1. Introduction

Vertically pointing measurements of a marine stratus cloud were obtained during the Atlantic Stratocumulus Transition Experiment (ASTEX) using the NOAA/Environmental Technology Laboratory’s (ETL) 35-GHz (K_a-band) Doppler cloud research radar. This radar is especially well suited for measurements of nonprecipitating and weakly precipitating cloud features by virtue of its excellent sensitivity (−30 dBZ at 10-km range), resolution (37.5-m resolution, 0.5° beamwidth, and 0.25-s sample time), and accuracy (±5 cm s⁻¹ under typical conditions). The short wavelength (8.6 mm) of this radar imparts an inherent sensitivity advantage over longer-wavelength radars for detection of the small hydrometeors that constitute clouds because the backscattering cross section of cloud droplets is proportional to λ⁻⁴ for Rayleigh scattering. Furthermore, the wavelength is sufficiently short that contributions to the observed reflectivity from clear-air refractive-index turbulent fluctuations are negligible compared with those from hydrometeors. Rainfall exceeding drizzle intensity, however, can cause attenuation of the signal at this wavelength. Attenuation from water vapor can also besignificant for radars of shorter wavelength than this. Examples of the NOAA/ETL radar’s ability to detect weak finescale cloud features and its use in deriving a variety of cloud microphysical parameters through the use of its Doppler moments measurements are given by Martner and Kropfli (1993) and Kropfli et al. (1994). Velocity and reflectivity data have been used by Frisch et al. (1995) to calculate total liquid using the pulse-pair processing of the Doppler data. In the present article, Doppler spectra, recorded in the radar’s time series mode with a height resolution of 150 m, are utilized to obtain drop-number density spectra as well as liquid and flux spectral distributions. The time series data needed for full spectral analysis could not be collected during the pulse-pair processing, so the results cannot be directly compared.

In the large-drop regime of liquid clouds, where fall velocities are much greater than the turbulent velocity, Doppler radar measurements of particle size spectra generally agree well with in situ measurements by aircraft (e.g., Rogers et al. 1993). In this regime, the precipitation spectral domain is easily separated by fall velocity from the cloud (or clear-air) spectral domain that is typically used by wind profilers to measure winds. This peak, near zero fall velocity, is then used to remove up-/downdrafts in the medium, and the corrected fall velocity spectrum is converted to drop size distribution (e.g., Atlas et al. 1973). In this regime, the reflectivity and fall velocity are the important measurables.

On the other hand, for small droplets the settling velocity (i.e., the terminal fall velocity of cloud droplets) is usually small compared with the turbulent velocity fluctuations in the medium. The droplets then serve mainly as tracers of the turbulent motion of the air, and their backscatter is not inherently distinguishable from the clear-air return. In this regime, the terminal fall velocity is not useful for deducing drop size. Turbulent movement of the droplets “smears” the fall velocity spectrum (Wakasugi et al. 1986, 1987; Gossard 1988, 1994; Gossard and Strauch 1990; Gossard et al. 1990; Sato et al. 1990; Currier et al. 1992; Rajopadhyaya et al. 1993), and the measured spectrum is, in fact, the convolution of the fall (settling) velocity with the turbulent velocity probability density function (PDF). Therefore, the measured spectrum must be deconvolved, and the spectral half-width to the e⁻¹ point, w_σ, replaces V_f (the terminal fall velocity in quiet air) as the important radar measurable, as illustrated in Fig. 1, where downward velocities are positive. We use w as a general symbol of vertical velocity. Figure 1 shows the normalized radar reflectivity spectrum from a hypothetical gamma droplet distribution in quiet air (solid, w_σ = 0) compared with corresponding theoretical reflectivity spectra when the population is embedded in an atmosphere with various levels of turbulence (dashed). Here w_M is the vertical velocity at the maximum of the gamma function (w_σ = 0). In this paper, we deal with a case of heavy stratus with some drizzle and deconvolve the measured spectra to recover the cloud spectrum and separate it from the drizzle regime. In this case, we find that the cloud and drizzle peaks are usually separablebecause of the very high spectral-velocity resolution (2.6 cm s⁻¹) in the radar spectra. [A 330-point fast Fourier transform (FFT) was used in the data reduction.] In the cloud regime, we use an interpolation formula proposed by Rogers et al. (1993) to represent the size versus fall velocity relationship between the small-droplet “Stokes” regime and the linear relationship in the larger-droplet regime.

2. The data

The data analyzed in this report were collected on Porto Santo Island in the Madeiras on 12 June 1992 from a 3-min time series beginning at 2109 UTC. Height profiles of Doppler cloud spectra were collected about every 0.2 s, with 150-m height spacing, providing about 1000 profiles of cloud spectra in the 3-min interval. The radar characteristics are summarized in Table 1. For the radar’s 0.5° beamwidth, it samples a volume of about 10⁴ m³ every 0.2 s in each 150 m of altitude. Each spectrum covers a number density range of at least seven orders of magnitude. Thus, short-wavelength radars provide a tool of great potential in studying and monitoring cloud microstructure. A three-channel microwave radiometer was collocated (about 20-m separation) with the radar to provide total liquid water above the site. The channels are coupled into a common antenna with 2.5° beamwidth. The offset paraboloid antenna is inside a shelter viewing a reflector outside oriented at 45° so that it reflects zenith radiation into the antenna.

The clouds herein described are of the stratus type found on the edge of subtropical high pressure regions of the globe, such as the Azores high, off the coast of southern California, Australia, Chile, and the Middle East. Heavy stratus and drizzle were of special interest to the radar analysis. The entry in the radar log for 2101 UTC read “10-dBz drizzle overhead.” The echo had weakened substantially by the time the time series data collection was implemented.

3. Analytical procedures

To illustrate the analytical procedures, we will assume a gamma function for the drop size distribution. The method is easily extended to other distributions, and Gossard (1994) analyzed a lognormal distribution for comparison with the gamma distribution. The difference in the functional forms of the spectra was found to be relatively unimportant (discrepancies typically less than 10%) in calculations of liquid water and flux. Figure 2 shows a schematic description of the analysis of the spectra. The top frame shows hypothetical spectra on a log N plot, where N is drop-number density and D is drop diameter in millimeters. For the deconvolution, it is convenient to parameterize the problem relative to the peak of the reflectivity spectrum. Then, assuming a gamma function for the drop size distribution,

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where α is an integer index; N_M is the number density at the maximum of the reflectivity spectrum function, defined below; and D_M is the corresponding diameter. If α = 0, then the distribution is the classical exponential and D_M = (6/3.67)D₀, where D₀ is the commonly used mass median diameter. The figure shows two different exponential regimes for cloud and for precipitation. The radar reflectivity factor is

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so the reflectivity spectrum is

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where S_M is the maximum value of the function (3) andD_M is the corresponding diameter; thus, S_M = N_MD⁻⁶_M. Drop size spectra for several α are shown in Fig. 2 (top), and the reflectivity spectra corresponding to the drop size spectra are shown in the bottom frame. The functions in (1) and (3) have thus been expressed in parameters defined at the maximum of (3) for convenience.

Integrating (2) and using (1), we find [Gossard 1994, Eq. (24)] that

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where S_QM is the spectral maximum (subscript M) of the quiet-air (subscript Q) spectrum we seek and S_M is the measured maximum. We define S_QM/S_M = DF as the deconvolution factor, and its inverse as the convolution factor CF. Equation (4) defines the function f(α) for future use. Note that w_M is defined as the size at the maximum or mode of the reflectivity spectrum, and we define D_M to be the corresponding diameter. When the relationship between size and fall velocity is linear, they occur simultaneously. If the radar resolves the spectral peaks, we note that the reflectivity factor Z and the spectral maximum S_M are quantities that a radar measures well. Therefore, even in the presence of up/downdrafts, small w_M’s can be deduced from (4). Because it depends on the ratio of Z to S, the calibration of the radar is not important to first order in deducing w_M (and therefore D_M).

The cloud liquid water LW is given by

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where ρ_w is the density of liquid water. The liquid spectrum (g m⁻³ mm⁻¹) is

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and the flux FLX (m⁻² s⁻¹) is

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(If D_M is in millimeters or micrometers, the units of ρ_w must be suitably chosen to retain the proper units of M and ρ_w.)

We note that the radar measures S_M, whereas it is the quiet-air value S_QM that appears in (4). To get the quiet-air spectrum, the measured spectrum and the turbulent velocity PDF must be deconvolved. The procedure is described by Gossard (1994). We will assume that S_QM has the form of the gamma function. The integral to be solved is

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where the i and j indices on w represent the integrations over the distribution function and over fallspeed, respectively, in the deconvolution. Here S_Q(w_j) is given by (3) after D is expressed as w using (5a) and (5b). In the case of (5b)), the conversion from D to w is implicit and most easily done numerically.

In general, the integrations are best carried out numerically. The velocity–size functions to use for w_j and D_j are given by (5a) and (5b). The diagrams given in Gossard (1994) were not detailed or accurate enough at small-drop diameters for the cloud stratus case studied here, so they are replotted as the diagrams in Figs. 3–6 with many values of w_σ in the range of small w_M. The results for the linear law are compared with those for the interpolation formula. They are shown in alternate figures and labeled at the top. Note that in the limit of very small drops, as V_f goes to zero, the deconvolved distribution tends to the turbulent PDF. If the PDF is Gaussian, Z/S_M is just π^0.5w_σ. It is convenient to introduce a deconvolution factor DF defined as DF = S_QM/S_M, where S_M is the maximum of the convolved spectrum and S_QM is the corresponding maximum of the spectrum in quiet air. Instead of plotting DF, as in Gossard (1994), we plot DF⁻¹ for convenience in interpolating graphical points. Plots are shown for the limiting case of α = 0 (exponential) and for the fairly extreme case of α = 2. These values bracket the lognormal distributions sometimes used.

The diagrams show that the size information is essentially lost for sizes smaller than about D_M = 0.04 mm when (5b)is used. [The more commonly used median diameter D₀ is (3.67/6)D_M = 0.024 mm for α = 0.] When the size D_M calculated from the observed Z_c/S_MC is smaller than about 0.04 mm, we have used 0.04 mm when calculating cloud liquid. This yields a minimum value because smaller droplet populations have greater liquid for the same reflectivity. Only radar gates 5 and 6 at 750- and 900-m heights, respectively, had cloud reflectivity modal sizes greater than D_M = 0.04 mm, so the median diameter of the cloud regime was typically less than D₀ = 0.024 mm (see section 4).

The analysis procedure is shown in Fig. 7. It shows the Doppler spectrum from the radar range gate 5 of the first averaged profile. It is chosen for illustration because the cloud and precipitation domains of the spectrum are well defined and clearly separable. The first raw spectrum is shown beside it for comparison. Note that the separation of the regimes is clearly evident even in the raw spectra.

To analyze a measured spectrum, the procedure is as follows.

1. As a first estimate, we consider the cloud peak in the spectrum to have zero fall velocity relative to the air. From the velocity difference between the precipitation and cloud peaks in the spectrum, find a first estimate of the D_M for the precipitation regime. We thus first assume that the velocity of the cloud peak represents the up-/downdrafts in the clear air and then refine the cloud “settling” velocity by iteration.

2. Find an optimum value of w_σ from a regression fit of a Gaussian (or exponential) PDF to spectral points on the up-velocity side of the cloud spectral peak (see insert in Fig. 7). The resulting log slope versus w² (for a Gaussian PDF) will, of course, depend somewhat on the range of points selected for the regression fit. In this example, the regression fit is slightly better for an exponential PDF than for a Gaussian PDF. (Note the curvature of the data points on the regression plot that reduces the correlation coefficient to 0.97, whereas the exponential PDF gives a correlation higher than 0.99. In general, we find that the agreement is not substantially better for one than the other.) To minimize the subjectivity in the selection of the points to be used in the regression analysis, the correlation coefficient should be optimized.

3. Integrate over the spectral peak of the precipitation regime to get its contribution to the total Z measured by the radar (including an extrapolation to zero drop size); then subtract it from the total Z measured by the radar to get the contribution to Z from the cloud regime.

4. Scale off the maximum value of the spectral density of the Z spectrum for the peak of the cloud regime (i.e., the S_M in Fig. 7) and calculate Z/S_M for the cloud regime. Enter this ratio value in the top frame of Fig. 3 or Fig. 5 (for α = 0 or α = 2) and find the D_M corresponding to the w_σ found in the regression analysis. This is the size scale of the cloud regime.

5. Using the above value of D_M, read the inverse deconvolution factor DF⁻¹ from the lower frame of the figure.

6. Having D_M and Z from both regimes, calculate the corresponding LWand FLX from (6a) and (6c). Similarly, total number density is found by integration of (2) using (1).

7. Make any higher-order corrections (usually negligible) necessary to account for the effect of the cloud-settling velocity on the velocity difference between the cloud and precipitation spectral peaks.

8. Convert the size scales D_M to the more commonly used mass-median drop size D₀ (e.g., see Gossard 1994) for comparison with data from other sources. [For α = 0, 3.67/D₀ = 6/D_M, etc.]

4. Discussion of the data

Each working profile in this paper consists of an average of 100 raw spectra, giving 10 averaged profiles with a sampling interval of about 20 s over the whole event. Figure 7 shows the first raw spectrum from gate 5 (right) and the first averaged spectrum from the same height. The smoothing and broadening of the averaged spectrum is a result of both turbulent smear and nonstationarity during the averaging interval. Figure 7 also defines many quantities for future use. An updraft of 0.55 m s⁻¹ is indicated by the cloud spectral displacement.

The data collection began after the main drizzle event had passed the radar. The time history of the radar reflectivity and of the total liquid water from microwave radiometer recordings is shown for gate 5 in Fig. 8. The radar record of liquid water decreases much more quickly after the end of the main drizzle event than does the radiometric record of liquid water. This is reasonable considering the much wider beamwidth of the radiometer. The radiometer averages over a horizontal area of 1600 m² at a height of 1 km, whereas the radar beam area is 64 m², so the radar’s spatial resolution is much greater. The circled points on the liquid water plot represent the liquid in the radar spectra.

The calculated drop size distributions for α = 0 and α = 2 are shown for gate 5 in Fig. 9 for (a) the cloud range (left-hand solid line), (b) the drizzle range (center solid line), and (c) the noise level [approximately D⁻⁶ from (2); right-hand solid curve]. The lower half of Fig. 9 shows the corresponding liquid and flux density spectra for the case of α = 0. Data reported by Noonkester (1984) measured with Particle Measuring Systems (PMS) probes aboard an aircraft in marine stratus off the coast of southern California are shown in the upper right-hand frame for comparison. Figure 10 shows the stacked drop size distributions for radar gates 2–7, with liquid and flux numerical values listed above each curve. Figures 11–13 show the height profiles of cloud and drizzle liquid, flux, and number density for α = 0 and α = 2. Figure 14 shows profiles of the mass median diameter and the radar reflectivity factor for α = 0. [Note that the median diameter D₀ is related to the modal diameter D_M by D₀ = (3.67/6)D_M (Gossard 1994). The median size is that commonly found in the literature. Recall that themodal diameter referred to here is the diameter at the maximum of the reflectivity spectrum, not the drop size spectrum.] For most gates, the cloud liquid represents a minimum estimate because the modal diameter was equal to, or less than, 0.04 mm [the smallest diameter reliably resolved when (5b) is used], and the smaller the drop size in the cloud regime, the greater is the liquid water.

It is evident that total liquid and flux are dominated by the cloud regime. However, the radar reflectivity is dominated by the precipitation (drizzle) regime at gates 5 and 6 (where the main cloud resides). Therefore, radar reflectivity should be a poor indicator of either liquid density or liquid flux under weak precipitation conditions. This has implications for Marshall–Palmer-type relationships.

No aircraft measurements were available in the Porto Santo area at the time of the radar measurements, so aircraft drop size/number density measurements made by Noonkester (1984) in climatically similar stratus clouds off the coast of southern California have been included in the top right-hand frame of Fig. 9. The dotted curve included on the 600-MSL spectrum in Fig. 10 was recorded by Noonkester at a height of 274 m above cloud base on 29 May 1981 in an air mass he classified as “marine.” (Noonkester also reported measurements in August that he classified as “continental.”) For the 29 May event, he states, “A northwest wind [onshore] was observed at the coast on each measurement day. Accordingly, a measurement region . . . was chosen to avoid areas likely to be downwind of coastal islands and the California coast. Flights were made toward the southwest from San Diego at 1-km elevation to find a region of extended homogeneous stratus clouds.” His measurements were made with two spectrometer probes manufactured by PMS, Inc. An ASSP-100 probe measured 32 radius increments in the range 0.23 μm ≤ r < 14.7 μm, and an OAP-200 probe measured 15 radius increments in the range 14.2 μm ≤ r < 150 μm. Noonkester’s measurements of liquid water near cloud base have also been included as the dotted curve in the top frame of Fig. 11, assuming our cloud base to be at 300 MSL. Noonkester states, “Because the increase in visibilities near the “cloud bases” was gradual along the descent, a definite [cloud base] could not be established. Therefore, [the base] was estimated and was used to establish the run elevations.” The run elevations above cloud base are indicated for the curves in the top right-hand frame of Fig. 9. Noonkester’s spectra agree well with the ASTEX radar spectra but suggest lower drop number densities for the California data and a cloud–drizzle transition drop size (0.04–0.05 mm), roughly half that in the ASTEX clouds (0.08–0.09 mm). Squires’s (1958) measurements of LW in Hawaiian dark stratus are indicated in Fig. 11.

5. Errors

We see from (6a) and (6c) that errors in LW and FLX depend most critically on D_M because it is raised to the third power. It is a noteworthy feature of this technique that D_M depends only on the ratio of Z to S_ZM [see (4)] and is therefore independent of attenuation and of the radar calibration. In fact, the drop size can be inferred from the uncalibrated Dopplerreflectivity spectrum, which is the fundamental quantity most effectively sensed by Doppler radars.

The LW and FLX, as well as total number density, are directly proportional to Z [see (6a) and (6c)], so errors in liquid and flux are directly proportional to errors in radar calibration. Thus, errors of roughly a factor of 2 are to be expected from this source, depending on the care exercised in calibrating the radars. However, we note that profiles of relative liquid and flux through clouds should be accurate. Refinements of the method might include incorporating the attenuation of 8-mm waves into the calculation of true Z when clouds are thick and dense and using radiometers to measure total path liquid to fine-tune the radar calibration. The attenuation to be expected in drizzle is about 0.06 dB km⁻¹ (e.g., Kerr 1951, p. 682), which is negligible for many purposes.

Experience shows that the proper identification of the cloud and drizzle regimes can be an important source of uncertainty. Sometimes identification is easy, as with gate 5 (upper frame of Fig. 15). At other times, it can be seriously subjective, as in the case of gate 6 (lower frame of Fig. 15), in which the image (noted on the curve) from the strong spectral line of the precipitation appears at an inconvenient location. The proper partitioning of the liquid between cloud and drizzle regimes then becomes difficult. Minimizing the artificial image is a radar systems problem. An image of a strong line in the spectrum is found equally spaced on the opposite side of zero. It results from imperfect matching of the in-phase and quadrature receiver channels. In very high quality systems, the images can be 30–40 dB weaker than the spectral line. In our data, the images are roughly 10 dB down.

As the size scale of the population becomes small, the cloud reflectivity spectrum departs less and less from the (Gaussian) PDF. Because the cloud drop size information depends on the departure in spectral shape from the Gaussian, the method becomes inaccurate as the drop size of the population becomes small; the method can then only be used to put bounds on the inferred quantities.

6. Conclusions

1. This technique for retrieval of drop-number density spectra at small drop sizes leads to size distributions that are reasonable, even extended into the “cloud” spectral regime.

2. When a Stokes range of drop size distributions [Eq. (5b)] is coupled with the linear range [Eq. (5a)] of fall velocity versus size, a limitation in size information is found at a diameter of about D_M = 40 μm (i.e., D₀ ≃ 24 μm), regardless of the turbulence intensity. It is a result of incorporating the Stokes domain, and does not exist if the linear relationship is extended to zero.

3. From the liquid and flux spectra (Fig. 9), we see that most of the liquid resides in the cloud regime for all gates (see Fig. 11); even the flux is dominated by the cloud regime. However, the radar reflectivity factor is dominated by the precipitation (drizzle) regime. Therefore, radar reflectivity is fundamentally a poor indicator of liquid water, and even of flux, when the precipitation is light.

4. The 8-mm radar allows an extension of the classical precipitation/drop size retrievals used by wind profilers because of 1) the 8-mm radar’s sensitivity to small drops, 2) its potentially very high velocity spectral resolution, and 3) its relativeinsensitivity to clear-air “contamination” possible in longer-wavelength radars.