396 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 14

Quantitative Interpretation of Laser Intensity Pro®les

R. R. ROGERS AND M.-F. LAMOUREUX Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada

L. R. BISSONNETTE Defence Research Establishment Valcartier, Courcelette, Quebec, Canada

R. M. PETERS Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania (Manuscript received 23 July 1996, in ®nal form 28 October 1996)

ABSTRACT The authors have used a commercially available laser ceilometer to measure vertical pro®les of the optical extinction in . This application requires special signal processing to correct the raw data for the effects of receiver noise, high-pass ®ltering, and the incomplete overlap of the transmitted beam with the receiver ®eld of view at close range. The calibration constant of the ceilometer, denoted by C, is determined from the pro®le of the corrected returned power in conditions of moderate attenuation in which the power is completely extin- guished over a distance on the order of 1 km. In this determination, the value of the backscatter-to-extinction ratio k of the scattering medium must be speci®ed and an allowance made for the effects of multiple scattering. These requirements impose an uncertainty on C that can amount to Ϯ50%. An alternative to determining the calibration constant is explained, which does not require specifying k, although it assumes that k is constant with height. Using this alternative approach, the authors have estimated many extinction pro®les in rain and compared them with re¯ectivity pro®les measured with a UHF boundary layer wind pro®ler. The values of the extinction coef®cient in the examples shown in this paper range from about 2 to 12 kmϪ1 and are generally larger than the values inferred from the radar re¯ectivity of the rain. The implication is that aerosol particles and cloud drops, which are not visible to the radar, are important in determining the optical extinction in rain in these examples.

1. Introduction before being extinguished by or cloud at higher altitudes. Since November 1992 a laser ceilometer has been We have made many comparisons of ceilometer pro- used continuously as part of the atmospheric remote ®les in rain with pro®les of radar re¯ectivity measured sensing facilities at McGill University. It provides data simultaneously by a UHF wind pro®ler. The question on the height of cloud base as a function of time, and also records at a rate of twice a minute the complete initially was whether the observed extinction of the ceil- vertical pro®le of the power received by the ceilometer, ometer signal was consistent with the extinction that in the same sense as an intensity pro®le measured by a could be estimated from the radar re¯ectivity of the rain. . Ceilometer ``signatures'' of rain, snow, fog, and To answer the question required quantitative interpre- haze are often recognizable by their different intensity tation of the ceilometer data using techniques developed pro®les and by the altitude reached before the signal for lidar data analysis. The conclusion we have reached diminishes to its noise level. The signal is typically is that the extinction in rain is determined more by fog, attenuated completely by clouds over a short distance, aerosols, and cloud droplets mixed in with the rain than but propagation distances through rain can be consid- by the raindrops themselvesÐno real surprise in ret- erable. Not uncommonly, the laser beam is able to pen- rospect. But along the way we have determined a sat- etrate a kilometer or two of rain in its upward course isfactory method of calibrating the ceilometer so that the measured intensity pro®les can be converted to pro- ®les of the atmospheric extinction coef®cient. Under appropriate conditions, a relatively inexpensive opera- Corresponding author address: R.R. Rogers, Atmospheric Sci- tional instrument can thus be used for quantitative es- ences, McGill University, 805 Sherbrooke St. W., Montreal H3A 2K6 Canada. timates of the atmospheric extinction and its variation E-mail: [email protected] with height and time.

᭧1997 American Meteorological Society

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TABLE 1. Laser ceilometer (Vaisala model CT 12K). TABLE 2. McGill University±NOAA boundary layer pro®ler.

Wavelength 0.904 ␮m Fixed parameters Peak power 40 W Frequency 915 MHz Pulse duration 0.135 ␮s Wavelength 32.8 cm Average power 5 mW Peak power 500 W Beamwidth 5 mrad Antenna aperture 1.8 m ϫ 1.8 m Vertical resolution 15.24 m (50 ft) Antenna type 64-element array Maximum range 4 km Number of beams 5 Pointing directions Vertical; 21Њ zenith angle at cardinal points This paper presents the background necessary for Beamwidth 9Њ quantitative analysis of ceilometer data and gives ex- Adjustable parameters and their typical values amples of results from one set of observations in rain. Pulse duration 0.7 ␮s These observations are typical of many others and are Interpulse period 61 ␮s Range resolution 105 m adequate to illustrate the principles of the analysis and Number of range samples 60 the results of the research. Number of coherent integrations 110 Number of spectral averages 50 Maximum radial velocity Ϯ 11msϪ1 2. The ceilometer and the radar Number of spectral points 64 The ceilometer is the Vaisala model CT-12K, the main operating characteristics of which are given in Table 1. Its laser source is a gallium arsenide semicon- description may be found in Rogers et al. (1994). The ductor diode operated in pulsed mode with a pulse en- dwell time required for a complete set of spectral mea- ergy of 6.6 ␮J and a pulse repetition frequency ranging surements in a given pointing direction is typically 35 from about 600 to 1100 Hz, automatically controlled to s. The radar is sensitive to scattering by maintain a constant average power output of 5 mW. The particles and by spatial ¯uctuations in the refractivity instrument is rigidly ®xed to the surface with the trans- of the optically clear air. At its long wavelength, atten- mitted beam pointed vertically. There is no provision uation by any atmospheric constituent is negligible and for pointing the beam in other directions. It is a bistatic the returned power can be converted to target re¯ectivity system with the transmitter and receiver separated by through the radar calibration factor. The radar sensitivity 31 cm. The transmitted beamwidth and the receiver ®eld is such that the minimum detectable signal at a range of view are both approximately 5 mrad (0.3Њ). Although of 1 km corresponds approximately to a re¯ectivity fac- the ceilometer is an operational instrument, intended tor Z in rain of Ϫ15 dBZ or a refractivity structure 2 Ϫ15 Ϫ2/3 mainly to give a continuous indication of the height of parameter (Cn ) in clear air of 10 m . Because the cloud base, our unit is equipped with software that pro- re¯ectivity of a particle with diameter D is proportional vides a complete record of the returned signal strength to D6/␭4 (a consequence of the Rayleigh scattering ap- as a function of height. Using as input what is called proximation) the radar is not able to detect clouds con- ``standard message #2'' in the technical manual for the sisting of droplets smaller than about 10 ␮m in diameter ceilometer (Vaisala 1989), the software integrates the or aerosols. received power for 30 s and generates data with an approximate range resolution of 15 m and a time res- 3. The lidar equation and inversion methods olution of 30 s. These data can be interpreted in terms of the vertical pro®les of the backscattering coef®cient The lidar equation for the ceilometer may be written and the extinction coef®cient using procedures devel- as (e.g., Zuev 1982) oped for lidar. Because the ceilometer runs continuously, zC22␤(z)t(z) p(z)ϵP(z)ϭ, (1) the data include pro®les through rain, snow, and haze, 22 as well as the sharp returns from cloud base. Although zz00 attenuation of the laser beam can be severe, penetration where p(z) is the power returned from altitude z, z0 is distances in rain of more than a kilometer are common. a ®xed reference altitude (1 km in our work), P(z)is The attenuation of snow is stronger, and that of cloud the range-adjusted power, ␤(z) the volume backscatter- still stronger. Scattering by aerosol particles is readily ing coef®cient, t2(z) the two-way transmittance, and C detectable in fair weather in conditions when the visi- the calibration constant of the ceilometer. The trans- bility is noticeably impaired, but is hard to detect on mittance is related to the extinction coef®cient ␥ by clear days. z The radar is a 915-MHz (33-cm wavelength) wind 2 pro®ler with a ®ve-beam, phased array antenna, which t (z) ϭ exp Ϫ2 ͵ ␥(r) dr . (2) []0 transmits a peak power of 500 W and is normally op- erated with a range resolution of 105 m. Its operating Although (1) is applicable in general, it is sometimes characteristics are given in Table 2; a more complete modi®ed to indicate the separate contributions of dif-

Unauthenticated | Downloaded 09/29/21 04:58 AM UTC 398 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 14 ferent coexisting atmospheric constituents (e.g., aerosol Strictly speaking, the backscattering coef®cient ␤ in particles and cloud drops); then ␤ is written as the sum (4) should be replaced by an effective coef®cient be- of the individual backscattering coef®cients and t2 as cause we are now including the effects of forward scat- the product of the individual transmittances. Because tering through the extinction reduction factor ␩. As a the operational use of the ceilometer depends on relative consequence of multiple scattering, many backscattering and not absolute power measurements, the calibration events are at angles close to but not exactly equal to constant C in (1) is generally unknown. 180Њ; hence the need to modify ␤. However, we assume Equation (2) is based on the assumption of single here that the phase function is suf®ciently ¯at near 180Њ scattering. In many lidar applications, however, multiple that the effective value of the backscattering coef®cient scattering has a signi®cant effect on the returned power. is approximately equal to the conventional ␤ de®ned Photons de¯ected from the beam by one scattering event for 180Њ. may be scattered back into the beam by another, causing The simplest solution of (4) is for a homogeneous the effective transmittance to be greater than the value medium, in which ␤, ␩, and ␥ are constant. Then A is given by (2). The importance of multiple scattering de- also constant and pends on the scattering phase function of the propa- A gation medium, particularly on the forward-scattering ␩␥ ϭϪ . (6) lobe, and on the characteristics of the receiver, especially 2 the ®eld of view. There is no exact equation to account To determine ␥, ␩ must be obtained from an independent for multiple scattering, but widely used is the approx- measurement or estimated from a scattering model. The imation of Platt (1973), in which the transmittance is advantages of this solution are that the calibration con- written as stant C is not required and no relation between ␤ and

z ␥ needs to be assumed. t2(z) ϭ exp Ϫ2 ␩(r)␥(r) dr , (3) A disadvantage of (6) is that a constant slope is no ͵ guarantee that ␩␥ is constant. This may be seen by []0 setting A(z) ϭ A, a constant, in (4). To proceed with a where ␩ Ͻ 1 is a factor that reduces the extinction term. solution then requires some assumption to eliminate one By considering the properties of the scatterers and the of the variables, ␤ or ␥. The usual approach is to assume particular lidar system, this factor can sometimes be a simple proportionality between these variables, approximated with reasonable accuracy. For example, for a receiver ®eld of view that is as broad as the width ␤(z) ϭ k(z)␥(z), (7) of the forward-scattering lobe of the phase function of where k, called the lidar ratio, depends on the scattering the scatterering medium, the factor ␩ is approximately medium and the wavelength. Using this assumption to 0.5. This follows from Babinet's principle that implies, express ␤ in terms of ␥ in (4) leads to as shown by Bohren and Huffman (1983, 107±111), that the contribution of diffraction to the extinction cross 1 dk ␩␥ Ϫ 2␩␥ ϭ A. (8) section of particles that are large compared with the (k/␩)␩␥ dz []΂΃␩ wavelength (i.e., their forward-scattering lobe) is ex- actly half the total extinction cross section. Assuming that k/␩ is constant and integrating upward Solutions of the lidar equation based on what is called a distance h from a reference altitude where ␩ ϭ ␩0 and the slope method proceed by substituting (3) in (1), ␥ ϭ ␥0, we obtain taking logarithms, and differentiating with respect to z. A␩␥00exp(Ah) This gives the result ␩(h)␥(h)ϭ . (9) A Ϫ 2␩␥00[exp(Ah) Ϫ 1] 1 d␤ Ϫ 2␩␥ ϭ A(z), (4) This shows that there are many pro®les of ␩␥ that satisfy ␤ dz (4) for A ϭ constant, depending on the boundary con- ditions ␩ and ␥ . Only if ␩␥ ϭ ␩ ␥ do we ®nd the where 0 0 0 0 solution (6). The more general solution (9) can be un- dP 1dS stable or even blow up for h suf®ciently large because A(z) ϭ ln ϭ . (5) dz P M dz the denominator decreases with increasing h and can 0 become zero or negative. This de®ciency can be avoided

In (5), P0 is a reference power (1 pW in our work), S by integrating downward from a reference altitude rather ϭ 10 log10(P/P0) is the range-adjusted power in , than upward (Klett 1981), though the pro®le of ␩␥ still and M ϭ 10 log10e ϭ 4.343. The measurable quantity, depends on the values ␩0 and ␥0 at the reference level, A(z), is the slope of the pro®le of the range-adjusted and these must either be assumed or obtained by some power measured on a logarithmic or scale. From other means. (4), the slope is seen to depend on both ␤(z) and A more general solution of the lidar equation, not ␩(z)␥(z). The inversion problem is to determine ␥(z) restricted to situations in which the slope parameter A from A(z). is constant, proceeds by differentiating (2) to obtain

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d 1 dt2 size and concentration both tend to increase with the lnt2 ϭϪ2␩␥ ϭ . (10) dz t2 dz relative humidity, so the optical properties of an aerosol, including the extinction coef®cient, also depend on the Then we set ␤ ϭ k␥ in (1) and use (10) for ␥ to get humidity. Data presented by Zuev (1982) and Weichel 2z2 ␩(z) (1990) show that for conditions near sea level at mid- dt2 ϭϪ 0 P(z)dz. (11) latitudes, the extinction coef®cient at our wavelength Ck(z) can increase from values in the order of 0.1 kmϪ1 at a Integrating this equation from z ϭ 0, where t2 ϭ 1, to relative humidity of 50% to values of 0.6±0.8 kmϪ1 at altitude h, where t2 ϭ t2(h), gives relative humidities near 95%. Because the aerosol con- tent of the atmosphere usually decreases with altitude, zP2 h (z)␩(z) t2(h)ϭ1Ϫ20 dz. (12) so does the extinction coef®cient, though there can be Ck͵ (z) altitudes of local maxima, caused by the trapping of 0 aerosols in layers. This expression is then substituted in (1) to give as the solution for ␥ c. Clouds P(h)z2/k(h) ␥(h) ϭ 0 . (13) h P(z)␩(z) Collis and Russell (1976) report values of the optical 2 C Ϫ 2zdz0 extinction in water clouds ranging from a little less than ͵k(z) Ϫ1 0 10 km for thin clouds to values an order of magnitude First derived by Hitschfeld and Bordan (1954) in a study greater for dense clouds. This range includes the ex- of radar attenuation by rain, (13) is more general than tinction coef®cients of certain cloud models used for the slope-method solutions but requires knowledge of scattering calculations, for example, those of Deir- the calibration constant C and suffers from the same mendjian (1969) and Zuev (1976). problem of instability as (9) because the denominator decreases with increasing h. Moreover, k and ␩ must be d. Precipitation speci®ed to solve for ␥. A stable solution may be ob- tained by integrating downward from a reference level Shipley et al. (1974) cite seven studies prior to their instead of upward from the ground, but then, in place own that reported the general power-law form of C, the solution depends on the value of ␥ at the ␥ ϭ aRb (14) reference level, which is generally unknown. The anal- r ysis in later sections of this paper rests mainly on ap- as the relationship between rainfall rate R and the optical plication of (6) and (13) to ceilometer observations. extinction due to rain ␥r, where a and b are parameters. The values reported for a ranged between 0.12 and 0.31 and for b between 0.59 and 0.82, when ␥ is expressed 4. Extinction and backscattering by various r in inverse kilometers and R in millimeters per hour. In atmospheric constituents their study, Shipley et al. estimated ␥r from lidar mea- a. Clear-air background surements in rain, allowing for the background aerosol extinction and for multiple scattering. The rain rate was At the wavelength of the ceilometer, the atmospheric measured independently with rain gauges. The values extinction in clear conditions is accounted for almost they found for a and b, including the variability among entirely by aerosol scattering. Contributions of molec- the samples, are a ϭ 0.16 Ϯ 0.04 and b ϭ 0.74 Ϯ 0.12, ular scattering and absorption, and of aerosol absorp- and thus comprise nearly the full range of values re- tion, are negligible in comparison. According to Weichel ported in the other studies. Using their average values (1990), a typical value of the clear-air background ex- for these parameters shows that ␥ increases from 0.16 tinction coef®cient at 0.9 m is about 0.15 kmϪ1. This r ␮ kmϪ1 for a light rain rate of 1 mm hϪ1 to 4.8 kmϪ1 for estimate is consistent with values reported by Collis the heavy rain of 100 mm hϪ1. (1966) and assumed by Shipley et al. (1974). It corre- Another way to determine the relation between ex- sponds to a meteorological visual range of about 14 km, tinction and rainfall rate is through model drop-size dis- based on an empirical formula relating the extinction tributions. One such model that is widely used is the coef®cient, wavelength, and visual range given by Wei- Marshall and Palmer (1948) exponential approximation chel (1990). In clean, cold, dry air, the extinction co- for widespread rain ef®cient can be less than 0.1 kmϪ1, and in hazy con- Ϫ1 Ϫ⌳D ditions it can be in the order of 1 km . N(D) ϭ N0e , (15) where N(D) dD is the number of drops per unit volume 3 Ϫ1 Ϫ3 b. Haze whose diameters are in dD, N0 ϭ 8 ϫ 10 mm m , and ⌳ is a function of rainfall rate R given by Air that is noticeably hazy contains more or larger aerosol particles than are present in clear air. Particle ⌳ϭ4.1RϪ0.21, (16)

Unauthenticated | Downloaded 09/29/21 04:58 AM UTC 400 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 14 where R is in millimeters per hour and ⌳ has units of Here Z is the radar re¯ectivity factor, which is related inverse millimeters. Because raindrops are much larger to the drop size distribution by than the wavelength, the geometric optics approxima- ϱ tion may be used so that the extinction cross section of Z ϭ N(D)DdD.6 a drop of diameter D is equal to (␲/2)D2. The volume ͵ 0 extinction coef®cient is then obtained by integrating over all drop sizes [For snow, N(D) is the distribution of melted diameters.] For R in millimeters per hour and Z in its conventional ϱ ␲␲2! units (mm6 mϪ3), typical values of the parameters in ␥ ϭ N(D)DdD2 ϭ N . (17) r 22͵ 0⌳3 (19) for rain are A ϭ 200 and B ϭ 1.6, and for snow 0 A ϭ 2000 and B ϭ 2.0. Raindrops do not, of course, extend to in®nitely large Combining (14) with (19) shows that, for rain, size, but the exponential function in (15) falls off fast ␦ enough to prevent unrealistically large drop sizes from ␥r ϭ ␣Z , (20) contributing signi®cantly to the integral in (17). Em- where ␣ ϭ aAϪb/B and ␦ ϭ b/B. For the typical values ploying (16) for ⌳ in (17) leads to a result with the of A and B, and for the average values of a and b given form of (14) but with a ϭ 0.37 and b ϭ 0.63, which by Shipley et al., we have for rain values differ from the averages reported by Shipley et Ϫ2 0.46 al. and predict a somewhat higher extinction for a given ␥r ϭ 1.4 ϫ 10 Z , (21a) rainfall rate. The difference may be explained by a ten- where ␥ is in inverse kilometers and Z is in conventional dency of (15) to overestimate the number of very small r units (mm6 mϪ3). For comparison, the relationship (17) drops in actual raindrop populations. for the Marshall±Palmer model drop size distribution, For snow, the extinction can, in general, be greater when combined with (19), gives than that of rain but there appears to be less known Ϫ2 0.39 about the dependence of the extinction on other param- ␥r ϭ 4.6 ϫ 10 Z . (21b) eters, such as precipitation rate. Based on observations Churchill (1992) computed the relation between ex- with a transmissometer, Warner and Gunn (1967) re- tinction and radar re¯ectivity for rain from drop size ported the linear relation distributions measured by airplanes ¯ying through hur-

␥s ϭ KR, (18) ricanes. He reported that

Ϫ1 Ϫ2 0.28 where ␥s (km ) is the extinction coef®cient of snow, ␥r ϭ 7.9 ϫ 10 Z . (21c) R is the precipitation rate expressed in terms of melted The differences among these equations give an idea of snow as millimeters of water per hour, and K ϭ 2.5 Ϫ1 Ϫ1 Ϫ1 the uncertainty of empirical relations between ␥ and Z km (mm h ) . The analysis leading to this estimate for measured and model drop size distributions. The included a correction for aerosol background extinction relationship considered most relevant for the present and its dependence on relative humidity. It did not allow study is (21a), based on the average of many light- for the contribution of multiple scattering to the trans- scattering observations. mittance, so the values of determined in these ex- ␥s For snow, combining (18) and (19) and using the periments may be too small by as much as a factor of typical values cited for A and B, and the Warner and 2. In a study of lidar measurements through the melting Gunn value for K, we obtain layer of snow, Sassen (1977) modeled the particle size Ϫ2 0.5 distribution and calculated the vertical pro®les of ␥ and ␥s ϭ 5.6 ϫ 10 Z . (22) the backscattering coef®cient through the melting lay- ␤ Care must be used in applying (22) because the re¯ec- er. His results for an assumed precipitation rate of 15 tivity of snow is often measured in terms of the equiv- mm hϪ1 show an extinction coef®cient in the snow of alent re¯ectivity factor Z , which is less than Z by the approximately 50 kmϪ1, which is not too far from the e factor 0.23 (see Drummond et al. 1996). In terms of Z , prediction of (18) for the same precipitation rate. Much e (22) becomes depends, of course, on the particle size distribution; and Ϫ2 0.5 for snow there is likely to be more variability in the ␥s ϭ 11.7 ϫ 10 Ze . (23) relation between ␥ and R than for rain, because of the Comparing (23) with (21a) shows that the extinction wide variations that can exist in the ice crystal habit coef®cient of snow is an order of magnitude greater and in the degree of aggregation of the crystals. than that of rain having the same radar re¯ectivity. We will ®nd it convenient later to regard the extinc- tion coef®cient of precipitation as a function of radar re¯ectivity instead of rainfall rate. To do this we employ e. Backscatter-to-extinction ratios empirical relationships between re¯ectivity and rain rate Solving the lidar equation usually requires making an of the form assumption about the lidar ratio k de®ned in (7). This Z ϭ ARB. (19) ratio is the same as the normalized backscattering phase

Unauthenticated | Downloaded 09/29/21 04:58 AM UTC JUNE 1997 ROGERS ET AL. 401 function of the scattering medium, namely, P(␲)/4␲, 0.082 srϪ1 at a rain rate of 0.5 mm hϪ1 to 0.058 srϪ1 at where P(␪) denotes the phase function for scattering 10 mm hϪ1. Accordingly, we have settled on 0.06 srϪ1 angle ␪. Estimates of k are available for many atmo- as the value to use for rain in our experiments. For snow, spheric constituents, either from light-scattering mea- Sassen's results suggest that 0.035 is appropriate. surements or from theoretical Mie scattering phase func- Ice-phase clouds and precipitation present a special tions computed for different cloud, haze, and rain mod- problem for quantitative measurements with a ceilom- els. eter. Crystals of plate-type habit have signi®cant aspect For aerosols, Spinhirne et al. (1980) reported many sensitivity for scattering, with a strong specular re¯ec- lidar estimates of k in the atmospheric boundary layer tion component. Because such crystals tend to fall with at a wavelength of 0.6943 ␮m. Their results indicate a a preferred horizontal orientation, the backscattered range of values between approximately 0.02 and 0.07 component measured in the vertical beam of a ceil- srϪ1 and an average of 0.05 srϪ1. For comparison we ometer may be strongly increased by specular re¯ection. may cite the values for two of the water haze models The lidar ratio k in such conditions may be much dif- of Deirmendjian (1969): for haze M, ␥ ϭ 0.1055 kmϪ1 ferent from the values summarized here, making ques- and k ϭ 0.019 srϪ1; for haze L, ␥ ϭ 0.0395 kmϪ1 and tionable any solution of the lidar equation that relies on k ϭ 0.010 srϪ1. These values are for a wavelength of an assumed value of k. Consequently, we have focused 0.7 ␮m, the nearest wavelength in Deirmendjian's tab- mainly on measurements in rain to avoid the uncertain- ulations to the ceilometer wavelength of 0.9 ␮m. It ap- ties of backscattering in ice-phase precipitation. pears reasonable to use 0.04 srϪ1 as a typical value of k for aerosols, but to recognize that there can be a vari- ability of at least Ϯ50% about this value. 5. Extinction in rain using the slope method According to the technical manual for our ceilometer Figure 1 is a plot of the power received by the ceil- (Vaisala 1989), k for clouds may be taken as 0.03 srϪ1, ometer as a function of time and height over a 1-h period although it is said to vary inversely with humidity in in moderately heavy rain, compared with the radar re- narrow limits around this value. Sassen et al. (1990), ¯ectivity pattern as measured in the vertical beam of for a wavelength of 0.5106 ␮m, give values of k for the wind pro®ler. The approximate height resolution of different cloud types: for example, 0.020 Ϯ 0.012 for the ceilometer data is 15 m and of the radar is 100 m. cirrus, 0.051 Ϯ 0.019 for altocumulus, and 0.024 Ϯ The time resolution of both sets of data is about 30 s. 0.015 for ice virga. The Deirmendjian liquid cloud mod- The ceilometer data are plotted in units of decibels rel- el C.1 has ␥ ϭ 16.73 kmϪ1 and k ϭ 0.05055 srϪ1 at a ative to a picowatt, or dBpW. The radar data are the wavelength of 0.7 ␮m. Grund and Eloranta (1990) il- re¯ectivity measured on a logarithmic scale in units of lustrate the time variability of k in a cirrus cloud ob- dBZ, de®ned by served over a 9-h period at a wavelength of 0.5106 ␮m. The values average close to 0.03 srϪ1 and vary between Z Ϫ1 ␨ ϭ 10 log , (24) approximately 0.02 and 0.04 sr . Bissonnette (1996) 10 Z calculated the pro®le of k in a model cloud whose prop- ΂΃0 erties vary with height in a way to simulate drop growth, where ␨ (dBZ) denotes the re¯ectivity, Z (mm6 mϪ3)is 6 Ϫ3 ®nding values generally in the range from 0.03 to 0.08 the re¯ectivity factor, and Z0 ϭ 1mm m . srϪ1 but changing rapidly and erratically with height. The radar pattern in this example shows precipitation There is thus good support for a value of k between at all altitudes below the maximum range plotted, 8 km. approximately 0.03 and 0.07 srϪ1 for clouds of different The band of high re¯ectivity just below 4 km is the kinds, but regarding k as constant with height, as is melting layer, which separates the snow above from the frequently done, may lead to erroneous conclusions. rain below. The re¯ectivity in the rain varies between In interpretating lidar measurements of precipitation approximately 25 and 45 dBZ. The ceilometer signal at 0.6328 ␮m, Sassen (1977) determined 0.035 srϪ1 as falls to its noise level at an altitude of about 1 km and the appropriate value of k for ice-phase precipitation is not plotted for a received power less than 10 dBpW. and 0.05 for water (though expressed in terms of a func- Thus, the ceilometer beam does not reach entirely tion he denoted by g11, which is related to k by g11 ϭ through the rain. 2k). Later, Sassen (1978) tabulated values of k for dif- This example was chosen because the plot suggests ferent types of precipitation particles, giving 0.05 srϪ1 a relationship between the ceilometer measurements and for raindrops, and generally smaller values for ice-form the intensity of the rain as indicated by the radar. The hydrometeors, with the exception of 0.095, the largest radar re¯ectivity of the rain twice exceeded 40 dBZ, value reported, for ``frozen near-spherical drops.'' Re- ®rst during a brief period near 1615 EST and later near cent unpublished calculations by L. Bissonnette (1996, 1635. The ®rst period was the more intense and included personal communication), for the wavelength of the a few measurements in excess of 45 dBZ. The period ceilometer and raindrops having the Marshall±Palmer of strongest signal measured by the ceilometer is ap- exponential distribution given by (15) show that k has proximately coincident with the strongest radar echo. a weak dependence on rainfall rate, decreasing from During this time, the ceilometer signal exceeds 40

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FIG. 1. Time±height patterns of power received by ceilometer (above) and radar re¯ectivity (below) in moderately heavy rain. Ceilometer power is in units of decibels relative to a picowatt (dBpW); re¯ectivity is in terms of equivalent re¯ectivity factor, in dBZ. dBpW and the attenuation is relatively strong, as in- pair is the radar re¯ectivity up to 3 km, showing the dicated by the shorter penetration of the ceilometer beam structure of the rain below the melting layer. The pro®les into the rain. Observations in rain do not always suggest are 2-min averages of the individual pro®les measured a relation between the radar and the ceilometer mea- in the vertical beam of the pro®ler. Averaging is in Z, surements; the example of Fig. 1 is fairly typical of not ␨, but the averages are converted to units of dBZ those cases in which some relation is evident. using (24). Measured re¯ectivities are erroneously low To illustrate the vertical pro®les of radar re¯ectivity at altitudes below 525 m because of incomplete recovery and the ceilometer signal in more detail, Fig. 2 compares of the radar receiver at close range. (This effect is ap- the pro®les at four selected times. The left side of each parent in the radar data of Fig. 1.) No attempt is made

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FIG. 2. Comparison of radar re¯ectivity pro®les and ceilometer pro®les at four selected times. Re¯ectivity is in dBZ; ceilometer power is the quantity S, which is the range-adjusted power, corrected for noise and the incomplete overlap of beams at close range measured on a decibel scale. Note that the height scales are different for the radar and the ceilometer. to correct for this error, and re¯ectivities are not plotted about 35 s, was as follows: a series of three measure- in Fig. 2 below 525 m. The antenna scanning cycle for ments in the verticalÐone east, three more vertical, one these measurements consisted of three pointing direc- north. A complete cycle thus required about 4.5 min, tionsÐnorth, east, and verticalÐand emphasized the during which there were six measurements in the ver- vertical. The sequence of measurements, each requiring tical beam. A given 2-min interval therefore contains

Unauthenticated | Downloaded 09/29/21 04:58 AM UTC 404 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 14 either two or three individual re¯ectivity pro®les, de- pending on the portion of the scanning cycle included in the interval. The right member of each pair of pro®les in Fig. 2 is the 2-min average pro®le of S de®ned in (5), that is, the range-adjusted power P(z) measured on a decibel scale. Three steps are required to convert a pro®le of the measured power to P(z): 1) correction for noise and the effect of high-pass ®ltering in the ceilometer re- ceiver, 2) compensation for the incomplete overlap of the transmitter beam and the receiver ®eld of view at close range, and 3) application of the range correction. These are explained in detail in the appendix. Brie¯y, the correction for noise and ®ltering at the ranges of interest is minor, tending to reduce the value of P for weak signals at distant ranges. More signi®cant is the compensation for incomplete overlap of the laser beam and the receiver ®eld of view. This correction has the effect of increasing the power at close range: the cor- rection is negligible beyond 1.5 km, but amounts to 0.3 dB at 1000 m, 1 dB at 360 m, 2 dB at 200 m, and 5 dB at 100 m. Because of the sensitivity of this correction to uncertainties in the beamwidth and the beam align- ment, it is imprudent to apply the correction at ranges closer than about 100 m. In fact, we decided not to attempt to correct the data below gate 8, at 122 m, where the correction is 4 dB. Measurements below this altitude are not plotted in the ®gure or used in the analysis. Finally, the range correction is applied simply by mul- FIG. 3. Time series of radar re¯ectivity at 630 m compared with tiplying the power by the square of the range in kilo- the slope of the ceilometer pro®le at that altitude. meters, which in effect establishes 1 km as the reference range. Figure 2a, for the time interval 1610±1612 EST, is optically homogeneous even in rain that appears essen- at the beginning of the strongest rain event in the record. tially uniform to the radar. The ceilometer is evidently The radar re¯ectivity decreases fairly regularly with sensitive to aerosols invisible to the radar that are in- height from about 43 to 34 dBZ over the altitude interval homogeneously mixed with the rain. But the fact re- plotted. The pro®le of range-adjusted ceilometer power mains that there is some association apparent between falls from a maximum of 26 dBpW at the closest range to its noise level at about 0.9 km. Its slope is not con- the rain pattern and the ceilometer measurements in Fig. stant, as it would be in a homogeneous medium, but 1. This association is illustrated more quantitatively in may be regarded as approximately constant over several Fig. 3, which compares the time series of the radar limited altitude intervals. In particular, the slope is steep- re¯ectivity at a ®xed altitude with the slope of the ceil- est in the interval above about 600 m, is least steep ometer pro®le at that altitude. There is only a limited between 350 and 500 m, and has an intermediate value range of altitudes where comparisons of this kind are at altitudes below 350 m. Figure 2b is for the time period possible, because of the lower limit of 525 m for the 1618±1620 EST, when the rain re¯ectivity is relatively radar and the upper limit of about 1 km in this set of weak and within 1 dB of 35 dBZ over the height interval observations for the ceilometer. The altitude chosen was plotted. The ceilometer power at close range is weaker 630 m, which corresponds to gate 6 for the radar and and the penetration distance farther than in Fig. 2a. Fig- gate 41 for the ceilometer. The slope dS/dz was deter- ure 2c, for a time 8 min later, shows the radar re¯ectivity mined by least squares ®tting the ceilometer data points again decreasing with height and a ceilometer pro®le at ®ve adjacent altitudes centered at 625 m, that is, for closely similar to that in Fig. 2b. The last pair of pro®les, gates 39±43, or the height interval 625 Ϯ 30 m. The for 1640±1642 EST, shows a nearly constant radar re- strongest negative slopes of approximately 30 dB kmϪ1 ¯ectivity and a ceilometer pro®le that actually increases coincide with the strongest radar re¯ectivity. Otherwise with altitude over a narrow height interval below 500 the correlation is not compelling. Assuming local ho- m. Any increase of this sort would be impossible in a mogeneity, the slope method allows the conversion of homogeneous medium. the slope to the extinction coef®cient. Taking the value It is clear from Fig. 2 that the atmosphere is not 0.5 for ␩ to allow for multiple scattering, (6) shows that

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attempted to determine this quantity empirically, from observations in which the ceilometer power appears to be completely attenuated. For such situations, there is an altitude H at which the returned power p falls to its noise level. At this altitude, the transmittance has fallen to zero or, more generally, to a small value ␧ determined by the receiver noise level. Then, from (12), 2zP2 H (z)␩(z) t2(H)ϭ␧ϭ1Ϫ0 dz, Ck͵ (z) 0 from which C is given by 2zP2 H (z)␩(z) Cϭ0 dz, (25) 1 Ϫ ␧ ͵ k(z) 0 In applying this result, we assume constant values of k and ␩ and set ␧ equal to zero. Then FIG. 4. Extinction coef®cient ␥ based on the slope method as a function of radar re¯ectivity for the same data as in Fig. 3. The top ␩ curve is a linear least squares ®t to the data. The bottom curve is the 2 C ϭ 2zI,0 (26) predicted relationship based on observations of Shipley et al. (1974); k the middle curve is the relationship expected for the Marshall±Palmer drop size distribution. where H I ϭ ͵ P(z) dz. (27) ␥ ϭϪ(1/M) dS/dz, and Fig. 3 implies that the extinction 0 Ϫ1 coef®cient at 630 m ranges between about 1 and 8 km . From the ceilometer measurements, we are able to es- Figure 4 is a scatter diagram of ␥ versus the re¯ec- timate I by numerical integration of P(z). To determine tivity factor for the data in Fig. 3. The extinction is C we then need to assume a value of ␩/k. So long as plotted on a logarithmic scale to facilitate comparison the sensitivity and the operating characteristics of the with power-law relations in the form of (20). The cor- ceilometer are steady, C may be regarded as constant relation coef®cient is 0.24 and the least squares linear independent of the scattering medium. However, ␩ and ®t is log10␥ ϭϪ0.572 ϩ 0.026␨, equivalent to the pow- 0.26 especially k are expected to have different values, de- er-law relation ␥ ϭ 0.268Z . Plotted for comparison pending on atmospheric conditions. Hence I is not con- is a line based on (21a), the predicted relation between stant but will vary with k and ␩, even for constant C. radar re¯ectivity and the extinction for rain only, ␥r One problem in evaluating (27) is to account for the based on the Shipley et al. (1974) values of a and b in contribution to the integral of the close ranges where P (14). The ceilometer points are all above the radar line cannot be measured because of insuf®cient overlap of and exceed the values expected for rain on average by the transmitted laser beam with the receiver ®eld of about a factor 3. However, the difference between the view. We extrapolate P to zero range by assuming that lines on Fig. 4 does gradually decrease with increasing the extinction coef®cient is constant between the ground re¯ectivity. For comparison, Fig. 4 also includes a curve and the lowest altitude where P is measured. Then, from based on (21b) for the Marshall±Palmer model distri- (2), the transmittance within this altitude range is given bution. It lies above the curve corresponding to Shipley by t2(z) ϭ exp(Ϫ2␩␥z). Let z denote the lowest altitude et al., but still below the best-®t line. The radar cali- 1 where P is measurable. Then, for z Ͻ z , on the as- bration constant is known only to an accuracy of about 1 sumption of constant ␥, Ϯ3 dB. This uncertainty is not suf®cient to explain the difference between the best-®t line and the theoretical P(z) eϪ2␩␥(zϪz1), lines. Hence, Fig. 4 strongly suggests that aerosols or ϭ P(z1) cloud drops invisible to the radar are dominating the extinction in this example, although the importance of and the contribution B to the integral over this range is z1 the rain does increase with increasing radar re¯ectivity. P(z1) B ϭ P(z) dz ϭ (e2␩␥z1 Ϫ 1). Beyond this qualitative statement, not much else can be ͵ 2␩␥ said because of the uncertain applicability of the slope 0 method in an inhomogeneous medium. Now if ␥ is suf®ciently small that 2␩␥z1 ϽϽ 1, then exp(2␩␥z1) ഠ 1 ϩ 2␩␥z1 and

6. Determination of the calibration constant P(z1) B ϭ (1 ϩ 2␩␥z Ϫ 1) ϭ P(z )z . (28) 2111 The general solution (13) of the lidar equation re- ␩␥ quires knowledge of the calibration constant C. We have Consequently, the integral I is evaluated in two parts:

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FIG. 6. Values of the calibration constant estimated from observa- FIG. 5. Ceilometer calibration constant C (pW km3 sr) based on tions on many days, mainly in rain. Eq. (26).

reduction in C. Deviations of ␧ from zero would also from the ground to z1 by setting P equal to P(z1), and lead to an underestimate of the calibration constant. from z1 upward on the basis of the measured pro®le of Figure 6 is a histogram of C values calculated from P, including the corrections explained in the appendix. measurements mainly in rain on 9 days between 27 April

In practice, we use z1 ϭ 122 m, corresponding to gate 8. 1994 and 19 January 1996. No trend was evident over Another problem is to decide on the upper limit of this period, so it seemed justi®ed to consider the data integration H in (25) and (27). Although P(z) is cor- all together. Because the measurable quantity is the in- rected for noise, some individual pro®les become irreg- tegral I, what is plotted is actually a histogram of the ular or noiselike when P falls below a value of about quantity 2(␩/k)I, where we used the rain values k ϭ 10 pW. Consequently, we have de®ned H as the altitude 0.06 srϪ1 and ␩ ϭ 0.5. The mean value of C for this where P diminishes to 10. sample is 1945 and the standard deviation 681. The Figure 5 is an example of a time sequence of estimates mode near 1000 pW km3 sr is accounted for by mea- of C based on (26). These are derived from the obser- surements on 17 January 1996, in which there was a vations in rain of 23 July 1995, used earlier in Figs. 1± surface fog layer that attenuated the signal completely 4. The calibration constant was calculated from each of within the lowest 200 m. In this situation, B computed the 2-min average pro®les of P over the 1-h period, from the approximation (28) underestimates the contri- including the four pro®les of Fig. 2. In applying (26) bution of the lowest altitudes to the integral I, explaining we used ␩ ϭ 0.5 and k ϭ 0.06. These values are ap- the small values of C. Also tending to bias C downward propriate for rain, but not necessarily for cloud or aer- are observations in conditions of weak attenuation in osols, though the results in section 5 point to these as shallow rain, in which the ceilometer may reach through the main cause of extinction. We will comment on the the rain to altitudes where the signal is below noise, possible effect of this inconsistency on the estimates of although attenuation is incomplete and ␧ in (25) is not the calibration constant later. truly zero. Attenuation of the beam by wet windows The values of C (pW km3 sr) are seen in Fig. 5 to also tends to reduce C. For all these reasons, the most range between approximately 1400 and 3200. The larg- meaningful values of C are probably the relatively large est values seem clearly to be associated with the period ones. Accordingly, we have settled on 2.5 ϫ 103 pW around 1615 EST corresponding to the heaviest rain and km3 sr as the appropriate value in situations where there the strongest signal on the ceilometer. There are many is no additional information on which to base an esti- possible reasons for the variability: 1) time variations mate. in the propagation medium that would affect k and ␩, The estimates of C given here are based on k ϭ 0.06 and hence I; 2) time variations in the transmitter or and ␩ ϭ 0.5, values which are thought appropriate for receiver that would affect the sensitivity of the ceil- rain at our wavelength. However, results in the previous ometer; 3) water on the ceilometer windows that would section give evidence that aerosol or cloud scattering impede transmission of the laser signal; and 4) devia- dominates the extinction, even in rain. The discussion tions of ␧ from zero caused by incomplete attenuation in section 4 indicates that k for these propagation media of the signal before the noise level is reached. None of is less than the value for rain, and typically 0.04 srϪ1. these reasons can be dismissed, though it is likely that As for ␩, it seems likely that the value for clouds and some are more important than others. Although the ceil- aerosols appropriate for the ceilometer will be close to ometer is equipped with a heater and blower to clear the rain value of 0.5 because the ®eld of view remains the windows automatically, they do get wet in rain and wide relative to the forward scattering phase functions, experiments show that this can reduce the returned pow- though these functions are broader for clouds and aer- er by as much as 2 dB. This translates to nearly a 40% osols than for rain. Consequently, the true value of ratio

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␩/kappearing in (26) is likely to be up to 50% larger dently of k, then (29a), which explicitly includes C, than the value employed in our calculations. This is should be used in solving for ␥; and 2) if, as in our another reason in favor of accepting 2.5 ϫ 103 rather work, it is only possible to infer the calibration constant than the mean value close to 2 ϫ 103 as the appropriate through I by assuming a value of k, then it is more value of the calibration constant. In accepting this value, appropriate to use (29b) or (29c) and avoid making the it should be acknowledged that there is still an uncer- assumption. However, while C might be expected to be tainty in the calibration factor amounting to perhaps fairly constant as long as the operating characteristics Ϯ50% owing to the sources of experimental and the- of the ceilometer remain steady, P(z) and I depend on oretical uncertainty mentioned earlier. properties of the propagation medium and on whether attenuation is complete before the noise level is reached. Therefore, a constant value of I is neither expected nor 7. Extinction pro®les by upward integration appropriate to use in (29). More appropriate is I deter- Having decided on a value of C, we can in principle mined for each individual pro®le, which will have a employ (13) to compute extinction pro®les from ceil- value that embodies k and ␩. Uncertainty still remains ometer measurements, assuming constant values for k because of the possibility that the signal may fall to the and ␩. Before proceeding, however, it is instructive to noise level before attenuation is complete, so this ap- substitute (26) for C in (13), assuming that k and ␩ are proach should be limited to conditions in which the constant. This shows that the extinction pro®le may be extinction appears to be complete over a relatively short written in any of the following forms: distance, but a distance long enough to permit an ac- curate numerical evaluation of the integral of P(z). P(h)z2 ␥(h) ϭ 0 (29a) Examples of extinction pro®les based on (29c) are h shown in Fig. 7. These four pro®les are for the same kC Ϫ 2z2␩ P(z) dz 0 ͵ observations as plotted in Fig. 2. To stay clear of the 0 singularity at h ϭ H, the integration was terminated at P(h) 10 gates (152 m) below H. In general, ␥ is relatively ϭ (29b) h constant up to 0.4 km, with a value ranging from 2 to 2␩ I Ϫ ͵ P(z) dz 4kmϪ1, and then increases with height. The values are []0 mainly larger than would be expected for rain alone, especially at the higher altitudes, indicating as in Fig. P(h) ϭ. (29c) 3 that cloud droplets and aerosol particles are important H in determining the extinction in rain. A possible ex- 2␩ ͵ P(z) dz planation for the general shape of the pro®les in Fig. h 7 is that the raindrops were entraining cloud droplets The version (29a) may be applied generally but requires into their wakes as they fell, causing an increase of knowing both C and k; (29b) depends on I in place of extinction with height in what was nominally the sub- C and k but is limited to situations in which the signal cloud layer. is strongly attenuated before reaching the noise level, As an approximate consistency check, we estimated so that ␧ ഠ 0; (29c) is equivalent to (29b) but written the total optical thickness for each of the 2-min average in a way to emphasize that P need only be integrated pro®les in the dataset. That is, for every computed pro- over the altitudes where it is measured, and that the ®le ␥(z), we calculated integration does not have to be extrapolated to zero h altitude as in the evaluation of I. All solutions require m ␶ ϭ ␥(z) dz, (30) that ␩ be speci®ed and independent of altitude. Because ͵ 0 they are based on the upward integration of the lidar equation, the solutions can become unstable for large where hm ϭ H Ϫ 152 m is the highest altitude where h. In particular, the denominators of (29b) and (29c) go ␥ is evaluated, using (29c). By truncating the data this to zero at h ϭ H. way, we risk underestimating the true value of ␶ but Basically, C is an intrinsic property of the ceilometer avoid the problem of instability in the calculation of ␥. that should not depend on the way it is measured. In At altitudes below z1 ϭ 122 m, we assumed ␥(z) ϭ principle, it could be determined from (1) by measuring ␥(z1). The time series of ␶ is plotted in Fig. 8. The values the power received from a target for which ␤ and t are range between 3.5 and 4.0 and average about 3.8. For known. However, our way of estimating C depends on any lidar system, the dynamic range of the receiver plac- measurements of I, which even for a constant C will es an upper limit on the value of ␶ that can be measured. vary depending on ␧,k,and ␩. Only if these quantities Advanced systems are capable of measuring values up are known can their effects be taken into account in to about 4 under conditions of single scattering. Multiple determining C. In view of these differences, the alter- scattering can increase this maximum by a factor of 1/␩. natives for determining ␥(h) may be interpreted as fol- Because multiple scattering is known to be signi®cant lows: 1) If the calibration constant is known indepen- for the ceilometer, the values in Fig. 8 seem plausible

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FIG. 7. Pro®les of extinction coef®cient from the data in Fig. 2.

considering its modest cost and performance. This is 8. Conclusions not proof that the computed extinction pro®les are cor- rect, but does show that the vertically integrated ex- With appropriate signal processing, ceilometer mea- tinctions are not inconsistent with the capabilities of the surements may be employed for quantitative estimates instrument. of the extinction pro®le. The estimates are only ap-

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Laboratory of the U.S. National Oceanic and Atmo- spheric Administration.

APPENDIX Correcting the Raw Ceilometer Data a. Correcting for noise and high-pass ®ltering The initial step in the quantitative analysis of ceil- ometer data is a semiempirical correction that compen- sates for receiver noise and adjusts for the effects of an A±C coupled ampli®er in the receiver, which acts as a high-pass ®lter with a time constant of approximately FIG. 8. Time series of optical depth ␶, estimated from extinction 54 ␮s. The algorithm for this correction was provided pro®les at 2-min intervals. by Dr. Jan LoÈnnqvist of the Vaisala company. It may be described by the following operations. proximate because of uncertainties in the lidar ratio k 1) Let Qi (pW) denote the measured power at the ith of the scattering medium and the ceilometer calibration gate, where i ϭ 1 to 250, corresponding to heights constant C. Also, the effects of multiple scattering can of 15.24i (m). The average power at the top 51 gates only be accounted for approximately. is computed to give an estimate of the noise level, A method is explained for estimating the extinction Q ϩ Q ϩ ´´´ ϩQ pro®le without direct knowledge of k and C, but it is av ϭ 200 201 250 . 51 limited to situations in which the ceilometer signal is completely attenuated over an altitude range suf®ciently 2) This average is subtracted from every individual long to permit accurate numerical evaluation of I as measured power: de®ned by (27). This requirement is most often satis®ed q Q av. in rain. The method still depends on a correction for i ϭ i Ϫ multiple scattering and the assumption that the lidar 3) All the resulting values that are positive are added ratio is constant with height. together: Comparisons with radar measurements in rain show 250 that the extinction is generally greater than expected ␹ ϭ q , ͸ i from rain alone. The difference is probably explained iϭ1 by cloud droplets and aerosol particles that are invisible to the radar but contribute signi®cantly to the extinction for all qi Ͼ 0. of the ceilometer signal. Optical depths inferred from 4) The quantity ␹ is divided by 600 and subtracted from the extinction pro®les in rain average just under 4.0, every qi: which appears to be consistent with the expected per- ␹ ␰ ϭ q Ϫ . formance capabilities of the ceilometer, when an allow- ii600 ance is made for multiple scattering.

The conclusion is that the ceilometer may be used for 5) Finally, an accumulating term is added to each ␰i to quantitative estimates of the extinction coef®cient, sub- give the power pi corrected for noise and high-pass ject to the uncertainties common to most lidar systems, ®ltering: plus an additional uncertainty caused by imprecise i knowledge of the calibration constant. ␰ ͸ j p ϭ ␰ ϩ jϭ1 , Acknowledgments. This work was supported by ii512 grants from the U.S. Of®ce of Naval Research and the where 512 is approximately the time constant of the Atmospheric Environment Service of Canada. We thank high-pass ®lter, expressed in units of microseconds, Jan LoÈnnqvist of the Vaisala company for taking an the round-trip time equivalent of the vertical reso- interest in our work and providing information on the lution of 15 m. After this step, any negative powers operation of the ceilometer. We are grateful to Guy Pot- are set to zero. vin for implementing the Vaisala corrections and help- ing with the ®nal stages of manuscript preparation. Ste- As an example of how the correction works, Fig. A1 phen A. Cohn, Phoebe Lam, and Sylvain Leblanc as- shows the pro®le of returned power for the 2-min period sisted in the early stages of the research. The UHF pro- 1626±1628 EST 23 July 1995, the data for which ap- ®ler is operated and maintained as a collaborative peared earlier in Fig. 2c. The pro®le corrected for noise project between McGill University and the Aeronomy and ®ltering is not much affected at low altitudes where

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FIG. A2. Schematic illustration of horizontal cross section through transmitter and receiver beams showing fractional overlap.

chord de®ning the overlap area and ␳1 and ␳2 are the distances from the beam centers to the chord, so that D

ϭ ␳1 ϩ ␳2. To evaluate the overlap function we make the sim- FIG. A1. Pro®les of raw ceilometer power (heavy) and power cor- plifying assumption r ϭ r ϭ r, so that ␳ ϭ ␳ ϭ ␳ rected for noise and high-pass ®ltering (light) for 1626±1628 EST 1 2 1 2 23 July 1995. The curves are essentially coincident at altitudes below and 2␳ ϭ D. The area A of the segment de®ned by the 500 m. chord of length L is given by LL A ϭ r2arcsin Ϫ (4r2Ϫ L 2). 1/2 the signal is relatively strong. The correction becomes 2r 4 signi®cant at the higher altitudes where the noise level ΂΃ is approached, and in fact it is the subtraction of the The total overlap area is twice A, so that the fraction F noise level in step 2 above that accounts for most of of the receiver beam that contains the transmitted beam the correction. is given by 1 LL b. Correcting for incomplete beam overlap F ϭ 2r2arcsin Ϫ (4r2Ϫ L 2), 1/2 ␲r2[]΂΃2r2 Because the ceilometer is a bistatic vertical-beam sys- where L is related to D and r by tem with a separation of 31 cm between the transmitter and receiver, the receiver ®eld of view does not include L ϭ (4r2 Ϫ D2)1/2, the transmitted laser beam at altitudes below about 60 m. From there to about 1.5 km the overlap is incomplete and the returned power is less than it would be if the transmitted beam were entirely contained in the ®eld of view. The correction for incomplete beam overlap is based on an approximate calculation of the fraction of the ®eld of view ®lled by the transmitted beam as a function of altitude. Figure A2 is a schematic depiction of a horizontal cross section of the transmitter and receiver beams at an altitude h where they have begun to overlap. Let ␪t and ␪r denote the half-angles of the transmitter and re- ceiver beams, respectively, so that the radii of the beams at h are given by r1 ϭ h␪t and r2 ϭ h␪r. According to Vaisala (1989), the beam half-widths are ␪t ϭ 2.5 mrad and ␪r ϭ 2.7 mrad. If D ϭ 31 cm denotes the distance between the beam axes, then the beams ®rst touch where r ϩ r ϭ D, corresponding to an altitude of 60 m. 1 2 FIG. A3. Correction factor (dB) for incomplete beam overlap. In Complete overlap is reached where r2 ϭ D ϩ r1,orat practice, the ceilometer data are only used at altitudes where this hϭ1.5 km. In the ®gure, L denotes the length of the correction is less than 4 dB.

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REFERENCES

Bissonnette, L. R., 1996: Multiple-scattering lidar equation. Appl. Opt., 35, 6449±6465. Bohren, C. F., and D. R. Huffman, 1983: Absorption and Scattering of Light by Small Particles. Wiley, 530 pp. Churchill, D. E., 1992: Vertical retrieval of solar and infrared irra- diances in the stratiform regions of EMEX cloud clusters. J. Appl. Meteor., 31, 1229±1247. Collis, R. T. H., 1966: Lidar: A new atmospheric probe. Quart. J. Roy. Meteor. Soc., 92, 220±230. , and P. B. Russell, 1976: Lidar measurement of particles and gases by elastic backscattering and differential absorption. Laser Monitoring of the Atmosphere, E. D. Hinkley, Ed., Springer- Verlag, 71±151. Deirmendjian, D., 1969: Electromagnetic Scattering on Spherical Po- lydispersions. Elsevier, 290 pp. Drummond, F. J., R. R. Rogers, S. A. Cohn, W. L. Ecklund, D. A. Carter, and J. S. Wilson, 1996: A new look at the melting layer. J. Atmos. Sci., 53, 759±769. Grund, E. J., and E. W. Eloranta, 1990: The 27±28 October 1986 FIRE IFO cirrus case study: Cloud optical properties determined by high spectral resolution lidar. Mon. Wea. Rev., 118, 2344± 2355. FIG. A4. Corrected pro®le from Fig. A1 showing additional cor- Hitschfeld, W., and J. Bordan, 1954: Errors inherent in the radar rections for incomplete beam overlap and for range-square adjust- measurement of rainfall at attenuating wavelengths. J. Meteor., ment. The curve labeled overlap indicates the effect of the overlap 11, 58±67. correction; that labeled range has both the overlap and range-squared Klett, J. D., 1981: Stable analytical inversion solution for processing correction and is the same as plotted in Fig. 2c. lidar returns. Appl. Opt., 17, 211±220. Marshall, J. S., and W. McK. Palmer, 1948: The distribution of rain- drops with size J. Meteor., 5, 165±166. and r is a function of altitude, Platt, C. M. R., 1973: Lidar and radiometric observations of cirrus clouds. J. Atmos. Sci., 30, 1191±1204. r ϭ h␪, Rogers, R. R., S. A. Cohn, W. L. Ecklund, J. S. Wilson, and D. A. with ␪ ഠ 2.6 mrad as a compromise between the trans- Carter, 1994: Experience from one year of operating a boundary- layer radar in the center of a large city. Ann. Geophys., 12, 529± mitter beamwidth and the receiver ®eld of view. 540. The correction consists of dividing the measured Sassen, K., 1977: Lidar observations of High Plains power by F or adding 10 log10(1/F) to the measured precipitation. J. Atmos. Sci., 34, 1444±1457. , 1978: Backscattering cross sections for hydrometeors: Mea- power in decibels. Figure A3 is a plot of 10 log10(1/F) as a function of height, which is the additive factor in surements at 6328 AÊ . Appl. Opt., 17, 804±806. , C. J. Grund, J. D. Spinhirne, M. M. Hardesty, and J. M. Alvarez, decibels to correct for incomplete beam overlap. We do 1990: The 27±28 October FIRE IFO cirrus case study: A ®ve not apply the correction at altitudes below 122 m (gate lidar overview of cloud structure and evolution. Mon. Wea. Rev., 8), where it amounts to about 4 dB, because the cor- 118, 2288±2311. rection rapidly increases at closer ranges and any in- Shipley, S. T., E. W. Eloranta, and J. A. Weinman, 1974: Measurement accuracies arising from uncertainties in the beamwidth of rainfall rates by lidar. J. Appl. Meteor., 13, 800±807. and the beam alignment are magni®ed. Spinhirne, J. D., J. A. Reagan, and B. M. Herman, 1980: Vertical distribution of aerosol extinction cross section and inference of Figure A4 is a plot of the corrected pro®le in Fig. aerosol imaginary index in the troposphere by lidar technique. A1, illustrating the further correction for the lack of J. Appl. Meteor., 19, 426±438. beam overlap. As expected, this correction ampli®es the Vaisala, 1989: Technical Manual, Laser Ceilometer CT12K. 304 pp. measurements at close range. This ®gure includes a third [Available from Vaisala Company, 100 Commerce Way, Wo- curve, which is the corrected power multiplied by (z/ burn, MA 01801.] 2 Warner, C., and K. L. S. Gunn, 1967: Measurement of snowfall by z0) , where z0 ϭ 1 km. This is the range-adjusted pro®le, optical attenuation. McGill University Stormy Weather Group which corresponds to P(z) in the theory. Scienti®c Rep. MW-51, 36 pp. [Available from Atmospheric The software provided by Dr. LoÈnnqvist of the Vaisala Sciences, McGill University, Montreal H3A 2K6, Canada.] company contains a subroutine that corrects empirically Weichel, H., 1990: Laser Beam Propagation in the Atmosphere. SPIE for the incomplete beam overlap. Although the Vaisala Optical Engineering Press, 98 pp. corrections agree to within 1 dB of the curve in Fig. Zuev, V. E., 1976: Laser-light transmission through the atmosphere. Laser Monitoring of the Atmoshphere, E. D. Hinkley, Ed., A3 at ranges beyond 122 m, we preferred to use our Springer-Verlag, 29±69. correction function because it is smooth and continuous, , 1982: Laser Beams in the Atmosphere. Consultants Bureau, unlike the Vaisala correction. 504 pp.

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