## 1. Introduction

Interaction of radiation with clouds depends, in part, on the size distribution of the cloud particles. For example, Wiscombe et al. (1984) show that absorption of shortwave radiation increases with the size of cloud particles. For infinitely thick clouds, Twomey and Bohren (1980) show that absorption in the shortwave by cloud is approximately proportional to the square root of the mean radius of cloud particles. Scattering by cloud particles is not negligible in computing radiation in the 8–13-*μ*m wavelength region through clouds (Toon et al. 1989). The intercomparison of radiative transfer models by Ellingson et al. (1991) and a study by Fu et al. (1997) also indicate that the computed infrared irradiance depends on the treatment of scattering by cloud particles.

Some earlier studies demonstrate that the size distribution of stratus cloud particles can be retrieved using radar reflectivity factor from millimeter-wave cloud radar and liquid water path estimated from microwave radiometer radiance measurements (White et al. 1991; Frisch et al. 1995, 1998; Sassen et al. 1999). These algorithms generally assume a specific three-parameter distribution for cloud particles such as a lognormal distribution. Assuming a known value for the geometric standard deviation and taking the number concentration to be constant with height, the algorithms retrieve the liquid water content at each radar sample volume. They derive the number concentration by integrating the liquid water content and matching it with the liquid water path obtained from a microwave radiometer. Given both retrieved liquid water content and number concentration, as well as the estimated geometric standard deviation for each radar sample volume, any moment of the three-parameter particle size distribution can be computed.

One source of error in these retrievals is the use of a climatological value for the geometric standard deviation (Miles et al. 2000). In many retrievals the retrieved size distribution is sensitive to the geometric standard deviation. Hence, if the climatological value assumed for the geometric standard deviation is not close to the actual stratus cloud value, there will be significant error in the retrieval.

This paper discusses an alternative retrieval method that incorporates the median radius of the cloud particle number concentration (count median radius; see Hinds 1982), as opposed to the geometric standard deviation. The advantage of this approach is that the extinction coefficients computed from retrieved size distribution are less sensitive to errors in median radii as opposed to geometric standard deviations. The algorithm retrieves vertical profiles of liquid water content, number concentration, and effective radius using vertical profiles of radar reflectivity factor, Doppler velocity, and estimated particle median radius, together with integrated liquid water path. The median radius at each height within the cloud is estimated from a statistical model that relates variance of the vertical velocities in stratus to the stratus particle median radius. The input to the statistical model is the variance of the mean Doppler velocity (i.e., the first moment of the Doppler spectrum) measured by a 35-GHz Doppler radar computed for a given time period. Since the algorithm does not use shortwave radiation measurements, the results it produces are independent of any accompanying shortwave radiation measurements.

## 2. Method

### a. Retrieval of number concentration and liquid water content

#### 1) The lognormal distribution

*n*th moment can be calculated (Hinds 1982) aswhere

*f*(

*r*) is the lognormal cloud particle number distribution defined as

*r*is the particle radius,

*N*is the particle number concentration,

*σ*

_{g}is the geometric standard deviation of the distribution, and

*r*

_{n}is the median radius. The median radius is defined as the radius for which half the particles are both smaller and larger than it.

#### 2) Frisch et al. (1995, 1998) algorithm

*Z*for the

_{i}*i*th sample volume is

*Z*

_{i}^{6}

*N*

_{i}*r*

^{6}

_{ni}

^{2}

*σ*

_{gi}

*q*, for the

_{i}*i*th radar sample volume iswhere

*ρ*

_{w}is the density of liquid water. We can eliminate

*r*

_{ni}by combining (3) and (4) to obtainSumming the liquid water contents

*q*over all radar sample volumes with height Δ

_{i}*z*yields the liquid water path,where

_{i}*n*

_{c}represents the number of in-cloud radar sample volumes. To obtain (6), we assume that

*N*

^{1/2}

_{i}^{2}

*σ*

_{gi}) is constant with height (Ovtchinnikov and Kogan 2000). When we assume that

*N*is constant with height and given all in-cloud radar reflectivity factors

_{i}*Z*, the cloud liquid water path

_{i}*Q,*and the geometric standard deviation

*σ*

_{g}, one can solve (6) for

*N*(White et al. 1991; Frisch et al. 1995, 1998). Then

*N*can subsequently be eliminated from (5) using (6) to obtain

#### 3) Proposed algorithm based on particle median radius estimation

*σ*

_{gi}from (3) and (4), as opposed to

*r*

_{ni}, we obtainWhen the liquid water contents

*q*

_{i}are summed over all radar sample volumes with height Δ

*z*

_{i}, the liquid water path

*Q*becomeswhere the number concentration

*N*is assumed to be constant with height. The particle median radius

*r*

_{ni}in (9) is estimated from the variance of the Doppler vertical velocity for each radar sample volume and hence varies with height. Eliminating

*N*using (8) and (9), we obtain

### b. Determination of the particle median radius

Considine and Curry (1996) relate the vertical velocity distribution of parcels either ascending or descending adiabatically to the particle size distribution by assuming that the vertical velocity uniquely determines the particle size. They also assume that at a fixed height within a cloud and over a sufficiently long period of time the distribution of the vertical velocity is Gaussian and the mean velocity is zero. Consequently, parcels that reach a certain height can have different water contents, depending on their vertical velocities. A parcel that has a larger vertical velocity at a given height must have traveled a longer distance from its place of origin within cloud as compared to a parcel having a smaller vertical velocity at the same height. Since they assume that the vertical velocity of a parcel is zero at the lifting condensation level of the parcel, a diluted parcel that has lifted adiabatically within cloud has smaller vertical velocity than a nondiluted parcel originating from their respective lifting condensation levels.

*r*

_{ni}and the variance of the vertical velocity are related as (see the appendix)In these equations

*ww*

*g*is the acceleration due to gravity, and

*T*is the air temperature in the parcel. The symbols Γ

_{e}, Γ

_{p}, and Γ

_{m}represent the environment, parcel, and moist adiabatic lapse rates, respectively;

*F*

_{D},

*F*

_{K}, and

*K*are defined in the appendix. Following Considine and Curry (1996), we assumed that

*X*equals 2.4 and neglected the radius of the particles in a parcel of zero vertical velocity. When we use 850 hPa for

*p,*273 K for

*T,*and 1.47 × 10

^{10}s m

^{−2}for

*F*

_{K}+

*F*

_{D}(Rogers and Yau 1989, p. 104);

*r*

_{ni}

*ww*

^{1/4}

_{i}

*r*

_{ni}is in micrometers and

*ww*

^{2}s

^{−2}. In this study, we used (14) to retrieve

*r*

_{ni}.

According to Stokes' law, the fall velocity of a 10-*μ*m radius spherical particle relative to air is approximately 1 cm s^{−1}. The variation in vertical air velocities of ±1 m s^{−1} in one stratus cloud event observed by Sassen et al. (1999) indicates that the fall velocity of cloud particles relative to air is negligible. These results suggest that apart from drizzle particles the vertical velocities of cloud particles are close to the velocity of air. Because the formation of stratus clouds is over scales of motion that encompass a radar sample volume and the statistical model of Considine and Curry (1996) is based on the distribution of vertical velocities that drives the vertical motion of parcels, and not individual cloud particles, we assume that the first moment of the Doppler radar spectrum, or the mean Doppler velocity *υ*_{d} represents *w* in the Considine and Curry (1996) study. We further assume that the mean Doppler velocity *υ*_{d} averaged over a 30-min interval is approximately zero, so that *υ*_{d}*υ*_{d}*ww**υ*_{d} as a measure of the vertical air motion that result from precipitating particles we limited the analysis to radar data with Doppler velocities between ±1 m s^{−1} and with reflectivity factors less than −20 dB*Z.*

The mean Doppler velocity *υ*_{d} averaged over the 30-min interval may depart from zero, and hence *υ*_{d}*υ*_{d}*ww**υ*_{d}*υ*_{d}*υ*_{d}*υ*_{d}*υ*_{d}*υ*_{d}*ww*

### c. Retrieval of effective radius

*r*

_{e}which is defined by Hansen and Travis (1974) asTherefore, we must relate

*N,*

*q*

_{i}, and

*r*

_{ni}to

*r*

_{ei}to use the retrieved results in radiative transfer studies. Again assuming a lognormal number distribution for

*f*(

*r*), the effective radius

*r*

_{ei}is

*r*

_{ei}

*r*

_{ni}

^{2}

*σ*

_{gi}

*σ*

_{gi}in (16), we obtain

## 3. Results

We used data collected at the Atmospheric Radiation Measurement (ARM) program (Stokes and Schwartz 1994) Southern Great Plains central facility (36.69°N, 97.48°W) to retrieve the size distribution of stratus cloud particles. We retrieved vertical profiles of cloud liquid water content, number concentration, and effective radius during two stratus cloud cases that occurred over the central facility. The first case was 30 April 1994 when a single layer stratus cloud passed over the site in the afternoon (Sassen et al. 1999). While the retrieval developed by Liljegren (1995) provided estimates of cloud liquid water path from microwave radiometer measurements, the University of Massachusetts 35-GHz Doppler radar provided radar reflectivity factor and Doppler velocity observations from 2030 to 2200 UTC. During the entire period, the reflectivity factors measured by the radar were less than −20 dB*Z,* suggesting the absence of drizzle. The general lack of drizzle during this period is also supported by surface observations that recorded no precipitation. Note that for approximately 6 min around 2130 UTC the cloud radar reflectivity factors were below the threshold of detection.

The second case took place on 3 December 1997 when a single layer stratus cloud covered the site. Reflectivity factors and Doppler velocities were measured by the ARM 35-GHz Doppler radar operated as a part of the Atmospheric Radiation Measurement program (Moran et al. 1998; Clothiaux et al. 2000). The clouds were thick and precipitating from 0000 to 0600 UTC. During this period, the radar detected hydrometers up to 8 km with a maximum reflectivity factor of approximately 10 dB*Z.* The clouds subsequently thinned and single layer stratus cloud covered the site from 1200 to 2400 UTC. During this period, radar reflectivity factors were less than −25 dB*Z* and no precipitation was recorded at the surface. The low radar reflectivity factors for these two cases also imply that these clouds were liquid water clouds (Mace and Sassen 2000).

The retrieved liquid water contents, particle effective radii, and number concentrations for the 30 April and 3 December cases are presented in Figs. 1 and 2, respectively. As Figs. 1 and 2 illustrate, cloud liquid water contents were higher on 30 April compared to 3 December, and in both cases the maximum liquid water content (≈1.0 g m^{−3} at 2100 UTC on 30 April and ≈0.4 g m^{−3} at 2100 UTC on 3 December) occurred when cloud top reached its maximum altitude. However, the same amount of liquid water content occurred near 1400 UTC when the cloud altitude is minimum on 3 December 1997. In the two cases the retrieved liquid water contents increased with height through the lower half or two-thirds of the stratus (Fig. 3), in agreement with both theoretical studies (Mason and Chien 1962; Khairoutdinov and Kogan 1999) and observations (Albrecht et al. 1985, 1990; Noonkester 1984; Stephens and Platt 1987). Averaging the liquid water content profiles from 2111 to 2142 UTC (apart from the break in cloud around 2130 UTC) on 30 April led to a maximum of 0.63 g m^{−3} at an altitude of 1.6 km above ground level, in agreement with the results of Sassen et al. (1999).

Sassen et al. (1999) report that in situ observations in the stratus with a forward scattering spectrometer probe (FSSP) from 2111 to 2142 UTC on 30 April led to an average value of 345 cm^{−3} for the number concentration that was fairly constant with height. The retrieved number concentration averaged over the same period was 338 cm^{−3}.

Since we assume a constant number concentration with height and the retrieved cloud liquid water increases with height, the retrieved particle effective radius increased with height (Fig. 3). This increase in the particle effective radius with height also agrees with observations (Nicholls 1984; Stephens and Platt 1987). The average volume radii, *r*_{vi}, which is the radius of a particle the mass of which equals the liquid water content divided by the number concentration, derived from the retrieval and in situ FSSP measurements made on 30 April increase with height and are comparable in magnitude (Fig. 4).

To further evaluate the reasonableness of the retrieved microphysical properties, we input them into two radiative transfer models and compared the results with observations. To this end we used delta two- and four-stream radiative transfer models (Toon et al. 1989; Liou et al. 1988) to compute the downward longwave and shortwave surface irradiance, respectively, using the retrieved number concentrations and effective radii as input to these models. We computed the optical properties of the cloud particles according to Mie theory using the retrieved effective radius and a geometric standard deviation of 1.42 to specify the parameters of a lognormal distribution.

In the radiative transfer calculations gaseous absorption was computed using the *k*-distribution tables with the correlated-*k* assumption developed by Kato et al. (1999) and Mlawer et al. (1997) for shortwave and longwave radiation, respectively. Rawinsondes provided atmospheric water vapor profiles for both case study periods, while we estimated the shortwave surface albedo for 3 December from the ratios of upward to downward pointing pyranometer measurements. The average surface albedo between 1800 and 2100 UTC on 3 December was 0.174, and we used this value for the surface albedo on 30 April because the upward surface irradiance was not measured on this day. We assumed the surface emissivity to be unity and used the surface air temperature for the surface temperature. Both the downward shortwave and downward longwave irradiances were computed at 1-min intervals over the two case study periods.

The resulting downward shortwave surface irradiances are well correlated with the measured irradiances between 2030 and 2200 UTC on 30 April (Fig. 5) and throughout the 8-h period from 1400 to 2200 UTC on 3 December (Fig. 6). On average, the modeled shortwave irradiance is greater by 21 W m^{−2} and smaller by 6.3 W m^{−2} than the measurements on 30 April and 3 December, respectively. These amounts correspond to 12% and 8% of the averaged measured downward shortwave irradiance for these two periods. The modeled longwave irradiance is smaller by 0.7 W m^{−2} than the measurements for the both days, which corresponds to 0.2% of the averaged measured downward longwave irradiance over the two periods.

For the comparison, we also used the effective radius of 5.4 *μ*m, which is the average effective radius for continental stratus clouds reported by Miles et al. (2000), to compute the downward shortwave and longwave surface irradiance. The liquid water was uniformly distributed from the observed cloud base to the top. The computed irradiance with the average effective radius is also well correlated with measurements, which implies that the variation of the shortwave surface irradiance is mainly caused by the variation in the liquid water path. On average, the modeled shortwave irradiance with the average effective radius is greater by 13 W m^{−2} and smaller by 23 W m^{−2} than the measurements on 30 April and 3 December, respectively. While Dong et al. (2000) reported a substantial variation in the effective radius over 2 yr at the site (4–14 *μ*m), using the effective radius of 5.4 *μ*m for computing downward irradiance does not introduce a significant bias error for these two cases because the retrieved effective radii are close to the average effective radius.

## 4. Discussion

When a cloud extends through *n*_{c} radar sample volumes, there are *n*_{c} + 1 equations in the retrieval consisting of *n*_{c} equations of (3) and one equation of (9). Assuming a lognormal distribution for each radar sample volume produces a total of 3*n*_{c} unknowns, that is, *N*_{i}, *σ*_{gi}, and *r*_{ni} for each radar sample volume. Therefore, we must make assumptions to reduce the number of unknowns from 3*n*_{c} to *n*_{c} + 1. Following White et al. (1991), we made the assumption that the number concentration *N*_{i} was constant with height. We then used the variance of the vertical Doppler velocity to obtain *r*_{ni} throughout the vertical extent of cloud and solved the resulting *n*_{c} + 1 equations for *N* and *σ*_{gi} (*i* = 1, … , *n*_{c}).

*q*

_{i}, effective radius

*r*

_{ei}, number concentration

*N,*and extinction coefficient

*β*

_{i}, are important to the radiation, we must quantify how errors in the estimates of the retrieved properties, and especially

*r*

_{ni}, propagate into estimates of these radiatively important quantities. In the following section, we estimated the error in retrieved values due to the error in the estimated median radius. For a comparison, we also estimated the error in retrieved values by the algorithm given by Frisch et al. (1995) due to the error in the geometric standard deviation. To estimate the error in the liquid water content due to the error in the median radius, we use (10) and take the derivative with respect to

*r*

_{ni}to obtainWhen we sum the error contributing to the

*i*th radar sample volume from other radar sample volumes (i.e., (∂

*q*

_{i}/∂

*r*

_{nk})Δ

*r*

_{nk},

*k*≠

*i*), the error in the liquid water content isWhen the fractional error in the median radius does not depend on height, the error in the liquid water content vanishes. The liquid water content retrieved by the algorithm given by Frisch et al. (1995) does not depend on the value of geometric standard deviation because of the assumption of the constant geometric standard deviation with height.

*r*

_{v}, the radius of a particle of which mass times the number concentration gives the liquid water content, asFor a lognormal distribution, the effective radius,

*r*

_{ei}, is

*r*

_{ei}

*r*

_{vi}

^{2}

*σ*

_{gi}

*N*and (10) for

*q*

_{i}in (24) and taking the derivative with respect to

*r*

_{ni}yieldsAgain, when we sum the error contributing to the

*i*th radar sample volume from other radar sample volumes, the error in the effective radius isWhen the fractional error in the median radius does not depend on height, (26) becomesTherefore, the fractional error in the

*r*

_{ei}is approximately half of the fractional error in

*r*

_{ni}when the fractional error in the median radius does not depend on height, hence the shape of the median radius vertical profile is correct. Similarly, substituting (6) for

*N*in (22), substituting the resulting equation in (23), and taking the derivative with respect to

*σ*

_{g}, we obtainHence, the fractional error in the effective radius retrieved by the algorithm given by Frisch et al. (1995) is 1.3 times greater than the fractional error in the geometric standard deviation when the geometric standard deviation is 1.4.

*r*

_{ni}, we take the derivative of

*N*with respect to

*r*

_{ni}in (9) and add the error due to all in-cloud radar sample volumes, the error in the number concentration isIf the fractional error in the median radius does not depend on height, (29) becomesThat is, the fractional error in the number concentration is approximately twice as large as the fractional error in

*r*

_{n}and opposite in sign. For comparison to the retrievals based on a constant value of

*σ*

_{g}we take the derivative of

*N*with respect to

*σ*

_{g}in (6) to obtainSince the value of 18 In

*σ*

_{g}is approximately 6 when

*σ*

_{g}is 1.4, the fractional error in

*N*is six times larger than the fractional error in

*σ*

_{g}.

*β*

*is given by (19), thenSubstituting (20) and (25) into (32) and summing the error due to all in-cloud radar sample volumes yieldsWhen we add the error due to all in-cloud radar sample volumes and if the fractional error in the median radius is independent with height, (33) becomesLikewise, the fractional error in the extinction coefficient as a result of error in the geometric standard deviation in the algorithm given by Frisch et al. (1995) isThe fractional error in*

_{i}*β*

_{i}is approximately half of the fractional error in

*r*

_{ni}if the fractional error in the median radius does not depend on height and about 1.3 times the fractional error in

*σ*

_{g}when

*σ*

_{g}is 1.4. Note that the error in

*q*

_{i}vanishes when the fractional error in

*r*

_{ni}is independent with height so that the magnitude of error in

*β*

_{i}is the same as the magnitude of error in

*r*

_{ei}.

In a study of in situ cloud measurements Miles et al. (2000) report that the median radius observed in stratus clouds varies from 1 to 10 *μ*m, while the geometric standard deviation varies from 1.2 to 2.2. Therefore, the variations in the median radius and geometric standard deviation are significant, implying that neither quantity should be set to a climatological average for the most accurate retrievals. However, using the effective radius of 5.4 *μ*m, which is the average effective radius of continental stratus clouds, for computing downward irradiance does not introduce a significant bias error because the retrieved effective radii for these two cases are close to the average effective radius. If the effective radius of stratus at given site does not change very much and if the value is known, using the value for the surface irradiance may give the unbiased irradiance when they are averaged over a certain period. However, Dong et al. (2000) reported a substantial variation in the effective radius at the site.

*r*

_{ni}, which is a function of

*D*

_{i}given by (12). The sensitivity of

*r*

_{ei}to

*D*

_{i}is given byThe sensitivity of

*r*

_{ni}to

*D*

_{i}iswhere

*r*

^{2}

_{n}

*D*= 0.96

*r*

_{0}= 0 is used to derive (37). Consequently, the fractional error in

*r*

_{ni}is half of the fractional error in

*D*

_{i}and the fractional error in

*r*

_{ei}due to the error in

*D*

_{i}is approximately one-fourth of the fractional error in

*D*

_{i}if the fractional error in

*r*

_{ni}does not depend on height. Considine and Curry (1996) show that

*D*is not a strong function of temperature but changes with

*ww*

*X.*While

*ww*

*X*was assumed to be 2.4 in the retrieval. The quantity

*X*can vary from zero to infinity depending on the lapse rate of the parcel relative to the environmental and adiabatic lapse rates. For realistic values of

*r*

_{ni},

*X*must be between 1 and 4 if the maximum of

*ww*

^{2}s

^{−2}, which was the case for the two study periods.

To verify the magnitude of the preceding error estimates we constructed a simple five-layer cloud model with layer median radii and geometric standard deviations given in Table 1. We then retrieved the effective radius, number concentration, and liquid water content for each layer with either the median radii or geometric standard deviations set to their perturbed values in Table 1. The case indicated by “*r*_{ni} perturbation” in Table 1 simulates the retrieval proposed in this paper and that by “*σ*_{g} set to 1.4” simulates the retrieval given by Frisch et al. (1995). We subsequently computed the extinction coefficients *β*_{i} using (19).

The results from this simple model are in agreement with the preceding error analysis. For example, when *r*_{n} is perturbed to a value approximately 20% smaller than the actual *r*_{n}, *r*_{ei} given by (16) decreases by approximately 10% and *N* given by (9) increases by approximately 80%. The resulting extinction coefficient increases by approximately 10%. Similarly, when *σ*_{g} is set to a value of 1.4 (a perturbation to a value approximately 20% greater than the actual *σ*_{g}), *r*_{e} decreases by approximately 15% and *N* given by (6) increases by approximately 120%. In this case the extinction coefficient increases by approximately 15%. All of these changes are what is expected based on (20)–(35).

*μ*m, a geometric standard deviation of 1.4, and a concentration of 1.0 cm

^{−3}to the case 1 value of

*r*

_{ni}and a geometric standard deviation of 1.4. We then retrieved

*N,*

*r*

_{ei}, and

*q*

_{i}both assuming the size distribution of the smaller particles. In both cases the retrieved number concentrations,

*N,*is less than its actual value. However, the number concentration obtained by setting

*σ*

_{g}to 1.4 is much smaller. To understand this result rewrite (6) and (9) asrespectively. For simplicity assume that

*r*

_{ni}is constant with height. Both sides of (38) have dimensions of one over length cubed, while the both sides of (39) have dimensions of one over length. By adding large particles to a single-mode lognormal distribution (as in case 2), the left-side denominator in (38) is augmented more than in (39), causing a larger change in

*N.*

We also investigated, in Table 2, sensitivities of retrieval values to instrumental errors to the true value given at the top of Table 2. Clothiaux et al. (1995) show that the one-way attenuation rate of 94-GHz radar signals is less than 5 dB km^{−1} when the liquid water content is less than 1 g m^{−3} and the mean radius is approximately 5 *μ*m. Attenuation of 35-GHz radar signals is about one-sixth of 94-GHz radar signals, which gives attenuation no more than 1 dB km^{−1} in the same cloud. In this study, first, we assumed the two-way attenuation of radar signal of 0.5 dB in each layer (case 1 in Table 2). Second, we assumed that radar reflectivity factors of all layers are 3 dB smaller than the true values due to the calibration error (case 2 in Table 2). In both cases the reflectivity factor is smaller than the reflectivity factor computed from the liquid water content. In order to tolerate these reductions in reflectivity factor keeping the liquid water path the same as in the algorithm, the size of particles must be reduced and the number concentration must be increased because reflectivity factor is more sensitive to the particle size. For both cases the extinction coefficient computed from the retrieved size distribution were increased due to increasing the number concentration because increasing the left side of (38) and (39) by attenuation or the calibration error increases *N* in the right side. Third, we increased the liquid water path by 10% (case 3 in Table 2). In all three cases, the optical thicknesses of the cloud integrated from extinction coefficients are greater than the true values. However, the optical thicknesses computed from the proposed retrieval results are affected less by these instrumental errors than those computed from the Frisch et al. (1995) retrieval.

One drawback of the proposed retrieval based on estimates of the median radius *r*_{ni} is that the geometric standard deviation becomes imaginary (i.e., ln^{2}*σ*_{g} < 0) when the retrieved effective radius is less than *r*_{ni}. In other words, when our estimate of the median radius is larger than the retrieved effective radius because of assumptions in the algorithm or instrumental errors or both, the geometric standard deviation is unphysical. Even when the retrieved standard deviation is unphysical, retrieved *N,* *q*_{i}, and *r*_{ei} can be reasonable such as shown in Table 2.

As mentioned before, the optical thickness computed from retrieved values for 3 December 1997 is greater than the optical thickness that would provide the observed surface irradiance. Based on the error analysis, possible reasons for this difference besides inhomogeneity of the cloud are that 1) estimated median radii are smaller than actual median radii especially where reflectivity factors are large so that right side of (33) is positive and 2) radar calibration is off so that actual reflectivity factors are higher than the measured values or the liquid water path estimated from the microwave radiometer is less than the actual value.

## 5. Conclusions

This paper demonstrates that number concentration and vertical profiles of liquid water content and effective radius can be retrieved using liquid water path estimates and vertical profiles of radar reflectivity factor and Doppler velocity. An important feature of the proposed retrieval is that the height dependence of the median radius *r*_{ni} of the cloud particle size distribution is estimated from vertical Doppler velocity measurements obtained by a 35-GHz cloud radar. In particular, the median radius is estimated from the variance of the vertical Doppler velocity measurements using the statistical model of Considine and Curry (1996). The Considine and Curry (1996) model assumes that the distribution of vertical velocities at a given height is Gaussian with a mean of zero and that the variance of the vertical velocity uniquely determines the median radius of the cloud particles. Moreover, the model implies that the cloud particle median radius is proportional to the fourth root of the vertical velocity variance if we neglect the radius of cloud particles in a parcel of zero vertical velocity. The proposed algorithm assumes that the cloud particle number concentration is constant with height.

We used two case study periods to evaluate the algorithm, a 1.5-h period on 30 April 1994 for which we have corroborating aircraft data, and an 8.0-h period on 3 December 1997. For the stratus case on 30 April the retrieved number concentration was 338 cm^{−3}, while in situ FSSP measurements reported a concentration of 345 cm^{−3} (Sassen et al. 1999). The retrieved and in situ average volume radii were also consistent with each other for this case.

Error analyses suggest that the error in the liquid water content vanishes and the magnitudes of the fractional error in the effective radius and shortwave extinction coefficient computed from retrieved cloud particle size distributions are half of the magnitudes of the fractional error in the estimated cloud particle median radius if the fractional error in the median radius is constant with height. Moreover, the algorithm is affected less by instrumental errors and the assumption of a lognormal distribution when the actual cloud particle size distribution deviates from lognormal as compared with algorithms that assume a prespecified value for *σ*_{g}.

We thank M. Poellot and Z. Wang for supplying processed FSSP data; Y. X. Hu, L. Smith, B. Lin and M. Ovtchinnikov for useful discussions; L. M. Hinkelman and X. Dong for useful comments; J. H. Mather for developing a delta four-stream model; and E. J. Mlawer for supplying the longwave *k*-distribution model. Data were obtained from the Atmospheric Radiation Measurement (ARM) Program sponsored by the U.S. Department of Energy, Office of Energy Research, Office of Health and Environmental Research, Environmental Sciences Division. S. Kato was supported by the NASA Clouds and the Earth's Radiant Energy System Grant NAG-1-1963. E. Clothiaux received support for this research from the Environmental Science Division of the U.S. Department of Energy (under Grant DE-FG02-90ER61071).

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# APPENDIX

## Median Radius Estimate from Vertical Velocity Variance

*w*inside clouds at a given height follows a Gaussian distributionwhere

*ww*

*w*aswhereIn these equations

*g*is the acceleration due to gravity,

*p*is the total atmospheric pressure,

*c*

_{p}is the specific heat capacity of dry air at constant pressure,

*L*

_{v}is the latent heat of vaporization of water,

*T*is the temperature of air in the parcel,

*e*

_{s}(

*T*) is the saturation vapor pressure at temperature

*T,*

*R*is the gas constant of dry air,

*ϵ*is the ratio of the molecular weight of water to the apparent molecular weight of dry air (0.622),

*k*

_{K}is the thermal conductivity of air,

*k*

_{D}is the diffusion coefficient of water vapor in air, and

*ρ*

_{w}is the density of liquid water. The parameters Γ

_{e}, Γ

_{p}, and Γ

_{m}represent the environment, parcel, and moist adiabatic lapse rates, respectively. The parameter

*X*is determined by the differences between ascending or descending parcel lapse rates and the moist adiabatic and environment lapse rates. Finally,

*r*

_{0}is the radius of the particles in a parcel for which the vertical velocity is zero.

*F*(

*r*), such thatSince

*r*> 0, if we let

*η*= [

*D*(

*r*

^{2}−

*r*

^{2}

_{0}

*z*is defined asSince erf(0) = 0 and erf(0.48) ≈ 0.5, the median radius isif

*r*

_{0}is set to 0. Considine and Curry (1996) assume that the vertical velocity of a parcel is zero at the lifting condensation level of the parcel, which is equivalent to assuming that particles have zero radius at the lifting condensation level. The median radius as a function of

*ww*

*r*

_{0}and

*X*is shown in Fig. A1. Here

*F*

_{K}+

*F*

_{D}to compute

*D*by (A9) is set to 1.47 × 10

^{10}s m

^{−2}(Rogers and Yau 1989, p. 104). The median radius is not a strong function of

*r*

_{0}for

*r*

_{0}smaller than 5

*μ*m except for small

*ww*

*X*increases, which means the lapse rate of the parcel approaches the moist adiabatic lapse rate, (A2) shows that the median radius becomes less dependent on

*ww*

Sensitivity study of retrieval algorithm. Numbers in parentheses indicate the assumed values

Sensitivity to instrumental errors. Numbers in parenthesis indicate the assumed values or inputs. A minus sign (−) indicates that the retrieved value is an imaginary number