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 NOAA Technical Memorandum ERL WPL-39 
 
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 THE USE OF RADAR FOR STUDIES OF CLOUDS 
 
 E. E. Gossard 
 
 Wave Propagation Laboratory 
 Boulder, Colorado 
 July 1978 
 
 noaa 
 
 NATIONAL OCEANIC AND / Environmental 
 
 ATMOSPHERIC ADMINISTRATION / Research Laboratories 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 LYRASIS Members and Sloan Foundation 
 
 http://archive.org/details/useofradarforstuOOgoss 
 
NOAA Technical Memorandum ERL WPL-39 
 
 THE USE OF RADAR FOR STUDIES OF CLOUDS 
 
 E. E. Gossard 
 
 Wave Propagation Laboratory 
 Boulder, Colorado 
 July 1978 
 
 I 
 
 .^QATMOS^,. 
 
 • ; 
 
 UNITED STATES 
 DEPARTMENT OF COMMERCE 
 
 Juanita M. Kreps, Secretary 
 
 NATIONAL OCEANIC AND 
 ATMOSPHERIC ADMINISTRATION 
 
 Richard A Frank. Administrator 
 
 Environmental Research 
 
 Laboratories 
 
 Wilmot N Hess Director 
 
 ^f nto? ^ 
 
NOTICE 
 
 The Environmental Research Laboratories do not approve, 
 recommend, or endorse any proprietary product or proprietary 
 material mentioned in this publication. No reference shall 
 be made to the Environmental Research Laboratories or to this 
 publication furnished by the Environmental Research Labora- 
 tories in any advertising or sales promotion which would in- 
 dicate or imply that the Environmental Research Laboratories 
 approve, recommend, or endorse any proprietary product or 
 proprietary material mentioned herein, or which has as its 
 purpose an intent to cause directly or indirectly the adver- 
 tised product to be used or purchased because of this Envi- 
 ronmental Research Laboratories publication. 
 
TABLE OF CONTENTS 
 
 1. INTRODUCTION 1 
 
 2. EFFECTS OF THE ATMOSPHERE ON RADAR OBSERVATION 5 
 
 Introduction 5 
 
 Reflection and Refraction 6 
 
 Absorption 6 
 
 Scattering 7 
 
 Scatter from clouds and precipitation 13 
 
 Scatter from the clear-air 17 
 
 3. VARIOUS RADARS AND THEIR POTENTIAL FOR CLOUD DETECTION 20 
 
 4. RADAR REFLECTIVITY AND CLOUD CHARACTERISTICS 22 
 
 Introduction 22 
 
 Drop-size distribution - some special cases 22 
 
 A general drop-size distribution function 24 
 
 Methods for measuring drop-size in-situ 28 
 
 Observation of drop-size distribution in natural clouds 29 
 
 Observations of drop-size distribution in man-made clouds .... 31 
 
 Observations of cloud reflectivity 33 
 
 Conclusions 35 
 
 5. RADAR ATTENUATION AND CLOUD OBSERVATION 36 
 
 Introduction 36 
 
 Attenuation of radar waves by small water drops 36 
 
 Optical extinction by small water spheres 38 
 
 A cloud parameter diagram 39 
 
 iii 
 
TABLE OF CONTENTS (cont'd) 
 
 Measuring Z and k 41 
 
 Analysis of errors 42 
 
 Conclusions 43 
 
 6. POLARIZATION AND SCATTERER CHARACTERIZATION 44 
 
 Introduction 44 
 
 Backscatter by non-spherical particles 46 
 
 Introduction 46 
 
 Theory 47 
 
 Geometry of scattering 51 
 
 Perfect sphere 54 
 
 Random orientation 55 
 
 Oblate spheroids 55 
 
 Prolate spheroids 57 
 
 Differential reflectivity of non-spherical particles 58 
 
 Attenuation by non-spherical particles 61 
 
 Scattering in an arbitrary direction 63 
 
 Conclusions and possible operational applications 64 
 
 APPENDIX 69 
 
 REFERENCES 113 
 
THE USE OF RADAR FOR STUDIES OF CLOUDS 
 
 E. E. Gossard 
 
 NOAA/ERL/Wave Propagation Laboratory 
 
 Boulder, Colorado 80302 
 
 1. INTRODUCTION 
 
 This report addresses the ways in which cloud detecting radars might 
 be of value at a weather observation station. From a survey of cloud 
 physicists and modelers, the following suggestions seem to be relevant: 
 
 a) If a cloud surveillance capability existed over a large area around 
 a weather station, the local forecaster would have a picture of the 
 distribution of solar insolation in the vicinity. This would clearly 
 aid in temperature nowcasting and short-term forecasting. It should 
 be of some aid in predicting where local storm development is most 
 likely and as input to some forecasting models. It should provide 
 
 a valuable supplement to satellite observations by providing details 
 of local cloud types and their three-dimensional structure. 
 
 b) In preparing flight plans for aircraft, and in landing operations, 
 
 it would often be of value to know the spatial distribution of clouds 
 and their density (especially when dealing with those aircraft re- 
 stricted to VFR) . 
 
c) When forecasting the dissipation of low stratus clouds or fog, the 
 forecaster would benefit by knowing the existence and distribution of 
 clouds above the fog layer. This is illustrated by Fig. 1- lb ob- 
 served by Petrocchi and Paulsen (1966) using a vertically pointing 
 TPQ-11 radar of 0.86 cm wavelength. There are two layers above the 
 fog in this figure. There is a clear trend toward intensification 
 
 of the important middle layer that would prevent dissipation of the 
 fog and alter the usual forecast of burn-off time. 
 
 d) Radars can also show the slope of advancing precipitation and thus 
 they can provide a precise prediction of the onset of precipitation. 
 This is seen in Fig. 1-2 which shows an approaching snowstorm descend- 
 ing at a rate of 5000 ft an hour. According to Petrocchi and Paulsen 
 (1966), trends in intensity can also provide insight into the duration 
 of a storm. 
 
 e) Cloud radars can reveal evidence of strong wind shear as shown in 
 Fig. 1-3. Such shear zones are of obvious interest around airports 
 and may also imply strong turbulence. 
 
 f) Given two frequencies, cloud radars may be able to monitor continuously 
 the mass median drop-size and liquid water density in evolving cloud 
 systems and thus aid in the prediction on the mesoscale of those that 
 will become precipitating storms. 
 
On the negative side, coverage by cloud radars seems to be rather 
 unpredictable. The radar "cloud" often does not correspond to the visual 
 cloud, and the radar echo cannot be related to the international cloud 
 classification system. 
 
 According to Plank, et al. (1954), of 89 periods during which echoes 
 were received slightly less than 50 percent of the echoes could be classed 
 as echoes from internationally defined clouds. Furthermore, only about 
 47 percent of the visual clouds were detectable on a 1.25 cm wavelength 
 radar. This is in good agreement with conclusions by Harper (1964) who 
 made measurements with a 0.86 cm wavelength radar at Malvern. He reported 
 that about 50 percent of high and low clouds and 75 percent of medium 
 clouds in southern England give detectable signals. His observations 
 indicate that radar is often unreliable as an indicator of true cloud 
 bases and tops. The radar observations often reveal fall streaks, suggesting 
 the presence of precipitation size particles and also suggest ice particles 
 in clouds to be a factor in radar detection. Some dense water fogs are 
 detectable by the fog tops are not defined by radar. 
 
 It seems reasonable to conclude that cloud detection radars can 
 potentially play a useful role in weather observation. However, at the 
 present stage they provide little quantitative information and the relation 
 of radar echoes to visual clouds remains obscure. It is probably important 
 to initiate some radar cloud research in order to clarify the relative im- 
 portance of cloud radars. Past work in this field has emphasized the cloud 
 
detection role of radars, and there is, probably little value in much 
 further research of this kind. However, the effects of clouds on attenuation 
 and depolarization have not been pursued as far as they might have been. 
 Past emphasis has been mainly on research at vertical incidence where 
 polarization experiments are very limited. However, even then some enhanced 
 returns apparently due to orientation of ice crystals were noted. 
 
 New requirements posed by satellite systems increase the availability 
 of radar components at very short wavelength and make it feasible to do 
 multiple radar, multiple polarization, multiple frequency experiments. 
 If combinations of reflectivity and attenuation observations can give 
 quantitative measures of cloud liquid water content, drop density, and 
 precipitation, or if polarization and/or cross-polarization can be used 
 to delineate state throughout volumes of cloud and storm systems, the 
 value of cloud radars at a weather station would be greatly enhanced. 
 Furthermore, two or more separated radars equipped with Doppler can 
 detect the radial component of the velocity of precipitation particles 
 within storms and (with suitable processing) are capable of displaying 
 the complete three dimensional particle motion field from which the 
 three dimensional air motion can be estimated. K-band radars can, in 
 principle, sense the particle fields at a much earlier stage in the 
 storms evolution. Thus multiple frequency, multiple radar systems offer 
 the hope of providing, not only information on the cloud microphysics , 
 but of also providing the cloud dynamics to complement the microphysics. 
 
 In what follows the measurement by radars of a) reflectivity, b) 
 attenuation and c) polarization will be examined in turn and specific 
 recommendations for their uses will be made. 
 
2. EFFECTS OF THE ATMOSPHERE ON RADAR OBSERVATIONS 
 
 Introduction 
 
 At microwave frequencies the refraction, absorption and scattering 
 properties of the nonionized clear atmosphere result mainly from the fact 
 that the water vapor molecule possesses an electric dipole moment. Of 
 lesser importance, except for wavelengths near 5 mm, is the magnetic dipole 
 moment of the oxygen molecule. Except near the absorption lines of water 
 vapor and oxygen, the refractive index for microwaves can be considered 
 to be essentially independent of frequency, and gaseous absorption can be 
 considered to be negligible. For most purposes radar frequencies are 
 deliberately chosen for which these conditions are satisfied. The real 
 part of the refractive index n is then related to the ambient tempera- 
 ture, pressure, and humidity (e.g., see Bean and Dutton, 1968) as 
 
 (n-1) x 10 6 = (77.6 P/T)(l - 4810 e/T 2 ) . (2-1) 
 
 In general the nonionized atmosphere can influence radar in several 
 ways: by a) reflection b) refraction c) absorption by atmospheric gases 
 d) absorption by hydrometeors e) scattering by dielectric inhomogeneities 
 and f) scattering by hydrometeors. 
 
Reflection and Refraction 
 
 Refraction and specular reflection have important effects on radar 
 systems only when the propagating wave is incident on stratified atmospheric 
 layers at very small angles of incidence. This case is important for 
 surveillance systems in which the radar points nearly horizontally in 
 the presence of atmospheric layers with large vertical gradients of 
 temperature and/or humidity. However reflection and refraction effects 
 seldom are important in radar studies of cloud structure or even the usual 
 applications of radar to investigate the structure of the clear air. 
 
 Absorption 
 
 At microwave frequencies, gaseous absorption occurs in the neighborhood 
 of water vapor and oxygen absorption lines in the spectrum. The absorption 
 spectra for oxygen and water vapor in the microwave band is shown in 
 Fig. 2-1 taken from Bean and Dutton (1968). Gaseous absorption is seldom 
 important in atmospheric studies by radar because radar frequencies are 
 usually chosen to minimize this kind of absorption. 
 
 On the other hand absorption by hydrometeors is important in many 
 meteorological applications. However, for cloud studies it only becomes 
 important at the very short wavelengths. For small spherical scatterers, 
 whose size is much less than a wavelength, the Rayleigh approximation is 
 applicable and the absorption crossection Q of a single sphere is given 
 by (e.g., see Batton, 1973) 
 
Q a = (TT 2 D 3 /A)[Im (- \^~±)] (2-2) 
 
 m + 2 
 
 where m = n-ik is the complex index at refraction of the scatterer, e.g., 
 
 water, A is wavelength, and D is diameter of the sphere. The attenuation 
 
 of the radar wave, due to hydrometeors, is the sum of their absorption and 
 
 the loss of energy scattered out of the beam. The attenuation by clouds 
 
 (as related to visibility) and precipitation (as related to rainfall rate) 
 
 is shown in Fig. 2-2. 
 
 Scattering 
 
 For meteorological applications, scattering by both hydrometeors and 
 by turbulence in the clear air is very important. A concise derivation 
 of the radar equation, which includes the radar system parameters, is 
 given in the Appendix. At this point it is necessary to introduce the 
 concepts of "reflectivity" resulting from scattering, but they will be 
 derived in terms of the incident electric field, and they will be presented 
 independent of radar systems. 
 
 The electric moment per unit volume induced in a scatterer is 
 
 (47T)" 1 E(£-e o ) (2-3) 
 
 where e is the (uniform) dielectric constant of the scatterer, c is the 
 ambient dielectric constant of the propagation medium, E is the electric 
 field in the scatterer. The total moment for a scatterer of volume V is 
 then 
 
f = (e-e o )(EV/47T) . (2-4) 
 
 The scattered field E at a distance r from a dipole of moment f is given 
 
 by 
 
 ,2 , . . 
 k f sxnijj 
 
 E = -5 exp(-i k • r ) (2-5) 
 
 s E r s ~s 
 
 o s 
 
 where k = 2tt/X is the wavenumber of the wave scattered in the direction 
 s s 
 
 r , and ty is the angle between the direction of polarization of the incident 
 
 electric field and the direction of k . 
 
 ~s 
 
 For a homogeneous sphere of dielectric constant £, the electric field 
 E induced in the sphere is (e.g., Gans, 1912) 
 
 E " E o (iTiAT' exp i(ut " to • V (2 " 6) 
 
 O 
 
 where E is the magnitude of the incident electric field and w is the 
 
 radio-frequency in radians per second. The vectors r and k are the 
 
 ~o ~o 
 
 range from the source and the wavenumber of the incident wave, and their 
 dot product gives its phase at the scatterer. Therefore 
 
 E s = {E o f $ TJTTT- }sin * ex P ^^-bs'Hs-bo'Ho)- (2 " 7) 
 
 s o 
 
 where E is the scattered field at r and V = (tt/6)D . 
 s s 
 
For atmospheric targets there are usually many scatterers within 
 the radar pulse volume. It is convenient to assume a monodispersive 
 drop size distribution whose drops are all of diameter D. This constraint 
 will be relaxed later. Letting N dr be the number of drops in the posi- 
 tion increment between r and r + dr , Eq. (2-7) becomes 
 ~o ~o ~o 
 
 E = E (k 2 /4TTr ) 3 KV sini|i / N(r ) exp i(cot-k -r -k -r )dr (2-8) 
 s oss r J v o K ~s ~s ~o -o ~o 
 
 where K = (e/e -1)/(e/e +2) = (e-e )/(e+2e ) and V = (4/3) tt(D/2) 3 is the 
 o o o o 
 
 volume of the spherical scatterer. But 
 
 k • r +k • r = k • (r +r ) + (k -k ) t = 2k • r + k • r 
 
 ~o ~o ~s ~s ~s -~s ~o ~o ~s ~o ~s ~o 
 
 where the definitions K = k -k and r = (r +r )/2 have been adopted. From 
 
 ~o ~s ~ ~s ~o 
 
 Fig. 2^3 
 
 K = 2k sin(6/2) (2-9) 
 
 where k = |k | = |k |. Recalling that average power is proportional to 
 E E*, where the overbar indicates time average, we have for the envelope 
 of the scattered power 
 
 P = P k A 9 1 K| 2 V 2 sin 2 ^ l/(4irr ) 2 = P (k 4 /r 2 ) | k| 2 (D/2) 6 sin 2 v I (2-10) 
 SOS s o s s 
 
 where I = |/N(r ) exp(-i K*r )dr | *" . Now suppose the number density 
 
 distribution function N(r ) to be composed of a deterministic and a random 
 ~o 
 
(in some sense) part, i.e., N = N(r ) + 6N(r ). The deterministic part 
 will in general lead to partial reflection from large scale gradients 
 of N within the radar beam, and it will be ignored in the present context. 
 Then 
 
 I = //6N(r 1 )6N(r 2 ) exp [ix- (r-j-i^) ]dr 1 dr 2 (2-lla) 
 
 I = //6N(r )5N(r +£)dr exp(-iK-Jt)d£ (2-llb) 
 
 ~o ~o - ~o ~ ~ 
 
 where % is the separation of a certain scatterer from some reference 
 position r in the scattering volume. It is convenient to define a 
 spatial correlation function 
 
 C(£) = -^— / 6N(r )6N(r +l)dr . (2-12) 
 
 6N 2 V V 
 
 To carry out the volume integration it is simplest to define a coordinate 
 system relative to the transmission direction, i.e., from transmitter to 
 the scattering volume V. Thus 6 is the angle to the scattering element dV 
 off the direction from transmitter to the center of the scattering volume, 
 £ is the separation of the scatterer from the volume center and <j> is 
 the azimuthal angle about the transmission direction. Therefore the 
 differential volume element d- . is 
 
 dv = l 2 dl sin9ded<t> (2-13) 
 
 10 
 
co 2TT -TT 
 
 I = 6N V / I C(£)d£ / / [exp(-iK£cos6)]sineded4> (2-l4a) 
 
 O TT 
 
 I = 6N 2 V — / ZC(l)s±nKi d£ . (2-l4b) 
 
 o 
 
 The choice of °° as an upper limit for 2- is justified by the assumption that 
 C(i) falls off fast enough so that the contribution for large 1 is negligible 
 
 A function often chosen for C(£) because of its mathematical simplicity 
 is 
 
 C(l) = exp(-W ) . (2-15) 
 
 Then 
 
 —j 87TJ1 3 
 
 I = 6N Z V 5— - , (2-16) 
 
 o 
 
 a result first obtained by Booker and Gordon (1950). 
 
 It is easy to establish the relation of scattered power to the 
 power spectrum E(k) of the scatterers by recalling the relation of E(x) 
 to the correlation function, i.e., the line spectrum 
 
 > 1 (k) =| / C(£) cosd di , (2-17) 
 
 o 
 
 where I is scalar separation of scatterers along a line. Choosing C(£) 
 exponential as before, 
 
 II 
 
o 
 
 However, Kovasnay et al. (1949) pointed out that the three dimensional 
 spectrum is related to the one dimensional spectrum of isotropic, homo- 
 geneous fluctuations such that 
 
 3<f> (K) 
 (j> 3 (K) = -K -^ . (2-19) 
 
 Therefore 
 
 ♦ 3 (K) = *&- — f^ • (2-20) 
 
 J ^ (1+£ Z K Z ) Z 
 O 
 
 Furthermore, if the scatterer concentration fluctuations are isotropic, 
 the spectrum of vector K is related to the spectrum of scalar K simply as 
 [see e.g., Bolgiano (1958)] 
 
 <J> (k) 7^2 I 3 
 Af„\ 3 ON o fn 91 v 
 
 <K<) = 5- ■ -9- 000 (2-21) 
 
 4^K 7T (1+JIk ) 
 
 o 
 Comparing (2-16) and (2-20,21) it is readily seen that 
 
 3 2 V K) 2 1 ^1 (K) 
 I = 87T-Vj)(K) = 2tt Z V -^ — = -2TT V ± -^ (2-22) 
 
 K 
 
 This remarkably simple relationship between the scattered power and the 
 spectrum of the scatterers is perfectly general as pointed out by Villars 
 
 12 
 
and Weisskopf (1954). Thus, for Kolmogoroff turbulence in the inertial 
 
 -5/3 
 subrange, one can simply insert the K dependence for (J), and ({>_ and 
 
 -11/3 
 arrive at a K dependence for I. 
 
 Scatter from Clouds and Precipitation 
 
 In calculating the scatter from cloud and precipitation particles 
 it is usually assumed that the scatterer concentration is random in the 
 sense that they are Poisson distributed, i.e., the particle concentration 
 in neighboring parcels is completely uncorrelated where the meaning of 
 "parcel" is an atmospheric volume of size approximately (A/2) . If the 
 
 numb 
 
 er of scatterers 6N(r )dr is independent of 6N(r )dr , the mean of 
 
 the product in (2-lla) is zero except when v.. = r • then. 
 
 I = / [6N(r)drT = / N(r)dr = N_ , (2-23) 
 
 T ' 
 
 the total number of drops in the volume. That the variance is equal to the 
 mean follows from the assumption of a Poisson distribution. 
 Then from Eq. (2-10) 
 
 P = P ^Yl l K l ( D / 2 ) N T (2-24) 
 
 ° (A7T) r s 
 
 which is just the product of the power scattered by a single drop and the 
 total number of drops. A scattering cross-section O is often defined as 
 
 13 
 
P o 
 
 P = -^ (2-25) 
 
 s 4irr Z 
 s 
 
 a = ATTk^ |K| 2 (f) 6 N T = (7T 5 /A 4 ) |K| 2 D 6 N t . (2-26) 
 
 Of course, in the case of cloud or precipitation particles the number 
 distribution is not adequately represented by a single diameter D as 
 assumed above. Instead the population of drops in the volume is repre- 
 sented by some type of size distribution N , where the number of drops 
 in the size interval D to D+dD is N dD. The total number of drops in 
 the volume is then 
 
 N T = / N D dD . (2-27) 
 
 o 
 
 The total reflectivity is therefore proportional to 
 
 Z T = / N D D 6 dD . (2-28) 
 
 o 
 
 In order to distinguish this quantity from the reflectivity n, it is called 
 the "reflectivity factor" Z (or Z when it is desirable to distinguish 
 total Z from its value for a limited range of drop sizes). Thus, defining 
 the reflectivity as 
 
 oo 5 oo 
 
 n = / N(D)o(D)dD = ^ |K| 2 / N(D)D 6 dD (2-29) 
 
 o X o 
 
 14 
 
we find 
 
 n = (tt 5 |k| 2 /a 4 )z t (2-30) 
 
 which is the usual expression for reflectivity from clouds and precipita- 
 tion found in all textbooks. 
 
 The possibility that number density in neighboring parcels is not 
 completely uncorrelated in precipitation was considered briefly by Goldstein 
 and by Seifert in Kerr (1951) and rejected as an important consideration. 
 However, it is not so evident that a coherent (Bragg) scatter is negligible 
 in clouds, where number densities can be as high as 1000 cm , and this 
 possibility was considered from a theoretical standpoint by Smith (1964) , 
 by Naito and Atlas (1964) and by Chernikov (1968). The subject was 
 apparently never pursued further because of the difficulty of measuring 
 cloud number density spatial spectra, which Eq. (2-22) shows to be the 
 decisive factor in the relative importance of incoherent vs. Bragg scatter 
 from clouds. Equation (2-24) shows that incoherent scattered power has 
 an inverse 4th power wavelength dependence, so short wavelength radars 
 favor the incoherent return. If the cloud water spatial spectrum is 
 
 approximately that of a passive scalar mixed by mechanical turbulence, 
 
 11/3 
 4>(k) should vary as approximately A , so the scattered power should 
 
 fall off only slightly with increasing wavelength. Until recently most 
 
 radars used for tropospheric weather observation had wavelengths of 
 
 30 cm or less and cloud returns might be expected to be dominated by 
 
 incoherent backscatter. With the modern use of radars at several meters 
 
 15 
 
wavelength for sounding the clear (and cloudy) atmosphere, the question 
 should be reexamined. 
 
 In the case of a cloud at saturation the question arises as to 
 whether condensation will lead to more drops or to growth of existing 
 drops or both. The answer probably depends on the number and kind of 
 condensation nuclei (CCN) present and the degree of super saturation. 
 The question can be largely avoided by dealing with liquid water content 
 rather than number density. Thus a distribution function VL for mass 
 of liquid water can be defined such that the total mass of liquid water 
 
 M„ 
 
 / Mp D dD (2-31) 
 
 and the correlation function analogous to (2-12) is 
 
 (I) = -~ / 6M(rj6M(r n +£)dr M (2-32) 
 
 6M 2 V 
 
 :o ~o ~ ~o 
 
 ,2 TI v 
 
 so Eq. (2-10) becomes 
 
 P = P k 4 9 1 Kl 2 p 2 sin 2 ^ l/(4Tir ) 2 (2-33) 
 
 SOS 11 s 
 
 where 
 
 I = 6M 2 V — flC(l)sinKZ di . (2-34) 
 
 16 
 
Scatter from the Clear Air 
 
 If we consider scatter from the molecules of the clear air, it is 
 evident that the parcel to parcel variations in the Poisson sense will 
 all be averaged out because the scale of molecular interaction is so 
 
 small (a few molecular diameters at most) while the volume of interest 
 
 3 
 is ^(A/2) . The only remaining contribution to the variance is that 
 
 resulting from processes capable of organizing the scatterer concentration 
 
 on scales of A/2 or larger such as atmospheric turbulence. If there is 
 
 variation of dielectric on this scale the incident electric field will 
 
 polarize small volumes dV which deviate from the average by 6e and each 
 
 small element will behave like a dipole of moment (E/4tt) 6edV. For small 
 
 deviations in £, 3K - (6e/e ) is the equivalent of changes in scatterer 
 
 number concentration 6NdV, so Eq. (2-10) gives 
 
 P s = p [ k s/( 4 ^ r s ) 2 ] sin2 X I (2-35) 
 
 -_ 
 
 where I = | / — ( r# J exp(-iK«r )dr 
 
 As before 
 
 I = 8tt 3 V(J5 £ (k) 
 
 where E(k) is now the spectrum of atmospheric dielectric fluctuations, 
 It is often more convenient to express the problem in terms of the 
 refractive index m given by 
 
 17 
 
/ye 
 
 where the permeability y is essentially unity for air. But m = n-ik - n 
 
 2 
 except near absorption lines in the spectrum, so £ - n and 6e - 2nSn. 
 
 Therefore 
 
 6e(r) 6e(r-£) = 4 6n(r) 6n(r-£) (2-37) 
 
 since £ - n - 1.0. Thus, using Eq. (2-20), we see that 
 
 » £ (k) = 4(J) n (K) 
 
 where $ (k) is the spectrum of n. Therefore 
 
 Thus 
 
 o 4 . 2 (J) (K) 
 P = P v ^- ^ 5__ . (2 _ 38) 
 
 s o -, 4 z 2 
 A r K 
 s 
 
 n = ^f sin 2 X ^V < 2 " 39 > 
 
 A 4 K 
 
 analogous to Eq. (2-30). 
 
 Another parameterization that has been found convenient uses the structure 
 
 2 
 parameter of refractive index C which is most easily defined in terms of 
 
 the structure function D (I) given by 
 
 18 
 
D(£) = [6n(r) - 6n(r-^)] 2 • (2-40) 
 
 In the inertial subrange of homogeneous, isotropic turbulence it is found 
 that 
 
 2 2/1 
 D (£) = C I ' (2-41) 
 
 2 
 where C is a proportionality constant called the structure parameter. 
 
 Ottersten (1969) shows that Tatarskii's (1961) exposition leads to the 
 
 conclusion that the one dimensional spectrum 
 
 2 -5/3 
 ) X (K) - (l/4)C n Z < 3/J 
 
 in the inertial subrange of homogeneous, isotropic turbulence. It has 
 further been demonstrated by Kovasznay et al. (1949) and Bolgiano (1958) that 
 the one dimensional scalar spectrum and the 3-dimensional scalar spectrum 
 are related by 
 
 d<MO 
 
 <J> 3 (K) = - K ^ . (2-42) 
 
 Thus ()).(<) = (1/4) C 2 K 5/3 = (3/5) <J>_(k), so 
 in J 
 
 ,5, ,27T.4 _ 2 -11/3 . 2. r „ .,. 
 
 n - it (t-) Ct~) c k sxn ^ • (2-43) 
 
 b a n 
 
 Therefore, three primary measures of atmospheric scatter are widely used — 
 
 2 
 Z, r\ and C depending on the nature of the scattering problem to be solved; 
 
 the scattering cross-section G is perhaps more widely used for single discrete 
 
 targets. Various existing radars will now be analyzed in terms of these 
 
 quantities. 
 
 19 
 
3. VARIOUS RADARS AND THEIR POTENTIAL FOR CLOUD DETECTION 
 
 It is of obvious interest to compare the Z values of various cloud 
 
 types with the minimum detectable Z of various radars. The starting point 
 
 of such a comparison is either Eq. (A24a) or (A24b) of the Appendix. Here 
 
 Eq. (A24b) will be used. Obviously, the minimum detectable Z depends on 
 
 the radar parameters P__, A , (P ) . and A. It also depends on the 
 r t e r mm r 
 
 range, r, and range resolution A. For a pulse radar, P^ and (P ) . are 
 6 6 F t v r'mm 
 
 usually expressed as peak powers although average powers are readily 
 used if the pulse-repetition period T and the pulse length x are known. 
 For single-pulse processing 
 
 (P ) . = k D T B (3-1) 
 
 rmin Be 
 
 if the noise is uniformly distributed over the receiver bandwidth B. Here 
 
 -23 
 k n = 1.375 x 10 is the Boltzman constant. The quality of the receiver 
 
 D 
 
 can be either expressed in terms of the "effective noise temperature," T , 
 or the receiver noise figure, F, related to the noise temperature by 
 
 T^ + 290' 
 290' 
 
 F = — (3-2) 
 
 F is often given in dB. 
 
 Table I presents the resulting calculations of Z . and n at a range 
 of 10 km for many radars including the 3.2 cm radar and the FM-CW radar 
 of the Wave Propagation Laboratory. Single pulse processing is not 
 
 20 
 
relevant to the FM-CW radar so a range cell size of 100 m and a sweep 
 
 length of 50 milliseconds was chosen in calculating (P ) . . 
 6 6 r mm 
 
 It is immediately evident that cumulonimbus, cumulus congestus and 
 Hawaiian orographic clouds should be detected by almost all radars at a 
 range of 10 km. However, fair weather cumulus and continental cumulus 
 should be detected only by the Defford and Wallops Island 10-cm radars 
 and by the 0.86 cm TPQ-11 and the CPS-9. Note that there is a fourth 
 power dependence of Z . on A, although r\ is independent of wavelength; 
 therefore, the short wavelength radars have a very great advantage in 
 the detection of targets consisting of small discrete spherical droplets. 
 
 Comparing Z values for the various clouds with the minimum detectable 
 values shown in Table I, we see that most clouds would not normally be 
 seen by the 3.2 cm wavelength radar listed in the Table at a range of 10 km 
 based on single pulse processing alone. However, if longer pulse- lengths 
 (e.g., 1-5 usee) were used and averages taken over many spectra, clouds 
 such as those observed at South Park should be detectable. If the velocity 
 spectra in clouds are narrow, the total processing gain to be realized 
 from Doppler processing and averaging may amount to as much as 10 dB. 
 
 21 
 
4. RADAR REFLECTIVITY AND CLOUD CHARACTERISTICS 
 
 Introduction 
 
 This section reviews past work on cloud detection by radar and 
 summarizes some conclusions regarding the detectability of various clouds 
 Some special drop size distributions are analyzed, and one quite general 
 form is discussed. 
 
 Drop-size distribution - some special cases 
 
 The results of measurements of cloud drop-size distributions prior 
 to 1949 have been summarized by Best (1951) . He concludes that the data 
 fit the relation 
 
 1 - F = exp[-(D/D Q ) n ] (4-1) 
 
 where F is the fraction of liquid water in drops of diameter less than D. 
 He shows that the mean of values of n found by all investigators is 3.3. 
 Furthermore, of the various kinds of mean, mode or median that might be 
 chosen as D , Best finds that the median drop diameter (the size at which 
 half the water is contained in larger drops) is most independent of n and 
 of minimum measurable diameter in the sample. 
 
 If the cloud drop-size distribution obeys a Marshall-Palmer expo- 
 
 -AD 
 nential, I\L = N e , where N^ dD is the number of drops of diameter between 
 D o D r 
 
 D and D + dD. The total mass of water is then given by 
 
 22 
 
Mj. « p £ N 2 / D 3 e" A ° dD = p £ N q ^ . (4-2) 
 
 o A 
 
 If D is mass median diameter, A = 3.67/D . Then the total number of drops 
 o o 
 
 per unit volume is given by N = N D (3.67) and the total liquid water 
 
 4 -4 
 content (LWC) is M= p tt N D /(3.67) . The water content in the drop- 
 size interval dD is 
 
 -3.*7D/D 
 
 M n dD = dM = p 7 N D e ° dD (4- 3a) 
 
 U bo 
 
 so the fractional water content in the interval dD is 
 
 ■3.67D/D 
 
 dM/N^ = [(3.67)76] (D/D ) e dD/D Q (for Marshall-Palmer). (4-4a) 
 
 This expression can be compared directly with dF, found from Eq. (1) to be 
 
 2 3 -0V D o ) 3 ' 3 
 dF = dM/M T =3.3 (D/D ) d d(D/D ) (for Best). (4-4b) 
 
 We are more interested in the fractional contribution to the total 
 reflectivity factor Z by drops of different size. Note that dM = p(tt/6) 
 D N dD and, from the definition of Z, 
 
 dZ = D 6 N D dD . (4-5a) 
 
 Therefore, 
 
 23 
 
dZ = p l (6/tt) D 3 dM . (4-5b) 
 
 So from Eqs. (4-4a) and (4-4b) we find expressions for the fractional contri- 
 bution to Z analogous to those for liquid water content M. They are plotted 
 in Fig. 4-1. The contribution to reflectivity factor is evidently concentrated 
 near the mass-median of the drop-size population. The total reflectivity 
 factor Z is thus related to the mass median drop-size as follows: 
 
 6! 7 6! M T D 3 
 Z = N '- — =- D = — =— (for Marshall-Palmer type exponential) (4-6a) 
 
 1 ° (3.67) 7 ° p 7T A" 5 
 Z T = r(y^- + 1) (6/tt) (M /p) D q 3 (for Best's distribution) (4-6b) 
 
 A general drop-size distribution function 
 
 The Marshall-Palmer and Best distributions are special cases (or 
 nearly so) of a more general function usually called (see Deirmendjian, 1969) 
 a "modified gamma distribution". It is given by 
 
 N dr = a r a exp(-br Y ) dr = a'D a exp(-b'D Y ) dD (4- 7a) 
 
 Ct+1 Y 
 
 where, here, r is drop radius and a' = a/2 , b' = b/2 . This distribution 
 
 is shown schematically in Fig. 4-2. 
 
 As seen above, the mass (or volume) median drop-size (D ) is a convenient 
 
 scaling length in radar scattering problems. In terms of D and G 
 
 24 
 
N D = a» D^ (D/D o ) a exp[-G(D/D o ) Y ] (4-7b) 
 
 Y 
 where G is readily identified as b(D /2) . Integration of N dD over 
 
 all D gives 
 
 N^a'Y" 1 D^ a G -C1*«)/Y r[(a + l)/Y] (4-8) 
 
 where N is the total number of drops per unit volume. The slope of the 
 
 distribution is zero at r = 0, at N = N and at r = °°. In Table II the 
 
 max 
 
 drop size radius or diameter associated with the maximum of N(r) is called 
 
 r or D . At N where dN/dD = 
 c c max 
 
 , a _ G 
 b = ~ Y = ~ 
 Yr c r o 
 
 whence it is readily seen that 
 
 1/Y 
 
 D = (^) D . 
 
 o a c 
 
 Similarly, 
 
 D 
 
 tt r l _3 
 
 MCD^ = P £ / D N p dn . (4-10) 
 
 o 
 
 Using Eq. (4- 7b) and integrating over all D 
 
 M T = | PTT a y" 1 G" (4+a)/Y r[(4 + a)/Y]2- (4+a) D 4+a . (4-11) 
 
 25 
 
loj ui nquia water contained in drops for which D < D is given by 
 
 x. 
 
 I( V " -Sr / ' * i4 * a - y)/y a"* dx (4-l 2a) 
 
 r '-— ) o 
 
 V 1 - e " x € + Sir — " x + 1] (4 " 12b) 
 
 Y 
 .1 n = 4+a-y is integral and where x = G(D/D ) . Note also that T(n+1) = n!. 
 
 G is uniquely determined for given values of a and y; it is found from that 
 
 value of x for which the mass M(D) = 1/2, so that half the volume (or mass 
 
 of liquid water) lies in sizes less than D . A plot of G is shown in 
 
 Figure 4-3. From (4- 7b) we see that the Marshall-Palmer distribution results 
 
 when a = 0, Y = 1. Then G = A = 3.67 and a' = N . It is therefore no 
 
 o 
 
 - 1 _ a -1 
 surprise that Eq. (4-8) then yields N = a' D (3.67) = y A whence we 
 
 identify N with a/2. Then Eq. (4-11) becomes 
 
 4 -4 -4 4 
 Hj, = y pir aG T(4) 2 D q (4-13) 
 
 in agreement with the above discussion of the Marshall-Palmer distribution. 
 
 The distribution of Best (Eq. 4-4b) does not reduce precisely to a special 
 case of Eq. (4- 7a). Note that G - 0.5 instead of unity when y = 3.3 and 
 a = 0.7. Also, in his discussion of Eq. (4-7a), Deirmendjian chooses to 
 constrain a to be integral. 
 
 Here, we are more interested in the reflectivity factor Z than in M 
 so, as in Eq. (4-6a) of the previous section, we note that dZ = D N dD = 
 (6/ptt)D dM. Using Eq. (4-.7b) and Eq. (4-9) and integrating over all D 
 
 26 
 
or 
 
 dZ = (6/pTT) M T r _1 [(4+a)/y] G~ 3/Y D 3 x ^ 7+a - Yj/Y e " X dx (4-14a) 
 
 __ 6M T D o r[(7 + a)/Y] 
 T pTrG 3/Y r[(4 + a)/y] 
 
 Some parameters that have been suggested for the modified gamma function 
 are shown in Table II for selected aerosols and hydrometers. The values 
 of r , N , a and Y are taken from Deirmendjian (1969) except as indicated, 
 but the values of G, NL, D , Z and a have been calculated from those drop- 
 size distributions from the relations (4-8), (4-9), (4-11). (The value 
 of r = 4 urn given by Deirmendjian for a Double-Corona cloud seems to be 
 an error. In our Table II we have assumed that it should have been r = 2 urn.) 
 For comparison with the Deirmendjian models, we have also calculated the 
 corresponding quantities from the Marshall-Palmer distribution for 
 rainfall assuming the coefficients found by Gunn and Marshall (1958) for 
 various rainfall rates. They find 
 
 Rain Snow 
 
 A(m _1 ) = 4.1 x 10 3 R" - 21 A(m _1 ) = 2.29 x 10 3 R -0-45 
 
 N (m" 4 ) = 0.08 x 10 8 N (m~ 4 ) = 0.058 x 10 S R -0-8 ' 
 
 o o 
 
 in the Marshall-Palmer distribution, where N^ = N e . All lengths are 
 
 'Do ft 
 
 in meters except rainfall rate R which is in mm hr . In Table II, it 
 has been assumed that R = 1, 4 and 16 mm hr for light, moderate and 
 
 heavy precipitation respectively. 
 
 27 
 
In the following paragraphs, observed cloud drop-size parameters 
 have been compiled and tabulated, and it will be seen that the Deirmendjian 
 parameters drastically underestimate the total liquid water content mea- 
 sured in natural clouds. Better in-situ measurements combined with good 
 radar measurements are obviously needed. 
 
 Methods for measuring drop- size in-situ 
 
 Most in-situ measurements of drop size distribution have been made 
 in powered aircraft, because the instruments required are complicated and 
 relatively heavy. Exceptions are the measurements made on Mt . Washington 
 and those made from a sailplane and reported by workers at the National 
 Center for Atmospheric Research (NCAR) near Boulder, CO. The advantage of 
 the latter systems is that the air stream is relatively undisturbed. 
 
 The conceptually simplest method used is direct photography if an 
 arrangement is achieved that allows an undisturbed sample of air to be 
 photographed. Unfortunately this is difficult and the method is expensive 
 requiring a great deal of time to collect and process enough data to 
 represent a significantly large sample. 
 
 One of the oldest methods is to expose slides to the air stream and 
 count individual drops and their sizes. Such slides may be coated with 
 grease (Weickmann and aufm Kampe, 1953) or with soot (e.g., Breed et al., 
 1976). This method is fairly accurate, but it is slow and laborious, 
 requiring the human measurement and counting of individual drops. The 
 photography and slide collection methods are most accurate for drops 
 larger than 100 urn. 
 
 28 
 
An earlier method used extensively by Diem (1948) and by Boucher (1952) 
 on Mt. Washington employed a cylindrical "impinger". It is probably 
 the least reliable of the widely used techniques. 
 
 Two recently developed methods that are now becoming widely used, 
 depend on the scattering properties of spheres for optical wavelengths. 
 A simple natural example of such effects are the halos that some clouds 
 produce around the solar disk. The instruments are called the Axial ly 
 Scattering Spectrometer Probe (ASSP) , best for particles 2-30 urn in 
 diameter; and the Forward Scattering Spectrometer Probe (FSSP) , best for 
 particles 3-45 urn in diameter. Such instruments aboard sailplanes probably 
 provide the best data that can be collected in quantity (Dye, 1973). Un- 
 fortunately the lack of power limits the mobility of the sailplane and 
 makes it difficult to sample different parts of a cloud system rapidly. 
 
 Observations of drop-size distribution in natural clouds 
 
 In 1953 Weickmann and aufm Kampe reported measurements of drop-size 
 spectra in cumulus clouds. Observations were made at many locations within 
 the clouds and the drop-size spectra from all observations were compiled 
 into average spectra for three cumulus types: fair weather cumulus (cumulus 
 humulis), cumulus congestus, and cumulonimbus. Photographs of typical 
 drop distributions are shown in Fig. 4-7. In addition to drop-size distri- 
 butions that allow the median diameter D to be calculated, the average 
 
 o 
 
 liquid water content M(g m ) and the average number of drops are given. 
 
 29 
 
The height distribution of cloud radii and of spectral spread are shown 
 in Fig. 4-5. The drop size reaches a maximum at near cloud base. The 
 total water content is concentrated in drops of much larger radius in 
 cumulus congestus than in fair weather cumulus as shown dramatically in 
 Fig. 4-6. This fact has important implications for cloud detecting radars. 
 However, some caution in accepting these data as typical is suggested by 
 the fact that the median diameters of the clouds investigated by Weickmann 
 and aufm Kampe are much larger than those of other investigators, so 
 the reflectivity factors may not be very typical. 
 
 Squires (1958) has compared clouds of maritime origin near Hawaii 
 with cumulus clouds of continental origin in Australia. Multiple exposure 
 samplers were used, and the drop- size concentrations showed spatial varia- 
 tions on scales down to 300 m or less. Clear patches were often found in 
 even dense clouds. However, characteristic differences were found. Squires 
 studied orographic clouds 600 m to 1500 m thick with updrafts of 0.1 to 
 0.25 m sec , dark stratus clouds about 300 m thick, and cumulus clouds 
 over the sea with updrafts at about 1ms . He compared the results of 
 the maritime cumulus with Australian continental cumulus of similar size 
 and found very significant differences. Maritime cumuli showed relatively 
 low drop concentrations (45 cm ) when compared with the continental cumuli 
 which contained about 228 drops cm . The liquid water content was similar 
 in both, so the drops in the marine clouds were significantly larger 
 than their continental counterparts. The cloud characteristics as 
 measured by Weickmann aufm Kampe and by Squires are compiled in Table 
 III. Table IV summarizes aircraft observations over Germany reported by 
 
 30 
 
Diem (1948). Figure 4-7 (from Boucher, 1952) shows cloud data collected 
 at Mt. Washington along with Diem's aircraft observations. 
 
 In experiments at South Park, near Denver, Colorado, Knollenberg's 
 Axially Scattering Spectrometer Probe (ASSP) was used to measure drop-size 
 distributions. Soot-coated slide samples were also taken. The results 
 were reported by Breed et al. (1976) in the Proceedings of the International 
 Cloud Physics Conference. The authors reported drop number density, mean 
 diameter D of the drops and total liquid water content for five cases 
 from both the ASSP and the slide observations. The results are shown in 
 Table V. Assuming a Marshall-Palmer exponential distribution, D has been 
 calculated from D, and Z has been calculated from Eq. (4-6a) . 
 
 It is clear from the various observations tabulated here that the 
 liquid water content of the Deirmendjian clouds is an underestimate of 
 what has been measured in natural clouds. This is most clearly true of 
 cumulus types, for which much observational data exist; there have been 
 few in-situ observations in the corona producing clouds. 
 
 Observations of drop-size distribution in manmade clouds 
 
 In recent years there has been concern about the ways in which man's 
 activity may affect the environment. One concern has centered on problems 
 of heat and moisture "pollution" and the resultant modification of local 
 climate. Thus cooling tower plumes associated with the various kinds of 
 power generating plants have been studied and their modification of the 
 environment examined. Huff (1972) and Agee (1971) have demonstrated that 
 
 31 
 
snowfall is enhanced downwind of power plants. The physical explanation 
 is not clear though mechanisms have been proposed (Hanna and Gifford, 1975) . 
 The drop size distribution within the plume is a critical factor in the 
 physics. It is exceptionally difficult to measure in a plume because 
 tower-mounted in- situ sensors are seldom feasible and aircraft-borne 
 penetrations are expensive and too brief to provide really satisfactory 
 data. It is therefore of interest to examine the utility of short wave- 
 length radars (e.g., 8.6 mm) to probe plume structure remotely. The 
 problem has also been studied by Ricks (1977) . 
 
 Few data sets are available, and we use here data acquired by 
 Pena (1977) from aircraft penetrations of a power plant plume at Keystone, 
 Pennsylvania. The same data set was used by Ricks. 
 
 The drop size spectra are listed in Table VI along with the conditions 
 under which the data were collected. We have used these data to calculate 
 by numerical integration the total number of drops (N ) , the total volume 
 of liquid particulates (VP) and the total reflectivity factor (Z ) . 
 Because of the importance of the large drops (Z « D ) , the distributions 
 have been extrapolated to infinite drop size by fitting an exponential 
 distribution to the measured distribution at the large-drop tail of the 
 spectrum. The results for Z values from manmade clouds are very similar 
 to the values found in natural clouds, for example in South Park (see 
 Table V), but they are almost 2 orders of magnitude larger than Z's 
 found from the Deirmendjian distribution for cumulus, for example. 
 
 32 
 
Observations of cloud reflectivity 
 
 Few radar observations of clouds are to be found in the literature; 
 most observations reported have been confined to precipitating systems, 
 primarily because of the radar wavelengths used. The status of cloud 
 observations using millimeter wavelength radars prior to 1964 has been 
 presented by Plank et al. (1954) and by Harper (1964) . At that stage of 
 technology radar was not an impressive tool for cloud observation. 
 
 Based on observations relating the percentage mass distribution by 
 size to drop-size distributions as measured at Mt . Washington Observatory, 
 Bartoff and Atlas (1951) undertook to relate median diameter to the radar 
 reflectivity factor Z. They arrived at the following relationship between 
 reflectivity factor, median diameter, and total water content M (where 
 the subscript T is dropped from now on) : 
 
 Z = 1.35 (6/tt) D 3 (M/p) x 10 12 (mm 6 m 3 ) (4-15) 
 
 where D is in meters and M is g m . Although Eq. (4-15) is empirical, 
 it is similar in form to (4-6b) . The difference between F(5/n+l) and 
 1.35 is probably not significant. Of course, D and M are not unrelated; 
 
 in fact, Atlas (1954) finds 
 
 D (m) = 26.5 x 10" 6 M 1 ' 3 
 
 Z = 0.048 M 2 (mm 6 m~ 3 ) (4-16) 
 
 Z = 139 D 6 x 10 24 (mm 6 m~ 3 ) . (4-171 
 
 J 3 
 
If radar reflectivity is used to estimate D and M, Atlas estimates 
 
 J o 
 
 that Eq. (4-15) gives a standard error of estimate of only 6.5% for D 
 and 21% for M. Equation (4-16) yields a standard error of 33% for M and 
 Eq. (8) a standard error of 16% for D . Apparently these relations might 
 be useful for some practical problems such as estimating visibility within 
 clouds, but they are not accurate enough to test the microphysics of 
 cloud models. 
 
 For advection fogs, Donaldson (1955) finds the following relationships 
 corresponding to the cloud Eqs. (4-15, 16, 17): 
 
 3 12 r 34% 
 
 Z - 3.26 D M x 10 < , (4-18) 
 
 o 
 
 53% 
 
 and 
 
 115% 
 > M 2 (4-19) 
 
 53% 
 
 91% 
 Z = 2.7 D x 10 < . (4-20) 
 
 o 
 
 48? 
 
 In Table I, the Atlas-Bartoff relationship, (Eq. (4-15), has been used 
 
 to compute Z for the Weickmann aufm Kampe data and the Squires data. These 
 
 6 * 
 
 may be compared with IN. AD. calculated from Squires' data. For comparison, 
 
 N D has also been computed, and it is clear that the Atlas-Bartoff equation 
 
 is better, though not dramatically better, and it might sometimes be useful 
 
 to substitute information about N for information about M when computing Z. 
 
 Squires' data are presented as drop size histograms in size increments of 
 typically AD = 10 urn. 
 
 34 
 
Conclusions 
 
 Experience gained in the 1950' s and 1960 's does not demonstrate 
 that radar reflectivity is a very useful quantity in the observation and 
 study of clouds. However, with the advances in radar technology in the 
 1970' s, and with the inclusion of dual- frequency attenuation measurements, 
 dual polarization and Doppler, the utility of radars for cloud investigations 
 should be re-evaluated. 
 
 There is a wide range in measured values of cloud drop size distri- 
 butions. This is partly due to the variety of measurement methods used, 
 but present evidence suggests that the large variation may be real. There 
 has probably been some bias because most experimental programs have been 
 aimed at storm and precipitation research, and aircraft penetrations have 
 tended to occur under conditions prior to or during precipitation conditions. 
 This may partly explain why larger drop sizes and higher Z values are 
 found from in- situ measurements than are deduced from the Deirmendjian 
 model. 
 
 35 
 
5. RADAR ATTENUATION AND CLOUD OBSERVATION 
 
 Introduction 
 
 The previous section addressed the question of useful experiments we 
 might do if we had a cloud detecting radar and considered what information 
 might be available from reflectivity measurements alone. In this section we 
 consider what information might be extracted if we had both reflectivity and 
 attenuation measurements available from radars with different wavelengths. 
 Such methods have been used with some limited success in studies of preci- 
 pitation (Goldhirsh and Katz, 1974), but the problem is different (and in 
 some ways simpler) if it were to be used with clouds. It could, in principle, 
 provide a spatial picture of the drop-size distribution in a developing 
 storm. 
 
 Attenuation of radar waves by small water drops 
 
 Attenuation of radar waves may result from gaseous absorption, particle 
 absorption (say by water droplets) or from particles scattering energy out of 
 the beam. If radar wavelengths are chosen which avoid the water vapor and 
 oxygen spectral lines (say 3.2 cm and 0.86 cm) gaseous absorption can be 
 considered small. If the water drops are very small compared with radar 
 wavelength, the Rayleigh approximation for the scattering and absorption cross- 
 sections gives (Battan, 1973 pg. 67): 
 
 (5-la) 
 
 <?s 
 
 * A 4 
 
 "a 
 
 = f K-K) D 3 
 
 
 36 
 
 (5-lb) 
 
2 2 
 where D is drop-size diameter, K = (m - l)/(m + 2) , m is the complex 
 
 index of refraction, A is radar wavelength and I(-K) is the imaginary part 
 of minus K shown plotted in Fig. 5-1. Ohviously, for very small D as 
 in clouds, the absorption cross-section is the important consideration. 
 
 Then k (the absorption coefficient) is the sum of the absorption cross- 
 sections of all the particles within a unit volume; so 
 
 2 °° 
 k = I Q a = j- K-K) / N D D 3 dD (5-2a) 
 
 vol o 
 
 where N dD is the number of drops in the size interval from D to D + dD. 
 
 However, note that total mass of liquid water 
 
 M = P t / N^ D 3 dD 
 O 
 
 k = |J I(-K) M (5-2b) 
 
 independent of the drop-size distribution. Thus, for water clouds, attenuation 
 provides a simple, direct measure of liquid water content, if temperature can 
 be estimated so that I(-K) is known. From Section 4 the reflectivity 
 factor Z is given by: 
 
 Z = N : =- D (for Marshall-Palmer exponential) (5-5a) 
 
 ° (3.67) 7 ° 
 
 Z = — r(1.91) D q M (for Best's cloud drop-size distribution) (5-3b) 
 
 37 
 
+3.67 D/D 
 
 where N = N e and D is the mass median diameter (i.e., the 
 
 o D o 
 
 drop-size above which half the liquid water is found) . From (5-2b) and 
 (5-3a,b) it is easy to verify that 
 
 D o 3 =^T(f^ff f(Best), D o 3 -^I(-K)%|^f (Marshall-Palmer). (5-4a,b) 
 
 If we consider the more general drop size distribution, Eq . (4-7a) 
 we see from Eq. (4-14b) that 
 
 D 3 = ptt/(6M_) G 3/Y r[(4+a)/Y]/r[(7+a)/Y] Z. (5-4c) 
 
 o T 
 
 Optical extinction by small water spheres 
 
 The extinction coefficient is defined by the relation I/I = exp(-0 r) 
 
 ext o ext 
 
 where I is the intensity of the transmitted light at range r and I is the 
 
 intensity at r = 0. If Q is the extinction efficiency of a drop, then 
 
 2 
 the extinction from a single drop is Q A, where A = TTa and a is the 
 6 e o o 
 
 radius of the crosssection of the spherical drop. For a distribution of 
 
 2 2 
 
 drops of uniform size the extinction coefficient is a = N Q TTa =NQ (tt/4)D 
 v ext x e o x e 
 
 where N is the number of drops per unit volume. For number density distribution 
 N D 
 
 OO 
 
 a = (tt/4) / Q D 2 N dD (5-5a) 
 
 ext J e D 
 
 o 
 
 (see Van de Hulst, 19^7, pg 129). In the Mie scattering limit of drops very 
 large compared with wavelength, which is the usual condition for optical 
 
 38 
 
wavelengths in the visible portion of the spectrum, Q ■+ 2 if the absorp- 
 tion is relatively small; then 
 
 a = (tt/2) / D 2 N dD. (5-5b) 
 
 ext D 
 
 o 
 
 It follows that 
 
 r(0.7) m t 
 
 a «i (0,7 - ) ~r (Best), o = 367 — |- (Marshall-Palmer) . (5-6a,b) 
 
 ext p D ext p D 
 
 o o 
 
 Again, the result may be calculated for the modified gamma function using 
 Eqs. (4-7b) and (5-5b) and we find 
 
 °ext = 3 P 1 M T g1/Y D 1 r K3+a)/Y]/r[(4-ta)/Y]. (5-6c) 
 
 Cloud parameter diagram (CLOUDPAD) 
 
 The assumption that 0=2 has been examined by Hodkinson (1966) and 
 
 Jiusto (1974). They find that it actually varies from 1 to 2 depending on 
 the detection angle and range to the scatterer. 
 
 From Eqs. (4-13), (4-l4b) , (5-2b) and (5-6c) it is clear that Z, k, 
 
 D , M and o are inter-related in a simple way if Y and a are known (as, 
 o' ext v j 
 
 for example is the drop size distribution is exponential). Atlas and 
 Ulbrich (1974) have made use of this fact and have constructed a "rain 
 parameter diagram" (RAINPAD) describing the relationships between the above 
 quantities and rain-rate R (mm hr ) assuming a Marshall-Palmer distribution 
 
 39 
 
in which A = 41 R (cm ). For particles of precipitation size, the 
 Rayleigh approximation is not necessarily valid. Atlas and Ulbrich there- 
 fore write M = lA where A = 10 k log e = 4.34 k is attenuation in dB per 
 unit distance; actually, they express A in dB km . In the Rayleigh 
 approximation 3=1. On the other hand, when the drops are very large 
 with respect to wavelength 3 = 3/2 for a monodisperse size distribution. 
 In general the functional behavior of 3 is quite complicated because 
 the full Mie scatter calculation is needed and the drop size distribution 
 must be known. Assuming a Marshall-Palmer distribution, Atlas and Ulbrich 
 made the necessary calculations for A = 3.22 cm and T = 10°C. Their results 
 are shown in Fig. 5-2. The inter-relationships are much simpler for clouds 
 because the Rayleigh approximation is valid. We have calculated the 
 corresponding "CLOUDPAD" shown in Fig. 5-3. On the cloud diagram R of 
 course is not relevant. In the cloud case, the diagram can be very general 
 if a dimensionless attenuation AA/I(-K) is defined and used as a parameter. 
 The diagram then applies to all temperatures in the Rayleigh size range. 
 In Fig. 5-3 an exponential drop size distribution was assumed but no 
 
 assumption of a value for N is necessary. 
 
 o 
 
 If Z and k can be observed by radar, some of the most important cloud 
 quantities such as liquid water content, drop size (D ) and extinction 
 coefficient can be monitored continuously in time and space about the radar. 
 If extinction coefficient can be obtained optically, the observational 
 errors can be estimated and the relevance of the assumed spectral functions 
 could then be judged. 
 
 40 
 
Measuring Z and k 
 
 Suppose we use two radars of different wavelengths, one having a 
 wavelength that is essentially unattenuated and the other heavily attenuated. 
 Then for the unattenuated radar the received power P is 
 
 rl 2 
 r 
 
 and from the other 
 
 C 2 
 P - = -=■ Z e 
 r2 2 
 r 
 
 r 
 ■2 / k dr 
 
 where r is range and C and C are known constants depending on the radar 
 parameters. Then it is easy to see that 
 
 and 
 
 Z = -fi- r 2 (5-7a) 
 
 L l 
 
 k = |d £n(P rl /P r2 )/dr . (5-7b) 
 
 From Z and k the cloud information is readily calculated from (5-4a,b) and 
 (5- 2b). 
 
 Unfortunately Eq. (5- 7b) depends on absolute calibrations of both 
 
 radars to get C n and C_, so the ratio P ,/P „ cannot be as accurate as 
 
 1 2 rl r2 
 
 comparative measurements made by a single radar. However, a single radar 
 can yield k if the gradients in Z are not large. To see this, write the 
 
 41 
 
integral over k from some range r to r + A where A is some small increment 
 of range over which backscattered powers are to be compared. Then 
 
 r +A 2 
 
 o P r 
 
 / k dr = - - In ~— . (5-8a) 
 
 If k and Z can be considered constant over the increment A, this gives 
 
 P r 2 
 r o 
 
 k = jj- in - ^ . (5-8b) 
 
 P r +A K^ 
 o 
 
 Now the ratio of powers applies to a single radar so only relative power 
 
 at different positions is needed, and relative powers can be measured 
 
 quite accurately. However, (5-8b) requires that Z be constant over the 
 
 increment A. Since A is approximately 1 km, this assumption would be 
 
 violated often in severe storms, but in a cloud system this assumption 
 
 may be better. 
 
 Analysis of errors 
 
 If we presume that careful calibration of the radars will allow absolute 
 power to be observed to an accuracy of ± 3 dB then Eq . (5-7a,b) and Eq . (5-4a,b) 
 lead to error bars on M shown in Fig. 5-4a. If relative powers for a single 
 radar have an accuracy of ± 0.3 dB Eq. (5-8a) gives the much smaller error 
 range shown in Fig. 5-4b. 
 
 However, it is to be remembered that the validity of (5-8b) depends 
 on Z remaining constant over the range of A. In severe storms there are 
 
 42 
 
many times when this approximation is not valid even when A is chosen as 
 small as 1 km. However, in the case of cloud systems that are more-or-less 
 stratiform, or of storms in the early stages of development, the approxima- 
 tion that Z is roughly constant over increments of a kilometer may be much 
 more realistic. It is likely that both (5-7b) and (5-8b) will be useful, 
 with (5-7b) used mainly through regions of large gradients of Z. 
 
 Figures (5-5a) and (5-5b) show the corresponding errors in D to be expected 
 for average reflectivities of -20 and -30 dBZ if Z can be estimated to 
 
 an accuracy of ± 3 dBZ and P /P . is known to ± 0.3 dB. We see that 
 
 o o 
 regardless of average reflectivity, the expected range of error is about 
 
 ± 25% which is probably not bad for many purposes since the range of D 
 
 is so great. 
 
 Conclusions 
 
 In some ways clouds are nicer to work with than precipitation because 
 the Rayleigh approximation can be used. 
 
 We see from Eq. (5-2a) that there is almost a factor of four difference 
 in the attenuation of 3.2 cm radars and 0.86 cm radars. This should be 
 enough difference to provide valuable information on spatial drop-size 
 distribution and on its evolution. If a 10 cm radar were used, its waves 
 could be considered as unattenuated and even better quantitative observations 
 could be made. This kind of information could be of great value in under- 
 standing the microphysics of nucleation and precipitation. As an aid to 
 
 43 
 
short-term forecasting at a Weather Service site, a forecaster might find 
 it of value to watch for a rapid growth in D " (by watching Z/k) as a 
 predictor of place and time of the onset of local precipitation. 
 
 POLARIZATION AND SCATTERER CHARACTERIZATION 
 
 Introduction 
 
 One of the impressive results of past radar cloud observations was the 
 conclusion that high reflectivities in clouds were often associated with the 
 presence of ice particles. More recently measurements of signals from satel- 
 lites received on two polarizations often show significant signal in the cross- 
 polarized channel even when there is little attenuation along the path. 
 These observations could result from the polarizing effects of ice particles 
 located somewhere on the path. Finally, cloud studies with radars have 
 always produced cases for which there was significant backscatter from the 
 apparently clear air. Some of these observations undoubtedly resulted 
 from backscatter from turbulent inhomogeneities in the dielectric structure 
 of clear air, but some of them might well result from enhanced backscatter 
 from particles having a favorable alignment relative to the radar's polar- 
 ization. The depolarization of satellite signals under conditions that 
 occasionally are visually "clear" suggests that such events may be 
 associated with ice crystals. 
 
 Additional support for this hypothesis comes from observations made 
 with the WPL FM-CW radar pointing vertically. Strong and consistent 
 
 kk 
 
returns were observed at Fraser, Colorado from particles identified as 
 ice blown off the peaks and ridges surrounding the site. An example is 
 shown in Figure 6-1, in which both cloud particles and dielectric 
 backscatter are shown. The dielectric backscatter is seen in the 
 breaking wave structure faintly visible at a height of 2 km. The 
 identification of the scattering particles was substantiated by simul- 
 taneous lidar observations. 
 
 Immediately we are presented with a mystery, of sorts, if we assume 
 
 an expression similar to (Eq. A17) to be valid, because K = (m -1)/ 
 
 2 
 (m +2) - 0.93 for water and only 0.20 for ice. The situation is even 
 
 more drastic for snow because K/p a constant and the density of snowflakes 
 
 is only about 0.05 that of ice. Thus K = 0.05 K. , and 
 
 snow ice 
 
 we would expect water drops to be much more effective scatterers than 
 ice or snow. A possible explanation lies in polarization effects and the 
 possibility that ice particles are often water coated. Growth and 
 nucleation processes may also be important because particles with ice 
 centers may grow to drop-sizes relatively large compared with coexisting 
 water drops. 
 
 These observations suggest that it is imperative that more information 
 be acquired on the inter-relationship between cloud types and radar 
 backscatter; they demonstrate the potential importance of radar as 
 an observational tool in a weather observing facility but emphasize 
 the gaps in present knowledge that prevent its use in a quantitative 
 and explicit way. 
 
 The observational results summarized above suggest an experimental 
 program in which radar polarization might be exploited to fill in some 
 
 45 
 
of the crucial information gaps 
 
 Backscatter by non-spherical particles 
 
 Introduction 
 
 There has been an upsurge of interest in methods of calculating the 
 polarization effects of non-spherical scatterers because of the anomalous 
 depolarization events that are now recognized to occur commonly on earth- 
 satellite paths. There are three primary ways in which the effect of 
 scatterer shape on incident radiation may be analyzed. The theory of 
 Gans (1912) is an amplification and extension of Lord Rayleigh's earlier 
 theory (1881) for scattering from small spheroids. With the availability 
 of computers, scattering theories based on Waterman's (1965) "extended integral 
 equation" technique have been used (e.g., Bringi and Seliga, 1977) to treat 
 non- spherical scatterers with diameters up to about 3A. It provides an 
 exact solution, but computers are generally needed for solution of the 
 transition, or T-matrix. Finally, numerical solution for the scattering 
 from bodies of arbitrary composition and shape is possible by dividing 
 the scattering volume into incremental volume elements and matching the 
 (complex) boundary conditions at all inter- volume-element boundaries. 
 Only the Gans method provides results parametrically simple enough to 
 make conceptually general statements about the effects of non-sphericity 
 on scattering. 
 
 46 
 
Theory 
 
 It is presumed in Gans theory that a uniform plane electromagnetic 
 wave intercepting an ellipsoid excites dipole moments that can be resolved 
 into three components along the orthogonal axes of the ellipsoid. The 
 length of the axes are presumed small compared with the wavelength, and 
 higher multi-pole moments are neglected. The Gans theory has been 
 adopted by Atlas, Kerker, and Hitchfeld (1953) to deduce some effects of 
 scatterer shape on radar backscatter. 
 
 The electric moment per unit volume induced in the scatterer is 
 
 2 2 
 E (m -m ) so the total moment is (see Section 2) 
 
 f = m 2 EV (m 2 /m 2 -l)/4^ (6-1) 
 
 where V is volume of the scatterer, m is its refractive index, m is 
 
 o 
 
 the refractive index of the propagation median, and E is the electric 
 
 field induced in the scatterer. Gans (1912) finds the induced field in 
 
 a spheroid to be related to the incident field E as 
 
 o 
 
 E - 
 E = ^ (6-2a) 
 
 m /m -1 
 
 1+ jS p 
 
 4tt 
 
 E 
 
 E n = Y^> ( 6 " 2b ) 
 
 n hi /m -1 
 
 1+ o p , 
 
 4tt 
 
 47 
 
E - 2 ° C 2 C6-2c) 
 
 m /m -1 
 
 i + J2 P . 
 
 4tt 
 
 where £ is the direction of the axis of rotation (or figure axis) of the 
 the ellipse and r\ and X, are the directions of two orthogonal diameters. 
 Therefore, from Eq. (6-1), 
 
 f r = (4fT) * m 2 g E r , f = (4tt) l m 2 g* E , f„ = (477) * m 2 g' E „ (6-3) 
 
 5 ; o 6 o? n o 6 on £ *• ' o 5 oC 
 
 where g is the complex quantity 
 
 . V[(m/m o r - 13 -3V T < -i X ,X' 
 
 g, g' = j = 4i L ' L e (6 ~ 4) 
 
 4TT + [(m/m ) -1]P,P» 
 
 where 
 
 P' = 2tt - P/2. 
 
 The quantities L, L', x> X'j an d g> g' are defined by the identities in 
 Eq. (6-4). For an oblate spheroid (flattened at the poles of the major 
 axis) Gans finds 
 
 s'-e^) 
 
 2 \ 1/2 
 P = ~ [ 1- [ ± -^- J arc sin e] (6-5a) 
 
 and for a prolate spheroid 
 
 ~f[h ..(»)-] 
 
 P = 4TT if f- I" T-T ) "I ( 6 " 5b ) 
 
 48 
 
2 2 1/2 
 here e = (a -b ) /a is the eccentricity and a and b are the semi axes of 
 
 the elliptical cross-section. 
 
 The scattered field E at distance r from a dipole of moment f is 
 
 s ' 
 
 given by 
 
 k s 2 f 
 
 E = = exp (-i k * r) (6-6) 
 
 m r 
 o 
 
 where k = 2tt/A. The r.f. phase factor, exp(-i k • r) , will be neglected 
 because we are interested in the signal magnitude. Also, the scattered 
 power P at r is related to the incident power P as 
 
 P a 
 
 P = — 2_ (6- 7] 
 
 2 
 4ffr 
 
 where a is the equivalent cross-section of a target isotropically scattering 
 
 the same power in the direction r as the real scatterer. Noting that 
 
 P , P , a E , E and that V = -r D is the volume of the equivalent 
 o s o s 6 x 
 
 sphere; Eqs. (6-3), (6-6), and (6-7) give 
 
 11 V 3 I A 4 [4 
 
 tt 5 D 6 (m 2 -l) 2 ,,5 
 
 e v 64tt _ 6 
 
 2 2 4 e 
 7T+(m -1)P,P'] A^ 
 
 D S, S» (6-8) 
 
 where a. is the cross-section of an individual drop scattering parallel 
 to the principle axis and a. ' is the scattering cross-section for the 
 two remaining orthogonal axes, and D is the diameter of a sphere of 
 equal volume. If we examine the limiting case as e -*■ 0, and expand both 
 the arc sin factor and the radical in Eq. (6- 5a) for small e, it is 
 
 49 
 
found that P ■*- P' -*■ 4tt/3 so that 
 
 .5 , i 2 , 1 2 5 
 
 * D 6ji_lL = I - D 6 |K| 2 (6 _ g) 
 
 A (m +2) A 
 
 in agreement with the well known result for Rayleigh scattering from small 
 
 spherical particles (e.g., Battan, 1973). 
 
 Of course, Eqs. (6-8) and (6-9) give the scattering cross-section for a 
 
 single particle only. The total scattered power is therefore proportional 
 
 to a = \ a. where the summation is over a volume presumed to be uniformly 
 
 vol 
 filled with scatterers. If the size distribution of particles is described 
 
 by N where N is the number of drops in the size interval D to D + dD, 
 
 Eq. (6-9) gives 
 
 5 
 a = — |KT / D b N D dD (6-10) 
 
 o 
 
 over the whole size range of spherical particles and for spheroidal 
 particles in general Eq. (6-8) gives 
 
 ^ =(^-) 2 4 / ■ { i ~ l)2 2 d V d ^- n ^ 
 
 \ / X [4Tr+(m Z -l)P,P'] U 
 
 and therefore from the definitions in Eq. (6-4) 
 
 5 
 
 4 ' "i/ 
 
 0,0' = 2-r J L,L'e" lX ' X ' D 6 N n dD (6-llb) 
 
 A 
 
 o 
 
 Seliga and Bringi (1976) define the quantity 
 
 Z DR = 10 log (a/a-) 
 
 50 
 
as the "differential reflectivity" where c, a' are given by Eqs. (6- 11a) 
 and (6- lib), and N is assumed to obey a Marshall-Palmer distribution, 
 i.e., N = N e ' o. Some insight is provided by first assuming a 
 uniform size distribution and assuming that all particles are oblate 
 spheroids with vertical axes, equal eccentricity and equal volume. Then 
 Eq. (6-8) gives 
 
 a/a' = g 2 /g' 2 
 
 Geometry of scattering 
 
 We first consider the radar backscatter problem, then generalize it 
 to the case of forward scatter because of the interest in scattering on 
 earth-satellite paths. The orientation of the axis of symmetry (figure 
 axis) of the ellipsoid is in the direction £ and two orthogonal minor 
 axes are chosen in the directions r\ and £. The geometry assumed is 
 shown in Figure 6-2 and is summarized as follows: 
 
 a) choose £ to be the direction of the rotational or figure axis 
 of the ellipsoid, 
 
 b) choose z to be the propagation direction of both the incident and 
 radar backscattered (with opposite sense) energy, 
 
 c) choose the x axis to be in the direction of the incident field 
 E where the zero subscript is dropped from now on. 
 
 d) without loss in generality, choose one minor axis E of the 
 ellipsoid to lie in the xy plane [i.e. , <j>' = <J> + 90°) and the 
 other to lie in the £z plane (i.e., (j>' = $, 6' = 6 - 90°). 
 
 51 
 
e) make 9 and 6 the angles between the £ direction and the x and z 
 
 axes respectively. <j) is the angle between x and the direction of 
 E. projected onto the xy plane. 8' and 6' are the angles between 
 the n. direction and the y and z axes respectively, and (J)' and cj>" 
 are the angles between x and the azimuthal directions of E 
 and E . 
 
 Dropping the zero subscript the projections of E onto the three axial 
 
 directions are 
 
 E^ = E cos9 = El, E = E cosG' = Em, E = E coscj)" = En 
 
 and the projections on x are 
 
 E r = E r cos6 = E r sin6 cose}), E = E cosG' = E cos6 coscj), E „ - -E„sinc 
 
 Similarly, on y and z, 
 
 E r = E r sin<5 sincj) E = E cos6 sincj) E „ = E^ cose; 
 
 yK K yn n y yc c 
 
 E r = E r cos6 E = -E sin6 E „ = 
 
 zC 5 zn n z? 
 
 2 2 2 2 2 
 
 E r = E sin 6 cos <b E = E cos 6 cos cf) E „ = E sin cb 
 
 x£ r xn T xc 
 
 2 2 
 
 E f. - E sin 6 coscj) sincj) E = E cos (S coscj) sincj) E „ = -E sincj) coscj) (6-12) 
 
 y £ yn y £ 
 
 E r = E sin6 cos6 coscj) E = -E cosS coscj) sin6 E r = 
 
 zC y zn v z? 
 
 52 
 
and 
 
 f -= m 2 g E _ = CLe" lX E r , f = m 2 g'E = CL'e" lX E . 
 xC o 5 xE, xE, xr\ o s XV] XT] 
 
 f _ = m 2 g*E „ = CL'e" lX 'E r etc 
 x£ o 6 x£ x£ 
 
 2 
 where C = 3 m V/4tt and V is the volume of the scatterer. Summing all 
 
 components along x, y, and z, 
 
 -iy 2 2 -iy' 2 2 -iy' 2 
 f = CE (Le A sin 6cos (J) + L'e A cos 6cos (J) + L'e A sin <J>) 
 
 CE 
 
 ]cLe" lX - L'e" lX ')£ 2 + L'e~ 1X '( (6-13a) 
 
 -iy 2 -iy' 2 -iy' 
 
 f = CE \Le A sin 6 sine}) coscj) + L'e cos 6 sincj) coscj) - L'e sine}) cose 
 
 CE |(Le" 1X - L'e" 1X ')£ 1 £ 2 } (6-13b) 
 
 f = CE coscj) cos6 sin6(Le" 1X -L'e" 1X ') = CE (Le" 1X -L'e" X ' )Jl A (6-13c] 
 
 where £.. = sin6coscf), £_ = sin6sincj), &_ = cos6 are the direction cosines of 
 E,_ projected onto the x, y, and z axis respectively. 
 
 If z is chosen to be the propagation direction, E = f = 0. If the 
 E vector lies on the x axis, the propagation plane is yz, and if the 
 x axis is horizontal the radiation is horizontally polarized. If xz is the 
 propagation plane, containing the E vector, the radar is vertically polarized 
 In either case, f is the cross-polarized component of the radiation. 
 
 53 
 
Of course, in reality we are not dealing with single scatterers but 
 with ensembles of scatterers oriented in some manner only describable 
 statistically. Atlas, Kerker, and Hitchfeld suggest three cases of 
 particular interest: (a) the case of random orientation, (b) the case 
 uniform orientation of oblate spheroids with figure axes vertical, and 
 (c) the case of prolates with their major axis arbitrarily oriented in 
 the horizontal plane. 
 From Eqs. (6-13a,b) 
 
 f x 2 .C 2 E 2 U 4 
 
 jLV 2iX + (l-2£ 1 2 + l 1 4 )(L')V 2iX + (H 1 2 -«, 1 4 )LL'e- 1Cx+X ' ) |(6-14 a ) 
 
 f 2 =G 2 E 2 CV 2 ) 2 LV 2i X-2LL'e- iCx+x,:i *U.-) 2 e- 2iX, | C 6-14b) 
 
 If an average is taken over a complete cycle of the incident wave propagating 
 such that L exp(-ix), L'exp(-ix') a exp i (cat - k z - X>X') 
 
 7^ 2 ^^M 2 ,^,^)^} 2 , 2(^ -^^LWL'le-^^'H (6-lSa) 
 .2„2 
 
 V-T 1 " ( "l £ 2 )2 |L|2 " 2|L||LMe- i(X - X,) + |L'| 2 (6-15b) 
 
 Perfect sphere 
 
 Return to Eqns. (6-4) and (6-5) and consider their limiting form for a 
 perfect sphere. Note that e = for a sphere. If the sin factor and 
 the radical in Eq. (6-5a) are expanded in series for small e, it is found 
 that P = P 1 +4tt/3 as e -> so that 
 
 r >4 
 
3V (m - 1) _ | 2 . I _ 
 
 3 m -1 _ 3„ ,, , , . 
 
 g = g* = = = a 1 —= 1— = a K (6-16aJ 
 
 4tt (m +2) (m +2) 
 
 for a sphere, i.e., Le~ 1X =K. Noting that I = I _ = if we choose a coordinate 
 system for the sphere such that £ = z, Eq. (6-14a) gives 
 
 f 2 = m Q 4 a 6 K 2 E 2 = (3/47T) ^VkV (6-16b) 
 
 in agreement with the classical result for the dipole moment of a sphere. 
 
 Random Orientation 
 
 If the orientation of the axis is entirely random 
 
 h 2 - I / £ i 2d£ i ■ W - h l £ >i ■ I etc - 
 
 i -i 5 
 
 Then 
 
 ■ 2„2 
 
 f x 2 " ¥" }l L | 2 + l5l L 'i 2 + TslMlL'l cos (X-X') (6-17a) 
 
 f y 2 = 4^-l^l L ! 2 - 2|M|L'|cos CX-X'D *|L'| 2 
 
 Oblate Spheroids 
 
 Suppose the scatterers are all oblate spheriods, oriented with their 
 figure axes vertical. The vertical direction is then £. Also, choose the 
 x axis to be horizontal. Since z is by definition the propagation direction, 
 the propagation plane is then zy; it must contain E,. (i.e., 9 = 90°) if £ is 
 to be vertical and x horizontal. Therefore, the projection of E r on x, is 
 &, = and on y is i ? = sin 6. 
 
 55 
 
Note that the polarization has not been specified. The Eqns . (6-15a) 
 and (6- 15b) were developed assuming E to be horizontal, lying along x. For 
 vertical polarization the x and y axes can be exchnaged. Then f and f 
 must be interchanged and also £ and £ . 
 Thus for horizontal polarization 
 
 2 2 
 
 ** = Hh |L'| 2 (6-18a) 
 
 f 2 = (6-18b) 
 
 and for vertical polarization 
 
 f x 2 = (6-19a) 
 
 f 2 = ^2^J|L| 2 sin 4 6 + |L«| 2 (l-2 sin 2 6 + sin 4 6) (6-19b) 
 
 2|L||L'|(sin 2 6- sin 4 6) cos( X -X') 
 
 Thus with horizontal polarization the E vector is perpendicular to 
 
 2 
 the y, z, and E, axes, so only f is non zero. Furthermore, since the x 
 
 axis of the spheroid is independent of 6, the moment f does not depend on 
 
 elevation angle. When the propagation plane yE, contains the E vector, the 
 
 polarization is said to be vertical. The projection of E onto the x axis 
 
 is then zero so f =0, but the projection onto E, clearly depends on the 
 
 angle between E, and the propagation direction z, so f depends strongly on 
 
 56 
 
Prolate Spheroids 
 
 Suppose the scatterers are prolate spheroids with their major axis 
 lying in the horizontal plane. Then £. is horizontal as is x by assumption. 
 
 Therefore the horizontal plane is xE, and £ , the projection of E». 
 onto x, is just £ = cos0, and £ , the projection of E». onto y, is £„ = sin6 
 sinS. 
 Thus 
 
 _____ 9 7T 2 
 
 .21 „ 2 sin 6 r . 2 QJQ sin 6 
 
 £. = 7 , £ 9 = — _=— / sin Od0 = — =— 
 
 1 2' 2 2n 
 
 -IT 
 
 TT 
 
 n 4 3 ,„ ,.2 sin 6 r 2 Q 2 QJQ 1 . 2 X 
 
 £ x = -, ^\ l 2 ) = 2tt I C0S 6 Sin 6de = g Sin 
 
 4 3.4. 
 £-, = -^ sm o; 
 
 so for horizontal polarization and horizontal prolates, 
 
 2 2 
 
 f x 2 = ^- | IM 2 + j |L| |L' |cosCx-X') + I |L'| 2 (6-20a) 
 
 2 2 
 
 f y 2 = S^ ¥ Sin2<5 ' L ' 2 -2|L||L'|cos(x-X') + |L'| 2 C6-20b] 
 
 and for vertical polarization and horizontal prolates 
 
 2 2 
 
 f v 2 = -9— 4 sin 2 6 |L| 2 -2|L||L'|cos(x-X')+|L'| 2 (6-21a) 
 
 2 2 
 
 f y 2 = ^- | |L| 2 sin 4 6 + (sin 2 6- | sin 4 6) | l| | L' | cos (x-X' ) 
 
 + (l-sin 2 6+| sin 4 6)|L' | 2 . (6-21M 
 
 5 7 
 
Differential reflectivity from non- spherical particles 
 
 Suppose a volume containing oblate spheroidal scatterers of common 
 size and shape is viewed simultaneously by radars of horizontal and vertical 
 polarization. Furthermore, assume that the axes of rotation of the spheroids 
 are all vertical. Then the ratio of powers in the two polarizations will be 
 given by the ratio of Eqs. (6-17a) and (6-18b). The ratio will in generally 
 vary dramatically with zenith angle. Examples are shown in Figs. 6-9 thru 6-12 
 parametric in the ratio a/b where a is the semi axis of rotation (principle 
 axis) and b is the orthogonal semi axial dimension. However, if the two 
 radars are viewing the volume horizontally (6 = 90°) a rather simple 
 relationship is obtained as shown in Figs. 6-3 and 6-4. For either water or 
 ice the ratio of the powers in the two polarizations depends mainly on the 
 ratio a/b with only a weak dependence on radar wavelength X or temperature. 
 Therefore, a simple measure of power backscattered on the two polarizations 
 provides a direct measure of drop shape. 
 
 In order to obtain information on actual drop-size, it is necessary 
 to use either a relationship between absolute reflectivity and drop size or 
 a relationship between drop shape and drop size. The latter relationship 
 has been investigated by Pruppacher and Beard (1970) . They studied water 
 drops falling at terminal velocity in a wind tunnel at 20° C at a pressure 
 of sea level in a nearly saturated environment. They obtained drop-shape 
 photographically, and examples of various size drops are shown in Figs. 6-5 
 and 6-6. At first glance the drops in Fig. 6-5, which are between 30 and 450 
 urn radii appear to be spherical. However, with magnification even drops as 
 small as 155.5 urn radius were found to be deformed. In summary, drops less 
 
 58 
 
than about 140 um (Reynold's No - 20) showed no deformation; drops such that 
 140 um < a < 500 Um were slightly, but measurably, deformed oblates conforming 
 fairly well to the relations obtained by Imai (1950) which can be combined 
 (Pruppacher and Beard) into the expression 
 
 i/b = [1-C9/16) a o PV t 2 /s] 1/2 (6-22) 
 
 where a is drop radius, in cm, p is saturated air density (1.19 x 10 g cm ), 
 o 
 
 V is terminal fall velocity in cm s , and s the surface tension of water 
 at 20° C (y 12. IS erg cm" 2 ). 
 
 Figure 6-6 shows drops in the size range 0.5 £ a £ 4.5 mm. It shows that 
 the shape deviates markedly from spherical in this range, but up to 1.5 mm 
 radius the shape is still well approximated by an oblate spheroid. At larger 
 sizes the bottom of the drop is flattened. The various experimental results 
 to be found in the literature were collected by Pruppacher and Beard and are 
 summarized in Fig. 6-7. In the size range 0.5 <_ a <_ 4.5 mm the deformation 
 a/b is linearly related to size a by the empirical expression 
 
 a/b = 1.030 - 0.124 a (6-23) 
 
 where a is in mm. 
 o 
 
 Pruppacher and Pitter (1971) have presented a method with more physical 
 basis for calculating drop shape but the method requires assumptions about 
 the appropriate pressure distribution over the drop and needs a high speed 
 computer. From the radar- observed deformations a/b it is easy to calculate 
 
 59 
 
effective values of drop radius a using Eqs. (6-22) and (6-23). Then, 
 using empirical fall velocity-size relations given by Gunn and Kinzer 
 (1949) or by Foote and DuToit (1969) , the fall velocity in quiet air can 
 be retrieved. Finally, with Doppler observation of total fall velocity, 
 the contribution from vertical air motion can in principle be deduced. 
 
 The discussion above has been based on the assumption that all drops in 
 the volume are of the same size and shape. Seliga and Bringi (1976) adopt 
 the more reasonable assumption that the drop sizes form a Marshall-Palmer 
 number distribution N = N exp-(AD ). The total backscattered power 
 
 P * I W D e 
 D=0 
 
 where the index "e" refers to a sphere of equal volume. So from Eq . 
 (6-10), 
 
 / D 6 S.'N HD 
 a' o e x D e Z' 
 ~- - ~ = I" (6-24) 
 
 / D S.N^dD 
 o J e l D e 
 
 where S, S' are defined by the identity, Eq . (6-8). Bringi and Seliga 
 (1976) call 10 log (Z'/Z) the "differential reflectivity" = Z__ although 
 "differential dBZ" would be a more appropriate term, as Z is conventionally 
 called reflectivity factor to distinguish it from the reflectivity T). 
 In their interpretation Z 1 is the value for horizontally polarized 
 incident radiation and Z is that for the vertically polarized wave. 
 When the distribution for N is inserted in Eq . (6-24), N cancels and 
 D disappears in the integration. Therefore, since the mass median 
 diameter D = 3.67/A, the quantity D is uniquely determined by the radar 
 
 power measurement Z /Z . Then, from an absolute measurement of either Z 
 
 H V n 
 
 60 
 
or Z , the value of N can be found. Using Pruppacher and Beard results to 
 relate a/b to D and thus calculate P, P* and S, S'; Seliga and Bringi 
 compute the integrals in Eq. (6-24) by numerical methods and obtain the results 
 shown in Fig. 6-8 (solid curves) where Z is defined above and where N is 
 
 UK O 
 
 related to the horizontal polarization reflectivity factor through the 
 quantity 10 log (Z h /N q ) . 
 
 It is noteworthy that essentially the same relationship between Z and 
 
 UK 
 
 size scale is found from Eqs. (6- 18a) and (6- 19b) by suitable definition of the 
 
 size scale (say 0,) to use in the uniform size model assumed in the use of 
 
 (6-18a) and (6-19b). The dashed curve shown in Fig. 6-8 for Z is found if the 
 
 UK 
 
 abcissa is D-, instead of the mass median diameter D , where AD, = 2.0. (Note 
 1 o' 1 
 
 that AD =3.67.) Thus, defining D n = 0.54D , the rather simple reasoning 
 o J ' 6 1 o' v & 
 
 that led to Figs. 6-4 and 6-5 can be used to deduce the drop-size scale D 
 from the differential reflectivity, effectively by-passing the numerical 
 integration of Seliga and Bringi and the shape-scale-fall velocity 
 relations of Pruppacher and Beard. 
 
 Attenuation by non-spherical particles 
 
 The total attenuation, Q , of radar waves by small particles is made 
 up of attenuation due to the scattering of energy out of the beam, Q , and 
 of attenuation due to absorption Q . That is, Q = Q + Q . In the 
 Rayleigh approximation for very small spherical particles (e.g., see Battan, 
 1973) 
 
 61 
 
128 *V \^\ 2 -_2j£ (A 2 |K| 2 (6 . 25) 
 
 s 3 i 4 2 o ^ 2 
 A |m +2 1 A 
 
 (-S— ® 
 
 In order to generalize these results to the case of non-spherical particles 
 return to Eqs. (6-4) and (6-5a,b) and consider their limiting form for the case of 
 a perfect sphere. In Eq. 6-5a) if the sin factor and the radical are 
 expanded in series for small e, it is found that P-*-4tt/3 as e->0 and Eq . (6- 16a) 
 is obtained. Eqs. (6-21) and (6-22) can then be written 
 
 8tt /2tt\ 4 , ,2 
 
 Q ^^l~) |g| 
 
 o 
 
 Q a - -^4 Im ^ 
 
 Am 
 o 
 
 Generalizing to the case of spheroids of random orientation 
 
 (f)\| g | 2 *2| g .[ 2 ) 
 
 Qr-^l' III-!-' -I-! i "■ > 
 
 9 m 
 o 
 
 2 
 Q a ■ — 2 Im(-g - 2g') . (6-28) 
 
 3 Am 
 o 
 
 We see from Eqs. (6-25) and (6-26) that Q « (V/A) 2 , whereas Q o « V/A. Therefore, 
 
 s a 
 
 when considering liquid cloud droplets and cm or mm radar wavelengths, Q a 
 should easily dominate Q because of the relatively small volume of each cloud 
 droplet. However, the imaginary part of the refractive index of ice is 
 
 62 
 
-4 
 very small, i.e., m - 1.78 -17.9 x 10 at -10°C, whereas for liquid water 
 
 m - 7.14 -i2.89 at 0°C for a 3.2 cm wavelength radar (Battan, 1973). 
 
 Obviously Eq. (6-27) can be used to calculate the attenuation in water clouds, 
 
 but for ice particles the scattering cross-section is the important component 
 
 and Eq. (6-28) must be used. 
 
 Scattering in an arbitrary direction 
 
 We have so far considered only backscatter because that is the case of 
 most interest in radar. However, in bistatic radar systems or in analysis of 
 the effect of spheroidal scatterers on oneway communications and telemetry 
 paths it is necessary to generalize the previous expressions. 
 
 If the scatterers are not spherical, the scattered electric field may 
 
 have components E , E , and E in all coordinate directions even though 
 x y z 
 
 the incident field is assumed to propagate in the z direction. The dipole 
 moments f , f , and f may exist along all three axes as shown in Figure 6-13. 
 We consider the scattering at an angle Y with the z axis and let the 
 angle Y lie in the xz plane. The total electric field associated with 
 the wave scattered at the angle Y is therefore the vector sum of the y 
 component and the components of E and E projected onto AB, the normal 
 to the propagation direction in the xz plane. The scattered field has 
 
 dipole moments f , f , and f , and the intensity of the power radiated is 
 
 x' y' z' j v 
 
 I(Y) = (2TT/A) 4 (f 2 + f 2 cos 2 Y + f 2 sin 2 y) (6-29a) 
 
 y x z 
 
 In the backscatter direction, for which Y = 180°, this reduces to simply 
 
 63 
 
I(y) = (2U/X) 4 (f 2 + f 2 ) (6-29b) 
 
 y x 
 
 as is to be expected. 
 
 Polarization effects in the forward scatter direction have recently 
 assumed great importance. The depolarization that results from non-spherical 
 scatterers imposes a major limitation on the possible use of polarization 
 diversity as a means of increasing channels on satellite-to-earth paths. 
 It is readily seen from the above, that both attenuation and depolarization 
 are simply calculated within the approximations contained in the Gans theory. 
 If the (spheroidal) shape of the scatterers is known; the depolarization and 
 the attenuation are intimately linked through Gans' factor g. 
 
 Imagine, then, the surprise of experimentalists when the phenomenon 
 of "anomalous depolarization" without attenuation was discovered on satellite 
 to earth paths. 
 
 The answer, of course, is to be found in the comparison of Eqs. (6-14), 
 (6-15), and (6-19). If we consider ice needles, it is immediately clear from 
 Eq. (6- 14b) and (6- 15a) that there will be significant cross-polarized scatter 
 
 from ice prolates unless 6=0. On the other hand the imaginary part of 
 
 -4 
 the refractive index of ice is very small (k - 8 x 10 ) so Eq. (6-19) will 
 
 yield very small attenuation. It seems clear (Cox et al., 1976) that the 
 
 depolarization results from ice particles in the atmosphere. 
 
 Conclusions and possible observational applications 
 
 1) When the incident electric vector is perpendicular to a plane 
 containing one of the spheroid axes, there is no dipole moment excited along 
 that axis so there will be no cross polarized component. 
 
 64 
 
2) Ice spheroids are relatively weak backscatterers compared with 
 water spheroids of comparable volume, and their backscatter and attenuation 
 depends only weakly on shape. On the other hand, water spheroids are 
 
 very shape- dependent. It is pointed out by Atlas, Kerker, and Hitchfeld, 
 
 (1953), that the shape effect should be completely negligible for snow. 
 
 2 2 
 They base their conclusions on the fact that (m -l)/(m +2) <* p where p 
 
 2 
 is density, and note that P is proportional to m -1. Since the density 
 
 of ice is about 0.9 and that of snow about on the order of magnitude less, 
 
 the very weak shape dependence of pure ice is further greatly reduced when 
 
 it occurs in the form of snow. However, for melting snow, when the flakes 
 
 are water coated, the reflectivity might become very shape- dependent . 
 
 3) Kerker, Langleben, and Gunn (1951) and Warner (1977) have made com- 
 putations on the basis of theory assumming an ice sphere to be surrounded by a 
 thin water film. They find that even a very thin water coating is sufficient 
 to make the scatterer behave like a water droplet of the same mass. 
 
 4) For oblates the backscattered power doesn't change with 6 for 
 horizontal polarization, but for vertically polarized transmission the 
 backscattered (parallel polarized) power decreases dramatically as 6 increases. 
 Thus the ratio of the two powers (difference in dB) is a measure of 
 scatterer shape if an RHI scan is used. This is also true if prolates (needles) 
 have their major axis horizontal but are randomly oriented in azimuth. 
 
 5) For prolate spheroids whose principal axes are vertical, the 
 backscatter increases as the zenith angle 6 increases. 
 
 6) The ratio of the backscattered power in vertical and horizontal 
 polarization and their phase difference is a sensitive measure of drop-shape. 
 If both radars view the volume at horizontal grazing angle (6 = 90°) especially 
 
 65 
 
simple relations are found as seen in Figs. 6-3 and 6-4. The ice phase differs 
 greatly from the water phase. Drop shape a/b is closely related to drop-size 
 so the dual polarization measurement is of significant potential value to the 
 meteorologist monitoring locally changing conditions. The drop-shape 
 deviates significantly from spherical only for drops greater than about 200 
 microns, so such a measurement may not be of much use in studies of non- 
 precipitating clouds. However, it should be a sensitive indicator of when 
 the first few large drops appear and permit the observer to monitor the 
 occurrence and growth rate of precipitation-size particles within clouds. 
 7) Polarization measurements can reveal to the meteorologist much 
 about conditions near the freezing level by exploiting the bright-band 
 phenomenon. As pointed out above, the reflectivity of snow above the 
 freezing level should be very small and independent of polarization at all 
 zenith angles 6. However, a region of high reflectivity occurs just below 
 the zero degree isotherm. Suggested reasons for the enhanced reflectivity 
 producing the bright band include (1) transition from ice to water (Austin 
 and Bemis, 1950) and (2) change in shape as particles melt (Wexler, 1955). 
 Based on depolarization measurements, Wexler concluded that shape changes 
 accounted for only about 1.5 dB of enhancement. Because flat oblates with 
 vertical axes would not be expected to produce depolarization, but should 
 produce significant aspect vs. reflectivity changes, an experiment should be 
 done using two polarizations and measuring the relative backscatter dependence 
 on elevation angle in an RHI scan. However, horizontally oriented prolates 
 (or needles) could cause significant depolarization, so measurements of 
 both the co-polarized and cross-polarized backscatter on both polarizations 
 would be of value to the meteorologist wishing to monitor the change of 
 
 66 
 
state and distribution of particles in his vicinity. The dependence of the 
 ratio of backscattered power in the vertical and horizontal polarizations on 
 zenith angle 6 and on wavelength A is shown in Figs. 6-9 thru 6-12. 
 
 8) For ice, k (the imaginary part of the refractive index) is very 
 small so X is small and attenuation due to absorption is very small. 
 Attenuation is almost entirely due to scattering. 
 
 9) Equations (6-18b) and (6- 19a) show that no cross-polarization results 
 from oblate spheroids whose axes are vertical for either horizontal or 
 vertical polarization. 
 
 10) Prolate spheroids whose major axes are randomly oriented in 
 horizontal planes produce significant cross polarization of the backscattered 
 signal. 
 
 11) When the electric vector is parallel to a plane containing the 
 major or minor axis of the spheroid only the dipole moment along that axis 
 is excited. 
 
 12) By monitoring the height region containing the freezing level, the 
 meteorologist could tell when an ice cloud begins to precipitate, because 
 the bright-band phenomenon depends on particles falling through the freezing 
 level. In fact the distinctive banding is partly a result of the change 
 
 in particle fall velocity as they change state from snow to liquid (see 
 
 Figure 6-14, taken from Lhermitte and Atlas, 1963). In Fig. 6-14 the radar was 
 
 of wavelength A = 3.2 cm and pointing vertically. The bright-band is often 
 
 more intense than would be expected simply from the change in state from 
 
 ice to snow. Note that Z increased by a factor of 30.6 between level 2 and 
 e J 
 
 level 3 in Figure 6-14. Referring to Eq. (6-9) and noting that |k| - 0.92 for 
 water and |k| * 0.197 for ice, we see that the change from ice to liquid can 
 
 67 
 
only account for a factor of about 5. This fact caused Lhermitte and Atlas 
 (1963) to propose that rapid aggregation of ice crystals into larger and larger 
 snovvf lakes proceeded down to level 3 within the bright-band. However, an 
 alternate explanation may be found in Figure 6-15, which shows the ratio of 
 backscattered power from water and ice particles. It shows that the 
 backscattered power increases by a factor of 41 if the scattering particle 
 changes from an ice oblate of axial ratio 0.1 (perhaps a snowflake) to a 
 liquid water oblate (as the snowflake melts) . 
 
 Likewise, Lhermitte and Atlas point out that the increase in fall 
 speed of the liquid rain can only account for about one-third of the observed 
 decrease in reflectivity, below the melting layer and they suggest that 
 drop break-up, in which each drop divides into 4-6 drops, must be occurring. 
 However, if the backscatter from an oblate (water coated) ice particle is 
 compared with a spherical water drop it is found that the latter is greater 
 by a factor of about 17 - again in good agreement with the observed difference 
 between levels 3 and 4. 
 
 These rough calculations assume equal volumes and densities for the 
 different particles - an assumption which cannot be easily justified, but 
 they indicate the potential importance of particle shape and suggest that 
 the subject of bright-band formation and morphology is not closed. Since 
 aggregation is an important mechanism of particle growth, and particle 
 break-up is an important control on drop-size, the proper interpretation of 
 the bright-band is important to the local forecaster. 
 
 68 
 
APPENDIX 
 
 The Radar Equation 
 
 The power intercepted and reradiated by a target is 
 
 P t G A 
 
 T 2 T (Al) 
 
 47rr 
 
 where P is transmitted power, G is gain of the transmitting antenna, and A,^ 
 is the target cross-sectional area. If the target scatters isotropically, 
 the power intercepted by the receiving antenna is 
 
 p t g t At A e 
 
 P r = 2 -S • ( A2 ) 
 
 4irr 47rr 
 
 The effective area A of the receiving antenna is related to its gain by 
 
 For a parabola whose aperture is A 
 
 8tt ? 
 
 G R = A P ^2 so A e " f A p • < A4 > 
 
 Early derivations of the radar equation assumed the power to be constant 
 between half -power points of the beam and zero elsewhere. For such a 
 "top hat" beam 
 
 69 
 
G n = 16/04) (A4a) 
 
 K 
 
 2 
 or just G D = 16/6 if the beam is conical with beamwidth 6 (rad) . However, 
 
 K 
 
 Probert-Jones (1962) studied the more realistic case of a Gaussian beam. 
 The gain of such an antenna is related to its beamwidths 8,<j) in radians, as 
 
 G R = k 2 -n 2 /Q<$> (A4b) 
 
 2 
 where k is a dimensionless constant near unity depending on how much of 
 
 the power from a particular antenna feed is intercepted by the dish. If 
 
 we consider a parabolic dish of diameter D, 
 
 9 = (J> = 1.56 A/D (A5) 
 
 for a top hat beam, and from Eq. (A4) and Eq. (A4b) 
 
 9 = <J> = 1.22 A/D (A5a) 
 
 for a Gaussian beam. 
 
 If we replace the geometrical target cross-sectional area with the 
 backscattering cross-section a (which is the ratio of the actual backradiated 
 power by the target to the power that would have been backscattered by an 
 isotropic scatter) and replace A in Eq. (A2) using Eq. (A3), we find for 
 a top hat beam 
 
 70 
 
64tt r 
 
 Radar Reflectivity and Backscatter 
 
 Separating the system parameters from target parameters and range, we 
 can write 
 
 P G 2 X 2 
 
 64it r r 
 
 If N is the total number of scatterers in a volume V, the total received 
 power is 
 
 K B 
 
 P = ^-r I O. . (A8) 
 
 r 4 . L . i 
 
 r i=l 
 
 If we choose to express the summation in terms of an average reflectivity 
 per unit volume, it can be written 
 
 P r = \ nv (A9) 
 
 r 
 
 where V is volume and the quantity 
 
 l N 
 n = £ I a (A10) 
 
 i=l 
 
 is the so-called radar reflectivity, 
 
 71 
 
If x is the pulse length, the effective pulse volume for distances much 
 greater than a pulse length is approximately 
 
 - =4^ (AH) 
 
 where 6 and <$> are the radar beamwidths of a top hat beam. Insertion into 
 Eq. (A9) yields, for the top hat beam, 
 
 P G 2 X 2 
 
 P _ _j n_ M£i (-A121 
 
 64tt r 
 
 For a Gaussian beam the integration over the intercepted volume should 
 include the gain function, so 
 
 P = ^-^ / ^H dV (A13) 
 
 r 64tt V r 
 
 which may be integrated to yield (Probert-Jones, 1962) 
 
 constant radar parameters target parameters 
 
 ^v 
 
 P = ^ (P. t A 2 G 2 8<f>) ^ (A14) 
 
 r 1024"/ £n2 to ^ 
 
 where G is defined by the following expression for gain with a Gaussian beam: 
 
 G(0,<jO = G exp 
 
 [(*•*)] 
 
 72 
 
Probert-Jones also took account of the fact that when the entire transmitted 
 beam is filled with targets the backscattered beam differs from the transmitted 
 beam pattern. He therefore assumed an equivalent cone and found the power 
 to be reduced by the factor (2 In 2)~ which appears in the constant factor 
 of Eq. (A14). 
 
 After eliminating G using (A3) and (A4a) , Eq. (A12) gives for a 
 "top hat" beam: 
 
 P = 0.0795 PA 4^ n (A15) 
 
 r t e Z 
 
 r 
 
 where A = ct/2 is the range cell depth. From Eqs . (A14) , (A3) and A4b) we 
 find for a Gaussian beam: 
 
 P = 0.0354 PA 4r n • (AlSa) 
 
 r t e 2 
 
 r 
 
 We see that calculated received power will be overestimated by 3.5 dB if 
 a "top hat" beam is assumed as was done prior to 1962. 
 
 Backscatter from Spherical Drops 
 
 For spherical targets a and n. can be related to the refractive index 
 of the particulate (say ice or water) . From Mie theory for backscattering 
 from a spherical drop 
 
 2 oo 
 
 O = ~~ \ I (-D 3 (2j + l) (a - b )|" (A16) 
 
 a j = l J - 1 
 
 73 
 
where 
 
 a is drop radius 
 
 a = 27Ta/A 
 
 a. are the coefficients of terms arising from induced magnetic dipoles. 
 ■* quadrapoles, etc. 
 
 b. are the coefficients of terms arising from induced electric dipoles. 
 ■* quadrapoles, etc. 
 
 j number of the term in the expansion. 
 
 The coefficients a., b. can be expressed in terms of Bessel and Hankel 
 3 3 
 
 functions of the 2nd kind with arguments a and refractive index 
 
 m = n - ik 
 
 where n is refractive index and k is the absorption coefficient. In the 
 Rayleigh approximation of a << X 
 
 ,2 2 . 2 ro ,6 6 ,2 _ 6 5 6 
 a. = La 6 |i-i| ■ { ^\ a — lK| 2 = 2 -^- |K| 2 (A17) 
 m + 2 A A 
 
 5 
 a = I a. = ^j |K j I N.D. in the Rayleigh limit (A18) 
 i 1 \ i 
 
 367^ M! Jl! ( A19) 
 
 A p i 
 
 where K = |(m - l)/(m + 2) | . |k| -0.93 for water, D is drop diameter, 
 p is density - 1.0 gm cm" and M = 0/6) pZN.D. is the mass of liquid water 
 per unit volume. 
 
 74 
 
Radar reflectivity 
 
 (Eq. 
 
 A10) 
 
 is thus 
 
 related 
 
 to th 
 
 3 radar refl 
 
 ectivity 
 
 factor, 
 
 
 
 
 
 
 
 
 
 Z = I N.D. 6 
 k l l 
 
 1 
 
 (where 
 in a 
 
 the 
 unit 
 
 summation is 
 volume) 
 
 over al! 
 
 size 
 
 increments 
 
 (A20) 
 
 as 
 
 it 5 |K| 2 
 
 Z 
 
 
 
 
 
 
 
 (A21) 
 
 SO 
 
 ,4 
 
 a n 
 
 
 
 
 
 
 
 
 (A22) 
 
 , 5 |K| 2 
 
 Since the total mass of 
 
 water in 
 
 the unit 
 
 volume is M = 
 
 (ff/6)p£N.D. 
 
 3 i. 
 we have 
 
 from (A20) 
 
 
 
 
 
 
 
 
 
 36M 2 
 
 2 2 
 p IT N 
 
 
 
 
 
 
 
 
 (A23) 
 
 if the volume is 
 
 assumed 
 
 to 
 
 ;ontain N drops of uni 
 
 form 
 
 5ize D so that Z = ND . 
 
 The reflectivity 
 
 factor 
 
 Z is 
 
 usua 
 
 lly expressed in 
 
 units 
 
 of (mm /m ) 
 
 intro- 
 
 18 
 ducing a factor of 10 ; 
 
 thus from Eqs. (A15) and 
 
 (A21) 
 
 
 
 Pr = 2 
 
 26 x 10 
 
 -"I 
 
 X 4 
 
 r 
 
 (top 
 
 hat beam) 
 
 
 and from Eqs. (A15a) and 
 
 A21) 
 
 
 
 
 
 
 Pr = 1 
 
 01 x 10 
 
 "17. 
 
 X 4 
 
 r 
 
 (Gaussian 
 
 jeam) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 75 
 
 
 
 
 
- 
 
 
 
 1 
 
 i~3slW!lMB ° 
 
 
 
 b 
 
 ^^^^^^^ 
 
 
 
 
 | i S^jg c 
 
 b 1-1 
 
 b 1-2 
 
 1 -3 
 
 Figure 1-1,2,3. Radar cloud observations obtained with a TPQ-11, 
 
 8.6 mm, vertically pointing radar. Figure illustrates temporal 
 changes within cloud structure and the influence of wind shear. 
 (Petrocci and Paulsen, 1972.) 
 
 76 
 
I 0.1 
 
 0001 
 
 1/X = 0.1 
 
 
 
 
 
 / 
 
 
 i 
 
 
 
 
 
 / 
 i 
 i 
 
 i 
 i ! 
 
 
 
 
 
 -£w- 
 
 i 
 i 
 / 
 
 \ 
 \ ! 
 
 \ 
 
 / 
 
 
 
 /J 
 
 ' Y" 
 
 /\J 
 
 '-- 
 
 
 7 
 
 /- 
 
 
 
 
 
 \=10 4 2 1.5 1.0 0.8 0.6 0.4 02 0.15 1cm 
 
 Figure 2-la. Theoretical values of atmospheric attenuation by oxygen 
 and uncondensed water vapor at sea level for a temperature of 20° 
 C. The solid curve gives the attenuation by water in an atmos- 
 phere containing 1 per cent water molecules (p=7.5 g/m 3 ) for 
 Ay/c? = 0.10 cm . _The dashed curve is the attenuation by oxygen 
 for ko/c = 0.02 cm l , where Ay is line width. (From Kerr, 1951). 
 
 200 
 
 100 
 80 
 60 
 
 40 
 20 
 
 10 
 8 
 6 
 
 4 
 2 
 
 
 
 
 
 (1 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 \ V 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ' 
 
 
 
 I 
 
 
 
 / 
 
 
 
 / 
 
 
 / 
 
 
 
 
 0.8 
 0.6 
 
 04 
 
 
 — \^ 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 l/\ = 4 | 6 
 X = 0.25 0.20 0.16 
 
 16 cm- 1 
 
 0.10 09 0.( 
 
 Figure 2-lb. Theoretical values of atmospheric attenuation by uncon- 
 densed water vapro in the millimeter region for a temperature of 
 of 20°C. The curve is for an atmosphere containing 1 per cent 
 water molecules (p = 7.5 g/m 3 ) for ISv/c = 0.11 cm 1 !, where Ay 
 is line width. (From Kerr, 1951). 
 
 77 
 
s 1 
 I 0.5 
 
 \ 
 
 5 
 
 
 i ; : 
 
 $ 
 
 ------ 
 
 
 — - 
 
 E 
 
 ._ 
 
 -- ■- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 \Ad) 
 
 
 
 
 
 
 
 
 
 
 
 
 -*\ - 
 
 d 
 
 : # 
 
 \r-z-z\— 
 
 — 
 
 
 — 
 
 
 
 
 
 _ Y 
 
 
 
 
 \N i 
 
 
 
 — 
 
 
 
 
 
 VX ! I 
 
 \\ 
 
 
 
 
 
 
 
 \\ 
 
 
 
 
 
 
 
 
 
 
 
 
 \ ! : \ ' 
 
 . 
 
 
 
 
 ^N^ 
 
 V ir^-Vi^ 
 
 
 
 -=p= 
 
 - t£V- | — -V — -n 
 
 
 
 
 
 \ i i \ 
 
 
 1\ 
 
 TV 
 
 — 
 
 
 w\ 
 
 ij 
 
 \ \ 
 
 
 — ■ 
 
 1 
 
 1 
 
 Wavelength in 
 
 Figure 2-2. Theoretical values of attenuation by rain and fog. Solid 
 curves show attenuation in rain of intensity. (a), 0.25 mm/hr 
 (drizzle); (b) , 1 mm/hr (light rain); (a), 4 mm/hr (moderate rain) 
 (d) , 16 mm/hr (heavy rain) . Dashed curves show attenuation in fog 
 or cloud; (e), 0.032 g/m 3 (visibility about 600 m) ; (/) , 0.32 g/m 1 
 
 (visibility about 120 m; (g) , 
 (From Kerr, 1951.) 
 
 2.3 g/m (visibility about 30 m) . 
 
 78 
 
-5 cd|cvj 
 
 
 CO 
 
 ii 
 
 
 ii 
 
 3Z* 
 
 rt 
 
 
 Pi 
 
 
 *J 
 
 
 C 
 
 
 u 
 
 
 u 
 
 
 ^ 
 
 
 +-> 
 
 
 
 <u 
 
 Ih 
 
 £1 
 
 0) 
 
 
 JO 
 
 > 
 
 E 
 
 "0 
 
 a 
 
 
 c 
 
 -t- 1 
 
 u 
 
 c 
 
 > 
 
 <u 
 
 CO 
 
 E 
 
 2 
 
 o 
 
 
 
 (/) 
 
 <D 
 
 3 
 
 
 o 
 
 a; 
 
 a; 
 
 3 
 
 CD 
 
 Td 
 
 ^ 
 
 
 4-> 
 
 >, 
 
 a) 
 
 Fh 
 
 * 
 
 ♦J 
 
 
 O 
 
 a 
 
 (- 
 
 • H 
 
 
 
 X 
 
 CO 
 
 l/> 
 
 M 
 
 c 
 
 
 o 
 
 bi) 
 
 •H 
 
 C 
 
 |J 
 
 
 cn 
 
 fH 
 
 
 <u 
 
 CD 
 
 +-> 
 
 5-< 
 
 79 
 
0.4 
 
 0.2 - 
 
 Fractional Contribution 
 to Liquid Water (M) 
 
 Empirical (Best) 
 
 From Marshall- Palmer 
 Exponential Dist. 
 
 0.4 
 
 1^ I 
 
 Fractional Contribution 
 to Reflectivity (Z) 
 
 From Best's Distribution 
 
 From Exponential 
 
 D/D, 
 
 Figure 4-1. Spectrum of fractional contribution to liquid water and frac- 
 tional contribution to radar reflectivity factor for Best's drop-size 
 distribution and for exponential distribution. 
 
 80 
 
8] 
 
Figure 4-3. Relationship between the coefficients of the modified gamma 
 distribution and the exponential factor G. 
 
 82 
 

 **>4%2§i Cumulus 
 
 V* 
 
 
 lis 
 
 *SS$i (a) 
 
 ht, o.©-.o «° 
 
 
 
 o U o 
 
 Cumulus 
 congestus 
 
 (b) 
 
 Cumulo- 
 nimbus 
 
 (c) 
 
 Figure 4-4. Samples of natural drop-size distributions in various clouds, 
 (Weickmann and aufm Kampe, 1953). 
 
 83 
 

 
 ~l 1 
 
 1 
 1 
 1 
 
 CIRCLES AROUND 
 INDICATE WIDTH 
 DROPLET SPECTR 
 
 < r 3~ ' i < « 
 
 DOTS 
 
 
 8 
 
 r 
 \ 
 
 \ 
 
 « < r r r, < 8 • 
 e<r r r,c, 2 . 
 
 1 
 
 
 o \>u 
 
 \ 
 
 \ 
 
 
 
 i 
 
 3 
 
 iJOOO 
 
 
 e ©^ 
 
 
 
 8) 1 © 
 
 • . <= / © 
 
 / 
 
 
 
 
 FIG 5 
 
 
 
 SQUARE RADIUS 
 
 Figure 4-5. Root -mean- square radii in cumuliform clouds versus height above 
 base. (Weickmann and aufm Kampe, 1953). 
 
 RADIUS IN MICRONS 
 
 Figure 4-6. Water-content spectra in fair-weather cumulus and cumulus 
 congestus. (Weickmann and aufm Kampe, 1953). 
 
 84 
 
LIQUID WATER CONTENT (gm~ 3 ) 
 Figure 4-7. Diagram showing the correlation between radar reflectivity 
 factor, the median volume diameter and the liquid water content for 
 clouds, based on 38 impinger observations at Mt . Washington N.H. , and 
 67 aircraft observations by Diem in German)'. (Boucher, 1952) 
 
 85 
 
0.2 
 
 2 2 
 Figure 5-1. Imaginary part of -K = - (m -l)/(ni +2) as a function of 
 
 wavelength and temperature. 
 
 86 
 
10.0 r- 
 
 1.0 
 
 0.1 = 
 
 0.01 
 
 0.001 
 
 0.0001 
 
 
 1 ' 1 ' 1 ' 1 1 
 
 1 ' 1 ' 1 
 
 
 
 
 
 E 
 
 / / / 
 
 • / / 
 
 <s / / 
 
 ° / / y 
 
 
 ; 
 
 = 
 
 ^^7/7/7 
 
 ^^yy^v^yy^y^y>^ 
 
 ^ 
 
 = 
 
 /^ /0 °o^l4// 
 
 '//^c^rs' Jzfsyl&s/ 
 
 - 
 
 
 
 
 
 ~ 
 
 
 K£y^yW^y% & 
 
 - 
 
 
 
 
 
 
 
 
 
 
 /~-^/ / /y y/^-&?/i 
 
 ^y^y^yy^f -^ 
 
 = 
 
 = 
 
 //^oyzS^^^^S 
 
 
 - 
 
 
 
 ysTsyvy*/ 
 
 
 
 ' • < \ //s /-y ZZ-~^/ y^/-^ ^^ 
 
 
 
 
 
 
 
 
 
 yt^/yM 
 
 
 z 
 
 ^y^^^^£^j^^^ 
 
 '^%& 
 
 - 
 
 ~ 
 
 y/yv^^y^^^>^^y^j^^yy^ 
 
 ?y<2> 
 
 - 
 
 - 
 
 '°^A^^^2^^^^P^ 
 
 
 - 
 
 - 
 
 
 
 - 
 
 _ 
 
 ^v 'yfy y^zsy yy^'jr' ^y/^y/ 
 
 
 - 
 
 
 
 
 
 
 
 
 
 _ 
 
 y> yy^y-^y^y^yj^^ysry/ 
 
 
 — 
 
 - 
 
 y^T^i^^y^^^yy^^/ 
 
 RAINPAD 
 
 
 
 s^-y^/^ <^y/^y^f * 
 
 
 
 z 
 
 ^^yyy^^yy^V £ 
 
 X= 3.22cm 
 
 - 
 
 
 '^yr y/ ^y<y / o 
 
 
 
 - 
 
 1 i 1 i 1 i 1 i 
 
 T=I0°C 
 
 = 
 
 - 
 
 !il, 
 
 - 
 
 10 <» 
 
 CO 
 
 CO 
 
 Q 
 
 I - 
 
 10 I 
 
 X 
 h- 
 O 
 
 I0 2 lu 
 
 10 
 
 4 C 
 
 10 20 30 40 50 60 
 REFLECTIVITY FACTOR (dBZ) 
 
 Figure 5-2. RAINPAD diagram of Atlas and Ulbrich (1974) graphically 
 
 demonstrating relationships between attenuation A, reflectivity factor 
 Z, total mass of liquid water M, rainfall rate R, and optical extinction 
 coefficient a. Calculations assume a Marshall-Palmer distribution and 
 fit a power law M = £AS to the Mie scattering solution. Temperature 
 and wavelength must be assumed. 
 
 87 
 
10 
 
 10' 
 
 10 
 
 -2 
 
 AX 
 
 10" 
 
 10 
 
 -4 
 
 — i i r 
 Cloudpad 
 
 i i 1 r 
 
 ^ <o ^ c^ <§>^W> 
 
 '°-60 -50 -40 -30 -20 -10 10 20 30 40 
 Reflectivity Factor (dBZ) 
 
 Figure 5-3. CLOUDPAD diagram in which Rayleigh scattering is assumed. Shows 
 relationships between attenuation A, reflectivity factor Z, total mass 
 of liquid water M , total number density N and optical extinction 
 
 coefficient a . When attenuation is plotted as the dimensionless 
 
 ext 
 quantity AX/I, diagram applies for al ^temperatures and wavelengths. 
 The quantity I = Imaginary part of - (m -l)/m +2). 
 
CO 
 
 E 
 
 10 In ?yl? 2 
 
 Figure 5-4a. Error in calculation of mass of liquid water assuming an error 
 of + 3 dB in measured power difference between radars of two different 
 wavelengths. 
 
 89 
 
CO 
 
 E 
 
 Range 10 km 
 
 Figure 5-4b. Error in calculation of mass of liquid water assuming an error 
 ^0.3 dB/km in measured attenuation between ranges r and r + A. 
 
 90 
 
14 
 
 12 
 
 10 
 
 E 
 
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 Error Bars if Reading Accuracy of 
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 Figure 5-5a. Error in calculation of median mass diameter for assumed 
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 Figure 5-5b. Error in calculation of median mass diameter for an assumed 
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 Figure 6-3. Relationship between drop-shape (a/b) and ratio of backscattered 
 power at vertical and horizontal polarizations for horizontal 
 propagation. 
 
 95 
 
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 PROLATES 
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 10 
 
 10 
 
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 Figure 6-4. Relationship between drop-shape (a/b) and phase difference 
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 Figure 6-9. Ratio of backscattered power in vertical and horizontal 
 
 polarizations for various zenith angles assuming spheroidal scatterers 
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 100 
 
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 Figure 6-10. Ratio of backscattered power in vertical and horizontal 
 
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 a/b. Conditions otherwise as indicated on figure. 
 
 101 
 
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 Figure 6-11. Ratio of backscattered power in vertical and horizontal 
 
 polarizations for various zenith angles assuming spheroidal scatterers 
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 102 
 
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 Figure 6-12. Ratio of backscattered power in vertical and horizontal 
 
 polarizations for various zenith angles assuming spheroidal scatterers 
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 103 
 
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 104 
 
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 Figure 6-14. Simultaneous profiles of reflectivity factor Z and root-mean- 
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 109 
 
Table IV 
 
 pfE^ 3 EN.D. f 
 
 Diem's cloud 
 
 types 
 
 (gr m" 
 
 0.32 
 
 3 ) 
 
 Z (mm m ) 
 I.l8xl0" 3 
 
 (m) 
 
 CU 
 
 11.2xl0~ 
 
 ou 2 
 
 
 0.87 
 
 
 2.76xl0 -2 
 
 20.6x10" 
 
 ■6C 
 
 
 0.09 
 
 
 3-53x10 
 
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 0.28 
 
 
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 NS 
 
 
 0.U0 
 
 
 l.UOxlO' 2 
 
 23.UxlO 
 
 ST 
 
 0.29 
 
 1 . 29x10 
 
 2U.lxl0" 
 
 110 
 

 
 Table 
 
 V. 
 
 
 
 
 
 South Park 
 
 Cumulus 
 
 
 N T (cm 3 ) 
 
 D (pa) 
 
 M (gr m" 3 ) 
 
 
 D q (pjn) 
 
 Z (mm m ) 
 
 687 
 
 8.1+ 
 
 .216 
 
 
 30.8 
 
 .0163 
 
 72 
 
 7.1+ 
 
 .015 
 
 
 27.2 
 
 7.8x10'^ 
 
 1+87 
 
 9-7 
 
 .231+ 
 
 
 35.6 
 
 2.7xl0" 2 
 
 565 
 
 9-3 
 
 .236 
 
 
 3fc.l 
 
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 l+6l 
 
 9-9 
 
 .233 
 
 
 36.3 
 
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 559 
 
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 35.5 
 
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 26.1 
 
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 30.5 
 
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 209 
 
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 29.7 
 
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 352 
 
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 111 
 
ft £ E 
 
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 in X X 
 
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 r-» f) h 
 
 on 
 
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 r-l ^H CM 
 
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 112 
 
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 Atlas, David, (1954). The estimation of cloud parameters by radar. 
 
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