### Height Transformations

Until recently, the Canadian Geodetic Survey (CGS) realized and maintained the vertical datum (CGVD28) using spirit levelling. As CGVD28 was the only vertical datum for years in Canada, there was no real need for height transformations, unless working across the US border where the Americans work in NAVD 88 or tying to a historical vertical datum. When a height transformation is required between two levelling-construct vertical datums, it is necessary to find in the working area benchmarks having heights in the two systems. Depending of the size of the area, the transformation can go from an offset to a complex surface. CGS does not produce vertical grid shift files for the transformation between vertical datums because the levelling lines are too sparse across Canada. Each height transformation is addressed on a case-by-case basis by contacting the Geodetic Information Services at CGS.

In 2013, CGS introduced a new vertical datum. CGVD2013 is a modern vertical datum realized by a geoid model, allowing compatibility with GNSS positioning technique. While precise heights can be readily obtained from GNSS surveys, these heights (*h*) are given with respect to an ellipsoid in a 3D geometric reference frame (e.g., NAD83(CSRS)). This is problematic as a large number of users are interested in elevations (*H*) with respect to mean sea level (MSL). The geoid model gives the geoid height (*N*), which is the separation between the ellipsoid and the MSL. CGS published several scientific geoid models since 1991 representing their own vertical datums. If a former geoid model (gravimetric or hybrid) was used, the transformation to a new geoid model is simply the difference between the two models (make sure that the two models are in the same geometric reference frame).

Thus, the transformation between two levelling-construct vertical datums necessitate in having common benchmarks in the two systems while the transformation between two geoid-construct vertical datums is simply the difference between the two geoid models. The challenge is for the transformation between a levelling-construct vertical datum and a geoid-construct vertical datum. CGS proposes three approaches that are described in the section ‘Transformation between CGVD28 and CGVD2013’.

### Transformation between NAD83(CSRS) and CGVD2013

This is a vertical transformation between the ellipsoidal and orthometric heights of a point. The orthometric height of a point in CGVD2013 (*H*_{CGVD2013}) is determined as:

H_{CGVD2013} = h_{NAD83(CSRS)} − N_{NAD83(CSRS)-CGG2013}

where *h*_{NAD83(CSRS)} and *N*_{NAD83(CSRS)-CGG2013} are ellipsoidal and geoid heights in the NAD83(CSRS) reference frame, respectively. The orthometric height will be at the same epoch as the epoch of the ellipsoidal height.

### Transformation between NAD83(CSRS) and CGVD28

Even though, CGS replaced CGVD28 with CGVD2013 in 2013 many users still work in the former vertical datum. In order to fill the gap for GNSS users working in CGVD28, CGS developed a hybrid geoid model named HTv2.0, allowing the direct transformation from NAD83(CSRS) ellipsoidal heights to CGVD28 heights:

H_{CGVD28} = h_{NAD83(CSRS)} (1997.0) − N_{NAD83(CSRS)-HTv2.0}

where *h*_{NAD83(CSRS)} *(1997.0)* is the NAD83(CSRS) ellipsoidal height for epoch 1997.0 and *N*_{NAD83(CSRS)-HTv2.0} is the HTv2.0 hybrid geoid height in the NAD83(CSRS) reference frame.

For the proper transformation to CGVD28, the ellipsoidal heights must be at the epoch of 1997.0 as the hybrid geoid model (HTv2.0) was developed using GPS constraints for the epoch 1997.0 at the benchmarks. This particular fact is important because published heights in CGVD28 do not change in time. This means that the vertical datum itself is moving at the same velocity as the terrain with respect to the ellipsoid.

### Transformation between CGVD28 and CGVD2013

CGVD28 and CGVD2013 do not have a direct tie between them. The former is a levelling-construct vertical datum, which gives you the height of a benchmark above the vertical datum while the latter is a geoid-construct vertical datum, which gives you the separation between the ellipsoid and the vertical datum in a 3D geometric reference frame. The one and only way to tie accurately these two vertical datums is by observing GNSS on benchmarks. Although, it is also possible to transform between these two datums using two approximate approaches.

**Measure ellipsoidal heights on existing benchmarks**

This approach allows the most accurate transformation between CGVD28 and CGVD2013 as long as the GNSS survey is done to geodetic standards and the benchmarks are stable. The main disadvantage of this approach is the requirement to collect data in the field. CGS recommends conducting GNSS surveys at a minimum of three benchmarks to verify stability and offset uncertainty.

The local offset (*β*) can be determined as following:

*β* = (*h*_{NAD83(CSRS)} – *N*_{NAD83(CSRS)-CGG2013A}) – *H*_{CGVD28}

where *h*_{NAD83(CSRS)} is the ellipsoidal height in NAD83(CSRS) observed by GNSS, *N*_{NAD83(CSRS)-CGG2013A} is the CGG2013A geoid height in NAD83(CSRS) and *H*_{CGVD28 }is the published height of the benchmark in CGVD28. Naturally, if the benchmarks are not GNSS friendly (i.e., against a foundation or restricted view of the sky), you can install temporary benchmarks and having the height differences measure by levelling.

**Use a national transformation model **

The national transformation model (Figure 5) uses the geoid height differences between the gravimetric geoid model CGG2013A and hybrid geoid model HTv2.0, which are realizations of CGVD2013 and CGVD28, respectively. However, HTv2.0 is not the true realization of CGVD28 as CGVD28 is actually realized by a network of benchmarks. HTv2.0 is only a close representation of CGVD28. Though, HTv2.0 became the *de facto* realization of CGVD28 for many GNSS users.

Thus, the offset (*β*) between CGVD28 and CGVD2013 can be determined approximately as follow:

β = N_{HTv2.0} – N_{CGG2013}

Knowing *β* at the point of interest, the orthometric height in CGVD2013 is given by:

H_{CGVD2013 }= H_{CGVD28} + β

With the national transformation model, one has to be careful for three items with respect to HTv2.0:

- HTv2.0 approximates CGVD28 where levelling lines are available. The separation can reach several centimetres in some regions.
- HTv2.0 generates a fictitious CGVD28 where there are no leveling lines; to confirm HTv2.0 one would have to extent the levelling network in these regions.
- HTv2.0 represents the separation between the ellipsoid and CGVD28 vertical datum for epoch 1997.0. If the epoch of CGVD28 in unknown, which is generally the case, it is technically impossible to determine the actual separation between the ellipsoid and the CGVD28 datum. As indicated above, the CGVD28 vertical datum is moving at the same velocity as the terrain with respect to the ellipsoid.

The national transformation model is the approach implemented in GPS-H.

Figure 5. The difference between heights in CGVD2013 (CGG2013A) and CGVD28 (HTv2.0).

**Use published elevations **

In order to ease the transition between CGVD28 and CGVD2013, CGS readjusted, with a series of constraints, the national first-order levelling network to conform as closely as possible to CGVD2013. This means that all benchmarks have a published elevation in CGVD2013. These heights are not necessarily accurate as they are derived from legacy levelling data and benchmarks may have move over the years. However, this approach can give you an approximation of the local offset (*β*) between CGVD28 and CGVD2013 within the periphery of the first-order levelling network.

β = H_{CGVD2013} – H_{CGVD28}