[Proj] +towgs84 approximation error

[Noel Zinn (cc)]

Jochem,

Yes, the (7) parameters will look very different if you estimate them on new 
data because a 7-parameter derivation in a small area is ill determined. 
Such transformations use more parameters than required to solve the problem. 
The result is the statistical insignificance of the translations.  By 
contrast 3-parameter transformation translations (and that's all you get, 
just 3 translations) with new or old data, add or subtract points from the 
adjustment, will look pretty much the same.  In fact, they have a physical 
(geodetic) interpretation.  They are translations at the center of the earth 
required to align the surface points.  The so-called "translations" that you 
are using are not that.  They are just numbers, insignificant numbers driven 
wildly by small errors in your surface point.  You should not call them 
translations.  You should call them "fudge factors".  However, derived in 
continental-and-up-sized areas, especially between world datums, they really 
are translations.

One solution to this problem is as you suggest, a rotation center on the 
surface (not the geocenter as in the 7-parameter transformation), the 
so-called Molodensky-Badekas (MB) transformation.  Unfortunately, MB has 
several variants, most of which (if not all) are irreversible, so this is 
not a popular alternative.

Another honest solution to this problem is the approach used with the 
multiple-regression equations of the 1980s.  BTW, they must still be 
popular, especially for badly distorted datums, because Survey Review has or 
will soon publish on the topic.  That solution is to start with the 
translations then add other parameters one by one and assess their 
significance in the adjustment.  Alternatively, throw them all in at once 
and then subtract the least significant, evaluate the determination of the 
adjustment, then subtract another if required, and so on.  You can program 
that procedure and you'll be surprised.

Seven parameters for a European country is bad adjustment practice.

Noel

Noel Zinn, Principal, Hydrometronics LLC
+1-832-539-1472 (office), +1-281-221-0051 (cell)
noel.zinn at hydrometronics.com (email)
http://www.hydrometronics.com (website)

-----Original Message----- 
From: Jochem
Sent: Thursday, March 23, 2017 5:13 PM
To: proj at lists.maptools.org
Subject: Re: [Proj] +towgs84 approximation error

Hi Noel,

You are right about the corrections between rotations and translations. This
will cause the parameters to look very different if you estimate them on new
data. However, besides that, I don't see what the problem is. You still get
precise transformated coordinates (and you don't always get that with
approximate formulas). Or do you see other problems?

You know the solution to this correlation problem by using a point in the
centre of the country as rotation pivot point and thus using a 10 parameter
transformation?
Section 2.4.3.2 of  Guidance Note 7, part 2
<http://www.iogp.org/pubs/373-07-2.pdf>   (these are approximated formulas
but of course a strict equivalent is possible).

Regards, Jochem



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