[Proj] +towgs84 approximation error

Thu Mar 23 16:53:14 EST 2017

```Hi Martin,

Actually, I think that it is mathematically perfectly possible to derive
exactly the same parameters from A to B by common points, as by combining
the parameters form A to WGS84 and the parameters form B to WGS84 (if you
used the same common points to estimate all the transformations).

An important condition is that you take the full covariance matrices of the
least-squares solutions into account. This is like the fact that you can
compute the average body length of a school by taking the average of the
average length of each class, as long as you weigh it with the number of
students in each class. I think it works for 7 parameter transformations
too. I could try if this actually works if you are interested.

Anyway, it is definitely possible to compute the parameters from A to B from
the parameters from A to WGS84 and from B to WGS84 in a way that A to B
gives the same result as A to WGS84 to B. For small angles this is simple
subtracting the parameters. For larger angles it is a bit more complicated
but still commonly known linear algebra. I derived the formulas once to make
a "calculator" for stacking and inverting transformations.

If you have different sets of common points between A and B, A and WGS84 and
B and WGS84, then you would indeed get 3 sets of inconsistent parameters.
However, more precise and consistent parameters could then be obtained by
least-squares adjustment of the 3 sets of parameters (ideally using the full
covariance matrices again). That you would have to store 3 sets of
parameters to relate 3 reference frames is utter nonsense. This is like
claiming that the height difference between the floor and the ceiling is
different from the sum of the height differences from the floor to the table
and from the table to the ceiling.

The part of ISO 19111 I don't understand (I'm good in least-squares theory,
but I bad in reading formal ISO norm texts. Could you explain that in normal
English?).

Now, there are two possibilities. Or what I just wrote is very interesting
for you, or I completely misunderstood that you meant. The third possibility
that I am wrong, I don't consider in this case ;-)

Regards, Jochem

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