PROCESSING PSEUDORANGES AND CARRIER PHASES - GPS Satellite Surveying - page 253

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7.4.2 Closed Solution

Th e closed-form point positioning solution has recently been treated in detail in
Gr afarend and Shan (2002) and Awange and Grafarend (2002a,b). The reader might
co nsult these publications for in-depth study of closed expressions, for derivations,
an d as a good source of additional references. Bancroft's (1985) solution is a very
ea rly, if not the first, contribution on this topic. We merely summarize the solution
us ing the notation of Goad (1998). In order to achieve compact expressions we define
th e following product of two arbitrary vectors g and h as

g T Mh

g , h ≡ (7.48)

wh ere M is the matrix

3 I 3

[25

O

M = (7.49)

O − 1

Lin
—
3.6
——
No
*PgE

The relevant terms of the pseudorange (7.42) are, if the interfrequency bias T i has

been applied,

cd t k = x i x k , P k +

− 1 ≤ i ≤ 4 (7.50)

Squaring both sides gives
x i

P i k − 2 x i cd t k =− x k · c 2 d t k (7.51) x i P k

[25

· − · x k + x k −

As can be verified, the four pseudorange equations can be written in the compact form

BM x k
cd t k

α − + Λ τ = 0 (7.52)

where

x k
cd t k

, x k
cd t k

1
2

Λ = (7.53)

x i
P k

, x i

1
2

i

α = (7.54)

P k

= α

4 T 1 2 3

α α α α (7.55)

T

τ = [1111] (7.56)