3 Lidar and Multiple Scatteringhome.ustc.edu.cn/~522hyl/%b2%ce%bf%bc%ce%c4%cf%d7/... · lidar,...

3

Lidar and Multiple Scattering

Luc R. Bissonnette

Defence Research & Development Canada – Valcartier, 2459 Pie-XI BlvdNorth, Val Bélair (Québec), Canada G3J 1X5([email protected])

The various types of lidar applications described in this book differ by thegoals pursued and the transceiver configurations designed to meet thesegoals. However, all lidars rely on the same basic physical process: scat-tering by discrete scatterers. The common framework for signal analysisand parameter retrieval is almost invariably the single scattering approx-imation. Under this approximation, the returned lidar signal originatesfrom a single scattering event at an angle determined by the chosentransmitter–receiver geometry but it is attenuated by all preceding andfollowing events. Actually, the extinction caused by multiple scatteringis fully taken into account in the single scattering lidar equation. Thisextinction process is, of course, not what is meant here by multiple scat-tering. Rather, we mean the part of the multiply scattered radiation thatactually ends up being collected by the receiver but that is still consideredlost by the single scattering models. In this chapter, we will review anddiscuss the effects multiple scattering has on the most common applica-tions, describe some calculation methods, and look at proposed ways ofcorrecting for or exploiting these additional contributions.

Multiple scattering is a widespread phenomenon in nature. Forinstance, it was extensively studied in the field of nuclear physics inwhich neutron transport plays a major role; see, for example, Davison[1], Case and Zweifel [2], Bell and Glasstone [3], and Williams [4].The governing equations are basically the same for neutron and opticalscattering. However, in lidar applications, the scattering medium is oftencharacterized by a scattering pattern highly peaked in the forward direc-tion, and the transmitters and receivers have generally narrow angular

44 Luc R. Bissonnette

apertures. For these reasons, the formulation of multiple scattering inlidar, while borrowing a great deal from neutron transport theory, hasdeveloped into a specialized field that is worth a separate chapter in abook like this one.

We briefly review in a first section some important lidar singlescattering methods that are useful to introduce and understand multi-ple scattering. Next, we look at how multiple scattering was historicallyfound to affect lidar returns and we present measurement results thatillustrate these findings.

We describe in a third section various calculation methods that havebeen proposed in recent years. Finally, we discuss in the fourth sectionthe corrections that can be applied to the single scattering solutions andwe report in Section 3.5 on progress made in the use of multiple scatteringas an additional source of retrievable information on particle properties.

Multiple scattering in lidar is a relatively recent research field stillin very active evolution. The work described here represents a view ofthe current state of the art. The scope of this short chapter is necessarilylimited and some particular developments may have been missed orleft out.

3.1 Pertinence of Multiple Scattering

To better recognize the pertinence of multiple scattering in lidar, it ishelpful to begin with a brief review of work accomplished in the realmof single scattering. Single scattering solutions are discussed at lengthelsewhere in this book. We recall here only the basic principles forsituations of moderate to high particle densities where multiple scatter-ing effects are likely to be more significant.

The single scattering backscatter lidar equation is presented inChapter 1. It is simply rewritten here in the form

Pss(z) = K(z)

z2β(z) exp[−2γ (z)], (3.1)

where Pss(z) is the single scattering power on the detector from range z,K(z) is the instrument function assumed given, β(z) is the backscatteringcoefficient, γ (z) = ∫ z

0 α(z′)dz is the optical depth, andα(z) is the extinc-tion coefficient. Even for cases in which the molecular contributions toα and β are negligible, Eq. (3.1) shows that the retrieval of either α or βconstitutes an underspecified problem: one equation for two unknowns.

3 Lidar and Multiple Scattering 45

Rearranging the terms in Eq. (3.1), defining

S(z) = Pss(z)z2/K(z), (3.2)

and differentiating with respect to z, we obtain the following ordinarynonlinear differential equation:

1

β(z)

dβ(z)

dz− 2α(z) = 1

S(z)

dS(z)

dz. (3.3)

For homogeneous media, we have the additional equation dβ/dz = 0which leads, upon substitution in Eq. (3.3), to the simple slope method

α(z) = − 1

2S(z)

dS(z)

dz, (3.4)

where S(z) depends on the measured power Pss(z) and the knownfunction K(z). The intricacies related to determining K(z) are discussedin other chapters. Note, however, that in the frequent occurrences ofconstant K(z) the solution is independent of K , which means that nocalibration is required.

For inhomogeneous media, it is customary to assume a relationbetween extinction and backscatter of the form

β(z) = Cαu(z), (3.5)

where both C and u are constants. While there is little theoretical groundto justify Eq. (3.5) in general, it does not constitute an exceedinglyrestrictive condition in many practical atmospheric situations [5]. UsingEq. (3.5) in (3.3), we have

u

α(z)

dα(z)

dz− 2α(z) = 1

S(z)

dS(z)

dz. (3.6)

Note that the constant C is factored out of Eq. (3.6). Equation (3.6) is anonlinear ordinary differential equation of elementary structure knownas the Bernoulli or homogeneous Riccati equation. The solution is

α(z) = S(z)1/u

S(zf )1/u/α(zf ) + 2

u

∫ zf

z

S(z′)1/udz′, (3.7)

where zf is the range at which the boundary value α(zf ) is specified.Note that the integral term in the denominator of Eq. (3.7) is positive


if zf > z but negative if zf < z. Note also that the solution is againindependent of K if K is independent of z. The major difficulty withthis solution is that it is unstable in media of moderate to high densityunless the boundary value is given at the far end of the measurable lidarreturn [6] where, however, it is less likely to be known. In addition, thesolution (3.7) rests on the validity of Eq. (3.5) which means that thesize distribution and composition of the scattering particles must changein a prescribed manner within the medium—the inhomogeneities beingsolely caused by fluctuations in number density. Further discussions onthis solution can be found in Klett [6], Fernald [5], and Bissonnette [7].

Despite the well-defined theoretical basis for solutions (3.4) and (3.7)and their variants, no general application method is possible because theuncertainties on the boundary value and the backscatter-to-extinctionrelation (3.5) always lead to particular situations. Two groups haveproposed expanded measurement techniques that provide additionallidar-derived independent data on α and β to resolve this problem.

A group at the University of Wisconsin [8–11, cf. also Chapter 5]has developed a high-spectral-resolution lidar transceiver that allowsdiscriminating between Mie and Rayleigh backscattering. The principleis that the Rayleigh backscatter is significantly Doppler broadened bythe large thermal velocities of the air molecules whereas the frequencycontent of the Mie backscatter is nearly unaffected by the slow par-ticle velocities. The two spectra are superposed with the narrow Miespectrum centered on the broadened Rayleigh spectrum. The separationtechnique, therefore, requires the use of a high rejection power notch fil-ter centered on the Mie spectrum or laser wavelength. From the knowncharacteristics of the filter, the transmitted and rejected fractions of theaerosol and molecular spectra can be calculated for each detection chan-nel, and the spectra eventually separated. Following the single scatteringapproximation, we have for the molecular backscatter

Pmss (z) = K(z)

z2Nm(z)

dσm(π, z)

dexp[−2γ (z)], (3.8)

where Nm(z) is the atmospheric molecular number density anddσm(π, z)/d is the differential Rayleigh or molecular scattering cross-section in the backward π direction. From the knowledge of theatmospheric temperature and pressure profiles, Nm(z) and dσm(π, z)/dcan be calculated and Eq. (3.8) only depends on α(z). Hence, combiningEqs. (3.1) and (3.8) allows the unambiguous determination of both α(z)

and β(z). The difficulty of the method is technical. In clouds of even


moderate densities, the aerosol backscattering coefficient is much largerthan Nmdσm(π, z)/d. Since the width of the spectrum from the cloudparticles is 4–5 times narrower than the molecular spectrum [9], it turnsout that the spectral density at the center can be orders of magnitudegreater than on the wings. Hence, to avoid contamination of the molec-ular signal by the particle backscatter, the rejection power of the filtermust be very high. Using an iodine cell, Piironen and Eloranta [11] haveachieved a rejection of ∼1:5000 for a bandwidth of 1.8 pm. A typicalbandwidth for the broadened Rayleigh spectrum is 8–10 pm.

Following the same line of thought of deriving additional independentmeasurements from their lidar, Ansmann et al. [12] proposed to measure,in addition to the elastic backscatter given by Eq. (3.1), the inelasticRaman backscatter off the nitrogen molecules of the atmosphere by useof a filter centered on the nitrogen Raman-shifted laser line. This givesin the single scattering approximation

PRss (z, λR) = K(z)

z2NN2(z)

dσRm (π, λR, z)

dexp[−γ (z, λ0) − γ (z, λR)],

(3.9)

where λ0 and λR are the laser and the N2-Raman-shifted wave-lengths, respectively, NN2(z) is the nitrogen molecular density,dσR

m (π, λR, z)/d is the differential N2-Raman cross section in thebackward direction, and γ (z, λ0) and γ (z, λR) are the optical depthsat the laser and Raman wavelengths, respectively. The nitrogen num-ber density and the molecular contributions to α(z, λ0) and α(z, λR) areobtainable from the atmospheric temperature and pressure profiles, andthe particle extinction coefficients at λ0 and λR are related by a simplepower law relation [12] that is well justified because λR and λ0 are closeto one another. That leaves only the aerosol contribution to α(z, λ0) asthe unknown in Eq. (3.9). Thus, by combining the elastic [Eq. (3.1)] andthe inelastic [Eq. (3.9)] returns, one can derive the profiles of α(z, λ0),α(z, λR) and β(z, λ0) unambiguously and with no further approxima-tions. Here, the wavelengths λR and λ0 are sufficiently distant for easyseparation, but high measurement precision is a requirement becausethe Raman signal is orders of magnitude less than the aerosol signal. Forcomparison, the N2 Raman cross section is smaller than the atmosphericRayleigh cross section by a factor of ∼1000.

Multiple scattering modifies the picture just described. Figure 3.1illustrates schematically the scattering events that contribute to the lidarreturn. The single scattering models take into account only the radiation


Fig. 3.1. Schematic diagram of multiple scattering in the lidar geometry. The thickgray line represents the outgoing laser pulse; the arrows, the scattering events; andthe two pairs of thin lines, the limits of two receiver fields of view, a narrow one fornear-single-scattering detection, and a wider one for multiple-scattering detection.

scattered once from the outgoing laser pulse back into the receiver, thebroken line arrow in Fig. 3.1. However, if the field of view is sufficientlywide and the mean free path between the scattering events sufficientlyshort for part of the scattered radiation to remain within the field of view,some of it will be re-scattered into the receiver as shown in Fig. 3.1. Inmost conventional applications, care is taken to keep the field of view asnarrow as possible to minimize the multiple scattering contributions butit can never be infinitely small to satisfy the single scattering condition.Therefore, some multiply scattered radiation is always present. In deal-ing with the conventional solutions (3.4) or (3.7), this extra signal wasnot of primary concern because of the greater uncertainties associatedwith the assumption of homogeneity or the specification of the bound-ary value and the backscatter-to-extinction ratio. With the advent of thehigh-spectral-resolution and Raman techniques that solve these prob-lems, multiple scattering becomes more pertinent. In moderate to densemedia, it can make a significant difference on the calculated solutions.

The obvious multiple scattering effect that is well depicted in Fig. 3.1is an increase in signal strength. The contributions from the off-axisscattering events are a net addition over the return predicted by the singlescattering models. The effect will clearly grow with the field of viewand the penetration depth as more and more scattered radiation fills thespace seen by the receiver. The measurement geometry is also a drivingfactor. For the same field of view and medium properties, the amount ofdiffused radiation within the collecting power of the receiver obviouslyincreases with the distance between the lidar and the medium boundary.A second driving parameter is particle size. It can be seen from thediagram of Fig. 3.1 that the number of light path segments that can be


found within a given receiver field of view increases in inverse proportionto the average angle of the forward scattering events in both the outgoingand return propagation legs. In other words, the amount of collectedmultiple scattering radiation grows inversely with the angular width ofthe forward peak of the scattering phase function. For particles of sizescomparable or larger than the lidar wavelength, the width of the peak isinversely proportional to the average particle size. Hence, the strengthof the multiply scattered lidar signal, and particularly its rate of increasewith field of view, depends on particle size in addition to particle density.

Another effect that is not illustrated in Fig. 3.1 is the incidence mul-tiple scattering has on the polarization state of the lidar signal. It isa well-known fact that the radiation backscattered at exactly 180◦ byspherical particles conserves the linear polarization of the original laserbeam. This property is extensively used to discriminate between particletypes as discussed by Sassen in Ref. [13] and in the chapter of this bookon polarization. The depolarization ratio �, defined as the ratio of thescattered field intensities in the perpendicular and parallel directions tothe original laser polarization and calculated by the exact Mie theory fora distribution of spherical water droplets [14], is plotted in Fig. 3.2 asa function of the scattering angle near 180◦. One clear feature is that �quickly jumps from 0 to 60% within 2◦ of the exact backscatter direc-tion. The same calculations show that � is less than 1% for all forwardangles less than 30◦. The diagram of Fig. 3.1 indicates that the contribut-ing multiple scattering paths have several forward scatterings at small

Fig. 3.2. Single scattering linear depolarization ratio� integrated over all azimuth anglescalculated for a model C2 Deirmendjian [15] cloud at 1.06 μm as a function of thescattering angle near the backscattering direction of 180◦.


angles and one backscattering at an angle close but not exactly equal to180◦. Given the steep rise of � near 180◦, we can expect depolarizationof the multiply scattered lidar returns. Since the calculated � is negligi-ble at the small forward angles, we further argue that the depolarizationarises almost exclusively from the single backscattering near 180◦. Thisis confirmed by the experimental work of Ryan et al. [16, 17] in sim-ulated laboratory water clouds; they found a linear depolarization ratioless than 1% for forward scattered light at optical depths up to 5 [16]but as large as 30% for backscattered light in the same conditions [17].We expect the degree of depolarization induced by multiple scatteringto depend on the same factors as those governing the increase in signalstrength.

In summary, multiple scattering in lidar manifests itself as greatersignal strength and alteration of polarization state. These effects dependon the measurement geometry, in particular the distance to the scatteringmedium and the physical penetration depth; on the system parameters,most importantly the receiver field of view; and on the medium proper-ties, i.e., the extinction coefficient, the angular scattering function, andthe optical depth. This constitutes a great challenge for modeling butmuch depends on it. First, the restoration of the solution accuracy ofsingle scattering retrievals in the case of contaminated measurements;and second, the exploitation of the information carried by the multiplescattering contributions. We discuss in the following sections the exist-ing experimental evidence of multiple scattering effects in lidar and theprogress made to understand, model and use the data.

3.2 Experimental Evidence

Multiple scattering in lidar has long been recognized. One of the firstpublished work that explicitly mentioned multiple scattering are the mea-surements by Milton et al. [18] of the reflectance of laser-illuminatedfair-weather cumulus clouds. Their application was not strictly speakinga lidar experiment because it did not involve ranging but the physicswas basically the same. They measured a reflectance of 2–3 times thevalue derived from single scattering calculations and they attributed thediscrepancy to multiple scattering. There followed some discussions[19, 20] centered on the use of a pulsed laser for the experiment but a cwsource for the calculations. The controversy had to do with differencesin depth of integration between the two configurations. Second-order


theoretical modeling by Anderson and Browell [21] showed that, if theeffective pulse length becomes comparable to the scattering mean freepath (= 1/α), the twice-scattered contribution amounts to more than20% of the singly scattered return [20]. It is obvious following thesediscussions that, despite the limitations of the models of the time, sec-ond and higher order scattering had to be taken into account to explainthe higher-than-expected cloud reflectance. This was later confirmed byPal et al. [22] in a laboratory-simulated water-droplet cloud.

Platt [23] carried out simultaneous lidar and radiometric measure-ments on cirrus clouds. He found that the optical depths of the measuredcirrus derived by conventional lidar methods was less than the true val-ues. He hypothesized that second- and higher-order scattering processeswere responsible for the observed stronger lidar signals. He proposed toredefine the optical depth in the lidar equation (3.1) by inserting a mul-tiplicative factor η to model the reduction of the extinction coefficient asfollows:

γ ′(z) =∫ zb+h

zb

ηα(z′) dz′, (3.10)

where α is the true extinction coefficient, zb is the range to cloud base,and h is the cloud physical thickness. The parameter η is not constant andPlatt argued that it should vary between ∼0.5 and 1. In Ref. [23], η wasestimated from the cloud optical thickness determined by radiometricobservations and theoretical modeling for the wavelength extrapolation.

The correction factor η is a simple way of representing the premiereffect of multiple scattering in lidar, namely, the increase in signalstrength. It is still widely used today. One conceptual weakness of the η

model is that multiple scattering does not affect only the optical depthγ but also the backscattering coefficient β of Eq. (3.1). There could besituations where the induced drop in the effective backscattering coef-ficient is greater than the gain caused by the added multiply scatteredcontributions, which would translate in a value of η greater than unity.

The second aspect of multiple scattering in lidar, i.e., the alterationof the polarization state of the returned signal, was also observed earlyon. Pal and Carswell [24] designed a lidar system to measure the back-scattering of a linearly polarized laser pulse in the parallel and transversepolarization directions simultaneously. They pointed their lidar at water-droplet stratocumulus clouds. They measured a significant perpendicularcomponent that continued to build up for 40–50 m beyond the rangewhere the parallel component reached its peak. Ratioing the two signal


intensities to calculate the linear depolarization ratio δ, they found a δ

that started at a low value of 1–2% at the base of the clouds and increasedmonotonically with penetration depth to reach values as high as 50% atmaximum range. The low depolarization at cloud base indicates thatthey were probing clouds of spherical water droplets. The subsequentincrease in δ was rightly attributed to multiple scattering contributions.Indeed, the number of forward scattering events increases and the aver-age angle of backscattering moves away from 180◦ with rising opticaldepth; both effects contribute to a greater proportion of depolarized orunpolarized light as can be inferred from Fig. 3.2. But there are otherfactors affecting δ. For example, Pal and Carswell [24] report resultsfrom a multilayered cloud deck which show a depolarization ratio thatdrops suddenly at the transition to a new layer before increasing again.This is explained by the localized dependence of the single scatteringreturn on the coefficient β compared with the more gradual buildup ofmultiple scattering which is a process of integration. Hence, multiple-scattering-induced depolarization is also a function of cloud structure.Very similar results on cross-polarized lidar returns are also reported byCohen [25].

The group atYork University pursued further their multiple scatteringlidar investigations on the basis of depolarization measurements. Theyproposed the following simple model [26]:

P‖ = P s‖ + Pm

u , (3.11)

P⊥ = Pmu , (3.12)

where P‖ and P⊥ are the return intensities measured in the paralleland perpendicular directions to the incident laser pulse polarization, thesuperscripts s andm stand for single and multiple scattering, and the sub-script u for unpolarized. The model assumes complete depolarization ofthe collected multiply scattered radiation. With the model of Eqs. (3.11)and (3.12), they were able to estimate the ratioM of the total to the singlescattering returns, i.e.,

M = Ptotal/Pss = (P‖ + P⊥)/(P‖ − P⊥), (3.13)

and the optical depth reduction γm = γ − γ ′ caused by multiplescattering, i.e.,

2γm(z) = ln[P‖/P s

‖] = ln

[P‖/(P‖ − P⊥)

]. (3.14)


They found values ofM of up to 3 and γm of up to 0.3–0.4 for penetrationdepths of 100–120 m in cumulus clouds.

The hypothesis of complete depolarization by multiple scatteringmade in Eqs. (3.11) and (3.12) was later found by the York group to beincorrect. In Ref. [22], the model was re-defined by assuming that partof the scattered radiation retains the original polarization of the emittedpulse. That polarized component was denoted Pm

‖ and Eq. (3.11) wasrewritten as follows:

P‖ = P s‖ + Pm

‖ + Pmu . (3.15)

To be able to isolate P s‖ in this case, they used a field stop to shield the

view of the outgoing laser beam. They found that the polarized multi-ple scattering component Pm

‖ amounted to ∼40% of the total multiplescattering signal (Pm

‖ + Pmu ). It is worth mentioning that the depolariza-

tion method described by Eqs. (3.12) and (3.15) only works under thehypothesis that the probed medium is composed of spherical particles.

The York group completed their depolarization analysis by a semi-nal paper [27] showing that the multiple scattering halo of lidar returnshas strongly preferred azimuthal polarization directions. The observedpatterns are very sensitive to the size of the scatterers.

Center-blocked field stops and depolarization were also used byAllenand Platt [28] to isolate the multiple scattering component of lidar returnsand measure it with greater resolution. They carried out experiments onmixed-phase clouds. Of particular interest is one instance where thedepolarization ratio clearly marks the transition between a region of icecrystals and one of spherical droplets. The ratio is shown to drop suddenlyat the interface and rise again monotonically by the action of multiplescattering. Our earlier comment on the requirement of purely sphericalparticles to relate unambiguously depolarization to multiple scatteringstill holds, but this result shows that depolarization may still be useful incomplex situations where the different particle types are not uniformlymixed.

Sassen and Petrilla [29] made measurements of the backscatter andlinear depolarization ratio δ from marine stratus clouds at differentreceiver fields of view. They observed a good deal of variability of thein-cloud data. However, their results show that, on average, δ increaseswith penetration depth from a low value of 2.5–4% at the base of theclouds, lower than the subcloud values of 3–6%, in agreement with thedata of the York group. Their observed δ’s pass through a maximum andthen tend to decrease toward the apparent cloud tops. The maximum δ


depends almost linearly on the field of view measured in units of thematched transmitter/receiver aperture. A typical maximum depolariza-tion ratio for a cloud at 450 m above ground level and a field of view of3 mrad is 20–30%. In addition, by varying the lidar elevation angle, theyfound that δ does not scale with the vertical or the slant range. This is yetanother confirmation that the depolarization induced by multiple scat-tering is not a local but an integral property that depends on the detailedcloud profile along the lidar path.

Werner et al. [30] studied multiple scattering in lidar as a source ofretrievable information on cloud microphysics. In accordance with theobservations made in the preceding paragraphs, they identify multiplefields of view, depolarization and pulse stretching as practical meansof measuring the multiple scattering contributions. The measurementof the multiply scattered lidar contributions have also been systemati-cally pursued by a lidar group in Canada [31–33]. They have developedmultiple-field-of-view (MFOV) depolarization lidars of various designs.

The diagram of a current MFOV lidar design [34] is shown in Fig. 3.3.It incorporates features that allow measurement of the main multiplescattering effects in the small-angle lidar geometry. The receiver field ofview is changed at the laser repetition frequency of 100 Hz by rotating a125-mm diameter aluminized glass disk, shown in an inset in Fig. 3.3,with apertures of different sizes etched at equidistant angular intervals.

Fig. 3.3. Diagram of an existing 1.06-μm multiple-field-of-view lidar. TM: telescopeoff-axis parabolic mirror; M: plane mirror; MFOV: multiple-field-of-view aperture disk;PCBS: polarizing cube beam splitter; F: narrow-band interference filter; A: attenuator;D1 & D2: silicon avalanche photodiodes; and inset: photograph of field-of-view aperturedisk.


The disk is positioned in the image plane of the main telescope mir-ror. The laser Q-switch is slaved to the disk rotation velocity to ensurethat the FOV apertures are in position on the lidar optical axis in syn-chronization with the laser pulses. The disk has 32 apertures defining32 FOVs between 0.1 and 12 mrad, full angle. A complete FOV scantakes 32/100 s during which time most clouds can be assumed to remainunchanged. After passage through the FOV aperture, the collected radia-tion is collimated, separated into parallel and perpendicular polarizationcomponents, and focused on 3-mm-diameter Si avalanche photodiodes.

Examples of multiply scattered depolarized returns from a conti-nental cloud deck are plotted in Fig. 3.4 as functions of height aboveground level. The lidar wavelength was 1.06 μm. The returns are trun-cated where the total signal drops to the smallest power resolved by thedetection system and the depolarization ratio where either of the parallelor perpendicular components reaches this limit. This explains why thedepolarization curves have shorter stretches than the backscatter curves.Figure 3.4 summarizes very well the experimental evidence of multiplescattering in lidar accumulated over the past 25–30 years and brieflydiscussed in the preceding paragraphs. We clearly see that the multiplescattering contributions grow with penetration depth from little differ-ence between the different fields of view at the base of the cloud to a

Fig. 3.4. Lidar returns (left) and linear depolarization ratios (right) as functions ofheight above ground for different receiver fields of view. Measurements at a wave-length of 1.06 μm from a continental stratus cloud deck. Fields of view from bottom totop curves are 0.52, 0.70, 0.96, 1.30, 1.79, 2.44, 3.89, and 6.20 mrad, half angle.


factor of ∼10 between the largest and smallest fields of view at maxi-mum depth. We note also in Fig. 3.4 that the return at 6.2 mrad penetrates80 m deeper into the cloud which cannot be explained by pulse stretch-ing for the geometry under which the data were collected. Therefore,there is a significant fading reduction caused by multiple scattering asobserved in the early work and modeled by Platt’s factor η. Measuringthe average slope on the far side of the maximum of the 6.2-mrad and0.52-mrad curves, we find η � 0.65, i.e., well within the 0.5 < η < 1bracket predicted by Platt [23].

The curves of the linear depolarization ratio δ of Fig. 3.4 reveal thesame basic features as already discussed. δ rises from a low value of 1–2%at cloud base to a maximum at a range that depends on the field of view.Except at the large field of view of 6.2 mrad, the maximum is not sharp,the curves mostly level off and no subsequent drop is observed within themeasurement precision of the instrument. The maximum value dependson the field of view but is everywhere less than 25%. This is comparablewith the findings of Sassen and Petrilla [29] which, however, showed amore pronounced depolarization fall beyond the maximum. Their mea-surements were for a thin cloud layer in which the maximum penetration,it was argued, coincided with the true cloud top. This, we have seen,is not the case for the experiment reported in Fig. 3.4. Note that thedepolarization ratio at 0.52 mrad, which matches the beam divergence,is everywhere less than 2%. Larger values of 5–10% were observed bySassen and Petrilla [29] for the matched conditions which, however, wereequal to 1 mrad in their case. From these findings and observations, weconclude that the linear depolarization ratio is indeed a clear indicatorof multiple scattering in the presence of spherical scatterers but, quan-titatively, δ does not appear to follow simple scaling laws. Modelingdepolarization requires that one take into account the complete meas-urement geometry and the detailed history of the lidar pulse throughthe inhomogeneities in particle concentration and size. With the addedcomplexity of non-spherical particles, it would seem that the exploitationof depolarization will be more complex than that of the total backscatterdependence on the field of view.

Another important aspect of multiple scattering in lidar that has notbeen addressed in most measurements reported above is time stretchingof the returned pulse. Range is generally calculated assuming that thezigzag multiple scattering paths of Fig. 3.1 have the same length as twicethe straight line to the backscattering event. If the receiver field-of-view(FOV) footprint, FOV · z, is less than the scattering mean free path equal


to 1/α, there are only a few forward scattering events contributing to thecollected return and they occur at angles of the order of the diffractionangle induced by the particles, i.e., ∼λ/de, where λ is the lidar wave-length and de is the effective particle diameter. Under this condition,the path length increase at the ith scattering is ∼i(λ2/d2

e )/(2α), where1/α is the “photon" mean free path and

√iλ/de is the polar angle of its

trajectory. Hence, after N forward scatterings, the total path increase �z

is given by

�z ∼N∑i=1

i

2α

λ2

d2e

∼ N(N + 1)

4α

λ2

d2e

. (3.16)

Taking N approximately equal to the optical depth γ ∼ α(z − zb) wherezb is the range to the boundary of the scattering medium and (z − zb) isthe penetration depth of the lidar pulse, we find

�z

z − zb∼ γ + 1

4

λ2

d2e

provided that FOV <(z − zb)

z

1

γ. (3.17)

Note that we have not made N equal to twice the optical depth aswould be expected for round-trip propagation because only about onehalf of the scattering events are small-angle diffraction scatterings. Forground-based applications on low-level clouds with (z − zb) ∼ 200 m,z ∼ 2 km, and γ ∼ 3, the condition on approximation (3.17) becomesFOV< 35 mrad, half angle. This condition is almost always satisfied inconventional ground-based lidar systems. Typical lidar wavelengths areof the order of 1 μm and the droplet effective diameter in water clouds is∼10–20 μm. Hence, the relative path increase �z/(z − zb) is less than∼0.01 at γ ∼ 3. In other words, for ground-based applications, pulsestretching induced by multiple scattering amounts to less than 1% of thepenetration depth and it is justified to neglect it. Werner et al. [30] meas-ured �z through a ground fog over a distance of 150 m. For ground fog,de ∼ 5 μm and the approximation (3.17) gives �z ∼ 6 m which agreesquite well with the value estimated from their plotted results.

The situation is quite different for long ranges. For example, theepochmaking LITE (Lidar In-space Technology Experiment) experi-ment [35] was conducted from an orbit altitude of 260 km. Thereforethe condition FOV < (z − zb)/(zγ ) for neglecting pulse stretching wascompletely violated for dense water clouds, even at the smallest 1.1-mradfield of view of the instrument. In such a geometry, the collected multi-ple scattering contributions arise from several events at small and large


Fig. 3.5. Four non-saturated pulses from dense marine stratocumulus clouds observedduring NASA’s LITE mission [35] of September 1994. Figure created by Anthony Davis(Los Alamos National Laboratory) with LITE data provided by Mark Vaughan (NASALangley Research Center).

scattering angles, all taking place within the receiver field of view. Sam-ple LITE returns from marine stratus clouds are reproduced in Fig. 3.5.They clearly show a signal above noise not only from a region belowthe actual cloud base but from negative altitudes, that is, if the range iscalculated from the time elapsed after the emission of the laser pulse.The multiply scattered in-cloud path length for these cases is shown inFig. 3.5 to be greater than 2.5–3 km for an actual thickness of ∼700 m.

Multiple scattering under receiver field-of-view footprints greaterthan the scattering mean free path invalidates range resolution basedon time of flight. Measurements in such conditions, although differentfrom conventional lidar, are still valid measurements that most certainlycontain information on medium properties. New instrument designs andmodeling tools are being proposed to exploit this situation. We willdiscuss those in the following sections.

3.3 Modeling

Multiple scattering in lidar is a radiative transfer problem. The radiativetransfer theory is defined and developed in several textbooks; see, for


instance, Chandrasekhar [36], Ishimaru [37], and Zege et al. [38]. Thefundamental quantity in radiative transfer theory is called the radianceI (t,R, n), where t is the time and R and n, the position and directionvectors, respectively. Bold symbols represent vectors and the superposedhat designates a unit vector. The radiance I (t,R, n) is the radiant fluxper unit solid angle and per unit area normal to n. To put it differently,the flux dP through an elementary area dS normal to n0 and within theelementary solid angle d is given by

dP = I (t,R, n)n · n0 dS d. (3.18)

I (t,R, n) has the units of Wm−2sr−1 and is in general a function of timeand six spatial and directional coordinates.

The equation governing the radiance I is the radiative transferequation written below in general non-stationary form:

1

c

∂

∂tI (t,R, n) + n · ∇RI (t,R, n) + α(t,R)I (t,R, n)

= αs(t,R)

∫4π

I (t,R, n′)p(t,R; n, n′

) dn′ + Q(t,R, n), (3.19)

where c is the speed of light, α is the extinction coefficient, αs is thescattering coefficient, p(t,R; n, n′

) is the scattering phase function, andQ(t,R, n) is the source/sink term. The phase function is normalizedsuch that ∫

4πp(t,R; n, n′

) dn′ = 1. (3.20)

Equation (3.19) is a heuristic model, it describes the conservationof the radiant flux through an elementary control volume. The particlepositions are assumed uncorrelated and their separation wide enoughto consider each particle to be in the far field with respect to the radi-ation scattered by its neighbors. Although diffraction and interferenceeffects are included in the calculation of the scattering and absorptionby a single particle, incoherent addition of powers instead of fields isused in constructing αs and p for the particle ensemble and in sum-ming for the radiance I . In other words, Eq. (3.19) does not follow fromthe rigorous first principles of Maxwell’s equations. However, Ishimaru[37] (Section 14.7) demonstrates that, under certain assumptions, thereexists a relationship between I and the mutual coherence function of thewave field.


Equation (3.19) does not take into account the polarization effects.However, the equation can be readily expanded to include polarization bysubstituting the Stokes vector for the radiance, and the Mueller matrix forthe phase function. The Stokes vector I is a four-dimensional vector thatdescribes the polarization state of I in terms of elementary polarizationcomponents. For example, denoting by the subscripts ‖,⊥,+ and − thepolarization states of I measured with linear polarizers oriented in theparallel, perpendicular and tilted at ±45◦ with respect to the scatteringplane, and by r and l the states measured through right-hand and left-hand circular polarizers, respectively, we can write the Stokes vectoras [39]

I = [(I‖ + I⊥), (I‖ − I⊥), (I+ − I−), (Ir − Il)]T, (3.21)

where the superscript T means the transpose operation. The fourelements are not all independent but related as follows:

(I‖ + I⊥)2 = (I‖ − I⊥)2 + (I+ − I−)2 + (Ir − Il)2. (3.22)

The 4 × 4 Mueller matrix is constructed from first principles in the samefashion as the phase function p. The necessary theoretical tools can befound in Bohren and Huffman [39].

Equation (3.19) is a complex integro-differential equation. No prac-tical general solution exists but numerous approximations have beenworked out to handle atmospheric and oceanic transmission problems.Chandrasekhar [36], Ishimaru [37], and Zege et al. [38] give the essen-tial derivation steps for most cases. We will not discuss further theseapplications in this chapter but concentrate on the lidar.

The lidar geometry is characterized by narrow beams and smallreceiver fields of view. In addition, the atmospheric or oceanic scat-tering phase functions at the popular lidar wavelengths are peaked in theforward direction. These particular conditions have led to special solu-tion methods of the radiative transfer equation (3.19). We review in thissection the most salient developments for modeling multiple scatteringin lidar.

3.3.1 Monte Carlo Methods

In view of the serious theoretical and computational difficulties of solvingEq. (3.19), Monte Carlo methods were considered from the beginningas a convenient alternative. The Monte Carlo procedure can be made


to model the same physical processes as the radiative transfer equationbut from a statistical approach. The continuous radiance function I isrepresented by a very large number of possible distinct trajectories. If oneknows the probability for each step in the sequence of events defininga trajectory, or a realization, a possible distinct trajectory can easilybe constructed. Then, from many such realizations, averages can becalculated to represent physical quantities, for example, the radiant fluxentering a receiver of given position, size and acceptance angle. For theradiative transfer problem, the probabilities are:

1. the probability of scattering or absorption

prob(scatt.) = αs/α, (3.23)

prob(abs.) = 1 − αs/α; (3.24)

2. the probability of scattering from direction n into direction n′ withinthe elementary solid angle dn′

prob(n → n′) dn′ = p(t,R; n, n′

) dn′; (3.25)

3. and the probability that the free propagation length from position Rin the new direction n′ before the next “collision” event is comprisedbetween l and l + dl

prob(l) dl = α(t,R + ln′) exp

[−∫ l

0α(t,R + xn′

) dx

]dl.

(3.26)

The needed collision parameters (whether scattering or absorption),photon direction and free path length are calculated by equating thecorresponding cumulative probability to a random number uniformlydistributed between 0 and 1.

The main advantages of the Monte Carlo approach are that it requiresfew simplifying approximations, that it allows separation of the contri-butions by scattering order, and that it can be extended to more complexmedia with relative ease. For example, to simulate propagation in thepresence of various types of scatterers, e.g., molecules, backgroundaerosols, water droplets and ice crystals, it suffices to add to (3.23)–(3.26)the corresponding probabilities for each species.

Radiative transfer Monte Carlo algorithms have been developedfirst to model solar transmission and backscattering by the atmosphere


and ocean waters. Plass and Kattawar [40–45] were among the firstresearchers to make systematic use of Monte Carlo simulations in thatfield.

The implementation of Monte Carlo calculations is rather straight-forward but for most radiative transfer problems in the backscattergeometry, techniques of variance reduction are necessary because theangular scattering is generally peaked in the forward direction whichmakes the probability of a series of events leading to the actual captureof a photon by a backscatter receiver extremely small. This leads to largefluctuations in the calculated averages that can be reduced by increas-ing the number of trajectories to unrealistic levels or, more practically,by lowering the variance with the help of computational and analyticalmeans.

Variance reduction can be fairly sophisticated. In their work alreadycited, Plass and Kattawar assigned a statistical weight to each photon,forced collisions within the physical domain of interest so as not towaste calculated trajectories, and then renormalized the photon weightto adjust for the imposed bias. That proved insufficient in lidar simula-tions because of the very small fields of view of conventional receivers.A significant improvement was achieved by adding analytic calculationsto the stochastic Monte Carlo calculations. This method is known as themethod of statistical estimation [46]. It is simply hinted in Plass andKattawar [47] but it is described as a main feature in Kunkel and Wein-man [48] and Poole et al. [49, 50]. The method consists in calculatinganalytically at each collision the probability that the photon would returndirectly to the receiver without further interactions. This probability isgiven by

prob(Rc) =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

(−nr · ncr )A

πL2p(t,Rc; nc, ncr )

× exp

[−∫ L

0α(t,Rr − xncr ) dx

]if (−nr · ncr ) ≥ cos(�),

0 if (−nr · ncr ) < cos(�),

(3.27)

where Rr is the position of the receiver, Rc is the position of thescattering or collision event, nr is the unit vector normal to the receiveraperture, ncr is the direction from the collision point to the receiver,i.e., ncr = (Rr − Rc)/|Rr − Rc|, nc is the unit vector giving the direc-tion of the photon prior to the collision, A is the receiver aperture area,L is the distance between the receiver and the scattering event given by


L = |Rr − Rc|, and� is the half-angle receiver field of view. The sum ofall probabilities for all collision events is used as the backscattered signalinstead of the sum of the very few photons which actually would scatterinto the receiver. This technique reduces considerably the variance of thecalculated signal because all collisions within the receiver field of viewcontribute. It was validated against Monte Carlo simulations performedwithout full application of the technique [49] and against measurementsin a cell of turbid water [50].

An additional type of a simple but efficient variance reduction methodwas introduced by Platt [51]. The rationale was to increase the numberof photon trajectories in the backward direction. As already discussed,the backscatter probability calculated from (3.25) is very small for mostparticulate media and lidar wavelengths of practical interest. Hence,there are very few “physical” photons contributing to the signal. To favormore backward trajectories, Platt [51] proposed to create an artificialphase function pa by folding the forward half of the true phase functionp into the backward direction as follows:

pa(t,R; n, n′) =

{Cp(t,R; n, n′

) for n · n′ ≥ 0,

Cp(t,R; n,−n′) for n · n′

< 0,(3.28)

where C is the normalization constant given by C−1 = 2∫

n·n′≥0

p(t,R; n, n′)dn′. To compensate for the artificially increased number

of backward trajectories, the scattered photon is weighted after eachcollision by the ratio

w = p(t,R; n, n′)/pa(t,R; n, n′

). (3.29)

Another method proposed by Bruscaglioni et al. [52, 53] consists indefining virtual cloud profiles by adapting the cloud depth or density toeach scattering order. The purpose is to optimize the statistics of a givenscattering order contribution by adjusting the cloud depth or density(or both) to have on average a number of scattering collisions equal tothe scattering order. For example, the virtual extinction coefficient forscattering order i could be

αi = Ciα with Ci = 2i/∫ L

0α(t,Rr + xnr ) dx, (3.30)

whereL is the physical depth of interest. The total signal is then obtainedby summing the contributions of the different scattering orders calculatedseparately each with a different virtual cloud coefficient αi . Of course,


the photon weights have to be renormalized by the ratio of the physicalto the virtual probability densities of the free path l, i.e., by using α

and αi in Eq. (3.26), respectively, with l derived from the virtual cloudcumulative probability. As there are different ways of defining scalinglaws of the type given by Eq. (3.30), the reader should consult the originalreferences [52, 53] for details.

Monte Carlo calculations of multiply scattered lidar returns are stillslow but they have become accessible to personal computers with the helpof the variance reduction techniques such as described above. They areadaptable to very complex problems with no or a minimum of simplifyingapproximations. No other solution method can match these capabilitiesand in some cases they are the only tool available. Another importantutility of Monte Carlo procedures that follows from their independenceon restrictive assumptions is that they can serve as numerical experimentsto validate analytical solutions and test inversion methods.

Efficient Monte Carlo codes for simulating lidar returns have beendescribed by Bruscaglioni et al. [53], Starkov et al. [54], and Winkerand Poole [55]. The three groups compared their calculations for a givenground-based cloud measurement scenario [56]; their results showedvery good agreement among themselves. That virtual experiment has ledto improvements for handling polarization and inhomogeneities in par-ticle concentration, shape and composition. Results have been presentedin numerous conferences and papers, particularly within the MUSCLE(MUltiple SCattering Lidar Experiments) group that was responsiblefor the original intercomparison reported in Ref. [56]. All the importantexperimental findings on multiple-scattering-induced signal increase anddepolarization, and their dependence on measurement geometry, opti-cal depth and medium scattering properties have been demonstrated byMonte Carlo calculations.

One drawback of the Monte Carlo approach is that the solutions arenumerical and specific to a given problem. Although there exist a numberof scaling relationships as discussed by Bruscaglioni et al. [52], one stillneeds to repeat the calculations for a whole set of values to obtain insighton the influence of a medium or instrument parameter.

3.3.2 Stochastic and Phenomenological Methods

It is possible to formulate the problem of multiple scattering in lidaranalytically, not in the framework of the radiative transfer equation as itwill be the case in the next section, but following the similar premise as


for the Monte Carlo approach. The same notion of random trajectoriesresulting from successive scattering and absorption events by randomlydistributed particles is used. The laws governing the propagation are thesame angular scattering and free path distribution functions given byEqs. (3.25) and (3.26).

Two main groups have used a formal stochastic approach to modelthe lidar return: Gillespie [57], and Oppel et al. [58] and Starkov et al.[54]. They consider the nth-order contribution of the power measured atthe receiver as the sum over the 3n-dimensional joint probability den-sity of trajectories made up of n segments, each defined by a free pathlength and a polar-azimuth heading. The joint probability density is con-structed from the elementary functions (3.25) and (3.26). The processis described by Starkov et al. [54] as a generalized Rayleigh’s randomwalk. For example, the end result in Gillespie [57] is a 3n-fold integralover a kernel consisting of the n-dimensional product of Eqs. (3.25) and(3.26) evaluated at the coordinate points of the integration variables timesfunctions that limit the domain of the integration variables. This expres-sion constitutes an exact multiple scattering lidar equation. The multipleintegral can only be computed numerically or by a Monte Carlo method,except for very special cases. For a homogeneous medium, Gillespie [57]was able to transform the integral from 3n to (3n − 4) dimensions. Thefinal expressions reached by these two groups are not reproduced herebecause they involve many intermediate definitions that would requiremore space than available with little additional useful information.

Basically similar expressions have been derived by employingmethods of integral calculus. We designate this approach here as phe-nomenological. In short, the scattered radiation out of an elementaryvolume is written as the outcome of a scattering event occurring insidethis volume on radiation coming from a preceding elementary volumeinside which the same type of scattering has taken place. For the nth-order scattering, there is a cascade of n such elementary processes. Thetotal contribution is obtained by integrating over all three variables defin-ing each of the n elementary processes. All integration variables are notindependent, they must satisfy a condition on the length of the totalpath. This constitutes the main practical difficulty of the approach asthe integration limits must be chosen from geometrical considerationsthat increase in complexity with n. In the stochastic models, the inte-gration limits are over the complete volume; in that case, the constraintstook the form of additional kernel functions. Handling the kernel func-tions probably constitutes a similar difficulty as defining the integration


limits. The stochastic expressions have the advantage of a more generalmathematical form. Both methods include time-dependent effects suchas pulse stretching since the trajectories are constrained by the time ofarrival.

A good example of the phenomenological approach is the modelof Cai and Liou [59]. They have derived their solution for the Stokesvector and, therefore, they can calculate any change in polarization stateinduced by multiple scattering.An adapted version for the received poweris reproduced here. According to their model, the power P (n)(τ ) con-tributed at the receiver by exactly n scatterings in a stationary mediumcharacterized by the extinction coefficient α(R) and the scattering func-tion p(R, θ) with θ = arccos(n · n′), and collected at time τ followingthe emission of the laser pulse, is given by

P (n)(τ ) = Pt

∫∫∫V1

dv1

∫∫∫V2

dv2 · · ·∫∫∫

Vn

dvn

× A

2πR21[1 − cos(ψm/2)]

α(Rn)p(Rn, θn)

4πR2n

×n−1∏i=1

αs(Ri)p(Ri , θi)

4π |Ri+1 − Ri |2 exp

{−∫ R1

0α

(R1

R1x

)dx

−∫ Rn

0α

(Rn − Rn

Rn

x

)dx

−n∑

i=2

∫ |Ri−Ri−1|

0α

(Ri−1 + Ri − Ri−1

|Ri − Ri−1|x)

dx

}, (3.31)

where Pt is the laser pulse power evenly distributed within the beamdivergence ψm,A is the receiver aperture area with diameter assumedmuch less than any other scale of the problem, and

τ = 1

c

[R1 +

n∑i=2

|Ri − Ri−1| + Rn

], (3.32)

θi = arccos

(Ri · (Ri+1 − Ri)

Ri |Ri+1 − Ri |)

for i < n, (3.33)

θn = arccos

(−Rn · (Rn − Rn−1)

Rn|Rn − Rn−1|), (3.34)


and c is the speed of light. In the spherical coordinates (R, φ,ψ), thevolume integrals are given by

∫∫∫Vi

dvi · · · =∫ 2π

0dφi

∫ ψ∗i

0dψi

∫ R∗i

R′∗i

R2i sin ψi dRi · · · . (3.35)

As already mentioned, the main difficulty is in the determination of theintegration limits. Some are straightforward, in particular ψ∗

1 = ψm/2,the beam divergence; ψ∗

n = ψr/2, the receiver field of view; R∗1 = cτ/2;

and R′∗i = Hb(φi, ψi) for i < n, the distance to cloud base along the

direction defined by Ri . The other ψ∗i ’s, R∗

i ’s and R′∗n are obtained by

finding the bounds of the integration volumes Vi’s inside which the con-dition defined by Eq. (3.32) for a fixed τ and fixed Rj ’s with j < i issatisfied. There follows a series of recurrence formulas where the R∗

i ’sand ψ∗

i ’s for i > 2, R′∗n and R∗

n depend on the values derived for the pre-ceding events in the cascade. Hence, the integration limits of Eq. (3.35)can be determined consecutively from i = 1 to n. The details can befound in Ref. [59].

A recent model by Samoilova [60] including polarization effects andderived from earlier less known Russian work is constructed in the samephenomenological fashion. Another model that also fits this categoryis that recently proposed by Eloranta [61]. Eloranta considers a sim-pler situation where all scatterings take place at small forward anglesexcept for one backscattering at an angle close to 180◦. Because theangles are small, the photon paths remain close to the lidar axis. Asa result, the increase in path length is negligible and the angular inte-gration limits of Eq. (3.35) are much simplified. Eloranta then modelsthe phase function p for the forward small angles by a Gaussian andassumes uniform backscattering. This enables him to carry out analyti-cally the integrals over the angles φi and ψi—they become convolutionsof Gaussian functions—and the resulting expression is reduced to ann-dimensional integral over range. The calculation of the solutions, ingeneral, still requires a numerical algorithm but at a much reduced cost.Eloranta goes one step further and shows that, under some special butpractical conditions, he can derive simple analytic formulas that providemuch physical insight into the dependence of multiple scattering on keymedium and instrument parameters. The Gaussian phase function mayappear restrictive but the strong diffraction peak of natural cloud and pre-cipitation particles at traditional lidar wavelengths is well represented by


a Gaussian and the resulting solutions are good first approximations foroptical depths less than 1–2.

The early second-order scattering work of Liou and Schotland [62],Anderson and Browell [21] and Eloranta [63] can be classified under thephenomenological approach. They followed basically the same deriva-tion steps as described above but for only two scatterings. Therefore, itwould seem that the problem formulation was already known 30 yearsago or at least 20 years ago for arbitrary scattering order. Progress inapplying these tools to correct for or exploit the multiple scattering con-tributions was slowed by the mathematical complexity of the integralsand the intensive calculations needed to derive numbers. In fact, thestochastic and phenomenological approaches offer little or no computa-tional edge over the Monte Carlo methods. However, the advantage ofhaving formal mathematical expressions is that simplified formulas arepossible for limit cases as demonstrated by Eloranta [61]. Even thoughthe accuracy of these simplified formulas may be poor, they are veryuseful for understanding observations and designing instruments.

3.3.3 QSA Approximation—General Theorem

A schematic diagram of multiple scattering in lidar is drawn in Fig. 3.6.The depicted situation is of common occurrence in applications. Theinstrument wavelength and the medium properties are such that scatteringat small angles θi is predominant. In addition, the footprint of the receiverfield of view � has a diameter less than the mean free path between thescatterings. Under such conditions, the trajectories that contribute mostto the received power are made up of small-angle forward scatterings onboth the outgoing and return propagation legs and a single backscatteringat an angle close to 180◦. This regime defines the Quasi-Small-Angle(QSA) approximation of radiative transfer.

Katsev et al. [64] have derived a general theorem that formallysimplifies the search for analytic, semianalytic or numerical solutionsof the multiple scattering lidar problem in the QSA approximation. Inshort, they have succeeded in using the Fourier space general solutionof the radiative transfer equation in the small-angle approximation limitto define an effective medium with the consequence of transforming theround trip lidar problem into a simpler one-way propagation problem.The mathematics is a little laborious but the result is well worth the effortof going through the main derivation steps.


Fig. 3.6. Schematic representation of a multiply scattered contribution to lidar return.T: receiver telescope; Ofov : aperture defining the receiver field of view; D: detector;zb: range to cloud base; z: range to backscatter event; θi : forward scattering angles;θb: backscattering angle; and �: half-angle receiver field of view.

In the framework of the QSA approximation and with the help of thediagram of Fig. 3.6, we decompose the lidar radiative transfer probleminto the forward propagation of the source radiance on the outgoing path,a diffusion reflection at the backscattering event, the forward propagationof the backscattered radiance on the return path, and the capture bythe receiver. All we need to assume at this stage is that there exists aGreen’s function solution for the radiance propagation. Mathematically,we formulate the problem described above by considering successively

• the source radiance multiply forward scattered to position R in thedirection n:

If (R, n) =∫

dR0

∫dn0Wsrc(R0, n0)Go(R, n; R0, n0), (3.36)

where Wsrc(R0, n0) is the normalized source radiance and Go is theGreen’s function solution of the radiative transfer equation specificto the outgoing problem;

• the radiance singly backscattered at R in the direction −nb:

Ib(R,−nb) =∫

dnαsb(R)pb(R; −nb, n)If (R, n), (3.37)


where αsb and pb are the scattering coefficient and phase function ofthe medium constituent responsible for the backscatter;

• the radiance backscattered at R that is subsequently forward scatteredon the return leg from R to a position R′′ and in the direction −n′′:

Ir(R′′,−n′′; R) =∫

dnbIb(R,−nb)Gr(R′′,−n′′; R,−nb),

(3.38)

where Gr is the Green’s function for the return problem (we distin-guish between Go and Gr because the wavelength can be differentin both cases as in inelastic Raman applications);

• and the collected power at the receiver originating from the locationR of the backscattering event:

P(R) =∫

dR′′∫

dn′′Wrec(R′′,−n′′

)Ir(R′′,−n′′; R), (3.39)

where Wrec is the receiver angular-spatial collection pattern.

Making use of the optical reciprocity principle

G(R′′,−n′′; R,−nb) = G(R, nb; R′′, n′′), (3.40)

defining a “receiver” source

Wrecsrc (R

′′, n′′) = Wrec(R′′,−n′′

), (3.41)

combining the expressions (3.36)–(3.39) and collecting the terms inlogical groups, we find for the received lidar power from position R

P(R) =∫

dnb

∫dnαsb(R)pb(R; −nb, n)

×∫

dR0

∫dn0Wsrc(R0, n0)Go(R, n; R0, n0)

×∫

dR′′∫

dn′′Wrec

src (R′′, n′′

)Gr(R, nb; R′′, n′′). (3.42)

Finally, if we define the source and receiver–source radiances

Isrc(R, n) =∫

dR0

∫dn0Wsrc(R0, n0)Go(R, n; R0, n0), (3.43)

I recsrc (R, n) =

∫dR0

∫dn0W

recsrc (R0, n0)Gr(R, n; R0, n0), (3.44)


as the multiply scattered radiances at position R originating from thetrue source Wsrc and the “receiver” source Wrec

src , respectively, we end upwith

P(R) =∫

dnb

∫dnαsb(R)pb(R; −nb, n)Isrc(R, n)I rec

src (R, nb).

(3.45)

Equation (3.45) reduces the lidar problem into two conceptually simplerpropagation problems, defined by Eqs. (3.43) and (3.44), connected byone backscattering event. This results from the QSA description of thelidar multiple scattering problem and the reciprocity principle of opticalpropagation given by Eq. (3.40).

Next, we turn to the solutions for Isrc(R, n) and I recsrc (R, n). In accor-

dance with the QSA approach, the forward propagation problem satisfiesthe conditions of the small-angle approximation (SAA). One importantsimplification is that the medium can be considered stratified. This isjustified by the smallness of the scattering angles, θi in Fig. 3.6, andthe narrow angular width of both the source and receiver functions Wsrc

and Wrecsrc . Thus, the R = zk + r dependence of the medium properties

is reduced to a z dependence and the direction vector n is approximatedby n � k + n⊥, where k is the unit vector along the z axis chosen tocoincide with the lidar axis, r is the component of the position vectornormal to k, and |n⊥| 1. Since |n⊥| 1, n⊥ is the vector scatter-ing angle, i.e., n⊥ � θ [i cosφ + j sin φ] where θ and φ are the polarand azimuth components of the radiance angular variable, respectively.The above approximations essentially amount to making sin θ � θ andcos θ � 1, and to neglecting path increase caused by beam spreading.Moreover, because n⊥ is small, the phase function for randomly orientedparticles depends only on the difference |n⊥ − n′

⊥| rather than on n⊥ andn′

⊥ independently. Under these conditions, the stationary version of theradiative transfer equation (3.19) with no source term becomes, in theforward direction,

∂

∂zI (z, r,n⊥) + n⊥ · ∇rI (z, r,n⊥) + α(z)I (z, r,n⊥)

= αs(z)

∫ ∞

−∞I (z, r,n′

⊥)p(z; |n⊥ − n′⊥|) dn′

⊥, (3.46)

where the integration limits on n′⊥ were extended to ±∞ because,

within the SAA,p(z; |n⊥ − n′⊥|)/p(z; 0) becomes negligibly small with

increasing |n⊥ − n′⊥|.


Both Isrc(R, n) and I recsrc (R, n) are solutions of Eq. (3.46) with their

respective phase functions po and pr . Because the integral term inEq. (3.46) is a convolution, it is feasible to obtain a formal solutionin the Fourier space. Following Ishimaru [37] (Section 13.2) or Zegeet al. [38] (Section 4.4.3), the solutions are

Isrc(z,q,p) = Wsrc(q,p + qz)Go(z,q,p), (3.47)

I recsrc (z,q,p) = W rec

src (q,p + qz)Gr(z,q,p), (3.48)

where the Fourier transforms are defined by

I (z,q,p) =∫ ∞

−∞dr∫ ∞

−∞dn⊥I (z, r,n⊥)e(−iq·r−ip·n⊥), (3.49)

Wsrc(q,p) =∫ ∞

−∞dr∫ ∞

−∞dn⊥Wsrc(0, r,n⊥)e(−iq·r−ip·n⊥), (3.50)

p(z,p) =∫ ∞

−∞dn⊥p(z,n⊥)e−ip·n⊥, (3.51)

with Wsrc standing for both Wsrc and W recsrc , and the Green’s function

solution G of the Fourier transformed Eq. (3.46) is

Go,r (z,q,p) = exp

{−∫ z

0[αo,r(z − ξ) − αs(o,r)(z − ξ)

× po,r (z − ξ,p + qξ)] dξ

}. (3.52)

The function Wsrc(0, r,n⊥) of Eq. (3.50) represents the boundary valuefor both sources Wsrc and Wrec

src , and the subscript (o, r) in Eq. (3.52)stands for either o or r .

Rewriting the expression (3.45) for the lidar signal P(R) in the SAAvariables, we have

P(z, r) =∫

dn⊥b

∫dn⊥αsb(z)pb(z; −k − n⊥b, k − n⊥)

× Isrc(z, r,n⊥)I recsrc (z, r,n⊥b). (3.53)

Since the angles n⊥ and n⊥b are small, the phase function pb dependson the difference |n⊥b − n⊥| only and we can write

pb(z; −k − n⊥b, k − n⊥) = pb(z;π − |n⊥b − n⊥|). (3.54)


Furthermore, we neglect in the SAA the path length increase caused bythe forward scattering zigzags and we equate the range z to the time τ

following the emission of the laser pulse, i.e., z = cτ/2. Hence, the lidarreturn measured at time τ , or from range z, is approximated by integrat-ing Eq. (3.53) over r in the plane z. Changing the angular integrationvariables from n⊥b and n⊥ to n⊥d = n⊥b − n⊥ and n⊥, we therefore havefor the lidar signal from range z

P (z) =∫

dr∫ ∞

−∞dn⊥d

∫ ∞

−∞dn⊥αsb(z)pb(z;π − |n⊥d |)Isrc(z, r,n⊥)

× I recsrc (z, r,n⊥ + n⊥d). (3.55)

We have extended in Eq. (3.55) the angular integration limits to ±∞because the function Isrc and I rec

src rapidly become negligible with increas-ing n⊥d and n⊥. Finally, for convenience, we rewrite Eq. (3.55) asfollows:

P(z) =∫ ∞

−∞dn⊥dαsb(z)pb(z;π − |n⊥d |)H(z,n⊥d), (3.56)

with

H(z,n⊥d) =∫

dr∫ ∞

−∞dn⊥Isrc(z, r,n⊥)I rec

src (z, r,n⊥ + n⊥d). (3.57)

Because the solutions Isrc and I recsrc are derived in Fourier space, we

Fourier transform Eq. (3.57) and apply the Parseval equality [65]. Wethus obtain for symmetric functions of p and n⊥

H (z,p) = 1

(2π)2

∫dqI ∗

src(z,q,p)I recsrc (z,q,p), (3.58)

where the ∗ indicates the complex conjugate of the function. The Parsevalequality states that the integral in physical space of the product of onefunction by the complex conjugate of another is equal to the integral inFourier space of the product of their corresponding transforms. Substi-tuting the solutions (3.47) and (3.48) for Isrc and I rec

src in Eq. (3.58), wehave

H (z,p) = 1

(2π)2

∫dqW ∗

src(q,p + qz)W recsrc (q,p + qz),

× G∗o(z,q,p)Gr(z,q,p). (3.59)


Using the general solution (3.52), we define an effective Green’s function

Ge(z,q,p) = G∗o(z,q,p)Gr(z,q,p)

= exp

{−∫ z

0[αe(z − ξ)−αe

s (z − ξ)pe(z − ξ,p + qξ)]dξ},

(3.60)

where

αe = αo + αr (3.61)

αes = αso + αsr (3.62)

pe = αsopo + αsrpr

αso + αsr

(3.63)

are the properties of the effective medium. In constructing the expres-sion (3.63) for the equivalent phase function pe, we have assumed thatpo(z; n⊥) is an even function of n⊥ so that p∗

o = po. This is always thecase for random orientation of scatterers. Note that for elastic backscat-tering, i.e., identical properties for the outgoing and return propagationlegs, the effective medium has twice the extinction and scattering coef-ficients but the same phase function as the true medium. Regrouping theterms inside the integral of Eq. (3.59), we define

I ∗e(z,q,p) = W ∗src(q,p + qz)Ge(z,q,p), (3.64)

W (z,q,p) = W recsrc (q,p + qz), (3.65)

where the equivalent radiance I e and receiver function W will be givenproper interpretations below. H (z,p) of Eq. (3.59) can thus be rewritten

H (z,p) = 1

(2π)2

∫dqW (z,q,p)I e∗(z,q,p). (3.66)

Now that we have obtained H in terms of the radiance solution forthe effective medium defined by Eqs. (3.61)–(3.63), we inverse Fouriertransform Eq. (3.66) to return to the physical space.Applying the Parsevalequality to the result, we find for symmetric functions of p and n⊥

H(z,n⊥d) =∫

dr∫ ∞

−∞dn⊥W(z, r,n⊥d + n⊥)I e(z, r,n⊥). (3.67)


Substituting Eqs. (3.67) for H(z,n⊥d) back into Eq. (3.56), we obtainthe following expression for the lidar signal from range z:

P(z) =∫

dr∫ ∞

−∞dn⊥d

∫ ∞

−∞dn⊥αsb(z)pb(z;π − |n⊥d |)

× W(z, r,n⊥d + n⊥)I e(z, r,n⊥). (3.68)

From the comparison of Eqs. (3.64) and (3.65) with the formalFourier space solution, Eq. (3.47) or (3.48), of the SAA radiative transferequation, the following interpretations for I e and W become obvious:

• I e(z, r,n⊥) is the source radiance forward propagated to position(z, r) from the true source function Wsrc(0, r,n⊥) in the effectivemedium defined by Eqs. (3.61)–(3.63), I e(z, r,n⊥) is called theequivalent or effective radiance; and

• W(z, r,n⊥) is the actual receiver function Wrec(z, r,n⊥) sinceW (z,q,p) of Eq. (3.65) is the vacuum (G = 1) forward-propagationFourier solution of the “receiver" source Wrec

src defined in Eq. (3.41)in terms of the true receiver pattern.

Equation (3.68) is the main result of this section. It implies that thecalculation of multiply scattered lidar returns can be accomplished inthe QSA regime by substituting the real medium by a fictitious mediumof effective properties given by Eqs. (3.61)–(3.63) in the outgoing path,of same angular backscattering function as the actual medium, and ofzero extinction and scattering in the return path. Hence, the round triplidar problem is replaced by the simpler problem of solving the for-ward propagation of a radiance beam in a properly specified effectivemedium. The return propagation is trivial as it takes place in vacuum.In other words, all forward scattering events of Fig. 3.6 are modeled tooccur on the outgoing propagation leg with reduced free path lengthsbecause αe

s is nearly twice greater than the individual αso and αsr . Thisis illustrated in Fig. 3.7 where no attempt was made to draw an exactlyequivalent trajectory because the equivalence exists on the average only.Individual trajectories such as depicted in Fig. 3.7 are virtual represen-tations. Finally, it should be noted that Eq. (3.68) allows for inelasticbackscattering; αo, αr , αso, αsr , αsb, po, pr , and pb can all be different.

The most common receivers have a pupil area�rec of dimension muchless than the field-of-view footprint at working ranges and a uniformangular profile delimited by a field of view � (half angle). With these


Fig. 3.7. Same as in Fig. 3.6 except that the multiple scatterings are drawn in theequivalent effective medium defined by Eqs. (3.61–3.63).

specifications, the function Wrec for r/z 1 has the trivial form

Wrec(z, r,n⊥) ={(�rec/z

2)δ(n⊥ − r/z) if r ≤ �z,

0 otherwise.(3.69)

Using Eq. (3.69) in (3.68), assuming axisymmetry and performing theintegral over the δ function, we obtain the following practical expressionfor the lidar return from range z and within field of view �:

P(z,�) = (2π)2�rec

z2

∫ �z

0r dr

∫ ∞

0n⊥dn⊥αsb(z)

× pb(z, π − |r/z − n⊥|)I e(z, r, n⊥). (3.70)

Katsev et al. [64] propose a formulation slightly different fromEq. (3.68). It is somewhat less intuitive but more efficient computa-tionally in cases of simple source and receiver profiles. They define anequivalent source profile by convoluting the true source and receiverfunctions as follows:

We(q,p + qz) = W ∗src(q,p + qz)W rec

src (q,p + qz). (3.71)

Then, the equivalent radiance is derived for the equivalent source andnot for the real source as it was done in Eq. (3.64), i.e.,

I e(z,q,p) = We(q,p + qz)Ge(z,q,p). (3.72)


The effective medium, however, is the same as before, i.e., Eqs. (3.61)–(3.63). Making use of Eqs. (3.47), (3.48), (3.60), (3.71) and (3.72) inEq. (3.58) yields

H (z,p) = 1

(2π)2

∫dq I e(z,q,p). (3.73)

The inverse transform is thus

H(z,n⊥d) = I e(z, r = 0,n⊥d), (3.74)

and Eq. (3.56) for the return signal P(z) becomes

P(z) =∫ ∞

−∞dn⊥dαsb(z)pb(z;π − |n⊥d |)I e(z, r = 0,n⊥d), (3.75)

or in axisymmetric conditions

P(z) = 2π∫ ∞

0dn⊥dαsb(z)pb(z;π − n⊥d)I e(z, r = 0, n⊥d). (3.76)

The receiver function, that appears explicitly in Eq. (3.68) or throughthe field of view � and angle r/z in Eq. (3.70), is embedded here inthe effective radiance. For example, a new I e(z, r = 0,n⊥d) has to becalculated for each different field of view but there is no integration overr compared with Eq. (3.70). In the physical space, the effective sourceprofile is calculated from the inverse Fourier transform of Eq. (3.71),which leads to a convolution. The effective medium is the same for bothI e and I e.

The solution given by Eq. (3.68) or (3.75) is general. For particularapplications, one needs to derive expressions for the Fourier transformsof the phase functions and of the true or effective source profile, and tosubstitute the results in Eq. (3.64) or (3.72). Then the functions I e and I e

have to be inverse Fourier transformed to calculate the physical effectiveradiances of Eq. (3.68) or (3.75). To this day, there are no known exactmethods for performing these tasks because of the complexity of thephase functions. Note, however, that there is no requirement to use theFourier approach to solve for I e or I e; any valid solution can be used inEq. (3.68) or (3.75).

The general result expressed by Eq. (3.68) or (3.75) constitutes asignificant simplification of the problem of multiple scattering in lidarwhere the QSA approximation is justified. In particular, the convolutionof the Green’s function in Eq. (3.42) that involves a fourfold integration


on the scattering angles has been eliminated. The problem becomes oneof forward propagation only. Compared with the stochastic and phe-nomenological methods of the preceding section, the number of nestedintegrations is reduced considerably for scattering orders of 7–8 neededat fields of view and optical depths of practical measurements.

3.3.4 QSA Approximation—A Neumann Series Solution

One early application of the QSA approach to calculate multiplyscattered lidar signals is the model of Weinman [66] and Shipley [67].Their basic definition of the problem is the same as illustrated in Fig. 3.6and used throughout the derivations of Subsection 3.3.3, namely: small-angle forward scatterings in both the outgoing and return paths and asingle backscattering event at an angle close to 180◦. Weinman alsoassumes, without demonstration, that the backscattered radiance cap-tured within the narrow fields of view of conventional lidar receiverspropagates basically in the same manner as the outgoing radiance. Thisworking hypothesis, made twenty years earlier, has nearly the same effecton calculations as the effective-medium theorem of Katsev et al. [64]discussed above. Another aspect of special interest to this chapter is thatWeinman makes use of the Neumann series that transforms the integro-differential radiative transfer equation into a set of purely differentialequations.

The Neumann series is the simple series

I =∞∑n=1

In. (3.77)

Equation (3.77) is substituted for I in the SAA radiative transferequation (3.46) and the individual In’s are chosen to satisfy

∂

∂zIn + n⊥ · ∇rIn + αIn

= (1 − δn1)αs

∫ ∞

−∞In−1(z, r,n′

⊥)p(z; n⊥ − n′⊥) dn′

⊥, (3.78)

with δn1 defined by

δij ={

1 if i = j,

0 if i �= j.(3.79)


The radiance In is interpreted as the contribution from the nth scatteringorder. The integral term in Eq. (3.78) becomes a known function of thesolutions obtained at the lower orders. The series begins with I1 whichis the trivial solution for the unscattered radiance. Hence, the Neumannseries solution method removes the difficulty of dealing with an integro-differential equation and it has the advantage of distinguishing betweenthe scattering orders. The algorithm proposed by Weinman [66] is a goodexample of its application.

Weinman further simplifies Eq. (3.78) by relating the variable r tothe scattering angle θ = arccos(n⊥ · n′

⊥) instead of treating r and n⊥as fully independent variables. More precisely, he sets 〈r · r〉 = �2θ2,where � is the scattering free path. Equation (3.78) is next reduced toa one-dimensional ordinary differential equation by Fourier transform-ing with respect to the transverse spatial coordinate. Finally, to makethe resulting solutions integrable analytically over the angular and trans-verse coordinates, Weinman models the phase function as a finite sum ofGaussian functions. The final expression for In has the form of (n − 1)nested integrations over the axial distance z which need to be performednumerically. The integration steps �z must be made small enough toensure negligible probability of a second scattering within �z. The com-plete expression is not reproduced here as many parameter, variable andfunction definitions are required for a satisfactory interpretation; theinterested reader can find the details in Ref. [66].

The computation time increases rapidly with increasing scatteringorder. However, for experiments conducted at fields of view chosenas small as possible to minimize multiple scattering, the number ofnecessary orders can be kept as low as 4–5 and the computation loadremains acceptable up to optical penetration depths of ∼3 encounteredwith real systems in real situations. Therefore, the method has definitepractical merit. It was successfully used by Wandinger [68] to estimatemultiple scattering effects in Raman and high-spectral-resolution lidars.Wandinger also showed in the same work that the method producesresults in excellent agreement with the MUSCLE group Monte Carlosimulations reported in [56].

3.3.5 QSA Approximation—Analytic Solutions

A great deal of theoretical work was carried out in Russia on ana-lytic and semianalytic solutions of the SAA radiative transfer equation.


A representative example of this effort applied to the lidar problem isreported by Zege et al. [69].

A first method begins with expanding in Taylor series, with respectto the angular variable n′

⊥, the radiance I (z, r,n′⊥) in the integral of

Eq. (3.46). Assuming that I is symmetric in n′⊥ and carrying out the

expansion up to the quadratic term, we have

I (z, r,n′⊥) = I (z, r,n⊥) + |n⊥ − n′

⊥|22

∇2n⊥I (z, r,n⊥) + · · · . (3.80)

For validity, the second term in Eq. (3.80) must be considerably smallerthan the leading term. Substituting Eq. (3.80) into (3.46), we find

∂

∂zI (z, r,n⊥) + n⊥ · ∇rI (z, r,n⊥) + [α(z) − αs(z)]I (z, r,n⊥)

= αs(z)β2(z)

2∇2

n⊥I (z, r,n⊥), (3.81)

where

β2(z) = 2π∫ ∞

0|n⊥ − n′

⊥|3p(z; |n⊥ − n′⊥|) d|n⊥ − n′

⊥|,� 2(1 − g), (3.82)

and g is the asymmetry factor defined byg = 2π∫ 1−1 cos θp(z, θ)d cos θ .

To be consistent with the second-order Taylor expansion, the cos |n⊥| thatwas approximated as unity in front of the z derivative in Eq. (3.46) mustbe reinstated in Eq. (3.81). We thus obtain using Eq. (3.82)[

1 − |n⊥|22

]∂

∂zI (z, r,n⊥) + n⊥ · ∇rI (z, r,n⊥) + [α(z) − αs(z)]

× I (z, r,n⊥) = αs(z)(1 − g)∇2n⊥I (z, r,n⊥). (3.83)

The angular scattering properties are all embedded in the asymmetryfactor. Further details on this approximation, called the small-anglediffusion approximation, can be found in Zege et al. [38, 69, 70].

Zege et al. derive in Ref. [69] an analytic solution of Eq. (3.83) byassuming a Gaussian radiance profile of the form

I (z, r,n⊥) = A(z) exp[−|n⊥ − B(z)r|2/C(z)]. (3.84)

Expressions for A,B and C are found by solving the ordinary dif-ferential equations obtained by substitution of Eq. (3.84) for I in


Eq. (3.83). Details and more references can be found in Zege et al.[38] (Sections 4.5.1–4.5.3).

The solution (3.84) was worked out by Zege et al. [69] for aneffective source profile and scattering medium defined by Eqs. (3.71)and (3.61)–(3.63), respectively. Substituting the resulting expressionI e(z, r = 0,n⊥) in Eq. (3.75) and assuming a multicomponent Gaus-sian model for the backscatter function pb, they obtained an analyticexpression for the multiply scattered lidar return. They compared theirpredictions with Monte Carlo calculations. All the features of the MonteCarlo simulations are well reproduced but the analytic solutions under-estimate the multiple scattering contributions by a factor of ∼2. Zegeet al. trace the origin of this bias to the assumed Gaussian profile ofEq. (3.84) that smooths too much the radiance near r = 0, particularlyfor narrow beams and peaked scattering phase functions. To verify this,they looked at the multiply forward scattered radiance from sources ofincreasing divergence. They found that the bias indeed decreases rapidlywith the divergence. The discrepancies are also less for lidar returns atvery large receiver fields of view.

Analytic solutions such as Eq. (3.84) have the obvious advantagesof providing instantaneous numbers and useful asymptotic formulas.However, in view of the important underestimation discussed in thepreceding paragraph, Zege et al. explored the option of computingI e(z, r = 0,n⊥) by inverse Fourier transforming numerically the solu-tions obtained through Eqs. (3.72), (3.71) and (3.60). The difficulty inthis case is to design an efficient multidimensional integration algorithmwhich amounts to choosing the optimal grid in the Fourier (q,p)-space.The task is somewhat facilitated by the analytic solution (3.84) whichprovides information on the relevant scales. In the case of Ref. [69],the choice was further aided by splitting the forward peak of the phasefunction into two components. This semianalytic solution method givesresults in excellent agreement with the Monte Carlo simulations of Ref.[56]. There is a non-negligible computational load but it is less than forthe phenomenological methods reviewed in this chapter. Furthermore,estimations of the contributions by scattering order can be obtained byexpanding in Taylor series the exponential term in Eq. (3.60). Tam andZardecki [71] developed such a solution in analytic form for a sourceof Gaussian-angular and δ-spatial profiles propagating in a uniformparticulate medium having a Gaussian phase function.

The analytic and semianalytic solutions derived by Zege et al. [69]for the one-way propagation of the effective radiance I e or I e coupled


with the general lidar Eq. (3.68) or (3.75) are an important step forwardin modeling multiple scattering in lidar. They not only constitute efficientcomputational tools but the general theoretical framework on which theyrest provides solid rigorous grounds for the advancement of the inverseproblem.

Tam [72] has worked out a two-stream model of the QSA radiativetransfer equation to calculate lidar returns. The forward-stream radiancesatisfies the radiative transfer Eq. (3.46) except for the phase functionthat is truncated to the forward hemisphere only. Tam solves the for-ward problem in Fourier space by expanding the phase function termof Eq. (3.52) as in his earlier work of Ref. [71]. He then simplifies thebackward-stream equation making the usual QSA approximations of apeaked forward phase function and a single backscattering but he stillneeds to solve a three-dimensional partial differential equation with acomplicated source term. In view of the effective medium theorem ofSubsection 3.3.3, this tedious task would no longer be necessary. This isone good example of the benefit that follows from the effective mediumtheorem.

3.3.6 QSA Approximation—A Semiempirical Solution

Two- and four-flux models have been used extensively in radiativetransfer problems. In general, applications have been limited to plane-parallel geometries. Following the work of Tam, Bissonnette [73]proposed a two-flux or two-stream model for narrow beams but with anengineering approach to make computations easier to handle arbitrarymedia. The radiance I is split into an unscattered Iu and a scatteredIs component. Both components are transformed into irradiances byintegrating over the forward and backward hemispheres as follows:

Uu =∫(2π)+

dn⊥Iu(z, r,n⊥) and

U±s =

∫(2π)±

dn⊥Is(z, r,n⊥). (3.85)

For simplicity, we assume an infinitely narrow source beam and we havefor Uu

Uu(z, r) = I0(z, r) exp[−γ (z)], (3.86)

where I0 is the source beam irradiance and γ is the optical depth. TheQSA radiative transfer equation (3.46) is also integrated over the forward


and backward hemispheres. The resulting equations contain the fluxfunctions

F±s =

∫(2π)±

n⊥Is(z, r,n⊥) dn⊥. (3.87)

F±s are the radiation fluxes in directions transverse to the beam axis. To

have a close system of equations, F±s must be related to U±

s . Here, themodel makes the empirical assumption that the lateral fluxes F±

s resultfrom a diffusion process defined by

F±s = −D±(z)∇rU

±s (z, r). (3.88)

The constitutive diffusion relation (3.88) and coefficients D± are notformally derived from the radiative transfer equation. Instead, by analogywith turbulent transport processes, it is postulated that D± are propor-tional to the product of the mean free path between the scattering eventsthat give rise to the lateral flux or transport of U±

s times the strength ofthis random “microscopic motion.” Empirical expressions for D± arederived from this postulate in Ref. [73]. With the help of the definingrelation (3.88), a closed set of two differential equations is obtained forU±

s . Because of the effective medium theorem of Subsection 3.3.3, itnow suffices in QSA lidar applications to solve for forward propagationalone. We therefore reproduce here only the forward flux equation,

∂

∂zU+

s − D+∇2r U

+s + (α − α+

s )U+s = α+

s I0(z, r) exp[−γ (z)], (3.89)

in which we have dropped the α−s U

−s term in accordance with

the QSA assumption of a single backscattering and where α±s =

αs

∫(2π)± dn⊥p(z,n⊥). Equation (3.89) has a general solution that

involves a twofold nested integration over z. For use in the lidar equa-tion (3.68) or (3.70), we need to reinstate the angular dependence thatwas integrated in the definition of U+

s . This is done in Ref. [73] in anad hoc fashion. Comparisons with laboratory measurements [74, 75] forboth transmission and backscattering show good agreement. On the otherhand, the MUSCLE intercomparisons of calculated multiply scatteredlidar returns [56] indicate that the simultaneous solutions of Eq. (3.89)and its return-stream counterpart give a resulting lidar signal that falls atthe limit of the data spread.

The model just outlined incorporates a good deal of empiricism andlacks the mathematical rigor of the preceding solution methods. How-ever, it has the main advantage of making the calculations affordable for


arbitrary media stratified perpendicularly to the lidar axis. The simplifi-cation to forward propagation only resulting from Eq. (3.70) improvesthe accuracy of the lidar solution and further reduces the requiredcomputational effort by more than one half. Short computation timesare important for retrieval applications where the search for an optimalsolution often involves iterations or repeated calculations.

3.3.7 Diffusion Limit

The diffusion limit in multiple scattering lidar has not been very muchexplored in the past because it essentially means that the photon originis lost and with it the intrinsic ranging property of traditional lidars.However, it has been suggested in recent studies [76, 77] that the off-axis returns can provide key bulk properties of dense diffusing clouds.Time and space simply take on different meanings that can be exploitedwith the diffusion theory.

We follow here the work of Davis et al. [77]. Since the off-axismeasurements of interest have to do with a cloud region large comparedwith the dimension of the source beam, the latter is modeled by deltafunctions of time, space and angular spread. We therefore seek a solutionof the homogeneous nonstationary radiative transfer equation (3.19) asa response to an impulse, which defines the Green’s function. The initialand boundary conditions are

G(t, z, r, n; 0, 0, 0, k)

=

⎧⎪⎨⎪⎩δ(t)δ(r)δ(n − k) for t > 0; z = 0; k · n ≥ 0,

0 for t > 0; z = �z; k · n ≤ 0,

0 for t = 0; r ∈ �2; 0 ≤ z ≤ �z,

(3.90)

where z = 0 is the base of the cloud and z = �z is the cloud thickness.In the following, we will drop the source coordinates in the Green’sfunction notation since they are constant throughout.

What we want to do here is derive, in the diffusion limit, a solutionfor G that satisfies the initial and boundary conditions (3.90). Accordingto Ishimaru [37] (Section 9.1), one main characteristic of the diffusionlimit is that G can be expanded as follows:

G(t, z, r, n) � 1

4π[J (t, z, r) + 3n · F(t, z, r)], (3.91)


where J and F are the radiant intensity and flux vector, respectively,

J (t, z, r) =∫

G(t, z, r, n) dn, (3.92)

F(t, z, r) =∫

nG(t, z, r, n) dn. (3.93)

Equation (3.91) is the expansion of G(t, z, r, n) about the condition ofuniform angular distribution given by G = J/4π , and hence the secondterm must be sufficiently smaller than the first, i.e., |F| J . From hereon, we will assume that this condition is satisfied and use the equal sign inEq. (3.91). The constitutive diffusion hypothesis assumed by Davis et al.[77] is

F = −1

3�t (z, r)∇RJ (t, z, r), (3.94)

where �t is the photon transport mean free path given by

�t = [α − αsg]−1, (3.95)

and g is the asymmetry factor. Substituting Eq. (3.94) for F in Eq. (3.91),we have

G(t, z, r, n) = 1

4π[1 − �t (z, r)n · ∇R]J (t, z, r). (3.96)

Therefore, the diffusion model of radiative transfer amounts to solvingfor the radiant intensityJ instead of the radiance. Integrating the radiativetransfer equation (3.19) over n, making the source termQ = 0, and usingthe constitutive relation (3.94), we obtain the following equation for J :

∂

∂tJ − ∇R · D∇RJ + c(α − αs)J = 0, (3.97)

where D is called the diffusion coefficient. It is given by

D = c�t

3= c

3(α − αsg). (3.98)

Equation (3.97) is the diffusion equation. In our search for asolution, we will assume in the following that the medium is homo-geneous, in other words, that D, α, αs and g are constants. As itis customary in mathematical physics, we transform the linear partial


differential equation (3.97) into an ordinary differential equation by useof a Fourier–Laplace transform, i.e.,

J (s, z,q) =∫∫∫

exp[−st + iq · r]J (t, z, r) dt dr. (3.99)

Equation (3.97) thus becomes the ordinary differential equation

d2

dz2J − J /L2 = 0 with L−2 = q2 + s/D + c(α − αs)/D.

(3.100)

The Fourier–Laplace transformation of the initial and boundary con-ditions (3.90) gives with the help of Eq. (3.96) and the assumption ofaxisymmetry with respect to the z axis the following mixed boundaryconditions for J (s, z,q):

1

2

[1 − �t

2

d

dz

]J (s, z,q) = 1 at z = 0,

1

2

[1 + �t

2

d

dz

]J (s, z,q) = 0 at z = �z.

(3.101)

The two-point boundary value problem of Eqs. (3.100) and (3.101) iseasily solved. However, the final expression for J (s, z,q) is quiteinvolved and the inverse transformation back into physical space is notpossible. Fortunately, the main observables can be expressed in terms ofJ (s, z,q).

In off-axis lidar applications, the accessible quantity is the reflectedflux at 180◦ from the base of the cloud defined as follows by Davis et al.[77]:

GR(t, r) =∫(2π)−

|k · n|G(t, 0, r, n) dn. (3.102)

Using Eq. (3.96) in (3.102), we obtain the following equation relatingGR to J :

GR(t, r) = 1

4

[1 + 2

3�t

∂

∂z

]J (t, z, r) at z = 0. (3.103)

Applying the Fourier–Laplace transformation to Eq. (3.103), we find

GR(s,q) = 1

4

[1 + 2

3�t

d

dz

]J (s, z,q) at z = 0. (3.104)


The boundary conditions (3.101) and the defining relations (3.103)and (3.104) are mathematically exact but they assume that the diffu-sion constitutive relations (3.91) and (3.94) remain applicable down toand including the transition boundary at z = 0. Obviously, the true radi-ance at z = 0 is not diffuse for all angles n. Since Eqs. (3.101), (3.103)and (3.104) contain the radiant intensity evaluated at z = 0, they cannotbe expected to model with full accuracy the true reflected flux. Con-sequently, researchers [2, 77] have chosen slightly different boundaryconditions than Eq. (3.101) because the collimated model of Eq. (3.90)is not compatible with the diffusion limit, and by extension they havealso modified the numerical constants of Eqs. (3.103) and (3.104). Theproposed alternate boundary conditions and relation between GR and J

are [77]

1

2

[1 − χ�t

d

dz

]J (s, z,q) = 1 at z = 0,

1

2

[1 + χ�t

d

dz

]J (s, z,q) = 0 at z = �z,

(3.105)

GR(s,q) = 1

2

[1 + χ�t

d

dz

]J (s, z,q) at z = 0, (3.106)

whereχ is an adjustable or free numerical factor ofO(1) to be determineda posteriori, for example by comparisons with Monte Carlo simulations.

From the definition of the Fourier–Laplace transform, Eq. (3.99), wehave the following relations between GR and a few observables:

GR(0, 0) =∫∫∫

GR(t, r) dt dr = R, (3.107)∣∣∣∣ ∂u

∂suGR(s, 0)

∣∣∣∣s=0

=∫∫∫

tuGR(t, r) dt dr = 〈tu〉R, (3.108)

∣∣∣∇q · ∇qGR(0,q)∣∣∣q=0

=∫∫∫

r2GR(t, r) dt dr = 〈r2〉R, (3.109)

where R is the space-time average cloud reflection, c〈t〉 is the mean pho-ton pathlength, c2〈t2〉 is the mean-square photon pathlength, and 〈r2〉 isthe mean-square horizontal transport length. Therefore, the measurablequantities R, 〈t〉, 〈t2〉 and 〈r2〉 can be written through Eqs. (3.106)–(3.109) in terms of the Fourier–Laplace solution J (s, z,q) of the diffuseintensity. The latter can be easily obtained in analytic form for a homo-geneous medium of sufficient density to justify the diffusion limit


approximation by solving the ordinary differential equation (3.100) withthe boundary conditions (3.105).

Davis et al. [77] and Love et al. [78] give the following asymptoticexpressions for the observables derived under αs = α or an albedoω = αs/α of unity:

R = �z

2χ�t + �z

= (1 − g)γ

2χ + (1 − g)γ, (3.110)

c〈t〉 = 2χ�z + corr. term, (3.111)

c2〈t2〉 = 4χ

5�2

z(1 − g)γ + corr. term, (3.112)

〈r2〉 = 8χ

3

�2z

(1 − g)γ+ corr. term, (3.113)

where γ = α�z is the single scattering cloud optical depth, and “corr.term” stands for a term of lesser magnitude than the shown leading term.The leading terms (without precise numerical constants) in Eqs. (3.110)–(3.113) can be derived from heuristic scaling arguments [76]. Althoughthey vanish for large γ , the correction terms are not negligible atcommonly observed values of γ , say, 5–50.

From any two of the four relations (3.110)–(3.113), the cloud bulkparameters �z and γ can in principle be retrieved from the reflectedoff-axis halo surrounding the laser beam measured with space-time reso-lution. These two cloud parameters are the most variable by far; for liquidstratiform clouds, one can confidently set g ≈ 0.85 [79] and χ ≈ 2/3[2]. Off-beam lidar inversions are briefly discussed in Subsection 3.5.3.

3.3.8 Summary

The direct problem of calculating multiply scattered lidar returns is nowwell understood and has been satisfactorily solved by a variety of tech-niques. The Monte Carlo methods have reached a high degree of maturity.They have the definite advantage of minimizing the number of neces-sary simplifying approximations and, for this reason, they provide easyvirtual experiments for testing other models and retrieval algorithms.Furthermore, except for the demand on computational resources, thereis no conceptual difficulty to set up Monte Carlo simulations for prob-lems with complex instrumentation specifications, special measurementgeometry and highly structured scattering properties. One disadvantageof Monte Carlo simulations is that the problems are treated one at a


time; solution trends can only be studied through large numbers of runs.The analytic and semianalytic models provide useful assistance in suchapplications. Despite the simplifying assumptions and the limited preci-sion and range, analytic expressions can readily exhibit many aspects of aproblem. In particular, the effective medium theorem of Subsection 3.3.3is a noteworthy achievement. It casts in rigorous mathematical terms asituation that was hypothesized in some form or other in many previousmodels.

The models reviewed in this section are representative of the stateof the art. They were built on several other contributions that are not allreferenced here, especially the early work in the former Soviet Union.The reader should consult Refs. [38, 64, 69, 70] for a more completebibliography on the latter.

3.4 Accounting for Multiple Scattering

The first obvious application of the multiple scattering models is estimat-ing errors on the parameters retrieved with single scattering algorithmsand working out corrections. As already mentioned in Section 3.2,Platt [23] introduced the parameter η, Eq. (3.10), to take into accountthe reduction in the extinction coefficient caused by multiple forwardscatterings. He used η to define the effective extinction and backscatter-to-extinction ratio, αe = ηα and ke = β/αe = k/η, respectively, andindicated [80] that the outputs of single scattering retrievals are actuallythe effective αe and ke instead of the true—meaning single scattering—extinction coefficient α and backscatter-to-extinction ratio k. If η were aconstant independent of range, it would be a simple matter to calculateits value for whole classes of problems and the corrections would bestraightforward. As it turns out, however, η is varying considerably withrange in a manner dependent on receiver field of view, range to cloud,wavelength, and angular scattering properties. Monte Carlo simulationsperformed by Platt [51] show that η increases very rapidly with pene-tration depth from a low value at the base of the cloud to a more or lessconstant value at optical depths greater than 2–3. The total increase istypically on the order of a factor of 2. In some instances of cirrus clouds,negative values of η, as low as −0.6, were found at cloud base. Thisillustrates once more that the interpretation of sole extinction reduc-tion implied by the η-formulation is not correct in general since thebackscattering coefficient is also affected. In summary, the correction


algorithms suggested by η can be misleading; η is not a simple factor buta function that varies by a significant amount not only from one appli-cation to another but within a same application. Nevertheless, for thespecific application of spaceborne cirrus measurements, Wiegner et al.[81] showed that a constant η can provide meaningful correction.

Wandinger [68] made a thorough study of the influence of multiplescattering on Raman and high-spectral-resolution lidar retrieval algo-rithms. Simulations carried out with the phenomenological model ofWeinman [66] showed that errors on the retrieved particle extinctioncoefficient can be as large as 50% at the base of a cloud, even for fieldsof view as low as 0.4 mrad (full angle). Generally, the errors drop below20% with further penetration depth. Backscattering coefficients are lessaffected with errors much below 20%. Wandinger went on to demonstratethat meaningful corrections of real measurements can be implementedby running direct-problem calculations of multiple scattering contribu-tions. To do this, one needs the size distribution, or the angular scatteringfunction, and the true extinction profile. The size distribution must beassumed, or at least bracketed within realistic limits, because it can-not be inferred from the conventional retrieval tools of the Raman andhigh-spectral-resolution methods. However, a corrected extinction pro-file, constrained by the assumed size distribution, can be derived throughiterations. The starting solution is the uncorrected extinction αe which isinputed along with the assumed size distribution in the chosen multiplescattering model to calculate the corresponding function η. A correctedα = αe/η is thus obtained and the direct-problem computations are rerunto calculate a new η. The reconstructed αe from the last solutions for αand η is then compared with the measured αe. If the differences aregreater than preset limits, the α profile is varied, according to a suitablesearch algorithm, at the input of the direct-problem model until properagreement is reached between the reconstructed and measured αe’s.

As a follow-on, Reichardt et al. [82] carried out several computationsbased on the same model ofWeinman in search of scaling laws that wouldallow estimating η from a standardized cloud. Their results show thatthis is possible, approximately, for such parameters as the transmitterwavelength and the receiver field of view in conditions of near homo-geneous clouds. However, simple scaling laws do not seem to work ingeneral.

In summary, corrections or adaptations of single scattering retrievalalgorithms to take into account multiple scattering effects are not straight-forward. As we have seen, there is a fair number of valid calculation


models of multiple scattering but the main inputs to drive these modelsare actually the medium properties we wish to correct for. Iterations toderive true values from “effective" values are possible but there is almostalways a missing input not available from the retrieval algorithms understudy, e.g., the phase function. However, because of the wide range ofplanned lidar-in-space applications in which multiple scattering cannotbe neglected, work is continuing at a steady pace to devise practicalretrieval algorithms to account for or even exploit multiple scattering,e.g., Winker [83].

3.5 Inverse Problem

Multiple scattering has mostly been considered in the past as a hindrancein single scattering retrieval methods. However, the modeling resultsof Section 3.3 show that the multiple scattering contributions containinformation on particle size that is not available in the single scatteringexpressions. We review in this section some of the recent efforts madeto solve the inverse problem, i.e., to retrieve at least part of this addi-tional information. There are a few promising results but a good deal oftheoretical development remains to be done.

3.5.1 Particle Size Distribution

One main characteristic of multiply scattered lidar signals is their depen-dence on the receiver field of view. This arises because the range ofscattering angles contributing to the collected signal widens with thefield of view. It is well known that the angular shape of the scatteringphase function depends on particle size. At small angles, the scattering ismainly caused by Fraunhofer diffraction which gives a one-to-one rela-tionship with the size of the scatterer. As discussed in Section 3.3,multiple scattering in the conventional lidar geometry of narrow sourcedivergence and receiver field of view is characterized by multiple small-angle forward scatterings. Hence, angularly resolved lidar returns arehighly dependent on the phase function forward peak and can in principlebe exploited for particle size retrieval.

We outline here one generic approach to solve for the particle sizedistribution. To keep the inverse problem linear, we assume double scat-tering only. Hence, the solution we seek will be applicable only tosmall optical depths which, for clouds of reasonable density, also means


small geometrical depths. We thus make the further assumption that thenormalized diameter density distribution,

f (ρ) = 1

N0(z)

d

dρN(z, ρ), (3.114)

is independent of the range z, where N(z, ρ) is the number of particlesof diameter ρ per unit volume and N0(z) is the total number density forall sizes.

Let the returned signal, denoted P(z,�j), be measured at Mdifferent fields of view, i.e., for j = 1,M. We model P(z,�j) inthe framework of the effective medium theorem of Subsection 3.3.3.Therefore, we have for second-order scatter

P(z,�j) = K

z2exp

[−2∫ z

zb

α(z′) dz′] ∫ z

zb

dz′2αs(z′)

×∫ φj (z

′)

02π sin φp(φ)αsbpb(z, π − z′φ/z) dφ, (3.115)

where K is the instrument constant, zb is the range to cloud base, p(φ)is the phase function for forward scattering assumed independent of z,αsbpb(z, π − z′φ/z) is the angular backscattering function responsiblefor the single backscattering event, and

tan φ = z

z − z′ tan �. (3.116)

We take αsbpb different from αsp to include Raman lidar, high-spectral-resolution lidar, reflection from ground or sea surface, etc.

Because our objective is to solve for the particle size distribu-tion, we express p(φ) as a sum over the individual diameters of thedistribution, i.e.,

p(φ) = 1

4

∫ ρmax

ρmin

dρf (ρ)πρ2

〈σs〉Qs(λ, ρ,m)P(φ, λ, ρ,m), (3.117)

where Qs(λ, ρ,m) and P(φ, λ, ρ,m) are, respectively, the scatteringefficiency and phase function at the wavelength λ for a particle of diam-eter ρ and refractive index m, and 〈σs〉 is the average particle scatteringcross-section given by

〈σs〉 = 1

4

∫ ρmax

ρmin

dρf (ρ)πρ2Qs(λ, ρ,m). (3.118)


We do not write pb(z, π − z′φ/z) in terms of f (ρ) because it does notnecessarily arise from the same particle distribution as indicated above.Actually, we consider pb(z, π − z′φ/z) as a known function. In caseswhere pb = p, we assume as a first approximation that pb � constantnear 180◦. For aerosol particles large compared with the wavelength forwhich this method is applicable, the magnitude of the phase functionvariations is much less in the near backward direction than in the nearforward direction. The smaller backward variations can subsequently betaken into account by iterations in which pb(z, π − z′φ/z) is evaluatedfrom the f (ρ) determined on the preceding iteration cycle.

We approximate the integral over ρ resulting from Eq. (3.117) as adiscrete sum of M terms as follows:

∫ ρmax

ρmin

dρf (ρ) · · · =M∑i=1

∫ ρi+1

ρi

dρf (ρ) · · · =M∑i=1

fi

∫ ρi+1

ρi

dρ · · · ,(3.119)

where fi is the average f over the ith interval. Within the intervals, wekeep the integration over ρ because the functions Qs and P can oscillaterapidly with ρ. Using Eqs. (3.117)–(3.119), we rewrite (3.115) as a linearequation

Pj = fiAij , (3.120)

where

Pj = P(z,�j+1) − P(z,�j), (3.121)

Aij = C(z)

∫ ρi+1

ρi

dρ ρ2∫ z

zb

dz′αs(z′)∫ φj+1(z

′)

φj (z′)

sin φ

× Qs(λ, ρ,m)P(φ, λ, ρ,m)pb(z, π − z′φ/z) dφ, (3.122)

C(z) = π2

〈σs〉K

z2αsb(z) exp

[−2∫ z

zb

α(z′) dz′]. (3.123)

Equation (3.120) is a linear equation that can be inverted for fi withwell-known constrained or regularized techniques [84] once the matrixelements of Aij are calculated. The αs(z

′) that enters the definition of Aij

can be estimated from an assumed value of the single scattering albedoand a backward Klett’s [6] solution which becomes independent of thefar end boundary value at the small penetration depths for which the


second-order expression (3.115) was derived. The elements of Aij areonly known to a multiplicative constant because of the unknown 〈σs〉,Kand αsb. Therefore, fi can be solved only in relative units. However,because of the Klett’s solution for α, the particle number density can beestimated a posteriori.

The matrix elements Aij can be obtained either with the exactMie-calculated Qs(λ, ρi,m) and P(φ, λ, ρi,m) in the case of spher-ical particles, or with the Fraunhofer expression for size parameters2πρi/λ � 1 since the significant contributions come from small-anglescatterings. Because of its simple analytic form, the Fraunhofer formulais useful in either case to select the ρ and � bin sizes that make thediagonal elements Aii of comparable magnitude.

Illustrative measurements and inversion results obtained by Royet al. [34] in laboratory-controlled water droplet clouds are plotted inFig. 3.8. The lidar returns were measured in the polarization direc-tion perpendicular to the source polarization. This had the advantage ofmaking pb(z, π) � 0 and, thus, eliminating the large single-scatteringcontribution. Results obtained at two optical depths, γ = 0.2 and 0.4,respectively, are shown. The agreement is good in the case of the smaller

Pperp

(z, Q

) (a

.u.)

Fig. 3.8. Lidar MFOV measurements (left) and corresponding retrieved droplet volumedensity distributions ρ3f (ρ) compared with in situ particle-sizer data (right). Pperp

is the collected return in the polarization direction perpendicular to that of the linearlypolarized laser source. The scattering medium is a water droplet cloud produced in a22-m long × 2.4-m × 2.4-m wide chamber located 100 m from the lidar. Solid circles:penetration depth of 6 m and γ = 0.2; solid squares: penetration depth of 10 m andγ = 0.4; and open circles: in situ measurements.


optical depth but only fair at γ = 0.4. This is consistent with the valid-ity domain of the second-order approximation made in the derivationof the linear equation (3.120). In both cases, the lidar-retrieved dis-tribution cuts off at smaller diameters than the in situ measurements.The probable cause is the limited field-of-view range of the instrumentfor the given measurement geometry. For example, large droplets atfar distances require very narrow fields of view whereas small parti-cles at short distances require wide fields of view as it can be easilyinferred from Eq. (3.116). For the results of Fig. 3.8, the smallest fieldsof view are still too wide for resolving the narrow angular spreadingdue the large droplets, hence the rapid cutoff of the retrieved distribu-tions at the large-diameter end. Following the same geometrical-factorargument, Roy et al. [85] have suggested that a spaceborne MFOV lidarreceiver of reasonable angular aperture could be used, because of thelong ranges, to characterize the size of submicrometer aerosol particlesin the atmospheric boundary layer.

An earlier application of a second-order scattering size retrievalmethod was published by Benayahu et al. [86]. They used two receiversin a bistatic configuration. One receiver had its axis shifted from the beamaxis so as not to collect singly scattered radiation. The idea is interestingbut the chosen instrument parameters and the measurement geometrymade the smallest scattering angle φ contributing to the multiple scat-tering channel of the order of 10◦. This is too large to collect enoughof the diffraction scatterings from typical cloud droplets. As a result,the sensitivity on droplet size is weak. A sample set of inversion resultsobtained for marine stratus clouds yielded an effective droplet diameter〈ρ3〉/〈ρ2〉 � 6.6 μm, which is 3–4 times smaller than most publishedmeasurement values for marine clouds.

Figure 3.8 shows that size information is indeed retrievable fromexperimental multiply scattered lidar data but the existing retrieval meth-ods are still limited in their application range. The results obtained todate are from early initiatives and better performances are expected withfuture developments. There is currently ongoing research to derive morerobust methods for application to cirrus clouds, e.g., Eloranta [87].

3.5.2 Extinction and Effective Particle Size

It is now well established that the multiply scattered lidar returns arefunctions of the medium extinction coefficient and angular scatteringproperties or particle size distribution. However, there are no simple


mathematical relations between these local properties and the observ-ables for multiple scattering of order greater than 2. Actually, theobservables are the results of many integrated interactions as illus-trated by the multiple integrals of the phenomenological models ofSubsection 3.3.2. This makes the inverse problem of retrieving the localextinction coefficient and particle size particularly difficult.

It has been proposed to use the powerful direct-problem tools tocarry out multi-dimensional searches of the medium parameters thatwould reproduce the measured signals. For example, Oppel et al. [88]have performed retrieval simulations for a medium of uniform extinctionand size distribution. The assumed measured signals were the paralleland perpendicularly polarized returns. The chosen calculation algorithmwas a variance-reduction Monte Carlo model. The search grid for thatexample was two-dimensional and made up of the extinction coeffi-cient and the modal radius of the size distribution; the distribution wasassumed to be of the Deirmendjian’s C1 type. The convergence crite-rion was the minimization of the root-mean-square distance between thecalculated and the “measured” signals. Oppel et al. [88] showed that anadaptive step size random search was significantly more efficient thana uniform search. They also found that the sensitivity of the minimiza-tion search is good for the extinction coefficient but low for the modalradius. The search-based inversion is conceptually simple and adaptableto different kinds of retrieval problems. However, it requires extensivecomputational resources and its practicality becomes questionable fornon-uniform media.

The concept of using direct-problem calculation methods in theinverse problem is also applicable to iterative solution algorithms. Tofix ideas, let the multiple scattering lidar equation be written

P(z,�) = Pss(z)M(z,�) = Pss(z)[1 + Fd(z,�) + Fg(z,�)],(3.124)

where � is the half-angle receiver field of view, Pss is the conventionalsingle scattering lidar expression given by Eq. (3.1), and M(z,�) isthe multiple-to-single scattering ratio split into components Fd and Fg,respectively, for diffraction scatterings alone and for all other scatteringsthat involve at least one geometrical optics scattering. Geometrical opticsmeans refraction and reflection. The QSA approximation of multiplesmall-angle forward scatterings and a single large-angle backscatter-ing is implied in Eq. (3.124). The separation of M into Fd and Fg hasthe advantage of consolidating the dependence on particle size almostexclusively in Fd .


The iteration principle is rather simple. If an intermediate solution forthe extinction coefficient and the particle size is known, the multiple-to-single scattering ratio M can be calculated by direct-problem methodsto correct the measured functions used in the single scattering inversionalgorithms, thus allowing the calculation of a refined solution. For exam-ple, the Klett algorithm, Eq. (3.7), could be rerun after redefining thesignal function S(z) as follows:

S(z) = P(z,�)

M(z,�)

z2

k(z), (3.125)

where P(z,�) is the measured multiply scattered lidar return, andM(z,�) and k(z) = β(z)/α(z) are the multiple-to-single scattering andthe backscatter-to-extinction ratios calculated with the intermediate solu-tions for extinction and particle size. It is assumed in Eq. (3.125) thatthe instrument function K is independent of z. Needed to implementsuch an iteration method are, in addition to a direct-problem calculationmodel for M(z,�), an algorithm for particle size and an initializationprocedure.

Veretennikov et al. [89] use the iteration concept described aboveto retrieve cloud extinction profiles from actual multiple-field-of-view(MFOV) lidar measurements. They avoid size retrieval by fixing theparticle effective radius. They initialize their extinction solution eitheron a reference value obtained with the slope method in cases wherean interval of constant extinction can be identified, or by the use of aregularization constant in lieu of a boundary value in the denominatorof the Klett expression (3.7). They compare retrievals obtained for dif-ferent fields of view and show that the iteration is generally stable. Theretrieved extinction profiles are in good agreement with what is expectedfrom continental stratus clouds but no independent measurements wereavailable.

There is no rigorous, general method of determining simultaneouslythe extinction coefficient and the effective or average particle diameter inoptically dense media. The difficulty resides with the size retrieval. Theparticle size dependence of multiply scattered lidar returns is a complexintegrated effect and we are missing a robust analytic inverse relationbetween the measured returns and the local particle size distribution.One preliminary semiempirical iterative technique has been describedby Bissonnette et al. [90].

The size retrieval algorithm proposed by Bissonnette et al. consistsin measuring the field-of-view spread of the multiply scattered returns,


quantifying this spread by a characteristic scale, and relating this scaleto the width of the forward peak of the scattering phase function or theeffective particle diameter, by means of a semiempirical model. Theexistence of a relation between the field-of-view scale and the particlesize rests on the now well-established premise that, for a receiver foot-print less than the scattering mean free path and for optical depth lessthan 3–4, the field-of-view dependence of P(z,�) is mostly driven bythe small-angle forward scatterings. The implementation of the methodrequires a direct-problem calculation model of the multiple scatteringfunction M(z,�) in Eq. (3.125) to run the iterations and an initializationprocedure.

The proposed initialization procedure is based on the approximationsthat at the onset of multiple scattering the receiver largest field of view�max is sufficiently wide to encompass all diffraction forward scatteringsand yet narrow enough for P(z,�) not to be significantly affected by thegeometrical optics scatterings, and that the smallest field of view �min

is small enough for P(z,�min) to approach the single-scattering returnPss(z). Under these conditions, it is easy to show, for instance usingEloranta’s analytical model [61], that

γ (z) � ln

[1 + 1

δb

P (z,�max) − P(z,�min)

P (z,�min)

], (3.126)

where δb is the ratio of the backscatter coefficient averaged over �max

to the single-scattering coefficient. δb arises because the backscatteringfunction pb is not necessarily uniform near 180◦. For Raman and high-spectral-resolution lidars, δb is conveniently equal to unity. For elasticbackscattering in low-level water clouds probed from the ground, wehave δb � 0.7. Monte Carlo simulations show that the approximation(3.126) is satisfied to within 20% in the interval

1.2 ≤ P(z,�max)/P (z,�min) ≤ 1.5. (3.127)

Therefore, the optical depth in the range defined by (3.127) can bedetermined from the relative strengths of the returns measured at �min

and �max , independently of a calibration constant. This γ (z) was suc-cessfully used in Ref. [90] to initialize the iterations for the simultaneousretrieval of the extinction coefficient and the effective particle diameter.A Klett solution applied to the modified signal S(z) of Eq. (3.125) wasused for the extinction coefficient and a semiempirical model for the par-ticle diameter. The direct-problem model needed to calculateM(z, θ) can


be any model that accepts the extinction and particle diameter profilesobtained from the previous-iteration solutions as inputs.

Tests on Monte-Carlo-simulated returns reported in Ref. [90] showthat the retrieved extinction coefficients are within 3–4% of the truevalues. The performance is not as good for the particle diameter. Whilethe solutions agree very well with the true effective diameters on average,the fluctuations are of the order of ±25%. In addition, there are limitson the retrievable sizes. This does not result from flaws in the solutionmethod but rather from the limited angular resolution of the MFOVreceiver: the field-of-view scale cannot be determined if it falls outsideof the interval [θmin, θmax].

Comparisons of lidar retrievals with actual field data are also shownin Ref. [90]. The experiment consisted of vertical lidar probings from afixed ground site and simultaneous in situ aircraft measurements in lowstratus clouds. Because of the variable and large separation (5–40 km)between the lidar and the aircraft, the comparisons were performed onlong time averages only. The results show good correlation between thelidar solutions and the aircraft measurements of the cloud liquid watercontent (LWC) and effective droplet diameter. However, there is a bias—lidar underestimation—varying between 15% and 30% depending onthe aircraft sensor. The analysis shows that the discrepancy cannot beattributed to the lidar alone.

Although there is still much analytical development to be made,the results of Ref. [90] demonstrate that multiple-scattering-based lidarretrieval can work. One advantage of multiple scattering retrieval isthe information gained on particle size which allows extrapolation tosecondary products like cloud liquid water content (LWC), extinctionat other wavelengths, cloud radiative properties, etc. The applicationsare limited to measurement geometries and medium properties thatsatisfy the QSA approximation. Optical depths are limited to 3–4,which fortunately corresponds to practical hardware limits, and thebounds on particle size are determined by the measurement geometryand the transceiver technical specifications. Developments in gate-able intensified imaging receivers could broaden the scope of MFOVdetection.

3.5.3 Bulk Properties

The diffusion limit of multiple scattering offers real opportunitiesof extending the usefulness of lidars. We have seen that the QSA


approximation, which constitutes the basis of the inversion methodsdescribed so far in this section, is no longer valid when the lidar footprintexceeds the scattering mean free path. A practical situation is that of aspaceborne lidar probing water clouds in the atmospheric boundary layer.However, there are still measurable reflected photons in such instancesand they return information on cloud properties. For dense clouds, thediffusion results of Eqs. (3.110)–(3.113) show that the cloud physicaland optical thicknesses and their average scattering properties, definedby the asymmetry factor, are in principle retrievable from space-timemeasurements of the reflected aureole.

Measuring the off-beam aureole poses some technical challenges.Unusually wide fields of view are required which complicates the rejec-tion of background radiation in favor of the faint diffusion-reflectedsignals. Two experimental systems have been built and tested so far:a high-speed gated/intensified imaging camera with a 60◦ full-anglefield of view for ground-based use [78], and a fiber bundle 8-ring MFOVreceiver with a 6◦ angular aperture for airborne use [91]. In the first case,Love et al. [78] obtained time resolved images from a stratus cloud deckat 1 km above ground, clearly showing a “wave” of diffusing light prop-agating radially from the central impact point of the laser source pulse;these authors used pairs of modified versions of Eqs. (3.111)–(3.113) toestimate the geometrical and optical thicknesses of the cloud deck. Inthe second case, Cahalan et al. [91] carried out preliminary measure-ments with the MFOV airborne lidar that reveal time delays between theouter- and central-ring signals from a thick stratus layer 7 km away; theirresults are compatible with the diffusion model of Eqs. (3.110)–(3.113).Finally, it is noteworthy that the LITE data in Fig. 3.5 have been usedin Eqs. (3.111) and (3.112) to infer with reasonable accuracy the opti-cal depth and physical thickness of the marine stratocumulus deck fromwhich the LITE returns originated [92].

Considering the growing interest [93] in space-based remote sens-ing of clouds, rapid developments are expected. Diffusion retrievalmay become in the future a significant contributor in the remotecharacterization of thick water clouds.

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