[Proj] datum conversion, estimating parameters

Noel Zinn ndzinn at houston.rr.com
Wed Mar 23 18:53:50 EST 2005


Given ECEF coordinates in both datums, solving for the 7 parameters is just
a least-squares problem.  Unfortunately, the 7 geocentric parameters are
highly correlated in a small area.  For example, at a single point it's
possible to solve for the 3 translations or for 2 rotations (any two if your
area of interest is Germany) and the scale factor and get the same results.
That's due to the correlations among the parameters.  It takes a large area
to reduce those correlations geometrically.  The area of Germany is less
than 0.1% of that of the world.  Consequently, the dillution of precision
(DOP) in solving for all 7 parameters in Germany with, say, 40 uniformly
distributed points is about 15.  If you're familiar with the DOP concept
from GPS, that's a big number.  Fewer points, more DOP.  In a country the
size of Australia 40 well distributed points yields a DOP of 3.  That's
something to work with.  And the Australians have done it!

In Europe NATO has been less cautious about about ill-conditioned
least-squares problems and has published 7-parameter shifts for a number of
countries.  This link => once
got one to their published list, but I see it's broken at the moment.  Maybe
it will pop back up, and maybe you'll find your unknown datum among them.
Be cautious.

Noel Zinn

----- Original Message ----- 
From: "Robert Jaeschke" <robert-j at gmx.de>
To: "PROJ.4 and general Projections Discussions" <proj at xserve.flids.com>
Sent: Wednesday, March 23, 2005 2:32 PM
Subject: Re: [Proj] datum conversion, estimating parameters

> Hello,
> On Tue, Mar 22, 2005 at 08:22:04PM -0600, Noel wrote:
>> To accomplish what you intend you need the ECEF (geocentric Cartesian)
>> coordinates for at least one point in both datums.
> Thank you. If i would have read the docs more carefully ... :-(
>> Differencing those ECEF
>> coordinates will give the the three translations (the first 3 of the 7
>> parameters, the rest of the parameters being zero), which is all you can
>> hope for with one point.  Solving for the other parameters requires many
>> more points over (and this is very important) a LARGE area in order to
>> achieve any statistical significance.  The problem is ill-conditioned
>> over
>> a small area.
> That's clear and makes the problem not easier: I have only data of
> Germany ...
>> This problem is NOT solved in projected space.
> My first thought was: Why not convert all values to geocent and then
> repeat my steps? Of course, I tried it. And of course it does not work.
> For conversion to geocent, level above sea is important and I don't have
> these values. And there are probably more reasons for that it's
> impossible,
> what I want. :-(
> Regards, Robert
> -- 
> "I don't even have an e-mail address. I have reached an age where my main
> purpose is not to receive messages."
> --- Umberto Eco, quoted in the New Yorker
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