Air Pollution Prediction Using Mat´ern Function Based ... - IEEE Xplore

2014 13th International Conference on Control, Automation, Robotics & Vision Marina Bay Sands, Singapore, 10-12th December 2014 (ICARCV 2014)

Th22.5

Air Pollution Prediction Using Mat´ern Function Based Extended Fractional Kalman Filtering S. Metia, S. D. Oduro and Q. P. Ha

H. Duc

Faculty of Engineering and IT University of Technology Sydney, Australia [email protected] [email protected] [email protected]

Office of Environment and Heritage NSW, Australia [email protected]

stations in most metropolitan areas. In order process big data associated with air quality modelling and control in a region, machine learning and artificial intelligence methods appear to be very promising. Voukantis et al. [27] applied the multilayer perceptron (MLP) approach, where a feedforward artificial neural network model is sued maps sets of input data onto a set of appropriate outputs for air quality modelling. In Wahid et al. [28], air pollutant levels were estimated by using radial basis function neural networks in a metamodelling framework. Since an artificial neural network (ANN) used for pollution profile estimation is generally trained in an offline manner, the network weights and biases are assumed constants after training and moreover, the neural network learning algorithms could not directly address the uncertainty of the prediction. Recently, Lu and Wang [11] have compared classical statistical models for resolving ozone variations in time series with MLP as well as support vector machine (SVM), a supervised learning model used to analyze data and recognize patterns for classification and regression analysis. They concluded MLP and SVM may suffer from local minima, over-fitting, which are also inherent in most ANN models. To better handle uncertainties, nonlinearities and improve estimation accuracy, taking into account the time-varying nature of the process, Kalman filtering remains a good option and has been used in air quality modelling, particularly for the inversion problem. Jorquera and Castro [9] adjusted emission inventory for Santiago to address the air quality inverse modelling by using a Kalman filter. They improved the emission inventory of particulate matter pollutants using observation data and tested its sensitivity. Similar study was done by Tang et al. [26] via inversion of CO emissions using an ensemble Kalman filter, where improvements were achieved by reducing uncertainty of the emission inventory of CO and Ozone (O3 ) for air quality management in Beijing. Within this framework, our aim is to estimate O3 , NO, and NO2 emissions over Sydney and surrounding areas by using Extended Kalman Filter (EKF) and Extended Fractional Kalman Filter (EFKF) algorithms. Performance of both algorithms is compared with each other in terms of estimation accuracy. This paper is organized as follows. After the introduction, Section II describes TAPM-CTM model, the a priori emission

Abstract—It is essential to maintain air quality standards and inform people when air pollutant concentrations exceed permissible limits. For example, ground-level ozone, a harmful gas formed by NOX and VOCs emitted from various sources, can be estimated through integration of observation data obtained from measurement sites and effective air-quality models. This paper addresses the problem of predicting air pollution emissions over urban and suburban areas using The Air Pollution Model with Chemical Transport Model (TAPM-CTM) coupled with the Extended Fractional Kalman Filter (EFKF) based on a Mat´ern covariance function. Here, the ozone concentration is predicted in the airshed of Sydney and surrounding areas, where the length scale parameter l is calculated using station coordinates. For improvement of the air quality prediction, the fractional order of the EFKF is tuned by using a Genetic Algorithm (GA). The proposed methodology is validated at monitoring stations and applied to obtain a spatial distribution of ozone over the region. Index Terms—Extended Kalman Filter; Extended Fractional Kalman Filter; Mat´ern covariance function; Ozone

I. I NTRODUCTION Ozone (O3 ) is a secondary pollutant gas that is naturally produced in the earth’s atmosphere. It is formed from the photochemical reaction of nitrogen oxides (NOX , mixture of NO and NO2 ) and volatile organic compounds (VOCs) in the presence of solar radiation, and also influenced by other factors, including meteorological and topographical. The stratosphere ozone is very useful as it could shield humans from the harmful influences of the sun’s ultraviolet rays. However, exposure to the troposphere ozone (ground level ozone) may be harmful rather than beneficial to living beings because it can damage living tissues. Nowadays, many cities and their surrounding areas are facing serious emission problems due to accumulative population, motor vehicles and industries. As a result, the number of cities with poor environmental quality continues to grow. There has been an increasing concern on the effect of air pollutants on human health. Many possible measures have been taken by the policy makers to get better air quality monitoring either by using direct measurement or simulation software. Ozone (O3 ), Nitrogen Oxide (NO), Nitrogen Dioxide (NO2 ), Sulphur Dioxide (SO2 ), Particulate Matter (PM2.5 & PM10 ) and temperature observation data in both urban and suburban environments are monitored at measurement

978-1-4799-5199-4/14/$31.00 ©2014 IEEE

758

et al. [22] for application to the estimation of air pollutant profiles [12]. The Mat´ern family of the covariance functions gives us more liberty to control the smoothness and the spatial and the temporal correlation of the process. The Mat´ern class covariance function is defined by Rasmussen and Williams [17]. These functions can be represented by: √ ν  √  1−ν 2ν 2ν 22 kν (τ ) = σ τ Kν τ , (1) Γ(ν) l l

inventory, observation data, and EKF algorithm implementation; results and discussions of EKF results are presented in Section III and conclusions are given in Section IV. II. M ETHODOLOGY In this section EKF and EFKF algorithms are derived together with mathematical models. Furthermore, details of the Mat´ern function are shown and implemented. In the simulation, results are compared between the EKF and the EFKF. In addition, chemical transport model, emission inventory and observation data are discussed in details.

where l and σ 2 are the length scale and magnitude hyper parameters controlling the overall correlation scale and variability of the process, Kν is a modified Bessel-function and ν a parameter controlling the smoothness of the process. The spectral density of the Mat´ern covariance function is obtained by:

A. The Air Pollution Model with Chemical Transport Model (TAPM-CTM) TAPM-CTM developed by the Commonwealth Scientific and Industrial Research Organization (CSIRO), has a 3D prognostic meteorological and air pollution model [7]. The meteorological component of TAPM-CTM predicts the flow of sea breezes and terrain-induced circulations, given the larger scale synoptic fields data. The model solves the momentum equations for wind velocity components, especially in the horizontal direction. The vertical component is related to the continuity equation in the terrain coordinate system. Similarly, the model solves scalar equations for potential virtual temperature, specific humidity of water vapour, water, rain water, cloud and bush fire pollution parameters. Pressure is determined from the sum of optional non-hydrostatic and hydrostatic components, and a Poisson equation is used to solve the non-hydrostatic component. Explicit cloud complex micro-physical processes are included in the model. The wind observations are also integrated into the momentum equations.

S(ω) = σ 2

2π 1/2 Γ(ν + 1/2) 2ν 2 λ (λ + ω 2 )−(ν+1/2) , Γ(ν)

(2)

√ where λ = 2ν/l is associated with the dominant pole of the process to be chosen. In this paper we selected ν = p + 1/2, where p is a non-negative integer. Thus, S(ω) ∝ (λ2 + ω 2 )−(p+1) ,

(3)

which is a rational function form to be rewritten as, S(ω) ∝ (λ + iω)−(p+1) (λ − iω)−(p+1) ,

(4)

from which we can extract the transfer function of a stable process as H(iω) = (λ + iω)−(p+1) . (5) The corresponding spectral density of the white noise process v(t) is 2σ 2 π 1/2 λ(2p+1) Γ(p + 1) q= . (6) Γ(p + 1/2)

B. A priori emission inventory In this paper, air pollutant emissions for the Greater Metropolitan Region in NSW Emissions Data Management System (EDMS v2.0) are used as a priori emission inventories. Emissions have been assigned to map coordinates for industrial and commercial controlled point sources, or 1-km by 1-km grid cells for natural, domestic, off-road and on-road area sources and industrial and commercial uncontrolled fugitive sources. Emissions are then calculated for each month, day of week and hour of day by using factors derived from the activity data. The base year of the inventory represents activities that took place in the 2008 calendar year. Emissions are projected from 2009 to 2036. It includes anthropogenic emissions for nitric oxide (NOX ), carbon monoxide (CO), sulphur dioxide (SO2 ), particle concentrations (PM10 and PM2.5) and volatile organic compounds (VOCs) emissions.

D. The length scale effect Here, the Mat´ern correlation function with smoothness parameter ν = 32 or ν = 52 is known as a higher-order autoregressive function. With p = 2, the value of ν is given 1 5 by ν = √ 2 + 2 =√ 2 and the value of λ is calculated by 2ν/l = 5/l, where l is the length scale. The λ = length scale is known as the correlation length of the Gaussian process. A large correlation length signifies an input with very smooth and predictable effect on simulator output while a small correlation length denotes an input with more variable and fine scale influence on the output. The length scale varies in different applications. In Solin and S¨ arkk¨ a [25], l = 0.1m was chosen for resonator maps for temperatures. Gneiting [3] used l = 600km for atmospheric data analysis. In Hartikainen and S¨ arkk¨ a [5], l = 10m was selected in their generated time series while Hiltunen et al. [6] adopted l = 5m and l = 10m in diffuse optical tomography (DOT) experiments. In soil dynamics modelling, Minasny and McBratney [13] picked l = 20m and l = 100m. In Minasny and McBratney [14], calculated l = 100m as a length scale parameter to analyze the soil properties, where the average distance between

C. EKF estimation scheme The Kalman Filter (KF), also known as linear quadratic estimation is a commonly used method to estimate the values of state variables of a dynamic system [10]. In a nonlinear system, the EKF can be used instead of the KF. The EKF is developed based on linearization about the current mean with a given covariance. The EKF estimation scheme has been developed on Mat´ern based covariance function by S¨ arkk¨ a

759

⎡

sampling points was around 100m and each point datum was referred to a support of 10m × 10m, the area in which bulked samples were collected. In our case, 20 stations are situated across the New South Wales and data are collected 24 hours a day. The average distance between stations is 72.625km which is calculated by using station coordinates. Thus, √ we use l = √ 72.625km as the length scale. As a result λ = 5/l = 5/72625. The corresponding Linear Time Invariant Stochastic Differential Equation (LTI SDE) model, with a triple pole at −λ, reads therefore ⎤ ⎡ ⎤ ⎡ 0 1 0 0 dx(t) ⎣ 0 0 1 ⎦ x(t) + ⎣ 0 ⎦ v(t), (7) = dt −λ3 −3λ2 −3λ 1

ΔΥ xk+1

n1 , . . ., nN are the orders of system equations and N is the system dimension. The nonlinear functions f (·) and h(·) can be linearized according to Taylor series, e.g. ∂f (˜ x) (˜ x − x) + W, (11) ∂x ˜ where W stands for the higher order terms neglected in the linearization using Taylor series. The nonlinear discrete order stochastic system is represented in the state-space form as follows: ΔΥ x˜k+1 = f (ˆ xk , uk ), (12) f (x) = f (˜ x) +

where the spectral density q of white noise v(t) is computed by (6). When parameters of a dynamic system is unknown, there is a problem for estimation using the EKF. To overcome it, the Extended Fractional Kalman Filter (EFKF) can be a better option, especially when environmental models are highly uncertain and nonlinear.

˜k+1 − x ˜k+1 = ΔΥ x P˜k

The EKF approach is used in networked systems for data fusion or fault detection in systems. But, it has a limitation in the estimation of unknown state variables to a higher level of accuracy. On the other hand the EFKF algorithm has been used for estimation of unknown state variables in complex systems where a fractional derivative can be used for their more accurate description. Mathematical elaboration on the Extended Fractional Kalman Filter (EFKF) has been conducted with simulation provided for a lossy network system by Sierociuk and Dzieli´ nski [23] and Sierociuk et al. [24]. They improved the estimation of packet losses in networks. In addition, it was shown that improvements were achieved not only for estimation in the case of packet losses but also for smoothing. Simulation on a cryptography system in subject to noises using the EFKF was studied by Sadeghian et al. [19], [20], [21] and Romanovas et al. [18]. They studied the transmission of signal in a noisy environment and improved the chaotic system to secure better communication. Further details of the EFKF and fractional systems were presented by Das [1], Das and Pan [2]. The generalized nonlinear discrete stochastic fractional order system in a state-space representation is given by ΔΥ xk+1 = f (xk , uk ) + wk , (8) k+1 

(−1)j Υj xk+1−j ,

(−1)j Υj x ˆk+1−j ,

(13)

=

(Fk−1 + Υ1 )Pk−1 (Fk−1 + Υ1 )T k  +Qk−1 + Υj Pk−j ΥTj ,

(14)

j=2

x ˆk = x˜k + Kk [yk − h(˜ xk )],

(15)

Pk = (I − Kk Hk )P˜k ,

(16)

with the initial conditions x0 ∈ RN , P0 = E[(ˆ x0 − x0 )(ˆ x0 − x0 )T ], where Kk = P˜k HkT (Hk P˜k HkT + Rk )−1 ,

∂f (x, uk−1 ) , Fk−1 = ∂x x=ˆ xk−1

∂h(x) , Hk = ∂x x=˜xk and the noise sequences vk and wk are assumed to be independent and zero mean. In addition, P˜k = E[(˜ xk − xk )(˜ x − xk )T ]

(17)

is the prediction of the estimation error covariance matrix. The covariance matrix of the output noise vk in (10). is defined as

(9)

Rk = E[vk vkT ],

(10)

whereas the covariance matrix of the system noise wk is given as Qk = E[wk wkT ], (19)

j=1

yk = h(xk ) + vk ,

k+1  j=1

E. EFKF estimation scheme

xk+1 = ΔΥ xk+1 −

⎤ Δn1 x1,k+1 ⎢ ⎥ .. =⎣ ⎦, . nN Δ xN,k+1

where xk is the state vector, uk is the system input, yk is the system output, wk is the system noise, vk is the output noise at the time instant k,   

 nN n1 ··· , Υk = diag k k

(18)

Furthermore, Pk = E[(ˆ xk − xk )(ˆ x − xk )T ] is the estimation error covariance matrix.

760

(20)

F. Fractional system identification 6400

The Genetic Algorithm (GA) based fractional system identification was presented in time domain by Poinot and Trigeassou [16]. They developed a fractional order heat transfer model in the state-space model using time domain analysis. Similarly, Hartley and Lorenzo [4] studied time domain analysis of fractional systems and developed a fractional mathematical model of transfer function. A rational system was matched with a fractional system by using a fitness function of a GA to tune the fractional order in the time domain. In Ionescu et al. [8], a different method was used to identify the system by varying the fractional value in an incremental form to avoid complex nonlinear identification calculation. The fitness function is defined as

M  J = (yi∗ − yi )2 , (21)

Beresfield

Northing km

6300

6200

Fig. 1.

s3.011 s2.026 s1.010

s3.712 s2.701 s1.703

s3.021 s2.032 s1.026

s3.053 s2.049 s1.055

Wollongong Kembla Grage Albion Park

300

350

400

Station coordinate across the New South Wales (NSW).

12

Ozone (pphm)

8

6 Ozone (pphm)

10

Station Data EKF EFKF

4 2 0 -2

0

20

40 Time(Hr)

60

80

6 4 2 0 0

Fig. 2.

150

300 450 Time(Hr)

600

750

Ozone level prediction of a station from 1st -31st January, 2008.

H. Station data uncertainty analysis Observed station data are the main inputs for the air quality modelling. The uncertainty in the observation data involves the measurement error, representativeness error and the error related to the instrument precision. We have implemented the Mat´ern covariance function in the extended fractional Kalman filter and results are presented in the next section. It can be seen that the EFKF ozone prediction can improve on the station data and therefore is more reliable since all the uncertainty such as noise, missing data and inaccurate data are removed from the profile.

TABLE I F RACTIONAL ORDER VALUES CALCULATED USING GA Temperature

Lindfield Rozelle Randwick Earlwood

Easting km

where M is number of data. is the real station data and yi is the estimated output data using GA in the Υk . The system order is initially defined by sni ±0.999 , where ni is the order of the ith equation of the system. From (7), it is interpreted that the system is a 3rd order system. Table I shows the value of fraction which is calculated by minimising (21). The aim is to find the optimal value of fractional order in yi using GA. Here, the GA toolbox is used in MATLAB environment to optimise the objective function (21). The initial order of (7) is considered as a string. These strings are represented by a random number as an initial population. After that search is carried out among this population. The evolution of the population elements is non-generational, meaning that the new elements replace the worst ones. The main different operators adopted in the GA are reproduction, crossover and mutation. One of the advantages of the EFKF is the connection of the estimation and the smoothing actions. The system model (7) is used to model the EFKF whereas GA is used to tune fractional values of the system.

NO2

Bargo

6150 250

yi∗

NO

Richmond Vineyard Prospect St. Marys Chullora Bringelly Liverpool Oakdale Macarthur

6250

i=1

Ozone (O3 )

Wallsend Newcastle

6350

III. R ESULTS AND DISCUSSIONS Figure 1 shows different station coordinate across the New South Wales (NSW). Average distance between stations are calculated as 72.615km. Figure 2, Figure 3 and Figure 4 show the profile of ozone(O3), NO and NO2 of Liverpool station

G. Observations Observation data for O3 , NO2 , CO, SO2 , PM10 and PM2.5 from 20 sites in Sydney and its surrounding areas are integrated to estimate pollutant profiles. These sites are maintained and monitored by the Office of Environment and Heritage of the New South Wales Government, and their hourly pollutant concentration observations are collected at monitoring stations. The 20 stations’ data are provided on an hourly basis subject to the air quality control procedure [15].

TABLE II M EAN SQUARED ERROR (MSE) FOR DIFFERENT ATMOSPHERIC PROFILES Atmospheric Parameters EKF EFKF

761

O3 0.7534 0.0592

NO 0.7607 0.3516

NO2 0.0427 0.0113

Temperature 5.1980 0.2265

16

12

4

Station Data EKF EFKF

3 NO (pphm)

14

2 1

NO (pphm)

0

10

-1

0

20

40 Time(Hr)

60

80

8 6 4 2 0 -2 0

Fig. 3.

150

600

750

NO level prediction of a station from 1st -31st January, 2008.

3

2

1.5 NO2 (pphm)

2.5

Station Data EKF EFKF

1

0.5

0

0

20

40 Time(Hr)

60

Fig. 6.

EKF Data (Ozone) Level Distribution

Fig. 7.

EFKF Data (Ozone) Level Distribution

80

1.5

2

NO (pphm)

300 450 Time(Hr)

1 0.5 0 0

Fig. 4.

150

300 450 Time(Hr)

600

750

NO2 level prediction of a station from 1st -31st January, 2008.

Fig. 5.

that the estimation closely follows the observations at the station data, which is the expected behaviour of the EKF and the EFKF. Table II shows the Mean squared error (MSE) of different profile using the EKF and the EFKF. It is interpreted that the NO2 data profile prediction is significantly improved with EFKF. MSE of the EFKF is less compared to that the EKF. There is also an undershoot present in the prediction of the EKF. Figures 2 and 3 show the EKF prediction below zero value but the EFKF removes these uncertainties from the estimation. Basically, the EFKF estimates a process by using a form of feedback control: the EFKF estimates the process state at some time and then obtains feedback in the form of

Station Data (Ozone) Level Distribution

data. The EKF and the EFKF are used to estimate the pollutant profile of station data. The detailed profile of the pollutant is zoomed in the respective figure. The obtained results show

762

[4] T.T. Hartley and C.F. Lorenzo, “Fractional-order system identification based on continuous order-distributions,” Signal Processing, 83: 2287– 2300, 2003. [5] J. Hartikainen and S. S¨ arkk¨ a, “Kalman filtering and smoothing solutions to temporal Gaussian process regression models,” Proceedings of IEEE International Workshop on Machine Learning for Signal Processing, pages 379–384, Kittil¨ a , Finland, 2010. [6] P. Hiltunen, S. S¨ arkk¨ a,I. Nissil¨ a, A. Lajunen and J. Lampinen, “State space regularization in the nonstationary inverse problem for diffuse optical tomography,” IOP Science, 27(2):025009, 2011. [7] P.J. Hurley, “The air pollution model (tapm) version4. part 1: Technical description,” http://www.cmar.csiro.au/research/tapm/docs/ [8] C.M. Ionescu, J.A.T. Machado and R.D. Keyser, “Modeling of the lung impedance using a fractional-order ladder network with constant phase elements,” IEEE Transactions on Biomedical Circuits and System, 5(1):83–89, 2011. [9] H. Jorquera and J. Castro, “ Analysis of urban pollution episodes by inverse modeling,” Atmospheric Environment, 44:42–54, 2010. [10] R.E. Kalman and R.C. Bucy, “New results in linear filtering and prediction theory,” ASME J. Basic Engineering, 83(1):95-108, 1961. [11] W.-Z. Lu and D. Wang,“Learning machines: Rationale and application in ground level ozone prediction,”Applied Soft Computing, 24:135-141, 2014. [12] S. Metia, S.D. Oduro, Q.P. Ha, H. Duc and M. Azzi, “Environmental time series analysis and estimation with extended kalman filtering,” IEEE First International Conference on Artificial Intelligence, Modelling & Simulation, pages 202–207, Kota Kinabalu, Sabah, Malaysia, 2013. [13] B. Minasny and A.B. McBratney, “The Mat´ern function as a general model for soil variograms,” Geoderma, 128(3-4):192–207, 2005. [14] B. Minasny and A.B. McBratney, “Spatial prediction of soil properties using EBLUP with the Mat´ern covariance function,”Geoderma, 140:324– 336, 2007. [15] On-line: http://www.environment.nsw.gov.au. [16] T. Poinot and J.-C. Trigeassou, “Identification of fractional systems using an output-error technique,” Nonlinear Dynamics, 38:133–154, 2004. [17] C.E. Rasmussen and C.K.I. Williams, “Gaussian processes for machine learning (adaptive computation and machine learning),”MIT Press, 2006. [18] M. Romanovas, A. Al-Jawad, L. Klingbeil, M. Traechtler and Y. Manoli, “Implicit fractional model order estimation using interacting multiple model Kalman filters,” 18th IFAC World Congress, pages 7767–7772, Milano, Italy, 2011. [19] H. Sadeghian, H. Salarieh, A. Alasty and A. Meghdari, “On the linearquadratic regulator problem in one-dimensional linear fractional stochastic systems,” Automatica, 50(1):282–286, 2014. [20] H. Sadeghian, H. Salarieh, A. Alasty and A. Meghdari, “On the fractional-order extended Kalman filter and its application to chaotic cryptography in noisy environment,” Applied Mathematical Modelling, 38(3):961–973, 2014. [21] H. Sadeghian, H. Salarieh, A. Alasty and A. Meghdari, “On the general Kalman filter for discrete time stochastic fractional systems,” Mechatronics, 23(7):764–771, 2013. [22] S. S¨ arkk¨ a, A. Solin and J. Hartikainen, “Spatio-temporal learning via infinite-dimensional bayesian filtering and smoothing,” IEEE Signal Processing Magazine, 30(4):51–61, 2013. [23] D. Sierociuk and A. Dzieli´ n ski, “Fractional Kalman filter algorithm for the states, parameters and order of fractional system estimation,”International Journal of Applied Mathematics and Computer Science, 16(1):129–140, 2006. [24] D. Sierociuk, I. Tejado and B.M. Vinagre, “Improved fractional Kalman filter and its application to estimation over lossy networks,” Signal Processing, 91(3):542–552, 2011. [25] A. Solin and S. S¨ arkk¨ a,“Infinite-dimensional Bayesian filtering for detection of quasi-periodic phenomena in spatio-temporal data,” Physical Review E, 88(5), 052909, 2013. [26] X. Tang, J. Zhu, Z.F. Wang, M. Wang, A. Gbaguidi, J. Li, M. Shao, G.Q. Tang and D.S. Ji, “Inversion of CO emissions over Beijing and its surrounding areas with ensemble Kalman filter,” Atmospheric Environment, 81, 676–686, 2013. [27] D. Voukantis, H. Niska, K. Karatzas, M. Riga, A. Damialis, “Forecasting daily pollen concentrations using data-driven modeling methods in Thessaloniki, Greece,” Atmospheric Environment, 44 5101–5111, 2010. [28] H. Wahid, Q.P. Ha, H. Duc and M. Azzi,“Neural network-based metamodelling approach for estimating spatial distribution of air pollutant levels,” Applied Soft Computing, 13(10):4087–4096, 2013.

(noisy) measurements. As such, the equations for the EFKF fall into two groups: time update equations (prediction) and measurement update equations (correction). The time update equations are responsible for projecting forward (in time) the current state and error covariance estimates to obtain the a priori estimates for the next time step. The measurement update equations are responsible for the feedback i.e. for incorporating a new measurement into the a priori estimate to obtain an improved a posteriori estimate. From Figure 4, it is interpreted that NO2 is always present in environment. The main source of NO2 is produced by plants, soil, water and motor vehicle exhaust. But O3 is formed during day time. It can be seen from Figure 2, that during night time O3 is negligible or has a low concentration. In case of NO, it is produced during day time and negligible or tends to zero during night time. NO2 and NO are formed during high-temperature combustion in the atmosphere, when oxygen combines with nitrogen. The exhaust gases from motor vehicles are major sources of NOX , as are the emissions from electrical power generation plants. Motor vehicle exhaust has more NO than NO2 , but once the NO is released into the atmosphere it quickly combines with oxygen in the air to form NO2 . That is the main reason NO is absent during night time. It can be observed in Figure 5, where station data (ozone) were used to plot in a spatial domain. All station data were estimated using the EKF and the EFKF and spatially distributed as shown in Figures 6 and 7, where estimation accuracy can be observed clearly in the northern area as compared to the distribution obtained from stations. It is shown in Table II that overall accuracy has been improved with MSE being reduced to 0.05 with EFKF from 0.75 with EKF when estimating Ozone. Similarly, NO, NO2 and Temperature are estimated with MSE values of the EFKF respectively 0.35, 0.01 and 0.22 as compared to those from the EKF with 0.76, 0.04 and 5.19. IV. C ONCLUSION We have presented the design of an Extended Fractional Kalman Filter based on Mat´ern covariance function to estimate spatial profiles of air pollutants. The methodology shows high accuracy of profile estimation of pollution and its robustness to noise owing to the advantage of the EFKF. These techniques can be applied to improve existing inventories and assessing their uncertainties. The improved inventories are useful for policy-making purposes. The EFKF is advantageous in the estimation process in terms of accuracy improvement as well as in profile smoothing with the incorporation of the Mat´ern covariance function as can be seen in results of comparison with measured station data from across the NSW. The proposed approach is promising towards a suitable solution to the inversion problem in air quality modelling and control. R EFERENCES [1] S. Das, Functional fractional calculus, Springer: Springer Briefs in applied sciences and technology, 2012. [2] S. Das S. and I. Pan, Fractional order signal processing : introductory concepts and applications, Springer-Verlag: Berlin Heidelberg, 2011. [3] T. Gneiting, “Correlation functions for atmospheric data analysis,” Quarterly Journal of the Royal Meteorological Society, 125:2449–2464, 1999.

763