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

[Noel Zinn (cc)]

Mikael,

> But I thought you extended the advice to people like me, who usually have 
> to choose between different published datum shifts.

Guilty, I confess, but there is a reason.  There are two ways to change a 
practice, from the top down (datum shift "derivers" in this case) and from 
the bottom up (datum shift "choosers" or "consumers" in this case).  Fashion 
is known to work both ways.  Haute Couture at the top and "street smarts" at 
the bottom.  Well, having observed geodetic practice on this topic during 
the last decade I despair.  Top down change isn't happening.  Authoritative 
agencies that should know better (NATO among them) are cranking out bad 
7-parameter datum transformations (e.g. for Cyprus, way too small by any 
standard).  The time has come for bottom-up change.  Datum transformation 
consumers need to be more discriminating.  I was purposely being 
provocative.

As I wrote previously, "small" can be defined quantitatively based on the 
dilution of precision of the 7-parameter adjustment.  Dilution of precision 
(DOP) is a GPS concept, a unitless number that is a function of the number 
of GPS satellites available and their distribution in space relative to the 
observer (it's related to the trace of the variance-covariance matrix 
mathematically).  DOP is the multiplier of random observation error into 
random coordinate error.  It needs to be as small as possible.  Well, 
7-parameter derivation adjustments have a DOP, too.  I call it P7DOP.  The 
number of survey points used corresponds to the number of satellites and the 
distribution in space corresponds to the area over which the survey points 
are spread.  Reduce either (number of survey points or the area) and DOP 
increases.  Not good.  An acceptable DOP is a judgment call, but at least 
it's a quantitative judgment.  Australia with 80 survey points gives a DOP 
of 2.2.  This means that if the average coordinate random error of the 
survey points used is 1 meter (could be better or worse), then the average 
parameter error is 2.2 meters.  I judge that to be acceptable.  Germany with 
80 survey points gives a DOP of 10.2 (unacceptable to me).  With just 20 
survey points the Germany DOP is 21 (even worse).  That's how it works. 
Most surveyors and navigators want their GPS DOPs to be under 3.

Your dissection of the OSGB 7-parameter shift is a little difficult for me 
to follow, but creative.  You begin by noting that a 3-parameter and a 
7-parameter transformation ought to agree at some point.  It's interesting 
to note that at a single point it's possible to derive many 3-parameter 
shifts, not just (dX, dY, dZ).  (rX, rY, dS), (rY, rZ, dS) and (dY, rX, rY) 
are among the many possibilities.  The derivation of a 7-parameter 
transformation at a single point is a singularity; can't be done (but I know 
that you didn't mean to imply that).

You then go on to detail the consequences of the 20.489 ppm dS (change in 
scale).  This is an enormous dS by 7-parameter standards and an omen of a 
questionable transformation.  Multiplied by the earth radius in Great 
Britain, this dS results in a height change of 130 meters.   Shouldn't that 
have been handled by the translations?  Or is this really a scale change 
between OSGB and WGS?  Not likely.  This dS is twice the ppm difference 
between the smallest foot (Clarke's) and largest foot (British 1936) in the 
EPSG Unit of Measure table.  So, it's physical reality is questionable, as 
is further reasoning derived from this dS frankly.  We can (and should) 
derive parameters with valid physical interpretations using the 
Molodensky-Badekas model.

The UK is a tough place to think about the interplay among the 7 parameters. 
There are six places in the world where it's easy: the two poles and the 
four intersections of the X and Y axes with the Equator.  Take one of them, 
0N/0E in the Gulf of Guinea, the intersection of Greenwich and the Equator. 
At that point dX is indistinguishable from dS (i.e. 100% correlated), dY is 
indistinguishable from rZ,  dZ is indistinguishable from rY,  and rX does 
nothing.  How large an area does it take to reduce those correlations 
enough?  Until you get to that size, deriving (and using in my opinion) a 
7-parameter transformation is poor geodetic practice.

Regards,
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)

--------------------------------------------------
From: "Mikael Rittri" <Mikael.Rittri at carmenta.com>
Sent: Friday, October 08, 2010 4:36 AM
To: "PROJ.4 and general Projections Discussions" <proj at lists.maptools.org>
Subject: Re: [Proj] Transformations in small areas - was Stereo 1970 (EPSG 
31700)

> 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.)
>
> From this kind of argument, I think one could estimate the
> threshold of "smallness" from the rotations and scale change
> of a 7-parameter transformation, together with its accuracy.
>
> A more general "smallness" threshold could perhaps be computed
> from the average rotations and scale changes among many typical
> 7-parameter transforms (and their accuracies).
>
> Best regards,
> Mikael Rittri
> Carmenta AB
> Sweden
> www.carmenta.com
>
> -----Original Message-----
> From: proj-bounces at lists.maptools.org 
> [mailto:proj-bounces at lists.maptools.org] On Behalf Of Noel Zinn (cc)
> Sent: den 8 oktober 2010 05:27
> To: PROJ.4 and general Projections Discussions
> Subject: [Proj] Transformations in small areas - was Stereo 1970 (EPSG 
> 31700)
>
> Thanks, Mikael.  I appreciate the exchange.
>
> [Complete message at 
> http://lists.maptools.org/pipermail/proj/2010-October/005429.html ]
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