## 1. Introduction

Supercooled water clouds that contain liquid water at temperatures below 0°C are a common occurrence around the globe (Hogan et al. 2004; Hu et al. 2010). To understand their role in the weather/climate system, accurate measurements of their macrophysical and microphysical properties are needed. One of the most basic properties is the vertically integrated liquid water content, which is referred to as the liquid water path (LWP). LWP can be retrieved from cloud radar (e.g., Matrosov et al. 2004), visible/near-infrared radiances (e.g., Minnis et al. 1995; Greenwald 2009; Heng et al. 2014), and thermal infrared radiance observations (e.g., Turner 2007), but the most common retrieval methods use microwave and submillimeter wave radiometer observations (e.g., Curry et al. 1990; Petty and Katsaros 1992; Lin and Rossow 1994; Westwater et al. 2001; Turner et al. 2007; Greenwald et al. 1993; O’Dell et al. 2008).

However, retrievals of LWP from satellite- and ground-based radiometer observations require that the underlying radiative transfer model used in the retrieval is accurate and does not suffer from systematic biases. In the microwave and submillimeter wave portion of the spectrum from roughly 0.5 to 300 GHz (henceforth referred to as the microwave for simplicity), the dielectric properties (i.e., refractive indices) of liquid water (and therefore its interaction with radiation) are temperature dependent. Historically, the temperature dependence of the refractive index has been measured in the laboratory, and as a result of a range of technical issues, the measurements have largely been at temperatures above 0°C. Many semiempirical models were fit to the laboratory data and used to extrapolate the laboratory-measured absorption to supercooled temperatures. However, several papers (e.g., Westwater et al. 2001; Lin et al. 2001; Ellison 2007; Kneifel et al. 2014) have illustrated that there are significant differences in the absorption coefficients predicted by these models at low temperatures (e.g., −20°C). These differences in the absorption coefficients translate directly into errors in the retrieved LWP, potentially as large as 50% or more depending on the frequencies and the temperature of the cloud (Kneifel et al. 2014, hereafter Kneifel14).

*ρ*

_{L}is the density of the liquid water,

*c*is the speed of light in a vacuum,

*ε*=

*ε*′ +

*iε*″ is the complex dielectric permittivity of the liquid water, and Im(

*x*) is the imaginary part of the complex number

*x*. Most liquid water absorption models in the microwave spectral region actually predict the complex permittivity

*ε*using various formulations of the Debye model (Debye 1929); empirically determined coefficients are used with the model to interpolate laboratory observations that were made at a small number of frequencies and temperatures to other frequencies and temperatures. Ellison (2007, hereafter Ellison07) and Kneifel14 provide detailed discussions of the differences among different absorption models used by the community. However, the key point is that all of these models use empirically determined coefficients to fit their assumed model to the data and to interpolate to the desired temperature and frequency.

Here, we have refit the empirical coefficients of one of the more recent liquid water absorption models using both the laboratory data used in the original fitting together with new field data that include observations of liquid water down to temperatures below −30°C. The datasets used in the analysis and the model we have elected to refit will be discussed, and the methodology used to determine the new model coefficients will be described. Our fitting method provides a quantitative estimate of the number of independent pieces of information in the data, as well as a full error covariance matrix that will be used to provide uncertainty estimates in the absorption coefficients that are derived from the new absorption model.

## 2. Datasets

### a. Laboratory data

Laboratory measurements of the absorption properties of liquid water in the microwave portion of the spectrum have been made for over a century, with some of the first quantitative measurements being made in the late 1880s (e.g., Tereschin 1889). A detailed historical review of the laboratory measurements of the dielectric permittivity of liquid water (ε) is given in Ellison et al. (1996, hereafter Ellison96). Ellison and his coauthors did a tremendous service for the community by carefully scrutinizing over 60 published laboratory datasets, evaluating the methods used and presenting the results in order to attempt to assign an uncertainty estimate for each, and then compiling all of this information into an easy-to-use table.

Many of these laboratory studies focused on either a single frequency and made observations over two or more temperatures or made observations at several frequencies at a single temperature. As our purpose is to improve the accuracy of liquid water dielectric models for use in quantifying absorption in the earth’s atmosphere, we chose only laboratory data collected at temperatures below 50°C. Furthermore, since the focus is on improving liquid water path retrievals from microwave radiometers, the laboratory data were also restricted to frequencies smaller than 500 GHz (the channels of virtually all microwave radiometers that are used for remote sensing are below this frequency). The full references and description of these data can be found in Ellison96.

There are 1130 points in our selected laboratory dataset. The frequencies in this selected dataset range from 0.465 to 480 GHz, with the median frequency at 9.4 GHz and the 25th and 75th percentiles of 4.1 and 23.8 GHz, respectively. The temperatures in this dataset range from −18.2° to 50.0°C, with a median temperature of 25.0°C and the 25th and 75th percentile values of 15.0° and 30.0°C, respectively. Note that there are only six laboratory observations at temperatures below −5°C; these observations were made by Bertolini et al. (1982) at a single frequency (9.61 GHz). Most of these measurements were made at temperatures above 0°C owing to the difficulty of keeping supercooled liquid water from freezing in the laboratory.

The uncertainties in the above-mentioned data, as estimated by Ellison96, range from as low as 0.15% to over 150%. We felt that the extremely small relative uncertainties were unrealistic, and to evaluate this, we applied our fitting methodology (explained in section 4) to these laboratory data to refit the Ellison07 model. We were only able to produce a fit similar to the Ellison07 model if we assumed that the uncertainty in the laboratory measurements was 5% or the value specified by Ellison96, whichever was larger. Thus, these are the uncertainty values that we applied to the laboratory data in the fitting of our new model (described below).

### b. Field data

The field observations used in this study, which were compiled by Kneifel14, were collected at three sites: a low-altitude (511 m MSL) mountainous site in southwest Germany (Wulfmeyer et al. 2008), a site in the German Alps at 2650 m MSL (Löhnert et al. 2011), and a very cold and dry site in the center of Greenland (3250 m MSL; Shupe et al. 2013). Observations at 31.4, 52.28, 90, and 150 GHz were collected at all three sites, and observations at 225 GHz were also collected at the Greenland site. All of the radiometers used in this study were built by the same manufacturer and used the same technique to stabilize the temperature of the radiometric components of the microwave radiometers to ensure stable operations (Rose et al. 2005). The radiometers that collected these observations were calibrated using both tip-curve calibrations (e.g., Han and Westwater 2000) and liquid nitrogen as a cold reference target (e.g., Maschwitz et al. 2013). Additional details of the sites and microwave radiometers can be found in Kneifel14.

_{x}and ν

_{y}simplifies toNote that there is no LWP dependence in Eq. (3). Kneifel14 demonstrated that the opacity ratios for different frequency combinations were very consistent from site to site, and that the observations from all three sites spanned the supercooled temperature range from 0° to −32°C. Kneifel14 combined ceilometer and cloud radar observations to determine cloud boundaries. Radiosonde data (from the low-altitude and Greenland sites) or numerical weather prediction analysis data (from the site in the Alps) were used together with the cloud boundaries to derive the temperature of the cloud (

*T*

_{cld}) and to estimate its uncertainty; additional details are described in Kneifel14.

To use the field data in this study, the opacity ratios of pairs of frequencies needed to be converted into a dataset of liquid water absorption coefficients

To account for uncertainty in the Stogryn95 model, we assumed that the error is 10% at 0°C and increases to 30% at −30°C, based on the previously mentioned studies. These errors were combined with the observational uncertainties in

## 3. Model background

The dielectric permittivity of a material describes how that substance interacts with an electromagnetic field. It depends strongly on the molecular structure of the matter itself and is often a function of wavelength, temperature, and pressure. For pure liquid water, the effect of variations in pressure on the dielectic permittivity is relatively small, and thus virtually all liquid water absorption models are concerned with predicting permittivity as a function of frequency and temperature, that is, *ε*(ν, *T*).

A wide variety of different interpolation functions has been fit to laboratory observations and used to predict ε. Many absorption models, such as Ellison07, Stogryn95, and Liebe et al. (1991, hereafter Liebe91), used a “double Debye” model to fit the observed laboratory data for frequencies less than 500 GHz. The primary challenge for these previous models is the lack of liquid water absorption observations for supercooled temperatures, and thus all of these models extrapolate to temperatures lower than 0°C. However, the recent model of Rosenkranz (2015, hereafter Rosenkranz15), which uses the Kneifel14 data as part of its development, uses a functional form that is different from the double-Debye model employed by others.

*T*is the temperature of the liquid (°C) and the coefficients

*s*

_{0},

*s*

_{1},

*s*

_{2}, and

*s*

_{3}were determined to be 8.7914 × 10

^{1}, −4.0440 × 10

^{−1}°C

^{−1}, 9.5873 × 10

^{−4}°C

^{−2}, and −1.3280 × 10

^{−6}°C

^{−3}, respectively. Meissner and Wentz (2004) demonstrated that different models of

*a*

_{1},

*b*

_{1},

*c*

_{1}, and

*d*

_{1}for the first Debye term;

*a*

_{2},

*b*

_{2},

*c*

_{2}, and

*d*

_{2}for the second Debye term; and the “critical” temperature

*t*

_{c}, which should be close to the glass transition temperature of pure water (Ellison07). In these calculations, the frequency ν is in hertz.

After the nine empirical coefficients are determined for the desired temperature and frequency, the liquid water absorption is computed using Eq. (1).

## 4. Fitting methodology

Since all models of liquid water absorption in the microwave are semiempirical, we wanted to use a technique that would provide both the uncertainties in the derived absorption coefficients and the information content used in the fitting procedure for each of the empirical coefficients. We elected to use the so-called optimal estimation framework (Rodgers 2000), which is an iterative Gauss–Newton retrieval technique, to derive the empirical coefficients (represented by the vector **X**). The objective of this study was to derive the optimal value of **X**, as well as the uncertainty in **X** as specified by the posterior covariance matrix _{X}.

A Bayesian approach like the optimal estimation framework requires prior information (**X**_{a}), which is used to constrain the solution and serve as the initial first guess. Here, the coefficients determined by Ellison07 were used as **X**_{a} (Table 1). To specify the error covariance of the prior (_{a}), we assume that there was a 25% uncertainty in each of these coefficients and that the errors are uncorrelated; this results in a diagonal matrix (note that assuming 100% uncertainty in _{a} did not change the results given below significantly, as will be discussed in section 5).

The coefficients for the double-Debye model of Ellison07 and TKC, the 1σ uncertainty in the TKC coefficients, the difference in the coefficients between the two models (%), and the DFS for each coefficient from the TKC fitting process. The total DFS was 7.0 out of a total possible of 9.

The forward model (*F*) is Ellison07’s formulation of the double-Debye model (section 3). The observational vector (**Y**) is constructed to contain both the field and laboratory observations (section 2; Turner 2015), and the uncertainties are assumed to be uncorrelated and are used to specify the observational covariance matrix _{Y}.

**X**

^{n+1}using the formulationwhere

**K**

_{n}is the Jacobian of

*F*computed at

**X**

^{n}, and T and superscript −1 denote the matrix operations transpose and inverse, respectively. The forward model

*F*is a combination of Eqs. (4), (5), and (1) (depending on whether the observational element in

**Y**is one of the laboratory measurements of permittivity or the field measurement of absorption). The error covariance of the solution

*N*= 9 (i.e., the dimension of the vector

**X**). In this study, the threshold for convergence is 1.8 (i.e., 5 times smaller than

*N*).

This framework required three iterations before the convergence criteria given in Eq. (13) was met. The coefficients for this new Turner–Kneifel–Cadeddu (TKC) model, along with the 1σ uncertainties in the TKC coefficients that come directly from the matrix

_{a}). If the posterior covariance

_{a}is close to zero, then the DFS will approach 0 and if

_{a}is “large,” then the DFS is maximized (although in this latter case, the retrieval may not converge because the prior would not provide any constraint).

## 5. Results

The first step in evaluating the new TKC model is to examine the DFS of the retrieval. The derived DFS for each of the nine TKC coefficients is provided in Fig. 1c and Table 1; the total DFS is 7.0, where the total possible is 9. Increasing the assumed uncertainty in the prior covariance from 25% to 100% increases the DFS by 8%, but it does not change the quality of fit to the data substantially. This analysis demonstrates that the new observations have high information content on seven of the nine coefficients, and thus the resulting uncertainties are small; this suggests that the large 44% and 18% changes in the *a*_{2} and *d*_{2} coefficients between the Ellison07 and TKC models (Table 1) are well supported by the data. However, for two of the coefficients (*b*_{2} and *c*_{2}), the observations provide very little information, and thus they are constrained to remain close to the assumed prior (i.e., the Ellison07 coefficients). We will investigate the impact of the uncertainties in the TKC coefficients (i.e., its posterior covariance matrix) in the next section.

The next step in evaluating the new TKC model is to see how it compares to other popular absorption models at different frequencies and temperatures. Figure 2 shows how this model compares against the Ellison07, Stogryn95, Liebe et al. (1993, henceforth Liebe93), and Rosenkranz15 models. The relative difference in the absorption coefficients from the TKC model depends strongly on frequency, and in general increases dramatically for temperatures below −10°C with the largest differences at temperatures below −30°C. The relative differences between the TKC and Ellison07 models are generally small (compared to the Stogryn95, Liebe93, and Rosenkranz15 models), although this might be expected since the TKC model (i) uses the same mathematical formulation as Ellison07 and (ii) the empirical coefficients were constrained to not differ from the Ellison07 by more than 25% (as a 1σ uncertainty). However, there are still some important differences between TKC and Ellison07 (Fig. 2a), such as the 30% difference in

How well does the new model agree with the observations used to fit the coefficients, relative to the Stogryn95, Ellison07, and Rosenkranz15 models? Table 2 shows the RMS difference between the laboratory-observed permittivity and the permittivity from the TKC, Rosenkranz15, Ellison07, and Stogryn95 models at four different temperatures. For the lowest temperature bin (0°C), the RMS differences in the permittivity coefficients between the TKC and Rosekranz15 models and observations are virtually identical, and are much lower than the differences between the observations and the other two models. For the higher temperatures (10°, 20°, and 40°C), the RMS differences in the real and imaginary parts of the permittivity are mixed; there is no clear indicator of which model ultimately fits the laboratory data better as not a single model has the lowest RMS difference in both ε′ and ε″ simultaneously for any of the three temperature bins. This is shown graphically in Fig. 3, where all four models provide qualitatively the same fit (i.e., are overlapping) to the laboratory data at both 0° and 40°C, although Table 2 does demonstrate that the RMS difference between the observations and models are different.

The RMS difference between the observed and computed permittivity coefficients in different temperature bins (each bin is ±1°C centered upon the indicated temperature). The analysis included all laboratory data between 0.1 and 500 GHz and less than 50°C. Cells with the lowest RMS difference value for each coefficient and temperature range are in bold.

The only laboratory dataset that spans a significant range of supercooled temperatures is the one collected by Bertolini et al. (1982). This dataset, which consists of 15 measurements at 9.61 GHz, has observations from −18.2° to 32.4°C. Figure 4 shows the modeled and measured real and imaginary parts of the permittivity. Again, the four models are quite similar with the Ellison07, Rosenkranz15, and TKC models almost identical, but at the coldest temperatures the Stogryn95 model underpredicts the value of ε″. However, the RMS differences between the observations and these models (Table 3) show that the TKC model fits the observations the best in both components, with RMS values markedly less for the TKC model than the Stogryn95, Ellison07, or Rosenkranz15 models.

The RMS difference between the observed and computed permittivity coefficients at 9.61 GHz, using the laboratory observations of Bertolini et al. (1982). There were 15 observations in this dataset that ranged from −18.2° to 32.4°C. Cells with the lowest RMS difference value for each coefficient are in bold.

The comparison of the models relative to the Kneifel14 field observations is shown in Fig. 5. At these supercooled temperatures, the TKC model clearly fits the observations the best over the entire temperature range and at all four frequencies. The other models demonstrate prominent errors, such as grossly overestimating the strength of

An advantage of using an optimal estimation framework to derive the TKC model coefficients is that a full error characterization is provided via the posterior covariance matrix

## 6. Impact on a LWP retrieval

To evaluate the impact of the new TKC absorption model on a retrieval algorithm, we analyzed all of the microwave radiometer observations at Summit, Greenland, in 2011 using two different liquid water absorption models. These observations were collected as part of the Integrated Characterization of Energy, Clouds, Atmospheric State, and Precipitation over Summit (ICECAPS) campaign (Shupe et al. 2013).

The original LWP retrievals performed on the ICECAPS dataset used the same absorption model used by the U.S. Department of Energy’s Atmospheric Radiation Measurement (ARM) Program (Ackerman and Stokes 2003), namely, the absorption model by Liebe91. This model is frequently used in the atmospheric science community. The Liebe91 model has similar liquid water absorption at supercooled temperatures to the Liebe93 model at lower frequencies (i.e., below 30 GHz), but it predicts significantly more absorption than Liebe93 at higher frequencies (dashed vs solid purple lines in Fig. 5). An optimal estimation framework was used to retrieve precipitable water vapor (PWV) and LWP from the MWR observations using the Liebe91 model (Turner et al. 2007; Cadeddu et al. 2013), although only observations at 23.8 and 90.0 GHz were used as input to the retrieval. The forward radiative transfer model, which includes the gaseous absorption models, is the monochromatic radiative transfer model (MonoRTM; Clough et al. 2005).

The retrieved PWV and LWP using the Liebe91 model were used as input into the same radiative transfer model to compute the downwelling brightness temperature (*T*_{b}) observed at the other microwave frequencies. The observed minus computed *T*_{b} results for 23.8, 31.4, 90, and 150 GHz are shown in Figs. 6a–d, respectively. These results were also aggregated into 5°C bins as a function of LWP (0–25, 25–50, 50–100, and 100–200 g m^{−2}), shown as colored symbols in Fig. 6. The LWP bins demonstrate that the bias in the computed *T*_{b} becomes progressively larger as the LWP increases, and that the bias is worse at colder temperatures (i.e., −20°C) than at warmer temperatures. Furthermore, the *T*_{b} bias changes sign as the frequency increases, from a positive *T*_{b} bias (the radiative transfer model having too little liquid water absorption in the calculation and hence underestimating the downwelling *T*_{b}) at 23.8 GHz to a negative *T*_{b} bias at 150 GHz.

However, when the TKC model is used in both the retrieval and forward radiative transfer model calculations, the results are much improved (Fig. 7). There is still a slight dependence of the *T*_{b} bias on the magnitude of the LWP at 23.8 and 31.4 GHz, but there is little-to-no dependence of the *T*_{b} bias on LWP at the higher frequencies. The TKC model somewhat overestimates the liquid water absorption at 23.8 and 31.4 GHz, leading to a negative *T*_{b} bias at these frequencies. Further, it slightly underestimates the liquid water absorption at 150 GHz, leading to a very small positive *T*_{b} bias. Nonetheless, the TKC absorption model provides appreciably better results both as a function of temperature and frequency than the Liebe91 used by the ARM program and many other groups.

## 7. Conclusions

A new liquid water absorption model has been developed for use in the microwave region of the spectrum (defined here as frequencies between 0.5 and 500 GHz). The TKC model uses the same double-Debye framework as suggested by Ellison07. The coefficients of the TKC model were derived using an optimal estimation framework using both laboratory and field observations. This retrieval framework allowed the information content that existed in the observations for each of the nine model coefficients to be determined, and to provide error bars on the predicted

The retrieval suggests that the observations had little information content for two of the nine empirically derived model coefficients, and thus the uncertainty in these two parameters defaulted to the assumed 25% level. However, the impact of the uncertainty in these two coefficients is small, as the uncertainty in *b*_{2} and *c*_{2} coefficients) were assumed. Furthermore, the TKC model provides the best spectral consistency with observations over the entire range of temperatures (from −35° to 50°C) than any of the other popular models currently used by most atmospheric scientists (e.g., Ellison07, Stogryn95, Liebe93, and Liebe91), and slightly better spectral consistency than a recently published model that used the same input dataset (i.e., Rosenkranz15). Thus, the TKC model provides an improved treatment of cloud liquid water absorption, especially for supercooled temperatures, and is suitable for use in other retrieval algorithms.

The new model was evaluated with the laboratory and field observations used to derive the model, and thus the agreement between the observations and TKC model calculations shown in Figs. 3–5 are really a consistency check and not a validation. New, independent observations are needed, especially at higher frequencies (above 225 GHz) and temperatures below 0°C, to evaluate the adequacy of the TKC model, especially since it differs substantially from the Rosenkranz15 model at those temperatures/frequencies.

There are potentially several impacts that result from this new model. First, by applying an optimal estimation framework in the fitting of the model coefficients, the uncertainty in the modeled absorption can be easily derived, which is useful for both sensitivity studies and data assimilation systems. Second, radiometric observations at frequencies above 150 GHz are being used to derive ice water path information (e.g., Evans et al. 2005), but separating the signal from the emission of supercooled liquid water from the ice scattering requires an accurate estimate of the liquid water absorption. This is particularly true as observations from these higher frequencies are being assimilated into numerical models in an attempt to better specify the ice water path in the simulations (Geer and Baordo 2014). Third, the signal from atmospheric radars can be significantly attenuated by liquid water absorption, and thus improvements in the knowledge of the absorption can reduce the errors in the attenuation correction. The changes in the liquid water absorption between the TKC model and the other absorption models at common radar wavelengths are shown in Table 4. Last, the improved liquid water absorption models yield more accurate retrievals of LWP from spaceborne microwave sensors by removing one source of systematic error.

The difference in liquid water absorption (%) between various models and the TKC model at supercooled temperatures and common radar wavelengths.

## Acknowledgments

This work was supported by the U.S. Department of Energy’s Atmospheric System Research (ASR) program by Grant DE-SC0008830 and by NOAA. The contributions to this study by S. Kneifel were supported by a postdoctoral fellowship from the German Academic Exchange Service (DAAD) and also partly funded by ASR. Effort at Argonne National Laboratory is supported by the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research, Atmospheric Radiation Measurement Infrastructure Basic Energy Sciences, under Contract DE-AC02-06CH11357. We thank Dušan Zrnić for his helpful comments on a draft of this manuscript. The excellent comments from three anonymous reviewers were also appreciated.

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