LEAST-SQUARES ADJUSTMENTS - GPS Satellite Surveying - page 120

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TABLE 4.2 Sequential Adjustment Models

Mixed Model Observation Model

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

f 2 ( 2 a , x a ) = o 2 a = f 2 ( x a )

N onlinear

model P 1 O

OP 2

P 1 O
OP 2

P = P =

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

Linear

1

model

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

B 1 P − 1 B 1 B 2 P − 1 B 2 P − 1

1

P − 1
2

M 1 = M 2 = M 1 = M 2 =

Normal
equation
elements

1 2

A 1 M − 1 A 2 M − 1 A 1 P 1 A 1 A 2 P 2 A 2 N 1 = A 1 N 2 = A 2 N 1 = N 2 =

[12

1 2

A 1 M − 1 A 2 M − 1 A 1 P 1 1 A 2 P 2 2 u 1 = w 1 u 2 = w 2 u 1 = u 2 =

1 2

v T Pv = v T Pv ∗ + ∆ v T Pv v T Pv = v T Pv ∗ + ∆ v T Pv

Lin
—
6.1
——
Nor
PgE

Minimum

v T Pv ∗ =− u 1 N − 1 u 1 + w 1 M − 1 v T Pv ∗ =− u 1 N − 1

T
1 P 1 1

u 1 +

w 1

1 1 1

v T Pv

( A 2 x ∗ + w 2 ) T ( A 2 x ∗ + 2 ) T

∆ v T Pv =

T ( A 2 x ∗ + w 2 )

∆ v T Pv =

T ( A 2 x ∗ + 2 )

x ∗ + ∆ x ∗ + ∆

x = x x = x

N − 1
1

N − 1
1

x ∗ =− x ∗ =−

u 1 u 1

E stimated

= M 2 + A 2 − 1 = P − 1

2

A 2 − 1

p arameters

A 2 N − 1 A 2 N − 1

T T +

1 1

A 2 T A 2 x ∗ + w 2 A 2 T A 2 x ∗ + 2

N − 1
1

N − 1
1

∆ x =− ∆ x =−

[12

v 1 = v 1 + ∆ v 1 v 1 = v 1 + ∆ v 1

B 1 M − 1 A 1 x ∗ + w 1

Estimated

v 1 =− P − 1

v 1 = A 1 x ∗ + 1

1

residuals

∆ v 1 =− P − 1

B 1 M − 1

A 1 ∆ x

∆ v 1 = A 1 ∆ x

1 1

E stimated
va riance of

v T Pv v T Pv

2
0

2
0

σ = σ =

r 1 + r 2 − u n 1 + n 2 − u

unit weight

Estimated

Q x = Q x ∗ + ∆ Q Q x = Q x ∗ + ∆ Q

p arameter

Q x ∗ = N − 1
1

Q x ∗ = N − 1
1

cofactor

∆ Q =− N − 1
1

A 2 TA 2 N − 1

∆ Q =− N − 1
1

A 2 TA 2 N − 1

matrix

1 1

Because, in most cases the P 1 matrix will be diagonal, no matrix inverse computation
is required. The size of the matrix T equals the number of observations in the second
group. Thus, if one observation is added at a time, only a 1

1 matrix must be inverted.
The residuals can be computed directly from the mathematical model as desired.
A third case pertains to the role of the weight matrix of the parameters. The weight
matrix expresses the quality of the information known about the observed parameters.

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