Bernese GPS Software Version 5.0 - Bernese GNSS Software

7. Parameter Estimation
p u × 1 vector of unknowns,
y n × 1 vector of observations,
P n × n positive definite weight matrix,
E(·) operator of expectation,
D(·) operator of dispersion,
σ 2 variance of unit weight (variance factor).
The observation equations of Section 2.3 may be written in this form. For n > u, the
equation system Ap = y is not consistent. With the addition of the residual vector v to
the observation vector y, one obtains a consistent but ambiguous system of equations, also
called system of observation equations:
y + v = Ap with E(v) = ∅ and D(v) = D(y) = σ 2 P −1 . (7.2)
Eqns. (7.1) and (7.2) are formally identical. E(v) = ∅, because E(y) = Ap, and D(v) =
D(y) follows from the law of error propagation.
The method of least-squares asks for restrictions for the observation equations (7.1) or (7.2).
The parameter estimates p should minimize the quadratic form
Ω(p) = 1
σ 2(y − Ap)⊤ P(y − Ap) (7.3)
where (y − Ap) ⊤ is the transposed matrix of (y − Ap). The introduction of the condition
that Ω(p) assumes a minimum is necessary to lead us from the ambiguous observation
equations (7.1) or (7.2) to an unambiguous normal equation system (NEQ system) for the
determination of p. The establishment of minimum values for Ω(p) leads to a system of u
equations dΩ(p)/dp = ∅, also called normal equations.
The following formulae summarize the Least-Squares Estimation in the Gauss-Markoff
Model:
Normal equations:
Estimates:
(A ⊤ P A) p = A ⊤ P y or N p = b (7.4)
of the vector of unknowns: p = (A ⊤ P A) −1 A ⊤ P y (7.5)
of the (variance-)covariance matrix: D(p) = σ 2 (A ⊤ P A) −1
(7.6)
of the observations: y = Ap (7.7)
of the residuals: v = y − y (7.8)
of the quadratic form: Ω = v ⊤ P v = y ⊤ Py − y ⊤ PAp(7.9)
of the variance of unit weight (variance factor): σ 2 = Ω/(n − u) (7.10)
Degree of freedom, redundancy:
Normal equation matrices:
f = n − u (7.11)
A ⊤ P A, A ⊤ P y, y ⊤ P y. (7.12)
This algorithm is used in the parameter estimation program GPSEST.
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