LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 123

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TABLE 4.5 Conditions on Parameters

Mixed Model with Conditions Observation Model with Conditions

f 1 ( 1 a , x a ) = o 1 a = f 1 ( x a )

N onlinear
model

P 1 P 1

g ( x a ) = o g ( x a ) = o

B 1 v 1 + A 1 x + w 1 = o
A 2 x + 2 = o

v 1 = A 1 x + 1
A 2 x + 2 = o

Linear
model

B 1 P − 1 B 1
N 1 = A 1 M − 1

P − 1
1
N 1 = A 1 P 1 A 1
u 1 =

M 1 = M 1 =

Normal
equation
elements

A 1

1

A 1 M − 1 A 1 P 1 1

u 1 = w 1

1

[12

v T Pv = v T Pv ∗ + ∆ v T Pv
v T Pv ∗ =− u 1 N − 1

v T Pv = v T Pv ∗ + ∆ v T Pv
v T Pv ∗ =− u 1 N − 1

M inimum
v T Pv

Lin
—
* 5 ——
No
PgE

u 1 + w 1 M − 1

T
1 P 1 1

w 1 u 1 +

1 1 1

= A 2 x ∗ + 2 T T A 2 x ∗ + 2 = A 2 x ∗ + 2 T T A 2 x ∗ + 2

v T Pv v T Pv ∆ ∆

x ∗ + ∆ x ∗ + ∆ x = x x = x

x ∗ =− N − 1
1

x ∗ =− N − 1
1

u 1 u 1

E stimated
p arameters

= A 2 N − 1 A 2 − 1 = A 2 N − 1 A 2 − 1 T T

1 1

A 2 T A 2 x ∗ + w 2 A 2 T A 2 x ∗ + 2

[12

N − 1
1

N − 1
1

∆ x =− ∆ x =−

v 1 + ∆ v 1 + ∆

v 1 = v 1 v 1 = v 1

B 1 M − 1 A 1 x ∗ + w 1

E stimated
residuals

v 1 =− P − 1

1

v 1 = A 1 x ∗ + 1

P − 1
1

B 1 M − 1 ∆ v 1 =− A 1 ∆ x ∆ v 1 = A 1 ∆ x

1

E stimated
variance v T Pv

r 1 +

v T Pv
n 1 +

2
0

2
0

σ = σ =

of unit
weight

r 2 − u n 2 − u

Estimated
parameter

Q x = Q x ∗ + ∆ Q Q x = Q x ∗ + ∆ Q

N − 1
1

N − 1
1

Q x ∗ = Q x ∗ =

cofactor
matrix

N − 1
1

A 2 TA 2 N − 1
1

N − 1
1

A 2 TA 2 N − 1
1

∆ Q =− ∆ Q =−