principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction...

Principal component analysis applied tomultiwavelength lidar aerosol backscatter andextinction measurements

David P. Donovan and Allan I. Carswell

The use of powerful Raman backscatter lidars enables one to measure the stratospheric aerosol extinctionprofile independently of the backscatter, thereby obtaining additional information to aid in retrieving thephysical characteristics of the sampled aerosol. We used principal component analysis to construct aself-consistent method for the retrieval of aerosol bulk physical and optical properties from multiwave-length elastic andyor inelastic Raman backscatter lidar signals. The procedure is applied to syntheticand actual lidar signals. We found that aerosol surface area and volume can be usefully estimated andthat the use of Raman-derived extinction data leads to a notable improvement in the accuracy of theestimations. © 1997 Optical Society of America

Key words: Aerosol, lidar, multiwavelength aerosol measurements.

1. Introduction

It is highly desirable to make quantitative inferencesabout the size distribution of stratospheric aerosol.For a variety of reasons, including the study of aero-sol microphysical processes and assessment of thepotential role of aerosol in heterogeneous chemistryaffecting ozone levels, it is crucial to be able to esti-mate direct physical quantities such as aerosol massmixing ratio, mean radius, and volume and surfacearea density.

One can make rough estimates of aerosol physicalquantities from single-wavelength lidar measure-ments by using altitude-dependent relationships be-tween the results of in situ aerosol size distributionmeasurements ~e.g., balloonborne optical particlecounter measurements! and inferred aerosol back-scatter ~e.g., Jager et al.1 and Jager and Hofmann2!.However, such an approach is necessarily limited tothe applicability of the size distributions used in con-structing the given relationships. For example, itmay be difficult to justify the use of an aerosolvolume-versus-backscatter relationship based on

The authors are with the Department of Physics and Astronomy,Institute for Space and Terrestrial Science, York University, 4700Keele Street, North York, Ontario M3J 1P3, Canada.

Received 20 February 1997; revised manuscript received 1 July1997.

0003-6935y97y369406-19$10.00y0© 1997 Optical Society of America

9406 APPLIED OPTICS y Vol. 36, No. 36 y 20 December 1997

mid-latitude size distribution measurements appliedto lidar backscatter measurements made at high lat-itudes during winter.

An alternative approach is to use solely the infor-mation contained in the lidar signals themselves andattempt to invert them in order to obtain relevantaerosol characteristics. According to Mie theory forthe backscatter at a given altitude and at wavelengthli we have

bp,i 5 *0

`

pr2Qbp,li~r!

dn~r!

drdr (1)

and for the extinction we have

ai 5 *0

`

pr2Qa,li~r!

dn~r!

drdr, (2)

where r is the particle radius, Qbp,li~r! is the Mie

backscattering efficiency, Qa,li~r! is the Mie extinc-

tion efficiency, and dn~r!ydr is the aerosol size distri-bution expressed as the number of particles per unitvolume between r and r 1 dr.

Equations ~1! and ~2! can be rewritten together as

gi 5 *0

`

Ki~r!dV~r!

drdr, (3)

where dV~r!ydr is the aerosol volume size distribu-tion; gi is either a backscatter or extinction measure-

ment; and Ki is the appropriate Mie backscatter orextinction kernel, which is equal to 3Qa,li

~r!y4r forextinction or 3Qbp,li

~r!y4r for backscatter. Qa,li~r!

and Qbp,li~r! are the Mie extinction and backscatter

efficiencies, respectively, for wavelength li. Here N~i 5 1 . . . N! is the number of extinction andyor back-scatter measurements. Having made a number ofmeasurements of the projection of dV~r!ydr on thekernels, one then attempts to reconstruct the sizedistribution by inverting the given set of measure-ments.

The possibility of using lidar measurements to es-timate the size distribution of stratospheric aerosolshas been previously discussed by a number of au-thors.3 Several approaches for extracting quantita-tive estimates of stratospheric aerosol physicalparameters using lidar data at two or more wave-lengths have been published. However, many ofthese methods contain various unrealistic assump-tions such as that the aerosol properties do not varywith height andyor that the aerosol size distributionshave a functional form ~e.g., single-mode log normal!,which may or may not be appropriate, depending onthe situation at hand.4–6 The inferred size distribu-tion can then be used to predict integral or bulk prop-erties such as aerosol surface area density or aerosolextinction at a given wavelength.

When one is limited to a finite number of extinctionand backscatter measurements it is generally not fea-sible to obtain accurate detailed information aboutthe entire aerosol size distribution.7 Because of theessential underdetermined nature of the problem,different retrieval methods can often lead to inferredsize distributions that differ markedly in their de-tails. However, it has been previously noted thatvarious bulk or integral properties of the aerosol sizedistribution such as aerosol total volume and surfacearea density and aerosol extinction and backscatter-ing at various wavelengths can be predicted withuseful accuracy, and the predictions are much lesssensitive to the retrieval method employed.8

Principal component or eigenvector analysis9,10 is aretrieval technique that is especially well suited forthe determination of aerosol integral properties.Principal component analysis allows one to assess theinformation content of a given kernel set and thesensitivity of the retrieved aerosol parameters tomeasurement errors. By using principal componentanalysis one can approximate any integral propertyof the aerosol size distribution by a linear combina-tion of the measured aerosol backscatter and extinc-tion coefficients themselves. In addition, principalcomponent retrievals, unlike many other inversiontechniques, do not require an initial guess of anykind. Principal component retrievals of aerosolphysical quantities have been carried out previouslyfor Stratospheric Aerosol and Gas Experiment~SAGE II! satellite extinction measurements.9,11

However, this technique appears not to have beenapplied previously in a self-consistent manner to thecase of lidar signal profiles.

Here we address the problem of inversion of mul-

tiwavelength lidar data to obtain aerosol physicalparameters from a general point of view, consideringonly the information available in the lidar measure-ments. With principal component or eigenvectoranalysis, one can consider the extent to which aerosolbackscatter and extinction can be used to predictaerosol physical and optical properties. From theseconsiderations a practical, self-consistent inversionroutine can be developed for the retrieval of height-resolved aerosol physical and optical parameters, in-dependent of any functional form assumptions. Theresults of this method applied to synthetic and actuallidar data are presented and discussed. Three casesare primarily considered: the case for which infor-mation is available from the RayleighyMie backscat-ter at 353, 532, and 1064 nm; the case for which theRayleighyMie backscatter returns are supplementedwith extinction information obtained with the vibra-tional Raman N2 backscatter12 at 385 nm ~from353-nm incident radiation!; and the case for whichthe Raman N2 backscatter at 608 nm ~from 532-nmincident radiation! can also be observed.

2. Methodology

A. Principal Component Analysis

Following Twomey9 dV~r!ydr can be written as

dV~r!

dr5 (

i51

i5N

yiKi~r! 1 c~r!, (4)

where c~r! is that part of dV~r!ydr orthogonal to Ki~r!~i.e., *0

` Ki~r!c~r! 5 0 for i 5 1 . . . N! and thus inac-cessible with the given set of measurements. Sub-stitution into Eq. ~3! then yields

gi 5 *0

`

(j51

N

Ki~r!Kj~r!yjdr 1 *0

`

Ki~r!c~r!dr. (5)

Using the fact that *0` Ki~r!c~r! 5 0 for i 5 1 . . . N one

then has

g# 5 ##y# , (6)

where ## is the N 3 N covariance matrix and g# and y#are column vectors of length N. Since ## is real,symmetric, and positive, its eigenvalues are all realand positive. Expanding ## in terms of its eigenvec-tors and eigenvalues, Eq. ~6! can then be written as

g# 5 ULUty# (7)

where U is a matrix containing as its columns theeigenvectors of ## and L is a matrix containing thecorresponding eigenvalues of ## as its diagonal ele-ments. This can then be inverted to give

y# 5 ##21g# 5 UL21Utg# . (8)

Moving from a continuous representation of the sizedistribution to a discrete one ~with the use of an

20 December 1997 y Vol. 36, No. 36 y APPLIED OPTICS 9407

appropriate numerical quadrature rule! the volumesize distribution solution vector is then given by

v# 5 Kty#

5 KtUL21Ut z g# . (9)

Here, v# is a vector whose elements vj contain thevolume of the aerosol between rj and rj 1 drj and thematrix K is defined so that its elements Ki, j 5 Ki~rj!.

Following Twomey9 Eq. ~9! can be rewritten in an-other form that makes its essential nature clearer.By defining

j# ; L21y2Utg# , (10)

where L21y2 is a matrix containing as its diagonalelements the reciprocal square roots of the eigenval-ues of ##. Equation ~9! then becomes

v# 5 KtUL21y2j#, (11)

which can be written equivalently as

v~r! 5 f# ~r!tj#, (12)

where

f# ~r! 5 L21y2UK# t~r!. (13)

The elements of f# ~r! are known as the orthonormalcharacteristic or principal component functions.

B. Integral or Bulk Quantities

In general, any integral property of the aerosol ~e.g.,scattering at another wavelength, total volume, sur-face area! can be expressed as ~see Ref. 8!

P 5 w# t~v# 1 c# ! (14)

where c# is a vector containing the parts of the truevolume distribution orthogonal to Ki~r! and w# is theweighting vector for the property in question. Forexample, in the case of aerosol volume wj 5 1, forsurface area wj 5 3yrj, and for aerosol extinction at agiven wavelength wj 5 3Qa,l~rj!y4rj. Using Eq. ~9!,Eq. ~14! can be rewritten as

P 5 a# tg# 1 w# tc# , (15)

where

a# t 5 w# tKtUL21Ut. (16)

Thus the inferred value of any integral parameter~a# tg# ! can be expressed as a linear combination of theN measurements without explicitly evaluating thesize distribution ~v# !. Once the coefficients ~the ele-ments of a# ! have been calculated for a given aerosolrefractive index, they can be stored and convenientlyapplied to a given data set.

C. Information Content and Error Magnification

Error magnification refers to how the uncertainty ineach of the initial measurements can be expected toinfluence the uncertainty in the aerosol parametersderived from the measurements themselves. In


many typical inversion problems even relativelysmall measurement uncertainties can lead to largeuncertainties in the final solution.

Two obvious areas of concern exist when one ap-plies an inversion technique to a given data set: Thecompleteness of the set of kernels used @the impor-tance of c in Eqs. ~4! and ~15!# and the sensitivity ofthe inferred aerosol parameters to the presence ofmeasurement errors ~i.e., the expected error magni-fication!. These two concerns depend on the natureof the kernels @the Ki~r!# involved in the given prob-lem. The more distinct the kernels, the higher theirinformation content and the solution vector and in-ferred bulk quantities obtained will be less sensitiveto measurement errors.

In practice, there is always an uncertainty associ-ated with the measurement of each gi. Thus Eq. ~9!must be modified to include the presence of an errorvector e# giving

v# 5 KtUL21Ut~g# 1 e# !. (17)

Then evidently the measurement error gives rise to asolution vector error given by

d# 5 KtUL21Ute# . (18)

The exact error magnification depends on the na-ture of the size distribution being sought; however,the error magnification is generally strongly influ-enced by the eigenvalues of the associated covariancematrix ##. It can be shown that the so-called worst-case magnification of the relative error ~i.e., the ratioof the percent error in the norm of v# to that of thenorm of g# ! is given by7

gmax 5 ~lmaxylmin!1y2, (19)

where lmax and lmin refer to the maximum and min-imum elements, respectively of L.

It can also be shown that the average error mag-nification is given by7

gavg < N21~lmaxylmin!1y2, (20)

where N is the number of measurements.The error magnifications can be very large if lmin is

small compared with lmax. Physically this meansthat one or more of the measurement kernels can beexpressed ~to within some limit! as a linear combina-tion of the remaining kernels and thus can effectivelycontribute no new information, depending on thelevel of measurement errors. It is common to reducethe degree of error magnification by deleting thesmaller eigenvalues, which is equivalent to truncat-ing the expansion of v# in terms of the characteristicfunctions @Eq. ~12!#.

As might be intuitively expected, the degree of er-ror magnification associated with integral quantitiesis generally much lower than that associated with theretrieval of the size distribution itself. Because thevalue of any integral parameter depends on theweighted integral throughout the retrieved size dis-tribution, one might expect that the errors are some-

what smoothed out. This notion can be put in amore quantitative form. The relative error magni-fication associated with a retrieved integral quantitycan be expressed as

gint 5 FSdPP D2 ug# u2

ue# u2G1y2 5 FSdPP D2 ug# u2

ue# *u2G1y2

5 5F(i51

N

~l21y2ei9 *0

`

w~r!fi~r!dr!2GS(i51

N

liji2D

F(i51

N

ji *0

`

w~r!fi~r!drG2S(i51

N

ei92D 6

1y2

,

(21)

where e# * 5 Ute# is the rotated error vector. Equation~21! is similar in form to the equation given byTwomey9 for the error magnification related to thesize distribution retrieval but is modified by the pres-ence of the integrals over the principal componentfunctions. The derivation of Eq. ~21! is given in Ap-pendix A.

For the conditions corresponding to the worst-caseerror magnification for v~r!~e192 5 ue# u2 and jN

2 5 uj#u2!,

gint,max 5 5lN

l1 3*0

`

w~r!f1~r!dr

*0

`

w~r!fN~r!dr42

61y2

5 gmax

*0

`

w~r!f1~r!dr

*0

`

w~r!fN~r!dr

. (22)

In practice, the principal component functions asso-ciated with the smaller eigenvalues tend to be muchmore oscillatory than the ones associated with thelarger eigenvalues, so that often,

*0

`

w~r!f1dr ,, *0

`

w~r!fN~r!dr,

and thus gint,max can be much smaller than gmax.

3. Application to Backscatter and ExtinctionMeasurements

Before the preceding methodology can be applied tothe inversion of actual lidar signals, one must con-sider the completeness and associated error magni-fications of the backscatter and extinction kernelsthat are potentially available. In this study threesituations involving lidar backscatter and extinc-tion measurements are considered and are listed inTable 1. These cases are based on the lidar wave-lengths used in our current measurement program.Case 1 considers a two-wavelength ~1064- and 532-nm! Nd:YAG system and the unabsorbed wave-

length of a hydrogen Raman-shifted XeCl ozonedifferential absorption lidar ~DIAL! system. In theconventional elastic ~RayleighyMie! DIAL mode theuseful wavelength is 353 nm, in the Raman modethe first Stokes N2 scattering at 385 nm ~from353-nm incident radiation! is also used ~case 2!.For the 532-nm Nd:YAG lidar the Raman returnsignal would be at 608 nm ~case 3!.

In addition to the three lidar cases, for compari-son, the case represented by the SAGE II satelliteextinction measurements, to which principal com-ponent analysis has been previously applied,9,11 isalso considered ~case 4!. Here we investigate theeffectiveness of the different measurement cases inestimating bulk aerosol properties by applyingprincipal component relationships for each caseto the inversion of optical measurements gener-ated using different synthetic aerosol size distribu-tions.

In general, principal component or eigenvectoranalysis allows one to predict aerosol physical prop-erties without making any restrictive assumptionsabout the size distribution. However, one must stillassume the shape and refractive index of the parti-cles in question. Fortunately, for stratospheric sul-fate aerosols, it seems reasonable to restrict theaerosol to being composed of spherical droplets ofH2OyH2SO4 in equilibrium with the ambient watervapor. Thus standard Mie-scattering theory is ap-plicable.

The kernels and corresponding eigenfunctions ~orcharacteristic functions! @Eq. ~13!# for each case ap-plied to spherical sulfate aerosol are shown in Figs. 1and 2, respectively. The refractive indices were cal-culated following the methods of Russell and Ha-mill13 and correspond to typical lower stratosphericconditions ~T 5 210 K and 5 3 1024 mbars partialpressure of water!. The extinction kernels for thelidar cases used here are the average values of theextinction at the transmitted and the receivedRaman-shifted wavelength. This average extinc-tion is the quantity most directly measured by Ra-man lidars.12

For the kernels shown in Fig. 1, keeping in mindthe expected logarithmic nature of stratosphericaerosol size distributions, a logarithmic interval spac-ing was used and the kernels ~as well as the variousweighting functions! were smoothed by integrating

Table 1. Measurement Cases Considered

Case System Measurements

1 Nd:YAG lidar bp~532 nm!, bp~1064 nm!

DIAL bp~353 nm!

2 Nd:YAG lidar bp~532 nm!, bp~1064 nm!

Raman DIAL bp~353 nm!, a~3531385 nm!

3 Raman Nd:YAG lidar bp~532 nm!, bp~1064 nm!, a~5321608 nm!

Raman DIAL bp~353 nm!, a~3531385 nm!

4 SAGE II a~385 nm!, a~453 nm!

a~525 nm!, a~1020 nm!


Fig. 1. ~a! Volume extinction and back-scatter kernels corresponding to the lidarcases and ~b! extinction kernels corre-sponding to the SAGE II case.

the Mie efficiencies over single-mode log normal sizedistributions of mode width 1.1. That is,

Ki~rj! 5

*0

`

pr2Qli~r9!

dn~rj, 1.1, r9!

dr9dr9

V~rj, 1.1!, (23)

where V~rj, 1.1! is the total volume of the particles inthe distribution and dn~rj, sW, r!ydr describes a lognormal size distribution with mode radius rj and

mode width equal to 1.1 @see Eq. ~25!#. The smooth-ing procedure reduces the kernel noise ~high-frequency oscillation present as a function of radius!,especially for the Mie backscatter and reduces thenumber of points required in subsequent calcula-tions. Performing such a procedure is consistentwith assuming that the aerosol size distributions towhich we are seeking to apply PCA can be well rep-resented by a series of the smoothing functions ~inthis case narrow, log normal distributions!.14 Dif-ferent smoothing methods ~including minimal

Fig. 2. Volume principal component functions @fi~r!# for ~a! case 1, ~b! case 2, ~c! case 3, ~d! case 4.


smoothing! were tried as well as linear grid spacings.As long as the width of the smoothing functions wasnot much greater than the distributions to which thederived principal component analysis bulk relation-ships were applied ~described in Subsection 3.A!,there was no substantial variation in their generalaccuracy. For the kernels shown here, an upper ra-dius of 50 mm was used. Different values for theupper radius ranging between 5 and 50 mm were alsofound to produce no great variations in the generalaccuracy of the obtained principal component rela-tionships.

It can be readily seen from Fig. 1~a! that the ker-nels for the lidar cases are distinct only between radiiof ;0.05 and ;5 mm. Hence, outside this intervallittle information about the aerosol size distributioncan be obtained. For the SAGE II case the kernelsare indistinct beyond ;1 mm. Hence the lidar situ-ations considered can be expected to yield better in-formation for larger particles.

The ratio of each eigenvalue to the smallest eigen-value ~liylmin! corresponding to the eigenvectorsshown in Fig. 2 are listed in Table 2. Calculationswere performed for a variety of refractive indicesranging from conditions corresponding to T 5 195 Kand 35% H2SO4 to T 5 250 K and 90% H2SO4 ~n385 51.42 2 1.47!. The variation of the magnitude of theeigenvalue with temperature and water partial pres-sure was generally less than 1% for the larger eigen-values, whereas variations of several tens of percentwere encountered for the smallest eigenvalues.

With regard to the estimation of aerosol surfacearea we found it useful to reformulate the kernels interms of the aerosol surface area distribution @dS~r!ydr#. By following the discussion of Subsection 2.A,one can estimate the aerosol surface area ~both the

Table 2. Eigenvalue Spectrum for Different Casesa

Case 1 123 5 1Case 2 86,000 45 5 1Case 3 137,000 7800 21 5 1Case 4 3740 188 34 1

aThe results correspond to 75% H2SO4 aerosol at a temperatureof 220 K.

distribution and the total surface area! from the aero-sol volume distribution. However, with regard tothe aerosol surface area, superior results were gen-erally obtained by reformulating the problem interms of the surface area distribution, i.e.,

gi 5 *0

`

Ksa,i~r!dS~r!

drdr, (24)

where dS~r!ydr is the aerosol surface area distribu-tion and Ksa,i is the appropriate surface area back-scatter or extinction kernel that ~unsmoothed! isequal to Qa,li

~r!y4 for extinction or Qbp,li~r!y4 for

backscatter. In a format similar to that in Figs. 1and 2 the smoothed surface area kernels as well asthe corresponding eigenfunctions are shown in Figs.3 and 4, respectively. The eigenvalue ratios corre-sponding to the eigenvectors shown in Fig. 4 arelisted in Table 3. The variation of the surface areaeigenvalue ratios with refractive index was found tobe similar to that found for the volume kernels.

If the set of kernels were complete, the end resultswe obtained by using either the surface area or vol-ume kernels would be identical. However, becausethe kernel sets are indeed not complete @i.e., c~r! inEq. ~4! is often significant#, the principal componentrelationships found between the backscatter and ex-tinction measurements and the aerosol properties arein general different, depending on the kernel formu-lation. Under the volume formulation, wsa~r! 5 3yr.Thus the predicted surface area is sensitive to errorsin the predicted volume distribution at small aerosolradii. For the surface area formulation, wsa~r! 5 1and the predicted surface area does not suffer fromthis same problem. As discussed further in Subsec-tion 3.A, posing the problem in terms of surface areaimproved the accuracy of the surface area retrievals,but for other quantities the volume formulation gen-erally gave more accurate results.

The principal component relations found in eachcase for aerosol volume, surface area, and extinction,along with the expected average error magnifications@defined here as ~¥i

N ai2!1y2y¥i

N ai!# are given in Table4. For example, following Table 4, for case 2 one canobtain the numerical value ~in mm2ycm3! of the prin-

Fig. 3. ~a! Surface area extinction andbackscatter kernels corresponding to thelidar cases and ~b! extinction kernels cor-responding to the SAGE II case.


Fig. 4. Surface area principal component functions @fi~r!# for ~a! case 1, ~b! case 2, ~c! case 3, ~d! case 4.

Table 3. Eigenvalue Spectrum for Different Cases Obtained withSurface Area Formulated Kernelsa

Case 1 375 8 1Case 2 12,000 155 4 1Case 3 34,300 924 204 3 1Case 4 2265 96 15 1

aThe results correspond to 75% H2SO4 aerosol at a temperatureof 220 K.

cipal component approximation for aerosol surfacearea density ~at T 5 220 K and 75% sulfate by mass!from

Sa < 1.96bp,353nm 1 6.02bp,532nm 2 4.59bp,1064nm

1 1.31a3531385nm,

Table 4. Principal Component Coefficients Derived with the Kernels Shown in Figs. 1 and 3 and the Full Number of Principal Components

Case 1 bp,353 nm bp,532 nm bp,1064 nm gint,avg

Volume ~m mm3 cm23! 1.40 1.46 14.59 1.2Sa ~m mm2 cm23! 19.55 20.67 228.52 3.4a353 nm 23.82 22.30 14.85 0.6a532 nm 20.75 27.02 10.45 1.0a1064 nm 1.09 14.20 5.42 0.8

Case 2 bp,353 nm bp,532 nm bp,1064 nm a3531385 nm gint,avg

Volume ~m mm3 cm23! 1.07 1.67 13.96 0.014 0.84Sa ~m mm2 cm23! 1.96 6.02 24.59 1.31 1.7a532 nm 12.36 23.82 4.27 0.32 1.04a1064 nm 3.89 13.70 7.54 20.12 0.64

Case 3 bp,353 nm bp,532 nm bp,1064 nm a3531385 nm a5321608 nm gint,avg

Volume ~m mm3 cm23! 20.27 0.37 15.73 0.077 4.70 0.8Sa ~m mm2 cm23! 20.35 8.08 28.42 1.50 4.79 2.3a1064 0.37 15.30 5.38 20.21 21.14 0.8

Case 4 a385 nm a453 nm a525 nm a1020 nm gint,avg

Volume ~m mm3 cm23! 0.31 20.35 0.055 0.32 1.7Sa ~m mm2 cm23! 2.82 22.56 1.05 0.45 2.6


Table 5. Principal Component Coefficients Derived with the Kernels shown in Figs. 1 and 3 and the First Two Principal Components

Case 1 bp,353 nm bp,532 nm bp,1064 nm gint,avg

Volume ~m mm3 cm23! 0.064 7.73 5.09 0.7Sa ~m mm2 cm23! 25.65 20.91 29.02 1.7a353 nm 21.92 6.64 1.31 0.8a532 nm 19.01 1.14 21.91 1.0a1064 nm 1.46 12.55 8.08 0.7

Case 2 bp,353 nm bp,532 nm bp,1064 nm a3531385 nm gint,avg

Volume ~m mm3 cm23! 3.52 1.82 0.66 0.011 0.4Sa ~m mm2 cm23! 0.91 1.29 1.18 1.41 0.5a532 nm 8.43 4.37 1.59 0.34 0.6a1064 nm 9.50 4.92 1.79 20.13 0.7

Case 3 bp,353 nm bp,532 nm bp,1064 nm a3531385 nm a5321608 nm gint,avg

Volume ~m mm3 cm23! 0.11 0.002 0.001 0.056 0.005 0.7Sa ~m mm2 cm23! 0.31 0.014 0.013 1.37 0.0039 0.8a1064 0.70 0.016 0.006 20.21 0.037 1.3

Case 4 a385 nm a453 nm a525 nm a1020 nm gint,avg

Volume ~m mm3 cm23! 0.003 0.052 0.083 0.098 0.6Sa ~m mm2 cm23! 0.51 0.49 0.45 0.26 1.1

where the backscatter coefficients are in units of sr21

km21 and the extinction coefficients are in units ofkm21. The surface area kernels were used to for-mulate the surface area relationships, whereas thevolume kernels were used to derive all other relation-ships.

In spite of the fact that the error magnifications weencountered using the full number of principal com-ponents are small when compared with the ones as-sociated with the retrieval of the shape of thedistribution itself, it may still be desirable to reducethe degree of error magnification by use of less thanthe full number of principal component functions topredict a given integral quantity. As an example,Table 5 shows the same principal component rela-tionships listed in Table 4, except with only the twomost significant principal component functions. Ingeneral, the error magnifications are lower thanthose for the cases when all the principal componentfunctions are used, especially for case 3 surface areaand for case 4.

Principal component relations similar to thoselisted in Tables 4 and 5 were found for various aerosolrefractive indices between T 5 195 K and 35% H2SO4and T 5 250 K and 90% H2SO4 by mass ~n385 51.42 2 1.47!. The relationship coefficients did notvary much for the SAGE II extinction measurements~of the order of 10% across the conditions considered!,whereas variations of greater than 100% for somecomponents were noted for the lidar cases. Thesevariations were most pronounced at sulfate percent-ages of less than ;50% and at lower temperatures~T , 205 K!. This is a result of the rapid change ofaerosol refractive indices for these conditions13 andthe fact that the shape of the backscatter kernels ismuch more dependent on the refractive index thanthe extinction kernels.

A. Application to Synthetic Size Distributions

Before applying principal component analysis to lidarprofiles, one should consider the accuracy to which

the total aerosol surface areas and volumes can bedetermined ~along with the associated error magni-fications! for the simpler case of single-point mea-surements. To do this, for each measurement case,we carried out principal component retrievals usingsynthetic scattering data generated for a wide varietyof single-mode, log normal size distributions @Eq.~25!#. Single-mode or multimode log normal distri-butions are often used to represent aerosol sizedistributions.15–18 In general, a single-mode, lognormal distribution has the form

dn~r!

dr5

N

Î2prln~sW!exp52

123lnS r

RmD

ln~sW!4

2

6 , (25)

where N is the total number of particles in the mode,Rm is the mode radius, and sW is the mode width.

The use of single-mode distributions here is notoverly restrictive. Because the estimates of the var-ious integral quantities are expressed as simple lin-ear combinations of the measurements themselves,the results of the retrievals carried out on single-mode distributions can be easily generalized to thecase of multimode distributions.

Figure 5 shows the relative difference @~retrieved 2true!ytrue# between the volume estimated when weused principal component relationships formulatedfrom different numbers of principal components ~in-cluding the appropriate coefficients listed in Tables 4and 5 for all and just two components, respectively!and that of the input ~or true! aerosol distribution.The results are shown as a function of mode radiusfor three different mode widths. For a mode radiusof 1.1 the error is quite variable with mode radius andis generally 650% for the lidar cases. For widerdistributions the maximum error decreases and theresults are smoothed to some degree. By comparingthe different cases, one can see that, in general, theaccuracy of the principal component relationships in-creases with the number of different measurement


Fig. 5. Relative difference between the volume predicted by principal component analysis and that of the input distribution as a functionof mode radii for three different mode widths.

parameters. Consistent with the remarks madeabove about the range of kernel sensitivities, the li-dar cases are more accurate over a wider range ofmode radii. It can also be noted that the inclusion ofextinction measurements in the lidar cases ~cases 2and 3! decreases the error of the approximation formode radii below a few tenths of a micrometer andthe accuracy above 1 mm is unaffected, because theextinction kernels are not sensitive to particles in thissize regime ~see Fig. 1!. Cases 2 and 3 do not showmuch loss ~if any! of accuracy when the number ofprincipal components used is reduced. For cases 3and 4 the range of accuracy of the principal compo-nent relationships is reduced somewhat when onlytwo components are used.

Figure 6 shows the average error magnification forretention of different numbers of principal compo-nents that we calculated assuming equal, relative,uncorrelated measurement errors. Here only the


error magnification associated with the retrievals forthe mode radius 5 1.55 cases are shown. From Fig.6 it can be seen that cases 1 and 2 have error mag-nifications below 1, even when all the principal com-ponents are used. For cases 3 and 4, however, errormagnifications can be large, except when only thefirst two components are used. This indicates that,depending on the level of measurement errors, itmight be appropriate to use only the first two com-ponents in these cases, even though some accuracymay be lost ~see Fig. 5 for cases 3 and 4!.

Corresponding to Fig. 5, Fig. 7 shows the results foraerosol surface area. Consistent with the results foraerosol volume, the lidar cases including extinctioninformation ~cases 2 and 3! perform well for moderadii from ;0.15 to 5 mm as does the SAGE II case.The lidar case in which only backscatter measure-ments are made ~case 1! does not compare well withthe other cases, though. The addition of even one

extinction measurement ~case 2! produces a greatimprovement compared with the results for case 1.For all cases no notable reduction in the accuracy ofthe principal component is seen as the number ofprincipal component functions used is reduced.

Similar to Fig. 6, Fig. 8 shows the average errormagnification for retention of different numbers ofprincipal components. Here cases 2 and 3 have lowerror magnifications, but using only three lidar back-scatter measurements ~case 1! shows poor resultswhen compared with the other cases. For cases 2and 3 low error magnifications are found, even whenall the principal component functions are used. Forcase 1 high levels of error magnification are found formode radii greater than ;0.5 mm, even when onlytwo components are used. For case 4 ~as for the casewith aerosol volume! the deletion of the last two prin-cipal component functions produces a large reductionin the error magnifications.

The surface area retrievals shown in Fig. 7 arebased on the relationships we found using the surfacearea kernels. Trials based on the relationshipsfound when we used the volume kernels were lessaccurate, with differences of as much as 100% fornarrow distributions. The results we obtained usingthe volume kernels for the aerosol total number den-sity ~not shown! were even worse. As mentionedabove, this relates to the nature of the weightingfunctions used to predict these quantities and theincompleteness of the kernels. As opposed to thesurface area formulation, where wsa~r! 5 1, under thevolume formulation, wsa~r! 5 3y4, and for aerosolnumber density, wn~r! 5 3y4pr3. Thus these quan-tities are more sensitive to errors in the retrieved sizedistribution at small aerosol radii, where the kernelsare less sensitive. However, by using the surfacearea formulation, we did not obtain notably more

Fig. 6. Average error magnifications for aerosol volume corre-sponding to the cases shown in Fig. 5 with a width parameter equalto 1.55.

accurate results for quantities other than surfacearea.

Of particular importance to the inversion of lidardata is the ability to account for the effects of aerosolextinction. For example, in case 1 one would bedealing not with a set of point backscatter measure-ments, but instead a set of elastic backscatter lidarprofiles, from which the aerosol backscatter valueswould have to be extracted. To do this, one must beable to predict the aerosol extinction accurately fromthe given measurement set. The relative differencesbetween the retrieved and actual aerosol extinctionat 1064 and 532 nm for log normal distributions ofwidth 1.1 for measurement cases 1, 2, and 3 areshown in Fig. 9, along with the appropriate errormagnification levels. For both wavelengths shown,the aerosol extinction is predicted to within ;20%with low relative error magnifications in the regionwhere the aerosol extinction would be largest ~seeFig. 1!. At 532 nm for case 3 the error is quite smallas expected owing to the addition of Raman extinc-tion data at 532 and 608 nm.

B. Summary

In the lidar cases, particularly for cases 2 and 3, it hasbeen demonstrated that for relatively low levels oferror magnification it is possible to predict aerosolvolume and surface area for fairly general aerosolsize distributions. In particular, it should be possi-ble to predict aerosol surface area and volume within;50% for aerosol distributions with mean radii be-tween ;0.1 and ;3 mm. These estimates can bemade without relying on the need for accurate ancil-lary aerosol data ~such as data from nearby opticalparticle counter soundings of the aerosol layer! ormaking any restrictive assumptions as to the natureof the aerosol size distribution. Moreover, these es-timates are formed from simple, linear combinationsof the measurement values and so, once the variouscoefficients have been computed, they can be readilyapplied to the data in question.

4. Application to Lidar Profiles

Applied to lidar aerosol sensing, principal componentanalysis is attractive from several viewpoints. Asnoted above, it is not necessary to make any assump-tions in the analysis about the size distribution of theaerosol. Also, the ability for one to express integralparameters, including aerosol extinction, as linearcombinations of the measurements using previouslygenerated coefficients is well suited to the inversionof lidar data. In a standard lidar situation one doesnot deal with a point set of backscatter andyor ex-tinction measurements but with a number of signalprofiles from which the aerosol extinction and back-scatter must be extracted.

The accurate retrieval of the aerosol backscatteroften depends on being able to account for aerosolextinction properly. This can be clearly seen by con-


Fig. 7. Relative difference between the surface area predicted using principal component analysis and that of the input distributions asa function of mode radii for three different mode widths.

sidering the form of the single-scattering lidar equa-tion for elastic backscattering19

P~z! }bp,M~z! 1 bp, A~z!

z2

3 expH22 *0

z

@aM~z9! 1 aA~z9!#dz9J . (26)

Here z is altitude and bp and a are the volume scat-tering and extinction coefficients, respectively. Adenotes the aerosol contribution and M denotes themolecular contribution, which can usually be ac-counted for independently by using auxiliary infor-mation ~e.g., local density profiles from radiosondesor those provided from meteorological analyses, towhich the lidar signals must be normalized at someaerosol-free altitude!. In Section 3, given a set of


point backscatter andyor extinction measurements,we demonstrated that aerosol integral parameterscould be usefully predicted using principal compo-nent analysis. It is important to consider whetherthis is indeed the case when one is starting from thelidar signals themselves. One must be able to ex-tract the aerosol backscatter information from thelidar signals in a reliably consistent manner.

In lidar backscatter retrievals one can compute theaerosol backscatter accounting for the aerosol extinc-tion at each of the given wavelengths ~for which nocorresponding Raman data are available! by an iter-ative procedure. Often this can be done by assum-ing a constant fixed relationship between the aerosolbackscatter and extinction at a single wavelength.In this paper we account for the aerosol extinction byusing multiwavelength principal component relation-ships. First the aerosol backscatter coefficients arecalculated assuming no aerosol extinction then, using

the resulting backscatter values, we can infer theextinction using the various principal component re-lationships. Once the extinction profile has beenpredicted, the backscatter profiles can be recalcu-lated. This process is repeated until the solutionconverges. By using this approach, we developed ageneral multiwavelength RayleighyMie andyor Ra-man lidar aerosol inversion procedure using principalcomponent relationships to account for the aerosol

Fig. 8. Average error magnifications for aerosol surface area cor-responding to the cases shown in Fig. 7 with a width parameterequal to 1.55.

Fig. 9. Relative difference between the aerosol extinction for ~topleft! 532 nm and ~top right! 1064 nm predicted by principal com-ponent analysis for cases 1, 2, and 3, and the corresponding errormagnifications ~bottom left and right, respectively!.

extinction and to predict aerosol volume and surfacearea.

A. Application to Simulated Data Sets

Figure 10 shows three sets of synthetic lidar signalscorresponding to the lidar cases described in Section3. We generated the signals using prescribed pro-files of atmospheric density, temperature, water va-por mixing ratio, and aerosol size distribution andnumber density profile. We then calculated theaerosol refractive-index profile by assuming that theH2OyH2SO4 droplets are in equilibrium with the am-bient water vapor.20 The enhancement of the signalin the RayleighyMie channels owing to the presenceof the aerosol layer between ;8- and 25-km altitudeis clearly visible. Here no simulated signal noisewas added and the simulated aerosol loading level ishigh to emphasize the effects of aerosol extinction.The corresponding retrieved aerosol scattering ratioprofiles @R 5 ~bp,A 1 bp,M!ybp,M# and extinction co-efficient profiles are shown in Fig. 11.

A demonstration of the ability of the retrievalmethod to account for the successful effects of aerosolextinction is presented in Fig. 12. Figure 12~a!shows the relative effect in the retrieved R profile ofneglecting aerosol extinction, whereas Figs. 12~b!–12~d! show the ratios between the retrieved and truescattering ratios for cases 1–3, respectively. Figure12~a! shows that applying no extinction correctionleads to a large maximum error of more than 35% ~offscale!, whereas for case 1 @Fig. 12~b!# the maximumerror is less than 10%. For cases 2 @Fig. 12~c!# and 3@Fig. 12~d!#, which use Raman backscatter extinctioninformation, the error is less than 5%.

Once the lidar profiles are inverted to yieldextinction-corrected aerosol backscatter and aerosolextinction coefficient profiles, one can estimate theaerosol physical properties by using the appropriateprincipal component derived relationships. As anexample, Figs. 13–15 show the predicted aerosol vol-ume v, surface area sa, and effective radius ~reff 53vysa! profiles derived from each respective lidar sig-nal set shown in Fig. 10.

The relative differences between the retrieved andtrue aerosol parameter profiles of Figs. 13–15 areshown in Fig. 16. By comparing the results for thedifferent integral quantities between the differentmeasurement cases, one finds the results are largelywhat would be expected on the basis of the analysiscarried out in Section 3. All cases retrieve the aero-sol volume quantity to within better than a factor of1.5. Surface area is retrieved only within a factor of4 in case 1 but within a factor of 1.5 for cases 2 and 3.Effective radius is not retrieved to any useful accu-racy in case 1, but it is within a factor of ;2 for case2, with even better results for case 3, especially foraerosol mean radii below 0.5 mm. The results shownin Figs. 13–15 are roughly representative of othersimulations that we carried out using various othersize distribution profiles.

The accuracy of the retrievals was not significantlyaltered when other temperature and water vapor pro-


Fig. 10. Synthetic lidar signals corre-sponding to ~a! case 1, ~b! case 2, ~c! case3.

files were considered. However, doubling or halvingthe water vapor mixing ratio used in the retrievals, ascompared with the profile used in generating the syn-thetic lidar signal profiles, did significantly changethe observed results. Figure 17 shows how the re-trieval error changed for case 2 when the water vapormixing ratio was doubled and halved as comparedwith the true water vapor profile. Here the surfacearea retrievals are fairly insensitive. However, thevolume retrievals were affected by as much as ;20%.It should be noted that for the simulations ~Figs.10–17! we used a cold temperature profile. That is,the lower stratospheric temperature profile we usedto generate the simulated signals often approached200 K to simulate cold Arctic conditions. At theselow temperatures the aerosol refractive indices aresensitive to the water vapor partial pressure. Othersimulations with warmer temperature profiles ~min-imum temperature greater than ;215–220 K!showed generally much less variation ~less than 5%!

when we varied the water vapor profile used in theretrievals with respect to the true profile.

The retrievals considered thus far are quite unre-alistic owing to the fact that no measurement noisewas added to the synthetic signals. The backscatterratios retrieved from a more realistic set of syntheticsignals at 353, 385, 532, and 1064 nm ~case 2! areshown in Fig. 18. The signal and background levelsare typical of those that have been encountered inpractice with the lidars located at the Network forDetection of Stratospheric Change monitoring sta-tion in the Canadian Arctic at Eureka, NorthwestTerritories. The signals used here represent ;4 h ofaveraging. The aerosol scattering ratio levels werechosen to be similar to those observed in the Arcticduring the 1993–1994 and 1994–1995 winters at theEureka site for the background sulfate aerosol layer.

The retrieved aerosol effective radius, surface area,and volume corresponding to the scattering ratio pro-files presented in Fig. 18 are shown in Fig. 19. The

Fig. 11. Scattering ratio and extinctionprofiles derived from the signals shownin Fig. 10 for ~a! case 1, ~b! case 2, ~c!case 3. Both the retrieved and trueprofiles are shown, but they are indis-tinguishable on this scale.


Fig. 12. Ratio of retrieved backscat-ter ratio R to true backscatter ratio for~a! no extinction correction, ~b! case 1,~c! case 2, ~d! case 3, corresponding tothe scattering ratios shown in Fig. 11.

volume retrievals are accurate to within a factor of1.5 throughout the aerosol layer. However, the

Fig. 13. Results of retrievals for aerosol volume, surface area, andeffective radius for the case 1 signals shown in Fig. 10~a!: dashedcurve, results of the principal component retrieval; solid curve,true profile.

Fig. 14. Results of retrievals for aerosol volume, surface area, andeffective radius for the case 2 signals shown in Fig. 10~b!: dashedcurve, results of the principal component retrieval; solid curve,true profile.

aerosol surface area is notably underestimated above;16 km, where the aerosol effective radius is below;0.1 mm. This is consistent with the results shown

Fig. 15. Results of retrievals for aerosol volume, surface area, andeffective radius for the case 3 signals shown in Fig. 10~c!: dashedcurve, results of the principal component retrieval; solid curve,true profile.

Fig. 16. Retrieval error for cases 1, 2, and 3 corresponding to theresults shown in Figs. 13–15.


in Fig. 7. The underestimation of surface areacauses an overestimation by a factor of ;3 in theretrieved aerosol effective radius show in Fig. 19.The results shown here illustrate that estimated sur-face area and mean radius should be treated withcaution if there is reason to suspect that the meanaerosol radius may actually be below ;0.1 mm.

In terms of the precision of the retrievals shown inFig. 19, the addition of realistic measurement uncer-tainty levels does not significantly induce serious un-certainty in the retrieved quantities until near 20 km,where the aerosol scattering ratio levels start to tendtoward 1.0. The large uncertainties in the retrievalsabove ;20 km are not so much the result of errormagnification but rather are the expected results oflarge relative uncertainties in the retrieved backscat-ters and extinction that can be expected at low R

Fig. 17. Sample effect of variation of water vapor mixing ratiowith respect to the true value for case 2 results shown in Fig. 14.

Fig. 18. Scattering ratios R at 353, 532, and 1064 nm and aerosolextinction at 385 nm retrieved from a set of noisy signals at 353,385, 532, and 1064 nm ~case 2!.


values. For a given measurement time and signalbackground level the effect of signal noise on theretrievals increases with decreasing aerosolamounts. For example, because bp,A } R 2 1, therelative error in bp,A is related to the error in R~which is the quantity most directly affected by signalnoise! by the relation

dbp, A

bp, A5

dRR 2 1

,

which can become large, even for small dR if R is closeto 1.

It should also be pointed out that in realistic situ-ations, when aerosol amounts are low and no Ramanlidar returns are available, significant errors in theaerosol backscatters ~leading to correspondinglylarge errors in the retrieved aerosol characteristics!can arise from uncertainties in the density profileused to account for the molecular backscatter andextinction,21 especially if the error in the density pro-file is large at the normalization range. However,when Raman returns are available ~cases 2 and 3!,the influence of errors in the density profile on theretrieved scattering ratio profiles is greatly reduced,in part because the Raman signal can serve as aproxy density profile.22 This is another benefit ofusing Raman data that is, in a sense, independent ofthe benefit incurred from accounting for the aerosolextinction.

B. Sample Application to Real Lidar Data

Lidar measurements at Eureka were carried out dur-ing the winters of 1993–1994, 1994–1995, 1995–1996and 1996–1997.23,24 Eureka is the location of aweather station and stratospheric observatory oper-ated by the Canadian Atmospheric Environment Ser-vice ~AES!. Two lidar systems are located atEureka, a XeCl excimer laser based Raman ozone~DIAL!25 and a Nd:YAG laser based elastic backscat-ter lidar operated by the Japanese MeteorologicalResearch Institute ~MRI! and the CommunicationsResearch Laboratory ~CRL!.26 The AESyISTS ~In-stitute for Space and Terrestrial Science! DIAL sys-

Fig. 19. Results of retrievals for aerosol effective number density,effective radius, surface area, and volume corresponding to Fig. 18:solid curve, true profiles; dashed curve, retrieval results; dash–dotcurves, plus or minus the estimated statistical error.

Fig. 20. Backscatter ratio R profiles and extinction profile measured at Eureka on ~left! January 22, ~middle! January 26, and ~right!February 6, 1994.

tem has an output of ;6 W at 353 nm and uses a1-m-diameter main receiver mirror, whereas theMRIyCRL Nd:YAG lidar has an output of ;2 W at532 nm and 3 W at 1064 nm. The excimer operatesat 300 Hz, whereas the Q-switched Nd:YAG fires at10 Hz. A 30-cm-diameter telescope is used to collectthe 532-nm backscatter, whereas a 50-cm-diametertelescope is used for the 1064-nm channel.

At this time access to the full set of multiwave-length lidar signals exists only for the 1993–1994winter, during which time the AESyISTS DIAL wasonly a two-channel system, that is, the Ramanbackscatter from molecular nitrogen at 332 and 385nm was not observed simultaneously with the elas-tic returns at 308 and 353 nm. However, occasion-ally we took measurements of the Ramanbackscatter by manually replacing the interferencefilters in the secondary optics module. Owing tothe crude method of obtaining the Raman data andother factors ~ranging from various technical prob-lems with both lidars to extended periods of badweather!, an extended time series of aerosol mea-surements obtained simultaneously ~or nearly so!at 353, 385, 532, and 1064 nm does not exist for thisfull measurement period. However, the 1993–1994 data from the AESyISTS DIAL and the MRIyCRL Nd:YAG lidar have provided a few sample datasets with which principal component retrievalscould be carried out.

Figure 20 shows three sample multiwavelengthbackscatter ratio R profiles for the 1993–1994 winter.As mentioned above, acquiring the 385-nm Ramanbackscatter data required a manual change of opticalfilters, thus the 385-nm signals were not measuredsimultaneously with the other signals. However, inthe cases shown here the 385-nm signals were ac-quired just shortly before andyor after the other datawere obtained. The density profiles used in the re-

trievals were provided by radiosondes flown at leasttwice daily from Eureka.

The results of aerosol integral property retrievals

Fig. 21. Results of retrievals for aerosol effective radius, surfacearea, and volume for 22 January 1994. The dash–dot curvesshow the statistical uncertainty ~1 standard deviation!.

Fig. 22. Results of retrievals for aerosol effective radius, surfacearea, and volume for 26 January 1994. The dash–dot curvesshow the statistical uncertainty ~1 standard deviation!.


corresponding to the R profiles of Fig. 20 are shown inFigs. 21–23, respectively. Here the lower altitudewas limited by the minimum opening altitude of thelidar chopper at that time. Unfortunately no ancil-lary data exist, such as coincident in situ aerosol sizedistribution measurements, with which to comparethe results. However, the retrieved quantities areconsistent with each other and with the results of thesimulations, which indicate that the retrieved aerosolproperties are likely reliable to within a factor of 1.5and which are consistent with previous in situ obser-vations of the aerosol size distribution within andaround the northern hemisphere polar vortex.27

5. Conclusions

The present application of the principal componentmultiwavelength inversion procedure to real datahas thus far been limited to the three cases presentedhere. However, the results are self-consistent andare in good agreement with the results of simulationsperformed on synthetic lidar signals and with exist-ing in situ data. It is hoped that future lidar mea-surements allow comparison of the lidar-derivedparameters directly with the results of in situ sen-sors. The research presented here demonstratesthat, with regard to stratospheric sulfate aerosol, onecan invert multiwavelength lidar data in a consistentfashion, accounting for aerosol extinction withoutmaking any a priori assumptions as to the nature ofthe aerosol size distribution being sought. Ofcourse, one must still assume that the kernels arevalid, which in this study means assuming that theaerosol is spherical and of known refractive index.However, this is not an overly restrictive assumptionto make for spherical sulfate aerosol.

It has also been demonstrated that principal com-ponent analysis can be applied to lidar backscatterand extinction measurements to account for the ef-fects of aerosol extinction and to deduce profiles ofaerosol volume, surface area, and effective radius aswell as aerosol extinction and backscatter at wave-lengths other than those of the lidar measurements.The estimates of aerosol physical properties, how-ever, are only reliable ~depending on the exact cir-

Fig. 23. Results of retrievals for aerosol effective radius, surfacearea, and volume for 6 February 1994. The dash–dot curves showthe statistical uncertainty ~1 standard deviation!.


cumstances! to within a factor of ;1.5–2. Bycomparing the various measurement situations in-vestigated, we found that the addition of a Ramanlidar extinction profile measurement at 353 nm to theRayleighyMie backscatter at 353, 532, and 1064 nmyields a useful improvement in the accuracy of theretrievals, especially in the determination of aerosolsurface area. The subsequent addition of a mea-surement of the Raman N2 backscatter at 608 nmwould also generally yield an improvement. The im-provement of the accuracy of the retrievals by theaddition of more backscatter andyor extinction mea-surements will of course be limited owing to the in-creasing interdependence of the kernels.7,9 The useof angular light scattering data ~from a bistatic lidar!would be helpful in providing more usefulinformation,28–31 however, this was not consideredhere.

Appendix A: Error Magnification of Integral Quantities

Here the question of the degree of error magnificationthat can be expected when attempting to retrieve anintegral quantity of the unknown aerosol size distri-bution is examined. In particular, Eq. ~21! ~the er-ror magnification for the retrieval of integralquantities! is derived.

The predicted value of any integral quantity isgiven by

P 5 *0

`

w~r!v~r!dr, (A1)

where w~r! is the appropriate weighting function forthe quantity in question. j# was previously definedas equal to L21y2Utg# so that the solution v~r! can beexpressed as

v~r! 5 (i51

N

jifi~r!, (A2)

where the elements of f# ~r! 5 L21y2UK# t~r! are theorthonormal characteristic or principal componentfunctions.

Using Eq. ~A2!, transforms Eq. ~A1! to

P 5 (i51

N

ji *0

`

w~r!fi~r!dr. (A3)

Evidently then, if the errors in ji are uncorrelated,the error in P will be given by

dP2 5 (i51

N

dji2F*

0

`

w~r!fi~r!drG2

. (A4)

Using the relationship between j# and g# , the principalcomponent error vector ~d#j! relates to the measure-ment error vector e# as

d#j 5 L21y2Ute#

5 L21y2e# *, (A5)

where e# * 5 Ute# is the rotated error vector that willhave the same norm as e# owing to the fact that U isorthogonal. The relative error in P can then begiven as

dPP

5 5(i51

N Fl21y2ei9 *0

`

w~r!fi~r!drG2

F(i51

N

ji *0

`

w~r!fi~r!drG2 61y2

. (A6)

To compare the relative error in P with the relativemeasurement errors the magnitude of the measure-ment vector g must be expressed in similar terms.Expressing g in terms of j# one has

g# 5 UL1y2j#. (A7)

So using the fact that U is an orthogonal matrix, oneobtains the magnitude of the measurement vector~ug# u2!:

ug# u2 5 j#tLj#

5 (i51

n

liji. (A8)

Thus using Eqs. ~A6! and ~A8! one obtains the errormagnification for a given integral quantity:

gint 5 FSdPP D2 ug# u2

ue# *u2G1y2

5 1H(i51

N Fl21y2ei9 *0

`

w~r!fi~r!drG2JS(i51

N

liji2D

F(i51

N

ji *0

`

w~r!fi~r!drG2S(i51

N

ei92D 2

1y2

.

(A9)

This relation is similar in form to the equation givenby Twomey9 for the error magnification related to thesize distribution retrieval but is modified by the pres-ence of the integrals over the weighted principal com-ponent functions.

The financial and technical support provided by theInstitute for Space and Terrestrial Science and theCanadian Atmospheric Environment Service isgratefully acknowledged. The financial assistanceof the Natural Sciences and Engineering ResearchCouncil of Canada is also acknowledged. The au-thors also thank the Japanese MRIyCRL lidar group,especially T. Shibata, for providing data obtained bytheir lidar at Eureka.

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principal component analysis applied to multiwavelength lidar aerosol backscatter and extinction...

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