### Transformation between NAD83(CSRS) and ITRF

NAD83(CSRS) and ITRF are 3D geometric reference frames (*f*, *l*, *h*). In the NAD83(CSRS) reference frame, which rotates along with the North American tectonic plate, the horizontal coordinates remain fairly consistent over time. In the ITRF, coordinates change over time, reflecting the counter clockwise movement of the North American plate (see Figure 3). ITRF comes in various realizations, which are captured by the latest year of observations included in the solution (ITRF[yy]yy, where yyyy is a 4-digit year (first two digits are dropped when year is before 2000)).

The transformation between 3D geometric reference frames is done by Helmert transformation using a set of 14 parameters representing three translations (*T _{x}*,

*T*,

_{y}*T*), three rotations (

_{z}*R*,

_{x}*R*,

_{y}*R*), one scale factor (

_{z}*S*) and their respective rate of change (

*T*,

_{x}*T*,

_{y}*T*

_{z, }*R*,

_{x}*R*,

_{y}*R*,

_{z}*S).*These Helmert parameters allows a mathematical transformation between 3D geometric reference frames. The International Earth Rotation and Reference Systems Service (IERS) calculates the transformation parameters between the different ITRF realizations.

For NAD83(CSRS), Canada and USA agreed in 1996 to define their respective North American Datum with respect to ITRF96 using a common set of seven parameters for the Helmert transformation (see the Canada-US reference). These parameters defined the realization of NAD83(CSRS) and unified it with the American NAD83(NSRS). Thus, by adding the agreed parameters for NAD83(CSRS) with the incremental parameters between the successive ITRF realizations, it allows the Helmert transformation from NAD83(CSRS) to any ITRF realizations (ITRF88, ITRF89, ITRF90, ITRF91, ITRF92, ITRF93, ITRF94, ITRF96, ITRF97, ITRF2000, ITRF2005, ITRF2008, and ITRF2014).

Figure 3. Rotation of the North-American plate in ITRF.

### Transformation between NAD83(CSRS) epochs

As previously mentioned, the static NAD83(CSRS) is anchored to the North American plate and therefore station coordinates do not change to reflect the horizontal motion of the Earth’s crust. Although this is the concept, residual trends and intra-plate motions remain in both the horizontal and vertical direction due to inaccuracy in plate motion model, compression along plate boundaries and glacial isostatic adjustment (post-glacial rebound). Since the parameters for the Helmert transformation cannot account for these residual effects, the Canadian Geodetic Survey implemented a grid-based approach to represent crustal motion Canada-wide. This grid is constructed from coordinates updated continuously and episodically at CACS and CBN stations over the past two decades. Figure 4 shows a map displaying the rates of horizontal and vertical motions across Canada for NAD83(CSRS). One can notice the maximum uplift around Hudson Bay, subsidence in the southern prairies and small horizontal displacement across Canada.

Most federal and provincial agencies have adopted NAD83(CSRS) for their geometric reference frame, but each agency did not adopt the same reference epoch as shown in Table 3. As horizontal and vertical velocities have rates from a few millimetres per year to over a centimetre per year, coordinates at different epochs may differ significantly. For example, the provinces of Quebec and Ontario works at epoch 1997 and 2010, respectively. This time separation can represent close to 13 cm in height difference for benchmarks just south of James Bay. For these same markers, the horizontal position difference would be about 3 cm and 23 cm for NAD83(CSRS) and ITRF, respectively. To ensure that coordinates are consistent at the highest precision level, they must be related to the same epoch.

Table 3. NAD83(CSRS) adopted epochs for Canada’s provincial geodetic agencies.

Province | Epoch |
---|---|

British Columbia (Mainland) | 2002 |

British Columbia (Vancouver Island) | 1997 |

Alberta | 2002 |

Saskatchewan | 1997 |

Manitoba | 2010 |

Ontario | 2010 |

Quebec | 1997 |

New Brunswick | 1997 |

Prince Edward Island | 1997 |

Nova Scotia | 2010 |

Newfoundland and Labrador | 2010 |

Nunavut | 2010 |

Northwest Territories | 2010 |

Yukon | 2002 |

Figure 4. GNSS estimated rates of 3D crustal motion in Canada for NAD83(CSRS) (horizontal in red and vertical in blue).