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