# [Proj] +towgs84 approximation error

Noel Zinn (cc) ndzinn at comcast.net
Thu Mar 23 18:44:12 EST 2017

```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.

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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