EPSG guidance note #7-2, https://epsg.org
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 additionl 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 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. First the velocities VN, VE and VU are interpolated for the required point from within the velocity grid. 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 where V describes the velocities of the ellipsoidal coordinates. 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. 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. Reverse For the reverse operation the same formula is used. As such the transformation is described as non-reversible.
Given a point with NAD83(CSRS)v6 ellipsoidal coordinates φ = 49°53'09.2927"N, λ = 99°54'41.0572"W, h = 373.795m at coordinate epoch 2010.00 is to be transformed to epoch 1997.00. Grid file: Canadian velocity grid v7.0 t2 = 1997.00 t1 = 2010.00 First 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