A new statistical approach to the alignment of time series

Summary. Much research in the Earth Sciences is centred on the search for similarities in waveforms or amongst sets of observations. For example, in seismology and palaeomagnetism, this matching of records is used to align several series of observations against one another or to compare one set of observations against a master series. This paper gives a general mathematical and statistical formulation of the problem of transforming, linearly or otherwise, the time-scale or depth-scale of one series of data relative to another. Existing approaches to this problem, involving visual matching or the use of correlation coefficients, are shown to have several serious deficiencies, and a new statistical procedure, using least-squares cubic splines, is presented. The new method provides not only a best estimate of the 'stretching function' defining the relative alignment of the two series of observations, but also a statement, by means of confidence regions, of the precision of this transformation. The new procedure is illustrated by analyses of artificially generated data and of palaeomagnetic observations from two cores from Lake Vuokonjarvi, Finland. It may be applied in a wide variety of situations, wherever the observations satisfy the general underlying mathematical model.