numerical study of particle-size distributions retrieved from angular light-scattering data using an...

Numerical study of particle-size distributions retrievedfrom angular light-scattering data using an evolution

strategy with the Fraunhofer approximation

Javier Vargas-Ubera,1,* Juan Jaime Sánchez-Escobar,2 J. Félix Aguilar,1 and David Michel Gale1

1Instituto Nacional de Astrofísica Óptica y Electrónica (INAOE), Luis E. Erro No. 1, Tonantzintla Puebla, 72840 México2Centro de Enseñanza Técnica Industrial (CETI), Nueva Escocia 1885, Frac. Providencia,

Guadalajara Jalisco, 44620 México

*Corresponding author: [email protected]

Received 20 November 2006; accepted 15 January 2007;posted 2 February 2007 (Doc. ID 77188); published 18 May 2007

An algorithm is presented based on an evolution strategy to retrieve a particle size distribution fromangular light-scattering data. The analyzed intensity patterns are generated using the Mie theory, andthe algorithm retrieves a series of known normal, gamma, and lognormal distributions by using theFraunhofer approximation. The distributions scan the interval of modal size parameters 100 � �� 150. The numerical results show that the evolution strategy can be successfully applied to solve thiskind of inverse problem, obtaining a more accurate solution than, for example, the Chin–Shifrin inversionmethod, and avoiding the use of a priori information concerning the domain of the distribution, commonlynecessary for reconstructing the particle size distribution when this analytical inversion method isused. © 2007 Optical Society of America

OCIS codes: 290.0290, 290.3200, 290.4020, 290.5850.

1. Introduction

The estimation of particle size is an important task ina large range of industrial processes and technologi-cal applications. A frequently used optical techniqueis the analysis of the scattering pattern produced bythe sample in the propagation direction. The estima-tion of a particle-size distribution (PSD) is an inverseproblem rigorously treated by means of the Mie the-ory [1]. However, when the size of the scatterer be-comes considerably larger than the wavelength of thelight, and the relative refractive index between thescatterer and the medium differs substantially fromunity, the dominant contribution to the near forward-scattered light can be described by the Fraunhoferapproximation of the scalar diffraction theory [1].

The Fraunhofer approximation represents an as-ymptotic limit of the Mie theory, independent of theoptical properties of the particles and the mediumand is numerically easy to manage. The problem is

solved either by a direct integral transform inversionmethod, known as the Chin–Shifrin method (CS)[2–8] or indirectly by using a numerical quadratureof the governing integral equation [8–10]. However,this asymptotic limit to the rigorous theory dependson initial information that must be provided by theuser. The most significant disadvantage in these in-version schemes, is that we must suppose a priori thedomain of the sought distribution. On the other hand,the evolutive algorithm obtains the PSD ignoring in-formation about the domain, since it works as a non-traditional optimization method [11,12], which isbased on the mechanisms of biological evolution.

An evolution strategy (ES) works with a set of po-tential solutions to the problem of interest, whereeach potential solution, called an individual, is rep-resented by a vector of real numbers. This is a goodrepresentation when the problem at hand deals withcontinuous parameters. The main objective is to finda PSD starting from an intensity pattern, reduced toobtain a vector of real numbers that encodes an ad-equate solution, evaluated according to a fitness func-tion that depends on the problem to be solved. The ES

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3602 APPLIED OPTICS � Vol. 46, No. 17 � 10 June 2007

begins searching for a solution within a population ofindividuals, each of which represents not only asearch point in the space of potential solutions to thegiven problem, but also may be a temporal containerof current knowledge about the laws of the environ-ment [13]. The starting population evolves towardsuccessively better regions of the search space bymeans of processes of recombination, mutation, andselection. The recombination mechanism allows formixing of parental information, which is passed on totheir descendants, while mutation introduces inno-vation into the population [14]. The environment de-livers a quality metric (fitness value) for each searchpoint, and the selection process favors those individ-uals with higher quality to reproduce more often thanweak individuals [13].

Thus our method retrieves a PSD from simulatedangular light-scattering data by solving an optimiza-tion problem in the real domain, where an objectivefunction [13] is minimized and the solution to thetotal scattered intensity equation is considered as aninverse problem.

In spite of its limitations, the Fraunhofer approxi-mation continues to be used in problems where theoptic properties of the particles and the medium thatcontains them are not so important, when the refrac-tive index of the particles, the medium, or both areunknown, or simply when we do not have sufficientcomputation power for the numeric treatment of theproblem. Numerical calculations using Mie theoryare slow when the sample includes large particlesizes. In such cases, Mie theory requires the evalua-tion of series, which for convergence has a number ofterms proportional [15] to the nondimensional sizeparameter � � ka, with k � 2�� the wavenumber, �the wavelength of light in the medium, which sur-rounds the particles, and a the radius. This results inapproximately 250 terms for � � 0.6328 �m anda � 25 �m, implying a factor of 4500 times more CPUtime, compared with calculations performed usingthe Fraunhofer approximation. Although the ES in-creases the CPU time by a factor of 30 compared withthe CS inversion method in the Fraunhofer approxi-mation, this is still relatively small when comparedwith the CPU time used by the Mie theory.

In this work we consider three unimodal normal,gamma, and lognormal distributions centered at�� 100 and �� 150 over a fixed interval of sizes, andwe use Mie theory to simulate an intensity patternI��, where � is the scattered angle. Then we retrievethe respective PSD by means of both the ES and theCS methods with the purpose of comparing the accu-racy with which each method recovers the proposeddistribution and the CPU time invested in each case.

In Section 2 we describe the formalism relating tothe Fraunhofer approximation and the respective CSinversion method. Section 3 presents a generalizeddescription of the algorithm implemented in thiswork. In Section 4 the main numerical results arepresented. Finally conclusions are presented in Sec-tion 5.

2. Fraunhofer Approximation and the Chin–ShifrinMethod

In the context of Mie theory the total scattered inten-sity I��, due to a proposed distribution f��, is givenby the following integral equation [1]:

I�� 0

�

I��, �, m�f��d�. (1)

This is a Fredholm integral of the first kind in whichthe kernel I��, �, m� represents the Mie-scattered in-tensity corresponding to a single particle of size pa-rameter � and relative refractive index m in theannular region defined by the forward-scattering an-gle � in the direction of propagation. f�� is the par-ticle density, such that f��d�, is the number ofparticles with sizes between � and � d�.

When the Fraunhofer approximation is used, thekernel does not depend on the relative refractive in-dex since I��, �� is modeled as the Airy pattern pro-duced by an opaque disk of radius a equivalent to thatof the particle of size parameter �. Hence from Eq. (1)the total intensity is given by [1]

I�� I0

k2F2 �0

��2J1

2��2 f��d�, (2)

where I0 is the incident beam intensity, J1, is theBessel function of the first kind and order one, and Fis the focal length of the receiver lens, as typicallyencountered in laser diffraction particle size analyz-ers. In this paper the integrals in Eqs. (1) and (2) arecalculated numerically using the trapezoidal rule inthe finite interval in which the proposed distributionis defined. These intensity patterns were generatedfor a common angular region 0.0573° � � � 10.886°,with angular sampling � � 0.0458° and 40 subdivi-sions in size �.

Equation (2) accepts an asymptotic analytic solu-tion, widely referred to in the literature as the CSmethod [4,7]:

f�� 2�k3F2

�2 �0

�

��J1��Y1��dd��3

I��I0

�d�,

(3)

where, Y1 is the Bessel function of the second kindand order one.

This solution is a direct integral transform inver-sion method in that the integrand directly containsthe measured intensity. Once again the integral inEq. (3) is calculated numerically by means of thetrapezoidal rule, and the derivative is calculated bythe finite differences method.

In the numerical evaluation of Eqs. (2) and (3), it isimportant to recognize the differences in both the sizeand angular domains because a priori we suppose

10 June 2007 � Vol. 46, No. 17 � APPLIED OPTICS 3603

that the major contribution to the integral occurs onlyfor a finite interval in sizes and for � in the range from�min to �max limited by the validity of the paraxialapproximation. The truncation by �max leads to a de-crease of the performance in the retrieval functionusing this method.

3. Proposed Algorithm

This recuperation method is based on the evolutivealgorithm described by Vázquez-Montiel et al. [14].To produce offspring we used the recombination andmutation operators such as those described in theprevious work. The selection operator was performedusing the configuration �� [13]. In this notation� represents the set of parents and � the set of off-spring, so the best individuals are chosen for the nextiteration from the set of parents and offspring; thusthe proposed algorithm produces an enlarged sam-pling because both parents and offspring have thesame chance of competing for survival. As a result adeterministic search is performed to find the solutionwithin the space of potential solutions.

To perform the proposed algorithm each individ-ual x�� is formed by the vector of elements called objectvariables x�,p with a respective strategy parameter �,p [14].

x�� x�,p, �,p�, where, p � 1, . . . , 5. (4)

Thus each individual represents the set of param-eters necessary to describe a PSD, given by means ofthe following assignation:

x�,1 mean size parameter,

x�,2 standard deviation,

x�,3 first size parameter,

x�,4 second size parameter,

x�,5 smallest size parameter, (5)

where “first size parameter” and “second size param-eter” correspond to the finite interval in which thedistribution is defined.

The genetic operators of recombination and muta-tion work with all the extended vector x�� from Eq. (4),because it contains the standard deviation for carry-ing out the mutation.

The process of evaluating the fitness of an individ-ual x�� consists of the following two steps [14]:

Y An intensity pattern I� is constructed from x�� bysubstituting only the first five elements of x�� from Eq.(4) to generate the distribution f��. Equation (2), isthen used to calculate the total scattered intensity.

Y Considering each element in the matrix I� as adimension in a Euclidian distance space, we assign asthe fitness the Euclidian distance between I� and the

reference intensity pattern represented by Iref. Thusthe fitness is given by

fitness � �i�1

j

�I�i� Irefi�2, (6)

where j is the number of elements in the intensitypattern.

We have chosen this fitness function because itevaluates the minimum distance between the opti-mized intensity and the simulated intensity for aspecific number of iterations. This function repre-sents a measure of the quality of the solution that thebest individual of a population contains in its struc-ture. In our numerical results, this particular fitnessfunction and the operators of recombination, muta-tion, and selection accomplish a mapping with a highcorrelation between the objective function and theproximity to the zone where the correct solution oroptimal solution resides.

Rechenberg [11] proposed a deterministic adjust-ment of strategy parameters during evolution calledthe “1�5 � success rule,” which reflects that, on av-erage, one output from every five mutations shouldcause an improvement in the objective function val-ues to achieve best convergence rates. If more than1�5 of the mutations are successful, the strategy pa-rameter � is increased, otherwise it is decreased. Theselection operator evaluates Eq. (6) for the total pop-ulation and chooses the best M individuals that willin turn form the new generation of solutions.

Our program was implemented in MATLAB program-ming language since it allows an algorithm to bewritten in the simplest form. We note that MATLAB iscalled an interpreter because it first interprets theoperations, and then the operations are performed.As a result we have a slow algorithm. However, acompiler such as Fortran provides a series of optimi-zations that can help speed up the compiled code. Ananalysis of this option is not intended to form part ofthis paper.

4. Numerical Results

Each numerical realization took approximately 30min to run on a PC with a Pentium IV processorunder the following conditions: A population equal toM � 50 is used throughout; the recombination oper-ator is applied to the set of parents to obtain 30% ofoffspring, while the mutation operator provides theremaining 70%. Thus the set of parents and offspringhas a similar number of individuals. The aptituderelated to each individual was determined by usingEq. (6). The selection operator chooses the best Mindividuals from the set of 2M parents and offspringfor the next iteration t � t 1 by means of a rankingprocedure. In each iteration we use the output of thebest individual to plot the behavior of the objectivefunction. This cycle is repeated until we obtain a goalvalue such that the objective function is minimized,or when a fixed number of iterations is reached. Thisstop criterion was fixed after testing different distri-


butions. In Fig. 1 we illustrate the typical behavior ofthe objective function for the three kinds of distribu-tion analyzed in this study.

The objective function never reaches the proposedgoal value of 10�9; hence we fixed the maximum num-ber of iterations at 500 as a threshold value and thegoal value 10�9 as our stop criterion in the algorithm.We observe good convergence of the three distribu-tions for 100 iterations. When the stop criterion isreached, our result is the best individual of the lastgeneration, represented by a vector, which corre-sponds to the PSD that best matches the analyzedintensity pattern.

Figure 2 contains several plots that compare thelogarithm of the optimized intensity pattern pro-duced by the ES with the theoretical intensity pat-terns calculated using Mie theory, normalized withtotal incident intensity. Figures 2(a)–2(c) correspondto the normal, lognormal, and gamma distributions,respectively, as used in Fig. 1. We can see that theoptimized intensity patterns, shown by the dottedcurves of Fig. 2, are in agreement with the values ofthe objective function. The optimized pattern showsevident discrepancies. This behavior suggests that fiterrors of the order of 10�5 or smaller will generate ahigh degree of similarity between intensity patterns,

as shown in Fig. 2. Nevertheless, in this work weconsidered 10�9 as a goal value to assure the bestresults.

A. Evolution Strategy and the Chin–Shifrin Method in theRecuperation of a Normal Distribution

The aim of this first part of our numerical study is toanalyze the behavior of the ES with respect to the CSinversion method when the intensity pattern is cal-culated by means of rigorous Mie theory, and therecuperation is realized with the Fraunhofer approx-imation. We choose two characteristic size rangeswhere the use of the Fraunhofer approximation isfeasible. For this work we have analyzed three dif-ferent kinds of distribution; however, for the presentdiscussion it is enough to consider only the resultsobtained for normal distributions since they illus-trate adequately the advantages of our algorithm incomparison with the results given by the CS method.

The two normal distributions analyzed cover therange 30 � � � 200, centered at �� 100, and 50� � � 250 centered at �� 150. We will refer to themas medium and large sizes, respectively. It has beenshown by several authors that the Fraunhofer ap-proximation describes well the scattering phenomenaproduced by spherical particles in this range of size

Fig. 1. Typical behavior in the minimization of the objective function for normal, gamma, and lognormal retrieved distributions in themedium size range.


parameters [2,6,16]. The intensity patterns were gen-erated with the same mesh and common angular re-gion used in the evaluation of the numerical integralscalculated in Section 2. The parameters that controlthe shape of the two normal distributions discussedhere are presented in Table 1.

For the two distributions mentioned above, we gen-erate an intensity pattern by means of Mie theory,then using the Fraunhofer approximation togetherwith our algorithm we obtain a first retrieved PSD,

and with the Fraunhofer approximation and the CSmethod we obtain a second retrieved PSD as shown inFig. 3. This figure illustrates adequately both thegood behavior of the ES in the recuperation and thepoor results reported by the CS inversion method.

The differences between the retrieved and pro-posed distributions can be considered as an indicatorof error in the recuperation, which is estimated bymeans of the standard deviation

s �1Ns

�i�1

Ns �frefi � f�i�2

frefi2 , (7)

where frefi and f�iare the i values of the proposed and

the retrieved distributions, respectively, and Ns is thenumber of points calculated. Both error values, thoserelating to the CS method and those associated withour algorithm are shown for the corresponding curvesby arrows in Fig. 3. Figure 3(a) represents the me-dium size range, and Fig. 3(b) represents the largesize range. Corresponding fit errors between the sim-ulated intensity patterns and the intensity patternsreported at the end of the optimization process in theES were 6.1934 � 10�5 and 1.4415 � 10�5 for themedium and large sizes, respectively. These resultswill be discussed further in Subsection 4.B.

Fig. 2. Comparison between the retrieved scattering pattern produced by the ES and the simulated intensity patterns calculated withMie theory. The plots correspond to the three distributions presented in Fig. 1: (a) normal, (b) lognormal, and (c) gamma.

Table 1. Shape Parameters of the Proposed Distribution Functionsa

PSD Proposed Normal Gamma Lognormal

Size ranges �� 0 A0 � �0

Medium 100 20 90 0.6 29.9 76 2.7 29.9Large 150 30 125 0.7 49.9 107 2.9 49.9

aA particle number N � 100 is used throughout. �� and � repre-sent the mean size parameter and the standard deviation in nor-mal distributions, respectively. and are the effective sizeparameter and the variance in gamma distributions. In the log-normal distributions, � is the standard deviation, and A0 is aquantity related with the mean size parameter of an equivalentnormal distribution through �� exp[ln(A0) �(�2�2)]. �0 deter-mines the smallest size parameter present in the gamma andlognormal distributions.


Figures 3(a) and 3(b) show that for both regionschosen for �, the ES generates superior results. Wenote also from Fig. 1 that the fit error between thecalculated and ES intensity patterns remains al-most constant from iteration number 100. This con-firms that our algorithm finds the closest solution tothe global optimum. Remember again that the ESmethod is capable of obtaining the correct resultswithout any a priori information regarding the dis-tribution interval, and that the possible discrepan-cies in the angular composition of the scatteredintensity do not affect the performance of the algo-rithm, as happens in general when the Fraunhoferapproximation is applied to solve the inverse prob-lem by quadratures, and again with the CS method.This last point is clearly observed with the CS re-cuperation in Fig. 3. We can conclude that in theparticular case of medium and large particles, theFraunhofer approximation implemented using anES algorithm obtains accurate results.

Finally the Fig. 3 results show the poor quality ofthe recuperation using the CS method. Thus, it is notrecomended to use this inversion method with theFraunhofer approximation for these size ranges.

B. Robustness of the Evolution Strategy for Retrieval of aParticle Size Distribution

In this second part of our numerical study, we showthe robustness of the ES implemented in our algo-

rithm with the Fraunhofer approximation. Given theintensity pattern corresponding to a known distribu-tion, our algorithm selects among the three kinds ofdistribution under consideration, finding that whichcan be satisfactorily fitted to this original intensitypattern. This procedure is carried out adjusting thesimulated intensity pattern, with the intensity pat-tern generated by each kind of distribution sequen-tially inside the algorithm evaluating the fit errorbetween both patterns. The corresponding retrieveddistributions are then compared with the proposedtheoretical distribution, and the recuperation errorbetween both distributions is obtained. The smallestvalues of both errors determines the best retrieveddistribution that the algorithm can obtain. The bestresult (smaller errors) coincides with the proposeddistributions.

To generate the simulated intensity pattern, weconsider only the gamma and lognormal distribu-tions, since the performance of the ES method withnormal distributions was discussed in Subsection 4.Aand produces similar results. Again we consider twointervals, similar with respect to the previous numer-ical simulations, representing medium and large par-ticle size ranges, and we employ the same domainand angular sampling parameters that we used inthe previous section.

Figure 4 shows the behavior of our algorithm forboth distributions. As can be seen, the performance of

Fig. 3. Comparison between the ES and the CS methods applied to the recuperation of a normal distribution. (a) Medium size region and(b) large size regions.


the ES is similar with respect to the previous numer-ical simulations. Figures 4(a) and 4(b) correspond tothe intensity patterns generated by gamma distribu-tions in both size ranges, while Figs. 4(c) and 4(d)correspond to the intensity patterns generated by thelognormal distributions.

The curves display an identical numerically re-trieved PSD, and we can see in Table 2 that the fiterror and the recuperation error are a minimum (inboldface) when the ES algorithm selects the gammadistribution. In Figs. 4(a) and 4(b) both normal dis-tributions (dotted curves) and lognormal distribu-

tions (circled curves) are retrieved from the intensitypattern generated by the above-mentioned gammadistribution. In the first case, the fit error and therecuperation error increase because when the EStransforms the fitting problem into an optimizationproblem the solution found is not near-optimal. Onthe other hand, when the proposed distributions areanalyzed with lognormal distributions our algorithmgives a better result. Here the recuperated lognormaldistributions are closer to the proposed distributionin the medium and large range; however, as men-tioned above, the minimal recuperation error is ob-tained when our algorithm analyzes the proposeddistribution with the gamma distribution. We con-clude that the ES is able to recognize the correctdistribution among the three possibilities of distribu-tions mentioned above.

We now look at the two lognormal distributionsshown in Figs. 4(c) and 4(d). The dotted curves showa numerically resolved size-distribution recoveryidentical to the normal distributions in both regionsas before. Recovery with the gamma distribution isacceptable in the medium region; however, for theregion of large particles, the retrieved distribution inFig. 4(d) is very poor. Table 3 shows the fitting and

Fig. 4. Recuperation with the ES from the intensity pattern generated by gamma and lognormal distributions in medium and large sizeranges. (a) and (b) correspond to normal, gamma, and lognormal distributions retrieved by the ES from the intensity pattern generatedby Mie theory with a gamma distribution at the medium and large sizes. (c) and (d) correspond to normal, gamma, and lognormaldistributions retrieved by the ES from the intensity pattern generated by Mie theory with a lognormal distribution at the medium and largesizes.

Table 2. Errors for Proposed Gamma Distributionsa

PSD Gamma-Medium Gamma-Large

Error I-fit PSD I-fit PSD

N 9.97 10�5 1.84 2.05 10�4 1.14G 7.92 10�5 1.02 1.73 10�4 0.67L–N 8.28 10�5 1.14 1.91 10�4 0.72

aColumn I-fit represents the fit errors between the simulatedintensity pattern with a gamma distribution and the intensitypattern reported by the ES to each kind of distribution. ColumnPSD represents the errors of recuperation between the proposedand the retrieved distributions.


recuperation errors corresponding to all curves ofFigs. 4(c) and 4(d).

Regarding the fit with a lognormal distribution, wecan see from Figs. 4(c) and 4(d) and Table 3 that thisdistribution (circles) generates superior results, withthe recuperation practically identical to the proposeddistribution. Again, in the particular case of a lognor-mal distribution used to generate an intensity pat-tern, the ES gives the best results, and it can becommented that the asymmetry of the distributionsin both ranges of sizes of particles does not affect theperformance of our algorithm.

From the above numerical simulations and forthese three particular distributions we can concludethat the ES is independent of the type of monomodaldistribution that we use. The ES can be adjusted witha distribution type unrelated to the proposed sizedistribution, though obviously with a lesser degree ofaccuracy in the results. However, this method canstill be of value when information regarding thesought distribution is not available, or when the onlyimportant task is to give a diagnosis of the size of theparticles and not the distribution type.

Finally, it is important to point out that our al-gorithm can analyze only distribution functionswith up to a maximum of six optimization variables.If our method uses more than six optimization vari-ables it cannot perform a search within the space ofpotential solutions. With a maximum of six vari-ables our algorithm converges to an accurate solu-tion at approximately the 100th generation in thesimplest situations and at approximately the 420thgeneration when the problem is more numericallycomplex. During each experiment our method re-quired more CPU time than the CS method, 30 minin our case, compared with 57 s for the CS method.When information about the sought distribution isunknown, it is preferable to consume more comput-ing time.

5. Conclusions

In this work we have presented a method based onevolutionary computation to retrieve particle-size dis-tributions from angular light-scattering data. Ourmethod analyzes intensity patterns generated usingMie theory, giving as a result a series of known mono-modal normal, gamma, and lognormal distributions by

means of the Fraunhofer approximation. The proposedmethod introduces two important characteristics whencompared to the traditional Chin–Shifrin method: Wehave obtained solid results that show that our programyields more accurate solutions compared to theChin–Shifrin method and does not require a prioriinformation about the domain of the particle-size dis-tribution that we are seeking. The only requirementat the beginning of each optimization process is forinformation related to the simulated intensity pat-tern and a group of possible candidates of distribu-tions in order to carry out the adjustment and obtainthe corresponding particle-size distribution.

For the particular case of the three kinds of distri-bution discussed in this work, the performance of ourprogram is not dependent on the type of monomodaldistribution that we are analyzing. Accurate resultsare obtained for both symmetrical and asymmetricaldistributions, and the method is able to identify thecorrect distribution from the three distribution typespresented. The results are less accurate but still sat-isfactory when our method is applied to a distributiontype unrelated to the proposed size distribution. Thiscan be important when information on the originaldistribution is unavailable or when only a diagnosis ofsizes present in the sample is required. We note thatpossible discrepancies in the angular composition ofthe scattered intensity do not affect the performance ofthe algorithm, unlike the case when the Fraunhoferapproximation is used to solve the inverse problem byquadratures or with the CS method.

We propose a threshold value of 500 for the numberof iterations and the goal value 10�9 as a general stopcriterion when this algorithm is used. These valuesguarantee the correct convergence in the objectivefunction, such that our algorithm can find the closestsolution to the optimum global.

This inversion method based on evolutionary com-putation demonstrates high precision and acceptablecomputing time, providing the evolutive operators areused with a correct objective function. However, wehave analyzed only distribution functions consideringup to six optimization variables as a maximum. Atpresent, this limitation does not allow us to retrievemore complex distributions, such as multimodal.

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Table 3. Errors for Proposed Lognormal Distributionsa

PSD Lognormal-Medium Lognormal-Large

Error I-fit PSD I-fit PSD

N 1.21 10�4 1.18 2.15 10�4 1.35G 1.46 10�4 1.24 1.85 10�4 9.59L–N 7.73 10�5 1.12 1.71 10�4 0.66

aColumn I-fit represents the fit errors between the simulatedintensity pattern with a lognormal distribution and the intensitypattern reported by the ES to each kind of distribution. ColumnPSD represents the errors of recuperation between the proposedand the retrieved distributions.


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numerical study of particle-size distributions retrieved from angular light-scattering data using an...

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