Atmospheric aerosols play an important role in manyatmospheric processes. Although only a minor con-stituent of the earth’s atmosphere, they have appre-ciable influence on the earth’s radiation budget, airquality, clouds, and precipitation as well as the chem-istry of the troposphere and stratosphere. Scatter-ing and absorption of incoming solar radiation andlong-wave terrestrial radiation by particles cause di-rect climate forcing, whereas indirect climate forcingcan be attributed to the influence of particles on thesize distribution of cloud droplets, thus changingtheir optical properties and lifetime.1–5 Many ef-ects are not well understood because of the multi-ude of influence factors and feedback mechanisms.or a better understanding of the importance of at-ospheric particles an investigation of the spatial
nd the temporal variability of their chemical and
The authors are with the Institute for Tropospheric Research,Permoserstrasse 15, 04318 Leipzig, Germany. The e-mail ad-dress for D. Muller is [email protected].
Received 2 July 1998; revised manuscript received 6 November1998.
2346 APPLIED OPTICS y Vol. 38, No. 12 y 20 April 1999
hysical properties—parameters describing theirean size, their volume or mass, surface-area, andumber concentrations, and their complex refractive
ndex—is needed.6–8
To gain information on aerosol particle parame-ters, two lidar systems and a data-retrieval schemehave been developed at the Institute for TroposphericResearch. The first unit is a transportable multiple-wavelength lidar.9 Two Nd:YAG and two dye lasersemit pulses simultaneously at 355, 400, 532, 710,800, and 1064 nm. The elastically backscatteredsignals at these six wavelengths—at 532 nm withpolarization discrimination—and the Raman signalsof nitrogen at 387 and 607 nm and of water vapor at660 nm are measured. Profiles of the particle back-scatter coefficients at the six emitted wavelengths,the particle extinction coefficient at 355 and 532 nm,the depolarization ratio at 532 nm, as well as thewater–vapor mixing ratio are derived from the de-tected signals. The second system is a stationarymultiple-wavelength Raman lidar. It consists of oneNd:YAG laser that emits at 355, 532, and 1064 nm.The elastically backscattered signals, again with po-larization discrimination at 532 nm, the nitrogen Ra-man signals at 387 and 607 nm, and the water–vaporRaman signal at 407 nm are detected. From thesesignals, backscatter coefficients at three wave-
aTl
m
itttst
lengths, extinction coefficients at 355 and 532 nm, thedepolarization ratio at 532 nm, and the water–vapormixing ratio are derived. An important feature ofthe two systems is use of the Raman-lidar techniquethat allows the independent determination of par-ticle backscatter and particle extinction coefficients.10
To derive physical particle properties, informationon the spectral behavior of the backscatter coefficientat six wavelengths and of the extinction coefficient attwo wavelengths is inverted. The optical data arerelated to the physical quantities by Fredholm inte-gral equations of the first kind:
b~l! 5 *0
`
Kb~r, m, l, s! f ~r!dr, (1)
a~l! 5 *0
`
Ka~r, m, l, s! f ~r!dr, (2)
where b~l! denotes the backscatter coefficient atwavelength l, a~l! is the respective extinction coeffi-cient, f ~r! describes the concentration of particles perradius interval dr and can be expressed in terms ofthe distribution of number, surface-area, or volumeconcentration. Kb~r, m, l, s! und Ka~r, m, l, s! arethe kernel efficiencies of backscatter and extinction,respectively. They depend on the radius r of theparticles, their complex refractive index m, the wave-length l of the interacting light, as well as the shapes of the particles.
Equations ~1! and ~2! cannot be solved analyticallyand require the application of specific mathematicalmethods.11–14 The numerical solution of theequations13–17 leads to the so-called ill-posed inverseproblem.18 The problem is characterized by incom-pleteness of the available information, the non-uniqueness of the solutions, and the noncontinuousdependence of the solutions on the available data.Additional difficulties emerging from lidar measure-ments—only a few optical data, mainly in terms ofbackscatter coefficients, are available, large measure-ment errors may occur, and the particle’s refractiveindex is an additional unknown—have so far prohib-ited any applicable procedure for the explicit retrievalof tropospheric particle properties from lidar data.
From the wide spectrum of applications and thecomplexity of the inverse problem with respect toEqs. ~1! and ~2!, a great number of different inversionlgorithms and solution approaches are offered.hese approaches include attempts to solve the prob-
em exactly or approximately analytically,19–36 theinversion method by Backus and Gilbert,37–39 least-squares-fit methods with constraints,40–58 numericfilter methods,12 nonlinear iterative methods,59–61
the principle of quasi-reversibility,62 the method ofregularization with constraints14,63–69 that is basedon earlier research40 and suggestions,11,13,70,71 meth-ods of nonlinear programming with physical con-straints,72,73 the principle of maximum entropy,74–76
the extreme value estimation,77,78 and the mollifierethod.79
As far as these methods are concerned, only a fewapplications to optical data derived from lidar mea-surements are known. The basic potential of themethod of inversion through regularization has beenshown.65,66 Other explicit inversion procedures forlidar applications that are being developed34,36,79 fo-cus on stratospheric aerosols that are much less vari-able in their physical and chemical properties thantropospheric aerosols. A few approaches are re-stricted to the sole reproduction of the measured op-tical data through comparison with theoreticallycalculated data. This reproduction is done either byadjustment of the parameters of a particle size dis-tribution of predetermined shape46,47,51,53,55 or adjust-ment of an arbitrarily shaped size distribution bystatistical methods.43,44 Under certain assump-tions, e.g., the combined use of extinction and back-scatter coefficients and a known complex refractiveindex, acceptable results may be derived. The dis-advantages of such methods are that usually an un-satisfying large number of different solutions existwith which the measurement data can be reproducedwithin error limits, that no differentiation betweenmathematically and physically caused errors is pos-sible, that an error estimation only on the basis ofextensive simulations is possible, and that a series ofa priori assumptions with respect to the solution arenecessary. Therefore the application of these meth-ods to experimental data has also been restricted tostratospheric particles for which the refractive indexand the shape of the size distribution are well known.
We propose here, based on evaluation of the prop-erties of the aforementioned approaches, the conceptof inversion with regularization with constraints asthe most favorable concept for the explicit retrieval oftropospheric particle parameters from lidar data.80
Regularization is performed by generalized cross-validation,81,82 a method that so far has not beenapplied to lidar data. The solution of Eqs. ~1! and ~2!s given for a mean complex refractive index and,aking into account the available wavelengths, inerms of the mean distribution of the volume concen-ration. The latter was chosen because of the moretable behavior of the inversion compared with, e.g.,he number concentration.43,44,66,68,69 The better
stability is explained by the fact that transformationto a higher-order representation shifts the maximumsensitivity of the used kernel efficiencies further intothe optically active size range of the investigated par-ticle size distributions.36,43,44,66,83
Special attention was paid to optimizing the algo-rithm for a small amount of optical data, to allowingfor large measurement errors that can be expectedfrom lidar data, and to studying the meaning of thecombined effect of extinction and backscatter data.
A mathematical description of the inversion prob-lem follows in Section 2. There also tools such asbase functions and smoothing are discussed. Themethod of regularization that we chose is described inSection 3. A few simulation results are summarizedin Section 4. An extended simulation study and the
20 April 1999 y Vol. 38, No. 12 y APPLIED OPTICS 2347
p
ItpEf
2
application of the method to lidar measurements arepresented in two forthcoming papers.
2. Inversion Problem
Equations ~1! and ~2! are formulated into a morespecific form:
gi~l! 5 *rmin
rmax
Ki~r, m, l, s!v~r!dr 1 eiexp~l!,
i 5 b, a; l 5 0.355, . . . , 1.064 mm, (3)
where gi~l! are the optical data. In the case of ex-erimental values these data have an error of ei
exp~l!.The i term denotes the kind of information, i.e.,whether it is backscatter b or extinction a. The dataare available at discrete wavelengths l, in this casethe six lidar wavelengths. The v~r! term is the vol-ume concentration distribution. The lower integra-tion limit is defined by rmin, and the upper limit isdetermined by rmax. The choice of respective valuesis described in Appendix A. Considering sphericalparticles, the volume representation of the kernelfunctions36,43,44,84 Ki~r, m, l, s! is calculated from therespective extinction and backscatter efficiencies84
Qi~r, m, l! and the geometric cross section pr2 as
Ki~r, m, l, s! 534r
Qi~r, m, l!. (4)
ntroduction of the subscript p 5 ~i, l! summarizeshe kind and the number of optical data. In theresent case the subscript takes values from 1 to 8.quations ~3! and ~4! are then written in modified
orm as
gp 5 *rmin
rmax
Kp~r, m!v~r!dr 1 epexp, (5)
Kp~r, m! 534r
Qp~r, m!. (6)
Equation ~5! must be explicitly solved for the volumeconcentration distribution. The mean complex re-fractive index implicitly follows from that procedure~see Appendix A!. The distribution v~r! is approxi-mated by a linear combination of base functions Bj~r!,also denoted as B-spline functions, and weight factorswj:
v~r! 5 (j
wjBj~r! 1 emath~r!. (7)
The right-hand side of Eq. ~7! contains the mathe-matical residual error emath~r! that is caused by theapproximation, and j is the subscript for the neces-sary base functions and weight factors. Base func-tions stabilize the inversion process28,37,38,66,68,69,85 inthe way that a physically reasonable subspace is se-lected from the mathematically acceptable solutionspace. In the next step one must find a sensible
348 APPLIED OPTICS y Vol. 38, No. 12 y 20 April 1999
approach to the shape and the number of the basefunctions.
It has been found, based on theoretical consider-ations85,86 as well as on simulation results with histo-gram columns, also called B-spline functions of zerodegree,43,44,50 triangle functions, known as B-splines offirst degree,66 and cubic functions, i.e., B-splines ofsecond degree,69 that inversion works best with trian-gle functions defined on a logarithmic-equidistantscale. Figure 1 illustrates the arrangement of thebase functions. The dotted curve shows a logarith-mic-normal distribution function that represents thetypical shape of a monomodal atmospheric particle sizedistribution. The triangle functions adequately takeaccount of this shape. The edges are approximatedfairly well by straight lines; the position of the maxi-mum is well identified.
The number of base functions was chosen to beeight. This amount was determined by physical aswell as mathematical constraints.68,81 On the onehand, it is possible in this way to reconstruct mono-modal and bimodal distributions, because an ade-quate representation of one mode needs a minimumof three base functions,80 and two base functions areneeded to determine the boundaries of the size dis-tribution in the inversion, i.e., rmin and rmax ~see Fig.1 and Appendix A!. On the other hand, the inver-sion problem remains unstable if too many base func-tions are used ~see Subsection 3.C!.44,87
The mathematical formulation of the base func-tions is
B1~r! 5 50 r , r1
1 2r 2 r1
r2 2 r1r1 # r # r2
0 r . r2
,
Fig. 1. Inversion window and base functions therein ~solid lines!.Dotted curve, monomodal, logarithmic-normal size distributionfunction.
I
t
w
T
w
Bj~r! 5 50 r # rj21
1 2rj 2 r
rj 2 rj21rj21 , r # rj
1 2r 2 rj
rj11 2 rj
rj , r # rj11
0 r . rj11
j 5 2, . . . ,7,
B8~r! 5 50 r , r7
1 2r8 2 rr8 2 r7
r7 # r # r8
0 r . r8
. (8)
The outer supports r1 and r8, corresponding to rminand rmax in Eq. ~5!, limit the position of the inversionwindow and cannot be determined a priori.68,83
Therefore a gliding inversion window with a variablewindow width must be used ~see Appendix A!.
Finally the weight factors must be determined.nserting Eq. ~7! into Eq. ~5! yields
gp 5 (j
Apj~m!wj 1 ep. (9)
Apj~m! is calculated from the respective kernel func-ion Kp~r, m! and the base function Bj~r! as
Apj~m! 5 *rmin
rmax
Kp~r, m!Bj~r!dr. (10)
ep 5 epexp 1 ep
math is the sum of experimental andmathematical errors. By writing the optical datainto a vector g 5 @gp#, the weight factors into a vector
5 @wj#, and the errors into a vector e 5 @ep#, onemay convert Eq. ~9! into a vector–matrix equation:
g 5 Aw 1 e. (11)
he matrix A 5 @Apj#, the elements of which are givenby Eq. ~10!, is called the weight matrix.
The solution of Eq. ~11! for the weight factors thatare investigated then follows:
w 5 A21g 1 e*, (12)
herein e* 5 2A21e describes the respective errorsand A21 denotes the inverse of matrix A. As exten-sively discussed in the literature14 the solution spacethat follows from Eq. ~12! still contains unstable, i.e.,strongly oscillating, solutions, although the opticaldata can be reproduced within error limits e. Anexplanation for this instability is given by the anal-ysis of the elements of A and A21 that show a highdynamic range across several orders of magnitude.The mechanism of inversion is the reconstruction ofthe investigated distribution by means of the eigen-vectors of A21. The contribution of the different eig-envectors is determined by the respectiveeigenvalues of A21. Because small eigenvalues of Acorrespond to large eigenvalues of A21, it follows thatin the case of erroneous data the error in the recon-structed distribution can be amplified by many ordersof magnitude if small eigenvalues are used for recon-
struction. At this stage regularization is introducedto suppress small eigenvalues and thus unstable so-lutions.
3. Regularization
A. Methodology
As a first step one demands that those solutions befound for which e in Eq. ~11! becomes a minimum, orto be precise, for which the distance between vectorAw and the vector of the optical data g becomessmaller than a predetermined value. This stepleads to the so-called minimization concept, also de-noted as the method of minimum distance.13,14 Thepenalty function e2 gives the maximum acceptabledistance between Aw and g and is defined by thesimple Euclidian norm i z i as
e2 $ iei2 5 iAw 2 gi2. (13)
Further physically reasonable constraints intensifythe minimization request in a second step. Theseconstraints are the demand for a smooth11,14,66,67,69
and positive solution67,88 as well as the reproducibil-ity of the input optical data. Use of the latter twoconstraints is described in Appendix A.
The constraint of smoothness is implemented inEq. ~13! through an additional penalty term G~v!, andthe new minimization problem is written as
e2 $ iAw 2 gi2 1 gG~v!. (14)
G~v! is a nonnegative scalar measure for the deviationof the inverted particle size distribution v~r! from therequested smoothness, and g is the so-called La-grange multiplier and is discussed extensively below.
The smoothness is mathematically defined by aquadratic combination of the weight factors wj andthus by a quadratic form of v~r! ~Ref. 14!:
G~v! 5 wTHw, (15)
where wT denotes the transpose of vector w and H isa band matrix with bandwidth one. Smoothing isperformed over three base functions of the respectiveinversion window.68 In the case of eight base func-tions H is written as14
The strength of smoothness is determined by the La-grange multiplier g; g can take values between 0 and`. Its mechanism is illustrated in Fig. 2. For g3` the minimization process leads to G~v! 5 0, i.e., aperfectly smooth solution v~r!. However, the result
20 April 1999 y Vol. 38, No. 12 y APPLIED OPTICS 2349
toe
tpgc
Gt~
tA
h
2
is always the same, independent of g. Accordingly alarge value for iAw 2 gi2 and in summary a largeresidual error follow. For g 5 0 there is no addi-ional smoothing other than that caused by the choicef base functions. Theoretically, if no experimentalrrors are introduced, the value for iAw 2 gi2 is
determined in this case only by round-off errors in thecalculation of the elements of A, w, and g. But theseround-off errors create the aforementioned oscilla-tions. Values of g that lie between create solutionsfor which the oscillating behavior is more or less pe-nalized by G~v! and thus suppressed. One choosesthat solution for which the complete penalty function,i.e., the sum iAw 2 gi2 1 gG~v!, takes a minimum.One must keep in mind that usually this v~r! thatminimizes iAw 2 gi2 does not simultaneously mini-mize G~v!, which means that the chosen solution withg Þ 0 gives larger values for the quadratic norm of theresiduum iAw 2 gi2 than the simple solution of theminimum distance given by Eq. ~13!, i.e., for g 5 0.
Figure 3 qualitatively illustrates the effect of g inhe inversion for a monomodal logarithmic-normalarticle size distribution.80 Curve ~D! shows theiven distribution from which the optical data werealculated. In the inversion, g values of ~A! 1022,
causing insufficient smoothing and thus oscillations,~B! 102, causing in this case appropriate smoothing,and ~C! 105, leading to strong oversmoothing, wereused. In Subsection 3.B a method is proposed todetermine the degree of necessary smoothing.
Mathematical implementation of the minimizationconcept in the inversion procedure is as follows.14
Inequality ~14! is replaced by an equation wherein~v! is expressed by Eq. ~15!. By use of the respec-ive transposed vectors, iAw 2 gi2 is written asAw 2 g!T ~Aw 2 g!. From Eq. ~14! then follows the
expression ~Aw 2 g!T ~Aw 2 g! 1 gwTHw of whichhe absolute minimum must be determined. WithT describing the transpose of A one finally obtains
the weight vector w of inversion by regularization:
w 5 ~ATA 1 gH!21ATg. (17)
Fig. 2. Qualitative illustration of the minimization concept:solid curve, penalty function iAw 2 gi2 of Eq. ~14!; dashed curve,penalty function G~v!; dotted curve, term iAw 2 gi2 1 gG~v! that
as a minimum in the range 0 # g , `.
350 APPLIED OPTICS y Vol. 38, No. 12 y 20 April 1999
With Eq. ~17! those eigenvalues of matrix A21 in Eq.~12! that lead to oscillations in the solution are sup-pressed. In other words, the inverse problem is sta-bilized by matrix ~ATA 1 gH!21 AT.
The strength of regularization depends on value g.The next step therefore is to choose the Lagrangemultiplier accordingly. In this context one mustkeep in mind that the range of values for g varies overseveral orders of magnitude66,69 in the inversion ofbackscatter and extinction coefficients.
B. Choice of Lagrange Multiplier
With respect to the concept of the best-possible objec-tive choice of the Lagrange multiplier a multitude ofmethods are offered.76,81,82,85,89 These methods dif-fer in requirements concerning the necessary numberand kind of input data, the maximum acceptable dataerror, as well as the choice of additional assumptionsthat lead to solutions, e.g., knowledge of the expectedmeasurement error or the relationship of the datawithin a data set. Of the four important ap-proaches, i.e., the method of maximum likelihood, themethod of minimum discrepancy, the Bayesian ap-proach, and the method of generalized cross-validation ~GCV!,16,17,81,82,85,86,90–92 the latter waschosen because it requires the fewest assumptions.
A few applications of GCV concerning optical datahave already been presented.67–69,89 GCV is theonly method that explicitly takes account of the rel-ative behavior of the respective data to one another.GCV needs neither an a priori estimation of theexpected error in the data, as is the case with themethods of minimum discrepancy and maximumlikelihood, nor an a priori assumption concerning thesolution andyor the statistical and systematic errorsas in the Bayesian approach. GCV shows a smallertendency toward oversmoothing compared with the
Fig. 3. Effect of the value of the Lagrange multiplier g on thequality of the inversion. Three different inversion results areshown for a given monomodal particle distribution @curve ~D!# andLagrange multipliers with values of ~A! g 5 1022, ~B! g 5 102, ~C!g 5 105.
tao
pl
I
pa
pfav
mit
w8tion2
o
methods of maximum likelihood and minimum dis-crepancy86 and does not react as sensitively towardstatistical and systematic errors81 that can well reachthe order of 20% in the case of lidar data. Theoret-ical approaches that a priori answer the question ofhe minimum number of needed data are not avail-ble. How many data are needed depends not onlyn the regularization method87,93,94 but also on the
measurement principle66,68,69,87,93 and thus on the de-gree of ill-posedness. Simulations with a variablenumber of backscatter and extinction data showedthat GCV may be applied to the limited availabledata set of six backscatter and two extinction coeffi-cients.80
In Subsection 3.C the mechanism of GCV as well asits mathematical formulation is described. A briefoverview of some properties of the parameter of GCVthat followed from respective simulations80 concernsthe choice of appropriate inversion windows, thenumber of base functions ~see Section 2!, as well asthe effect of the combination of backscatter and ex-tinction data. A detailed derivation of GCV is givenelsewhere.81,82
C. Generalized Cross-Validation
One starts with a set of k data points of data vector g@see Eq. ~3!#. Let w~p!~g! be the weight vector’s de-
endence on the sought Lagrange multiplier that fol-ows from Eq. ~17! when the pth data point ~p 5
1, . . . , k! in g is omitted. GCV shows that if thevalue for g is suitably chosen, the pth component@Aw~p!~g!#p of Aw~p!~g! should approximately predictthe value of gp. This prediction is performed for allavailable k data points. For a sufficiently largerange of the Lagrange multiplier its variation yieldsone g for which reconstruction is most possible for allk data points. The minimization problem that fol-lows81,95,96 leads to a closed expression for calculationof the parameter of GCV PGCV ~also denoted as theGCV parameter!:
PGCV~g! 5
1ki@I 2 M~g!#gi2
$1ktrace@I 2 M~g!#%2
3 min. (18)
describes the unit matrix. The influence matrix,
M~g! 5 A@ATA 1 gH#21AT, (19)
is directly related to the kernel matrix A ~Ref. 85! aswell as to the matrix @ATA 1 gH#21 AT of Eq. ~17!. A
hysical interpretation of the GCV is that it rejectsll these eigenvalues of A, the eigenvectors of which
cause high error amplification but contribute onlyinsignificantly to reconstruction of the sought parti-cle distribution.
In Fig. 4, ~a! and ~b! illustrate the GCV principle interms of the qualitative behavior of the GCV param-eter depending on the Lagrange multiplier g. Sixbackscatter and two extinction data were used for theinversion.80 In Fig. 4, ~a! shows the complete rangecovered by the Lagrange multiplier. For small g val-
ues the GCV parameter at first decreases. In theprocess, strong variations occur, indicating instabil-ity in Eq. ~18!. These variations vanish with a cer-tain value of the Lagrange multiplier, and a stablerange occurs within which the global minimum islocated. This minimum is indicated by a verticalline. In Fig. 4, ~b! shows curve ~a! on a stronglyenhanced resolution of the ordinate axis. Afterpassing the minimum, the GCV parameter again in-creases.
Different measurement methods cause differentcurve shapes of the GCV parameter.68,82,93 This factindicates that the curve contains additional informa-tion and may support determination of the solution.Figures 5–7 present properties of GCV deducted fromsimulations.80 Figure 5 shows that the behavior ofthe GCV parameter depends on the position of theinversion window. If the supports are unsuitablypositioned with respect to the sought distribution@Fig. 5~a!#, the GCV parameter shows slight oscilla-tions for in this case g values between 10 and 106 @Fig.5~c!, solid curve#. The oscillations are stronger for
article distributions with small mode widths thanor wide distributions. In the case of a more favor-ble interval division @Fig. 5~b!# one observes strongariations of PGCV for g , 10 but no oscillations for g
. 10 @Fig. 5~c!, dashed curve#. This behavior per-its one to infer the extent to which the respective
nversion window is suited to the representation ofhe investigated particle distribution.
An increase in the number of base functions thatould not contradict the applicability of GCV ~Refs.1 and 87! shifts the variations in the GCV parame-er into the investigated global minimum. Figure 6llustrates this behavior for the example of 16 insteadf 8 base functions. Because of this behavior theumber of base functions was kept small ~see Section!.
Fig. 4. Qualitative behavior of the GCV parameter depending onthe Lagrange multiplier g for the case of the inversion of a particlesize distribution from six backscatter and two extinction coeffi-cients: vertical line, global mimimum; ~a! behavior of the GCVparameter in the entire range covered by the Lagrange multiplier;~b! area around the global minimum on a strongly enhanced res-lution.
20 April 1999 y Vol. 38, No. 12 y APPLIED OPTICS 2351
awTtigttutib
it
a
2
Figure 7 reflects the behavior of the GCV parame-ter if only backscatter data or a combination of back-scatter and extinction data are used. The threecurves present the GCV parameter’s dependence onthe Lagrange multiplier if the information for inver-sion consists of six backscatter and two extinctioncoefficients ~solid curve!, seven backscatter coeffi-cients and one extinction coefficient ~dashed curve!,and eight backscatter coefficients ~dotted curve!. Inll three cases an inversion window was chosen withhich the distribution could be reproduced best.he GCV curve changes its behavior in such a wayhat if the extinction information is omitted the steepncrease in the GCV parameter after passing thelobal minimum no longer appears. A possible in-erpretation of this observation is that the value ofhe Lagrange multiplier, which is necessary for reg-larization, becomes increasingly inaccurate if ex-inction information is gradually omitted. This factndicates that the combination of extinction andackscatter information is necessary for a successful
Fig. 5. Behavior of the GCV parameter depending on the Lagcoefficients if the position of the inversion window, expressed in term~a! interval division with which the structures of the ~dotted curvposition of the inversion window as well as the interval division, ~shown in ~a! and ~b!. Vertical lines indicate the global minimum
Fig. 6. Behavior of the GCV parameter depending on the La-grange multiplier g in the case of 16 base functions for the inver-sion of six backscatter and two extinction coefficients.
352 APPLIED OPTICS y Vol. 38, No. 12 y 20 April 1999
nversion, which has been proved in other simula-ions.80
4. Results
The capabilities of the inversion algorithm in retriev-ing effective radius, volume, surface-area, and num-ber concentrations, as well as the mean complexrefractive index were tested in extensive simula-tions80 and will be presented in a forthcoming paper.Here only some important results are summarized.
Backscatter coefficients at 355, 400, 532, 710, 800,and 1064 nm as well as extinction coefficients at 355and 532 nm were created from monomodal and bi-modal logarithmic-normal distributions by means ofa Mie code.84 The chosen parameters for mode ra-dius, mode width, and complex refractive index tookaccount of the variability of atmospheric particle sizedistributions.80
multiplier g in the case of six backscatter and two extinctionthe positions of the supports @vertical lines in ~a! and ~b!#, is varied:ght particle distribution can only barely be reproduced, ~b! idealV parameter depending on the Lagrange multiplier for the cases
Fig. 7. Behavior of the GCV parameter depending on the La-grange multiplier g in the inversion of ~solid curve! six backscatternd two extinction coefficients, ~dashed curve! seven backscatter
coefficients and one extinction coefficient, and ~dotted curve! eightbackscatter coefficients. Vertical lines denote the global mini-mum.
ranges of
e! souc! GC.
Tm
s
rm
av4
mdbtme
irr5cg5of6f
Otfm
Extinction information proved to be necessary forthe stability of the algorithm, i.e., use of six backscat-ter and two extinction data permits one to retrievereliable results for the investigated particle parame-ters. In this case the optical data may have errors asgreat as 20%. Successive elimination of extinctionresults in deterioration of the results to the point ofempty solution spaces.
Figure 8 shows a simulation example for two dif-ferent size distributions, one monomodal logarithmic-normal ~A! and one bimodal logarithmic-normal ~B!.
he mode radius is 0.1 mm for distribution ~A! with aode width of 1.8. Mode 1 of distribution ~B! has a
mode radius of 0.1 and a mode width of 1.6 and mode2 a mode radius of 0.5 mm and a mode width of 1.6.The optical backscatter and extinction data fromthese two distributions had round-off errors of lessthan 0.1%. The refractive index was assumedknown and wavelength independent. For distribu-tion ~A! it was taken to be 1.5–0.01i; for distribution~B! the value was chosen to be 1.4–0.001i. The in-verted size distributions are shown with the inputdistributions.
The effective radii of 0.24 and 0.23 mm ~A! and 0.60and 0.67 mm ~B!, volume concentrations of 0.83 and0.85 mm3ycm3 ~A! and 74 and 79 mm3ycm3 ~B!,urface-area concentrations of 10.5 and 11 mm2ycm3
~A! and 369 and 354 mm2ycm3 ~B!, and number con-centrations of 42 and 43 cm23 ~A! and 783 and 379cm23 ~B! for the input and the inverted distributions,espectively, have been found. In the case of theonomodal distribution ~A! there is very good agree-
ment between the given and the inverted quantities.Effective radius, volume, surface-area, and numberconcentrations are reproduced with a less than 5%error. For the bimodal distribution ~B! the resultsre slightly worse. The error for the effective radius,olume, and surface-area concentrations is between% and 12%; the number concentration deviated by
Fig. 8. Inversion results for ~A! a monomodal and ~B! a bimodaldistribution. The given distributions are shown as solid curves,the distributions from the inversion as dashed curves.
ore than half of the value from the given one. Thiseviation can be traced back to the small number ofase functions and the resulting insufficient resolu-ion that causes approximation errors around theaxima of the distribution. The bimodality, how-
ver, is resolved by the algorithm.Considering all performed simulations under the
nclusion of measurement errors and of an unknownefractive index resulted in errors in the derived pa-ameters that, in the worst case, are approximately0% for effective radius, volume, and surface-areaoncentrations. Number concentrations show thereatest uncertainties with, in some cases, more than0%. However, on average the overall errors werenly half as large. With respect to the complex re-ractive index its real part deviated by less than0.05, and its imaginary part by less than 650%,
rom the correct value.
5. Conclusions
The method of inversion with regularization has beenchosen for routine calculation of physical particleproperties from lidar measurements of backscatterand extinction coefficients at several wavelengths.The solutions are given in terms of approximatingvolume concentration distributions from which meanparticle parameters, i.e., effective radius, volume,surface-area, and number concentrations, as well asthe mean complex refractive index, may be retrieved.To obtain reliable results, a combination of backscat-ter and extinction data is necessary. The algorithmcan handle realistic data errors as high as 20%, asexpected from lidar measurements.
The advantage of this method over former algo-rithms is that it does not need a priori assumptionson the shape of the sought particle size distributionsor the complex refractive index. The solution spaceis determined when constraints are applied, in theorder of regularization through the smoothness of thesought size distributions, the demand for positiveparticle distributions, and an agreement between theoptical input data and the data calculated from theretrieved particle size distributions. The strength ofregularization is determined by GCV. GCV permitsone to evaluate experimental data without relying ona priori performed simulations. It can be shownthat GCV is capable of dealing with the small numberof eight optical data. The entire algorithm is math-ematically well founded and offers various methodsfor future improvement, including elaborate methodsfor error estimations.
The algorithm will be employed in routine dataevaluation. Aerosol closure experiments, such asAerosol Characterization Experiment 2 ~ACE 2, Por-tugalyNorth Atlantic, JuneyJuly 1997!, the Linden-berg Aerosol Characterization Experiment ~LACE98, Germany, JulyyAugust 1998!, and the IndianOcean Experiment ~INDOEX, MaldivesyIndian
cean, January–March 1999!, are of special impor-ance because the availability of additional data setsrom different remote sensing and in situ measure-ent techniques at the field site of the six-
20 April 1999 y Vol. 38, No. 12 y APPLIED OPTICS 2353
wst
oisaomwdpc
tw
d
ti
2
wavelength lidar permits one to cross-check theresults. The algorithm will be extended toward theretrieval of multimodal particle distributions. Themain emphasis will be on exploiting the behaviorcharacteristics of the GCV parameter. This behav-ior aids in selecting a priori an appropriate inversion
indow and in evaluating the stability of the inver-ion. Use of a neural network for interpretation ofhe curves is planned.97,98 Currently only cases of
spherical particles are considered for this inversionscheme. Any nonsphericity effect will be anothersource of error. For a description of these errors,models are necessary that permit us to calculate thebackscattering and the extinction behavior and thedifference with respect to spherical scatterers. Re-spective algorithms have been developed99,100 andmight be incorporated into future versions of the al-gorithm.
Appendix A
The basic steps of the computer code are described~see Fig. 9!:
~1! Choose a gliding inversion window with a vari-able window width according to the given limits forrmin and rmax. The lower window limit is set at val-ues of rmin 5 0.01, 0.05, 0.1, 0.15, and 0.2 mm. Herene must observe that at the wavelengths used thenformation content of the kernel functions with re-pect to particles smaller than 0.1 mm is rather smallnd may therefore show significant errors. For eachf these values upper limits of rmax 5 1 to rmax 5 10m are allotted in steps of 1 mm. Thus altogether 50indows are given. The supports within the win-ows are logarithmic-equidistant. The inversion iserformed for every window. According to Eq. ~8!reate base functions Bj~r!, j 5 1, . . . , 8 within these
inversion windows, from which the sought distribu-
Fig. 9. Calculation steps for inversion with regularization.
354 APPLIED OPTICS y Vol. 38, No. 12 y 20 April 1999
tion is to be reconstructed. Triangle functions de-fined on logarithmic size intervals are used. Thenumber of base functions is equal to the amount ofavailable optical data.
~2! Determine the kernel functions Kp~m, r! for alldata points p 5 ~b, a; l! with the help of backscatterand extinction efficiencies Qp~m, r! @see Eq. ~6!#. Theefficiencies are stored in a data bank.
~3! Create the kernel matrix Apj~m! according toEq. ~10!. The trapezoidal rule with a step width of0.001 mm in the respective integral is used to calcu-late the elements of the matrix. These calculationsare time-consuming, and the matrices in the inver-sion must always be in the same format. Thereforethey are stored in a data bank for the inversion win-dows defined above as well as a multitude of complexrefractive indices and can be read in as required.For real parts of the refractive index, values of 1.33and 1.35–1.8 in stepwidths of 0.025 were used. Forthe imaginary part, values of 0, 1025, 1024, 1023, 5 31023, as well as 0.01–0.1 in stepwidths of 0.01, and0.1–0.7 in stepwidths of 0.1 were chosen.
~4! Calculate the weight factors wj from Eq. ~17!.The GCV method is used for determining the suitablevalue of the Lagrange multiplier g:
~a! Calculate 1000 values for g in logarithmic-equidistant steps within a predetermined range@see Eq. ~A1!#. In the simulations it was shownthat a suitable value for g fluctuates across sev-eral orders of magnitude.80 This variability isnot only determined by the sought particle dis-tribution but also by the complex refractive indexof the particles as well as the available numberand combination of data. With the followingmethod66 the variation range of g is fixed:
g 5 g0 expF1k (
pln~ATA!ppG , (A1)
g0 5 1023, . . . , 105.
Simulations showed that for g0 a range of valuesbetween 1023 and 105 is sufficient. The neces-sary scaling is performed by means of the ele-ments along the main diagonal of the matrixATA.
~b! Determine the GCV parameter for everyLagrange multiplier according to Eq. ~18!.
~c! Within the stable range of the GCV curvedetermine the global minimum of the GCV pa-rameter and thus the sought value of g.
~5! Calculate the approximated volume distribu-ion @see Eq. ~7!# from the sum of the respectiveeighted base functions.~6! Repeat steps ~2!–~5! for all given inversion win-
ows.~7! Check the 50 distributions that are obtained in
his way with the aid of additional constraints accord-ng to the following scheme:
watdpBptctt
disciplinary global research programme?” Contrib. Atmos.
~a! Are there only positive weight factorswithin the respective inversion window? If yes,the solution will be rejected because the windowfor the investigated distribution is too small.
~b! For the remaining inversion windows de-termine the upper and lower limit of the radiusrange in which the weight factors have taken onnegative values.
~c! Are there again positive weight factors be-yond these limits? If yes, no suitable Lagrangemultiplier was found. The respective solution isrejected because there is instability in the inver-sion algorithm.
~d! Accept all remaining solutions, i.e., thosewith a joined area of positive weight factors andnegative values toward small and large particleradii, and set all negative weight factors to zero.The ideal case presents solutions for which onlythe first and the last weight factors are negative.In this case the boundaries of the particle distri-bution can be determined relatively exactly, andthe errors in the derived mean particle proper-ties are rather small.
~e! From these distributions compute the opti-cal data and compare them with the input datathrough calculation of the mean quadratic devi-ation.
~8! As an inversion result choose either the solutionith the smallest deviation between backcalculatednd input optical data or take the mean value of allhe solutions for which the deviation in the opticalata is smaller than a given value. The latter de-ends on the application and is fixed individually.oth approaches yield acceptable solutions of theroblem. In summary, this final step determineshe investigated mean distribution of the volume con-entration and the mean complex refractive indexhat has been assumed for the respective distribu-ion.
References1. R. J. Charlson, S. E. Schwartz, J. M. Hales, R. D. Cess, J. A.
Coakley, Jr., J. E. Hansen, and D. J. Hofmann, “Climateforcing by anthropogenic aerosols,” Science 255, 423–430~1992!.
2. J. T. Kiel and B. P. Briegleb, “The relative roles of sulfateaerosols and greenhouse gases,” Science 260, 311–314 ~1993!.
3. J. E. Penner, R. E. Dickinson, and C. A. O’Neill, “Effects ofaerosols from biomass burning on the global radiation bud-get,” Science 256, 1432–1434 ~1992!.
4. R. J. Charlson and J. Heintzenberg, eds., Aerosol Forcing ofClimate ~Wiley, Chichester, 1995!.
5. T. R. Karl, P. D. Jones, R. W. Knight, G. Kukla, N. Plummer,V. Razovayev, K. P. Gallo, J. Lindseay, R. J. Charlson, andT. C. Peterson, “A new perspective on recent global warming,”Bull. Am. Meteorol. Soc. 74, 1007–1023 ~1993!.
6. P. V. Hobbs and B. J. Huebert, eds., “Atmospheric aerosols.A new focus of the International Global Atmospheric Chem-istry Project ~IGAC!,” ~IGAC Core Project Office, Massachu-setts Institute of Technology, Cambridge, Mass., 1996!.
7. J. Heintzenberg, H.-F. Graf, R. J. Charlson, and P. Warneck,“Climate forcing and the physico-chemical life cycle of theatmospheric aerosol—Why do we need an integrated inter-
Phys. 69, No. 2, 261–271 ~1996!.8. P. K. Quinn, T. L. Anderson, T. S. Bates, R. Dlugi, J. Heint-
zenberg, W. von Hoyningen-Huene, M. Kulmala, P. B. Russell,and E. Swietlicki, “Closure in tropospheric aerosol-climateresearch: a review and future needs for addressing aerosoldirect shortwave radiative forcing,” Contrib. Atmos. Phys. 69,No. 4, 547–577 ~1996!.
9. D. Muller, D. Althausen, U. Wandinger, and A. Ansmann,“Multiple-wavelength aerosol lidar,” in Advances in Atmo-spheric Remote Sensing with Lidar, A. Ansmann, R. Neuber,P. Rairoux, and U. Wandinger, eds. ~Springer-Verlag, Berlin,1996!.
10. A. Ansmann, U. Wandinger, M. Riebesell, C. Weitkamp, andW. Michaelis, “Independent measurement of extinction andbackscatter profiles in cirrus clouds by using a combinedRaman elastic-backscatter lidar,” Appl. Opt. 31, 7113–7131~1992!.
11. D. L. Phillips, “A technique for the numerical solution ofcertain integral equations of the first kind,” J. Assoc. Comput.Mach. 9, 84–97 ~1962!.
12. S. Twomey, “The application of numerical filtering to thesolution of integral equations encountered in indirect sensingmeasurements,” J. Franklin Inst. 279, 95–109 ~1965!.
13. A. N. Tikhonov and V. Y. Arsenin, eds., Solution of Ill-PosedProblems ~Wiley, New York, 1977!.
14. S. Twomey, ed., Introduction to the Mathematics of Inversionin Remote Sensing and Indirect Measurements ~Elsevier, Am-sterdam, 1977!.
15. V. E. Zuev and I. E. Naats, eds., Inverse Problems of LidarSensing of the Atmosphere ~Springer-Verlag, Berlin, 1983!.
16. P. C. Sabatier, “Basic concepts and methods of inverse prob-lems,” in Basic Methods of Tomography and Inverse Prob-lems, P. C. Sabatier, ed. ~Hilger, Bristol, UK, 1987!.
17. A. K. Louis, ed., Inverse und schlecht gestellte Probleme~Teubner, Stuttgart, 1989!.
18. J. Hadamard, “Sur les problemes aux derivees parielies etleur signification physique,” Bull. Univ. Princeton 49–52~1902!.
19. M. A. Box and B. H. J. McKellar, “Analytic inversion of mul-tispectral extinction data in the anomalous diffraction ap-proximation,” Opt. Lett. 3, 91–93 ~1978!.
20. A. L. Fymat, “Analytical inversions in remote sensing of par-ticle size distributions. 1: Multispectral extinctions in theanomalous diffraction approximation,” Appl. Opt. 17, 1675–1676 ~1978!.
21. J. G. McWhirther and E. R. Pike, “On the numerical inversionof the Laplace transform and similar Fredholm integral equa-tions of the first kind,” J. Phys. A 11, 1729–1745 ~1978!.
22. M. A. Box and B. H. J. McKellar, “Relationship between twoanalytic inversion formulas for multispectral extinctiondata,” Appl. Opt. 18, 3599–3601 ~1979!.
23. C. B. Smith, “Inversion of the anomalous diffraction approx-imation for variable complex index of refraction near unity,”Appl. Opt. 21, 3363–3366 ~1982!.
24. C. D. Capps, R. L. Henning, and G. M. Hess, “Analytic inver-sion of remote-sensing data,” Appl. Opt. 21, 3581–3587~1982!.
25. J. D. Klett, “Anomalous diffraction model for inversion ofmultispectral extinction data including absorption effects,”Appl. Opt. 23, 4499–4508 ~1984!.
26. M. Bertero, C. De Mol, and E. R. Pike, “Particle size distri-bution from spectral turbidity: a singular-system analysis,”Inverse Problems 2, 247–258 ~1986!.
27. G. Viera and M. A. Box, “Information content analysis ofaerosol remote sensing experiments using singular functiontheory. 1: Extinction measurements,” Appl. Opt. 26, 1312–1327 ~1987!.
20 April 1999 y Vol. 38, No. 12 y APPLIED OPTICS 2355
28. A. Ben-David, B. M. Herman, and J. A. Reagan, “Inverse “Aerosol size distribution measurements using a multispec-
2
problem and the pseudoempirical orthogonal function methodof solution. 1: Theory,” Appl. Opt. 27, 1235–1242 ~1988!.
29. A. Ben-David, B. M. Herman, and J. A. Reagan, “Inverseproblem and the pseudoempirical orthogonal function methodof solution. 2: Use,” Appl. Opt. 27, 1243–1254 ~1988!.
30. B. P. Curry, “Constrained eigenfunction method for the in-version of remote sensing data: application to particle sizedetermination from light scattering measurements,” Appl.Opt. 28, 1345–1355 ~1989!.
31. G. P. Box, K. M. Sealey, and M. A. Box, “Inversion of Mieextinction measurements using analytic eigenfunction theo-ry,” J. Atmos. Sci. 49, 2074–2081 ~1992!.
32. B. T. Evans and G. R. Fournier, “Approximations of polydis-persed extinction,” Appl. Opt. 35, 3281–3285 ~1996!.
33. G. P. Box, “Effects of smoothing and measurement-wavelength range on the accuracy of analytic eigenfunctioninversions,” Appl. Opt. 34, 7787–7792 ~1995!.
34. D. P. Donovan and A. I. Carswell, “Retrieval of stratosphericaerosol physical properties using multiwavelength lidar back-scatter and extinction measurements,” in Advances in Atmo-spheric Remote Sensing with Lidar, A. Ansmann, R. Neuber,P. Rairoux, and U. Wandinger, eds. ~Springer-Verlag, Berlin,1996!.
35. J. Wang and F. R. Hallett, “Spherical particle size determi-nation by analytical inversion of the UV–visible–NIR extinc-tion spectrum,” Appl. Opt. 35, 193–197 ~1996!.
36. D. P. Donovan and A. I. Carswell, “Principal component anal-ysis applied to multiwavelength lidar aerosol backscatter andextinction measurements,” Appl. Opt. 36, 9406–9424 ~1997!.
37. G. Backus and F. Gilbert, “The resolving power of gross earthdata,” Geophys. J. R. Astron. Soc. 266, 169–205 ~1968!.
38. G. Backus and F. Gilbert, “Uniqueness of inaccurate grossearth data,” Philos. Trans. R. Soc. London Ser. A 266, 123–192 ~1970!.
39. E. R. Westwater and A. Cohen, “Application of Backus–Gilbert inversion technique to determination of aerosol sizedistributions from optical scattering measurements,” Appl.Opt. 12, 1341–1348 ~1973!.
40. S. Twomey, “On the numerical solution of Fredholm integralequations of the first kind by inversion of linear system pro-duced by quadrature,” Assoc. Comput. Mach. 10, 97–101~1963!.
41. M. T. Chahine, “Determination of the temperature profile inan atmosphere from its outgoing radiance,” J. Opt. Soc. Am.58, 1634–1637 ~1968!.
42. R. J. Hanson, “A numerical method for solving Fredholmintegral equations of the first kind using singular values,”SIAM J. Numer. Anal. 8, 616–622 ~1971!.
43. J. Heintzenberg, H. Muller, H. Quenzel, and E. Thomalla,“Information content of optical data with respect to aero-sol properties: numerical studies with a randomizedminimization-search-technique inversion algorithm,” Appl.Opt. 20, 1308–1315 ~1981!.
44. H. Muller, “Die Bestimmbarkeit der atmospharischen Aero-solgroßenverteilung mit Hilfe eines 4-Wellenlangen-Lidars,”Ph.D. dissertation ~Wissenschaftliche Mitteilung Nr. 44, Uni-versitat Munchen, Munchen, Germany, 1981!.
45. E. F. Maher and N. M. Laird, “EM algorithm reconstructionof particle size distributions from diffusion battery data,” J.Aerosol Sci. 16, 557–570 ~1985!.
46. P. Rairoux, “Mesures par lidar de la pollution atmospheriqueet des parametres meteorologiques,” Ph.D. dissertation~These 955, Ecole Polytechnique Federale de Lausanne, Lau-sanne, Switzerland, 1991!.
47. J. Kolenda, B. Mielke, P. Rairoux, B. Stein, D. Weidauer, J. P.Wolf, L. Woste, F. Castagnoli, M. Del Guasta, M. Morandi,V. M. Sacco, L. Stefanutti, V. Venturi, and L. Zuccagnoli,
356 APPLIED OPTICS y Vol. 38, No. 12 y 20 April 1999
tral lidar-system,” in Lidar for Remote Sensing, R. J.Becherer, R. M. Hardesty, and J. P. Meyzonette, eds., Proc.SPIE 1714, 208–219 ~1992!.
48. C. Dellago and H. Horvath, “On the accuracy of the sizedistribution information obtained from light extinction andscattering measurements—I. Basic considerations andmodels,” J. Aerosol Sci. 24, 129–141 ~1993!.
49. H. Horvath and C. Dellago, “On the accuracy of the sizedistribution information obtained from light extinction andscattering measurements—II. Case studies,” J. Aerosol Sci.24, 143–154 ~1993!.
50. W. von Hoyningen-Huene and M. Wendisch, “Variability ofaerosol optical parameters by advective processes,” Atmos.Environ. 28, 923–933 ~1994!.
51. G. Beyerle, R. Neuber, O. Schrems, F. Wittrock, and B. Knud-sen, “Multiwavelength lidar measurements of stratosphericaerosols above Spitsbergen during winter 1992y93,” Geophys.Res. Lett. 21, 57–60 ~1994!.
52. G. Feingold and C. J. Grund, “Feasibility of using multiwave-length lidar measurements to measure cloud condensationnuclei,” J. Atmos. Oceanic Technol. 11, 543–1558 ~1994!.
53. B. Stein, M. Del Guasta, J. Kolenda, M. Morandi, P. Rairoux,L. Stefanutti, J. P. Wolf, and L. Woste, “Stratospheric aerosolsize distribution from multispectral lidar measurements atSodankyla during EASOE,” Geophys. Res. Lett. 21, 1311–1314 ~1994!.
54. M. Wendisch and W. von Hoyningen-Huene, “Possibility ofrefractive index determination of atmospheric aerosol parti-cles by ground-based solar extinction and scattering measure-ments,” Atmos. Environ. 28, 785–792 ~1994!.
55. U. Wandinger, A. Ansmann, J. Reichardt, and T. Deshler,“Determination of stratospheric aerosol microphysical prop-erties from independent extinction and backscattering mea-surements with a Raman lidar,” Appl. Opt. 34, 8315–8329~1995!.
56. F. Ferri, A. Bassini, and E. Paganini, “Modified version of theChahine algorithm to invert spectral extinction data for par-ticle sizing,” Appl. Opt. 34, 5829–5839 ~1995!.
57. F. Sun and J. W. L. Lewis, “Simplex deconvolutions ofparticle-size distribution functions from optical measure-ments,” Appl. Opt. 34, 8437–8446 ~1995!.
58. M. J. Post, “A graphical technique for retrieving size distri-bution parameters from multiple measurements: visualiza-tion and error analysis,” J. Atmos. Oceanic Technol. 13, 863–873 ~1996!.
59. L. Landweber, “An iteration formula for Fredholm integralequations of the first kind,” Am. J. Math. 73, 615–624 ~1975!.
60. S. Twomey, “Comparison of constrained linear inversion andan iterative nonlinear algorithm applied to the indirect esti-mation of particle size distribution,” J. Comput. Phys. 18,188–200 ~1975!.
61. O. V. Dubovik, T. V. Lapyonok, and S. L. Oshchepkov, “Im-proved technique for data inversion: optical sizing of multi-component aerosols,” Appl. Opt. 34, 8422–8436 ~1995!.
62. R. Lattes and J. L. Lions, eds., The Method of Quasi-reversibility—Applications to Partial Differential Equations~Elsevier, New York, 1969!.
63. M. D. King, D. M. Byrne, B. M. Herman, and J. A. Reagan,“Aerosol size distributions obtained by inversion of spectraloptical depth measurements,” J. Atmos. Sci. 35, 2153–2167~1978!.
64. V. V. Veretennikov, V. S. Kozlov, I. E. Naats, and V. Ya.Fadeev, “Optical studies of smoke aerosols: an inversionmethod and its applications,” Opt. Lett. 4, 411–413 ~1979!.
65. A. P. Ivanov, F. P. Osipenko, A. P. Chaykovskiy, and V. N.Shcherbakov, “Study of the aerosol optical properties and
microstructure by the method of multiwave sounding,” Izv. 83. K. S. Shifrin and I. G. Zolotov, “Spectral attenuation and
Atmos. Oceanic Phys. 22, 633–639 ~1986!.
66. P. Qing, H. Nakane, Y. Sasano, and S. Kitamura, “Numericalsimulation of the retrieval of aerosol size distribution frommultiwavelength laser radar measurements,” Appl. Opt. 28,5259–5265 ~1989!.
67. J. K. Wolfenbarger and J. H. Seinfeld, “Inversion of aerosolsize distribution data,” J. Aerosol Sci. 21, 227–247 ~1990!.
68. J. K. Wolfenbarger and J. H. Seinfeld, “Regularized solutionsto the aerosol data inversion problem,” SIAM J. Sci. Stat.Comput. 12, 342–361 ~1991!.
69. U. Amato, M. F. Carfora, V. Cuomo, and C. Serio, “Objectivealgorithms for the aerosol problem,” Appl. Opt. 34, 5442–5452~1995!.
70. A. N. Tikhonov, “Solution of incorrectly formulated problemsand the regularization method,” Sov. Math. Dokl. 5, 1035–1038 ~1963!.
71. R. J. Hanson and J. L. Phillips, “An adaptive numericalmethod for solving linear Fredholm integral equations of thefirst kind,” Numer. Math. 24, 291–307 ~1975!.
72. D. W. Cooper and L. A. Spielmann, “Data inversion usingnonlinear programming with physical constraints: aerosolsize distribution measurement by impactors,” Atmos. Envi-ron. 10, 723–729 ~1976!.
73. S. K. Friedlander, ed., Smoke, Dust and Haze: Fundamen-tals of Aerosol Behavior ~Wiley, New York, 1977!.
74. E. T. Jaynes, ed., Papers on Probability, Statistics and Sta-tistical Physics ~Synthese Library, Kluwer Academic, Dordre-cht, The Netherlands, 1983!.
75. E. Yee, “On the interpretation of diffusion battery data,” J.Aerosol Sci. 20, 797–811 ~1989!.
76. U. Amato and W. Hughes, “Maximum entropy regularizationof Fredholm integral equations of the first kind,” InverseProblems 7, 793–808 ~1991!.
77. P. Paatero, T. Raunemaa, and R. L. Dod, “Composition char-acteristics of carbonaceous particle samples, analyzed byEVE deconvolution method,” J. Aerosol Sci. 19, 1223–1226~1988!.
78. P. Paatero, “Extreme value estimation, a method for regular-izing ill-posed inversion problems,” in Ill-posed Problems inNatural Sciences, Proceedings of the International Conferencein Moscow, August 1991, A. N. Tikhonov, ed. ~VSP, Utrecht,The Netherlands, 1991!.
79. C. Bockmann and J. Niebsch, “A mollifier method for aerosolsize distribution,” in Advances in Atmospheric Remote Sens-ing with Lidar, A. Ansmann, R. Neuber, P. Rairoux, and U.Wandinger, eds. ~Springer-Verlag, Berlin, 1996!.
80. D. Muller, “Entwicklung eines Inversionsalgorithmus zurBestimmung mikrophysikalischer Partikelparameter des at-mospharischen Aerosols aus kombinierten Mehr-wellenlangen- und Raman-Lidarmessungen,” Ph.D.dissertation ~Universitat Leipzig, Leipzig, Germany, 1997!.
81. G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,”Technometrics 21, 215–223 ~1979!.
82. P. Craven and G. Wahba, “Smoothing noisy data with splinefunctions: estimating the correct degree of smoothing by themethod of generalized cross-validation,” Numer. Math. 31,377–403 ~1979!.
84. C. F. Bohren and D. R. Huffman, Absorption and Scatteringof Light by Small Particles ~Wiley, New York, 1983!.
85. F. O’Sullivan, “A statistical perspective on ill-posed inverseproblems,” Stat. Sci. 1, 502–527 ~1986!.
86. S.-Å. Gustafson, “Regularizing a Volterra integral equationproblem by means of convex programming,” Report ~Univer-sity of Stavanger, Stavanger, Norway, 1991!.
87. D. Nychka, G. Wahba, S. Goldfarb, and T. Pugh, “Cross-validated spline methods for the estimation of three-dimensional tumor size distributions from observations ontwo-dimensional cross sections,” J. Am. Stat. Soc. 79, 832–846 ~1984!.
88. V. S. Bashurova, K. P. Koutzenogil, A. Y. Pusep, and N. V.Shokirev, “Determination of atmospheric aerosol size distri-bution functions from screen diffusion battery data: math-ematical aspects,” J. Aerosol. Sci. 22, 373–388 ~1991!.
89. D. Lesnic, L. Elliot, and D. B. Ingham, “An inversion methodfor the determination of the particle size distribution fromdiffusion battery measurements,” J. Aerosol Sci. 26, 797–812~1995!.
90. R. R. Picard and R. D. Cook, “Cross-validation of regressionmodels,” J. Am. Stat. Assoc. 79, 575–583 ~1984!.
91. A. R. Davies and R. S. Anderssen, “Optimization in the reg-ularization of ill-posed problems,” Research Report CMA-R30-85 ~Centre for Mathematical Analysis, The AustralianNational University, Canberra, Australia, 1985!.
92. T. Lyche and K. Mørken, “A data reduction strategy forsplines with applications to the approximation of functionsand data,” J. Numer. Anal. 8, 185–208 ~1988!.
93. G. Wahba, Department of Statistics, University ofWisconsin–Madison, Madison, Wisc. 53706-1685 ~personalcommunication, 1996!.
94. P. Paatero, Department of Physics, University of Helsinki,SF-00014 Helsinki, Finland ~personal communication, 1996!.
95. D. M. Allen, “The relationship between variable selection anddata augmentation and a method for prediction,” Technomet-rics 16, 125–127 ~1974!.
96. A. S. Householder, ed., The Theory of Matrices in NumericalAnalysis ~Blaisdell, New York, 1964!.
97. S. Kitamura and P. Qing, “Neural network application tosolve Fredholm integral equations of the first kind,” in Pro-ceedings of International Joint Conference on Neural Net-works, Washington, D.C., 18–22 June 1989 ~Institute ofElectrical and Electronic Engineers, Service Center, Piscat-away, N.J., 1989!, p. 589.
98. A. Ishimaru, R. J. Marks II, L. Tsang, C. M. Lam, and D. C.Park, and S. Kitamura, “Particle-size distribution determi-nation using optical sensing and neural networks,” Opt. Lett.15, 1221–1223 ~1990!.
99. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by nonspherical par-ticles: a review,” J. Quant. Spectrosc. Radiat. Transfer 55,535–575 ~1996!.
100. T. Rother, German Remote Sensing Data Center, GermanAerospace Research Establishment, D-17235 Neustrelitz,Germany ~personal communication, 1998!.
20 April 1999 y Vol. 38, No. 12 y APPLIED OPTICS 2357
