SINEX - Solution (Software/technique) INdependent EXchange Format
Version 2.00 (May 24, 2002)
A P P E N D I X II
S U M M A R Y O F F O R M U L A S A N D
T H E I R C O N N E C T I O N T O S I N E X
This document gives a short summary of the basic formulas used for least squares
adjustment and it gives instructions which vector or matrix of the normal
equation system belongs to the individual SINEX blocks.
SUMMARY OF LEAST SQUARES ADJUSTMENT FORMULAS
You have n_obs linearized observation equations
(1) v = A dx - l
where
n_obs number of observations
v residual vector
A Jacobian matrix
dx corrections for the unknowns x concerning the apriori values x0
i.e. x = x0 + dx
with
n_unk number of unknowns
l vector 'observed' minus 'computed with apriori values'.
P denotes the weight matrix for the observations.
The goal of least square adjustment is to minimize the square sum of residuals:
(2) v' P v = min
where v' is the transposed vector of v.
This condition leads to the so called normal equation
(3) A' P A dx = A' P l
with normal equation matrix
(4) N = A' P A
and the vector of the right hand side of the normal equation
(5) b = A' P l .
The resulting unknown parameters can be determined with
(6) x = x0 + inv(A' P A) A' P l = x0 + inv(N) b
where inv stands for the inverse matrix and x0 are the apriori values.
The residuals can be computed with equation (1) and the aposteriori variance
factor is then
(7) s0 = (v' P v) / (n_obs - n_unk) .
The weighted square sum o fthe vector l (= observed minus computed) can be
obtained with
(8) l'Pl = v'Pv + dx' b
= v'Pv + dx' A' P l .
The variance-covariance matrix of the unknowns results in
(9) K = s0 inv(N).
If you introduce constraints as pseudo-observations with n_constr linearized
observation equations
(10) v_c = H dx - h
with
n_constr number of constraints as pseudo-observations
v_c residuals over the constraints
H Jacobian matrix for pseudo-observation equations
h vector 'observed' minus 'computed' for the constraints.
P_c denotes the weight matrix for your pseudo-observations.
The least square methods lead to the normal equation for the pseudo-observations
(11) H' P_c H dx = H' P_c h
with normal equation matrix of constraints
(12) N_constr = H' P_c H
and vector of the right hand side of normal equation for constraints
(13) b_constr = H' P_c h.
The complete normal equation system for the constrained solution can easily be
computed:
(14) (A' P A + H' P_c H) dx = A' P l + H' P_c h
with the constrained normal equation matrix
(15) N_total = A' P A + H' P_c H = N + N_constr
and the vector of the right hand side of the constrained normal equation system
(16) b_total = A' P l + H' P_c h = b + b_constr .
The unknown parameters of the constrained solution can be computed with
(17) x_c = x0 + inv(N_total) b_total .
Ater computing the residuals over the constraints with equation (10) the
weighted square sum of residuals of the constrained normal equation system can
be obtained with
(18) v'Pv + v_c' P_c v_c
and the number of degrees of freedom of the constrained normal equation system
is
(19) dof = n_obs + n_constr - n_unk .
The aposteriori variance-factor for the constrained normal equation system is
then
(20) s0_c = (v' P v + v_c' P_c v_c) / dof .
The variance-covariance matrix for the unknowns of this constrained normal
equation system can be computed with
(21) K_xx = s0_c inv(N_total)
And the variance-covariance matrix for the constraints is
(22) K_constr = s0_c inv(N_constr) .
IMPLEMENTATION IN SINEX
The different elements belonging to the normal equations can be stored in SINEX
files in the following way:
SOLUTION/STATISTICS block:
n_unk = NUMBER OF UNKNOWNS
n_obs = NUMBER OF OBSERVATIONS
(20) s0_c = VARIANCE FACTOR
(18) v' P v + v_c' P_c v_c = SQUARE SUM OF RESIDUALS (VTPV)
(19) dof = NUMBER OF DEGREES OF FREEDOM
SOLUTION/ESTIMATE block:
(17) x_c in field "Parameter Estimate"
SOLUTION/APRIORI block:
x0 in field "Parameter Apriori"
SOLUTION/MATRIX_ESTIMATE block:
(21) Type COVA: K_xx
Type CORR: correlation matrix of K_xx
(15) Type INFO: N_total = N + N_constr
SOLUTION/MATRIX_APRIORI block:
(22) Type COVA: K_constr
Type CORR: correlation matrix of K_constr
(12) Type INFO: N_constr
SOLUTION/NORMAL_EQUATION_VECTOR block:
(5) b = A' P l
SOLUTION/NORMAL_EQUATION_MATRIX block:
(4) N = A' P A