EPSG:1114

Geographic3D Offset by velocity grid (NTv2_Vel)

Attributes

Data source: EPSG

Information source: EPSG guidance note #7-2, https://epsg.org

Revision date: 2024-03-16

Remarks: Interpolation within the grid is in the horizontal component of the source CRS. NRCan uses biquadratic interpolation.

Formula

In Canada each version of NAD83(CSRS) is defined by a transformations from successive realizations of ITRF. The NAD83(CSRS) versions represent a single conventional reference system. They differ only through the inclusion of additional data and any changes in the ITRF adjustment technique which for many practical purposes are not significant. However in practice deformation between the frame epochs, mostly due to post-glacial isostatic adjustment, does lead to differences that may be significant for some purposes. The Canadian velocity grid in NTv2_Vel format is used not only as a point motion operation within one of the realizations but also as a transformation between NAD83(CSRS) versions at different epochs. It is considered to be reversible.

First the velocities are interpolated for the required point from within the velocity grid where V describes the velocities of the ellipsoidal coordinates, positive towards the future. These are usually given as a rate in linear units (say millimetres per year) resolved into north, east and up components VN, VE and VU. NRCan uses biquadratic interpolation. Then the north and east components are converted into latitude and longitude components by:
	Vφ = VN / (rho + h)
	Vλ = VE / [(nu + h) cos φ] 
where 
	rho is the radius of curvature of the CRS's ellipsoid in the plane of the meridian at latitude * 
	rho = a(1 – e2)/(1 – e2sin2*)3/2
	nu is the radius of curvature of the ellipsoid perpendicular to the meridian at latitude * 
	nu = a /(1 – e2sin2*)1/2
The up component VU is in the geometric domain so Vh = VU.

Then the ellipsoidal coordinates of a point at coordinate epoch t1 may be calculated at any other coordinate epoch t2 from:
φ(t2) = φ(t1) + (t2 - t1) * Vφ
λ(t2) = λ(t1) + (t2 – t1) * Vλ
h(t2) = h(t1) + (t2 – t1) * Vh

In general application of the method, t1 and and t2 would be expected to be user-input parameter values. For transformations between NAD83(CSRS)v2 and later realizations using EPSG Dataset coordinate operation method code 1114, t1 and and t2 are given in the Datum.Anchor_Epoch attribute of the source CRS and target CRS respectively.

Reverse
In principle a velocity grid is not reversible where it contains discontinuities. NRCan considers that the Canada v6 and v7 velocity grids are reversible because for most of Canada the changes in the velocity vector over short distances are small and the grid spacing is relatively large.

For the reverse operation the same formula is used but the sign of t2 – t1 is reversed. The velocity grid is interpolated at the new location coordinates. Iteration using the output coordinates may be required until there is no significant change in the interpolated velocities, although in practice this may not be necessary.

Example

Given a point with NAD83(CSRS)v6 ellipsoidal coordinates 
	φ =   49°53'09.2927"N, 	λ = 99°54'41.0572"W, 	h = 373.795m 
is to be transformed to NAD83(CSRS)v3.
	Grid file: Canadian velocity grid v7.0

First, t1 and t2 may be obtained through the datum.anchor_epoch attribute values for the datums of the source and target CRSs respectively.
	t1 (for NAD83(CSRS)v6) = 2010.00
	t2 (for NAD83(CSRS)v3) = 1997.00
(Should this epoch data not be available this way, it should be user-input).

Then the velocities VN, VE and VU are interpolated from within the velocity grid at location (φ, λ):
	VN	=	–1.00 mm/yr
	VE	=	  2.46 mm/yr
	VU	=	 –1.85 mm/yr

Then (after conversion to appropriate units):
	Vφ = –1.5690696E-10 rad/yr
	Vλ =   5.9740384E-10 rad/yr
	Vh = –0.00185 m / yr
and
	φ = 0.870673461 + (1997.00–2010.00) * –1.5690696E-10 = 0.870673463 radians
	λ = –1.743782974 + (1997.00–2010.00) * 5.9740384E-10 = –1.743782982 radians
	h = 373.795 + (1997.00–2010.00) * –0.00185 = 373.819 m

	φ =   49°53'09.2931"N, 	λ = 99°54'41.0588"W, 	h = 373.819m
	t = 1997.0

For the reverse transformation from coordinate epoch 1997.00 to coordinate epoch 2010.00, first the velocity grid is interpolated at t=1997 coordinates of 
	φ' =   49°53'09.2931"N, 	λ' = 99°54'41.0588"W, 	h' = 373.819m 
VN	=	–1.00 mm/yr
VE	=	  2.46 mm/yr
VU	=	 –1.85 mm/yr

from which
	φ =   49°53'09.2927"N, 	λ = 99°54'41.0572"W, 	h = 373.795m
	t = 2010.00

In this example the interpolated velocities are as for the forward transformation and no further iteration is required.
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