Circular Error Probable approximations for weapons testing

Circular Error Probable (CEP) is defined as “the radius of a circle centered at the target or mean point of impact within which the probability of impact is 0.5“.

CEP is used significantly in measuring weapons systems’ impact accuracy and some navigation systems accuracy. Approximations for CEP include non-parametric and parametric methods.

Non-parametric approximations make no assumptions about the underlying distribution of the impacts. One such method is simply to consider the sample median for CEP. The non-parametric methods can provide good results when the sample size is greater than 30, but are of limited use to flight testing especially when considering missile tests where the sample size usually not greater than 15.

Parametric approximations make an underlying assumption that the impacts are normally distributed. The CEP can then be computed either by closed form integral calculations, or by numerical calculation of integrals, or by algebraic approximations of the bivariate normal function.
The latest method can give very fast results, practical for flight tests and is very accurate in certain areas of the parametric space. Three parameters to consider for the impact distribution are bias, eccentricity and correlation. These parameters are of importance and should not be overlooked as they might provide some insight on the characteristics of the weapons system under test. Various algebraic approximations exist and they are discribed extensively in literature.

In case the distributions are zero mean Gaussians (bias=0) the main parameter to be considered is eccentricity and a simplified approximation can be used:

References:

[1] An examination of CEP approximation techniques
[2] AFOTEC Technical Journal CEP
[3] Use of Circular Error Probability in Target Detection

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