Gravimetric Forward and Inverse Modeling Methods of the Crustal Density Structures and the Crust-Mantle Interface - Planetary Exploration and Science: Recent Results and Advances - page 62

The numerical coefficients f Fl ( i )

n , m ,Fu ( i ) n , m : i D 0, 1, :::, I g in Eq. ( 4.4 )aregivenby

n C 2
k

n C 2

X

L . kC1Ci /
n;m
R kC1

. 1/ k
k C 1 C i

Fl .i/

n;m D

; (4.5)

k D 0

and

n C 2
k

. 1/ k
k C 1 C i

X

nC2

U .kC1Ci/
n;m
R kC1

F u .i/

n;m D

: (4.6)

kD0

The terms P m D n L n , m Y n , m and P m D n U n , m Y n , m in Eqs. ( 4.5 )and( 4.6 )definethe
spherical lower-bound and upper-bound laterally distributed radial density variation
functions L n and U n of degree n , respectively. These spherical functions and
their higher-order terms f L ( k C 1 C i )
n

, U ( k C 1 C i )
n

: k D 0, 1, :::; i D 1, 2, :::, I g are

defined by

“

8
<

D U ; 0 D kC L 0 P n .t/d 0

4
2nC1

dž

X

n

L .kC1/

D

n;m Y n;m ./ i D 0

mDn

L . kC1Ci /
n

./ D (4.7)

“

:

LJ 0 a i 0 D kC1Ci

0 P n .t/d 0

4
2nC1

L

dž

X

n

L .kC1Ci/
n;m

D Y n;m ./ i D 1; 2;:::I

mDn

and

“

8
<

D U ; 0 D kC U 0 P n .t/d 0

4
2nC1

dž

X

n

U .kC1/ D

n;m Y n;m ./ i D 0

mDn

U .kC1Ci/
n

./ D : (4.8)

“

:

LJ 0 a i 0 D kC1Ci
U

0 P n .t/d 0

4
2n

C 1

dž

n

X

U .kC1Ci/
n;m

D Y n;m ./ i D 1;2;:::;I

m D n