Error in loop closure algorithm?

[John Halleck]

On Thu, 12 Aug 2004, Mike McCombe wrote:

> Date: Thu, 12 Aug 2004 10:00:19 +0100
> From: Mike McCombe <mikemccombe at btinternet.com>
> To: John Halleck <John.Halleck at utah.edu>, Olly Betts <olly at survex.com>
> Cc: Survex User Group <survex at survex.com>
> Subject: Re: Error in loop closure algorithm?
> 
> 
> ----- Original Message ----- 
> From: "John Halleck" <John.Halleck at utah.edu>
> To: "Olly Betts" <olly at survex.com>
> Cc: "Mike McCombe" <mikemccombe at btinternet.com>; "Survex User Group"
> <survex at survex.com>
> Sent: Thursday, August 12, 2004 12:06 AM
> Subject: Re: Error in loop closure algorithm?
> 
> > > On Wed, Aug 11, 2004 at 11:04:43PM +0100, Mike McCombe wrote:
> > > > Survex moves not only the survey legs in the "bad" loop but also
> > > > adjusts the connecting leg (i.e. test.4 - test.5) and moves the
> > > > station in the "good" loop (i.e. test.4) where the connecting leg
> > > > joins. The "error" is only 1cm which wouldn't show in most cave
> > > > surveys (with me using the instruments!) . However, the error is still
> > > > 1cm if you scale the loops down to, say, 1m per leg - which becomes
> > > > quite noticeable when the loops are small and generally well-closed.
> >
> >   Basicly, Least Squares, can smear a blunder all over everything.
> >   Although whether or not it is very visible varies.
> 
> I appreciate the significance of the "LMS vs Blunder" religious argument but
> disagree with John's use of the word "everything". LMS algorithms distribute
> errors according to the covariances of the observations. The covariance

  Not quite...  the apriori covariance matrix, not the actual covariances,
  which are unknown.

> between survey legs in loops which are not coupled is zero (i.e. they are
> statistically independent) - hence errors in this example should not
> propagate between the two loops.

  I argue with your assumption that these are statistically independent.

  If you really think that because shots in the one loop are conceptutually
  independent of the other, and the apriori covariance matrix has them
  independent, that the solution has them independent (the apostori computation)
  I believe you are mistaken.

  If people here want me to, and think that there is something to be
  gained from it, I could work a small problem for you in detail.



  But let me first try a thought experiment as less work.

  Consider the figure 8:

  A     B     C
  .-----.-----.
  |     |     .
  .-----.-----.
  D     E     F

  There are three loops here.
  The left one, the right one, and the one all the way around.
  (Yes, people only naturally see the two... but mathematically
  there are the three...)

  You appear to be claiming that Least Squares will have that
  shot affect the left loop (that it is obviously in) but
  not the loop all the way around (that it is also in).


  Another way to look at it is.
  
  The legs B-A, A-D, D-E can be combined into one equivalent shot.
    (Which some programs actually do to cut down computation.)

  The legs  B-C, C-F, F-E can also be combined into a single
  virtual shot of appropriate weight.

   So now we have three measurements from B to E.  (The combined
   left leg, the combined right leg, and the shot B-E.)

   So you are, in effect, claiming that if there are three
   measurements of the same quantity being adjusted via least
   squares that it can magicly adjust two of them without
   affecting the third.