Introduction

Numerous studies have confirmed the fundamental role of clouds in the earth's radiation budget and climate system (e.g., Wielicki et al. 1995). Ice clouds have been identified as one of the most uncertain components of this system (e.g., Stephens et al. 1990; Tian and Ramanathan 2002). These clouds are often several kilometers thick (Kosarev and Mazin 1991), but in many cases they are optically “nonblack” with visible optical thicknesses less than about 4. An adequate quantitative description of these clouds in models is very important because they regularly cover up to 40% of the globe (Raschke 1993). Ice clouds influence the radiation field of the earth–atmosphere system in both solar (shortwave) and thermal IR (longwave) bands. The relative significance of the ice cloud role in these two bands (i.e., the so-called greenhouse-vs-albedo effect) leads to differing atmospheric cooling and heating rates, as well as to different impacts on surface and top of the atmosphere energy budgets.

Cloud radiative properties are determined by their macrophysical parameters, such as geometrical thickness, altitude, layering, and horizontal extent, together with their microphysical parameters. The radiatively important microphysical properties of ice clouds include ice water content (IWC), characteristic cloud particle size (e.g., effective radius, median, or mean sizes), and also particle shape (habit). The wide interest in clouds' role in the climate system has stimulated development of satellite- and ground-based remote sensing methods to retrieve these radiatively important microphysical properties. For cloud retrievals, satellite methods usually rely on passive measurements provided by different visible and infrared channels of instruments, such as the Moderate Resolution Imaging Spectroradiometer (MODIS), Advanced Very High Resolution Radiometer (AVHRR), Visible Infrared Scanner (VIRS), and Clouds and the Earth's Radiant Energy System (CERES). The passive measurements are able to provide only vertically integrated or layer mean parameters and generally are not suitable for multilayer cloud scenes. However, for the adequate cloud representation in models, it is necessary to know vertical profiles of the cloud radiative properties such as heating/cooling rates and radiative fluxes.

Most ground-based cloud retrieval methods employ suites of different vertically pointed remote sensors. Cloud radars commonly operating at Ka (at frequencies around 35 GHz) or W (at frequencies around 90 GHz) bands are often a centerpiece of these suites. A combination of a cloud radar and an IR spectrometer/radiometer allows retrievals of layer mean values of IWC and the cloud particle characteristic size (Matrosov et al. 1992; Mace et al. 1998) or vertical profiles of IWC and the characteristic size (Matrosov 1999). Combinations of lidar and radar measurements are also used for retrievals of microphysical profiles (Donovan and van Lammeren 2001; Wang and Sassen 2001), though the attenuation of lidar signals causes limitations of the applicability of lidar–radar approaches.

Recently, new ice cloud microphysical methods based on Doppler radar–only measurements have been suggested (Matrosov and Heymsfield 2000; Matrosov et al. 2002; Mace et al. 2002). These methods use vertical profiles of radar reflectivity and Doppler velocity to retrieve profiles of IWC and the cloud particle characteristic size. As compared with the methods that use combinations of microwave (e.g., radar) and optical (e.g., lidar or/and IR radiometer) instruments, the applicability of these new methods is extended to multilayer and optically thick clouds. Problems associated with a mismatch of fields of view of different instruments are nonexistent for the radar-only approaches. On the other hand, the use of Doppler information requires time averaging, and so the effective time resolution of these methods is relatively low (e.g., 20 min or so). Radars also often do not detect clouds or portions of clouds with populations of very small particles. This limitation is dependent on radar sensitivity and the implications of the radar missing small particles changes, depending on the applications.

Accurate modeling of the ice cloud impact on the earth's radiation budget and climate requires parameterization of cloud optical properties, such as the extinction coefficient, optical thickness, phase function, and single-scattering albedo, in terms of retrieved or assumed cloud microphysical parameters. A number of parameterization schemes have been suggested in the last few years (e.g., Ebert and Curry 1992; Fu 1996; Yang and Liou 1998; Key et al. 2002). Some of the cloud optical properties, however, can be retrieved directly from remote sensing measurements. The retrieved optical properties can then be incorporated directly into the radiation schemes of different models. Long-term remote sensing retrievals of cloud optical properties also permit a straightforward assessment of the frequency distributions and other important statistics of these cloud properties.

An extension of the Doppler radar–only ice cloud microphysical retrieval method, suggested recently by Matrosov et al. (2002), also allows retrievals of the vertical profiles of the cloud extinction coefficient. Radars operate at a wavelength that is vastly different from optical wavelengths and are not usually associated with any kind of optical measurements. Therefore, for the purpose of practical use, it is necessary to understand and quantify the accuracy of the optical product obtained from radar measurements. Although direct measurements of cloud-specific extinction is not readily available, measurements of its integral (i.e., the cloud optical thickness) can be made in a rather straightforward way using optical measurements. This study compares radar- and optically (i.e., radiometrically) derived optical thicknesses of ice clouds observed during the year-long Surface Heat Budget of the Arctic Ocean (SHEBA) experiment (Uttal et al. 2002) and estimates the accuracy of radar retrievals.

Radar retrievals of the cloud extinction coefficient and optical thickness

Using (2) and the particle bulk density assumption, one can obtain approximations for the factor X (Matrosov et al. 2002). The Locatelli and Hobbs (1974) particle mass–size relation leads to the following bulk density assumption:

ρ⁻³D^−1.1(3)

The solid ice density (0.9 g cm⁻³) is used when (3) provides values greater than 0.9 g cm⁻³. The assumption (3) was shown to be generally appropriate for cirrus (Brown and Francis 1995), though some later microphysical studies including Arctic clouds (e.g., Korolev and Strapp 2002) indicate that it might somewhat overestimate particle masses. Assuming the spherical cross sections for smaller particles when (3) provides values of A that exceed those for spheres, the following approximation for the coefficient X can be obtained:

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Substituting (4) in (1) will result in Z_e (mm⁶ m⁻³), α (m⁻¹), and D₀ (μm). For typical values of particle median sizes (D₀ < 300 μm), the approximation (4) is generally within 40% of the one obtained by Matrosov et al. (2002), using an older m–A–D assumption.

Estimates of α from Doppler radar measurements are done in two steps. First, the profile of D₀ is obtained from the time-averaged vertical profile of Doppler velocity measurements, using the approach presented by Matrosov et al. (2002), and then the extinction coefficient profile is calculated from (1) using the profile of the radar reflectivity. A typical averaging time is about 20 min. As a result of such averaging residual vertical air motion is usually significantly less than the reflectivity-weighted particle fall velocities. However, clouds with strong vertical air updrafts or downdrafts may require longer averaging times (Matrosov et al. 2002), and, in some extreme situations, Doppler velocity averaging would not produce satisfactory estimates of cloud particle terminal fall velocities (i.e., D₀ profiles).

Figure 1 shows an example of retrievals for one of the ice clouds observed during the SHEBA field experiment. A geometrically thick ice cloud was observed over a period of more than 18 h on 28–29 April 1998. Twenty-minute averages of the radar reflectivity and Doppler velocity observed in this cloud by the 8-mm-wavelength cloud radar (MMCR) are depicted in Figs. 1a and Fig. 1b. Figures 1c and 1d show the retrievals of the extinction coefficient and particle median volume size, D₀ (in terms of “maximum chord length” equivalent spherical diameters). Microphysical retrievals for this case were previously shown to compare well with aircraft measurements (Matrosov et al. 2002).

Uncertainties in the different retrieval assumptions coupled with the uncertainties of radar measurements and estimates of D₀ from the Doppler velocity data could result in about a factor of 2 (or even more) uncertainty (i.e., +100, −50%) in the extinction coefficient retrievals (Matrosov et al. 2002). These theoretical estimates of retrieval uncertainties might seem rather large but the radar assessment of the cloud extinction is still useful because it can be obtained for multilayer clouds when optical sensors are either blocked by the lower liquid cloud layers or their signals are severely attenuated. Passive satellite measurements cannot properly handle multilevel clouds either. Another important advantage of radar measurements is that they may be able to provide the extinction of the ice component of the nonprecipitating, mixed-phase clouds if contributions of ice particles to Doppler radar moments are much larger than contributions by water drops that are often much smaller than ice particles.

Uncertainties of the radar estimates of cloud optical thickness, τ, which is the vertical integral of α, are likely to be smaller than those for the extinction coefficient, because in the course of integration there is a partial compensation of errors in α that have opposite signs. The quality of optical parameter retrievals using radar measurements, however, can also be assessed by direct comparisons with results of measurements taken by optical instruments. Such comparisons can only be performed for single-layer ice clouds that are unobstructed as “seen” by such instruments. The SHEBA dataset provided a convenient opportunity for such comparisons.

Remote sensing retrievals of cloud extinction profiles from lidar measurements can be done for relatively thin clouds. These retrievals are possible with the use of specialized lidars, such as the Raman lidar or the University of Wisconsin high-spectral-resolution lidar (HSRL), when the aerosol (cloud) extinction profile can be directly obtained (Piironen and Eloranta 1994). During SHEBA such lidars were not available. The backscatter data from the available depolarization and backscatter unattended lidar (DABUL) have not been absolutely calibrated. Optical thickness estimates, however, could be performed in a relatively straightforward way using the radiometric measurements. Note also that the radiometric approach to estimate cloud optical thickness has an advantage over the lidar-based approaches in the sense that it can be applied to optically thicker ice clouds because, for typical particle sizes, the extinction optical thickness is about 2 times the absorption optical thickness.

Estimations of cloud optical thickness from AERI

Measurements of the atmospheric emitted radiance interferometer (AERI) provided a means to estimate cloud absorption optical thickness τ_a at the SHEBA ice station. This instrument provided high-resolution (1 cm⁻¹) spectral measurements of the downwelling radiance in a range from 500 (20 μm) to 3300 (3 μm) cm⁻¹ every 10 min. The AERI data in the IR “window” centered at 900 cm⁻¹ (±12.5 cm⁻¹) were used for calculating τ_a based on the following expression for the brightness temperature of the downwelling radiation T_b (Matrosov et al. 1998):

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where B is the Planck function for 900 cm⁻¹, P_a is the transmittance of the atmosphere between the cloud and the ground, δϵ is the term correcting cloud emissivity for scattering effects, and T_e and T_a are the cloud- and the atmosphere-emitting temperatures. The first term in (5) describes the radiation of the cloud attenuated by the atmospheric layer, the second term shows the radiation contribution from this layer, and R_g accounts for the ground radiation reflected by the cloud. The ground term R_g does not contribute significantly to the total downwelling radiation (Matrosov et al. 1998), and it is usually neglected.

Matrosov et al. (1998) suggested a procedure to find T_e and T_a from vertical temperature profiles (known from a nearby radiosonde sounding) and radar-measured cloud boundaries. Note that T_e differs from the midcloud temperature and it is approximately equal to the thermodynamic temperature at a certain level inside the cloud. The ratio of the cloud absorption optical thickness from the cloud base to this level, τ₀, and the total absorption optical thickness, τ_a, depends on τ_a and can be approximated for τ_a < 6 as τ₀/τ_a ≈ (0.006τ²_a − 0.09τ_a + 0.5) (Matrosov et al. 1998). For the optically thickest ice clouds, where the radiometric assessment of τ_a is still possible, the effective emitting cloud temperature T_e is approximately equal to the thermodynamic temperature at a cloud level where the absorption optical thickness from the cloud base is about 1. The effective atmosphere-emitting temperature T_a is mostly determined by the vertical profile of the water vapor. For typical water vapor profiles, T_a is approximately equal to the thermodynamic temperature at an altitude of about 1 km above the ground.

Atmospheric transmittance P_a is determined mostly by the water vapor. Because water vapor tends to be concentered in the lower atmosphere, it is assumed that most of the total vertically integrated water vapor amount (WVA) is in the layer between the cloud and the ground. A simplified relation suggested by Matrosov et al. (1998) for midlatitude conditions,

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was found also to be valid for the relatively dry conditions that usually are present in the Arctic. This expression was verified by comparing the cloud-free 900 cm⁻¹ AERI measurements with the prediction from (5), where only the second term is retained and WVA values are obtained from the dual-channel microwave radiometer (31.8 and 23.4 GHz) measurements or from radiosonde soundings. The differences between calculated and measured brightness temperatures were, as a rule, within a few kelvins.

The emissivity correction term δϵ is defined as the difference between cloud emissivity when accounting for scattering and the cloud emissivity under the assumption of a “pure absorption” regime. The scattering effects are twofold. On one hand, scattering increases the downward radiation by redirecting photons from other directions. On the other hand, it removes photons from the downward direction through volume scattering. The first effect enhances the downwelling radiation and the second effect reduces it. These two effects tend to approximately balance each other in the thermal IR radiation region. Often, scattering effects are neglected altogether (i.e., Inoue 1985). Modeling of the term δϵ was performed using the discrete ordinate algorithm for radiative transfer calculation (Stamnes et al. 1988).

The calculations were performed using the 48-stream algorithm version for three ice cloud particle models with equal volume spherical diameters of 22, 55, and 190 μm, which approximately correspond to the small (C20), medium (CS), and large (CU) ice particle clouds, according to the nomenclature presented by Minnis et al. (1993). The spherical model is considered here because ice particles sampled in Arctic ice clouds are mostly irregular particles and the fraction of pristine single hexagonal crystal is very small (Korolev et al. 1999).

Figure 2 shows the correction δϵ as a function of the total (extinction) cloud optical thickness, τ. This correction is the largest for the smallest particle model (C20). The particle size dependence of this correction is approximately inversely proportional to the square root of particle size. An overall approximation of δϵ(τ) can be achieved by a cubic polynomial function as shown in Fig. 2. In the Environmental Technology Laboratory (ETL)'s retrieval procedure, the term δϵ is estimated based on the layer mean value of the cloud particle size obtained from Doppler velocity measurements (Matrosov et al. 2002). Note, however, that this correction is rather small, and the pure absorption assumption is generally a good approximation for retrieving cloud optical thickness in the thermal IR range where ice absorption is significant.

The radiometric approach described above provides the absorption optical thickness τ_a. It is assumed that τ_a and the total visible extinction optical thickness τ are related as τ ≈ 2τ_a. This assumption has been validated by experimental comparisons presented by Matrosov et al. (1998) and by theoretical considerations for the large (in comparison with the wavelength) particle regime (i.e., the geometrical optics regime) where extinction efficiency is approaching 2 and the absorption efficiency is approaching 1. Note that some model studies using the hexagonal ice crystals resulted in a factor of 2.1 for the ratio τ/τ_a (Fu and Liou 1993), and some other observational estimates (Platt et al. 1987) showed this ratio in the range of values from 1.8 to 2.3. The variability of this ratio can be explained, in part, by the fact that the smallest cloud particles are not in the geometrical optics regime for the thermal infrared wavelengths. Such particles may actually be missed by the radar because of radar sensitivity limitations. Given inevitable retrievals errors, a value of 2 is assumed for this ratio for the purpose of this work.

Practically, radiometric estimations of cloud optical thickness are possible for unobstructed ice cloud layers with τ_a less than about 3, which corresponds to a cloud emissivity of about 0.95. For larger values of absorption optical thickness clouds become essentially “black,” and the radiation saturation effects prevent retrievals. Note that lidar signals can become practically extinct in clouds already at the level of about τ_a ≈ 1.5, which corresponds to the round-trip attenuation of about exp(−2τ) = exp(−6), assuming τ = 2τ_a. This illustrates the fact that the radiometric approach for estimating cloud optical thicknesses is applicable to a wider range of ice clouds than any method that uses lidar measurements. It should be noted, however, that for larger particles and fields of view, multiple scattering and other factors may lead to an increased penetration of lidar signals. During SHEBA, the radiometric approach was applied to the ice clouds with no intervening liquid layers (as identified by the dual-channel microwave radiometer measurements). The AERI data were available only for the first 8 months (November 1997–June 1998) out of the yearlong SHEBA period (Uttal et al. 2002).

Comparisons of radar- and radiometer-derived optical thicknesses

The radiometer approach to derive optical thickness described in section 3 uses some radar-based information, such as the location of cloud boundaries for cloud temperature estimates, and radar-derived particle size information for estimates of the scattering correction δϵ. This information, however, presents relatively minor adjustments to purely radiometrical estimates, and the optical thickness values derived using the radar and radiometer (i.e., AERI and microwave radiometer) data are essentially independent. Hereinafter, optical thicknesses derived using the method described in section 2 are referred to as τ from radar data and results of the approach from section 3 are referred to as τ from radiometric data.

The radiometric approach (section 3) presents a relatively straightforward and robust way to derive cloud optical thickness. The radar method (section 2) is a novel approach that suggests using a nonoptical instrument to retrieve cloud optical parameters. Although possible retrieval errors of the radar method were estimated (Matrosov et al. 2002), direct comparisons with optical (radiometric) estimates of τ from radiometer data will provide a better insight into the potential usefulness of the radar method. Relatively long-term SHEBA observations also provide possibilities of statistical analysis of radar retrievals of τ.

Figure 3 shows histograms of the normalized frequency distributions of the cloud optical thicknesses derived from collocated radar and radiometric measurements made with vertically pointed instruments during the SHEBA period. Very tenuous ice clouds with τ < 0.1 were excluded from the further analysis because retrieval errors for such clouds are rather high (greater than 100%). The frequency distributions in Figs. 3a and 3b are similar, though there are some minor differences indicating that, for smaller values of τ, the radar results may be negatively biased in comparison with radiometric data. Generally, both approaches show a gradual decrease in the occurrence frequency as the cloud optical thickness increases. The data are presented for clouds when both retrievals were available (i.e., for ice clouds that were not obstructed by liquid).

A scatterplot of cloud optical thicknesses retrieved from radiometric and radar measurements is shown in Fig. 4. The correlation coefficient characterizing data scatter is about 0.62. The mean bias of the radar data from the radiometric estimates is about −14%, and the corresponding relative standard deviation (RSD) is about 77%. The RSD characterizes the total uncertainty of radar retrievals of cloud optical thickness relative to the radiometrically derived values. Figure 5 shows the biases and standard deviations of radar estimates as a function of cloud optical thickness τ. It can be seen that uncertainties of radar retrievals generally decrease with τ. It is likely that the relative contributions of small particles to the total optical thickness are larger for smaller values of τ, which results in less accurate radar estimates of τ because of the radar-poor sensitivity to small particle populations.

The fact that the data scatter in Fig. 4 (and Fig. 6) is significant does not disqualify the radar approach for estimating τ. Indeed, the RSD value of 77% represents an average retrieval uncertainty to be about a factor of 2 (i.e., +100%, −50%). Similar or slightly smaller uncertainties are associated with remote (and also in situ) estimates of such cloud microphysical parameters as ice water content (Matrosov et al. 2002). The radar-based retrievals of cloud optical thickness have an advantage over optical measurements because they are applicable to multilayer cloud scenes, which are common in cloud remote sensing. Once the associated uncertainties are realized, optical thickness estimates provided by radar can be a very useful input for different models.

Measured radar moments are weighted by the product ρ²D⁶ (Matrosov and Heymsfield 2000), where ρ is the bulk density and D is the particle size. For smaller particles with a nearly solid ice density, it results in weighting by the 6th moment of the particle size distribution (PSD). For large particles, this product is approximately proportional to the 3.8th moment of the PSD because the bulk density of such particles is proportional to D^−1.1, according to the Locatelli and Hobbs (1974) relation mentioned above. There is a transition zone where the proportionality changes from the 6th moment to the 3.8th moment. Thus, larger cloud particles dominate radar signals. The smaller particles are accounted for by the assumption of the shape of the PSD. However, the smallest particle populations might not be adequately described by the exponential PSD assumed here. Extinction, on the other hand, is much more sensitive to smaller particles than radar moments. The role of small particles is greater for cloud extinction/optical thickness than for cloud ice water content/ice water path (IWP). An inadequate representation of small particles in radar-based retrievals is one possible explanation of the negative bias of radar retrievals of τ in comparison with radiometric estimates. Another possible explanation is the uncertainty of the coefficients in the density size and m–A–D relations, which provide a connection between microphysical and optical (i.e., extinction) properties. Potentially, it is possible to “tune” these coefficients to eliminate the existing bias. For example, tuning the value of the coefficient (3a) could result in the unbiased radar estimates of τ. However, the existing bias is not very significant given the expected overall retrieval uncertainties, and we will leave a subject of tuning m–A–D relations for further studies when more data become available.

Relations between cloud microphysical and optical properties

Matrosov and Snider (1995) used a relation similar to (9) but for the absorption optical thickness τ_a. Based on the analysis of remote sensing data for many one-layer ice cloud observational cases, they found that a′ varies approximately between 0.008 and 0.012 and b′ varies between about 1.6 and 2.2 if the IWC-weighted vertical average of median mass particle sizes expressed in micrometers is used as D_c, and the units of IWP are in grams per meter squared. The superscripts (′) in a′ and b′ make a distinction that they refer to τ_a and not τ in (9). With the assumption that τ = 2τ_a and using D₀ as D_c, the variability of the parameterization coefficients in (9) is 0.016–0.024 for a and 3.6–4.9 for b. Mean values of these coefficients are 0.02 and 4.2, respectively. It was assumed here that the median mass particle size is on average 10%–20% smaller than the median volume particle size (Matrosov et al. 1998). For multilayer ice clouds, D_c represents an average value through all of the ice cloud layers.

Note that according to (9) for the population of small particles when the second term in (9) is dominant, optical thickness is approximately inversely proportional to characteristic particle size. For population of larger particles, the dependence of τ on particle size becomes progressively smaller and a value of IWP approximately determines cloud optical thickness. This is because the particle size–bulk density dependence results in both IWP and τ becoming approximately proportional to the same moment of the PSD for larger particles.

Because the Doppler radar–based method (Matrosov et al. 2002) allows retrievals of vertical profiles of IWC and D₀, predictions of optical thickness using (9) can be compared with the radiometric measurements of τ. This will represent another comparison of independently estimated optical thicknesses. Figure 6 shows the scatterplot of the radiometrically derived optical thickness and the results obtained from microphysical retrievals using

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It can be seen from comparing Figs. 4 and 6 that the data scatter is similar. The bias of the optical thickness estimates from the cloud microphysics is −17% and the RSD is about 79%, which is slightly greater than from direct extinction estimates from radar data. The overall closeness of the results in Figs. 4 and 6 can be explained by the similarity of the approaches and assumptions used to retrieve cloud extinction as described in section 2, and cloud microphysical properties as described by Matrosov et al. (2002). Both approaches use the same input information, that is, the vertical profiles of the radar reflectivity and the Doppler velocity. This means that cloud optical thickness estimates can be obtained from Doppler radar measurements directly or from previously retrieved microphysical parameters using (10). An advantage of the direct approach, which is presented in section 2, is that it also provides a relatively straightforward way of estimating extinction profiles. Another advantage of the direct radar–only approach is that it handles multilayer clouds better. Note that (8), with coefficients from (9), can also be used for obtaining extinction profiles from known microphysical profiles (i.e., IWC and D₀).

Data collected from several balloonborne replicators in midlatitude cirrus clouds were used to verify the relations between cloud optical thickness and IWP. The replicator (Miloshevich and Heymsfield 1997) provides high-quality PSD measurements with a resolution of about 2 μm from about 10 μm to 2 mm in particle size. In comparison with the traditional two-dimensional cloud (2D-C) probes, replicators provide much better measurements for smaller particles. In situ estimates of the cloud extinction coefficient α are calculated using replicator data on the particle cross-sectional area A assuming an extinction efficiency of 2, and the particle mass (i.e., IWC) is calculated using the habit-dependent mass–size relations. The details of such calculations are given by Heymsfield et al. (2002). The optical thickness and IWP values are then obtained by integrating α and IWC values.

Figure 7 shows τ/IWP ratios from replicator data as a function of the mean layer value of particle median size D₀. The data were obtained from several Lagrangian aircraft spiral descents and replicator-balloon ascents during the First International Satellite Cloud Climatology Project (ISCCP) Research Experiment (FIRE-II) at Coffeyville, Kansas; and several Atmospheric Radiation Program (ARM) intensive observation periods (IOPs) near Lamont, Oklahoma (Heymsfield and Miloshevich 2003). Though the Lagrangian data do not provide optical thickness and IWP estimates in the vertical direction, they still present a valid comparison for the relation (10) because all of the profiles are considered along the ascent line and no direct retrievals from vertically pointed radar data are involved in this comparison.

It can be seen that there is generally good agreement between τ/IWP ratio values from in situ measurements and those predicted by the relation (10). There is a distinct flattening of the τ/IWP ratio dependence on the layer mean particle size for larger values of D₀, though values from (9) are a little lower than in situ data. On the smaller particle end, that the agreement is better and a clear trend for the in situ τ/IWP ratios to increase as D₀ diminishes is obvious. Though qualitative agreement is good, more data are obviously necessary to get more reliable quantitative comparisons.

Relations between radar-derived and optical cloud particle characteristic sizes

A convenient relation between D₀ and D_eff can also be obtained using (1) and the general equation for the radar reflectivity from Matrosov et al. (2002):

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Combining (1), (15), and the definition (13) results in

D_effXρ⁻¹_iG⁻¹D₀(16)

Using the approximations for X and G, which correspond to the assumptions made in this study, one can get D_eff ≈ 18 D^0.3₀ (for D₀ ≳ 75 μm). A correspondence between the two different expressions for the radar reflectivity [i.e., (1) and (15)] is discussed in the appendix.

Conclusions

It has been demonstrated that the measurements taken with a ground-based vertically pointed 8-mm-wavelength cloud radar can be effectively used for retrieving optical properties of ice clouds. The suggested remote sensing method uses measurements of the Doppler velocity and radar reflectivity to derive vertical profiles of the cloud extinction coefficient, α. Vertically integrating the extinction coefficient provides estimates of cloud visible optical thickness τ, which is one of the fundamental parameters describing the radiative impact of clouds. This method is applicable to multilayer ice clouds with no strong vertical air motions. The ability of the suggested method to handle multilayer cloud situations gives it an important advantage over other existing ground-based and satellite approaches. Potentially, this approach can provide the extinction/optical thickness of the ice component in mixed-phase clouds if ice particles dominate radar moments. However, the applicability of this approach to mixed-phase clouds needs to be investigated in more detail.

The radar approach allows direct estimates of cloud extinction and optical thickness from measurements, and its results can be used as model input, bypassing some intermediate model parameterizations that relate optical and microphysical cloud properties. Simple theoretical considerations suggest that the uncertainty of the extinction coefficient retrievals could exceed a factor of 2, though the accuracy of the optical thickness estimates is generally better because of some partial error cancellations as a result of the vertical integration.

The accuracy of the radar-based optical thickness retrievals was also estimated by comparing collocated radar- and radiometer-derived values of τ. Absorption optical thickness τ_a from radiometer measurements was derived using downwelling brightness temperatures in the thermal infrared band. Estimates of the water vapor amount from the dual-channel microwave radiometer were used to account for the intervening atmospheric layer. Scattering and nonisothermic effects were also accounted for when deriving τ_a from radiometer measurements. Visible optical thicknesses from radiometer data were estimated assuming τ = 2τ_a. These radiometer-based optical thicknesses provided the robust dataset for evaluating radar-based retrievals.

Comparisons of radar-based and radiometer-based optical thickness retrievals were performed using the yearlong dataset obtained the SHEBA experiment. The radar and radiometers were essentially collocated and vertically pointed. The relative standard deviation (RSD) of radar retrievals of τ with respect to the radiometric ones was 77%, with a negative bias of −14%. One possible explanation for this negative bias is an inherent low sensitivity of radar measurements to very small particles that still may contribute to the cloud optical thickness. Uncertainties in the coefficients of the adopted m–A–D and the particle size–density relations can also contribute to this bias because the extinction retrieval results depend on these relations. Judging from the overall relatively small bias between radar- and radiometer-derived optical thicknesses, the adopted relations seem to be appropriate in describing Arctic ice cloud properties. More studies are needed to establish how well this approach for extinction retrievals and the particular m–A–D and the particle size–density relations assumed here will work under different conditions.

Taking the RSD value as a measure of uncertainty for the radar-based retrievals of cloud optical thickness results in about factor of 2 (i.e., +100%, −50%) accuracy of the radar retrievals of τ. Given a rather wide range of applicability of this radar-based method, this should be considered as a positive result for a nonoptical instrument, such as radar. Similar accuracy of the optical thickness estimates can also be achieved by first deriving the IWP and layer mean cloud particle characteristic size D_c from radar-only measurements and then relating them to the cloud optical thickness using a τ–IWP–D_c relation.

An average τ–IWP–D_c relation was derived earlier based on the multisensor ground-based retrievals results for one-layer ice clouds. It was shown that this relation is in good agreement with the radar-only cloud retrievals. Comparisons with in situ cloud measurements taken during the cloud particle replicator ascents and aircraft Lagrangian descents also indicated the validity of this relation.

It was shown that Doppler radar–derived effective particle sizes are significantly greater than effective sizes used in some models because of definition differences. The correspondence between different definitions for ice cloud particles with changing bulk density was offered.