Geoscience Reference
In-Depth Information
ZZ
0
=
Y
0
ZZ
=
X
Y
0
X
0
Y
0
Y
P
Y
0
0
Y
0
X
X
0
X
0
X
Fig. 7.6. Geocentric coordinate systems
X
0
Y
0
Z
0
(celestial) and
XY Z
(terrestrial)
Besides depending on the station coordinates and the time, they also contain
the orbital and gravitational parameters.
Every observation furnishes an equation of type (7-46). Provided we have
a sucient number of such observation equations, we can solve them for the
station coordinates
X
P
,Y
P
,Z
P
, for the orbital parameters
a
0
,e
0
, etc., of the
reference ellipse, and for a certain number of gravitational parameters
C
nm
and
S
nm
. This is the principle of the
orbital method
. In practice, differential
formulas will be applied to determine corrections to assume approximate val-
ues by means of a least-squares adjustment. Therefore, the actual analytical
developments are from the outset directed toward obtaining differential for-
mulas corresponding to (7-46). The substitutions indicated above are, thus,
consistently performed in terms of the corresponding differential expressions.
In this way we are able to operate with linear equations and to employ that
ecient tool of linear analysis, matrix calculus. Simple though the princi-
ple of this procedure is, the details when written out are nevertheless so
complicated that the reader must again be referred to the literature, e.g.,
Kaula (1966 a), Montenbruck and Gill (2001). Computer formula manipula-
tion is also used.
Besides these analytical problems, which have been satisfactorily solved,
the geodetic application of (7-46) raises diculties similar in principle to
those involved in the determination of zonal harmonics by means of (7-24),
but even more serious in practice. Strictly speaking, an infinite number of
unknowns,
C
nm
,S
nm
, etc., are to be determined from a finite number of
observations. In order to get a definite solution, it must be assumed that the
effect of higher-degree terms is negligibly small. But even then there are very
many unknowns: coordinates of the observing stations, parameters of the
