Difference between revisions of "Sighting a Weapon"

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(Sighting Precision)
(Sighting Precision: thunder storms... temp save)
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== Sighting Precision ==
 
== Sighting Precision ==
  
In the Section on [[http://ballistipedia.com/index.php?title=What_is_Precision%3F#Precision_in_Shooting Precision_in_Shooting]] we define the precision of the shooting system by the equation:<br />
+
In the Section on [http://ballistipedia.com/index.php?title=What_is_Precision%3F#Precision_in_Shooting Precision in Shooting] we define the precision of the shooting system by the equation:<br />
  
 
&nbsp;&nbsp;&nbsp;&nbsp;<math>
 
&nbsp;&nbsp;&nbsp;&nbsp;<math>
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However this equation assumes that the weapon has been sighted once and that the sighting holds for the weapon. Let's consider some different scenarios with multiple sighting adjustments. We will take as a given:
 
However this equation assumes that the weapon has been sighted once and that the sighting holds for the weapon. Let's consider some different scenarios with multiple sighting adjustments. We will take as a given:
  
* The value of <math>\sigma</math> for how individual shots are dispersed around the COI from previous experiments. (eg we shot 100 shots one one target to get the "right" number for <math>\sigma</math>).
+
* The value of <math>\sigma</math> for the Rayleigh Distribution for how individual shots are dispersed around the COI from previous experiments. (eg we shot 100 shots one one target to get the "right" number for <math>\sigma</math>).
  
* We'll use multiple targets in the experiment. The front target will be for our sighter groups and the back target to record the total shots over all groups. In other words we'll shoot a group of some sort, adjust the sights, and change the front target being sure to overlay it perfectly on the back target. Then we'll shoot again. 
+
* We'll use multiple targets in the experiment. The left target will be for our sighter groups and the right target to record shots after each sighting adjustment. In other words we'll shoot a group of some sort, adjust the sights, and change the left target, then shoot one shot at the right target.  
  
* We'll shoot <math>m</math> sighter targets with <math>n</math> shots per target such that <math>m >> n</math>. The effective consequence here is that we can asume that all the shots are independent
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* We'll shoot <math>m</math> sighter targets with <math>n</math> shots per target.  
  
* The overall precision of the shots on the back target will thus be:
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* After each sighter target we'll take one shot at the right target.
 +
 
 +
* The overall precision of the shots on the right target will thus be:
 
&nbsp;&nbsp;&nbsp;&nbsp;<math>
 
&nbsp;&nbsp;&nbsp;&nbsp;<math>
 
\sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 + \sigma_{Sighting}^2
 
\sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 + \sigma_{Sighting}^2
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'''Scenario 1''' We shot one sighting shot per target.   
 
'''Scenario 1''' We shot one sighting shot per target.   
  
The overall precision of the shots on the back target will thus be:
+
For a value of \sigma_{Sighting}^2 we start with XXXX which is the relationship for the variance of individual shots about one shot. can use the value from the Rayleigh Distribution for the variance based on the sigma value.
&nbsp;&nbsp;&nbsp;&nbsp;<math>
 
\sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 + \sigma_{Sighting}^2
 
</math>
 
  
For a value of  \sigma_{Sighting}^2 we start with XXXX which is the relationship for the variance of individual shots about one shot. can use the value from the Rayleigh Distribution for the variance based on the sigma value.
 
  
 +
where <math>\sigma</math>is the term fit from the Raleigh distribution
  
 +
'''Scenario 2''' We shoot </math>n</math> shots per sighter target.
  
where <math>\sigma</math>is the term fit from the Raleigh distribution
+
'''Scenario 3''' We shoot </math>n</math> shots per sighter target and consult Carnac for the true average <math>POA^*</math> for each group.
  
'''Scenario 2''' The weapon has been previously sighted. We know the <math>\sigma</math> value for how shots are dispersed around the COI from previous experiments. Now we position two targets one in front of the other. The front target will be for our sighter groups and the back target to record the total shots over all groups. We take our <math>n</math> sighting shots, and adjust the sights per the adjustment. The front target is replaced and the sighting process is repeated. We do this <math>m</math> times so that <math>m \gg n</math>. This means that the shots on the back target will essentially be randomly distributed and include additional dispersion due to the sighting variation.
+
'''Scenario 4''' We shoot </math>n</math> shots per sighter target and consult Carnac for the true average <math>POI^*</math> for each group.

Revision as of 19:47, 3 June 2015

"Sighting" a weapon involves trying to align its sighting device so that the Point Of Aim (POA) coincides with the Center Of Impact (COI). The COI is from a sample of the infinite number of possible shots, the population of shots. The statistical challenge with sighting a weapon is that given any shot dispersion, and every weapon has some, then we can only estimate the COI. In principle we only know that increasing the number of shots (samples) will improve our estimate of the COI.


Sighting Process

As Einstein said, "Everything is relative." So for the purpose of this wiki we need to develop some frame of reference around which we can analyze situations. The sighting process nails down the theoretical perspective.

Assuming that "perfect" horizontal and vertical sight adjustments can be made for the fixed target distance used for sighting, the process is to:

  • Take \(n\) shots at target
  • Measure the horizontal and vertical distance from each shot to the center of the target.
  • Calculate the mean horizontal, \(\bar{h}\), and vertical, \(\bar{v}\), distances from the sample COI to the target center
  • Adjust the Horizontal and vertical sights perfectly to correct for the error measured.

It is also assumed that the weapon can maintain this "zero" until the weapon is sighted again. So we're ignoring the real world problems of the weapon getting banged around, undergoing temperature changes, and a zillion other real world details that effect the ballistics of the weapon over time.

The only thing that we can measure is the COI for the shot pattern, and hence this is generally our point of reference since the wiki is primarily focused on precision not accuracy.


Sighting Analysis Assuming Rayleigh Distribution

To analyze the sighting bias we continue to use the symmetric bivariate normal model of shot impacts.

First using Cartesian coordinates, we assume that both the measurements along horizontal and vertical axis have a Normal Distribution with the same variance of \(\sigma^2\) and that the distributions are both independent and uncorrelated. Thus:

    \(h \sim \mathcal{N}(\mu_h, \sigma^2)\) and \(v \sim \mathcal{N}(\mu_v, \sigma^2)\)

From the \(n\) "sighter" shots and the mean horizontal, \(\bar{h}\), and vertical, \(\bar{v}\), distances have been calculated.

Sighting Accuracy

Sighting Bias

Breaking the sighting error into parts assuming each error source is independent and normally distributed we have:

Sighting Precision

In the Section on Precision in Shooting we define the precision of the shooting system by the equation:

    \( \sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 \)

However this equation assumes that the weapon has been sighted once and that the sighting holds for the weapon. Let's consider some different scenarios with multiple sighting adjustments. We will take as a given:

  • The value of \(\sigma\) for the Rayleigh Distribution for how individual shots are dispersed around the COI from previous experiments. (eg we shot 100 shots one one target to get the "right" number for \(\sigma\)).
  • We'll use multiple targets in the experiment. The left target will be for our sighter groups and the right target to record shots after each sighting adjustment. In other words we'll shoot a group of some sort, adjust the sights, and change the left target, then shoot one shot at the right target.
  • We'll shoot \(m\) sighter targets with \(n\) shots per target.
  • After each sighter target we'll take one shot at the right target.
  • The overall precision of the shots on the right target will thus be:

    \( \sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 + \sigma_{Sighting}^2 \)

What we want to understand is how \sigma_{Sighting}^2 changes with different sighting process.

Scenario 1 We shot one sighting shot per target.

For a value of \sigma_{Sighting}^2 we start with XXXX which is the relationship for the variance of individual shots about one shot. can use the value from the Rayleigh Distribution for the variance based on the sigma value.


where \(\sigma\)is the term fit from the Raleigh distribution

Scenario 2 We shoot </math>n</math> shots per sighter target.

Scenario 3 We shoot </math>n</math> shots per sighter target and consult Carnac for the true average \(POA^*\) for each group.

Scenario 4 We shoot </math>n</math> shots per sighter target and consult Carnac for the true average \(POI^*\) for each group.