GEODETIC REFERENCE SYSTEMS - GPS Satellite Surveying - page 46

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Subscripts will be used when needed to clarify the use of symbols. For example,
the differencing operation x 1 . The same

convention is followed for other differences. A more complete notation for the local
geodetic coordinates is (n 1 ,e 1 ,u 1 ) instead of (n, e, u) , to emphasize that these com-
ponents refer to the geodetic horizon at P 1 . Similarly, a more unambiguous notation
is (

∆ in (2.95) implies ∆ x ≡ ∆ x 12 = x 2 −

β 1 ,ϑ 1 ) , to emphasize that these
observables are taken at station P 1 with foresight P 2 . For slant distance, the subscripts
do not matter because s

α 12 , β 12 ,ϑ 12 ) instead of just ( α , β ,ϑ) or even ( α 1 ,

s 21 .
Changing the sign of e in (2.95) and combining the rotation matrices R 2 and R 3
one obtains

= s 1 = s 12 =

w = R (ϕ, λ ) ∆ x (2.97)

with

[46

− sin ϕ cos λ − sin ϕ sin λ cos ϕ

R = − sin λ cos λ 0 (2.98)

Lin
—
-0.
——
Nor
PgE

cos ϕ cos λ cos ϕ sin λ sin ϕ

Su bstituting (2.97) and (2.98) into (2.92) to (2.94) gives expressions for the geodetic
ob servables as functions of the geocentric Cartesian coordinate differences and the
ge odetic position of P 1 :

tan − 1

− sin λ 1 ∆ x + cos λ 1 ∆ y

α 1 = (2.99)

− sin ϕ 1 cos λ 1 ∆ x − sin ϕ 1 sin λ 1 ∆ y + cos ϕ 1 ∆ z

[46

sin − 1 cos ϕ 1 cos

λ 1 ∆ x + cos ϕ 1 sin λ 1 ∆ y + sin ϕ 1 ∆ z

β 1 = (2.100)

∆ x 2 + ∆ y 2 + ∆ z 2

= ∆

x 2

+ ∆

y 2

+ ∆

z 2 s (2.101)

Equations (2.99) to (2.101) are the backbone of the 3D geodetic model. Other
ob servations such as horizontal angles, heights, and height differences—even GPS
ve ctors—can be readily implemented. Equation (2.100) assumes that the vertical an-
gl e has been corrected for refraction. One should take note of the fact how little math-
em atics is required to derive these equations. Differential geometry is not required,
an d neither is the geodesic line.

2.3.5.1 Partial Derivatives Because (2.99) to (2.101) expressed the geodetic
observables explicitly as a function of the coordinates, the observation equation ad-
justment model

a = f ( x a ) can be readily used. The 3D nonlinear model has the

general form

α 1 = α (x 1 ,y 1 ,z 1 ,x 2 ,y 2 ,z 2 ) (2.102)