International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004
filtering and smoothing for time discrete processes (sec-
tion 2).
The SSA is the well known Kalman filtering and smooth-
ing published by R.E.Kalman in 1960 (Kalman, 1960) and
discussed in numerous textbooks from different points of
view (Maybeck, 1979a, Oksendal, 1993). Equivalent later
formulations in terms of sequential least-squares can be
found in (Teunissen, 2001). The SSA has been success-
fully applied to precise navigation for surveying applica-
tions (Scherzinger, 1997).
We borrow the state-space name from the state-space rep-
resentation of a dynamical system. A state vector is a min-
imal set of variables whose values are able to describe a
system. The optimal solution to the prediction-filtering-
smoothing (section 2) is obtained through one of the recur-
sive algorithms of the Kalman filter type.
In the prediction-filter cycle, the most important entity is
the state vector. All the rest are subordinated parameters.
In a way, the state vector dominates the scene which, in
some situations, may represent a problem. One example
is the difficulty in the feedback of the results of adaptive
Kalman filter steps to a correct scaling ofthe inertial obser-
vations (angular rates and linear accelerations) in the iner-
tial navigation equations. (In the network approach (NA),
this reduces to a classical estimation of variance compo-
nents). Another example of the weaknesses of the SSA
is the estimation of gravity error states in the inertial nav-
igation equations. We may estimate the gravity error of
our gravity model in better or worse ways, depending on
a number of instrumental, modelling and mission related
factors. But we cannot impose that the gravity error esti-
mated at time 4, at point z, is the same as the gravity error
estimated at a later time / at point z 1f x9 — x, —the so-
called cross-over points— as discussed in (Térmens and
Colomina, 2003, Térmens and Colomina, 2004).
4 THE NETWORK APPROACH
In geomatics, a network is a set of instruments, observa-
tions and parameters that are inter-related through mathe-
matical models. The mathematical models are the obser-
vation equations. To solve the network is to perform an
optimal estimation of its parameters in the sense of least-
squares; i.e., the expectation of the parameters and their
covariance is known. Moreover, their covariance is mini-
mal (Koch, 1995). The network approach exhibits superior
performance when the connectivity that observations cre-
ate between the unknown parameters is high.
In the network approach, our network will be solved in a
grand, single adjustment step where all parameters, time
dependent and independent, will be simultaneously esti-
mated. This is giving us some hint on how to implement
the network approach for time dependent networks in a
computer programme. We discuss this in sections 6 and 7.
An [unknown] random variable —a time independent par-
ameter— is to the classical network approach what an [un-
180
known] stochastic process —a time dependent parameter—
is to the state-space approach. In the following, the names
“time dependent parameter" and "stochastic process" will
be used indistinctly.
Note that the state-space approach can be used, as well, for
the estimation of time independent parameters as they can
be modeled as stochastic constant processes. A stochastic
constant takes the same value c over time. c may or may
not be known before the estimation process; but once it
is estimated it will not change over the time period where
the stochastic process is defined. An example of a random
constant is a GPS ambiguity —integer or real— in a phase
observation equation.
Note, as well, that a stochastic dynamic model (stochastic
differential equation) can be transformed into a set of sto-
chastic difference equations. Then, the family of stochas-
tic difference equations can be seen as a set of observation
equations and the network approach can be used. To dis-
cretize a stochastic dynamic model, we propose the differ-
ence methods (it is the “natural” way to do it). We are
aware of limitations and/or inferior performances of the
numerical difference methods for the solution of ODEs.
However, the comparative analysis between difference met-
hods and other more sophisticated numerical methods (vari-
ational methods, multiple shooting, ...) is usually done
in the context of deterministic ODE (Stoer and Bulirsh,
1992). But, while the extension or generalization of the
difference methods for deterministic ODE to the SDE is
straightforward, the extension of the other mentioned meth-
ods is less obvious. In future investigations we will explore
these numerical issues. Further, we refer the reader to the
specific literature on the numerical solution of SDE (Kloe-
den and Platen, 1999).
5 COMPARATIVE ANALYSIS
In the previous sections we have looked at the SSA and the
NA as different approaches to, essentially, solve the same
problem. Before we introduce and discuss time dependent
networks we summarize their main advantages and disad-
vantages from a geomatic perspective.
NETWORK APPROACH
e Advantages:
I. Support for connectivity of parameters regard-
less of time.
2. Support for both traditional networks and for SDE.
3. Possibility to compute the covariance of a lim-
ited number of selected parameters.
4. Variance component estimation.
e Disadvantages:
. . 9
|. Large system of linear equations.”
2The matrices are essentially of the band-bordered type and we can
apply sparse matrix techniques, fill-in reduction techniques and memory
paging to solve the system of linear equations.
Inte
ST
Th
op
pe
sh
pr
sta
pri
m.
m
tic
