Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 www.sepub.org/scea Use of Kalman Flerng and Parcle Flerng n a Benzene Leachae Transpor Model Shoou-Yuh Chang, and Skdar Laf * Dep. of Cvl Engneerng, Norh Carolna A&T Sae Unversy, Professor * Dep. of Cvl Engneerng, Norh Carolna A&T Sae Unversy, Research Asssan 60 E Marke S., 438 McNar Hall, NC 74 chang@nca.edu; * rajb_90@yahoo.com Absrac Groundwaer conamnaon s one of he major envronmenal rsks relaed o landflls. Basc nformaon abou he behavor of polluans n sol-groundwaer s needed n order o evaluae he mgraon of leachae from landflls and o esablsh effcen groundwaer monorng sysems. However, n pracce, s very dffcul o ge exac subsurface daa. Thus, modelng he behavor of polluans durng he flow of leachae hrough sol s mporan n predcng he fae of he polluans. In hs sudy, a one-dmensonal ranspor model wh advecon and dsperson was used as he deermnsc model of benzene leachae ranspor from an ndusral landfll. A parcle flerng (PF wh sequenal mporance resamplng (SIR fler and dscree Kalman flerng (KF were proposed o mprove he predcon of he benzene plume ranspor. A radonal roo mean square error (RMSE of benzene concenraon s used o compare he effecveness of he KF, PF, and a convenonal numercal model. The resuls showed ha Kalman flerng ouperformed Parcle flerng n he nal me seps. Boh KF and PF can reduce he error up o 80% n comparson o a convenonal numercal approach. Keywords Leachae Transpor; Groundwaer; Kalman Flerng; Parcle Flerng Inroducon Landflls ha receve muncpal andndusral wase, are one of he mos severe sources of ground waer polluon due o leachae, he lqud produced when waer percolaes hrough sold wase landflls. Some of he more common sol conamnans are chlornaed hydrocarbons (CFH, heavy meals, Mehyl Terary Buyl Eher (MTBE, znc, arsenc, and benzene. Waer and sol conamnaon are mporan pahways of concern for ransmsson of benzene. In he U.S. alone, here are approxmaely 00,000 dfferen ses wh benzene sol or groundwaer conamnaon (Benzene: Encyclopeda 009. Subsurface conamnan ranspor models play an mporan role n explanng how a conamnaon plume evolves, by evaluang he lkely behavor of sysems for conamnan remedaon, predcng how he conamnan wll behave n he fuure, and assessng he rsk of he conamnan accuraly. Mahemacal deermnsc models are wdely used n he subsurface conamnan ranspor process. The predcons by hese models may devae from he real feld value. These unavodable predcon errors may arse from naccurae assumpon of dfferen crera and parameers, space and me lms of numercal schemes, and boundary condons. Wh he sysem model alone, s very dffcul o predc he rue dynamc sae of he polluan. Therefore, observaonal daa s needed o gude he deermnsc sysem model o assmlae he rue sae of he conamnan. The avalable daabases are growng, bu ypcally ncomplee and conan measuremen unceranes. Use of hese daabases for verfcaon of dynamc, mulvarable models s dffcul wh he radonal qualave and deermnsc models. Flerng s he mos applcable o deal wh a low dmensonal sysem wh a well known model and a dense daa base, for whch hghly accurae shor-erm forecass are requred (Schrader and Moore, 977. Alhough convenonal numercal models and esmaon echnques may somemes provde good soluons for many waer qualy problems, flerng echnques combned wh radonal numercal approaches can provde cos-effecve soluons where he consrucon of observaon wells s very cosly and somemes mpossble. Conamnan sae esmaon can be consdered as an opmal flerng problem whn a Bayesan framework. The mos well-known Bayesan sae-esmaon algorhm s he KF whch s very effecve for lnear sysems wh Gaussan dsrbuon. PF s generally appled for nonlnear and non Gaussan cases. Snce several unknown and random hydro-geologcal facors and parameers are 49
` www.sepub.org/scea Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 assocaed wh subsurface conamnan mgraon, he sysem behavor canno be caegorzed no a specfc lnear or nonlnear sysem. Therefore, boh flerng echnques n conjuncon wh a radonal deermnsc numercal model are examned n hs sudy. Boh KF and PF are generally appled n sgnal processng and engneerng, as well as n bology, bochemsry, srucure modelng, geoscences, mmunology, maerals scence, chemcal process modelng, pharmacology, oxcology, and socal scence. For he las hree decades, KF and PF have been appled n surface and subsurface hydrologc sysems and waer qualy modelng ( Van Geer, 98; Cosby e al., 984; Whehead and Hornberger, 984; Yu e al., 989; Graham and McLaughln, 989; Yangxao e al., 99; Zou and Parr, 995; Ferrares and Marnell, 996; Harroun e al., 997; Porer e al., 000;Walker e al., 00; McLaughln, 00; Cheng, 00; Chang and Jn, 005; Rozos and Kousoyanns, 0; L e al., 0; Panzer e al., 03. The objecves of hs sudy are o examne he effecveness of KF and PF n subsurface leachae ranspor modelng and o compare he performance of KF, PF, and a convenonal numercal model n a one dmensonal leachae model. Mehodology The one-dmensonal form of he advecon-dsperson equaon for benzene leachng n sauraed, homogeneous, soropc maerals n unform flow as shown n s descrbed by he followng paral dfferenal equaon (PDE: C D y C V C = ( R y R y where C = solue concenraon, mg/l; = me, d; y = caresan coordnae drecon along he flow lne, m; Dy = dsperson coeffcens n y drecon, m /d; V = lnear velocy of flow feld n he y drecon, m/d; R = reardaon facor, dmensonless. The boundary condons of he one-dmensonal mass ranspor equaon wh an nsananeous pon source are expressed as (a = 0 (Schwarz and Zhang 994: C = 0, a y = 0 and y wh he nal condon (a = 0: M= M0, C=C0, a y = 0; C=0, a 0 < y< ; The analycal soluon for he governng paral dfferenal equaon (PDE s gven by (Schwarz and Zhang, 994: ( y v / R M 0 4Dy/R C( y, e ηa 4πDyR = ( where η s he porosy of sol medum, and A s he cross-seconal area of he one-dmensonal model. M = M 0 and C = C 0 a = 0 y Leachae flow drecon, Vy y = 0 y = L FIG. MASS TRANSPORT FROM A POINT SOURCE Solvng he one-dmensonal subsurface ranspor equaon numercally by forward me cenered-space (FTCS dfference mehod; he followng soluon can be used, C,+ = λc, + λc, + λ3c +, (3 where C,= vecor of polluan concenraon a node and me ; C,+= vecor of polluan concenraon a node and me +; C+= vecor of polluan concenraon a node + and me ; D y V D λ = + ; λ = y ; R R R D y V λ = 3 ; R R = space co-ordnae of node; = me sep co-ordnae. Equaon (3 can be rewren n he followng saespace form: x = + Ax (4 where x+ and x are he sae varables descrbed as vecors of conamnan concenraon a all nodes n he problem doman a me (+ and, respecvely; A s he sae ranson marx whch gves a fne dfference scheme o move a me sep o he nex. One-dmensonal leachae conamnan plumes can be 50
Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 www.sepub.org/scea smulaed by assumng ha he sysem s sochasc. Sochasc processes are used for modelng complcaed real-world ranspor processes wh unceran sources of errors. In order o analyze and make nference abou a dynamc sysem, a leas wo models are requred; a model descrbng he evoluon of he sae wh me (he sysem model and, a model relang he nosy measuremens o he sae (he measuremen model needed o analyze a dynamc sysem (Arulampalam e al., 00. The above advecon-dsperson model of conamnan ranspor s a funcon of he reardaon facor, dsperson and velocy. In he PDE model, s assumed ha he velocy remans consan n he y drecon, he reardaon facor (R s consan, he sol layer s homogenous, and sohermal condon prevals. Furhermore, he sol layer s assumed o be compleely sauraed and benzene s assumed o be a non-reacve conamnan. However, n he feld none of hose ncdences can ake place. Agan he boundary condons may no preval hroughou he ranspor process. Subsurface consss of heerogeneous layers, each exhbng dfferen properes. Conamnan ranspor modelng canno accoun for all of hose phenomena. Ths s why he ranspor process should be smulaed as a sochasc sysem. Our dealsc model conans error wh respec o he prevously menoned assumpons whch conflc wh he acual rue feld value. Agan he paral dfferenal equaon (PDE s solely he funcon of a few parameers lke advecon, dsperson and reardaon. Many oher parameers and facors,.e., degradaon, adsorpon, volaly of solue, sol exure and so on, affec conamnan ranspor n he feld. Any uncerany analyss deals wh random varables. The specfc values of he random varables canno be obaned. Only he sascal naure of he random sysem can be found. In our model, a sandard procedure of sgnal processng (dynamc sae esmaon s followed (Welch and Bshop, 995. Alhough errors vary spaally and emporally, here s no defne mahemacal funcon o relae he errors wh space and me. Therefore, a probablsc approach s followed o ake hose no accoun. To smulae our process or predcon equaon, a random error was njeced no he deermnsc model; whch s he sysem error p. Snce hs error or nose arses from process fne dfference operaor A (sae ranson marx, called he process nose. Ths can be esmaed by he dfference beween he model predcon and he opmal esmae of he rue value. The sysem error p s assumed o have covarance marx Q. Snce he ranspor process occurs n a geographcally specfc area, p s a funcon of he hydrogeology specfc o he area where he ranspor of he polluan akes place, meanng ha he sysem error mus be correlaed regonally. Whou he rue value, he sysem error canno be esmaed. However, he probablsc naure of he error can be esmaed by repeve model calbraon. For example, one can perform he polluan ranspor expermen n a parcular feld by varyng he flow parameers and nal npus. Wh hose resuls, he devaon of expermenal resuls can be predced easly from deermnsc model resuls whch ndcae he sysem error p. By calculang he sandard devaon of sysem errors, he sysem error covarance marx Q (for a geologcally specfc feld can be easly obaned. Ths knd of calbraon s called off-lne samplng. Conwell e al. (997, Chang and Jn (005, and Webser and Olver (99 used a me-ndependen Gaussan sysem error wh σ sys = 8 mg/l. In hs sudy, σ sys =0 mg/l s used as he sandard devaon of he sysem error. Snce our one-dmensonal ranspor model deals wh only one horzonal pon, here s no praccal scope o mplemen regonal nose n he expermenal space doman. The Gaussan dsrbuon or he normal dsrbuon s he mos wdely used famly of dsrbuons n sascs and many sascal ess are based on he assumpon of normaly. When he sysem s hghly nonlnear, here s greaer endency for he sysem o be non-gaussan. Any conamnan ranspor sysem wh an nsananeous polluan source provdes an exponenal concenraon profle. However, our sysem s no hghly nonlnear. Therefore, s more reasonable o assume he nose as Gaussan han oher dsrbuons. Consequenly, a me ndependen Gaussan sysem error wh σ sys = 0 mg/l s njeced no he deermnsc model o consruc he sysem model. The process or sysem equaon can be expressed as, x = + = + Ax p, 034,,,,,... m (5 Here, m = 50 and p are he model sysem error and process nose, respecvely. The model error p s aken from a Gaussan dsrbuon whch has a zero mean and sandard devaon of 0 mg/l. The expeced value of error s zero. E{ p } = 0 and E{ p p T l } = Qδ. The l subscrp of error vecor p denoes he me ndex. Here δ l s a Drac s dela funcon havng a value eher 0 or. δ l = 0f l and δ l = 0f = l. For our case, = l whch gves E { p p } Q. Therefore, Q T = 5
www.sepub.org/scea Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 s a posve-defne marx. For our case, s a n n dagonal marx, and n s he number of nodes n space, where n = 0. Q = dag [ σ ] n n ; σ s he sandard devaon of sysem nose a node and =,,..., 0. We assume σ σ = 0 for our model. = sys To assmlae he acual feld nformaon opmally by flerng approach, observaon daa s requred. Feld measuremens (observaons are usually que lmed and may be used only o calbrae numercal models and o esmae dfferen flow parameers. Observaons n he feld may no be he rue esmae of he sae. For example, wo ypes of observed error can be ncluded here. One s nsrumenal error; dfferen nsrumens servng he same purpose mgh provde dfferen resuls a he same me. Agan dfferen person can measure dfferen value wh he same nsrumen. Therefore, he observaon s affeced wh nose or error whch occurs randomly. Observaonal error can be obaned from he analyss of hsorcal daa and measuremen calbraon. To smulae hs phenomenon, we njeced a random error wh Gaussan dsrbuon havng a sandard devaon of 5 mg/l. Ths s defned as an observaon error or observaon nose, and can be also ermed as measuremen nose. Pnder (973 used observaons n a ground waer modelng sudy of a plume of dssolved Chromum n a sand and gravel aqufer. A compuer model was used o yeld a reasonable reproducon of he observed conamnan plume. The soluon seres was ermed as measuremen daa. Chang and Jn (005 used an assumed rue value and a numercal random scheme o creae he measuremen daa n subsurface conamnan ranspor modelng; as well regonal measuremen nose where he hghes measuremen nose covarance elemen value was 9 mg/l. Observaon represens he rue feld daa, f here s no measuremen or observaon error. In oher words, observaon mus represen he rue sae wh a ceran amoun of accuracy dependng on measuremen accuracy. To underake he smulaon process and o be logscally accepable, an analycal soluon of he governng PDE s aken as he rue value of he conamnan sae. The measuremen nose s added o he analycal resul o make observaon or measuremen models. Alhough n he feld, we usually have fewer observaon locaons wh respec o all smulaon grd pons. For hs sudy, smulaed observaons are used a each of he sudy nodes. Therefore, he observaon daa paern marx H becomes a 0 0 deny marx. For hs model, only one observaon pon n space s adequae. The observaon or measuremen equaon can be expressed as, z = T Hx + O, = 034,,,,,... m (6 Observaon z s a vecor havng he observed sae values of all nodes a me sep. The superscrp T over sae vecor x, denoes he rue value of he sae. Observaon z can be smulaed by daa paern marx H and observaon error O. If we have he same daa a all he nodes we are dealng wh, H wll be an deny marx havng n n dmenson, where n s he oal number of nodes n he model doman. The error vecor O s assumed o be whe nose havng a covarance marx R. The observaon error vecor O s consruced by a vecor havng he elemens from a Gaussan dsrbuon wh a zero mean and a sandard devaon of 5 mg/l. R s a posve-defne marx. For our case, s n n dagonal marx, and n s he number of nodes n space where, n=0. R = dag [ ζ ] n n ; ζ s he sandard devaon of observaon nose a node and =,,..., 0. The deermnaon of he process nose covarance Q s usually very dffcul as ypcally he ably o drecly observe he process esmaed s absen. Somemes, a relavely smple (poor process model can produce accepable resuls f one njecs enough uncerany no he process va he selecon of Q (Welch and Bshop, 995. Process nose covarance Q s he funcon of he hydrogeology of he feld where he polluan ranspor akes place. The devaon of he deermnsc model from he rue sae depends on he accuracy of he parameers used and he heerogeney of he subsurface. If he sol layer s unform and he flow parameers reman consan durng he ranspor, he sysem error p wll be close o zero. Then he process nose covarance Q wll become neglgble. However, he real feld Q mus have some numercal values oher han zero due o he random behavor of he feld geology and flow parameers. To mplemen flerng, he sascal srucure of he process nose mus be deermned for he specfc feld where he conamnan ranspor akes place. An expermen can be desgned o measure Q and provde a parcular marx. Snce s beyond our capacy o predc he process nose covarance Q expermenally, a spaally ndependen Gaussan process nose wh 0 mg/l sandard devaon s seleced o underake he smulaon process. Agan, Q s somemes changes dynamcally durng fler operaon n order o adjus o dfferen dynamcs. However, s a common 5
Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 www.sepub.org/scea pracce o assume ha Q s consan n dynamc sae esmaon problems (Welch and Bshop, 995. In hs sudy, Q s kep consan hroughou he flerng operaon. The measuremen nose covarance R s generally possble and praccal o calculae. The measuremen nose covarance can be predced by examnng he performance of he measurng devce and measurng procedures. On-lne sae esmaon s done by applyng wo flerng approaches. I s usual o ake some off-lne sample measuremens n order o deermne he varance of he measuremen nose whle operang he fler. In hs sudy, measuremen nose O s aken from a Gaussan dsrbuon wh 5 mg/l sandard devaon. The correspondng nose covarance marx s consruced before he operaon of flers. The measuremen nose covarance R s kep consan hroughou he flerng operaon. Daa Assmlaon wh Kalman Flerng Consder a rue subsurface sae varable x. A KF scheme consders he esmae x E o be a lnear combnaon of predced value x P and he observed value x O. x = k x + k x (7 E P O P O Here kp and ko are weghng coeffcens for he x P and x O respecvely. The esmae x E s consdered o be opmal f he mean square error (MSE of x E s mnmum. E MSE = E( x x (8 where E s he expecaon value operaor. If he assumpon of an unbased esmae requremen s sasfed, hen MSEO kp = (9 ( MSEP + MSE O MSEP ko = (0 ( MSE + MSE The opmal esmae accordng o KF wll hen be P x = x + k (x x ( E P O P O Now consder and recall he process equaon and observaon equaon respecvely, x = + = + Ax p, 034,,,,,...m T z = Hx + O, = 034,,,,,...m Usng he basc dea of equaon (7, he opmal esmaor by KF s x + ( + = x + ( + k + (z+ Hx + ( ( where, x+(+ s he esmaed value afer he KF O adjusmen, and x+(- s he sae before he KF adjusmen,.e. he predced value from he model. The marx k+ s deermned by k = P ( H T + + T (HP + ( H + R + (3 where P+ s he opmal esmae error covarance marx and can be calculaed by P ( + = P ( P ( H (HP ( H + R HP ( T + + + T + + (4 T P + ( = AP ( + A + Q (5 Here K+ called he Kalman opmal gan or Kalman fler deermnes how much he esmaed value can gan usng hs flerng sysem from he observaons. Equaon (5, (6 and equaon ( o (5 are he sx basc equaons of KF. To predc he opmal sae by KF, x+ of equaon (5 s used and x s subsued by x+ n equaon (6 o ge z+. Then usng Equaon (5, (4, (3, and ( sequenally, he opmal esmaor x+(+ s esmaed. Ths value of x+ wll be used o predc nex me sep sae of x (.e. x+ by means of he above equaons. Ths recursve operaon wll connue up o he expeced me sep. Daa Assmlaon by PF wh SIR (Sequenal Imporance Re-samplng Parcle flers are sequenal Mone Carlo mehods ha esmae sysem saes from a sae space model gven he measuremen sequence. The probably dsrbuon funcon of he sysem sae can be nferred. The basc dea of PF s o approxmae he dsrbuon, p ( x z : usng a se of random samples called parcles. Le z: = (z, =,, be he measuremen sequence and assume ha he pror dsrbuon p(xo s known, he poseror probably can be obaned sequenally by predcon and updae. The predcon and updae equaons are as follows: p( x z = p( x x p( x z dx (6 : : p( x z : p( z x p( x x : = (7 p( z z : The above equaons are he opmal soluons from a Bayesan perspecve o he non-lnear sae esmaon problem. One lmaon s ha he evoluon of he poseror densy generally canno be deermned analycally. Thus, some approxmaons mus be made. PF approxmaes he poseror probably dsrbuon, p ( x z : by a se of supporng random samples (parcles x, =,,..., N, wh assocaed weghs 53
www.sepub.org/scea Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 N w : : δ = p(x z w (x x (8 where δ (x s an ndcaor funcon known as Drac s dela funcon whch s equal o uny f x = 0 and oherwse equal o zero. The weghs sum o zero. The flered sae s aken as he mean of poseror densy. The nex ssue s how o deermne he weghs of he parcles. Snce p ( x z : s no n he form of a radonal probably densy funcon, weghs canno be assgned by drec samplng, bu deermned usng mporance samplng. An mporance densy q( x k Z : k s defned from whch samples drawn. Thus he weghs are defned as: p( x z : w (9 q( x z : If he mporance densy s seleced appropraely and only dependen on he curren observaon, z and he pas sae, x he weghs can be updaed as follows(arulampalam e al., 00: p( z x p( x x w w (0 q( x x,z Wh hese parcles and assocaed weghs, he opmal sae can be esmaed by a normalzed summaon. The mean of he saes can be approxmaed by x = N = w x. To mplemen PF, generally wo mplemenaon ssues are consdered (Chen e al., 004; he frs of whch s degeneracy and he oher s how o choose mporance densy. Afer some eraons degeneracy occurs, when only one parcle has sgnfcan wegh and all oher parcles wegh zero. Thus consderable compuaonal effor wll have been spen on updang parcles whose conrbuon o he approxmaon of p ( x z : s neglgble. Re-samplng can be used o elmnae hose parcles wh small weghs hereby focusng on parcles wh large weghs. Re-samplng generaes se x k, =.... N wh Pr ( j x = x = w. Here j s he parcle ndex afer resamplng. The paren relaonshp s denoed, paren ( j =. The weghs are rese o be /N as he samples are ndependen and dencally dsrbued and hen drawn from a dscree densy funcon. By re-samplng, hose parcles wh zero wegh wll be dscarded. q( x x, z s used as he mporance densy. Usng q( x, ( x z = p x x yelds a smple form o updae he weghs accordng o equaon (0: w w ( z x. A PF wh hs mporance densy and re-samplng sep s called a sequenal mporance re-samplng (SIR fler. A sandard SIR PF uses he pror dsrbuon as he mporance densy. Fgure descrbes he mehodology followed n hs smulaon process. The mahemacal mechansc model of conamnan ranspor s represened by he paral dfferenal equaon expressed n equaon (whch s dscrezed by he FTCS mehod o ge dfference equaons. These equaons can be regarded as general sae space models wh he dfference varables defnng he saes. The sae space model represened n equaon (4 s ncorporaed wh ndependen and dencally dsrbued process nose o ge he process or sysem equaon. The analycal soluon of he ranspor model s ncorporaed wh ndependen and dencally dsrbued measuremen or observaon nose o smulae he observaon equaon. To run he flerng four npus are needed: process equaon, process nose varance, observaon, and measuremen nose varance. The sascal naure of process nose and observaon nose for our model s descrbed earler. Wh hose four npus, boh KF and PF echnques are mplemened o ge he opmal sae. Process nose Process nose varance Forward me cenral space dfference mehod Process or sysem equaon One dmensonal conamnan ranspor model Analycal soluon of ranspor model Measuremen or observaon equaon Applcaon of Kalman flerng and Parcle flerng Esmaed opmal concenraon Measuremen nose Measuremen nose varance FIG. AN OVERVIEW OF THE PROCEDURE FOLLOWED IN THE STUDY 54
Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 www.sepub.org/scea Illusraon Example A deskop sudy s underaken o smulae a synhec ndusral leachae ranspor feld. In hs sudy, one-dmensonal leachng of benzene from an ndusral landfll se s smulaed by KF and PF daa assmlaon scheme. Fgure 3 shows ha vercally 0 nodes wh 0.0 m spacng are aken n model space doman. Each me nerval s aken as 0.0 day. Those space and me nerval are chosen afer sably analyss of he numercal soluon of he problem. Y axs (Leachae flow drecon Indusral Landfll 3 4 5 6 7 8 9 0 FIG. 3 SCHEMATIC DIAGRAM OF BENZENE LEACHATE FLOW SHOWING 0 DISCRETE NODES Here λ s called dffuson number. In hs model, he benzene conamnan mass a he source, Mo, s 6.5 gm y = 0 and = 0 n a column wh a cross-seconal area of m. The longudnal dspersvy, α, s 0.0 meers. A waer flux of 0. m 3 /d s mananed connuously a y = 0. The porosy, η, of he leachng sol medum s 0.5, whch gves oruosy of porous medum, Γ = 0.63. Flow velocy, V, s calculaed as 0.8 m/day vercally downward. Molecular dffuson, Dm =.44E-06 m /day, gves he dsperson of he conamnan, Dy = 0.00800 m /day. Fracon organc Carbon = 0.000, sol sold specfc gravy =.65, organc carbon/waer paron coeffcen = 83 L/Kg, bulk densy =.99 g/cm 3, and reardaon facor =. are used n he sudy. The reardaon facor (R for benzene leachng no he specfed sol medum s calculaed by usng an EPA on-lne ool for se assessmen. For ndusral landfll leachae, ypcal concenraon of benzene s abou 000 mg/l and maxmum concenraon may reach 39000 mg/l (Tchobanoglous e al., 993. Inal concenraon (Co s calculaed as abou 985 mg/l for pulse or nsananeous source whch s njeced a he op node (.e. node no.. If we deal wh n dscree nodes n he space, he dmenson of he sae vecor wll be n. Here n = 0. Therefore, he nal sae vecor xo s a 0 vecor wh Co n he frs cell and zero n he nne oher cells. The Flerng Effecveness Examnaon The effecveness of flerng and deermnsc model s measured by comparng he model predced resuls wh he rue value. Snce all nose dsrbuons are assumed o be normally dsrbued, he roo mean square error (RMSE s used as he effecveness parameer. T RMSE( = C E(, C (, ( n where RMSE ( = error (mg/l a me sep ; CE (, = expeced value of concenraon a node a me sep ; C T (, = observed value of concenraon a node a me sep ; n = oal number of nodes. Resuls and Dscussons A he frs sep of he expermen, he concenraon profle s consruced whou observaon correcon. To deal wh he rue feld, he sae ranson marx, A, should be dfferen for each me sep. Snce all parameers used n hs expermenal scheme are kep consan wh me, he sae ranson marx does no change wh me. Tha means A = A- = = A = A. The sae ranson marx changes he sae from one me sep o he nex. The expermenal scheme s desgned o use 0 dscree nodes wh spacng n he space doman and 50 me seps wh magnude n he space doman. In hs way, he conamnan sae s esmaed by consrucng 50 sae vecors. Along he melne, he peak n he concenraon profle moves along from he frs node o he lower nodes. Ths ndcaes ha he velocy drecon of polluan ranspor s from he op layer of sol o he lower layer meanng he ranspor drecon along he y axs as depced n Fgure 3. Agan, he concenraon profles gve bell-shaped curves (Fgure 4 and 5, ndcang he nsananeous or pulse npu of he conamnan. In our model, no nal error of he sae s njeced. The same nal concenraon s provded for he deermnsc (FTCS model, KF, and PF. Therefore, he nal opmal error covarance marx (Po s a zero 55
www.sepub.org/scea Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 marx. Under hese condons where Q and R reman consan hroughou he KF operaon, boh he esmaon error covarance P and he Kalman gan K, quckly sablzes. The KF algorhm s processed wh s basc sx equaons as descrbed n he mehodology. The KF requres sx npus a he begnnng of he operaon of he fler, namely, nal sae vecor xo, process nose P, measuremen nose O, process nose covarance Q, measuremen nose covarance R, and sae ranson marx A. The probablsc analyss has been done by recursve calculaon of s sx equaons whch are n he marx form. For daa assmlaon by PF, 500 random samples are drawn from he pror densy funcon n each me sep. Then, by usng he observaon model develops a poseror densy funcon of he sae. Fnally, he mean value of he probably dsrbuon gves he opmal sae value. The basc npus of PF are as same as hose of KF. Frs, he deermnsc model was run for verfcaon and comparson purposes. The profle gven s a smooh curve whch represens he heorecal deermnsc behavor of he soluon process. The expermenal scheme was desgned n such a way ha, a each me sep, boh KF and PF combne he sysem model and observaon model o smulaneously provde he nex predcon. In conras o he smooh curve of he FTCS resul, KF and PF gve concenraon profles wh rregular shape whch represens more realsc feld behavor. Due o he randomness and unceranes of he rue feld, s concenraon profle generally gves oscllaed curves raher han smooh ones. The comparson beween he deermnsc model and he wo flerng model resuls s shown n Fgure 4 o 7. The expermen shows ha he benzene concenraon s subsanally reduced n he sudy space doman a hgher me seps. I s dffcul o compare dfferen mehods a lower levels of concenraon. Therefore, comparson s done by akng up o 5 mesep resuls. The comparson s consruced on he bass of fxed space and fxed me crera. Fgure 4 and 5 show he benzene concenraon profles a fxed locaons whle Fgure 6 and 7 show he profles a fxed me seps. Agan he model was run wh and whou numercal dsperson correcons. Fgure 4 shows ha he rue peak concenraon a node 3 s abou 900 mg/l and ha of he FTCS mehod s abou 670 mg/l. Therefore, he peak concenraon dfference beween he rue value and he FTCS mehod s abou 30 mg/l n he case whou any numercal dsperson correcon. For FTCS, he peak s locaed a almos he same me sep as he rue value. On he oher hand, Fgure 5 shows ha he peak concenraon for FTCS s almos he same as he rue value n he case wh numercal dsperson correcon. The dfference beween wo values s abou 50 mg/l. Bu he peak occurs wh a me-lag of 3 me seps. The predcon resulng from flerng, whch s correced by observaon daa, seems closer o he rue value han ha from he deermnsc FTCS model. I s very dffcul o dfferenae whch flerng mehod, KF or PF, performs beer a hs fxed node. Bu boh flerng resuls f well wh he rue value whereas he FTCS resuls devae more from he rue sae han ha of flerng. Fgure 6 and 7 depc he comparson of he deermnsc and flerng models wh he rue sae afer a 0.5 day wh and whou numercal dsperson correcons respecvely. Fgure 6 shows ha he peak value of he benzene concenraon profle akes place a he 6 h node and he dfference beween he rue value and FTCS mehod s abou 00 mg/l. On he oher hand, Fgure 7 shows ha he peak value akes place a he 5 h node n he case of he FTCS mehod; he dfference beween he rue value and FTCS mehod s more han 00 mg/l. Thus, he model wh dsperson correcon shows more dscrepancy beween rue value and FTCS value han he model whou dsperson correcon. Agan, lke he expermenal resuls a he fxed dsance, he smulaon resuls from flerng maches well wh he rue feld resuls a he same mesep. The deermnsc model provdes more devaed resuls from he rue feld han ha of flerng. Thus, flerng ouperforms he deermnsc model quanavely. Concenraon, mg/l 900 800 700 600 500 400 300 00 00 Parcle fler Kalman fler True value FTCS 0 0 5 0 5 0 5 Tme (*0.0 day FIG. 4 BENZENE CONCENTRATION PROFILE AT NODE 3 WITHOUT NUMERICAL DISPERSION CORRECTION 56
Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 www.sepub.org/scea Concenraon, mg/l 900 800 700 600 500 400 300 00 00 Parcle fler Kalman fler True value FTCS 0 0 5 0 5 0 5 Tme (*0.0 day FIG. 5 BENZENE CONCENTRATION PROFILE AT NODE 3 WITH NUMERICAL DISPERSION CORRECTION Concenraon, mg/l 600 500 400 300 00 00 Parcle fler Kalman fler True value FTCS 0 0 3 4 5 6 7 8 9 0 Dsance (*0.0 m Acually, he sysem model s affeced by numercal dsperson correcon, however, he observaon model s no. Therefore, he observaon model remans almos he same n boh cases, wh and whou numercal dsperson correcon. In addon, he flerng procedure provdes more wegh on observaon han he sysem. For hs reason, he flerng resuls are closer o observaon, whch reveals he rue sae accordngly. Snce he model whou numercal dsperson correcon ouperforms ha wh ha correcon, he former model was chosen o examne he flerng effecveness by RMSE (roo mean square error evaluaon. As shown n Fgure 8, RMSE value for he PF predcon gradually decreases wh he ncrease of me seps. The RMSE vs. me sep graph for he PF predcon akes a hyperbolc shape whch expresses a good mergng endency oward he rue value. Such a shape sgnfes he successful daa assmlaon. The error sars wh 35 mg/l a me sep and deceases o 0 mg/l a me sep. In hs way, PF predcon reduces he error from 30% o 80% from me sep o me sep, respecvely. On he oher hand, KF provdes conssen performance abou RMSE. RMSE for KF predcon vares from 40 mg/l o 4 mg/l. KF reduces hs error from 70% o 90%. In case of KF, ha means he range of error reducon vares whn a 0% range whereas error reducon of PF vares whn a 50% range. For he frs several me seps, he predcon error of KF s 30% o 50% less han ha of PF. FIG. 6 BENZENE CONCENTRATION PROFILE AFTER 0.5 DAY WITHOUT NUMERICAL DISPERSION CORRECTION 700 600 500 Parcle fler Kalman fler True value FTCS RMSE, mg/l 00 80 60 40 0 00 Parcle fler Kalman fler FTCS Concenraon, mg/l 400 300 00 00 0 3 4 5 6 7 8 9 0 Dsance (*0.0 m 80 60 40 0 0 5 0 5 0 5 Tme, (*0.0 d FIG. 8 PREDICTION ERROR (ROOT-MEAN-SQUARE ERROR FOR DATA ASSIMILATION WITH DETERMINISTIC MODEL (FTCS, KF AND PF FIG. 7 BENZENE CONCENTRATION PROFILE AFTER 0.5 DAY WITH NUMERICAL DISPERSION CORRECTION Concluson The resuls of hs sudy show ha esmaon 57
www.sepub.org/scea Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 echnques usng KF and PF offer a more accurae soluon han convenonal numercal approach for model developmen wh nosy and ncomplee daa. Alhough -D and 3-D models become more complcaed n mechansm expresson and n compuaonal mplemenaon, he conclusons obaned from hs smple one dmensonal model can be appled o he models wh hgher dmensons, because mahemacal sysems reman he same. The rue value of he sae akes place n a space for only one me due o he randomness of he feld behavor. I s mpossble o relae all hydro geologcal parameers mahemacally n a deermnsc model so ha can gve he rue soluon. Therefore, nsead of he rue value, he bes engneerng approach n conamnan ranspor modelng s o ge he soluon as close o he rue value. Snce a rue esmae of he conamnan sae canno be acheved, any possble esmae ha s he closes o he rue esmae can be reaed as a beer soluon for he conamnan ranspor sysem. For benzene ranspor, boh KF and PF can provde accurae resuls for he llusraed case. In hs sudy, he conamnan ranspor problem s reaed as a dynamc sae esmaon problem. The Bayesan approach, lke KF and PF, s based on he probablsc sae space formulaon and requres updae of nformaon on recep of new measuremens (observaons. The radonal deermnsc model does no requre eher probablsc formulaon of he sae or he updae of he sae wh new measuremens. Therefore, he program algorhm becomes more complex n he case of flerng approach han ha of he convenonal deermnsc model. As a resul, flerng requres more compuaonal effor compared o he convenonal numercal soluon. The expermen s desgned and consruced n such a way ha every me he program s run, dfferen random numbers are generaed o smulae he random behavor of he subsurface ranspor plume. The RMSE predcon curve does no vary sgnfcanly and he qualave shape remans he same for dfferen run. The RMSE predcon s evaluaed wh one me run. A me sep 5, he esmaed RMSE of he numercal soluon s 7.5 mes greaer han ha of KF and 3.5 mes greaer han ha of PF. I s ndcaed ha KF and PF ouperform he numercal soluon. The dscree KF has sx equaons ha sequenally provde he recursve esmaon of he sae n a marx form. Ths provdes a rgorous general framework for he dynamc sae esmaon problem wh lnear sysems and Gaussan dsrbuon. On he oher hand, parcle flers are approprae o handle general dynamc sae space models and do no rely on he assumpons of lneary or a Gaussan poseror densy. However, PF needs he probablsc formulaon of he sae. The problem algorhm and formulaon s more complex n case of PF han ha of KF. Therefore, PF requres more compuaonal effors han ha of KF. For hs one dmensonal case, KF ouperforms PF. On an average, KF can reduce he error from 70% o 90% and PF can reduce he error from 30% o 80% as compared o he convenonal numercal approach. ACKNOWLEDGMENTS Ths work was sponsored by he Deparmen of Energy Samuel Masse Char of Excellence Program under Gran No. DE-NA000078. The vews and conclusons conaned heren are hose of he wrers and should no be nerpreed as necessarly represenng he offcal polces or endorsemens, eher expressed or mpled, of he fundng agency. Noaon The followng symbols are used n hs paper: A = cross-seconal area; A = sae ranson marx a me sep ; C = solue concenraon, mgl -3 ; CE (, = expeced value of concenraon a node a me sep ; C T (, = observed value of concenraon a node a me sep ; C,= vecor of polluan concenraon a node and me ; Dy = dsperson coeffcens n y drecon, m s - ; E = expecaon value operaor; H = observaon daa paern marx; K+ = Kalman opmal gan or Kalman fler a me sep +; MSE = mean square error; N = oal number of random samples, (here N = 500; n = oal number of nodes; O = observaon error a me sep ; p = model sysem error or process nose; P+ = opmal esmae error covarance marx a me sep +; p ( x z : = condonal probably of x gven he observaon sequence z: = (z, =,,, ; Q = model sysem error covarance marx a me sep ; R = reardaon facor, dmensonless; RMSE = Error (mg/l a me sep ; R = observaon nose covarance a me sep ; = me, s; V = lnear velocy of flow feld n he y drecon, ms - ; w = assocaed wegh of x ; x= he sae varable descrbed as a vecor of conamnan concenraon of all nodes a me ; x = supporng random samples (parcles regardng o each elemen of sae vecor x, =,,, N; x E = esmaed sae varable by KF; x O = observed value; x P = predced value of sae vecor; xx = opmal mean sae afer 58
Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 www.sepub.org/scea normalzed summaon; y = caresan coordnae drecon along he flow lne, m; z = vecor havng he observed values of sae of all nodes a me sep ; η = porosy of sol medum; σ = sandard devaon of process nose a node ; σ sys =sandard devaon of he sysem error; δ (x =Drac s dela funcon; ζ = sandard devaon of observaon nose a node. REFERENCES Arulampalam, S., S. Maskell, N.Gordon, and T. Clapp. A uoral on Parcle Flers for on-lne non-lnear/ non-gaussan Bayesan Trackng. IEEE Transacons on Sgnal Processng 50(00:74-88. Chang, S.Y., and A. Jn. Kalman Fler wh regonal nose o mprove accuracy of conamnan ranspor models. Journal of Envronmenal Engneerng 3(005: 97-98. Chen, T., J. Morrs, and E. Marn. Parcle Flers for he esmaon of a sae space model. Proc. European Symposum on Compuer-Aded Process Engneerng 4. Compuer-Aded Chemcal Engneerng, Lsbon, Porugal, 8(004, 63-68. Chen, T., J. Morrs, and E. Marn. Parcle Flers for sae and parameer esmaon n bach processes. Journal of Process Conrol 5(005: 665-673. Cheng, X. Kalman Fler scheme for hree- dmensonal subsurface ranspor smulaon wh a connuous npu. MS hess, Norh Carolna A&T Sae Unversy, Greensboro, NC, 000. Conwell, P. M., S. E. Sllman, and L. Zheng. Desgn of a pezomeer nework for esmaon of he varogram of he hydraulc graden: The role of he nsrumen. Waer Resource Research 33((997: 489-49. Cosby, B. J., G. M. Hornberger, and M. G. Kelly. Idenfcaon of phoosynhess-lgh models for aquac sysems II: Applcaon o a macrophye domnaed sream. Ecologcal Modelng 3(984: 5-5. Encyclopeda. Benzene: Encyclopeda. Las modfed Augus 7, 009. hp://en.allexpers.com/e/b/be/ benzene. hml. Ferrares, M., and A. Marnell. An exended formulaon of he negraed fne dfference mehod for ground waer flow and ranspor. Journal of Hydrology 75 (996: 453-47. Graham, W., and D. McLaughln. "Sochasc analyss of non-saonary subsurface solue ranspor:. Condonal momens." Waer Resources Research 5( (989, 33-355. Harroun, K.EI, D. Ouazar, L.C. Wrobel, and A. H. D. Cheng. Aqufer parameer esmaon by exended Kalman Fler and boundary elemens. Engneerng Analyss wh boundary Elemens 9(3(997:,3-37. L, L., H. Zhou, H.-J. Hendrcks Franssen, and J. J. Gómez-Hernández. "Modelng ransen groundwaer flow by couplng ensemble Kalman flerng and upscalng." Waer Resources Research 48((0: W0537. McLaughln, D. An negraed approach o hydrologc daa assmlaon: nerpolaon, smoohng andforecasng. Advances n Waer Resources 5(00: 75-86. Panzer, M., M. Rva, A. Guadagnn, and S. P. Neuman. "Daa assmlaon and parameer esmaon va ensemble Kalman fler coupled wh sochasc momen equaons of ransen groundwaer flow." Waer Resources Research 49(3(03: 334-344. Pnder, G. A Galerkn-fne elemen smulaon of Groundwaer conamnaon on Long Island, New York. Waer Resources Research 9(973: 957-669. Porer, D., G. Bruce, W. Jones, P. Huyakorn, L. Hamm, and G. Flach. Daa fuson modelng for groundwaer sysems. Journal of Conamnan Hydrology 43(000: 303-335. Rozos, E., and D. Kousoyanns. Benefs from usng Kalman fler n forward and nverse groundwaer modellng. Proc., European Geoscences Unon General Assembly, Geophyscal Research Absracs, Venna, 3 (0: 0-.Schrader, B. P., and S. F. Moore. Kalman flerng n waer qualy modelng: heory vs. Pracce. Proc., 9 h conference on Wner Smulaon, Gaersburg, Maryland, (977: 504-50. Schwarz, F. W., and H. Zhang. Fundamenals of groundwaer. New York: John Wley & Sons, Inc., 994. Tchobanoglous, G., H. Thesen, and S. A. Vgl. Inegraed sold wase managemen. New York: McGraw-Hll, 993. Van Geer, F.C. (98. An equaon based heorecal approach o nework desgn for ground waer levels usng Kalman Fler. Inernaonal Assocaon of Hydrologcal Scence 36(98: 4-50. Walker, J. P., G. R. Wllgoose, and J. D. Kalma. "Onedmensonal sol mosure profle rereval by assmlaon of near-surface observaons: a comparson 59
www.sepub.org/scea Sudy of Cvl Engneerng and Archecure (SCEA Volume Issue 3, Sepember 03 of rereval algorhms." Advances n Waer Resources 4(6 (00, 63-650. Webser, R., and M. A. Olver. Sample adequaely o esmae varograms for sol properes. Journal of Sol Scence 43 (99, 77-9. Welch, G., and G. Bshop, (995. An nroducon o he Kalman Fler. Unversy of Norh Carolna, Deparmen of Compuer Scence, TR 95-04 (995. Las accessed March 0, 009. hp://www.cs.unc.edu/ ~racker /ref/s00. Whehead, P. G., and G. M. Hornberger. Modelng Algal behavor n he rver Thames. Waer Resource Research 8(8 (98: 945-953. Yangxao, Z., C. B. M. Te Sroe, and F. C. Van Geer. Usng Kalman Fler o mprove and quanfy he uncerany of numercal ground waer smulaon: applcaon o monorng nework desgn. Waer Resource Research 78(99: 995-006. Yu, Y.S., M. Hedar, and W. Guang- Te. Opmal esmaon of conamnan ranspor n ground waer. Waer Resources Bullen 5 (989: 95-300. Zou, S., and A. Parr. Opmal esmaon of wodmensonal conamnan ranspor. Ground Waer 33( (995: 39-35. 60