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Reconstruction of particle-size distributions fromlight-scattering patterns using three inversion methods

Javier Vargas-Ubera, J. Félix Aguilar, and David Michel Gale

By means of a numerical study we show particle-size distributions retrieved with the Chin–Shifrin,Phillips–Twomey, and singular value decomposition methods. Synthesized intensity data are generatedusing Mie theory, corresponding to unimodal normal, gamma, and lognormal distributions of sphericalparticles, covering the size parameter range from 1 to 250. Our results show the advantages anddisadvantages of each method, as well as the range of applicability for the Fraunhofer approximation ascompared to rigorous Mie theory. © 2007 Optical Society of America

OCIS codes: 290.0290, 290.3200, 290.4020, 290.5850.

1. Introduction

Light scattering as a diagnostic tool has proved tobe a suitable technique for resolving real problemsof increasing complexity in the area of particle siz-ing. This technique has inherent problems, how-ever, owing to the need to perform a large numberof numerical calculations to resolve the inverseproblem generated by the rigorous solution forspherical particles as described by Mie theory1 andthe need to input the refractive indices of the ma-terials being measured.

An alternative formalism to Mie theory is theFraunhofer approximation, which is based on scalardiffraction theory. This approximation is valid forparticles of radius greater than 2.5 �m when ana-lyzed with visible light, wavelength � � 0.5 �m.2Jones3 proposed that, for nonabsorbent particles witha radius greater than 1.5 �m, a refractive indexgreater than 1.3 and for errors below 20%, the Fraun-hofer approximation is valid for visible wavelengths.

Although Mie theory is valid in the region of radiiand wavelengths where the Fraunhofer approxima-tion may be used, it is important to compare bothtechniques and obtain a criterion that allows us todecide when it is appropriate to use Fraunhofer since

numerical calculations using Mie theory become ex-tremely slow as the particle size increases. Addition-ally one should consider the more general case of aconglomerate of particles of varying sizes, whichmore closely resembles real-world sizing tasks. Oneshould thus determine if the validity criterion de-pends on the type of distribution and the inversionmethod used to retrieve the particle-size distribution(PSD).

We use normal, gamma, and lognormal distribu-tions to numerically simulate an intensity patternusing Mie theory. For the inverse problem of PSDrecovery, we use the Chin–Shifrin (CS) method4–9 andthe singular value decomposition (SVD) technique,9–12

when working with the Fraunhofer approximation,and the Phillips–Twomey (PT) method12,13 and theSVD technique when using Mie theory. These in-version methods are widely used at present. Sto-chastic numerical techniques such as the MonteCarlo method14 and genetic algorithms15,16 havebeen reported elsewhere and are not considered inthis work.

In the Section 2 we present some general consid-erations of importance to the subsequent numericalsimulations. The kernels for the Fraunhofer approx-imation and Mie theory are presented in Section 3,and the CS, SVD, and PT numerical techniques areintroduced. Section 4 shows the results relating to thecritical size interval that defines the validity of theFraunhofer approximation that applies typically tothese monomodal distributions with the CS and SVDinversion methods. In Section 5 results are given forMie theory by using SVD and PT methods. The reg-ularization parameter, which affects the performanceof the PT method, is discussed in Section 6, and a

The authors are with the Instituto Nacional de AstrofísicaÓptica y Electrónica (INAOE), Luis E. Erro No. 1, TonantzintlaPuebla 72840, México. J. Vargas-Ubera’s e-mail address is [email protected].

Received 10 July 2006; accepted 28 August 2006; posted 12September 2006 (Doc. ID 72843); published 15 December 2006.

0003-6935/06/010124-09$15.00/0© 2007 Optical Society of America

124 APPLIED OPTICS � Vol. 46, No. 1 � 1 January 2007

comparison between Mie and Fraunhofer with theirrespective inversion methods for large particles ispresented in Section 7. Our conclusions are presentedin Section 8.

2. General Considerations

The gamma and lognormal distributions used in thiswork are those used frequently to model PSDs inaerosols.17 The gamma distribution is defined as

fG�� N�� 0��1�b��3

�bt��1�b��2��1�b��2� exp��0 � �

bt � for � � �0

0 for � � �0

,

(1)

where � � kr is a nondimensional quantity knownas the size parameter for a particle of radius r,k � 2�� is the wavenumber, and � is the wavelengthof light in the medium that surrounds the particles.Quantities t and b are the effective size parameterand the variance, respectively, and control the shapeof the distribution.5 Finally, �0 determines the small-est size parameter present in the distribution, N isthe total number of particles, and � is the gammafunction.

The lognormal distribution is defined as

fL�� N

�� 0�2exp��ln�� 0

A0

2 �2

�, (2)

where N and �0 are as defined for the gamma distri-bution, � is the standard deviation of the distribution,and A0 is a quantity related to the mean size param-eter �� of a normal distribution through

� � exp�ln�A0� �2

2 �. (3)

A third distribution used in this work is the normaldistribution, which facilitates interpretation of theresults when attempting to establish a critical inter-val of validity for the Fraunhofer approximation. Itserves as a reference point with respect to the gammaand lognormal distributions, which are in generalasymmetric, and it helps us to evaluate the quality ofthe retrieval for each inversion method used. Thenormal distribution is defined as

fN�� N

2exp��

12��

2�, (4)

where �, �, and N are the mean size parameter, thestandard deviation, and the number of particles, re-spectively.

In the rigorous simulations of intensity we use therelative refractive index m � np�nm � 1.5, where npand nm are the indices of refraction of the particle andthe surrounding medium, respectively.

The total scattered intensity, I��, attributable to agiven proposed distribution f ��, is defined by thefollowing integral equation1:

I�� 0

I��, ��f��d�, (5)

where � is the forward-scattering angle in the direc-tion of propagation. The kernel, I��, ��, represents thescattered intensity in the annular region defined byangle �, corresponding to a single particle with sizeparameter � and is calculated according to Mie the-ory. f �� is the particle density, such that f ��d� isthe number of particles with sizes between � and� � d�.

The integral in Eq. (5) is calculated numerically byusing the trapezoidal rule in the finite interval inwhich the proposed distribution is defined. If the dis-tribution of particles f�� is discretized into M sizeintervals, then for the most general case the totalintensity of Eq. (5) is reduced to a system of linearequations that gives rise to a matrix equation of theform

y � Ax, (6)

where y is the total intensity vector due to the con-tribution of each size interval at a given angle �, A isa matrix that contains the scattering coefficients aij,calculated according to Mie theory or Fraunhofer ap-proximation as appropriate, and x is the unknownvector that contains the fraction of particles in eachsize interval.

Normally in such problems matrix A is almost sin-gular or ill-conditioned, and a direct inversion usingcommon methods such as lower triangular–upper tri-angular (LU) decomposition or Gaussian elimination,usually produces nonphysical solutions. For this rea-son one must turn to numerical inversion methodsadequate for this type of matrix.

3. Kernel and Inversion Methods

In the Fraunhofer approximation the kernel I��, �� ismodeled as the Airy pattern produced by an opaquedisk of radius r equal to that of the particle. Hencefrom Eq. (5) the total intensity is given by1

I�� I0

k2F2 �0

�2J1

2��2 f��d�, (7)

where I0 is the incident beam intensity, J1 is a Besselfunction of the first kind, first order, and F is the focallength of the receiver lens, as typically encounteredin laser diffraction particle size analyzers. This equa-tion accepts an asymptotic analytic solution, widely

1 January 2007 � Vol. 46, No. 1 � APPLIED OPTICS 125

referred to in the literature as the CS method6,9

f�� 2k3F2

�2 �0

��J1��Y1��dd��3

I��I0

�d�,

(8)

where, Y1 is a Bessel function of the second kind, firstorder. This solution has differences in the samplingand size dominions, and this leads to serious prob-lems in the retrieval of the function. More precisely,angle � can acquire values only in a small interval,limited by the validity of the paraxial approximation,up to a maximum angle �max, and this has importantimplications in the discretization of the integral. Fur-ther problems are encountered with the angular sam-pling interval �� and the minimum sampling angle�min, and furthermore, this solution is applicable onlyfor particles whose radii are several times greaterthan the analyzing wavelength.

The second inversion method, SVD, is applicable toboth Fraunhofer and Mie theories. It consists of re-solving the inverse problem posed by Eq. (6) using thefactorization of range p of matrix A�m � n� into twoorthonormal matrices, U�m � p�, V�n � p�, and adiagonal matrix P�p � p�, such that

A � UPVT, (9)

Q � A�1 � VP�1UT. (10)

All the diagonal elements of matrix P are positiveand are known as singular values. The degree of sin-gularity in A is determined by the condition number,the ratio between the largest and the smallest singu-lar values. Furthermore, if the pseudoinverse matrixof A generated by SVD is Q, then the solution to theinverse problem presented in Eq. (6) is

x � Qy. (11)

To introduce the third inversion method, we nowconsider the kernel for spherical particles withinthe context of Mie theory. From Eq. (5) the totalintensity is1

I�� 0

I0�i1 � i2�

2k2R2 f��d�, (12)

where i1 and i2 refer to the intensity of light vibratingperpendicular and parallel to the reference plane,respectively. Information about particle size and rel-ative refractive index m is contained in these inten-sity contributions. R is the distance from the particleto the plane of observation.

Here it is important that we apply an adequateinversion method so as to obtain the maximum effi-ciency in calculating the inverse matrix that resolvesthe inherent ill-conditioned behavior in the inverseproblem expounded by Eq. (12). A common strategyto obtain a stable solution is to incorporate a prioriinformation pertaining to the desired result. A widelyused technique, known as the Tikhonov regulariza-tion12 and also referred to as the PT method,13 offersthis possibility.

The PT method postulates a minimization prob-lem, which in a least-squares sense can be written inthe form

��2 � �2�Bx�2 → min. (13)

Here, � � Ax � y is the residual vector, �·� denotes theEuclidian norm, and B is a matrix operator designedto reduce the ill-conditioned behavior of matrix A.Generally, B is a finite-differential operator of sec-ond order that has a smoothing effect on x andis included as a priori information about the dis-tribution sought. The � parameter represents anadjustable regularization parameter or Lagrangemultiplier; it is a real and positive number, and it isalso a weight factor for the included a priori infor-mation.

For a given value of �, from Eq. (13) one achieves aunique solution given by13

x � �ATA � �H��1ATy, (14)

where H � BTB serves as the regularization matrixto suppress spurious oscillations that appear in thesolution vector of the basic linear system [Eq. (6)],which is ill-conditioned.

Finally, the inverse matrix �ATA � �H��1 may becalculated by using standard techniques; for thiswork we use LU decomposition.

Although � must be determined to solve the inverseproblem, an initial proposal for the value of the reg-ularization parameter is suggested,12 since � is con-stant in the minimization problem of Eq. (13). Thisinitial value is given by

�0 �Tr�ATA�Tr�H�

, (15)

since this choice tends to make the two parts of theminimization have comparable weights. We can ad-just from there until we reach a good match betweenthe proposed and the retrieved distributions. Detailsof the proper selection of � are discussed in Section 6.

4. Validity Range in the Fraunhofer Approximation:Comparison between the Chin–Shifrin Method and theSingular Value Decomposition Technique

Taking as a premise the results of Hodkinson2 andJones,3 we consider a critical interval for the study of


the behavior of the Fraunhofer approximation, usingfive characteristic size ranges for each of the threedistribution types. The five normal distributions arecentered at size parameters � � 50, 45, 40, 35, and30, with a standard deviation � 15 used through-out. The corresponding gamma and lognormal distri-butions have modal peaks in the same vicinity as thatof the normal distributions. The distribution param-eters are shown in Table 1. The size dominion for alldistributions is 1 � � � 100, with the same samplinginterval �� 1. It is sufficient to consider just thedistributions centered at the extremes of the modalsize interval since they illustrate adequately the be-havior of the intensity patterns and the recuperationtendency.

Figure 1(a) shows a normal distribution centeredat � � 50 with a standard deviation of � 15, to-gether with gamma and lognormal distributions cen-tered within the same interval close to � � 50. Figure1(b) compares the logarithm of the intensity patterns

(normalized with total incident intensity) calculatedby using Fraunhofer and Mie theory [Eqs. (7) and(12)] for just the normal distribution, since thegamma and lognormal distributions generate identi-cal patterns. Figure 1(c) shows the three distribu-tions centered at � � 30. The resulting intensitypatterns, again for only the normal distribution, areshown in Fig. 1(d).

For the following analysis we consider the six dis-tributions of Figs. 1(a) and 1(c) as representative ofthe extremes of interval that we consider critical forthe validity of the Fraunhofer approximation. Thuson referring to the critical interval, we refer to themodal values in the interval of size parameter thatdelimits these distributions.

The intensity patterns calculated using the Fraun-hofer approximation and Mie theory for a single dis-tribution are in agreement for small angles but showdiscrepancies at larger angles. The angular region inwhich these discrepancies become evident depends tosome extent on where the distribution is centered.For example, for distributions centered at � � 50, theagreement begins to fail beyond � � 4°; for distri-butions centered at � � 30, the discrepancy occursbeyond � � 5°. This behavior can be explained byconsidering that the Fraunhofer approximation an-ticipates an inverse functional dependence betweenparticle size and scattering angle. Furthermore theangular region in which both formalisms give identi-cal results is that governed by the paraxial approxi-mation sin � � �, implicit in the characteristic Airypattern.

The similarity between intensity patterns pro-duced by the three distributions defined within a sim-ilar region of particle size eliminates any hypothesisthat the critical region of validity of the Fraunhoferapproximation is dependent on the type of distribu-tion we use. This conclusion is further borne out byconsidering the same three distributions in regionsbeyond the critical size interval. For distributionsdefined within the interval 1 � � � 60, with modalpeak in the interval 10 � � � 30, at least 68% (andoccasionally up to 100%) of the particles that make upthe distribution have a size parameter � � 45. Theresult is different intensity patterns for each theory.If we shift the modal peak and the size interval forthe distributions to greater sizes with respect to thecritical interval, the intensity patterns coincide towithin an angular interval 0° � � � 4°, for distribu-tions defined within the interval 30 � � � 170;whereas for distributions defined within the interval50 � � � 250 the intensity patterns coincide for0° � � � 2.5°.

The previous discussion enables us to justify ourselection of the interval 1 � � � 100, with corre-sponding modal values between 30 � � � 50 as beingthat which encompasses the limit of validity of theFraunhofer approximation. In the following para-graphs we analyze the retrieved distributions ob-tained in this region.

In Figure 2 we compare the retrieved distributionsobtained from the Fraunhofer approximation by us-

Table 1. Parameters of the Proposed Distribution Functionsa

Normal Gamma Lognormal

� � t b A0 �50 15 55 0.5 52 2.245 15 55 0.7 45 2.840 15 50 0.8 42 3.035 15 45 0.8 37 3.030 15 40 1.1 33 3.8

aThe size dominion 1 � � � 100 is used throughout. Sampling�� 1; minimum sampling angle �min 0.057°; angular sampling�� 0.057°; and maximum sampling angle �max 10°.

Fig. 1. (a), (c) Normal, gamma, and lognormal distributions cen-tered at � � 50 and � � 30, respectively. (b), (d) Intensity patternsfor the normal distribution of (a) and (c), respectively.


ing the CS and the SVD inversion methods. Figures2(a) and 2(d) correspond to normal distributions,Figs. 2(b) and 2(e) are for gamma distributions, andFigs. 2(c) and 2(f) for lognormal distributions. Eachpair of functions corresponds once again to the ex-tremes of the critical size interval. We observe thatfor all distribution types SVD inversion generatessuperior results compared with CS.

It is well known that when working with theFraunhofer approximation the retrieved distribu-tions are strongly dependent on the angular com-position of the scattered intensity, and to applynumerical techniques the sampling interval mustbe finite, �min � � � �max. Furthermore within thisinterval the aforementioned function must be dis-cretized into a finite number of points. In the profilesdiscussed in this section, we use the same values of�min � 0.057°, �max � 10°, and �� 0.057° throughout.

We find that the value of �max is critical for ade-quate retrieval of distributions using CS inversion.One must sample up to a value of �max that coincideswith the angular region in which the intensity pat-terns produced by Mie and Fraunhofer formalismsbegin to differ [see Figs. 1(b) and 1(d)]. If this is notthe case then the retrieved function bears no resem-blance whatsoever to that expected. Despite consid-eration of an adequate value for �max, in accordancewith the paraxial condition that gives rise to theFraunhofer approximation, we observe that the re-trieved function continues to present considerablenoise in the region corresponding to small particlesize, for each of the three distribution types. Thisnoise may be attributed to an incompatibility be-tween the intervals of Eqs. (7) and (8) as mentionedpreviously. For this reason, we reject the employmentof the retrieval method using the CS inversion in oursubsequent analysis of the critical interval of validity

of the Fraunhofer approximation, making a finalcomment that this inversion method is suitable in theregion of large particle sizes.

A retrieval using the SVD method also depends onthe correct selection of �max, although in this case thedependence is not so clearly evident. If the recuper-ation is carried out considering the totality of singu-lar values, we obtain a result similar to that obtainedwhen CS inversion is employed with an inadequatevalue of �max. The retrieved functions improve dra-matically, however, when we begin to eliminate (re-place with zeros) the smallest singular values in thelinear system of equations. We may interpret this asa diminishment in �max and a simultaneous improve-ment in the conditioning of the matrix to be inverted,since if for the same distribution we calculate theintensity by using an adequate value of �max, then thenumber of singular values that must be eliminated isless than in the previous case, and the condition num-ber of the matrix is reduced. The same behavior ofimprovement in the matrix conditioning is observedwhen the sampling ratio �� diminishes; however, ananalysis of this effect is not intended to form part ofthis paper.

Although no formula exists for determining thenumber of singular values that must be removed, theretrieved functions obtained by varying this param-eter to obtain the optimum results show that the SVDmethod is adequate for our present analysis. For thethree types of distribution, with modal peaks in theregion � � 50, acceptable results are obtained byusing SVD [see Figs. 2(a)–2(c)]. Meanwhile distribu-tions with modal peaks in the region � � 30 exhibitnoise [see Figs. 2(d)–2(f)], which is manifested as os-cillations superimposed upon the proposed distribu-tion. This behavior is attributable in large part to theinadequacy of the Fraunhofer approximation in theinversion process for the given size interval and, to alesser extent, to the asymmetry of the proposed dis-tributions.

Since the behavior is similar for each type of dis-tribution, both in the intensity patterns and in theinversions, we continue our analysis in this sectiononly with normal distributions. At the same time wefurther limit the interval of modal size in the distri-butions so as to obtain an indicative limiting value forthe Fraunhofer approximation. We consider a seriesof six normal distributions centered at � � 48, 47, 46,44, 43, 42, since by observation we note that in thisregion of mean size parameter, the Fraunhofer ap-proximation loses its validity. Each distribution hasthe same sampling as before (see the caption of Ta-ble 1), and the SVD inversion method is employedthroughout. The retrieval of this series of distribu-tions using SVD is shown in Fig. 3.

As an indicator of error between the proposed andthe retrieved distributions, we take the standard de-viation

s �1Ns

�i�1

Ns �fi � xi�2

fi2 , (16)

Fig. 2. Recovered distributions using the Fraunhofer approxima-tion with the SVD and CS inversion methods. Graphs (a), (b), and (c)correspond to normal, gamma, and lognormal distributions, respec-tively, with modal peaks at � � 50. Graphs (d), (e), and (f) as above,for � � 30.


where fi and xi, are the i values of the proposed andthe retrieved distributions, respectively, and Ns is thenumber of points calculated. The error values are0.86, 0.92, 1.06, 1.59, 1.94, and 2.32 for Figs. 3(a)–3(f),respectively.

These results allow us to confirm that for theFraunhofer approximation a critical interval existsfor the modal peak of distributions centered be-tween size parameter 42 � � � 48. We can considerthe distribution centered at � � 46 [Fig. 3(c)], asbest representing the threshold value of validity forthe Fraunhofer approximation, applied to recuperatesmooth distributions of types such as normal, gamma,and lognormal. We can conclude that, for monomodaldistributions with modal size parameters greater thanthe threshold value �� 46� with errors less thanunity and employment of the SVD inversion method, itis clearly convenient to use the Fraunhofer approxi-mation. Conversely, for monomodal distributions withmodal size parameters below this threshold value, wepropose the use of Mie theory.

5. Mie Theory: Comparison between Singular ValueDecomposition and Phillips–Twomey

For distribution retrieval using Mie theory, we useboth the SVD and PT inversion methods, with thethree distribution pairs considered at the extremes ofthe Fraunhofer interval of validity as defined in theprevious section. The plots are shown in Fig. 4, withassociated errors in Table 2. Retrieval using SVD isclearly inferior, exhibiting incorrectly centered modalpeaks and excessive noise, while the PT methodshows evident superiority. For the latter we used aregularization parameter � � 10�18, selected in agree-ment with Eq. (15). In Section 6 we discuss the im-portance of this parameter to obtain optimal results.

The poor performance of SVD with Mie theory maybe largely attributed to the instability of the scatteringmatrix as reflected by a large condition number, whichcan be used as a measure of the degree of ill-posednessin the inverse problem. The matrices generated in thenumerical calculations of Fig. 4 produce a conditionnumber of 1.56 � 1021, which when combined withthe asymmetry of the proposed distributions makessecond-order normalization a necessity, as in the caseof the PT method. In other words, first-order normal-ization methods such as SVD12 cease to be adequate inhighly unstable inversion problems. For example, inthe distributions of Section 4 (Fig. 3), where the scat-tering matrices were constructed using the Fraunhoferapproximation, the SVD method was successful, sincehere the condition number was 2 orders of magnitudelower �5.85 � 1019�.

The second reason for the poor performance of theSVD method with Mie theory is the asymmetricshape of the proposed distributions. For example,despite using the same condition number through-out, the symmetric distribution of Fig. 4(a) producesbetter results than the asymmetric distribution ofFig. 4(f).

Fig. 3. Normal distributions recovered using the Fraunhofer ap-proximation with the SVD inversion method in the critical sizeinterval.

Table 2. Error in the Retrieval of Fig. 4 using PT and SVD Methodswith Mie Theory

Figure DistributionModalPeak

ErrorPT

ErrorSVD

4(a) Normal 50 1.54 2.04(b) Gamma 50 1.42 4.724(c) Lognormal 50 1.21 4.754(d) Normal 30 1.17 9.254(e) Gamma 30 4.08 10.054(f) Lognormal 30 4.37 8.57

Fig. 4. Distributions recovered using Mie theory with the PT andSVD inversion methods. The regularization parameter � � 10�18 isused throughout. Graphs (a), (d), (b), (e), and (c), (f) correspond tothe normal, gamma and lognormal distributions, respectively.


With regard to the use of the PT method with Mietheory, the asymmetry in the distributions has littleeffect on the ability of this method to recover theproposed distributions. Although the error for PT isalmost tripled for the distributions with the highestasymmetry [gamma and lognormal with modal peak� � 30; see Figs. 4(e) and 4(f), respectively], the re-moval of the negative (nonphysical) values from theanalysis reduces the error to unity or less.

From the results presented in this section, we maythus conclude that the SVD inversion method is lessthan adequate when used with Mie theory, while thePT method, for the three kinds of asymmetrical dis-tributions considered here, gives acceptable results,especially if we discard nonphysical (negative) valuesin the retrieved profile. Based on these observations,we expand our analysis in the following sections toregions of large and small particle size, exclusively forone kind of distribution.

6. Phillips–Twomey Method and the OptimalRegularization Parameter

When we use Mie theory with the PT method, animportant aspect to be taken into account for solv-ing the inverse problem is the proper choice of theregularization parameter to obtain optimal resultsin the retrieved distribution. Two different approachescommonly exist for making this selection. The first isbased on an iterative process and can be applied in anysituation since no estimation of noise level in the inputdata is needed. This iterative procedure is robust andgives good estimations of the sought distribution. Inthe second approach more information is needed andno iteration is required. However, it is necessary toassume some restrictions on both the expected solutionand the noise of the measurements since they affectthe quality of the solution. Thus the application of thislast approach may be more difficult.

As already mentioned in Section 3, we use theiterative procedure for the selection of the regular-ization parameter. Taking �0 given by Eq. (15) as aninitial value guarantees that we are on the path to-ward a practical solution. The selection of this initialvalue accelerates the convergence of the iterative pro-cess and reduces the number of steps to reach anoptimal value �opt. We propose that �opt is reachedwhen the error between the recovered and the pro-posed distribution is a minimum.

Figure 5 shows the evolution of parameter � for twogamma distributions of several particle sizes. Weshow two different distributions to illustrate the chal-lenges involved in the procedure for proper selectionof �opt. The first is representative of distributions inthe region of large particles where the Fraunhoferapproximation can be applied. The second is for theregime of small particle sizes where Mie Theory isadequate. We have explored the solutions obtainedfor different values of � but report here only the dis-tributions that correspond to the initial and optimalvalues of the regularization parameters.

For the larger particles [Fig. 5(a)], the regulariza-tion parameter has an initial value given by �0 �

5.9938 � 10�11, represented by the symbol A in Fig.5(b). This gives a retrieved distribution with an error of1.6057, appearing in Fig. 5(a) as a dashed curve. Afterseveral steps an optimal value �opt � 5.9938 � 10�15 isobtained as indicated by the symbol � in Fig. 5(b). Thisgives a retrieved distribution with an error of 0.2797,corresponding to the thin solid curve in Fig. 5(a). How-ever, this last plot is not well distinguished since itcoincides with the proposed distribution.

For the small particles [Fig. 5(c)], the initial valueis �0 � 9.0564 � 10�21, also represented by A in Fig.5(d), and the corresponding retrieved distribution hasan error of 17.9142. The optimal value of the regu-larization parameter is �opt � 2.0 � 10�22 representedby the symbol � in Fig. 5(d), with an error betweenthe proposed and the retrieved distribution given by14.3582. In this case also the retrieved distributions[Fig. 5(c)] are represented by a dashed curve for theinitial value of �, and a thin curve for the optimalvalue of �.

We can see that �opt provides adequate results forlarge particles with good agreement between the pro-posed and the recovered distributions, since the erroris less than unity. However, for the region of smallparticle size, the reconstructed distribution is de-graded in spite of selecting the optimal value of �.This is typical in a very narrow distribution, where,as the distribution gets wider, the recovery improves.Similar results are obtained with narrow normal andlognormal distributions.

Since the scattering matrix condition number usedin the SVD method can be used as a measure of thedegree of ill-posedness, we have tried to find a rela-

Fig. 5. Evolution of the regularization parameter. Plots (a) and (c)are the distributions retrieved with values of �0 and �opt, respec-tively. In (b) the values of �0 � 5.9938 � 10�11 (A) and �opt

� 5.9938 � 10�15 �� generate the distributions of the large parti-cles shown in (a). Similarly in (d), the values �0 � 9.0564 � 10�21 (A)and �opt � 2.0 � 10�22 �� generate the distributions for the smallparticles shown in (c).


tionship between this condition number and the op-timal regularization parameter but without success.

In summary, we show that the application of thisiterative procedure to select the regularization pa-rameter gives optimal values. In general, good esti-mations of the retrieved distributions are obtained byusing the PT method. The process works well forbroad distributions with large particles, but failssomewhat for narrow distributions skewed towardsmall particle sizes. The performance of this methoddemands a considerable amount of computing time.

7. Comparison of Mie Theory and the FraunhoferApproximation

We now consider the comparison of the reconstructionof a normal distribution of large particles using Mietheory with the PT method and that reconstructedwith the Fraunhofer approximation using both SVDand CS methods. The proposed distribution is definedin the interval 50 � � � 250, centered at � � 150and deviation � 30. The intensity pattern is calcu-lated according to Mie theory, with m � 1.5 and� � 0.6328 �m, for the angular resolution �� shownin Table 1. This kind of distribution is intentionallychosen with the object of highlighting the unneces-sary applicability of the Mie theory for very largespherical particles.

Figure 6 shows the performance of the three meth-ods. The proposed distribution is represented by thecontinuous thick curve. The reconstructed size distri-bution with cross marks corresponds to Mie theorywith the PT method, using the optimum value ofregularization parameter, � � 1.0 � 10�12. Obviously,with a very small value of error equal to 0.1598 andpractically identical to the proposed distribution, thisreconstruction can be considered acceptable.

The plots shown by the dashed curve and the dottedcurve correspond to the Fraunhofer approximationwith the SVD and the CS methods, respectively. Qual-

itatively speaking, the plot for SVD fits better to theproposed distribution, with an error of 0.9759, com-pared with an error of 2.5809 for CS. The peak of theplot for CS is better aligned to that of the proposeddistribution, while the SVD peak is lightly shifted to-ward smaller �’s . Additionally, the plot for CS is verynoisy in the small size region, and this may be associ-ated with the truncation of the numeric integral atsome cutoff angle �max, while the negative number den-sity is due to the missing data at small angles of thescattering pattern. Meanwhile the plot for SVD showsa slight disagreement with the proposed distributionat extremes of small and large particle sizes. This ispossibly due to the discretization of the data in thescattering matrix and may also be due to a bad selec-tion of the angular sampling interval. These incompat-ible conditions for the Fraunhofer approximation withboth methods result in a poor inversion performancecompared with the Mie theory using the PT method.Nevertheless, we maintain that the Fraunhofer ap-proximation has a practical use for some applications,i.e., when the problem involves particles with un-known refractive indices and has practical measure-ment limitations, or simply when the main objective isto reduce computing time.

Finally, we state that Mie theory with PT inversionprovides superior results when compared with theFraunhofer approximation using either CS or SVDalgorithms, the first combination of methods exhibit-ing the lowest error. However, it must also be notedthat retrieval using Mie theory requires a factor of4500 more CPU time than the method using theFraunhofer approximation. For this reason, we sug-gest using Mie theory with the PT method only fordistributions under the threshold value for the valid-ity of the Fraunhofer approximation.

8. Conclusions

For smooth, normal, gamma, and lognormal, mono-modal particle-size distributions composed of sphericalparticles, with real relative refractive indices given bym � 1.5, it is appropriate to use the Fraunhofer ap-proximation beyond a certain critical size interval.This interval corresponds to the modal peak of thedistribution centered between 42 � � � 48. We con-sider the distribution centered at � � 46 as bestrepresenting the threshold value for the validity of theFraunhofer approximation. Additionally, the methodof singular value decomposition generates better re-sults than the Chin–Shifrin method when solvingthe inverse problem with the Fraunhofer approxi-mation. In this part of the work all the distributionswere considered in the size dominium between 1� � � 100.

The singular value decomposition inversion methodproduces unacceptable results when used with Mietheory, while the Phillips–Twomey method gives thebest results for all three kinds of distribution consid-ered here. The asymmetric shape of the distributiondoes not affect the efficiency of the Phillips–Twomeymethod. We also note that a larger sampling intervalhas no effect on the performance of this method.

Fig. 6. Comparison between Mie theory and the Fraunhofer ap-proximation in the region of large particle sizes.


The principal factor that affects the success of thePhillips–Twomey method is the regularization pa-rameter. The application of an iterative procedure toselect this parameter gives an optimal value thatproduces a good estimation of the retrieved distribu-tion. This optimal value produces an error of lessthan 1 for broad distributions with large particles,but the retrieved distribution is not so accurate fornarrow particle distributions skewed toward smallsizes. The performance of this method demands aconsiderable amount of computing time.

With regard to comparison of the Mie and Fraun-hofer formalisms within the region of large particlesizes where the use of the Fraunhofer approxi-mation is perfectly valid, we can conclude that thecombination of Mie theory with the Phillips–Twomeymethod results in a better approach compared withsingular value decomposition and the Chin–Shifrinmethod using the Fraunhofer approximation. The onlydisadvantage of Mie theory in this case is an increasein CPU time by 4 orders of magnitude. Hence we rec-ommend using the Fraunhofer approximation with thesingular value decomposition inversion method.

This study was conducted while J. Vargas-Uberawas at the Benemérita Universidad Autónoma dePuebla, Facultad de Ciencias Físico-Matemáticas.The permission to undertake doctoral studies isgratefully acknowledged. The authors thank theMexican National Council for Science and Technol-ogy (Conacyt) for a doctoral fellowship and acknowl-edge the partial support received from INAOE andthe Mexican Institute of Petroleum (IMP).

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