Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 219

Table 5.5 Computation of the three corrections for horizontal directions

Primary
quantities
in
relations

General
numerical
value ( 00 )

Three
corrections Meaning of reduction

First-
order

Second-
order

Third- and
fourth-order

ʴ 1 Reduction of the

observed direction
value to the ellip-
soidal normal

ʾ, ʷ 0.05-0.1 Applied Applied Handle with

discretion

ʴ 2 Reduction of the

observed direction
value to the normal
section direction
on the ellipsoid

H 2 0.01-0.7 Applied Applied Handle with

discretion

ʴ 3 Reduction of the nor-

mal section direc-
tion value to the
geodesic direction

S 0.001-0.007 Applied Not applied Not applied

Fig. 5.34 Schematic view
of triangulation network

5.5.3 Reduction of the Observed Zenith Distance

Zenith distance is needed for calculating the ellipsoidal height differences between
two neighboring points on the Earth's surface by means of trigonometric leveling.
When we observe at a triangulation point with respect to the plumb line, we get the
astronomical zenith distance z 1 . When the height difference is computed on the
ellipsoid referring to the ellipsoidal normal, the geodetic zenith distance z is used.
There are discrepancies between these two zenith distances due to the effect of the
deflection of the vertical, hence reduction is needed. Let z-z 1 ﾼ ε , then:

z ﾼ z 1 þ ε , ð 5 : 59 Þ

where is known as the correction for deflection of the vertical of the observed

zenith distance. Below are derivations of the expression of

ε

ε .