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<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>
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