The CPR aboard the CloudSat satellite is a nadir-looking W-band radar. Table 1 outlines the specifications of the CPR. The detailed vertical structure of hydrometeors can be derived from 94 GHz radar reflectivity from the CPR. The GMI aboard the GPM core satellite is a conically scanning microwave radiometer. As shown in Table 1, the GMI channels span a wide frequency range from 10 to 183 GHz (Newell et al., 2015). In this study, brightness temperature (Tb) at frequencies of 89 GHz and higher is used since these frequencies are sensitive to the microwave scattering by frozen hydrometeors. The GMI has the highest spatial resolution among space-borne passive microwave sensors equipped with frequencies above 166 GHz. The inclined orbit of the GPM satellite has occasional orbital overlaps with polar-orbiting satellites including CloudSat, allowing for simultaneous observations at various locations from time to time.

For the evaluation of the combined GMI and CPR algorithm being developed, we utilize a match-up observation dataset from GPM/GMI, Dual-frequency Precipitation Radar (DPR), and CloudSat/CPR (Turk et al., 2021). This dataset collects data when GPM and CloudSat fly over the same location within a time difference of 15 min. This dataset consists of observations from GMI, DPR, and CPR along CloudSat's ground tracks as well as the collocated ECMWF atmospheric state variable data (ECMWF-AUX). For comparison, also used are an existing cloud and precipitation product (2C-ICE and 2C-RAIN) derived from CPR and Cloud-Aerosol LIdar with Orthogonal Polarization Lidar (CALIOP) aboard the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) satellite (Deng et al., 2015). Detailed information regarding the products and parameters used in this study is provided in Table 2. The comparison and evaluation of the current algorithm with these datasets will be discussed in Sects. 4 and 5.

In this study, an a priori database constituted of cloud and atmospheric variables is constructed with global cloud-resolving simulations from the Nonhydrostatic ICosahedral Atmospheric Model (NICAM). The development of NICAM, initially begun by Tomita and Satoh (2004), is currently maintained by the Atmosphere and Ocean Research Institute (AORI) at the University of Tokyo, the Japan Agency for Marine-Earth Science and Technology (JAMSTEC), and the RIKEN Advanced Institute for Computational Science (RIKEN/AICS). NICAM has spawned numerous studies on tropical atmospheric dynamics (Miura et al., 2007; Miyakawa et al., 2014; Nakano et al., 2015). NICAM outputs of an Madden–Julian Oscillation (MJO) event offered a test bed for the assessment of cloud microphysical schemes in comparison with satellite observations (Masunaga et al., 2008). Technical details about the NICAM can be found in Satoh et al. (2008, 2014). The version of NICAM simulations adopted in this study was run using a single-moment microphysical scheme with a horizontal resolution of 14 km and a vertical resolution of 38 layers.

The Joint Simulator for Satellite Sensors (J-sim) (Hashino et al., 2013, 2016) is used for forward simulations of satellite observations in this study. J-sim, being developed by Japan Aerospace Exploration Agency (JAXA), contains radar and microwave radiometer modules based on the Satellite Data Simulator Unit (SDSU) (Masunaga et al., 2010), which are employed for simulating observations compatible with GPM/GMI, DPR, and Cloud Sat/CPR. J-sim allows various microphysical assumptions to be tested, such as particle size distribution (PSD) and particle shape in the forward radiative transfer calculations (for details, see Sect. 3.1). Technical details of J-sim are described in Hashino et al. (2013, 2016).

The flowchart of DNN training is shown in Fig. 3. As mentioned earlier, the frozen hydrometeor datasets of the cloud-resolving model (NICAM) are used as the reference data, and the observations simulated by the forward model (J-sim) from the NICAM data are input to the DNN training. The training dataset and procedure are described below in some detail.

The bottom half of Fig. 1 shows the main flow of the OEM for finale retrieval and uncertainty evaluation. The OEM is a Bayesian method that finds a solution which maximizes the given posteriori possibility density function ppostX|Y (Rodgers, 2000). Here, the state vector X is defined by combining vertical profiles of IWC and Nt, and the measurement vector Y is the CPR Ze and GMI Tb above 89 GHz. 10X=log(IWC)1⋮log(IWC)nlog(Nt)1⋮log(Nt)n,Y=Ze1⋮ZenTb89⋮Tb183±3 Here, n is the number of cloud ice layers. IWC and Nt profiles are set to be in a logarithmic form to avoid negative estimates. The microwave radiative transfer at 89 GHz and higher frequencies is significantly affected by water vapor. Ideally, the water vapor profile should also be included in the state vector X of Eq. (10) and optimized within the OEM framework. From a technical perspective, however, optimizing both the ice-particle and water vapor profiles is computationally demanding, as the amount of information provided by satellite observations (i.e., the dimension of the observation vector Y in Eq. 10) is too limited relative to the number of unknown parameters to be retrieved (i.e., the dimension of the state vector X). This imbalance can cause convergence issues in the retrieval. For thick ice clouds such as deep convective clouds, scattering signals from ice particles are expected to dominate brightness temperature despite the considerable absorption and emission signals from water vapor. Therefore, the water vapor profile from ECMWF-AUX is used as a fixed input, and only the ice cloud profile is optimized. The fidelity of the ECMWF-AUX water vapor profile is discussed in Sect. 4.1.

Assuming a Gaussian possibility density function, the cost function JOEM to be minimized by OEM is written as the following equation. 11JOEM=Y-FXTSe-1Y-FX+X-XaTSa-1X-Xa, where FX is the satellite observation simulated by the forward model (J-sim) from the state vector X; Xa is a priori state, given by the initial value estimated by DNN; and Sa is the covariance error matrix for the a priori state. For the sake of mathematical consistency, Sa in theory should be specified using error covariances calculated from the a priori dataset (i.e., the DNN training data). However, the error correlations derived from NICAM may not be fully reliable, as NICAM is a limited representation of cloud statistics in the real atmosphere. Hence Sa is tuned in a simplistic manner without referring to the a priori dataset. The elements of Sa including off-diagonal terms are defined as (Sa)kl=σa2exp⁡-dklL to take into account the level-to-level correlations (Delanoë and Hogan, 2008; Rodgers, 2000). Here, dij is the distance between k and l layers. Although several values of σa and L are tested and have little impact on the retrieval results, the combination σa=0.5 and L=3.5 km, which gives the most favorable performance, is selected. Se is the covariance error matrix for measurements which take into account not only sensor-derived measurement errors but also forward simulation-derived errors such as the uncertainty due to particle shape assumptions. The measurement error of CPR reflectivity is set to be 2.5 dBZ according to previous studies (Deng et al., 2010). A previous study (Kulie et al., 2010) reported that the uncertainty in the high-frequency Tb due to particle shape assumptions is 5.15 K at 166 GHz. In addition, since the GMI footprint is larger than the CPR footprint, errors caused by the non-uniform beam filling (NUBF) effect should be considered. The measurement error of GMI Tb is set sufficiently large value of 4 K, and the off-diagonal terms of Se are assumed to be zero.

The Gauss–Newton iteration method is used as the algorithm for finding the minimum value of the cost function in Eq. (11), and the state vector of the ith iteration Xi is repeatedly updated until convergence according to the following equation: 12Xi+1=Xi+Sa-1+HiTSe-1Hi-1HiTSe-1Y-FXi-Sa-1Xi-Xa, 13H=∂Ze1∂IWC0…∂Ze1∂IWCn⋮⋱⋮∂Zen∂IWC0…∂Zen∂IWCn∂Ze1∂Nt0…∂Ze1∂Ntn⋮⋱⋮∂Zen∂Nt0…∂Zen∂Nt0∂Tb89∂IWC0…∂Tb89∂IWCn⋮⋱⋮∂Tb183∂IWC0…∂Tb183∂IWCn∂Tb89∂Nt0…∂Tb89∂Ntn⋮⋱⋮∂Tb183∂Nt0…∂Tb183∂Ntn. The Jacobian matrix H is calculated using the finite difference method by applying forward simulations to IWC and Nt profiles being perturbed in each layer. The convergence is evaluated using the χ2 test (Rodgers, 2000) to obtain the final-retrieved state vector. OEM offers the retrieval errors defined by the trace of the following matrix S defined below. 14S=Sa-1+HTSe-1H-1 It should be noted that since Sa and Se are built upon simple assumptions, the retrieval error calculated by Eq. (14) is valid only approximately.

Figure 5 shows an example retrieval for a given CPR and GMI match-up observation. The solid black lines in Fig. 5a and b are the CPR reflectivity and GMI ΔTb used as the input, and Fig. 5c and d plot the DNN-based initial estimates of IWC and Nt profiles and the iteration process by OEM. Figure 5a also shows the CPR reflectivity and GMI ΔTb simulated by the forward model using the DNN and OEM estimates as the input. The DNN yields the estimates that are roughly, if not perfectly, consistent with the satellite observations, suggesting that the DNN performs well as an initial value estimator. The OEM, refining the DNN estimate, estimates the frozen hydrometer profiles in better agreement with both the CPR reflectivity profile and GMI Tb. A statistical evaluation of the algorithm performance will be discussed in Sects. 4 and 5.

In this section, the present algorithm is applied to actual match-up observations from CloudSat and GPM satellites (Turk et al., 2021). Figure 6a and b show a snapshot of simultaneous observations of CPR and GMI on 18 March 2016, containing a mature tropical convective system. Observed GMI Tb is plotted with solid lines, and the simulated clear-sky Tb from atmospheric data ECMWF-AUX is plotted with dashed lines. The GMI Tb and the simulated clear-sky Tb are in good agreement in the clear-sky regions (latitudes <-11°), showing the fidelity of the temperature and humidity sounding in use. A sensitivity experiment has been conducted (not shown) with the ECMWF-AUX humidity profile replaced with a saturated or supersaturated water vapor profile within the cloud-masked regions. This change was found to have no significant impact on the ice cloud retrieval results. Therefore, the ECMWF-AUX humidity profile is used as input even within cloudy regions in this study. The ΔTb for each channel is the difference between GMI Tb and simulated clear-sky Tb.

Figure 6c plots GMI 166GHz ΔTb with a dotted black line and vertically integrated CPR reflectivity Zeint, defined as follows (Kulie et al., 2010), with a blue line: 15Zeint=∫HFLHCTZehdh. Here, HFL and HCT are the freezing level and cloud-top height, respectively. Care needs to be taken, however, when comparing Zeint with the corresponding GMI ΔTb. While the CPR is a nadir-looking radar, GMI observations have a slanted viewing angle of about 52.8° at Earth's surface. As a result, the layer of cloud ice aloft producing a depression of GMI Tb is horizontally offset from the CPR profile matched up to the surface geolocation. To reduce the error due to this misalignment, ΔTb is shifted so that the correlation of horizontal pattern between ΔTb and Ze_int becomes the highest (shown in red line). As described in Sect. 3.3, the errors caused by the NUBF effect are already considered in the covariance matrix Se. The CPR reflectivity and the shifted GMI ΔTb are input to the algorithm to retrieve the frozen hydrometer profile. The retrieved IWC, Nt, and Dm profiles obtained from the current algorithm are shown in Fig. 7.

The OEM also provides the retrieval errors by Eq. (14). Figure 8a and b show the retrieval error of IWC and Nt on the logarithmic scale. Figure 8c and f are example profiles of the IWC and Nt retrieval errors extracted from a latitude of -6°. To assess the performance of the GMI-CPR synergy, the retrieval errors are compared between the CPR-only (blue line) and combined-use cases (red line), respectively. The combined-use case has smaller errors than the CPR-only case in all layers, confirming a positive impact of adding GMI observations to CPR measurements.

Figure 8d and g plot the reduction of errors when each GMI channel is added to the CPR-only observation one by one. The 89 GHz Tb contributes mainly to the reduction of retrieval errors in the lower layers, while 183±3 GHz Tb mainly reduce errors in the upper layers. The error reduction in the upper layers is exclusively owing to 183±3 GHz Tb only, with the contribution of other frequencies being minimal. The 166 and 183±7 GHz Tb contributes across all layers from the upper to the lower layers. Figure 8e and h show the sensitivity of each GMI high-frequency channel to IWC and Nt in each layer using the Jacobian matrix in Eq. (13). The peak of error reduction shown in Fig. 8d and g is consistent with the peak of sensitivity shown in Fig. 8e and h for each channel. As noted earlier, the true values of Sa and Se are unknown, so the retrieval errors theoretically calculated by Eq. (14) may not be accurate in practice. Given the uncertainty in Sa and Se, the quantitative interpretation of error reductions in Fig. 8 requires caution. Nevertheless, the sensitivity analysis results in Fig. 8d, e, g, and h meet physical expectations, and thus the error estimates above are considered to be valid from a qualitative perspective.

In situ data to validate cloud physical parameters are limited in availability. The algorithm performance is therefore tested using measurables (Ze and Tb) instead of retrieved variables (IWC, Nt, and Dm). To this end, measurables are reproduced with forward simulations using the retrieved frozen hydrometer parameters as the input for comparison with actual observations. This comparison is performed using 10 match-up observations of CPR and GMI, including the case shown in Figs. 6 and 7. The 2C-RAIN product is used for cloud liquid water and rain water beneath the cloud-ice layer. As far as the layer of liquid cloud and rain is optically thick for microwave radiation as typical of heavily raining clouds, high-frequency Tb becomes far less sensitive to liquid water path (LWP) than to IWP (Masunaga, 2022), and the uncertainty resulting from the liquid component is negligible.

Figure 9a and b show an example of simulated CPR radar reflectivity in the solid-phase layer and GMI Tb from the current algorithm estimates. Compared to the actual observation shown in Fig. 6, the spatial structure of radar reflectivity and the horizontal distribution of Tb are reproduced well. Figure 9c and d are scatter plots of the simulated and actual observations for the 10 match-up cases. The simulated radar reflectivity and Tb at high-frequency channels are both unbiased overall against the actual observations. This result assures self-consistency of the current algorithm.

DPR carried by the GPM core observatory yields simultaneous observations with GMI radiometry, providing additional data to assess the algorithm performance. GPM/DPR, a suite of Ku- and Ka-band radars, are sensitive to large frozen hydrometeors such as snow and graupel inside of deep convective clouds, having information independent of CPR and GMI observations. This study uses DPR reflectivity above freezing level to test the cloud ice estimates from the present algorithm. Similarly to Fig. 9, the Ku- and Ka-band radar reflectivity is simulated from the current algorithm estimates of frozen hydrometers (Fig. 10b and d) for comparison with the actual DPR observations ((a) and (c)). The current estimates of cloud ice reproduce the overall distribution of observed Ku and Ka radar reflectivity. As shown in Fig. 10e and f, the simulated DPR reflectivity exhibits no systematic bias against the actual DPR observation for the 10 match-up cases despite the significant spread.

This section discusses the assumptions of the particle model optimal for this synergistic algorithm. CPR reflectivity mainly captures the backscattering properties of particles, while GMI Tb values mainly observe the scattering and absorption properties. A combined use of these two independent information has the potential to constrain uncertainties in the assumptions of the particle model. To test this, the reproducibility of CPR and GMI observations is evaluated with different non-spherical particle models listed in Table 3. Only the particle model that consistently represents all the backscattering, absorption, and scattering properties would allow the algorithm to find the solution (frozen hydrometer profile) that accords with both CPR and GMI observations. Figure 11a–f compare the simulated reflectivity and Tb with the actual CPR and GMI observations for the six representative particle models (long column, thin plate, 4-bullet rosette, sector snowflake, dendrite snowflake and soft sphere). The CPR reflectivity is well reproduced regardless of the particle model assumptions by optimizing IWC and Nt (that is, the PSD parameters λ and N0) in our algorithm. On the other hand, the simulated Tb is clearly lower than the observed Tb for cold Tb values, except for the dendrite snowflake and soft sphere cases. These results indicate two points: (1) the soft sphere and dendrite snowflake are the optimal particle models for cold Tb values among the six models tested here, and (2) CPR observations alone are not sufficient to simultaneously constrain the uncertainties in the PSD and particle models.

Figure 11g plots the difference between the simulated and actual Tb as a function of IWP for each particle model assumption. Since the IWP estimate varies with the particle model assumptions, the horizontal axis is substituted by Ze_int in Eq. (15). Simulated Tb is much lower than the observation for most non-spherical particle models at large Ze_int (IWP), whereas for the dendrite snowflake and soft sphere, Tb bias is relatively small for the whole range of Ze_int. The findings imply that for clouds with very large IWP such as deep convective clouds, the dendrite snowflake and soft sphere may be the most appropriate particle models. It should be noted, however, that for more frequently occurring clouds with moderate to low IWP, differences in Tb bias among particle models are small, making it difficult to identify a clearly preferable model. A previous study (Fig. 11 in Kulie et al., 2010) shows a similar figure to Fig. 11g and also reported that most non-spherical particle models except the dendrite snowflake exhibit excessive scattering (negative biases to actual observation) for large IWPs. Kulie et al. (2010) first converted CPR reflectivity to IWC, assuming Liu's non-spherical model (Liu, 2008), and then simulated Tb from this IWC, assuming the same particle model to compare with the actual SSMIS 157 GHz Tb. Their study assumed a fixed PSD when simulating Tb, so the failure to reproduce SSMIS Tb values may be caused by an inappropriate PSD assumption rather than the particle model. However, even our algorithm, which optimizes PSD parameters by OEM, cannot find the solution that is simultaneously consistent with the CPR and GMI observations for non-spherical particle model except for dendrite snowflake.

While the results in Fig. 11g indicate that the soft sphere assumption best reproduces satellite observations, several previous studies have reported that non-spherical particles are essential in reproducing realistic scattering signals instead of soft sphere particles (Ekelund et al., 2020; Kuo et al., 2016; Olson et al., 2016; Kulie et al., 2010). This apparent inconsistency may be explained by a few possible hypotheses as follows.

The validity of soft spheres may vary largely with the particle density model (m–D relation) in use. The present work adopts the m–D relationship from Heymsfield et al. (2013), which is different from the soft sphere model used in previous studies. The current finding does not imply that all soft sphere models, if any, are superior to non-spherical particle models used in previous studies.

The soft sphere model best fits observations for very large IWPs, suggesting that it may only be well suited for certain cloud types such as deep tropical convective clouds. This is physically reasonable because an appreciable amount of graupel is formed by riming in deep convection. The scattering properties of graupel are likely better approximated by soft spheres than snowflakes and ice crystals. In contrast, the soft sphere assumption may be less appropriate for stratiform precipitation investigated in Olson et al. (2016), where scattering is likely dominated by aggregated snow particles. Such differences in cloud microphysics between convective and stratiform clouds may explain the contrasting results between the present and previous studies.