A UNIFIED APPROACH TO STATIC AND DYNAMIC MODELLING IN PHOTOGRAMMETRY 
AND REMOTE SENSING 
I. Colomina, M. Blazquez 
Institute of Geomatics 
Generalitat de Catalunya & Universitat Politécnica de Catalunya 
Castelldefels, Spain 
KEY WORDS: calibration, orientation, static, dynamic, modelling, estimation, GPS/INS, networks. 
ABSTRACT: 
Modern photogrammetry and, more generally, the current technology for Earth observation are dependent on various 
forms of data processing. After the sensing or acquisition step, the data are available in digital format and all what 
has to be done is to calibrate, to orient and to extract georeferenced information. In this context, data processing for 
trajectory determination, sensor calibration and sensor orientation follows various patterns, all of them particular cases 
of the general time dependent parameter estimation problem defined by the equation f t) + v(t), z(t), &(t) = 0. 
where f is the mathematical functional model, t is the time, £(¢) is the time dependent observation vector, v(t) is a 
white-noise generalized process vector, x(t) is the parameter vector and 4 (t) the time derivative of z(t). A number of 
different approaches to estimate parameters x(t) from data /(t) has been developed according to the particular form of 
the above model equation. € + v = f(x), f(£ + v, qz) = 0, f (t, &(t) + v(t), x(t) = 0 and At) = FQ, £0) + v(t), z(t) 
are examples of model equations leading to network and Kalman filter/smoother solution strategies. Although these 
two procedures have proven to be well suited to their respective model equation structure, the paper discusses some 
of their limitations and alternatives, particularly for time dependent problems. 
The proposed family of methods uses 
numerical techniques that integrate the rigorous least-squares method and the finite difference methods for the solution 
of the Boundary-Value problem of Ordinary Differential Equations. Although we do not claim that this has to substitute 
existing, proven techniques, the paper indicates how hybrid static and dynamic data processing can be easily integrated 
with this new approach. 
1 INTRODUCTION 
Nowadays, trajectory determination ! for navigation, geode- 
tic positioning and remote sensing orientation is mainly 
based on two parameter estimation methodologies: least- 
squares network adjustment —the network approach (NA)— 
and Kalman filtering and smoothing —the state-space ap- 
proach (SSA). It is known that Kalman filtering is a gen- 
eral form of sequential least-squares. However, in practice, 
there is no much connection between the two approaches 
other than some output estimated parameters following the 
network approach being used as input observations for a 
second estimation step following the state-space approach. 
And vice versa. It must be mentioned that the GPS re- 
search related community has since long been faced to 
the problem of making a decision between classical least- 
squares, Kalman filtering and smoothing and some inter- 
mediate approaches (Xu, 2003). The dilemma holds for 
both the processing of moving object trajectories and for 
the processing of stationary or quasi-stationary objects. For 
the family of problems just mentioned (static, quasi-static 
and kinematic) there are examples of successful applica- 
tion of both the state space approach and of the network 
approach. To illustrate the statement, we cite two "clas- 
sics" that have had and still have a significant impact in ge- 
omatics in the past decade. The GLOBK system (Herring, 
2003) uses Kalman filtering and has been successfully ap- 
  
! In this paper trajectory determination is understood as the determina- 
tion of a time series of positions, velocities and attitudes. 
plied to time-dependent precise networks for deformation 
monitoring originating from VLBI and GPS. At the oppo- 
site end, the GPS aircraft trajectories for Earth observation 
applications like aerial triangulation or LIDAR aerial sur- 
veys were determined under the network approach (Frie, 
1990). 
The goal of the ongoing research behind this paper is not 
to devise a “unified” algorithm that package both classi- 
cal least-squares and state-space estimation in “one.” The 
approach is rather pragmatic — numerical, algorithmic and 
software oriented— as the theories of least-squares estima- 
tion (Koch, 1995) and state-space estimation (Maybeck, 
1979a, Maybeck, 1979b) are well established. The actual 
goal is to interpret stochastic dynamic models —i.e., dif- 
ferential or difference equations— and their time depen- 
dent unknown parameters —4À.e., stochastic processes— in 
a way that, for the time dependent parameter estimation 
problem, both the network approach and the state-space 
approach are applicable. We do not claim that both ap- 
proaches be fully interchangeable. We do claim that in 
some circumstances, it might be advantageous to apply the 
network approach to the estimation of time dependent pa- 
rameters. As well, we claim that time dependent problems 
in geomatics do not necessarily require a SSA treatment. 
In addition to the numerical, algorithmic, software data 
modelling and software use potential advantages of a uni- 
fied approach, there are a number of estimation problems 
that might benefit from it. They include the modelling