by Bruce Daniel

HEX Systems Pty Ltd

18^{th} June 2012

## Introduction

Shared-target shooting is a practice whereby two or more shooters fire on a single target. With traditional targets and manual marking this is a successful method, and is often known as ‘Bisley style’ shooting, named after the Bisley range in Surrey, England, where competitions are often held in this way. Each shot is manually marked and shooters take turns to fire.

With the advent of electronic targets (ETs), shooters and target manufacturers are now exploring the possibilities for random shooting on shared targets. The main advantage of such an arrangement would be the reduced capital cost of the ET system, because fewer targets are required for any given number of shooters.

While there are a number of problems with shared-target shooting, this article focuses on the two most important ones, both of which arise when two or more shots are fired almost simultaneously. The first is the increased risk of ‘collisions’ – when two shots from different shooters strike the target within so small a time interval that one or both cannot be detected. The second problem is the increased risk of a ‘crossed shot’ – when one projectile overtakes another during flight, resulting in the shots being assigned to the shooters incorrectly.

Designers are looking at ways to reduce the risks of collisions and crossed shots to levels that would make shared-target shooting viable.

This article looks at the principles of shared-target shooting, the issues of collisions and crossed shots, and the potential solutions.

## Principles of shared-target shooting on ETs

ETs typically employ an array of acoustic sensors mounted in the target. These sensors detect the high-energy acoustic wave generated by a passing projectile. The time difference between the various sensors detecting the acoustic wave is used to determine the position of the shot.

ETs used for shared-target shooting must have additionally some means of identifying which shooter fired each shot that falls on the target. This is most important when two or more shots occur almost simultaneously – too close to be discerned by human observers. To date this has been done with ‘muzzle blast detection’, which detects the timing of the shots fired from the rifles and correlates them with the impacts on the target.

In its simplest form, an ET with muzzle blast detection would assign the first shot that strikes the target to the first shooter that fired. This method is based on the assumption that the two projectiles in question have identical flight times (i.e.: identical average speed over the distance).

## Collisions

As a projectile passes through a target, the high energy acoustic wave strikes the rubber faces, the sensors, the target frame and other hardware, causing them to vibrate for a short period of time thereafter. The sensors themselves resonate with the sudden burst of energy, as do the associated electronics. During this resonance, or ‘ringing’ time, the target is unable to detect another projectile that may be following. The electronics must ignore sensor signals during this ringing time. We call this the ‘deaf’ period; typically it is about 0.03 s (seconds).

If a second projectile were to strike a target during the deaf period following a first projectile, the second shot would not be detected and the shot would be lost.

How likely are such collisions to occur in practice? Taking as an example two shooters sharing a single target, each firing at random at an average rate of one shot per minute, the probability of at least one ‘collision’ (that is, two shots hitting the target within 0.03 s of each other) during a 3-hour shoot is about 16%, or a chance of 1 in 6. With three shooters sharing the target, the probability is about 42% (or 2 in 5); with four shooters it is about 66%, or a better-than-even chance.

Put another way, over a year of shooting (say 40 sessions at 3 hours), with two shooters one could expect on average about 7 collisions; with three shooters about 22; and with four shooters about 43.

The risk of collision can be reduced to some degree through design by reducing the ringing time, but it cannot be eliminated. The risk of collision has nothing to do with the muzzle blast detection system, the shooting distance or variations in projectile velocity – it is purely a matter of when the bullets hit the target.

## Crossed shots

As we said earlier, a simple shared-target ET with muzzle blast detection works on the assumption that any two projectiles have identical flight times. Unfortunately, in reality this is rarely the case.

Even with a consistent batch of ammunition, fired sequentially from a single rifle, there will always be random variations in velocity. And many other variables may be involved – between individual rifles, between types and batches of ammunition, with random variations within a batch of ammunition, and of course between class and calibre of rifle. With any of these variations, it is quite possible that a faster projectile may overtake a slower one during flight when fired almost simultaneously.

Fullbore projectile muzzle velocities commonly vary in the range 2,800 to 3,100 fps (feet per second). Over a distance of 900 m this difference in velocity amounts to a difference in flight time of about 0.3 s.

Let’s now take an example where two shooters, firing over 900 m, are sharing a single target that has a muzzle blast detection system. Shooter 1 fires at a muzzle velocity of 2,800 fps, and Shooter 2 at 3,100 fps. If Shooter 2 fires 0.2 s *after* Shooter 1, his shot would arrive at the target 0.1 s *before *Shooter 1’s shot. The system would register both shots (because the deaf period following the first impact would have passed) and would proceed to *incorrectly attribute* the first-received shot to Shooter 1 – a ‘crossed shot’. And no one would be the wiser. In practice, this might mean one shooter being credited with another’s inner four!

In this particular example, a ‘protection interval’ of at least 0.3 s would be needed to eliminate the risk of crossed shots. The effect of the protection interval is that, for any two muzzle blasts occurring less than 0.3 s apart, both shots may be registered but the system must at least notify the users that the shots are “too close to call”.

Once again, however, the real world is not so kind. Fullbore projectiles *do* have muzzle velocities outside the range 2,800 to 3,100 fps; even with precision equipment there are still ‘outliers’; and other calibres most certainly do have different velocity ranges. This means that even with a more conservative protection interval of 0.5 s there still remains a risk that near-simultaneous shots will occasionally be falsely attributed. A 0.5 s protection interval would be inadequate if for example two different calibres of rifle were firing side by side.

It is worth pointing out that the risk of crossed shots due to velocity variations is most pronounced at longer ranges, which is why we have used 900 m in our example. At shorter ranges, the protection interval can be significantly reduced. For example, over 300 m the difference in time of flight between 2,800 and 3,100 fps is about 0.05 s. In contrast, the risk of *collisions* – that is, as we discussed earlier, when a second shot arrives during a deaf period – is independent of the range.

And what are the chances that two shots will be fired within say 0.5 s of each other? Following our earlier example, with two shooters each firing randomly on a single target over 900 m at an average of one shot per minute, the probability of at least one “too close to call” event (i.e.: within a 0.5 s protection interval) during a three hour shoot would be about 95%, and on average 3 events would be expected. With three shooters one could expect about 9 events in 3 hours; and with four shooters, about 18 events in 3 hours!

What this means is that, with four shooters firing on a single target, near-simultaneous shot events (i.e.: within an interval of 0.5 s) will occur 18 times on average. And since each event involves two shots, that’s about 36 questionable shot results from a total of 720 shots over three hours! But if the system did not notify shooters of these events, there would be 36 opportunities in one day to swap someone’s bullseye five for someone else’s inner four!

It is worth clarifying – the events we are talking about here are where there is a *risk* of a crossed shot, not a known crossed shot event. But it is still very important because shooters need to feel confident that the shots they are given actually belong to them. And they need to be informed whenever there is reasonable doubt.

To complete the picture, at the shorter range of 300 m, using a protection interval of 0.07 s, over 3 hours, with two shooters the probability of an event would be about 34%, or 1 in 3; with three shooters it would be about 72%, or better than even; with four shooters it would be about 92%, or a likelihood of two or more events during a three hour shoot. Clearly the likelihood is much lower at 300 m than at 900 m, but it’s still significant nonetheless.

By the look of these numbers, if you’re using a shared-target ET system and you’re unfamiliar with the experience of receiving “too close to call” warnings, it may be time to start asking questions about how your system works!

## Some real-world data

All of the above numbers are based on probability analysis. The maths is not exactly simple but it’s very well established. And we happen to have a PhD in Applied Mathematics on our team, so we are very confident of our figures!

The Appendix at the end of this article sets out the statistical principles on which the theoretical probability values stated in this article are based. They may be independently verified by any suitably qualified person.

Nevertheless, we wanted to validate our calculations against real-world data, so we looked at the shot data logged by our own HEXTA-001 systems in real use. While our shot detection system has a resolution better than 0.5 microseconds (necessary for accurate position determination), it keeps logs of shot events only to the nearest 1 second. So we were only able to compare our log data with a theoretical probability calculation using a 1 s protection interval – a bit large, but valid for the purposes of comparison.

We examined the data logs from a number of shooting sessions in which two targets were used simultaneously – not as shared targets, but as two lanes in parallel. Each set of logs covered a day’s shooting – typically about three hours, including breaks. We recorded the number of instances where the ‘time stamps’ of shots on both targets were identical, meaning that the shots had occurred in the same 1 second. Because the shooters in the two lanes were shooting randomly and independently, it was reasonable to assume that, had they been shooting on a *single* target, the results would have been similar. In this way we were able to simulate the random shot behaviour of two-shooter shared-target shooting, with a protection interval of 1 s, while being absolutely sure which shooting position owned the shots.

It is important to note that there is an error in this experimental method, namely that the logs only indicate events where two shots fall *in the same 1 second of time*, which is different to events where two shots fall *within 1 second of each other*. For example, two shots may be very close together in time but fall in different (consecutive) seconds. Because the experimental method does not include such events, it will have a tendency to underestimate the true number of events that occurred.

The data covered five different days of shooting. We selected these logs randomly from our archive of log data. The total number of shots per day ranged from 200 to 420, and averaged 290 per day (or, on average, 145 shots per lane per day). The number of ‘simultaneous’ events identified per day ranged from 2 to 10, and averaged 5 per day – that is, 5 instances of shots occurring within the same 1 second.

By comparison, the *theoretical* event rate under the same conditions is 3.8 per day.

So, even though the experimental method has a tendency to underestimate the number of events, the actual number of events it recorded was greater than the theoretical number. Such a discrepancy is reasonable considering that the sample space (in this case, the number of days shooting) was quite small. Nevertheless, it does give an ‘order of magnitude’ indication that the theoretical predictions are valid.

## Where does all this leave shooters?

In our designs, we at HEX Systems have always striven to *eliminate* the chance of error – or at least to minimise it – wherever possible. Our efforts in this regard were redoubled following the events of the Delhi Commonwealth Games.

We therefore do not advocate *random* shared-target shooting, *except* in situations were the users are fully aware of the risks involved and have taken the decision to accept them. And such a situation can only be possible if the ET system and the manufacturer *inform* them of the risks.

The only other way to mitigate these risks in shared-target shooting is to employ a system of shot management in which shooters are queued to shoot in turn – something closer to Bisley-style shooting. This is certainly not out of the question, but it comes with a whole range of issues of its own!

## Appendix – Explanation of theoretical probability calculations

*This Appendix sets out the statistical principles on which the theoretical probability values stated in this article are based.*

The goal is to calculate how many times a shooter has to shoot to achieve a particular probability P of collision. Let’s assume that every shooter shoots at random intervals, but on average these intervals are A seconds apart. The third necessary parameter is the collision interval B (seconds) which is the minimum time between any two bullets reaching the target that results in no collision event. Note that under normal conditions A is significantly greater than B.

In case of two shooters let’s assume that one of them shoots independently and the other one creates potential collisions. The shots of the first one can be marked on a time line as points. Each point corresponds to the moment when a bullet hits the target. The segments are relatively far apart at random intervals. In order for the second shooter’s bullet to create a collision event it has to hit the target later than B seconds before a point and earlier than B seconds after. So the moment of impact has to lie within a 2B segment surrounding one of the points on the timeline. For every shot (bullet hit) of the second shooter there is one (on average) impact point on the timeline on the interval of A seconds. Therefore, for any single shot of the second shooter the probability of creating a collision is 2B/A. And the probability of successful shot without collision is P_{s} = 1 – 2B/A.

Example: A = 60 s; B = 0.02 s; Ps = 1 – 0.04 / 60 ≈ 0.9993 or 1499 in 1500 chance of success.

An easy way to combine probabilities for multiple shots is to calculate the probability of a completely successful set of shots, i.e. no collisions at any time. If we know the probability for N shots P_{N} we can calculate one for N+1 as P_{N+1} = P_{N} P_{s} because each consecutive shot event is independent of previous ones. Therefore, probability P_{N} = (P_{s}) ^{N}

Example: A = 60 s; B = 0.02 s; N = 600; P600 = (1 – 0.04 / 60) 600 ≈ 0.67 or 2 in 3 chance of success.

Calculating how many shots the second shooter has to make to achieve the probability P of at least one collision is equivalent to calculating the number of shots to get 1 – P probability of all successful shots. So we would like to know to which power N we have to raise P_{s} to get 1 – P:

Example: A = 60 s; B = 0.02 s; P = 0.5; N = ln(1 – 0.5) / ln (1 – 0.04 / 60) ≈ 1039.

If the number of shooters M is greater than two we can split them up in pairs. The number of unique pairs is the number of combinations of 2 out of a set of M or:

This number for 2 shooters is 1, for 3 is 3, for 4 is 6, etc.

The collision events for each pair can be considered independent and therefore the formula for the probability of success at each shot will be (P_{S})^{C(M,2)}, i.e. P_{s} to the power of C(M,2). And the formula for N when the number of shooters is M becomes:

Example: A = 60 s; B = 0.02 s; P = 0.5; M = 4; N = ln(1 – 0.5) / (6 * ln (1 – 0.04 / 60)) ≈ 173

It is also useful to know how many collisions we would expect to see on average in a given time period. To calculate this value we can use the probability of success Ps in a single shot interval. The probability of seeing a collision is (1 – P_{S}) and the expected number of collisions per interval equals this value. The expected number of collisions in a longer time period can be calculated as a sum of the expected number of collisions in all single shot intervals over this period.

To explain further: this works similarly to calculating the expected number of times to get heads in coin tosses. Each time a coin is tossed the probability of getting heads is 0.5. So, the average number of times we would see heads in a coin toss is 0.5. Then if the coin is tossed 100 times the expected average number of times of seeing heads is 0.5 * 100 = 50. In case of the roll of a die, the probability of rolling a six for a single die throw is ⅙. And if the die is rolled 600 times we would expect to roll six 600/6 = 100 times on average.

In case of shooting, if we look at a time interval of T seconds, there will be T/A single shot intervals. And the expected average number of collisions will be

for 2 shooters or for the more general case of M shooters. Again the collision events are independent for every unique pair of shooters and the expected average values can be added together. This, for example, is similar to the case when several people roll dice simultaneously several times. The expected number of times to roll six is ⅙ times the number of people times the number of times they rolled the dice together.

## Leave a Reply