The IMPACTS field campaign, conducted during the winters of 2020, 2022, and 2023, was a comprehensive observational program designed to investigate the microphysical and dynamical processes of snowstorms along the eastern United States. Coordinated measurements were made using NASA's ER-2 high-altitude aircraft and the P-3 cloud-sampling aircraft. The ER-2 carried a set of remote-sensing instruments, including high-frequency microwave radiometers and W-, Ku-, and Ka-band Doppler radars, while the P-3 carried in situ microphysical probes and meteorological sensors for detailed measurements of cloud and precipitation particles. The integration of these datasets offers a valuable testbed for validating and improving a multi-sensor algorithm to retrieve ice hydrometeors.

Table 1 lists the IMPACTS flight cases analyzed in this study, including observation dates, scenes (ocean or land), the dominant cloud–precipitation type, and the number of microwave radiometer footprints. Only cases with coincident ER-2 and P-3 observations that have a temporal difference between the two platforms within 15 min are selected. A match-up is adopted when the P-3 flight track is located within the ER-2 microwave radiometer footprint and all the necessary parameters for algorithm input and validation are available. The cloud–precipitation types are classified using W-band echo-top temperature (<-20 °C for deep, ≥-20 °C for shallow) (Aoki et al., 2026), presence or absence of strong updrafts in the W-band Doppler velocity (convective or stratiform), and near-surface air temperature (>0 °C for rainfall, <0 °C for snowfall).

The combined retrieval algorithm for ice hydrometeors employs the Joint Simulator for Satellite Sensors (J-sim) developed by the Japan Aerospace Exploration Agency (JAXA) (Hashino et al., 2013, 2016) for forward simulations. J-sim contains the Satellite Data Simulator Unit (SDSU) (Masunaga et al., 2010), which enables simulation of a wide range of microwave satellite observations, including W-band Doppler radar, Ku- and Ka-band radar, and microwave radiometers (Ohara et al., 2023; Ohara and Masunaga, 2025). J-sim is also applicable to analogous airborne sensors such as CRS, CoSMIR, and HIWRAP.

As ancillary data for the forward calculations, vertical profiles of temperature, water vapor, and pressure, as well as surface temperature and sea surface wind speed, are taken from the ERA5 hourly reanalysis data (Hersbach et al., 2023). Within cloudy and precipitating regions (Zw>-30 dBZ), relative humidity is set to 100 % instead of using ERA5 values. These variables are treated as fixed inputs in the retrieval algorithm described in the next section.

Land surface emissivity is modeled by the NESDIS microwave land emissivity model (Weng et al., 2001) implemented in the radiative transfer model in use (J-Sim). The emissivity is calculated assuming a vegetation fraction of 100 %, soil moisture fraction of 50 %, surface temperature from ERA5 hourly data, and zero snow depth in this study. These simplified assumptions may introduce errors in surface emissivity, particularly under conditions such as snow cover, which could affect primarily low-frequency brightness temperatures. However, the high-frequency channels (≥ 89 GHz) used in this study are more strongly influenced by water vapor absorption and cloud/precipitation scattering, and thus have reduced sensitivity to the surface, suggesting that the impact is relatively small.

The vertical profile of liquid water content (LWC) is estimated from the HIWRAP Ku-band radar reflectivity using the ZKu–LWC relationship (Masunaga, 2022) derived under the assumption of the Marshall and Palmer (1948) raindrop size distribution and the Rayleigh-scattering approximation. The liquid particle size distribution is assumed to follow the exponential form Nl(D)=N0,lexp⁡-λlD of Marshall and Palmer (1948), where N0,l=8000[m-3mm-1], and λl varies as a function of LWC. Under these assumptions and the Rayleigh approximation, LWC=3.4×10-3ZKu4/7 is obtained (see Chapter 10.2.1 of Masunaga, 2022). The uncertainty associated with this LWC estimate is discussed in the Supplement. In the present retrieval framework, liquid and ice particles are treated separately based on temperature from ERA5 hourly data, and thus mixed-phase conditions are not considered. The treatment of supercooled liquid water is a potentially critical limitation of the current approach and should be addressed in future work.

The structure of the retrieval algorithm is illustrated in Fig. 4. The overall framework is based on O25. In the first step, the algorithm rapidly estimates an initial guess of ice-hydrometeor profiles reasonably close to the solution, using an existing training dataset. In the second step, these initial values are refined through an optimal estimation (OE) method that updates the solution so it is further consistent with the multi-sensor observations along with estimated retrieval uncertainties. Finally, the terminal fall velocity of cloud and precipitation particles and the vertical air motion are estimated using the retrieved cloud-ice parameters and the observed W-band Doppler velocity (Vd).

This section briefly revisits the algorithm framework described in O25 and highlights the improvements implemented in this study. The major updates include: (1) the incorporation of a mixed-habit representation of nonspherical ice particles (Sect. 3.2); (2) an improved initial estimation method (Sect. 3.3); (3) the extension of the OE framework to include ZKu and ZKa profiles in addition to Zw profile and high-frequency Tb (Sect. 3.3); and (4) the estimation of terminal fall velocity and vertical air motion (Sect. 3.3). Finally, Sect. 3.4 presents theoretical sensitivity experiments to demonstrate the difference in scattering characteristics captured by high-frequency Tb, Zw, ZKu and ZKa measurements, as well as the expected synergistic benefits among these sensors.

The particle size distribution (PSD) is one of the key microphysical properties that governs both bulk cloud physical quantities (Eqs. 1–3) and the microwave scattering characteristics. In many previous studies, the PSD has been commonly approximated using analytical functions such as exponential, gamma, or lognormal distributions for computational convenience (Austin et al., 2009; Deng et al., 2010). Ice particles gradually grow as they fall through the cloud, evolving from small pristine crystals into larger aggregates, snowflakes, or graupel through processes such as vapor deposition, aggregation, and riming. To account for this vertical growth, several studies have introduced temperature-dependent PSD formulations (Austin et al., 2009; Heymsfield et al., 2013). Following O25, the present study assumes the temperature-dependent gamma PSD proposed by Heymsfield et al. (2013) expressed as 8N(D)=N0Dμexp⁡(-λD)μ=-14.09-0.248T(T<-61[°C]),μ=-0.59-0.030T(T≥-61[°C]) here N0 is the intercept parameter, μ is the dispersion parameter, λ is the slope parameter, and D denotes the maximum particle dimension. In the retrieval framework, μ is assumed as a function of air temperature (T) according to Eq. (8), while N0 and λ are free parameters to be optimized in the algorithm.

Another key microphysical property is the particle habit. In the present retrieval framework, the particle habit is prescribed in advance instead of being treated as a retrievable parameter. The particle habit assumption strongly affects both the microphysical relationships, such as the mass–dimension m(D) and area-ratio Ar(D) relationships, and the microwave single-scattering properties (SSPs). Selecting an appropriate particle habit is essential for achieving accurate ice-hydrometeor retrievals. In previous studies, the m(D) and Ar(D) relationships for various particle models have commonly been expressed in power-law form, mD=aDb and ArD=αDβ, with the coefficients (a,b,α and β) varying according to the assumed habit (Eriksson et al., 2018; Heymsfield et al, 2013; Liu, 2008). While SSPs for spherical particles can be computed exactly using Mie theory, the SSPs of nonspherical particles in the microwave regime are typically obtained from precomputed databases generated using the Discrete Dipole Approximation (DDA) (Draine and Flatau, 1994; Liu, 2008; Okamoto, 2002).

Table 4 lists the power-law coefficients (a,b,α and β) of m(D) and Ar (D) for each particle habit assumed in this study. The total of 13 particle-habit models are evaluated, consisting of (1) the soft-sphere model based on the m(D) and Ar(D) relationships reported by Heymsfield et al. (2013), with SSPs computed using Mie theory; (2) 11 nonspherical particle models from Liu (2008), each with distinct m(D), Ar (D), and DDA-derived SSPs; and (3) a temperature-dependent mixed-habit configuration described below. Whereas O25 assumed a single particle habit independent of environmental conditions, previous observational and laboratory studies have shown that the dominant ice-particle habit in nature is primarily controlled by temperature and secondarily by ice supersaturation. Under highly supersaturated conditions at temperatures below -40 °C, bullet rosettes are known to dominate (Bailey and Hallett, 2004, 2009), and aircraft observations have confirmed the prevalence of rosettes in cirrus at similar temperatures (Lawson et al., 2019). For temperatures above -40 °C, the habit distribution becomes increasingly complex. Bullet rosettes originating in upper levels may grow into plate-like polycrystals as they fall, while dendrite and sector snowflakes tend to dominate at temperatures warmer than -20 °C (Bailey and Hallett, 2009). Inspired by this physical background, this study defines a temperature-dependent mixed-habit configuration: bullet rosettes are assumed for all particles at temperatures below -40 °C, while the fraction of bullet rosettes (Fr) linearly decreases as the fraction of dendrite/sector snowflakes (Fs) increases with temperature between -40 and 0 °C. This mixing ratio (Fr and Fs) is expressed by the following equation. 9Fr=1(T≤-40[°C])Fr=T-40-40<T≤0[°C],Fs=1-Fr The impact of these habit assumptions on retrieval performance is discussed in Sect. 4.3.

This subsection describes the retrieval algorithm illustrated in Fig. 4. The first step, marked by the blue dashed box in Fig. 4, corresponds to the initial estimation. In O25, the initial values of cloud-ice profiles were estimated using a machine-learning approach trained on reference data derived from a cloud-resolving model, the Nonhydrostatic ICosahedral Atmospheric Model (NICAM), over tropical oceans. Simulated satellite observations from the reference data using J-sim were used as input to the machine-learning system. This was proved to be a workable approach but is limited by the model performance such as the extent to which simulated cloud properties represent the variability in nature. In this study, the training strategy is modified and simplified. First, a wide range of cloud temperatures and IWC values are prescribed, and the corresponding Zw is simulated using the forward model (J-sim). Because a temperature-dependent gamma PSD with parameters μ(T) and λT are assumed here (Heymsfield et al., 2013), the Zw–IWC relationship depends on the cloud temperature. Using this dataset, a regression model is trained to predict IWC from Zw and cloud temperature. Once IWC is retrieved, the associated Nt can be diagnosed from the μ(T) and λT. Applying this method to observed Zw profiles from CRS and temperature profiles from the ERA5 hourly reanalysis provides initial guess of the vertical profiles of IWC and Nt. However, this initial estimation still contains errors arising from signal attenuation of Zw within the cloud and the assumption that λ depends solely on temperature. These errors are corrected within OE framework in the second step of the algorithm, shown in the red dashed box of Fig. 4. In this step, Zw attenuation is explicitly accounted for, and λ is treated as a free parameter.

In the OE framework, the initial profiles of IWC and Nt are optimized by using CRS Zw, CoSMIR Tb, HIWRAP ZKu and ZKa. The details of the OE implementation are based on the theoretical background given by Rodgers, (2000) as described in O25. This subsection provides a brief overview and modifications introduced in this study. The vertical distribution of ice-phase hydrometeors to be retrieved is represented by the state vector X, and the observational inputs are represented by the measurement vector Y. The retrieval problem is expressed as the maximization of the posterior probability density function ppostX|Y, which is equivalent to minimizing the cost function shown in Eq. (11) under the assumption of Gaussian probability density function. 10X=log⁡IWC1⋮log⁡IWCnlog⁡Nt1⋮log⁡Ntn,Y=ZW1⋮ZWnTb89⋮Tb183±3ZKu1⋮ZKunZKa1⋮ZKan11J=Y-FXTSe-1Y-FX+X-XaTSa-1X-Xa The state vector X contains the vertical profiles of IWC and Nt, where n denotes the number of cloud layers. In O25, the measurement vector Y consisted only of the Tb and Zw vertical profiles, whereas the vertical profiles of ZKu and ZKa are additionally included in this study. F(X) is the simulated measurement vector (Tb, Zw, ZKu, ZKa) from the state vector X using a forward model (J-sim). The a priori state vector Xa corresponds to the initial estimate of the ice profile, and the error covariance matrices Sa and Se are defined in the same manner as in O25. Details of the specifications of Sa and Se are described in Sect. 3.3 of O25. The cost function is minimized iteratively using the Gauss–Newton method Eq. (12). 12Xi+1=Xi+Sa-1+HiTSe-1Hi-1[HiTSe-1(Y-FXi)-Sa-1Xi-Xa],13Hi=∂Zw1∂log⁡IWC0⋯∂Zw1∂log⁡IWCn∂Zw1∂log⁡Nt0⋯∂Zw1∂log⁡Ntn⋮⋱⋮⋮⋱⋮∂Zwn∂log⁡IWC0⋯∂Zwn∂log⁡IWCn∂Zwn∂log⁡Nt0⋯∂Zwn∂log⁡Ntn∂Tb89∂log⁡IWC0⋯∂Tb89∂log⁡IWCn∂Tb89∂log⁡Nt0⋯∂Tb89∂log⁡Ntn⋮⋱⋮⋮⋱⋮∂Tb183±3∂log⁡IWC0⋯∂Tb183±3∂log⁡IWCn∂Tb183±3∂log⁡Nt0⋯∂Tb183±3∂log⁡Ntn∂ZKa1∂log⁡IWC0⋯∂ZKa1∂log⁡IWCn∂ZKa1∂log⁡Nt0⋯∂ZKa1∂log⁡Ntn⋮⋱⋮⋮⋱⋮∂ZKan∂log⁡IWC0⋯∂ZKan∂log⁡IWCn∂ZKa1∂log⁡Nt0⋯∂ZKan∂log⁡Ntn∂ZKu1∂log⁡IWC0⋯∂ZKu1∂log⁡IWCn∂ZKu1∂log⁡Nt0⋯∂ZKu1∂log⁡Ntn⋮⋱⋮⋮⋱⋮∂ZKun∂log⁡IWC0⋯∂ZKun∂log⁡IWCn∂ZKun∂log⁡Nt0⋯∂ZKun∂log⁡Ntn The Jacobian matrix Hi, which represents the sensitivity of each observation to perturbations in IWC and Nt at each layer, is computed by repeated forward simulations and is updated each time the state vector is updated. The iterations are continued until the χ2 convergence criterion of Rodgers (2000) is satisfied, yielding the final retrieved state vector Xfin. The uncertainty of the retrieval is then calculated from the final Jacobian matrix Hfin using Eq. (14). 14S=Sa-1+HfinTSe-1Hfin-1. Using the retrieved IWC and Nt together with μ(T) assumed in Eq. (8), the PSD can be diagnosed as Nret(D). For the comparison with probe measurements, NretD is substituted into Eqs. (1)–(7) to calculate the cloud microphysical parameters (IWC, Nt, Dm, and Vt) for particles larger than 100 µm to ensure consistency with the probe data. Here, the mD and Ar(D) relationships use the coefficients corresponding to the assumed particle habit (Table 4). Finally, the vertical air motion W is estimated by subtracting the retrieved terminal fall velocity Vt from the observed Doppler velocity Vd.

This subsection discusses qualitatively the differences in the scattering signals detected by the multiple microwave sensors used as inputs to the algorithm as well as the expected synergy effects. The observed high-frequency Tb by passive microwave radiometers are sensitive to the extinction (absorption and scattering) by ice hydrometeors, while radar reflectivity mainly reflects their backscattering properties. Figure 5 shows contour lines of the 13 GHz backscattering coefficient σb13 (green), the 94 GHz backscattering coefficient σb94 (blue), and the 183±7 GHz extinction coefficient σe183±7 (orange), obtained by varying IWC and Nt over a wide range. The bullet-rosette particles are assumed here. In the upper-left part of the figure, all three coefficients exhibit nearly parallel contour patterns, indicating that the observed scattering information is largely redundant across sensors. In contrast, in the lower-right region – where IWC is large and Nt is small, meaning that the particle size becomes larger – the contour patterns differ substantially. In particular, the contour lines of the σe183±7 are nearly orthogonal to those of the σb13 and σb94 coefficients. This difference in contour patterns arises because 13 and 94 GHz wavelength remain within the Rayleigh-scattering regime for such particle sizes, while the shorter wavelength of 183 GHz interacts with the particles in Mie-scattering regime.

The near orthogonality of the two contours indicate that measurements from the two sensors carry independent information in retrieving Nt and IWC simultaneously (Pfreundschuh et al., 2020). In our retrieval framework, these complementary sensitivities are explicitly incorporated through the Jacobian matrix Hi in Eq. (13) to optimize the state vector X. For large ice hydrometeors, the combined use of radar and radiometer observations is expected to reduce the retrieval uncertainty in cloud microphysical properties that cannot be constrained by single sensor.

In this section, the performance of the retrieval algorithm is assessed through a comparison with in situ observations. Figure 6a–c show the retrieved vertical cross-sections of IWC, Nt, and Dm for the deep stratiform snowfall event over land on 5 February 2020 listed in Table 1. The retrieval was conducted under the assumption of a mixed rosette–snowflake particle habit as described in Sect. 3.2. The dashed line in each panel indicates the 0 °C level, and the solid gray line marks the flight altitude of the P-3 aircraft carrying the probe instruments, enabling a direct comparison at this altitude. Figure 6d–f show the comparison between the retrieved values (orange) and the probe observations (blue) of IWC, Nt, and Dm. Overall, the retrieved values exhibit good agreement with the in situ probe measurements in magnitude and variability.

Figure 7 is as Fig. 6 but for the deep convective rainfall case over ocean on 25 January 2020 shown in Fig. 2. In this case, a consistent bias is evident between 19:20 and 19:26 UTC while the overall variability is still captured. One possible explanation for this bias is that the in-situ measurements were obtained near the cloud base or cloud edges, where the influence of spatiotemporal mismatch is more substantial. Another possible explanation is that the bias arises from assumptions regarding particle habit and other microphysical properties. For example, the assumed mixed rosette–snowflake habit, where rosette particles formed in the upper layer fall and grow into snowflakes, may not be appropriate in updraft regions such as those in the cases shown in Figs. 2 and 7. Rather, snow and graupel particles formed at lower levels can be transported upward and mixed with particles generated aloft within an updraft. The impact of particle habit assumptions on retrieval accuracy will be discussed in the next section.

Figure 8 shows scatter plots comparing the retrieved values with the probe measurements for all cases listed in Table 1. Figure 8a–c correspond to the results for IWC, Nt, and Dm, respectively. The color scale indicates the number of data points contained in each bin. Although the scatter plots exhibit a considerable spread, mean bias relative to the probe data (in logarithmic scale) is small. The mean ratios of the retrieved values to the probe data are 1.01, 0.97, and 1.05 for IWC, Nt, and Dm, respectively. Deng et al. (2013) reported that the flight-mean ratios of retrieved IWC to in-situ values for the existing cloud-ice products 2C-ICE, DARDAR, and 2B-CWC-RVOD were 1.12, 1.59, and 1.02, and that the flight-mean ratios for the effective radius re were 1.05, 1.18, and 1.61, respectively. These values are comparable to the present results. Possible sources of the spread in Fig. 8a–c include differences in spatial resolution between the P-3 in situ probes and the ER-2 measurements as shown in Fig. 2b in addition to the uncertainties in assumptions (PSD and ice habit etc.) built in the algorithm. Additional sources of error may include uncertainties in the assumed water vapor and LWC estimates, as well as the temporal mismatch between the ER-2 and P-3 observations. However, supplementary analyses (see Supplement) indicate that these factors have a limited impact on the ice-hydrometeor retrieval accuracy for the IMPACTS cases.

The impact of the assumed ice particle habit on the retrieval accuracy is examined. Retrievals are performed repeatedly for each habit listed in Table 4 one after another, and the results are compared with the in-situ measurements in the same manner as in Fig. 8. Figure 9 summarizes the retrieval bias and RMSE for IWC, Nt, and Dm under each habit assumption. Uncertainties in particle habit assumptions can lead to retrieval biases of up to roughly two orders of magnitude in IWC and Nt. It is evident that the retrieval accuracy for IWC and Nt is higher for bullet rosettes, snowflakes, soft spheres, or the mixed rosette-snowflake habit particles (denoted as “-b ros”, “-snow”, “Sphere” and “Mix”) than for column-like or plate-like particles (denoted as “-col” and “-pl”). Among these, the mixed habit assumption provides the best performance overall. It should be noted that this conclusion is derived only from the cases observed during the IMPACTS field campaign. This mixed-habit assumption may not necessarily be appropriate for other cloud environments, for example, tropical convective systems in which strong updrafts could mix various type of ice particles originating from many different temperature layers. To address this limitation, future work will use a global satellite radar–radiometer matchup dataset (Turk et al., 2021; Aoki et al., 2026) to evaluate whether the same habit assumptions remain valid across different regions, seasons, and cloud types.

It is necessary to examine the self-consistency of the algorithm, that is, whether the synthetic observations simulated from the retrieved IWC, Nt, and Dm reproduce the actual ER-2 observations. Figure 10 compares the simulated and observed ER-2 measurements of CRS Zw, CoSMIR high-frequency Tb, and HIWRAP ZKa and ZKu for all cases listed in Table 1. Only the solid-hydrometeor layers (temperatures below -3 °C) are used in the radar reflectivity comparison. The color scale indicates the number of samples within each bin. An excellent agreement is generally achieved for all sensors (CRS, CoSMIR, and HIWRAP). This result demonstrates that the algorithm is able to retrieve ice-hydrometeor profiles that are physically consistent with passive and active microwave observations over a wide frequency range from 13 to 183 GHz.

While Fig. 9 shows that the mixed particle habit yields the best agreement with the in-situ measurements, an important scientific question is whether the available observations alone are sufficient to constrain particle habit. O25 demonstrated that the simultaneous reproducibility of Zw and high-frequency Tb depends on the assumed particle habit. It also showed that the uncertainty in particle habit can be effectively reduced particularly for tropical clouds with large IWP. However, for the IMPACTS cases, no clear differences in reproducibility were found among the different habit assumptions (shown in the Supplement) unlike in O25. This finding suggests that, for midlatitude snowstorms, prescribing the particle habit assumption consistent with in situ observations may be a more effective approach than attempting to constrain it solely from remote sensing measurements. This issue will be further investigated in future work when extending the algorithm to a global framework including both tropical and midlatitude systems.

This section extends the evaluation to validate the Zw-weighted terminal fall velocity (Vt) using not only the probe-measured PSD (Nprobe(D)) but also area ratio distribution (Arprobe(D)). This implies that agreement in terminal velocity requires not only consistency in the particle size distribution N(D) but also the validity of the assumed particle model Ar(D). Since Nret(D) can be diagnosed from the retrieved IWC, Nt, and Dm, and the assumed particle model (Table 4) provides the mD and Ar(D) relationship, Vt can be computed for both the retrieval results and the probe measurements using Eqs. (4)–(7). Here, the cloud retrieved values assuming the mixed rosette–snowflake habit, which exhibited the best overall performance in Fig. 9, is used to calculate Vt for comparison with the probe measurements.

Figure 11a–c show the vertical profiles of the Vt diagnosed from the retrieved cloud parameters for the two cases presented in Figs. 6 and 7. The velocity is defined as upward positive. Near the cloud top, the terminal fall velocity approaches 0 m s⁻¹ because the ice particles are small, whereas Vt increases as the particles grow larger in the deeper cloud layers. At the altitudes indicated by the gray lines, the retrieved Vt values are compared with the probe-measured Vt as shown in Fig. 11b and d. The comparison in Fig. 11b shows good agreement, while Fig. 11d exhibits biases in some regions. This is likely because the retrieval accuracy of IWC, Nt, and Dm – that is, of the diagnosed Nret(D) – is lower for the case in Fig. 7 (Fig. 11d) than for that in Fig. 6 (Fig. 11b). Figure 11e presents statistical comparisons of Vt for all observation cases. Both the retrieved and probe-measured Vtvalues are concentrated around 1 m s⁻¹, making the correlation difficult to evaluate, but the mean bias and RMSE is very small (-0.02 and -0.30 m s⁻¹), indicating good overall agreement between the retrievals and probe measurement.

An additional validation for Vt and vertical air motion W is also conducted using the Doppler velocity (Vd) observed by the CRS. Since Doppler measurements provide vertical profiles, they allow evaluations at various altitudes beyond a specific level as in the in-situ probe measurements. Figure 11f shows the histogram of the difference between the observed Vd and the retrieved Vt, which corresponds to vertical air motion W in orange. For comparison, the histogram of observed Vd is shown in blue. Only solid-phase data (temperatures below -3 °C) are used. Since both updrafts and downdrafts are present, the mean and mode (i.e., the most frequent value) of W should ideally approach 0 m s⁻¹ when a sufficient number of samples are available. As shown in Fig. 11f, the mean value of Vd is -0.92 m s⁻¹, whereas the mean value of W is 0.02 m s⁻¹, which is sufficiently close to zero. This indicates that particle fall velocity and vertical air motion are reasonably separated from the Doppler velocity. The width of the W histogram (standard deviation) is slightly smaller than that of Vd, although the difference is small. This may suggest that the variability in Vd is primarily explained by variations in W rather than Vt, but it may also reflect limited accuracy in the estimation of Vt.

This section discusses the improvement in retrieval accuracy achieved by multiple microwave sensors employed in tandem. Figure 12 compares the probe measurements (black lines) with the retrievals obtained using Zw only (blue), the combined use of Zw and Tb (green), and the combined use of Zw, Tb, ZKu, and ZKa (orange) for the land case on 25 January 2020. These retrievals are obtained by restricting the measurement vector Y, Jacobian matrix H, and measurement-error covariance matrix Se (see Sect. 3.3) to contain the selected components only of Zw, Zw+Tb, and Zw+Tb+ZKu+ZKa.

Retrievals using Zw only (blue lines) exhibit noticeable biases relative to the probe observations. In particular, the retrieved Dm remains nearly constant and fails to reproduce the variations observed by the probe measurements. This is likely because, in the optimal estimation using Zw only, the errors and uncertainties in the initial state Xa can hardly be corrected. In estimating Xa, attenuation effects are not considered, and the PSD parameters μ and λ are assumed to depend only on temperature. Therefore, the initial estimate contains errors due to attenuation as well as uncertainties in the assumed PSD parameters. When only Zw is used in the subsequent OE framework, attenuation-related errors can be corrected, whereas the uncertainty in λ assumed in the initial estimation cannot be effectively constrained. As a result, the retrieval accuracy of Nt and Dm remains limited when using Zw alone. In contrast, when other microwave sensor observations are incorporated (green and orange lines), the retrieval accuracy for Nt and Dm improves in several regions (e.g., between 20:36 and 20:43 UTC or between 21:08 and 22:15 UTC). This improvement demonstrates that additional information from Tb, ZKu and ZKa effectively constrains the uncertainties in Nt and Dm that cannot be resolved by Zw only.

Where does the combined use of multiple sensors most effectively improve the retrieval accuracy? The sensitivity experiment shown in Fig. 5 suggests that the synergistic effect is particularly efficient for large IWCs and small Nts, or resultingly large particle sizes. In the deeper cloud layers, water deposition and riming tend to increase IWC, while aggregation decreases Nt and increases Dm. Therefore, multi-sensor synergy is expected to enhance retrieval accuracy in deep ice-cloud layers.

Figure 13 shows the improvement in retrieval accuracy by the combined use of multiple sensors relative to the radar-only case as a function of the ice layer depth from the cloud top. The vertical axis represents the reduction in the retrieval error (on a logarithmic scale) against the in-situ probe observations when CoSMIR (Tb) or HIWRAP (ZKu and ZKa) measurements are incorporated. Positive values indicate an improvement in retrieval accuracy due to the combined use of sensors, whereas negative values indicate a degradation in accuracy. The horizontal axis shows IWP_above, a proxy for the depth of the cloud layer, defined as the vertically integrated IWC between the cloud top and the flight altitude of the probe measurement. Despite a considerable scatter, the mean values of bias reduction shown in solid lines indicate that the combined use of multiple sensors improves the retrieval accuracy of Nt and Dm, particularly within the deeper cloud layers (large IWP_above). This result supports the improved retrieval accuracy brought by additional instruments as exemplified in Fig. 12. The accuracy improvement with Zw+Tb+ZKu+ZKa is comparable to that with Zw+Tb. This may indicate that, as discussed in Fig. 5, high-frequency Tb contains substantial scattering information independent of Zw. Alternatively, it may simply reflect the limited sensitivity of ZKu and ZKa to ice microphysics for the midlatitude snowstorm cases in the IMPACTS campaign and for the altitude range sampled by the in-situ observations. The addition of Ku- and Ka-band radar observations is expected to provide greater benefit for cloud systems such as tropical deep convection that contain larger ice hydrometeors (e.g., hail and graupel). Future work will further investigate the synergy of multi-sensor observations by applying the algorithm to global satellite coincidence datasets.