NavList:

About DeWit, USNO Nautical Almanac, and Compac Data, Least squares algorithm for n LOPs SR:

 

"This rings a bell. Checking my copy of AstroNavPC, from 2001, I find that I had penciled "A = C and B = 0" into the margin of p. 68. Is that it? "

Yes, an example:

 

DeWit/USNO Nautical Almanac/Compac Data, Least squares algorithm for n LOPs

 

GHA	DEC	HO	BO	LO	LHA	HC	Z	p
85.0000	0.0000	60.7917	15.6178	-60.0000	25.0000	60.7911	240.0004	0.0006
35.0000	0.0000	60.7917	15.6178	-60.0000	335.0000	60.7911	119.9996	0.0006
60.0000	44.8267	60.7917	15.6178	-60.0000	0.0000	60.7911	360.0000	0.0006

 

 

Estimate position at time of fix:

Befix [deg] = 15.6178

Lefix [deg] = -60.0000

 

Least Squares information:

nObservations = 3

AA = 1.5000

BB = -0.0000

CC = 1.5000

DD = 0.0000

EE = 0.0000

FF = 0.0000

G = 2.2500

 

Error:

S = 0.0000

sigma = 0.0601 nm

sigmaB = 0.0490

sigmaL = 0.0490

 

Ellipse:

Prob = 0.9500

k = 2.4477

theta = 0.6585

a = 0.1201

b = 0.1201

 

Improved position at time of fix:

dB [deg] = 0.0000

dL [deg] = 0.0000

DO [deg] = 0.0000 nm

BI [deg] = 15.6178

LI [deg] = -60.0000

 

iteraciones = 7

 

…

The text then says- "statistical theory shows that the estimated position has a probability P of lying within" such a confidence ellipse.

…

As I see it, the confidence ellipse is intended to be plotted around that estimated position, so the estimated position will therefore ALWAYS be placed at the exact centre of such a confidence ellipse!

…

What would be nice to have, if it was possible, is the probability that such a confidence-ellipse will contain the TRUE position, not the estimated position.

…

 

This is the example in the “Compact Data 2001-2005”. The ellipse is centered in the fix not in the DR or initial position.

If we choose as an initial position one far from the fix, the ellipse axis, a,b,  tends to infinite.

The solution for 4 iterations and a 50% ellipse is:

I think the problem is the original text, the book uses the words: estimated position, (DR or initial for the iterative process), and improved estimated position.

 

Confidende ellipse is not new in AstronavPC & Compac Data, widely used in statistics:

http://en.wikipedia.org/wiki/Confidence_region

http://pierremarie.gagey.perso.sfr.fr/EllipseConfiance-a.htm

 

The confidende ellipse is a statistical technique that plots the region where the real position, not the calculated fix, could it be under certain probability, based under the observed phenomena. Historically the bisector of the azimuths determined the most probable position in a cocked-hat, and can result a cocked-hat. Other technique is to obtain the symmedian point also called Lemoine point or Grebe point,

http://en.wikipedia.org/wiki/Symmedian

With a two-body fix, we can draw a confidence ellipse again based on the standard deviation of observations as an input. It's an ellipse about the same size as the "overlap box" for the error bands around each LOP. And with a three-body fix, this answers your concern about a confidence ellipse being drawn too small when the LOPs just happen by chance to coincide in a small triangle. That small triangle should not imply a small confidence ellipse, and it doesn't if the s.d. is an input rather than calculated from the observations themselves.

An ellipse has 3 dof, degrees of freedom, thus without any assumption is not possible to plot it.

An example of a rapid fix, the algorithm is very robust.

 

See [NavList 5168] Series of Sun sights in relatively rapid succession

 

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Andrés Ruiz

Navigational Algorithms

https://sites.google.com/site/navigationalalgorithms/