INVERSION OF TWO-DIMENSIONAL SPHEROIDAL PARTICLE DISTRIBUTIONSFROM BACKSCATTER, EXTINCTION AND DEPOLARIZATION PROFILES VIA

REGULARIZATION

Christine Bockmann and L. Osterloh

Universitat Potsdam, Institut fur Mathematik, Am Neuen Palais 10, 14469 Potsdam, Germany, Email:bockmann@uni-potsdam.de

ABSTRACT

We consider an ensemble of spheroidal particles charac-terized by the complex refractive index, the aspect ratioand the size parameter of volume equivalent spheres, i.e.,the aspect ratio is the additional microphysical parameterwhich goes beyond the conventional Mie theory. We pro-pose a two-dimensional (2D) model to retrieve spheroidalparticle distributions, depending not only on the size pa-rameter but also on the aspect ratio, and microphysicalparticle properties from Raman lidar and depolarizationprofiles in contrast to conventional procedures. Since theproblem is ill-posed in nature we use an appropriate iter-ative regularization technique to retrieve the 2D particledistribution.

1. INTRODUCTION

Studying the influence of non-spherical cloud and aerosolparticles on the radiation budget of Earth’s atmosphere isof growing importance in remote sensing. Polar strato-spheric clouds and Cirrus clouds, for example, may con-tain large populations of non-spherical ice crystals whichhave a major impact on how those clouds scatter radi-ation. Saharan dust storms as well as volcanic erup-tions are other sources of non-spherical aerosol particleswhich are important for a better understanding of the di-rect and indirect climate effects of such global events.There exist essential differences in the light scattering be-havior between spherical and non-spherical particles. Wecan observe an increasing side scattering behavior andthe appearance of a backscattering depolarization if non-spherical particles are considered, see e.g. [1] and [2].Such spheroidal particles are used as microphysical mod-els in recent measurement campaigns like AERONET,EARLINET, and SAMUM to study the effect of non-spherical particles on different measurement scenariosand to estimate their influence on the radiative forcing,see e.g. [3, 4, 5, 6]. We investigated the role that depolar-ization profiles play in this inversion process to retrievethe particle distribution.

(a) oblate (b) prolate

Figure 1: Two spheroids. On the left, an oblate spheroid witha = 0.5, on the right, a prolate spheroid with a = 1.5.

2. SPHEROIDAL PARTICLES

Let us denote by rv the vertical semi-axis (rotating axis)of a spheroid and by rh the horizontal one. This meansfor the aspect ratio a = rv/rh. A spheroid is created byrotating the curve defined by

r(ϑ) = rv

(sin2 ϑ+

r2h

r2v

cos2 ϑ

)−1/2

, ϑ ∈ [0, 2π).

(1)See Fig. 1 for examples of oblate and prolate spheroids.

The inversion will be performed with the following dataΓj , j ∈ {bsca,hv, ext}: the total particle backscatterat 355, 532 and 1064 nm and cross-polarization compo-nent of particle backscatter at 355, 532 and 1064 nm aswell as particle extinction coefficients at 355 and 532 nm(3bsca+3hv+2ext wavelengths λ). Since we will assumer to denote the radius of the volume equivalent sphere,the volume distribution v(r, a) = 4πr3

3 n(r, a) is basicallystill valid by construction. Thus the volume distributioncan be used since it is less prone to numerical instabili-ties than the size distribution n(r, a). We can write thiswith the spheroidal efficiencies Qj from the scatteringdatabase for spheroidal particles [7], refractive index m

0 0.2 0.4 0.6 0.8 1

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

00.2

0.40.6

0.81

0

0.1

0.2

0.3

0.4

0.5

Figure 2: Some particular two-dimensional B-spline surfaces oforders 1, 2 and 4.

and surface S as

Γj(λ) =

amax∫amin

rmax∫rmin

3S

16πr3Qj(r, λ,m, a)v(r, a)drda.

(2)Thus, the integral over the size parameter is integratedagain over a parameter representing the different aspectratios. This means that we are here looking at 2D dis-tributions, for more details see [8]. In contrast to [9]we do not use a particular separation product statementv(r, a) = v1(r)v2(a). The retrieving problem of the dis-tribution from the 2D Fredholm system of first kind Eq. 2is ill-posed in nature and, therefore, needs particular reg-ularization methods.

In the first step, for the discretization of Eq. 2 by colloca-tion, one needs an extension of B-splines to two dimen-sions which are called the B-spline surfaces [10]. Insteadof one, one has now two vectors of knots, one for eachdimension. For illustration, see Fig. 2. In the secondstep, the resulting ill-conditioned linear equation systemis solved by an iterative regularization technique, see [11]and [12].

2.1. Extinction efficiencies

First, we can find the effect, like for spheres, here that theefficiencies are smoother the larger the imaginary part ofthe refractive index becomes. Note how the differencesin the optical efficiencies highly depend on the refractiveindex of the examined particles, see Fig. 3. For the non-absorbing case, m = 1.5, one can see a distinct ripplestructure in the extinction efficiency. For the absorbingcase with m = 1.5 + 0.01i, however, this is not the case.Second, indeed, judging from that surface plot the ex-tinction efficiency is more or less identical for all aspectratios. This is a very important fact that we need to con-sider; it becomes apparent that the extinction efficiency,

5 10 15 20 25 30 35 0.67

1

1.50

1

2

3

4

5

Qext(s)

s

a

Qext(s)

(a) m = 1.5

5 10 15 20 25 30 35 0.67

1

1.50

0.51

1.52

2.53

3.5

Qext(s)

s

a

Qext(s)

(b) m = 1.5 + 0.01i

Figure 3: Spheroidal extinction efficiencies for two differentrefractive indices with s = πr/λ.

and thus the extinction coefficients, become independentof the particle shape for cases with higher absorption. Forthe inversion that means that the extinction data itself willnot and cannot contain any viable information about theshape of the particles if the absorption is above a certainlevel.

2.2. Microphysical properties

We propose to extend the concept of microphysical prop-erties to spheroidal particles, additionally integrating overthe aspect ratio. Thus, we obtain the total surface-areaconcentration at, with G the geometrical cross-section,

at = 4

amax∫amin

rmax∫rmin

G(r, a)n(r, a)drda (3)

and the total volume concentration vt,

vt =4π

3

amax∫amin

rmax∫rmin

r3n(r, a)drda, (4)

while the effective radius is reff = 3 vtat . We will also in-troduce some new parameters to describe the variabilitywith respect to the aspect ratio. For one, we will intro-duce the mean aspect ratio, defined by

µa =

∫ amax

amina∫ rmax

rminn(r, a)drda

Nt, (5)

as well as the aspect ratio width, which is defined by

σa =

∫ amax

amin(a− µa)2

∫ rmax

rminn(r, a)drda

Nt, (6)

with Nt =∫ amax

amin

∫ rmax

rminn(r, a)drda the total number

concentration. We have introduced these quantities asthey will give us a good estimate as to the behavior of the2D distribution with respect to the aspect ratio. As onecan see, the mean aspect ratio µa is basically an estimatefor the “central” aspect ratio of the distribution, while theaspect ratio width represents a value that describes howmuch the values deviate from the mean value. An im-portant piece of data is the fraction of the total amount ofthe distribution that can be attributed to oblate and prolatespheroids and spheres. For that purpose, we introduce aparameter ξ, which describes a small deviation of the as-pect ratio from the spherical value 1. Note that this doesnot mean we expect these particles to behave like spheres;it is instead used to introduce a measure for the retrievalof the particle shape. We set ξ = 0.1. The fraction quan-tities are given by

vsphere =

1+ξ∫1−ξ

rmax∫rmin

v(r, a)drda/vt, (7)

voblate(prolate) =

1−ξ(amax)∫amin(1+ξ)

rmax∫rmin

v(r, a)drda/vt. (8)

We have voblate+vsphere+vprolate = 1. The vparticletype

describes the fraction of the distribution that is made upof that particle type.

3. SIMULATION RESULTS

To test the validity of the proposed algorithm, we tested,firstly, it on data simulated with spherical particles. Wehave used data gained from the usual forward modelassuming the (spherical) log-normal distribution withrmed = 0.1µm, σ = 1.6, Nt = 1 withm = 1.5+0.01i atextinction wavelengths of 355 and 532 nm and backscat-ter wavelengths at 355, 532 and 1064 nm.

For the results, let us look at Fig. 4. We have performedthe inversion without any error on the data. First, let uslook at Fig. 4(b), 4(d) and 4(f), in which we have usedthe exact same forward data as in the spherical case. In-terestingly, while the aspect-ratio-integrated distributionva(r) in Figure 4(d) can be calculated quite exactly, thesize-integrated distribution over the aspect ratios vr(a) asshown in 4(f) is very far from correct (of course, the truesolution here just consists of a delta peak at a = 1).

For the other three figures on the left, Fig. 4(a), 4(c)and 4(e), we have also incorporated depolarization infor-mation. This means that in addition to the data points

0

0.05

0.1

0.15

0.2

a

r (µm)

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

(a) with depolarization data

0

0.01

0.02

0.03

0.04

0.05

0.06

a

r (µm)

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1

(b) without depolarization data

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

v a(r

) (µ

m3 /(

cm3 µm

))

r (µm)

solutionreconstruction

(c) with depolarization data

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.2 0.4 0.6 0.8 1

v a(r

) (µ

m3 /(

cm3 µm

))

r (µm)

solutionreconstruction

(d) without depolarization data

0

0.01

0.02

0.03

0.04

0.05

0.06

0.6 0.8 1 1.2 1.4 1.6

v r(a

) (µ

m3 /c

m3 )

a

reconstruction

(e) with depolarization data

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.6 0.8 1 1.2 1.4 1.6

v r(a

) (µ

m3 /c

m3 )

a

reconstruction

(f) without depolarization data

Figure 4: Inverting data with the spheroidal model gained froma spherical forward model and a complex refractive index ofm = 1.5+ 0.01i. For (e) and (f), the exact solution consist of adelta peak at a = 1.

355ext, 355bsca, 532ext, 532bsca and 1064bsca we haveincluded 355hv = 532hv = 1064hv = 0, as we knowthat the simulated spherical particles will not result in anydepolarized backscatter. As one can see from the figures,while the reconstruction of va(r) is nearly identical, thereconstruction of vr(a) works much better here in thiscase. In Tab. 1, the reconstructions of the microphysi-cal properties as defined in Eqs. 3-7 are given. Just asfor spheres, the reconstruction of Nt still is problematic,while at and vt can be reconstructed nearly perfectly bothincluding and excluding depolarization data. The qual-ity is different for µa and σa and the vparticletype proper-ties, though. These quantities can be only reconstructedwhen the depolarization wavelengths are considered asone would expect.

This leads to a very interesting proposition, which we willfurther investigate in other examples; while informationabout the sizes of the particles are mostly included in theextinction and total backscatter data, the shape of the par-ticles (here represented by the aspect ratio) can only bereliably determined with knowledge about the depolar-ization.

Secondly, we made quite a lot sensitivity studies with nu-merical simulations under different points of view. Weshow only one aspect here in using a prolate ensemble,for more details see [8]. Since the degree of ill-posednessis higher for spheroidal particles, see [8], more a-prioriinformation is useful and very often necessary during

Table 1: Two-dimensional distribution retrieval with the spher-ical forward model and the spheroidal backward model.

property exact value retrieval retrievalwith hv without hv

reff 0.17 0.16 0.16Nt 1 3.4 3.8at 0.2 0.22 0.22vt 0.011 0.011 0.011

voblate 0% 8% 21%vsphere 100% 76% 28%vprolate 0% 16% 51%µa 1 1.01 1.13σa 0 0.083 0.35

Table 2: Reconstruction errors for the 2D case with a prolateensemble. The numbers are the mean reconstruction errors for100 error runs. The columns 2-4 and 5-7 show the reconstruc-tion quality with and without depolarization data, respectively.

error level 1% 5% 15% 1% 5% 15%reff 5% 7% 22% 5% 7% 24%vt 3% 4% 12% 5% 7% 22%µa 6% 10% 23% 20% 23% 25%

the inversion process, here we use three depolarizationprofiles additionally to compare this effect as mentionedabove. Whereas the effective radius is not influenced somuch, it is the case for vt and in particular for µa, seeTab. 2.

4. SUMMARY

In this work, a model used for forward calculation andinversion of aerosol optical properties that is based onMie theory and used for spheres has been extended towork on spheroids, where the underlying kernel functionswere exchanged by database values that have been cal-culated via a T-matrix method. We investigated a new2D model to retrieve microphysical properties from op-tical Raman lidar and, additionally, depolarization datafor non-spherical particles. From a mathematical pointof view the problem is more ill-posed than for sphericalparticles. We found, retrieving non-spherical 2D parti-cle distributions without using depolarization data failsmore or less. Depolarization profiles are very valuableand urgently necessary, in particular, if prolate particlesare present.

ACKNOWLEDGMENTS

This work was supported by the European Union seventhframework program Marie-Curie actions through ITARSproject 289923.

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