The Geoid and Different Height Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 148

There will inevitably be two gravity values that correspond to the gravity field of
the actual Earth and that of the normal ellipsoid, namely the true value of gravity
g and normal gravity value ʳ , the difference in value between g and ʳ , i.e., g ʳ is

the gravity anomaly.

4.2.3 Disturbing Potential

After introducing the normal ellipsoid, there exist two values of gravity potential
for any arbitrary points on the Earth: the true (measured) potential W of the real
Earth and the normal gravity potential U. There is a difference in value between the
two potentials. Subtracting the normal gravity potential from the true potential W of
the real Earth, the result is defined as T, the disturbing potential. Namely:

T ¼ W U,

or

W ¼ U þ T, ð4:26Þ

which means that the Earth's gravity potential is equal to the sum of the normal
gravity potential and the disturbing potential.
It follows from ( 4.13 ) that:

T ¼ V E V N þ Q E Q N :

The subscript E is for the actual Earth, and the subscript N for the normal
ellipsoid. When the normal ellipsoid is chosen, its axis of rotation is made coinci-
dent with the spin axis of the actual Earth and with equal angular velocity; thus
Q E ¼ Q N . It follows that:

T ¼ V E V N : ð 4 : 27 Þ

That is to say, the disturbing potential can be interpreted as the difference in
gravitational potential caused by the differences in mass distribution between the
Earth and the normal ellipsoid. The mass difference between the Earth and the
normal ellipsoid is called the disturbing mass (where the mass refers to the mass
difference of each point rather than the difference in total mass). Consequently, the
disturbing potential is the gravitational potential of the body constituted by the
disturbing masses. The gravitational potential of a body can be expressed as a
spherical harmonics series.
From ( 4.27 ) we can see that the disturbing potential can be obtained by
subtracting the normal gravitational potential from the Earth's gravitational poten-
tial. Equation ( 4.19 ) describing the Earth's gravitational potential is obtained under