518 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Bockmann et al.

Microphysical aerosol parameters frommultiwavelength lidar

Christine Bockmann and Irina Mironova*

Institute of Mathematics, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany

Detlef Muller

Institute for Tropospheric Research, Permoserstrasse 15, 04318 Leipzig, Germany

Lars Schneidenbach

Institute for Computer Science, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany

Remo Nessler

Laboratory for Air and Soil Pollution, Swiss Federal Institute of Technology, Switzerland, and Laboratory ofAtmospheric Chemistry, Paul Scherrer Institut, Switzerland

Received July 9, 2004; accepted September 2, 2004

The hybrid regularization technique developed at the Institute of Mathematics of Potsdam University (IMP) isused to derive microphysical properties such as effective radius, surface-area concentration, and volume con-centration, as well as the single-scattering albedo and a mean complex refractive index, from multiwavelengthlidar measurements. We present the continuation of investigations of the IMP method. Theoretical studiesof the degree of ill-posedness of the underlying model, simulation results with respect to the analysis of theretrieval error of microphysical particle properties from multiwavelength lidar data, and a comparison of re-sults for different numbers of backscatter and extinction coefficients are presented. Our analysis shows thatthe backscatter operator has a smaller degree of ill-posedness than the operator for extinction. This fact un-derlines the importance of backscatter data. Moreover, the degree of ill-posedness increases with increasingparticle absorption, i.e., depends on the imaginary part of the refractive index and does not depend signifi-cantly on the real part. Furthermore, an extensive simulation study was carried out for logarithmic-normalsize distributions with different median radii, mode widths, and real and imaginary parts of refractive indices.The errors of the retrieved particle properties obtained from the inversion of three backscatter (355, 532, and1064 nm) and two extinction (355 and 532 nm) coefficients were compared with the uncertainties for the caseof six backscatter (400, 710, 800 nm, additionally) and the same two extinction coefficients. For known com-plex refractive index and up to 20% normally distributed noise, we found that the retrieval errors for effectiveradius, surface-area concentration, and volume concentration stay below approximately 15% in both cases.Simulations were also made with unknown complex refractive index. In that case the integrated parametersstay below approximately 30%, and the imaginary part of the refractive index stays below 35% for input noiseup to 10% in both cases. In general, the quality of the retrieved aerosol parameters depends strongly on theimaginary part owing to the degree of ill-posedness. It is shown that under certain constraints a minimumdata set of three backscatter coefficients and two extinction coefficients is sufficient for a successful inversion.The IMP algorithm was finally tested for a measurement case. © 2005 Optical Society of America

OCIS codes: 010.0010, 010.1110, 100.0100, 100.3190, 280.0280, 290.0290.

1. INTRODUCTIONAtmospheric aerosol particles influence the Earth’s radia-tion balance both directly by scattering and absorbing so-lar radiation and indirectly by acting as cloud condensa-tion nuclei. For that reason, the chemical and physicalproperties of aerosols are needed to estimate and predictdirect and indirect climate forcing.1 Information on par-ticle extinction and backscatter coefficients at multiplewavelengths can be delivered by instruments of differentlidar systems.2–4 With this optical information, micro-physical properties of aerosols can be inverted. The in-version problem in a mathematical sense is nonlinear and

1084-7529/2005/030518-11$15.00 ©

ill-posed. Its solution requires the application of appro-priate mathematical regularization methods; see Refs.5–7.

In this contribution we considered general mathemati-cal aspects concerning the inversion of multiwavelengthlidar data. Two questions were in the focus of our stud-ies; i.e., what is the stability of the regularization methodconcerning the retrieval of the number and volume distri-bution of particle size distributions, and second, what isthe degree of ill-posedness of the underlying mathemati-cal model in dependence of the refractive index?

In a second step we present the performance character-

2005 Optical Society of America

Bockmann et al. Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. A 519

istics of the inversion algorithm developed at the Instituteof Mathematics of Potsdam University (IMP) with respectto simulated optical data. Sets of backscatter coefficientprofiles at 355, 532, and 1064 nm and of extinction coeffi-cient profiles at 355 and 532 nm are the standard outputof advanced aerosol Raman lidars based on a singleNd:YAG laser (emitted wavelengths: 355, 532, and 1064nm); e.g., many of the lidar stations in the EuropeanAerosol Research Lidar Network,2,8 (EARLINET) in prin-ciple, can measure this data set. During the EARLINETproject (2000–2003), 22 lidar stations in 13 Europeancountries delivered routine observations of optical aerosolproperties. For this reason, there is great interest to ob-tain detailed characterization of the physical particleproperties from that enormous data set. One lidar sta-tion, which is at the Institute for Tropospheric Research[(IfT) Leipzig, Germany], additionally has a second de-vice, namely, a multiwavelength Mie–Raman lidar thatprovides backscatter coefficients at six wavelengths andextinction coefficients at two wavelengths.3 This lidardelivers three additional backscatter coefficients at 400,710 and 800 nm.

Therefore, in our extensive simulation study of retriev-ing microphysical properties from lidar observations, wepaid attention to the number of backscatter and extinc-tion coefficients used. For this reason, first we investi-gated data combinations of three backscatter and two ex-tinction coefficients, and then we compared our retrievalresults with the same coefficients plus the additionalthree backscatter coefficients. Starting from a variety ofsize distributions and refractive indices, we calculated thebackscatter and extinction coefficients at the wavelengthsmentioned above by using a Mie-scattering code.9 Theparticle parameters inverted from the synthetic lidar datawere then compared with the input data.

Since all experimental data sets suffer from measure-ment errors, we tested the stability of the inversion withrespect to erroneous lidar data by adding noise to the syn-thetic lidar data before inverting them. We have chosendifferent levels of normally distributed noise, namely, 5%,10%, 15% and 20%, for our simulation analysis.

The results of measurements by different investigatorsgive us an impression about the possible variability ofaerosols in the atmosphere. Refractive indices vary con-siderably for different aerosol species. The real and theimaginary parts of black carbon, for example, range from'1.75 to '1.96 and from '0.44 to '0.66, respectively, forwavelengths of visible light.10–12 For organic carbon, onthe other hand, real parts of the refractive index between1.44 and 1.55 are found in the literature.13,14 Sulfatespecies relevant for the atmosphere are reported to havereal parts ranging from '1.43 to '1.53.10,11,13,15,16 OnlyRef. 15 lists imaginary parts different from zero for sul-fates, namely, of the order of 1028. For sea-salt anddustlike particles, refractive indices around 1.5 1 13 1028i and 1.53 1 8 3 1023i in the visible range, re-

spectively, are given in the literature.15,17

The size of aerosol particles ranges from a few nano-meters to several hundred micrometers. However, par-ticle size distributions are very different for differentkinds of aerosols. Some illustrative examples are pollens(10–100 mm), sea salt (1–10 mm), or diesel soot ('0.1

mm).18 Aerosol size distributions are often divided intodifferent size classes. Frequently used is the classifica-tion into transient nuclei (or Aitken), accumulation, andcoarse modes; the first two modes together are referred toas the fine mode. The limit between the fine-particle andthe coarse modes ranges from 1 to 2.5 mm in diameter, de-pending on the definition, whereas 100 nm is a commonboundary between the nuclei and the accumulationmodes.

Monomodal logarithmic-normal distributions are usedto describe the particle-number size distributions:

n~r ! 5Nt

r

1

A2p ln sexpF20.5

~ ln r 2 ln rmed!2

ln2 sG , (1)

where Nt , s, and rmed denote the total number concentra-tion, the mode width, and the median radius, respectively.

The detectable aerosol size range is limited for mea-surements with lidar systems.19 The reason is that onlyparticles with radii of the same order of magnitude as themeasurement wavelengths (355 to 1064 nm) possessparticle-size-dependent scattering efficiencies. There-fore, e.g., pollens cannot be observed with lidar systems.For that reason, for our extensive simulations we choosereal parts in a range from 1.4 to 1.8, imaginary parts from0 to 0.5, median radii from 0.05 to 0.3 mm, and modewidths from 1.4 to 1.8.

This paper is organized as follows. After describingthe mathematical background in Section 2, we discuss ourresults of the theoretical mathematical aspects in Section3. In Section 4 we study several measurement situationswith simulated noiseless and noisy data. Finally, wepresent the retrieval results from a measurement case.Section 5 summarizes our conclusions. Appendix Acloses the paper with a description of the developed soft-ware package.

2. MATHEMATICAL BACKGROUNDThis paper is a continuation of the investigations of theIMP technique. Thus we only briefly summarize themain points of the algorithm. A detailed review of thehybrid regularization technique is given in Ref. 5.

The IMP inversion algorithm allows us to derive micro-physical particle properties from multiwavelength lidarmeasurements. The mathematical model that relatesthe optical and the physical particle parameters consistsof a Fredholm system of two integral equations of the firstkind for the backscatter and extinction coefficients:

G~l, z ! 5 Er0

r1

Kp/extn ~r, l, m, s !n~r, z !dr

5 Er0

r1

pr2Qp/ext~r, l, m !n~r, z !dr, (2)

where r denotes the particle radius, m is the complex re-fractive index, s is the shape of the particles, r0 and r1represent suitable lower and upper limits, respectively, ofrealistic particle radii, l is the measurement wavelength,z is the height at which the scattering process occurs, n isthe particle number distribution, and Kp

n is the backscat-ter and Kext

n is the extinction number kernel. The ker-

520 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Bockmann et al.

nel functions reflect shape, size, and material compositionof the particles. In the framework of this paper, homoge-neous spherical particles, i.e., Mie scattering theory,20 areused under the assumption that small particles in a firstapproximation behave as spherical scatterers.9 G standsfor the backscatter coefficient b or the extinction coeffi-cient, a depending on the measurement data.

The following formulas hold for back- and extinctionscatter efficiencies of homogeneous spheres9:

Qp 51

k2r2U(

n51

`

~2n 1 1 !~21 !n~an 2 bn!U2

,

Qext 52

k2r2 (n51

`

~2n 1 1 !Re~an 1 bn!, (3)

where k is the wave number defined by k 5 2p/l and anand bn are the coefficients that can be derived from theboundary conditions for the tangential components of thewaves.9

Equation (2) can be reformulated into a more specificform:

G~l, z ! 5 Er0

r1

Kp/extv ~r, l, m !v~r, z !dr

5 Er0

r1 3

4rQp/ext~r, l, m !v~r, z !dr, (4)

where the v(r, z) term is the volume distribution we arelooking for, instead of the number distribution n(r, z) inEq. (2). Kp/ext

v are the volume kernel functions for back-scatter and extinction, respectively. Throughout the pa-per we assume that the height z is kept at a fixed valueand therefore can be omitted. The determination of thevolume distribution v as well as the number distributionn from a small number of backscatter and extinction mea-surements is an ill-posed inverse problem.

Figures 1(a) and 1(b) show the number kernel functionsof Eq. (2) for backscatter and extinction, respectively.Figures 1(c) and 1(d) show the volume kernel functions ofEq. (4) for backscatter and extinction, respectively. Therefractive index is always m 5 1.5 1 0i. Very roughlyspeaking, the problem is more ill-posed if the surfaceshape of the kernel function is smoother.21 In the do-main of interest between 300 and 1100 nm for the wave-length and from 0.1 to 2 mm for the particle radius, thevolume kernel is not as smooth as the number kernel.For that reason, we prefer to invert the operators in Eq.(4). More details about the degree of ill-posedness arepresented in Section 3.

Moreover, because the refractive index m in the volumekernels Kp/ext

v is an unknown quantity, too, the problem isa highly nonlinear one. In solving such problems with-out the introduction of appropriate mathematical toolssuch as a suitable discretization and regularization, theputative results would be highly oscillating; see Ref. 22.

The inversion process consists of four steps. The firstregularization step in the IMP method is performed withdiscretization, in which the investigated distributionfunction is approximated with variable B-spline func-tions. The projection dimension (number of base func-

tion) and the order of the B splines used serve, roughlyspeaking, as regularization parameters. In the secondstep, regularization is controlled by the level of truncatedsingular-value decomposition (SVD) performed during thesolution process of the resulting linear equation system.To reduce the computer time, we used a collocation pro-jection. For more details, see Refs. 5 and 23. Third, thehighly nonlinear problem of the complex refractive indexas a second unknown is handled by introducing a grid ofwavelength- and size-independent mean complex refrac-tive indices and by enclosing the area of possible real/imaginary-part combinations through inversion and back-calculation of optical data. Finally, the mean andintegral properties of the particles such as total surface-area concentration at , total volume concentration vt , andeffective radius reff are calculated from the inverted par-ticle volume distribution by the formulas

at 5 3E v~r !

rdr, vt 5 E v~r !dr, reff 5 3

vt

at.

(5)

Moreover, the inverted particle volume distribution andthe (inverted or known) refractive index are used to cal-culate the scattering and extinction coefficients, sscat anda, from which the single-scattering albedo v, defined asv 5 sscat /a, is computed.

3. DEGREE OF ILL-POSEDNESSThis section deals with a theoretical study of the degree ofill-posedness of the operators [Eq. (4)]. Such investiga-tions in the framework of lidar applications are made forthe first time, to our knowledge. The theoretical resultsare very useful because we found a dependence on the re-fractive index and the findings show the importance of thebackscatter coefficients in retrieving the particle size dis-tribution.

First, for a better understanding, we explain someterms. Every square-integrable kernel of a linear inte-gral operator such as Eq. (4) has a singular-value expan-sion (SVE) that is a mean convergent expansion of theform

K~r, l! 5 (i51

`

m iui~r !vi~l!,

r P Ir 5 @r0 , r1#, l P Il 5 @l0 , l1#, (6)

where $ui , vi% are the left and right singular functions ofthe kernel and m i are the singular values.24 The behav-ior of the singular values and functions is strongly linkedto the properties of the kernel. Roughly speaking, thesmoother the kernel, the faster the singular values m i de-cay to zero, where smoothness is measured by the numberof continuous partial derivatives of the kernel.25,26 Thesmaller the m i , the more oscillations in the singular func-tions ui and vi occur; see Fig. 2. Under some assump-tions, m i 5 o@i2( p11/2)#, i → `, where p is the number ofcontinuous partial derivatives with respect to the secondvariable (the wavelength).

First, we observe the operators [Eq. (4)] only veryroughly. The inversion of the Mie backscatter volume

Bockmann et al. Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. A 521

Fig. 1. (a) Backscatter and (b) extinction number kernel functions Kp/extn for (m1 5 1.5 1 0.0i) and (c), (e) backscatter and (d), (f) ex-

tinction volume kernel functions Kp/extv for different refractive indices: (c), (d) without absorption (m1 5 1.5 1 0.0i) and (e), (f) with

strong absorption (m2 5 1.5 1 0.5i).

kernel is potentially interesting because the kernel ishighly oscillating in the case of weak particle absorption;i.e., the imaginary part of the refractive index is small;see Fig. 1(c). The strong oscillating component of thekernel suggests that the classic instability, which is

caused by the smoothing out of fast oscillatory compo-nents in the solution space, may not occur. One can ex-pect that p is only a small value. However, in practice,the oscillation of the kernel occurs on such a small scalethat the particle distribution would need to be computed

522 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Bockmann et al.

on an extremely fine quadrature grid. This fact producesa new problem if there is noise in the data. On the otherhand, if the absorption is strong, i.e., the imaginary partof the refractive index is large, see Fig. 1(e), the volumebackscatter kernel is smooth. In contrast to the back-scatter volume kernel, the Mie extinction volume kernelis very smooth in both cases with and without absorption;see Figs. 1(d) and 1(f). Therefore one expects larger val-ues for p.

Moreover, ill-posed problems are often divided into dif-ferent classes with respect to the degree of ill-posedness.We present those degrees for the operators [Eq. (4)] laterin this paper. Typically, the singular values follow a har-monic progression m i . i2t or a geometric progressionm i . exp(2t i), where t is a positive real constant. Thedecay rate of the singular values m i is so fundamental forthe behavior of ill-posed problems that it makes sense touse this decay rate to characterize the degree of ill-posedness of the problem. References 22 and 27 give thefollowing definition: If there exists a positive real num-ber t such that the singular values satisfy m i 5 O(i2t),then t is called the degree of ill-posedness. The problem

Fig. 2. Qualitative approximations to six right singular func-tions vi for i 5 1, 5, 10, 15, 20, 25 of the volume backscatter ker-nel, (a) m1 5 1.5 1 0.0i and (b) m2 5 1.5 1 0.5i. We see thetypical behavior of increasing oscillations.

is characterized as mildly or moderately ill-posed if t< 1 or t . 1, respectively. On the other hand, if m i5 O@exp(2t i)#, i.e., the singular values decay very rap-idly, then the problem is termed severely ill-posed.

SVE is a powerful analysis tool, but unfortunately ananalytical solution exists only for a limited number ofcases. However, approximations to SVE can always becomputed numerically when Eqs. (2) and (4) are dis-cretized by means of the Galerkin method followed bycomputation of the SVD of the matrix obtained in thisway.28 The same statement applies to the analytical de-termination of the numbers p or t, respectively, for the li-dar kernels in Eqs. (2) and (4). If at all possible, it is ahard task. For that reason, we decide to deal with nu-merical approximations for the number t, too.

Before we determine numerically the degree of ill-posedness for the operators [Eq. (4)], we give some refer-ences as to how to compute the singular values and thevalues t numerically. One chooses orthonormal bases$f1 ,...,fn% and $ c1 ,...,cn% in the spaces L2(Il) andL2(Ir), respectively, and defines a matrix A with elements

~A !ij 5 ^f i , Kc j&, i, j 5 1,...,n. (7)

The expression ^•, •& denotes the scalar product in the realspace L2(Il) and K represents the integral operator.Then the SVD of A immediately gives approximations tothe SVE of the kernel. Let A P Rn3n be a square matrix.The SVD of A is a decomposition of the form

A 5 UDVT 5 (i51

n

s iuiviT , (8)

where U 5 (u1 ,...,un), V 5 (v1 ,...,vn) P Rn3n are ma-trices with orthonormal columns and the diagonal matrixD 5 diag( s1 ,...,sn) has nonnegative diagonal elementsthat appear in nonincreasing order such that s1 > s2> ... > sn > 0. The numbers s i . 0 are the singular

values of A, whereas the vectors ui and vi are the left andright singular vectors of A, respectively. In connectionwith discrete ill-posed problems, two characteristic fea-tures of the SVD are very often found. First, the singularvalues decay gradually to zero with no particular gap inthe spectra. An increase of the dimension of A will onlyincrease the number of small singular values as, for ex-ample, in our application to lidar data; see Fig. 3. Thatmeans almost all information of the operator is containedin the first few summands of Eq. (6); i.e., the discretiza-tion dimension of Eq. (4) can be kept low. Second, the leftand right singular vectors tend to have more sign changesin their elements as the index i increases, i.e., s i de-creases; see Fig. 2. This means that only the first fewsingular vectors are needed for retrieving a good numeri-cal approximation of the volume distribution without highoscillations. Both features are consequences of the factthat the SVD of A is closely related to the SVE of the un-derlying kernel.24,29,30

The singular values m j of K are approximated by the al-gebraic singular values s j of A. In detail, the n singularvalues s i

(n) of An,n are approximations to the first n sin-gular values m i of the kernel. Moreover, we introduce thefunctions

Bockmann et al. Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. A 523

Fig. 3. Approximations to the singular values and to the degree of ill-posedness of the volume backscatter and extinction kernels Kp/extv ,

(a), (b) without absorption (m1 5 1.5 1 0.0i) and (c), (d) with strong absorption (m2 5 1.5 1 0.5i). The values c(n), n 5 10, 15, 20, 25,30 are the condition numbers of the resulting coefficient matrices; see Eq. (7). The numbers follow from Galerkin discretization independence on the discretization dimension n.

n n

uj~r ! 5 (i51

~U !ijc i~r !, j 5 1,...,n,

vj~l! 5 (i51

n

~V !ijf i~l!, j 5 1,...,n. (9)

Those functions are approximations to the first n left andn right singular functions of the kernel. We compute thedouble integrals in Eq. (7) by Simpson’s numericalquadrature scheme. In that case we can expect that thequadrature errors do not exceed the approximation errorscaused by the base functions. With increasing dimensionn, the singular values s i

(n) (where n is the number of basefunctions) are increasingly better approximations to thetrue singular values m i .28,31 In other words, it holds that

s i~n ! < s i

~n11 ! < m i ,

0 < m i 2 s i~n ! < ~ iKiL2

2 2 iAiF2 !1/2 5.. Dn ,

i 5 1,...,n. (10)

with

(i51

~m i 2 s i~n !!2 < Dn

2. (11)

The true singular values m i of K are bounded by the com-puted singular values s i

(n) as follows: s i(n) < m i

< @(s i(n))2 1 Dn

2 #1/2. If Dn tends to 0 for n → `, the ap-proximated singular values s i

(n) converge uniformly in nto the true singular values m i ; see Figs. 3(a)–3(d).

Now the intervals Ir and Il are each divided into n sub-intervals $Ir

(i)% and $Il(i)% of the same lengths hr and hl ,

respectively, and the basis functions are then given by

c i~r ! 5 H hr21/2 : r P Ir

~i ! i 5 1,...,n

0: otherwise,

f i~l! 5 H hl21/2 : l P Il

~i ! , i 5 1,...,n

0: otherwise.

(12)

Let the lower and upper limits of particle radius and laserwavelength r0 5 0.001 mm, r1 5 5 mm, l0 5 300 nm,and l1 5 1100 nm, which are real-life domains with re-

524 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Bockmann et al.

spect to application in research with lidar; see Section 1.Then the formulas give second-order approximations tothe singular values m i ; see Fig. 3.

Applying the theoretical study, we determined the con-dition numbers24 c(n) 5 iAn,ni iAn,n

21 i (n 5 10, 15, 20,25, 30) and, by a numerically weighted nonlinear least-squares fit, an approximation to the degree of ill-posedness for the two volume kernels of Eq. (4) for twoqualitatively different refractive indices m1 5 1.5 andm2 5 1.5 1 0.5i; see Figs. 3(a)–3(d). In general, the vol-ume kernels are moderately ill-posed, since t is between2.25 and 9.10. A closer look shows that the degree of ill-posedness of the extinction volume kernel is higher, seeFigs. 3(b) and 3(d), than that of the backscatter volumekernel; see Figs. 3(a) and 3(c). Moreover, if the absorp-tion of the particles is strong, then the degree of ill-posedness grows rapidly; see Figs. 3(c) and 3(d). Realiz-ing the logarithmic scale in Fig. 3(d), one can see that thesingular values are almost located on a straight line.Therefore this extinction volume kernel with strong par-ticle absorption is almost severely ill-posed. As one ex-pects, the condition numbers grow with n, and they showthe same behavior as the degree for fixed n. The matri-ces are always highly ill-conditioned.

Figures 2(a) and 2(b) show qualitatively six approxima-tions of the right singular functions vi , i 5 1, 5, 10, 15,20, 25, of the volume backscatter kernel and the typicalbehavior that the higher the index i, the more compo-nents with high frequencies are present in vi . Owing tothe higher degree of ill-posedness, it seems somewhatlikely that the behavior of the oscillations is more un-structured in the case of strong particle absorption; seeFig. 2(b).

Finally, with respect to the evaluation of lidar measure-ments it is necessary to know how the degree of ill-posedness depends on the real and imaginary parts of therefractive index of the particles, m 5 mR 1 mIi. Thisdependence is shown in Fig. 4. On the one hand, there isno significant influence of the real part in the real-life do-main between 1.3 and 1.7; see Fig. 4(a). On the otherhand, the imaginary part in the domain between 0 and0.5, which determines particle absorption, has a very im-portant influence on the degree of ill-posedness. The de-gree increases rapidly between mI 5 0 and mI 5 0.25;see Fig. 4(b).

4. SIMULATION AND MEASUREMENTRESULTSWe have divided our extensive simulation study into twoparts. First, we assume that the refractive index isknown for the inversion of the synthetic lidar data. Theinvestigation is concentrated on an error analysis of theretrieval of microphysical parameters. The second partof the investigation focuses on the retrieval errors in thecase where the refractive index is assumed unknown forthe inversion. Monomodal logarithmic-normal distribu-tions are used for both cases to describe the particle-number size distributions; see Eq. (1). In the last part ofthis section we present the result on a measurement case.

The total number concentration Nt was set to 1 for allsize distributions used for the simulation study, since itonly scales the values of all derived optical properties inan equal way. The values s and rmed as well as the cor-responding integral properties are given in Table 1.

To give an impression of how the simulation works, wepresent in Fig. 5 a flow chart for the second case with un-known refractive index. For the first case with known in-dex, the process is similar.

A. Simulations with Known Refractive IndexBy combining the size distributions given in Table 1 withthe real parts 1.4, 1.55, and 1.7 and the imaginary parts0.0, 0.005, 0.01, 0.05, 0.1, and 0.5 of the refractive index,we calculated 90 synthetic lidar data sets for the ‘‘3 1 2case’’ and the ‘‘6 1 2 case.’’ The 3 1 2 case stands forbackscatter coefficients at l 5 355, 532, and 1064 nm andextinction coefficients at l 5 355 and 532 nm, and the 61 2 case stands for backscatter coefficients at l 5 355,400, 532, 710, 800, and 1064 nm and extinction coeffi-cients at l 5 355 and 532 nm. For the inversion the syn-thetic lidar data were imposed with 0% (exact case), 5%,10%, 15%, and 20% of normally distributed noise. Foreach noise level, ten different inversions were performedto increase the statistical significance.

Figure 6 shows the mean relative errors of the invertedparameters as a function of the noise level for the inver-sion with known refractive index. The uncertainty barscorrespond to the standard deviation. In the noiselesscase and for 6 1 2 coefficients, the errors are approxi-mately 2.5% for effective radius, surface-area concentra-

Fig. 4. Degree of ill-posedness of the backscatter and extinction volume kernels in dependence on the real and imaginary parts of therefractive index. There is (a) no significant influence of the real part but (b) significant influence of the imaginary part.

Bockmann et al. Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. A 525

tion, and volume concentration and approximately 0.2%for the single-scattering albedo. The single-scattering al-bedo is mainly influenced by the imaginary part of the re-fractive index, which is known in the present case. Forthe case of 3 1 2 coefficients, the errors slightly increasebut stay below 5% and 0.4%. With increasing noise onthe data, the retrieval errors grow. In the case of 20%noise, the uncertainty is approximately 15% and 2% forthe integrated microphysical properties and the single-scattering albedo, respectively, for both the 3 1 2 caseand the 6 1 2 case. It can be seen that—as soon as thelidar data become erroneous—the three additional back-

Fig. 5. Simulation scheme for the case of unknown refractive in-dex. ssa, single-scattering albedo.

Fig. 6. Mean relative errors of the particle properties derivedwith known refractive index for the 3 1 2 case (circles) and the6 1 2 case (squares). The uncertainty bars correspond to thestandard deviation.

Table 1. Log-Normal Distributions Used for theSimulations

Properties Values

Median Radiusrmed (mm)

0.05 0.1 0.1 0.3 0.1

Mode width s 1.8 1.4 1.8 1.6 1.6Effective radius reff (mm) 0.12 0.13 0.24 0.52 0.17Surface-area concentration

at (mm2 cm23)0.0632 0.158 0.251 1.757 0.196

Volume concentrationvt (mm3 cm23)

0.0025 0.007 0.020 0.304 0.011

scatter coefficients of the 6 1 2 case do not significantlylower the errors of the inversion results.

Figure 7 shows, for the example of the volume concen-tration, another feature of the retrieval error, namely,that it depends strongly on the imaginary part of the re-fractive index. It stays below 3% up to an imaginary partof '0.15, after which it rapidly increases. After reachinga maximum of '14% for an imaginary part of '0.25, itdecreases again and levels off at '8%. In comparingFigs. 4(b) and 7, one notices that the shape of the curvesis similar; i.e., there exists, as one suspects, a correlationbetween the degree of ill-posedness and the retrieval er-ror. The break between 0.2 and 0.25 occurs very clearlyin both curves. Therefore one has to be careful about in-verting optical data for cases of highly absorbing par-ticles.

Finally, we have to remark that we could not find a sig-nificant influence of different real parts of the refractiveindex on the results in the simulation study. This findingcorrelates with the behavior of the degree of ill-posednessin Fig. 4(a).

B. Simulations with Unknown Refractive IndexSince the retrieval errors do not depend significantly onthe real part of the refractive index, we used for this partof the simulation study only one unknown real part,namely, 1.55. Noise levels of 0%, 5%, and 10% were in-vestigated. We consider the 6 1 2 case first. In thenoiseless case the retrieval errors vary around 10% for ef-fective radius, volume concentration, and the imaginarypart of the mean refractive index. For the single-scattering albedo the error is less than 3%; see Figs. 8 and9. For a 10% noise level the retrieval errors are between20% and 30% for effective radius, volume concentration,and imaginary part of the refractive index and around 6%for the single-scattering albedo. For the 3 1 2 case andnoiseless data, the retrieval errors vary from 10%–20%for effective radius, volume concentration, and the imagi-nary part of the refractive index. For the single-scattering albedo the error again is less than 3%; see Figs.8 and 9. For a noise level of 10% the retrieval errors are20%–30% for effective radius and the volume concentra-tion, whereas it is approximately 35% for the imaginarypart. The error for the single-scattering albedo is around6%.

Fig. 7. Dependence of retrieval errors on the imaginary part ofthe refractive index without input noise.

526 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Bockmann et al.

We find the same result with respect to the importanceof the three additional backscatter coefficients as in theprevious case with known refractive index. As soon asthe lidar data are erroneous, the three additional back-scatter coefficients of the 6 1 2 case do not significantlylower the errors of the inversion results. But it must beemphasized that for the case of unknown refractive indexthe noise level has to be limited rather well below 10% orelse the hybrid regularization method would become un-stable again.

C. Results from Measurement DataThe performance of the algorithm was finally tested for aset of experimental data. The data were obtained fromRaman lidar observations of an aerosol plume observed atthe Institute for Tropospheric Research (IfT) in Leipzig,Germany, on 8 April 2002. A detailed description of thismeasurement is found in Ref. 32. The aerosol Ramanlidar4 provides backscatter coefficients at 355, 532, and1064 nm and extinction coefficients at 355 and 532 nm.Examples of the optical profiles are presented in Ref. 33.

For the inversion we selected optical data in the heightrange of 1800–2700 m. Optical depth in this heightrange was 0.28 at 355 nm and 0.13 at 532 nm. The par-ticle angstrom exponent describing the extinction spec-trum within the wavelength interval 355–532 nm was ap-proximately 1.9. The mean extinction coefficient was0.306 km21 at 355 nm and 0.141 km21 at 532 nm. Thebackscatter coefficients were 0.006 km21 sr21 at 355 nm,0.003 km21 sr21 at 532 nm, and 0.001 km21 sr21 at 1064nm wavelengths.

Figure 10 presents the solution domain of the refrac-tive index. The result with respect to the imaginary part

Fig. 8. Mean relative errors of the properties inverted with un-known refractive index for the 3 1 2 case (circles) and the 61 2 case (squares). The uncertainty bars correspond to thestandard deviation.

Fig. 9. Mean relative errors of the inverted refractive indices forthe 3 1 2 case (circles) and the 6 1 2 case (squares). The un-certainty bars correspond to the standard deviation.

of the refractive index is visualized on the horizontal axis.The trace shows the retrieval error for different values ofthe imaginary part assumed in the inversion. There is aclear minimum range around 0.005i. This value was ac-cepted as the result. Table 2 summarizes the results forselected parameters. Microphysical parameters obtainedwith another inversion scheme used at IfT, see Ref. 33,show comparable values for total volume concentration(33 6 11 mm3/cm3) and the imaginary part of the refrac-tive index (0.003 6 0.003). Although the effective ra-dius (0.19 6 0.04 mm) determined with the IfT algorithmis larger, the single-scattering albedo, which is a key pa-rameter regarding radiative forcing of aerosol particles,agrees very well (0.97 6 0.03); see Table 2.

From this comparison we may conclude that the IMPmethod can be used for the retrieval of microphysical pa-rameters on the basis of a minimum set of measurementwavelengths, i.e., backscatter coefficients at three wave-lengths and extinction coefficients at two wavelengths.

5. CONCLUSIONOur extensive mathematical and simulation analysis ofthe IMP hybrid regularization technique shows that it ispossible to derive microphysical aerosol parameters byusing a data set that consists of a minimum of particlebackscatter coefficients at three wavelengths (355, 532,and 1064 nm) and extinction coefficients at two wave-lengths (355 and 532 nm).

Fig. 10. Solution domain of the refractive index for the mea-surement case.

Table 2. Retrieval Results for the MeasurementCase from April 8, 2002, Carried Out with the

Raman Lidar at IfTa

Properties Results

Mean refractive indexReal part (minimum solutionb) 1.48 6 0.02 (1.454)Imaginary part (minimum

solution)0.0053 6 0.0025 (0.0037)

Effective radius 0.07 6 0.002 mmTotal volume concentration 42 6 3 mm3/cm3

Single-scattering albedo (at 532 nm) 0.96 6 0.02

a Measurement time was 1823–1922 UTC.b For an explanation, see Appendix A.

Bockmann et al. Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. A 527

Fig. 11. Screen shot of the developed software. The plot in front shows the result browser with results from a simulation. The plotin the background provides a partial view on the main window.

Our theoretical investigations show that the backscat-ter coefficients are very important and useful in the inver-sion process, because of the better degree of ill-posednessof the underlying equations and, therefore cannot beomitted. Although the degree for the extinction coeffi-cients is worse, we found that these coefficients stabilizethe inversion process. Additionally, our theoretical ex-aminations prove our findings of the simulations, that theinversion process becomes more complicated in cases withstrongly absorbing particles.

In the realistic simulation case with unknown refrac-tive index and 3 1 2 coefficients, we assumed correctdata as well as errors up to 10%. Uncertainties in the re-trieved size parameters (effective radius, volume concen-tration) are approximately 15% and 25%, respectively,those of the real part of the refractive index are 2% and3%, respectively, and those of the imaginary part are 15%and 35%, respectively. The single-scattering albedo wasretrieved to an uncertainty of 3% and 6%, respectively.The inversion of an experimental data set showed perfor-mance characteristics of the inversion algorithm similarto those found during the simulations with syntheticdata. For that reason, we assume that the code workedwell under these realistic conditions.

APPENDIX AWe developed a software package (alpha version) to calcu-late microphysical aerosol parameters from measure-ments or simulated data. At the beginning of a simula-tion input, every parameter (wavelength, refractiveindex, type of aerosol, number distribution, noise, grid pa-rameter; see also Fig. 5) is fed into the code through the

parameter interface. Noise can be added to syntheticdata, and multiple runs are possible. Any amount of cal-culations can be specified in order to keep the computerbusy over several days. The software writes checkpointsregularly, so the calculation state can be recovered in thecase of a crash. At any time the calculation can bestopped and the current state can be saved so that the cal-culations can be resumed later. If the calculation is fin-ished, a result browser provides a visual evaluation of theresults. Figure 11 shows a screen shot with results of acalculation on a 40 3 40 grid for the complex refractiveindex. In the case of such a calculated grid (case withunknown refractive index), the result is shown in terms ofa matrix-like result plot, such as the one shown in theleft-hand side of Fig. 11. The right-hand plot insert de-picts the volume distribution function (minimum solu-tion) from the corresponding minimum point in the ma-trix. This minimum point is defined as the best gridpoint in the following sense: After backcalculation of theretrieved regularized solutions at each grid point, the bestone is that point where the absolute or relative error inrelation to the input data is smallest. A number of pointsfrom the matrix (the solution domain; see also Fig. 10)can be specified, from which then the mean refractive in-dex, the microphysical parameters, and the single-scattering albedo are recalculated.

ACKNOWLEDGMENTSFinancial support of this work by the European Commis-sion under grant EVR1-CT-1999-40003 and by the Helm-holtz Association within the Virtual Institute under grantVH-VI-100 is gratefully acknowledged.

528 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Bockmann et al.

Corresponding author Christine Bockmann can bereached by phone, 49 331 977 1743; fax, 49 331 977 1001;or e-mail, bockmann@rz.uni-potsdam.de.

*Current address, Department of Earth Physics, Insti-tute of Physics, University of St. Petersburg, Uljan-ovskaja 1, 198504 St. Petersburg, Russia.

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