Wednesday, June 22, 2022

The physics of determining the distance from the gun to a microphone, and why it is nearly IMPOSSIBLE for the IDF to have shot Shireen (UPDATE)

As we've seen, all of the "investigations" of Shireen Abu Akleh's death that blame the IDF hinge on a single piece of evidence: audio analysis of the bullet sounds on different videos to determine the  distance from the shooter to the camera. 

Every single analysis makes the same wrong assumption: that the estimates by the experts consulted has some wiggle room that would allow the IDF to be within the range of the source of the gunfire.

Every investigation noted that the IDF was outside the range of what they had asked their experts to determine.  Every one fudges the data from the experts to indict the IDF. Crucially, not one of them went back to these experts and asked whether their calculations could possibly support the IDF shooting Abu Akleh.

Here is a short lesson of the physics and assumptions made by the experts to calculate the distances.

The guns used by both sides shoot bullets much faster than the speed of sound. Objects that travel faster than the speed of sound create a shockwave (in the case of aircraft, a sonic boom) that can be heard by those near the path of the object. This is an illustration of a shockwave for something traveling 1.4 times the speed of sound:

Most bullets travel significantly faster than the speed of sound. Here's a photo of the shockwave from a bullet.

The easiest way to picture this is to think about the wake of a speedboat on a lake.  The faster the bullet, the narrower the "wake." When the wake passes by one's ear or a microphone, you hear a clap sound. If the bullet is not shot in the general direction of the ear or microphone, no shockwave sound is heard at all.

The muzzle of the gun also produces a sound when the bullet is fired, the "bang." That bang travels at exactly the speed of sound from the gun, at the same speed in all directions. 

By measuring the difference in time between the "clap" and the "bang," we can calculate the distance of the gun.

Everyone agrees that Abu Akleh was killed by a 5.56mm bullet. Everyone agrees that both the IDF and militants in Jenin use weapons (M4s and M16s) that use those bullets. 

To determine the distance of the gun to the microphone, we need to know a few things:

* The speed of sound.
* The speed of the bullet between source and where the sounds are heard.
* The time gap between the sound of the shockwave to the sound of the gunfire.

Assuming that the listener/microphone is reasonably close to the bullet path, this gives a very good approximation of the distance. 

We know the speed of sound at various temperatures. 

We know the time gap from the videos - between roughly 295 ms and 310 ms.  Here are the last two gunshots from the first set of gunshots from audio analysis tool Audacity:

The speed of the bullet is variable, depending on the gun type. And keep in mind that since bullets sllow down, we want to know the average bullet speed at that distance for this calculation, not the muzzle speed which is always faster. This chart shows the speed for 5.56 mm bullets using various types of guns at various distances.  

The speed at 100 yards would be roughly the average speed of the bullet that traveled a total of 200 yards/meters, so it is a good approximation. Also, the slowdown slope over distance is pretty linear so we can take a good guess that an M16 with a muzzle speed of 960 m/s would have an average speed of about 880 m/s over 200 meters, and an M4 with a muzzle speed of 905 m/s would have an average speed of 824 m/s. 

This is the data that was used to determine the distance of the shooter to the microphones, with each investigation using somewhat different assumptions on bullet speed - but all of them came out with a range of between 155 meters and 195 meters for the distance to the gun from the microphone.

Rob Maher, one of the experts consulted by Bellingcat, CNN and the New York Times, emailed me the formula and his assumptions of the range from the CNN article, so you can do this yourself with a spreadsheet:

Measured time-of-arrival difference between shock wave and muzzle blast:  Time_D

  Speed of sound:  c

  Bullet average speed:  V

  Distance of firearm to target:  D

      Time_D = D/c - D/V  = D*{(1/c) - (1/V)}

  Solving for D, we get

      D = Time_D / {(1/c) - (1/V)}

So with Time_D = 0.306, c= 347, and V= 762, we get D = 195 meters.

Or with Time_D = 0.306, c= 347, and V= 884, we get D = 175 meters.

As we've shown, the IDF was most definitely outside those ranges.  (This modified NYT graphic uses yards and a more generous range.)

The experts chose the most expansive ranges they could to account for all the variables, and even then, the IDF is well outside the possible range.

Which means that the IDF couldn't have fired the shots that killed Shireen Abu Akleh - unless they moved within range when no one was looking, or there was a hurricane level wind gust going south at the moment of the gunfire, or if the IDF uses a gun with a much slower bullet velocity than of any known gun that uses 5.56mm bullets.

To be very accurate, we also need to know:

* The temperature
* The wind speed
* The distance from the ear/microphone to the path of the bullet. The further away they are, the longer it takes the shockwave passing by to hit the listener, and that makes the calculations a bit more complicated.

Those factors cannot possibly explain the discrepancy between the ranges calculated by the experts and the IDF position.* (see update) Yet not one of these analyses bothered going back to the experts and asked them if there was some other X factor that could explain the discrepancies.

One other important fact: You will notice that there is quite a difference between the time gaps of the two gunshots I placed in my graphic above - 295 ms and 306 ms. Assuming that the source of the gunfire is the same position (which is reasonable), this indicates that the shooter was not shooting each bullet with the same trajectory. If he was shooting wildly, and the direction of the bullets changed by a degree or two, that would explain such a discrepancy - the distance from the shockwave to the camera would change if the bullets were being sprayed across a larger area, and that would easily account for such a discrepancy in gunfire from the same gun.

Dr. Maher was nice enough to illustrate this for me as well:

Such a wild shooting pattern would be far more likely from Jenin terrorists than from the professional soldiers of the IDF, in my opinion. (Indeed, we saw bullets sprayed on the tree, on Abu Akleh and Ali Samoudi, in a radius that is far larger than that of a trained shooter aiming at a target.)

(I had also asked Dr, Maher if a bullet stopped by a tree or person would have a different time gap, but the answer is no - the shockwave "wake" would continue on to the microphone exactly the same way even if the bullet stopped somewhat short. If it stopped too short, there would be no shockwave sound at all.)

The media didn't ask the experts whether there was any factor that could account for the IDF shooting Abu Akleh at this distance. They didn't ask about the discrepancies between the audio gaps. They were trying to fit the data to their preconceived verdict that the IDF fired the weapons.

The liberal elements in the media often disparage the more right-wing media as not believing in science. This is science. And all the supposedly objective media - Bellingcat, the New York Times, the Washington Post, CNN and AP - literally ignored the science and fudged the data to make it appear that the IDF was in the range indicated by the audio forensics.  

It wasn't.

Every one of these media outlets tried to hide the science that would exonerate the IDF according to their own experts.That is a scandal. 

UPDATE: Rootclaim tweets that the distance from the mic to the bullet may indeed put the IDF guns in range.

 the calculation doesn’t account for the distance from the arriving bullet to the microphone. 
The bullets are known to have hit the journalists and a tree that were 10-15 meters from the camera. That means 30-45 milliseconds should be added to the 300-310 ms delay measured in the soundtrack. 
This chart shows the distance to the shooter, as a function of the average bullet velocity. The blue line is the erroneous calculation and the red line shows the corrected distance (using 10m correction).Image
Most rifles used by both sides are around the 800 m/s range (after deceleration in air), meaning an increase of ~20 meters, to a range of ~200 meters, which better matches the IDF’s location. 
We publish this specific finding ahead of our full analysis as it is widely used in public discourse, causing confusion. However, we generally advise against using a single piece of evidence as a “smoking gun”, since they rarely are (as demonstrated here…). 
Only a detailed probabilistic inference, using all evidence, and accounting for all possible sources of error, can provide a reliable assessment of a hypothesis’ likelihood. We’ll be publishing our full analysis soon. 
The math here is a little beyond me, but I had asked Dr. Maher about this specifically. Here are excerpts of his response on how to calculate the distance when the mic is further away:

If more precise geometry is to be used, you might choose to take into account the Mach angle of the bullet’s shock wave near the microphone.  If the bullet has slowed to, say, some velocity between 690-790 meters per second as it reached the area of the microphone, the Mach angle is then somewhere between 26 and 30 degrees.  Here is a plan view sketch of what I am referring to.

If you have a good prediction of the bullet’s speed and its trajectory, AND if you assume the bullet passed by without striking anything before passing the microphone, you could use the time of the shock wave arrival to back-track where the bullet was at various times before and after the shock wave arrival.  I haven’t done any of that work in this case because of all of the unknowns about speeds, trajectories, and positions. In other words, I assumed the microphone was sufficiently close to the bullet’s trajectory that the shock wave propagation time to the microphone was negligible.

10-15 meters is not negligible, but there are still a lot of unknowns that make the calculations difficult.

And more recently he wrote something that indicates that Rootclaim's numbers may be in the right ballpark:

My sketch below is intended to show a simple plan view, with the firearm at the left and the bullet traveling to the right.  The black circles labeled 1 and 2 are indicating possible microphone locations.  We don’t actually know the microphone location relative to the bullet path.  The circular arcs are depicting the path of the muzzle blast sound moving outward from the gun.  The red lines are depicting the ballistic shock wave of the bullet at two different moments.



Assuming the two microphone positions are roughly the same distance from the firearm, the time-of-arrival of the muzzle blast at each position will be essentially the same.  However, the arrival of the ballistic shock wave will be different at the two positions.

The arrival of the shock wave at position 1 will be essentially the time it takes the supersonic bullet to travel from the gun to position X, which is essentially position 1.

The ballistic shock wave arrival at position 2, however, will be delayed because the timing consists of the time required for the bullet to travel to position X-Δ, plus the time it takes the shock wave to travel at the speed of sound from X-Δ to position 2.

For example, if I imagine a scenario which X is 180 meters from the gun, the average bullet velocity is 884 m/s, and the two possible microphone positions 1 and 2 are separated by 5 meters, then Δ=2.9 meters, the distance from X- Δ to position 2 is about 5.8 meters, and so the delay between the shock wave and the muzzle blast is about 321 milliseconds at position 1, but 308 milliseconds at position 2.  Since we don’t actually know the relative position of the microphone and the bullet’s trajectory, we have over 10 milliseconds of uncertainty due to the 5 meters uncertainty of trajectory difference.  If we do the same scenario but with an average bullet velocity of 762 m/s, the shock wave-muzzle blast delay is 289 ms at position 1 and 276 ms at position 2 (13 ms difference).

Keep in mind that a bullet that was shot a few meters over the heads of the individuals making the recording has this same sort of timing adjustment. 

I hope this helps give an idea of why there is uncertainty in the analysis, and why I tried to be very clear with all of the reporters and with you about the need to combine this acoustic  evaluation with physical evidence, witness accounts, and all the rest.

I was a bit overconfident in my analysis above, assuming that the distance to the mic was not great enough to affect the calculations mush. It appears I was wrong, and that there is a significant additional uncertainty to the distance calculations. I believe that the angle of the bullets to the target relative to the microphone would also be a factor - a bullet from the southeast would pass somewhat closer to the mic to the north.

If Rootclaim's numbers are correct, and up to 20 meters can be added to the calculations, then rifles with slower bullets could barely include the IDF position. Al Jazeera identified the IDF guns as M4s which do have a slower muzzle speed than M16s. 

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