GEODETIC REFERENCE SYSTEMS - GPS Satellite Surveying - page 47

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β 1 = β (x 1 ,y 1 ,z 1 ,x 2 ,y 2 ,z 2 ) (2.103)

s = s (x 1 ,y 1 ,z 1 ,x 2 ,y 2 ,z 2 ) (2.104)

Th e observables and parameters are , respec-

tiv ely. To find the elements of the design matrix, we require the total partial derivatives
wi th respect to the parameters. The general form is

{α 1 , β 1 ,s } and { x 1 ,y 1 ,z 1 ,x 2 ,y 2 ,z 2 }

dx 1
dy 1
dz 1
···
dx 2
dy 2
dz 2

d α 1 g 11 g 12 g 13 g 14 g 15 g 16 d x 1

···
d x 2

=

β 1
ds

=

[ G 1 : G 2 ]

d g 21 g 22 g 23 : g 24 g 25 g 26

g 31 g 32 g 33 g 34 g 35 g 36

[47

(2.105)

Lin
—
0.3
——
No
*PgE

[ dx i dy i dz i ] T . The partial derivatives are listed in Table 2.4. This
particular form of the partial derivatives follows from those of Wolf (1963), after
some additional algebraic manipulations.

with d x i =

TABLE 2.4 Partial Derivatives with Respect to Cartesian Coordinates

∂ α 1

∂x 1 =−

− sin ϕ 1 cos λ 1 sin α 1 + sin λ 1 cos α 1

[47

g 11 = g 14 = (a)

s cos β 1

∂ α 1
∂y 1 =−

− sin ϕ 1 sin λ 1 sin α 1 − cos λ 1 cos α 1
s cos β 1

g 12 = g 15 = (b)

∂ α 1
∂z 1 =−

cos ϕ 1 sin α 1
s cos β 1

g 13 = g 16 = (c)

∂ β 1
∂x 1 =− g 24 =

− s cos ϕ 1 cos λ 1 + sin β 1 ∆ x
s 2 cos β 1

g 21 = (d)

∂ β 1

∂y 1 =− g 25 =

− s cos ϕ 1 sin λ 1 + sin β 1 ∆ x

g 22 = (e)

s 2 cos

β 1

∂ β 1

∂z 1 =− g 26 =

− s sin ϕ 1 + sin β 1 ∆ z

g 23 = (f)

s 2 cos β 1

∂s
∂x 1 =−

−∆ x

g 31 = g 34 = (g)

s

∂s
∂y 1 =−

−∆ y

g 32 = g 35 = (h)

s

∂s
∂z 1 =− g 36 =

−∆ z

g 33 = (i)

s