## The library of rough hailstone scattering coefficients

Hail is one of the most destructive and costly high-impact weather phenomena. Hail detection and size estimation is a crucial part of the National Weather Service (NWC) mission. For short-term nowcasting and warning, the NWS uses its network of dual-polarization weather radars (WSR-88D) as well as reports by the public and storm spotters. Dual polarization weather radar has been used to detect hail for over 30 years. Specific polarimetric signatures are associated with hail, and these signatures are discussed in the reference of this library.

The library of hail scattering coefficients aims to fill the paucity of rough hail scattering coefficients. The hailstone scattering is typically approximated by spheroidal models. This leaves negative differential reflectivity Z_{DR} and low correlation coefficient ρ_{HV} signatures unexplained. Most of the low ρ_{HV} were explained as caused by hail shape approximation using spheroids. Currently, studies considering the scattering of naturally occurring hail are sparse. Therefore, creating a spheroid-based hail model with surface added roughness is the logical step forward. Spheroid-based rough hail models have the advantage over any other irregular model. Their shape, roughness, and size can be mathematically described and created to follow observed patterns in nature. Each of the models in the library is different from all other models as surface roughness is generated for each hailstone model separately.

Electromagnetic (EM) modeling of hailstones was done using the commercially available software WIPL-D Pro. It uses the Method of Moments (MoM) computational EM tool designed to analyze 3D metallic and dielectric structures. A hailstone is represented as an object composed of interconnected plates that define a boundary between two different materials. An appropriate refractive index is assigned to the inner and the outer medium setting boundary conditions. From the incident electric field and boundary conditions for both the electric and magnetic field, the matrix of equivalent induced surface currents can be computed. Equivalent induced currents are determined for each surface element and approximated by a product of unknown coefficient and higher-order polynomial basis function. The matrix system is solved to determine the “unknown” coefficients. These coefficients with the basis function are then used to determine the hailstone’s scattered EM field.

The library covers "dry" (D) and "wet" (W) hailstones with the equatorial diameter ranging between 5 to 100 mm. The equatorial diameter for rough hailstones is the average of the angular diameters in the equatorial plane. In the case of wet hailstone models, the water coating is added to the ice core for which the equatorial diameter is defined. Therefore, the equatorial diameter of the hailstone with water coating will be slightly larger than the value recorded. The water coating is determined using the maximal surface water content prior to shedding following according to (Rasmussen and Haymsfield, 1987) given by:

*m _{liquid H2O}* = 0.268 + 0.1389

*m*

_{ice}The filename is generated in the `꙱A##_d###_R##.cft`

format, where the ꙱ defines the type of the hailstone (D or W), A stands for axis ratio, while number signs (#) are used to denote axis ratio (range 0.1 to 9.9), diameter in mm (5 to 100) and roughness as a percentage of equatorial radius (2 to 14%). In the library, two types of surface roughness are defined: random roughness R, and modified random roughness M. The difference between these types is the maximum value of surface protrusion which is defined by the percentage of the equatorial radius (for R type) or a local radius (for M type). Therefore, hailstones with R type protrusions are always within the [-*r _{equator}*,

*r*] multiplied by the percentage number in the name, whereas protrusions for M type are within [

_{equator}*-r(ϕ, θ)*,

*r(ϕ, θ)*]multiplied by the percentage number denoted in the scatterer name.

Therefore DA07_d050_R02.cft contains scattering coefficients for 50 mm equatorial diameter hailstone with axis ratio 0.7 and randomly oriented surface the maximal roughness of 2% of the equatorial radius at each surface point. If we considered DA07_d050_M02.cft maximal roughness would be 2% of the local radius at each point. Thus maximal protrusion is 1 mm at the equator and 0.7 mm at the poles of M type roughness hailstone.

Each scattering coefficients (.cft) file is organized as a tab-separated file with a header. The header contains the frequency for which scattering coefficients are calculated, followed by the scattering coefficients. Table 1 represents how and what values for each scatterer are stored.

Frequency ### GHz Header | |||||||||
---|---|---|---|---|---|---|---|---|---|

ϕ | θ | Re(s_{vv}) |
Im(s_{vv}) |
Re(s_{hv}) |
Im(s_{hv}) |
Re(s_{vh}) |
Im(s_{vh}) |
Re(s_{hh}) |
Im(s_{hh}) |

Variables used in the file, defined by Table 1 are defined for the coordinate system in which equatorial plane of the hailstone is in the xOz and the symmetry axis is aligned with y axis. The azimuth angle ϕ is defined in the xOy plane in counter-clockwise direction from x axis, the elevation angle θ is defined as the angle between the xOy plane and the z axis. The elevation angle is given in the -90° to 90° range. Remaining variables are real (Re() ) and imaginary (Im() ) components of the scattering coefficients. Namely, s_{hh} is the horizontal (H) scattering for H incident, s_{hv} H scattering for V incident (depolarization for V incident field), s_{vh} V scattering for H incident (depolarization for H incident field), s_{hh} H scattering for H incidence.

### References to the library and its application

Mirkovic, D., D. S. Zrnic, V. Melnikov, P. Zhang, 2021: Effects of Rough Hail Scattering on Polarimetric Variables. *IEEE Transactions on Geoscience and Remote Sensing*, **59**, https://doi.org/10.1109/TGRS.2021.3091907.

Mirkovic, D., Zrnic, D.S., 2023: Library of rough hailstone backscattering coefficients at 2.8 GHz. *Sci Data* **10**, 456, https://doi.org/10.1038/s41597-023-02352-3

Mirkovic, Djordje; Zrnic, Dusan, 2022: Scattering coefficients of rough hailstones from computational electromagnetic models (NCEI Accession 0254298). NOAA National Centers for Environmental Information. Dataset. https://doi.org/10.25921/hftw-pr46.

### Usage notes

Coefficients are calculated at 2.8 GHz for dielectric permittivities of ice and water. εr-ice = 3.17 − j0.0017 and εr−H_{2}O=78.3−j12.1. Scattering coefficients are in units of a meter. The library data can be accessed in text or binary format. Whereas the library format is described and straightforward for the text files, loading the binary files is best achieved using the MATLAB command “dataBIN = fread(read,[65341,10],‘float’);” where the [65341,10] sets the proper data format (table format).