Difference between revisions of "Sighting a Weapon"
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* Measure the horizontal and vertical distance from each shot to the center of the target. | * Measure the horizontal and vertical distance from each shot to the center of the target. | ||
* Calculate the mean horizontal, <math>\bar{h}</math>, and vertical, <math>\bar{v}</math>, distances from the sample COI to the target center | * Calculate the mean horizontal, <math>\bar{h}</math>, and vertical, <math>\bar{v}</math>, distances from the sample COI to the target center | ||
| − | * Adjust the Horizontal and vertical sights perfectly to correct for the error. | + | * 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. | 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. | ||
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== Sighting Accuracy == | == Sighting Accuracy == | ||
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=== Sighting Bias === | === Sighting Bias === | ||
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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 /> | ||
| + | |||
| + | <math> | ||
| + | \sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 | ||
| + | </math> | ||
| + | |||
| + | 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 overall precision of the shots on the back target will thus be: | ||
| + | <math> | ||
| + | \sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 + \sigma_{Sighting}^2 | ||
| + | </math> | ||
| + | |||
| + | What we want to understand is how \sigma_{Sighting}^2 changes with different sighting process. | ||
| + | |||
| + | '''Scenario 1''' 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 one sighting shots, and adjust the sights. The front target is replaced and the sighting process is repeated. We do this <math>m</math> times. This means that the shots on the back target will include additional dispersion due to the sighting variation. What is the precision of the overall measurement for the shots? | ||
| + | |||
| + | The overall precision of the shots on the back target will thus be: | ||
| + | <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''' 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. | ||
Revision as of 19:16, 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 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 \(\sigma\)).
- The overall precision of the shots on the back 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 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 one sighting shots, and adjust the sights. The front target is replaced and the sighting process is repeated. We do this \(m\) times. This means that the shots on the back target will include additional dispersion due to the sighting variation. What is the precision of the overall measurement for the shots?
The overall precision of the shots on the back target will thus be: \( \sigma_{System}^2 = \sigma_{Weapon}^2 + \sigma_{Ammunition}^2 + \sigma_{Shooter}^2 + \sigma_{Sighting}^2 \)
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 The weapon has been previously sighted. We know the \(\sigma\) 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 \(n\) sighting shots, and adjust the sights per the adjustment. The front target is replaced and the sighting process is repeated. We do this \(m\) times so that \(m \gg n\). This means that the shots on the back target will essentially be randomly distributed and include additional dispersion due to the sighting variation.
