Determining signal penetration depth is important to determine the ability of a spaceborne Doppler radar system to detect the portions of convective storms where most of the convective transport occurs. Here we will focus on quantifying the impact of signal penetration on the characterization of updraft mass flux (M_F), which is central to cumulus parameterization schemes in large-scale models (e.g., Arakawa and Schubert, 1974; Tiedtke, 1989).

The observational record of convective updraft properties is sparse. Under shallow convective cloud conditions, vertically pointing lidar, and radar systems have been used to characterize the sub-cloud and cloud layer dynamics (Lamer et al., 2015; Lamer and Kolias, 2015; Lareau et al., 2018; Endo et al., 2019). Only recently has the information from both these been merged to provide a comprehensive view of the dynamical field in and around shallow convective cloud systems (Zhu et al., 2021). In deep convection, limited aircraft observations and profiling radar techniques are available (e.g., LeMone and Zipser, 1980; Heymsfield et al., 2010; Williams, 2012; Heymsfield et al., 2013; Kumar et al., 2015; Wang et al., 2020). Based on the airborne Doppler radar observations, the peak updraft values are often above 10-km altitude (Heymsfield et al., 2010; Heymsfield et al., 2013). Here, direct sampling of the numerical simulations is used to construct a more comprehensive benchmark of convective updraft properties. A conservative threshold of 2 m s⁻¹ is used to identify a model grid point that contains convective updrafts (Houze, 1997). We track information about the fractional area coverage in the entire domain of the simulation ( α U ) and the mean air velocity ( V U ) of these convective updrafts at each height through the atmospheric column and compute the mass flux ( M F ) as M F ( Z ) = ρ ⋅ α U ⋅ V U (1) where ρ is the air density in kg m⁻³ and Z is the model grid level (i.e., height).

The corresponding profiles for the different convective cases are shown in Figure 2. Figure 2C indicates that mass flux through the simulated cloud systems generally peaks at or above 6–8 km height except for the RICO shallow convection case which is confined below 4 km, and the KWAJEX case that peaks at 3 km but exhibits a secondary maximum at 11 km. Thus, a spaceborne Doppler radar should be able to penetrate down to ∼6 km above the surface to capture most of the shape and the peak of the convective mass flux profile in deep convective systems.

Using the radar forward simulator, we further evaluate the impact of signal penetration on the characterization of convective updraft mass flux. Since the radar simulator used in this study accounts for frequency but lacks representation for multiple scattering effects (Battaglia et al., 2011; Battaglia et al., 2015), a conservative criterion based on signal-to-noise ratio (SNR) is used to estimate the penetration depth of the radar signal under different atmospheric conditions. Figures 3A,B, respectively, show the fraction of convective updrafts (V_AIR >2 ms⁻¹) with SNR > +5 dB as seen by a 35 and 94 GHz radar as a function of height. Starting at 14 km, both radar systems detect 100% of the scenes for all simulations. Moving downwards to ∼12 km, the intense MC3E and KWAJEX deep convective cases produce strong signal attenuation at 94 GHz and, as a result, a decrease in the fraction of updrafts is detected. The 35-GHz system is more resilient and only begins being affected by signal attenuation in these storms at ∼9 km. Due to signal attenuation which is generally strongest in the convective core, which is responsible for the bulk of the vertical transport of energy and moisture, even a small loss in the fraction of echoes detected could induce a large underestimation of the mass flux. Assuming that detection of 80% of convective updrafts is sufficient to capture the bulk of the mass flux occurring at each height, we estimate that a 94 GHz radar could be used to characterize the mass flux profile of deep convective systems from cloud top through ∼11 km and of weaker convective storms from cloud top through 7 km. Since this height is generally higher than the location of the mass flux peak, we conclude that a 94 GHz radar system alone would not be appropriate to monitor the mass flux of deep convective cloud systems. As for a 35 GHz radar, the intense continental convection (MC3E) case limits its “effective penetration” to 9 km height above the ground. In all other cases, the 35 GHz radar will be able to capture the peak of the convective mass flux and, in the case of weaker convective storms, penetrate much lower. Thus, a 35 GHz radar would be appropriate to monitor the mass flux peak in all, but the strongest deep convective systems.

The IFOV of spaceborne radars is effectively the projection of the radar sampling volume on Earth’s surface. IFOV is modulated by four main factors: 1) frequency (f), which inversely correlates with IFOV (ceteris paribus), 2) antenna size (D) which directly acts to increase IFOV, 3) number of antennas used for the Doppler velocity estimation, which indirectly acts to increase IFOV by decreasing antenna sizes, and 4) distance from the Earth (H_SAT), which directly acts to increase IFOV. A commonly used approximate relationship for the radar IFOV is I F O V ≅ 0.369 H S A T   [ k m ] D   [ m ] ⋅ f [ G H z ] (2)

The IFOVs for six spaceborne Doppler radar architectures considered in this study are listed in Table 1. The average orbit height is the same (H_SAT = 400 km) for all the architectures. This results in the 94 GHz radars having an overall smaller IFOV. Number of antennas comes next, with non-DPCA systems having overall smaller IFOV than DPCA systems.

The relationship between the updraft chord length (UCL) and the spaceborne radar IFOV is very important to determine the ability of a spaceborne Doppler radar system to resolve, and thus characterize, convective updraft properties. The spaceborne Doppler radar MDV measurements are the result of the convolution of true updraft properties with the IFOV. In the case of under sampling (IFOV > UCL), the estimated MDV is expected to underestimate the updraft magnitude and overestimate the updraft size. In previous studies, the impact of the radar range resolution and IFOV on shallow cloud properties (vertical and horizontal coverage and LWP) was demonstrated (e.g., Battaglia et al., 2020b; Lamer et al., 2020). Here we will assess the impact of sampling geometry on our ability to characterize updraft mass flux (M_F) and its components.

The observational record of convective updraft chord UCLs is sparse and measurements from limited aircraft observations and profiling radar techniques (e.g., LeMone and Zipser, 1980; Williams, 2012; Kumar et al., 2015; Lamer et al., 2015; Wang et al., 2020) are challenging to consolidate due to their limited sampling of individual storms and the strong dependency of their results on the instrument/platform sampling geometry and strategy used. Nevertheless, most reports of deep convective updraft cores document them as being less than 5 km, with their distribution peaking around 2–3 km (Wang et al., 2020) while shallow convective updraft cores were reported to be 100–500 m wide (Lamer et al., 2015).

Once again, direct sampling of the high-resolution model outputs is used to derive additional statistics of the properties of the convective updrafts. Spatially coherent convective updrafts are identified as contiguous updraft regions with air motion larger than 2 m s⁻¹. We track information about the fractional area coverage ( α U ) and the mean air velocity ( V U ) of convective updrafts of different chord lengths and compute their mass flux ( M F ) as M F ( U C L ) = ρ ⋅ α U ⋅ V U (3)

The distribution of α U , V U , and M F as a function of the UCL and the cumulative distribution of the contribution of updrafts with different UCL to the total M_F are shown in Figure 4. The UCL bins are 0.25 km wide, with center values from 0.25 to 10 km. Shallow convection is characterized by the narrowest UCL’s with only a small fraction of them exceeding 500 m. Deep convection simulations exhibit a broader distribution of UCLs especially for the more intense cases (MC3E, KWAJEX, Figure 4A). The mean V_U increases with the UCL, suggesting that broader updrafts are also characterized by stronger updraft magnitudes. This relationship between UCL and V_U can explain the contribution to the total M_F by updrafts with different UCL. Updrafts with UCL larger than 1.5 km equally contribute to the total updraft mass flux occurring in deep convective storms (Figure 4C). In weaker convective systems like TRMM-LBA and RICO, smaller convective updrafts (1.8 and 0.5 km UCL, respectively) are seen to be responsible for the bulk of the transport since larger updrafts do not seem to be systematically exhibiting stronger velocities.

The cumulative fraction of M F ( U C L ) allows us to determine which updraft sizes together contribute to 50% of the convective updraft mass flux. In the intense deep convective systems, that would be updrafts larger than ∼6 km; in weakly organized oceanic deep convective systems, that would be updrafts larger than ∼3 km; in shallow convective cloud systems, that would be updrafts <375 m. This behavior drives a need to design radar architectures that have an IFOV ≤ 3 km to monitor the bulk of the mass flux in deep convective systems.

Figure 1 allows us to visualize the IFOV achieved by the 6 radar architectures relative to the simulated cloud scenes noting that none of these architectures meet the criteria established for monitoring the mass flux of shallow convective clouds. On the other hand, five of the radar architectures meet the criteria established for monitoring the mass flux of deep convective clouds (radar 1, 2, 3, 5, and 6) except for when significant attenuation occurs.

To further evaluate the impact of IFOV on the characterization of convective updraft mass flux, we perform forward simulations where only the sampling geometry is considered. In effect, we turn on the radar instrument model and estimate the resulting vertical air motion. From those motions, at each radar height, the area fraction, the magnitude of updrafts of velocity >2 m s⁻¹, and their mass flux are computed. Using the same model swaths, the same convective updraft parameters are estimated at each model height using direct sampling at the native model resolution. The differences of these updraft properties as derived by the radar IFOV and the direct model sampling at each height are normalized by the model direct sampling value (i.e., relative errors). The relative errors in the updraft properties from all heights are used to compile the relative error distributions for different convective scenes and radar systems (Figure 5). As expected, the 1.0 km IFOV provides the best agreement between the model output and the forward-simulated radar observations for all three convective updraft parameters with most of the relative error values within ±20%. A 2.5-km IFOV results in broader relative error distributions in α U and V U (±35%). The M_F relative error distribution is centered around zero; however, in some cases, relative errors up to 50% in the M_F are estimated. The impact of the radar sampling volume on the convective updraft parameters is more drastic at 5 km IFOV with errors up to 100%.

In all cases, non-uniform beam filling (under sampling of the model dynamics) is responsible for the observed errors. Updraft features smaller than the IFOV presenting weak radar reflectivity go undetected (thus causing a negative α U error) while those presenting large radar reflectivity appear horizontally smoothed (thus causing a positive α U error). Beyond distorting the fractional area of convective updrafts, non-uniform beam filling also tends to cause an underestimation of the velocity of small convective updraft features surrounded by most downdrafts or clear air (thus causing a negative V U bias). Due to the way we defined convective updrafts (i.e., MDV >2 ms⁻¹) this negative V U bias may lead to convective updrafts being misclassified as non-convective (weak) updraft, thus taking only the strongest convective updrafts into consideration and yielding an overall positive error in the distribution of V U and M_F. It is important to note that these results are based only on a small number of simulated cases and that the relative error magnitude depends on the convection type. Exploring these convective type-based errors should be the focus of future studies.

To quantify the impact of platform motion on the MDV measured by spaceborne radars, we performed forward simulations of the weakly organized convective cloud scene of TRMM-LBA, setting V A I R and V_SED to 0 m s⁻¹. Figure 6 shows results for radar 5, which is a non-DPCA 13-GHz system with a 4 m antenna. Similar results are obtained from forward-simulations of the other two non-DCPA systems (radar 2 and radar 6; not shown). As highlighted by Figure 6B, platform motion alone can introduce MDV biases on the order of 30 m s⁻¹ in this highly heterogenous cloud scene. The MDV biases due to non-uniform beam filling (NUBF) will occur at the edges of every convective cloud (where large horizontal gradients of radar reflectivity occur) and within the periphery of all convective cores since a vertically oriented area of high radar reflectivity is one of their characteristic radar features. Although the use of the along-track radar reflectivity gradient can be used to correct for most of the NUBF-induced velocity bias, considerable residual errors from the application of an imperfect correction could complicate the detection of convective updrafts and could lead to false detections. Considering that this bias is nearly double the magnitude of the strongest dynamical features simulated by the model and the proximity to the location of the actual convective updrafts, we suggest that a non-DPCA system would be difficult to use for dynamical studies in such complex cloud scenes.

The total MDV error budget ( σ M D V ) for a spaceborne Doppler radar is affected by three main factors 1) intrinsic noise (spectral broadening) introduced by the platform motion ( σ B ) , 2) outstanding error in correcting MDV biases caused by non-uniform beam filling ( σ N U B F ) , and 3) outstanding error due to uncertainty in the antenna pointing characterization or alternatively error in the estimation of the horizontal wind when off-nadir pointing is needed. The MDV total error budget is given by the following expression: σ M D V = σ B 2 + σ N U B F 2 + σ P 2 (4)

Figure 7 shows the individual contribution of these factors in the 6 radar architectures under consideration in the current study. The relationship of σ B with signal-to-noise ratio (SNR) becomes evident (Figure 7A); for any of the radar configurations σ B is systematically lower for atmospheric features of higher radar reflectivity. The dependency of the MDV uncertainty with SNR is inherent to all the radar architectures; however, the DPCA radar architectures (1, 3, and 4) have negligible σ B   errors, as with any radar on a non-moving platform. On the other hand, the non-DPCA radars (2, 5, and 6) have much higher σ B errors. This is clearly illustrated in the comparison of the two 94-GHz radar system. Despite its larger antenna and higher sensitivity, the EarthCARE CPR (radar 2) has a much higher σ B contribution to the overall MDV error budget than the smaller antenna, which is a less sensitive DPCA 94-GHz radar.

In addition to SNR, the σ B for the non-DPCA radar architectures depends on the normalized spectrum width (Kollias et al., 2014). This explains the difference in their curves. Noticeably, for the same radar frequency, σ B is lower for the radar with larger antenna. This is illustrated in the case of two of the 13-GHz radars (5 and 6). For a 20 dBZ echo, the σ B is 1.2 and 0.28 ms⁻¹ for radars 5 and 6, respectively. In addition to having a negligible σ B , the use of a pair of antennas by the DPCA radars allows them to avoid the non-uniform beam filling MDV biases, something that is not the case for the non-DPCA radars. The along-track gradient of the radar reflectivity has been suggested to correct the NUBF velocity bias; however, the correction depends on the detailed distribution of the radar reflectivity within the radar sampling volume that is not known (Sy et al., 2014; Kollias et al., 2018). Thus, there is a residual, unbiased velocity error from the NUBF correction ( σ N U B F )   that increases the total MDV error budget ( σ M D V ) . The residual error from the NUBF correction is proportional to the square of the radar IFOV ( σ N U B F ∝ I F O V 2 ) . In Battaglia et al. (2020b), an estimate for σ N U B F was provided for a gentle along-track radar reflectivity gradients of 3 dBZkm⁻¹ (non-convective conditions, Kollias et al., 2014). These estimates should be considered as a lower bound   σ N U B F estimate. An along-track radar reflectivity gradient of 15 dBZkm^-1 (convective conditions) is used to provide an upper bound   σ N U B F estimate. The upper and the lower bound σ N U B F estimates are shown in Figure 7B. For the EarthCARE CPR, sampling convection is expected to be very challenging, thus, the lower bound   σ N U B F estimate is more relevant (Kollias et al., 2018). Due to its very narrow IFOV, the   σ N U B F is small compared to the σ B   term. On the other hand, the   σ N U B F term dominates the MDV error budget σ M D V for the two non-DPCA 13-GHz radars (5 and 6). The   σ N U B F term is the dominant term in the σ M D V budget even for a 6.0-m antenna size.

Finally, the σ P term represents uncertainty in the MDV introduced by antenna pointing uncertainties related to thermal distortions and vibrations of the antenna structure and/or uncertainty in the estimation of the horizontal wind when off-nadir pointing is needed. An uncertainty of 10–15 μrad in the knowledge of the spaceborne radar antenna pointing at an altitude of 400 km corresponds to an MDV uncertainty of 0.08–0.11 ms⁻¹ (Battaglia and Kollias, 2015). For the DPCA radar architectures studied here, a 2 off-nadir forward (along-track) pointing is required to minimize the vertical extend of the surface echo (Beauchamp et al., 2021). In this case, a 5 ms⁻¹ uncertainty in the knowledge of the horizontal wind will introduce ∼0.2 ms⁻¹ uncertainty in the MDV estimate (grey dashed line in Figure 7).

Summing these error sources allows us to conclude that DPCA radar configurations are overall more accurate (in terms of MDV) than non-DPCA systems especially in highly heterogenous conditions. It is worth noting that the MDV uncertainty depicted in Figure 7 estimates for “best-estimate” MDV produced at the highest resolution available. For relatively homogeneous scenes such as stratiform cloud conditions, it may be acceptable to perform additional along-track averaging (Kollias et al., 2014), apply noise-filtering techniques (Sy et al., 2014) or rely on conditional sampling (Protat and Williams, 2011) to produce a coarser but high precision MDV “best-estimate”. These techniques have been shown to lead to reduction of the MDV uncertainty by as much as a factor of 2 In forward simulations of the EarthCARE satellite (here radar 2).

The MDV measured by a nadir-looking spaceborne Doppler radar represents the sum of vertical air motion ( V AIR )   and the reflectivity-weighted hydrometeor sedimentation velocity (V_SED): M D V =   V S E D + V A I R (5)

Separating the contributions of these two terms is a necessary step for using MDV for dynamical studies which are associated with V AIR and microphysical studies which are associated with hydrometeor properties that impact V_SED (e.g., Kollias et al., 2002; Zhu et al., 2021). The condition | V a i r | < 2   m s − 1 is often used to separate stratiform and convective cloud conditions (Houze, 1997). In stratiform clouds, (e.g., frontal stratiform precipitation, stratiform regions of convective systems, ice clouds), the horizontal microphysical variability is moderate, and the vertical air velocity is much smaller than the hydrometeor sedimentation velocity. When these conditions are satisfied, the MDV can be related to the shape of the particle size distribution (PSD), to relevant moments (e.g., rainfall rate), and, under certain conditions, allow us to study the microphysical processes that influence their evolution (e.g., Protat and Williams, 2011; Kalesse and Kollias, 2013). The extent to which such inferences can be made with reasonable uncertainty, depends on the magnitude of σ M D V .

A combination of experimental and theoretical relationships between MDV and microphysical variables are presented here to illustrate the impact of the MDV uncertainty on microphysical variables in stratiform conditions. The formulation of analytical relationships between radar observables and microphysical variables requires a mathematical representation for the particle size distribution (PSD). The gamma distribution first introduced by Ulbrich (1983) and Willis (1984), and its normalization introduced by Testud et al. (2001) has been widely used to describe the PSD: N ( N 0 ∗ , D m , μ ) = N 0 ∗ f ( μ ) ( D D m ) μ ⁡ exp ( − ( 4 + μ ) D D m ) (6) where f ( μ ) = Γ ( 4 ) ( 4 + μ ) 4 + μ 4 4 Γ ( 4 + μ )

The three parameters ( N 0 ∗ , D m , μ ) have the following meanings: D_m is the volume-weighted mean diameter (defined as the ratio of the 4th to the 3rd moment of the PSD), N 0 ∗ is the intercept parameter of the exponential distribution that has the same water content as D_m, and μ describes the PSD shape. The MDV is independent of N 0 ∗ and for liquid phase hydrometeors (drizzle and raindrops), is a function of only D_m and μ In addition, the MDV is the reflectivity-weighted PSD sedimentation velocity; thus, the relationship between MDV and (D_m, μ ) depends also on the selected radar frequency. The relationship between MDV and D_m is shown in Figure 8 for four different μ values (−2, 0, 3, and 12) at 94 GHz radar (A), 35 GHz (B), and 13 GHz (C). Plotted under these analytical relationships are MDV and D_m estimates from the two-dimensional video disdrometer (2DVDs) observations (analysis details described in Section 2.1). Overall, the experimental data and the theoretical relationships agree and two distinct regimes emerge for each radar frequency.

The D_m estimation is often based on a combination of radar observables, however, here, we assume that MDV is the only available radar measurement. Under this assumption, the error in the D_m estimation is controlled by the rate of change of D_m with MDV and the MDV measurement error: δ ( D m ) ≈ ∂ ( D m ) ∂ ( M D V ) δ ( M D V ) (7)

At W-band, two different ∂ ( D m ) / ∂ ( M D V ) regimes are present with very different slope values, one for D_m values lower than 0.8 mm where ∂ ( D m ) / ∂ ( M D V ) ≈ 0.26   m m / m s − 1 and another one for D_m values higher than 0.8 mm where ∂ ( D m ) / ∂ ( M D V ) ≈ 0.78   m m / m s − 1 . The lower the ∂ ( D m ) / ∂ ( M D V )   value, lower the uncertainty in the D_m retrieval for a set MDV uncertainty. For example, an MDV uncertainty of 0.2 ms⁻¹ translates to an error of 0.05 and 0.15 mm, respectively, for D_m values less 0.8 than and greater than 0.8 mm. The range of D_m values where the slope ∂ ( D m ) / ∂ ( M D V ) is low increases at lower radar frequencies, suggesting that lower frequency radars are preferred for retrieving D_m from MDV measurements. On the other hand, at high D_m values (>1.75 mm) the slope ∂ ( D m ) / ∂ ( M D V ) at 94-GHz smaller thus suggesting that the 94-GHz MDV measurements will exhibit larger dynamic (sensitivity) range to D_m changes in high-D_m regimes. These differences are due to differences in the scattering by raindrops at the different radar frequencies (Kollias et al., 2002; Kollias et al., 2016). The complimentary use of MDV estimates at different radar frequencies can provide a strong constraint for D_m estimation (Giangrande et al., 2012; Matrosov, 2017).

The determination of the rime fraction in ice particles is another example where the use of the MDV can provide a strong constraint (Mason et al., 2018; Oue et al., 2021). In a recent study, Kneifel and Moisseev (2020) analyzed a large dataset of surface-based radar and in-situ observations and derived an experimental relationship between MDV and rime mass fraction (FR). The average relationship between MDV and RF for three radar frequencies (94-GHz, 35-GHz, and 13-GHz) is shown in Figure 8D. Although there is a considerable spread, the experimentally derived relationships offer a first order relationship for converting MDV errors to FR errors. If we focus on the FR range of 20–80%, an MDV uncertainty of 0.2 ms⁻¹ translates to an error of 14, 12, and 10% in FR at 94-, 35-, and 13 GHz respectively. This assessment ignores the fact that the shape of the particle size distribution (D_m) will also affect the MDV magnitude; thus, here it is assumed that this information is provided by other measurements (e.g., dual-wavelength radar measurements, Pfitzenmaier et al., 2019).