Establishment of Geodetic Coordinate Systems - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 342

X N new 2

ﾼ minimum, ð 7 : 30 Þ

one can obtain the ellipsoidal elements that will be a best fit for an area of interest
such as a new ﾼ a old +da, f new ﾼ f old +df, and the positioning elements of the new

ellipsoid Δ X 0 , Δ Y 0 , Δ Z 0 . Substituting the solution back into ( 7.28 ) will produce the

values of ʾ new , ʷ new , and N new at any arbitrary astro-geodetic point, certainly includ-

ing ʷ 0 , and N 0 at the geodetic origin.

Since

ʾ 0 ,

, and N are correlated, theoretically ( 7.29 ) is equivalent to ( 7.30 ).
However, it can be seen from ( 7.28 ) that, by changing the ellipsoidal elements,
the value of

ʾ , ʷ

remains unchanged, which indicates that the variations in deflections
of the vertical with the ellipsoidal elements are insignificant. Therefore, solving the
arc measurement equation according to the conditions in ( 7.29 ) will yield a result of
lower accuracy. Besides, considering that the change of N is milder than that of

ʷ

,
the effect of the local anomalies will be comparatively less. Thus, in practice, we
usually allow

ʾ , ʷ

N new ﾼ

minimum. When adopting the normal height system, cor-
respondingly, the condition of

∑

∑ ʶ new 2

minimum should be satisfied.
We have to point out that a nation, although with vast territory, is still confined to
a small proportion of land with respect to the whole Earth. Hence, the ellipsoidal
elements obtained by solving the arc measurement equations based on the survey-
ing data from one country are often dramatically different from those based on data
worldwide. One point is that the semimajor axis and flattening of the Earth ellipsoid
based merely on China's astro-geodetic data were posited as approximately
6,378,670 m and 1:292.0, respectively. Therefore, in the establishment of the
Xi'an Geodetic Coordinate System 1980 of China, these two parameters of the
size of the Earth ellipsoid were left out. The a and f values adopted were those
recommended at IUGG1975. In this case, solving the arc measurement equation
becomes a matter of determining the position and orientation of the ellipsoid.
Hence, the astronomical-geodetic orientation means providing the arc measure-
ment equations at the original astro-geodetic point, which can be written as:

ﾼ

N new ﾼ cos B old cos L old Δ X 0 þ cos B old sin L old Δ Y 0 þ sin B old Δ Z 0

Δ

f old 1

e old sin 2 B old sin 2 B old Δ

N old
a old

M old

e old sin 2 B old 1 a þ f þ N old :

1

ð 7 : 31 Þ

minimum, we calculate the difference in position between
the old and new ellipsoid centers

Based on N new ﾼ

∑

Δ X 0 , Δ Y 0 , Δ Z 0 , and substitute them into the

following equation: