7.4.2 Introduction to the Kalman Filter-Smoother

PREDICTION, FILTERING AND SMOOTHING

For several applications of GPS the parameters of interest (usually position in the case of "kinematic" GPS), or the dominant system errors (for example, the atmospheric refraction), or both, are time-varying. In addition, the time variation is more or less predictable. For such applications, the data processing techniques that are the most efficient and optimal, and therefore the most appropriate, are those based on the principles of Least Squares prediction, filtering and smoothing.

Least Squares filtering has its origins in electrical signal processing, and consequently the literature mainly refers to signal processing and communication engineering applications. The Kalman filter is perhaps the best known of the techniques that have gained wide acceptance across the whole spectrum of physical and engineering sciences. The synergy between classical least squares procedures used in geodesy (section 7.1.1), on the one hand, and the new filtering techniques on the other hand, has been recognised for over a decade. However, it was only with the advent of GPS that considerable interest has been aroused by geodesists in Least Squares filtering as a data processing tool. Kalman filtering is now used in a wide variety of GPS applications, and not restricted to moving platform scenarios, but include static positioning as well. In particular, Kalman filtering algorithms for the resolution of the cycle ambiguities and the detection and correction of cycle slips have been developed, and continue to be investigated.

The three concepts of prediction, filtering and smoothing are closely related and are best illustrated through an example (see Figure Below), in this case a moving vehicle for which the parameters of interest are its instantaneous position at some time t. The process of computing the vehicle's position in real-time (that is, observations are taken at time t_k, position required at t_k) will be referred to as filtering. The computation of the expected position of the vehicle at some subsequent time tk, based on the last measurements at t_k-1 is properly termed prediction, while the estimation of where the vehicle was (say at time t_k), once all the measurements are post-processed to time t_k+1, is referred to as smoothing.

The concepts of Prediction, Filtering and Smoothing.

The following development is taken from MERMINOD & RIZOS (1988). Assuming an initial state vector and a kinematic model, the prediction of parameters is first presented, independently of any other consideration. This prediction is then merged with measurements and the derivation of an optimal filter based on Least Squares principles is presented. The term "optimal" is precisely defined in this context: the filtered parameters must be unbiased and the trace of their variance-covariance matrix must be a minimum. This algorithm is called the "Basic Least Squares Filter", and its most important features can be demonstrated using only an estimate of the state vector at one epoch, a kinematic model and a set of observations at a later epoch. The equivalence of the Kalman form is then demonstrated. Relations involving the filtered estimates are presented in order to permit comparisons with results from classical Least Squares adjustments to be made.

The concept of smoothing is perhaps the most difficult to grasp. However, it suffices to say that the initial estimate may be somehow improved by future measurements and the equations for smoothing can be obtained in a rigorous manner, as a logical extension of the previous developments. As the parameters at different epochs are related through a kinematic model, the filtered estimate is no longer optimal as soon as later measurements become available. A new estimate that includes the contributions of the later measurements can be computed. Improving previous estimates via a new measurement is therefore smoothing and, as in the case of filtering, is referred to as "optimal smoothing" if optimal estimation methods are employed. Although the three procedures are separate, and can be applied independently, they may also be applied sequentially :

• The prediction step: based on past positioning information together with a kinematic model, the expected position and its precision at the next epoch of measurement is computed. The kinematic model is composed, as is the measurement model, of functional and stochastic components (section 7.1.1). Thus four models must be considered and, given a particular application and a certain data type, the filter design process is therefore one of selecting the appropriate models.
  
• The adjustment or filtering step: is a classical adjustment, except that a fairly good apriori estimate of the parameters is already provided from the prediction step. Basically, the resulting parameter estimates are weighted combinations of predicted quantities and measurement data. The Kalman filter is a particular form of the general Least Squares filter.
  
• The smoothing step: by which all the measurements are reprocessed after the last measurement has been made and the filtering step has been completed.

All the processes described are recursive in nature, and hence only two epochs t₀ and t are considered. This reduces the use of indices and hence contributes to the clarity of the mathematical expressions. An extension to more epochs is obvious.

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© Chris Rizos, SNAP-UNSW, 1999