quasi-analytical determination of noise-induced error limits in lidar retrieval of aerosol...

Quasi-analytical determination of noise-inducederror limits in lidar retrieval of aerosolbackscatter coefficient by the elastic,

two-component algorithm

Michaël Sicard,1,2,* Adolfo Comerón,1 Francisco Rocadenbosch,1,2

Alejandro Rodríguez,1 and Constantino Muñoz1

1Remote Sensing Laboratory (RSLAB), Department of Signal Theory and Communications,Universitat Politènica de Catalunya

2 Institut d’Estudis Espacials de Catalunya—Centre de Recerca de l’Aeronàuticai de l’Espai / Universitat Politènica de Catalunya

*[email protected]

Received 19 June 2008; revised 18 November 2008; accepted 20 November 2008;posted 21 November 2008 (Doc. ID 97645); published 7 January 2009

The elastic, two-component algorithm is the most common inversion method for retrieving the aerosolbackscatter coefficient from ground- or space-based backscatter lidar systems. A quasi-analytical formu-lation of the statistical error associated to the aerosol backscatter coefficient caused by the use of real,noise-corrupted lidar signals in the two-component algorithm is presented. The error expression dependson the signal-to-noise ratio along the inversion path and takes into account “instantaneous” effects, theeffect of the signal-to-noise ratio at the range where the aerosol backscatter coefficient is being computed,as well as “memory” effects, namely, both the effect of the signal-to-noise ratio in the cell where the in-version is started and the cumulative effect of the noise between that cell and the actual cell where theaerosol backscatter coefficient is evaluated. An example is shown to illustrate how the “instantaneous”effect is reduced when averaging the noise-contaminated signal over a number of cells around the rangewhere the inversion is started. © 2009 Optical Society of America

OCIS codes: 010.1100, 010.1110, 010.3640, 280.1100, 280.1350, 280.3640.

1. Introduction

While it is recognized that atmospheric aerosols havean important effect on the Earth’s radiation budget,their quantitative contribution to the surface radia-tive forcing still has one of the largest uncertaintyvalues among those of the different forcing factors[1]. In recent years a big effort has been devoted toreduce this uncertainty through systematic mea-surements by, generally speaking, in situ and remotesensing techniques. In this respect, lidars offer theunique capability of providing height-resolved aero-

sol measurements, which is a necessary feature toimprove the assessment of the aerosol effect inthe radiation balance. Aerosol lidar networksoperating in a coordinated manner, and frequentlyassociated to other in situ or remote sensing instru-mentation, are implemented or underway to providecontinental-scale four-dimensional (space–time) in-formation on aerosol distributions (e.g., seeRefs. [2,3]). It is expected that the spatial coverageis extended through existing actions to integratethe regional aerosol lidar networks into a global sys-tem [4]. At the same time, space-based lidars, like theCloud-Aerosol Lidar and Infrared Pathfinder Satel-lite Observations (CALIPSO) [5,6] mission, are al-ready providing global, albeit nonsimultaneous,

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176 APPLIED OPTICS / Vol. 48, No. 2 / 10 January 2009

information on aerosol and cloud distributions, andthis capability will be increased with future systems(e.g., Refs. [7,8]).In this global lidar aerosol sensing scenario of co-

operation between space-based instruments andground-based coordinated networks, data quality isa requirement for lidar measurements to have asignificant effect in atmospheric models and in theretrieval of aerosol microphysical properties. There-fore, there is no doubt that ground-based, elastic,backscatter lidar systems, still the most common li-dar systems used at the present time, and space-based, elastic, backscatter systems have to be fullycharacterized in terms of error budget. The tradi-tional way to retrieve the aerosol backscatter coeffi-cient from signals from such systems goes throughthe use of the so-called two-component algorithm[9–11]. Prior to the results from Refs. [9–11], severalauthors had significantly contributed to the estab-lishment of the first solution of the elastic lidar equa-tion known as the single-component algorithm [12–14], which was deduced from the first measurementsof rain intensity, equivalent to aerosol concentrationin lidar terms, by radar in 1954 [15]. The stable ana-lytical solution (still in the single-component version)of the lidar equation was formulated for the first timein Ref. [16]. Since then, many studies have investi-gated the effect of both assumptions necessary tosolve the two-component algorithm: the boundaryvalue [17–21] and the extinction/backscatter ratioprofile [20,22,23].

When dealing with real, noise-corrupted lidar sig-nals, it is important to be able to quantify the errorcaused, not only by the two previous assumptions,but also by the effect of noise on the used algorithm.Comerón et al. [24] formulated the statistical errorcaused by noise contaminating real-life signals onthe total backscatter coefficient retrieval by thesingle-component algorithm. However, the single-component algorithm is not widely used in practicesince it does not distinguish between moleculesand aerosols. The purpose of this paper is to general-ize the results from Ref. [24] and to propose a quasi-analytical formulation of the statistical error limitsof the aerosol backscatter coefficient retrieval forthe two-component algorithm due to noise-corruptedsignals.

Section 2 examines the differences between thesingle- and two-component algorithms. Section 3gives the quasi-analytical formulation of the statisti-cal error provoked by noise-corrupted signals onthe backscatter coefficient retrieval with the two-component algorithm. Section 4 illustrates with realdata the result found in Section 3. Conclusions arepresented in the last section.

2. Single-Component and Two-Component Algorithms

In the late sixties the single-component algorithm[12–14] was formulated in terms of backscatter coef-ficient β as a function of the range R and the receivedpower as

βðRÞ ¼ βmR2PðRÞR2

mPðRmÞ þ 2βmRRmR SðzÞz2PðzÞdz

; ð1Þ

where PðRÞ is the power received from range R, Rm isthe range from which the inversion is started, βm ¼βðRmÞ is the boundary condition, and SðRÞ ¼ αðRÞ=βðRÞ, the so-called total lidar ratio, is the ratio be-tween the atmosphere total extinction coefficientαðRÞ and the total backscatter coefficient βðRÞ. Fol-lowing the results from Ref. [16], Eq. (1) is a reformu-lation of Eq. (9) from Ref. [14] after switching theintegration bounds R and Rm of the integral in thenumerator, and changing the sign in front of it.

In the two-component algorithm [9–11] the effectsof the molecules and of the aerosols are consideredseparately:

βðRÞ ¼ βaðRÞ þ βRðRÞ ¼βmR2PðRÞ exp

�2RRmR ½SaðzÞ � 8π=3�βRðzÞdz

�

R2mPðRmÞ þ 2βm

RRmR SaðzÞz2PðzÞ exp

�2RRmz ½SaðxÞ � 8π=3�βRðxÞdx

�dz

; ð2Þ

where βaðRÞ represents the aerosol backscatter coef-ficient, βRðRÞ the Rayleigh (molecular) backscattercoefficient (which can be assumed to be known),andSaðRÞ ¼ αaðRÞ=βaðRÞ the aerosol lidar ratio givenby the ratio between the aerosol extinction coefficientαaðRÞ and the aerosol backscatter coefficient βaðRÞ.This is indeed the solution that is usually employedby the lidar community to retrieve the aerosol back-scatter coefficient when dealing with ground-basedbackscatter elastic lidars pointing in a single direc-tion and no Raman measurements are available.Even though the single-component solution as for-mulated in Eq. (1) can lead mathematically to thesame results, in practice it requires the knowledgeof the total lidar ratio, SðRÞ, which is harder to guessthan that of aerosols and less useful in practice.

10 January 2009 / Vol. 48, No. 2 / APPLIED OPTICS 177

3. Retrieval in the Presence of Noise

A. General Expression

In Ref. [24] the retrieval in the presence of noise wasconsidered for the single-component algorithm. Inwhat follows the results are generalized for thetwo-component algorithm case. In a general real si-tuation the photoreceiver output is corrupted bynoise and will be proportional to

P̂ðRÞ ¼ PðRÞ þ nðRÞ; ð3Þ

where nðRÞ represents the instantaneous “powernoise” affecting the measurement at range R. Asin Ref. [24] it is assumed that in general nðRÞ canbe made a zero-mean process that can therefore takepositive and negative values, by subtracting the biasterms from the signal at the output of the photorecei-ver. The magnitude retrieved with the algorithm gi-ven by Eq. (2) is then β̂ðRÞ. By substituting Eq. (3)into the expression of β̂ðRÞ, one obtains after goingthrough some algebra

1

β̂ðRÞ ¼�

1βðRÞ þ

1βm

R2m nðRmÞR2PðRÞ exp

��2

ZRm

R½SaðzÞ � 8π=3�βaðzÞdz

�

þ2RRmR SaðzÞz2nðzÞ exp

�2RRmz ½SaðxÞ � 8π=3�βRðxÞdx

�dz

R2PðRÞ exp�2RRmR ½SaðzÞ � 8π=3�βRðzÞdzg

�PðRÞ

PðRÞ þ nðRÞ : ð4Þ

At this point we compare Eq. (4) with theequivalent one in the error analysis for the single-component algorithm, Eq. (4) in Ref. [24], renamedβ̂scðRÞ in order to avoid confusion and which is re-peated here for the reader’s convenience with thevariable change CðzÞ ¼ 1=SðzÞ:

1

β̂scðRÞ¼

�1

βðRÞ þ1βm

R2mnðRmÞR2PðRÞ

þ 2

R2PðRÞZ

Rm

RSðzÞz2nðzÞdz

�PðRÞ

PðRÞ þ nðRÞ :

ð5Þ

We note that both expressions are formally identi-cal if we take into account the correspondences sta-ted in Table 1 between Eqs. (5) (single-component)and (4) (two-component).With these correspondences and drawing on the

results of Ref. [24], we can write in the case of thetwo-component inversion

β̂ðRÞ ¼ β̂aðRÞ þ βRðRÞ

¼�1þ nðRÞ

PðRÞ� βðRÞ1þ ζmðRÞ þ ζiðRÞ

; ð6Þ

where

ζmðRÞ ¼βðRÞβm

R2m nðRmÞR2PðRÞ exp

�2Z

Rm

R½SaðzÞ

� 8π=3�βRðzÞdz�; ð7Þ

ζiðRÞ¼ 2βðRÞexp��2

ZRm

R½SaðzÞ�8π=3�βRðzÞdz

�

RRmR z2SaðzÞnðzÞexp

�2RRmz ½SaðxÞ�8π=3�βRðxÞdx

�dz

R2PðRÞ :

ð8Þ

B. Standard Deviation of the Noise Terms

If nðRÞ is a zero-mean process, nðRÞ=PðRÞ, then ζmðRÞand ζiðRÞ are zero-mean random variables for a givenR as, for all their apparent complexity, their definingexpressions, Eqs. (7) and (8), are but linear opera-tions on nðRÞ. The standard deviation of nðRÞ=PðRÞis obviously σnðRÞ=PðRÞ, with σnðRÞ being the stan-dard deviation of nðRÞ, and clearly represents thenoise-to-signal ratio at range R.

The term ζmðRÞ represents the effect of the noise inthe boundary cell where the inversion is started.Calling σnm ¼ σnðRmÞ and using a procedure parallelto that employed in Ref. [24], the standard deviationof ζmðRÞ can be cast as

σζmðRÞ ¼σnm

PðRmÞexp

�−2

ZRm

RSaðzÞβðzÞdz

�; ð9Þ

which shows the stabilizing effect of the backwardalgorithm also in the effect of noise in that the expo-nential function will be progressively lesser than 1


for decreasing values ofR < Rm. This effect, however,will be small for optically thin atmospheres.The term ζiðRÞ corresponds to the effect of noise

integrated along the retrieval path. Under the rea-sonable assumption that noises in different resolu-tion cells are uncorrelated, which will occur if Δτ ≥12B, with Δτ the sampling period of the acquisitionsystem and B the photoreceiver bandwidth, we canapproximate [24]

hnðz1Þnðz2Þi ¼ σ2nðz1ÞΔRδðz1 � z2Þ; ð10Þ

and reasoning along the same lines as in Ref. [24] thestandard deviation of ζiðRÞ can be shown to be

σζiðRÞ ¼2βðRÞR2PðRÞ exp

��2

ZRm

R½SaðzÞ � 8π=3�βRðzÞdz

�

×�ΔR

ZRm

RS2aðzÞσ2nðzÞz4 exp

�4Z

Rm

z½SaðxÞ

� 8π=3�βRðxÞdx�dz

�1=2

: ð11Þ

This equation can alternatively be written

σζiðRÞ ¼2βm

R2mPðRmÞ

exp��2

ZRm

RSaðzÞβðzÞdz

�

×�ΔR

ZRm

RS2aðzÞσ2nðzÞz4 exp

�4Z

Rm

z½SaðxÞ

� 8π=3�βRðxÞdx�dz

�1=2

: ð12Þ

C. Error Bounds

Calling, for the sake of notation simplicity, ηðRÞ ¼nðRÞ=PðRÞ and ζðRÞ ¼ ζmðRÞ þ ζiðRÞ, the estimatedaerosol backscatter coefficient is, from Eq. (6),

β̂aðRÞ ¼ βðRÞ 1þ ηðRÞ1þ ζðRÞ � βRðRÞ: ð13Þ

We call

1þ ηðRÞ1þ ζðRÞ ¼ 1þ lðRÞ: ð14Þ

Note that themean of lðRÞ is not zero, whichmakesβ̂aðRÞ biased.The interval ðβaðRÞ; βaðRÞ þ luβðRÞÞ in which β̂aðRÞ

is found with a certain probability pu is calculated by

finding the value lu that defines the interval ð0; luÞ inwhich the random variable lðRÞ given by Eq. (14) isfound with probability pu. The probability pu is cal-culated as the integral of the joint probability densityfunction of ηðRÞ and ζðRÞ over the shadowed area inFig. 1, limited by the lines

1þ ηðRÞ1þ ζðRÞ ¼ 1;

1þ ηðRÞ1þ ζðRÞ ¼ 1þ lu: ð15Þ

Under the approximation represented by Eq. (10),ζmðRÞ and ζiðRÞ can be considered uncorrelated, be-cause the weight of any random variable nðzÞdz inthe integral in Eq. (8) is infinitesimally small. Froma physical point of view this approximation meansthat we consider negligible the effect of the noisein a resolution cell as compared to the linear combi-nation of noises in many other resolution cells. Wewill assume in addition henceforth that nðRÞ is aGaussian stochastic process to a good degree of ap-proximation. Actually the noise sources of a photore-ceiver can be broadly classified into shot noise andthermal noise. Thermal noise can be representedby a Gaussian process, and, if the rate rs (photons=sor charge carriers=s) of a shot-noise source is high en-ough compared to the system bandwidth B, shotnoise can also be approximated by a Gaussian pro-cess [25]. We will assume that the shot-noise sources,including the flow of incoming photons, satisfy thiscondition. Hence the random variables ζmðRÞ andζiðRÞ follow a joint Gaussian law because they are ob-tained as the result of applying linear operators tothe basic Gaussian process nðRÞ [26], and theirsum ζðRÞ is also a Gaussian random variable. Its

standard deviation is σζ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ2ζiðRÞ þ σ2ζmðRÞ

q, and

its mean is zero because ζmðRÞ and ζiðRÞ are uncor-related zero mean.

For similar reasons, ηðRÞ and ζðRÞ can be treatedas uncorrelated, zero-mean random (except in theparticular but virtually irrelevant case that R ¼ Rm)and jointly Gaussian variables. Since they are uncor-related, their joint probability density function isgiven by

Table 1. Correspondence of Terms in the Solution of β̂ðRÞ between theSingle-Component and the Two-Component Algorithms

Single-Component Eq. (5) Two-Component Eq. (4)

nðRÞ ↔nðRÞ expf2 RRmR ½SaðRÞ − 8π=3�βRðzÞdzg

PðRÞ ↔PðRÞ expf2 RRmR ½SaðRÞ − 8π=3�βRðzÞdzg

SðRÞ ↔SaðRÞ

Fig. 1. Integration domain for calculating pu.


f ηζðη; ζÞ ¼1

2 πσησζexp

�−

η22 σ2η

�exp

�� ζ22σ2ζ

�: ð16Þ

According to the standard deviation values dis-cussed in Subsection 3.B, ση ¼ σnðRÞ=PðRÞ (the expli-cit dependence on R of ση is dropped in the notationfor simplicity). The probability pu is then given by

pu ¼ 12 πσησζ

Z∞

�1exp

�� ζ22 σ2ζ

�Z ζð1þluÞþlu

ζ

× exp�� η22 σ2η

�dηdζ; ð17Þ

which can also be written as

pu ¼ 1

2ffiffiffiffiffiffiffi2 π

pσζ

Z∞

�1exp

�� ζ22 σ2ζ

��erf

�ζð1þ luÞ þ luffiffiffi2

pση

�

� erf� ζffiffiffi

2p

ση

��dζ: ð18Þ

Given a value, Eq. (18) can be solved for the corre-sponding lu value.Likewise it can be shown that the interval ðβaðRÞ �

llβðRÞ; βaðRÞÞ in which β̂a lies with probability pl isfound by solving for ll the equation

pl ¼1

2ffiffiffiffiffiffiffi2 π

pσζ

Z∞

�1exp

�� ζ22 σ2ζ

��erf

� ζffiffiffi2

pση

�

� erf�ζð1� llÞ � llffiffiffi

2p

ση

��dζ: ð19Þ

It can be shown that if σζ ≪ 1 and lu ≪ 1, ll ≪ 1,

then pu ≈12 erf ½lu=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðσ2η þ σ2ζ Þ

q� and pl ≈

12 erf ½ll=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðσ2η þ σ2ζ Þq

�, in agreement with the classical(perturbational) error propagation method, in whichassuming a small value of σζ we would have approxi-mated ð1þ ηÞ=ð1þ ζÞ ≈ 1þ η� ζ.4. Example

The error limits resulting from the calculation of thestandard deviations of ηðRÞ and ζðRÞ in the case ofthe two-component algorithm are illustrated byusing the same input data as in Ref. [24]. The photo-receiver electrical bandwidth was 10MHz, and theoutput was digitized at B ¼ 20 megasamples=s(hence Δτ ¼ 1

2B), yielding a range resolution ΔR ¼7:5m. Other technical details about the instrumentused can be found in Ref. [24]. The lidar profile wasacquired at the wavelength of 1064nm. Figure 2shows the noise-contaminated lidar range-correctedsignal between 600 and 6180m. Several aerosollayers are clearly observed until 4900m approxi-mately; we assume that the atmosphere is purelymolecular above that distance. As in the last figureof Ref. [24], we explored two cases depending on thenumber of resolution cells,N, used to average the va-lues of P̂ðRÞ around Rm and to force P̂ðRmÞ to this

average value: N ¼ 1 (no average) and N ¼ 17 (aver-age between Rm − 60 and Rm þ 60m). A constantaerosol lidar ratio value of Sa ¼ 50 sr was used toperform the inversion, and βm was taken accordingto the molecular atmosphere model proposed inRef. [27] for the 1064nm wavelength under standardconditions of pressure and temperature. Note how-ever that we are not emphasizing the physical char-acteristics of the atmosphere for this particular case—whether the aerosol lidar ratio accurately corre-sponds to the actual situation or if corrections toβm should be made to take into account departuresof the actual atmospheric conditions from the stan-dard ones—but rather the mathematical aspects ofthe effect of noise on the inversion. In this respect,note also that although a constant lidar ratio is usedin the example, the formulation allows for a functionwith an arbitrary dependence on range that could beemployed if by some means information on the lidarratio is available.

The standard deviations of η, ζm, and ζi—ση, σζm ,and σζi—are represented in Fig. 3. Note that thenumber of cells N only affects σζm. As the range de-creases, the signal-to-noise ratio increases, thereforeση decreases, as shown in the figure. Instead, by aver-aging over N ¼ 17 cells around Rm, σζm decreases bya factor of 4 to 5 compared to the initial value whenno average is performed (N ¼ 1), to a value around0.05. In that case, the approximation σζm ≪ 1 canbe made as Fig. 4 also shows. As expected, the termσζi is very small compared to the other two standarddeviations.

Figure 4 shows the aerosol backscatter coefficientand its error bounds in a 68% confidence intervalfound by setting pu ¼ pl ¼ 34% (to maintain a com-mon criterion with the probability of one standarddeviation of a Gaussian probability law below andabove the “true” value, usually employed to defineerror bars in classical error-propagation approaches)forN ¼ 1 (Fig. 4(a)) andN ¼ 17 (Fig. 4(b)). The “true”

Fig. 2. Lidar range-corrected signal between 600 and 6180m for4000 pulses integrated at 1064nm.


value has been taken as the central profile in the fa-mily of 17 profiles obtained when the inversion start-ing range is varied from 6000 to 6120mwithN ¼ 17.In Fig. 4(a) the asymmetrical behavior of the errorbounds is clearly visible, because the value of σζforN ¼ 1 varies around 0.25, which cannot be consid-ered small compared to 1. However, when N ¼ 17(Fig. 4(b)) the upper and lower error bounds are al-most perfectly symmetric, since σζ ≈ 0:05 ≪ 1, andclassical error propagation is justified. In the casepresented here, the error bars on the aerosol back-scatter coefficient are lower than 15% below 5000m and lower than 7% below 4000m, where the sig-nal-to-noise ratio is around 11 and 42, respectively.When averaging the values of P̂ðRÞ around Rm to re-duce the effect of noise in the cell from which the in-version is started, the effective value of σnm is dividedby

ffiffiffiffiffiN

p. However, when proceeding in this manner

one cannot indefinitely increase N to decrease σnmas the maximum allowable number of averaged cellsis limited by the dependence on R of the receivedpower being well approximated by a linear lawthrough the range covered by the N cells [24]. Inpractice the linear approximation will be safely satis-fied if NΔR does not exceed a few hundreds of me-ters. More sophisticated approaches, like adjustinga theoretical molecular backscatter profile to themeasured one, can be used to filter out noise affectingthe starting cell [28].

5. Conclusions

Expressions have been derived for the calculation ofstatistical error bounds for the aerosol backscattercoefficient retrieved by the two-component lidar in-version method. The expressions depend on thesignal-to-noise ratio along the inversion path andtake into account “instantaneous” effects (the effectof the signal-to-noise ratio at the range where theaerosol backscatter coefficient is being computed)as well as “memory” effects, namely, both the effect

of the signal-to-noise ratio in the cell where the inver-sion is started and the cumulative effect of the noisebetween that cell and the actual cell where the aero-sol backscatter coefficient is evaluated. The standarddeviations of the random variables representingthese effects allow assessment of their relative im-portance. The utilization of these expressions hasbeen illustrated with an example on real lidar data.Although the example presented has been done onthe two-component algorithm (backward) and ondata obtained from a ground-based, upward lookinglidar, the formulation developed in this paper is alsovalid for the forward solution of the elastic lidar in-version and for any line of sight. They are also validfor signal-to-noise ratio situations in which aperturbational classical error propagation approachwould not be valid (although they reduce to error pro-pagation results for low enough standard deviationsof the noise terms). It has been shown how the effectof the signal-to-noise ratio at the range where the in-version is started can be reduced by averaging the

Fig. 4. Lower (dotted curve) and upper (dashed curve) estimated68% confidence interval for the inverted backscatter coefficient:(a) N ¼ 1 and (b) N ¼ 17 ½Rm − 60;Rm þ 60m�.

Fig. 3. Standard deviations, ση, σζm , and σζi , of η (solid curve), ζm(dashed and dotted curves), and ζi (dash-dotted curve), respec-tively. The axes legend of σζi (dash-dotted curve) is on the left sideof the figure. The axes legends of ση and σζm are on the right side ofthe figure.


noise-contaminated signal over a number of cellsaround that range.The method presented in this paper permits as-

sessment in a quantitative manner the statistical er-ror produced on the retrieved aerosol backscattercoefficient when inverting noise-corrupted lidar sig-nals. With the quasi-analytical formulas supplied inthis paper, the calculation of the error produced onthe aerosol backscatter coefficient retrieval fromnoise-corrupted lidar signals only requires a simpleprogram dedicated mainly to solve Eqs. (18) and (19).

The work presented here was supported by theEuropean Union and FEDER funds under the EAR-LINET-ASOS project (EU Coordination Action, con-tract 025991 (RICA)); by the European Space Agencyunder the project 21487/08/NL/HE; and by the MI-CINN (Spanish Ministry of Science and Innovation)and FEDER funds under the project TEC2006-07850/TCM, and the Complementary Actions CGL20007-28871-E/CLI, CGL2006-26149-E/CLI, andCGL2008-01330-E/CLI. M. Sicard is grateful to theMICINN for the Ramón y Cajal position he holds. Fi-nally, the authors wish to thank both referees for thehistorical insights they provided to reference prop-erly the elastic, single wavelength lidar inversionmethods.

References1. P. Forster, V. Ramaswamy, P. Artaxo, T. Berntsen, R. Betts,

D. W. Fahey, J. Haywood, J. Lean, D. C. Lowe, G. Myhre,J. Nganga, R. Prinn, G. Raga, M. Schulz, and R. Van Dorland,“Changes in atmospheric constituents and in radiative for-cing,” in Climate Change 2007: The Physical Science Basis.Contribution of Working Group I to the Fourth Assessment Re-port of the Intergovernmental Panel on Climate Change,S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis,K. B. Averyt, M. Tignor, and H. L. Miller, eds. (CambridgeUniv. Press, 2007), pp. 129–234.

2. J. Bösenberg, A. Ansmann, J. M. Baldasano, D. Balis, C. Böck-mann, B. Calpini, A. Chaikovsky, P. Flamant, A. Hågård,V. Mitev, A. Papayannis, J. Pelon, D. Resendes, J. Schneider,N. Spinelli, T. Trickl, G. Vaughan, G. Visconti, andM.Wiegner,“EARLINET: A European aerosol research lidar network,” inAdvances in Laser Remote Sensing of the Atmosphere,A. Dabas, C. Loth, and J. Pelon, eds. (Edition Ecole Polytech-nique, 2001), pp. 155–158.

3. E. J. Welton, J. R. Campbell, J. D. Spinhirne, and V. S. Scott,“Global monitoring of clouds and aerosols using a network ofmicro-pulse lidar systems,” Proc. SPIE 4153, 151–158 (2001).

4. R. Hoff, G. Pappalardo, and J. Bösenberg, “GAWAerosol LidarObservations Network (GALION),” paper presented at theSymposium on Recent Developments in Atmospheric Applica-tions of Radar and Lidar of the 88th annual meeting of theAmerican Meteorological Society, New Orleans, Louisiana,US, 20–24 July 2008.

5. D. Winker, J. Pelon, and M. McCormick, “Initial results fromCALIPSO,” in Reviewed and Revised Papers Presented at the23rd International Laser Radar Conference, C. Nagasawa andN. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006),pp. 991–994.

6. D. Winker, W. Hunt, and M. McGill, “Initial performance as-sessment of CALIOP,” Geophys. Res. Lett. 34, L19803 (2007).

7. P. Dubock, M. Endmann, and P. Ingmann, “Progress withADM-Aeolus, the Spaceborne Doppler Wind Lidar ADM,” in

Reviewed and Revised Papers Presented at the 23rd Interna-tional Laser Radar Conference, C. Nagasawa andN. Sugimoto,eds. (Tokyo Metropolitan Univ., 2006), pp. 1011–1014.

8. A. Hélière, J. L. Vézy, A. Lefebvre, W. Leibrandt, C. C. Lin,T. Wehr, T. Kimura, and H. Kumagai, “The ESA EarthCAREMission: mission concept and lidar instrument pre-development,” in Reviewed and Revised Papers Presented atthe 23rd International Laser Radar Conference, C. Nagasawaand N. Sugimoto, eds. (Tokyo Metropolitan Univ., 2006),pp. 1041–1044.

9. F. G. Fernald, B. M. Herman, and J. A. Reagan, “Determina-tion of aerosol height distribution by lidar,” J. Appl. Meteorol.11, 482–489 (1972).

10. F. G. Fernald, “Comments on the analysis of atmospheric lidarobservations,” in Proceedings of the 11th International LaserRadar Conference, NASA Conf. Publ. 2228, 213–215 (1982).

11. F. G. Fernald, “Analysis of atmospheric lidar observations:some comments,” Appl. Opt. 23, 652–653 (1984).

12. E. W. Barrett and O. Ben-Dov, “Application of the lidar to airpollutionmeasurements,” J. Appl.Meteorol. 6, 500–515 (1967).

13. W. Viezee, E. E. Uthe, and R. T. H. Collis, “Lidar observationsof airfield approach conditions: an exploratory study,” J. Appl.Meteorol. 8, 274–283 (1969).

14. P. A. Davis, “The analysis of lidar signatures of cirrus clouds,”Appl. Opt. 8, 2099–2102 (1969).

15. W. Hitschfeld and J. Bordan, “Errors inherent in the radarmeasurement of rainfall at attenuating wavelengths,” J.Atmos. Sci. 11, 58–67 (1954).

16. J. D. Klett, “Stable analytical inversion solution for processinglidar returns,” Appl. Opt. 20, 211–220 (1981).

17. Y. Sasano and H. Nakane, “Significance of the extinction/backscatter ratio and the boundary value term in the solutionfor the two-component lidar equation,” Appl. Opt. 23, 11–13(1984).

18. Q. Jinhuan, “Sensitivity of lidar equation solution to boundaryvalues and determination of the values,” Adv. Atmos. Sci. 5,229–241 (1988).

19. M. Matsumoto and N. Takeuchi, “Effects of misestimated far-end boundary values on two common lidar inversion solu-tions,” Appl. Opt. 33, 6451–6456 (1994).

20. L. R. Bissonnette, “Sensitivity analysis of lidar inversion algo-rithms,” Appl. Opt. 25, 2122–2125 (1986).

21. F. Rocadenbosch and A. Comerón, “Error analysis for the lidarbackward inversion algorithm,” Appl. Opt. 38, 4461–4474(1999).

22. H. G. Hughes, J. A. Ferguson, and D. H. Stephens, “Sensitivityof a lidar inversion algorithm to parameters relating atmo-spheric backscatter and extinction,” Appl. Opt. 24, 1609–1613(1985).

23. Y. Sasano, E. V. Browell, and S. Ismail, “Error caused by usinga constant extinction/backscattering ratio in the lidar solu-tion,” Appl. Opt. 24, 3929–3932 (1985).

24. A. Comerón, F. Rocadenbosch, M. A. López, A. Rodríguez,C. Muñoz, D. García-Vizcaíno, and M. Sicard, “Effects of noiseon lidar data inversion with the backward algorithm,” Appl.Opt. 43, 2572–2577 (2004).

25. A. Papoulis, Probability, Random Variables and StochasticProcesses,” (McGraw-Hill, 1965), Sect. 16–5.

26. A. Papoulis, Probability, Random Variables and StochasticProcesses,” (McGraw-Hill, 1965), Sect. 8.4.

27. B. A. Bodhaine, N. B. Wood, E. G. Dutton, and J. R. Slusser,“On Rayleigh optical depth calculations,” J. Atmos. OceanTechnol. 16, 1854–1861 (1999).

28. T. Trickl, Forschungszentrum Karlsruhe, Institut für Meteor-ologie und Klimaforschung (IMK-IFU), Kreuzeckbahnstrasse19, D-82467 Garmisch-Partenkirchen, Germany, private com-munication, 1 October 2008.


quasi-analytical determination of noise-induced error limits in lidar retrieval of aerosol...

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