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<p>Hello Jochem</p>
<p>I do not know the details about how data are selected for
computing the 7-parameters, but I would presume that the set of
stations in "A to WGS84 to B" is not necessarily the same than
in "A to B". For example an operation from A to B may be
designed for a smaller geographic area than an operation
involving WGS84.</p>
<p>Before to talk about derived CRS, one note about ISO 19111
terminology. They define two kinds of coordinate operations:</p>
<ul>
<li>A <b>conversion</b> is a coordinate operation defined only
by mathematics and arbitrary parameters. Theoretically, the
precision of coordinate conversion is infinite (ignoring
limitations of floating point arithmetic). All map projections
are conversions. Change of axis order, unit of measurement,
etc. are also conversions.</li>
<li>A <b>transformation</b> is a coordinate operation derived
from measurements. Regardless the computer accuracy,
coordinate transformations have a limited accuracy because of
stochastic errors. All datum shifts are transformations.<br>
</li>
</ul>
<p>So an affine operation for example can be either an conversion
or a transformation, depending for which purpose it is used:</p>
<ul>
<li>An affine operation used for changing the unit of
measurement, or for changing the false easting or false
northing parameters of a map projection, is a <i>conversion</i>.
There is no error associated to that operation.</li>
<li>An affine operation used for applying a datum shift on
geocentric coordinates is a <i>transformation</i>. There is
an error associated to that operation no matter the accuracy
of the mathematics behind it.<br>
</li>
</ul>
<p>A derived CRS in ISO 19111 is a Coordinate Reference System
defined by another CRS (named the "base CRS") that we use as a
starting point, then a <b>conversion</b> that we apply <i>by
definition</i> on the coordinates in the base CRS in order to
get the coordinates in the derived CRS. Some key points:</p>
<ul>
<li>The base CRS and derived CRS are related by a coordinate
conversion, not a transformation.</li>
<li>The above coordinate conversion is part of the derived CRS
definition. In WKT 2 format, the parameters are specified
inside the CRS definition.<br>
</li>
<li>The above coordinate operation is the only possible
coordinate operation from the base CRS to the derived CRS.<br>
</li>
</ul>
<p>By contrast, a change of datum between two CRS has the
following properties:</p>
<ul>
<li>Datum changes are transformation, not conversion.</li>
<li>Datum change parameters are not part of CRS definition. In
WKT 2 format, the operation is specified in a separated WKT
element independent of the CRS definition (except for
BOUNDCRS, which is a compromise done for early-binding
implementations).<br>
</li>
<li>Many different coordinate operations can exist for the same
pair of CRS. In USA, there is about 80 datum shift operations
from NAD27 to WGS84.<br>
</li>
</ul>
<p>To make a parallel with Proj.4, the current Proj.4
implementation is designed as if all CRS were derived CRS since
the +towgs84 parameter is part of CRS definition. This is not
the reality (many CRS were designed before WGS84). In
late-binding implementations of map projection libraries,
whether a coordinate operation is used for the definition of a
derived CRS or not make a difference in:</p>
<ul>
<li>Where the parameter is defined (in the CRS or in a separated
construct).</li>
<li>The cardinality of the coordinate operation(s) associated to
that CRS (1:1 or 0:∞).</li>
<li>Whether the coordinate operation choice may depend on the
geographic area of the coordinates that we want to transform.<br>
</li>
<li>Whether the operation may have stochastic errors.<br>
</li>
</ul>
<p>That said, I mentioned the derived CRS because I saw a
statement saying that the CRS would be <i>by definition</i> an
other CRS with the coordinate operation defined by the 7
parameters. But I still a little bit surprised by a scenario
where a derived CRS would be defined by Bursa-Wolf parameters.</p>
<p> Martin</p>
<p><br>
</p>
<p>Le 23/03/2017 à 22:53, Jochem a écrit :</p>
</div>
<blockquote cite="mid:1490305994297-5313952.post@n6.nabble.com"
type="cite">
<pre wrap="">Hi Martin,
Actually, I think that it is mathematically perfectly possible to derive
exactly the same parameters from A to B by common points, as by combining
the parameters form A to WGS84 and the parameters form B to WGS84 (if you
used the same common points to estimate all the transformations).
An important condition is that you take the full covariance matrices of the
least-squares solutions into account. This is like the fact that you can
compute the average body length of a school by taking the average of the
average length of each class, as long as you weigh it with the number of
students in each class. I think it works for 7 parameter transformations
too. I could try if this actually works if you are interested.
Anyway, it is definitely possible to compute the parameters from A to B from
the parameters from A to WGS84 and from B to WGS84 in a way that A to B
gives the same result as A to WGS84 to B. For small angles this is simple
subtracting the parameters. For larger angles it is a bit more complicated
but still commonly known linear algebra. I derived the formulas once to make
a "calculator" for stacking and inverting transformations.
If you have different sets of common points between A and B, A and WGS84 and
B and WGS84, then you would indeed get 3 sets of inconsistent parameters.
However, more precise and consistent parameters could then be obtained by
least-squares adjustment of the 3 sets of parameters (ideally using the full
covariance matrices again). That you would have to store 3 sets of
parameters to relate 3 reference frames is utter nonsense. This is like
claiming that the height difference between the floor and the ceiling is
different from the sum of the height differences from the floor to the table
and from the table to the ceiling.
The part of ISO 19111 I don't understand (I'm good in least-squares theory,
but I bad in reading formal ISO norm texts. Could you explain that in normal
English?).
Now, there are two possibilities. Or what I just wrote is very interesting
for you, or I completely misunderstood that you meant. The third possibility
that I am wrong, I don't consider in this case ;-)
Regards, Jochem</pre>
</blockquote>
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