Geoscience Reference
In-Depth Information
Note that there is no
S
n
0
,sincesin0
λ
= 0. In these formulas we have used
the abbreviation
=
2
π
λ
=0
π
(1-85)
ϑ
=0
σ
for the integral over the unit sphere. The expression
dσ
=sin
ϑdϑdλ
(1-86)
denotes the surface element of the unit sphere.
Now we turn to the determination of the coecients
a
nm
and
b
nm
in
(1-81). Multiplying both sides of the equation by a certain
R
sr
(
ϑ, λ
)and
integrating over the unit sphere gives
f
(
ϑ, λ
)
[
R
sr
(
ϑ, λ
)]
2
dσ ,
R
sr
(
ϑ, λ
)
dσ
=
a
sr
(1-87)
σ
σ
since in the double integral on the right-hand side all terms except the one
with
n
=
s
,
m
=
r
will vanish according to the orthogonality relations (1-
83). The integral on the right-hand side has the value given in (1-84), so
that
a
sr
is determined. In a similar way we find
b
sr
by multiplying (1-81)
by
S
sr
(
ϑ, λ
) and integrating over the unit sphere. The result is
a
n
0
=
2
n
+1
4
π
f
(
ϑ, λ
)
P
n
(cos
ϑ
)
dσ
;
σ
f
(
ϑ, λ
)
⎫
⎬
⎭
a
nm
=
2
n
+1
2
π
(
n
m
)!
(
n
+
m
)!
−
R
nm
(
ϑ, λ
)
dσ
(1-88)
σ
(
m
=0)
.
f
(
ϑ, λ
)
b
nm
=
2
n
+1
2
π
(
n − m
)!
(
n
+
m
)!
S
nm
(
ϑ, λ
)
dσ
σ
The coecients
a
nm
and
b
nm
can, thus, be determined by integration.
We note that the Laplace spherical harmonics
Y
n
(
ϑ, λ
) in (1-81) may
also be found directly by the formula
Y
n
(
ϑ, λ
)=
2
n
+1
4
π
2
π
π
f
(
ϑ
,λ
)
P
n
(cos
ψ
)sin
ϑ
dϑ
dλ
,
(1-89)
λ
=0
ϑ
=0
where
ψ
is the spherical distance between the points
P
, represented by
ϑ, λ
,
and
P
, represented by
ϑ
,λ
(Fig. 1.7), so that
cos
ψ
=cos
ϑ
cos
ϑ
+sin
ϑ
sin
ϑ
cos(
λ
−
λ
)
.
(1-90)
Later, when being acquainted with Sect. 1.11, Eq. (1-89) may be verified by
straightforward computation, substituting
P
n
(cos
ψ
) from the decomposition
formula (1-105).
