temperature profiling by rayleigh-scattering lidar

8
Temperature profiling by Rayleigh-scattering lidar R. L. Schwiesow and L. Lading A lidar with high spectral resolution can be used to measure atmospheric temperature profiles according to conservative performance calculations. The technique analyzed relies on determining the temperature- dependent Rayleigh-scattering linewidth with two stabilized Michelson interferometers in parallel. From ratios of four integrated flux values from two photomultipliers, one can determine temperature profiles within a 1 K standard deviation to 5 km with 50-m height resolution in 11/4 min using a laser of 1-W average power and a telescope of 30-cm diam. 1. Orientation and Results Is it feasible to measure atmospheric temperature profiles with a lidar that determines the Rayleigh- scattering linewidth? In this paper we analyze a new method that uses a Michelson interferometer and con- clude that a lidar using reasonable component param- eters can obtain temperature data to 5 km with a + 1 K uncertainty and 50-m height resolution in an averaging time of 74 sec, even under daytime conditions. Our purpose is to present a method for remote temperature measurement using laser backscatter and to describe the linewidth measurement technique in enough detail that one can evaluate the validity of the analysis and extend the results to related atmospheric remote- sensing problems or to other lidar parameters. Atmospheric researchers and forecasters need tem- perature profiles to correlate with meteorological events. For example, urban pollution trapping depends on the location, strength, and persistence of temperature in- versions. The severity of convective storms is related to the temperature lapse rate, and cloud ceiling and visibility are affected by the temperature profile as well as by humidity variables. Fiocco and co-workers", 2 made a remote temperature measurement that used the Rayleigh-scattering line- width. They employed Fabry-Perot spectral analysis to determine the temperature at 4-km height averaged over a 2-km height interval and achieved a measure- ment accuracy of a few degrees kelvin with a 1-h aver- aging period at night. Lading et al. 3 discussed tem- perature profiling using Michelson interferometric Ronald Schwiesow is with NOAA Environmental Research Labo- ratories, Boulder, Colorado 80303; Lars Lading is with Riso National Laboratory, Postbox 49, DK-4000Roskilde, Denmark. Received 12 January 1981. measurement of the Rayleigh linewidth but did not analyze the performance in detail. Johnson 4 presented aspects of a coherent (heterodyne) lidar for temperature sensing by means of Rayleigh linewidth. Other ways to measure temperature remotely include rotational Raman scattering and differential absorption, which we will be able to compare with the Rayleigh-scattering method after seeing its quantitative features. II. Analysis The linewidth of Rayleigh scattering depends on temperature because the spectral distribution is caused by the Doppler shift from moving molecules, and the translational velocity distribution is linked to temper- ature by the Maxwell-Boltzmann relationship. The temperature profiling lidar presented here depends on molecular scattering, although aerosol scattering and sky background light are present as interference. Because many important meteorologicalphenomena are based in the lowest 5 km of the atmosphere, we ar- bitrarily chose to evaluate the lidar performance at 5-km height. The analysis is divided into four topics: line shapes; interferometric measurements; temperature sensitivity; and signal levels. A. Line Shapes The scattering physics involved in the interaction of light near 0.5-,um wavelength with the atmosphere (dominated by homonuclear diatomic molecules) is not completely understood. If the molecules are nonin- teracting and randomly distributed, the backscatter spectrum is Gaussian, 2 as we review later. Fluctuations in density lead to additional spectral components. 5 Density fluctuations are composed of adiabatic (pres- sure) fluctuations, which propagate and give rise to frequency-shifted Brillouin peaks, and isobaric (en- tropy) fluctuations, which result in the Landau-Placzek unshifted band. Extensions of the hydrodynamic 1972 APPLIED OPTICS/ Vol. 20, No. 11 / 1 June 1981

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Page 1: Temperature profiling by Rayleigh-scattering lidar

Temperature profiling by Rayleigh-scattering lidar

R. L. Schwiesow and L. Lading

A lidar with high spectral resolution can be used to measure atmospheric temperature profiles according toconservative performance calculations. The technique analyzed relies on determining the temperature-dependent Rayleigh-scattering linewidth with two stabilized Michelson interferometers in parallel. Fromratios of four integrated flux values from two photomultipliers, one can determine temperature profileswithin a 1 K standard deviation to 5 km with 50-m height resolution in 11/4 min using a laser of 1-W averagepower and a telescope of 30-cm diam.

1. Orientation and Results

Is it feasible to measure atmospheric temperatureprofiles with a lidar that determines the Rayleigh-scattering linewidth? In this paper we analyze a newmethod that uses a Michelson interferometer and con-clude that a lidar using reasonable component param-eters can obtain temperature data to 5 km with a + 1 Kuncertainty and 50-m height resolution in an averagingtime of 74 sec, even under daytime conditions. Ourpurpose is to present a method for remote temperaturemeasurement using laser backscatter and to describethe linewidth measurement technique in enough detailthat one can evaluate the validity of the analysis andextend the results to related atmospheric remote-sensing problems or to other lidar parameters.

Atmospheric researchers and forecasters need tem-perature profiles to correlate with meteorological events.For example, urban pollution trapping depends on thelocation, strength, and persistence of temperature in-versions. The severity of convective storms is relatedto the temperature lapse rate, and cloud ceiling andvisibility are affected by the temperature profile as wellas by humidity variables.

Fiocco and co-workers", 2 made a remote temperaturemeasurement that used the Rayleigh-scattering line-width. They employed Fabry-Perot spectral analysisto determine the temperature at 4-km height averagedover a 2-km height interval and achieved a measure-ment accuracy of a few degrees kelvin with a 1-h aver-aging period at night. Lading et al.

3 discussed tem-perature profiling using Michelson interferometric

Ronald Schwiesow is with NOAA Environmental Research Labo-ratories, Boulder, Colorado 80303; Lars Lading is with Riso NationalLaboratory, Postbox 49, DK-4000 Roskilde, Denmark.

Received 12 January 1981.

measurement of the Rayleigh linewidth but did notanalyze the performance in detail. Johnson 4 presentedaspects of a coherent (heterodyne) lidar for temperaturesensing by means of Rayleigh linewidth. Other waysto measure temperature remotely include rotationalRaman scattering and differential absorption, which wewill be able to compare with the Rayleigh-scatteringmethod after seeing its quantitative features.

II. Analysis

The linewidth of Rayleigh scattering depends ontemperature because the spectral distribution is causedby the Doppler shift from moving molecules, and thetranslational velocity distribution is linked to temper-ature by the Maxwell-Boltzmann relationship. Thetemperature profiling lidar presented here depends onmolecular scattering, although aerosol scattering andsky background light are present as interference.

Because many important meteorological phenomenaare based in the lowest 5 km of the atmosphere, we ar-bitrarily chose to evaluate the lidar performance at 5-kmheight. The analysis is divided into four topics: lineshapes; interferometric measurements; temperaturesensitivity; and signal levels.

A. Line Shapes

The scattering physics involved in the interaction oflight near 0.5-,um wavelength with the atmosphere(dominated by homonuclear diatomic molecules) is notcompletely understood. If the molecules are nonin-teracting and randomly distributed, the backscatterspectrum is Gaussian,2 as we review later. Fluctuationsin density lead to additional spectral components.5Density fluctuations are composed of adiabatic (pres-sure) fluctuations, which propagate and give rise tofrequency-shifted Brillouin peaks, and isobaric (en-tropy) fluctuations, which result in the Landau-Placzekunshifted band. Extensions of the hydrodynamic

1972 APPLIED OPTICS / Vol. 20, No. 11 / 1 June 1981

Page 2: Temperature profiling by Rayleigh-scattering lidar

theory 6 and kinetic model7 8 and comparisons of the twoapproaches9 show that corrections to the Gaussian lineshape are necessary at the accuracies we wish toachieve.

Corrections to the Gaussian line shape in backscatterare of the order of 5% or less of the peak molecularscattering intensity.'103 Although the linewidth of theGaussian component depends only on temperature, thesmall corrections can depend on density as well astemperature. This means that an iterative procedurewill be necessary to achieve high accuracy. From atemperature estimate based on a Gaussian approxi-mation and a density estimate constrained by the sur-face pressure, the spectral corrections can be calculatedto yield a better temperature estimate, which can beused to refine the corrections, and so on.

The point is that whatever the microdynamics of thescattering centers, the backscattered linewidth averagedover times much longer than the mean time betweenmolecular collisions is a quantity that depends only onbulk thermodynamic parameters. The linewidth isprimarily a function of temperature with a small den-sity-dependent correction that can be estimated fromexisting theories or from temperature-density calibra-tion. For the instrumental accuracy purposes of thisstudy, a Gaussian approximation is appropriate. Foractual temperature measurements a refined line shapeis needed. Detailed line shape calculations will be donein a separate study.

The Maxwell distribution law of velocities in threedimensions 1 4 can be integrated over components in twodimensions and used in the expression for the nonre-lativistic Doppler shift to yield a backscattered Rayleighspectrum given by

I(w) = NIAI 2 (c2/47rW2bT)P12 exp[-(wo - W)

2c 2 I4W2bTj. (1)

Variable N is the number of scattering molecules in-volved, A is the amplitude of the electric field at thereceiver scattered from each molecule, w and wo are theangular frequencies of the received and transmittedlight, c is the speed of light, T is the (kinetic) tempera-ture of the molecules, and b is a constant factor timesthe mass of an individual molecule. In a gas composedof a mixture of types of molecules, the velocity distri-bution function for each type of molecule is indepen-dent,1 5 and the spectrum of air is a sum of terms (1) foreach component. We can approximate the Rayleighspectrum for air by a single term of form (1), where themass-dependent factor is the number-density-weightedaverage of the molecular masses, and the error in theapproximation is <0.1% of the maximum signal. Theanalysis of temperature sensitivity to follow is not de-pendent on the exact shape of the spectrum, so thatapproximation (1) is completely satisfactory for ourpurpose. Of course, calculation of the temperature,rather than temperature differences, from the spectrumand first principles would require a more accuraterepresentation of the Rayleigh spectrum of a mixed gas.Using the weighted average molecular mass, b can betaken as a constant in the troposphere with value b =5.74 X 102 m2 sec- 2 K-1.

2L

.150

0

00

a.

L

00 1

Delay dM½

2

Fig. 1. Envelope of the interferogram for a Gaussian Rayleighscattering line when an ideal Michelson interferometer is used.Output of the interferometer as a function of optical delay variesbetween the two limiting lines plotted. Delay is given in terms of thedimensionless variable dM 1 12 , where d is an optical path differenceand M is proportional to temperature. For a fixed delay, an increasein temperature increases the minimum signal or decreases the max-imum; i.e., an increase in temperature increases the decay of the in-terferogram contrast as the delay increases (narrows the interfero-gram) because the Rayleigh spectrum becomes wider. Incident signalat the input is 2L. Light not transmitted is reflected from an ideal

interferometer.

Rayleigh scattering gives a Gaussian line shape in theapproximation (1). The characteristic frequency scaleis a full spectral width at the exp(-1) points of thespectrum of approximately Av = 3.38 GHz at 296 K fora 488-nm lidar wavelength. For simplicity, we take thespectral shape of the sky background filtered by an in-terference prefilter to be a Gaussian much wider thanthe Rayleigh linewidth and the aerosol scatteringspectrum to be approximately a Gaussian much nar-rower than the molecular scattering linewidth.

B. Michelson Interferometry

Two features of Michelson interferometry are espe-cially useful for a temperature profiling lidar. TheMichelson interferometer makes efficient use of inci-dent photons because it transmits one half of the inci-dent optical power to the detector, on the average, andthe characteristic width of the interferogram envelopeis itself a measure of the second moment or width of thescattered spectrum, which is directly determined by thetemperature. A Michelson interferometer divides theamplitude of an incident signal into two parts, delaysone with respect to the other, and recombines delayedand undelayed parts of the signal at a detector.' 6 Foratmospheric lidar applications the input signal is nearlymonochromatic, its spectrum being a convolution of thetransmitted spectrum and the scattered spectra dis-cussed above, and has a nearly diffraction-limited an-gular field. The output of the interferometer as afunction of optical path difference 2d introduced be-tween delayed and undelayed parts of the signal is

I(d) = ( 4)NJAl 212 + exp(-4d2 w'bT/c4 )X [exp(+i2dwo/c) + exp(-i2do/c)I1. (2)

This is a relatively rapid cosine damped by a Gaussianenvelope. The envelope of the interferogram is shownin Fig. 1. Because the temperature dependence appearsonly in the envelope, it is only useful to sample the in-terferometer output at extrema of the oscillatory factor

1 June 1981 / Vol. 20, No. 11 / APPLIED OPTICS 1973

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0

a.

Z+L+BI

Z+L

ziocI2

Delay dM /2

Fig. 2. Envelope of a composite interferogram for a signal consistingof a comparatively broadband sky background of intensity 2Z andspectral bandwidth constant Y > 16M, Rayleigh scattering of in-tensity 2L, and bandwidth constant M (proportional to temperature),and narrowband aerosol scattering of intensity 2B 2L and band-width constant C < 10-4 M. Contribution of the background to in-terference minima is appproximately the constant value Z for alldelays of >0.2, and the contribution of the aerosol signal to the minimais -0 for delays of <3. The study evaluates a linewidth measurementtechnique measuring interferometer output at minima near a delayof dM 112 1 and dM"12

- 3, first in the absence of Rayleigh andaerosol scattering to determine Z and relative channel sensitivity andsecond with Rayleigh and aerosol backscatter to determine M (andthereby temperature) with more precision than a +20 K initial

guess.

or at path differences given by d = mc/4vo, where m isan integer and vo is the frequency of the incident light.Even m corresponds to an interference maximum andodd m to a minimum.

The count rate for Rayleigh-scattered photons atextrema of the output of the interferometer is given insimplified form as

S,(d,M) = L[1 + (-1)m exp(-d2M)]. (3)

Coefficient L is an intensity normalization factor thatis one-half of the photon count rate at the input to theinterferometer, and variable M is proportional to thesquare of the width of the Rayleigh spectrum and totemperature. M is given by

M = (4W2b/c4)T. (4)

For higher temperature T (wider spectra or larger M),the envelope function in Eq. (3) decays more rapidlywith d. To give some scale to the interferogram, thevalue of d at the exp(-1) point of the envelope for T =296 K and a wavelength of 488 nm is d[exp(-1)] = 2.82cm. This is a practically achievable value.

Because of the assumed Gaussian line shapes ofbackground and aerosol signals, the interferograms ofthese contributions are, in analogy to Eq. (3),

Sb(d,Y) = Z[1 + (-l)m exp(-d2Y)], (5)

S 0(d,C) = B[1 + (-l)m exp(-d2 C)1. (6)

Variable Y is proportional to the square of the prefilterbandpass, and C is proportional to the square of theaerosol scattering linewidth, so that for a practicalsystem Y >> M >> C. By superposition, the interfero-gram at the output of a lidar receiving all three signalcomponents is approximately a photon count rate of

S(d) = Sr + Sb + Sa. (7)

The envelope of this interferogram is sketched in Fig.2.

C. Temperature Sensitivity of the Interferogram

The pure Rayleigh interferogram in Eq. (3), con-taining unknowns M and L, can be determined inprinciple from two signal measurements, which couldbe photon counts for extrema at optical path differencesd, and d2, for example. The system analyzed for thispaper consists of two separate Michelson interferome-ters, with delays d, and d2, operating in parallel with acommon prefilter. Making the interferogram mea-surements at two delays simultaneously causes the de-termination of M and L to be independent of fluctua-tions in laser transmitter power and independent ofatmospheric nonstationarity over time periods shorterthan the measurement (integration) time in such pa-rameters as temperature, aerosol backscatter, skybackground, and path attenuation. The more realisticinterferogram of Eq. (7) is more complicated than Eq.(3), but the parameters in Eq. (7) may still be deter-mined with measurements at two different optical de-lays in a manner detailed below.

For purposes of performance analysis, we assumemeasurements at extrema of the interferogram. Tomake such measurements, each interferometer isservo-stabilized to the desired optical path difference.The distance between successive extrema in the inter-ferogram is c/4vo = 0.12 Aim, which indicates the preci-sion required in fixing the delays. The position of theextrema will shift with changes in laser frequency, so theservo must also track laser frequency fluctuations.Optical stabilization techniques and the effect of sta-bilization errors on accuracy are important and will bediscussed in a later paper. This analysis concentrateson the accuracy limitations inherent in the signal sta-tistics as they are viewed from the point of view of in-terferometry.

By setting the parallel Michelson interferometers totwo different well-chosen minima of the interferogram,the aerosol contribution may be ignored approximatelyand the effect of the background simplified. In addi-tion, there is a noise (uncertainty) advantage to workingat a minimum rather than a maximum if the shape ofthe interferogram envelope function is known to withina scaling factor,17 and the noise associated with theaerosol signal is essentially eliminated. The signal S(d)in Eq. (7) is almost independent of the aerosol compo-nent Sa at a minimum because Sa (d) changes veryslowly with d if C is small. The linewidth of the aerosolcomponent is mostly determined by the laser linewidthif the vertical wind is less than a few meters per second.For a laser linewidth of •10-2 times the Rayleigh width(or •34 MHz), the exponential factor in the aerosolenvelope, 1 - exp(-d 2 C), is •9 X 10-4 if the delay is dS 3/M1/2. Then if the total aerosol signal is the sameorder as the total Rayleigh signal (B - L), Sa is negli-gible for delays restricted to d S 3/Ml 2. The back-ground component Sb is almost a constant Sb Z ford > 1/M1/2 if the prefilter bandwidth is larger than four

1974 APPLIED OPTICS / Vol. 20, No. 11 / 1 June 1981

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times the Rayleigh width (or 20.01 nm). The ap-proximation Sb Z follows directly from the fact thatfor Y 2 16M, the exponential term in the backgroundenvelope, exp(-d 2Y), is •10-6. These limitations onthe laser linewidth, prefilter bandwidth, aerosol back-scatter coefficient, and optical delay are reasonable.

For C, B, Y, and d satisfying the restrictions, thesignals in the two Michelsons are

S1 = L[1 - exp(-d',M)] + Z + cor1,

gS2 = L[ - exp(-d'M)] + Z + cor2. (8)

The correction terms (cor) include Brillouin scatteringeffects and terms of the order of 10-3L resulting fromthe approximations Sb Z and Sa 0 0 at interferenceminima. Since the Michelson interferometers and theirseparate photomultipliers are not strictly identical,relative gain factor g allows for the differences in sen-sitivity between the two parallel analysis channels.Instead of using notation d for the delay at an extre-mum, Eqs. (8) use d, and d2 to denote specific minimaof the interferogram. The correction terms may beignored for calculating the sensitivity of the lidar totemperature changes, but corrections must be knownto determine M (and hence temperature) accuratelywithout separate calibration. Although a detailedanalysis of the correction terms is not an objective of thispaper, corrections to the background term depend onY a known instrumental variable. Corrections to theaerosol term involve the instrumental variable C andan estimate of count rate B to _10-3 times the accuracyin L. Brillouin corrections require an estimate oftemperature and height of each measured region of theatmosphere and may be improved by iteration usingincreasingly accurate temperature estimates.

The solution of system (8) requires four data values,which are simply four total photon counts (integratedsignal flux values) that may be collected over a numberof laser pulses. Two of the photon count summationsare collected from the range gate corresponding to theheight of interest, one sum from each interferometer.The other two total count values are from sky back-ground only, which is measured in the time betweenlaser pulses. The background data, which can be col-lected over longer times than a range gate, determineZ and g for all range gates. Of course the relative col-lection times for Rayleigh backscatter and backgroundcounts must be known accurately in order to scaleproperly Z to L.

The best optical delay is near that d for which thesensitivity of the count rate to changes in temperatureis a maximum. The other important influence on thechoice of d is the noise level (variance of the count rate).Because the count rate at interference minima is aslowly varying function of d, noise level considerationswill shift the optimum d only slightly from the maxi-mum-sensitivity d. The sensitivity of the interferogramminima is, from Eq. (8),

6S/oM = Ld 2 exp(-d 2 M), (9)

which is a maximum for

dMl = 1. (10)

Because the sensitivity in Eq. (9) does not change sign,it is obvious that a good estimate for L and M from Eq.(8) will be obtained if d, is set near the maximum sen-sitivity position and d2 is chosen to provide a referenceintensity that is nearly independent of temperature.Figure 1 implies that even for d 2 only as large as d2M112= 3, the signal S2 is almost independent of M. FromEq. (9), the sensitivity for d 2 = 3M-1/2 is -0.3% of themaximum sensitivity. Therefore, a well-chosen set ofdelays is near

d = /Ml/ 2 , d 2 = 3/M/2.

A useful way to analyze the data is to form theratio

R = (S1 - Z)I(9S 2 - Z) = - exp(-d2M)]/[1 - exp(-dM], (11)which follows from Eq. (8). Ratio R is directly ob-tainable at each range gate in terms of the four photoncount summations, which come from the output of eachof two minimum-stabilized Michelson interferometersunder each of the two conditions of range-gated Ray-leigh backscatter plus background and sky backgroundalone. Although R is a scalar determined from directdigital data, Eq. (11) is transcendental and is best solvedby iterative techniques.

The standard deviation of the data ratio is obtainedfrom Poisson statistics, which apply to the uncertaintyin the value of a photon count signal. Speckle effectsare not important if the signal is averaged over a numberof shots and if the field is a few times diffraction-limited.If N is a measured total photon count and if A(value)represents the standard deviation of the quantity, AN= N'12. For Z << L (an approximation justified in thenext section) the root of the sum of the squares of frac-tional uncertainties for AR is

AR 1.02L-1/2. (12)

This value comes from assuming that the fractionalerror of a product is approximately the root of the sumof the squares of the fractional errors of the factors (ifN >> 1) in the form

AR/R = [A(S - Z)/(S1 - Z)12 + [(gS 2 - Z(gS 2 -Z)121/2

[(AS1 /S02 + (A52 /S2 )2 ]"2 (1/S 1 + 1/S2)1/2

by Poisson statistics. For the assumed delays, one cancalculate R at the minima and express and 2 interms of L to get (12). The alternative form of (12) formaxima of the interferogram shows that even if B = 0(no aerosol scattering) the standard deviation of the.ratio derived from maxima data is larger than that de-rived from interference minima data, although thefractional uncertainty AR/R is larger at a minimum.

The standard deviation of the temperature estimatecan be evaluated by calculating RIOM from Eq. (11).Expressed as a finite difference, AR/AM is R/aM; andevaluated at dM' 1/2 = 1 and d2M'/2 = 3 with (12), thepartial derivative allows the fractional uncertainty ornormalized standard deviation to be expressed as

AT/T = AM/M 2.77L-1/2. (13)

1 June 1981 / Vol. 20, No. 11 / APPLIED OPTICS 1975

Page 5: Temperature profiling by Rayleigh-scattering lidar

The photon count from molecular scatter L is one halfof the total photons present at the entrance to each in-terferometer, which in turn is one half of the numberincident at the parallel-interferometer spectral analysispackage. From the relation between L and signalphotoelectrons received, one can obtain a required levelof total signal photons 4L for a desired standard de-viation ATIT in Eq. (13).

D. Signal Levels

A simplified lidar equation in form

N, = Etanlat7 exp(-2aZ)/(hvH 2) (14)

helps organize the dependence of a real system on lidarvariables. The number of detected signal photons N8depends on time-integrated laser energy Et, which is -1J for a cavity-dumped argon-ion laser operating at488-nm wavelength with a 1-W average power for 1 sec,for example. Both calculations of the Rayleigh-scat-tering cross section of air18 and measurements' 9 on N2give a value for the differential cross section a of 8.66 X10-32 m2 sr' 5% at this wavelength. The standard 2 0

number density n at an altitude H of 5 km is 1.53 X 1025m-3 ; the atmospheric attenuation coefficient a varieswidely around 0.03 km'1, so that the two-way trans-mission for a 5-km vertical range is -0.75. For the lidarsystem itself, we assume an overall quantum efficiency1 of 0.15 (photomultiplier at 20% and optics at 75%), atelescope area a of 7.07 X 10-2 m2 (30-cm primary di-ameter), spectral prefilter transmission t of 0.6, and aheight resolution of 50 m. At 488-nm wavelength theconversion between number of photons and energy(hv)-l is 2.457 X 1018 photons J-1. Overall, one expectsa return N8 of -3.1 X 104 photoelectrons J-1 for thelidar outlined here. The uncertainty in this estimatedepends on some of the arbitrary parameter valueschosen.

Radiance of the zenith sky depends on aerosol load-ing, solar position, and other variables. From an eval-uation of existing data, Hughes2' concludes that theo-retical values for luminance ranging from 2.86 X 104 lmm-2 sr-1 at 4.6° from the sun to 4.55 X 102 lm m 2 sr-1

at 84.20 from the sun are valid estimates for viewingfrom the surface. If the radiance of the blue sky variesas X-4, the luminance data imply a radiance at 488-nmwavelength of 6.4 X 10-1 W m-2 sr-' nm-1 close to thesun and 1.0 X 10-2 W m-2 sr'1 nm1 farther away.Measurements22 in a desert with high albedo rangedbetween 5.2 X 10-' W m-2 sr-' nm', 100 from the sun,and 1.9 X 10-1 W m-2 sr-' nm1, 900 away. Calcula-tions23 for sky brightness at 550-nm wavelength rangebetween 1.2 X 10-1 W m- 2 sr-' nm'1 for a 250-km vi-sual range atmosphere and low surface reflectivity and5.5 X 10-' W m-2 sr-1 nm'1 for 4.5-km surface visualrange and high surface reflectivity, all values at 200above the sun. Because a zenith pointing lidar pointsno closer than 180 to the sun at midlatitudes, a value of2.5 X 101 W m-2 sr-' nm-1 to within a factor of 2 is areasonable sky radiance to use for system performanceestimates.

The prototype temperature profiling lidar uses acommon telescope with a polarization coded beamswitch for transmitter and receiver. If the transceiveris diffraction-limited, the lidar has an angular field of1.2 X 10-1 sr. Beam spreading caused by atmosphericrefractive-index inhomogeneities will widen the fieldor reduce the coherent aperture diameter of the tele-scope,24 but the lidar is immune to beam wander effects.An increase in the field of view of 25 times (angular di-vergence 5 times larger or coherent aperture 6-cm diam)to cover beam spread for a vertically pointing systemwith 30-cm aperture is reasonable because the structurefunction for refractive-index inhomogeneities decreasesrapidly with height. The lidar will detect -2.0 back-ground photoelectrons per pulse with the wider field ofview, a pulse period of i- corresponding to of 0.333 X10-6 sec, a spectral prefilter bandwidth of 5 nm, and a,t, and q from the signal analysis. This small valuejustifies the approximation Z << L in the previous sec-tion.

The small sky background is observed because thelidar has an unusually small field of view, which in turnis possible because lasers with sufficiently narrowlinewidths for the temperature measurement applica-tion also are diffraction-limited (single spatial mode),and because the congruent transmit and receive opticsreduce pointing accuracy problems compared with thoseof separate beam systems. Sky background data areused in the signal analysis to compare the gains of twointerferometer channels, so artificial background lightmust be introduced if natural sky background is insuf-ficient.

Aerosol backscatter is variable in space and timebecause it depends on air mass history, local sources,humidity, and other effects. Elterman2 5 deducedaerosol attenuation coefficients at 550-nm wavelengthas a function of height by measuring aerosol scattering.Using his mean profile from 105 individual profiles andhis normalized phase function extrapolated from 135to 1800, one can determine an aerosol backscattercoefficient of Oa (5) = 0.90 X 10-6 m-1 sr-' at 5-km al-titude and extrapolate to a value of Oa (0) = 1.52 X 10-6m-1 sr'1 at the surface. Elterman's estimated uncer-tainty is -'20%. For comparison, the molecularbackscatter coefficient at 5 km from the values assumedfor this study is ,m (5) = 1.33 X 10-6 m'1 sr'1 ±5% witha standard number density and at the surface is flm (0)= 2.33 X 10-6 m- 1 sr-1 ±10%16 for the average of midlat-itude summer and winter atmospheres26 at 488 nm.Although the aerosol data were taken at slightly longerwavelength than the Rayleigh scattering calculations,it is apparent that the ratio of aerosol to molecularbackscatter coefficient is -1 in a large number of cases.This observation justifies the approximation BIL -1used earlier in the analysis of the interferogram to ne-glect the contribution of the aerosol component to in-tensity minima.

A widely used model of aerosol scattering coeffi-cients,26 based on calculations from attenuation andaerosol size distribution measurements rather thanscattering measurements, gives larger aerosol-to-mo-

1976 APPLIED OPTICS / Vol. 20, No. 11 / 1 June 1981

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Table 1. Assumed Lidar System Parameters

Height (range) H 5 kmHeight resolution 1 50 mTelescope diameter d 30 cmAverage laser power P 1 WOptical and detector quantum efficiency i7 0.15Prefilter transmission t 0.6Prefilter spectral width w 5 nmLaser wavelength X 488 nm

lecular backscatter coefficient ratios at the surface.The model values at 488-nm wavelength are BIL =/a (0)/lm (0) = 2.9 at the surface for clear conditions(surface visibility2 6 of 23 km) and a (0)/3m(0) = 14 forhazy conditions (surface visibility, 5 km). Althoughlarger than Elterman's measurements at the surface, themodel gives a smaller value, fa (5)/13m(5) = 0.16, at 5-kmaltitude. All indications are that the aerosol scatteringshould not be a problem in the interferometric analysisof the Rayleigh linewidth because the aerosol-to-mo-lecular backscattered signal ratio is S 10 and is smallerat higher altitudes where the linewidth measurementis more difficult.

Rotational Raman scattering also occurs near theRayleigh line, but its narrowband cross section is smallenough to be negligible compared with that of Rayleighscattering. The rotational Raman scattering crosssection of N2, for example2 7 (integrated over Stokes andanti-Stokes bands and both polarizations), is 1.64 X10-33 m 2 sr-1 8% at 488 nm, which is more than 50times smaller than the Rayleigh backscatter cross sec-tion. Furthermore, the width of the rotational Ramanband2 8 is -7.10 X 103 GHz, which is more than 2 X 103wider than the Rayleigh width, and the intensity at thecenter of the rotational Raman spectrum (zero fre-quency shift or the location of the Rayleigh line) is muchless than the maximum Raman signal. The peaks of theRaman bands are -170 GHz away from the 3.38-GHzwide Rayleigh line. The rotational Raman signal in-tegrated over the Rayleigh linewidth is <10-5 times theintegrated Rayleigh signal and may be neglected. Thesignal outside the Rayleigh line is wide enough to betreated as (low intensity) background light.

Assumed values for prototype lidar system parame-ters are summarized in Table I. These values werechosen to be compatible with commercially availabletechnology but not necessarily off-the-shelf units. Theatmospheric parameters assumed for performancecalculation are collected in Table II. The correlationlength is not the one usually used in atmospheric scin-tillation work, which includes beam wander, because asingle-ended lidar is sensitive only to beam spread.

Ill. Results

A. Temperature Calculations

The temperature in any chosen range gate can beobtained from the ratio of measured photon countsusing Eqs. (11) and (4). Fortunately, the ratio in Eq.(11) is a well-behaved function of M, and an analyticexpression for the first derivative of R (M) is easily ob-tained. Newton's method29 is an effective iterative

procedure for finding M (and T) given a ratio R in Eq.(11). With a simple desk calculator program of ap-proximately fifty steps, temperature converges to avalue with <0.01 K uncertainty after five iterations orfewer if the initial guess is within ±20 K of the finaltemperature value. Basic calculations for obtaining thetemperature from Rayleigh scattering by means of twostabilized Michelson interferometers are quite directand simple.

B. Comparison With Theoretical Results

Lading and Skov Jensen30 have discussed the theoryof various close-to-optimum estimators for the spectralwidth of a narrowband optical signal. The parallelMichelson approach analyzed in this study is essentiallythe two-arm interferometer 2 that Lading and SkovJensen found to have good performance in principle.Aspects of the optimum estimator apparent in thesystem discussed here include two delays to avoid thenecessity of absolute calibration of the lidar sensitivityat the 0.1% or better level and operation at interferenceminima to reduce absolute variance. More precisemaximum likelihood estimation theory30 provides re-

.fined values for d in Eq. (10) and the uncertaintycoefficient in Eq. (13). Differences between the theo-retical values and those obtained by our operationalapproximations are so small as to be of no practicalimportance.

C. PerformanceThe required measurement time for a practical lidar

to obtain a desired temperature accuracy can be esti-mated with the help of Eq. (13). For a standard de-viation of 1 K at a temperature of -273 K, photonsequivalent to 2.29 X 106 photoelectrons must be inci-dent on the input to the interferometer package. If theexample lidar obtains 3.1 X 104 photoelectrons J'from 5-km range, a measurement time of -74 sec wouldbe required with 1-W average transmitter power. It isnot helpful to assign an uncertainty to the measurementtime because many variables affecting the calculationare assumed values for an example.

Performance of 74 sec to measure temperature to a1 K standard deviation from 5-km altitude with 50-mheight resolution using a lidar of 1-W output and 30-cmtelescope diam can be extrapolated to other measure-ment requirements. For example, 0.5 K resolution

Table 11. Assumed Atmospheric Parameters

Atmospheric attenuation coefficient a 0.03 km'INumber density n at 5 km 1.53 X 125 m- 3

Differential Rayleigh backscatter 8.66 X 10-32 m2 sr' icoefficient 5%

Coherent aperture (correlation length) 6 cmfor beam spread

Zenith sky brightness 2.5 X 10-1 W m- 2 sr 1

Aerosol/molecular scattering ratio 1-10fl.Iflrn

Differential rotational Raman 1.64 X 10-33 m2 sr-' backscatter coefficient 8%

Rotational Raman linewidth 7.10 X 103 GHz

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Page 7: Temperature profiling by Rayleigh-scattering lidar

would require four times as long, or -5 min, because Lgoes as (T/AT) 2 in Eq. (13). Measurement time in-creases with the square of maximum height from Eq.(14). However, the temperature resolution with apractical lidar will not be better at heights below theselected maximum if the maximum is within the nearfield of the focused telescope optics, because the fieldstop of the near diffraction-limited system approxi-mately cancels the H 2 advantage of closer ranges.Measurement time for a chosen temperature resolutiondecreases with increasing average transmitter laserpower and with increasing telescope collector area.Thus, 0.5 K resolution to 2.5-km altitude would requireonly 74-sec measurement time with a 1-W 30-cm systemor only 20 sec with a 2-W 42-cm lidar.

D. Comparison with Results from Other Techniques

Temperature profiling using rotational Ramanscattering from the atmosphere has been demonstratedby Cooney3l and co-workers.283233 Compared with theRayleigh scattering cross section, the Raman crosssection is much smaller and distributed much morewidely in frequency as discussed in Sec. II.D. Analyzingthe Raman spectrum with spectral filters narrower thanthe total Rajuan linewidth further reduces the signallevel. The larger spectral width of the Raman linescompared with the Rayleigh linewidth means that theRaman technique is more sensitive to background lightthan the Rayleigh. On the other hand, Rayleigh line-width lidar requires a comparatively more monochro-matic transmitter than a Raman lidar, so that higheraverage transmitter powers are more easily obtained fora Raman-based system.

It is more difficult to make quantitative comparisonsbetween the Rayleigh linewidth temperature profilingtechnique and differential absorption34 proposals forprofiling than between Rayleigh and Raman. Range-resolved differential absorption relies on accurate dif-ferences of transmission values at two different rangesand two wavelengths and on temperature-dependentpopulations of molecular energy levels. In contrast, theRayleigh scattering approach considers the linewidthof the full signal from a given range gate (not signaldifferences) and analyzes kinetic temperature directlyfrom the received spectrum (without the analysis ofmolecular absorption spectra and energy levels).

The fact that Raman temperature profiling lidar hasbeen demonstrated, coupled with a comparison ofbackscatter cross sections, lends support to the per-formance estimates resulting from this study. A Ray-leigh scattering lidar is an interesting developmentbecause it brings spectral and spatial resolution valuespreviously used in IR cw Doppler lidars to the visiblespectral region.

IV. Conclusions and Directions

Two Michelson interferometers in parallel can pro-vide a measure of the Rayleigh scattering linewidth,which in turn indicates the temperature of the scatter-ing region. The performance analysis shows that aground-based system with -W average transmitted

power and a 30-cm telescope, for example, can remotelysense the temperature at 5-km altitude with a 50-mheight resolution (±25 m) and a 1 K standard deviationof the temperature estimate with a measurement timeof -75 sec. You might find it interesting to check thevalidity of the performance calculations and their ap-plicability to temperature profiling. The high angularresolution and corresponding reduction in backgroundsignal achievable with a transceiver arrangement (neardiffraction-limited optics) may be applicable to othertypes of lidars as well.

This study of temperature profiling by Rayleighscattering suggests additional research. A theoreti-cal-computational problem is to determine the cor-rection terms in Eq. (8) so that temperature may becalculated directly from R without the necessity of in-strumental calibration. Calibration of a particularinstrument (prefilter width and laser linewidth) as afunction of known temperature and pressure in a testcell, for various aerosol scattering levels, would be acomplicated substitute for knowledge of the correctionterms.

One experimental development problem is to deter-mine the most cost- and energy-effective laser with thelinewidth and pulse length characteristics specified inthe analysis. Candidates include cavity-dumped argon,,narrowed copper vapor, and dye oscillator-amplifiersystems. Another development opportunity is thestabilization of fixed-delay Michelson interferometersfor this application.

Fiocco et al.' showed that a Rayleigh linewidth lidarcan remotely measure temperature. This study pre-sents an efficient30 high spectral resolution measure-ment technique and analyzes its performance in termsof standard deviation of the temperature estimate andmeasurement time with a lidar of modest parametervalues. If achieved, lidar measurements of temperatureto 1 K at 5 km in 1'/4 min with a compact lidar shouldhave wide applicability in atmospheric physics researchand possibly in specialized operational forecasting.

We thank A. Skov Jensen for careful listening thathelped refine the fixed-delay interferometer approach.G. Fiocco's enthusiasm encouraged us to undertake thestudy.References1. G. Fiocco, G. Benedetti-Michelangeli, K. Maischberger, and E.

Madonna, Nature London Phys. Sci. 229, 78 (1971).2. G. Benedetti-Michelangeli and G. Fiocco, in Structure and Dy-

namics of the Upper Atmosphere, F. Verniani, Ed. (Elsevier,Amsterdam, 1974), pp. 211-219.

3. L. Lading, A. Skov Jensen, R. Schwiesow, C. Fog, and E. Ras-mussen, in Abstracts Ninth International Laser Radar Confer-ence (AMS, Boston, 1979), p. 244.

4. L. A. Johnson, "Coherent Lidar as a Tool for Remote Tempera-ture Sensing in the Troposphere," NOAA Tech. Memo. ERL-WPL-41 (1979).

5. I. L. Fabelinskii, Molecular Scattering of Light (Plenum, NewYork, 1968), pp. 81-100.

6. R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966).7. A. Sugawara and S. Yip, Phys. Fluids 10, 1911 (1967).8. G. Tenti, C. D. Boley, and R. C. Desai, Can. J. Phys. 52, 285

(1974).

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9. C. M. Hammond, Jr., and T. A. Wiggins, J. Chem. Phys. 65,2788(1976).

10. S. Yip and M. Nelkin, Phys. Rev. 135, A1241 (1964).11. S. Yip, J. Acoust. Soc. Am. 49, 941 (1971).12. T. J. Greytak and G. B. Benedek, Phys. Rev. Lett. 17, 179

(1966).13. R. P. Sandoval and R. L. Armstrong, Phys. Rev. A. 13, 752

(1976).14. See, for example, D. F. Eggers, Jr., N. W. Gregory, G. D. Halsey,

Jr., and B. S. Rabinovitch, Physical Chemistry (Wiley, New York,1964), pp. 138-155.

15. S. Chapman and T. G. Cowling, The Mathematical Theory ofNon- Uniform Gases (Cambridge U.P., London, 1970), pp. 45,82.

16. M. Born and E. Wolf, Principles of Optics (Pergamon, London,1975), pp. 300-323.

17. A. Skov Jensen, Optimum Strategies of Parameter Estimationfor a Poisson-Distributed Signal, Riso-M-1959 (Riso NationalLaboratory, DK-4000 Roskilde, Denmark, 1977), pp. 7-8.

18. R. Penndorf, J. Opt. Soc. Am. 47, 176 (1957).19. R. R. Rudder and D. R. Bach, J. Opt. Soc. Am. 58, 1260 (1968).20. Committee on Extension to the Standard Atmosphere, U.S.

Standard Atmosphere, 1962 (Supt. Documents, U.S. GPO,Washington, D.C., 1962), p. 67.

21. J. V. Hughes, Appl. Opt. 3, 1135 (1964).22. K. Ya. Kondratev, Ed., Radiation Characteristics of the Atmo-

sphere and the Earth's Surface, NASA TT 71-58003 (U.S. GPO,Washington, D.C., 1973), pp. 334-335.

23. W. L. Wolfe and G. J. Zissis, Eds., Infrared Handbook (Office ofNaval Research, Washington, D.C., 1978), pp. 4-30-4-36.

24. R. L. Schwiesow and R. F. Calfee, Appl. Opt. 18, 3911 (1979).25. L. Elterman, "An Atlas of Aerosol Attenuation and Extinction

Profiles for the Troposphere and Stratosphere," AFCRL-66-828(1966).

26. R. A. McClatchey, R. W. Fenn, J. E. A. Selby, F. E. Volz, and J.S. Garing, "Optical Properties of the Atmosphere (Revised),"AFCRL-71-0279 (1971).

27. C. M. Penney, R. L. St. Peters, and M. Lapp, J. Opt. Soc. Am. 64,712 (1974).

28. A. Cohen, J. A. Cooney, and K. N. Geller, Appl. Opt. 15, 2896(1976).

29. See, for example, I. S. Sokolnikoff and R. M. Redheffer, Mathe-matics of Physics and Modern Engineering (McGraw-Hill, NewYork, 1966), pp. 652-660.

30. L. Lading and A. Skov Jensen, Appl. Opt. 19, 2750 (1980).31. J. Cooney, J. Appl. Meteorol. 11, 108 (1972).32. J. A. Cooney and M. Pina, Appl. Opt. 15, 602 (1976).33. R. Gill, K. Geller, J. Farina, J. Cooney, and A. Cohen, J. Avpl.

Meteorol. 18, 225 (1979).34. J. B. Mason, Appl. Opt. 14, 76 (1975).

4,238,148 9 Dec. 1980 (Cl. 354-112)Three-dimensional photographic technique.J. S. COURTNEY-PRATT. Assigned to Bell Telephone Labora-tories, Inc. Filed 14 Jan. 1980.

Caulfield and Somerstein (Applied Optics 16, 774, 1977) were evidently thefirst to obtain monaxial 3-D images by illuminating the object with a pulse oflight shorter than it takes the light to travel the depth of the object. If thenthe returned (line) image is recorded in a streak camera, the depth contoursof the object can be inferred from the distortions of the line. The improvementmade here is that the slit is replaced by a fly's eye array of lenses. J.R.M-A.

4,238,529 9 Dec. 1980 (Cl. 427-160)Long wavelength x-ray diffraction crystal.A. SICIGNANO, W. P. ZINGARO, deceased, AND J. ZINGARO,executrix. Assigned to North American Philips Corp. Filed 2 Aug.1979.

This invention describes a means of constructing an x-ray diffraction crystalusing Langmuir-Blodgett multilayer pseudocrystals as the dispersive element.The crystal comprises several metal soap reflecting planes supported by a glasssubstrate. The layers are monomolecular and of alternating divalence weights.The device is recommended for use at wavelengths longward of 50 A. R.H.

4,239,339 16 Dec. 1980 (Cl. 350-184)Varifocal objective for photographic camera.W. WAGNER. Assigned to Jos. Schneider GmbH & Co.; OptischeWerke Kreuznach. Filed 9 July 1979 (in Germany 10 July 1978).

A complex f/4.5 2:1 zoom lens is described, suitable for use on a 6 X 6 or 6 X41/2 -cm camera. The fixed front member contains four airspaced elements; thenegative variator has three elements; the negative compensator is a singlet; andthe fixed rear member has seven elements, making fifteen elements in all. Thesurface radii are limited by a set of inequalities. Three examples are given forthe zoom portion and two for the fixed rear portion. The actual focal lengthand angular field coverage are not stated. R.K.

4,239,340 16 Dec. 1980 (Cl. 350-214)Conversion lens system.S. OGINO. Assigned to Minolta Camera K.K. Filed 2 Oct. 1978 (inJapan 4 Oct 1977).

A seven-element telenegative system is described for use behind a normalf/2.8 or f/3.5 lens to double its focal length. The seven elements which are al-ternately negative and positive may be airspaced, or two adjacent elements maybe cemented. Excellent aberration correction is claimed. Four examples aregiven. R.K.

4,239,963 16 Dec. 1980 (Cl. 250-231 R)Fiber-optic accelerometer.R. R. AUGUST, V. H. STRAHAN, K. A. JAMES, and D. K. NICH-OLS. Assigned to Rockwell International Corp. Filed 26 July1978.

The accelerometer described in this disclosure consists of a flexible opticalfiber, affixed with one end to a support and moving freely at the other end, whichmay be thickened to provide higher inertia. Light enters the fiber at the fixedend and emerges from the free end, reaching there an array of light detectors,which array may be either linear or 2-D. While many prior-art accelerometersgive an analog output, this one gives a digital output, with a concomitant savingsof analog-to-digital conversion hardware. J.R.M-A.

4,240,697 23 Dec. 1980 (Cl. 350-183)Lens system having selectively shiftable focal length.E. TAKANO. Assigned to Canon K.K. Filed 3 June 1976 (in Japan5 June 1975).

A very complex zoom lens is described having a 17:1 focal length range. Theactual focal lengths can be altered by interchanging two rear relay systems. Thefour embodiments have these focal lengths: (1) 16.4-277 mm, 51 surfaces; (2)28.1-477 mm, 48 surfaces; (3) 12.3-209 mm, 54 surfaces; and (4) 21.0-356 mm,45 surfaces. The relative aperture is not stated. R.K.

Patents continued from page 1971

4,238,142 9 Dec. 1980 (Cl. 351-7)Method and apparatus for examining and photographing theocular fundus.W. RICHARDS, B. GROLMAN, and J. W. KANTORSKI. Assignedto American Optical Corp. Filed 18 Dec. 1978.

The ophthalmoscope is aligned by using a rather narrow pencil of light andfocusing on the (light-insensitive) optic disk. If done in a darkened room, thisallows the pupil to dilate, even without the use of mydriatics. After focusing,the restricting aperture is removed and at the same time a flash photographof the whole fundus is taken. J.R.M-A.

4,240,698 23 Dec. 1980 (Cl. 350-184)Large aperture zoom lens.S. TSUJI and Y. SATO. Assigned to Canon K.K. Filed 17 July 1978(in Japan 22 July 1977).

A complex zoom lens is described for use on a Super-8 movie camera. Thefocal length ranges from 7.6 to 58 mm at f/1.4. Three embodiments are givencontaining fifteen to seventeen elements plus a beam splitter and a rear window,making thirty-two to thirty-four surfaces in all. R.K.

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