## 1. Introduction

Supercooled liquid water (SLW; i.e., liquid water at temperatures below 0°C) is a frequent constituent of mixed-phase (e.g., Boudala et al. 2004) and snow-bearing clouds (Battaglia and Delanoë 2013). SLW inside clouds can be found down to −40°C (Heymsfield et al. 1991), and is frequently below −30°C (Turner 2005; Shupe 2011). SLW plays a fundamental role in cold cloud microphysical properties [e.g., Arctic mixed-phase clouds (Morrison et al. 2012)]; it also strongly influences the cloud radiative properties (Turner et al. 2007) that can cause dramatic changes of the surface energy budget (Bennartz et al. 2013).

Quantitative estimates of path integrated SLW (SLWP) are typically obtained from spaceborne or ground-based microwave radiometers (MWRs). Typically, the frequency range between 20 and 35 GHz is used for SLWP retrievals. Unlike methods in the infrared, MWRs operating at this low-frequency region do not suffer from saturation effects or hydrometeor scattering, and thus it is possible to derive SLWP even from convective or snow-bearing clouds. However, in some regions like the Arctic, the sensitivity to very small amounts of SLWP is of particular interest since a cloud with an SLWP amount from 10 to 50 g m^{−2} dramatically changes its short- and longwave radiative properties (Turner et al. 2007). One possible way to significantly improve the sensitivity and accuracy of ground-based SLWP retrievals is to augment the combination of the standard low-frequency channels with higher-frequency channels up to 200 GHz (Crewell and Löhnert 2003; Cadeddu and Turner 2011) as long as the high-frequency channels are not saturated by water vapor or disturbed by snowfall scattering (Kneifel et al. 2010).

A fundamental prerequisite for any liquid water retrieval in the MW is an accurate absorption model, which is essentially a model of the complex permittivity of liquid water. Standard permittivity models approximate existing laboratory observations using interpolation functions that are based on a simplified physical model of the absorption process in the MW. The longstanding but still unsolved major issue for the supercooled temperature region is that there is only one laboratory dataset in which the liquid water absorption was measured down to −18°C, and this dataset only included observations at a single frequency of 9.61 GHz (Bertolini et al. 1982). For higher frequencies (i.e., between 10 and 1000 GHz), there are no laboratory observations of liquid water absorption for temperatures below −6°C (Ellison 2007). As a consequence, all currently available permittivity models extrapolate into the supercooled temperature region. Without additional data points in the low-temperature region, the uncertainties in the model extrapolations naturally increase with decreasing temperature.

The uncertainties in modeling the absorption properties of SLW directly transfer to uncertainties in the SLWP retrievals. Lipton et al. (1999) simulated synthetic MWR observations for artificial single-layer liquid clouds for both a spaceborne and a ground-based sensor. They compared the permittivity model from Ray (1972) and two different versions of the model described in Liebe et al. (1991). Standard retrieval coefficients for a ground-based two-channel MWR (20.6 and 31.6 GHz) and the spaceborne Special Sensor Microwave Imager (SSM/I; 19.35–85 GHz) were then applied to the simulated brightness temperatures *T*_{B} that were computed with the different permittivity models. The resulting maximum SLWP difference for a 1-km-thick cloud with a cloud-top temperature of −20°C was 42% for the SSM/I sensor retrieval when applying the simulated *T*_{B} from the three permittivity models; for the ground-based MWR retrieval and a cloud-top temperature of −30°C, the influence on the retrieved SLWP differed by 64%. In a more recent study by Cadeddu and Turner (2011), ground-based MWR observations between 23 and 170 GHz were compared with radiative transfer simulations that used four different permittivity models (Liebe et al. 1991, 1993; Ellison 2006; Stogryn et al. 1995) and the SLWP estimate from a collocated infrared interferometer using the method outlined by Turner (2005). For clouds warmer than −15°C the SLWP bias between the infrared interferometer and the MW retrieval using different frequency combinations up to 150 GHz was found to be negligible. For clouds colder than −15°C, the MW retrieval errors using frequencies up to 90 GHz did not exceed ~25% for their considered models [except Liebe et al. (1991), which leads to larger deviations]. However, the SLWP retrieval error introduced by the models if including frequencies up to 170 GHz was found to be as large as 50% for the Stogryn model and as high as 80% for Liebe et al. (1991). The findings from Lipton et al. (1999) and Cadeddu and Turner (2011) might seem to be in some aspects contradictory; however, the selection of permittivity models [e.g., the older and stronger deviating model from Ray (1972) was not included in the study by Cadeddu and Turner (2011)] and the approach for estimating the potential SLWP errors were different. Despite the difference in model selection and methodology, both studies reveal that retrieval errors strongly increase for cloud temperatures lower than −15°C, particularly if frequencies higher than 35 GHz are included.

In absence of reliable laboratory data, different approaches have been developed to utilize MWR observations from supercooled clouds to evaluate the various permittivity models and their extrapolations to supercooled temperatures. If the properties of the dry atmosphere, water vapor profile, and the amount of SLWP can be well characterized, the absorption coefficient can be directly derived from MWR observations. Cadeddu and Turner (2011) found with such an approach that all four considered permittivity models significantly overestimated the SLW absorption coefficient between 90 and 170 GHz. This overestimation leads directly to an increasing underestimation of SLWP with decreasing temperatures by any MWR retrieval. The analysis was, however, limited to clouds with SLWP lower than 60 g m^{−2} because of saturation effects in the infrared spectrometer at higher SLWP values.

If no independent SLWP estimates are available, the MWR can still be used for validation even though the absolute values of the absorption coefficient cannot directly be derived. Mätzler et al. (2010) presented an approach to utilize the differences in the temporal variability of the dry atmosphere, water vapor, and liquid water to separate the part of the signal that is solely due to SLW. Taking the ratio at two different frequencies of the typically much faster opacity changes due to SLW, the measured opacity ratio is equal to the ratio of the SLW mass absorption coefficient. Mätzler et al. (2010) applied this method to supercooled clouds down to −27°C and the relatively narrow frequency range between 21 and 31 GHz. Their measured opacity ratios were found to fit best by the Stogryn model; however, the influence of correlated water vapor fluctuations at these specific frequencies is relatively large because of the inclusion of frequencies close to the water vapor absorption line at 22.24 GHz.

In this study, we apply the basic method developed by Mätzler et al. (2010) to observations collected over a much wider frequency range between 31 and 225 GHz and temperatures down to −33°C, and compare the observational results with six different permittivity models. Furthermore, we combine observations from different MWRs located at three observational sites: an Arctic location at 3250 m altitude at Summit Station in central Greenland, a high-altitude site (2650 m) in the German Alps, and a wintertime dataset from a low-altitude mountain site (511 m) in southwest Germany. In section 2, we briefly describe the six permittivity models used in this study, compare their predicted mass absorption coefficients, and estimate the related SLWP error introduced by the deviating absorption models. Relevant information about the different sites, instrumentation, and MWR calibration is given in section 3. Furthermore, we describe in this section the methods used to derive the opacity ratios. The observed opacity ratios are compared to the model predictions at 10 frequency combinations in section 4. In addition to the opacity ratio comparison, we discuss a method to estimate the mass absorption coefficient for 31.4, 52.28, 150, and 225 GHz using predictions of the mass absorption coefficient at 90 GHz from a reference model. The conclusions of the study and an outlook for future studies are given in section 5.

## 2. Permittivity models for liquid water

*d*smaller than 50

*μ*m), the size parameter

*x*=

*πd*/

*λ*(where

*λ*is the wavelength) is smaller than 0.1, and thus we can use the Rayleigh approximation to calculate the mass absorption coefficient

*α*

_{L}:

*α*

_{L}is dependent on the frequency

*ν*of the external electromagnetic field, the liquid water density

*ρ*

_{L}, and the complex permittivity of the fluid

*ϵ*=

*ϵ*′ +

*i*

*ϵ*″ with

*ϵ*′ and

*ϵ*″ being the real and imaginary part of

*ϵ*, respectively;

*c*

_{l}is the vacuum speed of light, and

*α*

_{L}apply different variants of the Debye function (Debye 1929) to interpolate the experimentally measured permittivity data. The complex Debye function is related to the underlying physical process of polarization due to an external electric field and describes the viscously damped thermal disordering of the molecular water dipoles after removal of the external field. This so-called relaxation process can be described with a characteristic relaxation time

*τ*for a given fluid and temperature. The absorption of energy from an external electromagnetic field is thus a maximum if the frequency of the external field approaches 1/(2

*πτ*). The basic formulation of the Debye function (single Debye) is as follows:

*ϵ*

_{s}is the permittivity for a static electric field whereas

*ϵ*

_{∞}represents the permittivity at “infinite” frequency. The parameters

*τ*,

*ϵ*

_{s}, and

*ϵ*

_{∞}are usually fitted to the experimental data as functions of temperature

*T*. The single Debye function (2) can be extended to include additional relaxation processes that can be caused by additional dipole structures in a solution. Theoretical simulations and experimental data indicate that pure liquid water comprises different polar structures implying more than one relaxation process. A detailed discussion of this topic and an analysis of the fit quality for different numbers of relaxation processes can be found in Ellison (2007). While the physical interpretation of additional relaxation processes is still controversial, most of the models used within this study use a double Debye function with two relaxation times

*τ*

_{1}and

*τ*

_{2}:

The double Debye function contains the additional relaxation time *τ*_{2} and a second fit parameter *ϵ*_{1}.

### a. Description of the permittivity models

In this study, we selected a broad range of published permittivity models for the comparison with our observations. The characteristics of the various models are summarized in Table 1. The valid temperature range of some models is limited to nonsupercooled conditions or temperatures higher than −20°C. However, in the supercooled temperature regime, all of them rely on a similarly sparse laboratory dataset. Based on the lack of laboratory data, every model can be seen as less or more trustworthy in the supercooled temperature regime. For this reason, we decided to compare all models in the same temperature range and accept the fact that some of them lie outside their specified temperature range. The model presented by Ray (1972) (RAY) is the oldest model in our comparison but was also the most widely used model before 1991. It provides interpolation formulas for a wide frequency range from 0 to 300 THz subdivided into seven spectral ranges. For the spectral range between 0.1- and 10-cm wavelengths (3–300 GHz), Ray used a single Debye function with a Cole–Cole modification (Cole and Cole 1941) to fit the relatively sparse experimental dataset.

Permittivity models for liquid water used within this study.

New experimental data motivated Liebe et al. (1991) to compile an updated permittivity model; this permittivity model is currently widely used in the atmospheric community. Liebe et al. introduced a double Debye function since new experimental data at frequencies larger than 100 GHz revealed the inability of the single Debye function to fit the higher-frequency data. In a later paper, Liebe et al. (1993) corrected the original function allowing the parameter *ϵ*_{∞} to be independent of temperature to avoid unrealistic results at frequencies above 100 GHz and temperatures between −20° and −40°C. The Liebe model (LIE) used in this study is the corrected version presented in Liebe et al. (1993); we further use their quadratic temperature-dependent fit for *τ*_{1} instead of the exponential fit.

The more recent models by Stogryn et al. (1995) (STO), Meissner and Wentz (2004) (MEI), and Ellison (2006) (ELL06) are all based on double Debye functions. The differences between the models arise from different datasets used to fit the Debye parameters and different definitions of the parameter functions. For example, in MEI and STO the parameter function for the first relaxation time *τ*_{1} contains a singularity at −45°C where *τ*_{1} becomes infinite. This behavior was motivated by the assumption that all supercooled water droplets will be transformed to solid ice by spontaneous phase transition at these low temperatures. While MEI also includes this singularity for the second relaxation time *τ*_{2}, the STO model assumes *τ*_{2} to be temperature independent, which has been mentioned to be rather unphysical by Mätzler et al. (2010).

For the most recent model by Ellison (2007) (ELL07), a comprehensive review of all available permittivity measurements was performed in combination with a comparison of different interpolation formulas. The most accurate interpolation formula for the laboratory dataset including measurements at microwave frequencies up to the far infrared region includes three Debye terms and two non-Debye terms for resonant effects in the far infrared.

A comparison of the six permittivity models in terms of mass absorption coefficient is shown in Fig. 1 for liquid water at 0° and −25°C. At 0°C and frequencies lower than 150 GHz, the various models agree relatively well. Even up to 300 GHz, only slight deviations of up to 15% can be found. The only exception is the Ray model, which starts to deviate at frequencies larger than 150 GHz, ultimately resulting in a 40% smaller mass absorption coefficient relative to the other models at 300 GHz. The situation is completely different in the supercooled temperature region. At −25°C, the models begin to substantially deviate from each other at frequencies above 30 GHz. The models show differences in mass absorption coefficients up to 70% at 300 GHz with the LIE, STO, MEI, and ELL06 models predicting significantly higher mass absorption values compared to RAY and ELL07 models. One reason for the increasing spread among the models with decreasing temperatures even at lower frequencies is the temperature dependence of the two relaxation processes. At 0°C, the absorption properties up to 100 GHz are dominated by the relatively well-known first relaxation process. At decreasing temperatures, the absorption properties become increasingly affected by the second relaxation process for which the temperature dependence is highly uncertain (Ellison 2007).

Basically, the reason for the large model deviations can be attributed to the extremely sparse experimental permittivity data in the supercooled temperature regime. While a large number of experimental datasets with liquid water measurements down to 0°C are available, the only dataset down to −18°C was collected at a single frequency (9.61 GHz) by Bertolini et al. (1982). For the higher-frequency range between 100 and 2000 GHz, only one laboratory dataset for the supercooled region down to −2°C is available from Ronne et al. (1997). This “no-data zone” for liquid water temperatures below −2°C and for frequencies above 20 GHz is the main source of uncertainty since all models have to extrapolate from the nonsupercooled temperatures into a widely unknown region of permittivity values.

### b. Implication of permittivity model uncertainties

From a more practical point of view, it is worthwhile to discuss the impact of the permittivity model uncertainties on radar attenuation and on SLWP retrievals from MWR. Attenuation becomes increasingly important for cloud radars operating at higher frequencies (e.g., W-band cloud radars). The radar one-way attenuation due to SLWP absorption predicted by the different absorption models is indicated by the right *y* axis in Fig. 1. Attenuation values [dB km^{−1} (g m^{−3})^{−1}] can be interpreted as the attenuation that would result from a 1-km-thick cloud with constant temperature and a constant liquid water content of 1 g m^{−3}. The related liquid water path (LWP) of 1 kg m^{−2} is relatively large and can usually only be found in convective clouds. For a 94-GHz cloud radar, all models predict a liquid water attenuation of about 4.5 dB at 0°C. However, for a supercooled cloud with a mean liquid water temperature of −25°C, the calculated attenuation values range from 3 dB for STO up to 6 dB for MEI.

_{init}of 30 g m

^{−2}. While the other atmospheric variables were kept constant, only the temperature of the cloud layer was varied between +5° and −35°C in steps of 5°C. The related

*T*

_{B}values were calculated with the RT4 radiative transfer model (Evans and Stephens 1995) and the different liquid water permittivity models. For small changes of SLWP (e.g., between 0 and 100 g m

^{−2}) the relation between

*T*

_{B}and SLWP is linear. With the calculated sensitivity ∂

*T*

_{B}/∂SLWP, derived for a certain permittivity model and constant cloud temperature, we can write

*T*

_{B}difference caused by two differing permittivity models for a constant SLWP

_{init}to an SLWP uncertainty. Assuming the LIE model as the truth, the error in retrieved SLWP caused by a differing permittivity model (e.g., RAY) can be calculated with

The relative SLWP errors

Impact of the permittivity model uncertainties on retrieved SLWP for a ground-based MWR as function of cloud temperature *T* in °C. The relative SLWP error has been estimated with a simplified physical single-frequency retrieval for (top) 31.4 and (bottom) 90 GHz, assuming a typical winter atmosphere at the UFS and a homogeneous 1-km-thick cloud layer with a SLWP of 30 g m^{−2}. The different colors denote the various permittivity models similar to Fig. 1. It should be noted that the LIE model has been taken as the truth, and thus the SLWP deviations associated with the other models only illustrate the sensitivity of the SLWP retrieval to permittivity model uncertainties but are not indicating the real accuracy of a specific model.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Impact of the permittivity model uncertainties on retrieved SLWP for a ground-based MWR as function of cloud temperature *T* in °C. The relative SLWP error has been estimated with a simplified physical single-frequency retrieval for (top) 31.4 and (bottom) 90 GHz, assuming a typical winter atmosphere at the UFS and a homogeneous 1-km-thick cloud layer with a SLWP of 30 g m^{−2}. The different colors denote the various permittivity models similar to Fig. 1. It should be noted that the LIE model has been taken as the truth, and thus the SLWP deviations associated with the other models only illustrate the sensitivity of the SLWP retrieval to permittivity model uncertainties but are not indicating the real accuracy of a specific model.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Impact of the permittivity model uncertainties on retrieved SLWP for a ground-based MWR as function of cloud temperature *T* in °C. The relative SLWP error has been estimated with a simplified physical single-frequency retrieval for (top) 31.4 and (bottom) 90 GHz, assuming a typical winter atmosphere at the UFS and a homogeneous 1-km-thick cloud layer with a SLWP of 30 g m^{−2}. The different colors denote the various permittivity models similar to Fig. 1. It should be noted that the LIE model has been taken as the truth, and thus the SLWP deviations associated with the other models only illustrate the sensitivity of the SLWP retrieval to permittivity model uncertainties but are not indicating the real accuracy of a specific model.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

## 3. Datasets and methods

The dataset used in this study is a combination of MWR measurements at 31, 52, 90, and 150 GHz from MWRs at three different sites, augmented by 225-GHz observations at one of the sites (Table 2). Auxiliary observations from cloud radar, lidar, and radio soundings (and/or numerical weather analysis data) are also available for all sites. These data are used to characterize the observed cloud types and atmospheric temperature and water vapor structure.

Instrument characteristics of the MWRs and frequency channels used in this study.

### a. Observational datasets

#### 1) Summit Station, Greenland (SMT)

In 2010, a large suite of instrumentation was deployed at Summit Station in the center of the Greenland Ice Sheet (72.6°N, 38.5°W; 3250 m MSL) as part of the Integrated Characterization of Energy, Clouds, Atmospheric State, and Precipitation at Summit (ICECAPS) project (Shupe et al. 2013). The primary objective of this project is to collect a multiyear dataset that can be used to investigate important cloud–atmosphere properties and processes, and how these processes impact both the surface mass and energy budgets. The ICECAPS instrument suite includes a millimeter-wave cloud radar (MMCR), two polarization sensitive lidars, a ceilometer, an infrared interferometer, a sodar, an X-band precipitation sensor, two launches of Vaisala, Inc., RS92 radiosondes per day, and two MWRs. These two MWRs are the Humidity and Temperature Profiler (HATPRO), which provides observations of downwelling radiance in the spectral bands from 21 to 31 GHz and from 51 to 58 GHz (Rose et al. 2005), and a high-frequency (HF) radiometer, which makes measurements at 90 and 150 GHz. The latter system is very similar to the Dual Polarization Radiometer (DPR) that was utilized in earlier field experiments (Turner et al. 2009). A third MWR, which was manufactured by the same vendor who made the HATPRO and HF radiometers, provides downwelling radiance measurements at 225 GHz, and was deployed by the Academia Sinica Institute of Astronomy and Astrophysics (ASIAA) to characterize the viewing conditions at Summit Station for the possible deployment of a submillimeter astronomical telescope. We used 15 months of data from August 2011 to June 2012 and September to December 2012 in this analysis.

#### 2) Mount Zugspitze, Germany (UFS)

The second dataset analyzed was collected at the Environmental Research Station Schneefernerhaus (47.42°N, 10.98°E; 2650 m MSL) at the Zugspitze in the German Alps. The east–west alignment of the mountain massif leads to frequent blocking situations under north and northwesterly flow, resulting in strong lifting along the northern side of the mountain. As a result of the strong adiabatic cooling and orographically induced turbulence, strong and fast liquid water fluctuations at temperatures down to −20°C are frequently observed on the lee side of the mountain where the UFS is located. We analyzed a 3-month wintertime period (December 2008–February 2009) collected during the Toward an Optimal Estimation-Based Snowfall Characterization Algorithm (TOSCA) campaign (Löhnert et al. 2011). The site was equipped with two MWRs [both manufactured by Radiometer Physics (RPG)]: a HATPRO radiometer (similar to the one at SMT) owned by the UFS and a DPR owned by the Ludwig Maximilians University of Munich that makes observations at 90 and 150 GHz (Turner et al. 2009). Up to 90 GHz, both systems use direct detection receivers; only the 150-GHz channels of the DPR are heterodyne systems. Each MWR is equipped with heated blowers and camera surveillance systems to ensure data quality and to exclude time periods that have artifacts (e.g., ice on the radomes). Auxiliary data from the UFS are provided by a zenith-pointing 35.5-GHz microwave cloud radar (MIRA; Metek GmbH) and a ceilometer (CL31; Vaisala). No radiosondes are operationally launched at the UFS, thus we use analyses from the Consortium for Small Scale Modelling’s (COSMO) operational weather forecast model COSMO-DE from the German Weather Service (2.8-km spatial and 1-h temporal resolution) for vertical thermodynamic profiles.

#### 3) Murg valley, Black forest, Germany (FKB)

The Atmospheric Radiation Measurement Program (ARM) Mobile Facility (AMF) was deployed as part of the Convective and Orographically Induced Precipitation Study (COPS; Wulfmeyer et al. 2008) and the general observation period (GOP; Crewell et al. 2008) in the Murg Valley (48.54°N, 8.40°E; 511 m MSL) in the Black Forest in southwest Germany from April to December 2007. We analyzed data from an autumn/winter period between October and December 2007 when thin layers of supercooled stratus clouds frequently occurred; this dataset complements the lower temperature data from SMT and UFS sites by providing measurements between 0° and −10°C. Although permittivity models do not deviate significantly in this temperature region, it is a valuable confirmation of our method since we expect that the observations will agree best with model predicted values in this “warm” temperature region. The FKB data also overlap in cloud temperature with data collected from the two other sites, providing an important consistency check between the three datasets. Two MWRs were deployed at the AMF site providing observations between 22 and 150 GHz: a HATPRO radiometer by the University of Cologne and the ARM two channel HF radiometer (Cadeddu et al. 2013) [both radiometers are very similar to those deployed at SMT (Table 2)]. From the large set of auxiliary observations provided by the AMF, our analysis used the observations from the 35-GHz cloud radar (MMCR), micropulse lidar (MPL), and the radiosonde ascents (Vaisala RS92) that were performed 4 times per day.

#### 4) Microwave radiometer calibration

All radiometers (HATPRO, HF, DPR, 225-GHz MWR) are equipped with a comprehensive temperature stabilization (better than 30 mK) for the radiometric components (Rose et al. 2005). Absolute calibrations are performed for all radiometers according to the manufacturers’ recommendations every few months using a liquid nitrogen target as a cold reference at ~77 K. The views of the liquid nitrogen target and the internal hot load are used to determine the system noise temperature, sensitivity (gain), and nonlinearity of the detector. Additionally, tip-curve calibrations (Han and Westwater 2000) were performed by all instruments at regular intervals; this calibration method also provides an absolute way to determine the gain and system noise of the radiometer. While the nonlinearity of the receiver can be assumed constant between two absolute calibrations, all radiometers use views of the internal ambient blackbody together with noise diodes (which were calibrated during the tip-curve process) every few minutes to update their gain and system noise calibrations.

### b. Opacity ratio method

Our method of computing the opacity ratios between two frequencies is similar to the technique presented by Mätzler et al. (2010). Thus, we will only describe the most relevant parts of the idea and the differing aspects of our approach.

*τ*can be written as a sum of the contributions from dry air

*τ*

_{D}, water vapor

*τ*

_{WV}, and—in case of nonprecipitating, pure liquid water clouds—liquid water of suspended cloud droplets

*τ*

_{L}:

*τ*

_{L}, are obtained for zenith viewing conditions by the integral

*ρ*

_{L}, the frequency- and temperature-dependent liquid water mass absorption coefficient

*α*

_{L}, and the vertical height

*z*.

*τ*is usually derived with the radiative transfer equation in zenith direction and assuming a nonscattering atmosphere using the concept of a mean radiative temperature

*T*

_{mr}of the atmosphere (Ulaby et al. 1986). This approach usually assumes the validity of the Rayleigh–Jeans approximation (RJA), which allows us to express the radiant intensity

*I*

_{ν}as a linear function of the equivalent blackbody temperature

*T*

_{BB}. However, as discussed, for example, in Han and Westwater (2000), the RJA is only valid if ħ

*c*≪

*k*

_{B}

*T*

_{BB}with the Planck constant ħ, the vacuum speed of light

*c*, and the Boltzmann constant

*k*

_{B}. For high frequencies like 225 GHz and very low temperatures like the cosmic background temperature

*T*

_{c}of 2.725 K, the RJA needs to be replaced by the inverse of the Planck function

*T*

_{B}using the inverse Planck function and thus we can derive the optical thickness directly in terms of radiant intensities with

*T*

_{mr}is defined similar to Han and Westwater (2000) as

*β*(

*z*) is the volume absorption coefficient of the atmospheric layer at height

*z*and

*τ*(0, ∞) denotes the opacity from ground level to the top of the atmosphere. The quantity

*T*

_{mr}has been directly derived from radiosonde profiles using an updated version of the monochromatic radiative transfer model (MonoRTM; Payne et al. 2011) for the SMT data. For the FKB and UFS sites,

*T*

_{mr}has been obtained from a statistical retrieval based on 2-m temperature and a long time series of radiosonde profiles (similar to, e.g., Mätzler and Morland 2009). The accuracy of

*T*

_{mr}using these methods is estimated to be better than 3 K, which has only a very small effect on the derived

*τ*of less than 1%. In fact, we found that because of the comparably high values of

*T*

_{mr}for our frequency range and atmospheric conditions,

*T*

_{mr}can also be derived with sufficient accuracy using the RJA in (8). However, the conversion of the radiometer intensities into

*T*

_{B}and the contribution of

*T*

_{c}should always be calculated with

*Z*to avoid too-high ice water contents that might contaminate the MWR data. A more detailed discussion of the influence of ice particles on the MWR observations is given later in this section. Approximately 75% of the analyzed cloud cases have cloud thicknesses less than 1 km and in 34% of the data the clouds are thinner than 500 m. The temperature within the cloud is assumed to be equal to a constant mean cloud temperature

*T*

_{cld}derived from the estimated cloud boundaries and temperature profile information. For the SMT site, we found several cases of thin, single-layer mixed-phase clouds close to the ground. For those cases, where the liquid water layer could be well identified to be only at the cloud top using collocated lidar observations,

*T*

_{cld}has been assigned to the cloud-top temperature. The assumption of a constant

*T*

_{cld}simplifies (6) to

*z*

_{cb}and cloud top

*z*

_{ct}. In addition to an independent estimate of LWP for deriving

*α*

_{L}, we also need to quantify the gaseous atmospheric contributions

*τ*

_{D}and

*τ*

_{WV}since only the sum of all atmospheric components can be derived as

*τ*using the MWR observations and (7).

A very elegant approach to disentangle the contribution from liquid cloud water and the gaseous constituents is to utilize their different temporal variability (Mätzler et al. 2010). Comparing *T*_{B} time series of ground-based MWR observations in clear-sky and cloudy situations reveals that cloud liquid water is the main reason for rapid changes in the observed *T*_{B}. The close correlation of fast opacity changes and liquid water fluctuations is illustrated in Fig. 3. A 10-min time period of derived opacities at 150 and 31.4 GHz with 4-s temporal resolution from the UFS site is shown as a scatterplot. The cloud considered here had a maximum thickness of 750 m with an average *T*_{cld} of −23°C; the cloud bottom was at the height of the UFS site. The measured opacities are nicely aligned along a straight line in the *τ*_{150} − *τ*_{31.4} space. To validate whether the opacity changes are due to liquid water or water vapor fluctuations within the cloud, we calculated the range of opacity values for LWP up to 270 g m^{−2} with (9) and for a constant *T*_{cld} using the RAY model. For this particular temperature and frequency combination, the slope of the predicted opacities from the RAY model fits the measured opacities extremely well.

Example of opacity variations at 150 and 31.4 GHz measured within a 10-min time period (4-s resolution) at the UFS site at 2150 UTC 14 Feb 2009. The fast opacity changes are mainly due to liquid water fluctuations (in the range of 200 g m^{−2}) with a 750-m-thick cloud layer (average cloud temperature *T*_{cld} = −23°C). The relationship between the opacities of the two channels predicted by the RAY model for solely liquid water fluctuations is shown by the dashed line. The solid line denotes the effect of only water vapor fluctuations predicted by the model from Rosenkranz (1998). The impacts of water vapor fluctuations in the range of up to 0.5 kg m^{−2} that are perfectly correlated (anticorrelated) with the liquid water fluctuations are indicated as dashed–dotted (dotted) lines.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Example of opacity variations at 150 and 31.4 GHz measured within a 10-min time period (4-s resolution) at the UFS site at 2150 UTC 14 Feb 2009. The fast opacity changes are mainly due to liquid water fluctuations (in the range of 200 g m^{−2}) with a 750-m-thick cloud layer (average cloud temperature *T*_{cld} = −23°C). The relationship between the opacities of the two channels predicted by the RAY model for solely liquid water fluctuations is shown by the dashed line. The solid line denotes the effect of only water vapor fluctuations predicted by the model from Rosenkranz (1998). The impacts of water vapor fluctuations in the range of up to 0.5 kg m^{−2} that are perfectly correlated (anticorrelated) with the liquid water fluctuations are indicated as dashed–dotted (dotted) lines.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Example of opacity variations at 150 and 31.4 GHz measured within a 10-min time period (4-s resolution) at the UFS site at 2150 UTC 14 Feb 2009. The fast opacity changes are mainly due to liquid water fluctuations (in the range of 200 g m^{−2}) with a 750-m-thick cloud layer (average cloud temperature *T*_{cld} = −23°C). The relationship between the opacities of the two channels predicted by the RAY model for solely liquid water fluctuations is shown by the dashed line. The solid line denotes the effect of only water vapor fluctuations predicted by the model from Rosenkranz (1998). The impacts of water vapor fluctuations in the range of up to 0.5 kg m^{−2} that are perfectly correlated (anticorrelated) with the liquid water fluctuations are indicated as dashed–dotted (dotted) lines.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Even though the typical time scale of water vapor variability can be assumed to be much longer than 10 min for nonconvective conditions, we have to consider the influence of water vapor fluctuations that might be correlated with the liquid water fluctuations because of internal cloud processes like, for example, evaporation or condensation of cloud droplets (Mätzler et al. 2010). The solid line in Fig. 3 has been obtained assuming solely water vapor fluctuations within the cloud (constant *T*_{cld}, constant pressure, no SLWP) using the water vapor absorption model from Rosenkranz (1998). The water vapor slope is much steeper compared to the liquid water slope, which is a necessary condition to be able to distinguish between water vapor and liquid water fluctuations. The dashed–dotted (dotted) lines in Fig. 3 also illustrate how the linear liquid water relationship would change if we assume that the liquid water fluctuations are perfectly correlated or anticorrelated with water vapor fluctuations in the range of 0.5 kg m^{−2} integrated water vapor content (IWV). The resulting uncertainty range due to water vapor fluctuations almost entirely covers the range of scatter in the data. However, it is likely that the scatter is also due to instrument noise or variability in cloud liquid water temperature.

Although we restricted our selection of clouds to nonprecipitating cases, we did not completely exclude mixed-phase clouds. Large ice and snow particles have been found to enhance the *T*_{B} observations at frequencies larger than 90 GHz because of scattering of the surface thermal emission back to the ground-based MWR (Kneifel et al. 2010). Thus, we also have to investigate the effect of potentially correlated ice water content (IWC) fluctuations on the derived opacities. Since the contribution of SLW to the cloud radar signal at 35 GHz can be neglected with respect to ice particles, we estimated the cloud IWC with the temperature dependent method provided in Hogan et al. (2006). The maximum ice water path (IWP) has been found to be 2 (5) g m^{−2} at the UFS (SMT) site. According to the results in Kneifel et al. (2010), Löhnert et al. (2011), and Kneifel (2011), the *T*_{B} enhancement due to the maximum IWP can be estimated to be 0.06 (0.3) K at 90 (150) GHz for the UFS dataset. For the largest IWP found in the SMT data, we can similarly estimate the maximum *T*_{B} enhancement to be 0.15 (0.75) K at 90 (150) GHz and 1.25 K at 225 GHz. Even though radar estimates of IWC are generally affected by large uncertainties, the *T*_{B} enhancements even at 225 GHz are in the range of radiometric noise and are thus not a critical source of error for this analysis.

*τ*to be only due to liquid water changes ΔLWP, we can write

Thus, the problem of determining *α*_{L} (*ν*, *T*_{cld}) reduces to the problem of finding an independent estimate of ΔLWP. MWR retrievals of LWP are not suitable since a certain permittivity model has already been assumed for the generation of the LWP retrieval itself. For thin clouds with small LWP up to 60 g m^{−2}, an independent measure of LWP can be derived from infrared spectroscopy (Cadeddu and Turner 2011; Turner 2005). However, the restriction to small LWP values limits the analysis to comparably low signals in the MWR observations, especially at 31 GHz.

*α*

_{L}cannot be derived in the described way. However, the MWR observations can be used to constrain existing permittivity models particularly if the MWR observations cover a wide spectral range. The fast opacity changes at two different frequencies can be used to derive the opacity ratio

The dependence of all variables on *T*_{cld} is not explicitly written for better readability. The ratio of fast opacity changes at two frequencies *α*_{L},

The MWR observations (only zenith direction) from all sites have been subdivided into 10-min periods. The temporal resolution of the MWRs ranges from 1 to 3 s. Thus, the data from the different MWRs have been first matched and then averaged onto 4-s time intervals. The slope *T*_{cld} and correlated water vapor fluctuations affect the opacities as well as instrument noise or uncertainties in the estimated *T*_{mr}. For linear regression problems with unknown error in both variables, Clarke and Van Gorder (2013) recently presented a method on how the quality of the fit can be assessed. The slope from a classical least squares fit can be biased depending on whether *τ*_{L}(*ν*_{1}) is fitted against *τ*_{L}(*ν*_{2}) or vice versa. A better estimate of the true slope can be obtained from the geometric mean of

A specific advantage of the opacity ratio method is that constant biases in the derived opacity values—for example, due to instrument calibration errors—do not affect the slope estimate. In contrast, an erroneous gain calibration would strongly affect

## 4. Results and discussion

We selected 10 frequency combinations between 31 and 225 GHz in a way to cover the entire observed spectral range but also to avoid frequencies within spectral regions of strong absorption bands. Thus, we only used the 31.4-GHz channel from the HATPRO instruments for K band (water vapor absorption line at 22.24 GHz) and only the 52.28-GHz channel in the V band (oxygen absorption band at 60 GHz). The measured opacity ratios for the three sites are presented together along with the various model estimates as function of *T*_{cld} down to −35°C in Figs. 4 and 5. The varying number of data points for the different frequency combinations is related to the dependence of the linear fit quality on the liquid water sensitivities in the different channels, as well as on the individual noise characteristics of each channel.

Ratios of fast opacity changes (similar to the example slope in Fig. 3) for the frequency combinations (top left) 150/90 GHz, (top right) 150/52 GHz, (middle left) 150/31 GHz, (middle right) 90/31 GHz, (bottom left) 90/52 GHz, and (bottom right) 52/31 GHz as function of average cloud temperature (°C). Measured opacity ratios are shown as colored dots from three sites: SMT (black), UFS (orange), and FKB (green). The colored lines indicate the ratio of liquid water mass absorption coefficients predicted by the six permittivity models (Table 1 and Fig. 1), which are equal to the opacity ratios for pure liquid water opacities. Error estimates for the measured opacity ratios have been determined for each data point. For better readability of the plots all observations including error bars have been binned into 2.5°C temperature bins (red filled circles). The error in the estimation of *T*_{cld} (red *x*-error bars) and the variability of derived opacity ratios (red *y*-error bars) is shown as the standard deviation of the errors within each bin with at least 20 data points.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Ratios of fast opacity changes (similar to the example slope in Fig. 3) for the frequency combinations (top left) 150/90 GHz, (top right) 150/52 GHz, (middle left) 150/31 GHz, (middle right) 90/31 GHz, (bottom left) 90/52 GHz, and (bottom right) 52/31 GHz as function of average cloud temperature (°C). Measured opacity ratios are shown as colored dots from three sites: SMT (black), UFS (orange), and FKB (green). The colored lines indicate the ratio of liquid water mass absorption coefficients predicted by the six permittivity models (Table 1 and Fig. 1), which are equal to the opacity ratios for pure liquid water opacities. Error estimates for the measured opacity ratios have been determined for each data point. For better readability of the plots all observations including error bars have been binned into 2.5°C temperature bins (red filled circles). The error in the estimation of *T*_{cld} (red *x*-error bars) and the variability of derived opacity ratios (red *y*-error bars) is shown as the standard deviation of the errors within each bin with at least 20 data points.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Ratios of fast opacity changes (similar to the example slope in Fig. 3) for the frequency combinations (top left) 150/90 GHz, (top right) 150/52 GHz, (middle left) 150/31 GHz, (middle right) 90/31 GHz, (bottom left) 90/52 GHz, and (bottom right) 52/31 GHz as function of average cloud temperature (°C). Measured opacity ratios are shown as colored dots from three sites: SMT (black), UFS (orange), and FKB (green). The colored lines indicate the ratio of liquid water mass absorption coefficients predicted by the six permittivity models (Table 1 and Fig. 1), which are equal to the opacity ratios for pure liquid water opacities. Error estimates for the measured opacity ratios have been determined for each data point. For better readability of the plots all observations including error bars have been binned into 2.5°C temperature bins (red filled circles). The error in the estimation of *T*_{cld} (red *x*-error bars) and the variability of derived opacity ratios (red *y*-error bars) is shown as the standard deviation of the errors within each bin with at least 20 data points.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

As in Fig. 4, but for opacity ratios including 225 GHz, which are only available from SMT site: (top left) 225/150 GHz, (top right) 225/90 GHz, (bottom left) 225/52 GHz, and (bottom right) 225/31 GHz.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

As in Fig. 4, but for opacity ratios including 225 GHz, which are only available from SMT site: (top left) 225/150 GHz, (top right) 225/90 GHz, (bottom left) 225/52 GHz, and (bottom right) 225/31 GHz.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

As in Fig. 4, but for opacity ratios including 225 GHz, which are only available from SMT site: (top left) 225/150 GHz, (top right) 225/90 GHz, (bottom left) 225/52 GHz, and (bottom right) 225/31 GHz.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

### a. Opacity ratios up to 150 GHz

For the six opacity ratios up to 150 GHz (150/90, 150/52, 150/31, 90/31, 90/52, and 52/31 GHz), observations from all three sites are available and are presented together in Fig. 4. The observations from the three sites and at all frequency combinations are found to be highly consistent with each other. This high consistency of the derived opacity ratios is strong evidence for the robustness of the method and the reliability of the derived values, given that the data come from different instruments, site locations, and cloud types.

Considering the different permittivity models, we find the smallest spread among the modeled opacity ratios at lower frequencies and higher temperatures. This behavior is expected from Fig. 1 as the simulated opacity ratios are equal to the ratios of mass absorption coefficients at the two frequencies considered and the corresponding temperature. At temperatures below −20°C the deviations between the models strongly increase. Some of the modeled opacity ratios reveal significant features like a strong increase in the ratio for temperatures lower than −25°C with the STO model; very different behavior is seen in the MEI model.

When comparing the observed ratios with the model estimates, we find that the range of ratios predicted by the permittivity models largely covers the range of data in the observed ratios. This might be interpreted as evidence that the general choice of a Debye model and the temperature-dependent Debye parameters is a reasonable concept for the supercooled region. In the “safe” temperature region around 0°C, the observations match the model estimates relatively well. The few outliers are most likely due to the increasing magnitude of correlated water vapor fluctuations at higher temperatures at the lower altitude of the FKB site. The lower-frequency ratios including the 52.28-GHz channel (90/52 and 52/31 GHz) show a larger number of outliers and scatter. This might be related to the general smaller *T*_{B} variation due to liquid water at lower frequencies but also due to nonnegligible influence of the oxygen absorption band at 60 GHz on the 52.28-GHz channel, which causes a higher sensitivity of this channel to temporal changes of the air temperature profile.

The observed ratios are binned into 2.5°C temperature bins *T*_{i} containing at least 20 observations per bin. The root-mean-square deviation (RMSD) is calculated with the bin-averaged observed ratios *γ*_{mod}(*T*_{i}) predicted by the permittivity models; for better readability the two frequency indices in the opacity ratios have been omitted in (12). The total number of valid temperature bins per frequency combination is *N*. To make the RMSD from different frequency combinations comparable, the RMSD is also normalized by the maximum difference of

NRMSD between observed opacity ratios (temperature binned) and the various permittivity model estimates as defined in (12) in percent. Boldface numbers indicate minimum NRMSD values for the different frequency combinations. The rightmost column (OBS) contains the mean standard deviation of the observed, temperature-binned opacity ratios normalized by the maximum difference of the ratios at each frequency combination similar to (12) in percent.

Considering the resulting NRMSD, one finds the lowest values for the MEI model at 150/90 GHz and 52/31 GHz; for the RAY model at 150/52 GHz, 150/31 GHz, and 90/52 GHz; and for the STO model at 90/31 GHz. The models LIE, ELL06, and ELL07 show larger NRMSDs for all six frequency combinations compared to the remaining models. The variability of NRMSD between the permittivity models for one specific frequency combination is also considerably different. While, for example, at 90/52 GHz the NRMSD values range only between 20.4% and 29.4%, the lowest value at 90/31 GHz is found with 5.3% for STO, which is over 3 times smaller compared to the largest value of 18.3% found for MEI. In addition to the NRMSD it is also important to compare the overall temperature dependence of the observations and the permittivity model estimates: Even though the MEI model shows the lowest NRMSD for three of the six frequency combinations, the general decrease of opacity ratios at temperatures below −20°C seems to be in contradiction to the observations. In a similar way this is true for the strong increase of opacity ratios for STO at temperatures below −20°C.

Based on either the NRMSDs or the more qualitative consideration of the temperature dependence, there does not appear to be a “golden” model that suitably fits the observations at all six frequency combinations in Fig. 4. For example, the RAY model provides a surprisingly good fit to the 150/31-GHz ratios (NRMSD of 3.9%) but it is systematically overestimating the ratios at 90/31 GHz. The newest ELL07 model is able to fit the data rather well at temperatures below −10°C but shows a tendency to overestimate the ratios at higher temperatures resulting in comparably large NRMSDs.

### b. Opacity ratios including 225 GHz

Opacity ratios that include observations at 225 GHz (225/150 GHz, 225/90 GHz, 225/52 GHz, 225/31 GHz) are only available from the SMT site in Greenland and are shown in a similar way as for the lower frequencies in Fig. 5. Similar to Fig. 4, the models increasingly deviate from each other with decreasing temperatures, but again the observed values are found to be in between the various model predictions. Interestingly, the STO model lies almost entirely outside the measured range at the four 225-GHz ratios, while it was within the range of the observation for the lower-frequency opacity ratios in Fig. 4. As a result, the STO model has the largest NRMSDs for the four 225-GHz ratios. Similar to the lower-frequency ratios, STO exhibits a strange “hook feature” with increasing ratios at the lowest temperatures. This unnatural behavior can be found in all frequency combinations with varying temperature dependence but it is most prominent in the 225/90-, 225/52-, and 225/31-GHz ratios. In contrast, the MEI model shows strongly decreasing opacity ratios at lower temperatures that cannot be found in the other models; this was also seen in the lower-frequency ratios in Fig. 4. The LIE and ELL06 models tend to overestimate the opacity ratios; however, this overestimate is more moderate and does not have the unnaturally strong increasing/decreasing features at the lowest-temperature region that other models have. Unlike all other models, the RAY model significantly underestimates the opacity ratios for these higher frequencies (Fig. 5). The ELL07 model fits all four observed opacity ratios reasonably well and provides the closest fit to the data at 225/52 GHz and 225/31 GHz with NRMSDs of 26% and 15.6%, respectively. At the two higher-frequency combinations of 225/150 and 225/90 GHz, the “bending up” of the ELL07 model at the lowest temperatures is not supported by the observations and thus the NRMSDs are slightly larger compared to the lowest values produced by MEI. However, the low number of observations at temperatures below −30°C and the comparably large uncertainties in the observed opacity ratios at particularly 225/150 and 225/90 GHz does prevent us from determining if the increase of opacity ratios at temperatures lower than −30°C and these specific frequency combinations is statistically significant.

### c. Implication for mass absorption coefficient

The opacity ratios at the 10 different frequency pairs clearly illustrate the need for the observations to cover a wide spectral range to evaluate the performance of permittivity models with opacity ratios. For example, a certain model might be able to fit a single opacity ratio over the temperature range well (e.g., the RAY model at 150/31 GHz), but its underlying mass absorption values might be completely erroneous. If we could identify a “golden model” that is able to fit all of the opacity ratios well over the supercooled temperature range, it would be unlikely that it predicts entirely incorrect mass absorption values.

*α*

_{L}can be determined if we are able to accurately specify the temperature dependence of a certain

*α*

_{L}(

*ν*

_{ref},

*T*) at a specific frequency

*ν*

_{ref}throughout the entire supercooled temperature region. If the reference frequency

*ν*

_{ref}is also included in the measured opacity ratios

*α*

_{L}(

*ν*,

*T*) with

Cadeddu and Turner (2011) found in their direct comparison of measured *α*_{L} with model prediction the STO model to fit their observations best, especially at 90 GHz. Our opacity ratio comparison (Fig. 4 and Table 3) revealed that STO fits the ratios at 90/31 GHz extremely well. It might seem to be contradictory, however, that STO fails to provide a good fit to the other two low-frequency ratios at 90/52 and 52/31 GHz. As discussed earlier, the 52.28-GHz channel observations are problematic because of the influences of the close oxygen band; this influence can also be found in the by a factor of 2 larger uncertainties of these opacity ratios. Our selection of the STO model at 90 GHz as our reference model *α*_{L}(90, *T*) is therefore mainly based on the findings in Cadeddu and Turner (2011) since model errors could easily cancel out for a single opacity ratio. Assuming an error of 10% for *α*_{L}(90, *T*) and using the error estimates for the bin-averaged observations of *γ*_{ν,90}, the resulting *α*_{L}(*ν*, *T*) and its associated error for 31.4, 52.28, 150, and 225 GHz can be calculated (Fig. 6).

Liquid water mass absorption coefficient (m^{2} kg^{−1}) as function of liquid water temperature (°C) for (top left) 225, (top right) 150, (middle left) 52.28, (middle right) 31.4, (bottom left) 90, and (bottom right) 23.8 GHz. The colored lines are the predicted values of the six permittivity models described in Table 1. The black filled circles (top two panels) are mass absorption coefficients derived according to (13) using the binned measured opacity ratios 225/90, 150/90, 52/90, and 90/31 GHz shown in Figs. 4 and 5 and the STO model at 90 GHz (light blue line in bottom-left panel) as reference model for *α*_{L}(90, *T*). Error bars in the *y* direction of the observations include the uncertainty of the derived opacity ratios and an assumed uncertainty of 10% for the reference model (STO) for *α*_{L}(90, *T*). The bottom-left panel shows the temperature dependence of the reference model (STO) at 90 GHz; the bottom-right panel illustrates the model spread at 23.8 GHz, which is often used in combination with 31.4 GHz for SLWP retrievals.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Liquid water mass absorption coefficient (m^{2} kg^{−1}) as function of liquid water temperature (°C) for (top left) 225, (top right) 150, (middle left) 52.28, (middle right) 31.4, (bottom left) 90, and (bottom right) 23.8 GHz. The colored lines are the predicted values of the six permittivity models described in Table 1. The black filled circles (top two panels) are mass absorption coefficients derived according to (13) using the binned measured opacity ratios 225/90, 150/90, 52/90, and 90/31 GHz shown in Figs. 4 and 5 and the STO model at 90 GHz (light blue line in bottom-left panel) as reference model for *α*_{L}(90, *T*). Error bars in the *y* direction of the observations include the uncertainty of the derived opacity ratios and an assumed uncertainty of 10% for the reference model (STO) for *α*_{L}(90, *T*). The bottom-left panel shows the temperature dependence of the reference model (STO) at 90 GHz; the bottom-right panel illustrates the model spread at 23.8 GHz, which is often used in combination with 31.4 GHz for SLWP retrievals.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

Liquid water mass absorption coefficient (m^{2} kg^{−1}) as function of liquid water temperature (°C) for (top left) 225, (top right) 150, (middle left) 52.28, (middle right) 31.4, (bottom left) 90, and (bottom right) 23.8 GHz. The colored lines are the predicted values of the six permittivity models described in Table 1. The black filled circles (top two panels) are mass absorption coefficients derived according to (13) using the binned measured opacity ratios 225/90, 150/90, 52/90, and 90/31 GHz shown in Figs. 4 and 5 and the STO model at 90 GHz (light blue line in bottom-left panel) as reference model for *α*_{L}(90, *T*). Error bars in the *y* direction of the observations include the uncertainty of the derived opacity ratios and an assumed uncertainty of 10% for the reference model (STO) for *α*_{L}(90, *T*). The bottom-left panel shows the temperature dependence of the reference model (STO) at 90 GHz; the bottom-right panel illustrates the model spread at 23.8 GHz, which is often used in combination with 31.4 GHz for SLWP retrievals.

Citation: Journal of Applied Meteorology and Climatology 53, 4; 10.1175/JAMC-D-13-0214.1

At 31.4 GHz, the STO model is found to fit the derived *α*_{L} extremely well, which is in agreement with the findings in Mätzler et al. (2010) and Cadeddu and Turner (2011). This consistent finding for 31.4 GHz is particularly important since 31.4 GHz is a common frequency used to retrieve SLWP. Considering the large deviations of the various permittivity models at 31.4 GHz, which can potentially lead to large SLWP retrieval biases (Fig. 2), the results suggests that the STO model should be preferred for temperatures down to −30°C in retrieval development and MW radiative transfer. Since our data are for temperatures above −30°C, we are unable to state which model is more accurate for cloud temperatures below this threshold. Relative to the STO model, all other models (especially MEI and RAY) increasingly overestimate *α*_{L}(31.4, *T*) with decreasing temperature. This implies that SLWP retrievals using other permittivity models than STO are likely to underestimate the true SLWP amounts of observed clouds at temperatures below −15°C.

At 52.28 GHz, the STO model is found to be still close to the derived *α*_{L}; however, as discussed before, the larger scattering in the 90/52-GHz ratio leads also to larger error bars of the derived *α*_{L} and the opacity ratios including 52.28 GHz should be interpreted carefully. In general, the observations provide an indication that the STO model might be the currently best choice for frequencies up to 90 GHz. At higher frequencies (150 and 225 GHz) the STO model is similar to the LIE, ELL06, and MEI models in that it increasingly overestimates the *α*_{L} values. The only model that fits the absolute values and the general temperature dependence of *α*_{L} at these higher frequencies is ELL07. Thus, at frequencies larger than 90 GHz up to 225 GHz, the ELL07 model seems to provide the most realistic estimates of *α*_{L}.

## 5. Conclusions

In this study, we applied a recently proposed method to utilize ground-based MWR observations of supercooled liquid clouds to evaluate the accuracy and spectral consistency of different permittivity models for supercooled liquid water. The different time scales of fluctuations in liquid water and atmospheric gases allow the derivation of liquid water opacities. Ratios of these opacities at two different frequencies are independent of SLWP and equal to the ratios of mass absorption coefficients that can be directly derived from permittivity models. We used observations between 31 and 225 GHz from different ground-based MWR located at three sites in Greenland, the German Alps, and southwest Germany to evaluate six permittivity models over a wide range of supercooled liquid water clouds. Cloud radar and lidar observations are used to estimate the cloud boundaries. Together with radiosondes and model analysis data, the average SLW temperature and its uncertainty range was derived. The final dataset^{1} contains a unique collection of supercooled clouds from different climatic regions including average cloud temperatures between −33° and +5°C.

The derived opacity ratios from the different sites–instruments show a remarkably high consistency to each other, which indicates the robustness of the method and the reliability of the derived opacity ratios. The comparison of the observed ratios with six permittivity models also reveals the following:

The range of opacity ratios predicted by the six permittivity models is generally able to cover the majority of observed ratios. The deviations between the models themselves and between models and observations are smallest at temperatures close to 0°C; they dramatically increase at cloud temperatures lower than approximately −15°C (depending on frequency combination).

Although certain models are found to provide almost a perfect fit to the observed opacity ratios at specific frequency combinations, no model was identified that provides a sufficiently accurate approximation to the observations over the entire spectral range and cloud temperature range.

The STO, LIE, and ELL06 models systematically overestimate the observed opacity ratios if frequencies higher than 90 GHz are included. The RAY model tends to underestimate the opacity ratios particularly for the ratios including 225 GHz. The most recent model by ELL07 is found to be the best approximation at temperatures below −10°C but tends to overestimate the opacity ratios at higher temperatures between −10° and +5°C.

The clear advantages of the opacity ratio method are its independence of an accurate and independent estimate of SLWP and its robustness in the face of possible MWR calibration offsets. A disadvantage of the approach is the loss of information about the absolute value of the mass absorption coefficient. Hence, the performance of the permittivity models can only be evaluated if a broad range of frequencies and temperatures is considered. For most applications (e.g., SLWP retrievals or estimation of cloud radar attenuation due to SLW), the absolute value of the mass absorption coefficient *α*_{L}(*ν*, *T*) is required.

By combining the measured opacity ratios with a reference *α*_{L} at 90 GHz (STO), which is based on the findings in Cadeddu and Turner (2011), the mass absorption coefficients at 31.4, 52.28, 150, and 225 GHz as function of temperature were derived. The resulting *α*_{L} are clearly limited by the uncertainties in the reference *α*_{L}(90, *T*) and the related uncertainties in the observed opacity ratios. Despite the remaining uncertainties of the method we can draw a few important conclusions about the applicability of the six permittivity models for frequencies between 31.4 and 225 GHz:

Consistent with our comparisons in the opacity ratio space, none of the investigated permittivity models were able to sufficiently approximate the observed mass absorption values at all four frequencies (31.4, 52.28, 150, and 225 GHz) over the entire temperature range.

For liquid water temperatures down to −30°C and frequencies lower than 90 GHz, the STO model has the best agreement with the observations. This finding is in general agreement with Mätzler et al. (2010) and Cadeddu and Turner (2011), and thus we suggest that the STO model be used for SLW retrieval development and MW radiative transfer in this specific frequency/temperature region.

For frequencies larger than 90 GHz (particularly 150 and 225 GHz), we find the ELL07 model superior; the majority of the remaining models tend to overestimate

*α*_{L}at these higher frequencies for temperatures down to −30°C.

Our study has shown the potential of ground-based MWR observations that cover a broad spectral range and a range of supercooled cloud temperature to evaluate liquid water permittivity models. However, to substantially improve the liquid water permittivity models in the supercooled temperature region and over a wide spectral range (e.g., 1–1000 GHz), more laboratory data are required. As shown in (1), the absorption index that can be derived from MWR observations is dependent both on the imaginary and real part of the complex permittivity. Only laboratory data are able to provide independent measurements of both parts of the complex permittivity to develop improved permittivity models. The ongoing activities to utilize higher MW frequencies in passive–active applications should lead to a renewed interest of the retrieval and cloud microphysics communities to perform extended laboratory studies in order to avoid systematic biases in current and future cloud retrieval products.

## Acknowledgments

The data from the UFS site used in this study were collected as part of the TOSCA campaign, which has been funded by the German Science Foundation (DFG) under Grant LO 901/3-1. The radiosonde, cloud radar, and lidar observations at the FKB site were collected by the U.S. Department of Energy ARM’s mobile facility as part of COPS; the MWR data at the FKB site have been collected during the GOP campaign, which has been funded by DFG under Grant WU 356/4-2. ICECAPS is supported by the U.S. National Science Foundation under Grants ARC-0856773, 0904152, and 0856559 as part of the Arctic Observing Network (AON) program, with additional instrumentation support provided by the NOAA Earth System Research Laboratory, ARM, and Environment Canada. ICECAPS data are available via the ARM data archive. Argonne National Laboratory’s work was supported by the U.S. Department of Energy, Office of Science, under Contract DE-AC02-06CH11357. Research done by Stefan Kneifel and Stephanie Redl has been carried out within the Hans Ertel Centre for Weather Research. Climate Monitoring Branch. Contributions from Emiliano Orlandi have been financed by the DFG priority program High Altitude and Long Range Research Aircraft [HALO (SPP 1294) Project: Using the HALO Microwave Package (HAMP) for cloud and precipitation research, GZ: CR 111/9-1]. The authors are also very grateful to Susanne Crewell (University of Cologne) for many fruitful discussions. Constructive comments from three anonymous reviewers are also gratefully recognized.

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The opacity ratio dataset together with a short description of the STO model and Interactive Data Language (IDL) routines for the STO and ELL07 model are available on request from the corresponding author.