Comparison of SCIPUFF Plume Prediction with Particle Filter Assimilated Prediction for Dipole Pride 26 Data

Comparison of SCIPUFF Plume Prediction with P articl e F ilter Assimilated Prediction for Dipole Pride 26 Data Gabriel T erejanu Departmen t of Computer Science a nd E ngineer ing University at Buffalo Buff alo, NY 14260 terejanu@buff alo.edu Y ang Chen g Departmen t of M echanical and Aerospace Engineer ing University at Buffalo Buff alo, NY 14260 cheng3 @buf falo.edu T arunraj Singh Departmen t of Mechanical and Aerospace Engineer ing University at Buffalo Buff alo, NY 1426 0 tsingh@eng.buffalo.edu Peter D. Scott Departmen t of Co mputer Science a nd E ngineer ing University at Buffalo Buff alo, NY 14260 peter@buf falo.edu Abstract —This paper presents the a pplication of a particle filter fo r data assimilation in the context of puff-b ased dis- persion models. Particle filters provide estimates of th e higher moments, a nd ar e well suited for str ongly nonlinear and/or non- Gaussian models. The Gaussian pu ff model SCIPUFF , is used in predicting the chemical concentration field after a chemical incident. This model is highly nonl inear and evolv es wi th variable state dimension and, after suffi cient time, high dimensionalit y . While th e particle fi lter forma lism naturally su pports variable state di mensionality high dimensi onality represents a challenge in selecting an adequate number of particles, especially for the Bo otstrap version. W e present an i mplementation of the Bootstrap particle filter and compare its p erf ormance with the SCIPUFF predictions. Both the model and th e Particle Filter are evaluated on th e Dip ole Pride 26 experimental data. Since there is n o av ailable ground truth, the d ata has been d ivided in two sets: traini ng and testing. W e show th at eve n with a modest number of particles, t he Bootstrap particle fi lter provides better estimates of th e concentration field compared with the process model, with out excessiv e incre ase in compu tational complexity . Keywords: Da ta Assimilatio n, Partic le Filter , Chem-Bio Field T est, Dispersion Model. I . I N T R O D U C T I O N There is an increase in requirement for accuracy and com- putational per forman ce in atmosph eric transpor t and diffusion models used in critical decision making in the context of chem- ical, biolog ical, radiolog ical, and nuclear (CBRN) inc idents. The output (field con centrations and dosages) of the disper sion models is u sed direc tly to guid e d ecision-makers, and as an input fo r higher fusio n levels, such as situation and thr eat assessment. Ther efore the accur acy o f the m odels as well as the time of d eliv ery of the fo recasts play s an im portant p art in decision m aking. Gaussian dispersion m odels h av e been extensively studied and used in assessing the impact of CBRN incidents. T hey have gain ed popu larity due to the ir straig htforward theoreti- cal appro ach, and due to their relativ ely low co mputatio nal complexity [1]. For a ccurate CBRN output, one ca nnot rely solely on mathematical models or on measurements record ed by the sensors on the field, because of their u ncertainties. Thus, for better estimates and lower unce rtainty , a fusion step, called Data Assi milation, is necessary to combine the model forecasts and the m easuremen ts. In Data Assimilation, the estimation o f the unkn own system states g iv en the underly ing dynam ics of the system and a set of observations may b e f ramed as a filter ing, smo othing or pred iction pr oblem. G i ven a fixed d iscrete time in terval, { t 1 , t 2 , . . . t N } , over w hich observations are available, the problem of filterin g is to find th e best state at time t k giv en all the o bservations pr ior to and inclu ding t k . Th e smoo thing problem is to find the b est state at tim e t k giv en all th e observations up to time t N , where t k ≤ t N . For t k > t N the pred iction problem is to forecast the state o f the system at time t k using all the observations in the given interval. For a lin ear system, under the assumption o f Ga ussian probab ility distributions, the pro blem of e stimating the states of the system has an exact closed f orm solu tion given b y the Ka lman Filter [2]. If the prob ability distributions are no n- Gaussian o r th e system is n onlinear, in genera l no closed-form solution is available and d ifferent assumptions and ap proxim a- tions h av e been made for qua si-optimal solu tions maintaining both acc uracy a nd tractab ility . The systems considered h ere are in gener al non linear, but the assum ption that the p rocess and observation u ncertainties can be ad equately modeled as Gaussian is made. The nonlinea r filtering prob lem has been extensi vely studied and successfully employed in m any applications, with various methods provid ed in the litera ture. Among the best und erstood and most frequen tly cited are th e Extended Kalman Filter (EKF), the Ensemble Kalman Filter (EnKF), and more recently the Un scented Kalma n Filter (UKF), and the Particle Filter (PF). Th e EKF is h istorically th e first, and still the mo st widely adopted approac h to no nlinear estimatio n problem s. It is based on the assumptio n over small tim e incr ements, nonlinear sy stem dynam ics can b e accurately mode led by a first-order T aylor series expan sion [3]. The PF uses a sam pling appro ach, estimating the p osterior probab ility distribution, including its higher order moments, by propag ating and u pdating a numb er of particles, without the assumption of Gau ssian statistics [4 ]. The variable a nd hig h dimensiona lity of the state vector, which p oses a prob lem to the stan dard n onlinear assimilation techn iques, can be dealt with usin g Particle Filters [5] . Daum, e t al. [6 ] showed that a carefully designed Particle Filter should mitigate the cu rse of dimensiona lity for cer tain filterin g p roblems. This pap er presents an implem entation of the Bootstrap Particle Filter to assimilate chemical con centration read ings of the Dipole Pride 26 experimen t [7 ] into the SCIPUFF dispersion m odel. T he Particle Filter takes into account the uncertainty d ue to th e d ata err ors in the observed meteorolo gy . The results show th at the particle filter , with a modest number of particles, provides better estimates of the conc entration field compare d with the proc ess mod el, under lying the impo rtance of m eteorolo gical d ata accuracy in CBRN incide nts. Data assimilation based on sampling tec hniques for CBRN incidents an d num erical weather pr ediction (NWP), such as Particle Filter [ 5], Unscented Kalman Filter [8], [9 ] or En- semble Kalman Filter [10], [1 1], b ecome mor e and more accessible as par allel co mputing be comes main stream with the in troductio n of multi-core processor s and mu ltiple CPU computer s. Sampling tech niques suc h as Mon te Carlo analysis [12], [1 3] or ensemble meth od [14 ], [15] h av e been used before to accoun t for uncertain ty due to data errors. The Dipole Pride 26 (DP26) field experiment and the SCIPUFF dispersio n model are presen ted in Section II . The Bootstrap Particle Filter is intro duced in Section III an d its implementatio n for this p articular p roblem is describe d in Section IV. Num erical results on the CBRN scenario are presented in Sectio n V an d th e conc lusions an d fu ture work are d iscussed in Sectio n VI . I I . D I P O L E P R I D E 2 6 A N D S C I P U F F The Dipole Pride 26 field exper iment has b een design ed to validate tr ansport and diffusion m odels at mesoscale distances [7]. Th e exper iment has b een con ducted at Y u cca Flat, Ne vada where gaseo us sulfur h exafluoride (SF6) has been re leased in a series of instrumented trials, o f which o nly 17 provid e useful puff dimension in formatio n. Three lines of sensors, Fig.1, each with 3 0 whole air- samplers have b een u sed to record average con centration s ev ery 15 min. The chemical sensors, known also as sequ ential bag samplers ( 1 2 b ags per senso r), are placed at 1 . 5 m above the gro und and sp acing along lin es is abou t 25 0 m. The total sampling time of the chemical sensor is 3 hr, hence total experimen tal du ration for each tria l to be mo nitored is 3 . 5 hr . This is ach iev ed by d elaying the acquisition of the last line of sen sors with 30 min. Six continu ous SF6 analyzer s, TGA-400 0, hav e been used to record high-fr equency v ariations of the gas co ncentration field , but their p lacement d oes not offer en ough resolution to capture the crosswind structure of the chemical plum e, and for th e p urpose of this paper the se readings h av e b een exclu ded fro m the study . Figure 1. Process Model: che mical dosage plot after 3 hr at 1 . 5 m Eight Meteoro logical Data (MEDA) stations were used to provide surface-b ased meteoro logical measureme nts and two pilot balloo n statio ns and on e radioso nde p rovided th e upper- air m eteorolog ical profiles. The meteoro logical measurem ents recorde d pr ovide in formation about the win d direction and speed, temperature, pressure and humidity . The v ariation of the wind field is g i ven b y the standard deviation of hourly wind speeds and directio ns recorded at the MEDA statio ns which are ab out 0 . 5 − 2 m s − 1 and 10 ◦ − 30 ◦ respectively [1 6]. In this p aper we ar e fo cusing o nly of trial n umber six, which to ok place in Nov 12, 1996 and wh ere a mass of 11 . 6 kg of SF6 had been release from 6 m he ight at the North dissemination site N2 as in Fig. 1. The dispersion model u sed to predict the ch emical co ncentratio ns and do se at the sensor locations is SCIPUFF [ 17]. SCIPUFF is a Lagran gian puff dispersion mo del that o utputs the che mical concen trations at specified locatio ns as cu mulative contr ibutions of Gaussian puffs. SCIPUFF is the dispersion eng ine in corpor ated into Hazard Pred iction and Assessment Capab ility (HP A C) tool, used by the Defen se Threat Reductio n Agen cy (DTRA) fo r situation assessment of CBRN incide nts. Besides the mean concentr ation field, SCIPUFF is also providing the u ncertainty in the conc entration value du e to the stoc hastic natu re of the turbulent d iffusion pr ocess. The simulation is driven b y the meteorolo gical data, which in this case is pr ovided by th e MED A station s, fro m wh ich a win d field is cr eated. The observed wind d ata are fit in a least squ are sen se, using variational methodo logy . An initial gr idded wind field is constru cted fr om the ob servation data by inter polation. Adjustments are then mad e to the initial 3D inter polated wind field vectors so as to satisfy con servation of mass in a way that also minimizes an integral fun ction of the d ifference between the initial a nd a djusted fields. Existing literatur e pr ovides evaluation stu dies of SCIPUFF with Dipole Prid e 26 [16 ], [18] , [19] . Th e results repo rted show that SCIPUFF prediction s are within a factor o f 2 of ob- servations ab out 50 % , wher e the mod el ev aluation was ba sed on max imum do sage anywhere on the samplin g lines [ 16]. The stu dies emp hasize th e impo rtance of wind field in the chemical concen tration p rediction accuracy and conclu de that SCIPUFF prediction per forman ce is comparable o r better than other dispersion models. I I I . B O O S T R A P P A RT C I L E F I LT E R In this section we focus our attention on seq uential state es- timation using sequential Mo nte Carlo ( SMC). SMC is k nown also as b ootstrap filtering, p article filtering, the c ondensatio n algorithm , interacting particle appro ximations and survival o f the fittest [20]. Consider the following non linear system, descr ibed by th e difference e quation a nd th e observation model: x k +1 = f ( x k ) + w k (1) z k = h ( x k ) + v k (2) Denote by Z k = { z i | 1 ≤ i ≤ k } the set of all observations up to time k , conditionally indep endent given the process with distribution p ( z k | x k ) . Also, assume that the state sequence x k is an un observed (hidde n) Markov p rocess with initial distribution p ( x 0 ) a nd tr ansition distribution p ( x k +1 | x k ) . Our aim is to estimate recursively the posterior distribution p ( x k | Z k ) a nd exp ectations of the fo rm [21 ]: E [ z k ] = E [ h ( x k )] = Z h ( x k ) p ( x k | Z k ) d x k (3) In par ticle filters, the posterior distribution p ( x k | Z k ) is approx imated with N weighted particles { x ( i ) k , w ( i ) k } N i =1 , giv en by p ( x k | Z k ) ≈ N X i =1 w ( i ) k δ x ( i ) k ( x k ) (4) where x ( i ) k are the particles drawn from the im portanc e function or p roposal distribution, w ( i ) k are the normalized importan ce weights tha t sum up to one an d δ x ( i ) k ( x k ) deno tes the Dirac-delta mass located in x ( i ) k . Thus the expectation of a known fu nction h ( x k ) with respect to p ( x k | Z k ) is then approx imated by Z h ( x k ) p ( x k | Z k ) d x k ≈ N X i =1 w ( i ) k h ( x ( i ) k ) (5) Suppose that we cann ot sample fro m the posterior d istribu- tion and u se a n importance samp ling app roach to sample from a prop osal distribution q ( x k | Z k ) . Hence, we can rec ursiv ely update th e weights: w i k +1 = w i k p ( z k +1 | x i k +1 ) p ( x i k +1 | x i k ) q ( x i k +1 | x i k , Z k +1 ) (6) After a few iterations all b ut one particle will ha ve negligible weights. Hence the algorithm s allots time to update a large number of weig hts with no effect in our sam pling, effect known as d egeneracy problem. This problem c an b e overcome by adding a r esampling strategy to Seq uential Impo rtance Sampling ( SIS) Algo rithm 1. Algorithm 1 Sequential Im portance Samp ling (SIS) algorithm 1: Draw x i k ∼ q ( x k | x i k − 1 , Z k ) for i = 1 . . . N 2: Compu te the impor tance weights w i k +1 = w i k p ( z k +1 | x i k +1 ) p ( x i k +1 | x i k ) q ( x i k +1 | x i k , Z k +1 ) 3: Norma lize the impor tance weights ¯ w i k = w i k P N i =1 w i k 4: Multiply/Discar d particles { x ( i ) k +1 } N i =1 with r espect to high/low impo rtance weights w ( i ) k +1 to obtain N new particles { x ( i ) k +1 } N i =1 with equal weights. The imp ortance fu nction p lays a sign ificant role in the particle filter . Usually , it is d ifficult to find a good pr oposal distribution especially in a high dimensiona l space. One ma y choose to approx imate it using different m ethods, thus differ- ent flav or of p article filter . One of the simplest imp ortance function is giv en by q ( x i k +1 | x i k , Z k +1 ) = p ( x i k +1 | x i k ) (7) This implementatio n is called th e Bootstrap Particle filter . By sub stituting (7) back into (6) the new weight update equation becomes: w i k +1 ∝ w i k p ( z k +1 | x i k +1 ) (8) The resamp ling step redu ces the sample imp overishment effect but introd uces n ew p ractical pro blems: it limits the op - portun ity to have a n efficient parallel alg orithm, and par ticles with high weigh ts ar e statistically selected m any times an d leads to a loss o f d iv ersity am ong th e par ticles [ 20]. I V . P A RT I C L E F I LT E R I M P L E M E N TA T I O N Three typ es of uncertain ties [2 2] are present in the m odel prediction s: model uncert ainty due to th e inaccurate rep- resentation of the chemical an d dyn amical processes, dat a uncertainty due to the errors in data used to drive the mod el and random t urb ulence of the atmosp here. The mod el unc ertainty is not completely red ucible, and in the SCIPUFF p rediction case it is estimated . The un certainty due to the variability o f the atmosphere cannot be f urther reduced an d th e un certainty due to data error s it is u sually high, mo re th an 5 0% of the total uncer tainty [23 ], but it can be minimize d. The d ata error s con sidered in th is pap er ar e meteorolo gical data: wind s peed and wind direction. The errors are due to the unrepresen tativ e sitting of the wind sensors in the field and sen sor accur acy and calibr ation [19]. This informa tion is rar ely av ailable and for this study a standar d deviation of 0 . 5 m s − 1 for the wind speed and 5 ◦ for the wind direction h as been consider ed. A FOR T RAN implementatio n of the Particle Filter h as been sp ecially crea ted for SCIPUFF to run in p arallel o n the cluster hosted at the Center fo r Computation al Research at Un iv ersity at Buffalo. This implementatio n of th e par ticle filter is designed to accoun t for the unce rtainty in the wind sensor r eadings while co ping with th e challenges o f the d ata assimilation f or the CBRN inciden ts using Gaussian puff models: variable dimensionality and high dimen sionality [5]. Compared with th e maximum d osage approach [1 6], the ev aluation method used in this p aper is to compare th e predicted d osage after every 15 min with the corr espondin g observed do sage. Since there is no g round truth, th e samples provid ed in the DP26 have been divided to two sets: t he training set used to perfor m the data assimilation, compo sed o f Line 1 and L ine 2 of sensors, and the test ing set , Line 3 of sensors, used fo r perfor mance ev aluation. Due to the spatial and tempora l distanc e betwee n the wind readings all the co nsidered independe nt rand om variables described by a Gaussian distribution with me an given by the n ominal sen sor r eading an d uncertainty given by the standard de viation mention ed above. Hen ce the particles, each representin g a SCIPUFF instance, are p ropag ated u sing wind field generated from data sampled from this distribution. Ea ch particle outpu ts a dose field d i j which depen ds on the wind field; here j is the sensor num ber, i is th e particle number and k is the time step. The estimated dosag e ˆ d j is given b y the following relation: ˆ d i ( k ) = N X i =1 w i k d i j ( k ) (9) Here w i k is gi ven by Eq.(8) . The conditio nal probability present in Eq.(8) is un known since the auth ors could not find infor- mation regarding th e uncertainty in the concentratio n readin gs for the whole air-samplers. The concentratio ns readings of the sensors have been assumed to be indepe ndent r andom variables due to the spatial and tempo ral distances. Th e likelihood have been appro ximated with a Gaussian fu nction giv en by : p ( d j ( k ) | d i j ( k )) = N  d i j ( k )   d j ( k ) , v ( k )  (10) Here d j ( k ) is the obser ved dosage at the j th sensor at time k and v ( k ) is the samp le variance of the difference b etween observed dosages and predicted ones over all the particles. Any predicted dosage less th an 1 ppt- hr (includin g zer o) has been set to 1 ppt- hr and all th e o bserved dosages less than 1 0 ppt-hr have been ignored . Hen ce the total n umber of dosage values to be compar ed is 47 . V . N U M E R I C A L R E S U LT S Both the pr ocess model predictions, u sing n ominal win d sensor readings, and the particle filter pred ictions with per- turbed wind field have been compar ed again st the obser ved dosages on the th ird lin e of sensors. T o ev aluate the p erform ance of the p article filter comp ared with the proc ess model, we consider the f ollowing statistical measures [2 4]: FB - fractio nal b ias, MG - ge ometric mean bias, NMSE - n ormalized mea n squar e err or, VG - geom etric variance, F A C2 - fraction of p rediction s with in a factor 2 of observations and F A C3 - fractio n of predictions within a factor 3 o f observations. FB = D o − D p 0 . 5( D o + D p ) (11) MG = exp( ln D o − ln D p ) (12) NMSE = ( D o − D p ) 2 D o D p (13) V G = exp  (ln D o − ln D p ) 2  (14) F A C2 = # D p such th at 1 2 ≤ D p D o ≤ 2 (15) F A C3 = # D p such th at 1 3 ≤ D p D o ≤ 3 (16) Here D o represents th e observed dosage an d D p the pre - dicted dosage. Since we are dealing with random sam ples, 50 Mon te Carlo ru ns have be en perf ormed in assessing the perfor mance of th e p article filter . Th e n umerical results ba sed on the p erforma nce metrics h av e b een tabulated in T able I . Overall th e particle filter provides improved estimates co m- pared to the pro cess model an d all the perfor mance m easures are better on a verage. Since the p redicted and observed values vary b y several or ders of magnitu de, MG , V G , F A C2 and F A C3 are more appro priate. While we do not see a significant T able I N U M E R I C A L R E S U LT S - 50 M O N T E C A R L O RU N S Proce ss Model Partic le Filter PF 95% CI FB 1 . 426 1 . 405 1 . 390 − 1 . 419 MG 0 . 456 0 . 443 0 . 402 − 0 . 484 NMSE 8 . 658 8 . 466 8 . 249 − 8 . 683 VG 86 . 75 61 . 20 53 . 65 − 68 . 74 F A C 2 6 . 38% 11 . 45 % 9 . 31% − 13 . 58% F A C 3 6 . 38% 22 . 17 % 19 . 45% − 24 . 89 % reduction in the geo metric bias, the geom etric variance and the fraction of factor 2 and 3 g i ve a significant impr ovement of the particle filter over the pro cess mo del. This is consistent with the scatter p lots shown in Fig.2 and Fig.3. The particle filter is able to alleviate the und er-prediction and over-prediction problem present in the dispersion models. The result reiterates the n eed o f accurate m eteorolog ical ob- servations and provides support for the use of data assimilation in the CBRN incidents. V I . C O N C L U S I O N The paper presents an implementation of th e Bootstrap Par - ticle Filter to correct the SCIPUFF concen tration pr edictions 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 Observed (ppt−hr) Predicted (ppt−hr) Scatterplot for Process Model Figure 2. Process Model with nominal wind field 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 Observed (ppt−hr) Estimated (ppt−hr) Scatterplot for Particle Filter Figure 3. P articl e Filter with pertur bed wind field (best run) using concen tration measuremen ts provide d by th e chemica l sensors deployed in the field. Due to the high uncertainty in the meteorolo gical inpu t, the CBR N d ispersion models should be accompan ied by a data assimilation step to account for this uncertainty and impr ove th e pr edictions. For the Dip ole Pr ide 26 the particle filter h as do ubled the num ber of predictio ns within a factor of 2 of the o bservations and it has tripled th e ones with in a factor of 3 o f the o bservations. A co mplete evaluation of th e particle filter on all the Dipole Pride 26 trials and simulations with unce rtainty for also the temperatur e, pressure and relative humidity , are planned as future work . A C K N O W L E D G M E N T This work was sup ported by the Defense Threat Reduction Agenc y (DTRA) under Contract No. W911NF-06-C-0162. The authors grate- fully ackno wledge the support and constructiv e suggestions of Dr . John Hannan of DTRA. The authors would also like to thank Dr . Joseph C. Chang and Dr . S tev e Hanna for their input regarding the Dipole P ride 26 dataset. R E F E R E N C E S [1] S. Arya, Air Polluti on Meteor ology and Dispersio n . Oxfor d Uni ve rsity Press, 1999. [2] R. Kalman, “A New Approach to Linear Filteri ng and Predicti on Problems, ” T rans.ASME - Journal of Basic E ngineeri ng , v ol. 82 (Serie s D), pp. 35–45, 1960. [3] J. Crassidis and J. Junkins, Optimal E stimation of Dynamic Systems . CRC Press, 2004. [4] B. Ristic , S. Arula mpalam, and N. 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