Natural Resources Canada.
2018-08-29
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For geodetic station Centennial Monument NCC100 with ellipsoidal coordinates referenced to NAD83(CSRS)v6 at coordinate epoch 2010.00:
lat = 45°25'45.714920"N
lon = 75°42'05.960075"W
h = 39.524 m
t = 2010.00 (years)
whose coordinates are required at coordinate epoch 2002.00.
First interpolate within the grid file for velocities at lat = 45°25'45.714920"N, lon = 75°42'05.960075"W:
VN = -0.00156 m/yr
VE = 0.00177 m/yr
V_up = 0.00202 m/yr
NAD83(CSRS)v6 uses the GRS 1980 ellipsoid for which a = 6378137.0m and e^2 = 0.006694380.
Then:
V_lat = -2.44979E-10 rad/yr
V_lon = 3.94759E-10 rad/yr
and
lat = 0.792892001 + (2002.00-2010.00) * -2.44979E-10 = 0.792892003 radians = 45°25'45.715324"N
lon = -1.321243139 + (2002.00-2010.00) * 3.94759E-10 = -1.321243142 radians = 75°42'05.960726"W
h = 39.524 + (2002.00-2010.00) * 0.00202 = 39.508m
t = 2002.00
urn:ogc:def:method:EPSG::1070
Point motion by grid (Canada NTv2_Vel)
Interpolation within the grid is in the horizontal component of the source CRS.
Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.
The NTv2_Vel grid contains point velocities resolved into north, east, and up components.
First the three velocities at a point are interpolated within the NTv2_Vel grid using bi-linear interpolation within the source and target CRS.
Then ellipsoidal coordinates of a point at coordinate epoch t1 may be calculated at any other coordinate epoch t2 from:
[Lat(t2)] = [Lat(t1)] + (t2-t1) * [V_lat]
[Lon(t2)] = [Lon(t1)] + (t2-t1) * [V_lon]
[h(t2)] = [h(t1)] + (t2-t1) * [V_up]
where V describes the velocities of the ellipsoidal coordinates. These are given as a rate in linear units (metres per year) resolved into north, east and up components VN, VE and Vh. The north and east components are converted into latitude and longitude components by:
V_lat = VN / (rho+h)
V_lon = VE / [(nu+h) cos(lat)]
where
rho is radius of curvature of the CRS's ellipsoid in the plane of the meridian at latitude lat
rho = a(1 – e^2)/(1 – e^2sin^2(lat))^3/2
nu is radius of curvature of the ellipsoid perpendicular to the meridian at latitude lat
nu = a /(1 – e^2sin^2(lat))^1/2