[Proj] Transformations in small areas - was Stereo 1970 (EPSG31700)

Clifford J Mugnier cjmce at lsu.edu
Fri Oct 8 11:44:07 EST 2010

In my PE&RS column on the Grids and Datums of Namibia, I quoted Professor Charles Merry, now retired from the University of Cape Town:

... "Turning now to Namibia, the results for this datum are summarized ... Although the rotations are not as large as those in Zimbabwe and are barely significant, they do serve to model distortions in the geodetic network and hence provide an improved fit between this network and the CTS.  Again it must be emphasized that these rotations have no physical interpretation.  As in all the countries investigated, except for South Africa, the scale factor plays no major role.  Although the seven-parameter set does improve the fit, it is by no means as remarkable an improvement as that experienced in Zimbabwe and it is debatable whether the extra effort is worth it.  Consequently, we recommend that (the-Ed.) three-parameter transformation shown: ?X = +616.6 m ±1.3 m, ?Y = +103.0 m  ±1.3 m, ?Z = -256.6 m ±1.3 m.  ... As in Zimbabwe, the Namibian networks suffer from significant distortions but in this case a seven-parameter transformation provides little improvement over a three-parameter transformation." 


I have used 7-parameter transformations from time-to-time in the past several decades of my practice, but when I have done so, I took particular notice of the distance of the country/region of interest from the origin point of the classical datum.  When close, I use the Bursa-Wolfe 7-parameter model (such as the distance of Trinidad & Tobago from La Canoa, Venezuela), and when a great distance from the datum origin (such as the distance of the City of Guayaquil, Ecuador from La Canoa, Venezuela), I have used the Molodensky model.  The differentiation allows scaling of the translation parameters.  There is zero difference in the accuracy of the resultant transformation, it just "looks" better when perusing the relative magnitudes of the translation parameters for other countries in the region.  Who cares about the "look?"  The local government's Federal Agency with jurisdiction for official approval.
One can see that there are diminishing returns in blindly increasing the area in order to justify a 7-parameter transformation.  Areas can increase so large that there are too many local distortions in varying meridianol chains to allow a single transformation of any type other than a "surface fit" such as implemented with a NadCon or NTv2 approach.  The generalization one can make for this is ... it depends ...  :-)


Clifford J. Mugnier, C.P., C.M.S.
Chief of Geodesy,
Center for GeoInformatics
Department of Civil Engineering 
Patrick F. Taylor Hall 3223A
Baton Rouge, LA  70803
Voice and Facsimile:  (225) 578-8536 [Academic] 
Voice and Facsimile:  (225) 578-4578 [Research] 
Cell: (225) 238-8975 [Academic & Research]
Honorary Life Member of the 
Louisiana Society of Professional Surveyors 
Fellow Emeritus of the ASPRS 
Member of the Americas Petroleum Survey Group


From: proj-bounces at lists.maptools.org on behalf of Noel Zinn (cc)
Sent: Fri 08-Oct-10 07:27
To: PROJ.4 and general Projections Discussions
Subject: Re: [Proj] Transformations in small areas - was Stereo 1970 (EPSG31700)


> 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

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.


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 <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

> 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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