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href="https://github.com/OSGeo/PROJ/edit/8.2/docs/source/usage/transformation.rst" class="fa fa-github"> Edit on GitHub</a> + </li> + </ul><div class="rst-breadcrumbs-buttons" role="navigation" aria-label="Sequential page navigation"> + <a href="projections.html" class="btn btn-neutral float-left" title="Cartographic projection" accesskey="p"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="ellipsoids.html" class="btn btn-neutral float-right" title="Ellipsoids" accesskey="n">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> + </div> + <hr/> +</div> + <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="geodetic-transformation"> +<span id="transformation"></span><h1>Geodetic transformation<a class="headerlink" href="#geodetic-transformation" title="Permalink to this headline">¶</a></h1> +<p>PROJ can do everything from the most simple projection to very complex +transformations across many reference frames. While originally developed as a +tool for cartographic projections, PROJ has over time evolved into a powerful +generic coordinate transformation engine that makes it possible to do both +large scale cartographic projections as well as coordinate transformation at a +geodetic high precision level. This chapter delves into the details of how +geodetic transformations of varying complexity can be performed.</p> +<p>In PROJ, two frameworks for geodetic transformations exists, the +<em>PROJ 4.x/5.x</em> / <strong class="program">cs2cs</strong> / <code class="xref c c-func docutils literal notranslate"><span class="pre">pj_transform()</span></code> +framework and the <em>transformation pipelines</em> framework. The first is the original, +and limited, framework for doing geodetic transforms in PROJ The latter is a +newer addition that aims to be a more complete transformation framework. Both are +described in the sections below. Large portions of the text are based on +<span id="id1">[<a class="reference internal" href="../zreferences.html#id13" title="Evers, K. and Knudsen, T. Transformation pipelines for PROJ.4. In FIG Working Week 2017 Proceedings. Helsinki, Finland, 2017. URL: http://www.fig.net/resources/proceedings/fig_proceedings/fig2017/papers/iss6b/ISS6B_evers_knudsen_9156.pdf.">EversKnudsen2017</a>]</span>.</p> +<p>Before describing the details of the two frameworks, let us first remark that +most cases of geodetic transformations can be expressed as a series of elementary +operations, the output of one operation being the input of the next. E.g. when +going from UTM zone 32, datum ED50, to UTM zone 32, datum ETRS89, one must, in the +simplest case, go through 5 steps:</p> +<ol class="arabic simple"> +<li><p>Back-project the UTM coordinates to geographic coordinates</p></li> +<li><p>Convert the geographic coordinates to 3D cartesian geocentric coordinates</p></li> +<li><p>Apply a Helmert transformation from ED50 to ETRS89</p></li> +<li><p>Convert back from cartesian to geographic coordinates</p></li> +<li><p>Finally project the geographic coordinates to UTM zone 32 planar coordinates.</p></li> +</ol> +<section id="transformation-pipelines"> +<h2>Transformation pipelines<a class="headerlink" href="#transformation-pipelines" title="Permalink to this headline">¶</a></h2> +<p>The homology between the above steps and a Unix shell style pipeline is evident. +It is there the main architectural inspiration behind the transformation pipeline +framework. The pipeline framework is realized by utilizing a special “projection”, +that takes as its user supplied arguments, a series of elementary operations, +which it strings together in order to implement the full transformation needed. +Additionally, a number of elementary geodetic operations, including Helmert +transformations, general high order polynomial shifts and the Molodensky +transformation are available as part of the pipeline framework. +In anticipation of upcoming support for full time-varying transformations, we +also introduce a 4D spatiotemporal data type, and a programming interface +(API) for handling this.</p> +<p>The Molodensky transformation converts directly from geodetic coordinates +in one datum, to geodetic coordinates in another datum, while the (typically more +accurate) Helmert transformation converts from 3D cartesian to 3D cartesian +coordinates. So when using the Helmert transformation one typically needs to do an +initial conversion from geodetic to cartesian coordinates, and a final conversion +the other way round, to arrive at the desired result. Fortunately, this three-step +compound transformation has the attractive characteristic that each step depends +only on the output of the immediately preceding step. Hence, we can build a +geodetic-to-geodetic Helmert transformation by tying together the outputs and inputs +of 3 steps (geodetic-to-cartesian → Helmert → cartesian-to-geodetic), pipeline style. +The pipeline driver, makes this kind of chained transformations possible. +The implementation is compact, consisting of just one pseudo-projection, called +<code class="docutils literal notranslate"><span class="pre">pipeline</span></code>, which takes as its arguments strings of elementary projections +(note: “projection” is the, slightly misleading, PROJ term used for any kind of +transformation). +The pipeline pseudo projection is supplemented by a number of elementary +transformations, all in all providing a framework for building high accuracy +solutions for a wide spectrum of geodetic tasks.</p> +<p>As a first example, let us take a look at the iconic +<em>geodetic → Cartesian → Helmert → geodetic</em> case (steps 2 to 4 in the example in +the introduction). In PROJ it can be implemented as</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=pipeline +step proj=cart ellps=intl +step proj=helmert convention=coordinate_frame + x=-81.0703 y=-89.3603 z=-115.7526 + rx=-0.48488 ry=-0.02436 rz=-0.41321 s=-0.540645 +step proj=cart inv ellps=GRS80 +</pre></div> +</div> +<p>The pipeline can be expanded at both ends to accommodate whatever coordinate type +is needed for input and output: In the example below, we transform from the +deprecated Danish System 45, a 2D system with some tension in the original defining +network, to UTM zone 33, ETRS89. The tension is reduced using a polynomial +transformation (the init=./s45b… step, s45b.pol is a file containing the +polynomial coefficients), taking the S45 coordinates to a technical coordinate +system (TC32), defined to represent “UTM zone 32 coordinates, as they would look if +the Helmert transformation between ED50 and ETRS89 was perfect”. The TC32 +coordinates are then converted back to geodetic(ED50) coordinates, using an +inverse UTM projection, further to cartesian(ED50), then to cartesian(ETRS89), +using the relevant Helmert transformation, and back to geodetic(ETRS89), before +finally being projected onto the UTM zone 33, ETRS89 system. All in all a 6 step +pipeline, implementing a transformation with centimeter level accuracy from a +deprecated system with decimeter level tensions.</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=pipeline +step init=./s45b.pol:s45b_tc32 +step proj=utm inv ellps=intl zone=32 +step proj=cart ellps=intl +step proj=helmert convention=coordinate_frame + x=-81.0703 y=-89.3603 z=-115.7526 + rx=-0.48488 ry=-0.02436 rz=-0.41321 s=-0.540645 +step proj=cart inv ellps=GRS80 +step proj=utm ellps=GRS80 zone=33 +</pre></div> +</div> +<p>With the pipeline framework spatiotemporal transformation is possible. This is +possible by leveraging the time dimension in PROJ that enables 4D coordinates +(three spatial components and one temporal component) to be passed through a +transformation pipeline. In the example below a transformation from ITRF93 to +ITRF2000 is defined. The temporal component is given as GPS weeks in the input +data, but the 14-parameter Helmert transform expects temporal units in decimalyears. +Hence the first step in the pipeline is the unitconvert pseudo-projection that makes +sure the correct units are passed along to the Helmert transform. +Most parameters of the Helmert transform are taken from <span id="id2">[<a class="reference internal" href="../zreferences.html#id3" title="Altamimi, Z., Sillard, P., and Boucher, C. ITRF2000: a new release of the International Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research: Solid Earth, 2002. doi:10.1029/2001JB000561.">Altamimi2002</a>]</span>, +except the epoch which is the epoch of the transformation. +The last step in the pipeline is converting the +coordinate timestamps back to GPS weeks.</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=pipeline +step proj=unitconvert t_in=gps_week t_out=decimalyear +step proj=helmert convention=coordinate_frame + x=0.0127 y=0.0065 z=-0.0209 s=0.00195 + rx=0.00039 ry=-0.00080 rz=0.00114 + dx=-0.0029 dy=-0.0002 dz=-0.0006 ds=0.00001 + drx=0.00011 dry=0.00019 drz=-0.00007 + t_epoch=1988.0 +step proj=unitconvert t_in=decimalyear t_out=gps_week +</pre></div> +</div> +</section> +<section id="proj-4-x-5-x-paradigm"> +<h2>PROJ 4.x/5.x paradigm<a class="headerlink" href="#proj-4-x-5-x-paradigm" title="Permalink to this headline">¶</a></h2> +<blockquote id="cs2cs-specific-options"> +<div><table class="docutils align-default"> +<colgroup> +<col style="width: 16%" /> +<col style="width: 84%" /> +</colgroup> +<thead> +<tr class="row-odd"><th class="head"><p>Parameter</p></th> +<th class="head"><p>Description</p></th> +</tr> +</thead> +<tbody> +<tr class="row-even"><td><p>+datum</p></td> +<td><p>Datum name (see <code class="docutils literal notranslate"><span class="pre">proj</span> <span class="pre">-ld</span></code>)</p></td> +</tr> +<tr class="row-odd"><td><p>+geoidgrids</p></td> +<td><p>Filename of GTX grid file to use for vertical datum transforms</p></td> +</tr> +<tr class="row-even"><td><p>+nadgrids</p></td> +<td><p>Filename of NTv2 grid file to use for datum transforms</p></td> +</tr> +<tr class="row-odd"><td><p>+towgs84</p></td> +<td><p>3 or 7 term datum transform parameters</p></td> +</tr> +<tr class="row-even"><td><p>+to_meter</p></td> +<td><p>Multiplier to convert map units to 1.0m</p></td> +</tr> +<tr class="row-odd"><td><p>+vto_meter</p></td> +<td><p>Vertical conversion to meters</p></td> +</tr> +</tbody> +</table> +</div></blockquote> +<div class="admonition warning"> +<p class="admonition-title">Warning</p> +<p>This section documents the behavior of PROJ 4.x and 5.x. In PROJ 6.x, +<strong class="program">cs2cs</strong> has been reworked to use <a class="reference internal" href="../development/reference/functions.html#c.proj_create_crs_to_crs" title="proj_create_crs_to_crs"><code class="xref c c-func docutils literal notranslate"><span class="pre">proj_create_crs_to_crs()</span></code></a> internally, +with <em>late binding</em> capabilities, and thus is no longer constrained to using +WGS84 as a pivot (also called as <em>early binding</em> method). +When <strong class="program">cs2cs</strong> of PROJ 6 is used with PROJ.4 expanded strings to describe the CRS, +including <code class="docutils literal notranslate"><span class="pre">+towgs84</span></code>, <code class="docutils literal notranslate"><span class="pre">+nadgrids</span></code> and <code class="docutils literal notranslate"><span class="pre">+geoidgrids</span></code>, it will generally give +the same results as earlier PROJ versions. When used with AUTHORITY:CODE +CRS descriptions, it may return different results.</p> +</div> +<p>The <em>cs2cs</em> framework in PROJ 4 and 5 delivers a subset of the geodetic transformations available +with the <em>pipeline</em> framework. Coordinate transformations done in this framework +were transformed in a two-step process with WGS84 as a pivot datum. That is, the +input coordinates are transformed to WGS84 geodetic coordinates and then transformed +from WGS84 coordinates to the specified output coordinate reference system, by +utilizing either the Helmert transform, datum shift grids or a combination of both. +Datum shifts can be described in a proj-string with the parameters <code class="docutils literal notranslate"><span class="pre">+towgs84</span></code>, +<code class="docutils literal notranslate"><span class="pre">+nadgrids</span></code> and <code class="docutils literal notranslate"><span class="pre">+geoidgrids</span></code>. +An inverse transform exists for all three and is applied if +specified in the input proj-string. The most common is <code class="docutils literal notranslate"><span class="pre">+towgs84</span></code>, which is used to +define a 3- or 7-parameter Helmert shift from the input reference frame to WGS84. +Exactly which realization of WGS84 is not specified, hence a fair amount of +uncertainty is introduced in this step of the transformation. With the +nadgrids +parameter a non-linear planar correction derived from interpolation in a +correction grid can be applied. Originally this was implemented as a means to +transform coordinates between the North American datums NAD27 and NAD83, but +corrections can be applied for any datum for which a correction grid exists. The +inverse transform for the horizontal grid shift is “dumb”, in the sense that the +correction grid is applied verbatim without taking into account that the inverse +operation is non-linear. Similar to the horizontal grid correction, <code class="docutils literal notranslate"><span class="pre">+geoidgrids</span></code> +can be used to perform grid corrections in the vertical component. +Both grid correction methods allow inclusion of more than one grid in the same +transformation</p> +<p>In contrast to the <em>transformation pipeline</em> framework, transformations with the +<em>cs2cs</em> framework in PROJ 4 and 5 were expressed as two separate proj-strings. One proj-string <em>to</em> +WGS84 and one <em>from</em> WGS84. Together they form the mapping from the source +coordinate reference system to the destination coordinate reference system. +When used with the <strong class="program">cs2cs</strong> the source and destination CRS’s are separated by the +special <code class="docutils literal notranslate"><span class="pre">+to</span></code> parameter.</p> +<p>The following example demonstrates converting from the Greek GGRS87 datum +to WGS84 with the <code class="docutils literal notranslate"><span class="pre">+towgs84</span></code> parameter.</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>cs2cs +proj=latlong +ellps=GRS80 +towgs84=-199.87,74.79,246.62 + +to +proj=latlong +datum=WGS84 +20 35 +20d0'5.467"E 35d0'9.575"N 0.000 +</pre></div> +</div> +<p>With PROJ 6, you can simply use the following:</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p>With PROJ 6, the order of coordinates for EPSG geographic coordinate +reference systems is latitude first, longitude second.</p> +</div> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>cs2cs "GGRS87" "WGS 84" +35 20 +35d0'9.575"N 20d0'5.467"E 0.000 + +cs2cs EPSG:4121 EPSG:4326 +35 20 +35d0'9.575"N 20d0'5.467"E 0.000 +</pre></div> +</div> +<p>The EPSG database provides this example for transforming from WGS72 to WGS84 +using an approximated 7 parameter transformation.</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>cs2cs +proj=latlong +ellps=WGS72 +towgs84=0,0,4.5,0,0,0.554,0.219 \ + +to +proj=latlong +datum=WGS84 +4 55 +4d0'0.554"E 55d0'0.09"N 0.000 +</pre></div> +</div> +<p>With PROJ 6, you can simply use the following (note the reversed order for +latitude and longitude)</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>cs2cs "WGS 72" "WGS 84" +55 4 +55d0'0.09"N 4d0'0.554"E 0.000 + +cs2cs EPSG:4322 EPSG:4326 +55 4 +55d0'0.09"N 4d0'0.554"E 0.000 +</pre></div> +</div> +</section> +<section id="grid-based-datum-adjustments"> +<h2>Grid Based Datum Adjustments<a class="headerlink" href="#grid-based-datum-adjustments" title="Permalink to this headline">¶</a></h2> +<p>In many places (notably North America and Australia) national geodetic +organizations provide grid shift files for converting between different datums, +such as NAD27 to NAD83. These grid shift files include a shift to be applied +at each grid location. Actually grid shifts are normally computed based on an +interpolation between the containing four grid points.</p> +<p>PROJ supports use of grid files for shifting between various reference frames. +The grid shift table formats are CTable, NTv1 (the old Canadian format), and NTv2 +(<code class="file docutils literal notranslate"><span class="pre">.gsb</span></code> - the new Canadian and Australian format).</p> +<p>The text in this section is based on the <em>cs2cs</em> framework. Gridshifting is off +course also possible with the <em>pipeline</em> framework. The major difference between the +two is that the <em>cs2cs</em> framework is limited to grid mappings to WGS84, whereas with +<em>transformation pipelines</em> it is possible to perform grid shifts between any two +reference frames, as long as a grid exists.</p> +<p>Use of grid shifts with <strong class="program">cs2cs</strong> is specified using the <code class="docutils literal notranslate"><span class="pre">+nadgrids</span></code> +keyword in a coordinate system definition. For example:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>% cs2cs +proj=latlong +ellps=clrk66 +nadgrids=ntv1_can.dat \ + +to +proj=latlong +ellps=GRS80 +datum=NAD83 << EOF +-111 50 +EOF +111d0'2.952"W 50d0'0.111"N 0.000 +</pre></div> +</div> +<p>In this case the <code class="file docutils literal notranslate"><span class="pre">/usr/local/share/proj/ntv1_can.dat</span></code> grid shift file was +loaded, and used to get a grid shift value for the selected point.</p> +<p>It is possible to list multiple grid shift files, in which case each will be +tried in turn till one is found that contains the point being transformed.</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>cs2cs +proj=latlong +ellps=clrk66 \ + +nadgrids=conus,alaska,hawaii,stgeorge,stlrnc,stpaul \ + +to +proj=latlong +ellps=GRS80 +datum=NAD83 << EOF +-111 44 +EOF +111d0'2.788"W 43d59'59.725"N 0.000 +</pre></div> +</div> +<section id="skipping-missing-grids"> +<h3>Skipping Missing Grids<a class="headerlink" href="#skipping-missing-grids" title="Permalink to this headline">¶</a></h3> +<p>The special prefix <code class="docutils literal notranslate"><span class="pre">@</span></code> may be prefixed to a grid to make it optional. If it +not found, the search will continue to the next grid. Normally any grid not +found will cause an error. For instance, the following would use the +<code class="file docutils literal notranslate"><span class="pre">ntv2_0.gsb</span></code> file if available, otherwise it would +fallback to using the <code class="file docutils literal notranslate"><span class="pre">ntv1_can.dat</span></code> file.</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>cs2cs +proj=latlong +ellps=clrk66 +nadgrids=@ntv2_0.gsb,ntv1_can.dat \ + +to +proj=latlong +ellps=GRS80 +datum=NAD83 << EOF +-111 50 +EOF +111d0'3.006"W 50d0'0.103"N 0.000 +</pre></div> +</div> +</section> +<section id="the-null-grid"> +<h3>The null Grid<a class="headerlink" href="#the-null-grid" title="Permalink to this headline">¶</a></h3> +<p>A special <code class="file docutils literal notranslate"><span class="pre">null</span></code> grid shift file is distributed with PROJ. +This file provides a zero shift for the whole world. It may be +listed at the end of a nadgrids file list if you want a zero shift to be +applied to points outside the valid region of all the other grids. Normally if +no grid is found that contains the point to be transformed an error will occur.</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>cs2cs +proj=latlong +ellps=clrk66 +nadgrids=conus,null \ + +to +proj=latlong +ellps=GRS80 +datum=NAD83 << EOF +-111 45 +EOF +111d0'3.006"W 50d0'0.103"N 0.000 + +cs2cs +proj=latlong +ellps=clrk66 +nadgrids=conus,null \ + +to +proj=latlong +ellps=GRS80 +datum=NAD83 << EOF +-111 44 +-111 55 +EOF +111d0'2.788"W 43d59'59.725"N 0.000 +111dW 55dN 0.000 +</pre></div> +</div> +<p>For more information see the chapter on <a class="reference internal" href="../resource_files.html#id6"><span class="std std-ref">Other transformation grids</span></a>.</p> +</section> +<section id="caveats"> +<h3>Caveats<a class="headerlink" href="#caveats" title="Permalink to this headline">¶</a></h3> +<ul class="simple"> +<li><p>Where grids overlap (such as conus and <code class="file docutils literal notranslate"><span class="pre">ntv1_can.dat</span></code> for instance) the first +found for a point will be used regardless of whether it is appropriate or +not. So, for instance, <code class="docutils literal notranslate"><span class="pre">+nadgrids=ntv1_can.dat</span></code>,conus would result in +the Canadian data being used for some areas in the northern United States +even though the conus data is the approved data to use for the area. +Careful selection of files and file order is necessary. In some cases +border spanning datasets may need to be pre-segmented into Canadian and +American points so they can be properly grid shifted</p></li> +<li><p>Additional detail on the grid shift being applied can be found by setting +the PROJ_DEBUG environment variable to a value. This will result in output +to stderr on what grid is used to shift points, the bounds of the various +grids loaded and so forth</p></li> +</ul> +</section> +</section> +</section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="projections.html" class="btn btn-neutral float-left" title="Cartographic projection" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="ellipsoids.html" class="btn btn-neutral float-right" title="Ellipsoids" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> + </div> + + <hr/> + + <div role="contentinfo"> + <p>© Copyright 1983-2022. + <span class="lastupdated">Last updated on 22 Mar 2022. + </span></p> + </div> + + Built with <a href="https://www.sphinx-doc.org/">Sphinx</a> using a + <a href="https://github.com/readthedocs/sphinx_rtd_theme">theme</a> + provided by <a href="https://readthedocs.org">Read the Docs</a>. + + +</footer> + </div> + </div> + </section> + </div> + <script> + jQuery(function () { + SphinxRtdTheme.Navigation.enable(true); + }); + </script> + +</body> +</html>
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