Table of Contents
The figure to the right depicts the effects of uneven mass distribution on the geoid. These deviations (mass anomalies)
cause the geoid surface to vary from that of the ellipsoid. In the case of mass excess, the geoid rises above the
ellipsoid; With deficiencies the geoid goes below the ellipsoid. The basic theory is that geoid undulation, deflections
of the vertical, and gravity anomalies are all caused by these mass anomalies. The gravity anomalies can be observed
and then used to compute the geometric deviation of the geoid from the ellipsoid.
Geoidal Undulation is the distance separation between the ellipsoid and the geoid. This distance is also often called geoidal separation and geoidal height. This distance is measured along the ellipsoidal prime vertical. It is positive (+) if the geoid is above the ellipsoid and negative (-) if the geoid is below the ellipsoid. Sir G. G. Stokes derived the following expression to compute the value of the undulation for a particular location.
where Ng is the geoid separation, the Stokes function is
and y (psi) is the angle subtended by a unit mass.
The practical evaluation of the integral in Stokes' formula is accomplished today using numerical techniques on a
computer with a fast Fourier transform. The National Geodetic Survey (NGS)
distributes software called GEOID99 that will predict the geoidal undulation at
a point. It can also be obtained over the Internet.
As an example illustrating the procedure, Stokes' formula can be rewritten as
For each compartment,, the gravity anomalies are replace by their average value
, and ck is computed as
where ck is obtained by integration over the compartment qk. If the Stokes' function S(y) is reasonably constant, it can be replace by its value S(yk) at the center of qk such that the final integral is simply the area Ak of the compartment, or
The advantage of the template method was that it allowed finer compartments nearer the point where the effects of the surrounding terrain are greater.
x = F - f h = (L - l) cos f
where x (xi) is the deflection of the vertical in the meridian, and h (eta) is the deflection of the vertical in the prime vertical. From the above equations we see that
f = F - x
l = L - h sec f
a = A - h tan f
where A is the astronomic azimuth, and a is the geodetic azimuth of a line.
In theory, the knowledge of gravity at a point of interest, and its astronomic latitude and longitude provide a basis for a global datum. Also astronomical coordinates can be converted to geodetic coordinates or vice-versa with the knowledge of the deflection of the vertical at the point. This creation of a global datum is a primary objective geodesy.
From previous discussions, it can be seen that for a point P on a system of level surfaces above an arbitrary point on the geoid P0, we have
where C is known as the geopotential number. To achieve good agreement with elevations in meters a unit of geopotential number is chosen to be 10 m2/s-2.
1 gpu = 10 m2/s-2 = 1000 gal-meters
Since g is approximately 9.8 m/s-2, values of gpu are 2% smaller than corresponding heights.
Def. Orthometric height is the distance along the curved plumb line from the geoid to a point P on the surface. Integrating the previous equation, we have
is the mean gravity along the plumb line.
Points on a level surface have the same dynamic height. However, dynamic height has no geometric significance in the real world.
where g (gamma) was defined in 1930 as
g = 978.0490 (1 + 0.0052884 sin2f - 0.0000059 sin2 2f) cm/sec2
for the international ellipsoid, and in its general form is
g = ge[1 + bsin2f - b1 sin22f]
This equation will be developed completely in a later lesson. Normal heights are independent of the route of surveying. However, like orthometric heights, they are not constants along a level surface.
The geographic coordinates, latitude and longitude, form two of the three spatial coordinates necessary to define a point on the earth's surface. The third coordinate is the orthometric height of the point. Latitude and longitude are referenced to the spheroid which approximates the geoid. On the other hand, orthometric height is referenced to the geoid, and is the length of the curved plumb line from the geoid (reference surface) to the point. The level bubble/compensator indicates the local direction of gravity to the geoid. The complication comes from the fact that the level surfaces are not parallel. Thus a significant, although admittedly small, error in a level loop misclosure is present in an errorless loop, and thus the determination of an elevation of a point is dependent upon the leveling route taken.
While the gravitational attraction varies by a small amount over the earth's surface, the centrifugal force caused by the rotation of the earth varies from zero at the poles to a maximum at the equator. Thus, level surfaces in the north-south direction are not parallel. To compensate for these differences geopotential numbers are used. As a potential difference, the geopotential number is independent of the particular leveling line used for relating the point to the geoid.A geopotential number is the potential difference between a point on the geoid and one on the surface. It is the same for all points on a level surface (equipotential surface), and is a natural measure of height. The geopotential number is measured in geopotential units (gpu) where the elevation of a point is dependent the leveling route taken. (1 gpu = 1000 gal-meters) From our discussions above the relationship between gpu and heights is
gpu = gH
Since g is approximately equal to 0.98 Kgal, gpu approximately equals 0.98H. The difference in gpu's between two points is
Dgpu12 = g2h2 - g1h1
From this equation, it is easy to see why first-order leveling requires gravity values at each benchmark.