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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
<title>CUCC Expo Surveying Handbook: Coordinate Systems</title>
<link rel="stylesheet" type="text/css" href="../../css/main2.css" />
</head>
<body>
<h2 id="tophead">CUCC Expo Surveying Handbook</h2>
<h1>Coordinate Systems</h1>
<p>
If you are not interested in the theoretical background, just jump down to the
<a href="#summary">summary</a>.
</p>
<p>
When dealing with geographical data like cave locations, you will
inevitably run into a whole zoo of coordinate systems with names like
WGS84, UTM, BMN and so on. While a thorough introduction is probably
more appropriate for a full course in geodesy, I'll try to summarise the most
important bits as far as they are relevant to us and as far as I understand
them myself.
</p>
<h2>Projections</h2>
<p>
In a first approximation the earth is a sphere. And unfortunately there are
some mathematical proofs showing that it's not possible to project the surface
of a sphere onto a 2D plane or map without distortions. People have still tried
hard and come up with a particular projection called the Transversal Mercator
projection, which has beneficial properties summarised as "locally there are
almost no distortions".
</p>
<p>
The non-transversal, standard Mercator projection essentially takes a cylinder
aligned with the rotational axis of the earth from north to south and wraps the
cylinder around the equator of the earth. Next all the important
landmarks are projected onto the cylinder by casting rays from the centre of
the earth through its surface and onto the cylinder. Once everything is mapped,
the cylinder is cut open and unwrapped onto a flat table and ready is your map.
This map will be very accurate and have very little distortions around the
equator, but the closer you get to the poles the more distortions will become
noticeable. In particular think of where the north and south poles will be
projected to.
</p>
<p>
The Transversal Mercator projection is very similar to the above, but instead
of aligning the cylinder with a north-south axis and intersecting earth along
the equator, it is tilted sideways, aligned with an east-west axis and
intersects earth in a circle for example along the 0-meridian through
Greenwich, through the poles, and somewhere through the Pacific. The rest is
done as before and once you cut the cylinder open and unwrap it, you'll get an
accurate map with little distortions exactly around the line of intersection,
which is called the "central meridian" of this particular Transversal Mercator
projection. Of course America and China would be heavily distorted with the
above choice of central meridian. So instead of doing just one of these
Transversal Mercator projections globally, the earth is divided into e.g.
60 zones and a different cylinder with a different central meridian is selected
for each zone. One particular definition of such zones has been internationally
standardised as Universal Transversal Mercator coordinates, but for the
entertainment of the local geodesists, different local coordinate systems and
"zones" have been defined for many countries. In Germany this is called
"Gauss-Kr&uuml;ger (GK)", in Austria there is a definition called
"Bundesmeldenetz (BMN)", and in the UK it is the "British National Grid (BNG)".
</p>
<p>
One more thing. Once you have your unwrapped cylinder you'll
have to define coordinates on this cylinder surface, your map. These are
usually metric coordinates, i.e. they specify how many metres you have to walk
north and east on the cylinder surface starting from a given origin. And
typically one starts the "easting" at for example the western boundary of a
zone and the "northing" at the equator. For a national Austrian grid, it
doesn't make sense to start at the equator and therefore some
"false easting" and "false northing" have been defined by omitting some
of the leading digits. This saves repeatedly typing all the
same prefixes over and over again.
</p>
<h2>Ellipsoids</h2>
<p>
Unfortunately the earth is not a sphere. A slightly more accurate
representation would be an ellipsoid, that is wider around the equator and
flatter at the poles. This has long been known and the Transversal Mercator
projection has been adapted to an ellipsoidal shape, so that it has even less
distortions. And of course, many clever people have come up
with many clever approximations of the ellipsoid. For example, the British
National Grid uses an ellipsoid defined by someone called Airy in 1830, and
Bessel has come up with a different ellipsoid in 1841. These were computed
by making accurate astronomical observations at different places within Europe.
In contrast, the more modern WGS84 ellipsoid has been defined by satellite
observations in more recent times.
</p>
<p>
The different ellipsoids not only vary in their major and minor axes, but
also the centre of the ellipsoids can be offset or the whole
ellipsoid can be rotated by a bit. So these offset and rotation parameters have
to be specified as well, and getting the ellipsoid parameters wrong would
typically result in coordinates that are around 500m off, which is unacceptable
for locating a cave entrance on the plateau. So we can't just ignore the
ellipsoids but have to get their definitions right.
</p>
<h2>Geoids</h2>
<p>
Unfortunately the earth is not an ellipsoid either, but rather something like
a potato. This is not so important for defining east and north coordinates,
but it is very important for defining altitudes. While one sensible definition
of altitudes would simply be the "height above ellipsoid", it actually makes
quite a bit of sense to rethink this definition and come up with something
different, called geoids (not to be confused with ellipsoids!).
</p>
<p>
Traditionally height was defined by "mean sea level", and in Austria they use
something called "Gebrauchsh&ouml;hen Adria", which is meant to be the height
above the Adriatic sea. Unfortunately you can only measure the mean sea level
along the coast and it becomes a bit more difficult in the mountains. So
starting from a single point defined as the mean sea level in Trieste in 1875
or so, the Austrians started to triangulate a grid of survey stations across
all of their empire. According to this triangulation they ended up with
several reference heights of certain peaks and so on, which is not necessarily
the real height above Adria anymore but includes some errors. Still, these
reference heights make up the "Gebrauchsh&ouml;he Adria", which literally means
something like "Used Height Adria".
</p>
<p>
As clinos are affected by gravity, so are the Austrian
triangulations, and it turns out that the mass of the continental
plates does indeed affect gravity. So if you simply approximate the mean
sea level by a "simple" ellipsoid such as the "height above ellipsoid" does,
then you end up with a completely different set of altitudes compared to
the triangulation results. It turns out that relative to the ellipsoid the
"mean sea level" at some point in the alps would be about 40m above the mean
sea level at some point along the coast, just because the heavy continental
crust would attract more water. The "Gebrauchsh&ouml;hen
Adria" have been defined with exactly this mass anomaly, and that's
what the Austrians use to this date.
</p>
<p>
Nowadays geodesists have come up with something called geoids. These geoids
define the shape of equipotential surfaces, i.e. the shape of the surfaces
along which a reference body would have the same potential energy in the
gravity field of the earth. So in a sense, the Austrians defined a small
portion of a geoid by measuring the gravity field and defining their
"Gebrauchsh&ouml;hen Adria" accordingly. In the meantime, some other geoids
have been defined and refined using satellite measurements and so on. There are
plenty of them available as huge "geoid height above ellipsoid"-tables in some
massive files (well, 4MB for the old, simple geoid models, 200MB for more
modern and accurate ones).
</p>
<p>
Most modern GPS receivers, at least most Garmin ones, will nowadays compute a
"height above sea level", and not a "height above ellipsoid". Unfortunately
at least Garmin devices do not allow to change this, and the bad news is that
in fact no one outside the Garmin Corporation really seems to know, how they
managed to approximate the geoid in their tiny little units with not very much
memory and computation power. But the good news is that the
differences between various geoids are usually in the range of 25cm, and the
Austrian "Gebrauchsh&ouml;hen Adria" make no difference there. In fact, as the
Bessel ellipsoid has been designed within Europe and adapted to the shape of
the alps, even the differences between the Bessel ellipsoid and the
"Gebrauchsh&ouml;hen Adria" are below 3.5m for most parts of Austria and about
40cm on the Schwarzmooskogel.
</p>
<h2>Converting Coordinates</h2>
<p>
Luckily all of the above is so horribly complicated, that people have long come
up with computer programs for converting these coordinate systems back
and forth. You just have to find an appropriate suite of software and learn how
to use it. And particularly the using part can still be quite complicated. For
the reasons detailed in the "Geoids" section above, I'd recommend converting
only the horizontal coordinates and keeping the altitude measurements from the
GPS.
</p>
<p>
I personally get along very well with Proj4, which is open source and free and
all that. It should also be packaged with all major Linux distributions and
installed on the expo computer. Unfortunately the current versions do not deal
very well with vertical datums (i.e. geoids), but we can ignore the geoids
anyway. To invoke it, you have to type in something like
</p>
<div style="background-color: #BDB"><pre>
cs2cs +from [+some +magic +parameters] \
+to [+some +more +magic +parameters]
</pre></div>
<p>
Then you type in the coordinates in the source format and you'll get
coordinates in the destination system, sometimes with x and y swapped back
and forth. The following table is intended to help you choose the right magic
parameters for your coordinate system:
</p>
<div style="background-color: #BDB"><table>
<tr><td>
Latitude-Longitude in WGS84 datum with heights above WGS84 ellipsoid:
<pre> +proj=latlon +ellps=WGS84 +datum=WGS84</pre>
</td></tr>
<tr><td>
Latitude-Longitude in WGS84 datum with heights above EGM96 geoid<sup>[<a name="ftnEGM96" href="#ftn.EGM96">1</a>]</sup>:
<pre> +proj=latlon +ellps=WGS84 +datum=WGS84 +geoidgrids=egm96_15.gtx</pre>
</td></tr>
<tr><td>
UTM coordinates in WGS84 datum with heights above EGM96 geoid<sup>[<a href="#ftn.EGM96">1</a>]</sup>:
<pre> +proj=utm +zone=33 +ellps=WGS84 +datum=WGS84 \
+geoidgrids=egm96_15.gtx</pre>
</td></tr>
<tr><td>
Austrian coordinates for our Loser data set<sup>[<a name="ftnBMN" href="#ftn.BMN">2</a>]</sup>:
<pre> +proj=tmerc +lat_0=0 +lon_0=13d20 +k=1 +x_0=0 +y_0=-5200000 \
+ellps=bessel +towgs84=577.326,90.129,463.919,5.137,1.474,5.297,2.4232</pre>
</td></tr>
</table>
</div>
<div class="footnote">
<p><sup>[<a name="ftn.EGM96" href="#ftnEGM96">1</a>]</sup>
Starting from version 4.8, the cs2cs program should have rudimentary support
for vertical datums. You might have to separately install the file egm96_15.gtx,
though. While this file strictly speaking only defines the EGM96 geoid, it can
serve as a good approximation to most other geoids, including the one used by
Garmin GPS receivers and the "Gebrauchsh&ouml;he Adria"
</p>
<p><sup>[<a name="ftn.BMN" href="#ftnBMN">2</a>]</sup>
There are a few different versions of the "+towgs84" part of the Austrian
coordinate system, which specifies the offset and rotation of the used Bessel
ellipsoid with respect to the WGS84 ellipsoid. According to an old table found
on this expo website, it should read "575,93,466,5.1,5.1,5.2,2.5", which is
clearly a mistyped version of the more commonly found definition
"575,93,466,5.1,1.6,5.2,2.5". Both of these seem slightly less accurate than
the "577.326,90.129,463.919,5.137,1.474,5.297,2.4232" proposed by various
other sources, but in the end it will only make a difference of
about a metre or so.
</p>
</div>
<h2><a name="summary">Summary</a></h2>
<p>
For all practical purposes I'd say, set your GPS receiver to UTM coordinates,
WGS84 ellipsoid, WGS84 datum. It will
usually spit out rather unspecific "heights above sea level", which are within
about 25cm of the heights in our data set. To convert the horizontal
coordinates from UTM zone 33 to our data set coordinates, use:
</p>
<div style="background-color: #BDB"><pre>
cs2cs +from +proj=utm +zone=33 +ellps=WGS84 +datum=WGS84 \
+to +proj=tmerc +lat_0=0 +lon_0=13d20 +k=1 \
+x_0=0 +y_0=-5200000 +ellps=bessel \
+towgs84=577.326,90.129,463.919,5.137,1.474,5.297,2.4232
</pre></div>
<p>
As an exercise you can try to convert the following between
latitude-longitude, UTM and data set coordinates:
</p>
<div style="background-color: #BDB"><table>
<tr><th>Point</th><th>lat-long WGS84</th><th>UTM WGS84</th><th>data set</th></tr>
<tr><td>161g</td><td>13d49'35.982"E 47d41'1.807"N&nbsp;&nbsp;&nbsp;</td><td>411941 5281827&nbsp;&nbsp;&nbsp;</td><td>37095.76 82912.23</td></tr>
<tr><td>204a</td><td>13.82146667 47.69093333</td><td>411563 5282622</td><td>36700.78 83698.97</td></tr>
<tr><td>2001-06&nbsp;&nbsp;&nbsp;</td><td>13.81911639 47.67609556</td><td>411362 5280976</td><td>36534.63 82048.14</td></tr>
<tr><td>2011-01</td><td>13.82701861 47.69979611</td><td>411995 5283601</td><td>37111.31 84686.99</td></tr>
</table></div>
<p><i>Olaf K&auml;hler, September 2012</i></p>
<hr />
</body>
</html>

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@ -133,14 +133,17 @@ it seems) write it down by hand on one of the A5 cave info sheets with all the
other details of your cave and put that in the surveys ringbinder file. </p>
<p>If you want to read about the nitty gritty of converting GPS coordinates to
the ones used by the Kataster system, you can do no better than read Wookey's
<a href="../../years/1996/gps.htm">Compass Points Article</a>. Briefly, this
says "it's horribly complicated and we don't really know how to do it
properly". Things have improved a little since those days, particularly as
without the fog of the SA variation it's now easy to find out whether your GPS
is set up right by just GPSsing a known point and comparing the results.
However, the main point of having a GPS fix on an entrance is so we can find it
again and be sure it is the same one!</p>
the ones used by the Kataster system, you can do no better than read the
short introduction to <a href="coord.htm">coordinate systems</a>, which briefly
says "it's horribly complicated and we use computer programs to do it properly".
(A rather outdated first attempt at this can also be found in Wookey's
<a href="../../years/1996/gps.htm">Compass Points Article</a> from 1996, which
briefly says "it's horribly complicated and we don't really know how to do it
properly".) Overall things have significantly improved since the early days,
particularly as without the fog of the SA variation it's now easy to find out
whether your GPS is set up right by just GPSsing a known point and comparing
the results. However, the main point of having a GPS fix on an entrance is
so we can find it again and be sure it is the same one!</p>
<hr />

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@ -46,6 +46,7 @@ detailed topics.</p>
<li>The <a href="lasers.htm">laser points</a></li>
</ul></li>
<li>A new cave: <a href="gps.htm">Getting a GPS fix</a></li>
<li>GPS for Surveying: <a href="coord.htm">Coordinate Systems</a> and how to convert between them</li>
<li>Base Camp: <a href="getin.htm">getting it in</a> to the computer</li>
<li>Base Camp: <a href="drawup.htm">drawing it up</a>, writing the description</li>
<!-- need to add blurb on QM quality etc. -->