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Atmospheric Environment 45 (2011) 4892e4897

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Atmospheric Environment

journal homepage: www.elsevier .com/locate/atmosenv

Inverse problem for aerosol particle size distribution using SPSO associated withmulti-lognormal distribution model

Yuan Yuan, Hong-Liang Yi*, Yong Shuai, Bin Liu, He-Ping Tan*

School of Energy Science and Engineering, Harbin Institute of Technology, 92, West Dazhi Street, Harbin 150001, PR China

a r t i c l e i n f o

Article history:Received 15 March 2011Received in revised form30 May 2011Accepted 3 June 2011

Keywords:AerosolParticle size distributionSPSOMulti-lognormal distribution model

* Corresponding authors. Tel.: þ86 451 86412308;E-mail addresses: yihongliang@hit.edu.cn (H.-L. Yi)

(H.-P. Tan).

1352-2310/$ e see front matter � 2011 Elsevier Ltd.doi:10.1016/j.atmosenv.2011.06.010

a b s t r a c t

A numerical study, used for particle size distributions of several classical aerosol types, is conducted inthis paper. By improving the log-normal distribution model in previous paper, we employ the stochasticparticle swarm optimization (SPSO) algorithm associated with a multi-lognormal distribution model. Thenonambiguity and robustness of SPSO with this model is validated. We recover particle size distributionsof two classical aerosol types using the measured AOT data (at wavelengths l¼ 0.400, 0.690, 0.870,1.020 mm) obtained from AERONET at different time. The results show that the model of this paperpresents characteristics of aerosol particle size distributions at different regions and time. And this modelshows good practicability to distinguish aerosol types and calculate particle size distributions.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Atmospheric aerosol is a multiphase system including solid,liquid and gas whose diameter is between 0.001 and 10 mm in air.Aerosol affects the Earth’s radiation budget by direct and indirecteffects and plays an important role in global climate systems. Thedirect effect is that aerosol particles scatter and absorb the solarradiation and surface infrared radiation (Schulz et al., 2006); theindirect effect is that aerosol particles act as cloud condensationnuclei and ice nuclei (Lohmann and Feichter, 2005). Meanwhile,aerosol is a very important correction in the progress of atmo-spheric radiative remote sensing, and is also the main factor forvisibility measurement based on scattering theory. So it is impor-tant to study scattering characteristics of aerosol particles (Shi,2007). In addition, the scattering and absorbing characteristics ofaerosols are important factors for laser transmission in atmosphere(Meehl et al., 1996; Le Treut et al., 1998). But the residence time ofaerosols in atmosphere is short, so their characteristics are signif-icantly different in space and time, and it is difficult to obtain thecharacteristics parameters of aerosols for the full spectrum directly.In certain region, if the type and size distribution of particles can be

fax: þ86 451 86413208., tanheping77@yahoo.com.cn

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obtained, the characteristics parameters for the full spectrum willbe calculated.

In recent years, various observational technologies and retrievalmethods have always been developed about by combined satellite-based and ground-based optical remote sensing to obtain theoptical characteristics of atmospheric aerosol. By the detectionmeans, such asMODIS (Schaap et al., 2008), MISR (Emili et al., 2010)and the AERONET sun-photometer global detection network (Radhiet al., 2010; Lee et al., 2010) et al, we can only measure directlyoptical properties of several bands, but cannot directly obtain thefull spectrum data. As the influence of atmospheric aerosols onterrestrial-atmospheric radiation balance system covers the fullspectrum, optical properties of the entire spectrum are thereforenecessary to have. Currently, we can only obtain these properties byindirect methods, that is, from known aerosol particle size distri-butions. Therefore, retrieving aerosol particle size distributions N(r)is necessary. Wang et al. (Wang et al., 2007) employed dampedGausseNewton iteration algorithm to retrieve aerosol particle sizedistributions combining the data obtained by sun-photometer.Wright et al. (Wright, 2000; Wright et al., 2002; Wright, 2007)analyzed the moments of the particle size distributions, anddeveloped the moment method to inverse N(r) using the opticalproperties of aerosol. Genetic algorithm (GA) have been studied tosolve reliably global optimal problems by Zuo et al. (Zuo et al., 2010)and the other random optimal method such as the stochasticparticle swarm optimization (SPSO) algorithm is also used in theinverse calculation of aerosol particle size distributions (Yuan et al.,

Nomenclature

a vector of the aerosol particle size distributionparameters

c1, c2 two acceleration coefficientsffitness objective functionM swarm sizeNi total aerosol number concentration for ith

lognormal distribution (cm�3)NP number of inverse parametersN(r) number density of radius r (cm�3)Ns number of samplesr particles radius (mm)w inertia weight coefficientYest inverse resultYexact exact value of the forward problem

Greek symbolsai geometric average diameter (mm)bi geometric standard deviationg measured errorl wavelength (mm)3rel relative errorsmði; aÞ inverse calculation values0ðiÞ measured data of the ith sample2 a standard normal distribution random variable

Table 1Influence of inverse accuracy on Ns and NP.

NP ¼ 4 NP ¼ 5

No SSA þSSA þSSA

Increasedparameter

ffitness Increasedparameter

ffitness Increasedparameters

ffitness

R1 7.33e�2 R1 9.383e�4 R1þ R2 4.414e�2R2 1.88e�2 R2 6.508e�7 R2þ R3 5.690e�2R3 7.95e�2 R3 1.887e�6 R3þ R1 1.413e�2

Y. Yuan et al. / Atmospheric Environment 45 (2011) 4892e4897 4893

2010) and the recognition of other particle size distributions (Wanget al., 2011; Wang et al., 2011).

In previous paper (Yuan et al., 2010), we investigated theapplication of the SPSO algorithm to retrieve aerosol particle sizedistributions associated with MIE scattering theory. The SPSOalgorithm has global convergence properties, and can be usedto solve the nonlinear problem successfully. A log-normal sizedistribution was employed to approximate the aerosol particle sizedistribution, and aerosol optical thicknesses (AOTs) and singlescattering albedos (SSAs) for several wavelengths were used in theinverse calculations. The SPSO algorithm was proved to be appli-cable to the inverse calculation of aerosol particle size distributionsand was robust. But as for the complex types, wide size distributionof aerosol particles, and multi peak value of the particle sizedistribution, the single log-normal distribution model has itslimitation and is not very accurate. Based on the single log-normal

0 500 1000 1500 2000 2500 3000

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

0.1

Best

fitn

ess

Generation

M=30M=50M=70

Fig. 1. Comparison of best fitness for different swarm size.

distribution model of previous paper, the size distribution ofdifferent kinds of aerosol particles are retrieved using multi-lognormal distribution model corresponding to different modesof aerosols. The multi-lognormal distribution model is validatedand analyzed by comparing of the data from AERONET with thesingle log-normal distribution model.

2. Methods

2.1. Forward and inverse problems

The retrieval calculation includes the forward and inverseproblems. The forward problem is to obtain the AOTs and SSAs byMIE scattering theory associated with the particle size distributionfunction. The inverse problem is to optimize the objective functionby the calculation of the forward problem from measured AOTsusing SPSO algorithm, and finally obtain the optimum solution. Thecalculation of the inverse problem adopts the SPSO algorithm, andthe detailed calculation progress and the validation can be seen inRefs. (Yuan et al., 2010; Tan et al., 2006).

In this paper, we use the multi-lognormal distribution model,then the size distribution N(r)is defined as:

NðrÞ ¼X3i¼1

Niffiffiffiffiffiffi2p

pbir

exp

"� ðlogðr=aiÞÞ2

2b2i

#: (1)

In Eq. (1) ai, bi, Ni are average diameter, standard deviation andnumber density of particles, respectively.

The inverse problem is to resolve the aerosol particle sizedistribution parameters from measured AOTs and SSAs; that is,assuming that AOTs and SSAs at several wavelengths are known, Ni,ai and bi can be determined by a retrieval algorithm. A key quantityin this determination is the objective function defined as follows:

ffitness ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXni¼1

½s0ðiÞ � smði; aÞ�2=nvuut (2)

where n is the number of computation samples; s0ðiÞ is themeasured data of the ith sample, and smði; aÞ is the correspondinginverse calculation value with a ¼ ðN1;N2;N3ÞT being a vectorof the aerosol particle size distribution parameters to be retrieved.Thus, the inverse problem reverts to searching for the vector

Table 2Complex refractive index of different wavelength for different aerosol types.

l Sool Marine Urban

n k n k n k

0.440 1.75 0.46 1.385 9.90e�5 1.530 0.0080.694 1.75 0.43 1.376 5.04e�8 1.530 0.0080.860 1.75 0.43 1.372 1.09e�6 1.520 0.0081.020 1.75 0.44 1.367 6.01e�5 1.520 0.008

0 2 4 6 8 100.0

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

2.5x10-4

3.0x10-4

3.5x10-4

4.0x10-4

Rel

ativ

e E

rror

Inverse Times

N1

N2

N3

Fig. 2. Relative error for different inverse time.

2 4 6 8 10

2.5

3.0

3.5

4.0

4.5

Rel

ativ

e E

rror

ε

Inverse Time

N1

N2

N3

Fig. 3. Relative error in different time for measurement error 5%.

Y. Yuan et al. / Atmospheric Environment 45 (2011) 4892e48974894

a that minimizes an as-yet-to-be-defined fitness value, and thisminimization procedure implements the retrieval algorithmmentioned above.

2.2. Chosen of inverse parameters

We investigate first the influence of different swarm sizes on theinverse problem. The convergence of the objective function isshown in Fig. 1 when the swarm sizes are M¼ 30, 50, 70 respec-tively. It can be seen from the figure that with increasing swarmsize, the convergence speed becomes faster; when the generationnumber reaches 3000, the convergence of M¼ 30 is much slowerthan the other two generations, and the calculation time increasestoo. Meanwhile, excessive swarm sizes will lead to longer compu-tation times because of the highly non-linear inverse problem. Asshown in the figure, increasing the swarm size will not greatlyaffect the convergence speed, and the convergence accuracy forM¼ 50 is similar to that for M¼ 70. Thus, considering the compu-tational accuracy and efficiency, we select a swarm size of M¼ 50and the value range for the inverse parameters is set to be withininterval ½10�6;106� in the following computations. Other parame-ters of the SPSO algorithm are set as: for global convergenceproperties and searching speed, we choose two acceleration coef-ficients c1 ¼ 1:8, c2 ¼ 1:8, and the inertia weight coefficientw ¼ 0 (Zeng et al., 2004). To analyze the implementation of theSPSO algorithm in solving the particle-size problem considered, thefollowing test cases will be examined.

Theoretically there are necessary nine parameters that are Ni, aiand bi of the three distributions. But NP is limited by Ns, and thereare only four measurement wavelengths (NP and Ns are denoted asthe number of inverse parameters and samples). It is impossible toretrieve all parameters. Therefore, NP is very important. In the nextcase, we discussed the influence of NP on the inverse accuracy. Thefeasibility of three parameters is studied in ref. (Wang et al., 2011).So in this paper, we will start to calculate from NP¼ 4. As shown inTable 1, except N1, N2 and N3, we add the inverse parameter Ri and

Table 3Inverse results with different measured errors using SPSO.

Parameter True value g ¼ 0 g ¼ 5

SPSO 3rel% SPSO

N1 105 100,000 0.00 103,057N2 0.2 0.2000 0.00 0.1922N3 0.1 0.100 0.00 0.1047

other values are considered as the known conditions. From thecalculation results of NP¼ 4 we can see that the accuracy of inverseresults is low (ffitness � 1� 10�2) with only the AOTs of fourwavelengths as the samples for forward problem; adding the SSAsof the same wavelengths as the samples, the accuracy can satisfyour requirement. For NP¼ 5 the accuracy is not good with the SSAsand AOTs as the samples. The reason is that the SSAs and AOTs ofthe same wavelength are not independent completely althoughthey are as the samples respectively, the independent samplenumber is still equal to the wavelength number. With NP>Ns thecalculation of the inverse problem is overdetermined, but withNP¼Ns the added SSAs play a certain role of increasing accuracy.

3. Numerical results and discussions

We analyze the variation of aerosols in two typical regions in thefollowing cases. According to the above analysis, we obtained therelation between NP and Ns: to ensure the calculation accuracy NPshould be no more than Ns. As there are only four measurementwavelengths and one of them is chosen to be verification wave-length, NP is determined to be three. For the characteristic ofaerosols, N1, N2 and N3 are chosen as the inverse parameters andother parameters are given as follows: average diameter ai ofdifferent distributions are set as the expectation diameters ofAltken, accumulation and coarse mode aerosols, and the standarddeviation is set as bi ¼ 0:307 according to Ref. (Shi, 2007).

3.1. Model validation

We discuss first the nonambiguity of the multi-lognormaldistribution model. In this case, we calculate the same problemten times with different initial values, and from the results weestimate the nonambiguity of the problem. The calculation condi-tions for this case are: the exact values of the forward problem areset as a ¼ ð0:1;0:1;0:01ÞT; Other parameters are set as: maximum

g ¼ 10 g ¼ 20

3rel% SPSO 3rel% SPSO 3rel%

3.06 106,995 7.00 112,178 12.23.9 0.1838 8.1 0.1689 15.64.7 0.1095 9.45 0.1187 18.7

Table 4AOTs obtained from AERONET in cases 1 and 2.

Case 1 Case 2

l 08.03.04 08.04.26 08.05.02 08.05.20 l 01.05.04 01.05.09 01.05.12 01.05.21

0.44 mm 0.580077 0.458438 0.618112 0.557095 0.44 mm 1.877169 0.265296 0.769965 0.5746340.69 mm 0.362060 0.318784 0.416288 0.357835 0.69 mm 1.818348 0.183749 0.568712 0.5718300.87 mm 0.250941 0.230496 0.288200 0.248439 0.87 mm 1.762104 0.134738 0.453727 0.5681571.02 mm 0.202799 0.187044 0.223295 0.199011 1.02 mm 1.713252 0.120415 0.408046 0.572225

Table 5Inverse results of case 1.

Case 1a Case 1b

ffitness N1 N2 N3 ffitness N1 N2 N3

SAMP1 5.49e�8 2.715 0.037 0.015 SAMP1 9.57e�10 1.136 0.034 0.012SAMP2 9.92e�12 2.024 0.041 0.014 SAMP2 6.57e�12 0.931 0.035 0.012SAMP3 8.47e�12 1.932 0.046 0.010 SAMP3 8.63e�12 0.901 0.036 0.011SAMP4 9.23�12 1.916 0.049 0.006 SAMP4 1.24e�11 0.899 0.037 0.009

Y. Yuan et al. / Atmospheric Environment 45 (2011) 4892e4897 4895

diameters of Altken, accumulation and coarse mode aerosolsare a1 ¼ 0:1 mm, a2 ¼ 1:0 mm, a3 ¼ 10:0 mm, respectively;b1 ¼ b2 ¼ b3 ¼ 0:307; selecting AOTs at the wavelengthsl1 ¼ 0:44 mm, l2 ¼ 0:69 mm, l3 ¼ 0:87 mm, l4 ¼ 1:02 mm as thecalculation samples; soot aerosol is chosen and the complexrefractive index is listed in Table 2. Other parameters of the inverseproblem are the same as above. For the sake of comparison, therelative error 3rel is defined as follows:

3rel ¼ 100� Yest � YexactYexact

: (3)

Here, Yest is the inverse result, Yexact is the exact value of the forwardproblem.

Fig. 2 shows the relative error 3rel of the ten times calculationsfor the maximum generation number of 3000. As seen in the figure,the multi-lognormal distribution model shows good nonambiguity,and the maximum relative error 3rel is lower than 5� 10�4. Whenthe computational accuracy is the highest, the relative error 3rel islower than 5� 10�5; for a ¼ ðN1;N2;N3ÞT, the 3rel of parameter N1is smallest and the 3rel of parameter N3 is largest, and the reason isthat the sensitivity of the three parameters to the forward problemis different.

The effect of measurement errors on the inverse computationalaccuracy is also considered. To demonstrate the effects ofmeasurement errors on the inverse parameters, random standarddeviations are added to the exact parameters computed from the

0.1 1 101E-5

1E-4

1E-3

0.01

0.1

1

10

Num

ber d

ensi

ty N

(r) (

cm-3

)

Particle radius

2008.3.04 2008.4.26 2008.5.02 2008.5.20

Fig. 4. Variation of aerosol particle size distribution with time in case 1.

direct solution. The following relation has been used in the presentinverse analysis:

Ymea ¼ Yexact þ s2 (4)

Here, 2 is a normal distribution random variable with zero meanand unit standard deviation. The standard deviation in themeasured AOT, s, for a g% measured error at 99% confidence, isdetermined as

s ¼ Yexact � g%2:576

(5)

The calculation conditions of this case are: the exact values ofthe forward problem are set as a ¼ ð105;0:2;0:1ÞT; Other param-eters are set as: a1 ¼ 0:1 mm, a2 ¼ 1:0 mm, a3 ¼ 10:0 mm;b1 ¼ b2 ¼ b3 ¼ 0:307; selecting AOTs at the wavelengthsl1 ¼ 0:44 mm, l2 ¼ 0:69 mm, l3 ¼ 0:87 mm, l4 ¼ 1:02 mm as thecalculation samples; soot aerosol is chosen. The calculation resultsare listed in Table 3. As shown in the table, the relative errors 3rel ofN1;N2;N3 are smaller than the measurement errors. Meanwhile,the 3rel of the three parameters are close, which shows that theinfluences of the three modes on the spectral optical thicknessesare almost same. As seen in Fig. 3, the 3rel of the 10 times calcula-tions change little for the measurement error of g ¼ 5. The aboveresults illustrate that the multi-lognormal distribution model isrobust.

3.2. Discussions on numerical results

To verify the applicability of the multi-lognormal distributionmodel to this problem, we analyze aerosol particle size distribu-tions of different regions. Firstly, the aerosol particle size distribu-tion of Taipei is considered in case 1 and in this region there mainlyis the marine aerosol. When the log-normal size distribution isadopted, the performance is not good enough. Meanwhile, terres-trial dust transport has little effect on this region so the particle sizedistribution changes little with time. The parameters of the calcu-lation are set as: a1 ¼ 0:05 mm, a2 ¼ 0:4 mm, a3 ¼ 3:3 mm andb1 ¼ b2 ¼ b3 ¼ 0:307; aerosol spectral optical thicknesses arefrom the ground observation data obtained from AERONET; asshown in Table 4, the AOTs at the wavelengths l1 ¼ 0:44 mm,l2 ¼ 0:69 mm, l3 ¼ 0:87 mm and l4 ¼ 1:02 mm are chosen. Thecomplex refractive index is listed in Table 2. We choose three AOTsas the samples of the inverse problem and the rest one acts as thevalidation for the inverse results. We calculate the particle size

Table 6Inverse results of case 2.

Case 2a Case 2b

ffitness N1 N2 N3 ffitness N1 N2 N3

SAMP1 9.25e�3 3.24e�7 1.68e�2 0.102 SAMP1 2.32e�9 1.24e�2 1.18e�2 9.93e�2SAMP2 1.63e�2 1.00e�8 1.01e�2 0.108 SAMP2 8.18e�12 6.95e�2 1.07e�2 1.01e�1SAMP3 1.57e�2 1.00e�8 1.26e�2 0.105 SAMP3 6.38e�12 6.82e�2 7.76e�2 1.05e�1SAMP4 1.14e�2 1.00e�8 5.07e�3 0.116 SAMP4 7.54e�5 4.77e�2 1.36e�3 1.16e�1

Y. Yuan et al. / Atmospheric Environment 45 (2011) 4892e48974896

distributions at different dates in this region. The results are shownin Table 5. SAMP1 to SAMP4 represent the inverse results that thecorresponding sample is discarded in the inverse computation (forexample, SAMP1 is the inverse result that sample one is discardedand samples two to four are selected). It can be seen that thecomputational accuracy of case 1 is very high, reaching 1� 10�9 incase 1a (4th, March, 2008) and case 1b (26th, April, 2008). Theresults of SAMP1 to SAMP4 are almost same for each case. Fig. 4shows the variation of the particle size distributions in a certaintime. We can see that the particle size distributions change little,and agree well with the above properties. Such agreements illus-trate that the multi-lognormal distribution model is suitable to themarine aerosol.

In previous paper, the log-normal size distribution model wasadopted to retrieve the aerosol particle size distribution of Beijing.We obtained good applicability in the log-normal distributionmodel, but the accuracy is lower than that of case 1 in this paper. Tovalidate the universality of themulti-lognormal distributionmodel,we also calculate the same case using the multi-lognormal distri-bution model. In case 2, we calculate the aerosol particle sizedistribution of Beijing. Firstly, we adopt the same parameters ascase 1. The results are shown in Table 6 (case 2a). The computa-tional accuracy of case 2a reaches only 1�10�2. It can be seen thatusing the parameters of themarine aerosol is not reasonable for theinverse calculation in this case. The reason is that aerosols in thisregion are mainly the urban aerosol. The characteristic is that theproportion of Altken mode is higher in the particle size distribu-tions. Other factors such as strong dust weather and human activityaffect the particle size distributions greatly, which lead to largevariety of the aerosol particle size distributions and types. For thecharacteristic of the urban aerosol, we adopt the complex refractiveindex of the urban aerosol as shown in Table 2, and other param-eters are the same as case 1, and AOTs are also obtained fromAERONET. The results are listed in Table 6. In case 2b, the calculationaccuracy is high. So for the multi-lognormal distribution model,distinguishing the aerosol type is very important. From Fig. 5, theaerosol particle size distributions vary greatly at different time inthis case, which agree well with the characteristic of urban

0.1 1 101E-5

1E-4

1E-3

0.01

0.1

1

10

Num

ber d

ensi

ty N

(r) (

cm-3

)

Particle radius

2001.5.04 2001.5.09 2001.5.12 2001.5.21

Fig. 5. Variation of aerosol particle size distribution with time in case 2.

aerosols. The number density of coarse mode aerosol increasesobviously on 21st May 2001, so we can see that it is dust weather.Thus, the multi-lognormal distribution model is also suitable forthe retrieved calculation of urban aerosols when the parametersare selected accurately.

4. Conclusions

In this paper, we employed the SPSO algorithm to retrieve theatmospheric aerosol particle size distribution using the multi-lognormal distribution model. Then the nonambiguity and theinfluence of measurement errors were analyzed, and from theresults, it can be seen that this model has good nonambiguity androbustness. Meanwhile, we retrieved the size distributions ofparticles in two typical regions. From the results we can see that theaccuracy of this model is high, and characteristics of differentaerosol types can be described exactly. For marine aerosols, theparticle size distributions vary little with time and are homoge-neous for different modes; for urban aerosols, the particle sizedistributions change obviously. Therefore, the multi-lognormaldistribution model is applicable to retrieval of aerosol particlesize distributions, and for different regions and time, we can obtaingood performance. It is very important to distinguish the aerosoltypes and adopt corresponding parameters. We can recognize theparticle size distributions of certain region easily associated withseveral AOTs using the multi-lognormal distribution model. Bycalculating the particle size distributions and distinguishing theaerosol type with this model, we can calculate the characteristicsparameters of aerosols in the full spectrum and the particle sizedistributions can also be applied to analyze other problems inatmospheric environment.

Acknowledgements

This work was supported by Program for New Century ExcellentTalents in University (NCET-09-0067), the key program of theNational Natural Science Foundation of China (Grant No.50930007) and the National Natural Science Foundation of China(Grant No. 50806018). A very special acknowledgement is madeto the editors and referees who made important comments toimprove this paper.

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