This file was created by scanning the printed publication. Errors identified by the software have been corrected; however, some errors may remain. On the Robustness of Data Assimilation Methods in Air Pollution Modeling X.F. Zhangl A.W. ~ e e m i n k ' Abstract The performance robustness properties of two data assimilation methods, Kriging ancl Kallnan filtering presented for the estimation of air pollution are investigated under perturbations of noise uncertainty ancl znodeP parameter uncertainty. The si~nulationresults are used to illustrate the advantages of Kalman filtering approach under various circumst ances. INTRODUCTION In Zhang et. a1 (1995) two d a t a assimilation methods, a Kriging approach and Kalman filtering are developed for the estimation of two dimensional air pollution problem. T h e deterministic system model and the statistics of system noise and measurement noise are assumed to be known exactly in Zlmng et. a1 (1 995). However, such an assutnption is often uilrealistic in applications, and noticeable deterioration of system performance may be caused by model errors and noise nncert ainty. In this paper we investigate the performance robustness properties of the two data assimilation methods proposed in Zhaug et. a1 (1995). T h e effects of hot 11 noise uncertainty and parameter uncertainty are examined. In Section 2, the stochastic system model is described, and in Section 3 t h e two d a t a assimilation methods are briefly reviewed. In Sec,tion 4, the influence of noise uncertainty on the Kalmau filtering approach is analysed, and a strategy to minimize the worst case performance is proposed. Then, in Section 5, s i n ulation results are presented, showing the advantages of t h e proposed robust Kalman filter design method compared with the conventio~lalKalman filter desigrr and the Kriging approach a t present of noise unc,rrtainty. Furthermore, the performance robustness is checked with respect t o uncertain transport speed of air pollution. Finally, Co~lclusionis given in Section 6. Department of Technical Malhenlatics and I n f o r ~ n a t i c s ,Delft IJniversity of Technology, Mekelweg 4, P. 0. Box 5031, 2600 CA Delft, The Netherlands THE STOCHASTIC DYNAMIC AIR POLLUTION MODEL The transport phenomena of air pollutant in the environment are usually described by an advection-diffusion equation where c is the concentration of the pollutant, t is time, u is the velocity field, D is the dispersion tensor including ~noleculardiffusion and dispersion. By introducing a numerical scheme (Heemink 1990, Van Eijkeren 1993) Eq. (1) can be represented by a discrete time state space model. where x(k) is an 11-vector state process taking corresponding values at model grid point at time instant k , n is the total number of model grid points at the is an r L x r~ dimension transient matrix pollutant concentration field, and oht ained by the numerical scheme discretizing the advection-diffi~sio~l Eq. ( I ) , is an n x q noise input matrix that interpolates q dimensional system noise process, with w ( k ) at 11 model grid points, r ( k ) is an m-vector ~neasure~nent and H are constant inatrices, w ( k ) measurement noise sequence v ( k ) . @, and v(k) and the initial state x(0) are assumed to be multiply independent and satisfied the following conditions: r where Q 2 0 is a positive semidefinite constant matrix, R > 0 is a positive definite constant matrix, hjr,= 1 if j = k and hjk = 0 if j # k. DATA ASSIMILATION METHODS The Kriging Approach Kriging is a method for the optimal estimation of a quantity y at an arbitrary location that has not been measured by using measurements taken at surrou~idingpoints. By changing the position of the point of esti~nation,it is possible to estimate the whole field of spatial variables. the estimation of the pollutants concentration value x at it11 grid point of G,,, and time instant k is given by: where .c is the total nu~nberof measurement stations. To determine the optimal choices of K ( i , I ) , K ( i , 2), ..., K ( i , s) of minimizing the esti~nationvariance for an arbitrary ith grid point of G,,,. We get (Zhang et. a1 1995) where yj,l denotes the value of the semi-variogram y(hj,l) of the deviation field. Hj,l denotes t lie value of the known covariance of m e a s ~ ~ r e m esnnoise. t The spatial correlation in the form of a semi-variogram is auto~naticallyestimated based on the available ~neasurernentdata at each time instant. The set of K ( i , j ) , j = 1, ..., s is obtained by Eq. (7) and then applied to Eq. (6). This produces an unbiased esti~nationwith minimu~nvariance of estimation error. The updated picture of the whole conc,entration field is obtained by adjusting x; throughout every grid point of the concentration field using Eq. (6). The Kalman Filter Approach Since the linear discrete-time system Eq. (2) and Eq. (3) is time-invariant, the Chandrasekhar-type algoritlml (Morf rt a1 1974) can be employed to get a steady state Kahnan filter as follows: where the gain matrix can be obtained by (Zhang et. a1 1995): with initial condition: The equations from Eq. (10) to Eq. (15) are iterated until: where 0 is determined by prespecified accuracy. Under the assumption that the dimension of system noise process q is much less than the dimension of system , Chandrasekhar-type algorithm provides a significant states 11, i.e, q << r ~ the co~nputationreduction (Heernink 1988,1990). A ROBUST FILTER DESIGN METHOD UNDER PERTURBATIONS OF NOISE UNCERTAINTY I11 general, it is a hard task to get suitable Q and R matrices which fit best the stochastic system model Eq. ( 2 ) with various observations. We suppose that the real steady state covariances of system noise and measurement noise are uncertain, and they can be described by: Q = Qo AQ, R = Ro AR. Further, we suppose that all information on AQ and A R is that the norm of AQ and A R are bounded by: 11 AQll 5 a , IlARll 5 9 . AS the optimal gain matrix I( of the Kalman filter is a function of Q, and R , the I( obtained by using Qo and Ro is in general no longer optimal due to the existence of uncertainties in Q and R matrices. First, we consider the prediction-type Kalman filter with gain matrix K p . Let e ( k 1) = z ( k ) - x(klk - 1) the actual estimation covariance denoted by NO(k 1lk) is determined by + + + + Sangsuk-Iarn (1990) prove that as k -+ oo Eq. ( 1 8 ) always converges to some constant No which satisfies the following algebraic Riccati equation From Eq. ( 1 9 ) we notice that the actual estimation covariance No is a 1110110tonic increasing function of the actual noise covariances Qo AQ and Ro A R. With Q := AQo aI, R := ARO+ P I where I is an identity matrix, we have Q 2 Qo AQ ,R 2 Ro AR. It is a reasonable choice to take the worst case covariance and R to design the Kaltnan filter which minimizes the upper bound of Kal~nanfilter performance, i.e + + + + + where I(,* is the optimal gain matrix of the Kal~nanfilter designed for the fixed pair of worst case noise covariances Qo aI and Ro /?I, and I i p could he any other gain matrix. We have the following saddle point inequality: + + Now, we consider the current estimation-type Kalman filter with gain matrix l i e . From Eq. (2), Eq. (3), and Eq. (8) we have the estimation error covariance of the form: Obviously, for any determined I<,, the actual estimation covariance P ( k ) is also a monotonic increasing function of Q AQ, and R AR. We have following saddle point inequality + + where K: is the optimal gain matrix of the current estimation-type Kalman filter designed using the fixed pair of worst case noise covariances Qo a I and Ro [3 I , and I(, could be any other gain matrix. From above discussion we can see that the selection of worst case noise covariances is a strategy to design a robust Kal~nanfilter (either prediction- type or current estimation-type) which mini~nizesthe upper bound of the Kalman filter performance. + + SIMULATION STUDY The simulation studies are aimed to examine the performance robustness of the two data assimilation methods with respect to noise uncertainty and model parameter uncertainty. The experimental region is a cycle area bounded in Q (see figure 1) . (a): True field (b) 9 Sample Locations 2 3 4 south (c) : Filtering results 2 3 south 4 4 2 3 south (d): Kriging results 2 3 south 4 Figure 1: (a): The concentration field at t=64; (b): "*" Observations, "." Model grid G,; ( c ) , (d): The eorespponding results of the Filtering and the Kriging approaches without model parameter and noise uncentainty. The concentration of the pollutant is denoted by z((,t ) where ( is the coy)' E 0. The true field is a stochastic process described by Eq. ordinate t = (g, (2). The deterministic part of this stochastic process is driven by the advectioudiffusion equation Eq (1) with a circular velocity u = u((;c,Y)') = 2n(- Y, g)' by I using the numerical approximation described in Heernink (1990) and V ~ I Eijkeren (1993). The initial condition of Eq. (1) consists of a constant background field and a small cosine-squared shape cone interpreted as a high density concentration area of local pollution. In our experiments, the simulated true field at t = 64 is shown in figure 1, (a). The regular model grid G,,, is shown in figure 1, (b). The systern noise ut is generated by w = (wl , ..., w,)' with zero mean and constant covariance Qo(i,j) = 2 x 10-'e -~ . ' a[s ( z t - . ~ ) ~ + ( ~ ~ - g ~ ) ~ ] measurement noise v is generated by Ro(i,j) = 1 x l ~ - ~ e - ~ [ ( " ' - " j ) ~ + ( ~ i - ~ ~ ) ~ ~ . The location of 9 observation stations is also plotted in figure 1 (b). When the deterministic model , the statistics of system noise and measurement noise are exactly known the reasonable good results of both data assimilation approaches are plotted in figure 1 (c), (d). Experiment 1 We assume that the second moments of statistics of the system noise and measure~nentnoise are uncertain. Let nominal filter use the optimal gain matrix corresponding to the nominal system noise covariance Qo and measurement noise covariance Ro, and robust filter use the robust gain matrix which min= Ro P I with a = 1 x imize the worst case of Q = Qo a1 and and /3 = 5 x First Q = Qo and R = Ro: The Root Mean Square (RMS) of the nominal filter and robust filter are plotted in (a) of figure 2. It can be seen that the RMS corresponding to robust filter is only slight higher than that of no~ninalfilter which is the optimal gain matrix in this case. S F C O T L ~ ~ T J Q = Qo cuI and R = Ro /?I: The RMS of the nominal filter and robust filter are plotted in (h) of figure 2. We can see that the RMS of robust filter is now 111uch lower than that of nominal filter. It is obvious from above simulation results the robust filter which minimize the worst case covariance of the estimation error makes the performance of the Kal~nanfilter much better from the point of view of robustness. To compare the two data assi~nilatiou methods the corresponding RMS of the Kriging approach are also plotted in figure 2 (a), (b). It can be seen that Kalman filter approach can provide more accurate result then The Kriging approach. + + x + + Experiment 2 We suppose that the Kalrnan filter is perturbed by uncertain deterministic system model. More exactly, it is assumed that the transport speed u which is a key model parameter is uncertain. Let the Kalman filter be designed by uo = (u5, uy) = 2 ~ ( - y , x ) , and let the true concentration field is run with + nu. u = uo Au = - 0 . 1 ~i.r. ~ 10% deviation. The true co~lcentratio~l field at t = 64 is plotted in (c) of figure 2. The correspo~ldingestimated field of the Kalman filter approach and the Kriging approach are plotted in (d) and (e) of figure 2,respectively. The RMS of the data assimilation procedures are plotted in (f) of figure 2. It is clear in this case that the performance robustness of the Kalman filter approach to tolerate model parameter uncertainty is much better than that of the Kriging approach. (a): RMS 0 (b): RMS 20 40 t (c): True field 60 0 20 40 60 t (d): Filtering results 4 2 south (e): Kriging results 3 south (1): RMS 4 40 t 60 10 1.5 - .- & 1 ' C 4 0.5 2 2 3 south 4 0 0 20 80 Figure 2: (a) and (b): The RMS of three methods with noise uncertainty. (c): Truth field at t=64; (d) and (e): The corresponding results of Filtering and Kriging approaches with 10% model parameter uncertainty; (f): RMS of both two approaches. CONCLUSION In this paper, the performance robustness of two data assimilation met hods under the perturbations of noise uncertainty and parameter uncertainty is investigated in two dimensional air pollution problem. The experimental results show the advantage of a robust Kalman filter design method with respect to noise uncertainty. Moreover, it is demonstrated in the simulation studies that the Kalman filter approach has much stronger performance robustness to tolerate model parameter uncertainty then the Kriging approach. Our current research focuses on applying Kalman filtering to " the evaluation of CH4 budget in Europe " in cooperation with RIVM air quality laboratory. ACKNOWLEDGMENTS This work has been carried out in cooperation with and with financial support from the RIVM. REFERENCES Heemink, A.W., 1988. Two-Dimensional Shallow Water Flow Identification. Applied Mat. Mod., 109-118. Heemink, A.W., 1990. Identification of Wind Stress on Shallow Water Flow Surfaces by Optimal Smoothing. Stochastic Hydrol. Hydraul. Vol. 4, 105-119. Morf, M., Sidhu, S.S., Kailath, T., 1974. Some New Algorithms for Recursive Estimation in Constant, Linear, Discrete-Time system. Trans. Autom. Control, Vol. AC-19, NO. 4, 315-323. Sangsuk-Iam, S., 1990. Analysis of discrete-time Kalman filtering under incorrect noise covariance, IEEE Transactions on Automatic Control, Vol. 35, 1304-1309. Van Eij keren, J.C.H., 1993. Backward Semi-Lagrangian Methods:an Adjoint Equation Method. In: Numerical Methods for Advection - Diffusion Problems, Vreugde hi1 C.B and Kora B Eds. Vieweg, Vol, 45, 215-241. Zhang, X.F., Van Eijkeren, J.C.H. and Heemink, A.W. 1995 Data Assi~nilation in Dynamic Environmental Pollution Modeling. RIVM report, No. 421503 006, Bilthoven, The Netherlands. BIOGRAPHICAL SKETCH Zhang X.F. is doing a Ph.D. project on the development of data assimilation methods for the estimation of air pollution. Heemink A.W. is a professor of faculty of technical mathematics and infor~natics,TU Delft.