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Molodensky transform

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New in version 5.0.0.

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The Molodensky transformation resembles a Helmert transform 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 [EversKnudsen2017].

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Alias

molodensky

Domain

3D

Input type

Geodetic coordinates (horizontal), meters (vertical)

output type

Geodetic coordinates (horizontal), meters (vertical)

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The Molodensky transform can be used to perform a datum shift from coordinate +\((\phi_1, \lambda_1, h_1)\) to \((\phi_2, \lambda_2, h_2)\) where the two +coordinates are referenced to different ellipsoids. This is based on three +assumptions:

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  1. The cartesian axes, \(X, Y, Z\), of the two ellipsoids are parallel.

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  2. The offset, \(\delta X, \delta Y, \delta Z\), between the two ellipsoid +are known.

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  3. The characteristics of the two ellipsoids, expressed as the difference in +semimajor axis (\(\delta a\)) and flattening (\(\delta f\)), are known.

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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.

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A derivation of the mathematical formulas for the Molodensky transform can be found +in [Deakin2004].

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Examples

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The abridged Molodensky:

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proj=molodensky a=6378160 rf=298.25 da=-23 df=-8.120449e-8  dx=-134 dy=-48 dz=149 abridged
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The same transformation using the standard Molodensky:

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proj=molodensky a=6378160 rf=298.25 da=-23 df=-8.120449e-8  dx=-134 dy=-48 dz=149
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Parameters

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Required

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++da=<value>
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Difference in semimajor axis of the defining ellipsoids.

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++df=<value>
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Difference in flattening of the defining ellipsoids.

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++dx=<value>
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Offset of the X-axes of the defining ellipsoids.

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++dy=<value>
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Offset of the Y-axes of the defining ellipsoids.

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++dz=<value>
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Offset of the Z-axes of the defining ellipsoids.

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++ellps=<value>
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The name of a built-in ellipsoid definition.

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See Ellipsoids for more information, or execute +proj -le for a list of built-in ellipsoid names.

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Defaults to “GRS80”.

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Optional

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++abridged
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Use the abridged version of the Molodensky transform.

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