Development of Simulation Models Supporting Next-Generation Airborne Weather Radar for High Ice Water Content Monitoring

In this article, a method of extending airborne weather radar modeling to incorporate high-ice-water-content (HIWC) conditions has been developed. A novel aspect is incorporating flight test measurement data, including forward-looking radar measurements and in situ microphysics probes data, into the model and part of the evaluations of modeling. The simulation models assume a dual-polarized, airborne forward-looking radar, while for single-polarized operations, the index-of-dispersion is included as a helpful indicator for HIWC detection. The radar system simulation models are useful for design evaluations for the next generation of airborne aviation hazard monitoring and incorporate HIWC hazard detection algorithms. Example applications of the simulator, such as hazard detection based on the simulated HIWC flight encounter scenarios based on specific numeric weather prediction (NWP) model outputs, are discussed.


I. INTRODUCTION
H IGH clouds with high ice water content (HIWC) represent a significant aeronautical hazard and a threat to aviation and jet engine operations at high altitudes [1], including commercial vehicles, rockets, supersonic vehicles, and other space exploration mission platforms. The development of the next generation of airborne aviation weather radar has an urgent need to detect and monitor HIWC conditions. Data analysis studies, flight campaigns, and modeling studies have been ongoing to address the HIWC challenge in the aviation and meteorology community [2], [3], [4], [5], [6], [7], [8], while the data collected from the commercial airborne radar is still limited. A physical knowledge-based model of radar observation of HIWC is needed for two reasons. First, it will guide the understanding and interpretation of radar measurements from existing flight campaigns. Second, it will guide the industry and manufacturers in designing, developing, and certifying the new airborne radar products. Numerous previous modeling efforts are related to airborne weather radar and cloud radars. For example, the authors in [9] established an initial framework for using numeric weather prediction (NWP) models in airborne aviation hazard estimation. The authors in [11] investigated the extension of airborne doppler weather radar simulation (ADWRS) for ice crystals, which was part of the standard tools for industry developers. In [13] and [27], the method of simulating ground-based polarimetric array radar using advanced MATLAB tools was first introduced. The authors in [15] extended the airborne radar sensor simulations to include antenna tilt control and terrain clutter effects. The authors in [10] provided another recent example of cloud radar simulation based on a cloud-resolving model. None of these developments, however, has provided a simulation model and complete procedure for airborne forward-looking HIWC detection that is evaluated with observations, especially for dual-polarized operations. A MATLAB-based design tool focusing on the airborne radar system, antenna, and waveform effects is not available from existing simulators. On the other hand, there is a significant amount of data from the previous flight campaigns [3], [4], [11], accompanied by phenomenology analysis of these data [5]. However, there still needs to be more connection between the scientific knowledge derived from these data and the application of the knowledge to airborne radar designs. Furthermore, lack of physical understanding and interpretation has hindered the acceptance of some practically appealing HIWC detection algorithms [2] by the aviation radar community since there is no model describing the origin of pulse-to-pulse variation of received signal amplitudes. This study introduces an improved airborne weather radar simulation tool and modeling method. The modeling starts from microphysical data of ice particles, and then, uses them to simulate the stochastic behavior of ice particles in a unit-sized, hypothetical radar resolution cell. Each radar pulse samples a Monte-Carlo realization of the particle states in the volume, according to the probability distributions of the relationship between ice-water-content (IWC) and radar variables and the radar sensor parameters. The complete radar scans and I/Q signals can then be generated from the NWP field grid of IWC This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ and the end-to-end system tool similar to [9], which includes the airborne aviation radar characteristics. The novel benefit of this approach is that a physics-based validated "radar observation model" provides the connection between the IWC and the radar measurements, which incorporates the randomness in the measurement caused by ice crystals. Such randomness may be useful for HIWC detection, but the impacts of such randomness have not been fully understood before. Furthermore, the new simulation tool adds flexibility to the overall system evaluation since it provides a unified interface to read the IWC field from the NWP model outputs for aviation encounter cases. It is able to use the 3-D data grid and antenna pattern, and then, translate the samples to radar spherical coordinates. In addition, we demonstrate an application of the simulation model with machine-learning (ML)-based HIWC detection and estimation cases using realistic radar signals produced from the system simulations.
This article is organized as follows: Section II discusses a physics-based, "Swerling-type" statistical model of volume targets that contain ice crystals. Section III summarizes the radar observation model that connects the measured microphysical data of ice crystals to the airborne weather radar observations. Section IV discusses airborne weather radar system modeling and presents examples and discussions of HIWC detection/estimation using simple ML algorithms. Finally, Section V concludes this article.

II. RADAR RESOLUTION VOLUME CONTAINING ICE PARTICLES
Radar observations of a radar resolution volume containing numerous ice particles can be modeled as a stochastic process. Each radar pulse samples a realization of random complex backward scattering radar cross-section (RCS) of the volume with size V (here, we assume V = 1 m 3 ) where V r is the complex received signal of volume target V , N is the total number of the ice particles in the volume, σ i is complex RCS of the ith particle for a radar operating at wavelength λ, σ i is determined by the particle's material, shape, thermal dynamic phase, and orientation, ω d is the Doppler frequency of the volume (assume it is the same for all particles in the volume for a forward-looking radar), t is time, and ψ i is the random phase, which is a combined effect of variations of particle position, orientation and shape, turbulence impact, etc., and uniformly distributes between 0 and 2π. For spheroid shape particles, σ i can be calculated using the Rayleigh-Gans scattering formula or T-matrix theory [16]. The backscattering amplitude f ( [17], [18]) of each particle i is calculated, and then, related to RCS by The volume backscattering RCS (σ) is expressed as σ = κV r V * r (we may assume κ is a constant value of one for simplicity). One of the fundamental backscattering observation variables of typical airborne weather radar is reflectivity factor Z, whose mean value is related to σ by where K w is the dielectric factor of water [18].σ represents the mean (or expected value E(·)) value of RCS. It can be shown thatσ Assuming the volume is only filled with ice particles, and if the particle size distribution (PSD) of the volume is available, a more useful expression for polarimetric reflectivity factor (i.e., horizontal and vertical polarizations) based on scattering amplitudes and integration is given by In (5), N (D) is the number distribution function of particles, which varies with diameter D, and usually has a unit of mm −1 m −3 . f hh,vv (π, D) represents the copolarized backwardscattering amplitude. The calculation of backscattering amplitudes is based on the assumption that the particle has a spheroid shape, with a specified axial ratio AR and canting angle. As a derived radar observation output, the differential reflectivity, Z dr , can then be obtained by calculating the ratio between Z h and Z v [18].
The pulse-to-pulse variation of the radar return signal power, which is related to the variance of the reflectivity factor, has been used in the "Swerling" algorithm for HIWC detection and estimation [11]. Based on (3), the variance of the reflectivity factor is given by in which and ξ = λ 8 /V 2 π 10 |K w | 4 is a coefficient related to wavelength. Based on the assumption that in (1), for any i = j, ψ i and ψ j are independent and uniformly distributed random variables, the Appendix shows that (7) leads to which indicates that the variance of RCS consists of mutual correlation terms i =j σ i σ j . Based on (3) and (5), the variance of reflectivity may be also expressed in terms of the PSD and scattering amplitudes as Here, η = ξ(4π) 2 is a constant. Within a single radar resolution volume, N (D) is assumed to be constant. The pulse-topulse variation of reflectivity is mainly caused by the random phase of scattering fields and various axial ratios and canting angles of particles. Equation (9) is only about the ice crystal "targets," and it is not related to any specific radar systems. The radar-derived IWC (RIWC) factor used in [11] can then be connected to the mean and variance of reflectivity as where η is a constant. The first term in the bracket can be shown to be closely related to Z h,v , and the second term is relatively small compared to the first term. If a dual-polarized radar is used, a potentially useful radar observation feature is the specific differential phase (K dp commonly in units of • km −1 ) of the volume, which is derived based on forward scattering amplitudes as The IWC is related with PSD through Here, IWC is a key ice content parameter affecting aviation safety. In (12), ρ e is the effective density of ice crystals. When ρ e has a unit of g/cm 3 and equal-volume diameter of ice particle D has a unit of μm, a useful relationship based on flight measurement data [1] is The aforementioned equation is a modified version of the BF-95 model [19]. It is assumed here that for the smaller size of particles, melting is relatively less significant. The density model in (13) may need to be adjusted somehow to "fit" specific measurement, for example, we often set ρ e as constant 0.917 g/cm 3 for particle diameter smaller than certain threshold, D t . The actual value of D t depends on the specific temperature of the environment, the m − D relationship used, and the measurement procedures.
Another useful variable, the total concentration of ice particles N t , is the summation of numbers of concentration for all sizes, and is given by A dual-polarized airborne weather radar would utilize the four observations variables (Z h , Z dr ) and (RIWC, K dp ) to detect HIWC conditions and estimate the ice content levels, in addition to other polarimetric radar variables. The challenge is developing a simulation model to describe the statistical relationships between IWC and these radar variables with unknown physical properties of particles in the volume. The solution based on the single-cell-Monte-Carlo (SCMC) method is discussed in the next section.

A. Development Procedure
The initial modeling has been based on flight test measurement data, including NASA's DC-8 campaign performed around the Gulf of Mexico, the NRC-Convair Campaign, and the French Falcon Campaign. The French Falcon 20 and National Research Council of Canada Convair (580) were used during the second HAIC-HIWC flight campaign based out of Cayenne (French Guiana) from 9 to 25 May 2015. Data from these aircraft represent the other two datasets used in this study. All the data collections were performed in 2015 [5]. More details about the datasets used in these campaigns are listed in Table I. Since NASA's flight campaign was the only one that included the forward-looking airborne weather radar data, we mainly used the NASA flight data to verify the radar scan returns. However, data from the other campaigns are essential for SCMC inputs.
The relationship between the radar reflectivity (which can be extended to other radar parameters) measured using Honeywell RDR4000 X-band airborne radar, the IWC measured by the second-generation IsoKinetic (IKP-2) probe, and PSDs measured by the 2-D stereo (2D-S) and precipitation imaging probe (PIP) are investigated. First, spatial matching and correlation are performed, in which the latitude, longitude, and elevation coordinates of each radar resolution volume are calculated based on the aircraft's location data and radar PPI scans. Then, these locations are compared with the location where the in situ probe data are obtained and correlated based on the nearest matching 3-D location. Second, only data where the difference in time between the radar data and probe data is less than 15 min are used.
The overall development procedure of the simulation tool, shown in Fig. 1, emphasizes the relationships between the IWC of clouds and the measured airborne radar observations. Our approach establishes a simplified microphysical model of ice crystals while evaluating these models with the best possible knowledge extracted from measurement data. A Monte Carlo method is used to address the uncertainty of the model parameters, and then, a software package is developed using the system simulation tools. The software supports the development of the future generation of airborne radar such as PARADOX (Polarimetric Airborne Radar Operating at X-band) and related standard developments [21].

1) Basic Assumptions:
The assumption of the SCMC simulation is that radar signatures can be generated by inserting randomly distributed hydrometeor particles in a single hypothetical radar resolution volume [9]. Single cell in SCMC represents a unit-sized radar resolution volume in which the hydrometers are distributed based on certain microphysical models. SCMC calculates the scattering amplitudes of all modeled particles using the T-Matrix, then computes the radar variables based on the scattering amplitudes using PSDs. Other characteristic parameters of these particles, such as the axis ratio (AR) and canting angles, are also included in the SCMC, as shown in Table I. For example, we assumed that particle sizes ranging from 0 to a certain percentage are associated with pure ice (no melting). The AR and Canting values are derived from averaged representative particle image sensor data from the flight campaigns (refer to Fig. 1). Monte Carlo simulations use uniformly distributed random AR and canting angles with the corresponding mean values. Atmospheric parameters such as temperature, which can influence the scattering properties of hydrometeors, are also part of the models. The computation of SCMC mainly relies on the Initial-Model-Database and the Relation-Mathematical-Model (Equations (1) to (12)). Including the ice crystal as a new type of hydrometeor and using the actual measured in situ probe data in the modeling process are novel aspects of this work.
2) Particle Size Distribution (PSD): For the NASA DC-8 campaign, the PSDs are measured for 10-1280 μm with a resolution of 10 μm using a 2-D stereo probe. Similarly, for 100-6400 μm, PIP measurements with the bin size of 100 μm are used in SCMC. Minor adjustments of PIP measured particle diameters (based on the maximum diameters) to equivolume diameters are needed here to use the BF-95 models properly. Based on sensor data correlation, example PSDs and the associated IWC values from the 2015 flight campaign are shown in Fig. 2(a). These example PSDs represent the campaigns shown in Table I. Some samples of these PSDs are able to be fit with the "double-Gamma" function, which contains six parameters (b 1 to b 6 ), as shown in (15), and illustrated in in Fig. 2(b). Note that according to the detailed study of the PSDs from campaigns, the number of modes required to fit the data could vary, so there is no single PSD model fitting that would be sufficient [6]. Therefore, in the SCMC simulations, the raw PSD data are sorted into a number of bins according to the measured IWC, and all PSDs measured at the corresponding time are used to compute the radar variables using the equations in Section II. The PSD fitting model is used only to extrapolate the PSD to very large IWC values when there is no measurement data.

3) Ice Crystal Material and Orientation:
The remaining important microphysical parameters of the ice crystals in radar volume are the electric permittivity (dielectric constant), density (or mass), axial ratio (AR), and canting angle. The effective dielectric constant ε e is calculated using Maxwell-Garnett mixing formula [22] for mixing ice, water, and air. The properties of such mixtures depend on the environment's temperatures. When the temperature is sufficiently low, ε e is dominated by the complex permittivity of pure ice, which is computed using equations in [23]. The density-size relation in (13) [20] is used in SCMC to compute (12). The AR and canting angles are assumed to follow uniform distribution, with recommended mean values in Table  I. Default ranges of these values are based on studies in [24], and existing PIP images. The actual canting angles would naturally depend on the wind and electrification conditions; therefore, the SCMC uses flexible configurations of these variables.

4) Monte-Carlo Simulation Runs and Outputs
: SCMC executes the relation-math-model in Fig. 1 using (3)-(13). The inputs are the measured PSDs, and the outputs are the radar variables and their statistics. Data quality control is done at this stage since there may be numeric errors and instabilities for a small fraction of PSD samples and T-Matrix outputs. Outliner outputs are discarded before further usage or processing. Example displays of the outputs from SCMC using the PSDs collected from the 2015 DC-8 flight campaign are shown in Fig. 3. Meanwhile, the similar simulation outputs from SCMC for the Convair and Falcon flight campaigns are shown in Figs. 4 and 5, respectively. These plots use binned data for each IWC value and each radar variable value and show the PSD sample count numbers as density colors. Note these plots are for statistics of radar variables for different radar resolution cells with different  PSDs. For example, in Fig. 3, each dot represents mean value of reflectivity for a particular PSD sample. The averaged bin values associated with all PSD samples for each IWC are plotted as red curves, and the cyan dashed line in Fig. 3(b) is the fitted trend line using polynomial fitting.  (13). Unit of the colorbar is probability. (b) Evaluation of the total concentration model in (14). For both plots, x-axis is measured IWC values, and y-axis are predicted IWC values and total concentration, respectively.
Comparing the simulated radar variables from the three different campaigns, we first notice that the Convair and Falcon flight measurement PSDs result in a higher range of reflectivity values than DC-8 data. The trend curves show different behaviors for different flight tests, which is because of the differences in the microphysics parameters (see Table I). The RIWC plots have similar trends among different flights and are in the identical range of the IWC values, similar to the results reported in [2]. The RIWCs do not show a linear regression relation with IWC but still display good indications of IWC levels. The most interesting radar variable is the K dp , which offers a statistically linear relationship between IWC and K dp based on regression analysis. The average curve of K dp starts to "bend down" for large IWC values due to the less available PSD samples, while the regression fit still shows a clear linear trend.
Initial self-consistency evaluations of SCMC modeling is depicted in Fig. 6. Fig. 6(a) shows the correlations between the IKP2 probe measured IWC and the IWC values predicted by SCMC using the modified BF-95 model and measured PSDs. Fig. 6(b) shows the computed total concentration N t of ice particles versus measured IWC values. The range of values of N t is comparable with other published results (such as [5]).

C. Distribution Density Functions From SCMC
Using PSDs derived from the in situ probe and the radar scattering models, SCMC outputs can be visualized in terms of histograms of radar variables in specific IWC ranges. The histograms include all the reflectivity samples and all the PSDs from SCMC outputs, that are within ±0.1g/m 3 of the indicated IWC values. Fig. 7 compares the histograms for IWC = 1g/m 3 and 2.3g/m 3 . The extracted probability density functions (PDFs) from the histograms using MATLAB's ksdensity function are illustrated in Fig. 8. The histograms show multiple peaks sometimes, such as IWC near 1g/m 3 , when the "bin size" is sufficiently small and the "fitting band-with" is high, including a peak centered around 10 dBz. From these PDFs, we can observe the trend of how the IWC affects the spreading of PDFs, which can be different for the different flight campaigns in different regions. For example, the "spreading" of the PDFs decreases with increasing IWC for the French Guiana campaign results but increases with IWC for the DC-8's Gulf-of-Mexico campaign, as shown in Fig. 8. Both types of "spreading versus IWC" trend are expected, due to the various PSD distributions of the campaign regions and microphysical property variations.

D. Comparison With Measured Radar Data
Quantitative evaluations by comparing SCMC simulation outputs with X-band airborne radar flight measurements are challenging because the data correlation procedure may result in differences in the sampled volume for different histograms, and the selection of data in the analysis based on the temperature, altitude, and spatial location criteria is mostly empirical at the current stage. Furthermore, the specific radar used in the campaign is not accurately modeled in SCMC. Nevertheless, qualitative comparisons through careful sensor data association and calibrations may still be performed. First, Fig. 9 contains example histograms of reflectivity factors and derivations from the 2015 flight campaign. These histograms are obtained through the time-space data correlation procedure described in Section III-A. The count of samples in the histograms is accumulated from all the radar resolution cells from all the collected scans during the campaign. Compared to Fig. 7 and the SCMC outputs, similar trends of increasing mean versus IWC can be seen. It is interesting to see the "double-peak" of the fitted PDF for high IWC value, which is identical to Fig. 7. These measured PDFs also show some differences compared to the SCMC simulations. Fig. 9(c) shows more clearly the increase of mean values of the distributions with higher IWC. Also, Fig. 9(d) shows the "dispersion" of the PDFs slightly increase, then vastly decreases with respect to IWC, which seems to be a combination of those two trends observed in Fig. 7. More analytical results are provided in the following sections.
Furthermore, Fig. 10(a) shows a comparison between the SCMC-simulated IWC-Z curve and the measurement curve derived from all the verified DC-8 flight campaign measurements. It compares the mean reflectivity values generated by the SCMC with the measured average reflectivity values for specific IWC values. Mean reflectivity values associated with each specific IWC value (±0.1 g/m 3 ) are used for the plot. It is seen that the for both simulations and measurements, reflectivity values start at around 22 dBz for IWC of 1 g/m 3 and increase to around 24 dBz for the IWC of 2.5 g/m 3 . Note the "cutoff" values for plotting are chosen to be higher than 1 g/m 3 and lower than 2.4 g/m 3 , which is because IWC less than 1 g/m 3 is usually not considered as hazardous and probe measurements for higher than 2.4 g/m 3 do not have sufficient statistical significance. From these comparisons, we may further confirm that mean reflectivity values versus IWC indeed have a clear increasing trend, but this relationship itself may not be sufficiently significant for the ML-based classification algorithm.
Comparison of RIWC estimations is then provided in Fig. 10(b). The RIWC from the 2015 DC-8 campaign radar measurements is computed based on the logarithm of the index of dispersion of the return signal amplitudes from each radar resolution cell, according to the descriptions in [2]. The RIWC of SCMC outputs is computed using (10), and then, averaged for all the PSD samples. Similar to reflectivity plots, the mean values of RIWC estimations from all estimations associated with specific IWC values are plotted. The trend of RIWC is identical to the reflectivity factor trend, while the range of values is correlated closer to the IWC values. Using RIWC values to estimate IWC directly may introduce an error of about ±0.5g/m 3 , as discussed in [2], which is verified with the RIWC estimations simulated from SCMC. From the results of Fig. 10, we may conclude that both reflectivity and RIWC can be useful features of HIWC, while an algorithm for HIWC detection may need to use them together with other features.

A. Overall Structure
The airborne weather radar system simulator is modified from PASIM [27], a weather radar simulation tool operating in the time domain, and was initially developed for ground-based weather radar system simulations. The expansion, which is called "airborne PASIM," uses the same modeling methods, while supporting airborne motions, geometries, and coordinate system translations. A detailed comparison between the original PASIM and the "airborne PASIM' are provided in Fig.  11. Besides the core simulation codes for I/Q data generation, which is based on the MATLAB phased array system toolbox, the airborne simulation system is different from ground radar PASIM as follows.
1) Incorporating SCMC simulation, which is a microphysicsbased "target signature generator." 2) Incorporating modules specifically designed for X-band transceivers, LFM waveforms, and pulse compressions. 3) Inclusion of aircraft motions, antenna pointing stabilization, and terrain clutter models. These elements are specific to airborne radars, and the theoretical details are provided in [14] and [15]. 4) The "truth" weather fields are from NASA's TASS model, with only the IWC field grid used, and a 3-D resampling algorithm is applied, which is discussed in [28]. The end-to-end airborne radar sensing simulation based on SCMC is summarized in Fig. 12.
The 3-D reference weather field is derived from the NASA's Terminal Area Simulation System (TASS) [29]. The TASS output field includes samples of IWC values and other output products such as radar reflectivity factor, wind vectors, and other meteorological parameters, which are arranged into 3-D Cartesian coordinate grids. The first step of airborne PASIM is to map these data to a radar-centered 3-D spherical coordinate system grids, which is achieved using nearest neighbor (NN) interpolation. Next, we find each intersection volume between the radar resolution volume and the TASS Cartesian grid volume and compute the average of all the IWC values of these enclosed grid volumes. For each IWC value in the polar radar grids, the SCMC is used to produce averaged polarimetric weather radar variables. These radar variables serve as the basis for generating I/Q data.
An example output of the airborne-PASIM simulation is shown in Fig. 13. Fig. 13(a) depicts the IWC field sampled from one of the "must-detect" cases extracted from the TASS model simulations and confined by a PPI scan field of view (FOV) of an airborne encounter scenario. The IWC data in the Cartesian grid have been converted into the polar radar grid. Fig. 13(b) shows the averaged reflectivity scan generated using the IWC field using SCMC. After ingesting this reflectivity  field, the system simulator produces I/Q signal samples for each resolution cell and then uses the pulse-pair method to estimate the reflectivity values. Simulated PPI scans from the airborne radar are shown in Fig. 13(c) and (d), respectively, using different antenna beamwidth values related to a similar type of aperture (dish type, similar to existing radar products). In Fig. 13(c), the effective 3-dB beamwidth of the antenna was assumed to be 1 • (which can be either physical beamwidth, or equivalent beamwidth after superresolution processing). The effective antenna beamwidth is 3.5 • in Fig. 13(d). The effect of the antenna beamwidth on spatial resolution can be seen clearly, which will, in turn, affect the HIWC detection performance.
Assuming that a full dual-polarized airborne radar is available, scans of the key dual-polarized radar variables are also generated for this scenario, as shown in Fig. 14. Interesting observations exist in such cases about the potential values of using polarimetric radar variables as indication features. For example, the estimated K dp radar scans [see Fig. 14(b)] and the "truth" K dp values [see Fig. 14(a)] further confirm that the relationship between IWC and K dp is K dp ≈ IWC, as mentioned in Fig. 3(b) for X-band radar. The values of Z dr are almost constant as shown in Fig. 14(c) and (d), around 0.44 to 0.45 dB, which is plausible based on the axial ratios defined in the SCMC simulation runs.
However, using Z dr as a feature for HIWC classification will be insignificant since Z dr has minor changes versus IWC.

B. Detection and Estimation of HIWC
The previous sections established an end-to-end framework for evaluating the algorithms for detecting and classifying HIWC conditions through airborne radar sensor measurements. We conclude that with single polarized radar variables, certain relationships with IWC can be applied, but the estimation errors may be large. In this section, we intend to provide an illustrative example of algorithm evaluation based on simple ML (i.e., artificial neural network, or ANN). The combination of flight test data and simulation data provides sufficiently large amounts of training and testing data for the evaluation. The biggest challenge is how to select and apply the features and use these features for quantitative predictions. Even though we can observe from the above analysis that radar measured variables show different relations with IWC levels, we use all the measurable and relevant radar variables in the ML algorithm as features. These features include reflectivity, differential reflectivity, specific differential phase, and RIWC for both polarizations. By doing so, we can better focus on the making the optimal detection and estimation results achievable. One of the issues is the usage of temperature as a feature variable, which is further investigated through the following experiment.
When the temperature is included in the feature variables, correlation coefficients of 0.94 or more were achieved for the training, cross-validation, and test datasets. A classification accuracy of better than 86% is achieved, as shown in Fig. 15(a) and (b). Note in Fig. 15(b), 1 represents IWC > 1 g/m 3 , and 0 represents IWC ≤ 1 g/m 3 . When the temperature is excluded from the feature variables, the correlation coefficients of around   0.92 is achieved, and classification accuracy of better than 85% is achieved, as shown in Fig. 16(a) and (b). The results verified that temperature is not a critical feature variable for HIWC detection, even it is important for a possible precondition for HIWC. Similar tests can be performed for different combinations of feature variables to attain further optimizations of ML algorithms.
To investigate the effectiveness of dual-polarized radar variables as features and provide examples of using the airborne-PASIM for system designs, we use Z h , K dp , and Z dr as input feature variables and use the same NWP weather field as in Fig. 13 as the test dataset. Z h , K dp , and Z dr from the SCMC outputs are used as the training datasets. The ANN is designed and optimized herein as a predictor-classifier for IWC values that assign each quantized IWC value as a class. Again, the reference IWC field for airborne-PASIM (the output variable for the test dataset) is taken from the TASS output IWC grid instead of actual field measurements. For each radar resolution cell spatially distributed overall, a PPI scan, reference radar variables are computed by averaging SCMC outputs, and I/Q signal samples of it are generated for both H and V channels based on an airborne encounter geometry. Z h , K dp , and Z dr are estimated using these I/Q data, and then, they are fed to the ANN (which is trained using SCMC outputs) to predict the IWC values. The results of IWC prediction are shown in Fig. 17(b). Compared to Fig. 17(a), a bias of IWC estimation for higher IWC values can be observed, while the overall result is promising and Fig. 17. Prediction of IWC levels using (Z h , K dp , and Z dr ) feature vector variables. (a) IWC values from TASS output field, (b) IWC values predicted by the ML algorithm (trained ANN using SCMC outputs). Unit of the colorbars is g/m 3 . validates the potential usage of polarimetric radar measurement for IWC estimation. A summary of root-mean-squared-error (RMSE) values of IWC estimation for this example is provided in Table II. Fig. 18 shows the ranking of the features used according to their importance. The ranking is calculated based on the univariate F-test ( [30]). It can be seen that for this example, the specific differential phase (K dp ) is the most important feature for estimating IWC. It is due to the linear relationship between IWC and K dp as seen in the SCMC outputs. The mean reflectivity values (Z h and Z v ) are of secondary importance, while Z dr and the variances of the reflectivity values (which include RIWCs) are less significant. This signifies the importance of K dp and the potential values of incorporating polarimetric airborne weather radar variables for HIWC detection and estimation.

V. CONCLUSION
This study attempts to develop a physics-based simulation tool and evaluation method for a future airborne weather radar, which may use dual-polarization measurements to detect and estimate HIWC conditions. The main contribution is the connection between the in situ probe-measured microphysical particle distributions and radar sensor measurements, as well as an "end-to-end" example of how to use radar measurements in ML-based algorithms. Dual-polarized radar variables, such as K dp , are found to be effective feature variables for IWC estimation. If the dual-polarization measurement is not available, on the other hand, effective HIWC indication using (Z v , RIWC), or (Z h , RIWC) may still be obtained. The physical model provides a statistical explanation of the radar variables and a physical interpretation of the RIWC when the PSD is fixed. The SCMC then provides statistical predictions of the radar variables when PSDs vary. The SCMC can then be incorporated into a full radar system simulation that is based on NWP-simulated weather fields as its inputs. Since the available field measurement data for various conditions, locations, and environments are still limited, future development will need to perform more flight campaigns, and include more forward-looking airborne radar data and possible dual-polarized measurement data.

APPENDIX MEAN AND VARIANCE OF REFLECTIVITY FACTORS
Suppose M is related to V r in (1) with Doppler frequency removed, it represents the complex voltage at radar base-band, resulting from N scattering sources with individual RCS of σ i and individual phase of ψ i . we have Considering random phases with uniform distribution from 0 to 2π between N particles, the mean of the RCS can be derived asσ Similarly, the variance of RCS can be written as Since we have After the expansion of the integrand of the aforementioned equation, any terms left with one or more exponential terms becomes zero as a result of the integration from 0 to 2π [ 2π 0 e jkθ dθ = 0]. The only terms that do not have exponential terms are obtained as a result of the product between two conjugate terms. As a result Yunish Shrestha received the Ph.D. degree in electrical and computer engineering from the University of Oklahoma, Norman, OK, USA, in 2022. He is currently working as a Systems Engineer with Veoneer, Pawtucket Blvd, Lowell, MA, USA. His research interests include radar system simulation and radar signal processing for multiple applications, application of machine learning algorithms to radar data to predict weather phenomena, applying machine learning techniques to radar sensing, waveform design for MIMO automotive radars, and deep learning techniques to detect and classify targets. He is currently a Presidential Professor with the School of Electrical and Computer Engineering, University of Oklahoma (OU), Norman, OK, USA. He was one of the founding faculty members of OU's Advanced Radar Research Center, served as a Technical Lead for the ARRC's multifunctional phased array radar development from 2008 to 2011, and is currently a Faculty Leader with the Intelligent Aerospace Radar Team. He is the Principal Investigator for the OU's Polarimetric Airborne Radar Operating at X-Band developments and leading the deployment missions related to radar and radios supporting advanced air mobility, autonomous vehicle systems, and air-surveillance services. He is one of the faculty fellow members of the Cooperative Institute for Severe and High Impact Weather Research and Operations, OU, and a representative of the OU at multiple Radio Technical Commission for Aeronautics special committees supporting industry standard developments for avionics.