Heights in Coordinate Systems

Increasingly, spatial data is 3D: our GPS traces include heights; we use LIDAR to create 3D models of buildings and cities; and we have Digital Elevation Models which capture the hills, mountains, and valleys of our terrain. Earlier, we described how coordinate systems tie data to physical locations on earth, allowing it to be combined and viewed in context. This process can work entirely in 2D (e.g., leading to lat/long or x/y coordinates), but it is particularly interesting when extended into the third dimension. Getting the extra details right — primarily knowing the vertical datum and units in use — is critical not only for combining data, but also for interpreting it correctly.

It turns out there are (at least) three significant options for dealing with heights in maps:

• Tidal. Sea level can be used to derive a baseline height, and then spirit levelling can be used to extend height measurements away from the water. Typically, mean sea level is used, but there are a surprising number of variants. These approaches are most useful for coastal applications (e.g., mapping boundaries that are relative to the land/water interface).

• Ellipsoid (Geodetic). GPS positions are initially calculated as geocentric coordinates — linear X, Y, and Z offsets from the center of the WGS84 ellipsoid. These are then converted into a latitude, longitude, and ellipsoid height. Height in this case means the perpendicular distance from a point to the ellipsoid (flattened sphere) used to model the earth’s shape. While convenient, ellipsoid heights don’t have the intuitive properties of heights: They are unrelated to sea level, and don’t capture the direction gravity pulls things downhill.

• Orthometric (Gravimetric). Orthometric heights are derived from ellipsoid heights and a model of the geoid. They measure heights above an equipotential surface, chosen to match global mean sea level or an existing vertical datum.

The main difference between tidal and orthometric heights is that local mean sea level isn’t the same “height” (in the potential energy sense) everywhere. Some parts of the ocean are persistently higher than others due to local factors including temperature and currents. Therefore, tidal datums are superior at measuring relative to the coastline and local sea level, and geoid-based vertical datums are superior at measuring heights consistently over large areas including both ocean and land.

It’s possible to convert between vertical datums based on different kinds of heights. This lets you overlay the corresponding data or take advantage of properties of a desired height system. Models of the geoid can be used to convert between ellipsoid and orthometric heights directly, but there are many details to get right. The easiest approach is to determine your source and destination vertical datums and then apply a recommended grid shift or procedure to make the conversion. For example:

• An EGM96 geoid grid is available to convert between WGS84 ellipsoid heights and tide-free EGM96/WGS84 orthometric heights, anywhere in the world.

• GEOID96/99/03 grids convert between NAD83 ellipsoid heights and NAVD88 orthometric heights in the United States.

• VERTCON converts between NGVD29 (a legacy vertical datum) and NAVD88 orthometric heights, also in the United States.

Coordinate systems and associated metadata play a huge role in making maps useful, and it’s important to keep track of these details. When working with data that includes heights, ensure you know what coordinate system it’s in, including both horizontal and vertical aspects, and be aware there are options to transform your data for analysis or combination with other kinds of data.