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Subject: Re: On LOPs
From: Jay Borseth (jaybo@XXX.XXX)
Date: Wed Jun 12 2002 - 19:44:40 EDT
(The thread that refuses to die.)
I've updated Pocket Stars to graph the 95% confidence ellipse when three
or more observations are enabled. For screenshots, see:
http://www.nomadelectronics.com/PocketPC/PocketStars/PocketStarsScreenSh
ots.htm and search for "Graph Page".
- Jay
-----Original Message-----
From: Navigation Mailing List
[mailto:NAVIGATION-L@XXX.XXX] On Behalf Of
WSMurdoch@XXX.XXX
Sent: Thursday, April 18, 2002 5:50 PM
To: NAVIGATION-L@XXX.XXX
Subject: Re: On LOPs
In a message dated 4/17/02 10:33:05 PM Eastern Daylight Time,
Gadus@XXX.XXX writes:
>I am still not convinced that an elliptical shape for the boundary is
>anything more than a convenient approximation when there are more than
>three LOPs but, that aside, the above equations make a lot of sense,
>even to this non-statistician.
I spoke to a statistician at work today who said the error contours were
elliptical so long as the error was normally distributed in two
dimensions and the two dimensions were at right angles to each other. I
am an engineer and do not pretend to understand.
>
>I don't understand. D, E and F all include the intercepts of the
>sights as terms in their calculation yet the size of the intercept is,
>in part, a result of the arbitrary choice of an assumed position.
Unless
>this is supposed to be the standard deviation of that part of the error
>in the estimated position which arises from projecting an azimuth
>(calculated onto the the nearest full degree) and an LOP from the
>assumed position to the estimated position, I don't see why the
>calculation includes the azimuth at all. Can anyone explain that?
>
I don't know either how the formula for S is derived, but this is my
guess for why it may be correct.
For each LOP, F is p1 squared and is something like the area of the
rectangle with the first estimated position and the closest point to it
on the LOP as diagonal corners. From that is first subtracted the
northerly distance from the first estimated position to the closest
point on the LOP times the easterly distance from the first estimated
position to the improved estimated position. Like F this has the units
of degrees squared. Then a similar term is subtracted. It is the
easterly distance from the first estimated position to the closest point
on the LOP times the northerly distance from the first estimated
position to the improved estimated position foreshortened by the cosine
of the latitude. It too has units of degrees squared. The remaining
area is related to the error and may be independent of the first
estimated position. I think azimuth finds its way into the figures
because the perpendicular to the LOP defines the clos! est point on the
LOP to first estimated position.
>
>But shouldn't "BF" be the assumed latitude and "LF" the assumed
>longitude?
>
You are right. It should have been (Long, Lat) rather than (Lat, Long).
Compact Data does it like (x, y). I did not catch it. With this first
found error corrected....
Bill Murdoch
**************************
These are the now once corrected formulas. (Any more mistakes in copying
and changing this to survive being typed and sent over the internet are
mine.)
From B.D. Yallop and C.Y. Hohenkerk, Compact Data for Navigation and
Astronomy for the Years 1991-1995.
If p1 and Z1 are the intercept and azimuth of the first observation, if
p2 and Z2 are the intercept and azimuth of the second observation, and
so on; form the summations
A = cos^2 Z1 +cos^2 Z2+...
B = cos Z1 sin Z1 + cos Z2 sin Z2 +...
C = sin^2 Z1 + sin^2 Z2 + ...
D = p1 cos Z1 + p2 cos Z2 + ...
E = p1 sin Z1 + p2 sin Z2 + ...
F = p1^2 + p2^2 + ...
As a check A + C should equal n, the number of sights.
G = A C - B^2
The improved estimated position is (Long, Lat), (LI, BI) made by
correcting the first estimated position (LF, BF)
LI = LF + dL
BI = BF + dB
where
dL = (A E - B D) / (G cos BF)
and
dB = (C D - B E) / G
The distance between the first estimated position and the improved
estimated position is
d = 60 sqrt(dL^2 cos^2 BF + dB^2)
If d exceeds about 20 nautical miles, set LF = LI and BF = BI then
repeat the calculation until d, the distance between the previous
estimate and the improved estimate, is less than about 20 nautical
miles.
If three or more position lines are obtained, an estimate of the error
may be calculated. The standard deviation of the estimated position
sigma in nautical miles is given by
sigma = 60 sqrt(S / (n-2))
where
S = F - D dB - E dL cos BF
sigma sub L = sigma sqrt(A / G)
sigma sub B = sigma sqrt(C / G)
In general as the number of observations increases the error in the
estimated position decreases. Statistical theory shows that the
estimated position has a probability P of lying within a confidence
ellipse which is specified by the lengths of its axes a and b and the
azimuth theta of the a-axis, where
tan 2 theta = 2 B / (A - C)
a = sigma k / sqrt(n / 2 + B / sin 2 theta)
b = sigma k / sqrt(n / 2 - B / sin 2 theta)
where
k = sqrt(-2 ln(1-P)) is a scale factor
Values of the scale factor for selected values of P are given in the
table below.
Probability, P 0.39 0.50 0.75 0.90 0.95
Scale factor, k 1.0 1.2 1.7 2.1 2.4
The usual confidence limit is 95%, that is P = 0.95.
The shape of the confidence ellipse depends only upon n and the
distribution of the observations in azimuth; whilst the size of the
ellipse, apart from the scale factor, depends upon the errors of
observation. The method assumes that the observations have equal
weight. The best results will be obtained when the observations are
equally spaced in azimuth. In such cases the effect of systematic
errors on the final calculated position will be minimized.
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