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I meant to say, a nice reference for the Gauss-Markov method is
Chapter 2 of Åke Björck, Numerical Methods in Matrix Computations. You
can probably see thm 2.11 in the free preview of the chapter

A couple of things to note about this least squares approach. 1) It
does not assume Gaussian errors. You still get the "best unbiased
linear estimator"
2) You get an unbiased estimate of the covariance of the errors in the
LOPs (in the version in the book they are identically distributed) as
residual squared over m-2 when you have m LOPS. So as usual in this
type of thing the square root of the excess of measurements over
unknowns is the important thing.  This residual vector of course
closely related to the navigators rule of thumb -how big is the cocked
hat. If you only have two LOPs of course you do not get an estimate of
the variance.

Bill

On Mon, 26 Nov 2018 at 08:47, Bill Lionheart  wrote:
>
> 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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