Optical rain gauge using a divergent beam Ting-i Wang, R. S. Lawrence, and M. K. Tsay We have shown that path-averaged rain rates can be obtained from the raindrop-induced amplitude scintil- lations of a divergent laser beam (spherical wave case). We found that the rain rate obtained from a diver- gent beam is less sensitive to drop-size distribution than that from a collimated beam. However, the path- weighting function is heavily weighted toward the receiving end in the spherical wave case, whereas in the plane wavecase, it is almost uniformly weighted along the optical path. The theory was confirmed by obser- vations on two optical paths, one using a collimated beam on a 200-m path, the other using a divergent beam on a 1000-mpath. The results for the longer path show a saturation effect for rain rates higher than 12 mm/h. 1. Introduction The extreme spatial variation of rain has long posed a problem for meteorological and hydrological studies in estimating area precipitation by using point rain gauges. Crane 1 performed an experiment to measure simultaneously the spatial variations of drop-size dis- tributions and rain rates using adjacent pieces of filter paper each with a 440-cm 2 collecting area. He pointed out that ". . . it is evident that an extreme variation is possible for adjacent samples. This points to the problem of using a very small sample size." Several investigations of optical and IR wave transmission through precipitation- 9 suggest that path-averaged rain parameters may be deduced from line-of-sight at- tenuation measurements. Such line-averaged optical measurements significantly reduce the random fluc- tuations noted in the rain rate measured by point sen- sors. We have developed a rain-measuring technique that uses the rainfall-induced irradiance (or amplitude) scintillations of a laser beam' 0 -' 3 instead of the atten- uation of the beam. In an earlier paper, 13 we presented a theory of a method for measuring path-averaged rain rates by using the standard deviation (or variance) of the rainfall-induced irradiance scintillation of a colli- mated laser beam (plane wave) detected by a single line detector. This technique was successfully confirmed by experiments. 13 From a practical viewpoint a di- vergent beam source (spherical wave) is easier to con- struct and use than a collimated beam (plane wave). In this paper, we generalize the technique to include the spherical wave case. The authors are with NOAA Environmental Research Laboratories, Wave Propagation Laboratory, Boulder, Colorado 80303. Received 28 April 1980. For a spherical wave incident on a spherical raindrop (Fig. 1) the scattered field can be approximated by two components: the field induced by diffraction of the sharp boundary of the drop and the light passing through the water drop. 14 The transmitted field is much weakerthan the diffractivefield if we assume that the wavelength of the incident wave is much shorter than the radius of the drop (a reasonable assumption in the optical and IR band), and that the change of re- fractive index from unity is not small (i.e., ka >> 1 and ka An >> 1, where k is the wave number a is the radius of the drop, and 1 + An is the refractive index of water). Thus the total scattered field is approximately equal to the diffractive field. The scattered field can be con- sidered as a spherical wave emitted from the center of the raindrop because we are only interested in the far- field solution. The interference at the receiving plane, caused by the incident wave and the scattered wave, gives the fine fringes (scintillations) shown in Fig. 1. We obtain the path-averaged rain rates by observingthe movement of the scintillations induced by falling rain- drops randomly located along the optical path. II. Theory Letting E 0 and Es denote the electric field of the in- cident and scattered waves, respectively, Van de Hulst1 4 gives exp(-ikr) expl-ik[(L -X) 2 + p 2 ]1/ 2 1 ikr ikL (1) X 2J1(flO) + exp(-ikL) ikL (2) where L is the path length between transmitter and receiver; r, x, and 0 are coordinates (as shown in Fig. 1); 1 November 1980 / Vol. 19, No. 21 / APPLIED OPTICS 3617
Transcript
Optical rain gauge using a divergent beam
Ting-i Wang, R. S. Lawrence, and M. K. Tsay
We have shown that path-averaged rain rates can be obtained from the raindrop-induced amplitude scintil-lations of a divergent laser beam (spherical wave case). We found that the rain rate obtained from a diver-gent beam is less sensitive to drop-size distribution than that from a collimated beam. However, the path-weighting function is heavily weighted toward the receiving end in the spherical wave case, whereas in theplane wave case, it is almost uniformly weighted along the optical path. The theory was confirmed by obser-vations on two optical paths, one using a collimated beam on a 200-m path, the other using a divergent beamon a 1000-m path. The results for the longer path show a saturation effect for rain rates higher than 12mm/h.
1. Introduction
The extreme spatial variation of rain has long poseda problem for meteorological and hydrological studiesin estimating area precipitation by using point raingauges. Crane1 performed an experiment to measuresimultaneously the spatial variations of drop-size dis-tributions and rain rates using adjacent pieces of filterpaper each with a 440-cm2 collecting area. He pointedout that ". . . it is evident that an extreme variation ispossible for adjacent samples. This points to theproblem of using a very small sample size." Severalinvestigations of optical and IR wave transmissionthrough precipitation- 9 suggest that path-averagedrain parameters may be deduced from line-of-sight at-tenuation measurements. Such line-averaged opticalmeasurements significantly reduce the random fluc-tuations noted in the rain rate measured by point sen-sors. We have developed a rain-measuring techniquethat uses the rainfall-induced irradiance (or amplitude)scintillations of a laser beam'0 -'3 instead of the atten-uation of the beam. In an earlier paper, 13 we presenteda theory of a method for measuring path-averaged rainrates by using the standard deviation (or variance) ofthe rainfall-induced irradiance scintillation of a colli-mated laser beam (plane wave) detected by a single linedetector. This technique was successfully confirmedby experiments.13 From a practical viewpoint a di-vergent beam source (spherical wave) is easier to con-struct and use than a collimated beam (plane wave). Inthis paper, we generalize the technique to include thespherical wave case.
The authors are with NOAA Environmental Research Laboratories,Wave Propagation Laboratory, Boulder, Colorado 80303.
Received 28 April 1980.
For a spherical wave incident on a spherical raindrop(Fig. 1) the scattered field can be approximated by twocomponents: the field induced by diffraction of thesharp boundary of the drop and the light passingthrough the water drop.14 The transmitted field ismuch weaker than the diffractive field if we assume thatthe wavelength of the incident wave is much shorterthan the radius of the drop (a reasonable assumptionin the optical and IR band), and that the change of re-fractive index from unity is not small (i.e., ka >> 1 andka An >> 1, where k is the wave number a is the radiusof the drop, and 1 + An is the refractive index of water).Thus the total scattered field is approximately equal tothe diffractive field. The scattered field can be con-sidered as a spherical wave emitted from the center ofthe raindrop because we are only interested in the far-field solution. The interference at the receiving plane,caused by the incident wave and the scattered wave,gives the fine fringes (scintillations) shown in Fig. 1.We obtain the path-averaged rain rates by observing themovement of the scintillations induced by falling rain-drops randomly located along the optical path.
II. Theory
Letting E0 and Es denote the electric field of the in-cident and scattered waves, respectively, Van de Hulst14
gives
exp(-ikr) expl-ik[(L -X)2 + p2]1/2 1
ikr ikL
(1)X 2J1(flO) +
exp(-ikL)ikL
(2)
where L is the path length between transmitter andreceiver; r, x, and 0 are coordinates (as shown in Fig. 1);
Fig. 1. Spherical wave incident on a spherical raindrop. Interfer-ence at the receiving plane, caused by the incident wave and scat-tered wave, gives the fine fringes. Angles 0 and 0 are defined as
0 = p/(L - x), and 0 = p/(L-x) + plx.
= ka = 2ira/X is the drop-size parameter, and a is thedrop radius; t = 271(m - 1), where m = 1.33, is the re-fractive index of water; J is the first-order Besselfunction of the first kind. The first term in the sum ofEq. (1) arises from diffraction, whereas the second termarises from the light that passes through the sphere.Because the raindrops are large compared with thewavelength (i.e., >> 1), the first term is the dominantterm; only it will be kept. This is equivalent to as-suming that refractive index of the sphere is infinite, i.e.,the sphere is considered to be opaque. Because we areinterested in only the far-field solution, where theFresnel-zone size (x)1/2 is much larger than the dropsize a and the longitudinal coordinate x is much largerthan the transverse coordinate p, we have
In deriving Eq. (3), we have used the approximations,valid for x >> p, that r - x p2/2x in the phase andr i x in the denominator. The first factor on theright-hand side describes the fringes of the interferencepattern caused by the interaction of the incident wavewith the scattered spherical wave that originates fromthe sphere. The second factor is the well-known far-field Airy-disk diffraction pattern of a sphere.
We use Tatarskii's approach1 5 for a wave that prop-agates in a random medium to calculate the covarianceand temporal power spectrum of the amplitude scin-tillations. We define the amplitude scintillation(normalized to the unscattered field E 0) as
where Re and Im are the real and imaginary part oper-ators, respectively. In the weak scattering case, 1E8/E 0 I<< 1, and, to first order, X is approximately equal toRe(E 3/E 0). If the receiver is located at the origin of
coordinates, the amplitude scintillation x(x,y,z) causedby a raindrop located at coordinates (x,y,z) is
x(x,y,z) = a sin [ k( 2 + ) 1[x(l - x/L)
X Jka(y2 + z 2)12/[x( - x/L)]j (5)(y2 + z2)1/2/(l - x/L)
For a horizontal line detector with an infinitesimalheight, the amplitude scintillation XI caused by a rain-drop located at coordinates (x,y,z) is the integration ofEq. (5) over y (the line detector is extended along they axis) from -1/2 to 1/2, where is the length of the linedetector. If we assume that the length is much longerthan a Fresnel-zone size for the path length L [i.e., I >>(XL)1/2; X = wavelength] and a << (L)1 2, the integra-tion can be accomplished by the method of stationaryphase with the result that
- (1 -x/L) [2 rx(l- x/L)]1/2
Xi = [ kz J HZJ
! XT = [2x(1 xL 4 ZIIY
;xIL) irJ~x(1-~ if IyI > 1/2. (6)
Closely following the derivation in Wang et al.,' 3 weobtain the variance of the amplitude scintillation de-tected by a single line detector:
a2 -- (Xl)
= 8.84 X 10-10 -1 fL dxh(x)(1 -x/L)2
X f daa-/ 2p(a,x), (7)
where ( ) denotes an ensemble average, h(x) is thetotal rain rate at path position x, and p (a,x) is theprobability-density function of the raindrop size at thatposition.
One problem in measuring the path-averaged rainrate from the variance of the amplitude scintillation isthat turbulent variations of refractive index producescintillations that contaminate the rain-induced scin-tillation, especially at frequencies below 500 Hz. Toseparate the contributions from turbulence and rain,a temporal-frequency filter must be applied to the signalbefore the variance measurement is made. Closelyfollowing the derivation in Wang et al., 13 afterstraightforward but tedious manipulations, we obtainthe filtered amplitude variance for the spherical wavecase of a single line detector as follows:
2p = 2.087r X 10-1 L- 1 J duh(u)(1 -)2
X Jadap(a,u) rfdf Jl(rfu (/8)°°o fh (Or fU2 /100)2
where u = x/L, f, = low cutoff frequency, and f2 = highcutoff frequency. For f < f2 < 6 kHz, the Besselfunction in Eq. (8) can be approximated by the first twoterms of its polynomial expansion, and Eq. (8) reducesto
To see how sensitive 2p is to drop size distribution, weassume that raindrops follow a Marshall-Palmer dis-tribution.16 For this case, we can show that
where f1 and f2 are in kilohertz, and Af = f2 - f, is thebandwidth of the filter in kilohertz. To investigatefurther the contribution of the second term (which is afunction of drop size distribution) in Eq. (12), we as-sume that the rain rate is uniform along the optical path,i.e., h(u) = h. Equation (12) can then be simplifiedto
a2 = 5.43 X 10- 9 L-1Afh
x [1 - 1.29 x 10-3 h0 21(f2 + f112 + f2)]. (13)
For comparison, we also put down the variance of theamplitude scintillation of a plane wave (Eq. (29) in Ref.13:
In comparing Eqs. (13) and (14), the contribution of thesecond term is much smaller for the spherical wave case.In other words, the measured variance of the amplitudescintillation of a divergent beam is less sensitive to dropsize distribution than that of a collimated beam.Therefore, a higher cutoff frequency can be used for thespherical wave case, which has the definite advantageof improving the rejection of turbulence contamination.If we use fl < f2 < 2 kHz, the second term in Eq. (13) is<5% for rain rates up to 250 mm/h, and this term canbe ignored. In this case, Eq. (12) is reduced to
C.p = 1.63 X 10-8 L-lAf f du(1 - ) 2h(u)
= 5.43 X 10- 9L1- 1Afh, (15)
where h is the path-averaged rain rate along the opticalpath Unlike the path-weighting function of the planewave case, which is uniformly distributed along theoptical path, the path-weighting function of a sphericalwave is heavily weighted toward the receiving end.(The transmitter is located at u = 1, and the receiver islocated at u = 0.) However, this derivation is only ap-plicable to the far-field case; it is not valid when the
detector is in the near field of raindrops. Nevertheless,it is safe to claim that for a spherical wave, the measuredrain rates are the average over the half of the opticalpath near the receiving end.
Ill. Experimental Results
Two optical paths (200 and 1000 m) were set up tomeasure path-averaged rain and snow rates for a 1-yearperiod starting in June 1977 at Table Mountain, a mesa12 km north of Boulder, Colo. Five tipping-bucket raingauges, each sending a pulse to a multichannel chartrecorder for each 0.254 mm of accumulated rainfall,were located under the path at 15, 190, 500, 700, and1000 m from the receiver. The rain gauges were labeledfrom 1 to 5, respectively. For the 200-m optical system,the transmitter is a He-Ne laser followed by an opticalsystem that expands the beam to produce a collimatedbeam 20 cm in diameter. For the 1000-m optical sys-tem, the laser beam was expanded to form a 2-m (ver-tical) X 0.3-m (horizontal) beam at the receiving end toovercome the vertical drift caused by the variabletemperature gradients of the atmosphere. The re-ceivers were horizontal line detectors 25 cm long and0.15 cm high. Interference filters with a passband of0.002 Am were used. Also multiple field stops were usedto exclude background light. The detected scintillationsignals were recorded on conventional analog stereo taperecorders for later processing.
The detected scintillation signals pass through a1-kHz narrow bandpass filter. The variances of theamplitude scintillations were recorded on chart re-corders. We used Eq. (14) to obtain the rain rates of the200-m path (collimated beam case) and Eq. (15) for the1000-m path (divergent beam case). Because we do nothave the absolute calibration for rain rates from theoptical measurements (see Ref. 12), we arbitrarily ex-panded the scale so that the optically measured rainrates had the same total amount of water content as thatof the tipping-bucket rain gauges. For the 200-m path,we compared the optical data with the average of raingauges 1 and 2. For the 1000-m path, we used raingauges 1 to 4 (5 was inoperative during this experiment).An hour of rain data was recorded during the earlymorning of 27 Aug. 1977. The comparison of the rainrates obtained for the 200-m and 1000-m paths is shownin Fig. 2. The agreement is excellent although, due tothe spatial averaging, the rain rate ofer 200 m has morefluctuations than that averaged over 1000 m. A scatterdiagram of the rain rates obtained for the two differentpaths is shown in Fig. 3. For rain rates less than -10mm/h, the linearity between the two measurements isagain excellent. A scatter diagram of the rain rate fromthe 1000-m path vs the average rain rates of the tip-ping-bucket rain gauges 1 to 4 is shown in Fig. 4. Theagreement is good although more scattered than in Fig.3. This is because the optical measurement is a spatiallyaveraged quantity, whereas the tipping-bucket raingauges are point measurements.
A mixture of heavy hail and rain data was recordedon the afternoon of 17 May 1978. A comparison of therain rates of the two paths is shown in Fig. 5. For rain
Fig. 2. Comparison of rain rates obtained for the 200- and 1000-mpaths 27 Aug. 1977. Because of spatial averaging, the rain rate over
200 m has more fluctuations than that averaged over 1000 m.
E X
cc
August 27, 1977
0 0 5 10 15
Rain Rate (200 m, mm/h)Fig. 3. Scatter diagram of the rain rates of the two different paths
as shown in Fig. 2. Solid line indicates the best linear regression.
rates <12 mm/h, the agreement between the two setsof rain rates is good. However, for heavy rain rates (h> 12 mm/h), there are apparently some saturation ef-fects on the 1000-m path (Fig. 6). We believe a satu-ration effect is due to the multiple-scattering of rain-drops. In an earlier paper,'0 we estimated the range ofvalidity of the single-scattering assumption by assumingthat the onset of important multiple-scattering effectsoccurs when the entire laser-beam cross section isblocked, on the average, by at least one falling raindrop.Then statistically each ray is scattered at least oncebefore it reaches the receiving plane. If we also assumethat raindrops follow a Marshall-Palmer distribution,the range limitation of the single-scattering theory ofa given rain rate h is [see Eq. (A8) of Ref. 10)
Lm = 5.9 X 10 3 h-06 8 5, (16)
or for a given pathlength L, the maximum rain rate is
_0 5 10 15Average of 4 Gauges (mm/h)
Fig. 4. Scatter diagram of the rain rates from the 1000-m path vsthe average rain rates of four tipping-bucket rain gauges located at15, 190, 500, and 700 n from the receiving end under the optical
path.
hm = 3.2 x 105L-1.4 6, (17)
where h is in mm/h and L in meters. Using Eq. (17), wehave hm = 13.3 mm/h for L = 1000 m. This agrees withour observed value of 12 mm/h. For a 200-m path, themaximum rain rate, measured accurately by the sin-gle-scattering theory, is 140 mm/h.IV. Conclusions
We have developed a theory, confirmed by experi-mental results, showing that path-weighted rain ratescan be obtained from the amplitude scintillation of adiverging laser beam. The instrument is simple andinexpensive, and the construction of the source is lesscritical than for the previous technique,"' 3 which re-quires a collimated bean. However, because of satu-ration, both scintillation techniques are limited to pathlengths shorter than 200 m for rain rates up to 140mm/h.
0c l 1 | I Books continued from page 3616stances. Interference microscopes for investigating transparent
Hail and Rain Rain micro-objects are also mentioned. Finally Chapter 10 deals with40 methods of enhancing the sensitivity of fringe measurements such
as multiple beam interferometry and photoelectric methods of mea-suring changes in path differences.
30 The book is written at a fairly elementary level and does not require
the reader to have much prior detailed theoretical knowledge to un-c 20 200 m derstand it. The small amount of mathematics is kept as simple asa: 11 / possible. There is a good deal of specific technical information about
A 1000 m individual instruments. The book contains 7 tables, 140 illustrations,10, - ,7 and 161 references to published literature. The presentation is
.- \ Amp careful, clear, and detailed; as a handbook it should prove useful to
a , engineers, physicists, and technicians engaged in making practical1500 1600 1700 use of interferometers.
MDT, May 17, 1978B. J. USCINSKI
Fig. 5. Comparison of rain rates obtained for the 200 and 1000-mpath of a mixture of heavy hail and rain 17 May 1978.
15 20 25 30 35 40 45Rain Rate (200 m)
Fig. 6. Scatter diagram of the rates of the two different paths shownin Fig. 5. Retrieved rain rates of the 1000-m path apparently show
some saturation effects for rain rates higher than 12 mm/h.
M. K. Tsay-on leave from the National CentralUniversity in Taiwan-was a visiting scientist with theWave Propagation Laboratory during 1978-1979.
References1. R. K. Crane, "Microwave Scattering Parameters for New England
Rain," Technical Report, Lincoln Laboratory, MIT, Lexington,Mass. (Oct. 1966).
2. D. Atlas, J. Meteorol. 10, 486 (1953).3. A. Arnulf and J. Bricard, J. Opt. Soc. Am. 47, 491 (1957).4. R. W. Wilson and A. A. Penzias, Nature 211, 1081 (1966).5. T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 45, 301 (1966).6. T. S. Chu and D. C. Hogg, Bell Syst. Tech. J. 47, 723 (1968).7. D. Atlas and C. W. Ubrich, J. Rech. Atmos. 8, 275 (1974).8. D. Atlas and C. W. Ubrich, at Seventeenth Radar Meteorology
Conference, Seattle, Wash., American Meteorological Societypreprints, p. 406 (1976).
9. D. C. Hogg and T. S. Chu, Proc. IEEE 63, 1308 (1975).10. Ting-i Wang and S. F. Clifford, J. Opt. Soc. Am. 65, 927
(1975).11. Ting-i Wang, G. Lerfald, R. S. Lawrence, and S. F. Clifford, Appl.
Opt. 16, 2236 (1977).12. Ting-i Wang and R. S. Lawrence, Appl. Opt. 16, 3176 (1977).13. Ting-i Wang, K. B. Earnshaw, and R. S. Lawrence, Appl. Opt. 17,
384 (1978).14. H. C. Van de Hulst, Light Scattering by Small Particles (Wiley,
New York, 1957).15. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on
16. J. S. Marshall and W McK. Palmer, J. Meteorol. 5, 165 (1948).
Neutron Interferometry. Edited by U. BONSE and H. RAUCH.Oxford University Press, New York, 1979. 488 pp. $58.00.
This book contains the proceedings of the international workshopon neutron interferometry held in June 1978 at the Max von Laue-Paul Langevin Institute in Grenoble-the first book on neutron in-terferometry to appear. Neutron interferometry is a very recentdevelopment; its first realization dates about 1974. It appears to bea powerful research tool, and these proceedings summarize very nicelythe state of the art of the equipment and some of the applications ofthe method.
The papers appearing in the book by forty-four researchers in thefield of interferometry are divided into three areas of interest:methods and instrumentation (ten papers), application (sixteen pa-pers), and related techniques of interferometry (eight papers). Thefirst group of papers deals with the basic problems of the constructionand performance of a neutron interferometer. Nothing is said aboutthe nature of the sources of neutrons (reactors) and what is desirableor undesirable about them. No doubt this section will greatly appealto experimentalists. However, it is the second group of papers(covering a little more than 200 pages) that the reviewer found mostinteresting. The question of what one can do with neutrons thatcannot be done more conveniently with electrons or various types ofelectromagnetic radiation seemed especially intriguing. As one mightanticipate magnetic properties of materials can be studied as well asspin effects of atoms and nuclei. But some unexpected phenomenaare also being examined, for example, the gravitational interactionon the microscopic level. The third set of papers deals with x-ray,electron, and optical interferometry and serves to contrast the varioustypes of interferometry.
This is an important book since it is at present the only one dealingwith a newly emerging technique that is sure to yield much valuablenew information as it matures. Consequently every college anduniversity library should include this book in its collection.
ALBERT C. CLAUS
Radio and Acoustic Holography. Edited by G. E. KORBUKOVand S. B. KULAKOV. Nauka Press, Leningrad, 1976. 144 pp. 61kop.
In this collection of ten papers dealing with present techniques inmicrowave and acoustic holography, about two-thirds of the book isconcerned with microwave holography.
The first paper considers the main stages involved in producingmicrowave holograms, i.e., the production, conversion, recording, andreconstruction processes. The use of amplitude-modulation, su-
