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Orthometric Height

Mean Sea Level and Orthometric Height

Mean sea level (MSL) is an average surface level of one or more among Earth's coastal bodies of water from which height can be measured. It maintains a surface of approximately equal gravity potential, i.e., potential energy required to overcome gravity. Gravitational force is a vector, with magnitude and direction. In order to characterize the gravity field, it is simpler to specirfy the gravitational potential, a scalar quantity, defined so that its spacial gradient (i.e., a vector specifying the rate of change along three orthogonal directions) equals the gravity vector at that point.

The geoid is a model of the Earth's equipotential surface. An equipotential surface is a closed, smooth surface surrounding the earth to which the direction of gravity is perpendicular at each point. It is a surface with a constant gravity potential. The geoid is usually within 1 meter of MSL. Physical surface of the Earth, known as terrain, is generally above the geoid. The gravity vector at any point on the Earth's surface is perpendicular to the geoid, not the ellipsoid or the terrain, although, in practice, the difference is small.

Height above MSL is a measure of the vertical distance of a location in reference to a historic mean sea level taken as a vertical datum (geoid model). This is known as orthometric height, \(H\), or altitude above MSL (AMSL):

\[ H_b \approx h_b - N(L_b, \lambda_b), \]

where \(N\) is the height of the geoid (also known as undulation) with respect to the ellipsoid (WGS 84 - 4,730,400m). The approximation is due to the geodetic height is measured normal to the ellipsoid, whereas the orthometric heigh is measured normal to the geoid. The following figure shows the various height illustrations.

Body Frame Body Frame

Geoid and Orthometric Height (Groves, p65)

The heights shown on topographic maps, for example, are orthometric heights. Calculation of the orthometric height of a point from GPS measurements is a two-step process:

  1. Determine the ellipsoidal/geodetic/curvilinear coordinates \((\phi, \lambda, h)\) from GPS measurements.
  2. Determine the geoidal/geoid height from a data base, and subtract it from the ellipsoidal/geodetic height, \(h\). The most accurate geoid model available at present is the WGS 84 geoid.