Analytical transform techniques to retrieve non-spherical particle size distribution

Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 287–297

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Analytical transform techniques to retrieve non-spherical particle size distribution Jian-Qi Zhao a,n, Jiangnan Li b a State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China b Canadian Centre for Climate Modelling and Analysis, Atmospheric Environment Service, University of Victoria, Victoria, British Columbia, Canada

a r t i c l e i n f o

abstract

Article history: Received 8 April 2013 Received in revised form 2 July 2013 Accepted 4 July 2013 Available online 12 July 2013

The measurement of particle size distribution (PSD) from the extinction spectra is a challenging problem, especially for non-spherical particles. In this work, the analytical transform techniques are developed to retrieve PSD from the measured scattering data for a variety of shapes of non-spherical particle including spheroids, cuboids, triangular prisms, hexagonal prisms and elliptical cylinders. In the retrieve calculation, all particles have aligned orientations and their extinction cross sections can be adequately calculated by using the anomalous diffraction theory (ADT). It is shown that for each type of the considered non-spherical particles, there exists an ADT transform pair between the size distribution and the extinction spectrum. The inverse formula of PSD from the related ADT transform is therefore established. This result provides a new approach to the solution of the inversion problem and has the potential to be used in finding solutions for more complicated particle shapes. It is found that the solution of inverse scattering for absorbing particles can be simplified to a particular solution in which the character of absorption becomes no longer present. For spheroids, triangular prisms, and elliptical cylinders, the retrieved PSD and the corresponding true PSD always belong to one family of function, due to the scaling relation. Therefore an improper choice of inversion parameters does not significantly affect the type of mode of PSD in a retrieval process. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Inverse problem Extinction spectrum Non-spherical particles Particle size distribution Light scattering

1. Introduction The determination of particle size distribution (PSD) of a system from multi spectral extinction coefficient measurements is an important and fundamental work, which has applications to various scientific and engineering fields, such as planetology and astrophysics, atmospheric optics, oceanography and limnology, colloidal chemistry, biophysics, and remote sensing. In general, under the single scattering approximation, this inverse problem can

n

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0022-4073/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jqsrt.2013.07.003

be expressed in terms of a Fredholm integral equation of the first kind: Z τðkÞ ¼

1 0

C ext ðk; m; a; μ; θÞf ðaÞ da;

ð1Þ

where k is the wavenumber of electromagnetic wave, τðkÞ is the measured extinction coefficient of a polydispersion of particles at wavenumber k, m is the complex refractive index, a,μ and θ are the characteristic size, the nonsphericity and the orientation parameter of a particle, respectively; and f(a) is the unknown PSD to be retrieved. The integral kernel function C ext ðk; m; a; μ; θÞ is the shape and size dependent extinction cross section of a particle,

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which can be determined by an experimental measurement or light scattering theory. For a non-spherical particle, it has been widely recognized that an exact solution of PSD by retrieval from the integral equation of Eq. (1) is very difficult or perhaps even impossible. This is mainly due to the irresolvable mathematical form of the integral kernel function for irregularshaped particles. Over the past several decades, there has been a lot of effort towards the developing of the approximate inversion algorithms which are based on the spherical particle assumption and Mie theory e.g., [1–4]. However in reality, most particles are non-spherical, such as ice crystals, raindrops and most types of aerosols [5,6]. Both the theoretical and experimental studies show that the non-spherical particles have significant difference in optical properties compared to the spherical particles [7,8]. Recently, a lot of attention has been paid to the establishment of numerical retrieval schemes, in which Cext for non-spherical particles can be calculated by some complex light scattering algorithms (such as, the T-matrix [9], FDTD [10], and DDA [11]). The integral equation is approximated by a finite system of linear equations. Then the PSD can be obtained by solving the linear equations e.g., [12–23]. These kind of numerical methods are limited by the following facts: (i) the incorporating of nonspherical scattering into the PSD retrieval requires large computational resources (such as memory and CPU time); (ii) the calculations often run into nonphysical oscillations or numerical instabilities. This limits the applications of the numerical retrieval methods. The analytical method, on the other hand can dramatically reduce the computing time and provide a rapid result for an inversion problem. In an analytical approach, the approximate light scattering method, such as the anomalous diffraction theory (ADT) [24], is often used to calculate Cext. Then PSD can be retrieved analytically with the help of the integral transform techniques. Since the analytical method is simple and easy to repeat the results, there have been a number of proposed inversion algorithms e.g., [25–35]. However, most of these solutions are limited to spherical particles [25–32]. A little progress has been made to the inversion solutions for non-spherical particles [33–35]. There lacks a general methodology and specific technology in handling the inverse problem for non-spherical particles due to its complexity. The purpose of this work is to develop a simple and effective analytical method for retrieval of PSD. The work contains two parts. In this part, most of our effort will focus on theoretical derivation. The application of the proposed new algorithm will be discussed in a subsequent paper. In following Section 2, the approximation method of ADT is introduced to obtain the extinction efficiency of particles. In Section 3, the analytical solutions of non-spherical inversion problem are shown in context of ADT for oriented spheroids, cuboids, triangular prisms, hexagonal prisms, and elliptical cylinders. In Section 4, the novel ADT integral transform pairs for a variety of non-spherical particle shapes are constructed. In Section 5, it will be shown that the formula of retrieval PSD can be much simplified by using the Schwarz integral representation. This enables the

proposed method to be more easily applied to realistic problems. In Section 6, the possible scaling relations in retrievals for non-spherical particles are given. In Section 7, we conclude with a brief discussion for the remaining limitations. 2. Extinction approximation An efficient calculation of the extinction cross section required in the integral equation is particularly important for the retrieval of PSD. ADT provides an analytical method to calculate the extinction cross section of non-spherical particles in an efficient way. In ADT, the index of refraction is assumed close to unity and the particle size parameter is large compared to the wavelength. This assumption implies that the refraction and reflection are negligible as a ray passes through a particle. Thus the presence of particle will only produces the change in the complex phase front of an incident plane wave over its geometrical shadowed area. In the framework of ADT, the forward scattering amplitude S(0) can be expressed as following [24]: 2

k ∬P ½1expðiφÞ dP; ð2Þ 2π pffiffiffiffiffiffiffi where i ¼ 1, φ ¼ ðm1ÞkΔz is the complex optical depth, k ¼ 2π=λ is the wave number to wavelength λ, m ¼ mr þ imi is the complex refractive index of the particle relative to the medium (its refractive index is generally assumed to be unity) with mi ≥0, Δz is a geometrical path length of a given ray through the considered particle. The integral is performed over the projected area P of the particle onto a plane perpendicular to the direction of light ray. The extinction cross section in ADT can be obtained from the forward scattering amplitude by using the optical theorem: Sð0Þ ¼

C ext ¼

4π 2

k

Re ½Sð0Þ;

ð3Þ

where Re indicates the real part of a complex quantity. It is convenient in mathematics to introduce a complex extinction efficiency Qad, Q ad ¼

2 ∬P ½1expðiφÞ dP: P

ð4Þ

The extinction efficiency is the real part of Qad, as Q ext ¼ Re ðQ ad Þ. Similarly, we define a complex extinction coefficient for a polydispersion of particles as Z 1 T ad ¼ PQ ad f ðaÞ da: ð5Þ 0

3. Formal solutions of the Fredholm integral equation For simplicity, it is assumed that all particles have aligned orientations and the angle dispersion can be ignored (note that this assumption is reasonable especially for colloidal suspensions with say an applied external magnetic field). PSD satisfies the natural boundary conditions of f ð0Þ ¼ 0 and f ð1Þ ¼ 0, and an f ðaÞ is absolutely

J.-Q. Zhao, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 287–297

integrable for n ¼ f0; 1; 2; 3g. Generally, the integral transform techniques play a key role in an inversion problem. In a complex plane κ, the Laplace transform pair to an f ðaÞ, which will be used in the following, is Z 1 W n ðκÞ ¼ expðiκaÞ½an f ðaÞ da; ð6Þ 0

an f ðaÞ ¼

1 2π

Z γ

W n ðκÞexpðiκaÞ dκ;

ð7Þ

where W n ðκÞ is defined on the upper complex κ plane and the contour path γ lies above any singularity. In Eq. (6), W n ðκÞ satisfies a recurrence relation dW n ðκÞ ¼ iW nþ1 ðκÞ: dκ

ð8Þ

Also there is a more sophisticated Struve transform of half integer order and its inverse [36,37], which will be applied to a1=2þn f ðaÞ: Z 1 wn ðκÞ ¼ ðκaÞ1=2 H0 ðκaÞ½a1=2þn f ðaÞ da; ðImðκÞ ¼ 0Þ;

Substituting Eq. (11) into Eq. (13), we obtain dW 0 ðκÞ 1 1 κ  W 0 ðκÞ ¼  W 0 ð0Þ þ ½T ðκÞT ad ð1Þ; dκ κ κ 4πχ ad

a

1=2þn

Z

1

f ðaÞ ¼

1=2

ðκaÞ

wn ðκÞY 0 ðκaÞ dκ;

ð10Þ

0

where Y 0 ðκaÞ and H0 ðκaÞ are the zero order Bessel function of the second kind, and the zero order Struve function, respectively. 3.1. Spheroids

where C is a constant. By taking a second derivative on both sides of Eq. (15) with respect to κ, we obtain W 2 ðκÞ ¼ 

 1 d  2 κ ½T ad ðκÞT ad ð1Þ ; 4πχκ dκ

Q ad ðκaÞ ¼ 24

exp ðiκaÞ 1exp ðiκaÞ 4 ; iκa ðiκaÞ2

ð11Þ

where κ is the complex effective wavenumber given by 2ðm1Þk κ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 cos 2 θ þ μ2 sin θ

ð12Þ

In Eq. (12), μ ¼ a=b is an aspect ratio, where a and b are the lengths of the semiaxis of rotation and other axes of the spheroid (μ 41 for prolates and μ o1 for oblates), θ is the angle between the symmetry axis and the direction of incident radiation. Eq. (11) is completely consistent with the traditional ADT result. Based on Eq. (5), the complex extinction coefficient of spheroids is Z 1 T ad ðκÞ ¼ χπa2 Q ad ðκaÞf ðaÞ da; ð13Þ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where χ ¼ μ2 cos 2 θ þ μ2 sin 2 θ. In Eq. (13) the dependence of T ad ðκÞ on μ, θ, and m is implicit. Letting κ ¼ ς þ iξ, we can define formally the real part of T ad ðκÞ as τad ðς; ξÞ ¼ Re ½T ad ðκÞ, τad ðς; ξÞ is the ordinary extinction coefficient. However, for notation consistency we use symbol τad ðκÞ instead of τad ðς; ξÞ. Though the formula of T ad ðκÞ is general different from (13) for different shapes of non-spherical particle, τad ðκÞ can always be defined in the same way.

ð16Þ

The application of inverse Laplace transform of Eq. (7) to W 2 ðκÞ, yields Z  1 d  2 κ ½T ad ðκÞT ad ð1Þ expðiκaÞ dκ: f ðaÞ ¼  2 2 8π χa γ κ dκ ð17Þ where the contour γ is traced out by the parametric Eq. (12) on the upper complex κ plane with the extension rule mðkÞ ¼ mn ðkÞ for k≥0. Without loss of generality, it can be assumed that γ starts at 1 þ is and ends at þ1 þ is, s ¼ lim fIm½κðkÞg≥0. On the other hand, for the limit of k-1 k-1, the asymptotic behavior of the Lorentz model can be used to determine the result of the imaginary part of the refractive index. This has been found in [39,40], lim mi ðkÞ ¼

Using the scalar wave approximation, Greeberg derived the formula of the complex extinction efficiency Qad of a spheroidal particle [38],

ð14Þ

R1 where W 0 ðκÞis defined by Eq. (6), T ad ð1Þ ¼ 2πχ 0 2 a f ðaÞ da. The solution of the differential equation of (14) is Z κ 1 κ W 0 ðκÞ ¼ W 0 ð0Þ þ ½T ðxÞT ad ð1Þ dx þ Cκ; ð15Þ 4πχ 0 ad

0

ð9Þ

289

k-1

α2 2k

3

:

ð18Þ

where α2 is a constant depending on the material. This limit shows that the imaginary part of the refractive index always approaches to zero at a high frequency regardless the particle is absorbing or not. Therefore, s ¼ 2 lim fIm ½κðkÞg∝k -0. k-1 Integrate the right-hand side of Eq. (17) by parts  1  þ1þi0 κ½T ad ðκÞT ad ð1Þexp ðiκaÞ jκκ ¼ ¼ 1þi0 8π 2 χa2 Z   1  2 2 ½T ad ðκÞT ad ð1Þ dκ 8π χa γ   Z  1 d exp ðiκaÞ1 κ2 ½T ad ðκÞT ad ð1Þ þ 2 2 dκ; dκ κ 8π χa γ

f ðaÞ ¼ 

ð19Þ where the first and second integrals in Eq. (19) vanish based on Eqs. (A1 and A2) shown in Appendix A. Moreover, from Eq. (11), Q ad ðκaÞcan be written as   4 d expðiκaÞ1 ; ð20Þ Q ad ðκaÞ ¼ Q ad ð1Þ þ 2 κ a dκ where we designate that Q ad ð1Þ ¼ 2, hereinafter the same. Thus the remaining third term in Eq. (18) becomes, f ðaÞ ¼

1 32π 2 χ

Z γ

½Q ad ðκaÞQ ad ð1Þ½T ad ðκÞT ad ð1Þκ2 dκ: ð21Þ

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3.2. Cuboids

with

The analytical formula for extinction efficiency of a cuboid, oriented at an arbitrary angle to the incident beam, has not yet been achieved [41]. Here we only consider a simple rectangular cuboid case that the incident light is parallel to two opposite faces of a cuboid. Suppose the length, height, and depth of a rectangular are a, b, and c, respectively. Furthermore, c¼ a is assumed. Using ADT, the complex extinction efficiency of cuboid can be obtained:   2η 1 2 2η 1  þ exp ðiκaÞ; Q ad ðκaÞ ¼ 2 þ ð22Þ 1 þ η ðiκaÞ 1 þ η 1 þ η ðiκaÞ

κ¼

pffiffiffi 3ðm1Þk ; 2 sin ð2π=3θÞ

ð31Þ

where θ is the angle between the direction of incident radiation and one side of a triangle. The related complex extinction coefficient T ad ðκÞ is obtained as Z 1 T ad ðκÞ ¼ χlaQ ext ðκaÞf ðaÞ da; ð32Þ 0

where χ ¼ sin ð2π=3θÞ. Substituting Eq. (30) into Eq. (32), we have iκ ½T ðκÞT ad ð1Þ; 2χl ad

where the effective wavenumber κ, and the parameter η are given respectively by

W 0 ðκÞ ¼ W 0 ð0Þ

ðm1ÞMinð tan θ; μ1 Þk ; sin θ Minð tan ðθÞ; μ1 Þ η ¼ 2 ; μ1  tan θ

where function W 0 ðκÞis defined by Eq. (6), and T ad ð1Þ ¼ R1 2χl 0 af ðaÞ da. Therefore,

κ¼

ð23Þ

ð24Þ

where 0 ≤θ ≤π=2, Min stands for the minimum, μ ¼ a=b is the aspect ratio, θ is the angle between the direction of incident radiation and the line perpendicular to the face that incident beam hits on. The complex extinction coefficient Tad can be expressed in terms of Q ad ðκaÞ as Z 1 T ad ðκÞ ¼ χa2 Q ad ðκaÞf ðaÞ da: ð25Þ 0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where χ ¼ 1 þ μ2 sin ½θ þ arctan ðμ1 Þ. Inserting Eq. (22) into Eq. (25) we obtain dW 1 ðκÞ W 1 ðκÞ W 1 ð0Þ ið1 þ ηÞ þη ¼η  ½T ad ðκÞT ad ð1Þ; ð26Þ dκ κ κ 2χ R1 where T ad ð1Þ ¼ 2χ 0 a2 f ðaÞ da, W 1 ðκÞ is defined by Eq. (6). The solution of this differential equation is Z 1 ið1 þ ηÞ κ ½T ad ðβκÞT ad ð1Þβη dβ: ð27Þ W 1 ðκÞ ¼ W 1 ð0Þ 2χ 0

Taking a derivative with respect to κ of both sides and using Eq. (8), we have ( Z ) 1 ð1 þ ηÞ d η κ ½T ad ðβκÞT ad ð1Þβ dβ : ð28Þ W 2 ðκÞ ¼  2χ dκ 0 Applying the inverse Laplace transform of Eq. (7) to W 2 ðκÞ, we obtain ð1 þ ηÞ f ðaÞ ¼  4πχa2

Z γ

d dκ

( Z ) 1 ½T ad ðβκÞT ad ð1Þβη dβ expðiκaÞ dκ: κ 0

ð29Þ 3.3. Triangular prisms For a triangular prism of the regular triangle base side a and the distance (between the two triangle bases) l, with the incident radiation parallel to the two triangle bases, the complex extinction efficiency can be obtained by ADT as [42]   1exp ðiκaÞ Q ad ðκaÞ ¼ 2 1 þ ð30Þ iκa

W 1 ðκÞ ¼ 

 1 d  κ½T ad ðκÞT ad ð1Þ : 2χl dκ

ð33Þ

ð34Þ

Applying the inverse Laplace transform of Eq. (7) to W 1 ðκÞ, we obtain Z  1 d  κ½T ad ðκÞT ad ð1Þ expðiκaÞ dκ: f ðaÞ ¼  ð35Þ 4πχla γ dκ

3.4. Hexagonal prisms It is difficult to obtain the complex extinction efficiency for an arbitrarily oriented hexagonal prism [43]. Here we only consider two special cases of orientation. Case 1. Flat incidence orientation For the case of a regular hexagonal base column with flat incidence orientation [42], the complex extinction efficiency and complex extinction coefficient are given by Q ad ðκaÞ ¼ 2exp ðiκaÞ þ

1exp ðiκaÞ ; iκa

ð36Þ

with pffiffiffi κ ¼ 3ðm1Þk:

ð37Þ

T ad ðκÞ is obtained as Z 1 T ad ðκÞ ¼ χlaQ ad ðκaÞf ðaÞ da;

ð38Þ

0

where l is the length of hexagon prism, χ ¼ 2. Substituting Eq. (36) into Eq. (38), we obtain dW 0 ðκÞ W 0 ðκÞ W 0 ð0Þ i þ ¼  ½T ad ðκÞT ad ð1Þ; ð39Þ dκ κ κ χl R1 where T ad ð1Þ ¼ 2χl 0 af ðaÞ da. The solution of Eq. (39) is Z 1 i ½T ðβκÞT ad ð1Þβ dβ; ð40Þ W 0 ðκÞ ¼ W 0 ð0Þ κ χl 0 ad then W 1 ðκÞ ¼ 

1 d χl dκ

( Z ) 1 κ ½T ad ðβκÞT ad ð1Þβ dβ : 0

ð41Þ

J.-Q. Zhao, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 287–297

Applying the inverse Laplace transform of Eq. (7) to W 1 ðκÞ, we obtain, f ðaÞ ¼ 

1 2πχla

Z γ

d dκ

( Z ) 1 ½T ad ðβκÞT ad ð1Þβ dβ expðiκaÞ dκ: κ 0

ð42Þ

extinction coefficient of a polydispersion of elliptic cylinders with a finite length l can be approximated by Z 1 T ad ðκÞ ¼ χ2laQ ad ðκaÞf ðaÞ da; ð51Þ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where χ ¼ μ1 sin θ cos 2 ϕ þ μ2 sin 2 ϕ, kl≫1and l sin θ≫a. In Eq. (51) the edge effect is neglected. From Eqs. (49) and (51), we obtain

Case 2. Edge-on incidence orientation For the case of a regular hexagonal base column with edge-on incidence orientation [42], the complex extinction efficiency and complex extinction coefficient are given by exp ðiκaÞexp ðiκa=2Þ Q ad ðκaÞ ¼ 24 ; iκa Z 1 T ad ðκÞ ¼ χlaQ ad ðκaÞf ðaÞ da;

ð43Þ ð44Þ

0

pffiffiffi where l is the length of hexagon prism, χ ¼ 3, κ is determined by κ ¼ 2ðm1Þk:

ð45Þ



1 2πχl

Z 0

κ

Z ½T ad ðyÞT ad ð1Þ dy ¼

W 0 ðκÞ ¼ W 0 ð2Nþ1 κÞ

i N n ∑ 2 κ½T ad ð2n κÞT ad ð1Þ: 4χl n ¼ 1

ð47Þ

Note that lim W 0 ð2Nþ1 κÞ ¼ 0due to the Riemann– N-1 Lebesgue lemma [44]. Finally, the inverse transform of Eq. (47) can be expressed by Z  i 1 f ðaÞ ¼ 2n κ½T ad ð2n κÞT ad ð1ÞexpðiκaÞ dκ : ∑ 8πχl n ¼ 1 γ ð48Þ

0

1

fi½J 0 ðκaÞ1 þ H0 ðκaÞgf ðaÞ da:

ð52Þ For an absorbing particle case, it is difficult in mathematics to invert PSD directly from the above Eq. (52). The solution can only be found under the non-absorbing condition of mi ¼ 0, as Z κ1=2 κ w0 ðκÞ ¼  ½τ ðxÞτad ð1Þ dx; ð53Þ 2πχl 0 ad where w0 ðκÞ is defined in Eq. (9), τad ðxÞ ¼ Re ½T ad ðxÞ is the real extinction coefficient. Applying the inverse Struve transform of Eq. (10) on w0 ðκÞ, we obtain

Substituting the Eq. (43) into Eq. (44) yields iκ ½T ðκÞT ad ð1Þ ¼ W 0 ðκÞW 0 ðκ=2Þ; ð46Þ 4χl ad R1 whereT ad ð1Þ ¼ 2χl 0 af ðaÞda. If κ is replaced by 2n κ for n ¼ 1; 2; 3…N, respectively, Eq. (46) can be replaced by a series of similar equations. By summation of the results from these series equations, we obtain

291

a1=2 f ðaÞ ¼

Z

1

0

  Z κ 1=2 κ ðκaÞ1=2 Y 0 ðκaÞ  ½τad ðxÞτad ð1Þ dx dκ; 2πχl 0

ð54Þ hence f ðaÞ ¼ 

1 2πχl

Z

1

0

Z κ  κaY 0 ðκrÞ ½τad ðxÞτad ð1Þ dx dκ:

ð55Þ

0

By a partial integration, Z 1 1 ½κY 1 ðκaÞ½τad ðκÞτad ð1Þ dκ; f ðaÞ ¼ 2πχl 0

ð56Þ

where Y 1 ðκaÞ is the first-order Bessel function of the second kind. Eq. (56) can be written as Z 1 1 ½Y 1 ðκaÞY 1 ð1Þ½τad ðκÞτad ð1Þκ dκ: ð57Þ f ðaÞ ¼ 2πχl 0 This formula generalizes the result obtained by McKellar [33].

3.5. Elliptical cylinders

4. ADT transform

For an elliptic infinite cylinder with its axis oriented at an angle θ with respect to the direction of incident radiation, its complex extinction efficiency can be obtained as

Although the analytical inversion solutions for nonspherical particles have been obtained in Section 3, we find that the inversion problem and related formal solutions can be directly represented by inversion transform between the size distribution and the extinction spectrum. We call this inversion transform as ADT transform, which is equivalent to the Laplace transform or the Struve transform. In the following the constituted ADT transform pairs are listed. For spheroids, Z 1 T ad ðκÞT ad ð1Þ ¼ ½Q ad ðκaÞQ ad ð1Þ½πχf ðaÞa2 da; ð58Þ

Q ad ðκaÞ ¼ π½H1 ðκaÞiJ 1 ðκaÞ;

ð49Þ

where H1 ðκaÞ and J 1 ðκaÞ are the first-order Struve and Bessel functions, respectively; κ is given by [45] κ¼

2ðm1Þk qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sin θ cos 2 ϕ þ μ2 sin 2 ϕ

ð50Þ

where the ratio μ ¼ a=b, a and b are size parameters of the semi-major axis and semi-minor axis for the elliptic cross section, ϕ is the angle between the semi-major axis and the plane defined by the direction of the incident radiation and the axis of the cylinder. Note, Eq. (49) reduces to the result of infinite cylinder if μ¼1 [46]. Then the complex

0

πχf ðaÞ ¼

1 32π

Z γ

½Q ad ðκaÞQ ad ð1Þ½T ad ðκÞT ad ð1Þκ2 dκ: ð59Þ

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J.-Q. Zhao, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 287–297

For cuboids,

Z

T ad ðκÞT ad ð1Þ ¼

χa2 f ðaÞ ¼

1

0

Z

ð1 þ ηÞ2 8π

γ

½Q ad ðκaÞQ ad ð1Þ½χa2 f ðaÞ da;

d dκ

ð60Þ

( Z ) 1 κ ½Q ad ðβκÞQ ad ð1Þβη dβ 0

( Z ) 1 d κ ½T ad ðβκÞT ad ð1Þβη dβ dκ:  dκ 0 For triangular prisms, Z 1 ½Q ad ðκaÞQ ad ð1Þ½χlaf ðaÞ da; T ad ðκÞT ad ð1Þ ¼

ð61Þ

ð62Þ

0

Z 

χlaf ðaÞ ¼

  1 d  κ½Q ad ðκaÞQ ad ð1Þ 8π γ dκ    d  κ½T ad ðxÞT ad ð1Þ  dκ: dκ

ð63Þ

For hexagonal prisms in case of flat incidence orientation, Z 1 T ad ðκÞT ad ð1Þ ¼ ½Q ad ðκaÞQ ad ð1Þ½χlaf ðaÞ da; ð64Þ 0

( Z ) Z 1 1 d κ χlaf ðaÞ ¼ ½Q ad ðβκÞQ ad ð1Þβ dβ 2π γ dκ 0 ( Z ) 1 d κ  ½T ad ðβκÞT ad ð1Þβ dβ dκ: dκ 0

ð65Þ

For hexagonal prisms in case of edge-on incidence orientation, Z 1 T ad ðκÞT ad ð1Þ ¼ ½Q ad ðκaÞQ ad ð1Þ½χlaf ðaÞ da; ð66Þ 0

χlaf ðaÞ ¼

1 32π

Z

d  dκ

γ

∑ f2n κ½T ad ð2n κÞT ad ð1Þg dκ:

ð67Þ

n¼1

For elliptical cylinders (without absorption), Z 1 τad ðκÞτad ð1Þ ¼ ½H 1 ðκaÞH 1 ð1Þ½2πlχf ðaÞa da;

ð68Þ

0

Z 2πlχf ðaÞ ¼ 0

1

½Y 1 ðκaÞY 1 ð1Þ½τad ðκÞτad ð1Þκ dκ:

ð70Þ

where ψðzÞis analytic in the upper half plane of z and satisfies the growth property 2 3 6 7 6 7 lim 6max jzj ¼ R jψðzÞj7 ¼ 0: 5 R-14 0 oImðzÞ

ð71Þ

From Eqs. (6), (14), (26), (33), (39) and (46), it is clearly shown that for absorbing particles, T ad ðκÞT ad ð1Þ, ½T ad ðβκÞT ad ð1Þ and ½T ad ð2n κÞT ad ð1Þare all analytic in ImðκÞ≥0. Then we can apply the Schwartz formula to the analytic function: Z 1 þ1 ½τad ðyςÞτad ð1Þ y dς; ð72Þ T ad ðyκÞT ad ð1Þ ¼ πi 1 yςyκ where y ¼ ð1; β; 2n Þ, lim Re ½T ad ðyκÞ.

ς ¼ Re ðκÞ,

and

τad ðyςÞ ¼

κ-ςþi0

5.1. Spheroids Substituting Eq. (72) with y¼1 into Eq. (17) and changing the order of integrations, we obtain  2   Z þ1  Z 1 1 d κ f ðaÞ ¼  2 2 expðiκaÞ dκ πi γ κ dκ ðςκÞ 8π χa 1 ð73Þ ½τad ðςÞτad ð1Þ dς: The quantity inside curly brackets can be evaluated by the Jordan's lemma and the residue theorem (see detail of Eq. (A4) in Appendix A),    2  Z 1 d κ d exp ðiςaÞ : expðiκaÞ dκ ¼ ¼ 2ς2 πi γ κ dκ ðςκÞ dς ς ð74Þ

d 1 ∑ f2n κ½Q ad ð2n κÞQ ad ð1Þg dκ n ¼ 1 1

of its real part by [47] Z 1 þ1 Re½ψðxÞ ψðzÞ ¼ dx; πi 1 xz

ð69Þ

In brief, there exists important relationship between PSD and the extinction spectrum, which provides the new representation of solution to the inverse problem with clear physical meaning. This may help to find retrieval solutions for particles with various complicated shapes. 5. Real inverse formulas In practice, only the real part of T ad ðκÞ can be directly measured. In a PSD retrieval formula, T ad ðκÞ can be replaced by its real part based on Schwarz integral formula, since Schwarz integral formula allows one to recover a complex function ψðzÞ from the boundary values

According to Eq. (20), the right-hand side of Eq. (74) can be evaluated as   d exp ðiςaÞ 1 2ς2 ¼  ðςaÞ2 ½Q ad ðςaÞQ ad ð1Þ þ 2: dς ς 2 ð75Þ Appling Eq. (75) to Eq. (73), we obtain Z þ1 1 f ðaÞ ¼ ½Q ad ðςaÞQ ad ð1Þ½τad ðςÞτad ð1Þς2 dς 16π 2 χ 1 Z þ1 1  2 2 ½τ ðςÞτad ð1Þ dς; ð76Þ 4π χa 1 ad where the second term in Eq. (76) can be proved to be zero (Eq. (A3) in Appendix A) It is easy to check the parity relations: Re½Q ad ðςaÞ ¼ Re ½Q ad ðςaÞ ¼ Q ext ðςaÞ, and τad ðςÞ ¼ τad ðςÞ. By taking the real part of both sides of Eq. (76), we obtain Z þ1 1 f ðaÞ ¼ ½Q ext ðςaÞQ ext ð1Þ½τad ðςÞτad ð1Þς2 dς; 8π 2 χ 0 ð77Þ where Q ext ð1Þ ¼ Re ½Q ad ð1Þ ¼ 2. By comparing Eqs. (77) and (21), it is found that the inverse scattering problem can be dramatically simplified

J.-Q. Zhao, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 287–297

as the result only depends on the real especial extinction coefficient, τad ðςÞ, which could be obtained from the function form of measurable ordinary extinction coefficient, τad ðκÞ, by setting Im ðκÞ ¼ 0. We would like to emphasize that the absorption by the particles is still permitted and likely to exist, so τad ðςÞ is not equal to the scattering coefficient in non-absorbing case. Eq. (77) indeed represents an inverse formula in which the imaginary component of the complex effective wavenumber does not appear. This is very interesting but not surprised because that f(a) is a mathematical function showing the distribution of particle size, and it is independent to the dielectric properties of particles.

Substituting Eq. (72) with y¼β into Eq. (29) and changing the order of integrations, we obtain,  Z    Z ð1 þ ηÞ þ1 1 d κ f ðaÞ ¼  expðiκaÞ dκ πi γ dκ ðςκÞ 4πχa2 1 (Z ) 1 η  ½τad ðβςÞτad ð1Þβ dβ dς: ð78Þ 0

By Eq. (A5) in Appendix A, above formula can be simplified to f ðaÞ ¼

ð1 þ ηÞ 2πχa2

Z þ1  1

ς

d expðiςaÞ dς

(Z

1

0

ð85Þ

5.4. Hexagonal prisms Case 1. Flat incidence orientation Substituting Eq. (72) (with y¼β) into Eq. (42), changing the order of integrations and using Eq. (A5) in Appendix A, we obtain 1 πχla

Z

þ1

1

)  (Z 1 d ς exp½iςa ½τad ðβςÞτad ð1Þβ dβ dς: dς 0

ð86Þ From Eq. (40), it is found that Z 1 lim ς ½τad ðβςÞτad ð1Þβ dβ ¼ 0:

ς- 7 1

ð87Þ

0

After integration by parts, Eq. (86) becomes ) Z þ1 ( Z 1 2 d ς ½τad ðβςÞτad ð1Þβdβ cos ðςaÞ dς: f ðaÞ ¼  πχla 0 dς 0 ð88Þ

) ½τad ðβςÞτad ð1Þβη dβ

dς:

Case 2. Edge-on incidence orientation ð79Þ

From Eq. (27), we have Z 1 ½T ad ðβςÞT ad ð1Þβη dβ ¼ 0: lim ς

ς- 7 1

By partial integration, f(a) can be rewritten as Z þ1  1 d  f ðaÞ ¼  ς½τad ðςÞτad ð1Þ cos ðςaÞ dς: πχla 0 dς

f ðaÞ ¼

5.2. Cuboids

293

ð80Þ

0

Integrating Eq. (79) by parts, and using Eq. (80) and the relation of τad ðβςÞ ¼ τad ðβςÞ, we obtain, " Z # Z 1 ð1 þ ηÞ þ1 d η ς f ðaÞ ¼  ½τad ðβςÞτad ð1Þβ dβ dς πχa2 0 0  cos ðςaÞ dς:

ð81Þ

Substituting Eq. (72), with y ¼ 2n , into Eq. (48) and through integrating term by term, we get Z þ1 i 1 ∑ f ðaÞ ¼ 2n ς½τad ð2n ςÞτad ð1ÞexpðiςaÞ dς: 4πχl n ¼ 1 1 ð89Þ n

n

Taking into account τad ð2 ςÞ ¼ τad ð2 ςÞ, the above equation can be reduced to Z þ1 1 1 f ðaÞ ¼ 2n ς½τad ð2n ςÞτad ð1Þ sin ðςaÞ dς: ð90Þ ∑ 2πχl n ¼ 1 0

6. Possible scaling relations 5.3. Triangular prisms Substituting Eq. (72) y¼1 into Eq. (35) and changing the order of integrations, we obtain    Z þ1  Z 1 1 d κ expðiκaÞ dκ ½τad ðςÞ f ðaÞ ¼  4πχla 1 πi γ dκ ςκ τad ð1Þ dς:

ð82Þ

Based on Eq. (A5) in Appendix A, the above equation becomes,  Z þ1  1 d f ðaÞ ¼ ς expðiςaÞ ½τad ðςÞτad ð1Þ dς: ð83Þ 2πχla 1 dς On the other hand, from Eq. (27), it is shown that lim fς½τad ðςÞτad ð1Þexp ½iςag ¼ 0:

ς- 7 1

ð84Þ

For PSD retrieval based on prior assumptions about particle shape and orientation, a natural question is to what extent will the difference in the result be if the incorrect inversion parameters are employed? In order to provide an answer, we examined a few of the numerical simulations. We assume the true PSD to be a typical bimodal gamma distribution: 2

pn un ðun aÞvn expðun aÞ; Γðv n þ 1Þ n¼1

f true ðaÞ ¼ f 0 ∑

ð91Þ

where f 0 ¼ 1cm3 , p1 ¼ 0.4, p2 ¼ 1  p1, u1 ¼4 μm  1, v1 ¼2, u2 ¼ 5 μm  1, and v2 ¼10. The extinction spectrums are generated in term of ADT with refractive indexes 1.33 and the size distribution of f true ðaÞ, for different types of non-spherical particles. Other input parameters required in the spectral extinction simulations are chosen as

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Fig. 1. Effects of inversion parameters on PSD retrieval for spheroids. The true PSD is a bimodal gamma distribution. The spectral extinction is simulated using parameters: ðμ; θÞ ¼ ð6=5; π=4Þ.

Fig. 2. Effects of inversion parameters on PSD retrieval for triangular prisms. The spectral extinction is simulated using parameter: θ ¼ π=4.

ðμ; θÞ ¼ ð6=5; π=4Þ for spheroids, θ¼π/4 for triangular prisms, ðμ; θ; ϕÞ ¼ ð4=3; π=4; π=4Þ for elliptical cylinders, ðμ; θÞ ¼ ð1; π=4Þ for cuboids. Then the retrievals are done based on the above derived inverse formulas for the four types of particles. We also deliberately let some false parameters be applied to the retrieval calculations. In Fig. 1, the retrieval results of spheroidal inversion for true PSD and the retrieved PSD (based on the false parameters) are shown. The false set of inversion parameters makes the inversion solutions different from the true solutions. However, the results of the true and retrieved PSD are similar in curve shapes and peak positions. Since the false parameters used are chosen arbitrarily, it can be concluded that the retrieved PSD are always similar in curve shapes and peak positions for different inversion parameters. The retrieval results for triangular prisms and elliptical cylinders are shown in Figs. 2 and 3, respectively. Also the true and retrieved PSD are similar to some extent. The retrieval results for cuboids are shown in Fig. 4. The results become more complicated than other shapes of particles. In some cases, the information of true PSD is lost as the curves of

Fig. 3. Effects of inversion parameters on PSD retrieval for elliptical cylinders. The spectral extinction is simulated using parameters: ðμ; θ; ϕÞ ¼ ð4=3; π=4; π=4Þ.

Fig. 4. Effects of inversion parameters on PSD retrieval for cuboids. The spectral extinction is simulated using parameters: ðμ; θÞ ¼ ð1; π=4Þ.

inversion become acute in the non-true mode of PSD, and a further rigorous theoretical analysis is needed. Based on the inverse formulas (57), (77), (81), and (85), we can find that the retrieved PSD is associated with the true PSD by the scaling relationship f ðaÞ ¼ sχ sκ n f true ðsκ aÞ; sχ ¼

χ true ; χ

sκ ¼

Re ðκÞ : Re ðκtrue Þ

ð92Þ ð93Þ

where n¼2 is for triangular prisms, n¼3 is for spheroids, elliptical cylinders, and also possibly for cuboids subject to the constraint η ¼ ηtrue . ftrue here means an arbitrary exact distribution, but not necessarily limited to the bimodal gamma distribution. On the other hand, Eq. (91) implies that the effects of nonsphericity and orientation cannot be separated in general, and the retrieved PSD and the true PSD generally belong to one family of functions. Therefore, in the retrieval process for spheroids, triangular prisms and elliptical cylinders, a different set of inversion parameters does not dramatically change the type of mode for PSD.

J.-Q. Zhao, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 287–297

In the above discussions, the spectral extinction data is all assumed to be error-free. However, the availability of the scaling relationships should be reexamined when the data is not error-free. In the following calculation, a random noise is added to the spectral extinction by employing a multiplicative factor of ½1 þ εRð0; 1Þ, where ε is a constant, representing the noise level, R(0,1) is the random number (zero mean and standard deviation of one). The effect of the noise on the retrieval of PSD for spheroids is illustrated in Fig. 5. It is clearly shown that for the same inversion parameters but at different noise levels, the corresponding inversion curves are well coincident and there is no obvious discrepancy in the retrieved PSDs. A similar conclusion can be drawn from Figs. 6–8 for triangular prisms, elliptical cylinders and cuboids, respectively. Therefore, the scaling relationship is always true for a noise level up to 10%.

Fig. 5. Effects of noises and inversion parameters on PSD retrieval for spheroids. The spectral extinction is simulated using parameters: ðμ; θÞ ¼ ð6=5; π=4Þ.

295

Fig. 7. Effects of noises and inversion parameters on PSD retrieval for elliptical cylinders. The spectral extinction is simulated using parameters: ðμ; θ; ϕÞ ¼ ð4=3; π=4; π=4Þ.

Fig. 8. Effects of noises and inversion parameters on PSD retrieval for cuboids. The spectral extinction are simulated using parameters: ðμ; θÞ ¼ ð1; π=4Þ under different noise levels.

7. Conclusions and limitations

Fig. 6. Effects of noises and inversion parameters on PSD retrieval for triangular prisms. The spectral extinction is simulated using parameter: θ ¼ π=4.

Based on ADT, several analytical inverse formulas of PSD have been derived by solving the integral–differential equation for the oriented absorbing spheroids, cuboids, triangular prisms, hexagonal prisms, and the oriented nonabsorbing elliptical cylinders. We have shown that: (i) for each of considered non-spherical shape particles, there exist an ADT transform pairs between the size distribution and the extinction spectrum, which help us to understand the physics of inverse problem. The ADT transform pair could potentially be used in finding the solutions for particles with more complicated shapes; (ii) the inverse scattering problem with absorption can be simplified to a particular inverse problem for which absorption seems to be no longer present; (iii) the effects of nonsphericity and orientation cannot be separated; (iv) the retrieved PSD and the true PSD are generally belong to one family of functions due to the scaling relations for spheroids, triangular prisms, and elliptical

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cylinders. Therefore, an improper choice of inversion parameters does not much affect the mode of the retrieved PSD. The inverse formulas based on ADT require the knowledge of extinction spectrum covering the entire wavelength range, but the practical extinction measurements are often made at finite wavenumbers. Therefore, an appropriate mapping technique is needed to provide suitable input data for inverse formula. We will report the progress in a subsequent work.

Acknowledgments

Z ¼ 4πχ 0

1

 expðiκaÞ1 κ ¼ þ1þi0 jκ ¼ 1þi0 f ðaÞ da ¼ 0: κ

(3) Since ½T ad ðκÞT ad ð1Þ is analytic on the upper half plane Re ðκÞ≥0, we can assert categorically that Z

Z

þ1

1

½T ad ðςÞT ad ð1Þdς ¼

γ

½T ad ðκÞT ad ð1Þ dκ:

Taking the real part of both sides of above equation and using the result of I2 gives Z þ1 ½τad ðςÞτad ð1Þ dς ¼ 0: I3 ¼ 1

This research is financially supported by the NSFC (Project 41075015) and the LASG Free Exploration Fund. Appendix A The following equations are used in the text: I1 ¼

lim

κ- 7 1þi0

½κ½T ad ðκÞT ad ð1Þ ¼ 0;

ðfor spheroidsÞ;

ðA1Þ

γ

Z I3 ¼

I4 ¼

½T ad ðκÞT ad ð1Þ dκ ¼ 0;

6 lim 4max jκj ¼ R

Z γ

½τad ðςÞτad ð1Þ dς ¼ 0;

ðfor spheroidsÞ;

1 πi

Z γ

ðA3Þ

   2  d κ d exp ðiςaÞ ; expðiκaÞ dκ ¼ 2ς2 κ dκ ðςκÞ dς ς   d κ d expðiκaÞ dκ ¼ 2ς ½expðiςaÞ: dκ ðςκÞ dς ðA5Þ

These equations can be proved as follows: (1) From Eqs. (11) and (13), we can obtain  Z 1 expðiκaÞaf ðaÞ da κ½T ad ðκÞT ad ð1Þ ¼ 4πχ i 0  Z 1 Z 1 1 1 f ðaÞ da expðiκaÞf ðaÞ da ; þ κ 0 κ 0 thus I1 ¼

lim

½κ½T ad ðκÞT ad ð1Þ ¼ 0:

κ- 7 1þi0

(2) From Eq. (11), we have Q ad ðκaÞQ ad ð1Þ ¼

  4 d exp ðiκaÞ1 ; κ a2 dκ

thus

Z

½T ad ðκÞT ad ð1Þ ¼ 4πχ

1 0

finally Z I 2 ¼ ½T ad ðκÞT ad ð1Þdκ: γ

ImðκÞ ≤0

ðA2Þ

þ1

1

1 πi

ðfor spheroidsÞ;

ðA4Þ I5 ¼

thus 2 R-1

Z I2 ¼

(4) Define a function of gðκÞ as  2  d κ 2 κ þ ¼ ; gðκÞ ¼ κ dκ ðςκÞ ðςκÞ ðςκÞ2

   d expðiκaÞ1 f ðaÞ da ; dκ κ

3 3 2 7 6 2 2 7 þ gðκÞ 5 ≤ lim 4 5-0: R-1 Rς ðRςÞ2

According to Jordan's lemma, we have # Z " 2 κ þ lim expðiκaÞ dκ ¼ 0; R-1 C  ðςκÞ ðςκÞ2 R where C  R is a large semicircle in the lower half plane, centered at the origin with infinite radius of R. Hence we can extend integration path γ to a closed contour which contains γ on the upper half κ plane and  C R on the lower half κ plane. The closed contour γ þ C R encloses clockwisely the singular point of κ ¼ ς. Now let us to evaluate I4 by Jordan's lemma and the residue theorem: # Z " 1 2 κ I4 ¼ expðiκaÞ dκ þ πi γ ςκ ðςκÞ2 " # 1 2 κ þ ¼ ∮γþC R expðiκaÞ dκ πi ςκ ðςκÞ2    2πi 2expðiκaÞ lim ðκςÞ ¼ κ-ς πi ςκ " #) d κexpðiκaÞ þlim ðκςÞ2 κ-ς dκ ðςκÞ2   d ¼ 2 2expðiςaÞ þ ðςexpðiςaÞ dς   2 d expðiςaÞ ¼ 2ς : dς ς (5) Finally I5 can be evaluated in a similar way as I4.   d κ expðiκaÞ dκ γ dκ ðςκÞ " # Z 1 1 κ þ ¼ expðiκaÞ dκ πi γ ðςκÞ ðςκÞ2

I5 ¼

1 πi

Z

J.-Q. Zhao, J. Li / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 287–297

" # 1 1 κ ∮γþC R þ expðiκaÞ dκ πi ðςκÞ ðςκÞ2    2πi expðiκaÞ lim ðκςÞ ¼ κ-ς πi ςκ " #) d 2 κexpðiκaÞ þlim ðκςÞ κ-ς dκ ðςκÞ2   d ¼ 2 expðiςaÞ þ ðςexpðiςaÞ dς d ¼ 2ς ½expðiςaÞ: dς

t¼

References [1] Phillips DL. A technique for the numerical solution of certain integral equations of the first kind. J Assoc Comput Mach 1962;9:84–97. [2] Twomey S. On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. J Assoc Comput Mach 1963;10:97–101. [3] Ramachandran G, Leith D. Extraction of aerosol-size distribution from multispectral light extinction data. Aerosol Sci Technol 1992;17:303–25. [4] Wang Y, Fan S, Feng X, Yan G, Guan Y. Regularized inversion method for retrieval of aerosol particle size distribution function in W1-2 space. Appl Opt 2006;45(28):7456–67. [5] Liou KN, Takano Y. Light scattering by nonspherical particles: remote sensing and climatic implications. Atmos Res 1994;31:271–98. [6] Nakajima T, Tanaka M, Yamano M, Arao K, Nakanishi Y. Aerosol optical characteristics in the yellow sand events observed in May, 1982 at Nagasaki—Part II models. J Meteorol Soc Jpn 1989;67: 279–91. [7] Mishchenko MI, Hovenier JW, Travis LD, editors. Light scattering by nonspherical particles. New York: Elsevier; 2000. [8] Gobbi GP, Barnaba F, Blumthaler M, Labow G, Herman JR. Observed effects of particles nonsphericity on the retrieval of marine and desert dust aerosol optical depth by lidar. Atmos Res 2002;61:1–14. [9] Mishchenko MI. Light scattering by randomly oriented axially symmetric particles. J Opt Soc Am A 1991;8:871–82. [10] Yang P, Liou KN. Finite-difference time domain method for light scattering by small ice crystals in three-dimensional space. J Opt Soc Am A 1996;13:2072–85. [11] Draine BT, Flatau PJ. Discrete-dipole approximation for calculations. J Opt Soc Am A 1994;11:1491–9. [12] Kahn R, West R, McDonald D, Rheingans B, Mishchenko MI. Sensitivity of multiangle remote sensing observations to aerosol sphericity. J Geophys Res 1997;102:16861–70. [13] Krotkov NA, Flittner DE, Krueger AJ, Kostinski A, Riley C, Rose W, et al. Effect of particle nonsphericity on satellite monitoring of drifting volcanic ash clouds. J Quant Spectrosc Radiat Transfer 1999;63:613–30. [14] Liu Y, Arnott WP, Hallet J. Particle size distribution retrieval from multispectral optical depth: influences of particle nonsphericity and refractive index,. J Geophys Res D 1999;104(24):31753–62. [15] Dubovik O, Holben BN, Lapyonok T, Sinyuk A, Mishchenko MI, Yang P, et al. Nonspherical aerosol retrieval method employing light scattering by spheroids. Geophys Res Lett 2002;29. http://dx.doi.org/ 10.1029/2001GL014506. [16] Mishchenko MI, Geogdzhayev IV, Liu L, Ogren JA, Lacis AA, Rossow WB, et al. Aerosol retrievals from AVHRR radiances: effects of particle nonsphericity and absorption and an updated long-term global climatology of aerosol properties. J Quant Spectrosc Radiat Transfer 2003;7580:953–72. [17] Herman M, Deuze´ JL, Marchand A, Roger B, Lallart P. Aerosol remote sensing from POLDER/ADEOS over the ocean: improved retrieval using a nonspherical particle model. J Geophys Res 2005;110: D10S02. http://dx.doi.org/10.1029/2004JD004798. [18] Kalashnikova OV, I. Sokolik IN. Importance of shapes and compositions of wind-blown dust particles for remote sensing at solar wavelengths. Geophys Res Lett 2002;29(10):1398. http://dx.doi.org/ 10.1029/2002GL014947.

297

[19] Dubovik O, Sinyuk A, Lapyonok T, Holben BN, Mishchenko M, Yang P, et al. Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust. J Geophys Res 2006;111:D11208. http://dx.doi.org/10.1029/2005JD006619. [20] Sun X, Tang H, Yuan G. Anomalous diffraction approximation method for retrieval of spherical and spheroidal particle size distributions in total light scattering. J Quant Spectrosc Radiat Transfer 2008;109:89–106. [21] Feng Q, Yang P, Kattawar GW, Hsu CN, Tsay SC, Laszlo I. Effects of particle nonsphericity and radiation polarization on retrieving dust properties from MODIS observations. J Aerosol Sci 2009;40:776–89. [22] Wang L, Sun X, Xing J. Retrieval of spheroidal particle size distribution using the approximate method in spectral extinction technique. Opt Commun 2012;285(7):1646–53. [23] Tang H, Lin JZ. Retrieval of spheroid particle size distribution from spectral extinction data in the independent mode using PCA approach. J Quant Spectrosc Radiat Transfer 2013;1115:78–92. [24] van de Hulst HC. Light scattering by small particles. New York: Dover; 1981. [25] Shifrin KS, Perelman AY. The determination of the spectrum of particles in a dispersed system from data on its transparency. I. The fundamental equation for the determination of the spectrum of the particles. Opt Spectrosc (USSR) 1963;15:285–9. [26] Box MA, McKellar BH. Analytic inversion of multispectral extinction data in the anomalous diffraction approximation. Opt Lett 1978;3: 91–3. [27] Shifrin KS, Perelman AY, Volgin VM. Calculations of particle-radius distribution density from the integral characteristics of the spectral attenuation coefficient. Opt Spectrosc (USSR) 1981;51:534–8. [28] Fymat AL. Analytical inversions in remote sensing of particle size distributions. 1. Multispectral extinctions in the anomalous diffraction approximation. Appl Opt 1978;17:1675–6. [29] Smith CB. Inversion of the anomalous diffraction approximation for variable complex index of refraction near unity. Appl Opt 1982;21: 3363–6. [30] Klett JD. Anomalous diffraction model for inversion of multispectral extinction data including absorption effects. Appl Opt 1984;23: 4499–508. [31] Wang J, Hallett FR. Spherical particle size determination by analytical inversion of the UV–visible–NIR extinction spectrum. Appl Opt 1996;35:193–7. [32] Franssens G, Mazière MD, Fonteyn D. Determination of the aerosol size distribution by analytic inversion of the extinction spectrum in the complex anomalous diffraction approximation. Appl Opt 2000;39:4214–36. [33] BHJ McKellar. Light-scattering determination of the size distribution of cylinders: an analytic approximation. J Opt Soc Am 1982;72(5): 671–2. [34] Zhao JQ. A new method for the determination of the spheroidal nonabsorbing particle size distribution. J Grad School Chin Acad Sci 2010;27(3):323–30. [35] Zhao JQ. Simple technique to evaluate effects of nonsphericity and orientation on particle size distribution retrieval. Int J Phys Sci 2011;6(31):7100–5. [36] Titchmash EC. A pair of inversion formulae. Proc London Math Soc 1923;23(2) (XXXIV–XXXV). [37] McKellar BHJ, Box MA, Love ER. Inversion of the Struve transform of half integer order. J Austral Math Soc, Ser B. 1983;25(2):161–74. [38] Greeberg JM. Scattering by nonspherical particles. J Appl Phys 1960;31:82–4. [39] Bohren CF, Huffman DR. Absorption and scattering of light by small particles. New York: Wiley; 1983. [40] Ku JC, Felske JD. Determination of refractive indices of Mie scatterers from Kramers-Kronig analysis of spectral extinction data. J Opt Soc Am A. 1986;3(5):617–23. [41] Napper DH. A diffraction theory approach to the total scattering by cubes. Kolloid Z. Z Polym 1967;218:41–6. [42] Chylek P, Klett JD. Absorption and scattering of electromagnetic radiation by prismatic columns: anomalous diffraction approximation. J Opt Soc Am A 1991;8:1713–20. [43] Sun W, Fu Q. Anomalous diffraction theory for arbitrarily oriented hexagonal crystals. J Quant Spectrosc Radiat Transfer 2001;63: 727–37. [44] Davies B. Integral transforms and their applications. Berlin: Springer-Verlag; 1978. [45] Fournier GR, Evans BTN. Approximation to extinction efficiency from randomly oriented circular and elliptical cylinders. Appl Opt 1996;35:4271–82.