Datums, map projections and coordinate systems

A coordinate is only meaningful when everyone agrees on what it is measured from. Two surveyors can locate the same monument perfectly and still report coordinates that differ by a metre or more if one uses an older datum, a different projection, or a different foot. As GNSS, GIS and digital design have made it easy to combine data from many sources, understanding reference systems has become one of the most practical skills in geomatics. This lesson explains the shape of the Earth and the surfaces used to describe it, horizontal and vertical datums in Canada and the United States, map projections and their scale distortions, the difference between ground and grid distances, and the modernization of the national reference systems now under way.

Three surfaces: terrain, geoid and ellipsoid

Surveyors work with three different surfaces. The terrain is the physical ground. The geoid is the equipotential surface of the Earth's gravity field that best fits mean sea level; it is the surface a still ocean would follow if extended under the continents, and it is the surface to which a level bubble is perpendicular everywhere. Because the Earth's mass is unevenly distributed, the geoid is bumpy, rising and falling by up to about 100 m relative to a smooth mathematical shape. The ellipsoid is that smooth shape: an ellipse rotated about the Earth's polar axis, slightly flattened at the poles, with a semi-major axis of about 6,378 km and a semi-minor axis about 21 km shorter. The GRS80 ellipsoid is used for NAD83 and is essentially identical to the WGS84 ellipsoid used by GPS.

Heights are measured from either surface. The ellipsoid height \(h\) is measured from the ellipsoid along the normal to it, and it is what GNSS measures directly. The orthometric height \(H\), the ordinary elevation, is measured from the geoid along the direction of gravity. The difference between them is the geoid height or geoid undulation \(N\), the height of the geoid above the ellipsoid:

Across most of Canada and the United States the geoid lies below the ellipsoid, so \(N\) is negative; in southern Alberta it is roughly −15 m. A GNSS receiver at the Coulee Ridge site therefore reports an ellipsoid height about 15 m smaller than the elevation. Converting between the two requires a geoid model, published by Natural Resources Canada for Canada.

Horizontal datums

A geodetic datum defines the origin, orientation and scale of the coordinate system and the ellipsoid used with it. In practice, a datum is realized by a network of monuments and permanent GNSS stations with published coordinates.

The North American Datum of 1927 (NAD27), based on the Clarke 1866 ellipsoid and a single origin in Kansas, was used for most mapping in the mid-twentieth century. It was replaced by the North American Datum of 1983 (NAD83), based on GRS80 and centred approximately at the Earth's centre of mass. Coordinates of the same point in NAD27 and NAD83 differ by tens to over a hundred metres, so the datum must be identified for any old coordinates.

NAD83 has been refined several times as GNSS made more accurate measurement possible, and each refinement, called a realization, is identified in parentheses. NAD83 is tied to the North American tectonic plate, so coordinates of points on stable parts of the continent change very little over time. Global frames such as the International Terrestrial Reference Frame (ITRF), and WGS84 which closely follows it, are tied to the Earth as a whole; in those frames, points in North America move by roughly 2 cm per year as the plate drifts. Coordinates in a global frame therefore need a date, called the epoch, as well as a datum.

Canada uses NAD83 on the Canadian Spatial Reference System, written NAD83(CSRS). Its current realization, version 8, was adopted in 2024, is based on a transformation from ITRF2020, and uses a reference epoch of 2010.0. Provincial agencies, such as Alberta's survey control program, maintain networks of control monuments and active GNSS stations tied to it. Natural Resources Canada's online precise point positioning service, CSRS-PPP, processes GNSS observations and returns coordinates in NAD83(CSRS) or ITRF.

Vertical datums

Canada's official vertical datum since 2013 is the Canadian Geodetic Vertical Datum of 2013 (CGVD2013). Unlike older datums, it is defined by a geoid model rather than a network of levelled benchmarks: an elevation in CGVD2013 is an ellipsoid height minus the geoid height from the official model. The current realization uses the CGG2013a geoid model, and Natural Resources Canada has indicated that a future realization will be based on a newer geoid model developed with the United States. The older datum, CGVD28, was realized by levelling and still underlies many municipal benchmark networks and older drawings. The two datums differ by amounts that vary from place to place, generally by up to a few tens of centimetres in southern Canada, so a benchmark's datum must always be confirmed.

Modernization of the reference systems

The United States and Canada, with Mexico, are working on a new generation of reference frames. The new North American Terrestrial Reference Frame of 2022 (NATRF2022) will be aligned with ITRF and will change NAD83 coordinates by roughly 1 to 2 m horizontally and up to a few metres in ellipsoid height. In the United States, the National Geodetic Survey released beta versions of the new frames and tools in September 2026 for a public testing period of about six months before final approval. Natural Resources Canada has said that it is participating and will announce any Canadian adoption in advance. For now NAD83(CSRS) and CGVD2013 remain official in Canada, but surveyors working near the border or with US data should watch these changes closely.

Map projections

Engineering computations are much easier on a flat grid than on an ellipsoid. A map projection is a mathematical transformation from latitude and longitude to grid coordinates, northing and easting. No projection can flatten a curved surface without distortion, but conformal projections preserve shape locally, so angles measured on the ground equal angles on the grid. Their scale varies from place to place, and that variation is described by the point scale factor \(k\): a distance on the ellipsoid multiplied by \(k\) gives the distance on the grid.

Two conformal projections dominate North American surveying. The transverse Mercator projection wraps a cylinder around the Earth touching along a central meridian, and it suits zones that are long north–south. The Lambert conformal conic projection uses a cone touching or cutting the Earth along standard parallels, and it suits zones that are long east–west. In both, the scale factor is less than 1 in the middle of the zone and greater than 1 near its edges, and zone widths are chosen so that distortion stays within acceptable limits.

The Universal Transverse Mercator (UTM) system divides the world into 60 zones, each 6° of longitude wide, with a scale factor of 0.9996 on the central meridian, a false easting of 500,000 m and, in the northern hemisphere, a false northing of 0. Lethbridge lies in UTM zone 12, whose central meridian is 111° W, about 130 km east of the city. For engineering work in Alberta, the province also defines the Alberta 3TM system, a transverse Mercator projection with narrower 3° zones and a central scale factor of 0.9999, which reduces distortion; many municipalities and provincial datasets use it.

Grid and ground distances

A horizontal distance measured with a total station at the project's elevation is a ground distance. To use it in grid computations, it must be reduced to the ellipsoid and then scaled to the projection. The elevation factor accounts for the height of the ground above the ellipsoid, and the grid scale factor accounts for the projection. Their product is the combined scale factor:

where \(R\) is the Earth's mean radius, about 6,371 km, and \(h\) is the ellipsoid height of the site. At high elevations the elevation factor is significant: at an ellipsoid height of 900 m, ground distances are about 141 parts per million longer than ellipsoid distances.

Grid distortion is the main reason many projects use a project coordinate system or a low-distortion projection. A low-distortion projection is designed, with its own central meridian or standard parallel and a scale factor chosen for the project's elevation, so that grid distances match ground distances to within a few parts per million over the area of interest. Alternatively, the national grid coordinates are scaled about a point by the inverse of the combined factor to produce ground-scaled coordinates. The Coulee Ridge traverse in lesson 7 used such a system, which is why its ground distances could be used directly. Either way, the parameters must be documented so that anyone can relate the project coordinates to the national system.

Convergence of meridians

On a projection, grid north is the same everywhere, but true north varies: meridians converge toward the pole. The angle between grid north and true north at a point is the convergence angle, which is zero on the central meridian and grows with distance from it. For a transverse Mercator projection it is approximately the difference in longitude from the central meridian times the sine of the latitude. At Lethbridge, about 1.83° west of the UTM zone 12 central meridian at latitude 49.7° N, the convergence is about \(1.83° \times \sin 49.7° \approx 1.4°\). Directions from astronomic or geodetic observations must be corrected by the convergence before they are used as grid azimuths, and software sign conventions should always be checked.

# Heights, scale factors and grid/ground conversion
import math

R = 6_371_000

# Orthometric height from GNSS ellipsoid height and geoid height
h, N = 897.53, -15.30
print(f"H = h - N = {h - N:.2f} m")

# Transverse Mercator point scale factor (spherical approximation)
def tm_scale(x_from_cm, k0):
    return k0 * (1 + x_from_cm**2 / (2 * R**2))

for x in (0, 50_000, 134_000, 180_000, 250_000):
    print(f"UTM scale {x/1000:5.0f} km from CM: {tm_scale(x, 0.9996):.6f}")

def combined(k, h_ell):
    ef = R / (R + h_ell)
    return ef, k * ef

k = tm_scale(134_000, 0.9996)
ef, cf = combined(k, 895)
print(f"UTM site 134 km from CM, h = 895 m: k={k:.6f}, EF={ef:.6f}, CF={cf:.6f} ({(cf - 1) * 1e6:+.0f} ppm)")
print(f"500.000 m ground = {500 * cf:.3f} m grid; ground from grid = grid / CF")

# Convergence angle for transverse Mercator (approximation)
dlon, lat = 1.83, 49.7
print(f"convergence ~ {dlon * math.sin(math.radians(lat)):.2f} degrees")

# The two feet again: a State Plane northing in feet
northing_ft = 1_000_000.00
print(f"US survey vs international foot difference at 1,000,000 ft: {northing_ft * (1200/3937 - 0.3048) / 0.3048:.2f} ft")

H = h - N = 912.83 m
UTM scale     0 km from CM: 0.999600
UTM scale    50 km from CM: 0.999631
UTM scale   134 km from CM: 0.999821
UTM scale   180 km from CM: 0.999999
UTM scale   250 km from CM: 1.000370
UTM site 134 km from CM, h = 895 m: k=0.999821, EF=0.999860, CF=0.999681 (-319 ppm)
500.000 m ground = 499.840 m grid; ground from grid = grid / CF
convergence ~ 1.40 degrees
US survey vs international foot difference at 1,000,000 ft: 2.00 ft

Coordinate system metadata

Every coordinate dataset should carry enough information for someone else to use it correctly: the horizontal datum and realization, the epoch where relevant, the projection or project system with its parameters, the vertical datum and geoid model, and the units, including which foot. GIS and design software identify coordinate systems by names and by registry codes, such as EPSG numbers, which helps but does not remove the need to check; software that silently applies the wrong transformation can move data by a metre or more without any warning. A short metadata statement on every plan and in every file avoids most of these problems.