retrieval of cloud optical parameters from space-based backscatter lidar data

9
Retrieval of cloud optical parameters from space-based backscatter lidar data Yuri S. Balin, Svetlana V. Samoilova, Margarita M. Krekova, and David M. Winker We present an approach to estimating the multiple-scattering ~MS! contribution to lidar return signals from clouds recorded from space that enables us to describe in more detail the return formation at the depth where first orders of scattering dominate. Estimates made have enabled us to propose a method for correcting solutions of single-scattering lidar equations for the MS contribution. We also describe an algorithm for reconstructing the profiles of the cloud scattering coefficient and the optical thickness t under conditions of a priori uncertainties. The approach proposed is illustrated with results for optical parameters of cirrus and stratiform clouds determined from return signals calculated by the Monte Carlo method as well as from return signals acquired with the American spaceborne lidar during the Lidar In-Space Technology Experiment ~LITE!. © 1999 Optical Society of America OCIS codes: 010.0010, 010.1290, 010.3640, 280.0280, 280.1310, 280.3640, 290.1310, 290.4210. 1. Introduction The interpretation of sensing data acquired with a spaceborne lidar on cloud fields has certain peculiar- ities that are caused on the one hand by the capabil- ities of lidars ~this topic was discussed earlier in Ref. 1! and on the other hand by the contribution that is due to a multiple scattering ~MS! background that may be quite large because the objects sounded may have large ranges. The enhanced MS contribution may occur naturally because of the large cross size of the scattering volume formed by a sounding beam on the upper cloud boundary, so it will compare with the physical thickness of the cloud layer under study. In this case the radiation scattered along the lateral direction does not escape the volume that is sounded and contributes to the formation of the total flux re- flected from the layer within the viewing cone of the lidar receiver. It has been determined that the MS level is governed by three factors, that is, by the distance to the volume sounded, the receiver’s field of view ~the product of these two quantities defines the laser beam’s diameter at the cloud top!, and the cloud’s optical parameters. 2 The MS contributions to the return signals estimated theoretically ~see, for example, Refs. 3 and 4! for space-based lidars have shown that the leading edge of the reflected signal is formed primarily because of the first and second or- ders of scattering, at t5 0.8 –1, where t is the optical thickness. At t [ @1, 1.5# the intensity of multiple scattering up to the fifth order increases, and the trailing edge of a return pulse, at t. 2, is formed by contributions from higher ~.5! orders of MS. At t5 1 the MS contribution to the total return signal is comparable with that from single scattering but in- creases to become the prevailing factor at t. 2. This circumstance results in an apparent downward movement of the lower cloud boundary, which thus causes its physical thickness to be overestimated. In general, in determining the optical parameters of a cloud from remote-sensing data one should solve a nonstationary radiation transfer equation. In some scattering media, such as cirrus clouds and op- tically thin water-droplet clouds, it may be acceptable first to eliminate the MS background from the return signal and then to process that signal by using a single-scattering approach ~see, e.g., Refs. 5 and 6!. However, such an approach is a complicated task, because the magnitude of the MS contribution de- pends on the optical properties and the size spectrum of cloud particles that are not known a priori, as they are, in general, the quantities that one wishes to determine from the return signals themselves. Be- sides, the calibrations that one may need to deter- Y. S. Balin, S. V. Samoilova, and M. M. Krekova are with the Institute of Atmospheric Optics, Siberian Branch of the Russian Academy of Sciences, 1 Academicheskii Avenue, 634055 Tomsk, Russia. The e-mail address for Y. S. Balin is [email protected]. tomsk.ru. D. M. Winker is with the NASA Langley Research Center, MS 475, Hampton, Virginia 23681. Received 30 September 1998; revised manuscript received 23 July 1999. 0003-6935y99y306365-09$15.00y0 © 1999 Optical Society of America 20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6365

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Page 1: Retrieval of Cloud Optical Parameters from Space-Based Backscatter Lidar Data

Retrieval of cloud optical parameters fromspace-based backscatter lidar data

Yuri S. Balin, Svetlana V. Samoilova, Margarita M. Krekova, and David M. Winker

We present an approach to estimating the multiple-scattering ~MS! contribution to lidar return signalsfrom clouds recorded from space that enables us to describe in more detail the return formation at thedepth where first orders of scattering dominate. Estimates made have enabled us to propose a methodfor correcting solutions of single-scattering lidar equations for the MS contribution. We also describe analgorithm for reconstructing the profiles of the cloud scattering coefficient and the optical thickness tunder conditions of a priori uncertainties. The approach proposed is illustrated with results for opticalparameters of cirrus and stratiform clouds determined from return signals calculated by the Monte Carlomethod as well as from return signals acquired with the American spaceborne lidar during the LidarIn-Space Technology Experiment ~LITE!. © 1999 Optical Society of America

OCIS codes: 010.0010, 010.1290, 010.3640, 280.0280, 280.1310, 280.3640, 290.1310, 290.4210.

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1. Introduction

The interpretation of sensing data acquired with aspaceborne lidar on cloud fields has certain peculiar-ities that are caused on the one hand by the capabil-ities of lidars ~this topic was discussed earlier in Ref.1! and on the other hand by the contribution that isdue to a multiple scattering ~MS! background thatmay be quite large because the objects sounded mayhave large ranges. The enhanced MS contributionmay occur naturally because of the large cross size ofthe scattering volume formed by a sounding beam onthe upper cloud boundary, so it will compare with thephysical thickness of the cloud layer under study.In this case the radiation scattered along the lateraldirection does not escape the volume that is soundedand contributes to the formation of the total flux re-flected from the layer within the viewing cone of thelidar receiver. It has been determined that the MSlevel is governed by three factors, that is, by thedistance to the volume sounded, the receiver’s field ofview ~the product of these two quantities defines the

Y. S. Balin, S. V. Samoilova, and M. M. Krekova are with theInstitute of Atmospheric Optics, Siberian Branch of the RussianAcademy of Sciences, 1 Academicheskii Avenue, 634055 Tomsk,Russia. The e-mail address for Y. S. Balin is [email protected]. D. M. Winker is with the NASA Langley ResearchCenter, MS 475, Hampton, Virginia 23681.

Received 30 September 1998; revised manuscript received 23July 1999.

0003-6935y99y306365-09$15.00y0© 1999 Optical Society of America

aser beam’s diameter at the cloud top!, and theloud’s optical parameters.2 The MS contributions

to the return signals estimated theoretically ~see, forxample, Refs. 3 and 4! for space-based lidars havehown that the leading edge of the reflected signal isormed primarily because of the first and second or-ers of scattering, at t 5 0.8–1, where t is the opticalhickness. At t [ @1, 1.5# the intensity of multiplecattering up to the fifth order increases, and therailing edge of a return pulse, at t . 2, is formed byontributions from higher ~.5! orders of MS. At t 5the MS contribution to the total return signal is

omparable with that from single scattering but in-reases to become the prevailing factor at t . 2.his circumstance results in an apparent downwardovement of the lower cloud boundary, which thus

auses its physical thickness to be overestimated.In general, in determining the optical parameters

f a cloud from remote-sensing data one should solvenonstationary radiation transfer equation. In

ome scattering media, such as cirrus clouds and op-ically thin water-droplet clouds, it may be acceptablerst to eliminate the MS background from the returnignal and then to process that signal by using aingle-scattering approach ~see, e.g., Refs. 5 and 6!.owever, such an approach is a complicated task,ecause the magnitude of the MS contribution de-ends on the optical properties and the size spectrumf cloud particles that are not known a priori, as theyre, in general, the quantities that one wishes toetermine from the return signals themselves. Be-ides, the calibrations that one may need to deter-

20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6365

Page 2: Retrieval of Cloud Optical Parameters from Space-Based Backscatter Lidar Data

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mine these properties can hardly be done from aspaceborne platform. Next, interpreting lidar data,even in the single-scattering approximation, is quitea complicated task. The great variety of atmo-spheric situations that may occur in lidar studiescalls for a large variety of data-processing techniques.In every case the technique developed is aimed atminimizing the negative effects of the informationdeficit and measurement errors on the final results.

In applications to laser sounding of clouds one maywrite the single-scatter lidar equation in a simpleform, assuming that clouds are a single-component,nonabsorbing medium:

P~1!~z! 5 Agp~z!b~z!expF22 *z0

z

b~z9!dz9GYz2

5 Agp~z!b~z!T2~z0, z!yz2, (1)

where P~1!~z! is the lidar return as a function of rangeZ in the single-scattering approximation, A is theinstrumental constant, z0 is the distance to the cloudupper boundary, gp~z! is the lidar ratio, b~z! is theextinction coefficient, and T2~z0, z! is the double-pathatmospheric transmission.

When one is trying to extract information on theoptical properties of clouds from lidar returns by solv-ing Eq. ~1! the errors are due mainly to the followingcauses:

~1! Multiple Scattering. The best way to estimatethis effect is by solution ~by the Monte Carlo method!of the nonstationary radiative transfer equation, as-suming realistic initial and boundary conditions. Acomprehensive overview of the methods that are ap-plicable to remote sensing can be found in Ref. 7.However, this approach has the drawback that it ispoorly suited to experimental automation purposes,especially when one is processing large amounts ofmeasurement data. This limitation recently stimu-lated the development of analytical approaches todescribe the MS effects ~see Refs. 8 and 9!.

~2! Uncertainties in the Boundary Conditions.he inversion methods require that some referencealues be set a priori. Those values are either b~z*!or local calibration6 or T2~z0, z*! in the case of inte-

gral calibration.5,10 By z* we denote the point wherehe calibration is being done. When one is process-ng actual lidar returns this point is chosen closer tohe far path’s end because the more informationbout the far portion of the path that is introduced,he higher is the stability of the solution obtained.

~3! Lidar Ratio Variability along the Sounding Path.Inasmuch as lidar equation ~1! involves two unknownquantities, gp~z! and b~z!, it is normally assumed that

p~z!ygp~z*! 5 1. This assumption works quite wellor processing return signals from clouds6 but is less

efficient in the case of an aerosol atmosphere.10,11 Tocompensate for the uncertainties one has to use eithermodel estimates of gp~z! or a wide scope of physicalffects ~for example, polarization or multifrequencyensing! during sounding. However, these effects areeyond the scope of this paper.

366 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

Strictly speaking, it is hardly possible to compen-ate for all the experimental and methodological er-ors. Thus, even if one manages to avoid errors thatre due to the MS effects ~for example, by applyingonte Carlo methods! the value of uncertainties that

re due to the range dependence of the lidar ratio andalibration errors will still be finite. However smallhese errors are the reconstruction errors will in-rease when t increases, thus causing the solution toiverge. This means that at unlimited t the problems ill posed. Thus, for example, if the range variationf the lidar ratio is ;30%, a high degree of accuracyn measuring the return signal itself or in estimatinghe MS background will not help. In this case theccuracy of reconstruction could be much more effi-iently improved by introduction a priori of informa-ion about the parameters sought when solutionethods developed for solving ill-posed problems

see, for example, Ref. 12! are applied. Regardless ofhe difficulties mentioned above, developments of al-orithms for handling actual lidar data acquired withpaceborne lidars from different clouds were recentlyeported in Refs. 13–15.

In the first part of this paper we derive analyticalstimates of the MS contribution to the spaceborneidar returns from clouds by making some qualitativeypotheses on the behavior of contributions from therst two orders of light scattering based on the theoryf double scattering.16 To assess the applicability

limits of this approach we carried out a numericalexperiment to compare the results obtained with thesignals calculated by the Monte Carlo method. Thesecond part of the paper deals with the inversiontechnique to retrieve the scattering coefficient profileb~z! from these signals. We also propose an algo-rithm for estimating the cloud optical thickness thatis needed to regularize the solution. We analyze theefficiency of the method proposed by modeling thetask numerically as well as by processing real lidarreturns. In the third part of the paper we presentsome results of applying this method to retrieval ofthe scattering coefficient profiles from the data ofspace-based sensing of water-droplet and cirrusclouds acquired within the framework of the LidarIn-Space Technology Experiment ~LITE! program.17

These data are also compared with the data collectedwith the Airborne Lidar Experiment18 ~ALEX! duringimultaneous underflights of the LITE.

2. Expressions for Estimating the Multiple-ScatteringContribution to Lidar Returns

The Monte Carlo method that is widely used for solv-ing the radiative transfer equation is an asymptoti-cally exact method. This method allows for both thewave and the quantum natures of light and is freefrom limitations on the forms of scattering phasefunction, multiplicity of photon interactions, andcomposition of the atmosphere as well as on its ex-tension and spatial homogeneity. When it is appliedto laser sensing problems the method provides forestimating the radiation flux from a given volumesounded, which is considered a point object as com-

Page 3: Retrieval of Cloud Optical Parameters from Space-Based Backscatter Lidar Data

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pared with the scattering medium as a whole.When we applied this method to our problem wemade use of the local estimate of the particles’ flux.19

Basic principles and detailed descriptions of con-struction of such estimates have been discussed inRefs. 19 and 20.

From this point of view the approximations that weintroduce below are, in a certain sense, semianalyti-cal. These approximations, on the one hand, arebased on the theory that gives an analytical descrip-tion of the lidar return in a double-scattering approx-imation,16 while, on the other hand, they use thesimilarity principle that has been derived for the sig-nals that are due to neighboring orders of scatteringfrom the Monte Carlo calculations for different cloudsand lidars. By the principle that we used to con-struct these approximations they differ from themodel representations of the MS background derivedby pure fitting with the Monte Carlo calculations~see, for example, Ref. 21!.

According to Ref. 16, the double-scattering signalay be presented in the form

P~2!~z! 5 d~z!P~1!~z!, (2)

where P~1!~z! is defined by Eq. ~1!:

d~z! 52pz2

b~z, p! *w*

x

sinw

Rb~x, w!b~x9, p 2 w!dxdw, (3)

here

R 5 R~z, x, w! 5 z2 2 ~2z 2 x!x sin2~wy2!

and

x9 5 x9~z, x, w! 5 x 1z~z 2 x!

z 2 x sin2~wy2!

are the parameters that determine the scattering ge-ometry, b~z, w! 5 b~z!g~z, w! is the directional scatter-ng coefficient, and g~z, w! is the normalized scatteringhase function. As the investigations presented inef. 16 showed, the function d~z! depends on the shapef the scattering phase function, on the optical depth ofhe medium sounded t~z!, and on the optical diameterf the scattering volume td~z! 5 b~z!z tan~w0!, where

w0 is the receiver’s field-of-view angle.For space-based sounding of a cloud at a distance z0

we have

d~z! 52pz2

gpF 1b~z! *

0

a1

G~w! *z0

z b~x!b~x9!

Rdxdw

11

b~z! *a1

a2

G~w! *x1

z b~x!b~x9!

Rdxdw

11

b~z! *a2

a3

G~w! *x2

z b~x!b~x9!

Rdxdw

1 *a3

p

G~w! *z0

z b~x!

x2 dxdwG , (4)

where

G~w! 5 g~w!g~p 2 w!sin~w!,

a1 5 2 arctanFz tan~w0y2!

z 2 z0G ,

a2 5 2 arctan2z 2 z0@1 1 cos~w0!#

z0 sin~w0!,

a3 5 p 2 arccosz 2 z0

~z 2 z0! 1 2Dz,

x1 5 x1~w! 5 z@1 2 tan~w0y2!cot~wy2!# ,

x2 5 x2~w! 5 zF1 2~z 2 z0!

z0cot~wy2!G ;

the angles a1, a2, and a3 and the functions x1~w! andx2~w! appear on passage in Eq. ~3! from multiple in-tegrals to integration over the surface, Dz is the rangeincrement, and the other parameters are defined asin Eq. ~3!. We obtained the relationship in Eq. ~4! byassuming that g~w! 5 g~z, w!, although this is not a

ecessary assumption in the double-scattering the-ry.The values of d~z! calculated for clouds with differ-

ent microstructures are depicted in Fig. 1, Model 2;the corresponding scattering coefficient profiles arepresented in Model 1. We assumed the lidar to be ata distance of 270 km from the Earth. We also as-sumed that the lidar transmitter isotropically emitswithin the cone of 2p@1 2 cos~c0!#, where c0 5 0.6mrad. The lidar return, in its turn, is assumed to becollected by a receiver within the cone of 2p@1 2cos~w0!#, where w0 5 1.6 mrad. These lidar param-eters correspond to those of a spaceborne LITE17 ~The

avelength of the incident radiation is l 5 532 nm;he diameter of the laser footprint on the cloud top isw 5 420 m!. The scattering phase function of aater-droplet cloud with the upper boundary at 6 kmbove the Earth’s surface ~Model1! was taken accord-

ing to Model C1 of Ref. 22. To set the form of thescattering phase function for cirrus clouds ~Model2!we used the results from Ref. 23 calculated by a geo-metrical optics approach to a description of light scat-tering from a polydispersion of randomly orientedcrystals of hexagonal shape. Cirrus clouds were as-

Fig. 1. Ratio of signals that are due to the first and the secondorders of MS d~z! for water-droplet ~Model1! and cirrus ~Model2!clouds. Curves 1, model profiles of the scattering coefficient;curves 2, d~z! that correspond to these profiles.

20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6367

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sumed, in our calculations, to be at a height of 10 kmabove the Earth’s surface.

Figures 2 and 3 illustrate some peculiarities in theformation of lidar return signals as a result of differ-ent orders of scattering from clouds of different mi-crophysical and optical properties. Figures 2~a! and~a! set out the functions P~S!~z! and P~i!~z! for i 5 1–5

calculated by the Monte Carlo method for cloudswhose b~z! profiles are shown in Fig. 1. The exper-imental conditions used in Fig. 2~a! were taken thesame as those above. We used lidar parametersfrom ALEX18 to calculate the data shown in Fig. 3~a!.

he flight height in this case was taken to be H 53.6 km, c0 5 0.6 mrad, w0 5 1.1 mrad, and dw 5 5.4

m. Analysis of these data leads to the following con-clusions:

Fig. 2. Comparison of the methods of calculating returns for aspaceborne lidar by use of low orders of MS. ~a! Monte Carlomethod. ~b! Method using an estimate of MS distribution: S,total signal; curves 1–5, signals that are due to first through fifthorders of scattering. Model profiles b~z! correspond to those inFig. 1.

Fig. 3. Comparison of the methods of calculating returns for anairborne lidar by use of low orders of MS. ~a! Monte Carlo meth-od: left, data from Ref. 4; right, our calculations. ~b! Methodusing an estimate of MS distribution: S, total signal; curves 1–5,signals that are due to first through fifth orders of scattering.

368 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

~1! No principal differences can be isolated in theformation of returns from water-droplet and cirrusclouds, regardless of the sounding geometry.

~2! The shapes of the curves are similar, althoughhe return signal that is due to the ith order of scat-

tering, P~i!~z!, has a maximum that is lower in am-plitude and shifted toward larger values of tcompared with those of P~i21!~z!. Inequality P~i!~z!, P~i21!~z! holds true only at small t; otherwise thereexists a value of t9 such that P~i!~z! . P~i21!~z! at t .t9. This qualitative information can be taken intoaccount in constructing the data-processing algo-rithms.

Let us assume that the estimates P~i!~z! at i . 2 areunctions of d~z! of the following form:

P~i!~z! 5 di~x!P~i21!~z!,

di~z! 5@d~z!#

~i 2 1!, (5)

where d~z! obeys condition ~4!. This representationmeets all the qualitative requirements for the ratiobetween P~i!~z! and P~i21!~z! that we noted above.The function di~z! allows one to account for the mi-crostructure and optical properties of a cloud, whichmakes it useful when one is reconstructing cloud pa-rameters from data on the background that are due toMS. The estimate of backscattered power from acloud with MS accounted for ~up to some nth order! iswritten as follows:

P~S!~z! 5 (i51

n

P~i!~z! 5 P~1!~z!exp@d~z!# 5 P~1!~z!DP~z!.

(6)

To study the reliability and the applicability limitsof the assumptions made, we carried out a numericalsimulation. The results for P~S!~z! and P~i!~z! for i 5–5 that we calculated by using Eqs. ~4!–~6! are pre-ented in Figs. 2~b! and 3~b!. The experimental con-itions are identical to those described for Figs. 2~a!nd 3~a!. One can see from these results that theeconstruction scheme based on Eqs. ~4!–~6! providesor a satisfactory estimation of the MS contribution tohe lidar returns at an optical depth t # 2 within anccuracy of 30%. At the same time, this approachay be helpful when one is correcting return signals

cquired from clouds with a spaceborne lidar for theackground noise that is due to MS needed to retrievecloud’s optical parameters. In what follows, we

how that the applicability limits of this approach toolving the inverse problem can be extended.

3. Algorithm for Reconstructing the ScatteringCoefficient Profile in Clouds

Applying the proposed approach to retrieval of thecloud’s optical parameters requires some a priori in-formation on the properties of these parameters toregularize the inverse problem solution and esti-mates of the MS contribution. In practice, only in-complete information of that kind is available. So it

Page 5: Retrieval of Cloud Optical Parameters from Space-Based Backscatter Lidar Data

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is necessary to develop techniques that may be effi-cient under conditions of that sort of information un-certainty. Without loss of generality we considerbelow the case when no a priori information is avail-able at all; otherwise its incorporation into the com-putation scheme is quite obvious.

Thus, according to Ref. 5, a solution to Eq. ~1! withespect to the profile b~z! that accounts for the MSontribution is as follows:

b~z! 5P~1!~z!z2

2εC~z0, z*! 1 2C~z, z*!, (7)

where

P~1!~z! 5 P~1!@b~z!, g~w!, z# 5 P~z!yDP~z!,

C~z1, z2! 5 *z1

z2

P~1!~z9!z92dz9; (8)

ε 5 ε~tε, z*! 51

exp@2t~z0, z*!# 2 1(9)

is a dimensionless parameter that is expressed bymeans of the optical thickness of the layer @z0, z*#, z*is an arbitrary point from @z0, zmax#, and P~z! is thelidar return. To make use of the scheme of Eqs.~7!–~9! we need the following data:

~1! Profile of the Scattering Coefficient @for use inEqs. ~8!#. Inasmuch as b~z! is also the value that weare seeking, let us make use of an iteration procedurestarting from the zeroth approximation ~either modelor calculated!:

We suppose that

b# ~ j! 5 b~ j21!;

then estimate from Eqs. ~8! that

P~1!,~ j! 5 P~1!~b# ~ j!, g!;

hen determine from Eqs. ~9! and ~7!

b~ j! 5 b@ε~t*!, P~1!,~ j!#. (10)

Here j is the number of iterations.

~2! Optical Thickness of the Layer @z0, z*#. Asnoted in Section 1, any method that is being used inthe inversion of lidar returns requires a referencevalue of the atmospheric parameters, and it is desir-able that such measurements be conducted at the farend of the path ~z* 5 zmax!. If no calibration mea-surements are feasible one is forced to use varioussimplifications and model representations. Thus,for example, in Ref. 24 a technique was studied inwhich it was assumed that the function b~z! is suffi-ciently smooth. In so doing we isolated a particularsolution from the parametric family @Eq. ~7!# thatminimizes the stabilizing Tikhonov functional.12 Inparticular, requiring the first derivative of the profile

b~z! to be smooth leads to the task of minimizing,with respect to parameter ε, the following functional:

F@bε~z!# 5 *z0

zmax

@bε~z! 2 bc~z!#2dz, (11)

where bc~z! is the profile reconstructed by the loga-rithmic derivative technique. As our calculationsshow, F degenerates as the MS background grows inmagnitude compared with P~1!~z!, and hence it has nominimum in the domain of physically meaningful tvalues. We propose to define functional F as thediscrepancy function between the maximum values ofthe scattering coefficient bmax:

F@b~z, tε!# 5 $bmax@tε~ j,k!# 2 bmax@tε

~ j,k21!#%2, (12)

tarting from zeroth approximation

tε~ j,0! 5 0.5~zmax 2 ze!bmax

5 0.5~zmax 2 ze!P~1!,~ j!~ze!ze

2

*ze

zmax

P~1!,~ j!~z9!z92dz9

,

where Ze is the point where the scattering coefficientreaches its maximum and k is the number of itera-tions when one is estimating the optical thickness.These iterations may be called internal and are per-formed with the assumption of some estimate of theMS background ~ j is fixed!. Note that, at z 5 ze, Eq.~12! coincides with the Eq. ~11!. As our calculationshowed, this function has a stable minimum at t thatorresponds to the value of b~z! that we are seeking.hus we have that, by minimizing function ~12! over, in combination with the procedure of Eqs. ~7!–~10!,e obtain the profile b~z! sought under conditions ofpriori uncertainty.As noted above, the value tmax is also related to the

MS orders that contribute to the return signal.Therefore it may be useful for achieving the correctestimates of P~1!~z! from Eq. ~2! at tmax , 1 or fromEq. ~6! at tmax $ 1.

~3! Defining the Cloud Type. Definition of theloud type means the choice of g~w! in Eq. ~8!. If nopecial methods for making this choice are used it isnly reasonable to distinguish between clouds by theistance to them and to use g~w! that is characteristicf cirrus clouds for clouds with upper boundaries ateights above 7 km while otherwise using the value of~w! for water-droplet clouds. In Ref. 25 it washown that use of a value of g~w! that is characteristicf crystal clouds while processing the return signalsrom water-droplet clouds yields a smoothened b~z!rofile that is close in value to that retrieved from theidar equation without accounting for MS. In thepposite case, when g~w! of water-droplet clouds is

used in processing returns from a crystal cloud, theretrieval of b~z! is already unstable at tmax 5 1. Thechoice of that or another form of the scattering phasefunction for water-droplet clouds does not essentiallyaffect the quality of b~z! reconstruction.

20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6369

Page 6: Retrieval of Cloud Optical Parameters from Space-Based Backscatter Lidar Data

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ea

asp

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6

Because any approximation of the MS backgroundintroduces errors into the single-scattering isolatedreturn, let us estimate the accuracy of reconstructingthe profile b~z! that results from incomplete account-ing for the MS contribution. Assume that b0~z! andb~z! were obtained according to Eqs. ~7!–~9! for theexact ε~tε!. Next, take the model

P~z! 5 P~1!~z!M~z!,

where M~z! is a nondecreasing function that de-scribes the MS background; it is always valid thatM~z! $ 1. The quantity b0~z! corresponds to thesolution when no MS is present, that is, M~z! [ 1:

nc

According to the mean-value theorem there exist z1 [@z0, zmax# and z2 [ @z, zmax# such that

st

then

M~z!

M~z2!#

b~z!

b0~z!#

M~z!

M~z1!,

*z0

z

M~z9!dz9

M~z2!#

t~z!

t0~z!#

*z0

z

M~z9!dz9

M~z1!# 1. (13)

hus the MS background in the return signal resultsn a distortion of the parameters that are recon-tructed, even if the calibration parameters arenown exactly. One should necessarily take this cir-umstance into account in interpreting the lidar data.

Figure 4 presents the profiles of the scattering co-fficient in a cloud of known type reconstructed at an

priori uncertainty relative to tε. Lidar returnsused in simulations @see Fig. 2~a!, curve S# were cal-culated by the Monte Carlo method. The functionsb~z! that we seek are shown by curves 1 in Figs. 4~a!nd 4~b!. Profile b~z! shown in Fig. 4~a! demon-trates the possibility of reconstructing b~z! by ap-lying the retrieval scheme of Eqs. ~7!–~10! when tε is

known exactly. From this point of view these pro-files may be considered the best fit achievable withthis method. Curve 2 in Fig. 4 shows the solution ofthe lidar equation obtained when MS was not takeninto account. This solution was taken as the initialapproximation in the iteration algorithm. The re-sult obtained with this algorithm after ten iterations

370 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

is shown by curve 3. Note that this method showedgood stability relative to the initial approximation, soonly approximately five iterations were sufficient forachieving convergence. Underestimation of the op-tical parameters even at an exact calibration is ex-plained by expressions ~13!. Figure 4~b! presentsb~z! profiles reconstructed at an unknown tε, the es-imate of which we found by minimizing functional12!. The curves in this figure are numbered in theame order as in Fig. 4~a!. The values of tε used inq. ~9! for calculating ε~t! in each case considered arelso shown in this figure.Analysis of the results obtained from the above

umerical simulation allows us to draw the followingonclusions:

~1! Use of the proposed algorithm permits recon-truction of a detailed structure in b~z! profiles atmax , 2, whereas at higher values of tmax the reso-

lution becomes poorer @see Fig. 4~a!#.~2! One may use this approach efficiently in esti-

mating tε under a priori uncertainty; the error doesnot exceed 30% @see Fig. 4~b!#.

Fig. 4. Comparison among methods of reconstructing scatteringcoefficients from the returns calculated by the Monte Carlo meth-od: 1, the exact profile of b~z!; 2, the profile reconstructed withoutMS taken into account; 3, the profile obtained after 10 iterations ata priori ~a! known and ~b! unknown tε @profiles P~z! correspond tocurves from Fig. 2~a!#.

b~z! 5P~1!~z!M~z!z2

2ε *z0

zmax

P~1!~z9!M~z9!z92dz9 1 2 *z

zmax

P~1!~z9!M~z9!z92dz9

.

b~z! 5P~1!~z!M~z!z2

2εM~z1! *z0

zmax

P~1!~z9!z92 dz9 1 2M~z2! *z

zmax

P~1!~z9!z92 dz9

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4. Retrieval of Cloud Optical Parameters from DataAcquired with a Spaceborne Lidar

The proposed algorithm was applied to retrieval ofthe scattering coefficient profiles from the data ob-tained with the American space-based lidar in theLITE program ~below we denote them L! and with aGerman airborne lidar ALEX ~denoted A below! dur-ng the LITE underflights on 14 September 1994 ~or-it number 79!. The specifications of the lidars thatere used are as follows:

L: wavelength, l 5 532 nm; H 5 272.5 km; w0 5.6 mrad; footprint diameter at 10-km height over thearth’s surface, dw 5 420 m; distance between the

successive laser footprints along the flight line, Dl 5800 m.

A: l 5 532 nm, H 5 13.6 km, w0 5 1.5 mrad, dw 55.4 m, Dl 5 20 m.

More-detailed information on the specifications ofthese lidars and some peculiarities of synchronizedmeasurements can be found in Ref. 18. These mea-surements were carried out synchronously over thenorthern part of Denmark. In these experimentsseveral cloud layers of different structure and densitywere detected at heights from 5 to 13 km. We chose,from these data, those that correspond to the flightline at 56.9° N latitude and 8–9.2° E longitude, whereboth cirrus ~at 8.9–12-km height; Fig. 5! and water-droplet ~at z 5 4.9–8 km; Fig. 6! clouds were ob-served. The observations with the L lidar weremade between 19:16:00 and 19:16:10 UTC; those withthe A lidar, between 19:10:48 and 19:04:22 UTC ~thepaceship and the aircraft flew in opposite direc-ions!. The array of data acquired with spaceborneidar L involves 100 lidar returns, each measured in00 m @Figs. 5~b! and 6~b! show the function S~z! 5

P~z!z2#. To make a comparison we composed an ar-ray of data acquired with the airborne lidar that alsoinvolves 100 return signals @Figs. 5~a! and 6~a!#, eachof which comprised an average of 21 shots within thefootprint of the L lidar. Thus the averaged returnsare centered in the 800-m range interval. Figures5~c! and 6~c! show the data for b~z! reconstructed bythe scheme of Eqs. ~7!–~10! for lidar A!. The recon-struction of b~z! profiles from data of the spacebornelidar was also accomplished with Eqs. ~7!–~10!. Fig-ures 5~d! and 6~d! show the results obtained after fiveiterations. In both of these cases no a priori infor-mation was used. We estimated the value of tmaxfrom lidar returns by minimizing functional equation~12!.

Figure 7 presents the data on tmax and bmax recon-structed for the clouds with lidar A ~curve 1! and withlidar L ~curve 2!. It can be seen that the data ob-tained from 8° to 8.8° E agree well in absolute valuesand in variations along the flight line. The en-hanced values of tmax and bmax obtained from air-borne data, compared with those obtained withspaceborne data, over the region at longitudes from8.8 to 9.2°E are explained by the presence of a dense

cloud 11–12 km below the aircraft that was absent inthe spaceborne case; see data L. All this is shown inmore detail in Fig. 8. Thus from Fig. 8~a! one cansee the functions S~z! averaged over five shots ~theregions between vertical lines in Fig. 5! and, from Fig.8~b!, the corresponding b~z! profiles reconstructed forthe same regions. The data for A and L are reduced

Fig. 5. Retrieval of scattering coefficient values of cirrus cloudsfrom data of synchronously flown airborne and spaceborne lidarson 14 September 1994 ~orbit number 79!: ~a! lidar A returns, ~b!lidar L returns, ~c! scattering coefficient retrieved from A data, ~d!scattering coefficient retrieved from L data. Here and in Figs. 6,8, and 9, rel. un. means relative units.

Fig. 6. Retrieval of scattering coefficient values of altostratusclouds from data of synchronously flown airborne and spacebornelidars on 14 September 1994 ~orbit number 79!: ~a! lidar A re-turns, ~b! lidar L returns, ~c! scattering coefficient retrieved from Adata, ~d! scattering coefficient retrieved from L data.

20 October 1999 y Vol. 38, No. 30 y APPLIED OPTICS 6371

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Forbt

6

to the same scale. Curves 1 in Fig. 8 show the dataobtained with the A lidar and curves 2 show thosewith the L lidar. Profiles b~z! reconstructed from Ldata without correction for the MS contribution arepresented by curves 3 for a comparison.

The right-hand part of Fig. 7 presents the data ontmax and bmax reconstructed for altostratus cloudsobserved at heights of 5.6–6.25 km ~see Fig. 6!.Here we also have isolated two portions, of threeshots each, from the data arrays to make a compar-ison. More-detailed information on these portions ispresented in Fig. 9, where the numbers at the curvesare the same as in Fig. 8. Part I of Fig. 9 involves thehorizontal boundary of a dense cloud, and there the Aand L data agree well, which means that the sound-ing beam penetrates this cloud. Part II of Fig. 9 alsoinvolve a dense cloud. The comparison made be-tween the airborne and the spaceborne lidar datademonstrates well the effects that are due to MS thatwe considered theoretically above. Thus signal L is

Fig. 7. Comparison of optical parameters of different clouds ob-tained by use of data from the spaceborne and airborne lidars.

Fig. 8. Comparison of scattering coefficients of cirrus clouds re-constructed by different methods for the regions shown in Figs. 5and 7: ~a! S~z! profiles, A data ~curves 1! and L data ~curves 2!.~b! The corresponding b~z! profiles for A ~curves 1! and L ~curves 2!present the data obtained with MS taken into account after thefifth iteration and ~curves 3! without accounting for MS.

372 APPLIED OPTICS y Vol. 38, No. 30 y 20 October 1999

wider than signal A, and its maximum is shifted tolower heights @Fig. 9~a!#. Taking MS into accountimproves the quality of the b~z! profile reconstructedfrom the L data in the upper part of the cloud ~wherehe A data can be considered reference data! andorrects the position of the upper cloud boundary,hus decreasing the geometric thickness of the cloudsee Fig. 9~b!#.

Analysis of the results of reconstruction of the b~z!profiles from data of a spaceborne lidar and theircomparison with the data acquired with the airbornelidar allow us to draw some conclusions about theprospects of using the space-based lidar systems aswell as about the problems in lidar data interpreta-tion to be addressed:

~1! The agreement between the A and L data oncirrus clouds may be considered good enough ~see

igs. 5, 7, and 8!. In this case the optical parametersf a cloud determined from A data may be consideredeference parameters because t # 1 and the soundingeam pierces the clouds, which removes doubts abouthe correctness of tmax estimates from lidar returns.

The cases of poor agreement between the data are, inour opinion, caused by poor synchronization betweenthe measurements rather than by the errors in dataprocessing. The data discussed above also confirmthe efficiency of using spaceborne lidars for soundingof cirrus clouds, because the MS contribution in thiscase is usually low and the algorithms of data pro-cessing proposed in this paper based on analyticalestimates of the MS and correction of lidar returns forit work well enough.

~2! Analysis of the interpretation of lidar returnsfrom dense water-droplet clouds ~Figs. 6, 7, and 9!shows the possibility of reconstructing the optical

Fig. 9. Comparison of the scattering coefficients of altostratusclouds reconstructed by different methods for the regions shown inFigs. 6 and 7: ~a! S~z! profiles, A data ~curves 1! and L data~curves 2!. ~b! The corresponding b~z! profiles for A ~curves 1! andL ~curves 2! present the data obtained with for MS taken intoaccount after the fifth iteration and ~curves 3! without accountingfor MS.

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Remote Sensing with Lidar, Selected papers of the 18th Inter-

properties of the clouds even in the presence of a highbackground from MS. At the same time, develop-ment of appropriate methods to extract informationon scattering media from the MS signals is needed,because the ability to use such methods would makeit possible to reconstruct scattering coefficient pro-files at larger optical thicknesses of a cloud; see Ref.26.

This research was funded in part by the RussianFoundation for Basic Research ~grant #98-05-64066!.

he authors are grateful to W. Renger of the Instituteor Atmospheric Physics, Deutsche Forschungsan-talt fur Luft und Raumfahrt, Oberpfaffenhofen, Ger-any who has kindly presented the data acquired

uring the ALEX airborne missions.

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