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`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 <strong>Asmuth</strong>, 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 <strong>Asmuth</strong>, 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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