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Evaluation of differential absorption radars in the 183 GHz band for profiling water vapour in ice clouds

Relative humidity (RH) measurements in ice clouds are essential for determining the ice crystals growth processes and rates. A differential absorption radar (DAR) system with several frequency channels within the 183.3 GHz water vapor absorption band is proposed for measuring RH within ice clouds. Here, the performance of a DAR system is evaluated by applying a DAR simulator to A-Train observations in combination with collocated European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis. Observations from the CloudSat W-band radar and from the CALIPSO lidar are converted first 5 into ice microphysical properties and then coupled with ECMWF temperature and relative humidity profiles in order to compute scattering properties at any frequency within the 183.3 GHz band. Self-similar Rayleigh Gans approximation is used to model the ice crystal scattering properties. The radar reflectivities are computed both for a space/air-borne and a ground-based DAR system by using appropriate radar receiver characteristics. Sets of multi-frequency synthetic observation of attenuated reflectivities are then exploited to retrieve profile of water vapour density by fitting the line shape at different levels. 10 days 10 of A-Train observations are used to test the measurement technique performance for different combination of tones when sampling ice clouds globally. Results show that that water vapour densities can be derived at the level that can enable ice process studies (i.e. better than 3%) both from a ground-based system (at the minute temporal scale and with circa 100 m vertical resolution) and from a space-borne system (at 500 m vertical resolution and with circa 5 km integration lengths) with four tones in the upper wing of the absorption line. A ground-based DAR system to be deployed at high latitude/high altitudes is 15 highly recommended to test the findings of this work in the field.


Introduction
Adequate understanding of the cloud and precipitation processes that contribute to Earth's water and energy cycle is required before significant progress can occur in our ability to predict future climate scenarios. This calls for a paradigm shift away from the current observing system that mainly capture snapshots of "states" to the next generation of observing systems that can observe both states and "processes" (Stephens et al., 2018).
Future space-borne cloud and precipitation radars are expected to be at the centre of such a revolution (The Decadal Survey, 2017), thus enhancing the view depicted in the past 20 years by the Tropical Rainfall Measuring Mission (TRMM) Ku-band Precipitation radar (Kummerow et al., 1998), the Global Precipitation Mission (GPM) Dualfrequency (Ku-Ka) Precipitation Radar (Skofronick-Jackson et al., 2016) and the CloudSat W-band Cloud Profiling Radar (Tanelli et al., 2007). While the first Doppler radar is expected to be launched on board the EarthCARE satellite in 2021 (Illingworth et al., 2015), innovative radar concepts have been studied in the past decade, ranging from multi-wavelength radars proposed, e.g. as payloads of the Aerosol/Cloud/Ecosystems (ACE) mission and the Polar Precipitation Measurement (PPM) mission, for microphysical studies, Joe et al., 2010;Durden et al., 2016;Tanelli et al., 2018), to Doppler radars for understanding cloud dynamics Illingworth et al., 2018;Battaglia et al., 2018;Kollias et al., 2018), and to constellations of radars in a CubeSat for advancing convective parameterizations (Peral et al., 2015;Haddad et al., 2017;Sy et al., 2017).
In parallel, radar systems operating at much higher frequencies such as the G band (110-300 GHz) have been proposed to study ice and snow microphysical properties (Hogan and Illingworth, 1999;. Furthermore, there is interest in exploring the possibility of profiling the water vapour in cloudy areas Millán et al., 2016;Roy et al., 2018) by using differential absorption radar (DAR) measurements near the 183.3 GHz water vapour absorption line. Water vapour is one of the most critical atmospheric variables for numerical weather prediction models (Millán et al., 2016) and profiles of humidity in cloudy areas are not adequately measured by current or planned systems, as stated by the World Meteorological Organization (Anderson, 2014;Nehrir et al., 2017). While Lebsock et al. (2015) theoretically investigated the possibility of profiling water vapour within the cloudy boundary layer in the presence of cumulus and stratocumulus clouds and of quantifying integrated column water vapour over ocean surfaces with a DAR system with channels on the upper wing of the 183.3 GHz absorption line, Millán et al. (2016) examined how the DAR technique can be applied to water vapour sounding in clouds at all levels by adopting multiple tones within the whole absorption band (140 to 200 GHz). A serious issue that must be considered is that international frequency allocations currently prohibit space-borne transmission at frequencies between 174.8 and 191.8 GHz due to reservation for passiveonly remote sensing. In contrast, allocations are more flexible for ground-based instruments.
Recently, the DAR technique within the G band has been demonstrated by Cooper et al. (2018): not only ground-based measurements of planetary boundary layer clouds have also been performed but an error model and an inversion algorithm have been developed for retrieving the water vapour profile (Roy et al., 2018). An initial assessment of the performances of such retrievals have been performed for boundary layer clouds.
This work aims to assess the potential of both space-borne and ground-based DAR systems, with a specific focus on water vapour profiling in ice cloud studies. When coupled with that of temperature the knowledge of the water vapour density in ice clouds has three benefits.
1. It allows us to derive the relative humidity (RH) with respect to ice (RH i ) and then to identify regions where depositional growth and sublimation processes are dominant (i.e. when the supersaturation is positive or negative in Fig. 1). Particle growth by deposition is an impor-tant growth process in cold environments, particularly when supercooled liquid water layers provide sufficient water vapour for rapid growth (i.e. in regions above the dashed blue line in Fig. 1a). Figure 1b shows, for instance, the growth rate of an initial 500 µm ice crystal for different RH i conditions, clearly highlighting how the ice crystal growth rate is affected by RH i .
DAR observations could complement polarimetric radar observations like differential reflectivity, which are particularly sensitive to depositional growth in temperature regions that favour growth of asymmetric particle shapes (e.g. Verlinde et al., 2013;Oue et al., 2016).
2. The detection and the description of supersaturation areas in high-level ice clouds could help us understand how ice crystal growth significantly enhances water mass fluxes due to sedimentation. This could have an impact on the dehydration of the air entering the lower stratosphere (Kärcher et al., 2014).
3. It may contribute to identifying ice crystal habits based on knowledge of the dominant growth in the different portions of the clouds, based on thermal and moisture conditions, as suggested by Bailey and Hallett (2009) (dominant habits reported at the top of Fig. 1). This identification may indeed be complicated by the fact that substantial changes in habit can occur due to vertical transport caused by convection or sedimentation with ice crystals experiencing temperature changes of tens of K across their lifetime. Since the shape and internal mass distribution of the ice particles affects their scattering properties, this has an immediate impact onto improving remote sensing retrievals.
The water vapour density for a given relative humidity is a strong function of temperature: for instance, for RH i = 100 % the water vapour density, ρ v , changes by more than 1 order of magnitude (from 4.85 to 0.34 g m −3 ; see x axis in Fig. 1) when moving from 0 to −30 • C. A knowledge of RH i within 5 %-7 % seems appropriate for identifying the relevant regimes in Fig. 1. Since RH i = ρ v /ρ s,i (with ρ s,i being the water vapour density in saturated condition with respect to ice), the uncertainty in RH i is affected by the uncertainty in the numerator and in the denominator. The uncertainty in the denominator is driven by the uncertainty in the temperature: an error of 1 K propagates into an 8 % to 10 % error in ρ s,i , with temperatures ranging from 0 to −30 • C. Uncertainties in current analyses of atmospheric temperatures are strongly region dependent, with large uncertainties over polar, oceanic and developing nations which lack frequent radio soundings (Langland et al., 2008). Temperature uncertainties on the order of less than 1 K or better are expected from reanalysis in the middle and upper troposphere and in regions where radiosonde observations are plentiful. Advances in hyperspectral microwave sounders promise to reduce errors in temperature profiling to 0.5 K (Blackwell et al., 2011; Aires et al.,  figure. (b) Temporal evolution of the diameter of a 500 µm crystal environment with different supersaturation RH i (as indicated in the legend) and at T = 260 K and p = 500 mb. The shading corresponds to a ±3 % perturbation in RH i . The rate of mass change is assumed to be driven by diffusional growth or sublimation (description provided in Field et al., 2008) with the Brown and Francis (1995) mass-size relationship.
2019), and this figure is certainly in reach for ground locations hosting a remote sensing observatory. This highlights that in order to retrieve useful information for ice cloud studies water vapour densities must be retrieved within ∼ 3 %-5 % or better -this in order to account for the previously mentioned additional uncertainty due to temperature -for a range of values between 0.5 and 5 g m −3 . The previous study by Millán et al. (2016) has clearly demonstrated that, when dealing with DAR profiling capabilities, the main roadblock for the use of space-borne DAR measurements in process studies is represented by the precision of the measurements with potential biases being generally much smaller (e.g. see their Fig. 7). The key science question that we aim to answer in this work is therefore whether or not it is possible to beat the noise of the measurement (by averaging and by including more tones) to the level at which water vapour profiling in ice clouds could help in refining our understanding of microphysical processes. Our strategy is therefore to exploit the novel retrieval model proposed in Roy et al. (2018) in assessing the precision of DAR techniques in profiling ice clouds both from a ground and a space-borne perspective. This will allow us to draw some conclusions on the potential of such observations for ice studies.
The paper is structured as follows: first the theory of water vapour retrieval with DAR is shortly revisited (Sect. 2). In Sect. 3 CloudSat observations are used to reconstruct realistic ice microphysics profiles that can be used as input in a forward model for simulating reflectivity profiles at any frequency in the G band. Sect. 4 presents an assessment of the performance of a DAR instrument based on a statistical analysis built on CloudSat climatology.
Conclusions and future work are presented in Sect. 5.

Theory of water vapour retrievals
Here the theory underpinning DAR, thoroughly covered in Lebsock et al. (2015), Millán et al. (2016) and Roy et al. (2018), is briefly revised. The measured reflectivity from a target with effective reflectivity Z e (r, f ) at a given range r is given by the following equation: where τ (0 → r, f ) is the one-way optical depth from the radar to the range r. The exponential term accounts for the radar attenuation due to the gases and the hydrometeors with the factor of 2 in the exponent accounting for the two-way path of the radar wave. Note that multiple-scattering effects Matrosov and Battaglia, 2009) will be neglected hereafter since they are minimized by the small radar footprints (less than 450 m for the space-borne configuration) and the low single-scattering albedo of the medium at frequencies in the vicinity of the absorption line. Multiple scattering could be an issue for tones located far away from the absorption centre when encountering heavily rimed particles and it could potentially be flagged by introducing a cross-polar channel to measure linear depolarization ratio (like proposed in Battaglia et al., 2007). As shown afterwards, such conditions, which certainly corresponds to CloudSat reflectivities exceeding 10 dBZ, are anyhow challenging for the DAR retrieval. Following Roy et al. (2018),

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A. Battaglia and P. Kollias: DAR for water vapour in ice clouds we consider the ratio of measured reflectivities at two ranges r 1 and r 2 = r 1 + r: where the r symbol corresponds to taking the mean value for ranges between r 1 and r 2 so that is the mean extinction coefficient for such ranges. This equation can be further simplified by separating the water vapour components from the other gases and introducing the water vapour absorption coefficient per unit mass, κ v , as follows: where we have assumed in the last step that the line shape κ v (f ) within the r layer can be approximated by its value at the mean temperature and pressure of the layer and we have conjoined the dry air and hydrometeor extinction. If we invert Eq.
(3) we can then write the following: and recombining Eqs. (5) and (4) we finally get the following equation: The DAR rationale is based on the idea that by performing measurements of the left-hand side of Eq. (6) at different frequencies it will be possible to fit the terms on the righthand side. The first term is directly proportional to the water vapour density via the line shape κ v (f ); the last two terms are related to the dry air plus hydrometeor attenuation and the effective reflectivity ratio at the two ranges (thus affected by the vertical variability). They can be assumed to vary weakly with frequency. Extinction of the supercooled droplet is indeed proportional to frequency (e.g. see Lhermitte, 1990) and ice crystals behave similarly, with a linear increase with frequency, as demonstrated in Fig. 2. Note that the ice crystal attenuation is mainly driven by scattering since the singlescattering albedo for all the cases illustrated here exceeds 0.95. Also the ice crystal attenuation is strongly dependent on the ice crystal type (i.e. on the scattering model) as already noticed in , but this dependence can be factored out in the differential method because of its distinctness from the absorption band spectral feature. Therefore, the last two terms are modelled in this study via a dependence which is linear with frequency. Since the line shape κ v (f ) is known at a given T and p then ρ v r can be derived by a least-squares fitting procedure that fits all three terms on the right in Eq. (6) to the measured γ terms. The procedure also allows the computation of errors for the retrieved fitted parameters and of a quality index for the fitting via the normalized χ 2 . Note that the quantities γ (f, r 1 , r 2 ) are not affected by absolute calibration, which makes the whole procedure immune to calibration errors. If only three tones are available (or the full range of tones is less than 10 GHz) then B is assumed to be equal 0 (as done in Roy et al., 2018). When only two tones are available, ρ v and its error can be directly computed from the following equation: as derived in Roy et al. (2018), where Z are the uncertainties in the reflectivity measurements at the given range and frequency. This shows that the maximum achievable absolute precision in water vapour density is fixed by the given spatial resolution (i.e. the value of r), by the difference in the line shape κ v between the two tones and by the precision of the reflectivity measurements. This has the consequence that, with the latter two factors being the same, lower values of absolute humidity will be measured with lower relative precision. Averaging over a longer path or time improves the precision because it increases r and the number of radar pulses (thus improves the precision of Z measurements), respectively. Adopting multiple tones allows us to improve the precision for a range of water vapour densities by finding the right balance between large differences in κ v and good precision in the reflectivity signal, i.e. good signal-to-noise ratio (SNR). When multiple tones are involved, the line-fitting retrieval routine implemented in this work gives an estimate of the measurement precision by generalizing Eq. (7).

Simulation of DAR profiles from CloudSat data
At present, no radar reflectivity measurements at multiple Gband tones are available that can be used to evaluate the performance of the technique. Our approach uses ice microphysical properties derived from space-borne sensors as input to a  (Tanelli et al., 2008). When integrated with the observations from the CALIPSO Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) (Winker et al., 2007) such observations can be used to retrieve ice microphysics. Here, retrievals adopting the DARDAR algorithm (Delanoë and Hogan, 2010, http://www.icare.univ-lille1.fr/projects/dardar/, last access: 20 June 2019) are used as input for the DAR modelling. ECMWF auxiliary data are used as input for temperature, pressure and relative humidity. The DAR forward model uses the millimetre-wave propagation model from Rosenkranz (1999) for gas attenuation, whereas the self-similar Rayleigh-Gans scattering model (Hogan and Westbrook, 2014) is adopted for computing the scattering properties of ice particles. This approach has the clear advantage that scattering properties can be computed at any frequency with practically no computational cost. The ice crystals model proposed by Leinonen and Szyrmer (2015) and labelled as model "A; LWP = 0.1 kg m −2 " is used to derive the parameters for the self-similar model by taking into account the internal structure of the aggregates. Tridon et al. (2019) have shown that the scattering properties generated Table 1. Technical specifications of the DAR space-borne system used in this study. The configuration here adopted is the one proposed in an on-going UK CEOI study (Duncan Robertson, personal communication, 2018 via this methodology generally fit triple frequency radar measurements and in situ measurements well. Noise is injected into the reflectivity measurements according to the following formula (see the appendix in Hogan et al., 2005): where N p is the number of transmitted radar pulses (e.g. in the space-borne configuration ∼ 4200 for an integration length of 5 km), τ s is the time between samples (i.e. the reciprocal of the pulse repetition frequency) and σ v is the spectral width of the Doppler spectrum. For space-borne systems the first term inside the bracket is practically always close to one because the Doppler spectral width is expected to exceed 2 m s −1 due to the large satellite velocity (see Eq. 6 in . The first term inside the square bracket needs to be at least one because the number of independent samples has to be smaller or equal to the number of samples. This implies that the so-called "time to independence" is of the order of 100 µs, thus smaller than the time between pulses (equal to 166.7 µs for a PRF = 6 kHz). The single pulse sensitivity is assumed to be −22 dBZ, a realistic value with current technology (see Table 1). For groundbased systems on the other hand we have assumed a spectral width equal to 1 m s −1 and a single pulse sensitivity of −50 dBZ at 1 km range with 1 s integration (see Table 2). The DAR system shown in Table 1  In addition to high-power sources (EIK), lower-power sources are available either using frequency multipliers Table 2. Specifics of the frequency-modulated continuous wave radar based on W-band power amplifier and GaAs Schottky diode frequency multiplication (Nils et al., 2017) for the ground-based simulation (Peter Huggard, personal communication, 2018

Case study
The methodology is demonstrated for a precipitating system observed by CloudSat over the Southern Ocean between Antarctica and South America on the 2 January 2007 at about 20:16 UTC. The system extends for roughly 1300 km with the temperature at the surface ranging from 281 K at the southern edge of the system to 274 K at the northern edge of the system. The CloudSat 94 GHz reflectivity as derived from the 2B-GEOPROF product (Mace et al., 2007) is shown in Fig. 3a. The zero isotherm clearly demarcates the ice vs. liquid transition. The co-located ECMWF reanalysis for the relative humidity field with respect to ice is depicted in Fig. 3b. In the glaciated region of the precipitating system, the synergy between the CloudSat radar and the CALIPSO lidar (Sassen et al., 2008) offers a unique perspective on the ice microphysics (Battaglia and Delanöe, 2013). The outputs of the DARDAR retrieval (Delanoë and Hogan, 2010) are shown in Fig. 3c and d.
These microphysical outputs are then used with lookup tables generated from scattering models to compute reflectivities at any frequency within the 183.3 GHz absorption line. Examples of two pairs of frequencies (187 and 200 GHz for the space-borne and 186.3 and 200 GHz for the groundbased configurations, respectively) are shown in Fig. 4. It is interesting to note how differently the two frequencies penetrate into the precipitating system, with the 187 GHz (186.3 GHz) severely attenuated by water vapour below 4 km (above 3 km) in the space-borne (ground-based) configuration. On the other hand, the 200 GHz is clearly attenuated in the region below 2 km at latitudes between −60 and −58 • , a combined result of large ice water and water vapour contents.
The profile at latitude −58.07 • (black arrow in Fig. 3a) is used here to demonstrate how to derive a water vapour profile in a three-step procedure (see Fig. 5): 1. an interval r is selected and the profiles of the quantity γ (f j , r) (see Eq. 6) are computed with their corresponding errors (computed from the estimated errors on the measured reflectivities via Eq. 8) at the different DAR frequencies f 1 , f 2 , etc. (continuous blue lines with bars in the small insets of Fig. 5); 2. the spectral dependence of the line shape κ v (f, p r , T r ) is derived at each level (dashed red lines in the small insets of Fig. 5) by using the average temperature and pressure of the layer and the gas absorption model; 3. a least-squares fitting procedure of the form expressed in Eq. (6) that accounts for the errors in γ (f j , r) allows us to retrieve estimates of the three fitting parameters (Â,B and ρ v r ). γ values that are too noisy are excluded from the fitting, e.g. at 2.76 km only four tones are considered for the space-borne configuration.
For the space-borne configuration the retrieval shows that a set of seven-tone DAR with frequencies on the upper wing of the 183.3 GHz band, as listed in the legend of Fig. 5a, can retrieve water vapour within the ice cloud with good precision (i.e. within 15 %) from 7.0 km (240 K) down to 2.5 km (268 K), with water vapour contents changing by more than 1 order of magnitude. The relative error in the retrieval of ρ v for the whole case study shown in Fig. 3 is reproduced in the Fig. 6a. Clearly there are two critical regions: (1) at low temperatures (≈ T < −30 • C) low values of ρ v limit the amplitude of the signal (e.g. compare the red curves between the top three small insets in Fig. 5); (2) at warm temperatures (≈ T > −10 • C) and large CloudSat reflectivities the cumulated attenuation tends to strongly reduce the SNR, and therefore (see Eq. 8) increase the uncertainty of the reflectivity measurements as a result of γ (f j , r). In both situations the retrieval errors become large, but such deterioration can be clearly identified by looking at the SNR of the different DAR channels and at the associated error induced in the estimated value of water vapour, ρ v r .
The same profile has also been used to analyse the performance of a ground-based instrument by assuming that the instrument is located at the −3 • C isothermal line and is looking upward. Again, tones in the upper wing of the absorption band are selected. The simulated reflectivities, shown in the bottom panels of Figs. 4-5, show strong attenuation in the lower troposphere with the tones close to the centre of the  Fig. 5. (b) Water vapour density as derived from ECMWF reanalysis with regions of constant relative humidity with respect to ice depicted as dashed lines. (c, d) Mean mass-weighted diameter of ice particle (c) and ice water content (d) as retrieved by the DARDAR product. line already reaching the noise level just above 2 km. The only tones that can penetrate deep into the clouds are the ones that do not have enough water vapour signal high up in the troposphere (e.g. the highest three tones at 2.68 km; see bottom small insets in Fig. 5). This demonstrates why, while the precision of the retrieval in the lower troposphere is excellent, it deteriorates quickly above 2.5 km. Figure 6b demonstrates the same thing for the whole event: the precision of the retrieval quickly worsens at 2 to 2.5 km above the ground, where temperatures decrease to values lower than −15 • C. On the other hand, by integrating for periods of the order of 1-2 min, ground-based system can achieve extremely accurate results for temperature between 0 and −15 • C.
Compared to the space-borne set-up things are expected to substantially improve when dealing with an airborne configuration (bottom panels). The key advantages are (1) better sensitivity because of closer distance to the target and a (2) slower platform speed that allows us to collect more pulses for the same integration length. As a result the precision and/or the resolution of the retrieval are significantly improved compared to the space-borne configuration (contrast Fig. 6a and c-d). The two bottom panels demonstrate the trade off between long and short pulses for the airborne mode. In the left panel a pulse which is 4 times shorter than that in the right panel is adopted (i.e. 120 vs. 480 m). As a result, the sensitivity is 12 dB (a factor of 16) better in the latter case, which translates to a much better precision of the retrieval. Thus, it is recommended to use pulses that match the required vertical resolution because SNR is a critical parameter for the precision of the measurement. Averaging more gates does not recover the same precision.
This case study highlights that sounding ice clouds by airborne or space-borne DAR systems is clearly advantageous with respect to ground systems because regions with low water vapour contents (thus low attenuation) are encountered first. This implies that tones close to the line centre can stay well above the minimum detection threshold (MDT) in the areas where they provide useful information (i.e. at low water vapour contents). The same is not true for ground-based geometry because, unless the temperature at the ground is very cold, large levels of attenuation are experienced by the radar pulse in the lower troposphere.  Table 1 for the scene shown in Fig. 3. The ground-based system is assumed to be located at the 270 K isotherm line drawn in Fig. 3a. Note the different ranges in the reflectivity colour bars of (a, b) and (c, d) driven by the better sensitivity achieved by the ground-based system at short ranges.

Statistical analysis from CloudSat climatology
The A-Train has provided the first global climatology of ice clouds with a detailed description of ice cloud occurrences, ice microphysics and ice radiative effects (Hong and Liu, 2015). The A-Train ice cloud dataset therefore represents an ideal test bed to investigate the potential of a DAR system for measuring relative humidity inside ice clouds. The methodology described in Sect. 3 has been applied to 10 d of CloudSat data (from 1 to 10 January 2007) to study the performances both of a space-borne and a ground-based DAR system with several channels within the 183.3 GHz absorption band. The ground-based system is assumed to look upward from the height corresponding to the 270 K isothermal level, as identified by the ECMWF reanalysis. For any profile with an ice water path exceeding 20 g m −2 the profile of water vapour is retrieved via the DAR technique and, by comparing such values with the assumed one (from ECMWF reanalysis), the relative error of ρ v is computed. Results are binned according to the CloudSat reflectivity values (above −10 and −25 dBZ for the space-borne and ground-based system, respectively) and the ambient temperatures (above 240 K). Figure 7 shows the fractional occurrence when the DAR systems provide ρ v with precision better than 3 % (i.e. a very valuable information). For the space-borne system there is an optimal region between −5 and 15 dBZ and for temperatures between 250 and 265 K. Results tends to worsen at temperatures close to 273 K and at very high W-band CloudSat reflectivities, which occur typically at higher temperatures (a result of the reduced number of tones with signal significantly above the noise floor) but also at very cold temperatures (a result of the reduced absorption for tones further away from the band centre) and low CloudSat reflectivities (a result of the reduced SNR).
For the ground-based system ( Fig. 7b and d) ρ v is optimally retrieved in the lower troposphere with the quality of the retrieval typically worsening with decreasing temperatures and decreasing reflectivities (due to the reduced SNRs). The only exception is at very large reflectivities, where nonlinearities of the right-hand term in Eq. (6) introduced by Mie and attenuation effects cause larger errors.
We have selected different combinations with 2, 3, 4 and 5 tones and we have analysed which combinations achieve the best retrieval performances. As a first step we have assumed that the sensitivity of the system does not change when in-  creasing the number of tones. This is the case if the duty cycle of the radar system could be increased accordingly and frequency diversity could be implemented. Otherwise the sensitivity of each channel decreases with √ N tones because of the reduction in the number of samples N p , and the effect will be discussed later. Results are summarized in Fig. 8. Clearly increasing the number of tones (all with the same sensitivities) is beneficial but the improvement when surpassing four tones is marginal (e.g. compare the four with the five and six tones). On the other hand, it is obvious that improving the SNR generally produces better results via a reduction of the noise in the reflectivity measurements according to Eq. (8). For instance, for the two-and four-tone curves the impact of the improvement corresponding to a variation by a factor of 2 in sensitivity (±3 dB) is illustrated in Fig. 8 by the shading. As a result there is indeed an improvement in water vapour profiling when using four vs. two channels. In fact there is the obvious advantage that with four tones it is possible to perform the three-parameter fit of Eq. (6), thus avoiding the biases introduced by frequency-dependent hy-drometeor scattering effects. This remains true even when considering DAR configurations with the same duty cycle. In that case, doubling the number of channels corresponds to averaging half the number of samples, which equates to a reduction of 1.5 dB in sensitivity (so roughly half the range currently shown by the shaded area). But the blue-shaded region remains well above the red-shaded region.

Discussion and conclusions
The potential of a multi-frequency differential absorption radar (DAR) system with several tones within the 183.3 GHz water vapour absorption band for profiling water vapour within ice clouds is assessed both for ground-based and space-borne configurations. Realistic ice profiles derived from A-Train observations are inputs of DAR simulations which are used to test the precision performances of water vapour retrievals based on fitting the line shape via a minimum least-squares fitting procedure.
Our findings can be summarized as follows.  . Fraction of retrieval points (y axis) having errors lower than a given threshold (x axis) for the space-borne configuration (a) and the ground-based configuration (b) with 2, 3, 4 and 5 tones. Only the combinations that achieve the best accuracies (as indicated in the legend) are reported. The shaded region indicates results when the sensitivity is increased or decreased by 3 dB. For the space-borne (ground-based) configuration the retrieval is applied only to points corresponding to CloudSat reflectivities exceeding −15 dBZ (−25 dBZ) and temperature exceeding 240 K.
1. With realistic minimum detection thresholds, DARs can provide useful information in thick ice-and mixedphase clouds and they can complement other techniques (e.g. water vapour DIALs, Nehrir et al., 2017). Fourtone DARs seem to be the right balance between complexity (i.e. number of channels) and retrieval performances. In the domain of CloudSat reflectivities above −15 dBZ and T > 240, some of the best four-tone combinations allow us to retrieve ρ v with precision better than 3 % in more than 25 % of the cases when ice is present, with the best results obtained for ice clouds with W-band reflectivities between −5 and 10 dBZ.
2. Ground-based DAR systems can provide excellent profiling of the warmer parts of ice clouds, where ρ v values exceed 1 g m −3 , but they become increasingly less precise when looking at the cold regions with low moisture. In such areas, things are expected to improve when colder ground temperature are considered. In this study we have simulated a scenario with a ground temperature of 270 K with global climatology. Of course the selection of the tones could be optimized for a specific location and time of the year based on the local cloud and temperature climatology. Also, scanning options could be considered to increase the differential absorption signal of channels far away from the centre of the band by increasing the path length.
3. Airborne or space-borne DAR systems are clearly advantageous with respect to ground systems when looking at regions with low water vapour contents because such regions are encountered first by the radar wave, and therefore are affected by less attenuation. This implies that tones close to the band centre can stay well above the MDT in the areas where they provide useful information (i.e. at low water vapour contents). The same is not true for ground-based geometry because, unless the temperature at the ground is very cold, large levels of attenuation are experienced by the radar tones close to the band centre in the lower troposphere.
4. Because SNR is a critical parameter for the precision of the measurement, the selection of the radar resolution should ideally match the vertical resolution required for the water vapour product.
5. The selection of the tones is driven by a tradeoff between differential signal and signal-to-noise ratio. Ideally the attenuation signal should be maximized but if the attenuation is too strong the signal becomes increasingly noisy and ultimately goes below the minimum sensitivity. For ground-based systems it would be ideal to have tones that can be adjusted depending on the atmospheric conditions and latitude and altitude location since with lower ground temperatures channels closer to the 183.3 GHz centre become increasingly useful.
6. The quality of the retrieval can be easily evaluated by considering retrieval errors and χ 2 values that are computed as part of the minimum least-squares fitting procedure.
7. Transmitting licenses are attainable for airborne and ground-based systems (e.g. in the UK, DAR tones within the following bands may be allowed: 173.85 to 182 GHz,185 to 190 GHz,191.8 to 195.75 GHz and 196.15 to 199.99 GHz, with other allowed windows below 173.85) but are currently much more unaccessible for space-borne systems since such bands are reserved to passive microwave radiometers. As a first step toward assessing the potential of the DAR concept for ice cloud studies and to properly evaluate its accuracy (via comparison with radio soundings) it is highly recommended to deploy a ground-based DAR system at high latitudes and high altitudes.
Data availability. This research is based on CloudSat data that are publicly available at http://www.cloudsat.cira.colostate.edu/ data-products (Cloudsat data processing center, 2019).
Author contributions. AB wrote the whole paper and developed the code for simulating the G-band DAR. PK contributed with the definition of the technology specifications and in the revision of the paper.
Competing interests. The authors declare that they have no conflict of interest.