influence of atmospheric and system parameters on multiple scattering in spaceborne backscatter...

Influence of atmospheric and system parameterson multiple scattering in spaceborne backscatterlidar measurements

Peter Völger, Andrew Y. S. Cheng, and Nobuo Sugimoto

We have simulated backscatter signals of spaceborne lidar systems with the help of a Monte Carlo model.Calculations were performed for various combinations of system parameters. As typical examples of atmo-spheric observation targets, two kinds of cirrus cloud and two kinds of aerosol were considered. Both totalmultiple scattering and the significance of individual higher scattering orders are discussed. For all cases,an approximate multiple scattering factor F was calculated that can be used to correct the single-scatteringlidar equation to account also for multiple scattering. © 2005 Optical Society of America

OCIS codes: 280.1310, 280.3640.

1. Introduction

Over the past decades, remote sensing by means oflidar has become an indispensable tool for monitoringthe atmosphere. Applications comprise observationsof aerosol layers, trace gases, wind speed, tempera-ture, and many other atmospheric characteristics. Byfar the most common are ground-based and airbornesystems but, more recently, systems for satellite-based observations have also been developed. One ofthe first was the so-called Lidar in-Space TechnologyExperiment (LITE) mission (see Ref. 1 for a detaileddescription) that was performed by NASA. It provedthat spaceborne lidar measurements are also techni-cally feasible. Additionally, the LITE provided valu-able observation data for studies of variousatmospheric phenomena. Among others, Winker andTrepte2 and Platt et al.3 used the LITE data to inves-tigate high-altitude clouds. Powell et al.4 studied thetransport of Saharan dust. Grant et al.5 and Hoff andStrawbridge6 looked into the detection of biomassburning and anthropogenically produced aerosols. Atabout the same time as the LITE was developed other

space lidar designs were also investigated. However,only the French–Russian Atmosphere par Lidar SurSaliout (ALISSA) system7 was finally employed forobservations.

Success of the LITE stimulated the development ofnew space lidars. Launch of the American–FrenchCloud-Aerosol Lidar and Infrared Pathfinder Satel-lite Observations (CALIPSO) is already scheduled forthe near future.8 Other designs have been or arecurrently being investigated by various groups.9–14

One significant characteristic of all satellitebornelidars is that the volume of the field-of-view (FOV)cone is much larger than for ground-based or air-borne systems. Hence, scattering in directions otherthan strictly forward or backward can contributemuch more to the detected backscatter signal as alarger volume of the FOV cone means that a photonis more likely to undergo further scattering insidethe cone. Therefore, even under clear atmosphericconditions the multiple-scattering (MS) portion ofthe total backscatter signal cannot be neglected.15

The actual amount of MS in backscatter signalsdepends on both instrumental and atmospheric pa-rameters. For a given situation it is possible toestimate the MS intensity with the help of modelsimulations. Several studies have been performedby various groups that generally discussed MS for aspecific lidar design and mostly focused on cloudobservations. One of the first such investigationswas presented by Wiegner16 who simulated cloudmeasurements for an early version of the Europeanatmospheric lidar (ATLID) system. Also within thecontext of the LITE, MS studies concentrated on

P. Völger is with the Swedish Institute for Space Physics, Box812, 98128 Kiruna, Sweden. A. Y. S. Cheng ([email protected])is with the Department of Physics and Materials Science, 83 TatChee Avenue, Kowloon, Hong Kong. N. Sugimoto is with the Na-tional Institute for Environmental Studies, 16-2 Onogawa,Tsukuba 305-0053, Japan.

Received 12 January 2004; revised manuscript received 4 May2004; accepted 26 May 2004.

0003-6935/05/061051-16$15.00/0© 2005 Optical Society of America

20 February 2005 � Vol. 44, No. 6 � APPLIED OPTICS 1051

implications for cloud measurements.3,17–19 MS insignals from aerosol layers was explicitly examinedonly for the specific design of the proposed JapaneseExperimental Lidar In-Space Equipment (ELISE)system, now abandoned.20 For the most recentspace lidar project, CALIPSO, MS has mainly beeninvestigated within the context of depolarizationmeasurements in clouds.21–23 Other space lidar

studies concentrated on problems in which MS wasfound to be unimportant.24–26

Here we present results of simulations that focuson the influence of instrument parameters on thecontribution of MS to the total backscatter signalunder given atmospheric conditions. In Section 2 webriefly explain how MS can be implemented in thelidar equation. In Section 3 we provide a short outline

Fig. 1. Phase functions (upper panel) and cumulative phase functions (lower panel) for cloud and aerosol types that we considered in thisstudy: solid curves, cirrus consisting of hexagonal columns (cirrus A); long dashed curves cirrus consisting of hexagonal plates (cirrus B);short dashed curves, aerosol with large mode radius (Haze L); dotted curves, aerosol with small mode radius (continental polluted).

1052 APPLIED OPTICS � Vol. 44, No. 6 � 20 February 2005

of the main features of the simulation model. Section4 describes the settings of input parameters for bothlidar and atmosphere that we used in our simula-tions. Thereafter follows a discussion of results witha focus on the influence of instrument parameters onthe MS contribution.

2. Background

The return signal in backscatter lidar measurementsis usually described by the so-called lidar equation

P1(R) � P0

C

R2 �(R)exp��2 �0

R

�(r)dr�, (1)

where P0 and P1 are the emitted and the detectedintensities, respectively; C is a constant that dependson the system parameters; � is the extinction; � is thebackscatter coefficient; and R is the distance at whichthe backscattering occurred. Obviously, this equationaccounts only for single scattering and neglects theinfluence of multiply scattered photons on the totaldetected intensity. For ground-based and airbornelidar applications this simplification is justified, asthe diameter of the observed volume is very small,typically less than 20 m. Hence, in optically ratherthin atmospheres (that are the most common objects

for lidar observations) multiply scattered photonsmostly leave the volume without contributing to thetotal backscatter signal. Exceptions are cirrus layersin which, because of the dominating forward scatter-ing, the MS portion of the total backscatter signal isalso of significance and must be considered in thesignal interpretation.27,28

For satelliteborne lidar systems, the distance tothose layers in the troposphere and stratosphere thatare to be investigated is much larger. Therefore, thediameter of the observed volume is also increasedconsiderably. As a consequence, photons are lesslikely to leave the volume after being scattered in adirection other than forward or backward. Thus theprobability that a photon hits the receiver and con-tributes to the detected signal, even after more thanone scattering in the atmosphere, is increased. Equa-tion (1) is then not sufficient to describe the totalbackscatter signal that becomes

Pt(R) ��i�1

�

Pi(R)

� P1(R) � Pms(R), (2)

where i is the scattering order. Parameter R is half of

Fig. 2. Ratio of MS to single-scattering intensities for cirrus Aassuming different FOV angles for a 650-km orbit height. Extinc-tion coefficients were set to 0.3 and 1.0 km�1.

Fig. 3. Ratios of intensities of individual higher scattering ordersto single-scattering intensities for cirrus A assuming 0.21 mrad asthe FOV angle for a 650-km orbit height. Extinctions were 0.3 and1.0 km�1.


the distance that a photon travels during the timebetween emission from the laser and detection on thereceiver. It is commonly associated with the maxi-mum range that was reached by a photon. However,it should be noted that, in the strict sense, this holdsonly for P1. The maximum range for multiply scat-tered photons is usually smaller (an exception is pho-tons that are scattered solely through 0° and 180°).This means that the MS signal detected for a specificR can stem from any range between 0 and R. Thisso-called pulse stretching severely influences back-scatter signals from optically thick clouds,29 to whichside scattering and multiple backscattering contrib-ute significantly. Both are negligible for the (opticallythin) cases that we consider in this study; see Section4 for a description.

The MS term complicates the interpretation ofbackscatter lidar signals considerably. It is possibleto give exact mathematical descriptions for all theterms in Eq. (2); however, no analytical solution isknown for inversion of the complete equation. To getaround this problem an approximate factor can beintroduced in the single scattering lidar equationthat accounts for the influence of MS on the lidarsignal.30

Pt(R) � P0

C

R2 �(R)exp��2�1 � F(R)� �0

R

�(r)dr,

(3)

with F being the approximate MS factor with valuesbetween 0 and 1. F accounts for the effect that MSincreases the backscatter signal in a way similar tothat of a reduced optical depth. From Eqs. (1) and (3)it follows that

F(R) �1

2 �0

R

�(r)dr

lnPt(R)P1(R). (4)

Both Pt and P1 must be known to estimate F. How-ever, P1 cannot be determined directly from lidarmeasurements. The only way to estimate the MSfactor is to perform model simulations for the atmo-spheric conditions that are of interest.

In cases when MS cannot be neglected, inversionsof the lidar signal by use of Eq. (1) (see, e.g., Refs. 31and 32) will yield erroneous results, as the equationdoes not account for MS. However, if F is known,inversions can be performed starting from Eq. (3) andby use of similar algorithms as in the case of Eq. (1).The introduction of F in Eq. (3) and subsequently inthe inversion tends to increase the retrieved extinc-tion coefficients. A detailed discussion of conse-quences for the inversion results is beyond the scopeof this paper and will be investigated in a futurestudy.

Fig. 4. MS factor F for cirrus A assuming different FOV angles. Orbit heights are 260 and 650 km; the extinction coefficients were 0.3and 1.0 km�1.


3. Simulation Model

All the lidar signals were simulated with a MonteCarlo model that was originally developed at the Uni-versity of Munich (for a model description see Refs. 33and 34). The model allows parameters to be set forboth the lidar system and the atmosphere accordingto the situation to be examined. Lidar parametersinclude, among others, pointing direction of the laserrelative to the receiver, FOV angle, and the distancebetween laser and telescope. The atmosphere mustbe divided into homogeneous layers with the pointingdirection of the receiver being the normal to theplanes between two adjacent layers. For each layer,the extinction coefficient, the single-scattering al-bedo, and the Mueller matrix that describes the an-gular scattering distribution must be given. Outputparameters are the total intensity and the individualintensities for the first n scattering orders with n tobe chosen by the user.

For results presented in this study we typicallycalculated photon paths for 100 million photons. On aSun Ultra SPARC-II with a 296-MHz processor thiscorresponds to a CPU time of approximately 12–20 h.Monte Carlo models generally need a relatively longcalculation time to yield results with acceptably lowvariance. However, their advantage is that such mod-els are physically exact. Hence, they describe the at-

mospheric scattering process without usingapproximations. Therefore, Monte Carlo models aregenerally applicable and not restricted to certainmeasurement situations.

4. Input Parameters

A. Lidar

For all the simulations we assumed that the systemis strictly monostatic, i.e., the direction of the emittedlaser pulse and the optical axis of the telescope coin-cide. Furthermore, the emitted wavelength was in allcases 532 nm; the range resolution of the signal was100 m.

MS is significantly influenced by the size and theshape of the observed volume of the atmosphere. Bothsize and shape depend on the FOV angle and the orbitheight of the lidar. The FOV angle was set to 0.13,0.21, 0.4, 0.7, 1.1, and 2.0 mrad full angle. The orbitheight varied between 150 and 1000 km. Althoughsome of these numbers seem unrealistic, we decidedto perform simulations for all the combinations ofboth parameters as this study was aimed at investi-gating the sensitivity of MS to lidar system parame-ters.

B. Atmosphere

To limit the number of simulations we investigatedonly cases that can be considered as representativefor two main targets of lidar observations, namely,cirrus clouds and aerosols layers. Both have signif-icantly different particle size distributions that re-sult in different features of the phase functions andthe cumulative phase function [defined as Cp�� 0

� p�'�sin �' d��, where p�� is the phase functionat scattering angle �]. Phase functions and cumula-tive phase functions of the examples in this study areshown in Fig. 1. They are described in more detail inthe following.

1. CirrusIce crystals in cirrus clouds vary in size between a

few and some hundreds of micrometers. Hence, theyare much larger than the wavelength of the emittedlaser pulse. As a consequence, the forward scatteringmaximum is generally prominent. Details of the an-gular scattering distribution depend on both theshape of the crystals and the size distribution. Forthis study we assumed that the cirrus cloud consistedof randomly oriented hexagonal crystals with slightlyrough surfaces. The mean radius of the size distribu-tion was 76.1 �m, which, according to Heymsfieldand Platt,35 corresponds to a temperature of �35 °C.Two cases were considered, one consisting of hexag-onal columns (cirrus A) and one of hexagonal plates(cirrus B). The Mueller matrices are taken from theCOPE (Cirrus Optical Properties, Enhanced version)data library.36 Obviously, because of the hexagonalshape of the ice crystals the phase functions showhalo features at 23° and 46° scattering angles [Fig.1(a)]. For plates the forward scattering maximum ismore prominent and scattering in all other directions

Fig. 5. Ratio of MS to single-scattering intensities for cirrus-Bassuming different FOV angles for 650-km orbit height. Extinctioncoefficients were set to 0.3 and 1.0 km�1.


is relatively smaller than for columns. The plot of thecumulative phase function [Fig. 1(b)] shows that mostof the total scattered intensity is concentrated in asmall cone around the forward direction of scattering.For columns, approximately 60% is within a 2° scat-tering angle; for plates it is as much as 80%. For allour cirrus simulations we assumed that the layer issituated between 8 and 10 km altitude. Above andbelow the cirrus only molecular scattering was con-sidered according to the U.S. Standard Atmosphere.

2. AerosolsFor this study we concentrated on aerosols in the

planetary boundary layer (PBL). As this is the alti-tude region with the highest concentration of aero-sols, we expect the strongest MS contribution to thetotal backscatter signal from the PBL. We consideredtwo types of aerosol. As a representative case for anaerosol with predominantly small particles we chosea mixture for a continental polluted region as de-scribed by Hess et al.37 Relative humidity was set to70%. The mixture consists of soot, water-soluble andnonsoluble particles with relative number densitiesof 0.686, 0.314, and 1.2 10�5, respectively. Accord-ing to Hess et al.37 all three components of themixture can be described by log-normal distributions.The mode radii are 0.012, 0.029, and 0.47 �m, andthe sigma parameters are 2.00, 2.24, and 2.51. Forsoot the refractive index was set to 1.76 � i0.44.For water-soluble and nonsoluble particles the re-

fractive indices were 1.40 � i6.3 10�3 and 1.51� i8 10�3. Most of the particles originate from di-rect anthropogenic emissions or from the formation ofparticles by gas-to-particle conversion.

The second example, a mixture with larger parti-cles, was described by Deirmendjian38 as the haze Ltype. Its size distribution is given by a modifiedgamma distribution with parameters of a� 4.9757 106, � 2, b � 15.1186, and � � 0.5. Theresulting mode radius is 0.07 �m. The refractive in-dex is 1.34 � i1.8 10�9. Phase functions and cumu-lative phase functions for both aerosol types areincluded in Figs. 1(a) and 1(b).

For our simulations we assumed the PBL to behomogeneous between the Earth’s surface and the2 km altitude. Above 2 km only molecular scatteringaccording to standard atmospheric conditions wasconsidered. We are aware that this rough profile over-simplifies real atmospheres. However, we decided toutilize this profile because it allows us to interpretchanges of the MS portion and of F in the layer easilyand to identify the sources of the changes.

5. Results and Discussion

Simulations were performed for each combination ofphase function (i.e., aerosol or cloud type), FOV angle,and orbit height. For each case, the total backscattersignal and the individual contributions of the firstfive individual scattering orders were calculated.

Fig. 6. MS factor F for cirrus B assuming different FOV angles. Orbit heights are 260 and 650 km; the extinction coefficients were 0.3and 1.0 km�1.


Then MS factor F, as defined in Eq. (4), was esti-mated. The following discussion concentrates onsome examples that are representative of our results.

It must be stressed that F is one among severalparameters that influence the eventual inversion ofthe lidar signal. Since Eqs. (1) and (3) are similar itcan be expected that error sources in the inversion ofa conventional lidar equation (e.g., lidar ratio, bound-ary value, signal noise; see Refs. 39–41) will alsoaffect the inversion of an approximate MS lidar equa-tion. Additionally, horizontal inhomogeneity of theatmosphere can induce considerable errors when anumber of lidar profiles are averaged to reduce signalnoise. This problem is particularly relevant in obser-vations of broken clouds.42 An in-depth discussion ofthe consequences of both inaccurate values for F and

other error sources on the inversion is beyond thescope of this study.

A. Cirrus

As the phase function for ice crystals is stronglypeaked in the forward direction, scattering will, inmost events, alter only slightly the direction in whichthe scattered photon is moving. Hence, photons willmost likely remain inside the FOV cone after a scat-tering event. This means that the total number ofphotons inside the FOV cone changes only slightlywhereas the number of unscattered photons de-creases exponentially, following the Beer–Lambertlaw. Consequently, the ratio of MS to single scatter-ing lidar signal converges toward a nearly exponen-tial increase. This characteristic is clearly visible in

Fig. 7. Ratio of MS to single-scattering intensity for a boundary layer with continental polluted aerosol (70% relative humidity). Linecoding as in Fig. 5.


Fig. 2 for the example of cirrus A. Convergence isfaster for higher extinction, since then the photon ismore likely to undergo another scattering inside theobserved volume. The drastic increase of the ratio atthe near end of the cloud layer at 10 km altitude is aresult of the large change in extinction coefficient atthe cloud boundary. (As already mentioned in Section4, we assumed only molecular scattering outside thecloud.) Separating individual scattering orders al-lowed us to examine the influence of higher orders ofscattering (Fig. 3). For an optically thinner case, thisconfirms the generally accepted assumption that thesecond scattering order dominates MS. Still, at thefar end of the cirrus, the third and fourth scatteringorders account for 30% and almost 10%, respectively,of all MS. At 1.0 km�1 extinction, second and thirdscattering orders account for only half of the total MS.This does not support the Reichardt and Reichardtassumption22 that intensities of the first three scat-tering orders are sufficient to describe the backscat-ter signal from cirrus as long as the extinctioncoefficient is less than 1.0 km�1. Figure 3 indicatesthat higher scattering orders can already become sig-nificant at extinction coefficients well below 1.0 km�1.

From the formula for the MS factor [Eq. (4)] itfollows that the nearly exponential increase of the

ratio of MS to single scattering must result in analmost constant F. The examples that are shown inFig. 4 confirm this, as in all the cases values for F arearound 0.6 after an initial increase in the first fewhundred meters. Only for the smallest FOV anglethat we considered, 0.13 mrad, does F slightly de-crease with a penetration depth at low orbit, indicat-ing that, in this case, convergence of the ratio of MSto single scattering to an exponential function isclearly not achieved. Figure 4 also reveals that alarger FOV angle generally results in slightly highervalues for F since a larger portion of photons, scat-tered away from the optical axis of the lidar, willcontribute to the total signal. On the other hand,changing the extinction coefficient or the orbit heighthas no effect on F. Spikes in F are not a result ofatmospheric features but are due to the stochasticnature of the Monte Carlo method used in the simu-lation model.

Simulations with the same parameter settingswere also performed for the second type of cirrusclouds, cirrus B. The resulting ratios of MS to singlescattering are presented in Fig. 5. Although absolutevalues are higher, the general features of the curvesare the same as for cirrus A, i.e., after an initiallystrong increase the ratio converges to an exponentialfunction. This also holds for the significance of indi-vidual higher scattering orders that are, therefore,omitted here. Higher ratios (by a factor of 2 at the farend of the cloud) result necessarily in higher valuesfor F (see Fig. 6), which are around 0.8 for cirrus B.With the exception of the absolute values of F, resultsfor both types of cirrus are similar. Also for cirrus Bour simulations showed that, for measurements witha small FOV angle from low orbits, F decreasesslightly with penetration depth but remains fairlyconstant for all other cases.

Figures 4 and 6 demonstrate that there is only aslight variation of F with FOV angle and orbit height.This is a consequence of the strong forward scatteringon ice crystals. Opening the FOV further has only aminor effect if the forward scattering peak is com-pletely or almost completely inside the FOV cone,since scattering through larger angles is weak (seeFig. 1). Similarly, changing the orbit height of thelidar has only a marginal effect. Hence, for a givencirrus type, the same MS correction factor can beused for most lidar designs. Only for combinations oflow orbit heights and small FOV angles does F showsome dependence on these parameters, since thenparts of the forward scattering cone lie outside theFOV cone.

B. Aerosols

Characteristics of phase functions for aerosols werealready described in Section 4. One main feature isthe relatively weak forward scattering but relativelystrong scattering through angles between 5 and 60°.This means that a rather large portion of photons isscattered in directions away from the optical axis ofthe lidar system. As a consequence the probabilitythat a photon leaves the cone inside the FOV after a

Fig. 8. Ratios of intensities of individual higher scattering ordersto single-scattering intensities for the aerosol mixture continentalpolluted assuming 1.1 mrad as the FOV angle for a 650-km orbitheight. Extinction was 0.3 or 1.0 km�1.


scattering event is high. Therefore, the ratio of MS tosingle scattering intensities increases much moreslowly than for the cirrus cases discussed above.

Figure 7 presents results from simulations forthree orbit heights (260, 450, and 650 km) obtainedwith a continental polluted aerosol mixture. The ratioof the MS to the single-scattering contribution to thelidar signal is shown as a function of the altitude overground. The extinction coefficient was 0.3 km�1 (leftcolumn) and 1.0 km�1 (right column). All the casesshow a strong increase of the ratio in the uppermostpart of the aerosol layer, i.e., at the near end of thelayer. Thereafter, for most examples the increase ofthe ratio of MS to single scattering becomes smalleras the penetration depth increases. Only for the com-bination of large FOV and large extinction coefficientdoes the behavior differ, as shown by the plots for

450- and 650-km orbit height and an extinction coef-ficient of 1.0 km�1. Here, for large FOV angles, theratio continues to increase approximately linearly oreven more strongly throughout the layer. The differ-ent functional dependence on penetration depth isdirectly related to the influence of contributions ofhigher scattering orders (photons that were scatteredmore than twice) to the total MS signal. A strongincrease of the ratio of MS to single scattering alsoindicates that intensities from higher scattering or-ders contribute significantly to the total backscattersignal. If the increase of the ratio slows down withpenetration depth, second-order scattering is thedominant contribution to the MS signal and higher-order scattering will mostly result in the photon leav-ing the FOV cone without contributing to thebackscatter signal. This can be confirmed when indi-

Fig. 9. MS factor F for various combinations of orbit height, FOV angle, and extinction coefficient: Line coding as in Fig. 6.


Fig. 10. MS factor F as a function of the FOV angle for aerosol mixture continental polluted.

Fig. 11. MS factor F as a function of orbit height for the aerosol mixture continental polluted.


vidual scattering orders are separated, as shown inFig. 8, which shows examples for an orbit height of650 km and a FOV angle of 1.1 mrad. Even for sucha large angle the second scattering order dominatesMS for an extinction coefficient of 0.3 km�1. As aconsequence, simpler, approximate models are suffi-cient for simulations. However, for the case withlarger extinction (lower panel in Fig. 8) higher scat-tering orders also become significant.

As a next step, we calculated the resulting MSfactors F. The values in Fig. 9 are for the same casespresented in Fig. 7. The most significant influence onF is due to the FOV angle; as it increases F alsoincreases and increases with the extinction coeffi-cient as well. The effect is more significant at the farend of the layer whereas differences are hardly no-ticeable at the near end. An increase in orbit heightgenerally results in larger F. Figure 9 also shows that

the maxima in F, which are apparent especially for alarge FOV, move to lower altitudes (i.e., larger pen-etration depths) as the orbit height increases. Forsmall FOV angles (e.g., 0.13 mrad) such maxima be-come visible only for very high orbits (650 km ormore).

To illustrate the influence of lidar parameters on F,Figs. 10 and 11 show the results in a different per-spective. Figure 10 demonstrates how F changes withthe FOV angle whereas the penetration depth is keptconstant. Obviously, F increases with the FOV anglein all cases. This is a result of the fact that an increasein FOV angle leads to a larger diameter of the volumeinside the FOV, hence increasing the probability thatphotons, which were scattered away from the opticalaxis of the lidar, remain inside the volume. The in-crease with FOV angle is strongest at small FOVangles and slows down as the FOV angle becomes

Fig. 12. Ratio of MS to single-scattering intensity for a boundary layer with aerosol type Haze L. Line coding as in Fig. 5.


larger. This characteristic is most obvious at the nearend of the aerosol layer. The functional dependence ofF on the orbit height is explicitly presented in Fig. 11.Generally, F increases with orbit height. The increaseis almost linear when the FOV angle is small. Whenthe FOV angle is larger, F increases strongly at smallorbit heights, whereas for large orbit heights the in-crease becomes smaller. This characteristic is mostsignificant at the near end of the layer. Increases of Fbecause of changes in orbit height result from similarreasons to increases that are due to changes in FOVangle, namely, a larger diameter of the FOV cone andhence a larger probability that scattered photons re-main inside the FOV.

Similar simulations were performed for the aerosolHaze L that represents an aerosol mixture with manyrelatively large particles. Figure 12 presents the re-trieved ratios of MS to single scattering as a functionof altitude for different FOV angles, orbit heights,and extinction coefficients. Similar to the continentalpolluted mixture, the ratio increases as each of thesethree parameters increases. Another similarity isthat the increase decreases with growing penetrationdepth when the extinction coefficient is small. How-ever, the ratio is generally larger for Haze L than it isfor the continental polluted mixture. This is an effectof stronger forward scattering for Haze L that en-hances the probability that scattered photons remaininside the cone of the FOV and, after another scat-tering, can contribute to the MS portion of the back-scatter signal. Similar to the continental pollutedaerosol, second-order scattering dominates MS forlow extinction coefficients also for Haze L, whereascombinations of high orbit and a large FOV anglelead to significant contributions of higher scatteringorders (see Fig. 13).

A consequence of higher ratios of MS to single scat-tering for Haze L is that MS factor F, shown in Fig.14, has larger values than for the continental pol-luted mixture. However, general characteristics aresimilar for both aerosols. Similar to the continentalpolluted aerosol, F has a maximum at or close to thenear end of the layer that is followed by a decreasethroughout the layer. The influence of the extinctioncoefficient becomes stronger as the penetration depthincreases and is most obvious for low orbits.

The explicit dependence of F on the FOV angle ispresented in Fig. 15, where F is shown at specificpenetration depths for certain extinction coefficientsand orbit heights. Since the amount of MS intensityincreases as the FOV angle increases, the same be-havior is also found for F. Moreover, F is most sensi-tive to changes in the FOV angle when the latter issmall; the curves become gradually flatter as the FOVangle increases. The reason is that, although the for-ward scattering peak of the phase function is rathermoderate, it still results in a large portion of scat-tered photons being concentrated around the opticalaxis. The deviation from a linear functional depen-dence of F on the FOV angle increases with both orbitheight and extinction coefficient. Similar to the con-

tinental polluted aerosol the dependence of F on orbitheight changes with FOV angle (Fig. 16). For thesmallest angle (0.13 mrad) the increase of F is linear.As the FOV is opened up, changes become larger,especially at low orbit heights, leading to an in-creased deviation from a linear dependence as theFOV angle increases. This characteristic is mostclearly visible at the near end of the layer, however,it is to a lesser extent still visible also at the far end.

6. Summary

Using a Monte Carlo model we simulated spacebornelidar measurements of cirrus clouds and aerosol lay-ers including multiple scattering. Our aim was toestimate the relative amount of total MS and indi-vidual higher scattering orders to determine a MSfactor F and to investigate the dependence of F oninstrument parameters for different atmospheric sit-uations. F can be used to modify the single-scatteringlidar equation in a way that the lidar equation alsoaccounts for MS. This is of importance for cases inwhich MS cannot be neglected since then inversionswith the conventional single-scattering lidar equa-tion yield erroneous results. Other factors that canintroduce errors in inversions (averaging varyingaerosol and cloud layers, wrong lidar ratio, etc.) werenot part of this study.

Fig. 13. Ratios of intensities of individual higher scattering or-ders to single-scattering intensities for aerosol Haze L assuming0.4 mrad as the FOV angle for a 650-km orbit height. Extinctionwas 0.3 or 1.0 km�1.


Two examples of cirrus clouds were considered. Ingeneral, results for both cirrus types show the samecharacteristics but differ in magnitude. The ratio ofMS to single scattering increased almost exponen-tially through most of the cloud, which is a sign thatonly a few photons left the observed volume after ascattering event. This is a consequence of strong for-ward scattering on ice crystals. In the case of extinc-tion coefficients around 0.3 km�1 most of the MSsignal is due to second-order scattering, however, at1.0 km�1 extinction; higher scattering orders alreadydominate the MS signal and cannot be neglected. Forboth types of cirrus, most combinations of FOV angleand orbit height result in similar values for F. Onlyfor small FOV angles (0.13 mrad) in combinationwith a low orbit (260 km or lower) does the MS factor

show a noticeable (however still small) increase withboth orbit height and FOV angle. The extinction co-efficient of the cirrus layer has no effect on the MSfactor. Changes of F with penetration depth were inthe range of only a few percent.

For aerosols the forward scattering maximum isless pronounced. Therefore photons are more likely toleave the FOV cone after scattering, and the ratio ofMS to single scattering is lower. Another conse-quence of less forward scattering is that higher scat-tering orders contribute less to the MS signal. TheMS factor F tends to be higher for aerosols with stron-ger forward scattering. In most cases the MS factordecreases with penetration depth, however, for largeFOV angles in the nearest few hundred meters of thelayer, F increases before a decrease is found. It is

Fig. 14. MS factor F for aerosol Haze L for various combinations of orbit height, FOV angle, and extinction coefficient. Line coding as inFig. 6.


Fig. 15. MS factor F as a function of the FOV angle for aerosol Haze L.

Fig. 16. MS factor F as a function of orbit height for aerosol Haze L.


generally larger when the FOV angle or the orbitheight is increased. The increase with FOV angle isapproximately linear for lower orbits. At higher or-bits the increase is strongest for small FOV angles.Similarly, the increase with orbit height is about lin-ear when the FOV angle is small. For large FOVangles, F increases most strongly at lower orbits.

The Monte Carlo model was developed at theMathematical Institute of the Ludwig-Maximilian-University in Munich, Germany. Muller matrices forice crystals were provided by M. Hess of DeutschesFernerkundungsdatenzentrum Oberpfaffenhofen,Germany. We thank M. H. Chan for help with thegraphics. This study was supported by a grant fromthe Research Grants Council of the Hong Kong SpecialAdministrative Region (Project CityU 1219/01P) andby a grant from National Natural Science Foundationof China and Research Grants Council of Hong KongJoint Research Scheme (Project CityU110/00).

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