Assimilation of sea-surface temperature into a hydrodynamic model of Port Phillip Bay, Australia
|
|
- Kelly Rice
- 6 years ago
- Views:
Transcription
1 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.
2 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
3 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. )
4 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) 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
5 Dte: 16 August :00:00, Depth: 1 m. 50' Currents (ms-1) 55' 38 S 5' 10' 15' 20' ' 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 temperture [degrees C] ) temperture [degrees C] b) temperture [degrees C] c) ruth 11 time [dys since ] ruth 11 time [dys since ] ruth time [dys since ] 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
6 rms error [degrees C] time [dys since ] 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, Ghil, M. nd P. Mlnotte-Rizoli (1991) Dt ssimiltion in meteorology nd ocenogrphy, Advnces in Geophysics, Vol 33, Houtekmer, P.L nd H.L. Mitchell (1998) Dt ssimiltion using n ensemble Klmn ilter technique, Monthly Wether Review, Vol 126, 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,
7
DATA ASSIMILATION IN A FLOOD MODELLING SYSTEM USING THE ENSEMBLE KALMAN FILTER
CMWRXVI DATA ASSIMILATION IN A FLOOD MODELLING SYSTEM USING THE ENSEMBLE KALMAN FILTER HENRIK MADSEN, JOHAN HARTNACK, JACOB V.T. SØRENSEN DHI Wter & Environment, Agern Allé 5, DK-2970 Hørsholm, Denmr ABSTRACT
More informationData Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading
Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationWeight interpolation for efficient data assimilation with the Local Ensemble Transform Kalman Filter
Weight interpoltion or eicient dt ssimiltion with the Locl Ensemble Trnsorm Klmn Filter Shu-Chih Yng 1,2, Eugeni Klny 1,3, Brin Hunt 1,3 nd Neill E. Bowler 4 1 Deprtment o Atmospheric nd Ocenic Science,
More informationPart I: Basic Concepts of Thermodynamics
Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry
More informationMonte Carlo method in solving numerical integration and differential equation
Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationCBE 291b - Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationIncorporating Ensemble Covariance in the Gridpoint Statistical Interpolation Variational Minimization: A Mathematical Framework
2990 M O N T H L Y W E A T H E R R E V I E W VOLUME 38 Incorporting Ensemble Covrince in the Gridpoint Sttisticl Interpoltion Vritionl Minimiztion: A Mthemticl Frmework XUGUANG WANG School of Meteorology,
More informationNew data structures to reduce data size and search time
New dt structures to reduce dt size nd serch time Tsuneo Kuwbr Deprtment of Informtion Sciences, Fculty of Science, Kngw University, Hirtsuk-shi, Jpn FIT2018 1D-1, No2, pp1-4 Copyright (c)2018 by The Institute
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationA027 Uncertainties in Local Anisotropy Estimation from Multi-offset VSP Data
A07 Uncertinties in Locl Anisotropy Estimtion from Multi-offset VSP Dt M. Asghrzdeh* (Curtin University), A. Bon (Curtin University), R. Pevzner (Curtin University), M. Urosevic (Curtin University) & B.
More informationNOTES AND CORRESPONDENCE. Ensemble Square Root Filters*
JULY 2003 NOES AND CORRESPONDENCE 1485 NOES AND CORRESPONDENCE Ensemble Squre Root Filters* MICHAEL K. IPPE Interntionl Reserch Institute or Climte Prediction, Plisdes, New Yor JEFFREY L. ANDERSON GFDL,
More informationSession 13
780.20 Session 3 (lst revised: Februry 25, 202) 3 3. 780.20 Session 3. Follow-ups to Session 2 Histogrms of Uniform Rndom Number Distributions. Here is typicl figure you might get when histogrmming uniform
More informationNon-Linear & Logistic Regression
Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationTests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More informationTopic 6b Finite Difference Approximations
/8/8 Course Instructor Dr. Rymond C. Rump Oice: A 7 Pone: (95) 747 6958 E Mil: rcrump@utep.edu Topic 6b Finite Dierence Approximtions EE 486/5 Computtionl Metods in EE Outline Wt re inite dierence pproximtions?
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More informationFig. 1. Open-Loop and Closed-Loop Systems with Plant Variations
ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses
More informationSynoptic Meteorology I: Finite Differences September Partial Derivatives (or, Why Do We Care About Finite Differences?
Synoptic Meteorology I: Finite Differences 16-18 September 2014 Prtil Derivtives (or, Why Do We Cre About Finite Differences?) With the exception of the idel gs lw, the equtions tht govern the evolution
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationp-adic Egyptian Fractions
p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS
ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS F. Tkeo 1 nd M. Sk 1 Hchinohe Ntionl College of Technology, Hchinohe, Jpn; Tohoku University, Sendi, Jpn Abstrct:
More informationA New Statistic Feature of the Short-Time Amplitude Spectrum Values for Human s Unvoiced Pronunciation
Xiodong Zhung A ew Sttistic Feture of the Short-Time Amplitude Spectrum Vlues for Humn s Unvoiced Pronuncition IAODOG ZHUAG 1 1. Qingdo University, Electronics & Informtion College, Qingdo, 6671 CHIA Abstrct:
More informationPurpose of the experiment
Newton s Lws II PES 6 Advnced Physics Lb I Purpose of the experiment Exmine two cses using Newton s Lws. Sttic ( = 0) Dynmic ( 0) fyi fyi Did you know tht the longest recorded flight of chicken is thirteen
More informationDriving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d
Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,
More informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationComputation and Modeling of the Air-Sea Heat and Momentum Fluxes. By Dr. Nasser H. Zaker
Computtion nd Modeling of the Air-e et nd Momentum Fluxes By Dr. Nsser. Zker ICTP June 00 1 Introduction The ocen receives energy through the ir-se interfce by exchnge of momentum nd het. The turbulent
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationNOTES ON HILBERT SPACE
NOTES ON HILBERT SPACE 1 DEFINITION: by Prof C-I Tn Deprtment of Physics Brown University A Hilbert spce is n inner product spce which, s metric spce, is complete We will not present n exhustive mthemticl
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More information3 Conservation Laws, Constitutive Relations, and Some Classical PDEs
3 Conservtion Lws, Constitutive Reltions, nd Some Clssicl PDEs As topic between the introduction of PDEs nd strting to consider wys to solve them, this section introduces conservtion of mss nd its differentil
More informationDissolved Oxygen in Streams: Parameter estimation for the Delta method
Dissolved Oxygen in Strems: Prmeter estimtion or the Delt method Rjesh Srivstv Associte Proessor, Deprtment o Civil Engineering, Indin Institute o Technology Knpur, Knpur 0806, Indi. Abstrct The Delt method
More informationClassical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011
Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,
More informationTHE INTERVAL LATTICE BOLTZMANN METHOD FOR TRANSIENT HEAT TRANSFER IN A SILICON THIN FILM
ROMAI J., v.9, no.2(2013), 173 179 THE INTERVAL LATTICE BOLTZMANN METHOD FOR TRANSIENT HEAT TRANSFER IN A SILICON THIN FILM Alicj Piseck-Belkhyt, Ann Korczk Institute of Computtionl Mechnics nd Engineering,
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationWhy symmetry? Symmetry is often argued from the requirement that the strain energy must be positive. (e.g. Generalized 3-D Hooke s law)
Why symmetry? Symmetry is oten rgued rom the requirement tht the strin energy must be positie. (e.g. Generlized -D Hooke s lw) One o the derities o energy principles is the Betti- Mxwell reciprocity theorem.
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More informationProbabilistic Investigation of Sensitivities of Advanced Test- Analysis Model Correlation Methods
Probbilistic Investigtion of Sensitivities of Advnced Test- Anlysis Model Correltion Methods Liz Bergmn, Mtthew S. Allen, nd Dniel C. Kmmer Dept. of Engineering Physics University of Wisconsin-Mdison Rndll
More informationIn-Class Problems 2 and 3: Projectile Motion Solutions. In-Class Problem 2: Throwing a Stone Down a Hill
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Deprtment of Physics Physics 8T Fll Term 4 In-Clss Problems nd 3: Projectile Motion Solutions We would like ech group to pply the problem solving strtegy with the
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationINVESTIGATION OF BURSA, ESKIKARAAGAC USING VERTICAL ELECTRICAL SOUNDING METHOD
INVESTIGATION OF BURSA, ESKIKARAAGAC USING VERTICAL ELECTRICAL SOUNDING METHOD Gökçen ERYILMAZ TÜRKKAN, Serdr KORKMAZ Uludg University, Civil Engineering Deprtment, Burs, Turkey geryilmz@uludg.edu.tr,
More informationMath Lecture 23
Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationThermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report
Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block
More informationBasic model for traffic interweave
Journl of Physics: Conference Series PAPER OPEN ACCESS Bsic model for trffic interweve To cite this rticle: Ding-wei Hung 25 J. Phys.: Conf. Ser. 633 227 Relted content - Bsic sciences gonize in Turkey!
More informationElectron Correlation Methods
Electron Correltion Methods HF method: electron-electron interction is replced by n verge interction E HF c E 0 E HF E 0 exct ground stte energy E HF HF energy for given bsis set HF Ec 0 - represents mesure
More informationTHERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION
XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More information2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More informationPredict Global Earth Temperature using Linier Regression
Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationCONTRIBUTION TO THE EXTENDED DYNAMIC PLANE SOURCE METHOD
CONTRIBUTION TO THE EXTENDED DYNAMIC PLANE SOURCE METHOD Svetozár Mlinrič Deprtment of Physics, Fculty of Nturl Sciences, Constntine the Philosopher University, Tr. A. Hlinku, SK-949 74 Nitr, Slovki Emil:
More informationLAMEPS Limited area ensemble forecasting in Norway, using targeted EPS
Limited re ensemle forecsting in Norwy, using trgeted Mrit H. Jensen, Inger-Lise Frogner* nd Ole Vignes, Norwegin Meteorologicl Institute, (*held the presenttion) At the Norwegin Meteorologicl Institute
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationChapter 2 Fundamental Concepts
Chpter 2 Fundmentl Concepts This chpter describes the fundmentl concepts in the theory of time series models In prticulr we introduce the concepts of stochstic process, men nd covrince function, sttionry
More informationSatellite Retrieval Data Assimilation
tellite etrievl Dt Assimiltion odgers C. D. Inverse Methods for Atmospheric ounding: Theor nd Prctice World cientific Pu. Co. Hckensck N.J. 2000 Chpter 3 nd Chpter 8 Dve uhl Artist depiction of NAA terr
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationChapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses
Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of
More informationMeasuring Electron Work Function in Metal
n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd
More informationMinimum Energy State of Plasmas with an Internal Transport Barrier
Minimum Energy Stte of Plsms with n Internl Trnsport Brrier T. Tmno ), I. Ktnum ), Y. Skmoto ) ) Formerly, Plsm Reserch Center, University of Tsukub, Tsukub, Ibrki, Jpn ) Plsm Reserch Center, University
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationDepartment of Chemical Engineering ChE-101: Approaches to Chemical Engineering Problem Solving MATLAB Tutorial VII
Tutoril VII: Liner Regression Lst updted 5/8/06 b G.G. Botte Deprtment of Chemicl Engineering ChE-0: Approches to Chemicl Engineering Problem Solving MATLAB Tutoril VII Liner Regression Using Lest Squre
More informationChapter 1. Chapter 1 1
Chpter Chpter : Signls nd Systems... 2. Introduction... 2.2 Signls... 3.2. Smpling... 4.2.2 Periodic Signls... 0.2.3 Discrete-Time Sinusoidl Signls... 2.2.4 Rel Exponentil Signls... 5.2.5 Complex Exponentil
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More informationEstimation of the particle concentration in hydraulic liquid by the in-line automatic particle counter based on the CMOS image sensor
Glyndŵr University Reserch Online Conference Presenttion Estimtion of the prticle concentrtion in hydrulic liquid by the in-line utomtic prticle counter bsed on the CMOS imge sensor Kornilin, D.V., Kudryvtsev,
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationCalculus - Activity 1 Rate of change of a function at a point.
Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus
More informationEntropy ISSN
Entropy 006, 8[], 50-6 50 Entropy ISSN 099-4300 www.mdpi.org/entropy/ ENTROPY GENERATION IN PRESSURE GRADIENT ASSISTED COUETTE FLOW WITH DIFFERENT THERMAL BOUNDARY CONDITIONS Abdul Aziz Deprtment of Mechnicl
More informationOrdinary Differential Equations- Boundary Value Problem
Ordinry Differentil Equtions- Boundry Vlue Problem Shooting method Runge Kutt method Computer-bsed solutions o BVPFD subroutine (Fortrn IMSL subroutine tht Solves (prmeterized) system of differentil equtions
More informationGenetic Programming. Outline. Evolutionary Strategies. Evolutionary strategies Genetic programming Summary
Outline Genetic Progrmming Evolutionry strtegies Genetic progrmming Summry Bsed on the mteril provided y Professor Michel Negnevitsky Evolutionry Strtegies An pproch simulting nturl evolution ws proposed
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationVorticity. curvature: shear: fluid elements moving in a straight line but at different speeds. t 1 t 2. ATM60, Shu-Hua Chen
Vorticity We hve previously discussed the ngulr velocity s mesure of rottion of body. This is suitble quntity for body tht retins its shpe but fluid cn distort nd we must consider two components to rottion:
More informationLECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry
LECTURE 14 Dr. Teres D. Golden University of North Texs Deprtment of Chemistry Quntittive Methods A. Quntittive Phse Anlysis Qulittive D phses by comprison with stndrd ptterns. Estimte of proportions of
More informationExperiment 9: DETERMINATION OF WEAK ACID IONIZATION CONSTANT & PROPERTIES OF A BUFFERED SOLUTION
Experiment 9: DETERMINATION OF WEAK ACID IONIZATION CONSTANT & PROPERTIES OF A BUFFERED SOLUTION Purpose: Prt I: The cid ioniztion constnt of wek cid is to be determined, nd the cid is identified ccordingly.
More informationTerminal Velocity and Raindrop Growth
Terminl Velocity nd Rindrop Growth Terminl velocity for rindrop represents blnce in which weight mss times grvity is equl to drg force. F 3 π3 ρ L g in which is drop rdius, g is grvittionl ccelertion,
More informationECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance
Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between
More informationThe Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.
Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,
More information