1. Introduction

Ground-based millimeter-wavelength radars operating in microwave transparency “windows” at K_a (near 35 GHz) and W (near 94 GHz) bands were first significantly employed for atmospheric research in the mid-1980s (e.g., Pasqualucci et al. 1983; Lhermitte 1987). These radars proved to be a very valuable tool for quantitative studies of atmospheric hydrometerors from the ground and aircraft (e.g., Mead et al. 1994; Kropfli et al. 1995). The first space missions using satellite-based millimeter-wavelength radars will be launched in the near future (Stephens et al. 2002; Iguchi et al. 2002).

The initial success of research using millimeter-wavelength radars in the United States and Europe led to the development of cloud radars that have been deployed at different locations on a permanent basis for routine unattended atmospheric observations. A good and well-known example of this is the development of the K_a-band millimeter-wavelength cloud radar (MMCR) for the U.S. Department of Energy's Atmospheric Radiation Measurement (ARM) Program (Moran et al. 1998). The vertically pointing ARM MMCR radars are deployed at several Cloud and Radiation Testbed (CART) sites including the tropical western Pacific, Oklahoma, Alaska, and Australia. They have proven to be invaluable remote sensing tools at the CART sites. The ARM MMCR radars were designed and constructed by the National Oceanic and Atmospheric Administration's Environmental Technology Laboratory (NOAA/ETL). One such MMCR radar is the core part of NOAA's Portable Cloud Observatory (NPCO), which is deployed on a temporary basis at various geographical locations during different field programs.

The MMCR and different millimeter-wavelength research radars were initially designed to provide detailed, long-term observations of nonprecipitating and weakly precipitating (e.g., drizzling) clouds. A number of cloud parameter retrieval algorithms have been developed to estimate cloud microphysical properties for both ice and liquid phase clouds. These algorithms use MMCR (or similar radar) data either alone (e.g., Matrosov et al. 2002a; Mace et al. 2002) or in combination with other instrument data (e.g., Matrosov et al. 1992; Mace et al. 1998; Matrosov 1999; Frisch et al. 1995; Wang and Sassen 2001; Donovan and van Lammeren 2001; Dong and Mace 2003) to retrieve either layer-averaged cloud properties or their vertical profiles. Due to strong attenuation of millimeter-wavelength radar signals and non–Rayleigh scattering effects by raindrops, vertically pointing ground-based cloud radar measurements in rain are not usually used for quantitative rainfall retrievals, though several rain retrieval algorithms for spaceborne radars (e.g., L'Ecuyer and Stephens 2002) and scanning attenuating radars at longer wavelengths (e.g., Testud et al. 2000) have been proposed. These algorithms usually require an estimate of path-integrated attenuation.

MMCR-type cloud radars collect routine vertically resolved information on the atmospheric column, and quantitative precipitation data could be a valuable addition to this information. Vertical rainfall data can be obtained from vertically pointing radars operating at S band (around 10 GHz) or from wind profilers (e.g., Williams 2002; White et al. 2000); however, deployment of an additional radar is not always feasible for a number of practical reasons. Extension of the use of cloud radars to retrieve rainfall parameters would be useful for many scientific applications, including cloud microphysical and atmospheric radiation studies, because phase transitions significantly influence the vertical profiles of heating rates. The vertically resolved information on rain will also be valuable for model validation efforts.

Some close relations exist between the rain attenuation at K_a band and rainfall rate (e.g., Aydin and Daisley 2002; Rincon et al. 2000). In this study such relations are analyzed for different types of rain, and an attenuation-based approach is suggested for retrievals of layer-average rainfall rate above the vertically pointing K_a-band radars. The applicability of this approach is illustrated with the data in rainfall events observed by the ETL NPCO K_a-band radar during the Cirrus Regional Study of Tropical Anvils and Cirrus Layers Florida Area Cirrus Experiment (CRYSTAL–FACE) in July 2002.

2. Attenuation and backscatter of K_a-band radiation in rain for vertical incidence

The specific attenuation of the electromagnetic radiation in rain, a, and the backscatter coefficient, η, are obtained by integration over the drop equal-volume spherical diameters, D, with respect to the drop size distribution (DSD) function N (Bohren and Huffman 1983):

View Expanded

where λ is the radiation wavelength, the maximum drop size in rain D_max is about 7 mm, and S₀(D) and S₁₈₀(D) are the dimensionless complex scattering amplitudes in the forward and backward directions, respectively. Although the amplitudes S₀(D) and S₁₈₀(D) generally depend on the polarization state of the incident radiation, the polarization dependence practically vanishes for vertically pointing rain measurements, and it is ignored in further considerations.

At K_a band, the size parameters of raindrops, x = πD/λ, are comparable with the wavelength, and the simple Rayleigh theory is not applicable for calculations of S₀(D) and S₁₈₀(D). The Mie theory is usually used for such calculations under the assumption of sphericity of raindrops. However, raindrops with D greater than about 0.5 mm are generally nonspherical, and calculations of attenuation and backscatter generally require expansions of traditional Mie theory series to compute scattering amplitudes. In this study, the T-matrix approach (Barber and Yeh 1975) was used for computations.

For the normal atmospheric conditions, Figs. 1a and 1b show the results of calculations of the specific attenuation coefficient at the MMCR frequency as a function of the rainfall rate, R, based on DSDs measured by the Joss–Waldvogel disdrometers (JWDs) at the Kendall–Tamiami Airport during the CRYSTAL–FACE experiment in July 2002 and during the Wallops Island experiment in February–April 2001 (Matrosov et al. 2002b). The Wallops and the CRYSTAL–FACE disdrometer datasets each consisted of about 3300 1-min-average DSDs collected over a period of several months. Rainfall rates were calculated using the relation between drop sizes and their fall velocities for normal conditions at sea level (Matrosov et al. 2002b). The drop shape model used here was that from Brandes et al. (2004), who presented a drop aspect ratio–size relation compiled from several recent studies of raindrop shapes. According to this relation, raindrops are on average less oblate than predicted by the equilibrium drop shape model of Pruppacher and Pitter (1971).

There is obvious general flattening of Z_e–R relations for R > 5 mm h⁻¹, which is mostly due to non-Rayleigh effects. As will be shown later, this effect is advantageous for the attenuation-based estimation of rainfall. Overall, the relations shown in Figs. 2a and 2b (e.g., Z_e = 407R^1.29 for CRYSTAL–FACE and Z_e = 324R^1.44 for Wallops at 15°C) are in general agreement with a relation at K_a band used by Meneghini et al. (1989): (Z_e = 355R^1.26).

For a purpose of illustration, Figs. 2c and 2d show the a–Z_e scatterplots calculated for the two considered DSD sets. The data scatter in the a–Z_e relations is much larger than that for a–R relations and is quite similar to the one for Z_e–R relations. The stability of the a–Z_e relations, and especially the constancy of the exponent in the best-fit a = f(Z_e) power law approximation is essential when applying the Hitschfeld and Bordan (1954) approach, which is sometimes used for retrieving the rainfall rate at attenuating radar wavelengths (e.g., Testud et al. 2000). It can be seen from the best-fit approximations in Figs. 2c and 2d that this exponent at K_a band changes quite significantly: from around 0.81 for the CRYSTAL–FACE data to about 0.73 for the Wallops data. Another reason that hampers the use of the Hitschfeld and Bordan approach for vertical rain profiling with ground-based K_a-band radars is the requirement for integrating attenuated radar reflectivities for the whole rain layer. As will be shown later, K_a-band cloud radar measurements in rain are often saturated near the ground and no attenuated reflectivity data are available when radar receivers are in the saturation regime. Here, the suggested approach to retrieve rainfall rate directly from the K_a-band attenuation estimates makes use of the linear stability of the a–R relations.

a. Temperature and drop shape dependence of a–R and Z_e–R relations

To evaluate the drop shape dependence of a–R and Z_e–R relations, calculations were also performed for equilibrium drop shape. These calculations revealed very small changes compared to the results shown in Figs. 1 and 2, thus indicating a weak dependence of radar parameters on drop shape at vertical incidence. The changes due to the drop shape model were typically within a few percent, and the a–R and Z_e–R relations for equilibrium shape are not presented here.

Figures 1a and 1b show the a–R relations calculated for two temperatures: 0° and 15°C. It can be seen that corresponding changes are very small and within the previously stated uncertainty due to DSD details. This negligible temperature dependence in attenuation at K_a band is in stark contrast to noticeable attenuation variability due to temperature changes at longer radar wavelengths. The temperature variations between 15° and 0°C at X band, for example, can result in about 10% changes in specific attenuation (Matrosov et al. 2002b). The difference in the attenuation temperature dependence at different radar frequencies can be explained by analyzing the extinction (i.e., attenuation) efficiency expansion with respect to size parameter, x (Bohren and Huffman 1983):

View Expanded

The first term in (5) is responsible for the Rayleigh-type effects and is dominant at larger λ. It decreases noticeably as temperature increases. The second and the third terms in (5) increase with temperature, and the contributions from these terms also increase as non-Rayleigh effects become more pronounced at higher radar frequencies. At K_a band, as temperature increases, a decrease in the first term is approximately offset by an increase in the second and third terms, resulting in a very weak temperature dependence. Though (5) is strictly applicable only for spherical particles, this explanation is still valid for slightly nonspherical drops because the effects of nonsphericity at vertical incidence are rather small.

The temperature variability of Z_e–R relations is generally more significant than that of a–R relations. A temperature increase from 0° to 15°C generally results in an increase in backscatter coefficients. For comparison, best-fit power law Z_e–R relations are shown in Figs. 2a and 2b for 0° and 15°C.

The data presented show that the attenuation–rainfall rate relations at K_a band are quite remarkable in the sense that they are nearly linear and do not exhibit significant variabilities due to details of DSD or temperature changes. These properties make these relations a convenient tool for attenuation-based approaches for estimating rainfall rates.

3. Retrievals of rainfall rate based on K_a-band attenuation estimates

Equation (8) assumes that the two-way attenuation effect of the rain layer is dominant compared to the changes in the actual reflectivity within the layer. This attenuation effect linearly increases with the rainfall rate and with the layer thickness. It can be seen from Figs. 1a and 1b that for R > 15 mm h⁻¹ and Δh = 0.5 km, the two-way attenuation is already about 4.2 dB. Since attenuation increases with rainfall rate, it is expected that estimates from (8) will be more accurate for higher rainfall rates. This is aided by the fact that, due to non-Rayleigh backscatter at K_a band, actual (nonattenuated) reflectivities on average increase with rainfall rate at a lower rate (compared to longer radar wavelengths) for higher values of R. Figure 3 illustrates this fact by showing details of the correspondences between Z_e and R at the higher rainfall rates.

It can be seen from Fig. 3 that data from both Wallops and CRYSTAL–FACE DSDs occupy the same scatter area. For R > 15 mm h⁻¹, the highest possible dynamic range of nonattenuated reflectivities is less than about 6–7 dB. For R > 25 mm h⁻¹, this dynamic range is typically less than 4 dB. Given this fact, for higher rainfall rates, the dominance of the decrease in reflectivity due to attenuation over the variability of nonattenuated reflectivity can easily be achieved for already modest layer thicknesses Δh. Furthermore, since a variability in a vertical profile of nonattenuated reflectivity is not expected typically to be higher than 2 dB km⁻¹ (Smyth and Illingworth 1997), the attenuation effects will dominate the possible variability in nonattenuated Z_e in the observed values of ΔZ_e at intervals Δh = 0.5 km beginning from the rainfall rates of about 7 mm h⁻¹.

The CRYSTAL–FACE experiment provided observational events that were used to illustrate both versions of the approach (i.e., with and without the reference reflectivity) for attenuation-based retrievals of rainfall rates from vertically pointing K_a-band cloud radar measurements. The ETL MMCR radar was deployed along with other ground-based instruments at the eastern ground validation site at the Kendall–Tamiami Airport. The NOAA Aeronomy Laboratory's Joss–Waldvogel disdrometer was practically collocated with the radar, and its data were used for comparisons with radar retrievals.

4. Assessment of attenuation-based rain retrieval errors

b. General assessment of retrieval errors

Using (8), it can be shown that the terms in (10) can be given as

View Expanded

where δZ_el is the difference between nonattenuated reflectivities in the beginning and the end of the interval Δh for which the estimate R_a is derived.

Figure 10 shows the relative retrieval errors for different values of δZ_el and Δh as a function of R_a. As before, δc/c was assumed to be 0.1. It can be seen that for given values of δZ_el and Δh the retrieval error for smaller values of R_a is determined mostly by the uncertainties in the nonattenuated reflectivity difference, and it diminishes as R_a increases as the attenuation effects are becoming progressively more dominate over this difference. As R_a increases, the role of the uncertainty of the coefficient c in the linear a–R relation is becoming increasingly more important and is becoming a dominant factor in the retrieval error for larger Δh and R_a values.

For Δh = 1 km (Fig. 10b), the relative retrieval error of R_a is better than 30% for R_a > 6 mm h⁻¹ if δZ_el = 1 dB and for R_a > 12 mm h⁻¹ if δZ_el = 2 dB. For Δh = 0.5 km (Fig. 10a) and smaller rainfall rates, the retrieval errors are roughly twice as high as for Δh = 1 km for the same values of δZ_el, though one would expect less variability of nonattenuated reflectivities at smaller distances Δh. For higher rainfall rates, the retrieval errors could be pretty reasonable, even for the 0.5-km resolution, especially in stratiform rains when δZ_el is expected to be small.

At the 1-km resolution (Fig. 10b), the retrieval errors are less than about 50% for all rain rates greater than approximately 17 mm h⁻¹, even for the uncertainty in the nonattenuated reflectivity difference δZ_el = 5 dB. It should be mentioned that for such rainfall rates, 5–6 dB is as much as this difference can possibly be due to non-Rayleigh scattering (see Fig. 3). In other words, for R_a > 17 mm h⁻¹ and Δh = 1 km, the maximum possible retrieval error is close to 50%. In reality the actual error will be smaller since the worst case scenario was considered in this example. Increasing Δh will further reduce the expected error (at the expense of resolution, though) since the main error contribution is proportional to 1/Δh, as can be seen from (12).

Figure 10c corresponds to Δh = 4.5 km and can be used to estimate retrieval errors for all of the rain layer-mean values, as described in section 3b. In this case δZ_el is the uncertainty in the reference reflectivity. It can be seen that even a rather high value of δZ_el = 5 dB results in rather reasonable retrieval errors for R_a > 6 mm h⁻¹. For larger rainfall rates R_a > 20 mm h⁻¹, the main contribution to the retrieval error comes from the uncertainty in the coefficient c in (3) and (11).

5. Conclusions

It has been demonstrated that vertically pointing K_a-band cloud radars can be successfully used for retrievals of low-vertical-resolution rainfall rate profiles and average rainfall rates in the rain layers. These retrievals are based on attenuation effects and utilize the general linearity of the rainfall rate (R)–specific attenuation coefficient (a) relations at K_a-band radar frequencies. The retrieval approaches are aided by the negligible temperature dependence of these relations and their low susceptibility to the details of the raindrop size distributions. The combined variability of a–R relations due to DSD details and temperature does not generally exceed about 10% for R > 5 mm h⁻¹ for both stratiform (i.e., below brightband) and convective (i.e., nonbrightband) rains.

In cases when the higher cloud reference reflectivity is known with an acceptable degree of certainty just prior and after the rain event and can be measured for the periods when the rain is present, an average rainfall rate for the entire rain layer, R_a, can be estimated in a rather straightforward way. A retrieval error of such estimates is mostly determined by the ratio of the uncertainty of the nonattenuated reference reflectivity during the rain event and the magnitude of the higher cloud reflectivity dip due to rain attenuation. Using ground-based K_a-band data from CRYSTAL–FACE, it was demonstrated that this approach can be successfully applied for convective shafts of warm rain quickly passing over the radar deployment site. An estimated retrieval error can be as low as about 15%–20% for moderate rainfall rates of around 10–15 mm h⁻¹. The reference approach is best when applied to short duration showers because the reference cloud is more likely to maintain a reasonably steady reference reflectivity.

Using the gradient approach, rainfall rates R_a greater than about 5–10 mm h⁻¹ can also be successfully retrieved with a spatial layer resolution of about 0.5–1 km in rain areas where radar echoes are neither in saturation nor in complete extinction. Vertical gradients of radar reflectivity due to attenuation in such layers should be greater than typical changes due to a natural variability of nonattenuated reflectivities. This requirement is alleviated by non-Rayleigh scattering effects, which cause the nonattenuated reflectivities to vary less with rainfall rate at K_a band compared to longer radar wavelengths that are typically used for radar rainfall studies. The errors of rainfall rate retrievals based on gradients are determined by the uncertainty in the linear a–R relation coefficient and by the uncertainty in the differences (δZ_el) between nonattenuated reflectivities, in the beginning and the end of the layer resolution interval Δh used for retrievals. The latter error source is usually dominant for Δh values that are less than about 1 km and is proportional to 1/Δh. The relative retrieval errors decrease as R_a increases. For δZ_el = 2 dB and Δh = 1 km the relative retrieval errors are about 35% for R_a ≈ 10 mm h⁻¹ and 20% for R_a ≈ 20 mm h⁻¹.

Retrievals of rainfall rates using the suggested gradient approach were demonstrated using the CRYSTAL–FACE data obtained in a heavy downpour during which radar signals were completely attenuated above about 3 km AGL. In the region where the radar receiver was not saturated, retrieval estimates of rainfall rates during this event were larger than 10 mm h⁻¹ with the maximum 500-m layer-averaged values exceeding 45 mm h⁻¹ at about 2.5 km AGL. Saturation of the radar signals at lower ranges AGL prevented retrievals near the ground. The radar-retrieved values were in good agreement with ground-based estimates from DSD recorded by the Joss–Waldvogel disdrometer a few minutes later when the rain observed by the radar aloft reached the ground. The retrieval uncertainties of radar estimates for these layer-mean values were estimated as 17%–28% for R_a ≥ 25 mm h⁻¹.

Though not a substitute for the methods used with vertically pointing longer-wavelength radars, the demonstrated approaches for rainfall retrievals can be used with vertically pointing K_a-band cloud radars that, like DOE ARM MMCR radars, routinely collect information in the atmospheric column at different test-bed sites. This would allow a more complete characterization of the atmospheric column above these sites by including rainfall periods. This is especially important for validating phase-transition processes in atmospheric column models. Though the proposed method suggests retrievals of rainfall rates aloft, it can be potentially generalized for estimating vertical structure of rain liquid water contents using relations between these contents and R. This generalization, however, is outside the scope of the current study.

Attenuation-based retrievals are limited to the altitudes where radar echoes are neither in saturation nor in complete extinction. The usable altitude range depends on the dynamic range, receiver gain, radar sensitivity, and some other radar parameters as well as on the rain intensity. Parameters of the precipitation measurement mode of cloud radars can be adjusted to better accommodate needs for attenuation-based rain rate retrievals and tuned to rain climatologies characteristic to particular sites.

One important advantage of the attenuation-based retrievals is that they do not require an absolute calibration of the radar. Inevitable uncertainties in the radar absolute calibration could reach a few decibels and can contribute significantly to errors in the retrievals that use measurements of the absolute radar reflectivity. The proposed method used reflectivity differences (in the logarithmic scale); thus, it avoids the calibration problem altogether as well as issues related to wet antennas/ radomes. The described retrieval approaches can also be useful for airborne measurements (Walsh et al. 2002) and prospective spaceborne missions such as the Global Precipitation Measurement (GPM), which will have a K_a-band radar aboard the core satellite.