Geoscience Reference
In-Depth Information
where
C
∗
still depends on the point
P
chosen as the pole
ψ
= 0. Second, we
average
C
∗
over the unit sphere:
2
π
π
1
4
π
C
∗
sin
ϑdϑdλ
λ
=0
ϑ
=0
(9-31)
8
π
2
2
π
π
2
π
1
∆
g
(
ϑ, λ
)∆
g
(
ϑ
,λ
)sin
ϑdϑdλdα.
=
λ
=0
ϑ
=0
α
=0
This is equal to the covariance function
C
(
ψ
), the symbol
M
in (9-6) now
being written explicitly:
8
π
2
2
π
π
2
π
1
∆
g
(
ϑ, λ
)∆
g
(
ϑ
,λ
)sin
ϑdϑdλdα.
C
(
ψ
)=
(9-32)
λ
=0
ϑ
=0
α
=0
The coordinates
ϑ
,λ
in this formula are understood to be related to
ϑ, λ
by (9-29) with
ψ
= constant, but to be arbitrary otherwise; this expresses
the fact that in (9-6) the average is extended over all pairs of points
P
and
P
for which
PP
=
ψ
= constant.
To compute the coecients
c
n
, substitute (9-32) into (9-27), obtaining
π
c
n
=
2
n
+1
2
C
(
ψ
)
P
n
(cos
ψ
)sin
ψdψ
ψ
=0
2
π
π
2
π
π
1
4
π
2
n
+1
4
π
(9-33)
∆
g
(
ϑ, λ
)∆
g
(
ϑ
,λ
)
=
·
λ
=0
ϑ
=0
α
=0
ψ
=0
·
P
n
(cos
ψ
)sin
ψdψdα
·
sin
ϑdϑdλ.
Consider first the integration with respect to
α
and
ψ
. According to (1-89),
we have
2
π
π
2
n
+1
4
π
∆
g
(
ϑ
,λ
)
P
n
(cos
ψ
)sin
ψdψdα
α
=0
ψ
=0
2
π
π
=
2
n
+1
4
π
∆
g
(
ϑ
,λ
)
P
n
(cos
ψ
)sin
ϑ
dϑ
dλ
=∆
g
n
(
ϑ, λ
)
,
(9-34)
the change of integration variables being evident. Hence (9-33) becomes
λ
=0
ϑ
=0
2
π
π
1
4
π
(9-35)
c
n
=
∆
g
(
ϑ, λ
)∆
g
n
(
ϑ, λ
)sin
ϑdϑdλ.
λ
=0
ϑ
=0
This may also be written
c
n
=
M
{
∆
g
∆
g
n
}
.
(9-36)
