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| author | PROJ deploybot <proj.bot@proj.bot> | 2022-03-22 20:00:06 +0000 |
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shifting utilizing a deformation model</a><ul> +<li class="toctree-l4"><a class="reference internal" href="#example">Example</a></li> +<li class="toctree-l4"><a class="reference internal" href="#parameters">Parameters</a></li> +<li class="toctree-l4"><a class="reference internal" href="#mathematical-description">Mathematical description</a></li> +<li class="toctree-l4"><a class="reference internal" href="#see-also">See also</a></li> +</ul> +</li> +<li class="toctree-l3"><a class="reference internal" href="geogoffset.html">Geographic offsets</a></li> +<li class="toctree-l3"><a class="reference internal" href="helmert.html">Helmert transform</a></li> +<li class="toctree-l3"><a class="reference internal" href="horner.html">Horner polynomial evaluation</a></li> +<li class="toctree-l3"><a class="reference internal" href="molodensky.html">Molodensky transform</a></li> +<li class="toctree-l3"><a class="reference internal" href="molobadekas.html">Molodensky-Badekas transform</a></li> +<li 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<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> + <div itemprop="articleBody"> + + <section id="kinematic-datum-shifting-utilizing-a-deformation-model"> +<span id="deformation"></span><h1>Kinematic datum shifting utilizing a deformation model<a class="headerlink" href="#kinematic-datum-shifting-utilizing-a-deformation-model" title="Permalink to this headline">¶</a></h1> +<div class="versionadded"> +<p><span class="versionmodified added">New in version 5.0.0.</span></p> +</div> +<p>Perform datum shifts means of a deformation/velocity model.</p> +<table class="docutils align-default"> +<colgroup> +<col style="width: 20%" /> +<col style="width: 80%" /> +</colgroup> +<tbody> +<tr class="row-odd"><td><p><strong>Alias</strong></p></td> +<td><p>deformation</p></td> +</tr> +<tr class="row-even"><td><p><strong>Input type</strong></p></td> +<td><p>Cartesian coordinates (spatial), decimalyears (temporal).</p></td> +</tr> +<tr class="row-odd"><td><p><strong>Output type</strong></p></td> +<td><p>Cartesian coordinates (spatial), decimalyears (temporal).</p></td> +</tr> +<tr class="row-even"><td><p><strong>Domain</strong></p></td> +<td><p>4D</p></td> +</tr> +<tr class="row-odd"><td><p><strong>Input type</strong></p></td> +<td><p>Geodetic coordinates</p></td> +</tr> +<tr class="row-even"><td><p><strong>Output type</strong></p></td> +<td><p>Geodetic coordinates</p></td> +</tr> +</tbody> +</table> +<p>The deformation operation is used to adjust coordinates for intraplate deformations. +Usually the transformation parameters for regional plate-fixed reference frames such as +the ETRS89 does not take intraplate deformation into account. It is assumed that +tectonic plate of the region is rigid. Often times this is true, but near the plate +boundary and in areas with post-glacial uplift the assumption breaks. Intraplate +deformations can be modelled and then applied to the coordinates so that +they represent the physical world better. In PROJ this is done with the deformation +operation.</p> +<p>The horizontal grid is stored in CTable2 format and the vertical grid is stored in the +GTX format. Both grids are expected to contain grid-values in units of +mm/year. GDAL both reads and writes both file formats. Using GDAL for +construction of new grids is recommended.</p> +<p>Starting with PROJ 7.0, use of a GeoTIFF format is recommended to store both +the horizontal and vertical velocities.</p> +<p>More complex deformations can be done with the <a class="reference internal" href="defmodel.html#defmodel"><span class="std std-ref">Multi-component time-based deformation model</span></a> transformation.</p> +<section id="example"> +<h2>Example<a class="headerlink" href="#example" title="Permalink to this headline">¶</a></h2> +<p>In <span id="id1">[<a class="reference internal" href="../../zreferences.html#id14" title="Häkli, P., Lidberg, M., Jivall, L., Nørbech, T., Tangen, O., Weber, M., Pihlak, P., Aleksejenko, I., and Paršeliunas, E. The NKG2008 GPS campaign – final transformation results and a new common Nordic reference frame. Journal of Geodetic Science, 6(1):1–33, 2016. doi:10.1515/jogs-2016-0001.">Hakli2016</a>]</span> coordinate transformation including a deformation model is described. +The paper describes how coordinates from the global ITRFxx frames are transformed to the +local Nordic realisations of ETRS89. Scandinavia is an area with significant post-glacial +rebound. The deformations from the post-glacial uplift is not accounted for in the +official ETRS89 transformations so in order to get accurate transformations in the Nordic +countries it is necessary to apply the deformation model. The transformation from ITRF2008 +to the Danish realisation of ETRS89 is in PROJ described as:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj = pipeline ellps = GRS80 + # ITRF2008@t_obs -> ITRF2000@t_obs +step init = ITRF2008:ITRF2000 + # ITRF2000@t_obs -> ETRF2000@t_obs +step proj=helmert t_epoch = 2000.0 convention=position_vector + x = 0.054 rx = 0.000891 drx = 8.1e-05 + y = 0.051 ry = 0.00539 dry = 0.00049 + z = -0.048 rz = -0.008712 drz = -0.000792 + # ETRF2000@t_obs -> NKG_ETRF00@2000.0 +step proj = deformation t_epoch = 2000.0 + grids = ./eur_nkg_nkgrf03vel_realigned.tif + inv + # NKG_ETRF@2000.0 -> ETRF92@2000.0 +step proj=helmert convention=position_vector s = -0.009420e + x = 0.03863 rx = 0.00617753 + y = 0.147 ry = 5.064e-05 + z = 0.02776 rz = 4.729e-05 + # ETRF92@2000.0 -> ETRF92@1994.704 +step proj = deformation dt = -5.296 + grids = ./eur_nkg_nkgrf03vel_realigned.tif +</pre></div> +</div> +<p>From this we can see that the transformation from ITRF2008 to the Danish realisation of +ETRS89 is a combination of Helmert transformations and adjustments with a deformation +model. The first use of the deformation operation is:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj = deformation t_epoch = 2000.0 grids = ./eur_nkg_nkgrf03vel_realigned.tif +</pre></div> +</div> +<p>Here we set the central epoch of the transformation, 2000.0. The observation epoch +is expected as part of the input coordinate tuple. The deformation model is +described by two grids, specified with <a class="reference internal" href="#cmdoption-arg-xy_grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+xy_grids</span></code></a> and <a class="reference internal" href="#cmdoption-arg-z_grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+z_grids</span></code></a>. +The first is the horizontal part of the model and the second is the vertical +component.</p> +</section> +<section id="parameters"> +<h2>Parameters<a class="headerlink" href="#parameters" title="Permalink to this headline">¶</a></h2> +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-xy_grids"> +<span id="cmdoption-arg-xy-grids"></span><span class="sig-name descname"><span class="pre">+xy_grids</span></span><span class="sig-prename descclassname"><span class="pre">=<list></span></span><a class="headerlink" href="#cmdoption-arg-xy_grids" title="Permalink to this definition">¶</a></dt> +<dd><p>Comma-separated list of grids to load. If a grid is prefixed by an <code class="docutils literal notranslate"><span class="pre">@</span></code> the +grid is considered optional and PROJ will the not complain if the grid is +not available.</p> +<p>Grids for the horizontal component of a deformation model is expected to be +in CTable2 format.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p><a class="reference internal" href="#cmdoption-arg-xy_grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+xy_grids</span></code></a> is mutually exclusive with <a class="reference internal" href="xyzgridshift.html#cmdoption-arg-grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+grids</span></code></a></p> +</div> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-z_grids"> +<span id="cmdoption-arg-z-grids"></span><span class="sig-name descname"><span class="pre">+z_grids</span></span><span class="sig-prename descclassname"><span class="pre">=<list></span></span><a class="headerlink" href="#cmdoption-arg-z_grids" title="Permalink to this definition">¶</a></dt> +<dd><p>Comma-separated list of grids to load. If a grid is prefixed by an <cite>@</cite> the +grid is considered optional and PROJ will the not complain if the grid is +not available.</p> +<p>Grids for the vertical component of a deformation model is expected to be +in either GTX format.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p><a class="reference internal" href="#cmdoption-arg-z_grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+z_grids</span></code></a> is mutually exclusive with <a class="reference internal" href="xyzgridshift.html#cmdoption-arg-grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+grids</span></code></a></p> +</div> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-grids"> +<span class="sig-name descname"><span class="pre">+grids</span></span><span class="sig-prename descclassname"><span class="pre">=<list></span></span><a class="headerlink" href="#cmdoption-arg-grids" title="Permalink to this definition">¶</a></dt> +<dd><div class="versionadded"> +<p><span class="versionmodified added">New in version 7.0.0.</span></p> +</div> +<p>Comma-separated list of grids to load. If a grid is prefixed by an <cite>@</cite> the +grid is considered optional and PROJ will the not complain if the grid is +not available.</p> +<p>Grids should be in GeoTIFF format with the first 3 components being +respectively the easting, northing and up velocities in mm/year. +Setting the Description and Unit Type GDAL band metadata items is strongly +recommended, so that gdalinfo reports:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>Band 1 Block=... Type=Float32, ColorInterp=Gray + Description = east_velocity + Unit Type: mm/year +Band 2 Block=... Type=Float32, ColorInterp=Undefined + Description = north_velocity + Unit Type: mm/year +Band 3 Block=... Type=Float32, ColorInterp=Undefined + Description = up_velocity + Unit Type: mm/year +</pre></div> +</div> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p><a class="reference internal" href="xyzgridshift.html#cmdoption-arg-grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+grids</span></code></a> is mutually exclusive with <a class="reference internal" href="#cmdoption-arg-xy_grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+xy_grids</span></code></a> +and <a class="reference internal" href="#cmdoption-arg-z_grids"><code class="xref std std-option docutils literal notranslate"><span class="pre">+z_grids</span></code></a></p> +</div> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-t_epoch"> +<span id="cmdoption-arg-t-epoch"></span><span class="sig-name descname"><span class="pre">+t_epoch</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-t_epoch" title="Permalink to this definition">¶</a></dt> +<dd><p>Central epoch of transformation given in decimalyears. Will be used in +conjunction with the observation time from the input coordinate to +determine <span class="math notranslate nohighlight">\(dt\)</span> as used in eq. <a class="reference internal" href="#equation-apply-velocity">(1)</a> below.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p><a class="reference internal" href="vgridshift.html#cmdoption-arg-t_epoch"><code class="xref std std-option docutils literal notranslate"><span class="pre">+t_epoch</span></code></a> is mutually exclusive with <a class="reference internal" href="#cmdoption-arg-dt"><code class="xref std std-option docutils literal notranslate"><span class="pre">+dt</span></code></a></p> +</div> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-dt"> +<span class="sig-name descname"><span class="pre">+dt</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-dt" title="Permalink to this definition">¶</a></dt> +<dd><div class="versionadded"> +<p><span class="versionmodified added">New in version 6.0.0.</span></p> +</div> +<p><span class="math notranslate nohighlight">\(dt\)</span> as used in eq. <a class="reference internal" href="#equation-apply-velocity">(1)</a> below. Is useful when +no observation time is available in the input coordinate or when +a deformation for a specific timespan needs to be applied in a +transformation. <span class="math notranslate nohighlight">\(dt\)</span> is given in units of decimalyears.</p> +<div class="admonition note"> +<p class="admonition-title">Note</p> +<p><a class="reference internal" href="#cmdoption-arg-dt"><code class="xref std std-option docutils literal notranslate"><span class="pre">+dt</span></code></a> is mutually exclusive with <a class="reference internal" href="vgridshift.html#cmdoption-arg-t_epoch"><code class="xref std std-option docutils literal notranslate"><span class="pre">+t_epoch</span></code></a></p> +</div> +</dd></dl> + +</section> +<section id="mathematical-description"> +<h2>Mathematical description<a class="headerlink" href="#mathematical-description" title="Permalink to this headline">¶</a></h2> +<p>Mathematically speaking, application of a deformation model is simple. The deformation model is +represented as a grid of velocities in three dimensions. Coordinate corrections are +applied in cartesian space. For a given coordinate, <span class="math notranslate nohighlight">\((X, Y, Z)\)</span>, velocities +<span class="math notranslate nohighlight">\((V_X, V_Y, V_Z)\)</span> can be interpolated from the gridded model. The time span +between <span class="math notranslate nohighlight">\(t_{obs}\)</span> and <span class="math notranslate nohighlight">\(t_c\)</span> determine the magnitude of the coordinate +correction as seen in eq. <a class="reference internal" href="#equation-apply-velocity">(1)</a> below.</p> +<div class="math notranslate nohighlight" id="equation-apply-velocity"> +<span class="eqno">(1)<a class="headerlink" href="#equation-apply-velocity" title="Permalink to this equation">¶</a></span>\[\begin{split}\begin{align} + \begin{pmatrix} + X \\ + Y \\ + Z \\ + \end{pmatrix}_B = + \begin{pmatrix} + X \\ + Y \\ + Z \\ + \end{pmatrix}_A + + (t_{obs} - t_c) + \begin{pmatrix} + V_X \\ + V_Y \\ + V_Z \\ + \end{pmatrix} +\end{align}\end{split}\]</div> +<p>Corrections are done in cartesian space.</p> +<p>Coordinates of the gridded model are in ENU (east, north, up) space because it +would otherwise require an enormous 3 dimensional grid to handle the corrections +in cartesian space. Keeping the correction in lat/long space reduces the +complexity of the grid significantly. Consequently though, the input coordinates +needs to be converted to lat/long space when searching for corrections in the +grid. This is done with the <a class="reference internal" href="../conversions/cart.html#cart"><span class="std std-ref">cart</span></a> operation. The converted grid +corrections can then be applied to the input coordinates in cartesian space. The +conversion from ENU space to cartesian space is done in the following way:</p> +<div class="math notranslate nohighlight" id="equation-enu2xyz"> +<span class="eqno">(2)<a class="headerlink" href="#equation-enu2xyz" title="Permalink to this equation">¶</a></span>\[\begin{split}\begin{align} + \begin{pmatrix} + X \\ + Y \\ + Z \\ + \end{pmatrix} = + \begin{pmatrix} + -\sin\phi \cos\lambda N - \sin\lambda E + \cos\phi \cos\lambda U \\ + -\sin\phi \sin\lambda N + \sin\lambda E + \cos\phi \sin\lambda U \\ + \cos\phi N + \sin\phi U \\ + \end{pmatrix} +\end{align}\end{split}\]</div> +<p>where <span class="math notranslate nohighlight">\(\phi\)</span> and <span class="math notranslate nohighlight">\(\lambda\)</span> are the latitude and longitude of the coordinate +that is searched for in the grid. <span class="math notranslate nohighlight">\((E, N, U)\)</span> are the grid values in ENU-space and +<span class="math notranslate nohighlight">\((X, Y, Z)\)</span> are the corrections converted to cartesian space.</p> +</section> +<section id="see-also"> +<h2>See also<a class="headerlink" href="#see-also" title="Permalink to this headline">¶</a></h2> +<ol class="arabic simple"> +<li><p><a class="reference internal" href="../../usage/differences.html#differences-deformation"><span class="std std-ref">Behavioural changes from version 5 to 6</span></a></p></li> +</ol> +</section> +</section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="defmodel.html" class="btn btn-neutral float-left" title="Multi-component time-based deformation model" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="geogoffset.html" class="btn btn-neutral float-right" title="Geographic offsets" 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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