Recent technological advancements have enabled the development of G-band (110–300 GHz) radars with high enough sensitivities for atmospheric remote sensing. There have been a number of successful ground-based and airborne demonstrators such as VIPR in USA , GRaCE in UK , CloudCube in USA , and more recently GRaWAC in Germany . As hypothesised by , the demonstrator instruments have shown that measurements in the G-band contain valuable information for a variety of atmospheric scenarios. In terms of humidity and liquid clouds, G-band data has been used successfully to profile water vapour and, when combined with other frequencies, to retrieve small amounts of liquid water in warm shallow clouds and improve microphysical retrievals in light rain . In the ice phase, and highlight that using a dual-frequency pair inclusive of G-band data allows sizing of smaller ice particles than has been possible using lower frequencies such as Ka and W bands.

Although there is currently no G-band cloud radar in space, with the highest frequency space-borne radars operating in the W-band (on-board CloudSat and the Doppler radar on EarthCARE ), the aforementioned findings suggest that a spaceborne G-band radar could be on the horizon. Indeed, GRaCE and CloudCube are intended as demonstrators for future satellite missions. Ideally such missions will exploit the benefits of multi-frequency observations; however it is also important to explore simpler methods, which may be accessible using a single frequency, or provide a robust first estimate of the microphysical state of the cloud before refinement using dual-frequency ratios (which typically have higher uncertainty). The success of EarthCARE's Doppler capability also motivates us to consider the information content of Doppler measurements, in addition to reflectivity.

Algorithms that currently exist to retrieve ice water content (IWC) and snowfall rate (S) from radar measurements can be of varying complexity, but the simplest methods are direct statistical relationships between reflectivity and ice population properties, i.e. Z–IWC or Z–S relationships. These relationships may be used directly to retrieve particle properties, or may form the initial estimate for an optimal estimation retrieval (this is the case, for example, in the EarthCARE cloud and precipitation microphysics (C-CLD) algorithm , where the estimate is subsequently refined using the Doppler velocity information).

However, it is well documented that the relationship between Z and microphysical parameters of interest is not unique, and varies considerably (e.g. ). show IWC as a function of Z at W-band, computed from a large number of in-situ particle size distributions. The IWC for a given reflectivity is spread over approximately an order of magnitude across the various cloud samples, with typical uncertainties on the order of a factor of two, and this variation is driven by variations in the breadth and shape of the particle size distribution (PSD), i.e., how wide or narrow the range of particle sizes is, and how the particle sizes are distributed around the mean. Retrieving S from reflectivity presents similar problems, with order of magnitude variability in the value of S for a given Z ( and references therein). Because Z and IWC (or Z and S) are proportional to different moments of the PSD, their interrelationship is sensitive to what that PSD shape and width is. Interestingly, the variability in the IWC–Z relationship decreases as Z increases, due to increased non-Rayleigh scattering and it is this phenomenon that we will exploit in the current study.

In this manuscript, we explore the usefulness of radar reflectivity and Doppler velocity in the G-band for estimating IWC and S, with a view towards a future spaceborne G-band radar. We consider theory and simulations, along with measurements from a ground-based G-band radar, to determine what information is required for accurate retrievals. We show that IWC and S can be retrieved with a single frequency G-band radar provided that the mass of a wavelength-sized particle is known or can be assumed, while the details of the PSD breadth and shape are not required. Two case studies are provided to illustrate the practical application of the theory, and to demonstrate that the retrievals are consistent with in-situ measurements of IWC and S. This work presents the first known retrievals of ice cloud and snowfall properties using a G-band radar, representing a major step forward in the use of high-frequency radar for atmospheric remote sensing. Unlike traditional optimal estimation approaches, which are computationally intensive but widely adopted, the method introduced here is both computationally efficient and robust, offering a practical alternative without compromising reliability.

Our first measurement parameter is the equivalent radar reflectivity factor Z: 1Z=1018λ4π5|Kwater|2∫0∞N(D)σr(D)dD,

and we denote the logarithmic equivalent as dBZ = 10log⁡10(Z). N(D) is the particle size distribution (PSD, m⁻⁴), such that there are N(D)dD ice particles with maximum dimension in the interval [D,D+dD], and σr(D) is the radar cross section (m²) of each ice particle of that size. λ is the free space wavelength, 1.5 mm at 200 GHz. The factor 1018 is present to convert the SI units of N, σr and λ to the conventional reflectivity unit of mm⁶ m⁻³. We choose to normalise Z using |Kwater|2=0.93, which is the value for liquid water at cm-wavelengths, following the approach of .

In addition to Z, we can also consider the mean Doppler velocity (MDV), which is the average of the vertical velocity of the particles v(D) (equal to the still-air fall speeds of the particles, plus any vertical air motion), weighted by their radar cross sections and PSDs, i.e. 2MDV=∫0∞N(D)σr(D)v(D)dD∫0∞N(D)σr(D)dD.

Our retrieval parameters are IWC and S, which can be written in terms of the PSD as: 3IWC=∫0∞N(D)m(D)dD4S=∫0∞N(D)m(D)v(D)dD, where m is the particle mass (in kg).

Note that the equations above give IWC with units of kg m⁻³ and S with units of kg m⁻³ m s⁻¹. Thus to express IWC in g m⁻³ and S in mm h⁻¹ (as we do in the following section) requires multiplication by 103 and 3600 respectively.

As we will see shortly, at high frequencies such as G-band, Z becomes almost directly proportional to IWC, while the product (Z×MDV) becomes nearly directly proportional to S. This is in stark contrast to the behaviour at lower frequencies, such as Ka-band, and is a result of the non-Rayleigh scattering which occurs when the particle becomes comparable in scale to the wavelength.

In this section, we perform simulations of IWC and S using Eqs. () and (). To illustrate the concept, we use the Large Plate Aggregate mixture from the ARTS scattering database , which combines pristine plate habits to cover the smaller sizes of the PSD with aggregates of plates to represent the larger sizes. The particles are randomly oriented in 3D, and details on their mass can be found in Sect.  and Table . We assume an exponential PSD: N(D)=N0exp⁡(-ΛD) where the slope parameter Λ controls the breadth of the distribution (and hence the average particle size), while the intercept parameter N0 scales the particle concentrations up and down to allow IWC to vary independently of particle size.

In the previous section, G-band scattering simulations showed that the ratios IWC/Z and S/(Z×MDV) approach a near constant value for values of Dm greater than ≈ 0.5 mm. In this section, we will consider the theory behind those results.

In the Rayleigh regime (i.e. where D≪λ), the radar cross section scales in proportion to the square of the volume of ice within the particle : 6σr=36π3λ-4m2ρice2ϵ-1ϵ+22cns, where m is the particle mass, ρice is the density of solid ice, and ϵ is the complex dielectric constant of solid ice. As a result, in that regime, we have: 7ZRayleigh=CRayleigh∫0∞N(D)cnsm(D)2dD, where 8CRayleigh=101836|Kice|20.93π2ρice2 and |Kice|2 is the dielectric factor for ice, which is ≈0.174 at cm and mm wavelengths . The dimensionless coefficient cns is a function of the shape of the particle, and its orientation relative to the polarisation of the radar, and is of order unity. For spherical particles cns=1. The Rayleigh Gans Approximation (RGA, see and references therein) also assumes that cns≈1, reflecting its assumption that the scatterer is composed of a weak dielectric, and hence that the electric field incident on any part of an ice particle is equal to the applied field from the radar. In reality, the coupling between neighbouring parts of the ice particle enhances the mean electric field , and increases cns to values larger than one. For example, suggest that for aggregates of non-spherical ice crystals, 1<cns<2. In what follows, we will assume that cns is independent of D and can be taken outside the integral in Eq. ().

Evidently ZRayleigh does not scale in proportion to IWC. To convert from one to the other, it is therefore necessary to independently estimate the shape and width of the PSD (and to know the relationship between m and D).

There is extensive literature (e.g. ) that suggests ice particle mass and maximum dimension are connected to one another statistically, via a power law relationship, i.e. m=aDb. Such power law relationships for aggregates and complex crystals are indicative of a geometry that is statistically fractal, with fractal dimension equal to b. For aggregates, the exponent b is typically ≈2, and this leads to ZRayleigh∝∫0∞D4N(D)dD, while IWC∝∫0∞D2N(D)dD. Again, this emphasises that at low frequencies (where scattering is close to the Rayleigh regime), Z is a significantly higher moment of the PSD than the ice water content or snowfall rate.

Equation () is appropriate for scattering by ice particles at centimetre wavelengths; however at millimetre wavelengths non-Rayleigh effects become increasingly important. In this case, the radar cross section has a more complex dependence on particle size. Physically, as the dimensions of the particle become comparable to the wavelength, interference begins to occur between the waves that are scattered from different parts of the particle. We express this non-Rayleigh behaviour via a dimensionless factor f which depends on the size of the particle relative to the wavelength: 9Z=CRayleighcns∫0∞N(D)m(D)2f(D/λ)dD. For fractal aggregates, used RGA to deduce the scaling of f when the particle is large enough compared to the wavelength, and found that it had a power law dependence, which for backscattering is: 10f≈cf4πRgλ-b. Here Rg is the radius of gyration, which as we will show later is proportional to D, and the non-dimensional coefficient cf is of order unity. used the discrete dipole approximation (DDA) to confirm that this power law scaling is still evident even when RGA is not strictly applicable (such as is the case for ice: e.g. ). Inserting this into Eq. (), we obtain: 11Z≈CRayleighcnscf4πcRg-bλb∫0∞N(D)m(D)2D-bdD, where cRg=(Rg/D), which as we will demonstrate later has a typical value of around 0.3. Equation () is valid provided that the particles dominating the reflectivity are sufficiently large relative to the wavelength. Based on our simulations in Figs.  and , this occurs when Dm≳0.5 mm. Inserting the mass-size relationship m=aDb into Eqs. () and (), we obtain our key result, valid in the regime where the particles dominating the radar measurements are comparable to, or larger than the wavelength: 12Z≈κmλ∫0∞N(D)aDbdD=κmλIWC, where the coefficient κ=CRayleighcnscf(4πcRg)-b and mλ=aλb is the mass of a wavelength-sized particle i.e. m(D=λ).

In the same way, we can show that 13(Z×MDV)≈CRayleighcnscf(4πcRg)-bλb∫0∞N(D)m(D)2D-bv(D)dD=κmλS. It also follows that the mean Doppler velocity is equal to the mass-weighted vertical velocity of the snowflakes.

The results in Eqs. () and () explain the near-constant values of IWC/Z and S/(Z×MDV) for large Dm at G-band in Figs. f and f. The key parameters in these relationships are therefore mλ and κ, and we now investigate these factors in depth.

From Eqs. () and (), we expect IWC/Z and S/(Z×MDV) to approach a constant value A=1/(κmλ) if Dm is large enough (note that the numerical values of the two ratios differ due to unit conversions, as discussed below). This is backed up by the simulations which showed that curves of IWC/Z and S/(Z×MDV) are increasingly independent of Dm if particle size is large enough compared to the wavelength, i.e. Dm≳0.5 mm in the G-band. We have seen that κ is composed of the coefficients cns, cf, cRg which we may expect to vary between scattering models as well as the exponent b of the mass-size relationship. The other key parameter is mλ which depends purely on the mass-size parameters a and b along with the (known) radar wavelength. In what follows we estimate the various components of κ, and tabulate a,b,mλ for a number of scattering models, to determine which parameter the results are most sensitive to, i.e. what properties we need to know to perform accurate retrievals.

Table  provides estimates for the mass-size parameters a and b, as well as the coefficients cns, cRg and cf for a number of different shape/scattering models. As well as the Large Plate Aggregate mixture, we consider three other mixtures from the ARTS database, namely the Large Block Aggregate mixture, Large Column Aggregate mixture, and the ICON snow mixture. These are described in more detail in . These different mixtures cover a range of different mass-size relationships, with the block aggregate mixture producing the heaviest particles for a given D, and the column aggregate mixture producing the lightest particles. Note that since we are mainly interested in particles larger than a few hundred micrometres in size, the values of a and b provided in Table  correspond to the aggregates in the mixture (see Table 5.3 in ), rather than the whole mixture of monomers and aggregates. The particles in all four ARTS mixtures are randomly oriented in 3D.

In addition to the ARTS mixtures, we also analysed data for three habits from the database of , which includes both unrimed and rimed aggregates of dendrites. Here we include results for the unrimed aggregates, along with aggregates with an effective liquid water path (ELWP) of 0.1 and 0.2 kg m⁻². The ELWP corresponds to what the liquid water path would be, if riming were 100 % efficient. In other words, larger ELWP corresponds to particles which are more heavily rimed. In contrast to the ARTS database, the data includes a large number of particle realisations scattered across a broad range of sizes. In our analysis, we took the particle properties and scattering data from this large ensemble, and rebinned them into uniform size bins. The values of a and b provided in the table for these models were then estimated by fitting a power law function to the average mass m in each D bin. Another difference in this database compared to the ARTS particles, is that the snowflakes are now preferentially oriented so that their short dimension is in the vertical and longer dimensions are in the horizontal. As we will see later, this has some relevance for the extent of non-Rayleigh scattering at a given D.

By inspection of the simulation data we can estimate A directly from the region of the IWC/Z curve where the values become nearly constant. In Fig. , we present simulations with the 7 different scattering models to show the effect of particle habit. This figure follows the same format as Figs. f and f. Figure a and b show results using the ARTS mixtures, and Fig. c and d use particles from the database. The shading of the lines was chosen based on the value of mλ for each habit (see Table ). The particles with smallest mλ (i.e the unrimed dendritic aggregates) are plotted using the lightest shade of grey, with darker shades representing habits with increasing mλ. It is clear that the different habits show similar qualitative behaviour, but different quantitative behaviour. The ratios are generally higher for lower mass particles, and lower for more massive particles. This suggests that mλ is a key sensitivity for the method presented here. In other words, knowledge of mλ is required in order to determine the magnitude of the asymptotic value. We note that for a given mλ, the ratios are different between the two databases. Since oriented particles have their mass distributed more widely in the horizontal and more narrowly in the vertical compared to particles with random orientation, a particle of a given mass will exhibit less non-Rayleigh scattering if it is oriented. This results in increased cf, thus increasing κ, as seen in Table .

Evidently, the curves are not perfectly flat. The plate and block aggregate mixtures from ARTS appear to approach an asymptotic limit more rapidly, whilst some of the others (such as the column aggregate mixture and the unrimed dendritic aggregates) show slight oscillations of IWC/Z and S/(Z×MDV) with Dm. In the following section, we hypothesise that this is related to the form of the function f, which captures the deviation from Rayleigh scattering, and in particular how closely f of the aggregate models follows the power law scaling assumed in Sect. .

Since, in practice, these curves are not perfectly flat, we choose a representative value at Dm=2 mm (i.e. centrally within the 1–3 mm interval used to estimate cns and cRg, as discussed below) and tabulate this as AIWC in Table .

For completeness, we also estimate the asymptotic value of S/(Z×MDV) which is tabulated as AS. Since our figures have S in units of mm h⁻¹, AS and AIWC are not numerically equal, but should be trivially related to each other by a unit conversion, and indeed we find these two independent estimates are consistent to within 6 %.

To work out the dimensionless non-spherical scattering coefficient cns, we calculated Z at 3 GHz (i.e. in the Rayleigh regime) using an exponential PSD, and compared it to results calculated using Rayleigh spheres of the same mass (i.e. Eq.  with cns=1). Dividing the two quantities gives cns. This was repeated for PSDs with different Dm, and an average of the values obtained in the interval Dm=1–3 mm is shown in Table .

Estimation of cRg=Rg/D is straightforward for the ARTS particles, since we have the full shape data of each particle, and from that we estimated Rg and D directly. Again, we computed a representative average value of this parameter. To do this, we first computed a mass-weighted average of cRg-b across the particle size distribution, and then inverted the result to obtain an overall value of cRg for the complete distribution: 14cRg,m=∫0∞N(D)m(D)cRg-bdDIWC-1/b. Repeating this process for PSDs with Dm=1–3 mm as before, we took a median of the resulting values and this is the figure shown in Table . The rationale for averaging cRg,m-b is that the coefficient appears in this (nonlinear) form in κ. cRg,m-b=∫0∞N(D)m(D)cRg-bdD/IWC. As for cns, we use the median value for Dm in the range 1–3 mm.

Since detailed shape data is not openly available for the particles from the database of , cRg cannot be calculated in the same way for those particles. However, suggest that cRg≈0.287, which is consistent with the values calculated for the ARTS particles, and this is the value we have used in our analysis.

After calculating cns and cRg, the only unknown coefficient is cf. To obtain cf, we can use the asymptotic values of A from our simulations, and divide this by the value of 1/κmλ that would be obtained if cf were equal to one. We used AIWC to determine the value of cf given in Table . This means calculating 1/κmλ using κ and mλ in the table is equivalent to AIWC in the table (allowing for a unit conversion of 103 to convert the SI units of IWC kg m⁻³ to the more convenient units of g m⁻³ used in Figs.  and ).

The values of cns and cRg are quite consistent across the various particle models, with values lying in the ranges 1.16±6 % and 0.31±10 % respectively. There is a higher variation (factor of 3.2) in cf. However the values are fairly consistent for the ARTS habits with values of 1.15–1.58, while the majority of the variation in cf comes from the particles in the rimed database. A larger value of cf=2.49 is found for the unrimed dendritic aggregates, and the largest value of cf=3.71 is found for rimed dendritic aggregates with ELWP = 0.1 kg m⁻². As riming increases to ELWP = 0.2 kg m⁻², cf decreases to 1.92.

The coefficients are used to calculate κ in Table , which is found to be very similar for all four ARTS mixtures considered, varying by only 25 %. However, mλ varies by a factor close to 4 for these mixtures. This implies that in some cases it may be suitable to assume a value for κ, meaning the only thing we need to know is mλ. In other words, we don't need to assume a particular scattering model, the only parameter required to perform a retrieval is the mass of a wavelength-sized particle. There is greater variability in κ for the particles from the rimed database. Considering all seven particle habits used here, κ varies by a factor of 3 (which is likely to be driven by the large variation in cf since the variability is small for the other coefficients). However mλ varies more strongly, by a factor of 6.5. Thus we suggest that mλ is the primary sensitivity in controlling what A is.

As discussed in Sect. , used RGA to demonstrate that for fractal aggregates large enough compared to the wavelength (i.e. 4πRgλ≳1), f is characterised by the power law given in Eq. (). We now test in more detail whether that approximation holds for realistic ice particle models from the scattering databases. The data in both databases were calculated using the numerically exact discrete dipole approximation. Provided the resolution of the discretisation is sufficient, the DDA provides an accurate solution to Maxwell's equations .

The red lines in Fig.  show f computed using Eq. (), for the particle habits outlined in Table . From the equations outlined in Sect. , we can write the non-Rayleigh radar cross section as: 15σr=36π3λ-4m2ρice2ϵ-1ϵ+22cnsf.

Thus we may also calculate f for realistic ice particle models by using σr and m from the databases, and cns from Table . Figure  shows f as a function of 4πRgλ, computed for the various particles in this study. In Fig. a–d, dots correspond to monomers from each ARTS mixture and crosses to the associated aggregates. At small values of 4πRgλ, f≈1 and these particles correspond to the monomer habits in the mixture, along with some of the smaller aggregates. The crossover to power law scaling for f begins at 4πRgλ≈1 for each of the ARTS habits, following the expected behaviour of . Figure e–g summarise the aggregates from the rimed database using statistical envelopes rather than individual symbols. For each logarithmic bin in 4πRgλ, a kernel density estimate of log⁡10f is used to derive smooth quantiles, which are shown as grey shaded bands, with the mean shown by the blue line. These panels reveal similar overall behaviour to the ARTS mixtures, but the crossover to power law scaling occurs at slightly larger values closer to 2. This shows that power law scaling is indeed evident in the scattering data for complex aggregates, rimed and unrimed, in the G-band, validating a cornerstone of our theoretical interpretation. Using the values of cRg in Table , we can see that the point where the crossover to power law scaling begins corresponds to D≈λ/4, at which size scattering from ice at the front and back of the particle would be out of phase in the backward direction. At 200 GHz (λ=1.5 mm) this critical diameter is 0.375 mm.

As outlined in Sect. , the power law scaling of f led to the results in Eqs. () and (), whereby an almost direct proportionality between Z and IWC, and Z×MDV and S was deduced. In other words, the power law scaling of f drives the flattening of the ratios IWC/Z and S/(Z×MDV) in the G-band. Although we determined in the previous section that mλ is the key parameter to determine the magnitude of the asymptotic value, the power-law behaviour of f is the factor which drives the flattening of IWC/Z in the first place. This influence is evident when comparing the data in Figs.  to . In Fig. , the data for the block and plate aggregate mixtures closely match the power law, and the ratios in Fig.  are the flattest for these habits. For example, in the range Dm=1–6 mm, the IWC/Z ratio varies by only 3 % and 11 % for the blocks and plates, while there are larger, more systematic deviations from the idealised power law curve for the large column aggregate mixture and the ICON snow mixture, resulting in larger variations in IWC/Z of 26 % and 27 %. The power law generally overestimates the column aggregate mixture data at larger values of 4πRgλ. For the ICON snow mixture, the power law underestimates the data at 4πRgλ=1.5, overestimates it at 4πRgλ=4.5, and underestimates it again around 4πRgλ=10. Because λ is fixed, and Rg∝D, these under/overestimations feed through to the scattering properties of different particle sizes, and as Dm is varied, these particle sizes are weighted to a greater or lesser extent in the value of Z and MDV, producing weak oscillations around the anticipated constant value of IWC/Z and S/(Z×MDV).

Analysis of in-situ observations has generally led to cloud PSDs being parameterised by either exponential or gamma distributions e.g. and . In this subsection, we consider the effect of PSD shape on our results. So far, we have assumed an exponential distribution shape. To investigate the sensitivity to this assumption, G-band simulations for plate aggregates are repeated using gamma PSDs N(D)=N0Dμexp⁡(-ΛD), where μ is a shape parameter. If μ=0, the gamma PSD reduces to an exponential PSD, while increasing μ decreases the small particle concentration, and produces a less disperse distribution. Negative values of μ are also possible, which give very broad distributions, including large numbers of small particles. Care is required when μ<0 since some moments of the distribution can become divergent – this is not the case for IWC, S or Z in the μ=-1 case shown here. The different coloured lines in Fig.  show results for representative values of μ, ranging from -1 to 5 (based on the analysis in ). Figure a and b show the ratios IWC/Z and S/(Z×MDV) for a range of Dm. Furthermore, Fig. c and d show the percentage difference of the ratios for nonzero μ values relative to our nominal simulations at μ=0 from Figs.  and .

It is clear that in the region of Dm we are interested in, the results are not very sensitive to the PSD shape parameter. Varying μ between -1 and 2 causes differences of less than 9 % for IWC, and less than 6 % for S, relative to using an exponential distribution. The differences are higher, with more oscillatory behaviour, for the largest value of μ (5), but remain below 12 %. This weak oscillatory variation is likely because as μ increases, the PSD becomes increasingly narrow, and dominated by a more restricted range of particle size. The scattering model comprises a single particle per size bin, and for this situation these will be aggregates of varying (random) configurations. The scattering properties fluctuate with size as a result of the random realisations, and the integrated values at high μ are more sensitive to these fluctuations, because of the narrower weighting in D space. In nature, these fluctuations may not be evident, because natural clouds probed by radar consist of an enormous ensemble of particles at all sizes. We therefore suspect this sensitivity to μ is an upper limit to the true sensitivity to PSD shape.

At very small Dm (below 0.3 mm) there is stronger sensitivity to the PSD shape, as the scattering becomes increasingly influenced by smaller particles in the Rayleigh regime. However, since our target in this work is Dm>0.5 mm, this is not a concern.

To illustrate the practicalities of retrieving IWC and S using Eqs. () and (), we now present measurements of two ice-phase clouds which passed over the Chilbolton Observatory in the UK, using the GRaCE 200 GHz cloud radar. This instrument has a 0.1° vertically-pointing beam, transmits around 80 mW output power in a pulse of variable duration, and records full Doppler spectra profiles, which can then be analysed to estimate Z and MDV. Further details can be found in . The aim of this section is to (a) demonstrate how retrievals can be made in practice, and (b) to provide some in-situ evidence (from a precipitation gauge in Case 1, and aircraft measurements in Case 2) that the S and IWC retrievals produced are realistic.

In this study, we demonstrate, for the first time, the near-linear relationship between G-band radar reflectivity and ice water content, leveraging the strong non-Rayleigh scattering characteristics unique to these frequencies. This simplifies retrieval methods significantly compared to lower-frequency radars. We explore the strong non-Rayleigh scattering produced by ice particles in the G-band, and present theory and simulations to show that measurements of Z and Z×MDV are almost directly proportional to IWC and S, respectively. This is in stark contrast to the behaviour that occurs at lower frequencies, and presents the opportunity for simple but accurate retrievals of cloud ice quantities. In principle, these retrievals could take place using G-band measurements alone, provided the profiles can be corrected for attenuation by water vapour (the dominant component), liquid water (if present) and ice. Alternatively the approach here can be incorporated into a more sophisticated multi-frequency retrieval. One approach would be to use the simple Z–IWC relationship as a robust first estimate of the water content profile, which is subsequently refined using the dual frequency ratio data.

We provide theoretical background and analysis to show that the near constant values of IWC/Z and S/(Z×MDV) are expected for fractal aggregates that are large compared to the wavelength. The behaviour is driven by the power law scaling of f, the dimensionless factor used to express non-Rayleigh scattering behaviour, and is consistent with the analysis of . Since this latter behaviour was originally derived using RGA (which, as and others have shown, does not capture the full physics of the scattering process), and since radar measurements of ice particles were not the intended application of Sorensen's study, it is necessary to turn to DDA scattering data on realistic ice particles to evaluate whether the anticipated scaling holds for real snowflakes. The simulations show that the power law scaling is indeed evident, and the expected linear relationship between IWC and Z is obtained. Some small fluctuations around this linear relationship are present (which are more evident when visualising their ratio as a function of Dm in Figs.  and ). Digging into the details on the non-Rayleigh function f indicates that this fluctuation is driven by deviations from the overall power law scaling of f. We speculate that these deviations are driven by variability in the particle geometry (e.g. different random realisations of aggregates) for different particle sizes. The direct proportionality of IWC and Z is most accurate when f of the assumed particle model closely follows this assumed power law scaling.

The expected magnitude, A, of the asymptotic value of IWC/Z is dependent on κ and mλ, but crucially is not sensitive to PSD parameters, and we show that the dominant factor controlling the magnitude of A is mλ, i.e. the mass of a wavelength-sized ice particle. If mλ is known (i.e. we can place a constraint on the mass-size relationship), then we expect retrievals to have uncertainties within ≈±30 % for IWC, and within ±15 % for S. These uncertainties become even narrower for Dm≳0.8 mm. The variability of κ between different particle scattering models is significantly smaller than the variability of mλ, varying by only 25 % for the four unrimed particle mixtures from the ARTS database. In contrast, mλ varies by a factor of almost 4 between the ARTS models. This suggests that it may be acceptable to fix the value of κ, leaving mλ as the only free parameter to select in the retrieval method. This in turn would mean there is no requirement to base the retrieval on the existing scattering database particle models - instead it is sufficient to know mλ from an appropriate mass-size relationship. We note that the unrimed and rimed dendritic aggregates from the database of have larger values of κ, which we propose is caused by the increase in cf resulting from particle orientation. However, κ for all seven particle models considered here varies by a factor of 3, which is still considerably less than the factor of 6.5 variability in mλ. In the future, we plan to examine the behaviour of different particle models and test the overall variability in κ and other parameters. Such an analysis would benefit from the availability of additional scattering data in the G-band.

The method is applicable in the regime where particles dominating the radar measurements are comparable to or larger than a quarter of a wavelength, and our data indicates this is satisfied for clouds where Dm⪆0.5 mm. Particles in this size range are frequent in low and mid-level ice clouds where 0<T<-20 °C . In high level cirrus clouds, the ice particles may be smaller than this, and the errors incurred in applying this method will be larger. The decline in accuracy for small particles is gradual and monotonic: at Dm=0.35 mm, IWC/Z and S/(Z×MDV) are typically a factor of 2 larger than the values in Table . At Dm=0.2 mm, IWC/Z is typically a factor of 6 larger than the values in Table , while S/(Z×MDV) is typically a factor of 4 larger. These deviations lead to underestimates of IWC and S. In addition, we have shown in Fig.  that there is increased sensitivity to the shape of the PSD at low Dm.

We applied the theory to two case studies, showing that the method gives feasible results. Firstly, we explored a case study from 7 March 2023 and compared values of retrieved S at low altitude to the precipitation rate measured at the ground, which showed similar results and a well correlated time series. Secondly, we looked at a case study from 28 February 2024. The availability of in-situ measurements on this day allowed comparison of the retrieved IWC to measurements made using the Nevzorov probe on board the FAAM aircraft. This comparison showed that the measurements generally fall within the distribution of the retrievals. This is promising evidence that a G-band spaceborne radar sampling vertical profiles of the kind that are currently being sampled by EarthCARE would provide valuable observations of the vertical profiles of ice in clouds and snow falling close to the surface. In the latter retrieval, A was calculated using κ=7×1010 mm⁶ kg⁻² and mλ using the mass-size relationship of , thus eliminating the requirement of a specific particle habit from a scattering database. Further in-situ validation experiments involving coincident radar measurements and aircraft-based particle sampling would significantly strengthen confidence in the retrieval method.

We have shown that the ratios IWC/Z and S/(Z×MDV) become flatter with increased frequency as one moves from C or Ka-band to W to G-band, and the value of Dm above which the ratios become almost constant also decreases at higher frequencies. In order for a direct retrieval method to be accurate at even smaller sizes (Dm<0.5 mm), radars operating at frequencies higher than 200 GHz could be considered. This may allow the retrieval of IWC in high level cirrus clouds where the particles are typically smaller . The trade-off is that gaseous attenuation becomes very large in the lower atmosphere at sub-millimetre wavelengths. However, this may be acceptable if the aim is to target the upper parts of the troposphere from space or airborne platforms, since the density of water vapour is much smaller, and therefore the gas attenuation in window regions is much more modest (0.5 dBkm-1 two-way at 340 GHz for an ice saturated atmosphere at -30 °C, 400 hPa). For example, we are currently developing a short-range 340 GHz radar for in-situ characterisation of snow; similar technology, with a larger antenna and transmit power, could be used to profile cirrus clouds from above.