decaf
Posts: 91
Joined: 8/1/2008 Status: offline

Very nice work, Rory. What Rory has derived here is the probability density function (PDF) for the "Rayleigh distribution". This can be found at wikipedia.org. This is the PDF for *ranges*, when the underlying distribution is 2D, independent, bivariate, normal Gaussian distribution, with zero means in both X and Y. For the CMO single bomb drop model X has a zero mean (unbiased), with a dispersion (call it sigma) Y has a zero mean (unbiased), with the same dispersion (note that both X and Y can be positive or negative, hence the zero mean) R is range, R = (X*X + Y*Y)^0.5 (from Pythagorean Theorem) (note: R is never, ever negative  so, expect a positive, nonzero mean) For those who remember their Integral Calculus, the 1D PDF derived from the 2D normal PDF yields the Rayleigh PDF, as seen in the equation and plots in wikipedia.org. The peak of the Rayleigh PDF can be gotten by taking the deriviative, setting it to zero, and solving for the range. Also the wikipedia provides the mean of the Rayleigh PDF, mu = sigma * (0.5*pi)^0.5 That mean is never zero (unless sigma = 0 > perfect bombs). Anyway, Rory, look at the PDF plots in the wikipedia.org. You will see the "annular" feature will never go away. However, you can get rid of the "annular" effect if you choose larger bins. Simply choose a bin size >= the mean of the Rayleigh PDF. And, I really enjoyed your stick analysis. The bomb error is now correlated in a stick. (And I sure don't know of any formula) But, over time we do get an approximate Rayleigh with a different kind of length scale. Kewl!
