The geometry of the earth - Physical Geodesy - page 213

Summarizing, we may say: (1) The method of zenith angles is theoreti-
cally rigorous but not in general suciently accurate; (2) the astrogeodetic
method using integration of vertical deflections is not theoretically rigorous
in this sense but still may be accurate enough.
Method 1 has been treated in Part II of this chapter, so method 2 war-
rants detailed considerations in the present Part III.

5.12 Reduction of the astronomical measurements

to the ellipsoid

Now we establish the relation between the natural coordinates Φ , Λ ,H and
the ellipsoidal coordinates ϕ, λ, h referring to an ellipsoid according to Hel-
mert's projection.
The ellipsoidal height h and the orthometric height H have been consid-
ered, e.g., in Sect. 4.6 (see also Fig. 5.4 and Eq. (5-91)). They are related
by h = H + N .
Thus, there remains the reduction of the astronomical coordinates Φ and
Λ to the ellipsoid and, if we also include the astronomical observation of the
azimuth, the astronomical azimuth A to the ellipsoid in order to obtain the
ellipsoidal coordinates ϕ and λ and the ellipsoidal azimuth α .
We introduce the auxiliary quantities

∆ ϕ =Φ − ϕ,

∆ λ =Λ λ,

∆ α = A − α.

− (5-92)

The reduction of Φ and Λ to the corresponding ellipsoidal coordinates ϕ and
λ is implicitly contained in Eq. (2-230):

ξ =Φ − ϕ =∆ ϕ,

(5-93)

η =(Λ − λ )cos ϕ =∆ λ cos ϕ,

where we have substituted the respective auxiliary quantities. Thus, the con-
version formulas from natural coordinates Φ , Λ ,H to ellipsoidal coordinates
ϕ, λ, h are

ϕ =Φ ξ,

λ =Λ − η/ cos ϕ,

−

(5-94)

h = H + N.

Now we turn to the reduction of the azimuth. Thus, the question is which
∆ α arises from ∆ ϕ and ∆ λ . The answer is found in Eq. (5-75), where we