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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B1. Istanbul 2004
2. Real-Time parameter estimation not feasible in
general.
STATE-SPACE APPROACH
e Advantages:
|. Real-Time parameter estimation capability.
2. The state vector dominates the scene.’ That is,
there is a clear definition of what the system is.
e Disadvantages:
1. Connectivity of parameters through static obser-
vation equations is not supported.
2. Filter divergence.
>
3. Computation of covariance matrices for all the
state vectors cannot be avoided.
The above list is by no means comprehensive but, in our
opinion, the only situation where the SSA is clearly su-
perior is real-time parameter estimation. This statement
should not be taken as a recommendation. In real life
problems, other factors may be taken into account. For in-
stance, in INS/GPS trajectory determination, a SSA based
software engine can be applied to both real-time and post-
processing computation modes. This aspect may be funda-
mental before making implementation decisions.
6 TIME DEPENDENT NETWORKS
A time dependent network is a network such that some of
its parameters are time dependent; i.e., that some of its pa-
rameters are stochastic processes. Analogously, we define
that to solve a time dependent network is to perform an
optimal estimation of its parameters which include some
stochastic processes. (However, this is easier said than un-
derstood and done. In this section we clarify the meaning
of the above statement and in section 7 we suggest some
implementation mechanisms.) We recall that optimality in
estimating a stochastic process means to estimate the best
expectation function path 2(1) in the sense of having min-
imal E (|z — ||?) as mentioned in section 2.
Note that we are asked to solve for more information in
time dependent networks that in time independent ones.
Accordingly, as it was to be expected, we will be given
more information before the estimation process. This new
information is the dynamic observation model for the ran-
dom process. If we now rename our traditional observa-
tion equations as the static observation model(s), then the
global picture of time dependent networks becomes clear
and clean.
An static observation model is an equation of the type
f(t.{+v,æ(t)) = 0 (1)
3For some models this advantage could be a disadvantage. See sec-
tion 3 for a related discussion.
181
where v is a normally distributed variable of null expecta-
tion. A dynamic observation model —or a stochastic dy-
namic model— is an equation of the type
Fi op) rwy, r(, 3:0) 0 (2)
where v(t) is a white noise process. In more global terms,
we will refer to the family of static observation equations
as the network static model. And to the family of dynamic
observation equations? as the network dynamic model. Typ-
ically, a particular dynamic model (2) will be given for
t € S’ where $' C S. Note that a dynamic observa-
tion equation may include time independent parameters
and that a static observation equation may include time de-
pendent parameters but not its derivatives. Note, as well,
that the static model may be of the form (1). This is not
only consistent with the concept of an static observation
equation but necessary when it contains a time dependent
parameter.
The dynamic model is a key component of a time depen-
dent network. Indeed, all what we know about z(4) before
solving the network is that x(£) is a stochastic process. In-
deed, the static model contributes to the determination of
x(t). However, without the dynamic model there is no “dy-
namics” in the process; i.e., we cannot guarantee that the
set {#(t)|t € S’} is a continuous path. In principle, strictly
speaking, mathematical continuity does not tell us much
about the roughness or smoothness of the solution path but
practical experience proves its effectiveness. (The lack of
dynamic modelling results, in practice, in somewhat rough
solutions for 4(t). A typical example of this is found in
the determination of GPS trajectories under the network
approach when compared with the same trajectory deter-
mined under the state-space approach which are, usually,
smoother.)
Note, last, that in practice, we do not have to compute the
auto-covariance function; we just have to provide a mech-
anism to compute it if requested.
We illustrate the above simple definition with two exam-
ples: a geodetic monitoring network and an airborne imag-
ing network (block) with INS/GPS aerial control. These
two examples are time dependent networks as they include
dynamic observation models and time dependent param-
eters. Note, for instance, that the orientation parameters
of a block can be seen as a set of time independent, unre-
lated parameters {mili =1...., n} or as a time dependent
parameter {p(t)|t € [a.b}.a.b € R}.
The airborne network (block) with INS/GPS aerial con-
trol is a time dependent network because its unknown ori-
entation parameters position, velocity and attitude depend
on the time. The “flight” is a stochastic process. This
one is a stochastic process over [fo. tj], where to and f;
are the initial and the final time of the flight respectively.
The stochastic process is just defined over a finite time pe-
riod and we cannot predict the system beyond /; because
An this paper no distinction is made between "equations" and "mod-
els" (both terms including the stochastic and functional components). We
will use both terms as appropriate to highlight the parallelism between
the dynamic and static aspects of the problem.
