Hail
To do hail justice, GWG has spent a lot of time on the concepts of hail here, and to that end, there's a table of contents here to outline the various concepts and principles of hail. To learn even more than what is supplied on this page, please schedule a technical consult to find out how GWG can help!
Table of Contents
Formation
Hailstones form in and around the updraft of severe convective storms. A fact not widely known, is that liquid water can exist in a "supercooled" state down to -40°C! At these temperatures, even the slightest disruption to the surface of a supercooled liquid water droplet will cause it to instantly freeze (such as collision of small ice particles with supercooled liquid water). As this supercooled liquid water is lofted by a storm, it runs into other particles within the storm. These collisions continually grow our embryotic hailstone until it is a size large enough to fall back out of the storm. Hailstone size can depend on many different parameters, however, of paramount importance is how long a growing hailstone can stay within this region of supercooled liquid water.
Hail size vs. Particle Size Distribution (PSD) vs. kinetic energy vs. hail probability
That's a lot of words! To understand hail as a true meteorological peril, it is worth digging into these concepts.
Hail Size
Although widely accepted and taught as the way to report hail, hail size is a terrible measure of potential for hail to do damage. Why? Not all hail is created equal! Some types of hail are more spongy and "splat" when they hit a surface. Other types of hail may be hard as a rock and generally as dense as pure ice. These hailstones tend to bounce and may show no sign of damage after hitting something. And finally, other hail may be somewhere in between that will crush, crack or split when it impacts a hard surface.
If you were to take a 3 different hailstones, one spongy, one of "medium" strength, and one of pure ice, and make them all the same size, which one do you think is going to cause more damage to whatever it hits? Hopefully, you said the "pure ice" hailstone. This alone, can be cause of some cases where we see that hail of the same size is reported at a location, but some structures/property had damage while others did not.
Particle Size Distribution (PSD)
In a microphysics class, I was once taught that the "particle size distribution is the holy grail of meteorology." I'll admit, at the time hearing that I didn't think much of it. Here, almost 20 years later, I'm a firm believer. Keeping in mind some of the challenges associated with linking maximum hail size to damage, one paramount concept neglected by picking a single hail size as representative of all hailstones is that it begins to lead people to think that other hail that took place in the same storm may not have caused damage. This is a really dangerous assumption!
Let's think of it this way, if a hailstorm drops hundreds of thousands of hailstones over your property, what is the chance that you find the one that's the biggest to measure? Even if you DO find the biggest hailstone, what about all those others you kicked out of the way to find the one big one? What is the chance the one big one hit your roof, or your windshield? By only selecting the maximum sized stone, we've ignored a plethora of hailstones which happen in higher quantities and may still be of damaging size! This is where knowing the size distribution of all of these hailstones becomes increasingly important.
If we look at this example particle size distribution, the blue arrow is pointing to 2" hail. Pretending that's the maximum size hail for an event, the statistical chances of you finding that hailstone are extremely small! However, if we look to the green arrow, there are somewhere between 10 and 100 TIMES more 1.5" hailstones than there are 2" hailstones! A structure or piece of property impacted by numerous 1.5" hail may see just as much, if not more, damage as a structure hit by a single 2" hailstone!
Kinetic energy
Anything that has mass and velocity has kinetic energy. Hail is no exception. In fact, in a perfect world, this energy is what is transferred to property and causes the damage. If a material cannot effectively dissipate the kinetic energy from the hail, catastrophic material failure occurs. In this sense, estimations of the kinetic energy from hail is the ideal what to predict damage to structures and property.
To accurately determine kinetic energy, one can start with first principles...however, here at GWG we don't want to put you through a math class! If one is willing to accept that we need to multiple the PSD (described above) by the fall speed of the hail stones, and accumulate them over some area, then it is possible to come up with an estimate of kinetic energy! Using this estimate, one can then see how much energy a hail storm actually imposes on structures/property.
Hail probability
When GWG discusses hail probability, it may be different than some expectations. For instance, if you were to hear "there's an 80% probability of hail over your house!" What does that statement mean to you? Does that mean there's an 80% chance of any sized hail at your house? Does one single hailstone of any size satisfy that probability?
Not surprisingly, GWG relates probabilities back to the PSD. To remove this ambiguity, GWG is able to discuss hail probability in terms of what size hail had what percent chance of impact a location over a specific time. PHEW! That is a mouthful. To put this into a concrete example, when necessary, GWG has methods to estimate the PSD from radar data and give an estimate similar to "during this storm, 1.5" hail had a 75% chance of impacting a 2,000 sq. ft. roof."
Detection/inference from radar
I (Scott) was once told that radar doesn't "detect", it "infers". While this is nuanced, it does carry a certain amount of weight to it, especially when we consider how hail "detection" algorithms operate. In general, hail detection algorithms do not rely on directly quantifying radar returns (such as [horizontal] reflectivity factor, differential reflectivity, [cross] correlation coefficient or specific differential phase), but instead rely on how and where these values are changing or observed.
Seen here is an image of 0.5 degree elevation reflectivity from the San Antonio radar during a hail event in April of 2016. Looking just at reflectivity alone isn't sufficient to accurately estimate the maximum hail size in a storm. However, if we introduce 2 polarimetric variables, differential reflectivity and (cross) correlation coefficient, a better picture begins to emerge. 
Here we can see that co-located with higher values of reflectivity, we see lower and even negative values of differential reflectivity. Also, in this same area, we see that correlation coefficient values are also low, less than 0.9 in some places. Looking at the overlap of these areas would indicate the presence of hail greater than 2".
So how did we arrive at this specific size of 2"? Newer, polarimetric radar algorithms focus more on the presence of liquid water, which acts as a marker for melting hail. By looking for signals in this meltwater, we are able to more accurately estimate hail sizes within a storm when compared to legacy algorithms (outlined below).
Other indicators of hail sizes within a storm are three-body scattering signatures (or "hail spikes they are sometimes called), and signs of resonance scattering. If we break down these into bite-sized pieces, they are easier to understand and help us paint a more accurate and scientifically reasonable estimate of the distributions of hail sizes within a storm.
Three-body scattering signatures, TBSS:
To understand TBSS, we first have to think of individual hailstones as small little antennas. These antennas interact with the electromagnetic waves from the radar, and some of that energy they emit (reflect) back to the radar, some of the energy is absorbed by the hailstone, and some of that energy is re-emitted in all directions from the hailstone.
Resonance scatterers:
Many automated algorithms claim skill in identifying large to giant hail (hail above 2" in diameter). However, there are many challenges to these claims. One of which being an effect called resonance.
Legacy algorithms
For the sake of clarity, what I mean by "legacy" here are algorithms which do not rely on dual-polarization radars. These algorithms have classically relied solely on how reflectivity changes with height. The elementary basis being, the higher the reflectivity higher above the ground, the greater probability of hail being present both aloft and at the surface.
Polarimetric algorithms
Polarimetric radars are incredibly sensitive to the presence of liquid water. This sensitivity forms the basis for the automatic discrimination of hail of various sizes with, for example, the polarimetric upgraded NEXRAD radars used by the National Weather Service (and other entities) in the United States.
Take me back to the learning resources!
References
American Meteorological Society, 2012: AMS Glossary. Accessed March 2023,
https://glossary.ametsoc.org/wiki/Hail
Markowski, P., and Y. Richardson, 2010: Mesoscale Meteorology in Midlatitudes. Wiley-Blackwell, 407 pp.
Ryzhkov, A.V., M. Kumjian, and S. Ganson, 2013: Polarimetric Radar Characteristcs of Melting Hail. Part I: Theoretical Simulations Using Spectral Microphysical Modeling. J. Appl. Meteor., 52, 2849-2870.