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Bob

To clarify your question you want to know how much each extra LOP
reduces the uncertainty. For example how much it reduces the area of
the within an elliptical probability contour for a fixed probability,
in terms of the azimuths and variances for each sight?

I think I can answer that, although most of the answer is in
Stansfield, Statistical theory of DF fixing, J or the IEE, 1947
together with the Gauss Markov theorem.

To fully answer your question though we need t decide the value of the
reduction in the size of the probability contour and if it is worth
the effort. You could compare for example with taking more
measurements of the same sights to reduce the variance as at least we
could compare the time cost.

If you use a computer to do the sight reduction calculate the least
squares point then there is no significant extra cost of using more
LOPs, it is just the time taken to measure the altitudes and record
them with the time.  If you want to compute the least squares point
and ellipse using ruler and compasses it turns out it is not so hard
to add extra lines of position. But this involves a neat trick in a
paper two colleagues and I recently submitted to JoN.

Bill Lionheart
On Sat, 24 Nov 2018 at 21:30, Bob Goethe  wrote:
>
> I have read with interest the discussion on the ever-troublesome cocked hat, 
and the difficulties associated with assigning a most probable point (or even 
a least improbable point).
>
> If one takes sights on two celestial objects - where he feels equally 
confident in the quality of the sights - and uses Frank's equation for 
determining the error elipse 
(http://fer3.com/arc/imgx/error-ellipse-ratio.jpg.thumb.jpg), to what extent 
can one say that he has pretty much what he needs to come up with as good a 
fix as can be had?
>
> That is to say, if one takes a sight on a third object, reduces and plots 
it, could he say that he has done more work with no significant probability 
that he knows more about his actual position than he had after plotting his 
initial two LOPs?
>
> And if a 3rd LOP represents a poor navigational return on time-invested, 
presumably a 4th LOP is even less worthwhile?
>
> Bob
>
> 

   

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