OCTOBER 1994 F E B V R E 1231Lidar Data Inversion for Cirrus Clouds: An Approach Based on a Statistical Analysis of In Situ Microphysical Measurements Guy FEBVRE LAMP, CNRS, Universit~ Blaise Pascal, Clermont-Ferrand, France (Manuscript received 17 February 1993, in final form 4 February 1994) ABSTRACT The problem of the lidar equation inversion lies in the fact that it requires a lidar calibration or else a referencevalue from the studied medium. This paper presents an approach to calibrate the lidar by calculating theconstant Ak (lidar constant A multiplied by the ratio of backscatter coefficient to extinction coefficient k). Thisapproach is based on statistical analysis of in situ measurements. This analysis demonstrates that the extinctioncoefficient has a typical probability distribution in cirrus clouds. The property of this distribution, as far as theattenuation of laser beam in the cloud, is used as a constraint to calculate the value of Ak. The validity of thismethod is discussed and results compared with two other inversion methods.1. Introduction Cirrus clouds have recently been recognized to havea great influence on weather and climate through theireffect on the radiative energy budget in the atmosphere(Liou 1986). To achieve a better understanding ofglobal climate change, a good description of the opticalproperties of these clouds is necessary. The lidar instrument is particularly adapted to study the opticalproperties of transparent or semitransparent clouds.Nevertheless, the inversion of the lidar equation is nottrivial. On the one hand, it requires either the calibration of the lidar or a reference value of the studiedatmosphere. On the other hand, we assume that thebackscatter coefficient and the extinction coefficientcan be linked according to a power law. From the Klett's algorithm (Klett 1981 ), which isbased on a logarithm solution of the lidar equation,Fergusson et al. (1983) and Mulders (1984) determinedthe reference value of the extinction coefficient usingmolecular scattering. Their approach is a very usefultechnique. Nevertheless a calibration is sometimespreferable when it is difficult to estimate a referencevalue of the atmosphere. For example, a calibrationbased on the strong extinction of cumulus cloud andthe invariant value of the backscatter to extinction ratioin water clouds has been developed by Elouragini(1991). For optically thin clouds like cirrus, a newconcept has been developed by Ruppersberg et al.(1990, 1992). This method, called "the shadow calibration,'' uses the laser beam attenuation throughmultilayered clouds. However, these methods are not Corresponding author address: Guy Febvre, LAMP, 24 Av. desLandais, Aubiere 63177, France.always appropriate for a systematic study of cirrusclouds. In this paper, we propose a new technique of lidarcalibration. This method is developed for cirrus studyfrom a vertically pointing 1.06-um-wavelength lidar.The sensitivity of this lidar does not allow the detectionof the molecular backscatter, so none of the methodsbased on molecular backscatter can be used. In our method we demonstrate with in situ measurements that the integrated probability density function (IPDF) of the extinction coefficient has an exponential form. With a cloud model, we show the effectof the laser attenuation.through the cloud on the IPDFshape. Application to one cloud case demonstrates howto implement the method. A comparison with twoother methods presents the advantages and the limitsof this new approach. The data used in this study were collected duringthe International Cirrus Experiment (ICE), which wascarded out to support the International Satellite CloudClimatology Project. This experiment took place in theNorth Sea during fall 1989 (Raschke et al. 1990).2. Presentation of the extinction coefficient distributiona. In situ measurements The in situ measurements come from a PMS 2D-Cparticle spectrometer probe (Knollenberg 1976)mounted on the Merlin aircraft operated by the Centred'Aviation Mrt~orologique of Mrt~o-France.. Themethod provides a size spectrum distributed over 30channels, each with a 25-~tm resolution (25-750-tzmsize range ), at 1-Hz frequency. The primary sizing parameter used for all particle images is the mean dic 1994 American Meteorological Society1232JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGYVOLUME 11 ameter. This diameter is ~he average of the image di mensions measured parallel and perpendicular, to the orientation of the photodiode array. The method of data procesing and accuracies have been discussed in detail in pr'~vious papers (Gayet et al. 1993a). The extinction coefficient can be calculated from the Size distribution by (Platt 1973) i=30 a = ~ ~rQeN~(a)a,2., (1) i=1With N~ the particle number of radius ai, i the clasiriumbe?, and Q; the extinction .efficiency equal to 2 inthe large-panicle approximation (radius greater than20 #m for considered lidar wavelength 1.06 t~m).b. Extinction coefficient distribution Figure ( gives an example ofmicrophysical data ofa cirrus .cloud sampled at 5900 m above sea level(-20.6-C) on 16 October 1989. This figure shows atime series of-the mean panicle size, as well as totalconcentration and extinction coefficient calculatedwith ( 1 ). 'The integral of' the probability density function(IPDF) of the extinction .coefficient is calculated according to these data. Figure 2 shows the IPDF in asemilog representation f0i' the. cloud portrayed on Fig.L The solid line represents the corresponding linearr~gression. The c0rr~lation' coefficient of 0.99 suggeststhat th.e linear function is a representative fit of theextinction coefficient IPDF in a semilog graph.30 50o ,: . . ' I4001 I I I12h20'30'' 12h24'30" Time (mn) FIG. 1. Time series of microphysical parameters measured in cirrusat 6-km altitude between 1220:30 and 1224:30 UTC 16 October1989. + + + 0.04 .... ~ ~ t0 Extinction coefficient (krn4)FIG. 2. Cumulative frequency of extinction coef~cieni obtained from the data presented in Fig. 1. To check ~he exponentiai behavior of the .'extinctioncoefficient IPDF, all cirrus sampled by the Merlin aircraft during ICE were analyzed. These data correspondto seven flights, which are divided into 59 cloud sequences. The.extinction coefficient !PDF is calculated for eachcloud sequence, and a linear regression is applied.: InTable 1, we i'eport, the corresponding ~correlation c0efficient and suitable microphysical parameters of thesampled 9irrus. Figure 3, representing the histogramof the correlation coefficient, shows that 75% of thestudied cases present a correlation coefficient greaterthan 0.97. As suggested by Lopez (1977), we first investigated the possibility that the probability den:fityfunction (PDF) would be described by a lognonnaldistribution, but the data revealed that an exponentialIPDF is much more accurate, Therefore, we assumethat the exponential relationship represents statisticallythe extinction coefficient IPDF in cirrus.c. DiscUssion1 ) ACCURACIES OF IN SiTU MEASUREMENTS It should be noted that the Merlin aircraft was :notequipped with a PMS 2D-P probe for the measurementofthe larger ic~ crystals. This may introduce only smallerror in particle sizing because no ice panicles around1 mm were sampled during the present study. Muchlarger uncertainties probably come from the lack ofreliable measurements of small ice particles (typicallyfrom 3 to 50 ~tm). Tentative estimates have been doneeither in extrapolating measured PMS 1D-C size spectra (Stephens 1980) or by using FSSP measurementsOCTOBER ! 994 F E B V R E 1233 TABLE 1. Characteristics of cloud sequences obtained in cirrus during ICE. They are reported as the date, the time of the sequences, thecloud length, the flight altitude (km) and the corresponding temperature (-C), the mean particle diameter (urn), and the correlation coefficientof the IPDF. Mean Time Cloud length Flight level Temperature diameterDate Number (UTC) (km) (km) (-C) (urn) Correlation coefficient15 September 1989 I 1043:30 9.2 5.7 -13.7 98 0.98 2 1050:30 19.9 6.7 -19.3 82 0.90 3 1053:30 105.3 7.6 -26.2 94 0.96 4 1109:45 29.8 7.9 -29.6 109 0.98 5 1115:15 67.0 7.9 -29.6 99 0.99 6 1150:00 7.5 7.9 -30.4 66 0.94 7 1153:15 17.0 7.9 -30.4 105 0.96 8 1155:30 18.9 7.9 -30.5 125 0.9820 September 1989 I 111 I:00 20.8 6.5 -16.3 92 0.98 2 1142:00 24.4 6.6 -17.2 101 0.99 3 1207:00 13.3 6.8 -18.7 171 0.98 4 1210:00 9.4 6.8 - 18.5 106 0.98 5 1212:30 13.2 6.8 - 18.2 72 0.98 6 1248:00 9.6 7.4 -22.8 80 0.94 7 1251:30 33.2 7.4 -22.9 52 0.98 8 1301:00 7.4 7.4 -22.9 45 0.97 9 1302:00 10.9 7.4 -22.9 54 0.99 9 October 1989 1 0804:45 9.7 6.7 -27.2 94 0.95 2 0806:00 23.4 6.7 -27.4 100 0.9510 October 1989 I 1328:00 154.8 6.7 -26.9 103 0.99 2 1356:00 30.7 6.7 -27.2 121 0.97 3 1401:00 26.9 6.8 -27.7 123 0.99 4 1405:00 8.4 7.1 -29.7 94 0.96 5 1406:15 10.2 7.3 -31.3 144 0.96 6 1410:15 5.8 7.3 -31.6 47 0.97 7 1411:15 9.7 7.3 -31.6 134 0.86 8 1421:00 15.5 7.3 -31.7 136 0.91 9 1423:30 54.2 7.1 -30.3 83 0.97 10 1525:30 5.9 6.7 -26.9 68 0.98 11 1527:30 19.8 6.7 -27.1 103 0.97 12 1530:00 123.6 6.7 -27.2 109 0.9912 October 1989 1 0949:15 23.4 6.7 -25.6 134 0.97 2 0955:30 31.4 6.7 -25.8 99 0.98 3 1000:00 17.5 6.7 -26.3 113 0.99 4 1002:15 15.8 6.7 -26.3 85 0.97 5 1004:45 36.8 6.7 -26.5 81 0.99 6 lOl 1:45 15.5 6.7 -26.6 69 0.97 7 1017:30 39.3 6.7 -26.5 123 0.99 8 1022:30 46.8 6.7 -26.3 116 0.95 9 1052:30 39.3 6.7 -26.4 138 0.99 10 1109:30 46.4 6.3 -23.3 120 0.99 11 1137:45 24.2 6.4 -23.4 116 0.99 12 1141:00 21.8 6.4 -23.4 126 0.96 13 1144:00 11.7 6.9 -27.1 106 0.9613 October 1989 1 1124:20 5.1 7.3 -32.1 34 0.98 2 1141:45 38.4 7.3 -30.3 106 0.99 3 1208:00 9.4 7.3 -30.8 59 0.97 4 1214:00 14.8 7.3 -30.4 120 0.99 5 1241:00 7.6 7.7 -33.8 33 0.9816 October 1989 I 0948:45 15 6.7 -2611 126 0.99 2 0951:15 35.9 6.8 -26.6 104 0.99 3 0959:30 64.3 6.8 -26.2 109 0.95 4 1008:00 15.2 6.8 -25.9 110 0.97 5 1010:00 47.2 6.8 -26.0 112 0.99 6 1038:30 30.7 7.4 -31.3 147 0.99 7 1153:00 34.8 7.7 -32.8 74 0.98 8 1159:30 26.1 7.2 -29.9 103 0.99 9 1220:30 34.8 5.9 -20.6 122 0.99 10 1230:15 52.1 6.8 -25.8 110 0.991234 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11252OCorrelation coefficient FIG. 3. Histogram of the correlation coefficient of the best linearfit of the extinction coefficient cumulative frequency. These data comefrom the 59 studied cases reported in Table 1.(Platt et al. 1989). Comparisons between FSSP (forward-scattering spectrometer probe) and 2D-C measurements made on the Merlin aircraft confirm thatthe FSSP is not a reliable probe for the measurementof ice particles with a complex shape [as already pointedout by Gardiner and Hallet (1985)]. Consequently,no measurement of small particles is considered in thesystematic data processing. Therefore, large particlescompared to the lidar wavelength are assumed, andthe calculation'of the extinction coefficient from (1)can assign a constant value to the extinction efficiencyQe. This equation is not applicable when the particleradius is lower than 20 ~tm. The absolute values of the extinction coefficient mustbe considered carefully because of the large uncertainties on the concentration measurement. These errorsare due to the difficult estimation of the probe samplingarea and detection response (Gayet and Soulage 1992).Nevertheless, the relative fluctuations of this parameterreflect the actual proportional changes in cloud microphysical characteristics. The linear characteristic ofIPDF in a semilog representation can be considered,but the slope and the intercept of the linear fit mustbe used carefully. The correlation of the IPDF linear fit must be usedcautiously when the mean diameter is small. Indeedthe uncertainties on the concentration measurementof small particles may be large and the hypothesis ofQe constant is not always reliable.2) CORRELATION COEFFICIENT In this study, the IPDF is plotted in a semilog graph.The linear behavior is isolated by a best linear fit. Because of this representation, the correlation coefficientnaturally presents a high value. The linearity is significant only for a correlation greater than 0.97. 3) CLOUD MORPHOLOGY The data sampling must have at least one hundredpoints for the statistical analysis to be correct. That isto say, the clouds like young cloud cells or contrails,with little spatial extent, are not very well represented.Consequently, the values of the correlation coefficientof the IPDF must be used carefully.d. Conclusion In conclusion, the extinction coefficient IPDF presents an exponential law in 75% of the studied casesof cirrus during the ICE experiment. The exponentialrelationship can be considered as a good representationof the extinction coefficient IPDF. However, cloudsthat are small or have particles with small mean diameter do not lend themselves to this method.3. Application to lidar equation inversiona. Equation and inversion From the scattering theory, the backscatter power Pat the altitude z is linked to the emitted laser energyby P(z) = Eo ~ ~r(Z)T2(Z), (2)where A is the lidar constant, fir the backscatter coefficient, Eo the emitted energy laser, and r the transmission of the medium. Here, P and z represent thelidar measurements, where z is the range between thebackscattered pulse and the receiver. Transmission r is defined as r2= exp[-2 ~( ae(z')dz']. (3) The true backscatter coefficient can be calculated from (2), if the lidar constant A and the attenuation r are known. To reduce the numbers of unknowns, the extinction coefficient is considered proportional to the backscatter coefficient ~r(Z) = kae(Z). (4) The lidar equation becomes Ak IP(Z) = E0 -~ lye(Z)T2(z). (5) In ice cloud, the phase function that governs the value of k varies with crystal habit, orientation, and particle sizes. Sassen ( 1978 ) gave values of k (4~rfi,./ae in his terminology) measured by an He-Ne laser in laboratory. His values ranged from 0.26 for ice crystal to 0.41 for aggregates. Platt et al. (1987, 1989) gave some values of k as function of the temperature. For clouds warmer than -40-C, the value of k is usually constant (0.013 sr-~ ), but a greater value (0.032 sr-~ )OCTOBER 1994 FEBVRE 1235has been determined at -62-C. These results show thatlarge uncertainties remain on the determination of kin cirrus clouds. Consequently, k is considered unknown but assumed to have a constant value withineach cloud investigated. To determine the extinction coefficient from (5),the values of A multiplied by k(Ak) must be known. Another way is the use of Klett's formalism (Klett1981 ). The differential lidar equation is dS 1 dl~r - 2~, (6) dz ~r dzwhere S(z) = ln[P(z)z2], (7)and the stable solution is exp[S(z) - S(zt)]ae(Z) = , (8) a,,(zt--'~ + 2 exp[S(z') - S(zt)ldz'where zt represents the altitude of the cloud top. Thissolution does not require any lidar calibration. However, a reference value of the extinction coefficient andthe value of k must be determined. A number of methods have been developed to determine these variables to invert the lidar equation.The simplest one uses the molecular backscatter fromboth sides of the cloud to calculate the cloud opticalthickness. This method associated with the Klett algorithm gives good results. Another way is the calculation of the optical depth with another instrument,for instance a radiometer. The next paragraphs describe how the exponentialdistribution of the extinction coefficient is used to determine the unknowns of the lidar equation.b. Specific properties of the extinction coefficient distribution We have seen in section 3a that the lidar measures ~the attenuated backscatter coefficient, and that the ex- ,~tinction coefficient (not attenuated) IPDF has an ex- 0~ponential form as seen from the in situ measurements. '~This section shows the extinction coefficient distribu- ~tion is sensitive to attenuation of the laser in the cloud.These characteristics can be shown with simulations ~of lidar measurements. A cloud is simulated assuming the extinction coefficient IPDF follows the same exponential relationshipat all levels in the cloud. Figure 4 represents the exponential relationship obtained from the in situ measurements shown on Fig. 1. The cloud is defined as agrid of 200 horizontal points on 50 levels. The distancebetween two horizontal points is equal to the distancebetween two lidar pulses (100 m). The distance between two vertical points is equal to 10 m, which issimilar to the lidar vertical resolution (6 m). Twohundred values of extinction coefficients are used fromthe data reported in Fig. 4. These values are set on allpoints of one level with a random process and thislevel is duplicated throughout the cloud. The modelleads to a cloud characterized by a horizontal heterogeneity, only because of the constant value of the extinction coefficient along a specific vertical. The attenuated backscatter coefficient is calculatedwith the lidar equation on all grid points. The corresponding IPDFs are obtained on three levels: base, 50,and 100 m above the cloud base (Fig. 5 ). This figureshows that the IPDF follows an exponential relationship at the cloud base, where the attenuation effect islow. On the contrary, no exponential form is observedat higher levels in the cloud. It should be noticed thatthe choice of the parameter value used in this model(intercept and slope parameters of the IPDF, horizontal, and vertical resolution) can affect the absolute values of the results portrayed in Fig. 5, but do not affectthe relative variations of the curves between the baseand the other levels. Consequently, our results suggest that the tPDF ofthe attenuated backscatter coefficient is sensitive to theattenuation effect. Indeed, upward-pointed lidar backscatter measurements follow an exponential distribution at the cloud base, but not at the top, although thetrue extinction coefficient distribution is exponentialthroughout the cloud. Our inversion method proposes to reproduce thisexponential distribution of the extinction coefficient indetermining the value of Ak.0.1 fI I0+ +Extinction coeffident (kin4) F~G. 4. Extinction coefficient cumulative frequency used in thecloud model. The equation is Fc = A exp(-Ba). The constants A= 1.16807 and B = -0.4864 come from the case in Fig. 2.1236 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 1110.01i ioi iI o ~OOm I[] +[] i+[] i +[] i + Attenuated backscatter coefficient (arbitrary units)FIG. 5. Cumulative frequency of the attenuated backscatter coefficient for three levels in the simulated cloud.C. Inversion method To solve the lidar equation, the constant Ak mustbe known. The determination of this value requires anadditional constraint as shown.in section 3a. This constraint is obtained considering the exponential behaviorof the IPDF. The correct value of the constant Ak givesthe correct attenuation correction, which in turn mustlead to the extinction coefficient IPDF observed within situ measurements. The following procedure is applied to determine thisconstant. (i) The extinction coefficient is calculated in allpoints of the cloud for a given value of the constantAk. (ii) At the chosen level, the IPDF is calculated anda linear regression on these data performed to obtainthe corresponding correlation coefficient. (iii) Steps' (i) and (ii) are repeated with differentvalues of Ak, in order to obtain the correlation coefficient as a function of the constant Ak. The maximumof this curve gives the value of Ak that best fits theconstraint. The chosen level must be in the upper part of thecloud so that the attenuation effect of the laser beamcan be measured. In the next section we present anapplication of this method. The value of Ak is determined at two levels in the cloud: the top and the middle.d. Application The lidar measurements came from the ALEX-F lidarmounted on. the Do228 aircraft operating at 1.06/~m.Each upward pulse had a vertical resolution equal toTABLE 2. Characteristics of the ALEX-F lidar mounted on l:he Do228 aircraft of the DLR.Laser type UTG 72 international laser systemsWavelength 1.06 #mEnergy per shot 400 mJBeam quality multimodeBeam divergence 2 mradShot frequency Single shot up to 1 HzTelescope type CassegrainPrimary mirror 35 cmFocus 400 cmField of view3 mradReceiver RCA 4517 photomultiplier and YAG 444 photodiodeDigitizer Biomotion 8100 (8 bit, 100 MHz)6 m. The main characteristics of the lidar are summarized in Table 2 (Moerl et al. 1981 ). Figure 6 representsthe attenuated backscatter coefficient cross section. Thiscloud is related to the cirrus case study carried out on13 October 1989 during the intensive field observationof ICE. The meteorological situation was characterizedby a cold front passing over the experiment area at midday. The examination of Fig. 6 shows that the cirruscloud top is around 8500 m (-37-C) and the cloudbase is around 6600 m (-20-C). The lidar measurements between 1123:00 and 1124:30 UTC will be considered later on. This cloud case has been chosen becauseit is the subject of another study concerning the microphysical and optical differences between natural cirrusand contrail (Gayet et al. 1993b). Before doing the inversion procedure, it is neces:~aryto check that the attenuated backscatter coefficient distribution is exponential at the cloud base, as we assumed in section 3b. In Fig. 7, the IPDF is representedin a semilog graph for three different levels inside the'cloud layer. The IPDF follows a quasi-exponential relationship at the cloud base, but not at higher levels.These results are in good agreement with those simu9.08.5~8.0~ 7.5~ 7.06.56.0 11h22 > 0.30 0.15 to 0.30 Arbitrary-- unit 0.05 to 0.15 I I tIlh23 11h24 11h25 11h26 Time UTC FIG. 6. Vertical cross section of the attenuated backscatter coefficient from the Do228/ALEX-F lidar between 1122:00 and1126:00 UTC 13 October 1989.OCTOBER 1994 F E B V R E 12370.10.01o.ool 0 0.1 0.2 0.3 0.4 0.5 Attenuated backscatter coefficient Arbitrary -traits 1 ~ [ I I level: 17.0km : 0.1 ...................... ! ......................................................................................... / 0.01 -~ ..................... i ........................ i ........................ i ........... '~' ......... i ......................0.001 0 0.1 0.2 0.3 0.4 0.5 Attenuated backscat~er coefficient Arbitrary unitsthose of the simulated IPDFs because of the verticalheterogeneity of natural cloud. To determine the value of Ak, we consider the hypothesis that the extinction coefficient IPDF followsan exponential law that is assumed valid for the wholecloud. Figure 8a shows constant Ak versus the correlation coefficient of the IPDF best linear fit, obtainedwith the data of the 8.0-km altitude. The maximumof this curve corresponds to a constant Ak equal to0.50. The same calculation is made at the cloud middle(7.4 km). In Fig. 8b, the result shows a value of Akequal to 0.42. Both of these values are used to computethe extinction coefficient on the whole cloud, as shownin Figs. 9a and 9b. i Cloud top~ 0.996 .................................................................................... 'i ........................................'d'~ 0.992 ...................................................................................'~ 0.988 ................................................................................. o.981 ...................................... 0.3 0.35 0.4 0.45 o.5 0.55 0.6 1 ,~ ! ! ! level!6.5km~ o.1 i! ................... ........................ ........................ i ...................... 0.01 - + ...................... ~ ........................... -~ .................................................................. + o.OOl ' I I I 0 0.1 0.2 0.3 0.4 0.5 Attenuated backscatter coefficient Arbitrary unitsFiG. 7. Cumulative frequency of the attenuated backscatter coefficient for three levels in the cirrus cloud presented in Fig. 6.lated in the model discussed in section 3b. The relativepositions of the experimental IPDF curves between thecloud base and the cloud top are quite different from Cloud middle ~0.996'd~ 0.992 ~0.988 r,.J 0.984 0.98 0.3 0.35 0,4 0.45 0.5 0.55 0.6 FIG. 8. Correlation coefficient of the best linear fit calculated fromthe extinction coefficient cumulative frequency as function of theconstant Ak: (a) top level, (b) middle level.1238 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11(a)8.0'7.57.0'6.5'6.0 ~m.,I.... ~ 0.05 to 0.5 ~I11h22 11h2311h24Time UTC!11h25 11h26radiometer measurement allows one to calculate theabsorption optical thickness in the infrared/5I~R fi'omthe cloud emissivity e ~~R = -ln(l - e). (9) For a given cloud particle distribution, the Mie theory links the optical thickness in the IR with the opticalthickness in the lidar wavelength: ax = -a~R, (10)where a is a proportional coefficient that is approximately equal to two in the range of cloud particles.This optical thickness calculated by the IR measurements is used to calculate the reference value in theKlett's method by the equation exp[S(zt)] [1 - exp(--2ab~R)] l~e(Zt)= z, , (11) exp(--2abx~R)2 ~ exp[S(z')]dz' o(b)9.0[8.58'-T~&7.5 -.[7.06.5 1 ":~~~ > 1.0 0.5 to 1.0 Km-1 0.05 to 0.56.04 11h22 11h23 11h24 11h25 11h26 Time UTCFIG. 9. As in Fig. 6 in terms of extinction coefficient calculated fromthe constants Ak determined in Fig. 8: (a) Ak = 0.50, (b) Ak = 0.42.where zt is the altitude of the cloud top and Zo thealtitude of the cloud base. The calculation of the extinction coefficient is made with the lidar measurements [P(z) and z] and from the PRT5 measuremerit (~).2) SHADOW CALIBRATION This method (Ruppersberg et al. 1990, 1992) isbased on the shading of a laser beam hitting upwardupon a cirrus cloud (labeled "ci" in equations hereafter) through a lower-lying cloud (labeled "llc"). Thelidar is calibrated by the shading. The method consists in calculating the true ci~tussignal in the shadowed region by determination ofAk.The hypothesis made is that the true cirrus signal hasthe same behavior in the shadowed region as in thenonshadowed region. The unknown (Ak) is determined by solving theequatione. Comparison Klett's algorithm associated with PRT5 measurements and the "shadow calibration" are used to compare the results of our calibration. These chosen methods do not use the aerosol backscatter. Furthermore,they have been developed for the same lidar, so thecalibration constants can be directly compared.1 ) KLETT'S METHOD As shown in section 3a, Klett's method is based onthe Bernouilli solution of the differential lidar equation(6). The necessary reference value is estimated froma PRT5 radiometer (Gratzki 1989). The simultaneous [ AZca ]xo [r2(x' AJnc) EiNT(X, Az-i) = 0, (12)where E is the sum of the relative lidar counts at theposition x and Az the geometrical depth of the considered cloud. The first term in the bracket correspondsto the cirrus shadowed by the contrail and the secondterm to the nonshadowed cirrus, where r is the shadowing cloud transmission. The authors give the calibration constant Ak with a20% accuracy. This method can be used with the Klett'sformalism or with a classical iterative method for thetransmission calculation. As the lidar measurement isthe only one required for the determination of the extinction coefficient of a multilayered cloud, the methodis particularly interesting.OCTOBER 1994 F E B V R E 1239f Results Klett's method is applied to the sequence describedin Fig. 6. Figure 10 shows the emissivity calculatedfrom PRT5 measurement as well as the deduced opticalthickness at 1.06 ~m versus time. Figures 11 show the optical thickness calculated withthe "PRT5 Klett's algorithm" versus the optical thickness obtained with our proposed method, first with Akequal to 0.50 and second with Ak equal to 0.42. Thelinear regression in the first case gives a slope equal to0.70 with a correlation coefficient equal to 0.96. in thesecond case slope equals 1.16 and correlation coefficientequals 0.93. The high values of these coefficients andthe slope close to I show that the two methods are ingood agreement. The shadow calibration has been made on the se quence displayed on Fig. 6 between 1123:00 and 1124:30 UTC. The shadowing region is located be tween 1123:00 and 1124:00 UTC and between 5.2 and 7.4 km. The constant Ak found with this method is equal to 0.42. This value is equal to the one found by our method in the cloud middle (Ruppersberg et al. 1990). The author o_,'- the shadow calibration analysis in dicates that this value of Ak leads to inadmissibly large values of the ext,_'nction coefficient (ae > 400 km-~) in the upper parts of the cloud. This problem of instability is discussed in section 4. These two comparisons show that the better cali bration is Ak equal to 0.42, even if some points present aberrant values of extinction coefficient.4. Discussion Our method is based on the following four hypotheses. O 0 1 -- 0 ~ I [ lib21 11h26Time (ran) Fir. 10. Time series of the emissivity and corresponding visibleoptical thickness between 1121:00 and 1126:00 UTC 13 October1989.(a) 21.61.2 J o ~ ........................ '~ ........................ i-O ...................... ~ .......................0 0.4 0.8 1.2 1.6 2 Optical depth (Klett's method)(b) 2.5 .... I ....I ....! ....I .... o~ 2 .............................................. o',.o ............. ~J ..................................................,~ 1.5 ~:~ I ............... J ...............O......J .......................... [ .......................~ --'"'"'"'""~'"'"~'~::5~'~[~'"6'-''' 0.5 0 0.5 ~ .5 2 2.5 Optical dep~ (~ett's method) F~G. 11. Comparison of the optical thickness calculated with ourAk values and ~th ~ett's method. The da~ come from ~e ~quencepresented in Fig. 9: (a) Ak = 0.50, (b) Ak = 0.42. (i) The extinction coefficient IPDF follows an exponential relationship, shown with in situ measurements. Up to now, we have no physical interpretation.However, we may recall that only 75% of the studiedcases during the ICE present such a feature. (ii) The extinction coefficient IPDF is sensitive tothe laser attenuation in the cloud. The depth of thecloud must be sufficient for this phenomenon to beobservable. Practically, an optical depth greater than0.2 is necessary. (iii) The value Ak is assumed constant in the studiedcloud. Moreover the statistical analysis requires a sampling of at least one hundred lidar pulses. The minimum distance to calculate Ak in a cirrus (7 km withan aircraft speed equal to 70 m s-~ and one lidar pulse1240 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 11per second) is too large for a representative study ofthe variation of k. (iv) The multiple scattering factor 7. is neglected (n= 1 ) because the introduction of factor n in the lidarequation increases the number of unknowns. It appearsin the lidar equation as a power ( < 1 ) of the transmission. The consequence is that if multiple scattering isassumed equal tO 1, the measured extinction coefficientis greater than the actual extinction coefficient. Platt(1980) showed that for cirrus .clouds the parameter vvaries from about 0.42 to 1 .according to the depth ofcloud penetration, the cloud optical depth, and thecloud extinction coefficient. It also varies according tothe distance between cloud and lidar and the laser receiver aperture. If the hypothesis n = 1 is used, extinction coefficient values are overestimated. Hence, weobserve unstable behavior of the method when opticalthickness is high. To treat the problem of inadmissiblelarge values of the extinction coefficient (observed withoptical thickness larger than 1.5), it is necessary to usemultiple scattering value lower than 1. It is not a goodsolution to calculate Ak in cloud top even if it reducesvery high values. Therefore, in our approach, the bettervalue of Ak is that determined in the middle level. Asn is equal to 1, the inversion requires caution that largevalues of the optical depth can have large errors. To limit the errors due to the two previous points,we can estimate the dispersion of the k and n valuesby plotting the emissivity e deduced from the PRT5measurements versus the integrated received lidarpower 3'' (Platt et al. 1989). These two parameters canbe expressed by (Platt et al. 1989) q/ Ak =-~n 1 -exp[2~aln(1 -,)], (13)where a is the ratio of visible extinction coefficient toIR absorption coefficient and ~ is the emissivity. If the corresponding plot of ~,' against e is fitted bythis equation, the dispersion of the plot represented bythe error bars gives information on the cloud heterogeneity and the variation of the suitable parameters.Figure 12 presents this processing for the sequence ofthe studied cloud (Gayet et al. 1993b). The resultsshow that the dispersion is lower than 10%. We canreasonably deduce that the parameters k and ~ do notvary much. The verification of the cloud homogeneity is necessary to eliminate some possible zones where k presents anomalous values, for example, contrail (Gayetet al. 1993b), specular reflection from oriented crystals(Rockwitz 1989), and others. This paper presents an iterative method of lidarequation inversion by using the extinction coefficientIPDF. The principle can be used to invert the lidarequation with the Klett algorithm. The constraint (extinction coefficient IPDF is exponential) can be usedto determine the reference value. Likewise, another0.250.20.150.10.050 0.1 0.2 0.3 0.4 0.5 0.6 EmissivityFIG. 12. Integrated received lidar power vs emissivity for the data from the cirrus cloud presented in Fig. 6.statistical constraint [e.g., the lognormal probabilitydistribution (Lopez 1977 )] can be used if it is justifiedby in situ measurements and if it is sensitive to thelaser beam attenuation in cloud. As a conclusion, this paper presents a new approachto calibrate a lidar. The technique uses a typical statistical behavior of the extinction coefficient, which issensitive to the laser beam attenuation. This behavioris used as a constraint in order to solve the lidar equation. The proposed technique gives, results in goodagreement with two comparative methods. Nevertheless larger use requires the following: (i) a study of the introduction of the multiple-scattering factor in the lidar equation, to make possiblethe study of optically thick clouds; (ii) a study to reduce the horizontal size of the slimpling, in order to estimate the variation of the parameter k. The main advantage is that only lidar measurementsare used, and the originality is that the cloud is considered as a statistical sample. Acknowledgments. This work was supported bygrants from CNRS/PNEDC No. 88/50/N/5080,DRET/Universit6 Blaise Pascal No. 89 1183, :andEPOCH/EUCREX-0002-c (MB). I would like tothank P. Wendling and G. H. Ruppersbergs of DLR,Oberpfaffenhofen, Germany. Thanks are also due toJ. F. Gayet for his helpful comments.REFERENCESElouragini, S., 1991: Etude des propri~t6s optiques et g6om6triques des cirrus par t61bdbtection optique active (lidar) et passive (radiombtrie). Ph.D. thesis, Universit~ Paris VI, 279 pp.Fergusson, J. A., and D. H. Stephens, 1983: Algorithm for inverting lidar returns. Appl. Opt., 22, 3673-3675.OCTOBER 1994 F E B V R E 1241Gardiner, B. A., and J. Hallet, 1985: Degradation of in-cloud droplet measurements in the presence of ice particle. J. Atrnos. Oceanic Technol., 2, 171-180.Gayet, J.-F., and R. G. Soulage, 1992: Observation of high ice particle concentrations in convective cells and cloud glaciation evolution. Quart. J. Roy. Meteor. $oc., 118, 177-190. , P. A. Brown, and F. Albers, 1993a: A comparison of in-cloud measurements obtained with six PMS 2D-C probes. J. Atmos. Oceanic Technol., 10, 180-194. , G. Febvre, G. Brogniez, and P. Wendling, 1993b: Microphysical and optical properties of cirrus and contrails; cloud field study on 13 October 1989. Proc. Sixth ICE/EUCREX Workshop, Stockholm, 30-39.Gratzki, A., 1989: Bestimmung der vertikalen Verteilung des Eis ehaltes in Cirruswolken aus kombinierten Lidar und Infrarot Radiometer Messungen. Ph.D. thesis, University of K61n, 100 pP.Klett, J. D., 1981: Stable analytical inversion solution for processing lidar returns. Appl. Opt., 22, 211-220.Knollenber, R. G., 1976: Three new instruments for cloud physics measurements: The 2D spectrometer, the forward scattering spectrometer probe and the active scattering aerosol spectrom eter. Preprints, Seventh Int. Cloud Physics Con f, Boulder, CO, Amer. Meteor. Soc., 554-561.Liou, K. N., 1986: Influence of cirrus cloud on weather and climate processes: A global perspective. Mon. Wea. Rev., I14, 1167 1199.Lopez, R. E., 1977: Some properties of convective plume and small fair-weather cumulus fields as measured by acoustic and lidar sounders. J. Appl. Meteor., 16, 861-865.Moerl, P., M. E. Reinhardt, W. Renger, X. Schellhase, 1981: The use of the airborne lidar system "ALEX-F" for aerosol tracing in the lower troposphere. Beitr. Phys. Atmos., 45, 403-410.Mulders, J. M., 1984: Algorithm for inverting lidar returns: Comment.Appl. Opt., 23, 2855-2856.Platt, C. M. R., 1973: Lidar and radiometric observations of cirrus clouds. J. Atmos. Sci., 30, 1191-1204.---, 1980: Remote sounding of high clouds. III: Monte Carlo cal culations of the multiple-scattered lidar return. J. Atmos. $ci., 38, 156-167. , and A. C. Dilley, 1981: Remote sounding of high clouds. IV: Observed temperature variations in cirrus optical properties. J. Atmos. Sci., 38, 1069-1082.---, J. D. Spinhirne, and W. D. Hart, 1989: Optical and micro physical properties of a cold cirrus cloud: Evidence for regions of small particles. J. Geophys. Res., 94, 11 151-11 164.--, J. C. Scott, and A. C. Dilley, 1987: Remote sounding of high clouds. Part VI: Optical properties of midlatitude and tropical cirrus. J. Atmos. Sci., 44, 729-747.Raschke, E., J. Schmetz, J. Heintzenberg, R. Kandel, and R. Saunders, 1990: The International Cirrus Experiment (ICE). A joint Eu ropean effort. ESA J., ~4, 193-199.Rockwitz, K. D., 1989: Scattering properties of horizontally oriented ice crystal columns in cirrus clouds. Part 1. Appl. Opt., 28, 4103 4110.Ruppersberg, G. H., and W. Renger, 1990: Shadow technique for improved inversion of lidar data to cirrus optical depth. Proc. Third ICE/EUCREX Workshop, Villeneuve d'Ascq, 57-59. , and , 1992: Shadow technique for improved inversion of lidar data to cirrus optical depth. European Space Agency Report, ESA-TT, 1263 pp.Sassen, K., 1978: Backscattering cross sections for hydrometeors measurements at 6328 angstrom. Appl. Opt., 117, 804-806.Stephens, G. L., 1980: Radiative properties of cirrus clouds in the infrared region. J. Atmos. $ci., 37, 435-446.