Assimiltion o se-surce temperture into hydrodynmic model o Port Phillip By, Austrli Mtthew R.J. urner 1, Jerey P. Wlker 1, Peter R. Oke 2 nd Rodger B. Gryson 1 1 Deprtment o Civil nd Environmentl Engineering, he University o Melbourne 2 CSIRO Mrine Reserch, Hobrt, smni Abstrct Hydrodynmic models cn predict sttes o interest to the costl engineer, however, due to uncertinties in the model physics, model prmeters, initil conditions, nd model orcing dt, lrge errors in prediction oten result. o counter this, n ensemble sequentil dt ssimiltion scheme hs been pplied to the Model or Esturies nd Costl Ocens (MECO), to constrin model predicted wter temperture with remotely sensed se-surce temperture observtions. his pper describes series o synthetic twin experiments tht contrst two ensemble sequentil dt ssimiltion schemes, the Ensemble Klmn Filter (EnKF) nd the Ensemble Squre Root Filter (EnSRF) in both one nd three dimensionl orms. he experiments show tht the ssimiltion gretly improves the model prediction. he three dimensionl orm outperorms the one dimensionl orm, nd tht the EnSRF outperormed the EnKF signiicntly in the one dimensionl orm but only mrginlly in the three dimensionl orm. 1 Introduction Dt ssimiltion is sttisticl technique, which combines model orecst nd n observtion to estimte the true stte o the phenomenon being predicted. his pper presents n introduction to the concepts o sequentil dt ssimiltion nd compres two well-known techniques rom the literture in costl ppliction. Synthetic surce temperture ields (SS) re ssimilted into costl hydrodynmic model nd show signiicnt improvement in the prediction cpbility o the model. he beneits o using dt ssimiltion re its bility to improve model prediction. his is o importnce in short rnge orecsting where prediction o uture stte is desired. In costl setting this could be where will be the loction o n lgl blooms most likely be, or wht concentrtion o suspended sediment should we expect t given loction two dys hence, while in ports setting improved prediction o wter levels or wve ields hve obvious beneits or the sety nd relibility o shipping. Other beneits o dt ssimiltion re tht n investigtion o the dt ssimiltion nlysis cn point to potentil model deiciencies. For instnce, i the ssimiltion lwys corrects the model in certin direction this is suggestive o poor model prmeteristion. hus dt ssimiltion provides eedbck tht ids model improvement. Accurte prediction o wter temperture is o prticulr importnce in ecologicl modelling, where temperture inluences growth prmeters. Unortuntely, ccurte prediction o wter temperture is not lwys possible, especilly in highly enclosed wter bodies where tmospheric exchnge is the dominnt driver o wter temperture. Errors in wter temperture prediction re due to our poor understnding nd conceptulistion o thermodynmics nd hydrodynmics, s well s uncertinties in initil conditions, model prmeters, nd model orcing dt. Model prediction errors cn be reduced by using dt ssimiltion. Dt ssimiltion combines the model predicted sttes with observtions bsed on their reltive uncertinties. he result o dt ssimiltion is new set o stte estimtes tht re closer to the truth nd hve lower level o uncertinty thn either dt set (model or observtions) individully. Dt ssimiltion techniques hve been widely pplied in meteorology nd ocenogrphy (Gill nd Mlnotte- Rizzoli, 1991), but ssimiltion o se-surce temperture (SS) into costl ocen models hs received r less ttention. Applying dt ssimiltion to costl models hs become incresingly ccessible due to recent dvnces in computing power nd the lunch o new stellite observing systems. However, while mny dt ssimiltion pproches exist in the literture, it is not cler which o these re best suited to by nd estury modelling. hereore, s well s introducing dt ssimiltion generlly, this pper explores the chrcteristics o two ensemble sequentil dt ssimiltion techniques the Ensemble Klmn Filter (EnKF) nd the Ensemble Squre Root Filter (EnSRF). he key dierence between these two dt E 0 25 km MELBOURNE " POR PHILLIP BAY Figure 1: Loction digrm or Port Phillip By, south estern Austrli.
ssimiltion techniques is their tretment o observtions. hese two techniques re contrsted in series o synthetic twin experiments in both one dimensionl (where only the verticl correltion o the wter column is considered or individul model cells nd the ssimiltion is perormed cell wise through the model domin) nd three dimensionl (where the sptil correltion o model sttes is considered nd the entire model domin is modiied by the ssimiltion in single step) orm. he experiments re undertken or Port Phillip By locted in south estern Austrli (Figure 1). 2 heoreticl Bckground It is well understood tht both model predictions nd observtions o physicl stte re oten prone to error. For models the mthemticl equtions representing the phenomenon re simpliiction o ctul processes; computing power limits the sptil nd temporl resolution nd uncertinties ssocited with boundry nd initil conditions ll combine to produce uncertinties in the model prediction. In the costl mrine setting two dt types re most commonly vilble; point source nd remotely sensed. While point mesurements re usully reltively ccurte t the locl scle, extending these dt introduces uncertinties o scle. In contrst, remotely sensed dt provide good sptil coverge, but only or shllow surce lyer t n instnt in time, nd re sensitive to tmospheric eects nd errors in the lgorithm used to relte the mesurements to the physicl stte being observed. While n estimte o the sptil nd temporl vrition in wter temperture cn be mde bsed on either model predictions or observtions lone, both re ected by dierent types o uncertinty. Observtions, lthough subject to errors my be ccurte: models give temporl stte estimtes or the entire domin which observtions cn not. Sequentil dt ssimiltion combines the model predictions nd observtions to chieve n improved estimte o the physicl stte. he lgorithm or sequentil dt ssimiltion is s ollows. Strting rom best estimte o the physicl stte, model run predicts the physicl stte in the uture. When n observtion becomes vilble the model is stopped. he model stte t this point becomes the orecst, or bckground ield. Bsed on the reltive uncertinty in nd the dierence between the observtion nd the orecst; nd the covrinces between the orecst nd observtion errors, correction is clculted or the model stte. his correction is dded to the orecst to give the nlysis. he model is then reinitilised using the nlysis, nd is run orwrd until nother observtion becomes vilble nd the process repeted. As the nme suggests, the observtions re sequentilly ssimilted into the model. he most well known sequentil dt ssimiltion technique is the stndrd Klmn Filter (Evensen 2003) given by x = x + K( d Hx ), (1) where x is vector o the model predictions, with superscripts nd denoting nlysis nd orecst respectively, d is vector o observtions nd H is mtrix tht mps the model stte x to the observtions d ; K is weighting mtrix known s the Klmn gin given by 1 K = PH ( HPH + R), (2) where P is the orecst error covrince mtrix tht quntiies the covrinces o the uncertinties o the model stte, nd R is the observtion error covrince mtrix tht quntiies the covrinces o the uncertinties ssocited with the observtions. he inluence o the Klmn gin on the nlysis cn be seen by considering n exmple where P nd R re sclrs. I the uncertinty ssocited with the model is less thn the uncertinty ssocited with the observtions, P << R, then ollowing eqution (2) K pproches zero nd in eqution (1) more relince is plced on the model orecst. Conversely, i the observtions re more certin thn the model orecst, R << P, K pproches unity nd more relince is plced on the observtions. An dvntge o the Klmn Filter over other sequentil dt ssimiltion techniques, such s direct insertion, is tht the sttisticl reltionship between stte elements enbles the ilter to updte not just those stte elements tht re observed, but lso other unobserved stte elements tht my be dierent vribles nd t dierent loctions. For instnce, stellites typiclly mesure SS, however using the Klmn Filter SS observtions cn be used to modiy ll other stte elements, including temperture t depth, slinity, se-level nd currents. Becuse the Klmn Filter is liner it does not del stisctorily with highly nonliner models. In n ttempt to overcome this limittion, Evensen (2003) introduced the Ensemble Klmn Filter (EnKF), whereby covrince error sttistics were obtined through the use o n ensemble o model orecsts. hus rther thn one model run being propgted through time n ensemble o model runs re mde. Ech run strts rom slightly dierent position, relecting the uncertinty ssocited with the model initil conditions nd ech model is orced by slightly dierent orcing dt relecting the uncertinty ssocited with the orcing dt. Model error is incorported too. he result is tht when n observtion becomes vilble or nlysis the ensemble o orecsts will hve spred representing the uncertinty ssocited with the orecst. his is illustrted in Figure 2 where ech solid dot represents
most respects similr to the EnKF, but does not require perturbed observtions. Ater the nlysis o observtions the nlysis error covrince should be P = + ( I KH) P ( I KH) KRK, (4) Figure 2 Grphic representtion o sequentil ssimiltion. Dots represent ensemble members, both initil/nlysed vlues (solid) nd orecst vlues (open). he reltive spred o n ensemble indictes its uncertinty. one ensemble member nd in combintion represent the uncertinty ssocited with prticulr model stte. Ech ensemble member is propgted through time by the model to give orecst (open dot) when n observtion becomes vilble. Due to the uncertinties in the model nd orcing conditions the uncertinty o the orecst hs spred. he ssimiltion reduces the spred o the ensemble members (solid dots) indicting reduction in uncertinty ssocited with the nlysed stte. he nlysed vlues re used to initite the next orecst nd the process repets itsel. In ensemble orm eqution (1) becomes X = X + P H ( HP H e e + R) 1 ( D HX ), (3) where X is n ensemble mtrix o n model stte relistions, X = [ x1, x2, x3,..., xn ] ; nd D is n ensemble mtrix o observtions. his ensemble is creted by dding n relistions o rndom perturbtions to the vector o observtions d. he orecst error covrince mtrix is pproximted or the model o ensemble predictions by X' X' P e =, (3) n 1 where X ' is mtrix o the ensemble perturbtions o X bout men x. An ensemble pproximtion o observtion error covrinces is lso possible, but hs well understood problems (Kepert 2004), nd is not employed here. Becuse o smpling error introduced through the use o perturbed observtions in D, the EnKF ormultion is expected to be less ccurte thn one tht does not require perturbed observtions (Whitker nd Hmill, 2002). In response to this Whitker nd Hmill (2002) proposed the Ensemble Squre Root Filter, which is in where P is the pre-nlysis orecst error covrince. I the observtions in eqution (3) re not perturbed the post-nlysis orecst error covrince P will be underestimted, s the lst term o eqution (4) disppers (Whitker nd Hmill, 2002). While the EnSRF does not use perturbed observtions it voids underestimting the post-nlysis orecst error covrince, P, by updting the perturbtion mtrix X ' nd the ensemble men x seprtely. he ensemble men is updted by the usul Klmn gin K given in eqution (2), while the perturbtions re updted by new gin mtrix ~ 1 K = P H (( HP H + R ) ( R 1 HP H + R + ). (5) In this pper we consider the sequentil ssimiltion ppliction in both the one nd three dimensionl orms. In the one dimensionl orm, ech observtion is ssimilted independently modiying the temperture o the wter column directly beneth it. Only the ensemble perturbtions rom the model cells beneth the observtion re used to generte the orecst error covrince mtrix. he dvntge o the one dimensionl pproch is tht by using single observtions the R nd HP H mtrices collpse to sclrs. his mens tht the computtionl cost o this ssimiltion orm is signiicntly less thn the three dimensionl ssimiltion orm. In contrst, the three dimensionl orm ssimiltes ll the observtions together to derive correction to wter temperture throughout the entire model domin. Solving this requires mtrix inversions o lrge mtrices (2000+ elements) but hs the dvntge tht ll the error covrince inormtion is shred throughout the domin. 3 Experimentl Setup he dt ssimiltion experiments described in this pper re illustrted through their ppliction to cse study. he cse study loction is Port Phillip By, shllow (~20m depth) highly enclosed bsin locted in south estern Austrli. here re three sources o heting nd cooling or the by. hese re the tidl exchnge o wter with the open se through the heds, tmospheric het luxes, nd riverine inputs o wter. O these the riverine inluence is miniml nd the tmospheric het luxes re the predominnt heting nd cooling mechnism. )
i!( # Long Ree 1!(!( LAVERON RAAF Hobsons By # # MOORABBIN AIRPOR Centrl # POIN WILSON FRANKSON AWS rom initilly clm conditions, nd the models run or period o 40 dys. he hydrodynmic modelling ws undertken using the CSIRO Model or Esturies nd Costl Ocens (MECO). MECO is inite dierence hydrodynmic model bsed on the three dimensionl equtions o momentum, continuity, nd conservtion o het nd slt, utilising the hydrosttic nd Boussinesq ssumptions (Wlker et l, 2002). 0 4.5 9 18 27 36 Kilometers Figure 3: Loction o monitoring sites (circles) nd wether sttions (tringles) used in the modelling. idl exchnge is limited by the nrrow entrnce t the heds. he synthetic dt ssimiltion experiments conducted here use twin experiment technique to exmine the improvement in prediction cpbility. he bsis o this technique is to compre how close degrded model pproches its true twin when observtions re ssimilted. he ollowing set-up is used. An initil truth model run ws undertken using tmospheric dt collected predominntly t Point Wilson (see Figure 3). Dt t Lverton were used or dt types not collected t Point Wilson. Snpshots o the surce (top 1m) lyer were extrcted t n intervl o two dys to crete set o stellite observtions. hese were degrded through the ddition o sptilly uncorrelted noise with stndrd devition o 0.3 C, which is in the rnge o the error ssocited with stellite observed SS (Brown nd Minnett, 1999). A second model run ws set up using tmospheric orcing predominntly rom Frnkston (Figure 3) with dt rom Moorbbin or dt types not collected t Frnkston. Evportion dt ws common to both model runs s only one sttion collected it. his dierent tmospheric orcing dt ws used to represent the typicl uncertinty ssocited with model orcing dt. he initil wter temperture ws set 1 C wrmer to represent the uncertinty ssocited with speciying model initil conditions. his second model run, termed the open loop run, indictes wht prediction perormnce would be expected i there ws no ssimiltion. Four dt ssimiltion runs were perormed bsed on the second model set-up, with ech run diering by the dt ssimiltion technique nd orm used. he ensemble initilistion nd propgtion ollowed the procedure outlined in urner et l (2005). In ll cses 14 ensemble members were used s trde-o between ccurcy nd computtionl time. In both the truth nd open loop runs n initil 10 dy model spinup ws undertken to produce relistic current ields he originl MECO thermodynmics ormultion ws used in ll simultions. his ormultion computes bulk vlues o the components o the energy blnce. he net het lux due to longwve nd shortwve rdition together with sensible nd ltent het lux is used to djust the surce lyer temperture in this ormultion. Heting eects due to shortwve rdition bsorption through the wter column re lso included. An exmple o model output indicting currents superimposed over wter elevtion is presented in Figure 4. his igure shows some o the chrcteristics o Port Phillip By which initited this study. he nrrow entrnce to the by constricts the low, s illustrted by the high velocities t the entrnce. his limits exchnge between Port Phillip By nd Bss Strit to the southern portion o the by; pproximtely the re in Figure 4 covered by the dense concentrtion o low rrows. While MECO llows or the use o sptilly vrying tmospheric inputs, sptilly uniorm inputs hve been used throughout this pper. he justiiction or this is tht the sptil extent is not so lrge s to wrrnt the complictions o generting sptilly vrying wind ields, nd other tmospheric inputs re not expected to vry signiicntly. 4 Results o illustrte the experiment results, time series o wter temperture hve been extrcted rom three comprison sites (see Figure 3) t dierent loctions within the wter column in ech cse (Figure 4). hese igures demonstrte tht the dt ssimiltion is ble to positively impct the deeper unobserved lyers o the wter column in ddition to the observed surce lyer. Ech igure compres the open loop nd ensemble men predicted by the dierent dt ssimiltion techniques with the truth. Consider irst the dierence between the truth nd the open loop in ech o the three cses. he initil dierence in wter temperture prediction is due to the 1 C rise mde to the degrded model initil condition. Over time the grphs mirror ech other nd. move closer. his is n eect o the initil condition error being diminished s result o little or no error in the open boundry nd tmospheric orcing, nd the sme model physics (heting nd cooling) used or both scenrios. While tmospheric
Dte: 16 August 2002 00:00:00, Depth: 1 m. 50' Currents (ms-1) 55' 38 S 5' 10' 15' 20' 0.5 0.2 0.1 0.05 5' 10' 15' 20' 25' 30' 35' 40' 45' 50' 55' 145 E 5' 10' 15' Figure 4: Smple o output rom MECO model. Shding indictes wter elevtion, while rrows indicte surce currents during n ebb tide. dt sets re tken rom dierent loctions they re not so dierent to produce wildly diering predictions. For ll three comprison loctions the introduction o observtions through dt ssimiltion signiicntly improves the model wter temperture orecst. Any initil discrepncy between the open loop nd the ensemble men is due to the ensemble genertion technique. he initil ssimiltion genertes signiicnt improvements in prediction with subsequent ssimiltion times resulting in smller corrections. As the observtion error is constnt throughout the ssimiltion period, the mount o improvement mde by the ssimiltion is dependent upon the uncertinty ssocited with the model prediction. his in turn is unction o the spred o the ensemble members bout the ensemble men nd is clculted using eqution (3). A comprison between the dierent iltering techniques nd open loop run is mde by contrsting the root men squred (RMS) dierence between the ensemble men orecst nd the truth or the entire model domin (Figure 5). his shows tht the initil verge orecst error is bout 0.8 C; signiicntly worse thn the prescribed observtion error o 0.3 C; the initil 1 C initil condition dierence ws reduced during the spin-up period. Moreover, ssimiltion signiicntly improved the model predictions both in terms o bsolute RMS nd reltive to the open loop. Contrsting the one dimensionl exmples; the EnSRF perormed signiicntly better thn the EnKF. he poor perormnce o the one dimensionl EnKF is due to smpling error in the perturbed observtions, which is mgniied by the one dimensionl orm. Using three dimensionl ssimiltion orm gve signiicntly better results thn the one dimensionl ssimiltion orm, consequence o more inormtion being vilble to the three dimensionl orm. Both techniques ppered eqully s eective in the three dimensionl orm, lthough the EnSRF perormed slightly better over the irst 10 dys o the ssimiltion period. Smpling error in the EnKF ppers to be reduced by verging over time. he improvements in prediction cn be clculted reltive to the open loop prediction (Figure 5) by Surce elevtion 0.75 0.5 0.25 0-0.25-0.5-0.75 temperture [degrees C] ) temperture [degrees C] b) temperture [degrees C] c) 15 14 13 12 ruth 11 time [dys since 01-01-1990] 14 13.5 13 12.5 12 11.5 ruth 11 time [dys since 01-01-1990] 13 12.5 12 11.5 11 10.5 10 ruth time [dys since 01-01-1990] Figure 4: ime series o predicted wter temperture or the open loop nd ssimilted model runs s compred to the truth run or () Long Ree monitoring loction or the surce (depth 1m); (b) Hobsons By monitoring loction or the middle (depth 5m); nd (c) Centrl monitoring loction or the bottom (depth 23m). dividing the dierence between the RMS error o the ssimiltion nd the RMS error o the open loop by the RMS error o the open loop prediction. Both three dimensionl ssimiltion techniques reched 90% improvement over the ssimiltion period, while the one dimensionl EnSRF verged 80-90% nd the one
rms error [degrees C] 0.8 0.6 0.4 0.2 0 time [dys since 01-01-1990] Figure 5: RMS dierence between ensemble men prediction nd truth or the entire model domin. dimensionl EnKF verged 60-70% over the ssimiltion period. 5 Discussions he results obtined through the twin experiments show tht the three dimensionl orm perormed signiicntly better thn the one dimensionl orm. his result is not unexpected. By using three dimensionl orm the ilter is ble use the sptil reltionships inherent in the model orecst to produce n improved updte. Although both ilter types perormed well in the three dimensionl orm, the EnSRF perormed slightly better. It hd better prediction or the irst 10 dys o the ssimiltion period nd s good prediction or the reminder o the period. Moreover, it predicted the estimte o RMS error slightly better thn the EnKF. his is due to the smll number o ensembles (14 in this ppliction), which in combintion with the smpling error inherent in the EnKF reduces its perormnce. With longer ssimiltion run or more ensemble members the EnKF nd EnSRF re expected to give equivlent results, which ws shown with the three dimensionl EnKF pproching the EnSRF over time (Figure 5). However, the number o ensemble members is the vrible which most signiicntly inluences computtionl costs, with the ssimiltion step being reltively inexpensive. Still, these tests cnnot be considered thorough enough to stte with certinty tht the EnSRF is the better choice. he experiments hve shown tht signiicnt improvements to model predictions o wter temperture re possible through the ssimiltion o surce lyer observtions. hese experiments however, hve been perormed in n rtiicil environment nd the sme level o perormnce cnnot be expected in rel cse. One o the min limittions o this study is tht the sme model equtions were used to crete the observtions nd truth dt s or the ssimiltion nd open loop orecsts. his mens the model ws conditioned to perorm well reltive to the truth. In relity, the physicl process occurring cnnot be modelled exctly nd in rel cse more errors will be introduced in this wy. 6 Conclusions wo sequentil dt ssimiltion techniques hve been compred in both one nd three dimensionl orms. While the three dimensionl orm is superior to the one dimensionl orm, the EnSRF perorms only slightly better thn the EnKF nd the EnSRF is less ected by dimensionlity. While the perormnce o both ilters is dmirble, n ppliction using rel observtions is necessry to better pprecite the improvements ctully relised rom dt ssimiltion. his pper hs demonstrted the signiicnt potentil o dt ssimiltion techniques to improve the relibility nd ccurcy o model prediction. All o the ssimiltion techniques used gve signiicnt improvement over the control without ssimiltion. he techniques described re, more generlly, pplicble to other costl mrine modelling settings nd will improve ny costl orecsting system. 7 Acknowledgements he work undertken in this pper hs been unded by University o Melbourne CSIRO Collbortive Grnt nd Melbourne Reserch Scholrship. he modelling dt ws provided by Melbourne Wter, Melbourne Ports Authority, Bureu o Meteorology nd MAFRI. 8 Reerences Brown, O.B. nd P.J. Minnett (1999) MODIS inrred se surce temperture lgorithm theoreticl bsis document, Version 2, University o Mimi. Evensen, G. (2003) he ensemble Klmn ilter: theoreticl ormultion nd prcticl implementtion, Ocen Dynmics, Vol 53, 343-367. Ghil, M. nd P. Mlnotte-Rizoli (1991) Dt ssimiltion in meteorology nd ocenogrphy, Advnces in Geophysics, Vol 33, 141-266. Houtekmer, P.L nd H.L. Mitchell (1998) Dt ssimiltion using n ensemble Klmn ilter technique, Monthly Wether Review, Vol 126, 796-811. Kepert, J.D. (2004) On ensemble representtion o the observtion-error covrince in the ensemble Klmn ilter, Ocen Dynmics, (In Press) urner, M.R.J., J.P. Wlker nd P.R. Oke, (In preprtion) Ensemble member genertion or sequentil dt ssimiltion, Ocen Modelling. Wlker, S.J., J.R. Wring, M. Herzeld nd P. Skov (2002) Model or Esturies nd Costl Ocens: User Mnul, Version 4.01, CSIRO Mrine, Hobrt. Whitker, J.S nd.h. Hmill (2002) Ensemble dt ssimiltion without perturbtions. Monthly Wether Review, Vol. 130, 1913 1924.