉÇ=Åçåíáåìçìëäó=áå=íáãÉ=Äó=ÇçáåÖ=ëçK=^=ëÉÅçåÇ=~ééêç~ÅÜ=ëí~êíë=íçéJÇçïå=~åÇ=~ééêçñáã~íÉë=θ =~ë=~å=ôj^=íê~åëÑÉê=ÑìåÅíáçåK=få=íÜ~í=Å~ëÉI=EOKPRF=áë=~ééêçñáã~íÉÇ=Äó=ÇáëÅêÉíÉ=ÅçåîçäìíáçåW==t∑∞∑h − d = Θ p ≡ Θ p ≡ Θ( B) p ≡ ( Θ ∗ p)(7.4)t t−i i i t−i t ti=−∞ i=0=ïÜÉêÉ=t =áë=~=ÇáëÅêÉíÉ=íáãÉ=áåÇÉñ=E t ∈ N FI Θ =áë=íÜÉ=íê~åëÑÉê=çê=ÄäçÅâ=êÉëéçåëÉ=ÑìåÅíáçå=~åÇ= B =áë=íÜÉ=Ä~Åâï~êÇJëÜáÑí=çéÉê~íçê=ÇÉÑáåÉÇ=ÄóW==bB pt= pt − b(7.5)=Θ =Å~å=ÄÉ=ïêáííÉå=ëóãÄçäáÅ~ääó=~ë=xëÉÉ=ëÉÅíáçå=OKOX=_çñ=~åÇ=gÉåâáåëI=NVTMzW=..…………………………………………………………………………………………….….

jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=−1Θ (B) = δ (B) ω(B)(7.6)=ïÜÉêÉ= δ (B) =áë=íÜÉ=~ìíçêÉÖêÉëëáîÉ=çéÉê~íçê=Eåçí=íç=ÄÉ=ÅçåÑìëÉÇ=ïáíÜ=íÜÉ=aáê~Å=ÇÉäí~=ÑìåÅíáçåFI=~åÇ= ω (B) =íÜÉ=ãçîáåÖ=~îÉê~ÖÉ=çéÉê~íçêK=qÜÉ=~Åíì~ä=Åçãéìí~íáçå=çÑ= ht=áë=ÇçåÉ=êÉÅìêëáîÉäó=~ëW==h − d − δ ( h − d) − ... − δ ( h − d) = ω p + ω p + ... + ω p(7.7)t 1 t−1 nr t−nr 0 t 1 t−1ns t−nsïÜÉêÉ= nr =~åÇ= ns =ÇÉÑáåÉ=íÜÉ=åìãÄÉê=çÑ=é~ê~ãÉíÉêë=çê=çêÇÉê=çÑ=íÜÉ=íê~åëÑÉê=ÑìåÅíáçåK=^äíÜçìÖÜ=ôj^=ãçÇÉäë=~êÉ=ìëÉÇ=áå=~=î~êáÉíó=çÑ=ëÅáÉåÅÉë=~åÇ=~êÉ=ìëì~ääó=éÉêÅÉáîÉÇ=~ë=ÚÄä~Åâ=ÄçñÛ=ãçÇÉäëI=ôuENF=ãçÇÉäë=ã~ó=ÄÉ=ëÉÉå=~ë=êÉéêÉëÉåíáåÖ=íÜÉ=éÜóëáÅë=çÑ=~=ëáãéäÉ=ëçáä=Åçäìãå=xhåçííÉêë=~åÇ=_áÉêâÉåëI=OMMMz=~åÇ=~å=ôuE nr F=ãçÇÉä=~ë=êÉéêÉëÉåíáåÖ=~=EäáåÉ~êF=ÇáëíêáÄìíÉÇ=ÖêçìåÇï~íÉê=ãçÇÉä=ïáíÜ= nr ÅÉääë=EëÉÉ=ëÉÅíáçå=OKPKNFK=qÜÉ=êÉÅìêëáîÉ=Åçãéìí~íáçå=çÑ=ETKTF=áë=Åçãéìí~íáçå~ääó=äÉëë=ÇÉã~åÇáåÖ=íÜ~å=íÜÉ=ÇáêÉÅí=Éî~äì~íáçå=çÑ=ETKPF=çê=ETKQF=~åÇ=~ë=ôj^=ãçÇÉäë=Ü~îÉ=~=äçåÖ=ÜáëíçêóI=íÜáë=ï~ë=éêçÄ~Ääó=~å=áãéçêí~åí=~Çî~åí~ÖÉK=táíÜ=ãçÇÉêå=íÉÅÜåçäçÖóI=ÜçïÉîÉêI=íÜáë=áë=åçí=~å=áëëìÉ=~åóãçêÉI=~åÇ=ÅçåëÉèìÉåíäó=~ë=~=íÜáêÇ=~ééêç~ÅÜI= Θ =~åÇLçê=θ =Å~å=~äëç=ÄÉ=~ééêçñáã~íÉÇ=áå=çíÜÉê=ï~óëK==7.2.3 The PIRFICT method and use of distribution functionssçå=^ëãìíÜ=Éí=~äK=xOMMOz=éêÉëÉåíÉÇ=~=åÉï=ãÉíÜçÇ=çÑ=íáãÉ=ëÉêáÉë=~å~äóëáëI=Å~ääÉÇ=íÜÉ=mfocf`q=ãÉíÜçÇK==^=ÇáÑÑÉêÉåÅÉ=ïáíÜ=ôfj^JíóéÉ=~åÇ=âáåÇêÉÇ=íáãÉ=ëÉêáÉë=ãçÇÉäëI=ïÜÉíÜÉê=ÇáëÅêÉíÉ=çê=Åçåíáåìçìë=íáãÉ=xÉKÖKI=_êçÅâïÉääI=OMMNX=vçìåÖ=~åÇ=d~êåáÉêI=OMMSzI=áë=íÜ~í=áí=áë=Ñçêãìä~íÉÇ=~ë=~=EÅçåíáåìçìë=íáãÉF=Åçåîçäìíáçå=áåíÉÖê~ä=ETKPFI=áåëíÉ~Ç=çÑ=~ë=~=íÉãéçê~ä=ÇáÑÑÉêÉåÅÉ=çê=ÇáÑÑÉêÉåíá~ä=Éèì~íáçåK=få=íÜáë=Ñçêãìä~íáçåI=θ =ã~ó=áå=éêáåÅáéäÉ=ÄÉ=~å=~êÄáíê~êó=Åçåíáåìçìë=ã~íÜÉã~íáÅ~ä=ÑìåÅíáçå=Ü~îáåÖ=~=âåçïå=áåíÉÖê~äK=qÜÉ=mfocf`q=ãÉíÜçÇ=Ü~ë=ëÉîÉê~ä=éê~ÅíáÅ~ä=~Çî~åí~ÖÉë=xsçå=^ëãìíÜ=Éí=~äKI=OMMOzI=~åÇ=Ñ~Åáäáí~íÉë=íÜÉ=áåíÉÖê~íáçå=çÑ=Ç~í~=~åÇ=éÜóëáÅ~ääóJÄ~ëÉÇ=ãÉíÜçÇë=áå=íïç=ï~óëK=cáêëíI=íÜÉ=éÜóëáÅ~ä=ÄÉÜ~îáçê=çÑ=~=ëóëíÉã=ã~ó=ÄÉ=áãéäÉãÉåíÉÇ=ÇáêÉÅíäó=áå=íÜÉ=ãçÇÉä=~ë=êÉëéçåëÉ=ÑìåÅíáçå=θ I=Ñçê=Éñ~ãéäÉ=~ë=~å~äóíáÅ=ëçäìíáçå=íç=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=áå=~=ëéÉÅáÑáÅ=ëáíì~íáçåK=bäÉãÉåí~êó=éÜóëáÅ~ä=éêáåÅáéäÉë=ã~ó=ÄÉ=ìëÉÇ=íç=êÉÇìÅÉ=íÜÉ=ÇÉÖêÉÉë=çÑ=ÑêÉÉÇçã=áå==íÜÉ=ãçÇÉäI=ÉKÖKI=Äó=äáãáíáåÖ=íÜÉ=ëáÖå=çÑ=íÜÉ=ÉÑÑÉÅí=íç=ÉáíÜÉê=éçëáíáîÉ=figure 7.2: Skewed impulse response function with mean2çê=åÉÖ~íáîÉ=çê=Äó=ëÜ~êáåÖ= µ and variance σ .===NRT=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=11Surface levelMGL before 96MGL after 96Screen16FP707816FP70771016FP7062Height (m+ref)916FP7122816FP706916FP706816FP706716FP706316FP706416FP706616FP7065117111111=0 500 1000 1500 2000 2500ëÉêáÉëK=lå=íÜÉ=çíÜÉê=Ü~åÇI=îáëì~äáò~íáçå=~äëç=~áÇë=Úîáëì~ä=ëóëíÉã=áÇÉåíáÑáÅ~íáçåÛI=~ë=ëé~íá~ä=é~ííÉêåë=áå=íÜÉ=ÖêçìåÇï~íÉê=ÜÉ~Çë=êÉîÉ~ä=ãìÅÜ=çÑ=íÜÉ=Ä~ëáÅ=ëíêìÅíìêÉ=~åÇ=ÑìåÅíáçåáåÖ=çÑ=~=ëóëíÉã=EÉKÖKI=ÑáÖìêÉ=TKQÄFK=^ë=ÉåîáêçåãÉåí~ä=ëóëíÉãë=~êÉ=ìëì~ääó=ÜÉíÉêçÖÉåÉçìë=~åÇ=Ü~îÉ=ä~óÉêë=ïáíÜ=î~êá~ÄäÉ=êÉëáëí~åÅÉI=ãçåáíçêÉÇ=äçÅ~íáçåë=~í=ÇáÑÑÉêÉåí=ÇÉéíÜë=Å~å=ÄÉ=ãçêÉ=çê=äÉëë=áëçä~íÉÇ=Ñêçã=É~ÅÜ=çíÜÉê=~åÇ=Çáëéä~ó=ÇáÑÑÉêÉåí=ÄÉÜ~îáçêK=qÜÉ=ÇÉéíÜ=çÑ=ïÉää=ëÅêÉÉåë=~åÇ=ãçåáíçêÉÇ=äçÅ~íáçåë=áë=íÜÉêÉÑçêÉ=áãéçêí~åí=íç=ìåÇÉêëí~åÇ=íÜÉ=çÄëÉêîÉÇ=ÄÉÜ~îáçêI=~åÇ=Oa=çê=éêÉÑÉê~Ääó=Pa=îáëì~äáò~íáçå=áë=åÉÉÇÉÇ=íç=áåíÉêéêÉí=Ç~í~=~åÇ=ãçÇÉäáåÖ=êÉëìäíë=ÅçêêÉÅíäóK=6121122Distance on transect (m)figure 7.4: Examples of visualization of series of groundwater level observations and theircharacteristics (a) regime curves (b) in a 2D cross-section.22=7.3.4 Main screen and modeling toolsqÜÉ=íáãÉ=ëÉêáÉë=ãçÇÉäáåÖ=íççäë=~åÇ=ãÉíÜçÇë=ÇáëÅìëëÉÇ=áå=íÜÉ=éêÉîáçìë=ëÉÅíáçå=~êÉ=~í=íÜÉ=íÜáêÇ=äÉîÉä=çÑ=íÜÉ=afht=ÜáÉê~êÅÜóI=~åÇ=Ñçêã=íÜÉ=ÜÉ~êí=çÑ=jÉåó~åíÜÉëK=qÜÉ=ëÉíJìé=çÑ=íÜÉ=ã~áå=ëÅêÉÉå=EÑáÖìêÉ=TKRF=Ñçääçïë=íÜÉ=ëÅÜÉã~íáò~íáçå=çÑ=~å=çéÉå=ëóëíÉã=~ë=ÇÉéáÅíÉÇ=áå=ÑáÖìêÉ=TKNI=ïÜáÅÜ=~ÇÇë=íç=íÜÉ=áåíìáíáîÉåÉëë=çÑ=íÜÉ=ìëÉê=áåíÉêÑ~ÅÉK=qÜÉ=ìééÉê=êáÖÜí=èì~Çê~åí=áë=ÇÉîçíÉÇ=íç=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉë=EíÜÉ=çìíéìíFI=íÜÉ=äçïÉê=êáÖÜí=èì~Çê~åí=íç=Éñéä~å~íçêó=ëÉêáÉë=EíÜÉ=áåéìíF=~åÇ=íÜÉ=äçïÉê=äÉÑí=èì~Çê~åí=Ñçê=ãçÇÉä=éêçéÉêíáÉë=~åÇ=êÉëìäíë=EíÜÉ=ëóëíÉãFK=qÜÉ=ÑçìêíÜ=~åÇ=Ñáå~ä=èì~Çê~åí=áå=íÜÉ=ìééÉê=äÉÑíJÜ~åÇ=ÅçêåÉê=Åçåí~áåë=~=ã~éI=áå=ïÜáÅÜ=ëé~íá~ä=ëÉäÉÅíáçåë=Å~å=ÄÉ=éÉêÑçêãÉÇ=~åÇ=Ç~í~=~êÉ=éêÉëÉåíÉÇ=áå=íçé=çê=ÅêçëëJëÉÅíáçå=îáÉï=EÑáÖìêÉ=TKQÄFK=^=ëÉé~ê~íÉ=ëÅêÉÉå=áë=ìëÉÇ=Ñçê=Pa=îáëì~äáò~íáçåK=få=éêáåÅáéäÉI=íÜÉ=ë~ãÉ=îáëì~äáò~íáçå=íççäë=~êÉ=ìëÉÇ=Ñçê=íÜÉ=Ç~í~=íÜÉãëÉäîÉë=~åÇ=Ñçê=ÅÜÉÅâáåÖ=~åÇ=áåíÉêéêÉíáåÖ=íáãÉ=ëÉêáÉë=ãçÇÉä=êÉëìäíëK=b~ÅÜ=íáãÉ=ëÉêáÉë=áë=ãçÇÉäÉÇ=áåÇÉéÉåÇÉåíäó=çÑ=íÜÉ=çíÜÉê=ëÉêáÉëK=pé~íá~ä=é~ííÉêåë=íÜ~í=ÉãÉêÖÉ=Ñêçã=íÜÉ=êÉëìäíë=çÑ=ãìäíáéäÉ=íáãÉ=ëÉêáÉë=íÜÉêÉÑçêÉ=óáÉäÇ=î~äì~ÄäÉ=~åÇ=áåÇÉéÉåÇÉåí=áåÑçêã~íáçå=çå=íÜÉ=ëé~íá~ä=ëíêìÅíìêÉ=~åÇ=ÑìåÅíáçåáåÖ=çÑ=~=ëóëíÉãI=~åÇ=Ñçê=ÅÜÉÅâáåÖ=íÜÉ=éä~ìëáÄáäáíó=çÑ=êÉëìäíë=xsçå=^ëãìíÜ=Éí=~äKI=OMMUzK=jÉåó~åíÜÉë=çÑÑÉêë=íÜÉ=éçëëáÄáäáíó=íç=ìëÉ=ÄçíÜ=mfocf`q=~åÇ=ôjÊNI= ns F=íóéÉ=êÉëéçåëÉ=ÑìåÅíáçåë=Ñçê=íáãÉ=ëÉêáÉë=~å~äóëáëK=qÜáë=~ääçïë=ÄçíÜ=ãÉíÜçÇë=íç=ÄÉ=Åçãé~êÉÇ=ÇáêÉÅíäóI=~ë=ï~ë=éìÄäáëÜÉÇ=áå=xsçå=^ëãìíÜ=Éí=~äKI=OMMOzK=fí=Å~å=~äëç=ëÉêîÉ=íç=~ëëÉë=íÜÉ=î~äáÇáíó=çÑ=íÜÉ=Åçåíáåìçìë=êÉëéçåëÉ=ÑìåÅíáçå=ìëÉÇK=få=~ÇÇáíáçåI=jÉåó~åíÜÉë=~äëç=Åçåí~áåë=ãÉíÜçÇë=çÑ=åçåJäáåÉ~ê=íáãÉ=ëÉêáÉë=ãçÇÉäáåÖI=ÑçääçïáåÖ=íÜÉ=íÜêÉëÜçäÇ=çê=qôpl=~ééêç~ÅÜ=xhåçííÉêë=~åÇ=aÉ=dççáàÉêI=NVVVzK=qÜÉëÉ=ãÉíÜçÇë=~êÉ=ìëÉÑìä=Ñçê=ãçÇÉäáåÖ=ëóëíÉãë=ïáíÜ=éÉêáçÇáÅ=êìåJçÑÑ=çê=éÉêáçÇáÅ~ääó=ìåÅçîÉêáåÖ=Çê~áå~ÖÉ=ãÉ~åëK=====NSN=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=T=NSO=figure 7.5: Main screen of Menyanthes, consisting of four quadrants. The set-up and coloring (dotmarkers) reflect the system schematization and colors in figure 7.1.7.3.5 Simultaneous time series analysislåÉ=çÑ=íÜÉ=ÑçêÉãçëí=~Çî~åí~ÖÉë=çÑ=íÜÉ=mfocf`q=ãÉíÜçÇ=áë=íÜ~í=áí=ÖêÉ~íäó=êÉÇìÅÉë=íÜÉ=~ãçìåí=çÑ=ÉÑÑçêí=åÉÉÇÉÇ=Ñçê=ãçÇÉä=áÇÉåíáÑáÅ~íáçåK=^ë=ëí~íÉÇ=áå=íÜÉ=áåíêçÇìÅíáçåI=íÜÉ=áíÉê~íáîÉ=_çñJgÉåâáåë=ëíóäÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉÇìêÉ=Ñçê=ÇÉÑáåáåÖ=íÜÉ=åìãÄÉê=çÑ=é~ê~ãÉíÉêë=áå=~å=ôj^=ãçÇÉä=Å~å=ÄÉ=îÉêó=âåçïäÉÇÖÉ=~åÇ=ä~Äçê=áåíÉåëáîÉ=xaÉ=dççáàÉê=Éí=~äKI=NVURzI=ïÜÉêÉ~ë=ÑìåÅíáçåë=äáâÉ=íÜÉ=ëÅ~äÉÇ=Ö~ãã~=ÇáëíêáÄìíáçå=~êÉ=îÉêó=ÑäÉñáÄäÉ=~åÇ=ã~ó=ÄÉ=ìëÉÇ=ÉÑÑÉÅíáîÉäó=íç=ãçÇÉä=íÜÉ=êÉëéçåëÉ=çÑ=~=ëóëíÉã=êÉÖ~êÇäÉëë=çÑ=íÜÉ=ÜóÇêçÖÉçäçÖáÅ=ëÉííáåÖ=çÑ=íÜÉ=çÄëÉêî~íáçå=ïÉääë=xsçå=^ëãìíÜ=Éí=~äKI=OMMOzK=qÜÉ=mfocf`q=ãÉíÜçÇ=~ääçïë=Ñçê=íÜÉ=ìëÉ=çÑ=~=ëí~åÇ~êÇ=êÉëéçåëÉ=ÑìåÅíáçå=Ñçê=~=ëéÉÅáÑáÅ=íóéÉ=çÑ=áåéìíI=~åÇ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=ÅçåëÉèìÉåíäó=êÉÇìÅÉë=íç=íÜÉ=éêçÄäÉã=çÑ=áÇÉåíáÑóáåÖ=ïÜáÅÜ=áåéìíë=çê=ëíêÉëëÉë=áåÑäìÉåÅÉ=íÜÉ=Çóå~ãáÅë=çÑ=~=ëóëíÉã=xsçå=^ëãìíÜ=Éí=~äKI=OMMUzK=få=Å~ëÉ=çÑ=ÖêçìåÇï~íÉê=ÜÉ~ÇëI=çê=~åó=ÉåîáêçåãÉåí~ä=î~êá~ÄäÉ=Ñçê=íÜ~í=ã~ííÉêI=ëíêÉëëÉë=íÉåÇ=íç=Ü~îÉ=~=ëé~íá~ä=ÉÑÑÉÅí=~åÇ=áåÑäìÉåÅÉ=~ää=äçÅ~íáçåë=ïáíÜáå=~=ÅÉêí~áå=êÉÖáçåK=^ëëÉëëãÉåí=çÑ=íÜÉ=êÉÖáçå=çÑ=áåÑäìÉåÅÉ=çÑ=~=ÅÉêí~áå=ëíêÉëë=EÑçê=áåëí~åÅÉ=ÖêçìåÇï~íÉê=Éñíê~ÅíáçåF=áë=áå=Ñ~Åí=çåÉ=çÑ=íÜÉ=éçëëáÄäÉ=~ééäáÅ~íáçåë=çÑ=íáãÉ=ëÉêáÉë=ãçÇÉäëK=rëáåÖ=íÜÉ=mfocf`q=ãÉíÜçÇI=ëíêÉëë=áÇÉåíáÑáÅ~íáçå=íÜÉêÉÑçêÉ=ÖÉåÉê~ääó=~ééäáÉë=íç=~ää=çÄëÉêî~íáçå=..…………………………………………………………………………………………….….

jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=ïÉääë=áå=~å=~êÉ~I=ã~âáåÖ=áí=~=Ä~íÅÜ=éêçÅÉëëK=få=ÑáÖìêÉ=TKSI=íÜÉ=ìëÉê=áåíÉêÑ~ÅÉ=çÑ=jÉåó~åíÜÉë=áë=ëÜçïå=ïÜáÅÜ=ëìééçêíë=íÜÉ=éêçÅÉëë=çÑ=ëáãìäí~åÉçìë=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=Ñçê=ãìäíáéäÉ=íáãÉ=ëÉêáÉëK=få=íÜÉ=Äçñ=ã~êâÉÇ=ïáíÜ=~=çåÉI=íÜÉêÉ=áë=~=äáëíÄçñ=ïáíÜ=äçÅ~íáçåë=çê=ÖêçìåÇï~íÉê=äÉîÉä=ëÉêáÉëK=^ää=ëÉííáåÖë=ã~ÇÉ=áå=íÜÉ=ìëÉê=áåíÉêÑ~ÅÉ=~ééäó=íç=íÜÉ=ëÉêáÉë=ëÉäÉÅíÉÇ=áå=íÜáë=äáëíÄçñK=få=íÜÉ=Äçñ=ã~êâÉÇ=ïáíÜ=~=íïçI=íÜÉêÉ=áë=~=äáëíÄçñ=ïáíÜ=ëíêÉëëÉë=çê=Éñéä~å~íçêó=ëÉêáÉëK=jÉåó~åíÜÉë=~ìíçã~íáÅ~ääó=ëÉäÉÅíë=íÜÉ=åÉ~êÉëí=ãÉíÉçêçäçÖáÅ=ëí~íáçåëI=~ë=éêÉÅáéáí~íáçå=~åÇ=Éî~éçê~íáçå=~äï~óë=áåÑäìÉåÅÉ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉäK=qÜÉ=Éñéä~å~íçêó=ëÉêáÉë=ëÜçïå=áå=äáëíÄçñ=íïç=~êÉ=åçí=åÉÅÉëë~êáäó=ìëÉÇ=Ñçê=~ää=äçÅ~íáçåë=áå=äáëíÄçñ=çåÉI=Äìí=áå=Åçåíê~ëíI=áí=áë=~=äáëí=çÑ=Éñéä~å~íçêó=ëÉêáÉë=ìëÉÇ=Ñçê=~åó=çÑ=íÜÉãK=mêçéÉêíáÉë=çÑ=Éñéä~å~íçêó=ëÉêáÉë=íÜ~í=~êÉ=ëÉíI=~êÉ=figure 7.6: User interface for ‘identifying’ multiple timeseries models simultaneously.ëÉí=Ñçê=~ää=äçÅ~íáçåë=ëÉäÉÅíÉÇ=áå=äáëíÄçñ=çåÉ=Ñçê=ïÜáÅÜ=íÜÉó=~êÉ=ìëÉÇK=^å=Éñ~ãéäÉ=áë=íÜÉ=íóéÉ=çÑ=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçå=ìëÉÇ=Ñçê=~=ÅÉêí~áå=ëíêÉëë=xëÉÉ=sçå=^ëãìíÜ=Éí=~äKI=OMMUzI=ïÜáÅÜ=Å~å=ÄÉ=ëÉäÉÅíÉÇ=áå=íÜÉ=ÇêçéJÇçïå=ãÉåì=íÜ~í=áë=ã~êâÉÇ=ïáíÜ=~=íÜêÉÉK=tÜÉå=~ééêçéêá~íÉ=ëÉííáåÖëI=Éñéä~å~íçêó=ëÉêáÉë=~åÇ=êÉëéçåëÉ=ÑìåÅíáçåë=~êÉ=ÅÜçëÉåI=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêë=Å~å=ÄÉ=çéíáãáòÉÇ=Äó=éêÉëëáåÖ=íÜÉ=ÚÉëíáã~íÉÛ=ÄìííçåI=~ÑíÉê=ïÜáÅÜ=íÜÉ=ãçÇÉäë=~êÉ=çéíáãáòÉÇ=çåÉJÄóJçåÉK=aÉí~áäë=çå=íÜÉ=çéíáãáò~íáçå=éêçÅÉÇìêÉ=~åÇ=åçáëÉ=ãçÇÉäI=ïÜáÅÜ=~êÉ=~äëç=Ä~ëÉÇ=çå=~=ÅçåíáåìçìëJíáãÉ=Ñçêãìä~íáçåI=~êÉ=éêÉëÉåíÉÇ=áå=xsçå=^ëãìíÜ=~åÇ=_áÉêâÉåëI=OMMRzK=qÜÉ=ãçÇÉä=áÇÉåíáÑáÅ~íáçå=éêçÅÉÇìêÉ=çÑ=_çñ=~åÇ=gÉåâáåë=áåÅäìÇÉë=Çá~ÖåçëíáÅ=ÅÜÉÅâáåÖ=çÑ=êÉëìäíëI=ìëáåÖ=ëí~íáëíáÅ~ä=ãÉíÜçÇë=~åÇ=ÅêáíÉêá~K=`çêêÉä~íáçåë=íÜ~í=~êÉ=î~äáÇ=Ñêçã=~=ëí~íáëíáÅ~ä=îáÉïéçáåí=Å~åI=çÑ=ÅçìêëÉI=ÄÉ=åçåJÅ~ìë~ä=çê=éÜóëáÅ~ääó=åçåJéä~ìëáÄäÉK=qÜÉ=Éñ~ãéäÉ=~ééäáÅ~íáçå=ïáää=ÇÉãçåëíê~íÉ=Üçï=éÜóëáÅ~ä=áåëáÖÜí=ã~ó=~äëç=ÄÉ=çÑ=ìëÉ=Ñçê=ÅÜÉÅâáåÖ=íÜÉ=éä~ìëáÄáäáíó=~åÇ=~ÇÉèì~Åó=çÑ=íáãÉ=ëÉêáÉë=~å~äóëáë=êÉëìäíëK====NSP=7.3.6 Menyanthes versus alternative softwareqÜÉ=ÅçãÄáå~íáçå=çÑ=íáãÉ=ëÉêáÉë=ãçÇÉäáåÖ=íççäë=~åÇ=íÜÉ=ÑçÅìë=çå=ÖÉçÜóÇêçäçÖáÅ=ëóëíÉãë=~åÇ=íÜÉáê=ÅçêêÉëéçåÇáåÖ=Ç~í~=ã~âÉ=jÉåó~åíÜÉë=ìåáèìÉ=áå=áíë=ëçêíI=~åÇ=Ü~êÇ=íç=Åçãé~êÉ=íç=çíÜÉê=ëçÑíï~êÉK=qÜÉêÉ=áëI=çÑ=ÅçìêëÉI=ëçÑíï~êÉ=íÜ~í=çîÉêä~éë=~åÇ=áë=Åçãé~ê~ÄäÉ=íç=é~êí=çÑ=íÜÉ=ÑìåÅíáçå~äáíó=çÑ=jÉåó~åíÜÉëK=eóÇêçÖÉçäçÖáÅ=ãçÇÉäáåÖ=ëçÑíï~êÉ=ëìÅÜ=~ë=sáëì~ä=jçÇÑäçï=EëÉÉ=ïïïKëïëíÉÅÜåçäçÖóKÅçãFI=dêçìåÇï~íÉê=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=T=jçÇÉäáåÖ=póëíÉã=~åÇ=dêçìåÇï~íÉê=sáëí~ë=EëÉÉ=ïïïKëÅáëçÑíï~êÉKÅçãF=ëçäîÉ=íÜÉ=ÇáÑÑÉêÉåíá~ä=Éèì~íáçå=Ñçê=ÖêçìåÇï~íÉê=ÑäçïI=ìëáåÖ=íÜÉ=ÑáåáíÉ=ÇáÑÑÉêÉåÅÉ=ãÉíÜçÇK=pçÑíï~êÉ=äáâÉ=ji^bj=EëÉÉ=ïïïKëÅáëçÑíï~êÉKÅçãF=çê=qqáã=EëÉÉ=ííáãKÖççÖäÉÅçÇÉKÅçãF=ìëÉ=íÜÉ=~å~äóíáÅ=ÉäÉãÉåí=ãÉíÜçÇ=Ñçê=íÜáëK=pÉÉI=ÉKÖKI=xoÉÖÖá~åá=~åÇ=pÅÜÉääÉâÉåëI=OMMRX=qçÇáåáI=OMMTz=Ñçê=ãçêÉ=áåÑçêã~íáçå=çå=ãçÇÉäë=íÜ~í=E~äëçF=ÇÉ~ä=ïáíÜ=ëìêÑ~ÅÉ=ï~íÉê=ÜóÇêçäçÖóK=jÉåó~åíÜÉë=ÇáÑÑÉêë=Ñêçã=ëìÅÜ=ãçÇÉäë=áå=íÜÉ=ëÉåëÉ=íÜ~í=áí=çåäó=ãçÇÉäë=íÜÉ=íÉãéçê~ä=~åÇ=åçí=íÜÉ=ëé~íá~ä=Çóå~ãáÅëI=~åÇ=áí=ãçÇÉäë=ÖêçìåÇï~íÉê=ÜÉ~Çë=áåëíÉ~Ç=çÑ=ÑäçïK=cçê=~ééäáÅ~íáçå=áå=éê~ÅíáÅÉI=~å=áãéçêí~åí=éêçéÉêíó=çÑ=jÉåó~åíÜÉë=áë=íÜ~í=áí=ÇçÉë=åçí=êÉèìáêÉ=~åó=ÖÉçÜóÇêçäçÖáÅ=ëÅÜÉã~íáò~íáçå=çê=é~ê~ãÉíÉêáò~íáçåK=qÜÉêÉ=~êÉ=çÑ=ÅçìêëÉ=~äëç=çíÜÉê=íáãÉ=ëÉêáÉë=ãçÇÉäáåÖ=é~Åâ~ÖÉë=äáâÉ=j~íä~ÄÛë=póëíÉã=fÇÉåíáÑáÅ~íáçå=qççäÄçñ=xiàìåÖI=OMNNzI=íÜÉ=`~éí~áå=qççäÄçñ=xq~óäçê=Éí=~äKI=OMMTzI=çê=o=x`êóÉê=~åÇ=`Ü~åI=OMMUzK=`çãé~êÉÇ=íç=íÜÉëÉI=íÜÉ=ìëÉê=áåíÉêÑ~ÅÉ=çÑ=jÉåó~åíÜÉë=áë=ãçêÉ=ÚÑêáÉåÇäóÛ=íç=åçåJëéÉÅá~äáëí=ÜóÇêçäÖÉçäçÖáëíëI=~ë=áíë=éêáã~êó=ÑçÅìë=áë=çå=íÜÉ=ÖêçìåÇï~íÉê=ëóëíÉã=áíëÉäÑI=~åÇ=åçí=çå=íÜÉ=ëí~íáëíáÅ~ä=~ëéÉÅíë=~åÇ=éêçéÉêíáÉë=çÑ=íÜÉ=ãçÇÉäë=~åÇ=Ç~í~K=^äëçI=jÉåó~åíÜÉë=ìåáèìÉäó=Åçåí~áåë=íÜÉ=mfocf`q=ãÉíÜçÇI=ïÜáÅÜ=Éå~ÄäÉë=íÜÉ=Ñ~ëí=~åÇ=É~ëó=~å~äóëáë=çÑ=ä~êÖÉ=åìãÄÉêë=çÑ=íáãÉ=ëÉêáÉë=ëáãìäí~åÉçìëäóK===7.4 Example applicationNSQ=7.4.1 Time series decompositionpí~åÇ~êÇ=íáãÉ=ëÉêáÉë=ãçÇÉäë=~êÉ=äáåÉ~êI=ïÜáÅÜ=ã~ó=éçëÉ=äáãáí~íáçåë=íç=íÜÉáê=ìë~ÄáäáíóK=iáåÉ~êáíóI=ÜçïÉîÉêI=~äëç=Ü~ë=~=åìãÄÉê=çÑ=~Çî~åí~ÖÉë=~åÇ=ã~åó=~ééäáÅ~íáçåë=~êÉ=Ä~ëÉÇ=çå=áíK=tÜÉêÉ~ë=Éèì~íáçå=EOKPRF=çåäó=~ééäáÉë=íç=íÜÉ=ëáåÖäÉ=áåéìí=Å~ëÉI=áí=Å~å=ÄÉ=êÉ~Çáäó=ÉñíÉåÇÉÇ=íç=ãìäíáéäÉ=áåéìíë=ìëáåÖ=ëìéÉêéçëáíáçåI=ïÜáÅÜ=áë=î~äáÇ=Ñçê=äáåÉ~ê=ëóëíÉãëK=få=Å~ëÉ=çÑ= n =áåéìíëI=íÜÉáê=ÅçãÄáåÉÇ=ÉÑÑÉÅí=áë=çÄí~áåÉÇ=ëáãéäó=Äó=ëìããáåÖ=íÜÉ=áåÇáîáÇì~ä=ÉÑÑÉÅíëW==nh( t) − d = ∑ ( θi∗ pi)( t)(7.10)i=1=^ë=ëí~íÉÇ=éêÉîáçìëäóI=íÜÉ=iθ =Ûë=áå=ETKNMF=~êÉ=åçí=åÉÅÉëë~êáäó=áåÇÉéÉåÇÉåíK=m~ê~ãÉíÉêë=Å~å=ÄÉ=ëÜ~êÉÇ=~Åêçëë=ëÉîÉê~ä=áåéìíë=áå=çêÇÉê=íç=êÉÇìÅÉ=íÜÉáê=åìãÄÉêK=^äëçI=Ñçê=ÇáÑÑÉêÉåí=íóéÉë=çÑ=áåéìíëI=ÇáÑÑÉêÉåí=êÉëéçåëÉ=ÑìåÅíáçåë=Å~å=ÄÉ=ìëÉÇ=xëÉÉ=sçå=^ëãìíÜ=Éí=~äKI=OMMUzK=^=ÅçåëÉèìÉåÅÉ=çÑ=ETKNMF=áë=íÜ~í=íáãÉ=ëÉêáÉë=ãçÇÉäë=Å~å=ÇÉÅçãéçëÉ=íáãÉ=ëÉêáÉë=áåíç=é~êíá~ä=ëÉêáÉë=ëÜçïáåÖ=íÜÉ=ÉÑÑÉÅíë=çÑ=áåÇáîáÇì~ä=áåéìíë=EÑáÖìêÉ=TKTFK=qÜáë=éêçéÉêíó=Éå~ÄäÉë=áåíÉêîÉåíáçå=~å~äóëáë=~ë=ÇÉëÅêáÄÉÇ=Äó=xeáéÉä=Éí=~äKI=NVTRX=eáéÉä=~åÇ=jÅiÉçÇI=NVVQI=`ÜK=NVzK=qáãÉ=ëÉêáÉë=~êÉ=ÇÉÅçãéçëÉÇ=áå=~å=ÚÉñéä~áåÉÇÛ=~åÇ=~å=ÚìåÉñéä~áåÉÇÛ=é~êí=EÉîÉå=Ñçê=íÜÉ=ëáåÖäÉ=áåéìí=Å~ëÉFI=~ääçïáåÖ=Ñçê=íêÉåÇ=ÇÉíÉÅíáçå=çê=íÜÉ=èì~åíáÑáÅ~íáçå=çÑ=~åíÜêçéçÖÉåáÅ=áåÑäìÉåÅÉë=ïÜÉå=íÜÉêÉ=áë=çåäó=çåÉ=å~íìê~ä=Ñ~Åíçê=ëìÅÜ=~ë=éêÉÅáéáí~íáçå=ëìêéäìë=xÉKÖKI=oçäÑI=NVUVX=dÉÜêÉäëI=NVVVX=váÜÇÉÖç=~åÇ=tÉÄÄI=OMNNzK=få=íÜÉ=Å~ëÉ=çÑ=ãìäíáéäÉ=áåéìíëI=íÜÉ=éêáã~êó=áåíÉêÉëí=áë=ìëì~ääó=íÜÉ=áãé~Åí=çÑ=çåÉ=çÑ=íÜÉëÉ=áåéìíëI=ëìÅÜ=~ë=éìãéáåÖ=xÉKÖKI=sçå=^ëãìíÜ=Éí=~äKI=OMMUX=e~êé=~åÇ=sÉëëÉäáåçîI=OMNNz=çê=ÜóÇêçäçÖáÅ=ãÉ~ëìêÉëK=jçêÉ=áå=ÖÉåÉê~äI=íáãÉ=ëÉêáÉë=~å~äóëáë=Å~å=êÉîÉ~ä=ãìÅÜ=çÑ=íÜÉ=..…………………………………………………………………………………………….….

jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=ÑìåÅíáçåáåÖ=çÑ=~=ëóëíÉã=~åÇ=çÑ=íÜÉ=áåíÉê~Åíáçåë=ÄÉíïÉÉå=áíë=é~êíë=xÉKÖKI=iÉÉ=~åÇ=iÉÉI=OMMMX=c~ÄÄêá=Éí=~äKI=OMNNzK=^å=~êÅÜÉíóéáÅ~ä=~ééäáÅ~íáçå=çÑ=íáãÉ=ëÉêáÉë=ãçÇÉäë=äáÉë=áå=íÜÉ=~êÉ~=çÑ=ÚÅçåíêçä=ÉåÖáåÉÉêáåÖÛI=ïÜÉêÉ=íÜÉ=éêçÄäÉã=áë=Üçï=íç=ã~åáéìä~íÉ=íÜÉ=áåéìíë=áåíç=~=ëóëíÉã=íç=çÄí~áå=íÜÉ=ÇÉëáêÉÇ=çìíéìíK=cçê=ÜóÇêçÖÉçäçÖáÅ=éìêéçëÉëI=íÜáë=ãÉ~åë=íÜ~í=íáãÉ=ëÉêáÉë=ãçÇÉäë=Å~å=ÄÉ=ìëÉÇ=íç=çéíáãáòÉ=Eçê=áÇÉåíáÑó=éêçÄäÉãë=ïáíÜF=íÜÉ=ï~íÉê=ã~å~ÖÉãÉåí=áå=~å=~êÉ~=xÉKÖKI=_áÇïÉää=Éí=~äKI=NVVNX=iÉÉ=Éí=~äKI=OMMRzK==7.4.2 Impact of hydrologic measures in a Dutch dune area^ë=~å=Éñ~ãéäÉI=~å=~ééäáÅ~íáçå=çÑ=jÉåó~åíÜÉë=áë=éêÉëÉåíÉÇ=áå=ïÜáÅÜ=íÜÉ=éêáã~êó=~áã=áë=íç=~ëëÉëë=íÜÉ=áãé~Åí=çÑ=ÜóÇêçäçÖáÅ=ãÉ~ëìêÉëK=líÜÉê=~ééäáÅ~íáçå=Éñ~ãéäÉë=áåÅäìÇÉ=íÜÉ=ÅÜ~ê~ÅíÉêáò~íáçå=çÑ=ÖêçìåÇï~íÉê=Çóå~ãáÅë=xsçå=^ëãìíÜ=~åÇ=håçííÉêëI=OMMQzI=íÜÉ=~ëëÉëëãÉåí=çÑ=íÜÉ=áåÑäìÉåÅÉ=çÑ=éìãéáåÖ=xsçå=^ëãìíÜ=Éí=~äKI=OMMUzI=íÜÉ=ÇÉîÉäçéãÉåí=çÑ=ã~éë=çÑ=íÜÉ=êáëâ=çÑ=ÉñíêÉãÉ=äÉîÉäë=xj~åòáçåÉ=Éí=~äKI=OMNMzI=íÜÉ=Éî~äì~íáçå=çÑ=ÉÑÑÉÅíë=çÑ=Åäáã~íáÅ=ÅÜ~åÖÉë=çå=ëí~Öå~åí=ëìêÑ~ÅÉ=ï~íÉê=xiÉÜëíÉå=Éí=~äKI=OMNNz=~åÇ=íêÉåÇ=ÇÉíÉÅíáçå=~åÇ=áãé~Åí=çÑ=ä~åÇ=ìëÉ=ÅÜ~åÖÉë=xváÜÇÉÖç=~åÇ=tÉÄÄI=OMNNzK=qÜÉ=~êÉ~=ìåÇÉê=ÅçåëáÇÉê~íáçå=ÜÉêÉ=áë=é~êí=çÑ=íÜÉ=^ãëíÉêÇ~ã=t~íÉê=pìééäó=ÇìåÉëI=ïÜáÅÜ=áë=íÜÉ=~êÉ~=íÜ~í=Ü~ë=ëìééäáÉÇ=íÜÉ=Åáíó=çÑ=^ãëíÉêÇ~ã=ïáíÜ=ï~íÉê=ëáåÅÉ=íÜÉ=åáåÉíÉÉåíÜ=ÅÉåíìêóK=qÜÉ=~êÉ~=Åçåí~áåë=Ñ~ÅáäáíáÉë=Ñçê=ëíçêáåÖ=~åÇ=áåÑáäíê~íáåÖ=ëìêÑ~ÅÉ=ï~íÉê=EÑáÖìêÉ=TKUFI=~åÇ=éìãéáåÖ=ïÉääë=Ñçê=ÖêçìåÇï~íÉê=Éñíê~ÅíáçåK=fí=áë=éÉêÜ~éë=íÜÉ=ãçëí=áåíÉåëáîÉäó=ãçåáíçêÉÇ=ÜóÇêçÖÉçäçÖáÅ=ëóëíÉã=ïçêäÇ=ïáÇÉK=qÜÉ=ÑçÅìë=áë=çå=íÜÉ=ëçìíÜÉêå=é~êíI=ïÜáÅÜ=áë=êÉä~íáîÉäó=ìåÇáëíìêÄÉÇ=Äó=íÜÉ=ï~íÉê=ëìééäó=~ÅíáîáíáÉëK=qïç=ã~áå=Å~å~äë=ïÉêÉ=ÇìÖ=áå====NSR=figure 7.7: Groundwater level series decomposed into partial series showing the effects of individualinputs. The figure shows results from one of the 438 locations in the example application.=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

`Ü~éíÉê=T=NSS=figure 7.8: Location of the 438 selected piezometer screens (red dots) and meteorologic stations (greendots). The Van Limburg Stirumkanaal (left) and Oosterkanaal (right) are indicated by a blue line,whereas a red arrow shows the location of the transects of figure 7.10.= íÜáë=é~êí=çÑ=íÜÉ=ÇìåÉ=~êÉ~I=~åÇ=Çê~áå=íÜÉ=ëóëíÉãK=qÜÉ=Å~å~ä=áå=íÜÉ=tÉëí=åÉ~ê=íÜÉ=ëÉ~=áë=Å~ääÉÇ=íÜÉ=Ús~å=iáãÄìêÖ=píáêìãâ~å~~äÛI=ïÜÉêÉ~ë=íÜÉ=Å~å~ä=áå=íÜÉ=b~ëí=åÉ~ê=íÜÉ=éçäÇÉê=~êÉ~=áë=~ééêçéêá~íÉäó=Å~ääÉÇ=íÜÉ=ÚlçëíÉêâ~å~~äÛ=çê=É~ëíÉêå=Å~å~äK=få=çêÇÉê=íç=áåÅêÉ~ëÉ=íÜÉ=å~íìê~ä=î~äìÉ=~åÇ=êÉëíçêÉ=íÜÉ=ÉÅçäçÖáÅ~ä=ÑìåÅíáçåáåÖ=çÑ=íÜÉ=~êÉ~I=íç=ê~áëÉ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉäë=~åÇ=áåÅêÉ~ëÉ=íÜÉ=~êÉ~=çÑ=EíÜêÉ~íÉåÉÇF=ïÉí=ÇìåÉ=ëä~Åâ=ÉÅçëóëíÉãëI=ÜóÇêçäçÖáÅ=ãÉ~ëìêÉë=ïÉêÉ=í~âÉå=áå=íÜÉ=éÉêáçÇ=Ñêçã=NVVR=íç=NVVSK=^ë=é~êí=çÑ=íÜÉëÉ=ãÉ~ëìêÉëI=íÜÉ=s~å=iáãÄìêÖ=píáêìãâ~å~~ä=ï~ë=Ä~ÅâÑáääÉÇ=ïáíÜ=ë~åÇ=~åÇ=íÜÉ=äÉîÉä=çÑ=íÜÉ=lçëíÉêâ~å~~ä=ï~ë=ê~áëÉÇ=Äó=RM=ÅÉåíáãÉíÉêëK=få=~ÇÇáíáçåI=íÜÉ=ÑÉï=éìãéáåÖ=ïÉääë=íÜ~í=ïÉêÉ=éêÉëÉåí=áå=íÜÉ=~êÉ~=ïÉêÉ=ÅäçëÉÇ=EéìãéáåÖ=ëí~íáçå=kççêÇïáàâ=~åÇ=~êíÉëá~å=ïÉääë=ÄçêÇÉêáåÖ=íÜÉ=lçëíÉêâ~å~~äFK=cáå~ääóI=~äëç=íÜÉ=äÉîÉä=áå=íÜÉ=ëíçê~ÖÉ=Å~å~äë=ï~ë=ê~áëÉÇK=få=ëéêáåÖíáãÉ=ÑçääçïáåÖ=íÜÉ=ÅçãéäÉíáçå=çÑ=íÜÉëÉ=ãÉ~ëìêÉë=áå=NVVTI=ÅçåëáÇÉê~ÄäÉ=Åêçé=Ç~ã~ÖÉ=ï~ë=êÉÅçêÇÉÇ=áå=íÜÉ=ÑäçïÉê=ÄìäÄ=ÑáÉäÇë=åÉáÖÜÄçêáåÖ=íÜÉ=ÇìåÉ=~êÉ~K=qÜÉ=Ç~ã~ÖÉ=ï~ë=Éëíáã~íÉÇ=íç=ÄÉ=~êçìåÇ=íÜêÉÉ=ãáääáçå=Ççää~êëI=~åÇ=~ë=íÜÉ=Å~ìëÉ=~ääÉÖÉÇäó=ï~ë=çñóÖÉå=ëÜçêí~ÖÉ=çê=ÚÇêçïåáåÖÛ=çÑ=íÜÉ=ÄìäÄëI=~=äáåâ=ï~ë=ã~ÇÉ=ïáíÜ=íÜÉ=ãÉ~ëìêÉë=í~âÉå=Äó=íÜÉ=ï~íÉê=ëìééäó=Åçãé~åóI=ïÜáÅÜ=ï~ë=Ää~ãÉÇ=Ñçê=íÜÉ=Ç~ã~ÖÉ=xÉKÖKI=läëíÜççêåI=OMMMzK=tÜáäÉ=éêÉîáçìë=~ííÉãéíë=íç=èì~åíáÑó=íÜÉ=ÉÑÑÉÅíë=çÑ=íÜÉëÉ=ãÉ~ëìêÉë=ìëáåÖ=ôfj^=ãçÇÉäë=Ñ~áäÉÇI=~=åÉï=~ííÉãéí=ï~ë=ã~ÇÉ=ïÜÉå=jÉåó~åíÜÉë=ÄÉÅ~ãÉ=~î~áä~ÄäÉK=^=ëìãã~êó=çÑ=íÜÉ=êÉëìäíë=çÑ=íÜáë=áåîÉëíáÖ~íáçå=~êÉ=éêÉëÉåíÉÇ=áå=íÜÉ=ÑçääçïáåÖI=ïÜáäÉ=êÉëíêáÅíáåÖ=çìêëÉäîÉë=íç=íÜÉ=ÉÑÑÉÅíë=çÑ=íÜÉ=ãÉ~ëìêÉë=çå=íÜÉ=ÖêçìåÇï~íÉê=ÜÉ~Çë=~í=íÜÉ=ãçåáíçêÉÇ=äçÅ~íáçåëK====..…………………………………………………………………………………………….….

jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=få=íÜÉ=~êÉ~=ïÜÉêÉ=íÜÉ=ãÉ~ëìêÉë=ïÉêÉ=í~âÉåI=SUM=äçÅ~íáçåë=EéáÉòçãÉíÉê=ëÅêÉÉåëF=~êÉ=ãçåáíçêÉÇ=Äó=íÜÉ=ï~íÉê=ëìééäó=Åçãé~åóK=mêÉÅáéáí~íáçå=Ç~í~=ï~ë=ÅçääÉÅíÉÇ=~í=ÑáîÉ=ÇáÑÑÉêÉåí=äçÅ~íáçåë=áå=~=íê~åëÉÅí=ÖçáåÖ=Ñêçã=íÜÉ=ëÉ~=íç=íÜÉ=éçäÇÉê=~êÉ~=EëÉÉ=ÑáÖìêÉ=TKUI=çåÉ=çÑ=ïÜáÅÜ=áë=ÚojPÛ=áå=ÑáÖìêÉ=TKTFK=oÉÑÉêÉåÅÉ=Éî~éçê~íáçå=Ç~í~=ï~ë=çÄí~áåÉÇ=Ñêçã=íÜÉ=ãÉíÉçêçäçÖáÅ=ëí~íáçå=ÚiÉáÇìáåÛ=çÑ=íÜÉ=ï~íÉê=ëìééäó=Åçãé~åó=EÚosMPÛ=áå=ÑáÖìêÉ=TKTFK=^é~êí=Ñêçã=íÜÉëÉ=~åÇ=íÜÉ=Ç~íÉë=çÑ=íÜÉ=ÜóÇêçäçÖáÅ=ãÉ~ëìêÉëI=åç=çíÜÉê=Ç~í~=ïÉêÉ=ìëÉÇK=eçïÉîÉêI=åçí=ÉîÉêó=~î~áä~ÄäÉ=íáãÉ=ëÉêáÉë=éêçîÉÇ=íç=ÄÉ=ìë~ÄäÉ=Ñçê=íÜÉ=~å~äóëáëK=cêçã=íÜÉëÉ=SUM=ëÅêÉÉåëI=ëÉêáÉë=ïÉêÉ=ëÉäÉÅíÉÇ=íÜ~í=Ü~Ç=ãÉ~ëìêÉãÉåíë=ÄçíÜ=ëçãÉ=íáãÉ=ÄÉÑçêÉ=~åÇ=~ÑíÉê=íÜÉ=ãÉ~ëìêÉëI=ëé~å=~=éÉêáçÇ=íÜ~í=áë=åçí=íçç=ëÜçêí=~åÇ=Åçåí~áå=~=ëìÑÑáÅáÉåí=åìãÄÉê=çÑ=ãÉ~ëìêÉãÉåíë=áå=íçí~äK=_~ëÉÇ=çå=íÜÉëÉ=ÅêáíÉêá~I=QPU=ëÅêÉÉåë=çÑ=NVQ=çÄëÉêî~íáçå=ïÉääë=ïÉêÉ=ëÉäÉÅíÉÇ=Ñçê=íÜÉ=~å~äóëáë=EÑáÖìêÉ=TKUFK=^ë=íÜÉ=~êÉ~=áë=êÉä~íáîÉäó=ìåÇáëíìêÄÉÇ=Äó=íÜÉ=ÅìêêÉåí=ï~íÉê=ëìééäó=~ÅíáîáíáÉëI=çåäó=íÜÉ=ÉÑÑÉÅíë=çÑ=å~íìê~ä=ëíêÉëëÉë=EéêÉÅáéáí~íáçå=~åÇ=Éî~éçê~íáçåF=~åÇ=íÜçëÉ=çÑ=íÜÉ=ãÉ~ëìêÉë=ïÉêÉ=í~âÉå=áåíç=~ÅÅçìåíK=b~ÅÜ=ãÉ~ëìêÉ=íÜÉêÉÄó=Ü~ë=~=ÇáÑÑÉêÉåí=ÉÑÑÉÅí=áå=ÇáÑÑÉêÉåí=é~êíë=çÑ=íÜÉ=ëóëíÉãI=Äìí=~ää=ãÉ~ëìêÉë=ïÉêÉ=í~âÉå=áå=~=êÉä~íáîÉäó=ëÜçêí=éÉêáçÇ=EÅçåëáÇÉêáåÖ=íÜÉ=äçåÖ=ãÉãçêó=çÑ=íÜÉ=ÇìåÉ=ëóëíÉãFK=qÜÉáê=ÉÑÑÉÅí=áë=íÜÉêÉÑçêÉ=ëíêçåÖäó=ÅçêêÉä~íÉÇI=~åÇ=áí=áë=åçí=éçëëáÄäÉ=íç=ëÉé~ê~íÉ=~åÇ=Éëíáã~íÉ=íÜÉ=áåÇáîáÇì~ä=ÉÑÑÉÅíë=ÇáêÉÅíäó=Ñêçã=íÜÉ=Ç~í~K=fí=áëI=ÜçïÉîÉêI=éçëëáÄäÉ=íç=Éëíáã~íÉ=íÜÉ=ÅçãÄáåÉÇ=ÉÑÑÉÅíë=çÑ=íÜÉ=ãÉ~ëìêÉëI=Äó=ãçÇÉäáåÖ=íÜÉã=~ë=~=ëÜáÑíÉÇI=ëáåÖäÉ=eÉ~îáëáÇÉ=ëíÉé=ÑìåÅíáçå= H ( t)çê=ëíÉé=íêÉåÇ=~ë=áåéìí=ëÉêáÉëK=qÜáë=ÑìåÅíáçå=ã~ó=ÄÉ=ÇÉÑáåÉÇ=ÄóW==tH ( t) = ∫ δ( τ − T)dτ(7.11)−∞=~åÇ=áí=ÅÜ~åÖÉë=áåëí~åí~åÉçìëäó=Ñêçã=òÉêç=íç=çåÉ=~í=íáãÉ=T I=Ñçê=ïÜáÅÜ=íÜÉ=OU íÜ =çÑ=cÉÄêì~êó=NVVR=áë=ÅÜçëÉå=ÜÉêÉ=EíÜÉ=Ç~íÉ=íÜ~í=íÜÉ=s~å=iáãÄìêÖ=píáêìãâ~å~~ä=ï~ë=Ä~ÅâÑáääÉÇFK=få=ïçêÇëI=~=ëíÉé=ÑìåÅíáçå=êÉéêÉëÉåíë=~=ëìÇÇÉå=~åÇ=ä~ëíáåÖ=ÅÜ~åÖÉ=áå=~å====NST=PrecipitationSurplusSystem(response)GroundwaterHeadStep ChangeEffectfigure 7.9: A step change in the input is transformed by the system into a sigmoid-shaped effect.When the change is not linked to one of the other input series, it has its own independent responsefunction.KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK=

`Ü~éíÉê=T=NSU=áåéìí=ëÉêáÉëK=qÜÉ=ÉÑÑÉÅí=çÑ=~=ëíÉé=ÑìåÅíáçå=çå=íÜÉ=çìíéìíI=ïÜáÅÜ=áë=Å~ääÉÇ=íÜÉ=ëíÉé=êÉëéçåëÉ=ÑìåÅíáçåI=áë=ÖÉåÉê~ääó=åçí=áåëí~åí~åÉçìë=Äìí=áí=Ü~ë=~=Öê~Çì~äI=ëáÖãçáÇJäáâÉ=ëÜ~éÉ=EÑáÖìêÉ=TKVFK=tÜÉå=íÜÉ=ëíÉé=ÅÜ~åÖÉ=áë=åçí=äáåâÉÇ=íç=çåÉ=çÑ=íÜÉ=çíÜÉê=áåéìí=ëÉêáÉë=E~ëI=ÉKÖKI=~=ëìÇÇÉå=ÅÜ~åÖÉ=áå=Éî~éçíê~åëéáê~íáçå=Äó=ÑçêÉëí=ÅäÉ~êáåÖ=ïçìäÇFI=áí=ëÜçìäÇ=ÄÉ=ÖáîÉå=áíë=çïå=áåÇÉéÉåÇÉåí=áãéìäëÉ=êÉëéçåëÉ=ÑìåÅíáçåI=Ñçê=ïÜáÅÜ=~=ëÅ~äÉÇ=Ö~ãã~=ÇáëíêáÄìíáçå=áë=ìëÉÇ=ÜÉêÉK=^å=Éñ~ãéäÉ=çÑ=êÉëìäíë=Ñçê=~=ëáåÖäÉ=ëÉêáÉë=ï~ë=~äêÉ~Çó=ëÜçïå=áå=ÑáÖìêÉ=TKTK=fí=ã~ó=ÄÉ=ëÉÉå=Ñêçã=íÜÉ=äçïÉê=Öê~éÜ=áå=íÜáë=ÑáÖìêÉ=íÜ~í=íÜÉ=êÉëéçåëÉ=çÑ=íÜÉ=ÇìåÉ=ëóëíÉã=áë=áåÇÉÉÇ=ëäçïI=~åÇ=íÜÉ=Éëíáã~íÉÇ=ÉÑÑÉÅí=çÑ=íÜÉ=ãÉ~ëìêÉë=Ü~ë=åçí=êÉ~ÅÜÉÇ=áíë=ëí~íáçå~êó=î~äìÉ=~ÑíÉê=U=óÉ~êëI=~í=íÜÉ=ÉåÇ=çÑ=íÜÉ=íáãÉ=ëÉêáÉëK=cçìê=çìíéìí=ëÅêÉÉåë=çÑ=jÉåó~åíÜÉë=~êÉ=ëÜçïå=áå=ÑáÖìêÉ=TKNMK=qÜÉó=Åçåí~áå=êÉëìäíë=Ñêçã=ãìäíáéäÉ=ëÉêáÉë=áå=Oa=ÅêçëëJëÉÅíáçåëK=få=jÉåó~åíÜÉëI=íÜÉ=ãçÇÉä=éÉêÑçêã~åÅÉ=áë=ÅÜ~ê~ÅíÉêáòÉÇ=áå=ëÉîÉê~ä=ï~óëI=çåÉ=çÑ=ïÜáÅÜ=áë=íÜÉ=Éñéä~áåÉÇ=î~êá~åÅÉ=éÉêÅÉåí~ÖÉ=EbsmFK=qÜÉ=bsm=áë=ÇÉÑáåÉÇ=~ëW==2 2σ h −σnEVP = *100%(7.12)2σ h=2h2nïÜÉêÉ= σ =~åÇ= σ ÇÉåçíÉ=íÜÉ=î~êá~åÅÉ=çÑ=íÜÉ=ÖêçìåÇï~íÉê=äÉîÉä=çÄëÉêî~íáçåë=~åÇ=Éêêçê=çê=êÉëáÇì~ä=ëÉêáÉë=êÉëéÉÅíáîÉäóK=qÜÉ=bsm=Ñçê=É~ÅÜ=ëÉêáÉë=áë=éäçííÉÇ=~ë=~=ÅçäçêÉÇ=Ççí=~í=íÜÉ=äçÅ~íáçå=çÑ=áíë=éáÉòçãÉíÉê=ëÅêÉÉå=áå=ÑáÖìêÉ=TKNM~K=qÜÉ=Éëíáã~íÉÇ=ÉÑÑÉÅí=çÑ=íÜÉ=ãÉ~ëìêÉëI=Ä~ëÉÇ=çå=~=ëÉäÉÅíáçå=çÑ=ëÅêÉÉåë=íÜ~í=~êÉ=éä~ÅÉÇ=ÜáÖÜÉê=íÜ~å=JNRãHêÉÑI=áë=éäçííÉÇ=áå=ÑáÖìêÉ=TKNMÄK=^ë=íÜÉ=ÉÑÑÉÅíë=~êÉ=ëíáää=åçåJëí~íáçå~êó=~åÇ=áåÅêÉ~ëáåÖI=íÜÉ=Éëíáã~íÉÇ=Ñáå~ä=ÉÑÑÉÅí=Å~å=çåäó=ÄÉ=ÑçìåÇ=íÜêçìÖÜ=Éñíê~éçä~íáçå=~åÇ=áë=íÜÉêÉÑçêÉ=áåÉñ~ÅíK=qÜÉ=Éëíáã~íÉÇ=ÉÑÑÉÅí=çå=PNJUJOMMPI=~í=íÜÉ=ÉåÇ=çÑ=íÜÉ=íáãÉ=ëÉêáÉëI=áë=ìëÉÇ=ÜÉêÉI=~äçåÖ=ïáíÜ=áíë=ÅçåÑáÇÉåÅÉ=áåíÉêî~ä=EHLJ= 2σ FK=kÉñí=íç=íÜ~íI=íÜÉ=î~äìÉë=çÑ=íÜÉ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉ=~åÇ=íÜÉ=Éëíáã~íÉÇ=Éî~éçê~íáçå=Ñ~Åíçê=~êÉ=ëÜçïå=áå=ÑáÖìêÉë=TKNMÅ=~åÇ=ÇI=êÉëéÉÅíáîÉäóK=få=~ää=ÑáÖìêÉë=TKNM~F=íç=TKNMÇFI=ëé~íá~ä=é~ííÉêåë=ÉãÉêÖÉ=Ñêçã=íÜÉ=áåÇáîáÇì~ä=êÉëìäíëK=qÜÉ=bsmÛëI=Ñçê=áåëí~åÅÉI=~êÉ=ÅäÉ~êäó=ÜáÖÜÉëí=áå=íÜÉ=ëÜ~ääçïÉê=ëÅêÉÉåë=E~åÇ=ê~åÖÉ=ÉîÉå=ìé=íç=VUKQBFI=ïÜáäÉ=íÜÉ=äçïÉëí=î~äìÉë=Å~å=ÄÉ=ÑçìåÇ=çå=íÜÉ=tÉëíÉêå=ëáÇÉ=åÉ~ê=íÜÉ=ëÉ~K=cêçã=íÜáë=ëáãéäÉ=é~ííÉêåI=ëÉîÉê~ä=ÅçåÅäìëáçåë=ã~ó=ÄÉ=Çê~ïåK=cáêëíI=íÜÉêÉ=ãìëí=ÄÉ=~=EëÉãáFÅçåÑáåáåÖ=ä~óÉê=~í=~=ÇÉéíÜ=çÑ=~Äçìí=JNM=íç=JOM=ãÉíÉêë=H=êÉÑ=EïÜáÅÜ=ÅçêêÉëéçåÇë=ïáíÜ=ëçáä=éêçÑáäÉ=Ç~í~FI=~ë=çíÜÉêïáëÉI=íÜÉ=äçïÉê=ëÅêÉÉåë=ïçìäÇ=åçí=ÄÉÜ~îÉ=ÇáÑÑÉêÉåíäó=íÜ~å=íÜÉ=ÜáÖÜÉê=ëÅêÉÉåë=K=cçê=íÜáë=êÉ~ëçåI=çåäó=íÜÉ=ëÜ~ääçïÉê=ëÅêÉÉåë=~êÉ=ÇÉéáÅíÉÇ=áå=ÑáÖìêÉ=TKNMÄFK=pÉÅçåÇI=äçï=bsmÛë=áåÇáÅ~íÉ=íÜ~í=íÜÉêÉ=~êÉ=Ñ~Åíçêë=ãáëëáåÖ=áå=íÜÉ=ãçÇÉäëI=~åÇ=íÜÉ=ëé~íá~ä=é~ííÉêå=ÅäÉ~êäó=éçáåíë=íç=íÜÉ=íáÇÉ=~ë=ÄÉáåÖ=çåÉ=çÑ=íÜÉãK=^=äçÖáÅ~ä=Å~åÇáÇ~íÉ=Ñçê=~=ëÉÅçåÇ=ãáëëáåÖ=Ñ~Åíçê=áë=íÜÉ=éìãéáåÖ=íÜ~í=áë=ÖçáåÖ=çå=áå=íÜÉ=kçêíÜÉêå=é~êí=çÑ=íÜÉ=~êÉ~K=qÜÉ=áåÑäìÉåÅÉ=çÑ=éìãéáåÖI=äáâÉ=íÜ~í=çÑ=íÜÉ=íáÇÉI=ëéêÉ~Çë=ãìÅÜ=Ñ~êíÜÉê=áå=~=ÅçåÑáåÉÇ=ä~óÉêI=Éñéä~áåáåÖ=ïÜó=íÜÉëÉ=Ñ~Åíçêë=~êÉ=åçí=åÉÅÉëë~êó=íç=ãçÇÉä=íÜÉ=Çóå~ãáÅë=áå=íÜÉ=ëÜ~ääçïÉêI=éÜêÉ~íáÅ=ëÅêÉÉåë=~ÇÉèì~íÉäóK=cêçã=íÜÉ=é~ííÉêå=íÜ~í=áë=îáëáÄäÉ=áå=íÜÉ=ÉÑÑÉÅíë=áå=ÑáÖìêÉ=TKNMÄFI=íÜÉ=áåÇáîáÇì~ä=ÉÑÑÉÅíë=çÑ=íÜÉ=Ä~ÅâÑáääáåÖ=çÑ=íÜÉ=s~å=iáãÄìêÖ=píáêìãâ~å~~ä=~åÇ=íÜÉ=êáëÉ=çÑ=íÜÉ=äÉîÉä=çÑ=íÜÉ=lçëíÉêâ~å~~ä=~êÉ=îáëáÄäÉK=qÜÉ=ÉÑÑÉÅí=çÑ=Ä~ÅâÑáääáåÖ=~=Å~å~ä=áë=äçÖáÅ~ääó=ÜáÖÜÉëí=åÉ~ê=íÜÉ=Å~å~ä=áíëÉäÑ=E~åÇ=ê~åÖÉë=ìé=íç=NãTQI=çå=PNJUJOMMPFK=páãáä~êäóI=íÜÉ=ÉÑÑÉÅíë=çÑ=~=êáëÉ=áå=Å~å~ä=ëí~ÖÉ=áë=ÜáÖÜÉëí=~í=íÜÉ=Å~å~äK=qÜÉ=ÉÑÑÉÅíë=Öê~Çì~ääó=Ñ~ÇÉ=áå=íÜÉ=ÇáêÉÅíáçå=çÑ=íÜÉ=ëÉ~=~åÇ=íÜÉ=éçäÇÉê=~êÉ~I=~åÇ=íçï~êÇë=íÜÉ=ãáÇÇäÉ=çÑ=íÜÉ=ÇìåÉ=~êÉ~K=qÜÉ=äçÅ~ä=Çê~áå~ÖÉ=Ä~ëÉ=ëÜçïë=~=Öê~ÇáÉåí=ÖçáåÖ=..…………………………………………………………………………………………….….

jÉåó~åíÜÉëW=ëçÑíï~êÉ=Ñçê=ÖêçìåÇï~íÉê=Ç~í~=figure 7.10: Model results along the transect displayed in figure 7.8, showing a) the explainedvariance percentages as colored dots at the locations of the piezometer screens. b) the estimated effectin screens higher than -15m+ref, on 31-8-2003. c) the local drainage base, d) the estimatedevaporation factor.= Ñêçã=íÜÉ=ëÉ~=íçï~êÇë=íÜÉ=E~ÅíáîÉäó=Çê~áåÉÇF=éçäÇÉê=~êÉ~K=kÉñí=íç=íÜ~íI=íÜÉ=éÜêÉ~íáÅ=ëÅêÉÉåë=áå=íÜÉ=ÅÉåíÉê=çÑ=íÜÉ=ÇìåÉ=ëóëíÉã=ÅäÉ~êäó=ëÜçï=~=ÜáÖÜÉê=Çê~áå~ÖÉ=Ä~ëÉK=qÜáë=ã~ó=ÄÉ=ÇìÉ=íç=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=éêÉÅáéáí~íáçå=ëìêéäìë=ÅêÉ~íÉë=~=ÚäÉåëÛ=çÑ=ÑêÉëÜ=ï~íÉê=íÜ~í=Ñäç~íë=çå=íçé=çÑ=íÜÉ=ë~äí=ï~íÉê=ÄÉåÉ~íÜK=qÜÉ=Éëíáã~íÉÇ=Éî~éçê~íáçå=Ñ~Åíçê=áë=ÅäÉ~êäó=äçïÉê=ÄÉåÉ~íÜ=íÜÉ=ÅçåÑáåáåÖ=ä~óÉêI=~åÇ=ÉîÉå=êÉ~ÅÜÉë=åÉÖ~íáîÉ=î~äìÉëK=^ë=íÜáë=áë=éÜóëáÅ~ääó=áãéä~ìëáÄäÉI=áí=ëìÖÖÉëíë=íÜ~í=íÜÉ=Éî~éçê~íáçå=Ñ~Åíçê=Éëíáã~íÉë=ÄÉäçï=íÜÉ=ÅçåÑáåáåÖ=ä~óÉê=~êÉ=Äá~ëÉÇI=éêçÄ~Ääó=ÇìÉ=íç=íÜÉ=Ñ~Åí=íÜ~í=íÜÉ=ÉÑÑÉÅíë=çÑ=éìãéáåÖ=~êÉ=åçí=í~âÉå=áåíç=~ÅÅçìåí=~åÇ=íÜÉ=ãçÇÉä=éÉêÑçêã~åÅÉ=áë=äçïK=^ë=ï~íÉê=ìë~ÖÉ=çÑíÉå=ëÜçïë=~=ëÉ~ëçå~ä=é~ííÉêåI=íÜÉ=ÉÑÑÉÅíë=çÑ=éìãéáåÖ=~êÉ=É~ëáäó=ÅçêêÉä~íÉÇ=ïáíÜ=íÜçëÉ=çÑ=Éî~éçê~íáçåK=få=~ää=ÑáÖìêÉëI=~äëç=ëÉîÉê~ä=ÚçìíäáÉêëÛ=~êÉ=îáëáÄäÉK=qÜÉëÉ=çìíäáÉêë=áå=é~êí=ÅçåÅÉêå=íÜÉ=ë~ãÉ=äçÅ~íáçåëI=~åÇ=íÜÉ=êÉëìäíë=~í=íÜÉëÉ=äçÅ~íáçåë=ëÜçìäÇ=ÄÉ=ÅÜÉÅâÉÇ=Ñçê=éä~ìëáÄáäáíóK=qÜÉ=êÉëìäíë=çÑ=íÜÉ=éÜêÉ~íáÅ=ëÅêÉÉåëI=ÜçïÉîÉêI=ïÜÉêÉ=íÜÉ=ãçÇÉä=éÉêÑçêã~åÅÉ=áë=ÜáÖÜI=~êÉ=ãìÅÜ=ãçêÉ=ÅçÜÉêÉåíK=^ÇÇáíáçå~ä=ÇÉí~áä=çå=íÜÉ=ãçÇÉäáåÖ=éêçÅÉÇìêÉI=íÜÉ=ãçÇÉä=çìíéìíI=áíë=áåíÉêéêÉí~íáçå=~åÇ=éä~ìëáÄáäáíó=áë=ÖáîÉå=áå=xsçå=^ëãìíÜ=Éí=~äKI=OMMUzK==kçíÉ=íÜ~í=íÜÉ=ÑáääáåÖ=áå=çÑ=~=Å~å~ä=çê=Çê~áå~ÖÉ=ãÉ~åë=Å~å=~äëç=ÄÉ=ëÉÉå=~ë=~=ÅÜ~åÖÉ=áå=ëóëíÉã=éêçéÉêíáÉëI=ã~âáåÖ=íÜÉ=ëóëíÉã=íáãÉJî~êá~åíK=få=éêáåÅáéäÉI=íáãÉJî~êá~åí=ëóëíÉãë====NSV=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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8pìãã~êó=~åÇ=ÅçåÅäìëáçåë==Chapter8 Summary and conclusions===Summary and conclusions===NTP=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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A= = = aÉêáî~íáçåë=AppendixDerivations===NVR=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

^ééÉåÇáñ=^==DerivationsNVS=A.1 Auto- and crosscorrelation functions of the residuals andinnovations=qÜÉ=~ìíçÅçêêÉä~íáçå=çÑ=íÜÉ=áååçî~íáçåë=Éèì~äë=EïáíÜ=íÛÛY=íÛY=íFW==tt '2 t ' 2n −α ( t−τ ) ασn −α ( t−τ)2ασ2E{ ν ( t) ν ( t ')} E{ e dW( τ ) e dW( τ )}=0ββ= ∫ ∫(A.1)=''ÇìÉ=íç=íÜÉ=éêçéÉêíáÉë=çÑ=íÜÉ=táÉåÉê=éêçÅÉëë=E E{dW( t)dW( t )}=0 if t ≠ t FK=rëáåÖ=ENNFI=íÜÉ=~ìíçÅçêêÉä~íáçå=çÑ=íÜÉ=êÉëáÇì~äë=Å~å=ÄÉ=ïêáííÉå=~ë=EÑçê=íÛY=íFW==t ''t 2−α ( t−t') 2ασn −α ( t−τ)E{ ñ ( t) ñ ( t ')} = E(e ) ñ ( t ') ñ ( t ')+ ñ ( t ') ∫ e }dW( τ )} (A.2)β=ïÜáÅÜ=ÖáîÉëW==E n t n tt−t'− ( − ') 2̃ ̃ (A.3){ ( ) ( ')} = e a t t σn=rëáåÖ=ENOFI=íÜÉ=ÅêçëëÅçêêÉä~íáçå=ÄÉíïÉÉå=êÉëáÇì~äë=~åÇ=áååçî~íáçåë=Éèì~äëI=Ñçê=~åó=íÛY=íW==−α∆tE{ ν ( t) ñ ( t ')} = E{ ñ ( t) ñ ( t ') − e ñ ( t − ∆t) ñ ( t ')}(A.4)=ïÜáÅÜ=ÖáîÉëI=ìëáåÖ=E^KPFW==−a( t−t ') 2 −α∆t −a( t−∆t −t') 2E{ ν ( t) ñ ( t ')} = e σn− e e σn= 0(A.5)=A.2 Relation between residual variance and individualinnovations=pí~êíáåÖ=ïáíÜ=EQKNUFI=ïÉ=Å~å=ïêáíÉ=íÜÉ=áååçî~íáçå=î~êá~åÅÉ=~ë=~å=ÉñéÉÅíÉÇ=î~äìÉ=~åÇ=ÖÉíW==2 12σn( Ψ) = ( )E{ ν ( t , )}2αtiΨ (A.6)− ∆ i1−e=..…………………………………………………………………………………………….….

= = = aÉêáî~íáçåë=2ìëáåÖ=ÉîÉêó= ν ( t i) =áåÇáîáÇì~ääóI= N =ëáåÖäÉ=ë~ãéäÉ=Éëíáã~íÉë=çÑ= ˆ σn( ti, Ψ)Å~å=ÄÉ=çÄí~áåÉÇ=ïáíÜW==2 1 2ˆ σn, t( Ψ) = ( ) ν ( t , )i2αt iΨ (A.7)− ∆ i1−e=åÉñíI=ïÉ=Å~å=ÖÉí=~=ãçêÉ=~ÅÅìê~íÉ=Éëíáã~íÉ=áÑ=ïÉ=~îÉê~ÖÉ=íÜÉ= N =Éëíáã~íÉë=çÑ=2ˆ σn( ti, Ψ)W==N1 2∑( ) ν ( t , )2iΨ− α∆ti2 i= 1 1−eˆ σn( Ψ ) =(A.8)N=A.3 SWSI (Sum of Weighted Squared Innovations) criterion=dáîÉå=íÜÉ=ÑçääçïáåÖ=äçÖ=äáâÉäáÜççÇ=ÑìåÅíáçåW==NN 22ν ( ti, Ψ)Λ{ Ψ | O} = −0.5N ln(2 π ) − 0.5∑ln{ σν( ∆ti, Ψ)} − 0.5∑(A.9)2i= 1 i=1 σν( ∆ti, Ψ)=ïÉ=Å~å=êÉéä~ÅÉ= σ2 ν ( ∆ t i, β ) =Äó= σ2 n( β ) ìëáåÖ=EQKNUF=~åÇ=ÖÉíW==NN 2−2α∆t2 ν ( t , )iiΨΛ{ Ψ | O} = −0.5Nln(2 π ) − 0.5∑ln{(1 − e ) σn} − 0.5∑(A.10)−2α∆ti2i= 1 i=1 (1 − e ) σn=2åÉñíI=ïÉ=ÖÉí=Äó=éä~ÅáåÖ= N / σ ( Ψ ) =áå=íÜÉ=ä~ëí=íÉêã=çìíëáÇÉ=íÜÉ=ëìãã~íáçå=ëáÖåW=nN−2α∆ti2{ | O} = −0.5Nln(2 π ) − 0.5∑ln{(1 − e ) σn} − ...i=1Λ ΨN1 2∑( ) ν ( t , )2iΨ− α∆tiN i=1 1−e0.52σ nN=~åÇ=êÉéä~ÅÉ=áå=ÄçíÜ=íÉêãëσ2 ( Ψ)Äó=E^KUF=íç=ÖÉí=W=nN1 2( , )N∑ ν t2j Ψ− α∆tj−2α∆tj=11−ei{ | } = −0.5 ln(2 π ) − 0.5 ln{(1 − ) } − 0.5∑(A.11)Λ Ψ O N eN (A.12)i=1N=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK===NVT=

^ééÉåÇáñ=^==NVU=få=íÜÉ=ä~ëí=íÉêã=ëÉîÉê~ä=áíÉãë=Ü~îÉ=íÜìë=ÄÉÉå=Éäáãáå~íÉÇK=_ÉÅ~ìëÉ=íÜÉ=ëìã=çÑ=ëÉîÉê~ä=äçÖ~êáíÜãë=áë=íÜÉ=äçÖ~êáíÜã=çÑ=íÜÉáê=éêçÇìÅíI=E^KNOF=Éèì~äëW==N1 2( , )N ∑ ν t2j Ψ− α∆tj−2α∆tj=11−eiΛ{ Ψ | O} = −0.5N ln(2 π ) − 0.5ln[ ∏{(1 − e ) }] − 0.5N(A.13)i=1N=^ë=áí=áë=Åçåëí~åí=Ñçê=~åó=ëÉí=çÑ=íÜÉ=ãçÇÉä=é~ê~ãÉíÉêë= Ψ I=íÜÉ=ëìãã~íáçå=íÉêã=Å~å=ÄÉ=éä~ÅÉÇ=çìíëáÇÉ=íÜÉ=éêçÇìÅí=ëáÖåI=ÖáîáåÖW=N1 2∑ ν ( t , )2j Ψ− α∆tjNj= 11−eN−2α∆tiΛ{ Ψ | O} = −0.5Nln(2 π ) − 0.5ln[{ } ∏{(1 − e )}] − ...(A.14)N=0.5NthïÜáÅÜ=Éèì~äëI=Äó=í~âáåÖ=~å= N =éçïÉê=êççí=íç=íÜÉ=éçïÉê=çÑ= N I=~åÇ=éä~ÅáåÖ= N =çìíëáÇÉ=íÜÉ=äçÖ~êáíÜã=N1 2∑ ν ( t , )2jΨ− α∆tjNj= 11−e−2α∆tiΛ{ Ψ | O} = −0.5N ln(2 π ) − 0.5Nln[ N∏(1 − e )] − ...N0.5N=thqÜÉ= NÅçåëí~åí=ÖáîÉå= ( i)=éçïÉê=êççí=çÑ=íÜÉ=éêçÇìÅí=íÉêã=áë=íÜÉ=ÖÉçãÉíêáÅ~ä=ãÉ~å=ïÜáÅÜ=áë=~äëç=n t =~åÇ=Å~å=ÄÉ=éä~ÅÉÇ=áåëáÇÉ=íÜÉ=ëìãã~íáçå=íÉêãW=N∑NN−2α∆ti(1 − )∏ei=1 2ν−2α∆tj( tj, Ψ)j=1 1−eΛ{ Ψ | O} = −0.5N ln(2 π ) − 0.5N ln[ ] − 0.5N(A.16)N=^ë=íÜÉ=Ñáêëí=~åÇ=ä~ëí=íÉêãë=çÑ=íÜÉ=äáâÉäáÜççÇ=ÑìåÅíáçå=~êÉ=åçï=Åçåëí~åíI=E^KNSF=Å~å=ÄÉ=ã~ñáãáòÉÇ=Äó=ãáåáãáòáåÖ=~=ëìã=çÑ=ïÉáÖÜíÉÇ=ëèì~êÉÇ=áååçî~íáçåëW==N−2α∆tNi(1 e )N ∏ −2 i=12= ∑ν−2α∆tjj=1 −S { Ψ | O} ( tj, Ψ )(A.17)1 e=i=1i=1(A.15)..…………………………………………………………………………………………….….

= = = aÉêáî~íáçåë=A.4 Amplitude and phase response of a scaled gammadistribution functionEÇÉêáî~íáçå=íÜ~åâë=íç=hÉÉë=j~~ëF==qÜÉ=~ãéäáíìÇÉ=~åÇ=éÜ~ëÉ=êÉëéçåëÉ=çÑ=~=ëÅ~äÉÇ=Ö~ãã~=ÇáëíêáÄìíáçå=ÑìåÅíáçå=Epd=ÇÑI=ëÉÉ=ëÉÅíáçå=OKQKNF=Å~å=ÄÉ=ÑçìåÇ=Äó=ÅçåîçäìíáåÖ=áí=ïáíÜ=~=Ü~êãçåáÅ=ëáÖå~ä= f ( t ) =ïáíÜ=íáãÉ= t =~åÇ=ÑêÉèìÉåÅó=ξ W==i tf ( t)= e ξ(A.18)ïÜáÅÜ=ÖáîÉëW==∞n n−1iξt ∞ξ(1 )( ) ea ii t aτξ τ τ− +− −aτn−1a∫ ∫(A.19)g( t) = e A e d τ = A ( aτ ) e daτΓ( n) Γ( n)0 0qÜÉ=Ö~ãã~=ÑìåÅíáçå= Γ( n)Éèì~äë=x^Äê~ãçïáíò=~åÇ=píÉÖìåI=NVSQzW==∞n n−1−ntΓ ( n) = k ∫ t e d t ( R n > 0, R k > 0)(A.20)0rëáåÖ=E^KOMFI==E^KNVF=Å~å=ÄÉ=ïêáííÉå=~ëW==iξte Γ( n)Ag( t)= A = eΓ( n) ξ (1 )n ξ (1 )n+ i + ia akÉñíI=Äó=êÉéä~ÅáåÖ=á=áå=íÜÉ=Ñáêëí=é~êí=çÑ=E^KONFI=ïÉ=Ü~îÉW==g( t)=A2ξ(1 + )2an2en ξiξ( t−atan )ξ aiξt(A.21)(A.22)bêÖçI=íÜÉ=~ãéäáíìÇÉ=êÉëéçåëÉ=çÑ=~=pdJíóéÉ=fo=ÑìåÅíáçå=íç=~=Ü~êãçåáÅ=ëáÖå~ä=E^KNUF=áëW==A(A.23)2 nξ(1 + )22a^åÇ=áíë=éÜ~ëÉ=êÉëéçåëÉ=áëW====NVV=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

^ééÉåÇáñ=^===n ξatanξ a(A.24)A.5 Impulse response of a system of linear reservoirsEÇÉêáî~íáçå=íÜ~åâë=íç=hÉÉë=j~~ëF==få=ã~íêáñ=åçí~íáçåI=íÜÉ=îÉÅíçê=çÑ=áãéìäëÉ=êÉéçåëÉë=çÑ=~=ëóëíÉã=çÑ= n =äáåÉ~ê=êÉëÉêîçáêë=áë=EëÉÉ=ëÉÅíáçå=OKPKNFW==1 ⎡1⎤θ = exp( −A t )c1 A⎢0 0⎥(A.25)⎣ ⎦ïÜÉêÉ=íÜÉ=ÉñéçåÉåíá~ä=áë=~=ëçJÅ~ääÉÇ=ã~íêáñ=ÉñéçåÉåíá~äK=låÉ=çÑ=íÜÉ=éçëëáÄäÉ=ÇÉÑáåáíáçåë=çÑ=~=ã~íêáñ=ÑìåÅíáçå= f ( A ) =áëW=== =−1f ( A) = Ef( Λ)E (A.26)=ïÜÉêÉ= E =áë=íÜÉ=ã~íêáñ=çÑ=ÉáÖÉåîÉÅíçêë=çÑ= A I=~åÇ= Λ =íÜÉ=ã~íêáñ=çÑ=ÉáÖÉåî~äìÉëK=fÑ=~ää=1 TêÉëÉêîçáêë=~êÉ=áÇÉåíáÅ~äI= A =áë=ëóããÉíêáÅ=~åÇ= E− = E K=`çãÄáåáåÖ=E^KOSF=~åÇ=E^KORF=áå=íÜ~í=Å~ëÉ=óáÉäÇëW===−λ1t⎡θ1 ⎤ ⎡e11 e12 . e e1n⎤ ⎡⎤ ⎡e11 e21 . en1 ⎤ ⎡1⎤⎢ λ2tθ⎥ ⎢2 1 e21 e22 . e⎥ ⎢⎥−⎢2nee12 e22 . e⎥ ⎢n20⎥⎢ ⎥ ⎢⎥= ⎢ ⎥ ⎢ ⎥ ⎢ ⎥. c1 A0. . . . ⎢.⎥(A.27)⎢ ⎥ ⎢ ⎥ ⎢ . . . . ⎥ ⎢ . ⎥⎢ ⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥ ⎢ ⎥θ1 2. e 1 2. 0n tnen en e −λ⎣ ⎦ ⎣ nn ⎦ ⎢ ⎥ enen e⎣⎦ ⎣ nn ⎦ ⎣ ⎦=_ÉÅ~ìëÉ=íÜÉ=ä~ëí=ÅçäìãåîÉÅíçê=ãçëíäó=Åçåí~áåë=òÉêçëI=íÜáë=Éèì~äëW==−λ1t⎡θ1 ⎤ ⎡e11 e12 . e e1n⎤ ⎡⎤ ⎡e11⎤⎢ λ2tθ⎥ ⎢2 1 e21 e22 . e⎥ ⎢⎥−⎢2nee⎥⎢ ⎥ ⎢⎥ 12= ⎢ ⎥ ⎢ ⎥. c1 A0. . . . ⎢.⎥(A.28)⎢ ⎥ ⎢ ⎥ ⎢ . ⎥⎢ ⎥ ⎢ ⎥ ⎢⎥ ⎢ ⎥θ1 2. e n tnen en e −λ⎣ ⎦ ⎣ nn ⎦ ⎢⎥ e⎣⎦ ⎣ 1n⎦=çê==OMM=..…………………………………………………………………………………………….….

= = = aÉêáî~íáçåë=−λ1t⎡θ1 ⎤ ⎡e11 e12 . e111 en ⎤ e⎢ λ2tθ⎥ ⎢2 1 e21 e22 . e⎥ ⎢− ⎥⎢ ⎥ 2n⎢ e12e ⎥= ⎢ ⎥⎢ . ⎥ c1 A ⎢0. . . . ⎥ ⎢.⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥θ1 2.1e n tnen en e −λnn ⎢ en ⎥⎡⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦(A.29)=ëç=íÜ~íW== =θ == =çêW==1[ e e . e ]i i1 i2inc1 A0i ij 1 jc1 A0j=1⎡e⎢⎢e⎢⎢⎢⎣eλ− 1t11 e−λ2t12e.−λ1en tn⎤⎥⎥⎥⎥⎥⎦(A.30)n1−λej tθ = ∑ e e(A.31)=få=Å~ëÉ=çÑ=åçåJáÇÉåíáÅ~ä=êÉëÉêîçáêëI=ïÉ=Ñáêëí=ìëÉ=~=ëáãáä~êáíó=íê~åëÑçêã=áå=çêÇÉê=íç=ÖÉí=~=ëóããÉíêáÅ~ä=ã~íêáñI=~ëW==−1A*= JAJ (A.32)=ïÜÉêÉ= J =áëW==⎡ A⎤1⎢⎥⎢ A2⎥J = ⎢. ⎥(A.33)⎢⎥⎢A ⎥⎣n ⎦=~åÇ= A =~åÇ= A * =~êÉ=ëáãáä~êK=_ÉÅ~ìëÉ= A * =áë=ëóããÉíêáÅI=E^KPOF=Å~å=ÄÉ=ïêáííÉå=~ëW==−1 −1 −1 −1 TA* = JAJ = JEΛE J = ( JE) Λ( JE) = ( JE) Λ( JE )(A.34)=ëç=íÜ~íW== =A = J −1 ( JE) Λ( JE) T J (A.35)====OMN=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

^ééÉåÇáñ=^==~åÇW==−1f { A} = J ( JE) f { Λ}( JE) T J (A.36)=få=çìê=êÉëÉêîçáê=Å~ëÉI=íÜáë=óáÉäÇëW==⎡1⎤⎢1 1 T0⎥−θ = J ( JE)exp{ −Λt}( JE)J ⎢ ⎥(A.37)c1 A⎢0. ⎥⎢ ⎥⎣0⎦ïÜáÅÜ=ã~ó=ÄÉ=ïêáííÉå=~ëW==n1ej tθi= ∑ α −λij(A.38)c A1 0 j=1=få=ïçêÇëI=íÜÉ=ëçäìíáçå=áë=~=ïÉáÖÜíÉÇ=ëìã=çÑ=ÉñéçåÉåíá~äëI=áå=ïÜáÅÜ=íÜÉ=ïÉáÖÜíë=α ÇÉéÉåÇ=çå=íÜÉ=ÉáÖÉåîÉÅíçêë=çÑ= A K=OMO=..…………………………………………………………………………………………….….

^ééÉåÇáñ=_==OMQ==..…………………………………………………………………………………………….….

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oÉÑÉêÉåÅÉë=Yihdego, Y., and J. A. Webb (2011), Modeling of bore hydrographs to determine the impactof climate and land-use change in a temperate subhumid region of southeasternAustralia, Hydrogeology Journal, 19, 877-887.Young, P. C., and K. J. Beven (1994), Data-based mechanistic modelling and the rainfall-flownon-linearity, Environmetrics, 5, 335-365.Young, P., S. Parkinson, and M. Lees (1996), Simplicity out of complexity in environmentalmodelling: Occam’s razor revisited, Journal of Applied Statistics, 23, 165-210.Young, P. C. (1998), Data-based mechanistic modelling of environmental, ecological,economic and engineering systems, Environmental Modelling & Software, 13, 105-122.Young, P. C., and H. Garnier (2006), Identification and estimation of continuous-time, databasedmechanistic (DBM) models for environmental systems, Environmental Modelling &Software, 21(8), 1055-1072, DOI : 10.1016/j.envsoft.2005.05.007.Yu, C., A. W. Warrick, and M. H. Conklin (1999), A moment method for analyzingbreakthrough curves of step inputs, Water Resources Research, 35(11), 3567-3572.Zadeh, L. A. (1956), On the identification problem, IRE Transactions on Circuit Theory, 3, 277-281.Ziemer, R. E., W. H. Tranter, and D. R. Fannin (1998), Signals and systems: continuous anddiscrete, Prentice-Hall, Upper Saddle River.Zwamborn, M. H. (1995), Modeling of the unsaturated zone for the benefit of groundwatermodels (in Dutch), Kiwa N.V., Nieuwegein.===ONR=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

AppendixCmìÄäáÅ~íáçåë=Publications and curriculumvitae===ONT=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

mìÄäáÅ~íáçåë=PublicationsONU=Peer reviewed publications (11)Von Asmuth, J. R., C. Maas, M. Knotters, M. F. P. Bierkens, M. Bakker, T. N. Olsthoorn, D. G.Cirkel, I. Leunk, F. Schaars, and D. C. Von Asmuth (submitted), Menyanthes: softwarefor hydrogeologic time series analysis, interfacing data with physical insight,Environmental Modelling & Software.Post, V. E. A. and J. R. Von Asmuth (in prep.), Hydraulic head measurements: Newtechnologies, classic pitfalls. Hydrogeology Journal.Lehsten, D., J. R. Von Asmuth, and M. Kleyer (2011), Simulation of Water Level Fluctuationsin Kettle Holes Using a Time Series Model, Wetlands, 31, 511-520, DOI 10.1007/s13157-011-0174-7.Manzione, R., M. Knotters, G. Heuvelink, J. Von Asmuth, and G. Camara (2010), Transferfunction-noise modeling and spatial interpolation to evaluate the risk of extreme(shallow) water-table levels in the Brazilian Cerrados, Hydrogeology Journal, 18(8), 1927-1937 .Von Asmuth, J. R., K. Maas, M. Bakker, and J. Petersen (2008), Modeling time series ofgroundwater head fluctuations subjected to multiple stresses, Ground Water, 46, doi:10.1111/j.1745-6584.2007.00382.x(1), 30-40.Bakker, M., K. Maas, and J. R. Von Asmuth (2008a), Calibration of transient groundwatermodels using time series analysis and moment matching, Water Resources Research,44(W04420), doi:10.1029/2007WR006239.Bakker, M., K. Maas, F. Schaars, and J. R. Von Asmuth (2007), Analytic modeling ofgroundwater dynamics with an approximate impulse response function for arealrecharge, Advances in Water Resources, 30(3, doi:10.1016/j.advwatres.2006.04.008), 493-504.Von Asmuth, J. R., and M. F. P. Bierkens (2005), Modeling irregularly spaced residual seriesas a continuous stochastic process, Water Resources Research, 41(12), W12404,doi:10.1029/2004WR003726.Von Asmuth, J. R., and M. Knotters (2004), Characterising spatial differences ingroundwater dynamics based on a system identification approach, Journal of Hydrology,296(1-4), 118-134.Witte, J. P. M., and J. R. Von Asmuth (2003), Do we really need phytosociological classes tocalibrate Ellenberg’s indicator values?, Journal of Vegetation Science, 14, 615-618.Von Asmuth, J. R., M. F. P. Bierkens, and K. Maas (2002c), Transfer function noise modelingin continuous time using predefined impulse response functions, Water ResourcesResearch, 38(12), 23_1-23_12.=Book chapters and conference proceedings (6)Manzione, R. L., M. Knotters, G. M. B. Heuvelink , J. R. Von Asmuth, and G. Câmara (2009),Predictive risk mapping of water table depths in a Brazilian Cerrado area, in Qualityaspects in spatial data mining, edited by Stein, A., Shi, W., and Bijker, W., CRC Press,Taylor & Francis Group, Boca Raton, Florida,Bakker, M., C. Maas, and J. R. Von Asmuth (2008b), Transient calibration of flow to ditcheswith entry resistance using measured moments of response functions, in Calibration andReliability in Groundwater Modelling: Credibility of Modeling Proceedings of ModelCARE2007 Conference, IAHS Publ. 320,..…………………………………………………………………………………………….….

mìÄäáÅ~íáçåë=OOM=KWR 2011.106, KWR Watercycle Research Institute, Nieuwegein.Von Asmuth, J. R., A. P. Grootjans, and S. Van der Schaaf (2011), Over de dynamiek vanpeilen en fluxen in vennen en veentjes. Eindrapport deel 2, OBN-onderzoek ‘Herstelvan biodiversiteit en landschapsecologische relaties in het natte zandlandschap’,Rapport nr. 2011/OBN147-2-NZ, Bosschap, bedrijfschap voor bos en natuur,Driebergen.Von Asmuth, J. R. (2010), Over de kwaliteit, frequentie en validatie van druksensorreeksen,Rapportnr. KWR 2010.001, KWR Watercycle Research Institute, Nieuwegein.Von Asmuth, J. R., S. Van der Schaaf, A. P. Grootjans, and C. Maas (2010), Weerstand enwegzijging in natte natuurgebieden, schatting via analyse van gemeten (grond)waterpeilen,Delft University of Technology, Delft.Leunk, I., K. J. Raat, and J. R. Von Asmuth (2010), Snel en nauwkeurig detecteren vanputverstopping met tijdreeksanalyse (Menyanthes), rapport BTO 2010.036(s), KWRWatercycle Research Institute, Nieuwegein.Maas, C., J. R. Von Asmuth, R. Agtersloot, F. Schaars, and P. Maas (2009), HetGrensmaasproject en de Vlaamse VHR-gebieden, een signalerings- en alarmeringsprotocol tenaanzien van de grondwaterstanden, KWR 08.075, KWR Watercycle Research Institute,Nieuwegein.Von Asmuth, J. R., C. Maas, and M. Knotters (2009), Handleiding Menyanthes, versie 1.9, KWRWatercycle Research Institute, Nieuwegein.Knotters, M., S. P. J. Van Delft, H. E. Keizer-Vlek, J. R. Von Asmuth, P. C. Jansen, F. P. Sival,and C. E. Van ‘t Klooster (2008), Evaluatie monitoring Deurnese Peel en Mariapeel.Kwantificering van effecten van maatregelen en advies over het monitoringplan, Alterra-Document2, Alterra, Wageningen.Maas, C., and J. R. Von Asmuth (2008), Advies Monitoring Grensmaas, KWR 08.020, KWRWatercycle Research Institute, Nieuwegein.Cirkel, D. G., C. Maas, and J. R. Von Asmuth (2007), Temporele variaties van nitraat inonttrokken grondwater. Ontwikkeling en toepassing van een tijdreeksmodel voor demodellering en voorspelling van nitraatconcentraties in onttrokken grondwater,Rapport KWR 07.003, Kiwa Water Research, Nieuwegein.Von Asmuth, J. R., M. Knotters, and C. Maas (2006), Tijdreeksanalyse voor (eco)hydrologen,achtergronddocumentatie en cursushandleiding, Kiwa Water Research/Alterra,Nieuwegein/Wageningen.Maas, C., D. G. Cirkel, and J. R. Von Asmuth (2005), Tijdreeksanalyse Voornes Duin, deel 1:Afzonderlijke peilbuizen, rapport KWR 05.055, Kiwa Water Research, Nieuwegein.Cirkel, D. G., and J. R. Von Asmuth (2005a), Hydrologische evaluatie antiverdrogingsmaatregelenTerwisscha, rapport nr. KWR 05.29, Kiwa Water Research,Nieuwegein.Cirkel, D. G., and J. R. Von Asmuth (2004), Effecten van de vernattingsmaatregelen in deAmsterdamse waterleidingduinen , rapport nr. KWR 03.092, Kiwa Water Research,Nieuwegein.Cirkel, D. G., C. Maas, and J. R. Von Asmuth (2004), What Els? Evaluatie van het voorlopigemeetnet voor Extreem Lage Stijghoogten van de provincie Noord-Brabant, KWR.04.073Kiwa Water Research, Nieuwegein.Maas, C., and J. R. Von Asmuth (2004), Tijdreeksanalyse Mander, Een onderzoek naar dehydrologische wisselwerking tussen de stuwwal van Ootmarsum en de Slenk vanReutum, Kiwa Water Research, Nieuwegein.Maas, C., and J. R. Von Asmuth (2003), Tijdreeksanalyse grondwaterstanden 1975-2002,..…………………………………………………………………………………………….….

mìÄäáÅ~íáçåë=Effectiviteit van het stand-still-beleid van de Provincie Limburg, Kiwa Water Research,Nieuwegein.Jansen, A. J. M., J. R. Von Asmuth, J. Bunnik, and A. C. Zuidhoff (2001), Vijf NB-wet-terreinenop het landgoed Twickel (Overijssel), rapportnr. KOA 01.049, Kiwa N.V., Nieuwegein.Von Asmuth, J. R. (2000b), Onderzoeksvoorstel grondwaterspiegeldynamica in ecologischperspectief, Kiwa N.V., Nieuwegein.Koppejan, H., P. J. M. Melman, J. R. Von Asmuth, and D. De Jong (1999), Standaardvoorschriftkwelderkartering in Nederland, rapport MDGAE-98-02, Meetkundige dienst, Delft.Von Asmuth, J. R., B. Van Gennip, and A. M. De Meulmeester (1999), P.Q.-onderzoekRottumerplaat en -oog, rapp.nr. MDGAE 9911, RWS-Meetkundige Dienst, Delft.Van Gennip, B., J. R. Von Asmuth, J. Cools, and M. Bakker (1998), De buitendijkse gebiedenlangs het Haringvliet en Hollandsch Diep, RWS - Meetkundige Dienst, Delft.Von Asmuth, J. R., M. Tinga, and J. A. M. Janssen (1998), Project Stroomlijnen WerkprocesVegetatiekartering, eindrapport fase 1., Intern rapport, RWS-Meetkundige Dienst, Delft.Von Asmuth, J. R., A. G. Knotters, and L. J. Schulpen (1997a), Arc Info Gebruikers Cursus.Cursus met beknopte cursushandleiding, RWS-Meetkundige Dienst, Delft.Von Asmuth, J. R., B. Gennip, and P. J. M. Melman (1997b), P.Q.-onderzoek 'Groene Strand'Terschelling, rapp.nr. MDGAT 9753, RWS-Meetkundige Dienst, Delft.Zonneveld, L. M. L., J. R. Von Asmuth, and J. A. M. Van Dongen (1997), Vegetatiekartering ‘deGrie’ Terschelling 1993, rapp.nr. MDGAT 9536, , RWS-Meetkundige Dienst, Delft.Reitsma, J. M., J. R. Von Asmuth, and E. R. Stenfert-Steehouwer (1996), De schorren van deWesterschelde 1990 / 1993, rapp.nr. MDGAT 9623, RWS-Meetkundige Dienst, Delft.Von Asmuth, J. R., and M. Tolman (1996), Vegetatiekartering Schiermonnikoog 1992, rapportageen ecologische interpretatie , rapp.nr. MDGAT 9603, RWS-Meetkundige Dienst, Delft.Von Asmuth, J. R. (1995a), Monitoring in de Millingerwaard, aanzet tot een vijf-jaarlijksevegetatiekartering en thematische / geometrische analyse m.b.v. GIS, Afstudeerrapport,Landbouwuniversiteit Wageningen, Wageningen.Von Asmuth, J. R. (1995b), Project Monitoring Rottum, Rottumeroog en -plaat als testcasevoor het vergelijken van vegetatiekaarten, rapp.nr. MDGAT 9548, RWS-MeetkundigeDienst, Delft.Von Asmuth, J. R. (1994), Disturbance, regrowth and replanting in the tropical rainforests of Rabi,Gabon, Afstudeerrapport, Landbouwuniversiteit Wageningen, Wageningen.===OON=KKÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁKÁK

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