diff --git a/handbook/survey/coord.htm b/handbook/survey/coord.htm new file mode 100644 index 000000000..ebd70c96e --- /dev/null +++ b/handbook/survey/coord.htm @@ -0,0 +1,286 @@ +<!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-Krüger (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 "Gebrauchshöhen 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 "Gebrauchshöhe 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 "Gebrauchshöhen +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 +"Gebrauchshöhen 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 "Gebrauchshöhen 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 +"Gebrauchshöhen 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 "Gebrauchshöhe 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 </td><td>411941 5281827 </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 </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 Kähler, September 2012</i></p> +<hr /> + +</body> +</html> + diff --git a/handbook/survey/gps.htm b/handbook/survey/gps.htm old mode 100644 new mode 100755 index f4ae80095..80f5a1cf1 --- a/handbook/survey/gps.htm +++ b/handbook/survey/gps.htm @@ -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 /> diff --git a/handbook/survey/index.htm b/handbook/survey/index.htm old mode 100644 new mode 100755 index ce58bb7e3..2ad2289bf --- a/handbook/survey/index.htm +++ b/handbook/survey/index.htm @@ -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. -->