The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 149

the conditions that the origin of coordinates is at the Earth's mass center; the
coordinate axis coincides with the Earth's three principal axes of inertia. Equation
( 4.23 ) of the normal gravitational potential, on the other hand, is valid when the
coordinate origin is at the center of the ellipsoid; one coordinate axis coincides with
the axis of rotation of the normal ellipsoid. When the normal ellipsoid is selected,
its center is made at the Earth's mass center, the coordinate axis is made coincident
with the Earth's principal axis of inertia, the Earth's axis of rotation coincides with
the spin axis of the normal ellipsoid, and the total mass of the normal ellipsoid is
equal to that of the Earth. Hence, the disturbing potential T is:

n

P nk cos

1 X

n

GM
ˁ

a
ˁ

a 0 nk cosk T ¼ ʻ þ b nk sin k ʻ ð ʸ Þ , ð 4 : 28 Þ

n¼2 k¼0

where a 0 nk is the difference in the expansion coefficients between the Earth's
gravitational potential and the normal gravitational potential.

4.3 Height Systems

4.3.1 Requirements for Selecting Height Systems

The height of a point on the Earth's surface can be determined by leveling,
trigonometric leveling, and GPS measurement. Whichever method is used, a
reference surface (zero-elevation surface) and reference line (the line along which
the height is measured) will be involved. The height of a point on the Earth's
surface is geometrically defined as the distance from the point along the reference
line to the reference surface. Different reference lines or reference surfaces for
heights will constitute different height systems. Obviously, the height of the same
Earth's surface point in different height systems also varies.
For the height system to be chosen, the following requirements need to be
fulfilled:

1. To represent the position of a point, the height of the point is required to be
unambiguous and independent of the leveling path.
2. In practice, when converted to the adopted height system, the corrections to the
measured height differences for points in a limited area should be very small so
that they can possibly be ignored while dealing with low-order leveling data.
3. From the geometric problem-solving perspective, the ellipsoidal height is the
sum of the measured height and the geoid height; thus it requires that the adopted
height system should make the method for determining the difference between
the geoid and the reference ellipsoid (normal ellipsoid) sufficiently rigorous and
convenient, as well as practical.