A Study on Determination of Particular Matter Concentration using Least Squares Approximation and Block Signal Processing

A Study on Determination of Particular Matter Concentration using Least Squares Approximation and Block Signal Processing

Hun Choi*

Dep. of Electronic Eng. Dong-eui University Busan, KOREA

AbstractSince atmospheric particles may adversely effects human life, the continuous and automatic monitoring of the particulate matter (PM) concentration is required. In this paper, we present a new method for determining mass concentration in the PM monitoring system using beta-ray absorption method. The linear least squares function approximation (LLSFA) and block signal processing (BSP) are used to reduce the noise in the measured data. By combining the LLSFA and the BPS, the proposed method can improve the determining performance of the particulate matter concentrations in the air pollutant monitoring system using the beta-ray absorption method. Simulation results with the computer generated data and the empirical data show the proposed method accurately determines the PM concentrations for real time.

KeywordsPM Concentations; Beta-ray absorption method; Linear least squares function approximation; Block signal processing

1. INTRODUCTION

   The fast economic growth the last decades has resulted in an increase of the sources of particulate air pollution not only in large metropolitan areas but also in medium-sized urban areas. Atmospheric particles originate from a variety of sources and possess a range of physical and chemical properties. Especially, fine particulates less than 10 and

   2.5 m size are referred to as PM10 and PM2.5, respectively. These particular matters (PM) have the most significant impact on human health because they can penetrate deep into

   the lung[1]. Over the last decades a number of studies conducted in many countries around the world observed associations between ambient particle concentrations (especially PM10 and PM2.5) and human health risk[2,3]. To monitor the PM concentrations in open air, on-line method collecting data and calculating mass concentration with the collected data for real time or near real time is required. It is well-known that beta-ray absorption method (BAM) is a good solution for determining the PM concentration[4]. In this method, a known amount of electron scattering and attenuation through a clean filter is compared with that of a dust-sampled filter. To determine mass concentration, a ratio of the number of detected beta particles passing through the clean filter and the dust-sampled filter is used. Generally, these data measured in air pollution monitoring such as PM10 and PM2.5 usually contained the additive noise (thermal noise, electric noise in the beta-ray detector, power supply noise, and etc.). This additive noise can

   This work was supported by Dong-eui University Grant(2015AA133). Hun Choi* : Corresponding Author

   Joosang Youn

   Det. of Multimedia Eng. Dong-eui University Busan, KOREA

   Fig. 1. Schematic diagram of the PM monitoring system using the BAM.

   cause unintended bias in calculating the PM concentrations. Therefore, to improve the reliabilities of the calculated concentrations, it is important to find the best way to reduce the additive noise[5].

   This paper presents a new method for determining the mass concentration in the PM monitoring system using the BAM. To reduce the additive noise in the measured data, the proposed method uses the linear least squares function approximation (LLSFA)[6,7] and the block signal processing (BSP)[8]. The time varying mass concentration is calculated with the denoised data. In addition, by combining the merits of the LLSFA and the BSP, the proposed BSP-LLSFA method gives both the excellent estimation of the mass concentration without processing delay and the reduced computational complexity. To evaluate the performance of the proposed method, computer simulations were performed with computer generated signals and a real measured signal as the input.

2. METHOLOGY

   1. Beta-ray abruption

      Fig. 1 shows the configuration of the PM monitoring system which uses BAM. Air is drawn into the inlet and deflected downwards into the acceleration jet of the impact unit. Because of their greater momentum, particles larger than 10 m cut point impact out and are retained in the middle

      plenum impaction chamber. Particles smaller than 10 m are carried upward by the air flow and down the vent tubes to the

      beta gages sampler. After traversing the inlet configuration, the PM10 and PM2.5 particles are deposited on a glass fibre filter tape. A low level of beta-rays emitted from the source and passes through the filter tape and deposited particles. The increase of particles collected in the tape causes a lower beta- ray measurement in the measuring chamber. Applying Beer- Lambert Law for these signals, we can get the concentrations of the PM (PM10 and PM2.5) as follows.

3. THE PROPOSED METHOD

   To analyze the PM concentration, the observed beta-ray intensity (C0 and C ) that used in (1) are modeled by

   M C

   BS

   Qt

   lnC0 C

   where MC is mass concentration [ g / m3 ], S is the area of deposited sample (PM10 or PM2.5) [ cm2 ], Q is a rate of flow

   Fig. 2. The intensity data segment, data segment index, data segment time index corresponding to the observed time index

   [l/min], t is a time of deposition [min], C0 is the initial beta- ray intensity [count], C is the beta-ray intensity after it passes

   through the sample [count], B is a coefficient for unit

   C(n) I (n) v(n)

   conversion.

   1. Linear Least Squares Function Approximation

   Noise is constantly present in many applications with the measurement systems. Even though all control parameters remain constant, the resultant outcomes vary. Therefore, a process of quantitatively estimating the trend of the outcomes, also known as curve fitting, becomes necessary. The curve fitting process fits equations of approximated curves to the raw observed data. Nevertheless, for a given set of data, the

   Where n is time index I(n) is the pure beta-ray intensity without noise, u(n) is the colored noise, and v(n) is the additive noise.

   v

   Assumption A1: The additive noise v(n) is white Gaussian random process with zero mean and variance 2 . Hence, the beta-ray intensity I(n) and the additive noise v(n) are statistically independent.

   Assumption A2: The variation of beta-ray intensity is smaller than that of noise ( 2 2 ).

   fitting curves of a given type are generally not unique. Thus, a I v

   curve with a minimal deviation (error) from all data points is desired. This best fitting can be obtained by the method of least squares. The method of least squares seeks to minimize the sum of the squares of the squares of the deviation between the approximating function value and the observed data value.

   Consider data segment that consists of the last L observed noisy beta-ray intensity, the k-th intensity data segment (Ck) is represented by

   Assume that the given N data set are (x1, y1), (x2, y2), , (xN,

   yN), where x is the independent variable(time) and y is the

   Ck [C(L(k 1) 1)

   C(L(k 1) 2)

   C(Lk)]T

   dependent variable(observed data). Let f(x) denote an unknown linear function (straight line) that approximates the observed data y. The best fit straight line to approximate the observed data in linear least squares sense is given by

   where k is an integer as the index of intensity data segment, it is related to the observed time index n such as n = kL.

   Fig. 2 shws the intensity data segment (Ck), data segment index (k), and data segment time index (Lk) corresponding to

   f (x) ax b

   the observed time index (n). The mean value of k-th intensity data segment is expressed as

   L

   NS S S S S S S 1

   a xy x y b xx y xy x

   Ck L Ck (i)

   NS xx Sx Sx NS xx Sx Sx

   i1

   k

   k

   1 L I (i) 1 L v (i)

   N N N N

   L i1 L i1

   xx

   i

   x

   i

   xy

   i

   i

   y

   i

   k

   k

   S x 2 S x S x y S y

   where I (i) and v (i) are the i-th pure beta-ray intensity data

   i1

   i1

   i1

   i1

   and the corresponding additive noise in the k-th intensity data

   where the subscript i means the time index of each variable (time and the corresponding observed data) in the given N data set.

   segment, respectively.

   Under assumption A1, the second term on the right hand side (r.h.s.) in (7) can be assumed to be zero when L is large. Therefore, we can obtain a mean value of the k-th intensity data segment as

   Ck

   1 L I

   k

   L i1

   (i)

   Using the result of (8), the mass concentration MC,k in k-th data segment time can be represented as

   M

   J k [ f(t

   i1

   k ,i

   ) y

   2

   k ,i ]

   f(tk ,i ) atk ,i b

   where a and b are the x-slop and y-intercept coefficients of the best fit straight line that minimizes the cost function of (12), respectively. From (13) , (12) can be rewritten as

   Fig. 3. The new notations for BPS-LLSFA

   M

   J k (a, b) (at

   i1

   k ,i

   • b y

     2

     k ,i )

     M C ,k

     BS ln C

     Qt

     k 1 C k

     In (14), the cost function is quadratic and convex. Therefore, the cost function has a global minimum solution. From (14), we can get the partial derivatives of the cost function with respect to a and b , and set the results to zeroes

     where tis the same as the length of data segment, i.e. as

     t L . The mass concentration M C ,k in (9) is more than zero,

     because all the parameters are positive and lnCk 1 Ck 0 for

     J k (a, b) 2M (at

     b y )t 0

     large t. Under assumption A2, however, the second term on the r.h.s. of (7) can not be neglected when a time of deposition

     a i1

     k ,i

     k ,i k ,i

     k

     t is not sufficiently large. Therefore, lnCk 1 C 0 is not

     valid any more. Generally, digital filtering technique gives a good solution for reducing noise in many applications with noisy data). But, it can lead to a serious processing delay

     J k (a, b) 2 M (at

     b i1

     k ,i

   • b y

     k ,i

     ) 0

     (filtering delay increasing with the length of filter) in the application with the beta-ray signals obtained by low sampling rate ( 1Hz). The LLSFA method is an alternative solution to

     From (15) and (16), we can get as follows

     overcome this problem. The LLSFA method using the BSP can reduce computational complexity and adversely effect of high frequency noise. The PM monitoring system requires the huge processing power due to their large number of the

     M

     at

     2

     k ,i

     i1

   • bt

     M

     i1

     k ,i

     t

     M

     i1

     k ,i

     yk ,i

     measured signal. Moreover, it suffers from high frequency

     M

     a t

     M M

     • b y

   noise. The block signal processing based on the linear least

   squares function approximation (BSP-LLSFA) method may

   k ,i

   i1

   i1

   i1

   k ,i

   be a good solution to overcome these problems. To apply the BSP-LLSFA method to the estimation of the PM concentration, we use new notations for independent (time index) and dependent (observed data) variables of (2) as

   The matrix expression of (17) and (18) can be rewritten as

   M

   1

   shown in Fig. 3. From Fig. 3, the data segment time index

   M t 2

   t

   M t y

   vector and mean value vector corresponding to the last M data

   a

   i 1

   k ,i

   i 1

   k ,i

   i 1

   k ,i

   k ,i

   segments are respectively expressed as

   b M

   M

   tk ,i M yk ,i

   t [ L(k M 1)

   L(k M 2)

   Lk]T

   i 1

   i 1

   k

   [ tk ,1

   tk ,2

   t ]

   T

   k ,M

   Solving for a and b , we can obtain as follows

   MSty St Sy

   Stt Sy S S

   ty t

   a

   b

   y [C C

   … C ]T

   MS S S MS S S

   tt t t

   tt t t

   k (k M 1)

   (k M 2) k

   [ yk ,1

   yk ,2

   T

   y ]

   k ,M

   t

   M M M M

   S

   S

   where tk and yk are column vectors with M elements. Using (10) and (11), the cost function for determining the best fit straight line to approximate the M data segments can be

   Stt

   2

   k ,i

   i1

   St t

   i1

   k ,i

   ty

   t

   i1

   k ,i

   yk ,i

   y

   y

   i1

   k ,i

   expressed as

   From above results, the PM concentration for the M data

   segments at the data segment time index Lk can be expressed as

   ~ KS ~ ~

   Type 2: Exponentially decreasing data

   M C , k Q~t ln Ck ,0 Ck

   C(n) I (n) v(n)

   ren v(n) for {x | 0 n 40000}

   Fig. 5. The performances of the proposed method for different types of input data (The used parameters: L = 10, M = 360, Lini = 1, and Mini = 18 (5% of M)).

   Fig. 4. Procedure of the proposed BSP-LLSFA method.

   Fig. 6. The performance of the proposed method for type 3 input data (The used parameters: L = 10, M = 360, Lini = 1, and Mini = 18 (5% of M)).

   where the observation noise v(n) is the white Gaussian random process with zero mean, it is added to the pure beta- ray intensity signal such that SNR 25dB . In (26), the data parameters were set to r 1750, 1105 . The sampling

   ~

   ratio of the generated data C(n) was 1Hz. The used real data is

   In (22),

   ~

   Ck ,0

   Ck ,0 ak [L(k M 1)] bk

   ~

   Ck ak (Lk) bk

   Ck

   and ~ are calculated using (13), and ~t is

   an actual data that was measured by Thermo Andersen continuous ambient particulate monitoring system (FH62C14). To calculate the mass concentration shown in (22), the used parameters for the proposed method were set to Q 1.68106 , L = 10, and M = 360. Fig. 5 shows the theoretical and simulated mass concentration curves for the

   the length of the observed time index corresponding to the data segment time index vector (tk), i.e., ~t ML . The

   detailed procedure of the proposed method is shown in Fig. 4.

4. SIMULATIONS

To evaluate the performance of the proposed BSP-LLSFA method, we carried out computer simulations with computer generated data and a real measured data as the input. We considered two types of the pure beta-ray intensity data I(n) of

(5) to generate the input data for the practical applications. Type 1: Linearly decreasing data

type 1 and type 2 input data. From the results of Fig. 5, the proposed method provides the best estimations of the theoretical mass concentrations for two types of the input data. The mach between theory and simulation is excellent during steady-state and good during the initial transient, where the over or under damped curve can be reduced by controlling initial parameters (Lini and Mini which shown in Fig. 4).

Considering that a new filter is fed from a filter tape supply reel, we performed a simulation with a type 3 input data given by (27).

Type 3: Combined linearly decreasing data considering the filter

tape change

C(n) I (n) v(n)

0.015n 1750 v(n)

for {x | 0 n 40000}

0.015n 1750

for

0 n 21599

I (n) 1750

for

21600 n 22199

0.010n 1972

for

22200 n 43200

Fig. 6 shows the estimation performance of the proposed method for type 3 input. Assume that the filter changes at 6 hour and the initialization procedure of measurement allows 10 minutes for the filter to zero. During this time, where the mass concentration is zero, the initial beta-ray intensity is measuring

Fig. 7. Performance comparison of reference (FH62C14) and the proposed method (The used parameters: L = 10, M = 360, Lini = 1, and Mini

= 36 (10% of M)).

VI. CONCLUSIONS

This paper presents a new mass concentration estimation algorithm for the particulate matter (PM10 and PM2.5) monitoring system using the beta-ray absorption method. By combining the merits of the linear least squares function approximation (LLSFA) and the block signal processing (BSP), the proposed BSP-LLSFA method gives both excellent estimation of the mass concentration without processing delay and the method gives reduced computational complexity. The initial transient estimation performance of the proposed method can be improved by controlling the parameters used in BSP. Several simulation results have shown the good estimation performance of the proposed method.

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   and it is set to

   ~ . From the result shown in Fig. 6, we can

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   C0

   find that the proposed method provides values of the theoretical mass concentration over 90% in approximately 8 minutes. The performance of the proposed method for the real input is shown in Fig. 7. Just like the results for the computer generated input, the performance of the proposed method is excellent.

   V. CONCLUSIONS

   This paper presents a new mass concentration estimation algorithm for the particulate matter (PM10 and PM2.5) monitoring system using the beta-ray absorption method. By combining the merits of the linear least squares function approximation (LLSFA) and the block signal processing (BSP), the proposed BSP-LLSFA method gives both excellent estimation of the mass concentration without processing delay and the method gives reduced computational complexity. The initial transient estimation performance of the proposed method can be improved by controlling the parameters used in BSP. Several simulation results have shown the good estimation performance of the proposed method.

   ACKNOWLEDGMENT

   This work was supported by Dong-eui University Grant (2015AA133).

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