Sect_7.4.1

7.4.1 Introduction to the Kalman Filter-Smoother

INTRODUCTION

In order to relate the various Least Squares procedures that are used in GPS data processing, an attempt is made to classify the Least Squares processes with respect to three criteria:

the available data: either measurements only, or, in addition, available apriori estimates of the parameters,
the processing mode: either batch (that is, all of the data at once), or step-by-step (sequential), and
the system state behaviour: either static or kinematic (or time-varying).

The most common example of a time-varying system that is encountered by surveyors is one in which the instantaneous position of a moving GPS antenna is to be estimated using the "kinematic GPS surveying" techniques (section 5.5.5). In the case of kinematic positioning, step-by-step procedures are the key to real-time applications. However, the two concepts should not be confused. The choice of the processing mode is not imposed by the system behaviour, but rather it is a matter of data organisation. Step-by-step estimation procedures can be very useful even in the static case. They have proven to be very efficient for upgrading geodetic networks as new survey data becomes available, as for example the incorporation of EDM measurements into a previously triangulated network (KINLYSIDE, 1992; HARVEY, 1994). Hence for the static positioning case, the possible combinations are (MERMINOD & RIZOS, 1988):

Estimation procedures appropriate for kinematic systems are (MERMINOD & RIZOS, 1988):

Considering a set of n observations and u parameters at each of e epochs, the change from a static system process to a kinematic system process can be summarised for the different cases:

Classical and Bayesian processes would involve an increase in the total number of parameters. However, a relation between the parameters at different epochs can be modelled. Typically, truncated time polynomials are used for such a purpose. The total number of parameters therefore is between u (static) and u . e (independent parameters at all epochs). Thus, even if n is smaller than u, it is possible to model the system with enough epochs and a limited complexity of the kinematic parameter model (for example, a moderate order of polynomial).
Step-Classical involves, by definition, the use of measurements only, and each step, epoch, or group of measurements, is independent of the others. Thus, n must be larger than u at all epochs. This procedure can be considered as a succession of classical adjustments. However the stringent requirements on the number of measurements makes this method of limited use for many applications other than real-time navigation.
Filter implies that the parameter set at one epoch can be related to the previous ones. Thus, n may be smaller than u. A filter can be defined as Bayesian sequential estimation in a kinematic environment. A filter with weights attributed only to measurements reduces to sequential least squares and one step of the filter is equivalent to Bayesian Least Squares. Filters are therefore considered here as the most general of the adjustment processes.

Back To Chapter 7 Contents / Next Topic / Previous Topic