The further development of numerical models forweather forecasting and climatic studies will requirean increase in both the vertical resolution and theprecision of measurements of physical parameters~temperature, humidity, surface fluxes!, cloud anderosol distribution, and cloud phase in the tropo-phere. Clouds and aerosols have a significant in-uence on the Earth’s radiative balance and inducearious kinds of climatic feedback.1–3 As such, cir-
rus clouds, which have a significant influence on theEarth’s radiative balance and are difficult to ob-serve,4 require particular attention.5 Clouds, aero-ols, and water vapor also control a large part of the
P. Chazette [email protected]! is with the Laboratoire desSciences du Climat et de l’Environnement, Laboratoire Mixte,Commissariat a l’Energie Atomique, Centre National de la Recher-che Scientifique, F-91191 Gif-sur-Yvette Cedex, France. J. Pelonand G. Megie are with the Service d’Aeronomie du Centre Nationalde la Recherche Scientifique, Institut Pierre-Simon-Laplace, Uni-versite Pierre et Marie Curie, B 102, 4 Place Jussieu, 75232 ParisCedex 05, France.
Received 25 May 2000; revised manuscript received 20 March2001.
3428 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001
radiative fluxes at the Earth’s surface and are ofcritical importance, as are turbulent surface fluxes, inquantifying the energy balance over land and ocean.An accurate evaluation of energy and water-vaporfluxes from the surface into the free troposphere isalso required for analysis of climate evolution.6 Foruch an analysis, determination of the structure ofhe boundary layer appears to be a key element.7–9
Thus, improved knowledge of the spatial distributionand the temporal evolution of cloud cover and theplanetary boundary layer appears to be the mainrequirement in present-day climatic studies.
Making measurements from space is certainly oneof the most appropriate ways in which one can obtainclimatologies of the properties enumerated above.Measurements of cloud properties are at presentmade by passive remote-sensing systems operating inthe visible, infrared, and microwave spectral ranges.In addition to cloud cover, temperature and humidityprofiles can be determined by passive sounders formeteorological and climatic studies.10–13 From suchsystems, the top altitudes of dense clouds can be de-termined with an accuracy ranging from 100 to1000 m that is related to an uncertainty of the orderof 1–2 K in the measurement of the radiative tem-perature, to the cloud amount, and to the verticaltemperature profile.14,15 This uncertainty could in-
uo
saEiN
oc
t
crease for high-altitude semitransparent cloud layersand for multilayer clouds.16–18 Passive radiometryalone can thus lead to important errors in determi-nation of the geometric and optical structures ofsemitransparent upper-level clouds. Furthermore,passive systems do not permit information on theatmospheric boundary-layer structure to be derived.
The need for better precision and space–time res-olution, linked to the study of the radiative effect andthe climatic effect of clouds and to surface–atmosphere interaction, thus requires a new genera-tion of spaceborne active instruments. Amongthose, visible–near-infrared backscatter lidaremerges as one of the most powerful and sensitivetools for the detection of atmospheric-scattering lay-ers with increased vertical and horizontal resolu-tion.7,19,20 The first demonstration mission that
sed a spaceborne lidar, the Lidar In-Space Technol-gy Experiment ~LITE!, was performed onboard the
Space Shuttle in 1994.21 The great potential of aspaceborne lidar for determination of improved cloudclimatology on a global scale was clearly demon-strated. In addition, coupling active and passivesystems can lead to significant improvement ofclimatic and meteorological parameters by includingboth the vertical and the horizontal information givenby the two systems, as in the case of atmospheric-parameter retrieval.22 Further long-duration mis-ions, such as the Pathfinder Instruments for Cloudnd Aerosol Spaceborne Observations–Climatologietendue des Nuages et des Aerosols ~PC! mission,
mplemented cooperatively by NASA and the Centreational d’Etudes Spatiales ~France!23 and the Earth
Radiation Mission as studied by the European SpaceAgency,24 have both been designed to take advantagef combined passive and active remote sensing forlimate studies.
Our purpose in this paper is to emphasize, usingdirect-inverse numerical simulations, the potentialcontribution of a spaceborne backscatter lidar systemto the detection of semitransparent clouds and atmo-spheric boundary layers in the case of a low, weaksignal-to-noise ratio ~SNR!. Indeed, such conditionsof low SNR are frequently encountered for space-borne systems, and specific analysis procedures,which would be different from the ones used forground-based or airborne systems, have to be devel-oped. Section 2 introduces the proposed methodol-ogy. The direct model that is used to carry out thesimulations is described in Section 3. A brief de-scription of atmospheric parameters that control thedirect lidar signal simulation is given in Section 4.In this paper, the inverse analysis method is referredto as a classic threshold-detection scheme ~presentedin Section 5!, which is directly applicable to denseclouds. However, as this scheme fails to retrieve thegeometric properties of semitransparent scatteringlayers when the SNR is less than 4, a new inversemethod of doing so has been developed that is basedon the improved signal-filtering technique describedin Section 5 below. The accuracy of the determina-tion of the geometric properties of semitransparent
cloud layers and of the atmospheric boundary layeras retrieved in the simulation are analyzed in Section6, and we discuss the influence of various factors suchas lidar vertical resolution and cloud properties interms of lidar SNR to further assess the potentialityof the method.
2. Methodology
Figure 1 is a schematic representation of the geomet-ric parameters of the atmospheric-scattering layersthat could be determined from spaceborne lidar mea-surements and are considered in the present analy-sis. These parameters include the geometricthickness DzCi of semitransparent cirrus clouds ~Ci!and the mean top altitudes ZCu, ZCi, and ZABL of,respectively, dense cumulus-type clouds, semitrans-parent cirrus-type clouds, and the atmosphericboundary layer ~ABL!.
To increase the measurement precision in terms ofsensitivity and of spatial and temporal resolution, theanalysis method should permit the determination ofthe structure parameters for the lowest possibleSNR. This is so in particular when one is consider-ing spaceborne measurements, as one has to find atrade-off among the available laser power, instru-ment size and mass, and system performance.
The logical steps taken toward determination ofthe parameters studied in terms of algorithm analy-sis are diagrammed in Fig. 2. The direct signal iscalculated first; the lidar equation ~detailed in Ap-pendix A! is used to simulate the raw lidar signal ona shot-to-shot basis. It is based on the use of bothsystem parameters and atmospheric-scattering pa-rameters. Suitable parameters for a spacebornebackscatter lidar are listed in Table 1, as taken fromprevious studies based on realistic system designs.25
We give such parameters as a basis for the presentanalysis to get representative values of SNR. Theyare not critical for this analysis, as most of the resultsare analyzed in terms of variable SNR.
First a synthetic atmosphere is considered, on the
Fig. 1. Schematic representation of the geometrical properties ofthe various scattering layers ~clouds and ABL!. The thicknessDzCi of semitransparent cirrus-type cloud layers ~Ci!, the meanaltitudes ZCu, ZCi, and ZABL of dense cumulus-type clouds, semi-ransparent cirrus-type clouds, and the ABL are shown.
20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3429
ttstacbg
mms
tfi
3
basis of which several statistical realizations of theraw lidar signal, including signal, background light,and system noise, as detailed in Table 1, are calcu-lated. Critical parameters of the atmospheric modelare described in Section 3, and the lidar equation isfurther adapted to the atmospheric targets as pre-sented in Appendix A and detailed in Section 4. Thealtitude of the top of a scattering layer ~clouds orABLs! is considered to vary from shot to shot to rep-resent atmospheric variability along the satellitetrack. Several laser shots are simulated, and, foreach shot, the related noise according to a normaldistribution is taken into account for a given SNR.In the inverse problem, the detection procedure isapplied to each manifestation of noise to test theperformance of the inverse algorithm.
3. Atmospheric Model
An atmospheric model for simulating lidar returnsfrom clouds and from the boundary layer has beendeveloped, based on the main variables ~altitude, op-ical thickness, scattering coefficient, backscatter ra-io! used in assessing lidar measurements of variouscattering layers in the atmosphere. It is used inhis paper for the study of dense clouds ~i.e., cumulusnd stratocumulus! and semitransparent clouds ~i.e.,irrus! as well as for the ABL. Rayleigh scatteringy molecules has been considered to define the back-round values when no particles are present. A
Fig. 2. Diagram of the various simulation steps involved in re-trieval of the geometrical parameter of scattering layers by aMonte Carlo statistical analysis.
Table 1. Suitable Parameters, System Noise, and Background Light fora Spaceborne Backscatter Lidar
Emitted energy 50 mJ ~5 W!Telescope diameter 50 cmField of view 0.1 mradOptical efficiency 0.4Quantum efficiency ~photomultiplier! 0.14Detector’s noise equivalent power 4.8 3 10217 W Hz21y2
Amplifier’s noise equivalent power 5 nV Hz21y2
Background light ~W nm21 sr21!Cloudy day 0.85Clear day 0.50Night 1027
430 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001
id-latitude standard pressure and temperatureodel was used to derive the vertical molecular den-
ity profile.26 Backscatter and extinction cross sec-tions from Nicolet27 were used to derive molecularbackscattering and extinction coefficients.
A. Dense Clouds
On average, approximately 50% of the Earth is cov-ered by dense clouds, e.g., cumulus, stratocumulus,and altostratus.28 This percentage varies with thetype of surface ~continents or oceans!, the altitude,he latitude, and the season. The backscatter coef-cients of these clouds range from 1 to 20 km21 sr21
in the visible spectrum.29 Extinction coefficients arein the 20–200 km21 range.6,29,30 These clouds canbe considered perfect scatterers of the laser radiation,which penetrates only a few tens of meters into suchclouds at both visible and near-infrared wavelengths.In the present simulation, the backscatter coefficientof 10 km21 sr21 that is typical of denser cumulus andstratocumulus clouds is used as a reference for theanalysis of dense clouds.
An important parameter in the simulation is thealtitude of the cloud, as the altitude defines the rangedomain to be explored. For example, in the tropicaland equatorial regions of the Earth, where the verti-cal extent of the troposphere can reach 18 km, therange for cloud detection will be much larger than atpolar latitudes where the tropopause height is onaverage 7 km. The occurrence of a dense cloud layerbetween the surface and the tropopause can be ob-served in a convective situation. Outside and abovethe convective core, cloud dynamics leads to reducednumbers of droplets or crystals, and cirrus clouds areobserved at upper altitudes that can extend overlarge areas. We have assumed that the vertical ex-tent of dense clouds is limited to 14 km in the absenceof cirrus clouds above.
B. Cirrus Clouds
Cirrus clouds, which like denser clouds are also sig-nificant, occur on average over 50% of continents and35% of oceans.28 The backscatter coefficients ofcirrus-type clouds in the visible spectrum are in the1023–1021 km21 sr21 range.26,31,32
Although it is of importance to the radiation bud-get, the altitude of the cirrus cloud is not highly sen-sitive to SNR for direct simulation, as spacebornemeasurements are considered to be made at an alti-tude of 700 km. The solid angle of detection thatcontrols the signal amplitude as a function of range~see Appendix A! is almost independent of the cirruscloud’s altitude. The most sensitive parameter fordirect simulation is the optical thickness of cirrusclouds. A typical cloud optical thickness of 0.15 hasbeen considered representative of the optically thincirrus clouds that have a significant effect on radia-tion.33 The geometric thickness of the cloud wastaken to be 600 m in the simulation and correspondsto a backscatter coefficient value of 6 3 1023 km21.The geometric thickness considered here is smallerthan the average value derived from climatolo-
26,32
c
t
uclmf
vdsoott
gies and corresponds to the lower range of theobserved values. In this analysis we thus consideras a first approximation that both the extinction co-efficient and the normalized phase function wp of theirrus cloud are constant throughout the cloud, so b 5
wptyDzCi. The value of wp is taken equal to 0.025sr21 in the visible, as observed for cold cirrusclouds.34,35 Assuming a priori that on a statisticalbasis the average backscattering coefficient ~and theextinction coefficient! vary linearly with geometrichickness, we can link the optical thickness t to the
geometric thickness of the cloud by
t < kDzCi2. (1)
Such a relation enables one simply to calculate theattenuation of the laser radiation within the clouds.The coefficient k was determined by Kneizys et al.26
from statistical studies of optically dense cirrusclouds as equal to 0.14 and by Sassen and Cho,32 frommore-recent observations of optically thin cirrusclouds as equal to 0.42.
As optical thickness is small, in-cloud transmissionis high and the most critical parameters for the studyare geometric thickness and backscattering coeffi-cients. Sensitivity to the selected values is dis-cussed in Section 7 below. The altitude range inwhich the cirrus must be detected is also an impor-tant parameter of this study. We consider the mostdemanding case, which corresponds to tropical andequatorial latitudes, for which cirrus clouds can befound at 8–18 km.
C. Atmospheric Boundary Layer
The average altitude of the ABL cannot be deter-mined from passive soundings, mainly because itstemperature is too close to the temperature of theEarth’s surface. When one is using active lidar sys-tems, the clean ABL observed above the ocean can beconsidered in a first approximation to be a semitrans-parent cloud that has an optical thickness less than0.2 in the visible and near infrared.36 The approach
sed here is similar to the one presented above forirrus clouds. The phase function for boundary-ayer aerosols will depend on the composition and on
oisture effects of the layers. Typically the valuesor wp range from ;0.01 to 0.05 sr21.6,36 This corre-
sponds to backscatter coefficients at 532 nm thatrange from 1023 to 1022 km21 sr21. There is, how-ever, a difference in terms of optical properties be-tween the ABL and the clouds that are related to thesize of the scattering particles. Water droplets inclouds are large, with a median diameter greaterthan 3 mm, so the backscatter coefficient varies onlyslowly with wavelength l. By contrast, in the case ofthe ABL a large fraction of the particles has a typicalsubmicrometric size distribution, and the scatteringproperties vary as l2a, where 0 , a , 2. We haveconsidered here low values of the dry backscattercoefficient for two types of marine atmosphericboundary layer, as observed during previous cam-paigns.36 Profiles shown in Fig. 3 have similar val-
ues of their backscattering coefficients near thesurface ~b 5 2.5 3 1023 km21 sr21 at 532 nm!, but theertical extent and the relative humidity effects areifferent. In a moist and unstable ABL, the back-cattering coefficient that is due to particles at the topf the ABL ~2 km! reaches a value approximately anrder of magnitude higher than near the surface. Inhe case of the ABL at radiative equilibrium, its ver-ical extent is small, 0.5–1 km.
4. Adaptation to a Spaceborne Orbiting Lidar System
The parameters defined in Section 2 are to be deter-mined through an analysis of the backscattered lidarsignal in space and time. This analysis depends onthe characteristics of the lidar system and on thegeometry of the measurements as described in Ap-pendix A. The lidar signal calculated from the lidarequation needs to account for additional constraints,such as those linked to multiple scattering and atmo-spheric variability, as described below.
A. Multiple-Scattering Effects
Lower tropospheric clouds as considered in Section 3are optically dense media in which the laser-emittedphotons experience strong multiple scattering. Inthis case multiple scattering leads to importantpulse-stretching effects, which will increase the ob-served backscattered intensity, depending on the geo-metric characteristics of the lidar system.37 Owingto high extinction in clouds and to the signal inducedby multiple scattering, only the top altitude of denseclouds can be determined. No other cloud layer canbe detected below.
In cirrus clouds, the extinction coefficient is muchsmaller than for dense clouds, and laser radiation canpenetrate these clouds with low attenuation. How-ever, cirrus clouds are composed of ice crystals, whichstrongly diffract and scatter light in the forward di-rection. Following Platt,38 one can identify the ef-fect of multiple scattering on backscattered laserradiation as increased transmission by reducing theoptical thickness by a factor of h ~see Appendix A!.
Fig. 3. Schematic representation of the ABL in the atmosphericmodel. The backscatter coefficient is plotted as a function of thealtitude for two cases: stable and unstable ABLs.
20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3431
aTamttt
t
cc
cteso
3
The value of h is smaller than 0.5 and depends on thecrystal shapes, the optical thickness, and the lidarsystem’s geometry.39,40 Residual scattering afterpropagation through optically thin cirrus cloud leadsto a signal that stays small enough not to disturbdetection of the cloud base for the selected lidar sys-tem’s parameters.39
With respect to the ABL, our simulation will con-centrate mainly on the determination of the layer’stop altitude, and the multiple-scattering effect willthus not be considered.
B. Noise
The precision obtained in the determination of agiven geometric parameter ~top or base altitude! ofthe scattering layer ~i! is a function of the scatteringintensity, which defines the SNR. It is also a func-tion of the variability of the top or base height to bemeasured. The precision can thus be characterizedby its variance s~i!2, which is expressed as
s~i!2 5 sm~i!2 1 sR~i!2, (2)
where sm is the uncertainty in the value of the geo-metric parameter. It is a function of the bias and thestandard deviation associated with the detection pro-cedure and the measurement method. sR is thestandard deviation associated with the space–timevariations of the layers that are being studied ~vari-bilities of both structural and optical parameters!.o minimize the influence of sm it is necessary tossume that the vertical resolution of the measure-ent is less than the characteristic value of the ver-
ical variation of the cloud layer. For example, inhe case of stratocumulus clouds the atmospheric ver-ical fluctuations sR that are due to entrainment at
cloud-top altitude are of the order of a few hundredmeters.41 Comparable values are obtained for ABLop height,7,9 and optically thick cirrus clouds but
with larger values are characteristic of thin cirrusclouds.35 Indeed, variability in cirrus base heightan be much larger, as crystals may fall out in largeells.42 However, this previous process does not cor-
respond to the cirrus type that we consider here.Clouds or ABL’s are indeed inhomogeneous media,
as dynamics mixes air masses of different scatteringproperties at scales that are characteristic of turbu-lence and organized motions. This inhomogeneity isreflected in the boundary height fluctuations butalso in the scattering intensity in the vertical andhorizontal dimensions. Fractal models have beenproposed that include nonstationarity and intermit-tency,43 which can allow the atmospheric variabilityto be described. In this paper we account only forthe atmospheric fluctuations, using a constant valueof the relative variance of the scattering coefficients.
For a stable, dense cloud layer, for which both sRand sm are small, the error is controlled by the mea-surement resolution Dz. Aiming at an equivalenttemperature error of 0.5 K at cloud top to minimizeerrors in the radiation budget for a dense cloud andassuming a temperature lapse rate of 26.5 Kykm
432 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001
should yield a resolution slightly better than 100 m,which is comparable with the variability sR. In thesimulation we assumed a vertical resolution of thelidar system of 100 m. We considered that the alti-tude fluctuations indicate uniform distribution in the~zi 2 Dzy2, zi 1 zDy2! range about the scatteringlayer mean altitude zi.
C. Signal-to-Noise Ratio Extinction in Cirrus Clouds
We have considered using diurnal measurementshere to study the influence of backscatter coefficient bon the biases associated with restitution of the struc-tural parameters. Assuming in a first-order approx-imation that the atmospheric contribution over acirrus cloud can be considered negligible, the firstcharacteristic point of the cirrus cloud is detectedwith a SNR that is expressed as ~Appendix A!
SNR9 ; b expS22bDzmwp
D . (3)
The value of the SNR at the bottom of the cloud isthen given according to Eq. ~1! by
SNRb 5 SNRt expS 22b2
am2wp2 1
2bDzmwp
D . (4)
This value depends on the path length and thus onthe angle between the lidar beam’s direction and thevertical direction as given by m ~i.e., the cosine of thisangle!. The variation of SNRb as a function of back-scatter coefficient b is shown in Fig. 4 for three valuesof m and a value of 3 for the SNR at the top of theloud. As one could expect, the SNR at the bottom ofhe cloud decreases rapidly when the backscatter co-fficient increases. The SNR dependence on the ob-erving angle is also important, leading to a reductionf SNRb by 50% when the angle varies from 0° to 45°,
for a mean backscatter coefficient value of 1022 km21
sr21. This value will in turn lead to an underesti-mation of the geometric parameters of the cloud.
Fig. 4. SNR at the cloud bottom ~SNRb! as a function of backscat-ter coefficient b and the lidar system’s field of view. A SNR of 3was assumed for the cloud-top signal.
b~
cvS
pdaatsaot
5. Statistical Analysis
Our statistical analysis is based on the use of a MonteCarlo method to estimate the precision of the resto-ration of the geometric characteristics of the scatter-ing layers. The calculated values are compared withthe initial values after evaluation of the probabilitiesof a false alarm and of inability to detect existingscattering layers. The biases and standard devia-tions of the values of the structure parameters to bedetermined are calculated.
The final distribution of the retrieved altitude isobtained from 4 3 104 random realizations and thusensures a normal distribution about the mean value.An example of a simulated signal is given in Fig. 5and corresponds to standard cirrus cloud ~see Section2! and the system parameters defined in Table 1.The SNR value is 3 at cloud top and ;2.5 at cloudbase.
Two methods of analysis that use signal-filteringmethods have been considered in this study. Thedirect method uses a standard procedure based on apredetermined threshold value for detection of scat-tering layers. We use it as a reference for densecloud analysis. A more-elaborate method is basedon the preliminary filtering of the lidar signal bycalculation of its variance in predetermined windows.First we describe both methods and then we comparethem, using the statistical properties of the detectedlidar signals.
A. Direct Method Based on Threshold Determination
A standard procedure for determining the existenceof a peak signal S for any altitude point i within agiven noise level consists in fixing a threshold value Xthat is a function of the SNR:
S~i! . X. (5)
For lidar measurements, the threshold value muste a function of the probability-density functionsPDFs! of the noise and of the cloud-layer backscatter
coefficient as determined over the whole expected al-
Fig. 5. Simulated lidar signal as a function of altitude in thepresence of a 600-m thin cirrus cloud layer. The mean signal assimulated from the atmospheric model and the noise fluctuationsare superimposed.
titude range, as the cloud-top altitude is unknown.In the following analysis we assume that these dis-tributions are normal. They are thus centered onthe mean value of the signal and of the backgroundnoise. Their values of half-width at half-maximumare the associated standard deviations of the noise~sS and sN!. The SNR is thus defined as the ratio ofthe average cloud backscattered signal to the stan-dard deviation of the noise of the cloud signal’s PDF,sS. This standard deviation is larger than that cal-ulated for the noise only, as it includes backscatterariability. These two distributions, assuming aNR of 3, are given in Fig. 6.The detection threshold, which has been arbitrarily
laced on the horizontal scale in Fig. 6, enables us toefine the areas that correspond to the probability offalse alarm and to the probability of no detection ofn existing event. The probability of a false detec-ion is equal to the PDF area of the scattered cloudignal situated below detection threshold. The prob-bility of no detection is calculated from the PDF areaf the mean noise situated above the same detectionhreshold.
B. Dense Clouds
The detection of the cloud-top altitude of a densecloud was considered. As discussed in Appendix Aand when the vertical resolution of 100 m is com-pared with the cloud extinction coefficient ~see Table2!, the lidar signal for a dense cloud corresponds to asingle peak. Therefore the procedure for detectingcloud-top altitude consists in determining the exis-tence of a single point in the lidar’s backscatteredsignal, located above the detection threshold, withinM 1 1 points of the domain that defines the expectedcloud range. The probability pn of no detection istherefore expressed according to the complementarynormal distribution function Fc as
pn 5 1 2 Fc~ xs!, (6)
Fig. 6. Representation of the PDFs of the cloud backscatteredlidar signal ~Cloud PDF! and of the noise level ~Noise PDF!.
20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3433
w
wcs
t
tl
O
wpScm
Table 2. Atmospheric Parameters Chosen for the Simulations
3
where
Fc 51
Î2p *xs
`
expSx2
2 Ddx, (7)
where xs is the centered value normalized to the sig-nal’s standard deviation, which corresponds to thethreshold value X as defined in Fig. 6. The proba-bility of a false detection, for M independent mea-surement points that are different from the cloud-topaltitude, is given by
Pf 5 1 2 ~1 2 pf!M, (8)
where pf 5 Fc~xN! is the probability that a pointithin the noise ~signal! will appear above the
threshold value ~false alarm!. xN is the centeredvalue normalized to the deviation of noise standardthat corresponds to the threshold value X. As Pfvaries with M, it depends on the altitude range in
hich scattering layers are to be detected, as dis-ussed in Section 2. For dense clouds the corre-ponding value of M is 140.Threshold value X must then be defined such that
he probabilities Pf and pn are minimal. These prob-abilities are inversely linked, so if one decreases, theother one increases. Thus the probability of totalerror Pe in cloud detection has been assumed to be thesum of the two error probabilities considered corre-lated variables such that Pe 5 Pf 1 pn 2 Pfpn. Fig-ure 7 shows the variation of this probability of totalerror for various values of the SNR as a function ofthreshold value X normalized to standard deviationsN of the noise level. This shows that the total prob-ability of error is less than 5%, only if the SNR islarger than 4. For the dense cloud considered inSection 2, the system parameters of Table 1, and a
Fig. 7. Variation of the total probability of error for the directmethod ~see text! as a function of the normalized threshold for fourvalues of the SNR.
Scattering Layer z ~km! Dz ~km
Cu 2Ci 10 0.6Planetary boundary layer 1,2
434 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001
vertical resolution of 100 m, the SNR as defined inAppendix A is 20. The value of Pf is then muchsmaller than 1%, and the cloud-top altitude ~cumulusype! is then retrieved with a standard deviation ofess than 50 m.
C. Semitransparent Cloud and Atmospheric BoundaryLayer
For a semitransparent layer or for the ABL, the num-ber of points that contribute to the backscatteredsignal is greater than 1 and the restoration of thethickness of these layers requires detection of thecollection of N representative independent points.
ne defines therefore the probability Pn of no detec-tion for q adjacent points in a semitransparent layerby the relation
Pn 5 1~q 2 1!!~N 2 q!!
~N 2 1!! )i51
q
@1 2 pn~i!#, (9)
here pn~i! is the probability of no detection for aoint i inside the cloud, which is a function of theNR. We now suppose that all points i inside theloud are associated with a SNR that is close to 3 andust be detected ~q 5 N! and that the probability Pf
of a false alarm is fixed to a value of 0.05 to facilitatea comparative study of the conditions for no detec-tion. For SNRs of ,4, the probability of no detectionis thus significant ~larger than 0.1! for thin cirrusclouds with thicknesses that range from 0.3 to 1 km.Reducing that probability would necessitate increas-ing the probability of a false alarm to a value largerthan 0.1, which is not acceptable.
In the case of ABL detection, the unstable ABL hascharacteristics comparable with those of the cirruscloud, although the SNR decreases more rapidly inthe depth of the layer. The backscattering coeffi-cient for the shallow ABL is approximately two timessmaller than for the cirrus cloud. Even when oneconsiders a vertical extent of 1 km, the SNR valuestays below 2. Therefore the same approach as forcirrus clouds could be used, but the error in determi-nation of the top of the ABL would be larger.
These results can be improved by the use of better-adjusted filtering. Indeed, as the signal variance isexpected to be increased by atmospheric fluctuationsin the scattering layer, this new procedure uses thevariance of the detected signal rather than the aver-age value, as is explained in Subsection 5.D.
D. Quadratic Method Based on the Filtering Technique
The quadratic method of detection is based on pre-liminary filtering of the lidar signal by calculation ofits variance in given altitude windows. After this
Location ~km! bmax ~km21 sr21! at 532 nm
0–14 100–14 6 3 1023
0–4 2.6 3 1023 and 6.6 3 1023
!
a
w
wnoc
Wb
P
first step, we apply a threshold value to the set ofpoints that define the optimal variance to identify thescattering layers and their boundaries. This valueis obviously a function of the SNR before filtering.
The altitude-dependent profile of the relative vari-ance Sf of the signal is calculated on a shot-to-shotbasis with an adapting filtering window that includes~2n 1 1! points. The relative variance at a givenltitude point k is then determined as
Sf~k! 51
2n 1 1 (i5k2n
k1n FS~i! 2 S#
sNG2
, (10)
where S# and sN are, respectively, the mean value ofthe detected signal and the calculated standard de-viation of the noise in an altitude range where onlynoise is expected to be present. Assuming that thesignal follows a normal distribution, variance Sf fol-lows a x2 distribution Fx2, with 2n 1 1 degrees offreedom, defined as
Fx2~ x! 51
2n11y2GSn 112D
z x2n expFx2G , (11)
here G is the gamma function:
G~ x! 5 *0
`
ux21 exp~2u!du, (12)
ith x . 0. To define the threshold value that weeed to identify the signature of the scattering layersn the lidar signal ~mean altitude and thickness!, weompared the estimated lidar signal variance, Sf,
with the noise level. The probability pf of false de-tection calculated according to the x2 statistical law isassociated with each point outside the layer. Tolimit the probability of false detection, a maximumvalue pfmax is attached to pf, and the signature of ascattering layer is then obtained if the following con-dition is fulfilled:
Sf .Fx2
21~1 2 pfmax!
2n 1 1, (13)
where Fx221 is the inverse distribution of Fx2.
For each type of scattering layer that is consideredin this analysis, the precision of this procedure wastested with a statistical Monte Carlo method thatrelies on a large number of randomly distributedmanifestations of the lidar signal. Thus the value ofthe optimal SNR, which corresponds to given valuesof the bias, the standard deviation, and the minimalprobability of rejection for the geometric parametersof the scattering layers, could be quantitatively de-termined. We now apply this method to semitrans-parent clouds ~thin cirrus clouds! and to the ABL.
e consider the case of dense clouds as being solvedy the direct method ~Section 4!.
6. Results
A. Semitransparent Clouds ~Thin Cirrus!
We optimize the quadratic filter, assuming a uniformprobability of signal return throughout the verticalextent of the scattering layer. The initial size of thefiltering window corresponds to five measurementpoints ~n 5 2 and Dz 5 100 m! to yield a minimalconfiguration of the filter that does not introduce sig-nificant bias in the determination of the cloud thick-ness. To assess the potentiality of the technique weperformed several sensitivity studies to evaluate theinfluence on the threshold value of the vertical reso-lution of the lidar measurement, of the SNR, and ofthe cloud’s optical properties which in turn determinethe penetration of the laser radiation within the cloudand thus the SNR at the bottom of the cloud.
1. Influence of the Threshold ValueTo optimize the detection threshold, we consider athin cirrus cloud with a mean thickness of 600 m.The detection of the cloud top is assumed to be carriedout with a SNR of 3. Figure 8 illustrates the varia-tion of the probability Pf of false detection and theprobability Pn of no detection according to the value ofpfmax, which defines the threshold value. The figureshows that, when pfmax is 5 3 1024, Pf , 0.03 and
n , 0.02. These values are a good compromisewhen one considers the SNR. This value of 5 3 1024
for the threshold level was thus adopted in furtherevaluation of the method. A typical result of thismethod with the adopted threshold value is shown inFig. 9. The cirrus cloud layer is detected with a SNRof 6 after signal filtering, compared with a SNR of 3for the initial profile.
2. Influence of the Lidar’s Vertical ResolutionWhen the vertical resolution of the lidar measure-ments is decreased, fewer data points are obtainedwith which to determine the signature of the scatter-ing layer. Therefore the probability of no detectionwill increase. This situation is represented in Fig.
Fig. 8. Variation of the probabilities of false detection and of nodetection as a function of the threshold value for a SNR of 3.
20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3435
it
3Iwtctmasg
wSilslvctoafiwf
3
10~a!, which shows that the probability of no detec-tion as defined above rapidly increases above 0.2 forDz 5 200 m if we consider a SNR of 3 for a thin cirruscloud as defined in Subsection 6.A.1. In contrast,the probability of false detection, which is repre-sented in the same figure, varies less rapidly with Dz.As expected, it decreases with the vertical resolutionof the measurements, because the number of pointsoutside the altitude range of the useful signal is in-versely proportional to Dz.
To assess further the influence Dz of the verticalresolution, one can calculate the bias and the stan-dard deviation of the ~measured! geometric thicknessof the cloud, using the statistical method describedabove. As shown in Fig. 10~b!, the bias is negative
Fig. 9. Representation of the altitude-dependent lidar signal af-ter filtering and of the threshold level in the case of detection of a600-m thin cirrus cloud.
Fig. 10. ~a! Probabilities of false detection and of no detection asfunctions of lidar vertical resolution Dz for a 600-m thin cirruscloud. ~b! Bias and standard deviation of the simulated measure-ments.
436 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001
and increases rapidly in absolute value to 200 m forDz 5 200 m and then decreases to 120 m when Dzreaches 250 m. This inverse variation of the depen-dence of the bias as a function of Dz is linked to thehigh value of the Pn of no detection probability and tothe limited number of points that are representativeof the cloud’s signature on the lidar signal. Thestandard deviation increases less rapidly when Dzincreases. Increasing from 100 m when Dz 5 50 m,t stays almost constant near 200 m when Dz is largerhan 150 m.
. Influence of the Signal-to-Noise Ration our previous calculation a conservative value of 3as considered for the value of the SNR. However,
he determination of the geometric parameters of theloud varies greatly with the SNR value. To analyzehis dependence, we show in Fig. 11 the bias of geo-etric thickness of the cloud and the standard devi-
tion of its mean height as functions of the SNR foreveral values of the width of the filtering window asiven by the number of points being used ~2n 1 1!.
These results are obtained with the threshold valueof 5 3 1024 previously adopted and with the verticalresolution of 100 m maintained for a thin cirruscloud’s mean thickness of 600 m. As the dependenceof the standard deviation on the number of points n ofthe filtering window is slight, the result are shownonly for n 5 2. In contrast, the variation of the bias
ith the width of the filtering window and with theNR and thus with the optical thickness of the cloud
s much larger. The bias can exceed 200 m in abso-ute value for SNRs smaller than 2. This analysishows that, in the case of a SNR greater than 3, theower value of the bias is obtained for n 5 2. If theertical extent of the filtering window is not suffi-iently large, several layers can be detected throughhe inversion method, although the cloud consistsnly of a single layer. To remove this ambiguity for
weak SNR requires increasing the width of theltering window and thus the number of pointsithin the window. The positive bias that will be
urther induced in the retrieved geometric properties
Fig. 11. Bias and standard deviation of the mean altitude of a600-m thin cirrus cloud as a function of SNR for several values ofthe extent of the filtering windows ~n!.
Dcst~os
of the cloud by such an increase will, however, becompensated for by a decrease in the probability offalse detection.
If we assume that a thin cirrus cloud is detected onthe lidar signal with the SNR at its top altitude near3, the SNR at the bottom is close to 2.5 ~Fig. 4!.
etermining the geometric thickness of thicker cirruslouds with the same lidar system and with the sametatistical result requires the SNR at the bottom ofhe cloud to be close to 2.5. Considering Eqs. ~3! and4! and hypothesis ~1!, we determine that the previ-us criterion is fulfilled for cirrus clouds with back-catter coefficients from 5 3 1023 to 1022 km21 sr21,
corresponding to optical thicknesses of 0.1–0.4. As;70% of the cirrus clouds have backscatter coeffi-cients of less than 1022 km21 sr21,26 and thus a geo-metric thickness of less than 1000 m, a large fractionof cirrus clouds will be observed without significantbias by use of the filtering window with n 5 2.
B. Atmospheric Boundary Layer
As the ABL can be considered to a first approximationa semitransparent scattering layer, the methodologydeveloped in Section 5 for cirrus clouds is directlyapplicable. However, a fundamental difference canbe identified because of the stronger dependence onaltitude of the signature of the ABL on the lidarsignal. Indeed, the top of the ABL, defined as theentrainment zone, corresponds to a sharp transitionbetween the mixed layer below and the free tropo-sphere, generally characterized by a large gradient inthe aerosol backscattering coefficient. This gradientis further enhanced, depending on the relative hu-midity profile, as high values of the relative humidityinduce particle growth45 and thus enhance the back-scatter coefficient.
The use of the filtering method to detect the alti-tude of the top of a semitransparent layer in theatmosphere can be strongly biased, depending on theoptical thickness of the layer. For the ABL, thismethod will permit retrieval of a mean altitude forthe top of the ABL that will correspond to a givenaltitude level inside the entrainment zone. For anunstable ABL, the signature on the lidar signal ismarked, and the transition between the mixed layerand the entrainment zone is also rapid. In this case,a mean altitude defined in the entrainment zone willbe a good approximation of the top altitude of theABL. In the case of a stable ABL, the mixed layerbelow is more nearly uniform, and the transition withthe free troposphere is less marked. The top of theboundary layer will then be retrieved with less pre-cision, also in relation to the difficulty in defining thisaltitude level properly.
The biases and standard deviations of the retrievedaltitude of the top of the ABL are represented in Fig.12 as functions of the SNR at the top. The proba-bility of no detection and the probability of false de-tection are also shown in the same figure. Becausethe ABL can be as thick as 2 km in mid-latitudes, theextent of the filtering window includes seven mea-surement points ~n 5 3!. One can observe that the
probability Pn of no detection and the standard devi-ation in the retrieved altitude increase rapidly forSNRs smaller than 3. This is especially true whenone is considering an unstable boundary layer. Thiseffect is linked to the altitude averaging that resultsfrom the use of an extended filtering window. How-ever, when the SNR is larger than 3 at the top of theABL, the ABL’s altitude is retrieved with a standarddeviation of less than 150 m in the case of an unstableABL. For a stable ABL the bias is larger because ofthe poorer definition of the entrainment zone on thelidar signal itself.
The previous result highlights a limitation of activesensors in the present effort to determine lower at-mospheric variables accurately. In contrast to semi-transparent clouds, which can be located at anyaltitude in the troposphere and even in the lowerstratosphere ~polar stratospheric clouds!, the ABLhas a geometric thickness that does not exceed 4 km.This thickness varies according to the convective anddynamic activity in the mixed layer, and a typicalABL profile can be defined as a function of the surfacetemperature and thus as a function of latitude.Therefore additional information can be obtainedfrom the location of the measurements that can beused further as initial conditions for retrieval. Inaddition, cumulus clouds are frequently observedclose to the top of the ABL. They can be identifiedfrom the lidar signal itself or from additional imagingsystems on the same platform and thus again provideadditional information to help in the retrieval proce-dure.
These preliminary results obtained for the deter-mination of the altitude of an ABL are similar inmagnitude to those obtained for thin cirrus clouds,especially with an unstable ABL. They thus dem-onstrate the ability of spaceborne lidar to retrieve animportant variable that has not yet been retrieved ona global scale, which is of importance for bothweather forecasting and climate modeling.
7. Conclusions
Following the successful flight of the LITE instrumentonboard the Space Shuttle, long-term operation of aspaceborne lidar for the three-dimensional study ofaerosol and clouds in the depth of the atmosphere is
Fig. 12. Bias, standard deviation, and probability of no detectionas a function of the SNR for the measurement of stable and un-stable ABLs.
20 July 2001 y Vol. 40, No. 21 y APPLIED OPTICS 3437
ebtdltaop
t
Rnmp
fi
3
now becoming a reality. The PC mission now underdevelopment in a joint effort by NASA and the CentreNational d’Etudes Spatiale is to fly in 2004. Whenthey are operated from space, lidar signals are gener-ally in the range of a low signal-to-noise ratio becauseof the inverse square dependence of the lidar signal onthe distance between the emitter and the scatteringzone. Therefore, specific methods have to be devel-oped to ensure accurate determination of the geomet-ric and optical properties of scattering layers in theatmosphere. These properties include dense clouds,semitransparent clouds, and the structure of the at-mospheric boundary layer. In this paper the detec-tion of such scattering layers was studied forconditions of a fairly low SNR ratio ~;3!, as could beexpected from spaceborne lidar characteristics. Ifclassic algorithms, based on threshold detection, canbe used even with such SNRs for retrieval of the alti-tude of the tops of dense clouds ~e.g., cumulus clouds!,new methods have to be developed and their accuracytested to retrieve the altitude of semitransparent lay-ers such as thin cirrus clouds, polar stratosphericclouds, and the ABL when the SNR does not exceedvalues of 3 for horizontal resolutions of a few kilome-ters.
As a preliminary study, a new inversion method hasbeen developed and tested, based on the use of filteringof a lidar signal by use of a simple variational filteringwindow. Numerical simulations based on an atmo-spheric model of lidar scattering in the atmosphereshow that the geometric properties of thin cirrusclouds and the altitude of the top of the boundary layercan be retrieved with standard deviations of less than100–150 m for vertical resolution of the lidar system inthe 50–100-m range and a SNR of 3. Such values arewell within the range of the characteristics of currentlyplanned spaceborne lidar systems after averaging overa few tens of shots. It thus can be inferred that suchsystems are adapted to global retrieval of the geomet-ric properties of atmospheric scattering layers on amesoscale, providing information on variables of greatimportance for further simplification of weather fore-cast and climate models. Further steps will now betaken to validate and improve this method by use ofexperimental databases as provided by airborne sys-tems that can indeed be considered simulators of fu-ture spaceborne lidar systems.
Appendix A. Lidar Equation Analysis
The geometric properties of the atmospheric layersconsidered in the present analysis are determinedfrom the equation for backscattered lidar signal S~v,z! for a measurement taken at an altitude z and awavelength l ~Ref. 45!:
S~l, z! 5C~l!
~ zs 2 z!2b~l, p, z!
3 expF22m *
z
zs
a~l, z9! z dz9G 1 N~l, z!,
(A1)
438 APPLIED OPTICS y Vol. 40, No. 21 y 20 July 2001
where C is an altitude-independent constant that isproportional to the detector’s quantum efficiency, theoptical efficiency of the lidar system, and the energyemitted by the laser system, zs is the altitude of thesatellite, b is the backscatter coefficient, and a is thextinction coefficient. m 5 cos u, where u is the angleetween the direction of the laser beam and the ver-ical. N~z! is the additive noise that is due to aetection process ~signal and detected backgroundight! and system noise. The atmospheric modelhat has been developed for this study takes intoccount the optical characteristics of the atmosphere,f dense clouds ~i.e., cumulus type!, and of semitrans-arent layers ~i.e., cirrus clouds and the ABL!.The SNR is defined as
SNR 5S~l, z! 2 N~l, z!
sN~l, z!, (A2)
where sn is the total detection noise. Signal noisearises from detection and is due to shot noise of thedetected signal and background light, to the intrinsicnoise of the detection system, and to atmosphericfluctuations. In the case of shot noise, the noisevariance is proportional to the average signal. It isadded to the background shot noise and to the intrin-sic noise to define the variance of system noise. Un-like other noise sources, atmospheric fluctuationscannot be treated as additive noise. Because of theexpression of the backscattered signal, these fluctu-ations are to be accounted for as an additional rela-tive variance.
For diurnal measurements, the contribution ofbackground skylight is important compared with theuseful lidar signal. Equations ~A1! and ~A2! leadhus to the expression
SNR ; b~l, p, z!expF22m *
z
zs
a~l, z9!dz9G . (A3)
1. Dense Clouds
In the visible and the near infrared, these cloudscan be considered perfect scatterers for the laserradiation. Therefore the lidar equation is applied,with total attenuation considered, over a timeequivalent to the sampling time and correspondingto the measurement’s vertical resolution Dz. The
ayleigh scattering contribution inside the cloud isegligible, as the Rayleigh scattering is 2 orders ofagnitude lower than the Mie scattering. The
article extinction ap and the backscatter phasefunction wp ~ratio of backscatter to extinction coef-
cients, wp 5 0.055 sr21! are taken to be constantinside the cloud.39
The backscattered signal at cloud-top altitude zn,
wcDwc
5. J. H. Seinfeld and S. N. Pandus, eds., Atmospheric Chemistry
1
measured at altitude zs of the satellite, for a cloudlayer of thickness Dz can therefore be written as25
S# ~l, zn! 5C~l!
Dz~ zs 2 zn!2 m z
b~l, p, zn!
2hdap~l, zn!F1
2 expS2hd
mz ap~l, zn!εDzDGT~l, zn, zs!
1 N~l, z!, (A4)
here T is the atmospheric transmission betweenloud-top and satellite altitudes. The altitude rangez corresponds to the time constant or spectral band-idth of the detection. ε ~0 , ε , 1! represents the
loud fraction represented in Dz. hd is an empiricalfactor defined by Platt that accounts for multiplescattering effects.39
In the case of a cloud with a large backscattercoefficient bp ~bp . 5 km21 sr21! and for Dz . 100 m,the exponential term tends toward 0. Equation ~A4!can then be rewritten as25
S# ~n, zn! 5C~n!
Dz~ zs 2 zn!2 m
b~n, p, zn!
2hda~n, zn!T~n, zn, zs!.
(A5)
The cloud therefore behaves as a reflector with areflection coefficient equal to
bp~n, p, zn!y2hdap~n, zn! 5 mwp~n!y2hd. (A6)
2. Semitransparent Layers: Cirrus Clouds and BoundaryLayer
For semitransparent layers the scattering medium isoptically thin and the exponential term is small. Afirst-order development of this term in Eq. ~A4! al-lows the standard lidar equation to be retrieved, sothe vertical profile of the backscattering coefficient ofcirrus cloud particles bp is given by
bp~n, p, z! 5 bpCimax, (A7)
where z is in the altitude range between the bottomand the top of the cloud or the ABL layer. The back-scattering coefficient is assumed to be equal to mo-lecular backscattering outside this range.
This paper represents contribution 0544 of the Labo-ratoire des Sciences du Climat et de l’Environnement.
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