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authorKristian Evers <kristianevers@gmail.com>2017-11-29 21:58:15 +0100
committerKristian Evers <kristianevers@gmail.com>2017-11-29 21:58:15 +0100
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Molodensky transform
================================================================================
-Perform a datum shift in geodetic coordinate space.
+The Molodensky transformation resembles a :ref:`Helmert` 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 parameterize, 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 [EversKnudsen2017]_.
+
++---------------------+----------------------------------------------------------+
+| **Input type** | Geodetic coordinates. |
++---------------------+----------------------------------------------------------+
+| **Output type** | Geodetic coordinates. |
++---------------------+----------------------------------------------------------+
+| **Options** |
++---------------------+----------------------------------------------------------+
+| `+da` | Difference in semimajor axis of the defining ellipsoids. |
++---------------------+----------------------------------------------------------+
+| `+df` | Difference in flattening of the defining ellipsoids. |
++---------------------+----------------------------------------------------------+
+| `+dx` | Offset of the X-axes of the defining ellipsoids. |
++---------------------+----------------------------------------------------------+
+| `+dy` | Offset of the Y-axes of the defining ellipsoids. |
++---------------------+----------------------------------------------------------+
+| `+dz` | Offset of the Z-axes of the defining ellipsoids. |
++---------------------+----------------------------------------------------------+
+| `+ellps` | Ellipsoid definition of source coordinates. |
+| | Any specification can be used (e.g. `+a`, `+rf`, etc). |
+|   | If not specified, default ellipsoid is used. |
++---------------------+----------------------------------------------------------+
+| `+abridged` | Use the abridged version of the Molodensky transform. |
+| | Optional. |
++---------------------+----------------------------------------------------------+
+
+The Molodensky transform can be used to perform a datum shift from coordinate
+:math:`(\phi_1, \lambda_1, h_1)` to :math:`(\phi_2, \lambda_2, h_2)` where the two
+coordinates are referenced to different ellipsoids. This is based on three
+assumptions:
+
+ 1. The cartesian axes, :math:`X, Y, Z`, of the two ellipsoids are parallel.
+ 2. The offset, :math:`\delta X, \delta Y, \delta Z`, between the two ellipsoid
+ are known.
+ 3. The characteristics of the two ellipsoids, expressed as the difference in
+ semimajor axis (:math:`\delta a`) and flattening (:math:`\delta f`), are known.
+
+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.
+
+A derivation of the mathematical formulas for the Molodensky transform can be found
+in [Deakin2004]_.
+
+
+Examples
+###############################################################################
+
+The abridged Molodensky::
+
+ proj=molodensky a=6378160 rf=298.25 da=-23 df=-8.120449e-8 dx=-134 dy=-48 dz=149 abridged
+
+The same transformation using the standard Molodensky::
+
+ proj=molodensky a=6378160 rf=298.25 da=-23 df=-8.120449e-8 dx=-134 dy=-48 dz=149
+