Wavelet-Based Multiresolution Analysis. of Wivenhoe Dam Water Temperatures. Don Percival

Transcription

1 Wavele-Based Muliresoluion Analysis of Wivenhoe Dam Waer Temperaures Don Percival Applied Physics Laboraory Deparmen of Saisics Universiy of Washingon Seale, Washingon, USA hp://faculy.washingon.edu/dbp Collaboraive effor wih Sarah Lennox, You-Gan Wang and Ross Darnell, wih suppor from Seqwaer personnel

2 Background: I Queensland Bulk Waer Supply Auhoriy (Seqwaer) manages cachmens, waer sorages and reamen services o ensure qualiy and quaniy of waer supplied o Souheas Queensland ongoing monioring program recenly upgraded wih permanen insallaion of verical profilers a Lake Wivenhoe dam each profiler moniors waer qualiy indicaors every wo hours a differen dephs a a fixed locaion (emperaure, ph,... ) leads o a unique opporuniy o sudy flucuaions in hese indicaors in a subropical dam as a funcion of ime and deph will concenrae on a 600+ day segmen of emperaure flucaions X recorded a dam wall (emperaure is regarded as imporan driver for oher waer qualiy indicaors)

3 Lake Wivenhoe Phoograph: Andrew Wakinson (Seqwaer) CSIRO. Undersanding waer qualiy measuremens 2

4 Degrees C X a Dephs of, 5, 0, 5 & 20 Meers

5 Background: II complicaed srucure boh across ime and down deph Q: how can we bes quanify variaions in daa? propose o simplify ask by breaking X ino componens capuring daily, subannual & annual (DSA) variaions can formulae precise definiions for each componen in erms of a wavele-based muliresoluion analysis (MRA) DSA componens are such ha hey are approximaely pairwise uncorrelaed sum up o original X exacly and based upon coefficiens ha decompose sample variance of X exacly across ime 4

6 Background: III some quesions our approach can help address:. how does variance change across ime? 2. are variaions from one day o he nex more prominen han variaions from, say, one monh o he nex? 3. how repeaable are variaions in annual cycle a each deph? 4. wha are he pairwise relaionships beween deph series over differen spans of ime (e.g., day-o-day, monh-o-monh)? resuling analysis mainly descripive, bu provides insigh ino componens needed for a formal saisical model 5

7 Overview of Remainder of Talk give overview of sandard wavele analysis describe adapaions for analysis of dam emperaures discuss preparaions o daa prior o analysis presen key resuls of our analysis (complee analysis documened in recenly compleed manuscrip) 6

8 Basic Descripion of Wavele Analysis: I le X be a column vecor wih elemens X, = 0,,..., N X represens ime series of N regularly sampled observaions, i.e., ime associaed wih X is is he ime a which X 0 was observed is he sampling ime beween adjacen observaions (e.g., = 2 hours for waer emperaure ime series) is ime index for elemen X wavele analysis of X is a linear ransformaion, expressed as fw = f WX, where f W is a marix ransforming X ino a vecor of maximal overlap discree wavele ransform (MODWT) coefficiens f W 7

9 Basic Descripion of Wavele Analysis: II f W conains wo ypes of MODWT coefficiens wavele coefficiens scaling coefficiens le s focus firs on wavele coefficiens, denoed by f W j, while each X in X has jus a ime index, each wavele coefficien f W j, in f W has wo indices: level index j, where j =, 2,..., J 0 ime index, where = 0,,..., N maximum level J 0 depends upon he paricular applicaion (for waer emperaure series, J 0 = 9 as discussed laer) 8

10 Basic Descripion of Wavele Analysis: III index for f W j, says ha formaion of his coefficien involves only pars of X cenered abou a paricular ime index j ells us how many values in X are in effec being used o form f W j, if j is small (large), f W j, depends mainly on a small (large) number of values from X anoher inerpreaion is ha j is an index for he inerval of frequencies f given by µ I j = 2 j+, 2 j f W j, is ha par of a localized Fourier decomposiion of X associaed wih frequencies f I j (localizaion dicaed by ) 9

11 Frequency Inervals I j When = 2 hours I 4 I 3 I 2 I f (cycles per day) 0

12 Basic Descripion of Wavele Analysis: IV le s now focus on scaling coefficiens e V J0, in f W each scaling coefficien also has a level index and a ime index, bu level index can assume only single value J 0 ime index on e V J0, has same inerpreaion as for f W j, inerval of frequencies associaed wih V e J0, is I 0 = 0, 2 J0+ union of I j, j = 0,,..., J 0, is [0, /(2 )], i.e., all physically meaningful frequencies in Fourier decomposiion of X scaling coefficiens capure localized low-frequency variaions in X, whereas wavele coefficiens do he same over frequency inervals I j, j =, 2,..., J 0

13 Frequency Inervals I j When = 2 hours & J 0 = 4 I 0 I 4 I 3 I 2 I f (cycles per day) 2

14 Wavele-based Analysis of Variance: I place all wavele coefficiens in f W associaed wih level j ino vecor f W j & all scaling coefficiens ino vecor e V J0 denoe he square of Euclidean norm of X as kxk 2 P X2 imporan propery of MODWT of X is ha k f Wk 2 = kxk 2 since f W is he union of f W, f W 2,..., f W J0 and e V J0, also have J 0 X j= k f W j k 2 + k e V J0 k 2 = kxk 2 can inerpre k f W j k 2 as par of kxk 2 aribuable o localized Fourier coefficiens associaed wih frequency inerval I j, and k e V J0 k 2 as being associaed wih low-frequency inerval I 0 3

15 Wavele-based Analysis of Variance: II le X = P X /N denoe sample mean of X, and consider is sample variance: ˆσ 2 X = N N X =0 can reexpress he above as J 0 ˆσ X 2 = X j= N kf W j k 2 + X X 2 = N kxk2 X 2 µ N k V e J0 k 2 X 2 J 0 X j= ˆσ 2 j + ˆσ2 0, where ˆσ 2 j and ˆσ2 0 are sample variances associaed wih f W j and ev J0 (sample mean of e V J0 is X also, whereas wavele coefficiens come from populaions wih zero means) 4

16 Wavele-based Analysis of Variance: III can break up sample variance of X ino J 0 + pars, J 0 of which (he ˆσ j 2 s) are aribuable o flucuaions in inervals of frequencies I j, and he las (ˆσ 0 2 ), o flucuaions over lowfrequency inerval I 0 refer o decomposiion of ˆσ 2 X J 0 ˆσ X 2 = X afforded by j= ˆσ 2 j + ˆσ2 0 as a wavele-based analysis of variance (ANOVA) 5

17 Wavele-based ANOVA for m Waer Temperaures o s are ˆσ 2 j s while x is ˆσ2 0 0 (will offer inerpreaion laer) x wavele & scaling variances o o o o o o o o o j 6

18 Muliresoluion Analysis (MRA) can use he MODWT coefficiens o obain a wavele-based addiive decomposiion known as a muliresoluion analysis sar wih fac ha X can be recovered from is MODWT coefficiens f W = f WX via synhesis equaion X = f W T f W pariioning boh W f and W f allows rewriing X = W ft W f as JX X = ed j + S e J0 j= e D j is a deail series depending jus on f W j and hose rows in fw used o creae f W j from X e D j capures par of X aribuable o flucuaions in I j e S J0 is a smooh series capuring low-frequency flucuaions 7

19 J 0 = 3 MRA for m Waer Temperaures Degrees C Degrees C Degrees C X es 3 ed 3 I 0 = [0, 0.75] cycles/day I 3 = (0.75,.5] cycles/day Degrees C ed 2 I 2 = (.5, 3] cycles/day Degrees C ed I = (3, 6] cycles/day 8

20 Adapaion of MRA for Waer Temperaures: I idea behind MRA is o break X up ino componens e D j and es J0 capuring differen aspecs of X (in saisical erms, componens should be approximaely pairwise uncorrelaed) J 0 usually chosen so ha e S J0 capures prominen large-scale (low-frequency) flucuaions in X seing J 0 = 9 means ha e S 9 capures flucuaions lower in frequency han 4.3 cycles/year empirically e S 9 is preferable o eiher e S 8 or e S 0 in capuring inerannual variaions e S 8 is arguably undersmoohed, conaining flucuaions beer ascribed o inra-annual variaions e S 0 is somewha oversmoohed, hence disoring he inerannual flucuaions 9

21 Degrees C J 0 = 8 Smooh for 0 m Waer Temperaures

22 Degrees C J 0 = 9 Smooh for 0 m Waer Temperaures

23 Degrees C J 0 = 0 Smooh for 0 m Waer Temperaures

24 Adapaion of MRA for Waer Temperaures: II wih J 0 = 9, would have 9 deail series and smooh e S 9 in usual wavele-based MRA, bu desirable o have decomposiion wih physically moivaed componens wo imporan physical drivers for dam waer emperaures are revoluion of earh abou sun (influences annual variaions) daily roaion of earh (influences diurnal variaions) seek addiive decomposiion isolaing hese variaions seing J 0 = 9 resuls in e S 9 capuring annual variaions any purely periodic daily variaion in ime series wih = 2 hours can be expressed exacly wih a Fourier decomposiion involving a consan and sines & cosines wih (a mos) 6 frequencies, namely, fundamenal frequency f = cycle/day and five harmonics f k = kf, k = 2, 3, 4, 5 and 6 cycles/day. 23

25 Frequency Inervals I j When = 2 hours I 3 I 2 I f (cycles per day) 24

26 Adapaion of MRA for Waer Temperaures: III daily flucuaions are capured primarily in D e, D2 e and D e 3 accordingly, le s define a daily componen as D = D e + D e 2 + D e 3 A = S e 9 defines he annual componen combine remaining deails ino a subannual componen S = D e 4 + D e D e 9, leading o he modified MRA X = D + S + A will refer o his modified MRA as he DSA decomposiion 25

27 Degrees C Degrees C Degrees C DSA Decomposiion for m Waer Temperaures D = daily = e D + e D 2 + e D S = subannual = e D 4 + e D 5 + e D 6 + e D 7 + e D 8 + e D 9 30 X A = annual = e S 9 26

28 Adapaion of ANOVA for Waer Temperaures: I can formulae ANOVA corresponding o DSA decomposiion in wo ways (one obvious, and he oher, no so obvious) obvious way is o jus add squared wavele coefficiens for each level involved in forming D and S elucidaion of saisical properies of combinaion requires model o sor ou relaive influence of squared coefficiens from differen f W j s (no easy o come by) no-so-obvious way is define a new ransform, say U = UX, wih associaed synhesis equaion X = U T U U conains hree ypes of coefficiens D, S and A, each having N elemens 27

29 Adapaion of ANOVA for Waer Temperaures: II coefficiens saisfy kdk 2 + ksk 2 + kak 2 = kxk 2, wih 3X kdk 2 = kw f 9X j k 2, ksk 2 = kw f j k 2, kak 2 = kv e J0 k 2 j= j=4 above leads o an ANOVA based upon he U ransform: ˆσ X 2 = N kdk2 + µ N ksk2 + N k Ak e 2 X 2 = ˆσ D 2 + ˆσ2 S + ˆσ2 A, where 3X 9X ˆσ D 2 = ˆσ j 2, ˆσ2 S = ˆσ j 2 and ˆσ2 A = ˆσ2 0 j= j=4 manipulaion of synhesis equaion X = U T U leads o DSA decomposiion X = D + S + A 28

30 Adapaion of ANOVA for Waer Temperaures: III have essenially collapsed 3N wavele coefficiens in f W, f W 2 and f W 3 ino N coefficiens D, from which can deermine D likewise, have collapsed 6N wavele coefficiens in f W 4, f W5,..., f W 9 ino N coefficiens S, from which can deermine S will refer o U as he DSA ransform D, S and A collecively as DSA ransform coefficiens D, S and A alone as daily, subannual & annual coefficiens elemens of D, S and A by D, S and A 29

31 Daa Preparaion monioring sysem a Wivenhoe Dam designed o measure waer emperaure and oher variables a dephs of, 2,..., 20 m every wo hours (will concenrae on, 5, 0, 5 and 20 m as represenaive dephs) proocol successfully adhered, wih he excepion of some gaps in he daa some jier in collecion imes (unlikely o impac analysis) can fill in gaps using a sochasic inerpolaion scheme also need o pay aenion o how o handle boundary condiions for MODWT and DSA ransforms 30

32 Degrees C 0 m Waer Temperaures wih Gaps

33 Degrees C Gap-filled 0 m Waer Temperaures

34 Degrees C Gap-filled 0 m Waer Temperaures

35 Degrees C Gap-filled 0 m Waer Temperaures

36 Degrees C Degrees C Degrees C DSA Decomposiion for m Waer Temperaures D = daily S = subannual A = annual 35

37 Degrees C Degrees C Degrees C DSA Decomposiion for 5 m Waer Temperaures D = daily S = subannual A = annual 36

38 Degrees C Degrees C Degrees C DSA Decomposiion for 0 m Waer Temperaures D = daily S = subannual A = annual 37

39 Degrees C Degrees C Degrees C DSA Decomposiion for 5 m Waer Temperaures D = daily S = subannual A = annual 38

40 Degrees C Degrees C Degrees C DSA Decomposiion for 20 m Waer Temperaures D = daily S = subannual A = annual 39

41 Relaive Imporance of Three Componens disance beween adjacen verical ick marks on all plos is 5 degrees Celsius in erms of overall (global) variabiliy in each X, annual componen A is obviously dominan daily componen D appears o conribue leas, bu here are local sreches over which i has greaer variabiliy han S can quanify relaive conribuion of D, S and A coefficiens o variance of each X globally using DSA-based ANOVA: where ˆσ 2 D = 3X j= ˆσ 2 X = ˆσ2 D + ˆσ2 S + ˆσ2 A, ˆσ 2 j, ˆσ2 S = 9X j=4 ˆσ 2 j and ˆσ2 A = ˆσ2 0 40

42 Wavele-based ANOVA for Waer Temperaures ˆσ 2 j s o lef (righ) of verical dashed line conribue o ˆσ2 D (ˆσ2 S ) wavele variances m 5 m 0 m 5 m 20 m j

43 DSA-based ANOVA of Waer Temperaure o m 5 m 0 m 5 m 20 m ˆσ 2 D ˆσ 2 S ˆσ 2 A ˆσ 2 X ˆσ D ˆσ S ˆσ A ˆσ X

44 Noes on ANOVA paern of ˆσ j 2 s a 5 and 20 m quie similar, as are hose for and 5 m (wih some divergence a j = 6, 7 & 8) paern for 0 m represens a ransiion beween shallower and deeper dephs gross paerns are largely he same across all dephs: increase from j = o j = 3, followed by a drop beween j = 3 & 4, and endency o increase afer ha fundamenal frequency of daily variaions evidenly more imporan han harmonics variance of annual coefficiens A is one or wo orders of magniude greaer han ha of subannual coefficiens S variance of S is greaer han ha of D by a leas a facor of 2 43

45 Degrees C Annual Componen for m Waer Temperaures

46 Degrees C Annual Componen for 5 m Waer Temperaures

47 Degrees C Annual Componen for 0 m Waer Temperaures

48 Degrees C Annual Componen for 5 m Waer Temperaures

49 Degrees C Annual Componen for 20 m Waer Temperaures

50 Noes on Annual Componens overall heigh decreases wih deph imes of peaks in boh 2008 and 2009 and of valley in 2008 arrive a laer imes he deeper he deph imes of peaks in 2008 occur beween sar of February and sar of March (span of one monh) imes of peaks in 2009 occur beween sar of January and sar of April (span of hree monhs) ime span beween 2008 & 2009 peaks increases wih deph (.5 monhs for & 5 m, increasing o 3 monhs a 20 m) imes of peaks for & 5 m closely linked in boh 2008 & 2009 annual componen in 2008 a each deph does no repea iself in

51 Degrees C squared 30-day Smoohed D 2 a m wih 95% CIs

52 Degrees C squared 30-day Smoohed D 2 a 5 m wih 95% CIs

53 Degrees C squared 30-day Smoohed D 2 a 0 m wih 95% CIs

54 Degrees C squared 30-day Smoohed D 2 a 5 m wih 95% CIs

55 Degrees C squared 30-day Smoohed D 2 a 20 m wih 95% CIs

56 Degrees C squared 30-day Smoohed S 2 a m wih 95% CIs

57 Degrees C squared 30-day Smoohed S 2 a 5 m wih 95% CIs

58 Degrees C squared 30-day Smoohed S 2 a 0 m wih 95% CIs

59 Degrees C squared 30-day Smoohed S 2 a 5 m wih 95% CIs

60 Degrees C squared 30-day Smoohed S 2 a 20 m wih 95% CIs

61 Degrees C squared 30-day Smoohed D 2 a, 5, 0, 5 & 20 Meers

62 Degrees C squared 30-day Smoohed S 2 a, 5, 0, 5 & 20 Meers

63 Noes on Time-Varying Variances of D and S variance of D a m is relaively sable across ime, bu no a lower dephs opposie paern holds for S: hree lower dephs have more homogeneous variances han wo shallower ones 62

64 Global Cross-Correlaions Beween DSA Coefficiens 5 ses of coefficiens in all (D, S and A a 5 dephs) here are 5 2 = 05 cross-correlaions o consider 75 are beween-ype cross-correlaions, i.e., involving differen ypes of coefficiens eiher a same deph or differen dephs beween-ype cross-correlaions generally small: 6 are beween 0. & 0.5, and remaining 69 are beween 0.03 and 0. lends credence o claim ha DSA ransform is separaing X ino differen ypes of coefficiens (D, S and A) ha are approximaely uncorrelaed remaining 30 cross-correlaions are wihin-ype 63

65 Global Wihin-Type Cross-Correlaions D m 5 m 0 m 5 m 5 m m m m S m 5 m 0 m 5 m 5 m m m m A m 5 m 0 m 5 m 5 m m m m X m 5 m 0 m 5 m 5 m m m m

66 Monh-by-monh Correlaions for D wih 95% CIs m 5 m 0 m 5 m 0 5 m 0 0 m 0 5 m 0 20 m 65

67 Monh-by-monh Correlaions for S wih 95% CIs m 5 m 0 m 5 m 0 5 m 0 0 m 0 5 m 0 20 m 66

68 Noes on Monh-by-Monh Cross-Correlaions cross-correlaions for D end o be smaller and less ime dependen han hose for S in paricular, small cross-correlaions beween D a m and deeper dephs indicae lile direc daily co-emporaneous variaions beween emperaures near surface and a deeper levels cross-correlaions for S have sreches of high correlaion, e.g., beween 5 and 20 m from Feb o Sep 2008, followed by gradual decline (period also associaed wih decreased variabiliy a boh dephs) periods of high correlaion well aligned wih known periods of sraificaion 67

69 Concluding Remarks: I bigges conribuor o variance of X is A (annual coefficiens) eviden from jus 600 days of daa ha A can vary considerably from year o year (migh be able o idenify explanaory covariaes from sparsely sampled hisorical daa) nex bigges conribuor is S (subannual coefficiens), while D (daily) is smalles S is more homogeneous in variabiliy a deeper dephs, whereas D is mos homogeneous a m (can inerpre in erms of influence of amospheric condiions) global saisics do no necessarily reflec localized paerns in various X, poining o advanages of curren sampling scheme and of localized measures such as he DSA ransform 68

70 Concluding Remarks: II DSA approach is largely descripive, bu addresses some quesions ha could be answered also by formal saisical models pleny of opporuniy for fuure work, including sudy of oher waer qualiy indicaors (chlorophyll-a, urbidiy, dissolved oxygen, specific conduciviy) colleced by he profiling sysem and heir relaionship o emperaure 69

71 Thanks o... Sarah Lennox for organizing his seminar numerous folks a CSIRO who made my visi possible Seqwaer for opporuniy o analyze heir daa 70