Modeling the PDF of shotgun pellets at a Skeet/Trap range

Who is up for an interesting intellectual exercise?

Let's model the distribution of shotgun pellets at skeet and trap ranges.  I've started developing a model to predict the probability density solely based on the geometries involved.  

What is the probability of finding a pellet during a field sampling event? 

The image below shows the preliminary output of a single shot fired from each skeet (blue) and trap (red) position.  I've also incorporated an algorithm to randomly select the number of pellets to strike the target, thus falling out of the trajectory early in the field. 

Distribution of pellets:  A unique pellet distribution pattern was built for each shooting position.  From each skeet position, I modeled one shot trajectory for each target from the HH and LH.  This allowed me to use a Normal (Gaussian) distribution of pellets between the two trajectories, at 125 yards from each skeet-shooting position.  This was replicated for trap-shooting at 175 from the position.






How many pellets will drop early after striking the target?

Using a 30" circle at 21 yards, 85% of pellets would travel through.  This provides a general spread of the pellets as they travel from the gun bore.  Pellet density as a function of radius was calculated based on these parameters.  By comparing the pellet density at radius to the profile area of a standard clay target (4.43 in^2), I'm able to estimate the number of pellets to strike the target as a function of the distance from the center of the shot cloud.  I've created a random number generator to select the distance from the center for each shot between 0 and 30 inches, which supplies the number of pellets to strike the target. 

Then I weighted the number generator to provide more center-strikes than edge-strikes. The histogram below shows that the most strikes will drop 70 pellets.  In reality, the number of pellets that will strike a clay target are dependent upon the size, shape, and overall length of the 'Shot String'.  Here, I'm developing based on a 2D planar model of the shot string.  This complexity of a shot string is not time or cost effective at this time.  However, it seems that a range between 0 and 3% of the total pellet count would be sufficient to model the breaking zone due to the inherent complexities of the (football shaped) shot-string.

Where will these pellets land?
It was estimated that 90% of the pellet's kinetic energy would dissipate after striking the target, allowing for approximately 80 horizontal feet of additional travel for skeet-shots fired from positions 1-7, and 125 feet from position 8, and 60 feet for all trap-shots.  These distances provided a based from which distributions were developed for pellets that would fall out early from the trajectory after striking the target
The current probability distribution is coming along nicely.  I'll continue to develop this model and compare it against field evaluations.  The next steps will be to use multiple sizes of shot and incorporate potential PAH exposure from the clay targets.  Thank you all for helping to build these pieces.  The latest update includes an option for multiple/overlapping ranges.  Below is a point distribution of 4 ranges.