: : 2. (0) T
N = |Nox Nos 1 = 15 +A1wP ww
NN. ho 10 A Tu Palas
Nyx = NO t ATP ay AL) ;
N= ND + AT UP oy AL a, » 20d
Na NO AT SP ayAlo -
The superscripts (0) indicate that, if new parameters
X (y) and t (,, are added, the column/row spaces of the orig-
inal Na ee and N,, matrices have to be extended by
zero vectors and the row spaces of the original vectors and
by zero elements accordingly.
The updating of the k - 1 stage normals can be described
as
: LA
INYAN} | | = |" +. (14)
[ L+Al
The addition of the term AN to the k - 1 normals will re-
sult in alterations of the matrix factors L and U, i.e.,
(U 4- ^ U) B = ran rt (15)
í Al
"T
2.2. Sequential treatment with Givens
Transformations
Sequential estimation with orthogonal transformations us-
ing QR decomposition is described in Lawson and Han-
son (1974). Both additions/deletions of column and row
vectors of the A matrix are discussed there. Householder
transformations as well as Givens rotations are used. Blais
(1983) recommended the application of Givens rotations
for the sequential treatment of surveying and photogram-
metry networks. Our approach uses the estimation model
(Equation 4a). Instead of obtaining the updated upper tri-
angular matrix (Equation 15) by means of Gauss factor-
ization of the normal equations, it applies Givens
transformations directly to the upper triangular matrix U
of the previous stage. At stage k-1 the reduced system
(Equation 9) takes the form
s 7 aM - d. (9)
í t
Adding one observation equation, including a set of new
parameters y, to this system results in stage k and gives
(with P @ = 1)
Uto. d
{= (16)
T
ak) y Ha)
in which
y isthe new parameter vector of length p,
al is the row vector with the coefficients of the new
Observation equation, and
lg is the right hand side of new the observation
equation.
Applying a series of orthogonal Givens transformations
G= 66,1 --> G, (17)
(n is the total number of system parameters)
to Equation (16) results in
Eu o|- » Mc» «d Jn
6/09 ic pred Ir, (182)
aw 11 gll
Ed. 1172 à) a
G| 0 Ip ml : (18b)
| Lay j 1 Tw } 1
The updated solution vector can be found by back-substi-
tution into
U (19)
Sc.» c. »*.»
Il
a
The sparsity patterns of both U and al, can be exploited
advantageously in order to speed-up computations.
If a covariance matrix of the parameters has to be updated
essentially the same approach can be followed as for the
updated parameters. Another option is to derive it from
the upper triangle U using Equation (14) in Gruen
(1985a).
Methods for the deletion of observations and the addition
and deletion of parameters are described in Golub (1969)
and Lawson, Hanson (1974). Some of these methods fit
nicely into the mechanization of the Givens approach. De-
letion of observations can be handled by introducing these
observation equations with negative weights into the
standard format (Equation 16). Complex arithmetic is
avoided in computations.
For the deletion of parameters one simply cuts out the cor-
responding columns of the upper triangle U and trans-
forms the remaining matrix to upper triangular form with
Givens matrices. The transformation of vector d is also
necessary.
The variance factor can be updated either through explicit
computation in Equation (3c) after the "new" residuals
have been determined or through a sequential approach
using the Givens transformations. Lawson, Hanson (1974,
page 6) have shown that
Q = (Py)! s dia, (20)
with d» being derived from the factorization of the obser-
vation Equation (4a) as
