Vertical Datums

Nowadays, heights of points could be reckoned with respect to an ellipsoid; in fact, we have already introduced this height as the ellipsoidal height, h (Section 2.1.2). However, this height does not correspond with our intuitive sense of height as a measure of vertical distance with respect to a level surface. Two points with the same ellipsoidal height may be at different levels in the sense that water would flow from one point to the other. Ellipsoidal heights are purely geometric quantities that have no connection to gravity potential; and, it is the gravity potential that determines which way water flows. An unperturbed lake surface comes closest to a physical manifestation of a level surface and mean sea level (often quoted as a reference for heights) is also reasonably close to a level surface. We may define a level surface simply as a surface on which the gravity potential is constant. Discounting friction, no work is done in moving an object along a level surface; water does not flow on a level surface; and all points on a level surface should be at the same height – at least, this is what we intuitively would like to understand by heights. The geoid is defined to be that level surface that closely approximates mean sea level (mean sea level deviates from the geoid by up to 2 m due to the varying pressure, salinity, temperature, wind setup, etc., of the oceans). There is still today considerable controversy about the realizability of the geoid as a definite surface, and the definition given here is correspondingly (and intentionally) vague.

A vertical datum, like a horizontal datum, requires an origin, but being one-dimensional, there is no orientation; and, the scale is inherent in the measuring apparatus (leveling rods). The origin is a point on the Earth’s surface where the height is a defined value (e.g., zero height at a coastal tide-gauge station). This origin is obviously accessible and satisfies the requirement for the definition of a datum. From this origin point, heights (height differences) can be measured to any other point using standard leveling procedures (which we do not discuss further). Traditionally, a point at mean sea level served as origin point, but it is not important what the absolute gravity potential is at this point, since one is interested only in height differences (potential differences) with respect to the origin. This is completely analogous to the traditional horizontal datum, where the origin point (e.g., located on the surface of the Earth) may have arbitrary coordinates, and all other points within the datum are tied to the origin in a relative way. Each vertical datum, being thus defined with respect to an arbitrary origin, is not tied to a global, internationally agreed upon, vertical datum. The latter, in fact, does not yet exist. Figure 3.4 shows the geometry of two local vertical datums each of whose origin is a station at mean sea level. In order to transform from one vertical datum to another requires knowing the gravity potential difference between these origin points. This difference is not zero because mean sea level is not exactly a level surface; differences in height between the origins typically are several decimeters.

P

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P₀

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vertical datum A vertical datum B

ellipsoid H_QB

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Figure 3.4. Two vertical datums with respect to mean sea level.

The heights that are measured and belong to a particular vertical datum ultimately are defined by differences in gravity potential with respect to the origin point. There are a number of options to scale the geopotential difference so that it represents a height difference (that is, with distance units). The most natural height (but not necessarily the most realizable height from a theoretical viewpoint) is the orthometric height, H, defined as the distance along the (curved) plumb line from the level surface through the datum origin to the point in question. With sufficient accuracy, we may neglect the curvature of the plumb line and approximate the orthometric height as a distance along the ellipsoidal normal. Analogous to Figure 3.2, we then have

H = −h N , (3.34)

where N is the distance from the ellipsoid to the level surface through the origin point. This is the geoid undulation only if the geoid passes through the origin point. Otherwise it is the geoid undulation plus the offset of the geoid from the origin point.

For North America, the National Geodetic Vertical Datum of 1929 (NGVD29) served both the U.S. and Canada for vertical control until the late 1980’s. The origin of NGVD29 was actually based on several defined heights of zero at 21 coastal (mean sea level) tide-gauge stations in the U.S. and 5 in Canada. This caused distortions in the network since, as noted above, mean sea level is not a level surface. Additional distortions were introduced because leveled heights were not corrected rigorously for the non-parallelism of the level surfaces. In 1988 a new vertical datum was introduced for the U.S., Canada, and Mexico, the North

American Vertical Datum of 1988 (NAVD88). Its origin is a single station with a defined height

(not zero) at Pointe-au-Père (Father’s Point), Rimouski, Québec. This eliminated the theoretical problem of defining a proper origin based on a single level surface. Also, the leveled heights were more rigorously corrected for the non-parallelism of the level surfaces. The origin point for NAVD88 coincides with the origin point for the International Great Lakes Datum of 1985 (IGLD85).

Chapter 4