Question 1 of 7: Coordinate Systems, Reference Frames and Datums

Question 1: Coordinate Systems, Reference Frames and Datums (20 marks — a 10, b 10)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Two realizations of NAD83 in Canadian practice — NAD83(original), the 1986 continental adjustment, and NAD83(CSRS+epoch), the modern space-geodetic realization tagged with an epoch — together with the global ITRF(epoch) used internationally.

Find. (a)(i) the main differences between the two NAD83 versions; (a)(ii) the meaning of “(CSRS+epoch)” and why the epoch matters; (b) the similarities and differences between NAD83(CSRS+epoch) and ITRF(epoch), with the order of magnitude of any offset.

(a)(i) NAD83(original) vs NAD83(CSRS). NAD83(original) — in Canada the 1986 adjustment, sometimes written NAD83(1986) — was defined largely from terrestrial observations (triangulation, trilateration, traverse) with only a limited amount of early Doppler satellite data, and it was held fixed to that 1980s solution; in Canada it was essentially a two-dimensional (horizontal) datum with static coordinates. Its internal consistency is only at the metre-to-decimetre level over continental distances, and it has no rigorous, time-dependent connection to a global frame. NAD83(CSRS) is the Canadian Spatial Reference System realization: it is defined by high-precision GNSS observations at the Canadian Active Control System (CACS) and Canadian Base Network (CBN) stations, rigorously connected to the ITRF through a published transformation, three-dimensional (it includes ellipsoidal heights) and internally consistent to about a centimetre, with station velocities so that coordinates refer to a stated epoch. It deliberately keeps the NAD83 origin, orientation and GRS80 ellipsoid, so like the original it is not exactly geocentric (the NAD83 origin is about 2 m from the geocentre). In practice the two versions differ in station coordinates by decimetres, reaching about a metre in places, so they are not interchangeable without a datum transformation (e.g. NRCan NTv2 grids).

(a)(ii) Meaning of “(CSRS+epoch)” and the importance of epoch. “CSRS” identifies the realization (the Canadian Spatial Reference System, tied to a specific ITRF solution — e.g. ITRF2005 or ITRF2008 — through a published 14-parameter transformation). The “epoch” is the reference date, written as a decimal year (e.g. 2010.0), at which the published coordinates are valid. Epoch is essential because the Earth’s crust is not rigid: although NAD83(CSRS) is fixed to the stable North American plate (which itself moves about 2 cm/yr in ITRF), residual motion within the frame remains — glacial isostatic adjustment of up to about 1 cm/yr vertically, plate-boundary deformation of several mm/yr or more in coastal British Columbia, and local subsidence. A point’s coordinates therefore change measurably with time, so a centimetre-level coordinate is meaningless without its epoch. To combine or compare observations taken at different times one propagates coordinates from one epoch to another with a crustal-velocity model (NRCan’s velocity grid): \(\mathbf{X}(t_2)=\mathbf{X}(t_1)+\mathbf{v}\,(t_2-t_1)\).

(b) NAD83(CSRS+epoch) vs ITRF(epoch). Similarities: both are three-dimensional, geocentric, Cartesian \((X,Y,Z)\) / ellipsoidal reference frames realized from space geodesy (GNSS, VLBI, SLR, DORIS); both are epoch-based and time-dependent; and NAD83(CSRS) is defined through a published 14-parameter (Helmert + rates) transformation from a specific ITRF, so their internal precision is comparable at the centimetre level. Differences: NAD83(CSRS) is attached to the stable interior of the North American plate, so points on that plate appear nearly stationary in it, whereas in ITRF the same points drift at the full plate velocity (~2 cm/yr); and the two frames have different origins/orientations, so coordinates for the same mark differ by roughly 1–2 m in position (the historic NAD83 non-geocentricity), with the discrepancy growing over time at the plate-motion rate if the epoch is not tracked. Practically: use NAD83(CSRS+epoch) for Canadian mapping and cadastre; convert to ITRF(epoch) for global or scientific work.