analysis of lidar backscatter profiles in optically thin clouds

Analysis of lidar backscatterprofiles in optically thin clouds

Stuart A. Young

The solution of the lidar equation for profiles of backscatter and extinction in optically thin clouds isconstrained by values of the cloud transmittance determined from the elastically scattered lidar signalsbelow and above the cloud. The method is extended to those cases in which an aerosol layer lies below orabove the cloud layer. Examples are given in both cases. An analytical expression for the average lidarratio in the cloud is derived for those cases in which molecular scattering is significant.Key words: Lidar, optical thickness, cloud backscatter profiles, lidar ratio, multiple scattering.

r 1995 Optical Society of America

1. Introduction

In an attempt to assess the impact of clouds on theglobal climate, many studies are being made of theradiative and microphysical properties of clouds aswell as their distribution globally and with height.Lidars are playing their part in this effort and havebeen operated at many different sites during the FirstInternational Satellite Cloud Climatology ProgramRegional Experiment in the U.S.; the InternationalCirrus Experiment in Europe and the ExperimentalCloud Lidar Pilot Study 1ECLIPS2, to which scientistson several continents contributed; and the Pilot Radia-tion Observation Experiment 1PROBE2. Other stud-ies have followed and are planned as part of theAtmospheric Radiation Program and the SouthernOcean Cloud Experiment 1SOCEX2.In this section the difficulties in solving the lidar

equation for profiles of backscatter and extinction inoptically thin clouds are considered. Methods forobtaining the solutions by using lidar-derived valuesof the cloud transmittance are covered in Section 2,and the results of applying these analytical methodsto both simulated and measured cloud signals arepresented in Section 4.Lidars can, in the appropriate circumstances, pro-

vide profiles of the volume backscatter coefficient, thevolume extinction coefficient, their height integrals,

The author is with the Division of Atmospheric Research, Com-monwealth Scientific and Industrial Research Organization, Pri-vate Mail Bag No. 1, Mordialloc Victoria 3195, Australia.Received 13 February 1995; revised manuscript received 5 June

1995.0003-6935@95@307019-13$06.00@0.

r 1995 Optical Society of America.

and the depolarization ratio that can be interpreted interms of the physical state of the cloud particles or ofthe degree of multiple scattering of radiation in thecloud. The altitude of the cloud base, and often thecloud top, can also be measured. The interpretationof lidar signals, however, is not unambiguous, asthere are two unknowns in the equation that de-scribes the lidar signal, and both may be affected bymultiple scattering.The lidar-signal voltage V1r2 produced by scattering

at a range r from the lidar 1and beyond the lidaroverlap range2 is

V1r2 5K

r23bM1r2 1 bC1r24TM

210, r2TC21rb, r2 1 Vo, 112

whereK is a system constant including output energy,receiver area, etc.; bM1r2 and bC1r2 are the molecularand cloud backscatter coefficients at range r, respec-tively; rb and rt are the ranges of the cloud base andtop, respectively; and Vo is an offset voltage thatincludes contributions from the sky background sig-nal, detector dark current, and amplifier and digitizeroffset voltages. Note that

TM10, r2 5 exp32 e0

r

sM1z2dz4 122

is the molecular transmittance from the lidar torange r, and

TC1rb, r2 5 exp32 erb

r

h1z2sC1z2dz4 132

20 October 1995 @ Vol. 34, No. 30 @ APPLIED OPTICS 7019

is the cloud transmittance from the cloud base torange r, where sM1z2 and sC1z2 are the molecular andcloud extinction coefficients, respectively, and thefactor h1z2 is an approximate representation of theeffects of multiple scattering in the cloud.In order to solve the lidar equation for backscatter

or extinction the raw lidar signal 3Eq. 1124 is usuallytransformed into a variable that removes the inverserange-squared dependence. Solutions are commonlyexpressed in terms of the variable,1

X1r2 5 3V1r2 2 Vo4r2, 142

or its logarithm,2,3

S1r2 5 ln X1r2. 152

The use of the logarithmic form creates problemswhere low signal-to-noise ratios produce negativevalues of X1r2, as in the regions found above highclouds. It is not acceptable to reject such valuesfrom analysis or merely to set S1r2 to zero, as this cancause biased results. Heavy smoothing to removethese negative values is unattractive, as it unnecessar-ily degrades the range resolution of the lidar in themain regions of interest inside the cloud, where thesignal is strong. For these reasons it is preferable touse the variable X1r2 during data analysis.The solution to the lidar equation for profiles of

extinction in haze and cloud has received extensivecoverage in the literature.1–11 Significant differencesbetween the analysis of lidar signals from thick cloudsand those from thin clouds exist. Where the cloud isoptically thick the contribution to backscatter andextinction by air molecules within the cloud is usuallyregarded as negligible, and the analysis is simplified.The rate of convergence of the solution profile to thetrue profile of extinction is more rapid for opticallythick clouds, where the lidar signal is influenced moreby the extinction process than by the backscatter thatdominates in optically thin clouds. These differ-ences have implications for the analysis of the lidarsignals from the different clouds.Solutions to the lidar equation in which there is one

dominant scatterer,2–4 as in the case of optically thickclouds, are usually achieved by omitting themolecular-scattering terms, setting h1r2 to unity, and assuming arelationship between backscatter and extinction ofthe form2,3

b1r2 5 k1s1r2k2. 162

Klett3 has shown that if the lidar is calibrated at somedistant reference point rm, then a stable solution forthe extinction profile can be written as

s1r25exp53S1r22S1rm24@k26

31@s1rm241 12@k22er

rm

exp53S1z22S1rm24@k26dz

.

172

7020 APPLIED OPTICS @ Vol. 34, No. 30 @ 20 October 1995

Provided that the cloud has sufficient optical thick-ness, any error in the boundary value will become lessand less significant as the integration in the denomi-nator proceeds, and the solution will approach thecorrect value as r goes from rm to rb. Similarly, errorsor noise in the value of S1rm2, which increase as r2increases, also become less significant as r approachesrb. The rate of convergence is strongly dependent onthe optical thickness of the cloud and has beenstudied by several authors.9–11 The solution 172 hasbeen used successfully in the analysis of lidar cloudsignals,12,13 usually by assigning a value of unity to k2.Generally, the more attenuating the cloud, the fasterthe convergence.Unfortunately, the above method is not as satisfac-

tory for optically thin clouds. Here three problemsarise. First, the optical thickness is too low to pro-duce rapid enough convergence of the solution tocorrect for errors in the estimate of the boundaryvalue or for noise or offset in the signal at thecalibration range. Second, the signal is produced byscattering from two species, and both must be consid-ered. Third, as pointed out in the short but incisivepaper by Fernald,1 in this case the extinction-to-backscatter ratio must be known if solutions for s1r2 orb1r2 are to be found. This ratio F 5 s@b has come tobe known as the lidar ratio, and this term is usedthroughout the paper. It is dependent on the size,shape, and refractive index of the cloud particles andvaries with cloud type and also, to some extent, withinany cloud because the size distribution of the cloudparticles varies with height.14 The lack of an accu-rate knowledge of the lidar ratio is a significantproblem in the interpretation of lidar measurementsof optically thin clouds that is not present in theanalysis of thick clouds. 3Note that neither F nor k1appears in Eq. 172.4In this paper, the term optically thin clouds refers

to clouds that can be penetrated by the lidar tomeasure a signal from the atmosphere on the far side.This definition may seem somewhat arbitrary, but forthe purposes of this paper it is quite a practical one, asit is just these clouds that generally have insufficientoptical depth to force a rapid convergence of thesolution.Methods to reduce the effects of boundary-value

errors and the uncertainty in the value of the lidarratio for clouds, FC, thereby allowing increased accu-racy in the analysis of optically thin clouds, are thefocus of this paper. Techniques for determining cloudoptical thickness are extended to those cases in whichthe atmosphere above or below the cloud contains anaerosol layer.

2. Methods

In this section algorithms for the analysis of lidarsignals from optically thin clouds are considered.As in the case of moderately turbid atmospheresconsidered by Fernald,1 solutions for optically thinclouds require an accurate calibration of the lidarsignal and a value of the lidar ratio for clouds.In the methods described below, the analysis makes

use of a reference signal that represents the lidarsignal measured in the absence of clouds. This maybe a modeled signal representing the lidar profileexpected from a purely molecular atmosphere, or itmay be an actual profile obtained under cloud-freeconditions. The following algorithms have been in-

Fig. 1. 1a2 Simulated lidar signal from a cirrus cloud. 1b2 Refer-ence signal from molecular atmosphere. 1c2 Retrieved backscatterprofile 13 10 26 m21 sr212.

cluded in software used for the automatic analysis ofdata acquired during several field experiments.Before analysis is begun, the lidar profile is subdi-

vided into three regions. An outer window is de-fined, bounded by ranges r1w and r2w, as shown in Fig.11a2, so that all acceptable data points lie within thiswindow. Within this outer window another innerwindow is defined, usually after prior inspection ofthe raw lidar data, so that all cloud signals in the dataset are above range r1 and below r2. These rangesneed not be set precisely at the cloud boundaries, asthey serve only as initial values for the cloud-

detection algorithm that follows, thereby allowing thecloud layer to vary considerably in height and thick-ness without the need to redefine r1 and r2 for eachprofile. The region from r1w to r1 is referred to belowas region 1, and the region from r2 to r2w is referred toas region 2. Regions 1 and 2 should be chosen insuch a way that they are free from any broken cloud orvariable aerosol layers. Cases in which this is notpossible are covered below.The first step in the analysis is the determination of

the cloud boundaries covered in Subsection 2.A.Next the lidar signal is calibrated, and a boundaryvalue is determined 1Subsection 2.B.2 by fitting themeasured signal to the reference signal in eitherregion 1 or region 2. The cloud transmittance, whichis needed for the calculation of the lidar ratio, isdetermined in Subsection 2.C., in which the case of acloud in a purely molecular atmosphere is consideredfirst. The method is then extended to the cases inwhich an aerosol layer is either above or below thecloud layer. In Subsection 2.D. an analytical expres-sion for the effective lidar ratio hFC in an opticallythin cloud is derived, allowing this parameter to beused between the cloud base and the cloud top duringthe analysis of the lidar profile.A two-component solution to the lidar equation

must be used for the analysis of lidar signals fromoptically thin clouds in order to correct for the signifi-cant scattering from air molecules. The algorithmsproposed by Fernald1 satisfy this requirement, asdoes the modified solution by Klett.8 The Fernaldalgorithms have the advantage of using the variableXrather than the noise-susceptible S, and this is impor-tant in the lidar system used here. In the following,the Fernald equations are modified to include theeffects of multiple scattering in the manner of Sassenand Cho.15 For the analysis of data at ranges be-tween the lidar and some distant calibration range rc,the modified equation is

where

FM 5 sM1r2@bM1r2 5 8p@3 192

is the lidar ratio for air molecules and

C1rc2 5 X1rc2@3bM1rc2 1 bC1rc24 1102

is a calibration term evaluated at the calibrationrange. Equation 1102 normalizes the range-squaredcorrected variable X1r2 with the total backscatter atthis range.Note that here the cloud backscatter bC and lidar

bM1r2 1 bC1r2 5

X1r2exp3221hFC 2 FM2 erc

r

bM1z2dz4C1rc2 1 2hFC e

rc

r

X1r2exp3221hFC 2 FM2erc

z

bM1s2ds4dz, 182


ratio FC are used, but the solution could be expressedas generally in terms of the aerosol backscatter bAand the lidar ratio FA and is used in this form below.An approximation is made here by assuming that h isconstant with range.

A. Determination of Cloud Boundaries

The automatic determination of the ranges of thecloud base and top has been discussed in the litera-ture.15–17 The methods use various algorithms thatsearch for a region in which the signal meets somecriterion, such as exceeding some threshold for asufficient number of consecutive points, or the deriva-tive of the signal with respect to range may beinterpreted to determine the boundaries.16 As thedata from the lidar used here often contain significantamounts of photon or electronic noise superimposedon the signal, the derivative method is often inappro-priate. Also, because the analysis algorithms re-quire that one know the cloud boundaries beforehand,the scattering-ratio method15 is not used, except in aspecial case discussed below. Alternative cloud-detection algorithms have been included in the analy-sis software and are summarized here.The range of the cloud base, rb, is determined by

extrapolating the reference cloud-free signal, which isrescaled by fitting it to the measured signal in region1, forward into the inner-window region 1r1 to r22 untilthe ratio of the measured signal to the extrapolatedsignal exceeds some preselected value for a certainpreselected number of points. The range of the cloudtop, rt, is determined similarly by extrapolating thereference signal, which is rescaled by fitting to themeasured signal in region 2, backward into the inner-window region. This technique is found to work wellwhen the signal-to-noise ratio is high, as in thesignals from low- and middle-altitude clouds.When the signal-to-noise ratio is low, as is often

found in high, optically thin cirrus clouds, a secondmethod is found to be preferable. The average andstandard deviations of the ratio of the measuredsignal to the reference signal are calculated over thewhole of region 1 and also of region 2. The cloud baseis defined as the first of a number of consecutivepoints beyond r1 where the ratio of the measured tothe reference signals exceeds the average ratio inregion 1 by a certain number of standard deviations.The cloud top is found similarly.The selection of the criteria for determining the

cloud boundaries is made after an initial examinationof the data. It has been found that three consecutivepoints are usually sufficient to ensure reliable detec-tion of clean signals from low clouds, whereas five toseven points may be needed in extremely noisy lidarsignals from weak, high clouds.

B. Signal Calibration and Determination of Boundary Valuesfor Solutions

The Klett algorithm 3Eq. 1724 is often used for theanalysis of optically thick clouds. The extinction-coefficient boundary value s1rm2 must be estimated in


the cloud near the last data point before the signal isextinguished by reference either to previous, indepen-dently measured values or to modeled data. It is notunusual for the initial value to be wrong by a factor ofbetween 2 and 10, and this error is compounded bythe small signal 3S1rm24 at this calibration range, withits associated large relative error. However, theseerrors rapidly lose significance as the denominator inEq. 172 becomes dominated by the integral term, whichis governed by the optical depth along the solutionpath, and the solution converges rapidly.In certain circumstances the determination of the

boundary value in the case of optically thin clouds iseasier than in the case of clouds that cannot bepenetrated by the lidar. Often the atmosphere abovea cloud layer is clear, and the scattering can beassumed to be molecular with small errors. At532 nm the error in neglecting the aerosol contribu-tion to scattering in the troposphere above the mixedlayer is often less than 1% to 5%.18 A reliableboundary value to initiate the solution can thus oftenbe chosen in the region above the cloud by simplyusing the molecular-scattering value. Because thelidar signal in this region is weak and often hassignificant superimposed noise, the following methodis adopted here.Let us consider the ideal case, in which the cloud

lies in a height region of the atmosphere whereaerosol scattering is negligible. The clear regionsbelow and above the cloud will be referred to as region1 and region 2, respectively, as before. A modeledsignal that represents the signal expected from apurely molecular atmosphere is generated. 1The mo-lecular-density profile needed is obtained in the pre-sent case from local radiosonde ascents, a climatologyof monthly mean radiosonde data gathered over manyyears,19 or satellite data.2 Themodeled signal is then

M1r2 5 bM1r2TM210, r2@r2. 1112

It is convenient to split the transmittance into twofactors, the first for the region between the lidar andsome calibration range and the second for the regionfrom the calibration range to the range r. Providedthat there is sufficient signal above the cloud, thecalibration range can be either above or below thecloud. Thus, for r . rc

M1r2 5 TM210, rc2bM1r2TM

21rc, r2@r2. 1122

From Eq. 112, the measured signal in the aerosol-freeregion above both rc and the cloud 1region 22 is

V1r2 5 3KTC21rb, rt2TM

210, rc2bM1r2TM21rc, r2@r24 1 Vo,

5 KTC21rb, rt2M1r2 1 Vo. 1132

If a linear regression of the measured signal V1r2against M1r2 is performed in region 2, both theunknown offset voltage Vo and the calibration factorC2 5 KTC

21rb, rt2 are determined. In particular, thecalibration factor, which is the product of the lidar-

system constant and the square of the cloud transmit-tance, can be determined with very much improvedaccuracy than if a single point were used. Note thatthis calibration factor is equivalent to C1rc2 as definedin Eq. 1102, with the boundary value in the denomina-tor set to bM. Signal calibration, offset removal, andboundary-value selection have all been accomplishedin this process.Alternatively, if calibration is performed in region

1, then the calibration factor is simply C1 5 K. Inpractice the measured signal is fitted to the modeledsignal in both region 1 and region 2 and to theregression coefficients used to calculate the cloudtransmittance as described below.

C. Determination of Cloud Transmittance

The cloud transmittance can be measured with amethod used in the past to determine the transmit-tance of smoke plumes.20 The attenuation of thelidar signal as it passes through the cloud is deter-mined by comparing the signal below and above thecloud. In the simple case of a cloud in a region ofinsignificant aerosol scattering discussed in Subsec-tion 2.B., the cloud transmittance is simply the ratioof the calibration factors obtained in region 2 andregion 1:

TC 5 1C2@C121@2 5 3KTC

21rb, rt2@K41@2. 1142

Note that the calibration factors C1 and C2 aredetermined through linear regression of the mea-sured signal to the modeled signal over the whole ofregions 1 and 2, respectively. This gives a betterestimate of the transmittance than if the calculationused values obtained only at the ranges rb and rt andalso allows an estimate of the uncertainty in the valueof TC to be made.The transmittance of a light beam through a cloud

is increased by multiple-scattering effects in whichphotons that have been scattered out of the samplingvolume of the lidar in one scattering event may bereturned to the lidar field of view in subsequentscattering events. Also, photons that are scatteredat shallow forward angles are not lost from the lidarbeam and may experience further scattering. Themagnitude of the multiple-scattering effect dependson the number density, size distribution, and shape ofthe cloud particles, and also on the lidar field of viewand the distance of the scattering volume from thelidar. The transmittance calculated by the use of theabove method includes the effects of multiple scatter-ing as shown in Eq. 132, with h assumed to be constant.

1. Cloud-Transmittance Calculation with anOverlying Aerosol LayerDuring the acquisition of data during the TropicalGlobal Atmosphere/Coupled Ocean-Atmosphere Re-sponse Experiment 21 the strong scattering from theMt. Pinatubo aerosol layer, which was overlying thehigh tropical cirrus clouds being studied, meant thatthe atmosphere above the clouds could not be de-

scribed simply by the use of molecular scattering.A new reference profile was needed for this andsimilar cases.When a large number of cloud-free profiles are

averaged and this average profile is fitted and res-caled to a model molecular profile above the aerosollayer, a new reference profile can be produced andused in place ofM1r2 in the determination of both cloudboundaries and cloud transmittance. Acorrect valueof the backscatter boundary value for initiating thesolution profiles can also be determined from thisprofile.The new reference profile also includes the effects of

aerosol scattering 1bA and TA2. This reference signalnow becomes 3cf. Eq. 11124

W1r2 5 3bM1r 2 1 bA1r24TM210, r2TA

210, r2@r2, 1152

whereas the measured signal is

V1r2 5 K3bM1r2 1 bA1r2 1 bC1r24

3 TM210, r2TA

210, r2TC21rb, r2@r2 1 Vo. 1162

Below the cloud layer bC is zero and TC2 is unity, so

a linear regression of the measured to referenceprofiles again produces values of K and Vo. Abovethe layer bC is also zero and TC

2 5 TC21rb, rt2 can be

found as before, with Eq. 1142 and the regressioncoefficients determined in the regions above andbelow the cloud.Cirrus cloud transmittance has also been measured

in the past by using the stratospheric aerosol layerproduced by the eruption of El Chichon as a calibra-tion target. However, the method of Hall et al.22differs from the present one in three ways. Theearlier method used an absolutely calibrated, pulsed,IR Doppler lidar, whereas in the present methodcalibration and background removal are effectivelyperformed on every profile. Also, the cirrus transmit-tance was determined in the earlier method by usingthe reduction in the apparent, integrated strato-spheric backscatter from its value when no cirrus waspresent, whereas here the calibration is performed byfitting the measured signal to the cloud-free referenceprofile over a large height range. Because the refer-ence and measured signals in regions 1 and 2 areproduced at visible wavelengths predominantly bymolecular scattering, which is relatively well defined,the present method of calibration is considered to beless susceptible to normalization errors when visiblelidars are used than if the reference and measuredsignals were compared only in terms of the aerosolcontribution to scattering.The most significant difference between the meth-

ods is that whereas Hall et al.22 only calibrated abovethe cirrus layer, here calibration is performed bothabove and below the layer by fitting to the referencesignal in these regions. At visible wavelengths, theatmospheric transmittance due to molecules andaerosols between the lidar and the cloud base 1often at12 to 14 km altitude2 can vary considerably. In


particular, during the tropical experiment just men-tioned, low, wispy cloud occasionally attenuated thelidar signal from the atmosphere above. When thelower bound of region 1 was set above this low cloud,the decrease in signal strength was common to bothregion 1 and region 2, so the correct value of cloudtransmittance was still given by Eq. 1142. This wouldnot have been the case had calibration been per-formed only on the far side of the cloud.

2. Cloud-Transmittance Calculation with anUnderlying Aerosol LayerDuring SOCEX1 in June–July 1993, the lidar wasbeing used to study marine stratocumulus cloudslocated at the top of a moist mixed layer on thenorthwest coast of Tasmania. The winter airmasswas clean, with low cloud condensation nuclei counts,and the resulting clouds were often optically thin,making it possible to determine TC and hFC. It hadbeen anticipated beforehand that the method wouldrequire further development for use with these cloudsbecause of the strong scattering from the moist aero-sols in the mixed layer. A marked step decrease inthe lidar profile was produced at the top of this layer.Obviously, if this step were not considered, a spuri-ously low value of the cloud transmittance wouldresult.Several different algorithms were tried, including

the use of a measured cloud-free reference profile asdescribed above. Successful application of this algo-rithm was limited by the occasional, significant shot-to-shot variations in the height distribution of theaerosol in the mixed layer. Successful results werefinally obtained with the approach described below.A value of C2 can be obtained, as before, by fitting

the measured signal to a reference, cloud-free signalin region 2. However, although a fit cannot be per-formed in region 1 1the mixed layer2, a value for C1 canbe obtained from the measured signal by solving thelidar equation in the forward direction until the cloudbase is reached. This solution requires a value of thelidar-system constant, which is obtained beforehandby solving the reference profile for the aerosol-scattering profile. This algorithm is now describedin more detail.The reference profile produced by averaging many

cloud-free signals is analyzed with Eq. 182 1with thecloud terms replaced by aerosol terms and h set tounity2. The aerosol-backscatter profile bA1r2 is calcu-lated right down to the minimum overlap range of thelidar, rmin. By using values of FA based on surfacemeasurements of aerosol-size distributions, the aero-sol transmittance between the calibration range andrmin can be calculated:

TA21rmin, rc2 5 exp322 e

rmin

rc

sA1z2dz4 . 1172

The averaged, cloud-free profile that is used as areference profile for the analysis of the cloud profilesis given by Eq. 1162, with bC set to zero and TC to


unity. It can be seen, then, that if this signal is fittedabove the mixed layer, where aerosol scattering isnegligible, to a model clear-air signal given by Eq. 1112,then the calibration factor is

C 5 KTA210, rc2. 1182

Because no analysis can be performed for ranges lessthan rmin, we calculate the quantity

KTA210, rmin2 5 C@TA

21rmin, rc2. 1192

If we assume that TA210, rmin2 varies little from shot to

shot, then the quantity defined in Eq. 1192 can be usedas a calibration constant for the analysis of subse-quent profiles. This is preferable to the assumptionof a constant boundary value for backscatter at somerange in the mixed layer.Each individual profile in the data file can now be

analyzed with this calibration. First, the measuredsignal is fitted to the reference profile in region 2 asbefore, and signal offset Vo and calibration factorKTA

210, rb2TC21rb, rt2 are determined. The aim now is

to solve the lidar equation in the forward directionstarting at rc 5 rmin in order to produce a value ofKTA

210, rb2. 1The accurate calibration of the signaland the low optical thickness ensure that the solutionwill not diverge greatly.2 The ratio of these twofactors gives the desired TC

21rb, rt2. We proceed asfollows. First the offset is removed from the mea-sured signal, Eq. 1162, and the result is divided byKTA

210, rmin2TM210, rmin2 and solved for bA1rmin2:

bA1rmin25 3X1rmin2@KTA210, rmin2TM

210, rmin242 bM1rmin2.

1202

Themolecular-transmittance factor is calculated fromthe molecular profile with Eq. 122. The aerosol-transmittance over the next range step can be esti-mated with the value of FA chosen above and multi-plied by KTA

210, rmin 2 to form KTA210, r2. The aerosol

backscatter at this next range, bA1r2, can then becalculated with Eq. 1202 1with rmin replaced by r2 andimproved by iteration.23 The scattering ratio

R1r2 5 3bM1r2 1 bA1r24@bM1r2 1212

is also calculated and used to test for the presence ofcloud.15 By stepping through the consecutive rangesin this fashion, the algorithm finds the cloud-baserange rb and also calculates C1 5 KTA

210, rb2. If allaerosol scattering above the cloud base is consideredas scattering from cloud particles, the cloud transmit-tance can then be calculated as described above. Weare now in a position to calculate the effective lidarratio.

D. Determination of the Effective Lidar Ratio

The lidar ratio for the cloud particles is, as definedabove, the ratio of the cloud extinction and backscat-ter coefficients. Both quantities vary with the cloud-particle-size distribution, so some variation in their

ratio is expected in the cloud. This variation ismeasurable with Raman lidars24 and high-spectral-resolution lidars,25 but cost and other limitationsmake these lidars relatively rare. The average valueof this ratio through the cloud is still, however, auseful quantity, as it allows solutions for b to bedetermined more accurately, especially for opticallythin clouds, where only small-to-moderate extinctioncorrections are made. It also provides informationon the cloud-particle-size distribution and the amountof multiple scattering.We proceed now to derive an analytic expression for

the quantity called here the effective lidar ratio byintegrating the two-component lidar equation fromcloud base to cloud top and using the cloud transmit-tance that was determined above. The method is anextension of the method of Platt6 to the case ofoptically thin clouds in an atmosphere in whichmolecular scattering is not negligible. 1Platt, how-ever, calculates the cloud transmittance with thelidar-radiometer technique rather than using thelidar itself.2 When this effective lidar ratio is used inthe analysis of a lidar profile, the solution is effec-tively constrained by the value of the cloud transmit-tance.It is convenient to introduce the effective cloud

optical thickness htC, defined as

htC1rb, r2 5 2ln3TC1rb, r24, 1222

5 h erb

r

sC1z2dz, 1232

5 hFC erb

r

bC1z2dz, 1242

If the assumption is made that aerosol backscatter isnegligible above the cloud base, Eq. 1162 can berearranged:

X1r2@3KTM210, rb2TA

210, rb2TM21rb, r24 2 bM1r2TC

21rb, r2

5 bC1r2TC21rb, r2. 1252

Integration is now performed from cloud base to cloudtop, making use of the relationships in Eqs. 1222 to 1242,to find

31@KTM210, rb2TA

210, rb24 erb

rt

X1z2@TM21rb, z2dz

2 erb

rt

bM1z2TC21rb, z2dz 5 31 2 TC

21rb, rt24@2hFC. 1262

Note that TC2 appears on both sides of the equation.

To resolve this difficulty, let us replace the producthsC1z2 with its average value through the cloud in thecalculation of TC

2 in the left-hand side of Eq. 1262.From Eq. 132we find that

TC21rb, z2 5 exp322hs1z 2 rb24. 1272

When this substitution into the LHS of Eq. 1262 ismade, the effective cloud lidar ratio can then befound:

hFC 5 0.531 2 TC21rb, rt24@G1rb, rt2, 1282

where

G1rb, rt25 31@KTM210, rb2TA

210, rb24erb

rt

X1z2@TM21rb, z2dz

2 erb

rt

bM1z2exp322hs1z 2 rb24dz. 1292

In practice, this value is further refined by the useof an iterative process in which the measured opticalthickness calculated by Eqs. 1142 and 1222 is comparedwith that calculated from Eq. 1242 with the profile ofbC1r2 obtained from Eq. 182. The first iteration stepgives

hFC,1 5 hFC1htC,m@htC,c2,

then subsequent iterations are performed until thecalculated optical thickness htC,c is acceptably close tothe measured value htC,m:

hFC,n 5 hFC,n21 1 1htC,m 2 htC,n212

3 1hFC,n21 2 hFC,n222@1htC,n21 2 htC,n222.

The capability of improving the estimate of the effec-tive lidar ratio by the use of iteration was included inthe analysis software because it was not known howmuch of an error would be introduced by the approxi-mation made in Eq. 1272. In practice it has beenfound that the value after iteration usually differslittle from the initial estimate.Estimates of the uncertainties in these retrieved

quantities can be expressed in terms of the uncertain-ties in the linear regression coefficients. The abso-lute uncertainty in ht is half the relative uncertaintyin T2 and is obtained from Eqs. 1142 and 1222 as

d1ht2 5 0.5dT2@T2 5 0.531dB1@B122 1 1dB2@B22

241@2,

1302

where dB1 and dB2 are the uncertainties in B1 and B2,respectively. From Eqs. 1282 and 1292 it can be seenthat the relative uncertainty in the determination ofhFC for thin clouds is dominated by the uncertainty inthe numerator:

d1hFC2@hFC . dT2@11 2 T22 . 0.5dT2@ht. 1312

Sassen and Cho15 also arrive at a value of the lidarratio averaged through the cloud layer. By assum-ing an appropriate value for the multiple-scatteringfactor h, and assuming purely molecular scatteringbelow the cloud, they vary the lidar ratio iteratively soas to maximize the agreement between the calculatedlidar backscatter and an assumed molecular backscat-


ter in the 500 m above the cirrus layer. It is consid-ered that the present method, which uses linearregression of the measured signal to a referencesignal over considerable height ranges above andbelow the cloud, can achieve amore precise determina-tion of the cloud transmittance and also, as a result,the effective lidar ratio. The success of the analyti-cal expression in estimating hFC also shows thatiteration is usually unnecessary when the presentmethod is used.Values of the lidar ratio for cirrus clouds at IR

wavelengths have been determined by Hall et al.,22who compared the integrated cirrus backscatter, cor-rected for attenuation, with the cirrus optical thick-ness determined from the apparent reduction in theintegrated backscatter of the El Chichon strato-spheric layer. Athin-cloud approximationwasmade,and the allowance for molecular scattering made herewas unnecessary at their IR wavelengths.With all these methods, the best results are ob-

tained when there is only one cloud layer. Obviously,the description of an average effective lidar ratio for aprofile containing several layers of different clouds isof limited physical use. Note, however, that if thelayers are sufficiently separated in height to allow theselection of reference regions above and below eachcloud layer, then the present method can provideeffective lidar ratios for each layer.

3. Practical Considerations

A. Digitizer Limitations

In order to determine the lidar ratio, the digitizermust measure accurately both the peak cloud signaland the weak signal above the cloud. This is notpossible when a digitizer of low amplitude resolution1e.g., 8 bits2 is used. This limitation can be overcomeby recording the signal on two digitizer channelssimultaneously but using gains that differ by a factorof 5 to 10, thus increasing the dynamic range of therecording system. During analysis, the signals fromboth channels can bemerged automatically by the useof a linear regression of those corresponding pointsthat are not clipped by the digitizer and also have asufficiently small relative error.

B. Signal-Induced Noise

It is important to warn here of a potential problemthat may cause spurious results when the cloudtransmittance is determined as described above.When some photomultiplier tubes are subjected to astrong, short light pulse, it is possible that the outputsignal for some period afterward will be spuriouslyincreased by the phenomenon known as signal-induced noise 1SIN2.26–30 The relatively weak signalfrom the clear region above a strongly scatteringcloud layer may be affected by SIN and will cause thecalculated transmittance to be too high.Where possible, the lidar profile above a strong

cloud layer should be compared with a cloud-freereference signal. SIN will appear as a spurious


enhancement of the signal that decreases with rangeat a rate different from the reference signal. Apossible solution is to reduce the strength of the lightpulse striking the photocathode by inserting a neutraldensity filter in the optical path and then increasingthe gain of the amplifiers.

4. Results of Analysis of Lidar Cloud Signals

The above algorithms have been used for analyzingcloud and aerosol lidar data that have been acquiredduring recent experiments 1ECLIPS, PROBE, SOCEX2.Many of the procedures in this program have beenautomated to facilitate the analysis of many thou-sands of lidar profiles. The program is largely con-trolled through menus that allow the analyst to selecta range of parameters used in the analysis of the data.These include outer- and inner-window ranges, cali-bration intervals, backscatter boundary values,smoothing interval, background-removal method,cloud-detection algorithm, and reference-atmosphereselection, among many others.The algorithms in the analysis software have been

tested with modeled profiles produced by the use of alidar-simulation program that I have written. Thisprogram allows the specification of the molecularatmosphere, aerosol and cloud profiles, and solarbackground, in addition to the lidar-specific param-eters such as detector, amplifier, and digitizer 1orphoton counter2, each with its own gain, bandwidth,noise, and resolution. The modeled signals presentthe analysis software with all the familiar difficultiesof signal noise and limitations in dynamic rangecausing either clipping or insufficient amplitude reso-lution, but the analyst has the advantage of beingable to check the output from the analysis softwareagainst a known input profile. Examples of theperformance of the analysis algorithms when testedagainst simulated signals are given below, along withexamples of the analysis of measured signals.

A. Simulated Optically Thin Cloud in a Clear Atmosphere

Asimulated lidar signal from a thin cirrus cloud in anotherwise purely molecular atmosphere is plotted inFig. 11a2, using the variable S1r2 defined in Eq. 152.The cloud base and top are at 9908 m and 10,568 mrespectively; peak backscatter is 7.0 3 1026 m21 sr21,and the optical thickness is 0.129, implying a transmit-tance of 0.878. The multiple-scattering factor andthe lidar ratio are set to be constant with height, witha product hFC 5 40 sr. The reference signal calcu-lated from the modeled molecular atmosphere isplotted in Fig. 11b2.For analysis, the outer-window ranges 1r1w and r2w2

were set at 5 and 20 km, and the inner-window ranges1r1 and r22were set at 9 and 12 km 3Fig. 11a24. For bestresults r1 and r2 should be set as close to the cloud aspossible, but generally sufficient leeway is left toensure that all clouds in the data set lie within theinner window. The ranges selected here are notoptimum but suffice to show the behavior of the

algorithm under typical analysis conditions. Whenthe second cloud-detection algorithm in Subsection2.A. was used, the cloud base rb was determined asthe first of five successive points in the inner windowwhere the ratio of the signal to the reference exceededthe average of this ratio in region 1 by one standarddeviation. The cloud top rt was determined in asimilar fashion by the use of the average ratio inregion 2. The algorithm retrieved the cloud base andtop ranges successfully.Cloud transmittance was determined with Eq. 1142,

and the results of the linear regressions in region 1and region 2 are presented in Table 1. The linearregression coefficients A, B, and R are, respectively,the offset, scale factor, and correlation coefficient.The effective optical thickness and the value of hFCare also listed and compared with themodeled values.The initial value of hFC calculated with Eq. 1282 is notchanged significantly after iteration. This is notsurprising in this case, as the cloud has a constantextinction throughout most of its depth and theapproximation in Eq. 1272 is valid. Satisfactory deter-minations of the cloud transmittance and effectiveoptical thickness have thus been achieved with thissimulated cloud signal, and the effective lidar ratiohFC has been determined within 5%. The errors inthese retrieved quantities result from the noise in thesimulated signal and lie within the range of the listeduncertainties given by Eqs. 1302 and 1312. Note thatthe above quantities were retrieved exactly when asimulated noise-free signal with a digitizer resolutionequal to the precision of the computer was analyzed.The corrected, integrated, attenuated cloud back-

scatter defined in Eq. 1292 was found to be nearly 11%less than the uncorrected integral of the total attenu-ated backscatter, implying that the derived value ofhFC would be overestimated by this amount if thecorrection formolecular scattering had not beenmade.The signal profile was then analyzed by the use of

the Fernald algorithm in Eq. 182 with the derivedvalue of hFC between rb and rt and a value of 50 sroutside this range. The retrieved backscatter profile3Fig. 11c24 shows a faithful reproduction of the origi-

Table 1. Analysis of a Modeled Cirrus Cloud in a ClearMolecular Atmosphere

Parameter Region 1 Region 2

Lower limitfor fit

5000 m 12,000 m

Upper limitfor fit

9000 m 20,000 m

A 122.2 6 2.12 3 1026 127.1 6 3.02 3 1027

B 12.51 6 0.012 3 1022 11.97 6 0.032 3 1022

R 0.9969 0.9484

Cloud Parameter Measured Modeled

T 0.885 6 0.006 0.887htC 0.122 6 0.007 0.129hFC 1sr2 38.1 6 2.1 40.0

nal backscatter profile. In this case the calibrationconstant was determined from the lower referenceregion 1r1w to r12, and the calibration height waschosen as the point below the cloud base. The back-scatter profile below the calibration height was re-trieved by the use of the backward-integration algo-rithm, whereas above this height the forward formwas used. Although the solution could have beeninitiated on the far side of the cloud with the analysissoftware, the present choice was made because thesignal-to-noise ratio is higher in the lower region andgives a more accurate system constant, and alsobecause this example shows that a forward solutioncan still give accurate results, provided that thesolution is constrained.

B. Measured Cirrus Cloud below the Mt. Pinatubo AerosolLayer

During the PROBE the Commonwealth Scientific andIndustrial Research Organization lidar was operatedat Kavieng 12.5° S, 150.48° E2 in Papua, New Guineafrom January to February of 1993. The referenceprofile 3Fig. 21b24, obtained on the afternoon of 4February 1993 during one of the few clear periods, isthe average of several thousand profiles and showsthe Mt. Pinatubo stratospheric aerosol layer quiteclearly.The lidar signal plotted in Fig. 21a2 shows the

stratospheric aerosol layer in addition to a cirrus

Fig. 2. 1a2 Lidar signal from a cirrus cloud below the stratosphericlayer. 1b2 Reference cloud-free signal. 1c2 Retrieved backscatterprofile 13 1026 m21 sr212.


layer between approximately 12 and 13 km. Thelidar receiver field of view was 2 mrad for thesemeasurements. Analysis of the reference profilesindicated that the aerosol backscatter in the regionbetween the base of the stratospheric layer and thetop of the cirrus layer was not negligible and alsovaried with height. To calculate the cloud transmit-tance then, it was necessary to use the methodsdescribed in Subsection 2.C.1.Cloud boundaries were determined as those heights

corresponding to the first of three points where thesignal exceeded the extrapolated fit by a ratio of 1.15for the cloud base and 1.25 for the cloud top. Thecloud base was thus determined as 12,115 m, and thecloud top was determined as 12,954 m. The regions1 and 2 used during analysis are defined in Table 2,which also lists the regression results and the derivedvalues of T, htC, and hFC. Good fits were obtainedboth above and below the cloud layer and resulted insmall uncertainties in T and hFC.The profile of aerosol backscatter 3Fig. 21c24 was

calculated with the system constant determined fromthe fit in region 1 with the derived value of hFCbetween the cloud base and top and a value of 50 sroutside the cloud. The solution profile does notdiverge either inside the cloud or throughout thestratospheric aerosol layer, even though the forwardsolution was initiated just below the cloud base, andillustrates how well it is constrained by the use of thepresent method.The correction for molecular scattering in the calcu-

lation of hFC for this profile is 29%. The value of23.5 sr obtained for hFC can be compared with previ-ous measurements for tropical cirrus. Platt et al.31obtained a value equivalent to 19 sr for tropical cirruswith mid-cloud temperatures between 240 and280 °C. Mid-cloud temperature for the present pro-file was 260 °C.

C. Simulated Stratocumulus Cloud above anAerosol Layer

The method outlined in Subsection 2.C.2 for theanalysis of optically thin clouds overlying a mixedlayer containing significant aerosol scattering is dem-onstrated here on the simulated lidar signal plotted in

Table 2. Analysis of a Measured Cirrus Cloud below anAerosol Layer


Lower limitfor fit

5500 m 14,000 m

Upper limitfor fit

11,000 m 33,000 m

A 12.17 6 2.882 3 1025 14.1 6 7.92 3 1026

B 19.10 6 0.032 3 1022 18.69 6 0.152 3 1022

R 0.9997 0.9580

Cloud ParameterT 0.9770 6 0.0085htC 0.0229 6 0.0087hFC 1sr2 23.5 6 1.4


Fig. 31a2. The cloud center is at a height of 1000 m,where the peak backscatter is 1.0 3 1024 m21 sr21 andthe effective lidar ratio is 20 sr. The cloud thicknessat half-maximum backscatter is 100 m. The simu-lated reference signal from the modeled aerosol layeris shown in Fig. 31b2. The mixed-layer height is950 m, and the lidar ratio and the aerosol backscatterare constant with height, with values of 50 sr and1.0 3 1026 m21 sr21, respectively. The plots of thesimulated lidar profiles are truncated at 3000 m,where the effects of noise become excessive, and thesolution is also truncated at the overlap range1250 m2.The algorithm described in Subsection 2.C.2, in

which the analysis is initiated at the minimum usefulrange with a known value of the system constant, wastested with the simulated profile. The referenceprofile was calibrated against a molecular atmo-sphere profile in region 2 and solved to calculateKTA

210, rmin2TM210, rmin2 and the constant C1 with Eqs.

1172–1212. Regression coefficients for region 2 areincluded with the value of C1 for the region 1 in Table3. As there was no linear regression performed inregion 1, the uncertainties in this region have beenset to zero.The modeled values of cloud transmittance, effec-

tive optical thickness, and hFC were all successfullyretrieved within the uncertainties calculated solely

Fig. 3. 1a2 Simulated lidar signal from a stratocumulus cloudabove the boundary-layer aerosol. 1b2 Simulated reference signalfrom a cloud-free atmosphere. 1c2 Retrieved backscatter profile13 1026 m21 sr212.

from the uncertainty in the fit in region 2. Thecorrection for molecular scattering in the calculationof hFC is only 2% in this case and is less than theuncertainties from other sources. The solution forthe backscatter profile 3Fig. 31c24 was then calculatedwith the derived value of hFC between cloud base andcloud top and with a value of 50 sr elsewhere andcorresponded exactly to the modeled profile withinthe limits of the simulated detector and digitizationnoise.

D. Measured Stratocumulus Cloud above anAerosol Layer

Last, the performance of the above algorithm isdemonstrated with an actual lidar signal from astratocumulus cloud. The profile plotted in Fig. 41a2was measured on 16 July 1993 during SOCEX at theCape Grim Baseline Air Pollution Station 140.68° S,155.69° E2. The stratocumulus cloud coverwas exten-sive, with occasional gaps that permitted themeasure-ment of cloud-free reference profiles. The referenceprofile 3Fig. 41b24was obtained by averaging 20 profilesin one such gap just 10 min before the cloud profilewas measured. The field of view of the lidar receiverwas 5 mrad and gave a minimum range for fulloverlap of 265 m.Analysis of this signal proceeded as with the simu-

lated signal described above. The reference cloud-free profile between 1200 and 2100 m was calibratedto a molecular profile measured by aircraft on thesame afternoon. With the system and aerosol trans-mittances thus determined, the cloud parameterswere derived and a profile of aerosol and cloudbackscatter 3Fig. 41c24was calculated as in the previousexample.The regression results and the derived cloud proper-

ties are summarized in Table 4. The molecular-scattering correction to hFC in this case is 4%. Thecloud base was detected at 802 m, and the cloud top at1051 m. The uncertainties in the cloud parameterswere calculated from the fit in region 2 alone and donot include any errors in the minimum range calibra-tion of the signal profile. Errors could have resultedfrom the use of incorrect values for the lidar ratio in

Table 3. Analysis of a Modeled Stratocumulus Cloud above anAerosol Layer a


Lower limit for fit 250 m 1100 mUpper limit for fit — 2250 mA 10.0 6 0.02 120.018 6 0.0192B 110.28 6 0.02 17.088 6 0.0412R 0.0 0.9799

Cloud Parameter Measured Modeled

T 0.8304 6 0.0024 0.8307htC 0.1859 6 0.0029 0.1855hFC 1sr2 20.03 6 0.26 20.0

aCalculation of tA began at the lower limit. Regression was notperformed in Region 1.

the mixed layer, or from changes in the aerosoltransmittance between the lidar and the calibrationrange 1265 m2. However, varying the lidar ratiobetween 30 and 50 sr alters TA

21rmin, rb2 by only 2%,and as the atmospheric conditions changed littlebetween the recording of the signal and the referenceprofiles, it is unlikely that FA would have changed bythis amount, and TA

210, rmin2 would have been fairlyconsistent also. As the mixed-layer aerosol opticalthickness was less than 0.03, the forward solution forthe backscatter profile would have diverged verylittle, so the subsequent calculation of C1 at the cloudbase should have little error.

Fig. 4. 1a2 Lidar signal from a stratocumulus cloud above theboundary layer. 1b2 Reference cloud-free signal. 1c2 Retrievedbackscatter profile 13 1026 m21 sr212.

Table 4. Analysis of a Measured Stratocumulus Cloud above an AerosolLayer a


Lower limit for fit 265 m 1250 mUpper limit for fit — 3000 mA 10.0 6 0.02 120.020 6 0.0152B 10.1779 6 0.02 10.1378 6 0.00392R 0.0 0.8277

Cloud ParameterT 0.880 6 0.012htC 0.128 6 0.014hFC 1sr2 13.6 6 1.36

aCalculation of tA began at the lower limit. Regression was notperformed in Region 1.


5. Discussion and Conclusions

The problem of the analysis of lidar signals fromoptically thin clouds has been addressed. Theseclouds have too low an optical thickness to forcesufficiently rapid convergence to the true profile ofextinction or backscatter by the use of the standardalgorithms.3 Accurate analysis of these data alsorequires that the boundary value of the solutionprofile be accurately known, that the scattering fromboth molecules and cloud particles be considered, andthat the lidar ratio for the cloud particles be known.1Methods for determining accurate boundary values

have been presented. Where the atmosphere out-side of the cloud can be considered to be clear, thelidar signal is fitted over a range of heights to areference profile calculated from a purely molecularatmosphere, thereby reducing the error in the calibra-tion constant and the signal offset. Where aerosollayers are present, the cloud profile is calibratedagainst a reference profile measured under cloud-freeconditions.An analytical expression for the effective lidar ratio

averaged through the cloud is derived for the case ofoptically thin clouds, where bothmolecular and cloud-particle scattering must be considered. This param-eter is calculated by integrating the two-componentlidar equation through the cloud. Both simulatedand measured lidar signals have shown that themolecular correction in Eq. 1292 can be significant.The cloud transmittance and effective optical thick-

ness have been successfully retrieved with the simu-lated signals frommodel clouds by fitting the signal toa reference atmosphere above and below the cloud.This method has been extended here to those cases inwhich an aerosol layer lies above the cloud or belowthe cloud and has been successfully demonstrated onsimulated signals to have retrieved the cloud param-eters of optical thickness and effective lidar ratio.Satisfactory results have then been achieved whenthese methods have been applied to actual cirruscloud profiles measured below a strong stratosphericaerosol layer and thin stratocumulus clouds above amoist marine mixed layer with significant aerosolscattering.Although the effective lidar ratio 1the product of the

multiple-scattering factor and the lidar ratio2 is as-sumed to be constant through the cloud, it is consid-ered that reliable profiles of cloud backscatter can stillbe produced for the optically thin clouds consideredhere. The backscatter solutions are constrained byusing a value of the effective lidar ratio that wascalculated with the optical thickness determined be-forehand.Reference has been made throughout this paper to

the effective optical thickness ht and effective lidarratio hFC in order to stress that measurements ofthese quantities with most lidars are usually scaled bya multiple-scattering factor h. This factor is a func-tion not only of cloud microphysics and distance fromthe lidar, but also of lidar geometry, which must beborne inmind when interpreting suchmeasurements.


The specification of the lidar receiver field of view,along with reported values of cloud optical thickness,lidar ratio, and extinction or backscatter profile,allows more useful comparisons with values obtainedwith different geometries to be made by referring tomodeled values of the multiple-scattering func-tion.32,33The accurate measurement of the cloud optical

thickness has been made possible with the limitedresolution of an 8-bit digitizer by recording the samesignal simultaneously on two digitizer channels withdifferent gains. A caution is given, however, aboutSIN that may be produced by the very strong signalsfrom low clouds. This would give spuriously lowvalues of optical thickness.Additional algorithms for the automatic determina-

tion of cloud base and cloud top have been suggestedand tested. The choice of which algorithm to use ismade after considering the signal-to-noise ratio in thelidar signal, the amount and homogeneity of anyaerosol scattering present, and the digitizer resolu-tion.

I am pleased to acknowledge the assistance of mycolleagues, Peter Manson and Graeme Patterson, inthe design, assembly, and testing of the new lidaroperated at Cape Grim during SOCEX. Martin Plattsuccessfully coordinated the participation in bothPROBE and SOCEX. The molecular-density profileused for the analysis of the example SOCEX lidarprofile was calculated from aircraft measurements byReinout Boers. The U.S. Department of Energy,through the Atmospheric Radiation Measurementprogram, supported the participation in PROBE inKavieng, Papua, New Guinea.

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analysis of lidar backscatter profiles in optically thin clouds

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