GEODETIC REFERENCE SYSTEMS - GPS Satellite Surveying - page 36

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a reference does have the advantage that some of the reductions (geoid undulation,
deflection of the vertical) can possibly be neglected, which is an important considera-
tion when the geoid is not known accurately. With today's advanced geodetic satellite
techniques, in particular GPS, and accurate knowledge of the geoid, one prefers so-
called global ellipsoids that fit the geoid globally (whose center of figure is at the
center of mass, and whose axes coincide with the directions of the ITRF). The rela-
tionship between the Cartesian coordinates (x) = (x,y,z) and the geodetic coordi-

nates (ϕ) = (ϕ, λ ,h) is according to B.9 to B.11,

x = (N + h) cos ϕ cos λ (2.66)

y = (N + h) cos ϕ sin λ (2.67)

e 2 )

z = [ N( 1 − + h ] sin ϕ (2.68)

[36

where the auxiliary quantities N and e are

a Lin

—
6.0
——
Lon
PgE

N = 1 (2.69)

e 2 sin 2 ϕ −

e 2 f 2 = 2 f − (2.70)

Th e transformation from (x) to (ϕ) is given in Appendix B. It is typically performed
ite ratively.
The expression (2.3) can be applied to transform between a local datum and a
ge ocentric datum provided the transformation parameters are known. It is best to
co ntact the responsible agency for the latest set of parameters because the transforma-
tio n parameters are continuously updated, particularly for older datums. For example,
th e large collection that includes probably all known datums is available through the
N ational Imagery and Mapping Agency, NIMA (2002). The NGS makes the trans-
fo rmation software regarding the NAD83 available at NGS (2002). Both agencies
pr ovide software that in some cases considers the geodetic network distortions and
cr ustal motions to achieve a more accurate transformation. A difficulty in using (2.3)
is that in the past, one dealt with a horizontal and vertical datum separately and that
th e respective connecting elements, the geoid undulations, might not be available.

[36

2. 3.3 Geoid Undulations and Deflections of the Vertical

One approach to estimate the geoid undulation is by measuring gravity or gravity
gradients at the surface of the earth. At least in principle, any observable that is a
function of the gravity field is suitable for determining the geoid. Low-earth orbiting
satellites (LEOs) have successfully been used to determine the large structure of the
geoid. Satellite-to-satellite tracking is being used to determine the temporal variations
of the gravity field, and thus the geoid. The reader may want to check the results
of the Gravity Recovery and Climate Experiment (GRACE) mission launched in
early 2002.