On the use of the Kalman filter in timescales

The Kalman filter is a very useful tool of estimation theory, successfully adopted in a wide variety of problems. As a recursive and optimal estimation technique, the Kalman filter seems to be the correct tool also for building precise timescales, and various attempts have been made in the past giving rise, for example, to the TA(NIST) timescale. Despite the promising expectations, a completely satisfactory implementation has never been found, due to the intrinsic non-observability of the clock time readings, which makes the clock estimation problem underdetermined. However, the case of the Kalman filter applied to the estimation of the difference between two clocks is different. In this case the problem is observable and the Kalman filter has proved to be a powerful tool. A new proposal with interesting results, concerning the definition of an independent timescale, came with the GPS composite clock, which is based on the Kalman filter and has been in use since 1990 in the GPS system. In the composite clock the indefinite growth of the covariance matrix due to the non-observability is controlled by the so-called ‘transparent variations’—squeezing operations on the covariance matrix that do not interfere with the estimation algorithm. A useful quantity, the implicit ensemble mean, is defined and the ‘corrected clocks’ (physical clocks minus their predicted bias) are shown to be observable with respect to this quantity. We have implemented the full composite clock and we discuss some of its advantages and criticalities. More recently, the Kalman filter is generating new interest, and a few groups are proposing new implementations. This paper gives an overview of what has been done and of what is currently under investigation, pointing out the peculiar advantages and the open questions in the application of this attractive technique to the generation of a timescale. 1. Introduction The Kalman filter is an important estimation technique largely used in control theory, signal analysis, navigation, traffic control, and in many other fields. Born as a dynamical version of the least squares method, it needs a model of the evolution in time of the physical system, an ‘a priori’ information from a previous estimation or measurement, some knowledge of the noises superimposed on the model, and the measures. All this information is processed, giving rise to a recursive estimation [1, 2]. The estimate is obtained by mixing two pieces of information. From the previous estimation, knowing the dynamical evolution of the system and also the noise that affects the reliability of the evolution, a prediction is made. From the fresh measurement and considering the noise superimposed on the measures, a second estimate is made. By combining these two partial estimates in a weighted average, the final estimate is obtained. The resulting estimate is linear and is optimal, in the least square sense, when the noises involved have a white Gaussian distribution. In the other cases it remains the best possible linear estimate. It is also necessary that each noise has zero mean to ensure unbiased estimates. Such features make the application of the Kalman filter to the processing of atomic clock data and, in particular, to the generation of a timescale very appealing. Atomic timescales are in fact the accumulation of time units as realized by the atomic standard, according to the definition of the SI second. Every physical realization is slightly different 0026-1394/03/SS0326+09$30.00 © 2003 BIPM and IOP Publishing Ltd Printed in the UK S326