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transform</li> + <li class="wy-breadcrumbs-aside"> + <a href="https://github.com/OSGeo/PROJ/edit/8.2/docs/source/operations/transformations/molodensky.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="horner.html" class="btn btn-neutral float-left" title="Horner polynomial evaluation" accesskey="p"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="molobadekas.html" class="btn btn-neutral float-right" title="Molodensky-Badekas transform" 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="molodensky-transform"> +<span id="molodensky"></span><h1>Molodensky transform<a class="headerlink" href="#molodensky-transform" 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>The Molodensky transformation resembles a <a class="reference internal" href="helmert.html#helmert"><span class="std std-ref">Helmert transform</span></a> with zero +rotations and a scale of unity, but converts directly from geodetic coordinates to +geodetic coordinates, without the intermediate shifts to and from cartesian +geocentric coordinates, associated with the Helmert transformation. +The Molodensky transformation is simple to implement and to parametrize, requiring +only the 3 shifts between the input and output frame, and the corresponding +differences between the semimajor axes and flattening parameters of the reference +ellipsoids. Due to its algorithmic simplicity, it was popular prior to the +ubiquity of digital computers. Today, it is mostly interesting for historical +reasons, but nevertheless indispensable due to the large amount of data that has +already been transformed that way <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> +<table class="docutils align-default"> +<colgroup> +<col style="width: 27%" /> +<col style="width: 73%" /> +</colgroup> +<tbody> +<tr class="row-odd"><td><p><strong>Alias</strong></p></td> +<td><p>molodensky</p></td> +</tr> +<tr class="row-even"><td><p><strong>Domain</strong></p></td> +<td><p>3D</p></td> +</tr> +<tr class="row-odd"><td><p><strong>Input type</strong></p></td> +<td><p>Geodetic coordinates (horizontal), meters (vertical)</p></td> +</tr> +<tr class="row-even"><td><p><strong>output type</strong></p></td> +<td><p>Geodetic coordinates (horizontal), meters (vertical)</p></td> +</tr> +</tbody> +</table> +<p>The Molodensky transform can be used to perform a datum shift from coordinate +<span class="math notranslate nohighlight">\((\phi_1, \lambda_1, h_1)\)</span> to <span class="math notranslate nohighlight">\((\phi_2, \lambda_2, h_2)\)</span> where the two +coordinates are referenced to different ellipsoids. This is based on three +assumptions:</p> +<blockquote> +<div><ol class="arabic simple"> +<li><p>The cartesian axes, <span class="math notranslate nohighlight">\(X, Y, Z\)</span>, of the two ellipsoids are parallel.</p></li> +<li><p>The offset, <span class="math notranslate nohighlight">\(\delta X, \delta Y, \delta Z\)</span>, between the two ellipsoid +are known.</p></li> +<li><p>The characteristics of the two ellipsoids, expressed as the difference in +semimajor axis (<span class="math notranslate nohighlight">\(\delta a\)</span>) and flattening (<span class="math notranslate nohighlight">\(\delta f\)</span>), are known.</p></li> +</ol> +</div></blockquote> +<p>The Molodensky transform is mostly used for transforming between old systems +dating back to the time before computers. The advantage of the Molodensky transform +is that it is fairly simple to compute by hand. The ease of computation come at the +cost of limited accuracy.</p> +<p>A derivation of the mathematical formulas for the Molodensky transform can be found +in <span id="id2">[<a class="reference internal" href="../../zreferences.html#id8" title="Deakin, R. E. The standard and abridged Molodensky coordinate transformation formulae. Technical Report, Department of Mathematical and Geospatial Sciences, RMIT University, Melborne, Australia, 2004. URL: http://www.mygeodesy.id.au/documents/Molodensky%20V2.pdf.">Deakin2004</a>]</span>.</p> +<section id="examples"> +<h2>Examples<a class="headerlink" href="#examples" title="Permalink to this headline">¶</a></h2> +<p>The abridged Molodensky:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=molodensky a=6378160 rf=298.25 da=-23 df=-8.120449e-8 dx=-134 dy=-48 dz=149 abridged +</pre></div> +</div> +<p>The same transformation using the standard Molodensky:</p> +<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>proj=molodensky a=6378160 rf=298.25 da=-23 df=-8.120449e-8 dx=-134 dy=-48 dz=149 +</pre></div> +</div> +</section> +<section id="parameters"> +<h2>Parameters<a class="headerlink" href="#parameters" title="Permalink to this headline">¶</a></h2> +<section id="required"> +<h3>Required<a class="headerlink" href="#required" title="Permalink to this headline">¶</a></h3> +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-da"> +<span class="sig-name descname"><span class="pre">+da</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-da" title="Permalink to this definition">¶</a></dt> +<dd><p>Difference in semimajor axis of the defining ellipsoids.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-df"> +<span class="sig-name descname"><span class="pre">+df</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-df" title="Permalink to this definition">¶</a></dt> +<dd><p>Difference in flattening of the defining ellipsoids.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-dx"> +<span class="sig-name descname"><span class="pre">+dx</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-dx" title="Permalink to this definition">¶</a></dt> +<dd><p>Offset of the X-axes of the defining ellipsoids.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-dy"> +<span class="sig-name descname"><span class="pre">+dy</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-dy" title="Permalink to this definition">¶</a></dt> +<dd><p>Offset of the Y-axes of the defining ellipsoids.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-dz"> +<span class="sig-name descname"><span class="pre">+dz</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-dz" title="Permalink to this definition">¶</a></dt> +<dd><p>Offset of the Z-axes of the defining ellipsoids.</p> +</dd></dl> + +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-ellps"> +<span class="sig-name descname"><span class="pre">+ellps</span></span><span class="sig-prename descclassname"><span class="pre">=<value></span></span><a class="headerlink" href="#cmdoption-arg-ellps" title="Permalink to this definition">¶</a></dt> +<dd><p>The name of a built-in ellipsoid definition.</p> +<p>See <a class="reference internal" href="../../usage/ellipsoids.html#ellipsoids"><span class="std std-ref">Ellipsoids</span></a> for more information, or execute +<a class="reference internal" href="../../apps/proj.html#cmdoption-proj-le"><code class="xref std std-option docutils literal notranslate"><span class="pre">proj</span> <span class="pre">-le</span></code></a> for a list of built-in ellipsoid names.</p> +<p><em>Defaults to “GRS80”.</em></p> +</dd></dl> + +</section> +<section id="optional"> +<h3>Optional<a class="headerlink" href="#optional" title="Permalink to this headline">¶</a></h3> +<dl class="std option"> +<dt class="sig sig-object std" id="cmdoption-arg-abridged"> +<span class="sig-name descname"><span class="pre">+abridged</span></span><span class="sig-prename descclassname"></span><a class="headerlink" href="#cmdoption-arg-abridged" title="Permalink to this definition">¶</a></dt> +<dd><p>Use the abridged version of the Molodensky transform.</p> +</dd></dl> + +</section> +</section> +</section> + + + </div> + </div> + <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> + <a href="horner.html" class="btn btn-neutral float-left" title="Horner polynomial evaluation" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="molobadekas.html" class="btn btn-neutral float-right" title="Molodensky-Badekas transform" 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 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