Vertical Offset by velocity grid (NRCan NTv2_Vel)

Formula

In Canada the realization of CGVD2013 is defined by application of a geoid model to the NAD83(CSRS) geographic 3D CRS. Although the vertical datum is static, deformation, mostly due to post-glacial isostatic adjustment, leads to changes in ellipsoidal height between the CSRS frame epochs that may be significant for some purposes. When the geoid model is applied to the ellipsoidal heights any change in ellipsoidal height manifests itself as a change in CGVD2013 gravity-related height. 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) realizations at different epochs. The transformation between CGVD2013 snapshots at different epochs can be made through a concatenated operation involving the geoid model defining CGVD2013 and the application of the velocity grid. This concatenation simplifies to:
H(t2) = H(t1) + (t2 – t1) vU.
where
vU is the vertical velocity interpolated from the velocity grid, in which positive velocity values represent land uplift, i.e. CSRS ellipsoidal height and CGVD2013 height increasing with time;.
t1 and t2 are the source (from) and target (to) coordinate epochs respectively, where t is positive towards the future.

In general application of the method, t1 and and t2 would be expected to be user-input parameter values. For transformations between CGVD2013 static snapshots [CGVD2013a(1997) height, CGVD2013a(2002) height and CGVD2013a(2010) height] using this coordinate operation method (code 1113), t1 and and t2 are given in the Datum.Anchor_Epoch attribute of the source CRS and target CRS respectively.

Reverse
In principle interpolation of a velocity grid is not reversible when it crosses 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 node spacing is relatively large. For the reverse calculation the same formula is used but with the sign of the difference in epoch reversed, i.e. the source and target coordinate epochs are transposed. Any change in NAD83(CSRS) horizontal coordinates between the two epochs will be insignificant for the interpolation of the velocity grid and any realization of CSRS may be used for this.

For the reverse calculation the same formula is used but with the sign of the difference in epoch reversed, i.e. the source and target coordinate epochs are transposed and the sign of (t2 - t1) changes.

Example

Given a point with NAD83(CSRS)v6 coordinates 49°53'09.293"N, 99°54'41.057"W and gravity related height H = 396.737 m in CGVD2013(CGG2013a) epoch 2010 whose gravity-related height is required in CGVD2013(CGG2013a) epoch 1997: 

First the velocity grid is interpolated at 49°53'09.293"N, 99°54'41.057"W for vU. This is found to be -1.85 mm/year = –0.00185m/year.

Then t1 and t2 may be obtained through the datum.anchor_epoch attribute values for the datums of the source and target CRSs respectively. (If this data is unavailable this way, it should be user-input).  

Then H(1997) = H(2010) + (1997.0 – 2010.0) * –0.00185)
	      = 396.737 + (–13.0 * –0.00185)
	      = 396.761 m

For the reverse calculation, the same point with gravity related height H = 396.761 m in CGVD2013(CGG2013a) epoch 2010 whose gravity-related height is required in CGVD2013(CGG2013a) epoch 2010, first the velocity grid is interpolated as above using the same geographical coordinates: 
	vU = –0.00185  m/year.

Then  t1 and t2 may be obtained through the datum.anchor_epoch attribute values for the datums of the source and target CRSs respectively.

Then H(2010) = H(1997) + (2010.0 – 1997.0) * vU:
	      = 396.761 + (13.00 * –0.00185)
	      = 396.737 m

EPSG:1113