# Glossary

## Terms

 Equipotential Surface (W) Orthometric height (H) Geopotential number (C) Normal Height (Hn) Geopotential number difference (ΔC) Dynamic height (Hd) Geoid (W0) Normal Orthometric height (Hno) Telluroid Geodetic or ellipsoidal height (h) Quasi-geoid Sea Surface Height (SSH) Vertical Datum Sea Surface Topography (SST) Reference ellipsoid Gravity (g) Mean Sea Level (MSL) Normal gravity (γ) Geoid height (N) Deflection of the vertical (ξ, η) Height anomaly (ζ)

## List of Figures

[Click on an image thumbnail to view a larger image, notice]

 Figure 1: Ellipsoidal, orthometric and geoid heights (h = H + N) [text version - figure 1] Figure 2: Dynamic and orthometric heights [text version - figure 2] Figure 3: Sea Surface Topography [text version - figure 3] Figure 4: Deflection of the vertical [text version - figure 4]

## Definitions

Equipotential Surface (W): It is a surface having a constant potential and being everywhere perpendicular to the direction of gravity. The equipotential surface is level, i.e., the water is at rest. It exists an infinity of equipotential surfaces. These surfaces do not intersect each other, but they converge towards the poles. Thus, the geometric distance between two equipotential surfaces is less at the poles than at the equator. Note: The mean sea level is not an equipotential surface because oceans have a permanent topography that is caused by temperature, salinity, currents, etc. (Unit: m2/s2)

Geopotential number (C): It is the potential difference between an equipotential surface (Wi) and a reference equipotential surface (W0) along a plumb line. The reference geopotential surface is usually the geoid or the vertical datum. (Unit: m2/s2)

Geopotential number difference (ΔC): It is the potential difference between two equipotential surfaces at two distinct locations at the earth surface (Wj22,h2) - Wi11,h1)). The geopotential number difference can be determined from levelling:

ΔCij = (ΔHij + ε)(gi +gj)/2

where ΔHij is the instrumental height difference between points j and i, g is the gravity and ε are the systematic corrections applied to the levelling measurement. (Unit: m2/s2)

Geoid (W0): Specific equipotential (level) surface, which defines best, in a least-square sense, the global mean sea level. It is the true zero surface to measure elevations. For practical purpose, the geoid can also be defined as the equipotential surface representing a national vertical datum. For example, NRCan will use the equipotential surface (62636856.0 m2/s2) representing best the coastal mean water level for North America as the new vertical reference system for Canada. (Unit: m2/s2)

Telluroid: Surface whose normal potential U is equal to the actual potential W at the Earth's surface along the ellipsoidal normal. The telluroid is not an equipotential surface. The telluroid was proposed by Molodenskii to avoid the complex determination of the topographical density and vertical gradient of gravity, which are necessary components in geoid modelling. (Unit: m2/s2)

Quasi-geoid: Surface parallel to the telluroid that is transferred to the mean sea level. The geoid and quasi-geoid are approximately the same surface over the oceans. However, the separation between the quasi-geoid and geoid can reach close to the metre level in the Canadian Rockies. (Unit: m2/s2)

Mean Sea Level (MSL): It is the arithmetic mean height of the sea in reference to a surface such as a chart datum, ellipsoid or geoid. It is determined from hourly observations over a 19-year cycle to average out the tidal lows and highs caused principally by the gravitational forces from the moon and sun. The mean sea level has small hills and valleys with respect to the geoid. It is the traditional zero elevation.

Vertical Datum: It is the reference surface for a height system, i.e., it is the zero elevation. The vertical datum is not necessarily an equipotential surface (e.g., CGVD28, ellipsoid and telluroid). A vertical datum is made of two components: a reference system and a reference frame. The former is its definition while the latter is its realization.

Reference ellipsoid: Mathematical representation of the Earth (e.g., GRS80). Its surface is defined as equipotential. An equipotential ellipsoid of revolution is defined by four constants:

a:  Semi major axis (m)
GM:  Geocentric gravitational constant (m3 s-2)
J2:  Dynamical form factor (it is related to the ellipsoid eccentricity)
ω:  Angular velocity (rad s-1)

Geoid height (N): It is the separation between the reference ellipsoid (e.g., GRS80) and the geoid. The distance is measured along the ellipsoidal normal. Geoid heights are tied to a 3-D reference frame such as NAD83(CSRS) or ITRF. Geoid heights allow the conversion of ellipsoidal heights (h) to orthometric heights (H): H = h - N. The geoid height is also called the geoid undulation. (Unit: m)

Height anomaly (ζ): It is the separation between the telluroid and the earth or ocean surface. Also it can express the separation between the ellipsoid and the quasi-geoid. Height anomalies allow the conversion of ellipsoidal heights (h) to normal heights (Hn): H = h - ζ.(Unit: m)

Orthometric height (H): It is the elevation of a point above the geoid. It is measured along the plumb line, which is perpendicular to the equipotential surfaces. (Unit: m)

Normal Height (Hn): It is the elevation of a point above the quasi-geoid or elevation of the telluroid above the ellipsoid. The difference between normal and orthometric heights is more significant at high elevations. (Unit: m)

Dynamic height (Hd): It is the potential difference between two equipotential surfaces along a plumb line scaled by a constant gravity value. For Canada and USA, the constant value is the normal gravity on the ellipsoid at latitude 45° (γ45°). Dynamic heights have no geometric meanings. They are mainly used for the management of large water basins (e.g., Great Lakes). The surface of a lake has a constant dynamic height because it is an equipotential surface. Because the equipotential surfaces converge towards the poles, the surface of the lake closer to a pole will have a lower orthometric height than the opposite end of the lake. (Unit: m)

Normal Orthometric height (Hno): This terminology is not proper, but it is used to define the type of heights currently available in Canada (CGVD28). These heights are neither orthometric nor normal, i.e., they are not compatible with the geoid or quasi-geoid. They are determined using normal gravity, but they are based on the formulation of orthometric heights. Normal orthometric heights are used in Canada because no actual gravity measurements were available at the time of the realization of CGVD28. The objective of the 1928 adjustment was the determination of the most accurate orthometric heights, which explains why they are commonly referred as orthometric heights. (Unit: m)

Geodetic or ellipsoidal height (h): Elevation of a point above the reference ellipsoid. The distance is measured along the ellipsoidal normal. (Unit: m)

Sea Surface Height (SSH): It is the distance measured along the ellipsoidal normal between the ocean surface and ellipsoid. It is equivalent to the ellipsoidal height of the ocean surface. Instantaneous SSH can be observed by satellite radar altimetry (e.g., Topex/Poseidon, ERS-1, Jason, etc). (Unit: m)

Sea Surface Topography (SST): It is the separation between the geoid and ocean surface. SST can be determined from Sea Surface Height (SSH) measured from satellite radar altimetry and a geoid height (N): SST = SSH - N. SST is positive if the ocean surface is above the geoid. It is equivalent to the orthometric height of the ocean surface. (Unit: m)

Gravity (g): It is the combination of the gravitational (mass) and centrifugal (rotation) forces. Overall, gravity increases from ~9.78 m/s2 at the equator to ~9.83 m/s2 at the poles. The elevation and mass density are components that influenced the local gravity value. (Unit: m/s2 or Gal; 1 m/s2 = 100 Gal = 0.1 kGal = 1x105 mGal)

Normal gravity (γ): It is an approximate gravity value determined from the parameters defining an equipotential ellipsoid of revolution, e.g., GRS67, GRS80. (Unit: m/s2 or Gal; 1 m/s2 = 100 Gal = 0.1 kGal = 1x105 mGal)

Deflection of the vertical (ξ, η): It is the angle between the directions of the plumb line and ellipsoidal normal. It has two components: a north-south component (ξ) and an east-west component (η). (Unit: arcsec)