NavList:
A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Andrés Ruiz
Date: 2010 Mar 3, 10:30 +0100
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-
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
---
Andrés Ruiz
Navigational Algorithms
https://sites.google.com/site/navigationalalgorithms/