The Square Root Ensemble Kalman Filter to Estimate the Concentration of Air Pollution

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1 International Conference on Mathematical Applications in Engineering (ICMAE August 200 Kuala Lumpur Malasia The Square Root Ensemble Kalman Filter to Estimate the Concentration of Air Pollution Erna Apriliani Department of Mathematics Institut Tenologi Sepuluh Nopember (ITS Surabaa Indonesia april@matematia.its.ac.id Didi Khusnul Arif Bandung Arr sanoo Department of Mathematics Institut Tenologi Sepuluh Nopember (ITS Surabaa Indonesia Abstract Kalman filter is an algorithm to estimate the state variable of dnamical stochastic sstem. The advantages of the Kalman filter are ensemble Kalman filter (EnKF Square root Ensemble Kalman Filter (SQRT-EnKF In the ensemble Kalman filter we need more computational time than the Kalman filter. The square root ensemble Kalman filter is proposed to eep the computational stabilit and the reduced ran ensemble Kalman filter is proposed to eep the computational stabilit and reduce the computational time of square root ensemble Kalman filter. We have applied Kalman filter to estimate the concentration of air pollution in the cit. Some area in cit is divided in n n grids some data are taen to estimate for n n position. Here we applied the SQRT-EnKF and EnKF algorithm to estimate the concentration of air pollution. We compare the accurac and computational time between those algorithms. Kewords : The ensemble Kalman Filter Square Root Matrices concentration of air pollution I. INTRODUCTION Kalman filter is an algorithm to estimate the state variable of the stochastic dnamical linear sstem. This algorithm combines the mathematical model with the measurement data [3]. Kalman filter has been applied in various problems such as the estimation of the water level in the river wave of ocean and the tide [5]. The estimation of pollution distribution in the groundwater []. The other side there are man modification of Kalman filter algorithm. This modification has been done to avoid the convergence of algorithm to reduce the computation time to decrease the error of estimation such as The Reduced Ran square Root Covariance Filter (the RRSQRT Filter[7]. One of the modification of Kalman filter is the Ensemble Kalman filter[2]. In this paper we appl the Ensemble Kalman filter and the square root Ensemble Kalman Filter to estimate the air pollution distribution. Before we appl those algorithm we we describe the mathematical model of air pollution we discretize respect to time and position such that we get a state space model we derive the algorithm of square root Ensemble Kalman filter for this model and then we do simulation. We compare the accurac and the computational time between the Ensemble Kalman filter (EnKF and Square Root Ensemble Kalman filter (SQRT EnKF. II. MATHEMATICAL MODEL OF AIR POLLUTION DISTRIBUTION The model of air pollution concentration is called Gauss difussion and it can be written as [46]: t U C D + U C D Where C is an pollution concentration D D is diffusion coefficient in and direction respectivel U is and direction respectivel. U wind velocit in Suppose we assume the diffusion coeffiencient and the wind velocit are constan so that we can be rewritten the eq.( as follows 2 2 C C D + D 2 U 2 + U t (2 Before we applied the ensemble Kalman filter to estimate the concentration of air pollution we discretize eq. (2 respect to space and time t and then we write in the state space form. The state space sstem of air pollution modeling is Or C + f( C X + f( X The model is not eactl the same with the real sstem we tae some assumptions that cannot be written in the model so that we write the sstem as X f( X + + Gw ( 3 ( /0/$ IEEE

2 Where X + Gw is the state variable in time + in C + this case the concentration of pollution matri sstem noise and sstem Gaussian white noise respectivel w N(0 Q. To mae correlation between the state which we will estimate and the measurement data we define the measurement equation as follow: z H v + ( 4 Where z H v is measurement data measurement matri state variable and measurement Gaussian white noise v N(0 R III. ENSEMBLE KALMAN FILTER Suppose we has a dnamic stochastic sstem Eq. (3 and measurement Eq. (4. B using Kalman filter we estimate the state variables of Eq. ( based on the data measurement Eq. (4. Kalman filter is one of data assimilation method because in Kalman filter we combine the model of sstem Eq. (3 and the data measurement Eq. (4. Ensemble Kalman filter is one of Kalman filter modifications. Kalman filter is an estimation method for linear dnamic stochastic sstem but Ensemble Kalman filter is an estimation method for non linear dnamic stochastic sstem based on measurement data. In the Kalman filter algorithm we give ust one vector for initial estimation 0 and the covariance of prediction step is determined from the equation [3]. But in the ensemble Kalman filter we generate N ens ensembles of initial estimation 0 N( 0 P 0 so that the ensemble Kalman filter need more computational time than Kalman filter. The algorithm of the ensemble Kalman filter is [2] a. Initial Estimation Generate the N-ensembles of initial estimation 0 [ N 0 N ] with 0 i ~ N( 0 P0. b. The prediction step ˆ f ( ˆ u + w (5 where w ~ N(0 Q is the ensemble of noise sstem Mean of prediction step estimation ˆ N ˆ N Error covariance of prediction step estimation P ( ( T ˆ ˆ ˆ ˆ N c. The Correction step Generate the ensemble of measurement data z z + v (6 Where v ~ N(0 R is an ensemble of measurement noise Kalman gain is defined as T T K P H HP H + R ( ( Estimation of correction step is ˆ ˆ + K z H (7 ˆ Mean of correction step estimation: ˆ N ˆ N + With error covariance P [ ] I K H P d. Substitute Eq. (7 in prediction step Eq. (5 e. Continuing until we get correction step estimation Eq. (7. The application of Ensemble Kalman filter in some cases become instable or divergence in due to round of error in matri covariance. To avoid this divergence we can use the square root Ensemble Kalman filter. The algorithm of SQRT-EnKF has same algorithm with EnKF for initial step and prediction step. In the correction step the SQRT-EnKF algorithm is below: Generated the N ensemble measurement data such as eq. (6 Define H ~ T T E v and C S S + E E S The ensemble correction estimation ~ T ˆ ( ˆ + S C z H After that we decomposed the matri singular value decomposition : C b using the [ U D V ] svd( C T Define matri M DU S decompose matri M [ U D V] svd( M The error of ensemble estimations are ~ T ˆ ( D / 2 V I D The ensemble of estimations are ˆ ~ + Repeat the prediction step and correction step such that we get the time step which desired.

3 IV. SIMULATION RESULTS Actuall it is difficult to get real data the concentration of pollution because the measurement tools is limited the otherwise we concern on the algorithm of Ensemble Kalman Filter Square Root Ensemble Kalman filter and the reduced ran Ensemble Kalman filter so that the measurement data is generated from MATLAB program that represents Eq (4. Here we divide an area into 00 point ( 0 grids in and position respectivel. We place five the measurement equation in five position and we estimate the concentration in 00 points / places in those area. We assume there is no source of pollution ecept the initial concentration Here we compare the accurac of estimation b using Ensemble Kalman filter with b using Square root Kalman filter. At this moment we cannot appl the reduced ran Ensemble Kalman filter. same result of estimation that is mean those algorithm have almost the same accurac. The concentration of pollution is decreasing when time is increasing because there is no source pollution ecept initial concentration. Figure 3. Estimation b SQRT EnKF 3D Figure. The Concentration Estimation Figure 4. Contour of estimation b EnKF Figure 2. The Concentration Estimation b EnKF 3D In this case Ensemble Kalman filter (EnKF and the Square root Ensemble Kalman Filter (SQRT EnKF give almost the

4 Table. The CPU time of algorithm Simulation ENKF SQRT-EnKF I II III IV V VI VII VIII IX X Average Figure 5. Contour of Estimation b SQRT EnKF After 0 time step we get that the concentration more less than the 3 time step the concentration of pollution after 0 time step is represented on figure 6. We repeat the simulation until 0 time and we compare the computational time and the accurac of those algorithm. From table. we now that the Ensemble Kalman filter need less computational time than the Square root Ensemble Kalman Filter. The accurac of those algorithm is represented on table 2 and shows that error almost same. Table 2. The Error of Estimation Simulation EnKF SQRT EnKF I II III IV V VI VII.42.8 VIII IX X Average Figure 6. The Concentration of Pollution after 0 time step V. CONCLUDING REMARKS From description and the simulation we conclude that: We can appl the Ensemble Kalman Filter and the Square Root Ensemble Kalman Filter to estimate the concentration of Pollution The Square Root Ensemble Kalman filter need more computational time than the Ensemble Kalman Filter but have the same accurac For the net research we will appl the reduce ran of covariance matri to Square Root Ensemble Kalman Filter and also we can tr for another places of measurement tools. ACKNOWLEDGMENT This paper is part of our research with the title Reduced ran and Ensemble of Kalman Filter. This research is funded b Hibah Bersaing research Grant 200.

5 REFERENCES [4] Hanea R. Data Assimilation Concept and The Kalman Filter Approach Bahan RWS TU Delft 2005 [] Apriliani Grounwater Pollution Estimation b Data Assimilation Method research report Hibah bersaing XII [2] Evensen G. Sequential Data Assimilation with a nonlinear quasigeostrophic model using Monte Carlo methods to forecast error statistic. J. Geophs Vol 99 page [3] Lewis L Fran. Optimal Estimation with an introduction to stochastic control theor John Wile and Sons New Yor 986. [5] Heemin A.W. Data Assimilation For Non Linear Tidal Models International Journal for Numerical Methods in Fluids 990 [6] Nevers N.D. 995 Air Pollution Control Engineering McGraw-Hill Inc New Yor