The geometry of the earth - Physical Geodesy - page 209

and the expression

δR ij = δA ij − δo i (5-77)

is immediately obtained.

Zenith angles
The zenith angle z ij as function of the local level coordinates is given in
(5-69). If n ij ,e ij ,u ij , the components of x ij , are substituted by (5-65), the
relation

cos z ij = u ij /s ij
= X ij cos Φ i cos Λ i + Y ij cos Φ i sin Λ i + Z ij sin Φ i

(5-78)

X 2

+ Y 2 + Z 2

ij ij ij

is obtained, where (5-70) and (5-72) have been used. After a lengthy deriva-
tion, the relation

δz ij = X ij cos z ij − s ij cos ϕ i cos λ i ( δX j − δX i )

s 2

ij sin z ij

+ Y ij cos z ij − s ij cos ϕ i sin λ i
s 2

( δY j − δY i )

ij sin z ij

(5-79)

+ Z ij cos z ij − s ij sin ϕ i ( δZ j − δZ i )

s 2

ij sin z ij

− cos α ij δ Φ i − cos ϕ i sin α ij δ Λ i

is obtained.
It is presupposed that the zenith angles are reduced to the chord of the
light path. This reduction may be modeled by

z ij = z ij meas + s ij

2 R k,

(5-80)

where z ij meas is the measured zenith angle, R is the mean radius of the
earth, and k is the coecient of refraction. For k either a standard value
may be substituted or the coecient of refraction is estimated as additional
unknown. In the case of estimation, there are several choices, e.g., one value
for k for all zenith angles or one value for a group of zenith angles or one
value per day. (It is known that measured zenith angles are “weaker” than
other observations, which can be taken into account by giving them lower
weights.)