NETWORK ADJUSTMENTS - GPS Satellite Surveying - page 305

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Combining (8.8) and (8.10) gives the linearized form

t + u + s u + Qu − x = o (8.11)

For the purpose of distinguishing various approaches, we call the transformation
(8.8) model 1. The seven transformation parameters ( ) can be

estimated by a least-squares. The Cartesian coordinates u and x are the observations.
Equation (8.11) represents a mixed model f (

∆ x, ∆ y, ∆ z, s, ε, ψ , ω

o . See Section 4.4 for addi-
tional explanations of the mixed model adjustment. Each station contributes three
equations to (8.8).
A variation of (8.8), called model 2, is

a , x a ) =

t + u 0 + ( 1 + s) R ( u − u 0 ) − x = o (8.12)

[30

where u 0 is the vector in the system (u) to a point located somewhere within the
network that is to be transformed. A likely choice for u 0 is the centroid. All other
notation is the same as in Equation (8.8). If one follows the same procedure as
described for the previous model, i.e., omitting second-order terms in scale and
rotation and their products, then (8.12) becomes

Lin
—
1.3
——
Lon
*PgE

t + u + s ( u − u 0 ) + Q ( u − u 0 ) − x = o (8.13)

Model 3 uses the same rotation point u 0 as model 2, but the rotations are about
the axes (n, e, u) of the local geodetic coordinate system at u 0 . The n axis is tangent
to the geodetic meridian, but the positive direction is toward the south; the e axis is
perpendicular to the meridian plane and is positive eastward. The u axis is along the
ellipsoidal normal with its positive direction upward, forming a right-handed system
with n and e . Similar to Equation (8.12), one obtains

[30

t + u 0 + ( 1 + s) M ( u − u 0 ) − x = o (8.14)

If ( λ 0 ,h 0 ) are the

geodetic coordinates for the point of rotation u 0 , it can be verified that the M matrix is

η , ξ , α ) denote positive rotations about the (n, e, u) axes and if (ϕ 0 ,

R 3 ( λ 0 ) R 2 ( 90

M = − ϕ 0 ) R 3 ( α ) R 2 ( ξ ) R 1 ( η ) R 2 ( 90 − ϕ 0 ) R 3 ( λ 0 ) (8.15)

Since the rotation angles ( η , ξ , α ) are differentially small, the matrix M simplifies to

M ( λ 0 ,ϕ 0 , η , ξ , α ) = α M α + ξ M ξ + η M η + I (8.16)

where

− λ 0

0 sin ϕ 0 cos ϕ 0 sin

M α = − sin ϕ 0 0 cos ϕ 0 cos λ 0 (8.17)

cos ϕ 0 sin λ 0 − cos ϕ 0 cos λ 0 0