DOCH Again

Hanno Ix has revived this topic by his recent post:

He claimed:

a) true location (TL) is not a normal distribution

b) probability of fix hitting TL is exceedingly small

Statement a) is unclear.

First a point is not a distribution. But the probability of the TL occuring at points in space is a
distribution.

We can use rectangular coordinate system and we will find:

p(x,y) = 1/Pi * exp( - (x^2 +y^2))

a Gaussian distribution

For the Probability Density distribution in xy plane with coordinates measured in lengths of σ .

In order to find probability multiply by area, dx dy.

Or we can use polar coordinate system in which case we have:

p(r) = 1/Pi *exp( - (r^2))

a Rayleigh distribution

For the Probability Density distribution in polar coordinates, again unit r is σ .

As noted by John Karl and Geoffrey Kolbe, this distribution has no hole, ala claim b) .

To find probability in annular region, multiply by area 2π r dr. (there is the hole for r =0).

and we should expect probability of fix being an exact point to be 0.

to compute average distance from fix to TL

integrate r * p(r) * 2Pir dr from 0 to infinity

you find r =.886 for
average r ( fix to TL)

in units of σ

John Karl's 2010 analysis is fine as far as it goes, and that is certainly further than is required by the practical navigator. Others have commented on how to think about the fact that we may not be inside the DOCH.

Not surprisingly the analysis can be made more complicted without improving the utility of the analysis. Not accounted for is the DR. That, like a sight, represents imperfect knowledge about our position and could be viewed as a different gaussian distribution along the LOP seperate from the distribution affecting the intercept.

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