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diff --git a/docs/source/usage/operations/transformations/deformation.rst b/docs/source/usage/operations/transformations/deformation.rst deleted file mode 100644 index 2489f18e..00000000 --- a/docs/source/usage/operations/transformations/deformation.rst +++ /dev/null @@ -1,148 +0,0 @@ -.. _deformation: - -================================================================================ -Kinematic datum shifting utilizing a deformation model -================================================================================ - -Perform datum shifts means of a deformation/velocity model. - -+-----------------+--------------------------------------------------------------------+ -| **Input type** | Cartesian coordinates. | -+-----------------+--------------------------------------------------------------------+ -| **Output type** | Cartesian coordinates. | -+-----------------+--------------------------------------------------------------------+ -| **Options** | -+-----------------+--------------------------------------------------------------------+ -| `xy_grids` | Comma-separated list of grids to load. | -+-----------------+--------------------------------------------------------------------+ -| `z_grids` | Comma-separated list of grids to load. | -+-----------------+--------------------------------------------------------------------+ -| `t_epoch` | Central epoch of transformation. [decimalyear]. Only used in | -| | spatiotemporal transformations. | -+-----------------+--------------------------------------------------------------------+ -| `t_obs` | Observation time of coordinate(s). Mostly useful in 2D and 3D | -| | transformations. [decimalyear]. *Optional*. | -+-----------------+--------------------------------------------------------------------+ - -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. Tntraplate -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. - -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. -Details about the formats can be found in the GDAL documentation. GDAL both reads and -writes both file formats. Using GDAL for construction of new grids is recommended. - -Example -------------------------------------------------------------------------------- - -In [Häkli2016]_ 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 ERTS89. Scandinavia is an area with significant post-glacial -rebound. The deformations from the post-glacial uplift is not accounted for in the -offcial ETRS89 transformations so in order to get accurate transformations in the Nordic -countries it is necesarry to apply the deformation model. The transformation from ITRF2008 -to the Danish realisation of ETRS89 is in PROJ described as:: - - 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 transpose - 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 - xy_grids = ./nkgrf03vel_realigned_xy.ct2 - z_grids = ./nkgrf03vel_realigned_z.gtx - # NKG_ETRF@2000.0 -> ETRF92@2000.0 - step proj=helmert transpose 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 t_epoch = 1994.704 t_obs = 2000.0 - xy_grids = ./nkgrf03vel_realigned_xy.ct2 - z_grids = ./nkgrf03vel_realigned_z.gtx - -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:: - - proj = deformation t_epoch = 2000.0 - xy_grids = ./nkgrf03vel_realigned_xy.ct2 - z_grids = ./nkgrf03vel_realigned_z.gtx - -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 *xy_grids* and *z_grids*. The first is -the horizontal part of the model and the second is the vertical component. - -Mathematical description -------------------------------------------------------------------------------- - -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, :math:`(X, Y, Z)`, velocities -:math:`(V_X, V_Y, V_Z)` can be interpolated from the gridded model. The time span -between :math:`t_c` and :math:`t_{obs}` determine the magnitude of the coordinate -correcton as seen in eq. :eq:`apply_velocity` below. - -.. math:: - :label: apply_velocity - - \begin{align} - \begin{pmatrix} - X \\ - Y \\ - Z \\ - \end{pmatrix}_B = - \begin{pmatrix} - X \\ - Y \\ - Z \\ - \end{pmatrix}_A + - (t_c - t_{obs}) - \begin{pmatrix} - V_X \\ - V_Y \\ - V_Z \\ - \end{pmatrix} - \end{align} - -Corrections are done in cartesian space. - -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. Consequentyly though, the input coordinates needs to be converted to -lat/long space when searching for corrections in the grid. This is done with *cart* -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: - -.. math:: - :label: enu2xyz - - \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} - -where :math:`\phi` and :math:`\lambda` are the latitude and longitude of the coordinate -that is searched for in the grid. :math:`(E, N, U)` are the grid values in ENU-space and -:math:`(X, Y, Z)` are the corrections converted to cartesian space. diff --git a/docs/source/usage/operations/transformations/helmert.rst b/docs/source/usage/operations/transformations/helmert.rst deleted file mode 100644 index 03e5db15..00000000 --- a/docs/source/usage/operations/transformations/helmert.rst +++ /dev/null @@ -1,388 +0,0 @@ -.. _helmert: - -================================================================================ -Helmert transform -================================================================================ - -The Helmert transformation changes coordinates from one reference frame to -anoether by means of 3-, 4-and 7-parameter shifts, or one of their 6-, 8- and -14-parameter kinematic counterparts. - -+-----------------+-------------------------------------------------------------------+ -| **Input type** | Cartesian coordinates (spatial), decimalyears (temporal). | -+-----------------+-------------------------------------------------------------------+ -| **Output type** | Cartesian coordinates (spatial), decimalyears (temporal). | -+-----------------+-------------------------------------------------------------------+ -| **Options** | -+----------------+--------------------------------------------------------------------+ -| `x` | Translation of the x-axis [m]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `y` | Translation of the y-axis [m]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `z` | Translation of the z-axis. [m]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `s` | Scale factor [ppm]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `rx` | X-axis rotation in the 3D Helmert [arc seconds]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `ry` | Y-axis rotatoin in the 3D Helmert [arc seconds]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `rz` | Z-axis rotation in the 3D Helmert [arc seconds]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `theta` | Rotation angle in the 2D Helmert. [arc seconds]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `dx` | Translation rate of the x-axis. [m/year]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `dy` | Translation rate of the y-axis. [m/year]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `dz` | Translation rate of the z-axis. [m/year]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `ds` | Scale rate factor [ppm/year]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `drx` | Rotation rate of the x-axis [arc seconds/year]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `dry` | Rotation rate of the y-axis [arc seconds/year]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `drz` | Rotation rate of the y-axis [arc seconds/year]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `t_epoch` | Central epoch of transformation. [decimalyear]. Only used in | -| | spatiotemporal transformations. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `t_obs` | Observation time of coordinate(s). Mostly useful in 2D and 3D | -| | transformations. [decimalyear]. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `exact` | Use exact transformation equations. *Optional*. | -+----------------+--------------------------------------------------------------------+ -| `transpose` | Transpose rotation matrix. *Optional*. | -+----------------+--------------------------------------------------------------------+ - -The Helmert transform, in all it's various incarnations, is used to perform reference -frame shifts. The transformation operates in cartesian space. It can be used to transform -planar coordinates from one datum to another, transform 3D cartesian -coordinates from one static reference frame to another or it can be used to do fully -kinematic transformations from global reference frames to local static frames. - -All of the parameters described in the table above are marked as optional. This is true -as long as at least one parameter is defined in the setup of the transformation. -The behaviour of the transformation depends on which parameters are used in the setup. -For instance, if a rate of change paramater is specified a kinematic version of the -transformation is used. - -The kinematic transformations require an observation time of the coordinate, as well -as a central epoch for the transformation. The latter is usually documented -alongside the rest of the transformation parameters for a given transformation. -The central eopch is controlled with the parameter `t_epoch`. The observation -time can either by stated as part of the coordinate when using PROJ's -4D-functionality or it can be controlled in the transformation setup by the -parameter `t_obs`. When `t_obs` is specified, all transformed coordinates are -treated as if they have the same observation time. - -Examples -+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ - -Transforming coordinates from NAD72 to NAD83 using the 4 parameter 2D Helmert: - -:: - - proj=helmert ellps=GRS80 x=-9597.3572 y=.6112 - s=0.304794780637 theta=-1.244048 - -Simplified transformations from ITRF2008/IGS08 to ETRS89 using 7 parameters: - -:: - - proj=helmert ellps=GRS80 x=0.67678 y=0.65495 z=-0.52827 - rx=-0.022742 ry=0.012667 rz=0.022704 s=-0.01070 - -Transformation from `ITRF2000@2017.0` to `ITRF93@2017.0` using 15 parameters: - -:: - - proj=helmert ellps=GRS80 - x=0.0127 y=0.0065 z=-0.0209 s=0.00195 - dx=-0.0029 dy=-0.0002 dz=-0.0006 ds=0.00001 - rx=-0.00039 ry=0.00080 rz=-0.00114 - drx=-0.00011 dry=-0.00019 drz=0.00007 - epoch=1988.0 tobs=2017.0 transpose - -Mathematical description -+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ - -In the notation used below, :math:`\hat{P}` is the rate of change of a given transformation -parameter :math:`P`. :math:`\dot{P}` is the kinematically adjusted version of :math:`P`, -described by - -.. math:: - :label: propagation - - \dot{P}= P + \hat{P}\left(t - t_{central}\right) - -where :math:`t` is the observation time of the coordinate and :math:`t_{central}` is -the central epoch of the transformation. Equation :eq:`propagation` can be used to -propagate all transformation parameters in time. - -Superscripts of vectors denote the reference frame the coordinates in the vector belong to. - - -2D Helmert -------------------------------------------------------------------------------- - -The simplest version of the Helmert transform is the 2D case. In the 2-dimensional -case only the horizontal coordinates are changed. The coordinates can be -translated, rotated and scale. Translation is controlled with the `x` and `y` -parameters. The rotation is determined by `theta` and the scale is controlled with -the `s` paramaters. - -.. note:: - - The scaling parameter `s` is unitless for the 2D Helmert, as opposed to the - 3D version where the scaling parameter is given in units of ppm. - -Mathematically the 2D Helmert is described as: - -.. math:: - :label: 4param - - \begin{align} - \begin{bmatrix} - X \\ - Y \\ - \end{bmatrix}^B = - \begin{bmatrix} - T_x \\ - T_y \\ - \end{bmatrix} + - s - \begin{bmatrix} - \hphantom{-}\cos \theta & \sin \theta \\ - -\sin \theta & \cos \theta \\ - \end{bmatrix} - \begin{bmatrix} - X \\ - Y \\ - \end{bmatrix}^A - \end{align} - - -:eq:`4param` can be extended to a time-varying kinematic version by -adjusting the parameters with :eq:`propagation` to :eq:`4param`, which yields -the kinematic 2D Helmert transform: - -.. math:: - :label: 8param - - \begin{align} - \begin{bmatrix} - X \\ - Y \\ - \end{bmatrix}^B = - \begin{bmatrix} - \dot{T_x} \\ - \dot{T_y} \\ - \end{bmatrix} + - s(t) - \begin{bmatrix} - \hphantom{-}\cos \dot{\theta} & \sin \dot{\theta} \\ - -\sin\ \dot{\theta} & \cos \dot{\theta} \\ - \end{bmatrix} - \begin{bmatrix} - X \\ - Y \\ - \end{bmatrix}^A - \end{align} - -All parameters in :eq:`8param` are determined by the use of :eq:`propagation`, -which applies the rate of change to each individual parameter for a given -timespan between :math:`t` and :math:`t_{central}`. - - -3D Helmert -------------------------------------------------------------------------------- - -The general form of the 3D Helmert is - -.. math:: - :label: general-helmert - - - \begin{align} - V^B = T + \left(1 + s \times 10^{-6}\right) \mathbf{R} V^A - \end{align} - -Where :math:`T` is a vector consisting of the three translation parameters, :math:`s` -is the scaling factor and :math:`\mathbf{R}` is a rotation matrix. :math:`V^A` and -:math:`V^B` are coordinate vectors, with :math:`V^A` being the input coordinate and -:math:`V^B` is the output coordinate. - -The rotation matrix is composed of three rotation matrices, one for each axis: - -.. math:: - - \begin{align} - \mathbf{R}_X &= \begin{bmatrix} 1 & 0 & 0\\ 0 & \cos\phi & -\sin\phi\\ 0 & \sin\phi & \cos\phi \end{bmatrix} - \end{align} - -.. math:: - - \begin{align} - \mathbf{R}_Y &= \begin{bmatrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta & 0 & \cos\theta \end{bmatrix} - \end{align} - -.. math:: - - \begin{align} - \mathbf{R}_Z &= \begin{bmatrix} \cos\psi & -\sin\psi & 0\\ \sin\psi & \cos\psi & 0\\ 0 & 0 & 1 \end{bmatrix} - \end{align} - -The three rotation matrices can be combined in one: - -.. math:: - - \begin{align} - \mathbf{R} = \mathbf{R_X} \mathbf{R_Y} \mathbf{R_Y} - \end{align} - -For :math:`\mathbf{R}`, this yields: - -.. math:: - :label: rot_exact - - \begin{bmatrix} - \cos\theta \cos\psi & -\cos\phi \sin\psi + \sin\phi \sin\theta \cos\psi & \sin\phi \sin\psi + \cos\phi \sin\theta \cos\psi \\ - \cos\theta\sin\psi & \cos\phi \cos\psi + \sin\phi \sin\theta \sin\psi & - \sin\phi \cos\psi + \cos\phi \sin\theta \sin\psi \\ - -\sin\theta & \sin\phi \cos\theta & \cos\phi \cos\theta \\ - \end{bmatrix} - - -Using the small angle approxition the rotation matrix can be simplified to - -.. math:: - :label: rot_approx - - \begin{align} \mathbf{R} = - \begin{bmatrix} - 1 & -R_z & R_y \\ - Rz & 1 & -R_x \\ - -Ry & R_x & 1 \\ - \end{bmatrix} - \end{align} - -Which allow us to express the most common version of the Helmert transform, -using the approximated rotation matrix: - - -.. math:: - :label: 7param - - \begin{align} - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^B = - \begin{bmatrix} - T_x \\ - T_y \\ - T_z \\ - \end{bmatrix} + - \left(1 + s \times 10^{-6}\right) - \begin{bmatrix} - 1 & -R_z & R_y \\ - Rz & 1 & -R_x \\ - -Ry & R_x & 1 \\ - \end{bmatrix} - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^A - \end{align} - -If the rotation matrix is transposed the transformation is effectively reversed. -This is cause for some confusion since there is no correct way of defining the -rotation matrix. Two conventions exists and they seem to be equally popular. -In PROJ the rotation matrix can be transposed by adding the `transpose` flag in -the transformation setup. - -Applying :eq:`propagation` we get the kinematic version of the approximated -3D Helmert: - -.. math:: - :label: 14param - - \begin{align} - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^B = - \begin{bmatrix} - \dot{T_x} \\ - \dot{T_y} \\ - \dot{T_z} \\ - \end{bmatrix} + - \left(1 + \dot{s} \times 10^{-6}\right) - \begin{bmatrix} - 1 & -\dot{R_z} & \dot{R_y} \\ - \dot{R_z} & 1 & -\dot{R_x} \\ - -\dot{R_y} & \dot{R_x} & 1 \\ - \end{bmatrix} - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^A - \end{align} - - - - -The Helmert transformation can be applied without using the rotation parameters, -in which case it becomes a simlpe translation of the origin of the coordinate -system. When using the Helmert in this version equation :eq:`general-helmert` -simplies to: - -.. math:: - :label: 3param - - \begin{align} - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^B = - \begin{bmatrix} - T_x \\ - T_y \\ - T_z \\ - \end{bmatrix} + - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^A - \end{align} - -That after application of :eq:`propagation` has the following kinematic -countpart: - -.. math:: - :label: 6param - - \begin{align} - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^B = - \begin{bmatrix} - \dot{T_x} \\ - \dot{T_y} \\ - \dot{T_z} \\ - \end{bmatrix} + - \begin{bmatrix} - X \\ - Y \\ - Z \\ - \end{bmatrix}^A - \end{align} diff --git a/docs/source/usage/operations/transformations/hgridshift.rst b/docs/source/usage/operations/transformations/hgridshift.rst deleted file mode 100644 index 7f3c9895..00000000 --- a/docs/source/usage/operations/transformations/hgridshift.rst +++ /dev/null @@ -1,39 +0,0 @@ -.. _hgridshift: - -================================================================================ -Horizontal grid shift -================================================================================ - -Change of horizontal datum by grid shift. - -+-----------------+--------------------------------------------------------------------+ -| **Input type** | Geodetic coordinates. | -+-----------------+--------------------------------------------------------------------+ -| **Output type** | Geodetic coordinates. | -+-----------------+--------------------------------------------------------------------+ -| **Options** | -+-----------------+--------------------------------------------------------------------+ -| `+grids` | Comma-separated list of grids to load. | -+-----------------+--------------------------------------------------------------------+ - -The horizontal grid shift is done by offsetting the planar input coordinates by -a specific amount determined by the loaded grids. The simplest use case of the -horizontal grid shift is applying a single grid:: - - +hgridshift +grids=nzgr2kgrid0005.gsb - - -More than one grid can be loaded at the same time, for instance in case the -dataset needs to be transformed spans several countries. In this example grids -of the continental US, Alaska and Canada is loaded at the same time:: - - +hgridshift +grids=@conus,@alaska,@ntv2_0.gsb,@ntv_can.dat - -The ``@`` in the above example states that the grid is optional, in case the grid -is not found in the PROJ search path. The list of grids is prioritized so that -grids in the start of the list takes presedence over the grids in the back of the -list. - -PROJ supports CTable2, NTv1 and NTv2 files for horizontal grid corrections. Details -about all three formats can be found in the GDAL documentation. GDAL reads and -writes all three formats. Using GDAL for construction of new grids is recommended. diff --git a/docs/source/usage/operations/transformations/index.rst b/docs/source/usage/operations/transformations/index.rst deleted file mode 100644 index 5112ffa2..00000000 --- a/docs/source/usage/operations/transformations/index.rst +++ /dev/null @@ -1,17 +0,0 @@ -.. _transformation_list: - -================================================================================ -Transformations -================================================================================ - -Transformations coordinate operation in which the two coordinate reference -systems are based on different datums. - -.. toctree:: - :maxdepth: 1 - - deformation - helmert - molodensky - hgridshift - vgridshift diff --git a/docs/source/usage/operations/transformations/molodensky.rst b/docs/source/usage/operations/transformations/molodensky.rst deleted file mode 100644 index d4ff3e79..00000000 --- a/docs/source/usage/operations/transformations/molodensky.rst +++ /dev/null @@ -1,74 +0,0 @@ -.. _molodensky: - -================================================================================ -Molodensky transform -================================================================================ - -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 - diff --git a/docs/source/usage/operations/transformations/vgridshift.rst b/docs/source/usage/operations/transformations/vgridshift.rst deleted file mode 100644 index e9f78310..00000000 --- a/docs/source/usage/operations/transformations/vgridshift.rst +++ /dev/null @@ -1,41 +0,0 @@ -.. _vgridshift: - -================================================================================ -Vertical grid shift -================================================================================ - -Change Vertical datum change by grid shift - -+-----------------+--------------------------------------------------------------------+ -| **Input type** | Geodetic coordinates (horizontal), meters (vertical). | -+-----------------+--------------------------------------------------------------------+ -| **Output type** | Geodetic coordinates (horizontal), meters (vertical). | -+-----------------+--------------------------------------------------------------------+ -| **Options** | -+-----------------+--------------------------------------------------------------------+ -| `+grids` | Comma-separated list of grids to load. | -+-----------------+--------------------------------------------------------------------+ - -The vertical grid shift is done by offsetting the vertical input coordinates by -a specific amount determined by the loaded grids. The simplest use case of the -horizontal grid shift is applying a single grid. Here we change the vertical -reference from the ellipsoid to the global geoid model, EGM96:: - - +vgridshift +grids=egm96_16.gtx - - -More than one grid can be loaded at the same time, for instance in the case where -a better geoid model than the global is available for a certain area. Here the -gridshift is set up so that the local DVR90 geoid model takes presedence over -the global model:: - - +vgridshift +grids=@dvr90.gtx,egm96_16.gtx - -The ``@`` in the above example states that the grid is optional, in case the grid -is not found in the PROJ search path. The list of grids is prioritized so that -grids in the start of the list takes presedence over the grids in the back of the -list. - -PROJ supports the GTX file format for vertical grid corrections. Details -about all the format can be found in the GDAL documentation. GDAL both reads and -writes the format. Using GDAL for construction of new grids is recommended. |