J11.3 STOCHASTIC EVENT RECONSTRUCTION OF ATMOSPHERIC CONTAMINANT DISPERSION

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1 J11.3 STOCHASTIC EVENT RECONSTRUCTION OF ATMOSPHERIC CONTAMINANT DISPERSION Inanc Senocak 1*, Nicolas W. Hengartner, Margaret B. Short 3, and Brent W. Daniel 1 Boise State University, Boise, ID, Los Alaos National Laboratory, Los Alaos, NM, 3 University of Alaska, Fairbanks, AK 1. INTRODUCTION Event reconstruction of cheical or biological (CB) agent dispersion into the atosphere is an iportant topic in hoeland security and environental onitoring, and it constitutes an inverse proble. In event reconstruction, also referred to as source characterization or source inversion in different studies, the ajor goal is to characterize the source of a containant or CB agent dispersion event in ters of release location and aount by using inforation fro a sensor network, which can be available in the for of tie-averaged concentration and wind easureents fro scattered locations. Early detection of the CB agents with quick and accurate reconstruction of the dispersion events is critical in organizing an eergency response. Once the dispersion event is characterized, forward projections can be perfored to analyze the extent of exposure to the containation. Event reconstruction of atospheric containant dispersion has received a growing interest in recent years. Several studies have appeared in solving the proble with different ethods. For instance, Keats et al. (7) presented a Bayesian inference ethod for deterining the source of a dispersion event within coplex urban environents. A sourcereceptor relationship was incorporated into the likelihood function by solving an adjoint equation for the scalar concentration, which was found to be efficient in decreasing the overall calculation tie. Thoson et al. (7) applied an inverse proble approach to locating a known gas source fro easureents of gas concentration and wind data. A search algorith with siulated annealing ethod was eployed to find the source location and its strength. The siulated annealing approach helps prevent the search algorith to converge onto a local iniu that ight surround the global iniu. Allen and Haupt (7) developed a source characterization ethod in which a forward dispersion odel is coupled with a backward receptor odel using a genetic algorith. A second-order closure integrated puff odel was used in the source characterization. The ethod was validated with both *Corresponding author address: Departent of Mechanical and Bioedical Engineering, Boise State University, Boise, ID ; phone: (8) , e-ail: senocak@boisestate.edu synthetic and experiental field data. Allen et al. [4] extended this ethod by considering the wind direction as an unknown paraeter. Instead of the puff odel, a siple Gaussian plue odel was considered in their study. Johannesson et al. (4) presented dynaic Bayesian odels using Monte Carlo ethods for target tracking and atospheric dispersion event reconstruction probles. Both the well established Markov chain Monte Carlo (MCMC) ethod and the sequential Monte Carlo ethod for dynaic probles are discussed in detail in their study. Chow et al. (6) and Neuann et al. (6) extended the Bayesian event reconstruction ethod of Johanneson et al. (4) for neighborhood scale atospheric dispersion events. Both coputationally intensive building resolving coputational fluid dynaics odels, and coputationally less intensive epirically based Gaussian puff odels were adopted in these studies, respectively. The results have shown that the Bayesian ethodology is efficient in delivering probabilistic answers to the event reconstruction proble. Depending on the coplexity of the forward odel, the Bayesian approach to inverse probles can be coputationally intensive in ters of the overall execution tie. In the event reconstruction proble, the dispersion odels are typically executed for any ties within the MCMC fraework in order to saple fro the posterior distribution. It should be noted that MCMC chains converge onto the source location quickly, but chains are actually executed longer in order to deliver results with uncertainty, which is iportant in eergency response operations. With siple fast-response dispersion odels, the overall execution tie for MCMC chains is less of a proble, but with high-fidelity odels the issue can be a liiting factor in eergency response situations. To address this issue, Marzouk et al. (7) reforulated the Bayesian approach to inverse probles by adopting polynoial chaos expansions to represent rando variables. In their study, a transient diffusion proble was considered. The results have shown that significant gains in coputational tie can be obtained by adopting the new schee over direct sapling. In what follows, a stochastic event reconstruction ethod is presented, extending the Bayesian inference fraework described in Johannesson et al. (4) and Chow et al. (6). Particularly, a probability odel is proposed to take into account

2 both zero and non-zero concentration observations that can be available fro a sensor network. The paraeters in the probability odel are calculated fro prior distributions, resulting in a ethod free fro tunable paraeters. Siple fast-running Gaussian plue dispersion odels are adopted as the forward odel in the Bayesian inference ethod. The event reconstruction ethod is validated using data fro a tracer dispersion experient (Erik and Lyck, ).. BAYESIAN FORMULATION The forward odeling proble can be defined as predicting the response of a syste, using a physical theory (forward odel) and syste paraeters. In the inverse odeling proble, an inference is ade on the values of the syste paraeters based on observations on the syste response (Tarantola, 5). Loosely speaking, inverse probles can be forulated as follows: F 1 ( d), (1) where d is a vector of observations, is a vector of odel paraeters, and the operator F is the forward odel that governs the syste response. Inverse probles can be ill-conditioned, because sall changes in d can lead to large changes in. Depending on the nature of inverse probles; both deterinistic and probabilistic approaches have been developed for solving the. Probabilistic approaches can be forulated within the context of Bayesian inference, which is pursued in the present study. The present event reconstruction proble requires estiating the odel paraeters () (e.g. release location, aount of aterial released, wind direction etc.) given the observed concentrations (d) fro a sensor network. Bayes' theore defines the posterior probability of a set of odel paraeters () given the observations (d) as follows (Gilks et al., 1996; Carlin et al., 1996): L( ) p ( d) =, () d) where d) is the posterior probability density, L(d ) is the likelihood function, ) is the prior probability density, and d) is the arginal probability density. The posterior probability density given in Eq. () defines the conditional probability density of forward odel paraeters (), given the observed data (d), Calculation of d) is central in Bayesian inference, and it can also be viewed as a solution to an inverse proble. Direct coputation of the posterior density, using Bayes' theore, necessitates the coputation of the arginal probability density d), given in Eq. (), which can be coputationally intensive to the point of being ipractical for ost applications. A practical approach to estiate the posterior probability density is to perfor MCMC sapling by noting the following (Metropolis et al., 1953; Gilks et al., 1996; Carlin et al., 1996) d) L( ). (3) Within this fraework, the observed data (d) enters the forulation only through the ratio of likelihood function. Specification of the likelihood function deserves attention, because it odels how the observations are acquired. For instance, the sensor network cannot quantify the concentration of tracers below its threshold specification for detection, and registers a zero concentration value. Hence, a likelihood function is needed that accounts for zero sensor readings when in fact the actual concentration is non-zero. Let be the odel, C the predicted concentration, ξ the concentration easured by an ideal sensor, and d is the concentration observed by an actual sensor. It is assued that the observations d are related to ξ as follows: αc with probability e, d = (4) αc ξ with probability 1 e, and ξ, given the odel, has a lognoral distribution with density 1 ( logξ log C ) ( ξ ) = exp. p (5) πξ In Eq. (4), it is assued that the probability of not detecting a plue can be calculated based on the predicted concentration C, and at the threshold concentration C th, the plue is detected with probability ½, fro which α can be coputed as ex α C th) = α = log. (6) Cth Then, the likelihood function can be calculated as follows: L( = + I[ d > ] d, ξ ) dξ = I[ d = ] ex αc [ 1 ex αc ) ] ) ξ ) dξ + ξ ) δ ( ξ ) dξ, d (7) where δ d is the Dirac delta-function. Therefore the likelihood function can be written as: L( = Ι[ d = ] ex αc ) + [ 1 ex αc )] (8) + Ι[ d > ] πd ( log d logc ) exp. In eergency response situations, the overall runtie for delivering answers is an iportant factor. Hence, fast running Gaussian plue dispersion odels are adopted as the forward odel to copute C. A Gaussian plue dispersion odel for unifor steady wind conditions can be written as follows (Panofsky and Dutton, 1984):

3 Q y C ( x, y, z) = exp π U y z y (9) ( z H ) ( z + H ) r r exp + exp z z where C is the concentration at a particular location, Q is the release rate, U is the ean wind speed, H r is the height of the release, x is the distance along the wind, y is the distance along the horizontal crosswind direction, and z is the distance along the vertical axis. Note that the release location is the origin for x, y and z directions. y and z are called the standard deviation in the horizontal crosswind and vertical directions, respectively. These two paraeters are also known as the Gaussian plue dispersion paraeters, and they are defined epirically for different stability conditions. For Pasquill C type stability, Briggs forulas for urban conditions paraeterize the standard deviations as follows (Panofsky and Dutton, 1984):.5 y =. x (1 +.4 x), (1) z =. x. Several forulas have been proposed for the standard deviations, and their particular fors reain to be proble specific. Results typically benefit fro adjusting the epirical paraeters. In the present study, soe of the epirical constants that appear in the above forulas are treated as stochastic paraeters within the Bayesian fraework as shown below:.5 y = C1 x (1 +.4 x), (11) z = C x, where C 1 and C are stochastic paraeters that replaces the epirical paraeters. and. in Eq. (1), respectively. This approach allows one to calibrate the deficiencies in the forward dispersion odel to certain extents, and helps iprove the event reconstruction results significantly. Furtherore, in the posterior distribution, these paraeters converge onto values that give better agreeent with the observed data, which can be used in perforing forward projections for the dispersion event under consideration. The prior probability density ter ) in Eq. (3) represents prior knowledge about the odel paraeters () before observing the data (d), which can be expressed by assigning probabilities to odel paraeters () based on bounds on certain physical properties or expert opinions. For instance, data regarding the probabilities of possible wind directions acting on a city ight be available fro previous eteorological studies. Defining a prior probability is subjective and depends on the proble at hand. In the event reconstruction proble, since the release can originate fro any location, one can assign a unifor prior probability to release location, and one ay also assue that low release rates ight be ore likely than high release rates. Hence, the following can be written Qin ) = r) Q) 1, (1) Q where r is the release location and Q is the release rate, and Q in is a user defined iniu eission rate that is of iportance for a given proble. It should be ephasized that the above for is proble specific, and it depends on the available prior inforation about a specific event. Various algoriths exist for MCMC sapling. In the present study, Metropolis algorith is adopted to siulate saples fro the posterior (Metropolis et al. 1953). The reader is referred to Gilks et al. (1996) and Carlin and Louis (1996) for a detailed explanation of the algorith. In the Metropolis algorith, a candidate state ( * ) is sapled fro a proposal distribution at each iteration, and the candidate state is accepted with probability N * * π n ( ) ρ (, ) = in,1, (13) n= 1 π n ( ) where N is the total nuber of sensors in the network and is the current state. When the MCMC algorith has converged, it is expected that the proposal distribution draw saples fro the target distribution, which is defined as π n ( ) = L( ). (14) The above equation is coputed using Eqs. (8) and (1). In Eq (8), the variance of the distribution ( ) is not specified, but it is calculated in the course of MCMC iterations fro an inverse gaa prior distribution. Y() Source at (,) X() Figure 1: Traces of four independent Markov chains converging onto the containant release location. Square arkers denote sapler/sensor locations colored with easured concentration levels in ng/ 3. Logarithic (base 1) values are shown. Clear arkers indicate sensors that registered zero concentration values. 3. RESULTS & DISCUSSION Environental sensor networks have been deployed in various cities, and specifics of these networks and actual data fro the sensor network are not publicly available. Hence, direct testing of event reconstruction ethods is not feasible. However,

4 tracer field experients designed for atospheric dispersion and air pollution studies can be utilized in evaluating the perforance of event reconstruction odels. A series of tracer experients were perfored in the Copenhagen area in 1978 and Concentrations of tracer sulphurhexafluoride (SF6) and eteorological conditions were easured and reported in Erik and Lyck (). For all the experients, tracer was released fro a tower with a height of 115. Saplers/sensors were placed -3 above the ground level along three crosswind arcs that are positioned -6 k away fro the tracer release point. The location of the saplers is shown in Fig. 1. The total sapling tie for the concentration easureents was 1 hour. In the tracer data corresponding to the experient perfored on October 19, the detection liit was given as 9 ng/ 3, and any value below this liit was indicated as zero. This value is used to set the sensor threshold value (C th) in Eq. (6) of the stochastic event reconstruction ethod. Out of 4 saplers, 7 saplers registered zero concentration values. In the following exaple, the tracer dispersion experient is reconstructed in ters of the following paraeters = ( x, y, H, Q, θ, U, C1, C, ), (15) where x and y are the spatial locations of the release, H is the release height, Q is the release rate, θ is the wind direction, U is the wind speed at the release height, C 1 and C are the stochastic ters in the turbulent diffusion paraeterization given in Eq. (11), and is the variance ter in Eq. (8). X Y X Y Q θ U H the location of the release is one of the priary goals. Probabilistic answers instead of deterinistic answers are preferred due to the nature of operations. Hence, posterior distributions need to be apped in ters of probabilities. The high diensionality in the stochastic inversion process requires a careful check using ulti-variate analysis. For that purpose the results are analyzed on a socalled trellis plot as shown in Fig.. The plots on the diagonal are the arginal probability distributions of the forward odel paraeters. The off diagonal plots are the joint posterior distributions of the forward odel paraeters. 9% and 5% contour lines are also overlayed on the joint posterior distributions. The true answer fro the field experient is highlighted with white colored circles on each subplot. Y() Source at (,) X() Figure 3: Probabilistic plue envelope for 95% confidence level. Concentration unit is ng/3 and logarithic (base 1) values are plotted. Clear arkers indicate sensors that registered zero concentration values Q θ U H Figure : Multi-variate analysis of event reconstruction paraeters. The white dot represents the true value fro. The outer and inner contour lines indicate 9% and 5%, respectively. Fig. 1 shows traces of four different Markov chains converging on the source location. The locations of the saplers are also indicated in this plot, and they are colored with the one hour averaged concentration easureents. Fig. 1 clearly shows that chains converge on the source location independent of the starting point. In typical eergency response operations, finding In atospheric dispersion events, it is iportant that eergency responders are provided with results that incorporate uncertainty involved in the proble. The present Bayesian inference fraework is convenient for addressing that need. Fig. 3 presents a probabilistic plue envelope with a confidence level of 95%. The plue envelope is generated by running a forward odel for each posterior saple and storing the concentrations on a vector at desired locations. Then, the concentration value corresponding to the 95 th percentile in the data is selected as the probabilistic plue envelope that gives a confidence level of 95%. As can be seen fro Fig. 3, the plue envelopes all the saplers/sensors. Hence, in analyzing this plot, one can have 95% confidence in assuing that the actual concentration is what the plot indicates or below. Fig. 4 provides a check of this assuption. Concentration data fro the probabilistic plue envelope is copared against the actual easureents fro the saplers on a scatter plot. Clearly, about 95% of the data is

5 overpredicted by the siulation that would allow safer decisions in case of harful dispersion events. Fig. 5 shows the histogras of the stochastic paraeters given in Eq. (11). The constant epirical paraeters given in Eq. (1) are also overlaid on this plot. As can be observed fro this plot, the stochastic ters converged onto different values. Use of Eq. (11) greatly iproved the event reconstruction results. As shown in Fig. 6, the event reconstruction ethod adopting Eq. (1) in the Gaussian plue odel is not successful, because the MCMC chain converges onto a zone that does not include the true source location. However, the event reconstruction ethod that adopts Eq. (11) is deeed successful, because the MCMC chain converges onto a zone that includes the true source location. 1 proble can be posed with any unknown paraeters. In the event reconstruction of Copenhagen tracer experient, up to 9 paraeters are treated as unknowns. It is found that stochastic treatent of the epirical paraeters of the Gaussian plue dispersion odel, iproves the event reconstruction results significantly. This approach allows the optiization of the dispersion odel according to the observations. In the posterior distributions, stochastic paraeters that appear in Eq. (11) converge on to values specific to the dispersion proble at hand, which can also be used in post-event dispersion projections log(coputation), 95% confidence sapler s detection liit probability C 1 probability C Figure 5: Histogra of stochastic paraeters given in Eq. (11). The dashed lines locate the constant epirical values given in Eq. (1) log(experient) Figure 4: Scatter plot of probabilistic plue envelope (95% confidence level) vs. easureents. 4. CONCLUSIONS A stochastic event reconstruction ethod for atospheric containant dispersion is presented. The ethod is based on Bayesian inference with MCMC sapling. The Bayesian approach provides a convenient fraework to propagate uncertainty to the final event reconstruction results. To address the fastresponse operational needs, siple fast-running Gaussian plue dispersion odels are adopted as the forward odel in the inverse proble. A probability odel is suggested to take into account both zero and non-zero concentration easureents that can be available fro a sensor network because of sensor s detection liit. Variance ter in the likelihood function is considered as unknown, and it is calculated fro an inverse gaa prior distribution, resulting in an event reconstruction ethod free fro tunable paraeters. In practice, the release location and release rates are of great iportance to the eergency responders. It is shown that the event reconstruction Y() MCMC chain adopting Eq. (1) Source at (,) MCMC chain adopting Eq. (11) X() Figure 6: Coparison of the traces of two MCMC chains. The red and blue chains belong to event reconstruction ethods that adopt Eq. (11) and Eq. (1) in the Gaussian plue odel, respectively The siulations have shown that the stochastic event reconstruction ethod is successful in capturing the true answers within the posterior distribution. Posterior distributions of the unknown paraeters were also used to generate probabilistic plue envelopes with specified confidence levels

6 ACKNOWLEDGMENTS The authors wish to thank Drs. Michael Willias and Michael Brown for helpful discussions. REFERERENCES Allen, C. T., and S. E. Haupt, 7: Source characterization with a genetic algorith-coupled dispersion-backward odel incorporating SCIPUFF. J. Appl. Meteor. 41, Allen, C. T., G. S. Young, and S. E. Haupt, 7: Iproving pollutant source characterization by better estiating wind direction with a genetic algorith. Atos. Environ., 41, Carlin, B. P., and T. A. Louis, 1996: Bayes and Epirical Bayes Methods for Data Analysis. Chapan & Hall. Chow, F. K., B. Kosovic, and S. T. Chan, 6: Source inversion for containant plue dispersion in urban environents using building-resolving siulations. 6 th Syposiu on the Urban Environent, Aerican Meteorological Society. Erik, S., and E. Lyck, : The Copenhagen tracer experients: Reporting of easureents. Riso National Laboratory, Riso-R-154(rev.1) (EN). Gilks, W. R., S. Richardson, and D. J. Spiegelhalter, 1996: Markov Chain Monte Carlo in Practice. Chapan & Hall. Marzouk, Y. M., H. N. Naj, and L. A. Rahn, 7: Stochastic spectral ethods for efficient Bayesian solution of inverse probles. accepted for publication in J. Cop. Phys. Johannesson, G., B. Hanley, and J. Nitao, 4: Dynaic Bayesian odels via Monte Carlo-An introduction with exaples. Lawrence Liverore National Laboratory, UCRL-TR Keats, A., E. Yee, and F. Lien, 7: Bayesian inference for source deterination with applications to a coplex urban environent. Atos. Environ., 41, Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, 1953: J. Che. Phys., 1, Neuann, S., L. Glascoe, B. Kosovic, K. Dyer, W. Hanley, and J. Nitao, 6: Event reconstruction for atospheric releases eploying urban puff odel UDM with stochastic inversion ethodology. 6 th Syposiu on the Urban Environent, Aerican Meteorological Society. Panofsky, H. A., and J. A. Dutton, 1984: Atospheric Turbulence. John Wiley. Tarantola, A., 5: Inverse Proble Theory and Methods for Model Paraeter Estiation. SIAM. Thoson, L. C., B. Hirst, G. Gibson, S. Gillespie, P. Jonathan, K. D. Skeldon, and M. J. Padgett, 7: An iproved algorith for locating a gas source using inverse ethods. Atos. Environ., 41,