Global Positioning System Reference
In-Depth Information
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The gravity field or functions of the gravity field are typically expressed in terms of
spherical harmonic expansions. For example, the expression for the geoid undulation
N
is (Lemoine et al., 1998, pp. 5-11),
a
r
n
∞
n
C
nm
cos
m
λ
P
nm
(
cos
GM
γ
λ +
S
nm
sin
m
N
=
θ
)
(2.71)
r
n
=
2
m
=
0
In
this equation the following notation is used:
N
Geoid undulation. There should not be cause for confusion using
the same symbol for the geoid undulation (2.71) and the radius of
curvature of the prime vertical (2.69); both notations are traditional in
the geodetic literature.
[37
λ
ϕ,
Latitude and longitude of station where the undulation is computed.
C
nm
, S
nm
Normalized spherical harmonic coefficients (geopotential coefficients),
of degree
n
and order
m
. A set degree and order 360 is currently
published by the Goddard Space Flight Center (GSFC, 2002). In this
notation,
C
nm
denotes the difference between the spherical harmonics
of the geopotential and the normal gravity field harmonics.
Lin
—
3.8
——
Lon
*PgE
P
n
m
(
cos
θ
)
Associated Legendre functions.
θ =
90
−
ϕ
is the colatitude.
r
Geocentric distance of the station.
GM
Product of the gravitational constant and the mass of the earth.
GM
is identical to
k
2
M
used elsewhere in this topic. Unfortunately, the
symbolism is not unique in the literature. We retain the symbols
typically used within the respective context.
[37
γ
Normal gravity. Details are given below.
a
Semimajor axis of the ellipsoid.
Geoid undulation computed from an expression like (2.71) refers to a geocentric
ellipsoid with semimajor axis
a
. The coefficients
C
nm
are computationally adjusted
to the specific flattening of the reference ellipsoid. The summation starts with
n
=
2.
Figure 2.12 shows a map of a global geoid.
There is a simple mathematical relationship between the geoid undulation and the
de
flection of the vertical. The deflections of the vertical are related to the undulations
as
follows (Heiskanen and Moritz, 1967, p. 112):
1
r
∂N
∂
ξ =−
(2.72)
θ
1
r
sin
∂N
∂
η =−
(2.73)
θ
λ
Di
fferentiating (2.71) gives
a
r
n
∞
n
P
nm
(
cos
C
nm
cos
m
λ
d
GM
γ
θ
)
λ +
S
nm
sin
m
ξ =−
(2.74)
r
2
d
θ
n
=
2
m
=
0
