3.1.1 RICO simulations

The simulations are based on data collected during the Rain in Cumulus over the Ocean (RICO) field study. RICO was a comprehensive field study of shallow cumulus convection which was located in the winter trade winds of the northwestern Atlantic ocean, just upwind of the islands of Antigua and Barbuda. An overview of the experiment is provided in Rauber et al. (2007). The focus of RICO was on the statistical character of the cloudy boundary layer, particularly on the characterization of precipitation in shallow cumulus. The cloud-resolving model used in this study is version 6.11.2 of the System for Atmospheric Modeling (SAM) described by Khairoutdinov and Randall (2003). Double-moment (Morrison et al., 2005) microphysics was used with a prescribed concentration of cloud droplets of 100 cm⁻³. The horizontal doubly periodic domain size was 57.6 km × 57.6 km with horizontal grid spacing of 100 m, while the vertical grid had 120 levels with the domain top at 4.8 km and constant 40 m grid spacing.

Two simulations have been performed. The setup of the first simulation closely follows the setup used in the study by vanZanten et al. (2011). The second simulation was for drier conditions, more characteristic of the midlatitudes than the subtropics. There is no straightforward way to scale the soundings and forcing profiles to accomplish that, so a very simple procedure was used. The original sea surface temperature of 299.8 K and the initial temperature of the sounding were reduced by 9.8 K. The water vapor sounding and prescribed large-scale vapor tendencies were scaled by a factor that was obtained to keep the relative humidity at the first model level constant. The other profiles, such as the horizontal wind and subsidence rate were not changed. As the result of the procedure, the freezing level moved from around 4.9 km down to a 3.4 km height. Each simulation was run for 2 d with a time step of 2 s.

Figure 5 shows one scene (out of the 65 considered in this study) illustrating the low-freezing-level simulations with the colors modulated by the total integrated c-LWP and the contour lines showing the r-LWP equal to 0.1 and 0.5 kg m⁻², (black and gray lines). Instantaneous fields of view (IFOVs) of planned and operated spaceborne radars are shown for reference on the right-hand side.

Figure 5Example of a SAM simulation output: cloud liquid water path (log ₁₀(LWP[kg m⁻²]) with contour lines of 10 and 500 g m⁻² rain LWPs in black and white, respectively, at native model resolution. The dashed black line corresponds to the ground track for the overpass shown in Figs. 7–8.

3.1.2 Forward radar simulator

Four steps are followed in order to produce radar observables from the model output.

1. Scattering properties of the medium are computed at the fine model resolution. Gas attenuation is computed according to the model from Rosenkranz (1998). Scattering properties of cloud droplets and raindrops are computed via Mie theory (e.g., see Lhermitte, 1990). An exponential drop size distribution is assumed for rain with $N_{\mathrm{0}\mathrm{r}}=\mathrm{8}\times {\mathrm{10}}^{\mathrm{6}}$ m⁻⁴ (Marshall and Palmer, 1948) whereas a Gamma distribution with μ=3 with an effective radius increasing from 3 to 15 µm from cloud base to cloud top according to Bennartz (2007) is adopted. The impact of the Marshall and Palmer assumption is discussed later.

2. The scattering properties are used as input of the radar forward model developed in Hogan and Battaglia (2008) for computing the attenuated and unattenuated reflectivity profiles (but with the multiple-scattering flag turned off) and of an Eddington approximation code (Kummerow, 1993) for computing T_B.

3. A surface echo is added. The surface normalized backscattering cross section, σ₀, is computed as a function of the 10 m wind speed and the sea surface temperature following Wu (1990). For the computation of T_B, surface emissivities are computed according to ocean Fresnel models.

4. The results produced in steps 2 and 3 are convolved with the radar range weighting functions with a top-hat pulse with a 500 m range resolution (see Lamer et al., 2020) and two-dimensional Gaussian antenna patterns with different instantaneous fields of view. Along-track integration of 500 m is carried out.

5. Random noise with a zero mean and 2 K (0.7 dB) standard deviation is added to the T_B values and the PIAs, while reflectivity noise is computed using the dependence on signal-to-noise ratio (SNR) and averaging pulses described in Hogan et al. (2005). Note that, as shown in Leinonen et al. (2017), the uncertainty in the PIA is dominated by the uncertainty in the clear-sky σ₀ and not by the uncertainty in the surface return (unless for cases close to full attenuation) which, at a high SNR and for CloudSat configuration, amounts to 0.17 dB (Lebsock and Suzuki, 2016). Leinonen et al. (2017) articulate that the uncertainty in the clear-sky σ₀ can be estimated from the standard deviation of the observed clear-sky surface cross sections, with an average value of 0.24 dB for CloudSat so that the total PIA precision is estimated to be 0.29 dB. In reality this is an optimistic assumption because the clear-sky σ₀ is likely more uncertain, especially if there are no clear-sky observations in the vicinity of the cloudy or rainy profile, which occurs e.g., when continuous stratocumulus cloud decks are present. In addition, even if such calibration points are available, winds are likely to change in the presence of precipitation due to local circulation (e.g., this feature is evident in the SAM simulations, not shown). σ₀ has a strong sensitivity to the 10 m wind speed (Fig. 6), whereas the sensitivity to SST is very weak (<0.08 dB K⁻¹, not shown). Therefore with 7–10 m s⁻¹ wind (which are characteristic values in our simulations), an uncertainty of 1 m s⁻¹ in the wind speed induces an uncertainty of roughly 0.4–0.6 dB in σ₀ (with much worse results obtained for lower wind speeds). Therefore we generally think that a PIA precision of 0.7 dB is more appropriate to be assumed in our discussion. Similarly nadir emissivity sensitivities are on the order of 0.2–0.3 % m⁻¹ s at all frequencies here considered. A 1 m s⁻¹ uncertainty propagates into a maximum 0.6–1 K uncertainty in the T_B values.

6. Since the sensitivity improves proportionally to the antenna gain, the single-pulse minimum detection signal (MDS) of each configuration is obtained by scaling the values provided in Table 1 for each frequency according to the scaling of the antenna footprint area. Forward-simulated reflectivities below the MDS are removed.

Figure 6Sensitivity of the normalized radar cross section, σ₀, to 10 m wind speed according to the Wu (1990) model.

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3.1.3 Example of simulated radar observables

Figure 7 indicates the integrated cloud and rain LWP, the brightness temperature T_B, and the path-integrated attenuation PIA along the dashed black line drawn in Fig. 5. The integrated measurements are shown for different horizontal resolutions (model 0.1 km and radar 1 and 4 km) and for different radar frequencies (Ku-, Ka-, W- and G-band). The horizontal transect of the cloud and rain LWP indicates the presence of two clusters of shallow precipitating clouds with the first one having the majority of its water in cloud-size droplets and the latter having the majority of its water in rain-size particles. Each cell is approximately 5 km wide and exhibits considerable variability with two to three cores spaced by 1.5–2 km. The integrated LWP is shown in the original model resolution and for two different radar footprints (1 and 4 km). Noticeably, the 1 km radar footprint is able to preserve the shallow precipitation spatial variability, while the 4 km footprint results in considerable smearing of the warm-rain spatial distribution. This is consistent with the discussion in Sect. 2. T_B clearly reacts to the emission of the precipitating clouds which warm the cold oceanic background (Eastman et al., 2019). When increasing frequency, the nadir ocean emissivity and the atmospheric gas optical thickness (mainly due to water vapor) rise, thus causing a significant warming of the baseline clear-sky T_B. This reduces the contrast between clear-sky and rainy cells, thus making T_B measurements less useful. On the other hand, optical thicknesses of the precipitating columns (which are driving the T_B warming) also increase with frequency, thus making the high-frequency channels the most sensitive to the presence of rain. For instance, Ku-band T_B values have a low baseline and a potentially large dynamic range, but only a few kelvins of enhancement are produced in this scene because of the low sensitivity of the Ku-band extinction to cloud and rain (see Battaglia et al., 2020a). Vice versa the W-band baseline T_B values are already quite warm due to a high ocean surface emissivity; furthermore they are strongly affected by the amount of water vapor in the atmosphere, with significantly lower clear-sky baselines for environments with a lower freezing level (e.g., ∼235 K for this and, similarly, for the scene shown in Fig. 2 compared to ∼255 K for the scenes with the highest freezing level heights). T_B values at 220 GHz (not shown) are already saturated in clear-sky conditions and are therefore not considered further in this paper.

Figure 7(a) Integrated cloud and liquid water path along the cross section shown in Fig. 5 at the native model resolution and when accounting for different footprint sizes, as indicated in the legend. (b) Brightness temperatures for different frequencies and footprint sizes as indicated in the legend. Random noise with a standard deviation of 2 K is added to the measurements. A nadir-looking geometry is assumed. (c) Two-way hydrometeor PIAs computed simulating SRT (surface reference technique) observations for different frequencies as indicated in the legend. Red, green, blue and magenta correspond to the Ku-, Ka-, W- and G-band, respectively.

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For the same cross section, vertical profiles of cloud and rain water content and reflectivities are shown in Fig. 8. The two rain cells located around 23 and 35 km are well profiled by all frequencies, but the configurations with the largest footprints (panels b and d) tend to fatten their structures. The increased level of attenuation at higher frequencies is clearly highlighted by the reduction in the surface return (panels e and f); at 220 GHz, the surface return and the 1.5 km lowest rain disappear below the MDS, corresponding to the heavily precipitating core at 35 km.

Figure 8(a) Vertical profile of r-LWC (rain liquid water content) for the cross section corresponding to the black line shown in Fig. 5 at the model resolution (100 m); the dashed black and white lines correspond to D_m equals 0.5 and 1.5 mm, respectively. (b) Vertical profile of c-LWC (cloud liquid water content) for the same cross section. The units of the color bar for both (a) and (b) corresponds to log ₁₀ of LWC in grams per cubic meter; so for instance −1 means 0.1 g m⁻³. (c–g) Simulated vertical profiles of reflectivities (color bar units: dBZ) for bands and horizontal resolutions as indicated in the text at the top each panel. Black lines correspond to contour levels of the antenna-weighted PIA from the top of the atmosphere.

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3.2.1 Impact of instrument footprint

Warm precipitating clouds are naturally very inhomogeneous as clearly highlighted in Fig. 5. This inhomogeneity and their inherent 3D structure is very challenging when considering spaceborne observations, with the instrument footprints naturally convolving (thus smoothing) the natural variability of the rain microphysics fields. For instance the sharp peaks in the c-LWP and r-LWP at the 100 m model native resolution are already reduced when moving to 1 km footprints and completely washed out when moving to 4 km (Fig. 7a). Similar effects can be seen in T_B values and PIAs (e.g.,  compare lines with the same colors and different styles in Fig. 7b and c) and also in the reflectivity profiles with rainy cells appearing less intense but with larger horizontal extent for the coarser-horizontal-resolution radar configurations (compare Fig. 8c and d).

NUBF effects are particularly detrimental when considering estimates of (two-way) PIA via the surface reference technique (hereafter designated PIA_SRT) and in the implementation of attenuation corrections for measured reflectivities within profiling algorithms (Durden, 2018, and reference therein). The PIA_SRT is derived by contrasting the surface return under rainfall and the surface return under clear-sky conditions (Nakamura, 1991; Meneghini et al., 2015). In equations

whereas the “true” PIA is obtained via averaging the integrated attenuation along each direction Ω within the footprint, pia(Ω):

where the integrals are extended to the full antenna pattern and G represents the antenna gain. If pia(Ω) is constant across the footprint then PIA_SRT=PIA, but in the presence of NUBF PIA_SRT always underestimates the true PIA, as a direct consequence of the concavity of the exponential function. This is clearly demonstrated in Fig. 9 for a Ka-band having a footprint of 1 and 4 km (panels a and b, respectively). Note the presence of negative values for PIA_SRT (y axis) in correspondence to small true PIAs due to the injection of noise in their estimates. The radar footprint can impact the PIA estimates in two different ways:

1. It can lead to considerable underestimation of the PIA due to NUBF effects (i.e., the spatial extent of the cloud covers only a fraction of the effective radar footprint that is the result of the instantaneous field of view and of the along-track integration). This effect is amplified for larger radar footprints (see departure from the 1:1 line in Fig. 9).

2. The range of the true PIA value decreases when coarser footprints are adopted due to the convolution with the broader two-way antenna gain function. As a result of the smaller measured PIA values, the relative error introduced by the measurement error increases. Thus the fraction of useful PIA measurements strongly increases with greater frequencies and with smaller footprints (see last column in Table 2).

Figure 9Illustrating the impact of NUBF on estimate of the hydrometeor PIA via the surface reference technique: density scatterplot of hydrometeor PIA vs. PIA_SRT for the whole dataset of RICO simulations for Ka-band radars with footprint of 1 km (a) and 4 km (b) and for footprints of 0.5 km at 94 GHz (c) and 220 GHz (d).

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Similar results are found at the other frequencies and are summarized in Table 2 where the slope coefficient of the fitting line (m_SRT), the correlation (ρ_SRT) and the maximum value of the PIA_SRT's are reported. With the same footprint, the NUBF effect tends to be more and more acute with increasing frequencies. For current configurations like for the CloudSat CPR or the GPM Ka, the value of m_SRT is significantly lower than 1. It is clear that proper corrections must be applied. While for the GPM DPR, studies are focused on identifying the pixels mostly affected by NUBF by using a combination of normal- and high-sensitivity scans (with flags introduced in the “Trigger” module; see Mroz et al., 2018) and on implementing corrections (Seto et al., 2015); to our knowledge, no mitigation is currently planned for warm-rain retrievals in the CloudSat algorithms currently in operation (Lebsock and L'Ecuyer, 2011; Leinonen et al., 2016). The same applies to the EarthCARE CPR (which is less affected, having a footprint substantially smaller) algorithms (Mason et al., 2017), now in preparation. A possible solution would be to identify regions highly affected by NUBF by producing PIA_SRT estimates at a much higher temporal frequency than cloud measurements (which is possible thanks to the typically higher SNR of the surface) and using the along-track PIA gradient as a proxy for NUBF (something similar is currently envisaged for the Doppler NUBF correction; e.g., see Kollias et al., 2014). The estimated gradient of PIA_SRT's could be used to partially correct for the bias. Figure 10 shows the relationship for a 94 GHz radar with a 0.5 km IFOV. There is still quite some spread around the median value (black line), as expected from the highly nonlinear impact of NUBF on PIA_SRT's (e.g., see example in Mroz et al., 2018). For larger footprints the spread of the biases around the median tends to increase (not shown). Most of the nonlinearities are expected to be particularly acute in the presence of footprints with considerable empty fractions. Future work should investigate the role of ancillary visible or infrared collocated images or of scanning or push-broom configurations with oversampled footprints to better quantify the footprint empty fraction.

Figure 10Density plot of PIA_SRT bias (with respect to the true PIA) as a function of the absolute value of the along-track PIA_SRT gradient for the entire dataset of simulations. A configuration for an EarthCARE-like configuration is considered (94 GHz radar with a 0.5 km IFOV). The black line corresponds to the modal values of the biases for a given absolute value of the PIA_SRT along-track reflectivity.

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3.2.2 Value of radiometric measurements vs. PIAs

In order to better assess the value of T_B a simple scenario is considered, with the inclusion in the environment of the case study of Fig. 5 of a cloud layer at roughly 2 km and a rain layer underneath with increasing c-LWP and r-LWP values. The corresponding brightness temperatures for cloud-only (crosses) and rain-only (lines) conditions are plotted in Fig. 11. Different microphysics with exponentially distributed DSDs are considered for the rain. In addition to the Marshall and Palmer configuration we have added another configuration with a constant intercept of $N_{\mathrm{0}}=\mathrm{5}\times {\mathrm{10}}^{\mathrm{7}}$ m⁻⁴ which is labeled as “Cumulus” since it resembles the one labeled with the same name in Lebsock and L'Ecuyer (2011). In addition the SAM model actually adopts a two-moment scheme where rain concentration and rain water content are both predicted (Morrison et al., 2005); such a scheme is approximated by a power law fit of the form ${\mathrm{\Lambda }}^{-l}=\mathrm{2.7}\times {\mathrm{10}}^{-\mathrm{4}}{\text{RWC}}^{\mathrm{0.3}}$, where Λ is the slope parameter in reciprocal meters and RWC is the water content in kilograms per cubic meter (and it will be referred to as “MO05”). From these plots the different response of the different frequencies is immediately clear: while the 13.6 GHz never saturates for an LWP smaller than 5 kg m⁻², the 36.5 and 94 GHz saturates at different c- and r-LWPs. The sensitivity of T_B to a change in cloud and rain LWPs is illustrated in Fig. 11b. The sensitivity to cloud is generally smaller than to rain, but such a difference becomes increasingly smaller with increasing frequencies, similarly for the differences between different rain microphysics. This is a direct consequence of the dependence of the extinction per unit mass on the characteristic size of the DSD (see Fig. A6 in Battaglia et al., 2020a).

Figure 11T_B (a) and T_B sensitivity (b) as a function of the c-LWP and r-LWP for a simple scenario with cloud-only and rain-only conditions. Different microphysics schemes are considered for rain (see text for details). Red, green, blue and magenta lines correspond to 13.6, 36.5, 94 and 220 GHz, respectively.

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The SAM simulation can reach very low cloud and rain LWPs (down to 10^−4.5 and 10^−5.5 kg m⁻², respectively). An indication of how the two quantities covary when considering quantities averaged over footprints on the order of 1 km is provided by the color image in Fig. 12. Figure 12 further illustrates where PIA and T_B can be useful as constraints for retrievals. In order to gauge the relevance of PIA and T_B measurements, the mean values of ΔT_B (defined as the enhancement of the T_B values with respect to their clear-sky reference values) and PIAs in correspondence to different pairs of r-LWPs and c-LWPs binned logarithmically within their range of variability have been computed. A threshold of 0.7 dB for hydrometeor PIAs and of 2 K for ΔT_B values has been used to roughly identify regions where those signals are sensitive to variations in c-LWPs and r-LWPs. The corresponding contour lines are shown in Fig. 12 for PIAs (dashed) and ΔT_B values (continuous lines) for the different frequencies (different colors). We can make the following remarks.

• Ka- and W-band ΔT_B values and PIAs (see green and blue curves) are potentially very useful, as is G-band PIA. Ku-band PIA appears effective only for heavily precipitating clouds (i.e.,  r-LWP ≥400 g m⁻²). The inclusion of increasing frequencies increases the retrieval potential towards smaller values of the c-LWP, but, even when considering the G-band, integral constraints are useful only for c-LWPs and r-LWPs exceeding roughly 30 and 15 g m⁻², respectively.

• For any given frequency, T_B is generally useful for smaller cloud and/or rain content compared to PIAs (if a precision of 2 K and 0.7 dB is assumed for T_B and PIAs, respectively). For instance at 35 GHz for nonraining (raining) clouds, T_B values are certainly useful for all instances of c-LWP >60 g m⁻² (r-LWP >20 g m⁻²), but PIAs are useful only when c-LWP >350 g m⁻² (r-LWP >100 g m⁻²). W-band (Ka-band) PIAs seem to have the same (slightly better) potential as Ka-band (than Ku-band) T_B values.

• T_B tends to saturate at larger LWPs than the corresponding PIAs (see also Fig. 2). The dynamic range of T_B values (defined as the difference between saturated and background T_B) is constantly reducing with increasing frequency (because of the increase in the ocean nadir emissivity and the water vapor optical thickness). At the G-band (not shown) it becomes so small that T_B values are practically of no use.

Figure 12Distribution of r-LWPs vs. c-LWPs with log ₁₀ of the occurrences modulated by the color bar. Footprints of 1 km are used for all instruments. The continuous lines (unless otherwise labeled) correspond to where T_B exceeds the background T_B by 3 K, whereas dashed lines correspond to where the hydrometeor PIAs exceed 1 dB. Red, green, blue and magenta lines correspond to 13.6, 36.5, 94 and 220 GHz, respectively.

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3.2.3 Potential of multifrequency retrievals of c-LWPs and r-LWPs

Here we want to investigate which frequency and which variable combination is more effective for retrieving column-integrated cloud and rain liquid water paths. The retrieval methodology is based on an ensemble Kalman smoother approach described by Grecu et al. (2018). The large database of RICO simulations is used to produce statistical relationships between observations and the state variables. Here the vector of unknowns is $\mathit{x}=[\text{c-LWP},\text{r-LWP}]$, while the vector of measurements can include both PIA_SRT's and T_B values and/or reflectivities measured at the “closest-to-the-surface” clutter-free bin; thus y_obs=[T_B…Z_ns…PIA…], which can contain multiple frequencies and/or multiple observables. Note that near-surface reflectivities, when multifrequency observations are available, have been suggested as variables with good potential to provide constraints on integral quantities (Durden, 2018).

Given a vector of observations y_obs, the retrieved vector of unknowns is computed as

where $\left\{\stackrel{\mathrm{̃}}{\mathit{y}}\right\}=[{\stackrel{\mathrm{̃}}{\mathit{y}}}_{\mathrm{1}},{\stackrel{\mathrm{̃}}{\mathit{y}}}_{\mathrm{2}},\mathrm{\dots }{\stackrel{\mathrm{̃}}{\mathit{y}}}_n]$ is a subset of the simulated observations of the training database within a normalized distance, δ (see Eq. 4), from y_obs of less than 1 and $\left\{\stackrel{\mathrm{̃}}{\mathit{x}}\right\}=[{\stackrel{\mathrm{̃}}{\mathit{x}}}_{\mathrm{1}},{\stackrel{\mathrm{̃}}{\mathit{x}}}_{\mathrm{2}},\mathrm{\dots }{\stackrel{\mathrm{̃}}{\mathit{x}}}_n]$ is the corresponding state variables. The operator 〈 〉 indicates the mean operation across the subset. The normalized distance between the observation vector and the j element of the subset of $\left\{\stackrel{\mathrm{̃}}{\mathit{y}}\right\}$ is defined as

where σ_k is 2 K for T_B and 1 dB for PIAs and depends on the SNR for Z^ns according to formula A9 in Hogan et al. (2005).

Absolute bias and root mean square errors for different combinations of observables at different frequencies with matched footprints of 1 km are plotted in Fig. 13. It is clear that Ka- and W-only retrievals perform poorly but with the W-band (Ka-band) performing better for the cloud (rain) component. Combining single-frequency PIAs and T_B values (circles) certainly improves the retrieval, but it is the combination of two frequencies that definitely improves simultaneously c-LWPs and r-LWPs, with the Ka–W combination performing better than the W–G combination (compare cyan and magenta circles). The inclusion of near-surface reflectivities also produces a significant improvement in the performances of both the c-LWPs and the r-LWPs (see dashed lines). The inclusion of the G-band in addition to the Ka and W seems to only marginally improve results (compare dashed cyan and black lines). Note that LWPs are expressed in logarithmic units, thus RMS differences of 0.1, 0.2 and 0.3 correspond to fractional errors of $+\mathrm{26}\mathit{\%}/-\mathrm{21}\mathit{\%}$, $+\mathrm{58}\mathit{\%}/-\mathrm{37}\mathit{\%}$ and $+\mathrm{100}\mathit{\%}/-\mathrm{50}\mathit{\%}$, respectively. So for instance the Ka-band–W-band pair achieves the best retrieval performances with an RMSE of better than 30 % for c-LWPs and r-LWPs exceeding 100 g m⁻². Similar conclusions (not shown) can be drawn when 2.5 km footprints are used.

Figure 13Bias (a, c) and RMSE (b, d) in retrievals of c-LWPs (a, b) and r-LWPs (c, d) in log ₁₀ units for the configuration with 1 km footprints. A value of 0.1, 0.2 and 0.3 corresponds to a factor of 1.26, 1.58 and 2 error, respectively. Continuous lines, crosses, circles and dashed lines correspond to PIA only; T_B only; PIA and T_B combined; and PIA, T_B and Z^ns combined, respectively. Green, blue, cyan, magenta and black lines correspond to Ka only, W only, Ka–W combination, W–G combination and Ka–W–G combination, respectively.

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