inverse problem for particle size distributions of atmospheric aerosols using stochastic particle...
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Journal of Quantitative Spectroscopy &Radiative Transfer
Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 2106–2114
0022-40
doi:10.1
� Corr
E-m
tanhepi
journal homepage: www.elsevier.com/locate/jqsrt
Inverse problem for particle size distributions of atmosphericaerosols using stochastic particle swarm optimization
Yuan Yuan, Hong-Liang Yi, Yong Shuai, Fu-Qiang Wang, 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 29 January 2010
Accepted 24 March 2010
Keywords:
Radiative properties
Stochastic particle swarm optimization
Aerosol optical thickness
MIE theory
73/$ - see front matter & 2010 Elsevier Ltd. A
016/j.jqsrt.2010.03.019
esponding author. Tel.: +86 451 86412308; fax:
ail addresses: [email protected] (H.-L. Y
[email protected] (H.-P. Tan).
a b s t r a c t
As a part of resolving optical properties in atmosphere radiative transfer calculations,
this paper focuses on obtaining aerosol optical thicknesses (AOTs) in the visible and
near infrared wave band through indirect method by gleaning the values of aerosol
particle size distribution parameters. Although various inverse techniques have been
applied to obtain values for these parameters, we choose a stochastic particle swarm
optimization (SPSO) algorithm to perform an inverse calculation. Computational
performances of different inverse methods are investigated and the influence of swarm
size on the inverse problem of computation particles is examined. Next, computational
efficiencies of various particle size distributions and the influences of the measured
errors on computational accuracy are compared. Finally, we recover particle size
distributions for atmospheric aerosols over Beijing using the measured AOT data (at
wavelengths l=0.400, 0.690, 0.870, and 1.020 mm) obtained from AERONET at different
times and then calculate other AOT values for this band based on the inverse results.
With calculations agreeing with measured data, the SPSO algorithm shows good
practicability.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
Atmospheric aerosols are suspensions of small solid orliquid particles in air. Examples include dust, soot, micro-organisms, spores and plant pollens, and cloud precipi-tates, hail, sleet, rain and snow, consisting of water andice, all of which play an important role in the environmentbecause they take part in many physical and chemicalprocesses [1,2]. Atmospheric aerosols can scatter andabsorb short-wave solar radiation and long-wave terres-trial radiation, and thus affect the radiation balance of theearth–troposphere system. Thus, atmospheric aerosols areof particular interest in study of atmospheric radiation.
ll rights reserved.
+86 451 86413208.
i),
A prominent characteristic of atmospheric aerosolparticles is the great temporal–spatial variability of theirphysical and chemical properties. It is generally believedthat the diameters of atmospheric aerosol particles are inthe range of a few nanometers (nm) to tens of microns(mm). Concentrations of atmospheric aerosols with dia-meters smaller than 1 mm range from several tens toseveral thousand cm�3, while that with diameters largerthan 1mm are commonly less than 1 cm�3. An atmo-spheric aerosol particle can be solid, droplet, or compositeparticles combining solid and liquid phases. Chemically,they can be either homogeneous or inhomogeneous.Shapes of atmospheric aerosols can vary from the verysimple spherical liquid drops to complex non-sphericalshapes. In general, life spans in the atmosphere extendfrom a few days to several weeks. Because residence timesof atmospheric aerosol in air are short and theirtemporal–spatial characteristics vary widely, there hasbeen till date a lack of sufficient data to study the various
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Y. Yuan et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 2106–2114 2107
influences on global and local climate systems of atmo-spheric aerosols of different origin and type in theatmosphere.
Countries worldwide have always been developingobservational technologies and retrieval methods broughtabout by combined satellite-based and ground-basedoptical remote sensing that has in recent years lead toenormous progress in understanding optical characteris-tics of atmospheric aerosol. One of the key applicationareas for satellite-based detectors, which include moder-ate and high resolution imaging spectro-radiometers(MODIS, HIRDLS) [3,4], advanced very high resolutionradiometer (AVHRR) [5,6], and multi-angle imagingspectrometers (MISR) [7], is to detect global, includingterrestrial, aerosol optical properties. With respect toground-based remote sensing, one important develop-ment is to have established the AERONET sun-photometerglobal detection network [8]. Aerosol optical propertiescan then be extracted from data obtained and usedto verify satellite-based remote sensing data [9,10].By these detection means, we can only measure directlyoptical properties of several bands, but cannotdirectly obtain the full spectrum data. As the influenceof atmospheric aerosols on the terrestrial–atmosphericradiation balance system covers the full spectrum,optical properties of the entire spectrum are thereforenecessary to be present. Currently, we can only obtainthese properties by indirect methods, that is, fromknown aerosol particle size distributions and its asso-ciated parameters, optical properties of the entirespectrum could be computed. However, obtaining theseparameters directly is also very difficult. Therefore,retrieving aerosol particle size distributions N(r) isnecessary. Various retrieving techniques have beenapplied to decrease the degree of ill-condition problem.These include regularization by truncated singular valueddecomposition of LIDAR data [11], linear and nonlineariterative techniques [12,13], damped Gauss–Newtoniteration algorithm [14], moment methods [15–17], andcomputed tomography [18]. Nevertheless, traditionalalgorithms used for these problems depend on initialvalues or derivatives which are difficult to resolveaccurately by numerical simulations. Random optimalmethods such as the genetic algorithm (GA) havebeen studied to solve reliably global optimal problems[19,20].
The particle swarm optimization (PSO) algorithm,which was introduced by Eberhart and Kennedy[21–23], is able to find global optimum solutions or goodapproximate solutions, usually without theoretical proof.This is solely due to its ability to explore the searchdomain with ‘jumps’ from one local solution to others,and thus the global optimum solution can be reached stepby step. As reported in Ref. [24], many kinds of problemsthat can be solved by GA are able to be solved identicallyby PSO, without suffering the difficulties encountered inGAs. The PSO algorithm has been studied extensively bymany researchers in recent years. Ozcan and Mohan [25]have proved that the PSO can guarantee convergence, butnot a global optimum. Van den Bergh [26] studied theglobal convergence (GC) and local convergence of basic
PSO and GCPSO, and he pointed out basic PSO could notguarantee global or local convergence.
To guarantee convergence to global optimum solu-tions, we propose in this paper a modified PSO algorithmwith a stochastic selection. To begin with, MIE scatteringtheory is used to calculate aerosol optical thickness (AOT),which in turn is used as an input for performing theinverse analysis at different wavelengths. A stochasticparticle swarm optimization (SPSO) algorithm is adoptedto minimize the objective function and estimate para-meters that characterize atmospheric aerosol particle sizedistributions.
2. Methods
2.1. Forward radiation problem
Aerosol particles are assumed for simplicity to beuniform and spherical. For a given aerosol particle sizedistribution, the calculation of the forward problem is todetermine vertical AOTs. The calculation procedure is asfollows.
The extinction efficiency factor Qext and scatteringefficiency factor Qsca of spherical particles are given asfollows, respectively
Qextðm,wÞ ¼ Ce
G¼
2
w2
X1n ¼ 1
ð2nþ1ÞRefanþbng ¼4
w2Re S0f g
ð1Þ
Qscaðm,wÞ ¼ Cs
G¼
2
w2
X1n ¼ 1
ð2nþ1Þ 9an92þ9bn9
2h i
ð2Þ
Here, m is the optical constant of the relevant particlesor equivalently the complex refractive index, m=n� ik,with n and k denoting the (single) refractive and absorp-tion indices, respectively; w is the scale parameter, givenby w=2pr/l, with r as the particle radius and l as thewavelength; G is the geometric projected area, G=pr2
(m2 or mm2); the complex numbers an and bn are Miescattering coefficients, functions of m and w; Ce and Cs areextinction and scattering cross-sectional areas, respec-tively. Re denotes the real part of a complex number.
The radiative properties of particle swarm are relatedto the particle’s optical constant, the concentration andthe size distribution of particles. The extinction coefficientb, scattering coefficient ss, and absorption coefficient k ofparticle swarm are defined as
b¼Z 1
0NðrÞCe dr¼ p
Z 10
r2NðrÞQeðrÞ dr ð3Þ
ss ¼
Z 10
NðrÞCs dr¼ pZ 1
0r2NðrÞQsðrÞ dr ð4Þ
k¼ b�ss ð5Þ
where N(r) is the number density distribution ofparticles with N0 of the number density of particles,N0 ¼
R10 N rð Þ dr.
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Values for AOTs are then given by
tAOD, l ¼ bkl¼ lpZ 1
0r2NðrÞQeðrÞ dr ð6Þ
Here, l is the path length of the particle system.Detailed derivations can be seen in Refs. [27,28].
In accordance with statistical results, n(r) is usuallyassumed to be one of three options: (i) the Junge
distribution (n(r)=br�a), (ii) the gamma distribution
nðrÞ ¼ rb expð�arÞ� �
, or (iii) the log-normal distribution
nðrÞ ¼ 1ffiffiffiffiffi2pp br exp � ðlogðr=aÞÞ2
2b2
h i� �. If the undetermined para-
meters (a and b) can be fixed in some manner, then thesize-distribution N(r) can be obtained by N(r)=N0n(r).
The inverse problem is to resolve the aerosol particlesize distribution parameters from measured AOTs; that is,assuming that AOTs at several wavelengths are known,N0, a, and b can be determined by a retrieval algorithm. Akey quantity in this determination is the objectivefunction defined as follows:
ffitness ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXn
i ¼ 1
½t0ðiÞ�tmði,aÞ�2=n
vuut ð7Þ
where n is the number of selected computation samples;t0(i) is the measured AOT of the ith sample and tmði,aÞ isthe corresponding inverse calculation value; anda¼ ðN0,a,bÞT is a vector of the aerosol particle sizedistribution parameters to be retrieved. Thus, the inverseproblem reverts to searching for the vector a thatminimizes an as-yet-to-be-defined fitness value, and thisminimization procedure implements the retrieval algo-rithm mentioned above.
2.2. SPSO algorithm
The basic idea behind PSO can be described explicitlyas follows: each individual in the swarm, referred to as a‘particle’, represents a potential solution; each particlemoves its position in the search domain and updates itsvelocity according to its own and its ‘neighbours’ flyingexperiences, aiming at a better position for itself subjectto satisfying certain fitness criteria.
According to the simple PSO model, which wasproposed by Eberhart and Kennedy [21], at generationt+1, the velocity Vi(t+1) for each dimension of the ithparticle can be updated as follows:
Viðtþ1Þ ¼wViðtÞþc1r1 PiðtÞ�XiðtÞ½ �þc2r2 PgðtÞ�XiðtÞ� �
ð8Þ
where w is the inertia weight coefficient; c1 and c2 are twopositive constants called acceleration coefficients; Pi andPg are local and global individual best locations, respec-tively; r1 and r2 are random numbers in the interval [0, 1].The new position Xi for ith particle can be expressed as
Xiðtþ1Þ ¼ XiðtÞþViðtþ1Þ ð9Þ
In the standard PSO, if 0owo1 and Vi(t+1)oVi(t), thereexists a fixed number such that if the generation numberexceeds it, the current global best position Pg of the swarmdoes not vary, and consequently, all components of Vi will besmaller than a given error value, and the particle’s positionremains unchanged. Even if a better solution exists in this
direction, the particle swarm may stop evolving beforefinding the global solution and fall into the so-calledpremature convergence. This is the reason why the standardPSO may fall into a local optimum solution.
To avoid premature convergence, we present the SPSOalgorithm [29]. Here, setting the inertia weight w=0 andsubstituting Eq. (8) into Eq. (9), we have
Xiðtþ1Þ ¼ XiðtÞþc1r1 PiðtÞ�XiðtÞ½ �þc2r2 PgðtÞ�XiðtÞ� �
: ð10Þ
To improve the global searching ability, Pg is main-tained as the historic best position, and an extra particlelabeled j with position Xj is generated randomly in thesearch domain. In this way, the following updatingprocedure is obtained:
Pj ¼ Xj
Pg ¼ argminðPi,PjÞ: ð11Þ
This means that if Pg=Pj, the random particle j islocated at the best position and the new random particlewill be sought repeatedly. Therefore, at least one particleis generated randomly in the search domain, thusimproving the global searching ability. More details ofthe method can be found in Refs. [29,30].
2.3. Computation procedures
The implementation of the SPSO approach for solvingthe inverse radiation problem can be carried out accord-ing to the following procedures:
�
Step 1: Input system data, and initialize a particleswarm; input system configuration, control para-meters such as the lower and upper bounds for theestimated radiative parameters, and the dimension nof each particle’s position space; randomly generate aninitial swarm of particles with random positions on n-dimensions in the solution space; set the index ofiteration t=0.
� Step 2: Calculate the fitness value; calculate the initialfitness for each particle by substituting the position ofeach particle into the forward radiation problem; thefitness value is set equal to the calculated value of theobjective (or fitness) function with all particle posi-tions as arguments.
� Step 3: Compare the fitness value for each particle witha priori best Pi; if the fitness value is lower than Pi, setthis value as the current Pi, and record the correspond-ing particle position.
� Step 4: Choose the particle associated with the best Piof all particles, and set the value as the current globalbest Pg.
� Step 5: Introduce a randomly selected particle into thepopulation, and update the particle position for eachparticle.
� Step 6: Check the stop criterion; If the pre-setmaximum number of generations is reached or if noimprovement to the best solution is obtained after agiven number of iterations Nt, then the process isterminated; otherwise, increment the iteration indext=t+1, and loop to Step 2.
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2.4. The performance of SPSO
100
1000
2000
3000
4000
5000
6000
Com
puta
tion
Tim
e (S
ec)
Swarm Size
SPSO
PSO
20 30 40 50
Fig. 2. Comparison of computation time of PSO and SPSO for different
swarm size.
1E-3
0.01
0.1
est F
itnes
s
30
40
70
100
Swarm Size
In this section, the performance of the proposed SPSO isinvestigated through comparisons with the standard GAand PSO methods. The standard PSO sets c1=c2=2.05 anduses a linearly varying inertia weight over generations,varying from 0.9 to 0.4. The c1 and c2 of SPSO are set to 2.0.The probabilities of mutation and crossover of GA are set to0.3 and 0.6. All three methods use a population size of 50.There are two termination criteria: (1) when the iterationaccuracy is below a level fixed at 10�10 and (2) when themaximum generation number of 1000 is reached.
As shown in Fig. 1, best fitness values for the SPSOalgorithm converge much faster than that for the standardPSO algorithm and GA. Moreover, the SPSO algorithm canarrive at the lowest best fitness values among the threemethods within a smaller number of generations. Asshown in Fig. 2, SPSO is less time-consuming than PSOgiven the same swarm size. Thus, the SPSO algorithm issuperior in terms of searching quality and speed inderiving results.
3. Numerical experiments
3.1. Theoretical simulation
We examine first the influence of different swarm sizeson the inverse problem of computation particles, whichdetermine the parameters for a SPSO. Fig. 3 displays theinfluence of swarm size on the descending rate of thefitness values. It can be seen that the fitness valueconverges faster with increasing swarm size, butcomputation time will increase. For the aerosol problemconsidered in this study, excessive swarm sizes will lead tolonger computation times because of the highly nonlinearinverse problem, and as shown in Fig. 3, increasing theswarm size will not greatly affect the computationalaccuracy. Thus, considering the computational accuracyand time, we select a swarm size of M=50 and the range of½xmin, xmax� ¼ ½0:001, 1000� in the following computations.Other parameters of the SPSO algorithm are set as c1=1.8,c2=1.8, and w=0. To analyze the implementation of the
01E-4
1E-3
0.01
0.1
1
Bes
t Fitn
ess
Generation
GA
PSO
SPSO
200 400 600 800 1000
Fig. 1. Comparison of best fitness of GA, PSO, and SPSO.
SPSO algorithm in solving the particle-size problemconsidered, we investigated the following test cases.
Case 1. Assuming that the particle size distribution fits alog-normal size distribution, gamma size distribution andlog-normal cross-section area distribution, we chooseaerosol particles of dust with complex refractive index m
obtained from Ref. [31]. Table 1 lists m values of dustaerosol particles at some reference wavelengths. The pathlength l of the particle system is set as 1, the fourreference wavelengths are chosen as 0.400, 0.690, 0.870,and 1.020 mm, the AOTs for the four wavelengths are
0
1E-5
1E-4B
Generation
500 1000 1500 2000 2500 3000
Fig. 3. Comparison of best fitness function for different swarm size.
Table 1Complex refractive index of different wavelength of the dust aerosol.
l n k l n k
0.400 1.53 0.008 1.800 1.33 0.008
0.694 1.53 0.008 2.250 1.22 0.009
0.860 1.52 0.008 2.700 1.18 0.013
1.300 1.46 0.008 3.200 1.22 0.01
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calculated by the MIE scattering theory, the convergenceaccuracy is set as 1.0�10�10, and the maximum numberof iterations is set at 3000. The change in fitness valueswith generation iterations is shown in Fig. 4. It can be seenthat the implementation of the SPSO algorithm in solvingthe inverse problem for particle size distributions iseffective. The convergence speed is fastest given a log-normal particle size distribution. Table 2 lists the inverseresults for the maximum generation number of 10 000. Asshown in the table, when the log-normal size distributionis adopted, the retrieval accuracy is better than the othertwo, similar in manner to the tendency in fitness values.
Case 2. A log-normal particle size distribution is alsoemployed in this case. To demonstrate the effects ofmeasurement errors on the inverse parameters, randomstandard deviations are added to the exact parameterscomputed from the direct solution. The following relationhas been used in the present inverse analysis:
Ymea ¼ YexactþsB
0
1E-6
1E-5
1E-4
1E-3
0.01
0.1
Bes
t Fitn
ess
Generation
Gamma size distribution
Log-normal size distribution
Log-normal cross-section
area distribution
500 1000 1500 2000 2500 3000
Fig. 4. Comparison of best fitness function for different distribution
function.
Table 3Inverse results with different measured errors using SPSO.
Parameter True value g=0 g=5
SPSO erel% SPSO
N0 10.0 9.9999 0.00 10.0955
a 0.05 0.0500 0.00 0.0498
b 0.7 0.7000 0.00 0.6990
Table 2Inverse results with different distribution functions using SPSO.
Log-normal size distribution Gamma size distribu
Parameter True value SPSO erel ð%Þ True value SPSO
N0 10.0 9.9999 0.00 100.0 100.0
a 0.05 0.0500 0.00 10.0 10.0
b 0.7 0.7000 0.00 0.3 0.3
Here, B is a normal distribution random variable withzero mean and unit standard deviation. The standarddeviation in the measured AOT, s, for a g% measured errorat 99% confidence, is determined as
s¼ Yexact � g%2:576
ð12Þ
For the sake of comparison, the relative error erel isdefined as follows:
erel ¼ 100�Yest�Yexact
Yexactð13Þ
From Table 3, it can be seen that without measurementerrors, the agreement between estimated and exact valuesof the inverse results is excellent. With increasingmeasured error g%, the relative error erel of the estima-tion increases. Meanwhile, the influence of measurementerrors on the three inverse parameters is different in log-normal size distributions. As seen in Fig. 5, the erel ofparameter N0 is largest in the inverse results and thereason is that N0 has little effect on the retrieved AOTresults; that is, the objective function is insensitive to N0
so the particle number density could not be reliablyestimated. In Fig. 5, we can also see that the erel ofparameter b is smallest in the inverse results and thereason is that b has much effect on the retrieved AOTresults. For g%=20%, the relative error erel of N0, a, and bare about 16.23%, 9.41%, and 2.56%, respectively.Therefore, even though the measurement error increasesconsiderably the relative errors erel of N0, a, and b are notlarge, especially for b, of which the relative error is only2.56%. Thus, even if there are measurement errors we canstill obtain good inverse results. And special attentionshould be given to parameter b in the inverse progress.
Case 3. The four reference wavelengths used in this paperare in the visible and near infrared band. Here, either allmeasured AOTs for these four wavelengths cannot beobtained or measurement errors of certain wavelengthsmay introduce large errors in the inverse results. For thisreason, we considered the effect of reducing the number
g=10 g=20
erel ð%Þ SPSO erel ð%Þ SPSO erel%
0.06 10.3840 3.84 11.6231 16.23
0.02 0.0489 2.18 0.0453 9.41
0.07 0.7017 0.24 0.7179 2.56
tion Log-normal cross-section area distribution
erel ð%Þ True value SPSO erel ð%Þ
47 0.05 0.10 0.1001 0.13
007 0.01 0.25 0.2501 0.03
002 0.06 0.10 0.1003 0.28
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of reference wavelengths on the inverse results. In Table 4,SAMP1 to SAMP4 represent the inverse results that thecorresponding sample is discarded in the inversecomputation (for example, SAMP1 is the inverse resultthat sample one is discarded and samples two to four areselected). From computational results, we can see that therelative error erel of SAMP1 is obviously larger than theothers, and so we continue to analyze SAMP1. Fig. 6 showsthe inverse results for SAMP1 with ten random initialvalues. As seen in this figure, the relative error erel
between estimated and exact values is 10%. The reasonis that the value of the AOT tAOT,l as described by Eq. (6) isdependent on the number density distribution N(r) andthe extinction efficiency factor Qext(l,r). Thus, if thetendencies of the extinction efficiency factor Qext(l,r) atdifferent wavelengths are too similar, they will not becompletely independent of each other. Hence, correlationsamong the samples are observed. Fig.7 shows thecomparison of Qext(l,r) with particle size at differentwavelengths. With the exception of the peak in theextinction efficiency factor for l=0.400 mm, the peaks inthe curves associated with the other three wavelengths lieat near positions, making the three samples dependent oncomputational results having multi-valued solutions as aconsequence. To verify this conclusion, we selected a fifth
0-2
0
2
4
6
8
10
12
14
16
18
Cal
cula
tion
Rel
ativ
e E
rror
Measured Error
N0
5 10 15 20
��
Fig. 5. Comparison of relative error erel with measured error g increased.
Table 4Inverse results with different selected wavelength using SPSO.
Parameter True value SAMP1 SAMP2
SPSO erel ð%Þ SPSO
N0 10.0 10.4444 4.44 9.9999
a 0.05 0.0487 2.62 0.0500
b 0.7 0.7063 0.89 0.7000
Parameter True value SAMP5 SAMP6
SPSO erel ð%Þ SPSO
N0 10.0 10.0008 0.01 10.0010
a 0.05 0.0500 0.01 0.0500
b 0.7 0.7000 0.00 0.7000
sample with its wavelength l5 far from those of the threechosen samples to replace samples two to four withthree others designated as SAMP5–SAMP7, respectively.We then selected a sixth sample with wavelengthl6=1.060 mm between l2 and l4. SAMP8 is the inverseresult of selecting samples two, four, and six. From theresults, we can see that the agreement between estimatedand exact values of the inverse results for SAMP5–SAMP7is excellent, although the inverse result for SAMP8 is stillnot converging well. Fig. 8 shows the inverse results ofSAMP8 for ten random initial values. The above resultsprove that correlations exist among the samples. Thus, anecessary condition to avoid correlations is that samplesshould be selected practicably.
As seen in Fig. 9, we show the influence of variations indistribution parameter b on AOT, proving the sensitivityof parameters on wavelength. Just like the distributionparameter b, the correlations of the other two parametersN0 and a on AOT are also rather strong. Thus, we can selectsamples without the need of considering this point.
3.2. Discussions on numerical results
We retrieve particle size distributions associated withdust aerosols from data obtained in Beijing, China, a
SAMP3 SAMP4
erel ð%Þ SPSO erel ð%Þ SPSO erel%
0.00 9.9998 0.00 10.0191 0.19
0.00 0.0500 0.00 0.0499 0.15
0.00 0.7000 0.00 0.7005 0.07
SAMP7 SAMP8
erel ð%Þ SPSO erel ð%Þ SPSO erel%
0.01 10.0006 0.01 10.3422 3.42
0.01 0.0500 0.00 0.0498 0.38
0.00 0.7000 0.00 0.7021 0.30
0
0.0
0.1
0.2
0.3
0.4
0.5
Rel
ativ
e E
rror
Inverse Times
N0
2 4 6 8 10
��
Fig. 6. Relative error for different time in SAMP1 (case 3).
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0.1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Qex
t
Radius
0.400
0.694
0.869
1.300
1 10
Fig. 7. Comparison of Qext with particle size at different wavelengths.
0
0.0
0.1
0.2
0.3
0.4
0.5
Rel
ativ
e E
rror
Inverse Times
No
2 4 6 8 10
�
�
Fig. 8. Relative error for different time in SAMP8 (case 3).
0.20
1
2
3
4
5
AO
T
distribution parameter �
0.400
0.670
0.870
1.020
0.4 0.6 0.8 1.0 1.2
Fig. 9. Correlation of distribution parameter and aerosol optical
thickness at different wavelengths.
Y. Yuan et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 2106–21142112
typical test region in many applications. Other unmea-sured AOTs are calculated based on the inverse results.Fig.10 shows measured AOTs at different times obtainedfrom AERONET data. Data from two days, April 24, 2002,and May 12, 2001, are used in the aerosol inversion. Thecomplex refractive index, m=n� ik, is also obtained fromAERONET data and shown in Table 5. The value range forthe inverse parameters is set to be within interval[0.001, 1000] and the settings of other inverse param-eters are the same as in the cases discussed above. Theinverse results are listed in Table 6. In case 4a (April,2002), with the exception of SAMP4, the inverse resultsare almost the same from the computations with differentsample combinations. When the four samples are com-bined, the result denoted by SAMP0 is the same as theothers. However, the computational accuracy of case 4areaches only 1�10�2. In case 4b, no matter what combi-nation of samples is computed, we obtain the same resultand the computational accuracy is very high.
From the results for case 4b, we find that SPSO isefficient and robust. We also find that this method canreduce the dependence on initial values and the searchrange. Although the range in cases analyzed in this papercan contain the value range of the aerosol particle sizedistribution parameters, we can still obtain good compu-tational results. In case 4a, we find that the reason for lowcalculation accuracy may be due to the difference in somedegree between the actual particle size distribution andlog-normal distributions or due to obvious errors in
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
AO
T
Wavelength
12 May, 2001
24 April, 2002
0.5 0.6 0.7 0.8 0.9 1.0 1.1
Fig. 10. Measured AOT in different time in Beijing.
Table 5Complex refractive index of different wavelength from AERONET.
Case 4a Case 4b
l n k l n k
0.400 1.53 0.008 0.400 1.33 0.008
0.694 1.53 0.008 0.694 1.22 0.009
0.860 1.52 0.008 0.860 1.18 0.013
1.020 1.46 0.008 1.020 1.22 0.01
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0.4
0.2
0.4
0.6
0.8
1.0
AO
T
Wavelength
Calculation
Measured
0.5 0.6 0.7 0.8 0.9 1.0 1.1
Fig. 11. Comparison of calculation and measured AOT of case 4b.
Table 6Inverse results of case 4.
Case 4a Case 4b
Accuracy N0 a b Accuracy N0 a b
SAMP1 2.25e�2 4.017 0.070 0.711 SAMP1 9.61e�8 0.0302 0.901 0.218
SAMP2 4.02e�2 5.687 0.072 0.649 SAMP2 9.66e�8 0.0301 0.900 0.219
SAMP3 4.97e�2 4.520 0.078 0.638 SAMP3 6.26e�8 0.0306 0.895 0.216
SAMP4 2.17e�2 0.638 0.236 0.080 SAMP4 8.51e�4 0.0351 0.833 0.207
SAMP0 4.60e�2 5.831 0.068 0.667 SAMP0 5.89e�4 0.0302 0.900 0.218
Y. Yuan et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 111 (2010) 2106–2114 2113
certain measured samples. Thus, in the retrieving proce-dure, there are no computational AOTs that are comple-tely consistent with measured ones. This shows that ineach computation we obtained an optimum fitness valuethat is not completely the same as the others. Similar tothe above results for case 3, the correlation betweenselected samples in SAMP4 is too strong, causing largerdissimilar results between SAMP4 and other combina-tions. However, the average relative error between thecomputed and measured AOT is smaller than 5%, and inengineering applications this is deemed acceptable.Finally, Fig. 11 presents a comparison of measured andinverse AOTs obtained from the inverse parameters atwavelength band of [0.400, 1.02] mm associated with case4b. We can see that in this band, calculations agree withmeasured data. Such agreements lay a good foundationfor further calculations of atmosphere radiative transferand aerosol radiative forcing.
4. Conclusions
In this paper, we investigated the application of theSPSO algorithm to retrieve the atmospheric aerosolparticle size distribution from measured AOTs that areobtained from AERONET data. To verify the applicabilityof the SPSO algorithm to this problem, we have performeda series of tests. Meanwhile, we compared the inverseeffect of different particles size distributions and foundthat for different distributions the computational results
could all be obtained exactly. We also compared differentsample combinations, finding that the SPSO algorithm isrobust and verifying also the effect of sample correlationon computational accuracy. Finally, by retrieving themeasured data we analyzed the practicability of the SPSOalgorithm to this problem and point out the variousfactors influencing the computation accuracy. Thus, thispaper serves as platform for further study on atmosphericaerosols.
Acknowledgements
This work was supported by Program for New CenturyExcellent Talents in University (NCET-09-0067), the keyprogram of the National Natural Science Foundation ofChina (Grant no. 50930007) and the National NaturalScience Foundation of China (Grant no. 50806018 and50636010). A very special acknowledgement is made tothe editors and referees who made important commentsto improve this paper.
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