Cloud-base water content measurement using singlewavelength laser-radar data

Ariel Cohen

Monochromatic backscattering laser-radar data are used for the determination on the number density ofcloud droplets within a cumulus cloud base. The method is based upon general properties of a cloud base

as derived from in situ measurements in a large variety of continental cumulus clouds. The backscatterlaser profile from the cloud base is analyzed, and points with equal optical-depth values are detected in

each profile. The method of detection requires no knowledge of the multiple scattering contribution, eventhough its effect is not neglected and is allowed to vary as a function of the optical depth. Corrections for

general clouds are suggested based upon lidar measurements and analysis of the backscattering profilecharacteristics.

1. Introduction

A remote determination of the number density ofclouds and aerosols in essentially real time over largeregions of the sky is important for progress in cloudphysics. Measurements in situ, by use of collectingapparatus transported in airplanes, provide the gen-eral properties of the average variation of the numberdensity in clouds and the main features of the drop-lets' size distribution. A remote-sensing technique,such as laser-radar scattering,2-5 has the advantage ofmeasuring each cloud at a very low cost and of pro-viding data that can be immediately interpreted. Onthe other hand, laser scattering irradiance is a func-tion of several atmospheric parameters, such as thenature of the particles (refractive index), sizes, andnumber density. Since from each altitude the laserradar provides merely one piece of information, itwas suggested that several scattering angles6 -7 orwavelengths- 9 be used in order to obtain a series ofindependent data from which the distribution func-tion could be inverted. The inversion procedures areusually based upon the assumption that the directfunctions (for single scattering) are known, and henceare valid only when multiple scattering is neglected.In this work we present an analysis that makes use ofbackscatter profiles from a single wavelength andthat does not neglect the multiple scattering effect.The assumptions are chosen to match the conclusionsof the in situ measurements mentioned above,

suggesting, for example, that the size distributionfunction is fairly invariant from one continental cu-mulus cloud base to the next.

A lidar echo from a dense cloud is shown (Figs. 2and 3) in order to suggest a correction to the present-ed theoretical method. This correction is due to theappearance of a thin layer in the cloud-base for whichthe assumption of constant number density vs heightdoes not hold.

The absolute calibration of the method requiresone simultaneous measurement by both the lidar sys-tem and a collecting apparatus. The results lead toabsolute values of the number density using the laserscattering information for any other cloud.

The use of the method for the analysis of aerosollayers other than hydrometers is also discussed.

II. Method

The analysis of laser-radar backscattering informa-tion from clouds involves the following unknowns:

(1) the profile of the relative size distributionfunction f(r,z) of the cloud doplets, where r is thewater droplet radius and z the height within thecloud;

(2) the profile of the number density of the parti-cles Nz);

(3) the optical depth within the cloud

TZ0, Z = f flf(Z)N(z)dz,

The author is with the Hebrew University of Jerusalem, Depart-ment of Atmospheric Sciences, Israel.

Received 10 July 1974.

where af(z) is the extinction cross section calculatedfor the distribution function f(rz) and is normalizedby N(z) = fSof*(rz)dr:

December 1975 / Vol. 14, No. 12 / APPLIED OPTICS 2873

II

10

9

_ a

1-0

n 7m00j 6C)

w0

I0w

3

2

LIQUID WATER CONTENT LWC,gm 3)0.5 1.0 1.5 2.0 2.5

Timefor i

Collisionhrs.)

E-

U

Cr

zw

J

Z

0l`)

-ja-0

0U

0 250 500 750 1000 1250CLOUD DROPLET CONCENTRATION (ncm-3)

Fig. 1. Cloud droplet concentration and liquid water content in a cumulus cloud (after Ref. 1). Right: the droplet size distribution. Allreported data are from measurements in the cloud.

us(z) = f a(r,x)f*(r,z)dr/ f*(rz)dr, (2a)

where o is the laser wavelength and f* is the abso-lute distribution function [it should be emphasizedthat the parameter T,,,z, is not an independent un-known and can be derived from the profiles of f(r,z)and N(z) ];

(4) the backscattering cross section lfM(z), whichincludes multiple-scattering effects:

p3A(z) = p3(z) M4zf*(rzi)],where

,f8(Z) = fo ,P(r,X, = rif*(r,z)dr/f f*(r,z)dr, (2b)

0(r,X,O = r) is the Mie scattering cross section for thesize parameter a 2-xr/X per unit solid angle at ascattering angle 0 = -, and M[z,f(r,zd] is the multi-ple scattering coefficient, which is M[z,f(rz ] =I1 forsingle scattering alone. Note that zo • zi • z, sincethe multiple-scattering contribution to the singlescattering from an altitude z (recorded at a time t[=2z/c, where c is the light velocity] after the laser shot)is coming from cloud layers below z.

The last two optical parameters T,zi and /3fm(z)of the cloud layers are dependent on the specific laserwavelength and therefore have little significance forcloud physicists. Nevertheless, they are mentionedhere due to their appearance in the laser-radar equa-tion:

1(z1) = A(z0)3f, M(z1) exp[-2 f %a(z)N(z)dz ]/z 1 2, (3)

where A(zo) = A*Io exp(-2roz). A* is the laser-radar efficiency coefficient; Io is the laser output; and70'Zo is the optical depth for the laser wavelength be-tween the system and the cloud base.

Quantitative analysis of Eq. (3) can therefore beachieved (with merely backscattering measurementsand one wavelength) by introducing a number of as-sumptions. In this work this is done by acceptingthe following properties of a cumulus cloud, based onseveral measurements in situ.'

(1) f(rz) = f(rz&. This assumption signifies thatwithin the cloud layers detected by the laser-radar

2874 APPLIED OPTICS Vol. 14, No. 12 / December 1975

IZ2/A T = 31N 0M(r) exp(-2T),(T = gOg), (4)

where M(T) is the multiple scattering coefficient(>1).

Since f(rz) is constant with height, of = af af andaf is independent of z:

T = a a1NOM (T) exp (- 2r). (5)

The optical depth r is also simplified to the form

Fig. 2. Laser-radar backscattering from a cumulus cloud base re-ceived by a Tektronix scope. The time-base axis (= height axis) isfrom right to left. The atmospheric molecular + aerosol backscat-tering is shown to the right and the cloud appears in the middle of

the picture. For the height scale, see Fig. 3.

(6)r =f zrfNdz = No1f(z -zogoo

and Eq. (5) can be written as

T = [af/(z - zo)]TM(7) exp(-2r) = [af/(z - z)]S, (7)

whereS rM(r) exp(-27).

a

Z

b

Z=4.87km Zmax=4. 5 km

:D

A_

C

-_4I'

1ZN

INNa

A_

etM

S .

S.

\ I

I = 4 ZZ,*=4 2km'

Fig. 3. The backscattering irradiance from a cumulus cloud asderived from Fig. 2. Curve a is an enlargement of Fig. 2, where I(relative units) is defined in Eq. (3). Curve b is defined in Eq. (9).

The new maximum of b appears at an altitude z = 4.5 km.

(the first few hundred meters of the cloud), the samesize distribution function is found (see Fig. 1).

(2) The multiple-scattering coefficient M[z,f*(r,z)]is a universal function of the optical depth (see, forexample, Ref. 10), for f(rnz) constant (from cloudto cloud).

(3) Within the cloud-base layers the number den-sity varies slowly with height, and thus N(z) = No(see Fig. 1).

Under these assumptions Eq. (3) can be written

(8)

T and z - zo are measurable by lidar for any cumuluscloud. Let T*(z) be T-(z -z):

T* = a exp(-2r)M(r). (9)

A typical curve of T*(z) derived from a laser-back-scattering measurement (Figs. 2, 3(a)) is shown inFig. 3(b). The main characteristic of the curve is theappearance of a maximum point. (The behavior ofthe function T*(z) is thus similar, for example, to theweighting functions appearing in the temperatureprofile analysis out of satellite measurements. SeeFig. 4. For the height where T*(z) reaches a maxi-mum value, dT*Idr = 0. Since r = Noofrz - zo),and since No, af, and z0 are constants for each cloud,it follows that

dT*/dr = 0 # dT*/dz = 0.

A maximum value in T* occurs at an optical depthr for which also S(r) is maximum (S = T*/af).

0

z

Fig. 4. Weighting functions of T*(z)-Eq. (9)-for M(T) = 1.Curve A represents a cloud in which the number density of thedroplets is greater (twice) than its value in cloud B. For such acloud (A), the maximum of T*(z) is shifted towards smaller clouddepth (half) as compared to cloud B. For the determination of zo,

zo*, and z2*, see Sec. IV.

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As a consequence, the maximum will always ap-pear for the same T independert of No. This leads tothe conclusion that for each cloud the height z forwhich the measured curve T* obeys dT*/dz = 0, theoptical depth from the cloud base to z has the samevalue.

111. Determination of Number Density and WaterContent

Let To(z) be the backscattering profile from a cu-mulus cloud for which f(r,z) and No are known, andlet T(z) be the backscattering profile from anothercumulus cloud. If we now locate the two altitudes z1in To(z) and Z2 in T(z) for which r is the same, NF/Nocan be determined:

T = 0oA0 (z1 - ZoO) = Ru,(Z2 - Z0,2),

and hence

N/K = Z - o,i)/(Z2 - Z0,2)- (10)

Equation (10) implies that the determination of NINo requires no knowledge of M(-), T, or f(r,z).

The water content of the cloud is directly propor-tional to N and the relative distribution function ofthe droplet size.

IV. Determination of z0 .

Assumption (3), (N(z) = NO), does not alwayshold, especially for a thin layer in the cloud base.Within this layer a better fit to the real cloud isachieved by letting N(z) increase from 0 at the cloud-base surface zoo to No at the upper surface of thelayer z2*, above which N(z) = No. For any profile ofNz) within the cloud, the average N can be definedas

N = flxdz/cz - o) 4r = gtN(z - zo). (a)

The elimination of Assumption (3) alters Eq. (7) tothe form

T(z) = [af/(z - z0)]r[N(z)/N}M(r) exp(-2 r). (12)

As a consequence, the smaller the thin layer men-tioned above, the better the accuracy achieved, sinceN(z)/N - 1 with (2* - zo*) -- 0. In the case thatZ2 -Z* qZ 0,

N(Z > 2*) - [!NO(Z2 * - Z*) + (Z - z2*)No] ( -zo)

-° {[ 2Z + 2) / -Z4)} (13)By substituting Eq. (13) into Eq. (12),

T(z > 2) ={a[ - (o Z2 )]r exp (-2 7)M(T).

(1lb)For low r values, M(r) is close to unity (see, for ex-

ample, Ref. 10). Therefore, a sharp maximum inT(z) is expected only when N(z) is changing rapidly(see Figs. 2 and 3(a)). To the first approximation,the cloud height at which a maximum in T(z), z > zo,appears (including z = z0*), or the altitude for whichthe profiles' derivative is significantly reduced, z =Z2 2*(including zo* = Z2*).

Fig. 5. Laser-radar backscattering from an aerosol layer abovethe mixing layer. The laser-radar system is directed vertically andthe height axis is from right to left. The backscattered light ismeasured in two polarizations: parallel to the linear polarizedlaser light (above) and normal (below). The height (above thelaser-radar) for which the maximum return signal is detected is1,000 m and the time base of the scope is set to 2 usec/cm => 3 kmrange for the whole picture. The pronounced depolarization ofthe laser light is explained by the existence of nonspherical parti-

cles in the aerosol and multiple scattering effects.

Fig. 6. Laser-radar backscattering from an altostratus cloud.The backscattered light is measured in two polarizations: parallelto the linear polarized laser light (above) and normal (below).The height axis is from right to left. It can be clearly seen that themaximum intensity is detected at different altitudes for each po-larization. This is due to the fact that for low values (near thecloud base), the multiple scattering can be neglected and the singlescattering contributes merely to the parallel component of thescattered light,

2876 APPLIED OPTICS / Vol. 14, No. 12 / December 1975

J!- K Mmr_� -

. - A� I - IAd I� Prw_ ;.. .... ..

.- .... '.

I , - Lj.,j.-.1

I -

It should be noted that the method can thus beused only if Z2* < ZMax, where zMa. corresponds to

T*(z) = (Z* + Z2*) T(z).

For those cases we define=O (Z * + Z2 ) (14)

V. Aerosol Layers

Figure 5 shows a ruby-laser radar return from anaerosol layer starting above the mixing layer. Polar-ization measurements indicate that multiple scatter-ing also occurs in that case (see discussion below, andFig. 6). The layer Z2* - zo* is much larger, but caneasily be determined (see Figs. 3 and 5). Assumingagain that above Z2* the number density is constant,N can be calculated in relative units from aerosol toaerosol. The main difference between an aerosollayer and the cloud base is that in the case of aerosolan assumption should also be made on the index ofrefraction.

VI. Discussion and ConclusionsThe behavior of the multiple scattering function

M(r) can be roughly estimated by the analysis of thedepolarization of the laser light inside the cloud (Fig.6). The depolarization takes place whenever multi-ple scattering occurs, since the single backscattering(0 = 180°) from spherical particles does not changethe polarization of the incident light. The multiple-scattering process depolarizes the light; therefore, thenormal component of the scattered light (with re-spect to the linear polarized incident laser light) isdue to the multiple scattering. The minimal contri-bution of the multiple scattering relative to singlescattering can thus be estimated by assuming that atleast a similar part of the parallel component of thescattered light is attributed to multiple scattering(for small , the parallel portion of the multiple scat-tering is much stronger than the normal component).M(r) is also dependent upon the laser-radar receiv-er's field of view: the narrower the field of view, theless multiple scattering is expected. The conse-quence is that the value of exp(-2-r) normally de-creases faster than the increase in M(r). Therefore,the analysis is always made on r exp(-2r)M or T*(Eq. (9)), for which a maximum value appears.

T* is plotted against z (since X is not known). Thederivative dT*/dr equals the derivative dT*/dz atthe point of maximum (=0) only if r can be expressedas a constant of the cloud (fNo) times the cloudlayers' thickness (z - z. Therefore, the methodpresented above does not apply for a varying concen-tration N(z) in the cloud, except for simple casessuch as the one mentioned in Sec. IV.

Probably one of the most attractive advantages ofthe method described, which provides the approxi-mate value for the number density, is the fact that itrequires only the detection of the point of maximum(zMa_) as the concentration changes from cloud tocloud. These heights are detected from relativeunits of T*, and no knowledge of the atmospheric ex-tinction below the cloud (or before the aerosol layer),or of the lidar parameters (A* and Io in Eq. (3)) isnecessary.

The author wishes to thank Vernon E. Derr of theWave Propagation Laboratory, Boulder, for his inval-uable comments, and J. Neumann and M. Graber forhelpful discussions. This research was partially sup-ported by the National Oceanic and Atmospheric Ad-ministration, U.S. Department of Commerce, undercontract 04-3-158-57, and by the U.S.-Israel Bina-tional Science Foundation.

References1. A. Gagin and J. Neumann, Rain Stimulation and Cloud Phys-

ics in Israel (The Hebrew University, Jerusalem, 1972), 63 p.2. F. F. Hall, Jr., in Laser Applications (Academic Press), New

York, 1974, Vol. 2.3. R. T. H. Collis, Appl. Opt. 9,1782 (1970).4. R. G. Strauch and A. Cohen, in Remote Sensing of the Tropo-

sphere V. E. Derr, Ed. (U.S. Gov. Print. Off., Washington,D.C., 1972), 23-1-23-35.

5. A. Cohen, J. Neumann, and W. Low, J. Appl. Meteorol. 8, 952(1969).

6. B. M. Herman, S. R. Browning, and J. A. Reagan, J. Atmos.Sci. 28, 763 (1971).

7. G. W. Grams, I. H. Blifford, B. G. Schuster, and J. J. de Luisi,J. Atmos. Sci. 29, 900 (1972).

8. A. Cohen, V. E. Derr, G. T. McNice, and R. E. Cupp, Appl.Opt. 12, 779 (1973).

9. E. R. Westwater and A. Cohen, Appl. Opt. 12, 1340 (1973).10. H. C. Van de Hulst, Light Scattering by Small Particles

(Wiley, New York, 1957).

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