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 61

where g t , g b , g i , g s ,and g c are, respectively, the gravitational attractions generated
by the topography and density contrasts of the ocean (bathymetry), ice, sediments,
and remaining anomalous density structures within the consolidated (crystalline)
crust.
The gravity disturbance ı g at a point ( r , ) is computed according to the
following expression (e.g., Heiskanen and Moritz 1967 ):

R
r

nC2

X

n

X

n

GM
R 2

ıg.r;/ D .n C 1/T n;m Y n;m ./; (4.2)

mDn

n D 0

where GM D 3986005 10 8 m 3 s 2 is the geocentric gravitational constant,
R D 6, 371 10 3 m is the Earth's mean radius, Y n , m are the (fully normalized)
surface spherical harmonic functions of degree n and order m , T n , m are the (fully
normalized) numerical coefficients which describe the disturbing gravity pote nt ial
T (i.e., the difference between the Earth's and normal gravity potentials), and n is
the maximum degree of spherical harmonics. The coefficients T n , m are obtained
from the coefficients of a global geopotential model (describing the Earth's
gravity field) after subtracting the spherical harmonic coefficients of the normal
gravity field. The 3-D position is defined in the spherical coordinate system
( r , ), where r is the spherical radius and D .';/ is the spherical direction
with the spherical latitude ® and longitude . The full spatial angle is denoted as
dž D f 0 D .' 0 ; 0 / W ' 0 2 Œ =2;=2 ^ 0 2 Œ0;2/ g . For gravity points situated
at the Earth's surface, the geocentric radius r is calculated as r Š R C H ,where H is
the topographic height.
Tenzer et al. ( 2012a ) developed and applied a uniform mathematical formalism
for computing the topographic and stripping gravity corrections of the Earth's
inner density structures. This numerical scheme utilizes the expression for the
gravitational attraction g (defined as a negative radial derivative of the respective
gravitational potential V ; i.e., g D @ V /@ r ) generated by an arbitrary volumetric
mass layer with a variable depth and thickness while having laterally distributed
vertical mass density variations. The gravity correction g at a point ( r , )is
computed using the following expression:

R
r

nC2

X

n

X

n

GM
R 2

g.r;/ D .n C 1/V n;m Y n;m ./; (4.3)

n D 0 mDn

where the potential coefficients V n , m read

X

I

Fl .i/

n;m :

3
2n C 1

1
¡ Earth

n;m Fu .i/

V n;m D (4.4)

iD0