ANALYSIS AND MODELING OF HYDROLOGIC TIME SERIES. Wasserhaushalt Time Series Analysis and Stochastic Modelling Spring Semester

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1 ANALYSIS AND MODELING OF HYDROLOGIC TIME SERIES Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8

2 ANALYSIS AND MODELING OF HYDROLOGIC TIME SERIES Defnon Wha a me ere? Leraure: Sala, J.D. 99, Ch. 9 n Handbook of Hydrology ed. By D.R. Madmen Connuou Tme Sere (e.g., rver flow, ar emperaure, amopherc preure) Dcree Tme ere (e.g., number of orm per year, floodng damage n a gven area n a gven me) Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8

3 ANALYSIS AND MODELING OF HYDROLOGIC TIME SERIES Purpoe The man am of me ere analy o characere he propere of he rajecory over me of a gven varable (e.g., reamflow a a gven e) Propere of me ere are of grea gnfcance n plannng, degnng and operaon of waer reource yem Buldng mahemacal model o generae ynhec hydrologc record o foreca hydrologcal even o fll n mng daa Waer reource degn Hydrologcal model of me ere evoluon Synhec generaon of daa Degn of waerwork and her operaon rule. The yem mu be analyzed under all poble naural and operang condon. Numercal mulaon (ochac) Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 3

4 ANALYSIS AND MODELING OF HYDROLOGIC TIME SERIES Eample: degn and operaon of mulpurpoe reervor In order o nfer a relable degn and operang rule, a conderable number of reamflow realaon requred. Horcal reamflow: Daa avalably Repreenave of all poble cenaro Long-erm rk/relably analy Synhec (generaed) reamflow: Long-erm mulaon Scenaro analy Relably analy Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 4

5 STOCHASTIC PROCESSES AND HYDROLOGIC TIME SERIES Hydrologc me ere, () A) Sochac proce An even ermed random or ochac when oucome or occurrence unceran. Sochac proce a collecon of random varable repreenng he evoluon of ome yem of random value over me. B) Sochac model Mahemacal model repreenng he rucure and he propere of a me ere. A) Idenfcaon of he ochac rucure of he me ere. TYPE: connuou v dcree, ngle v mulple, correlaed v uncorrelaed, nermency, aonary v non-aonary, COMPONENTS: deermnc v ochac (perence), rend and hf, eaonaly B) Model emaon INFERENCE: ype and componen of he me ere dcae he model rucure, (e.g., correlaon lnked o degree of model perence componen) CALIBRATION: parameer emaon Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 5

6 EXAMPLES OF TIME SERIES 5 Whe Noe Auocorrelaed Long-memory [-] [-] [-] Wh a rend [-] Wh a hf [-] Tme Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 6

7 EXAMPLE OF HYDROLOGIC TIME SERIES / a) Connuou me and varable o Waer level, Dcharge o Temperaure o Sol moure o b) Dcree me and connuou varable o Annual peak flow o Mean monhly flow o Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 7

8 EXAMPLE OF HYDROLOGIC TIME SERIES / c) Connuou me and dcree varable o Threhold eceedance o Ranorm arrval o d) Dcree me and dcree varable Counng ere (e.g., number of rany day per year, number of overflow per monh) Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 8

9 STOCHASTIC PROCESSES ÎT T { },...,, k, k ÎT k Enemble One random realzaon of one rajecory T Þ ( ) one realaon of he ochac proce All realzaon of he proce Tme ere: one poron of he rajecory ou of many poble realzaon Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 9

10 STOCHASTIC PROCESSES A ochac proce he ENSEMBLE of all he poble rajecore (.e., realzaon of emporal equence) of he random varable The random varable ha pecfc propere (e.g., a acal drbuon, acal momen, ereme value, ec.), whch have o be preerved by he ochac model of he proce. Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8

11 CHARACTERISTIC VARIABLES OF A STOCHASTIC PROCESS Random varable Sac (Momen), ( ) ( ), Momen are me-dependen Mean of a random varable (epeced value) Mean condonal o me r-h order momen condonal o me µ + ò ( ) ( ) ( ) f ( ) d( ) mµ ( - µ [ () ) ] [ ( ) ] µ ( / ) m( ) + µ µ r ( ) - [ ] ò ( ) r ) ( ) - m( ) f ( ( ) ) ( d( ) Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8

12 CHARACTERISTIC VARIABLES OF A STOCHASTIC PROCESS Second order cenral momen Varance ( ) + ò [ ] ( ) - m µ ( ) f ( ( ) ) d( ) ( ) - Co-Varance -, g g + + ò ò, µ d( ) d( - - ( ) [ ( ) - m( )][ ( ) - m µ ( )] f ( ( ), ( )) ( ) E[ ( ( ) - m( ))( ( ) - m µ ( ))] / µ Correlaon coeffcen ( ), ( + ) r g ( / ) ( ) / ( ) ( + ) Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 )

13 PROPERTIES OF STOCHASTIC PROCESSES /4 Saonary I mple ha he probablc mechanm governng he ochac proce doe no change wh me.!! STATISTICS OF THE PROCESS ARE TIME INVARIANT Mahemacally peakng: Obervaon a me and a me,,...,, 3 N +, +, 3 + N,..., + Ther jon probably funcon f (,,..., ), 3 N (, ) f,..., +, + 3+ N + The wo N-dmenonal pdf are acally dencal. Snce τ doe no affec f( ), f() no a funcon of me. N-dmenonal pdf Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 3

14 PROPERTIES OF STOCHASTIC PROCESSES /4 If he acal deny hold for k (momen) < M k-h order rcly aonary In pracce a nd order weak aonary verfed o hold o conder he proce aonary. Th mple: Mean à Varance à Covarance à ( / ) µ µ ( / ) ( ) ( ) ( ) g r g / Tme Invaran Eample of non-aonary procee are comng from clmae change, land ue change, anhropogenc mpac. Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 4

15 PROPERTIES OF STOCHASTIC PROCESSES 3/4 Ergodcy I mple ha one (oberved) equence of realzaon repreenave of he enre enemble Sample compared o he Enemble populaon + µ ò f ( ) d mˆ ò T - T ( ) d Mean of he enemble Mean of he ample Ergodcy µ mˆ.e., µ + ò f - T ( ) d lm ò T T ( ) d [ ˆ ] P µ m f T Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 5

16 PROPERTIES OF STOCHASTIC PROCESSES 3/4 Varance ˆ + ò - T [ ] - µ f ( ) T ò ( ( ) - mˆ ) d d ˆ Co-Varance g g + + ò ò (, dd - - ( ) [ ) - µ ][ ( ) - µ ] f (, ) T T ( ) ( ( + ) - mˆ )( ( ) mˆ ) ò - d g ( ) g( ) Ergodcy canno be proved and herefore aumed o hold a workng hypohe, o allowng o compue nd order ac from oberved ample. Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 6

17 SAMPLE STATISTICS OF A TIME SERIES / Overall ample ac of a ere Mean Varance N N å N N - å( - ) N: ample ze Sandard devaon Coeffcen of varaon C V Auocovarance and auocorrelaon coeffcen C N å - N k ( - )( ) k + k - r k C k C k ³ he lag For k C Covarance mplfe o he varance!! Under ergodcy aumpon hee ac repreen he proce ac. Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 7

18 SAMPLE STATISTICS OF A TIME SERIES / The correlogram (correlaon coeffcen a funcon of he lag) can be ued a ndcaor of he ype of ochac model uable o repreen he me ere. Auocorrelaon Lag Shor memory Dfferen degree of perence of he proce Auocorrelaon.5 Auocorellaon funcon Lag Removng Ba correcon Seaonaly Long memory Auocorrelaon.5 Uncorrelaed proce Auocorellaon funcon Lag Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 8

19 TIME SERIES MODEL Bac Srucure Tme ere d + h Sochac proce Deermnc componen Sochac componen No random Funcon of he proce memory -, -, Funcon of he proce conrol varable Predcable Random characer Saonary and ndependen proce I provde he modellng of he unpredcable componen of he proce Memory : perence of he gnal, e.g., - nfluence r ( ) ¹ Oher deermnc componen are: eaonaly, rend, hf I realaon are uncorrelaed r µ h ( ) ( ) h µ h h ( ) " ÎT Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 9

20 TIME SERIES MODEL Eample of model formulaon Annual Rver Flow f + h - Year Dependence on he prevou year flow I accoun for he varably of meeorologcal condon The model o formulaed mu preerve he acal propere of he oberved proce: Mean, Varance, Auocorrelaon, pdf, Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8

21 TIME SERIES MODEL Eample of model conrucon / Eample: Auoregreve model (Annual Temperaure me ere) Sandardze he me ere y - µ Compue he ample auocorrelaon funcon of y and compare wh heorecal auocorrelaon of AR() proce. Correlogram. r 3 Compung annual value of andardze emperaure ung he model y f y - + e e Whe noe (ndependen, aonary and normally drbued) f r Lag- auocorrelaon. 4 Check for model adequacy. Compue he auocorrelaon funcon of he redual and compare wh auocorrelaon of whe noe. Compue drbuon of redual. e r( ) y -f y- e Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8

22 TIME SERIES MODEL Eample of model conrucon / [ C] Orgnal Sere Sandardzed derended Auocorrelaon.5 f -.7 Auocorellaon funcon heorecal auocorrelaon of model [-] Lag Auocorrelaon Tme [Year] Auocorellaon funcon.5 Auocorrelaon of redual Frequency Lag Lag Redual ε- Redual Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8

23 HYDROLOGICAL ANALYSIS OF TIME SERIES Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 3

24 MODELLING OF SINGLE STATIONARY SERIES Lnear AuoRegreve Model / AR model epre he curren value of a varable a he weghed um of a preagned number of pa value and a varae ha compleely random. pa value f + h - AR() model Componen repreenng he memory of he proce wh repec o prevou occurrence (proce perence) Random varae, N(, h) e.g., monhly flow, ep. If FEB, - JAN Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 4

25 MODELLING OF SINGLE STATIONARY SERIES Lnear AuoRegreve Model / Generalaon: AR(p) model p pa value f... f - - p - p f h Memory componen of he p prevou occurrence (Markovan rucure) Random varae o f f,...,, f p are parameer, wegh of he prevou ae o o h a aonary, ndependen (uncorrelaed), normally drbued random varae à whe noe a aonary ochac proce, derved from a proce ha ha been reaed o remove rend, hf and eaonaly. Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 5

26 EXAMPLE OF SYNTHETIC GENERATION wh AR() / Varable: annual rver flow Mean: 47 mm Sandard devaon: 95.8 mm Lag- Auocorrelaon coeffcen:.34 r Generaon of for n year Model AR(), +, +, ~ aonary and normally drbued nal value for he fr generaon ep 47 AR(): ~ f ~ - + h Gauan E [h ] ~ ~ f + r Generang equaon: - e f r ( ) ( [h ] h - r ~ - r ) Sandard Gauan Varae N (,) Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 6

27 EXAMPLE OF SYNTHETIC GENERATION wh AR() / Subung wh he above aumpon: ( - ) + r + r - - e - r) + r - + ( - r e ( -.34) e Compung: Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 7

28 MODELLING OF SINGLE PERIODIC SERIES Non-aonary nduced by eaonaly e.g., rver flow à monhly perence AR(p) p åf - + h Snce a aonary varable Perodc AR(p) y - µ - µ y + E [ h ] p y,, µ + å f h, - h h [ ] Uncorrelaed normal varae Mean and andard devaon for eaon Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 8

29 Eample of monhly perence, rver flow, perence due o effec of orage and uburface flow. ( ) Q Q b Q Q Thoma-Ferng model of monhly flow monh à equaon ( ) h Q Q b Q Q If we do apply mple lnear regreon: ( ) b y y - + å å - - n n n ny y b ø ö ç è æ - ø ö ç è æ - - å å å n n n ny y n ny y r y b r MODELLING OF MONTHLY FLOWS /3 Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 9

30 MODELLING OF MONTHLY FLOWS /3 Thoma-Ferng model con d Sandard error of emae: h - r Y 3 Combnng, and 3 Q r - Q Smlare (deny) wh perodc AR(p) wh p: ( ) Q - Q + e r PARAMETERS Q Ø Average flow for monh y y ( -, - - µ ) µ h, µ + f - +, - f r 36 parameer: Q,, r Q - r Noe erm e Ø Average flow for monh Ø Lag- Auocorrelaon coeff. for monh Ø Varance of flow for monh Ø Gauan random varae - Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 3

31 MODELLING OF MONTHLY FLOWS 3/3 Thoma-Ferng model con d The model can be ealy formulaed n andard form q - q u u r u + e r Noe: negave value of u poble - - For kewed oberved ample one can ranform he orgnal ere wh a logarhmc ranformaon: y - µ y, y ln u u r y u - + e - r ( ) q Noe: µ y, ¹ Hgher order perence e.g., monh q y,, y, Q ( Q - Q ) + b ( Q - Q ) + e - r Q + b r Q Q Noe he mulple correlaon coeffcen: -, - Q Waerhauhal Tme Sere Analy and Sochac Modellng Sprng Semeer 8 3