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Unfortunately, this theoretical advantage is in most cases balanced by
the practical disadvantage that the relevant series converge very slowly. In
certain cases, however, the convergence is good. Then the use of spherical
harmonics is very convenient practically; we consider such a case in the next
section.
The spherical-harmonic expansion of the gravity anomalies ∆ g may be
writtenindifferentways,suchas
g ( ϑ, λ )=
g n ( ϑ, λ ) ,
(9-14)
n =2
where ∆ g n ( ϑ, λ ) is the Laplace surface harmonic of degree n ; or, more ex-
plicitly,
g ( ϑ, λ )=
n
a nm R nm ( ϑ, λ )+ b nm S nm ( ϑ, λ ) ,
(9-15)
n =2
m =0
where
R nm ( ϑ, λ )= P nm (cos ϑ )cos mλ ,
(9-16)
S nm ( ϑ, λ )= P nm (cos ϑ )sin
are the conventional spherical harmonics; or in terms of fully normalized
harmonics (see Sect. 1.10):
g ( ϑ, λ )=
n
a nm ¯
S nm ( ϑ, λ ) .
R nm ( ϑ, λ )+ b nm ¯
(9-17)
n =2
m =0
Here ϑ is the polar distance (complement of geocentric latitude) and λ is the
longitude.
Let us now find the average products of two Laplace harmonics
n
a nm ¯
S nm ( ϑ, λ ) .
R nm ( ϑ, λ )+ b nm ¯
g n ( ϑ, λ )=
(9-18)
m =0
These average products are
2 π
π
1
4 π
g n g n }
g n ( ϑ, λ )∆ g n ( ϑ, λ )sin ϑdϑdλ,
M
{
=
(9-19)
λ =0
ϑ =0
since the averaging is extended over the whole earth, that is, over the whole
unit sphere. Take first n = n , which gives the average square of the Laplace
harmonic of degree n :
2 π
π
g n ( ϑ, λ ) 2 sin ϑdϑdλ.
1
4 π
g 2
M
{
n }
=
(9-20)
λ =0
ϑ =0
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