NavList:

Recently Lars Bergman and I spent some time digesting formulas given in the HMNAO Compact Data publication that Lars has in his possession. After a few false starts we were able to derive all the formulas that appear in the attached scan. The publication states "If three or more position lines are obtained an estimate of the error in position may be calculated".

The recent discussion on the thin triangle can serve to demonstrate the use of these formulas with the results summarized below

Offsets 11.1NM North, 20.5 East
(Note that because the distance to the offset point is greater than 20NM it is advised to repeat the calculation centred on the  offset point but as Frank instructs us "don't worry about longitude scaling" this can be avoided.)

Here's a summary of further results for the thin triangle

Estimated position error, σ : 1.89NM
Standard deviation in longitude σ_L : 1.69NM (this is more accurately described as standard deviation in departure).
Standard deviation in latitude σ_B : 1.58NM

95% Confidence ellipse has
Semi-major axis : 4.64 NM
Semi-minor axis : 3.25 NM in the direction of 39.2° azimuth

Using the results obtained in my Journal of Navigation paper Probabilities in a Gaussian Cocked Hat (https://doi.org/10.1017/S0373463319000110) and assuming a gaussian error distribution with standard deviation σ as above, the probability that the thin triangle encloses the observer's true position is 34.7%.

I attach a spreadsheet that computes the quantities described here.

Robin Stuart