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From: Herbert Prinz (no email)
Date: Wed Nov 03 2004 - 11:19:54 EST
Alexandre Eremenko wrote:
> As I understand from your last messages on
> the topics, these authors only discuss a Least Square algorithm,
> but DO NOT discuss why the simple averaging of the altitudes
> can sometimes be wrong, that is they do not discuss non-linearity
> of the altitude changes?
Dear Alex,
That's correct. Why would they? Simple averaging of the altitudes is always
wrong. I mention the papers in that paragraph because they are examples for
the rigorous method of "averaging". Linearity of of the altitude does not come
into play here at all.
You seem to be arguing that because the altitude function is linear in
reasonable intervals, averaging the altitudes is permissible. Not so.
Consider an overdetermined system of linear equations. The equations are
linear, by definition! Ok? You cannot solve this system by any kind of simple
averaging.
M * x = a
The correct method is to multiply with the transposed matrix M_t and solve the
system that results from that.
(M_t * M) * x = M_t * a
The algorithm in the N.A. that I mentioned several times is an exact
equivalent to this method. The reason why the N.A. procedure is iterative
(just like the original St. Hilaire intercept method!) is that the LOPs aren't
straight lines. Whether an iteration is necessary in actual practice depends
on how good an approximation our linear system was. So you see, non linear
equations are perfectly acceptable. They are NOT the source of the problem.
All I am saying in my message of 2004-10-09, third paragraph, is that if you
have more than two observations for a fix, you treat the problem as an
overdetermined system. In my message of 2004-10-19, esp. paragraphs starting
with "Second,...", and "Third, ...", I elaborated in more detail on the
reasons for this. I say it again:
One problem with averaging groups of sights individually before reducing them
is that you minimize separately the residuals within each group of
observations, instead of their total sum. I asked you whether you can justify
this statistically and you did not address the problem. I find it all the more
surprising that since then, you have claimed twice to have refuted my
"exaggerated claim" that averaging was a thing of the past.
I have not seen a refutation, yet.
Best regards
Herbert Prinz
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