Surfaces where W(x, y, z) is a constant are considered equipotential surfaces, a.k.a. geopotential surfaces.
dW = g dn = -g dH
Example: Assume two level surfaces are 200 meters apart at the equator, i.e. dH = 200 m, then dW = 195,609.8 m2/sec2. Using this
potential and the value of g at the poles
(983.221 gal), the separation of the same level surfaces at poles is dH =
195,609.8/983.221 = 198.950 m. That means, this particular potential surface has a difference in separation of 1.050 m from the equator to the poles.
The effect on differential leveling can be seen in the Figure 4. Assume that the two lines are equipotential surfaces. If a level line was measured from A1 to A2, and then from A2 northward to B2, the difference in elevation would be A1A2. However, if the level line was started at B2 and continued to B1 and then A1, the elevation difference would be B1B2. Thus the level line went from A1 to A2 to B2 to B1 and finally back to A1. The misclosure of the loop would be A1A2 - B1B2.
As another example, assume a differential leveling line going up a hill as shown in the figure to the right. The leveling
surfaces created by the turning points are depicted as 1 through 4 with surface gravity being gA, g2, g3, and gB. Also, let
dHA2, dH23, and dH3B represent the observed differences in elevation and dH'A2, dH'23, and dH'3B represent the
corresponding differences between the level surfaces with the intersecting gravity at point B1, B2, and B3 be symbolized by
g'1, g'2, and g'3.
Since the gravity potential is constant for a equipotential surface, this means that
Spherical Harmonic Expansion of the Gravitational Potential
In equation (1), the density
function r is unknown and thus the gravitational potential of the earth cannot be computed.
However, in exterior space it is possible to create an approximate representation using spherical harmonics.
The distance 1/l can be expressed as
for the figure to the right where P is the attracted mass and P' is the attracting mass, m. If the above equation is arranged in a series, the results are a set of zonal harmonics, Pn cos(y) which are known as Legendre polynomials. Developing these functions results in the terms
Pnm (cos f) cos ml and Pnm (cos f) sin ml
which are known as surface harmonics terms. After substituting the spherical harmonic expansion into 1/l, the following function is obtained
for k = 1, m = 0 for k = 2, m != 0.
For n = 0 the integration yields the potential of the earth's mass M concentrated at the center of mass. If the semi-major axis is introduced, and the integrals are pulled out as Cnm and Snm (harmonic coefficients) the gravitational potential can be rewritten as
The harmonic coefficients Cnm and Snm are mass integrals, and describe the distribution of mass inside the earth, a is the earth's semi-major axis, and Pnm are the associated Legendre functions. If rotational symmetry is assumed, only zonal coefficients which are dependent on latitude are present. The formula for this condition is
In satellite geodesy, the coefficients Jn = -Cn, Jnm = Cnm, and Knm = -Snm are generally used.
The most significant part of spherical harmonics is their geometrical meaning. The surface harmonics divide the earth into regions. The effects of low zonal harmonics on the earth are shown in Figure 7. The term J2 = -C2 is known as the dynamic form factor and depicts the flattening of the earth. Likewise J3 shows the pear-shaped earth (See Figure 8.) which was verified early on with satellite observations. The superposition of these terms leads to a more realistic representation of the gravitational potential, gravity anomalies, and geoid undulations. These coefficients are part of the broadcast data from a GPS receiver.
The coefficient J2 delivers the flattening of of the mean (best fitting) reference ellipsoid as
where
EGM96 was developed by setting n and m to 360 which results in a 15' grid on the Earth's surface.
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