Time-dependent transformations

Introduction

PROJ offers several coordinate operation methods that can deal with time-dependent transformations:

  • 15-parameter Helmert operation, with the translation rates (dx, dy, dz), the scale rate (dz), the rotation rates (drx, dry, drz) and the central epoch of transformation (t_epoch). Such transformations are typically used between a static CRS (bound to a tectonic plate) and a dynamic CRS (such as one of the ITRF or WGS 84 realizations).

    To transform Sydney coordinates at epoch 2026.0 from ITRF2014 to GDA2020, you can use for example:

    echo -33.8623 151.2077 0 2026 | cs2cs -d 8 ITRF2014 GDA2020

-33.86230294 151.20769882 0.00101330 2026

    or equivalently using the PROJ pipeline returned by projinfo ITRF2014 GDA2020:

    echo -33.8623 151.2077 0 2026 | cct -d 8 \
  +proj=pipeline \
  +step +proj=axisswap +order=2,1 \
  +step +proj=unitconvert +xy_in=deg +xy_out=rad \
  +step +proj=cart +ellps=GRS80 \
  +step +proj=helmert +x=0 +y=0 +z=0 +rx=0 +ry=0 +rz=0 +s=0 +dx=0 +dy=0 +dz=0 \
        +drx=0.00150379 +dry=0.00118346 +drz=0.00120716 +ds=0 +t_epoch=2020 \
        +convention=coordinate_frame \
  +step +inv +proj=cart +ellps=GRS80 \
  +step +proj=unitconvert +xy_in=rad +xy_out=deg \
  +step +proj=axisswap +order=2,1

-33.86230294    151.20769882    0.00101330     2026.0000

    and the reverse transformation:

    echo -33.86230294 151.20769882 0.00101330 2026 | cs2cs -d 8 GDA2020 ITRF2014

-33.86230000 151.20770000 0.00000002 2026

    Note that in the reverse transformation the epoch is also specified as seemingly being an input information, whereas it is the target epoch for ITRF2014. This is an historic PROJ behavior before the introduction of a more generic framework to deal with coordinate epochs. Since PROJ 9.4, you can also qualify a dynamic CRS with a coordinate epoch using the syntax "<CRS name>@<epoch>" or "<CRS code>@<epoch>"

    For example, re-using above use cases

    echo -33.8623 151.2077 | cs2cs -d 8 ITRF2014@2026.0 GDA2020

-33.86230294 151.20769882 0.00101330

    echo -33.86230294 151.20769882 0.00101330 | cs2cs -d 8 EPSG:7844 EPSG:9000@2026.0

-33.86230000 151.20770000 0.00000002

  • Point motion operation between two epochs of a dynamic CRS using a velocity grid-based transformation, with the Kinematic datum shifting utilizing a deformation model operator.

    For example, to transform coordinates in NAD83(CSRS)v7 CRS from epoch 2010.0 to epoch 2002.0

    echo 60 -100 0 | PROJ_NETWORK=ON cs2cs -d 8 "NAD83(CSRS)v7 @ 2010.0" "NAD83(CSRS)v7 @ 2002.0"

60.00000011 -100.00000045 -0.06764700

    or equivalently using the PROJ pipeline returned by projinfo "NAD83(CSRS)v7 @ 2010.0" "NAD83(CSRS)v7 @ 2002.0" --spatial-test intersects

    echo 60 -100 0 | PROJ_NETWORK=ON  cct -d 8 \
  +proj=pipeline \
  +step +proj=set +v_4=2010 \
  +step +proj=axisswap +order=2,1 \
  +step +proj=unitconvert +xy_in=deg +z_in=m +xy_out=rad +z_out=m \
  +step +proj=cart +ellps=GRS80 \
  +step +proj=set +v_4=2010 +omit_fwd \
  +step +proj=deformation +dt=-8 +grids=ca_nrc_NAD83v70VG.tif +ellps=GRS80 \
  +step +proj=set +v_4=2002 +omit_inv \
  +step +inv +proj=cart +ellps=GRS80 \
  +step +proj=unitconvert +xy_in=rad +xy_out=deg \
  +step +proj=axisswap +order=2,1 \
  +step +proj=set +v_4=2002

60.00000011 -100.00000045 -0.06764700 2002.0000

  • Using a multi-component time-based deformation model with the Multi-component time-based deformation model operator.

    For example, to transform coordinates from ITRF96 at epoch 2026.0 to NZGD2000

    $ echo -41.2924 174.7787 0 2026 | PROJ_NETWORK=ON cs2cs -d 8 ITRF96 NZGD2000

-41.29240855 174.77870601 0.00000000 2026

    or (since PROJ 9.9)

    $ echo -41.2924 174.7787 0 | PROJ_NETWORK=ON cs2cs -d 8 ITRF96@2026 NZGD2000

-41.29240855 174.77870601 0.00000000

    or equivalently using the PROJ pipeline returned by projinfo ITRF96@2026 NZGD2000 --spatial-test intersects

    echo -41.2924 174.7787 0 | PROJ_NETWORK=ON cct -d 8 \
  +proj=pipeline \
  +step +proj=axisswap +order=2,1 \
  +step +proj=unitconvert +xy_in=deg +xy_out=rad \
  +step +proj=set +v_4=2026 \
  +step +inv +proj=defmodel +model=nz_linz_nzgd2000-20180701.json \
  +step +proj=set +v_4=2026 \
  +step +proj=unitconvert +xy_in=rad +xy_out=deg \
  +step +proj=axisswap +order=2,1

-41.29240855    174.77870601    0.00000000     2026.0000

Other types of time-dependent transformations are not yet supported when PROJ must infer the transformation pipelines:

  • time-specific Helmert transformations

  • transformations between two dynamic CRS at different epochs

  • plate motion models (as available in ITRF2008, ITRF2014 and ITRF2020).

Through C API

Since PROJ 9.9, the most convenient way, that will work with all above mentioned coordinate operations, is to attach an epoch to a dynamic CRS, forming a osgeo::proj::coordinates::CoordinateMetadata instance, either by passing a string of the form "<CRS string> @ <epoch>" to proj_create() or proj_create_crs_to_crs(). Or when operating on PJ* instance, to use proj_coordinate_metadata_create() from an existing CRS instance, and pass it to proj_create_crs_to_crs_from_pj() or proj_create_operations()

PJ_CONTEXT *ctxt = proj_context_create();
PJ *transf = proj_create_crs_to_crs(C, "ITRF2014@2026.0", "GDA2020", nullptr);
PROJ_COORD coord;
coord.xyz.y = -33.8623;  // latitude in degree
coord.xyz.y = 151.2077;  // longitude in degree
coord.xyz.z = 0;         // ellipsoidal height in meter
PROJ_COORD coord_in_gda2020 = proj_trans(transf, PJ_FWD, coord);
proj_destroy(transf);
proj_context_destroy(ctxt);

When using the proj_create_operations() functionality, proj_coordoperation_requires_per_coordinate_input_time() can be used on the returned operations to check if they have a time dependency.

Note that with PROJ 9.3 or earlier for 15-parameter Helmert or, PROJ 9.8 or earlier, for multi-component time-based deformation model transformations, you must set the epoch in the PROJ_COORD::xyzt::t or PROJ_COORD::v[3] member since CoordinateMetadata is not available.