backscatter error bounds for the elastic lidar two-component inversion algorithm

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012 4791

Backscatter Error Bounds for the Elastic LidarTwo-Component Inversion Algorithm

Francesc Rocadenbosch, Member, IEEE, Stephen Frasier, Senior Member, IEEE, Dhiraj Kumar,Diego Lange, Eduard Gregorio, and Michaël Sicard

Abstract—Total backscatter-coefficient inversion error boundsfor the two-component lidar inversion algorithm (so-calledFernald’s or Klett–Fernald–Sasano’s method) are derived in an-alytical form in response to the following three error sources:1) the measurement noise; 2) the user uncertainty in thebackscatter-coefficient calibration; and 3) the aerosol extinction-to-backscatter ratio. The following two different types of er-ror bounds are presented: 1) approximate error bounds usingfirst-order error propagation and 2) exact error bounds using atotal-increment method. Both error bounds are formulated in ex-plicit analytical form, which is of advantage for practical physicalsensitivity analysis and computational implementation. A MonteCarlo approach is used to validate the error bounds at 355-, 532-,and 1064-nm wavelengths.

Index Terms—Backscatter coefficient, Fernald algorithm, inver-sion, lidar, signal processing.

I. INTRODUCTION

E LASTIC-BACKSCATTER lidars are laser remote-sensinginstruments that are widely used as range-resolved

atmospheric probes [1]. Examples are found in ground-basedaerosol observation networks such as the European AerosolResearch Lidar Network (EARLINET) and the Micro-PulseLidar Network (MPLNET) [2] and in space missions such asthe Lidar In-Space Technology Experiment (LITE) [3] and,more recently, onboard the Cloud-Aerosol Lidar and Infrared

Manuscript received May 25, 2011; revised October 27, 2011 andFebruary 17, 2012; accepted March 17, 2012. Date of publication May 30,2012; date of current version October 24, 2012. This work was supportedin part by the European Union through the Aerosols, Clouds, and TraceGases Research Infrastructure Network (ACTRIS) Project under Contract262254, the European Space Agency under Contract 21487/08/NL/HE, andthe Spanish Ministry of Science and Innovation (MICINN) and European Re-gional Development (FEDER) Funds under Grant TEC2009-09106 and throughthe Chemistry–Aerosol Mediterranean Experiment (ChArMEx) Project un-der Complementary Actions/Grants CGL2011-13580-E/CLI and CGL2008-01330-E/CLI. The work of D. Kumar and D. Lange was supported by theGeneralitat de Catalunya/Agència de Gestió d’Ajuts Universitaris i de Recerca(AGAUR) and Spanish Ministry of Foreign Affairs and Cooperation (MAEC-AECID), respectively, through their pre-Ph.D. fellowships.

F. Rocadenbosch, D. Kumar, D. Lange, and M. Sicard are with the RemoteSensing Laboratory (RSLab), Department of Signal Theory and Communi-cations, Universitat Politècnica de Catalunya, 08034 Barcelona, Spain, andalso with the Institute for Space Studies of Catalonia–Aeronautics and SpaceResearch Center, Universitat Politècnica de Catalunya, 08860 Barcelona, Spain(e-mail: [email protected]).

S. Frasier is with the Microwave Remote Sensing Laboratory (MIRSL), De-partment of Electrical and Computer Engineering, University of Massachusetts,Amherst, MA 01003 USA.

E. Gregorio is with the Department of Agroforestry Engineering, Universitatde Lleida, 25001 Lleida, Spain, and also with the Remote Sensing Labora-tory (RSLab), Department of Signal Theory and Communications, UniversitatPolitècnica de Catalunya, 08034 Barcelona, Spain.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TGRS.2012.2194501

Pathfinder Satellite Observations (CALIPSO) satellite [Na-tional Aeronautics and Space Administration (NASA)–CentreNational d’Etudes Spatiales (CNES), 2006] [4].

The lidar equation is inherently underdetermined, becauseit contains two unknowns (the atmospheric extinction and thebackscatter coefficient) but only a single observable (the op-tical power returned as a function of time). Backscatter lidarsprovide only range-resolved profiles of attenuated backscat-ter signal [5]–[7]. This underdetermination is in contrast toother schemes such as elastic Raman systems, high-spectral-resolution lidars (HSRLs) [8], and variational multianglebackscatter-lidar retrievals [9], [10], all of which enable inde-pendent inversion of both aerosol extinction and backscattercoefficients [11].

Building on previous works, including the work of Hitschfeldand Bordan (1951) [12], Barret and Ben-Dov [13], Viezee et al.[14], Davis [15], Fernald [16], Collis and Russell [5], and Kohl[17], in 1981, Klett presented a stable inversion algorithm toinvert the elastic single-scattering lidar equation that assumes aone-component atmosphere [18], where there is no separationbetween aerosol and molecular components. In 1984, Fernaldpresented the two-component version of the algorithm [19],which Klett reformulated in a unified approach [20]. BothKlett’s (KLT) one-component algorithm and Fernald’s two-component algorithm [also known as Klett–Fernald–Sasano’smethod (KFS)] require additional inputs to resolve the under-determination of the lidar equation. They are a provision of:1) a boundary condition and 2) a range-dependent extinction-to-backscatter ratio. The boundary condition usually consistsof a known or presumed value of the extinction or backscattercoefficient at the far end of the range profile. This value isused as an absolute calibration for retrieving extinction orbackscatter coefficients at lesser ranges. Henceforth, we sim-ply refer to this approach as the calibration. The extinction-to-backscatter ratio may include both molecular and aerosoleffects, or it may include aerosol effects only. Many authorsuse the term “lidar ratio” to refer to the aerosol-only extinction-to-backscatter ratio. In this paper, we will make the distinctionbetween the “total” lidar ratio (including molecular effects) andthe aerosol-only lidar ratio when necessary.

Methods of assessing the calibration for the KLT one-component inversion algorithm were proposed by Klett [21],[22] and for the two-component algorithm by Sasano andNakane [23]. Several authors have carried out sensitivity studieswith regard to uncertainties in the lidar ratio [24], the impactof assuming a range-independent lidar ratio [25], uncertaintiesin the calibration [26], [27], and the forward/backward sta-bility of these inversion methods as a function of the opticaldepth [28].

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4792 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012

Although, historically, this was not the case, currently, bothone- and two-component inversion algorithms are usually for-mulated in backscatter-coefficient form. The backscatter coeffi-cient is always the preferred quantity for retrieval, because theextinction coefficient is estimated by multiplying the profileof the backscatter coefficient by the assumed extinction-to-backscatter ratio profile used as input to the retrieval. Errorsin the assumed lidar ratio may result in larger error-propagatederrors [6], particularly in situations of a complex layering ofaerosols [29]. Kunz [30] and Kovalev [31], [32] have proposedalternative variants (not the object of this paper) that allowtrustworthy extinction retrievals, where the far-end calibrationis replaced by the optical depth of the sounding path or bya near-end calibration and a nephelometer measurement. Thesynergetic combination of a backscatter lidar with a sun pho-tometer is also extensively used [33]. Furthermore, optimalestimation [34] and adaptive filtering [35], [36] methods offerthe possibility of incorporating different relevant information(such as optical thickness or spectral radiance measurements[37]) into the lidar inversion problem and to provide inversion-error indicators. These advanced methods, which usually findapplications in the context of global space-borne measurementsare, however, more complex.

Although, from a purely mathematical analysis, both the one-and the two-component algorithms yield equivalent solutions,the two-component algorithm is always the preferred approach.This case is because the KFS algorithm enables the use ofthe aerosol-only lidar ratio, a parameter that characterizes themicrophysical aerosol properties [38]. In contrast, KLT requiresa total lidar ratio, including molecular effects. From a physicalpoint of view, the assumption of a constant total lidar ratio isnot justified under relatively clear atmospheres. However, foroptically thick atmospheres, the aerosol component becomesdominant, and the total lidar ratio reduces to the aerosol lidarratio, which gave rise to the first applications of the one-component algorithm in the 1980s.

This paper concentrates on the two-component backscatter-coefficient inversion algorithm and is the fifth in a series[39]–[42] from the Remote Sensing Laboratory, Department ofSignal Theory and Communications, Universitat Politècnica deCatalunya related to study the behavior and error sensitivityof the one- and two-component algorithms. This paper firstcontributes a comprehensive analytical approach in explicitmathematical form that merges into a single body all thefollowing main error sources involved in the KFS inversion ofthe aerosol backscatter coefficient: 1) systematic errors due touncertainties in the calibration; 2) systematic errors due to arange-dependent aerosol lidar ratio; 3) random errors due to afinite signal-to-noise ratio (SNR) in the optoelectronic receiverof the lidar system at all ranges, except for the calibration; and4) random errors due to a finite SNR at the calibration range.The latter two error sources are separately considered, becauseit was shown in [40] that source 4 dominates.

Errors in the backscatter-coefficient calibration (error source1) and in the assumed lidar ratio (error source 2) are system-atic errors, because they induce biases in the retrieval of thebackscatter coefficient once they are encountered. These errorsare in contrast to the random errors induced by noise (errorsources 3 and 4). Although it is common to treat random errorsas drawn from independent Gaussian distributions with stan-

dard deviations adding in the mean square, systematic errorsmust separately be treated. These errors in the input parametersto the retrieval are more appropriately described by a worst casedeviation from their nominal value, assuming that input errorsmay uniformly be distributed between these worst case limits.A similar approach for the Raman lidar inversion algorithm isdescribed in [43].

This paper explicitly finds the backscatter-coefficient errorbounds for the KFS algorithm in both approximate and exactform. The following two different sets of explicit error boundsare introduced: 1) first-order derivative error bounds (approx-imate), which are the KFS counterpart of those found for theKLT algorithm in [41]), and 2) total-increment error bounds(exact) for the dominant error sources (sources 1, 2, and 4).These characteristics are new to the state of the art in the lidarcommunity.

This paper is organized as follows. In Section II, the KFSinversion algorithm is reviewed and reformulated in both back-ward and forward form. In Section III, first-order error boundsare presented. Then, in Section IV, total-increment (i.e., exact)error bounds are obtained for the dominant error sources. InSection V, both first-order and total-increment error boundsare cross examined and validated using a Monte Carlo (MC)method for the random component at wavelengths of 355 nm[ultraviolet (UV)], 532 nm [visible (VIS)], and 1064 nm[near infrared (NIR)]. Finally, concluding remarks are given inSection VI.

II. REVIEW OF THE KFS TWO-COMPONENT ALGORITHM

A. Review of the KFS Algorithm

The KFS inversion algorithm is formulated in backwardbackscatter-coefficient form as (1), which is shown at thebottom of the next page, where P (R) is the single-scatteringoptical-return lidar power, R is the range along sight, Saer(R)and Smol = 8π/3 are the aerosol and the molecular (Rayleigh)lidar ratios, respectively, βaer(R) and βmol(R) are theaerosol and molecular backscatter components, and Rm (R ≤Rm) is the calibration range. In (1), note that, despite thetwo-component separation, the term β(Rm) = βaer(Rm) +βmol(Rm) represents the total backscatter coefficient. In prac-tical tropospheric applications, the calibration range is usuallychosen in an atmospheric molecular reference range aloft,where the aerosol backscatter component becomes negligi-ble (βaer(Rm) � βmol(Rm)), and consequently, β(Rm) ≈βmol(Rm).

B. Modified Backward KFS Form

In this section, the aerosol and the molecular backscattercoefficients are assimilated into the total backscatter coefficient,β(R) = βaer(R) + βmol(R), and errors on the molecularbackscatter coefficient are neglected so that

dβaer(R) = dβ(R). (2)

This approach is justified, because the molecular compo-nent can be assumed to be very well known. In practice,the atmospheric molecular component is estimated from localtemperature/pressure radio-sounding measurements or a U.S.standard atmosphere model, given ground-level temperature

ROCADENBOSCH et al.: BACKSCATTER ERROR BOUNDS FOR LIDAR TWO-COMPONENT INVERSION ALGORITHM 4793

and pressure data [44]. Therefore, when calibrating in an at-mospheric layer dominated by molecular scattering, we have

βN = β(RN ) = βaer(RN ) + βmol(RN ) ≈ βmol(RN ). (3)

By introducing the discrete range, Rj = Rmin + (j −1)ΔR, j = 1 . . . N , where ΔR is the spatial resolution of thelidar data, and N is the number of range samples (or cells) tobe inverted, (1) can be rewritten in discrete form as

βj(βN , �S, �U) =βN Uj Fj(�S)

UN + 2βN Hj(�S, �U)(4)

where Uj , Fj , and Hj are the shorthand for U(Rj), F (Rj),and H(Rj), which are auxiliary functions that were evaluatedfor each range, and the vector �S is the range-dependent aerosollidar ratio Saer(Rj). The auxiliary functions Uj , Fj and Hj aredefined as

Uj =R2jP (Rj) (5)

Fj(�S) = exp[2Gj (�S)

](6)

where

Gj(�S) =

⎧⎨⎩

N∑i=j

wi

(Saeri − Smol

i

)βmoli j < N

0 j = N

(7)

Hj(�S, �U) =N∑i=j

wiSaeri UiFi(�S). (8)

In (7) and (8), wi, i = 1 . . . N denote generic integrationweights (e.g., wi = h = 1, i = 1 . . . N − 1;wN = 0 in the caseof rectangle integration, which requires N ≥ 2 points). Thenotation βj(βN , �S, �U) is a reminder that the total backscattercoefficient inverted at the range cell Rj depends on the totalbackscatter coefficient at the far-range calibration βN , the user-input range-dependent aerosol lidar ratio �S, and the range-corrected power �U . In the following section, the superscript“aer” for the aerosol lidar ratio is omitted; therefore, �S refers to�Saer, and the “aerosol lidar ratio” is simply addressed as “thelidar ratio.”

C. Comparison With the KLT One-Component Algorithm

When comparing KLT versus KFS in [41, eq. (5) and (6)]with (4), (6), and (8), the KLT-to-KFS correspondence (seeTable I) is obtained. The Uj into UjFj(�S) relationship agreeswith previous published results [42, Table 1], and Gj(�S, �U)

into Hj(�S, �U, �F ) is a new relationship, completing thetransformation.

TABLE IKLT-TO-KFS TRANSFORMATION RELATIONSHIPS. IN BOTH

ALGORITHMS, βj STANDS FOR THE TOTAL (AEROSOL PLUS

MOLECULAR) BACKSCATTER COEFFICIENT AT THE RANGE CELL Rj

D. Forward Case

In the forward-integration form of the KFS algorithm (i.e.,the calibration range located at the near end of the inver-sion range), the far-end calibration at R = RN is replaced bythe near-end calibration R = R1, i.e., βN → β1, in (4), and∑N

i=j(.) is replaced by −∑j

i=1(.) in all subsequent formulas.In so doing, (4) for the forward case becomes

βj(β1, �S, �U) =β1UjF

Fj (�S)

U1 + 2β1HFj (�S, �U)

(9)

where FFj (�S) = exp(2GF

j ) after (6), and GFj and HF

j are

defined following (7) and (8), but replacing∑N

i=j(.) by

−∑j

i=1(.), as aforementioned. This case leads to the well-known classic forward form, including a minus sign in frontof the factor of 2 in the denominator and in the exponentialarguments of (1). Note also that the minus sign that arisesfrom the aforementioned change of summations is algebraicallyequivalent to substituting Saer → −Saer and Smol → −Smol

into the KFS backward form of (4), which also accounts for theopposite signs of the backscatter-to-lidar-ratio derivatives of theforward/backward form (see Section IV-B).

III. FIRST-ORDER BACKSCATTER-COEFFICIENT

ERROR BOUNDS

This section parallels [41, Sec. 7], where the backscatter-coefficient error bounds are computed from the superpositionof error sources 1–4 (Section I) using a first-order derivativeapproach. Following [41, eq. (6)], we have⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩|dβj | =

∣∣∣ ∂βj

∂βNdβN

∣∣∣+ N∑k=1

∣∣∣ ∂βj

∂SkdSk

∣∣∣+

N−1∑k=1

∣∣∣ ∂βj

∂PkdPk

∣∣∣+ ∣∣∣ ∂βj

∂PNdPN

∣∣∣ ; j < N

|dβj | = |dβN |; j = N

(10)

βaer(R) =

[R2P (R)

]exp

{2

Rm∫R

[Saer(u)− Smol

]βmol(u)du

}

[R2mP (Rm)]

βaer(Rm)+βmol(Rm)+ 2

Rm∫R

Saer(u) [u2P (u)] exp

{2

Rm∫u

[Saer(v)− Smol]βmol(v)dv

}du

− βmol(R) (1)


where dβj is the total backscatter coefficient error at rangeRj , and dβN , dSk, dPk, and dPN , respectively, stand for errorsources (1)–(4).

For the case j < N , the terms ∂βj/∂βN , ∂βj/∂Pk, and∂βj/∂PN can readily be computed based on [41, eqs. (6), (8),and (9)] and the function substitutions indicated in the KLT-to-KFS transformation Table I. However, this procedure cannotbe followed when computing the errors due to the lidar ratio∂βj/∂Sk, because the KFS auxiliary functions Fj and Hj (6)(8) also depend on the lidar ratio. This case is revisited in theAppendix.

The case j = N in (10) expresses the assumed error onthe backscatter-coefficient calibration. Finally, the terms thatcomprise (10) and denoted as εj,1−4 are detailed in Table II.

The treatment of systematic and random errors is explainedas follows. The relationship between the backscatter-coefficientretrieval error (δεj,1) due to the calibration error (δβN

) shownin (36) is a straightforward modification of (27), where low-ercase deltas have been used to denote systematic errors. Therelationship between the retrieval error (σεj,3) due to noise inrange cells 1 . . .N− 1 (σUk

) shown in (41) is obtained from(32) by treating the range-corrected random noises (dUk) asindependent Gaussian random variables with standard deviationσUk

. The retrieval error due to noise in the calibration cell isgiven by (43) and is similarly derived using the approximationshown, which is described in [41] (Sec. 3.2 and Table 1,p. 3386). At this point, note that GN = 0 and FN = 1 [(6) (7)]and that, when considering error sources 3 and 4, d(UkFk) =FkdUk, k = 1 . . . N , because the only “fluctuating” variabledue to noise is the range-corrected power Uk. A few commentsare in order.

First, the relative impacts on the retrieval of the backscatter-coefficient calibration error (δεj,1) and the standard deviationof the noise at the calibration cell (σεj,4) may be compared byevaluating their ratio as

δεj,1σεj,4

≈∣∣∣∣UN

βN

∣∣∣∣ δβN

σUN

= SNRNεβNr . (11)

Here, εβNr = δβN

/βN is the relative error in the backscatter-coefficient calibration, and SNRN is the SNR at the calibrationrange, R = RN . Thus, it is not necessary to carry out separatesimulations to evaluate the separate impacts of error sources 1and 4. Although it was shown in [41] that the relative impactsof these different error sources are related by an equation that isanalogous to (11), it should not be interpreted that a systematicerror can be derived from the random error, or vice versa.

Second, with regard to errors due to the measurement noise(σεj,3 and σεj,4), based on (41), the backscatter-coefficient erroron the jth range cell is inversely proportional to both the SNRat that cell, SNRj = Uj/σUj

, and a “cross-cell SNR,” definedas SNRj,k = Uj/σUk

. A similar dependence was found in[40] and [41] and, earlier, by Knauss [45], who predicted aninverse SNR sensitivity. With regard to σεj,4 , (43) can be rewrit-ten as σεj,4 ≈ |(β2

jUN/βNUjFj)|(1/SNRN ) (see similarjustification steps in [41, p. 3383]. It emerges that a finite SNRat the calibration range propagates errors to all the range cells.

With regard to error due to a range-dependent lidar ratio(δεj,2), as a first approximation, we define a systematic lidar-

TABLE IITOTAL BACKSCATTER-COEFFICIENT ERROR-PROPAGATED TERMS FOR

THE KFS BACKWARD INVERSION ALGORITHM IN RESPONSE TO ERROR

SOURCES 1–4 (SEE SECTION III). FOR THE CASE j = N , THE TOTAL

BACKSCATTER-COEFFICIENT ERROR IS DIRECTLY THE CALIBRATION

ERROR. FOR THE KFS FORWARD ALGORITHM, CONSIDER

THE CHANGES IN SECTION II-D

ratio relative error p that relates the lidar-ratio error to the truerange-dependent atmospheric lidar ratio as [41]

dS(R) = pS(R) ⇔ dSk = pSk. (12)

Equivalently, the atmospheric lidar ratio is assumed to lie withinS(R)(1± |p|). The error bound computation uses the first-order series expansion of (4) around p. Toward this end, (4)is rewritten as a function of lidar-ratio perturbation p as

βj(p) =βN Uj Fj(p)

UN + 2βN Hj(p)(13)


where the incremental auxiliary function Fj(p) is related toGj(p) through (6), and Gj(p) and Hj(p) in (7) and (8) become

Gj(p) =

{(1 + p)Ij,1 −Kj ; j < N0 j = N

(14)

where

Ij,1 =

N∑i=j

wiβmoli Saer

i , Kj =

N∑i=j

wiβmoli Smol

i (15)

Hj(p) = (1 + p)

N∑i=j

wiSaeri UiFi(p). (16)

Based on (14) (15) and (6), Fj(p) takes the form

Fj(p) = exp [2Gj(p)] = Fj(0) exp(2pIj,1). (17)

Finally, the backscatter-coefficient error is obtained after first-order series expansion as

δεj,2 ≈∣∣∣∣∣(

∂βj

∂p

∣∣∣∣p=0

)p

∣∣∣∣∣ (18)

where the superscript “S” denotes “due to the lidar ratio.” (18)is computed by substituting the proportionality condition of(12) into the general expression of the propagated lidar-ratioerror εj,2 [(28)–(31)]. The result is summarized in Table III andyields symmetrical error bounds.

IV. TOTAL-INCREMENT BACKSCATTER-COEFFICIENT

ERROR BOUNDS

Total-increment error bounds stand for infinite-order or,equivalently, exact error bounds. The procedure is concep-tually simple, because it reduces to compute the total errorβj(x±Δx)− βj(x), where x is the variable of interest. In thefollowing discussion, Δx refers to a generic input error, whichmay be large and is therefore not expressed as a differentialamount ΔβN (backscatter-coefficient calibration error), ΔSk

(lidar-ratio error), or ΔUk, k = 1 . . . N (range-corrected noise-induced error; see Section III).

The first-order error propagation approach in Table II isjust a perturbational approach that simply scales the inputerrors by partial derivatives to estimate the total backscatter-coefficient error. In contrast, under low SNRs or when theuser’s uncertainty of the algorithm inputs ([x−Δx, x+Δx])is comparatively large, first-order derivative error bounds failto correctly estimate the backscatter-coefficient error. There-fore, total-increment error bounds provide a convenient way ofcomputing exact upper and lower error bounds (usually withasymmetrical amplitudes around the true backscatter value) inexplicit form.

A. Error Source 1: Error Due to the Backscatter-CoefficientCalibration (ΔβN )

Based on (27), it emerges that the derivative of the in-verted backscatter coefficient with respect to the backscatter-coefficient calibration is always positive, (∂βj/∂βN ) > 0,

TABLE IIIFIRST-ORDER ERROR BOUNDS FOR THE KFS BACKWARD INVERSION

ALGORITHM IN RESPONSE TO ERROR SOURCES 1–4 (SEE SECTION III).FOR THE CASE j = N , THE BACKSCATTER-COEFFICIENT ERROR BOUND

IS DIRECTLY THE CALIBRATION ERROR BOUND. LOWERCASE DELTAS

DENOTE SYSTEMATIC ERRORS, AND LOWERCASE SIGMAS DENOTE THE

STANDARD DEVIATION OF RANDOM ERRORS. FOR THE KFS FORWARD

ALGORITHM, CONSIDER THE CHANGES IN SECTION II-D

because βj , βN , Uj , UN , and Fj are positive-definite magni-tudes. As a result, βj(βN ±ΔβN ) = βj ±Δβj (the plus andminus signs are one-to-one maintained) and the total-incrementerror bounds of (46) result.


B. Error Source 2: Error Due to the Range-DependentLidar Ratio

Based on (4), the incremented backscatter-coefficient func-tion can be expressed as

βj(�S +Δ�S) =βN Uj Fj(�S +Δ�S)

UN + 2βN Hj(�S +Δ�S, �U)

Δ�S=p�S−→

βj(p) =βN Uj Fj(p)

UN + 2βN Hj(p, �U). (19)

The lidar-ratio increment Δ�S is related to the lidar ratio �Sthrough the relative error p so that Δ�S = p�S [(12)]. As aresult, the incremental term (�S +Δ�S) (equivalently, �S(1 + p))becomes only a function of the scalar relative error p, and(19) reduces to (13). Incremental auxiliary functions Fj(p) andHj(p, �U) can be computed from (6) and (8), respectively.

The sign of the backscatter-coefficient’s derivative with re-spect to the lidar-ratio relative error ∂βj/∂p at each particularrange Rj determines whether the upper and lower backscatter-coefficient error bounds at each range cell are, respectively, ob-tained from βj(p), i.e., βj(�S +Δ�S), and βj(−p), i.e., βj(�S −Δ�S), or with opposite signs. For the backward integrationcase, this derivative is obtained following a somewhat lengthybut similar development to (∂βj/∂p)|p=0 in (18) and (37).Formally

∂βBj

∂p= 2βB

j (p)IBj,1 −2βB

j (p)2

UjFBj (p)

×[IBj,2(p) + 2(1 + p)IBj,3(p)

]j < N (20)

where IBj,1−3 is given by (38)–(40) in Table III. The resultis identical for the forward integration case (j > 1) with su-perscript “F” (forward) instead of superscript “B” (backward).Note that forward integrals IFj,1−3 must include a minus signaccording to Section II.D.

We note that a more elegant and physically- rooted wayof identifying the sign of the backscatter-coefficient derivativeto the lidar-ratio relative error is to recall that, in forward(backward) integration, the inverted backscatter coefficientat any range increases (decreases) with the lidar ratio (seeFig. 1). This property is the basis of the two-point lidar-ratioestimation method in an aerosol layer aloft using combinedforward/backward integration ([11, p. 7123] and as detailedin [46]). The derivative of the backscatter coefficient withrespect to the lidar ratio is obviously zero at the calibrationpoint. In summary

∂βFj

∂p> 0,

∂βBj

∂p< 0, ∀p, j,

∂βFj

∂p

∣∣∣∣∣j=1

=0,∂βB

j

∂p

∣∣∣∣∣j=N

=0, ∀p

(21)

which is a condition that applies to any range Rj . Therefore,βj(�S ±Δ�S) = βj ±Δβj in the forward case, whereas βj(�S ±Δ�S) = βj ∓Δβj in the backward case (see Table IV).

TABLE IVTOTAL-INCREMENT ERROR BOUNDS FOR THE KFS BACKWARD

INVERSION ALGORITHM IN RESPONSE TO ERROR SOURCES 1–4 (SEE

SECTION IV). SUPERINDICES “U” AND “L” STAND FOR THE “UPPER”AND “LOWER” ERROR BOUNDS, RESPECTIVELY. εuj,1−4 AND εlj,1−4 ARE

POSITIVE-DEFINITE ERROR AMPLITUDES. FOR THE CASE j = N , THE

UPPER/LOWER BACKSCATTER-COEFFICIENT ERROR BOUNDS ARE

DIRECTLY GIVEN BY THOSE OF THE CALIBRATION ERROR. “SYS” AND

“RAND” STAND FOR “SYSTEMATIC” AND “RANDOM” ERRORS,RESPECTIVELY. FOR THE KFS FORWARD ALGORITHM,

CONSIDER THE CHANGES IN SECTION II-B


Fig. 1. Behavior of the forward and backward forms of the KFS-invertedbackscatter coefficient for several values of the aerosol lidar ratio. Lidar ratiosvary from 20 sr to 80 sr (in steps of 10 sr) in a simulated backscatter profileof an elevated dust layer. The calibration range is at 4 km. For R < Rcal,the inversion is through backward integration, and the upper (lower) profilecorresponds to the smallest (largest) lidar ratio. For R > Rcal, the inversion isthrough forward propagation, and the upper (lower) profile corresponds to thelargest (smallest) lidar ratio. Simulation wavelength: 355 nm.

C. Error Sources 3 and 4: Errors Due to theMeasurement Noise

As discussed in Section I, the impact of measurement noisein the KFS algorithm has been studied in [42]. Althoughexact backscatter-coefficient error bounds that satisfy a constantconfidence level are analytically given, its formulation is inimplicit form. This case means that, given a confidence level,two auxiliary integrals [42, eqs. (7) and (10)] and two integralequations [42, eqs. (15) and (16)] must be solved for eachrange of interest. This approach yields two error bounds, whichare later used to compute the upper and lower backscatter-coefficient error bounds.

The explicit formulation of total-increment error bounds ishampered by the fact that the measurement noise is usuallyuncorrelated with range, i.e., each range cell along the inver-sion range contributes independent error amounts ΔUj , j =1 . . . N − 1. This condition leads to the superposition of N − 1noise sources, that is, to an (N − 1)-D problem, impedingany explicit formulation of the total-increment error bounds inTable IV.

However, because of the comparatively larger impact of errorsource 4 (see NIR grounds and the results in [42, Sec.1.3]),the first-order error bound σεj,3 , as given by (41), represents avery good approximation of an already small quantity. One finalremark is that the first term of the error-propagated backscatter-coefficient derivative εj,3 ≈ (βj/Uj)dUj [(32)] represents thetotal increment βj(�U +Δ�U)− βj(�U) if Hj(

−→S ,

−→U ) [(8)] is as-

sumed to be nearly independent of fluctuations in �U . This is, in-deed, the case, because range-corrected power fluctuations tendto smooth out with range during forward/backward integration.

Finally, the error due to the measurement noise at the calibra-tion cell is analogous to the noise measurement in Section IV-A,except that, now, the derivative of the inverted backscattercoefficient to the power at the calibration range is alwaysnegative, (∂βj/∂PN ) < 0 [(33)]. The error bounds are givenin (51), Table IV.

V. DISCUSSION

First-order error bounds (Table III) and total-increment errorbounds (Table IV) are validated here using a multiwavelengthMC approach at wavelengths of 355 nm (UV), 532 nm (VIS)and 1064 nm (NIR). In the MC simulation, for each wavelength,a set of 100 profiles of the aerosol backscatter coefficient hasbeen inverted, given 100 noisy lidar power returns realizedfrom a synthetic backscatter atmospheric profile and a range-dependent SNR profile (Fig. 2).

To make the simulation more realistic, the shape of theprofile of the aerosol backscatter coefficient has been obtainedfrom a 532-nm inversion of a measurement record that wasobtained with the RSLAB lidar (slant path, 54◦ elevation angle).The 355- and 1064-nm aerosol backscatter components havebeen extrapolated from the inverted backscatter coefficient at532 nm, assuming a λ−1 spectral dependency. The molecularbackscatter component follows a U.S. standard atmospheremodel [44] (15 ◦C and 1013.15-hPa ground-level conditions)and a λ−4 spectral dependency. A mean total extinction, α ≈2× 10−4 m−1, at 532 nm, which corresponds to a total opticaldepth, τ ≈ 1.2, over the slant sounding path, is simulated. Tostudy error sources 1–4 in identical simulation conditions, awavelength-independent lidar ratio, Saer = 50 sr, is used, andthe simulated measurement noise level is adjusted to ensure anSNR of 5 at the maximum range (a relatively modest figurein practice) in all three lidar channels. The inversion intervalranges from Rmin = 0.2 km to a maximum range, Rmax =6 km. The calibration range is chosen at Rcal = Rmax = 6 km,where the lidar return is dominated by molecular scattering.The atmospheric boundary layer, characterized by significantaerosol backscatter, ends at an approximately 5-km range.

Lidar system parameters that were used for the simulationare based on the new multispectral elastic Raman lidar (MRL)of the RSLAB (40/130/130-mJ energy at 355/532/1064-nmwavelength, respectively; 3.6-ns pulsewidth; Nd:YAG lasersource; 35.5-cm aperture; 3.9-m focal-length telescope). UVand VIS channels are photo-multiplier tube based with anapproximate reception channel noise-equivalent power (NEP),NEP355,532 = 7.7× 10−15[W · Hz−1/2], whereas the NIRchannel is avalanche photo diode based with an approximatechannel NEP1064 = 9.3× 10−13[W · Hz−(1/2)]. The SNRmodel used is described in [47, Annex A] and assimilatessignal-shot photo-induced, dark-shot, and thermal noisecomponents into a single range-dependent noise-equivalentvariance.

Backscatter-coefficient plots are visible wavelength normal-ized (VIS normalized) to aid intercomparison at the threewavelengths. Thus, the UV and NIR profiles of the invertedbackscatter coefficient are scaled by the (532 nm/355 nm)−1

and (532 nm/1064 nm)−1 factors, respectively. A VIS-normalized plot of Fig. 2(a) would appear with UV, VIS, andNIR traces, all coincident (figure not shown).

A. Error Sources 3 and 4: Errors Due to theMeasurement Noise

Noise in All Range Cells, Except for the Calibration Cell(σεj,3 in Table III and ε

u/lj,3 in Table IV): According to the

superposition principle, the simulation runs with SNR(R)for R �= Rcal (see Fig. 2(b)) and with all other error sources


Fig. 2. Simulated lidar signals. (a) Aerosol backscatter-coefficient atmospheric profiles (solid trace) and related molecular (Rayleigh) levels (dashed). (b) Noisyrange-corrected power returns (solid) and related SNR profiles for each channel: UV: 355 nm, blue; VIS: 532 nm, green; and NIR: 1064 nm, red.

Fig. 3. Analysis of aerosol backscatter-coefficient error due to noise corrupting all range cells, except for the calibration cell (error source 3) for the SNR profilein Fig. 2(b). (a) Envelopes of the aerosol backscatter coefficient from MC inversion (100 realizations) with superimposed first-order error bounds (vertical errorbars) at 3σ. (b) Amplitude of the backscatter-coefficient error bound as a function of range: Comparison of MC error bounds (noisy traces) and first-order errorbounds (solid traces). Both (a) and (b) are VIS normalized. UV: 355 nm, blue; VIS: 532 nm, green; and NIR: 1064 nm, red).

inactive, i.e., SNR(Rcal) → ∞ (no noise on the return powerat the calibration cell; error source 4), perfect backscatter-coefficient calibration (error source 1), and known atmosphericlidar ratio (error source 2).

Fig. 3(a) plots the envelopes of the family of the MC-invertedprofiles of the aerosol backscatter coefficient along with first-order error bounds [(41) in Table III] computed at 3σ (errorbounds are plotted as vertical bars that were centered at theinput “true” profile of the atmospheric backscatter coefficient),whereas Fig. 3(b) compares their error amplitudes. The erroramplitudes represent the difference between the upper andlower backscatter-coefficient error bounds and the true profileof the atmospheric backscatter coefficient. In Fig. 3(b), theupper and lower MC error bounds superimpose and appearas a single noisy trace at each wavelength. Because of thefirst-order series expansion, first-order error bounds are alwayssymmetric. In addition, Fig. 3(b) shows perfect agreementbetween both MC and first-order error bounds at all wave-lengths. This result is of advantage to approximate the total-

increment error bound εu/lj,3 (not found for this error source) as

εu/lj,3 ≈ 3σεj,3 in Table IV.

Fig. 3 shows that errors increase with range in response toa progressively decreasing range-dependent SNR [Fig. 2(b)]and also increase toward the UV. One explanation for this caseis that the σεj,3 term (βj/Uj)σUj

= βj/SNRj in (41) (seeTable III) is inversely proportional to the SNR and directlyproportional to the backscatter coefficient. Toward the UV,σεj,3 increases due to the higher scattering in this band and alower SNR [Fig. 2(b)]. As mentioned in Section IV.C, the termσHU,j [(42)] becomes numerically much lower, because noiseaverages out when integrating.

Noise in the Calibration Cell (σεj,4 in Table III and εu/lj,4

in Table IV): Simulation conditions are analogous to the con-ditions that were used for error source 3, except that now,SNR(Rcal) = SNRN = 5, and SNR(R) → ∞, R �= Rcal.Fig. 4(a) shows that the effects of the measurement noise at thecalibration cell propagate down to all the inversion cells andare comparatively larger in the NIR. Thus, in the NIR, errors


Fig. 4. Analysis of aerosol backscatter-coefficient error due to noise at the calibration range (error source 4) for SNR(Rcal) = 5, SNR(R) → ∞, andR �= Rcal. (a) As shown in Fig. 3(a). (b) First-order error-bound amplitudes at 3σ (thick traces) and asymmetrical MC error-bound amplitudes (thin traces),where solid and dashed traces correspond to upper and lower MC error bounds, respectively. The total-increment error bounds perfectly match the upper and lowerMC error bounds and superimpose with them. Both (a) and (b) are VIS normalized. UV: 355 nm, blue; VIS: 532 nm, green; and NIR: 1064 nm, red.

tend to progressively amplify backward with range (up to ap-proximately 1.8 km), whereas in the UV, they reduce backwardwith range (see the analogous behavior for error source 1 inSection V-B). Fig. 4(b) shows fairly good agreement be-tween first-order error bounds (σεj,4 in Table III) and MCerror bounds, evidenced by the first-order error bounds fallingin between upper and lower MC error bounds. In contrastto what happened when studying error source 3, MC errorbounds are no longer symmetric. One explanation for thiscase is that noise at the calibration range tends to be thedominant error source (σεj,4 ≥ σεj,3 ) over the whole inver-sion range, hence causing that larger backscatter-coefficienterrors cease to be Gaussian distributed [40]. By comparingFigs. 3(b) and 4(b), the impact of noise at the calibrationrange is more prominent toward the NIR. Thus, in the UV,σεj,4 ≈ σεj,3 (this distinguishing feature was not identifiedin earlier work, because it was conducted at 1064 nm). Amathematical hint for this case comes from the ratio be-tween these two noise-induced error sources, σεj,4/σεj,3) ≈(βj/βN )(1/Fj)(σUN

/σUj) [(41)–(43)], where, through ex-

periment, it has been found that (2β2j /UjFj)

2σ2HU,j �

(βj/Uj)2σ2

Ujin (41). Because of the higher molecular com-

ponent in the UV, the ratio βj/βN (recall that β stands for the“total” backscatter coefficient and βN is calibrated in a purelymolecular reference range; see Section II.A) is much smaller inthe UV than in the NIR, thus enabling σεj,3 and σεj,4 to become

comparable in the UV. The total-increment error bounds εu/lj,4 at3σ (Table IV) perfectly match the upper and lower MC errorbounds in Fig. 4(b) and superimpose with them.

Superposition of Error Sources 3 and 4 (σεj,3−4in Table III

and εu/lj,3−4 in Table IV): First-order error bounds (σεj,3−4

) and

total-increment error bounds (εulj,3−4) are compared with the

implicit integral error bounds from previously published results(Section IV.C). All three types of error bounds are computedat 3σ (p = 99.73% probability that an inverted backscatter-coefficient realization falls within the error bounds). To com-pute first-order error bounds 3σεj,3−4

, Table III is used. To

compute total-increment error bounds, εu/lj,3 [approximation in

(50)], Table IV is used, and obviously, the exact equation (51)with ΔUN = 3σUN

is used to compute εu/lj,4 .

Because upper and lower integral error bounds must besolved for each range cell and the solutions become numericallyill conditioned for dense atmospheres (approximately τ > 2),they have only been computed for a discrete set of six ranges,from 1 km to 6 km, equispaced at 1 km. In nearly all the sim-ulation runs, the upper and lower MC error bounds computedwith 100 lidar signal realizations coincided with the integralerror bounds (i.e., the exact theoretical reference). Thus, theMC error bounds can be considered reliable bounds of the3-σ inverted backscatter-coefficient population and, therefore,equivalent trustworthy extrapolations of the integral “exact”error bounds over all the range cells.

The multiwavelength performance of both first-order andtotal-increment error bounds with reference to the implicitintegral error bounds is shown in Fig. 5. Fig. 5(a) shows acomparatively poorer but still fairly good performance of thefirst-order error bounds, which give error bound amplitudesin between those of the MC error bounds or slightly closerto the MC lower error bound (the upper MC error bound inthe NIR falls below the implicit integral error bound as aconsequence of the natural statistical dispersion in this specificsimulation run and wavelength). Fig. 5(b) shows that total-increment error bounds give virtually identical estimates withthe implicit integral error bounds, with the advantage of beingformulated in explicit form, being simpler to compute, andproviding range-resolved information. The mean backscatter-relative error between both types of error bounds is below 1.7%in the UV, 0.6% in the VIS, and 0.5% in the NIR, with thisdifference being due only to the approximation in (50). Thespectral behavior of Fig. 5 is analogous to the spectral behaviorof Fig. 4(b).

B. Errors Due to the Backscatter-Coefficient Calibration(δεj.1 in Table III and ε

u/lj,1 in Table IV)

By virtue of the relationship established in (11), the behaviorof this error source is qualitatively similar to the noise at the


Fig. 5. Superposition of error sources 3 and 4: Error amplitude plots that compare the total-increment and first-order error bounds with implicit integral errorbounds. (a) Performance of first-order error bounds. Crosses and circles denote implicit integral upper and lower error bound amplitudes at 3σ, respectively, noisythin solid and dashed traces denote MC upper and lower error amplitudes, respectively, and solid thick traces denote first-order error-bound amplitudes at 3σ. (b)Performance of total-increment error bounds: Crosses and circles denote implicit integral upper and lower error-bound amplitudes at 3σ, respectively, whereassolid lines denote MC upper and lower error-bound amplitudes. The total-increment error bounds perfectly match the MC error bounds and superimpose withthem. The plots in (a) and (b) are VIS normalized. UV: 355 nm, blue; VIS: 532 nm, green; and NIR: 1064 nm, red.

Fig. 6. Analysis of aerosol backscatter-coefficient calibration error (errorsource 1): Same description as in Fig. 3(a). The family of inverted backscatter-coefficient profiles is in response to a step-function profile of the atmosphericaerosol backscatter coefficient that simulates the atmospheric boundary layer(R ≤ 5 km). The range of calibration errors is ±30% about the nominalbackscatter Rayleigh level at the calibration range (Rcal = Rmax = 6 km).Plots are VIS normalized. UV: 355 nm, blue; VIS: 532 nm, green; and NIR:1064 nm, red.

calibration range (error source 4; see Fig. 4); hence, analogousplots are retrieved (figure not shown), which are scaled bya multiplicative factor. For example, a relative backscatter-calibration error of δβN

= 0.1βN yields the plot in Fig. 4(a)(SNRN = 5), with error bounds scaled by a factor of 0.5/3(the dividing factor 3 is because, in Fig. 4(a), error envelopesare plotted at 3σ). Therefore, similar simulation conclusionsapply; in particular, the backscatter-coefficient calibration errorbecomes dominant in the NIR. This case is best corroboratedin Fig. 6, which uses a step-function atmospheric backscatter-coefficient profile, with a 1-km falling edge between 4 and5 km simulating the end of the boundary layer, and an relative

backscatter-calibration error, εβNr = 0.3. In the mixing layer

(0.2–4 km range) the error bound amplitudes can be rankedNIR > VIS > UV, as expected.

C. Errors Due to the Lidar Ratio (δεj,2 in Table III and

εu/lj,2 in Table IV)

Simulation conditions for this case assume noiseless powerlidar returns [SNR(R) → ∞ in all range cells], perfectbackscatter-coefficient calibration, and lidar-ratio errors definedby a relative error figure p. During the tests, (20) and (21)always gave the same signs, as expected.

Fig. 7 shows the performance of the total-increment errorbounds, which perfectly match the simulated error deviations.For small-to-moderate errors (p = ±30%, figure not shown),the total-increment upper and lower error bounds tend to sym-metrically distribute around the “true” atmospheric backscattercoefficient, i.e., similar upper and lower error amplitudes. Thisis no longer the case for large errors (p = ±90%). The invertedbackscatter-coefficient error bounds and their asymmetry in-crease toward the UV, which reinforces the fact that lidar ratiouncertainties become more critical toward the UV.

VI. CONCLUSION

Two different types of backscatter-coefficient inversion er-ror bounds have been formulated: 1) first-order error bounds(Section III) and 2) total-increment error bounds (Section IV).Both types of error bounds have analytically been formulatedin explicit form for the two-component KFS lidar inversionalgorithm subject to error sources 1–4. The error bounds havebeen validated using an MC method.

First-order error bounds are obtained using the classic error-propagation approach. They are symmetric about the truevalue, with amplitude lying between those of the upper andlower MC error bounds. Their amplitudes encompass most ofthe inverted backscatter-coefficient profiles in practical situa-tions (SNR ≥ 5, lidar-ratio relative error strength, p ≤ 30%;


Fig. 7. Analysis of lidar-ratio errors (error source 2). (a) Aerosol backscatter-coefficient envelopes with the superimposed total-increment error bounds (verticalerror bars): Relative error, p = ±90%, about the nominal lidar ratio of Saer = 50 sr. (b) Backscatter-coefficient error-bound amplitudes associated with (a). Thesolid trace denotes upper and lower error-bound amplitudes. The total-increment error bounds perfectly match error bounds and superimpose with them. Both (a)and (b) are VIS normalized. UV: 355 nm, blue; VIS: 532 nm, green; and NIR: 1064 nm, red.

Section V). However, strictly speaking, first-order error boundsare still approximate. With larger errors (lower SNRs and/orhigher uncertainties), upper and lower MC error bounds be-come progressively asymmetric, a property that first-order errorbounds cannot reflect.

It has been shown that, when the random error source followsa Gaussian distribution, the total-increment error bounds thatwere computed at 3σ coincide with 3-σ statistical confidencelevels and therefore provide the exact result in explicit ana-lytical form. The total-increment error bound associated witherror source 3 was explicitly not found because of the multidi-mensionality of the problem. However, it is well approximatedby the 3-σ first-order error bound as ε

u/lj,3 ≈ 3σεj,3 . Similarly,

when the uncertainty of a systematic error source is assumed tobe uniform (the usual case for error sources 1 and 2 when nofurther a priori information is available), the total-incrementerror bound gives the total error span on the MC invertedbackscatter-coefficient profiles.

Similar to the KLT algorithm, the effect of noise at thecalibration cell dominates (particularly toward the NIR) overthe effect of the noise from all other range cells. Althoughfundamentally different, error sources 1 and 4 yield similareffects on the retrieval through (11); thus, error sources 2 and4 are of most concern. With regard to their spectral behavior,uncertainties in the lidar ratio largely dominate the UV errorbounds, whereas the backscatter-coefficient calibration is thedominant error source in the NIR. The explicit analytical errorbound formulation summarized in Tables III and IV is, to thebest of our knowledge, new in the state of the art of lidarinversion algorithms.

APPENDIX AERROR PROPAGATION DUE TO THE

(RANGE-DEPENDENT) LIDAR RATIO

In (10), the term εj,2 =∑N

k=1(∂βj/∂Sk)dSk expressesthe backpropagated backscatter-coefficient error due to range-dependent lidar-ratio errors dSk. To derive the error bounds, wedepart from the modified KFS form of (4) and express the lidar-

ratio-induced backscatter-coefficient error εj,2 as a function ofpartial derivatives of Fj and Hj as

εj,2 =

N∑k=1

∂βj

∂SkdSk

=

N∑k=1

∂βj

∂Fj

∂Fj

∂SkdSk

+N∑

k=1

∂βj

∂Hj

(∂Hj

∂Sk+

N∑p=1

∂Hj

∂Fp

∂Fp

∂Sk

)dSk (22)

where

∂βj

∂Fj=

βj

Fj

∂βj

∂Hj= −

2β2j

UjFj. (23)

Next, the Fj(�S) and Hj(�S, �U, �F ) dependency on the lidar ratiois expanded. The dependency of Fj(�S) on the lidar ratio is

∂Fj

∂Sk=

∂Fj

∂Gj

∂Gj

∂Sk,∂Fj

∂Gj=2Fj , and

∂Gj

∂Sk=

{0 k < jwkβ

molk k ≥ j

(24)where (6) and (7) have been used. The dependency ofHj(�S, �U, �F ) on the lidar ratio is

∂Hj

∂Sk=

{0 k < jwkUkFk k ≥ j

∂Hj

∂Fp=

{0 p < jwpSpUp p ≥ j

(25)

where the definition of Hj in (8) has been used.Based on these expressions, (22) can be rewritten in terms of

the low-level derivatives (i.e., ∂βj/∂Fj , ∂βj/∂Hj , ∂Fj/∂Gj ,∂Gj/∂Sk, and ∂Hj/∂Sk, ∂Hj/∂Fp), resulting in

εj,2 =∂βj

∂Fj

∂Fj

∂Gj

N∑k=j

∂Gj

∂SkdSk +

∂βj

∂Hj

N∑k=j

∂Hj

∂SkdSk


+∂βj

∂Hj

N∑p=j

⎛⎝∂Hj

∂Fp

∂Fp

∂Gp

N∑k=p

∂Gp

∂SkdSk

⎞⎠ . (26)

Finally, by substituting (23)–(25) into (26), the sought-after equation [(28) in Table II] is obtained. Theauxiliary integral terms dIj,1−3 [(29)–(31) in Table II]can immediately be identified with the terms in (26),i.e.,

∑Nk=j(∂Gj/∂Sk)dSk,

∑Nk=j(∂Hj/∂Sk)dSk, and

(1/2)∑N

p=j(∂Hj/∂Fp)(∂Fp/∂Gp)∑N

k=p(∂Gp/∂Sk)dSk),respectively.

As a result, not only lidar ratio errors but also an integratedversion of these errors propagate backward through �F (�S)

and �H(�S).

ACKNOWLEDGMENT

The authors would like to thank the three anonymousreviewers who helped in improving this paper toward atwo-component atmosphere, a multiwavelength approach, andcareful structure of this paper.

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Francesc Rocadenbosch (M’10) received the B.S.and Ph.D. degrees in telecommunications engineer-ing from the Universitat Politècnica de Catalunya(UPC), Barcelona, Spain, in 1991 and 1996, respec-tively, and the M.B.A. degree from the University ofBarcelona, Barcelona, in 2001.

From 1991 to 1992, he was with the University ofLas Palmas de Gran Canaria, Las Palmas de GranCanaria, Spain, working on microwave systems. In1993, he joined the Electromagnetics and PhotonicsEngineering (EEF) Group, Department of Signal

Theory and Communications (TSC), UPC, where he has been an AssociateProfessor since 1997 and also a Joint Representative of the Remote SensingLaboratory (RSLab), where he has been steering the development of the UPCactivities on multispectral elastic Raman lidar (laser radar) since 1993. He is aReviewer of well-known international journals such as Applied Optics, OpticsLetters, and Optical Engineering. His research interests include lidar remotesensing, related signal processing, and, more recently, its application to offshorewind farms (KIC-InnoEnergy strategic plan).

Dr. Rocadenbosch is a Reviewer of the IEEE TRANSACTIONS ON GEO-SCIENCE AND REMOTE SENSING. He is the recipient of the Salvà i CampilloAward for the Best Research Project from the Catalan Association of Telecom-munications Engineers in 1998 as a member of the RSLab Lidar Group,the Telecommunications National Award in 2003 as a Member of the EEFGroup, TSC Department, UPC, and the Best IEEE Reviewer Award from theTransactions on Geoscience and Remote Sensing in 2010.

Stephen Frasier (M’94–SM’04) received the B.E.E.degree from the University of Delaware, Newark, in1987 and the Ph.D. degree in electrical and computerengineering from the University of Massachusetts,Amherst, in 1994.

From 1987 to 1990, he was with SciTec, Inc.,Princeton, NJ, where he worked on the analysisof electromagnetic and optical signatures of rocketplumes, the evaluation of laser detection systems,and the development of data acquisition systemsfor airborne IR sensor tests. In 1990, he joined

the Microwave Remote Sensing Laboratory, University of Massachusetts,where his graduate work involved the development and application of digital-beamforming phased-array radar for oceanographic research applications. In1997, he joined the faculty of the University of Massachusetts, where he iscurrently a Professor and a Codirector of the Microwave Remote Sensing

Laboratory, Department of Electrical and Computer Engineering. His researchinterests include microwave imaging, interferometric techniques, and applica-tions in radar oceanography and radar meteorology. He currently leads radarresearch programs that study ocean surface winds and advanced radar methodsfor estimating winds in the atmospheric boundary layer.

Dr. Frasier is a Senior Member of the Geoscience and Remote SensingSociety, the International Scientific Radio Union (URSI) Commission F, theAmerican Geophysical Union, and the American Meteorological Society.

Dhiraj Kumar received the B.S. degree in physicsfrom Dr. B. R. Ambedkar University, Agra, India,in 1999, the M.S. degree in applied optics from theIndian Institute of Technology (IIT), Delhi, India,in 2003, and the dual Master in Research on Infor-mation and Communication Technologies (MERIT)degree from the Universitat Politècnica de Catalunya(UPC), Barcelona, Spain, in collaboration with thePolitecnico di Torino, Torino, Italy, in 2007. He iscurrently working toward the Ph.D. degree in theRemote Sensing Laboratory (RSLab), Department of

Signal Theory and Communications, UPC.His research interests include multispectral lidar remote sensing, optical

system design and analysis, and related system integration.

Diego Lange received the B.S. degree in electronicsengineering from the San Simón University (UMSS),Cochabamba, Bolivia, in 2005 and the M.S. degree inelectronics engineering from the Universitat Politèc-nica de Catalunya (UPC), Barcelona, Spain, in 2009.He is currently working toward the Ph.D. degree inthe Remote Sensing Lab. (RSLab), Department ofSignal Theory and Communications, UPC.

His research interests include multispectral lidarsignal processing, error-bound estimation, and adap-tive inversion of optical parameters.

Eduard Gregorio received the B.S. degree in me-chanical engineering from the University of Lleida(UdL), Lleida, Spain, in 2003 and the M.S. degree inindustrial engineering from the Universitat Politèc-nica de Catalunya (UPC), Terrassa, Spain, in 2005.He is currently working toward the Ph.D. degree onremote sensing of pesticide spray drift using lidarsystems in the Department of Agroforestry Engineer-ing, Universitat de Lleida and in collaboration withthe Department of Signal Theory and Communica-tions, Universitat Politècnica de Catalunya.

Since 2009, he has been with the Department of Agroforestry Engineering,University of Lleida, where he is currently an Assistant Professor and a Memberof the Research Group on Precision Agriculture, AgroICT and Agrotechnology.His research interests include lidar remote sensing, electrical machines, andprecision agriculture.

Michaël Sicard received the Ph.D. degree in phys-ical methods for remote sensing from the InstitutPierre-Simon-Laplace, Paris, France.

He is currently an Associate Professor with theRemote Sensing Laboratory, Department of SignalTheory and Communications, Universitat Politèc-nica de Catalunya (UPC), Barcelona, Spain. In 2001,after a short stay with the Centre d’Applicationset de Recherches en Télédétection, University ofSherbrooke, Sherbrooke, Canada, he joined UPC,with a European Space Agency External Fellowship

Grant in 2002 and followed on with a Ramón-y-Cajal Spanish contract in 2004,which includes aerosol optical and microphysical characterization due to datafusion, aerosol transport models, and atmospheric boundary layer study andmodeling. He is a Coordinator of the Spanish/Portuguese Lidar Network and theSpanish contact point of the international project Chemistry–Aerosol Mediter-ranean Experiment (ChArMEx). He has published more than 30 papers in peer-reviewed journals and more than 70 international conference proceedings. Hisresearch interests include, on a technical point of view, the development of lidartechniques and instruments in all their aspects for aerosols observations and thestudy of aerosols.

backscatter error bounds for the elastic lidar two-component inversion algorithm

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