[Proj] Transformations in small areas - was Stereo 1970 (EPSG 31700)

[Mikael Rittri]

Hello Noel,

> I appreciate the exchange.

So do I. What you say is quite interesting. 

> ... my advice to use 3-parameter translations in a small area ...

Okay, I can accept this as an advice to people who derive new 
datum shifts, like yourself. As long as you can estimate the
error, and the error is acceptable, why not?  But I thought you
extended the advice to people like me, who usually have to choose
between different published datum shifts.  I agree that a 7-parameter
datum shift is not necessarily better than a 3-parameter one, and
the accuracies quoted by EPSG are often hard or impossible to compare.
But I don't see that I should avoid using a published datum shift,
just because it uses 7 parameters in a "small" area.

> 7-parameter derivations in small frontiers are ill conditioned (my thesis)
> ... 
> because 7-parameter transformations are no more "accurate" in a small 
> area than a 3-parameter translation derived from the same data set.

I am sure you are right, for a given value of "small". 
    What surprised me is when you said that Romania and even Germany 
are small in this sense.  Romania is about 700 km in diameter, 
Germany is 800 km, while Australia (which you said is large enough)
is 3800 km. So you are saying that the threshold for "smallness" is 
somewhere between 800 and 3800 km. 
   I would have expected the threshold to be more like 100 km.  

My example of the 3-parameter transform for OSGB 1936 may, as you
say, not be the best possible 3-parameter transform for this datum.

> Tfm Code 1039 provides us the opportunity to test my assertion...

That would be quite interesting, but a bit of work as you say.
But I think I can predict roughly how good the best 3-parameter
transformation could be. 
    The idea is that a 3-parameter and a 7-parameter transform 
for the same area ought to agree exactly on at least one point.  
Around this fixed point, the advantage of the 7-parameter transform 
is that it can supply a rotation and a scale change.  Well, three 
rotations in 3D space, but they should correspond to a single 
rotation around an oblique axis through the fixed point.  
I think this single rotation would be about as large as the 
three basic rotations, but I could be wrong.  Anyway, in the 
OSGB example, I think the main improvement of the 7-parameter 
transform comes from the scale change, not the rotations (since
they are fairly small), and the scale change is -20.489 ppm.  
If the best possible 3-parameter transform for OSGB agrees with 
the given 7-parameter transform in the middle of Great Britain, 
then the maximal radius is about 540 km, and 540 km * 20.489 ppm =
= 11 meters.
    So, the best possible 3-parameter transform for OSGB has to 
deviate from the given 7-parameter transform by up to 11 meters,
(either at Land's End in the southwest or the Orkney Islands in 
the north).  Since the 7-parameter transform is claimed to be at 
most 5 meters wrong, this means that the best 3-parameter transform
is worse.  (In the best possible case, the 5-meter error would 
occur in both Land's End and Orkney, in a direction that makes 
the 3-parameter transform wrong by only 6 meters. But that's 
optimistic.)