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diff --git a/_sources/operations/transformations/deformation.rst.txt b/_sources/operations/transformations/deformation.rst.txt new file mode 100644 index 00000000..84b87949 --- /dev/null +++ b/_sources/operations/transformations/deformation.rst.txt @@ -0,0 +1,229 @@ +.. _deformation: + +================================================================================ +Kinematic datum shifting utilizing a deformation model +================================================================================ + +.. versionadded:: 5.0.0 + +Perform datum shifts means of a deformation/velocity model. + ++-----------------+--------------------------------------------------------------------+ +| **Alias** | deformation | ++-----------------+--------------------------------------------------------------------+ +| **Input type** | Cartesian coordinates (spatial), decimalyears (temporal). | ++-----------------+--------------------------------------------------------------------+ +| **Output type** | Cartesian coordinates (spatial), decimalyears (temporal). | ++-----------------+--------------------------------------------------------------------+ +| **Domain** | 4D | ++-----------------+--------------------------------------------------------------------+ +| **Input type** | Geodetic coordinates | ++-----------------+--------------------------------------------------------------------+ +| **Output type** | Geodetic coordinates | ++-----------------+--------------------------------------------------------------------+ + + +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. Intraplate +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. GDAL both reads and writes both file formats. Using GDAL for +construction of new grids is recommended. + +Starting with PROJ 7.0, use of a GeoTIFF format is recommended to store both +the horizontal and vertical velocities. + +More complex deformations can be done with the :ref:`Multi-component time-based deformation model <defmodel>` transformation. + +Example +------------------------------------------------------------------------------- + +In :cite:`Hakli2016` 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 ETRS89. Scandinavia is an area with significant post-glacial +rebound. The deformations from the post-glacial uplift is not accounted for in the +official ETRS89 transformations so in order to get accurate transformations in the Nordic +countries it is necessary 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 convention=position_vector + 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 + grids = ./eur_nkg_nkgrf03vel_realigned.tif + inv + # NKG_ETRF@2000.0 -> ETRF92@2000.0 + step proj=helmert convention=position_vector 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 dt = -5.296 + grids = ./eur_nkg_nkgrf03vel_realigned.tif + +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 grids = ./eur_nkg_nkgrf03vel_realigned.tif + +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 :option:`+xy_grids` and :option:`+z_grids`. +The first is the horizontal part of the model and the second is the vertical +component. + +Parameters +------------------------------------------------------------------------------- + +.. option:: +xy_grids=<list> + + Comma-separated list of grids to load. If a grid is prefixed by an ``@`` the + grid is considered optional and PROJ will the not complain if the grid is + not available. + + Grids for the horizontal component of a deformation model is expected to be + in CTable2 format. + + .. note:: :option:`+xy_grids` is mutually exclusive with :option:`+grids` + +.. option:: +z_grids=<list> + + Comma-separated list of grids to load. If a grid is prefixed by an `@` the + grid is considered optional and PROJ will the not complain if the grid is + not available. + + Grids for the vertical component of a deformation model is expected to be + in either GTX format. + + .. note:: :option:`+z_grids` is mutually exclusive with :option:`+grids` + +.. option:: +grids=<list> + + .. versionadded:: 7.0.0 + + Comma-separated list of grids to load. If a grid is prefixed by an `@` the + grid is considered optional and PROJ will the not complain if the grid is + not available. + + Grids should be in GeoTIFF format with the first 3 components being + respectively the easting, northing and up velocities in mm/year. + Setting the Description and Unit Type GDAL band metadata items is strongly + recommended, so that gdalinfo reports: + + :: + + Band 1 Block=... Type=Float32, ColorInterp=Gray + Description = east_velocity + Unit Type: mm/year + Band 2 Block=... Type=Float32, ColorInterp=Undefined + Description = north_velocity + Unit Type: mm/year + Band 3 Block=... Type=Float32, ColorInterp=Undefined + Description = up_velocity + Unit Type: mm/year + + .. note:: :option:`+grids` is mutually exclusive with :option:`+xy_grids` + and :option:`+z_grids` + +.. option:: +t_epoch=<value> + + Central epoch of transformation given in decimalyears. Will be used in + conjunction with the observation time from the input coordinate to + determine :math:`dt` as used in eq. :eq:`apply_velocity` below. + + .. note:: :option:`+t_epoch` is mutually exclusive with :option:`+dt` + +.. option:: +dt=<value> + + .. versionadded:: 6.0.0 + + :math:`dt` as used in eq. :eq:`apply_velocity` below. Is useful when + no observation time is available in the input coordinate or when + a deformation for a specific timespan needs to be applied in a + transformation. :math:`dt` is given in units of decimalyears. + + .. note:: :option:`+dt` is mutually exclusive with :option:`+t_epoch` + +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_{obs}` and :math:`t_c` determine the magnitude of the coordinate +correction 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_{obs} - t_c) + \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. Consequently though, the input coordinates +needs to be converted to lat/long space when searching for corrections in the +grid. This is done with the :ref:`cart<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. + + +See also +----------------------------------------------------------------------------- + +#. :ref:`Behavioural changes from version 5 to 6<differences_deformation>` |