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The literature on geophysical time-series analysis is so extensive that to review even one topic, such as volcanic tremor series, is a major task (e.g. Konstantinou & Schlindwein 2002). The purpose of this paper is not to attempt such a review but rather to outline some new tools for nonstationary and nonlinear time-series analysis that have been developed and used successfully in other areas of the environmental sciences and appear to have good potential for application in a volcanological or wider geophysical context. These stochastic methods of time-series analysis have the advantage that they all exploit powerful recursive (sequential updating) methods of estimation that facilitate the analysis of data generated by nonstationary and nonlinear systems (e.g. Young 1984).

This paper starts by reviewing briefly some of the model-based methods of time-series analysis, signal extraction and forecasting that have appeared in the statistical and time-series analysis literature over many years and then proceeds to describe in more detail one approach that has attracted considerable interest over the past two decades. This is based on the concept of an ‘unobserved component’ model and it exploits recursive estimation for the purposes of estimating time-variable parameters in nonstationary systems. It is shown that such an approach allows for various, practically useful procedures in time-series analysis: signal extraction; interpolation over gaps in time series; and forecasting or backcasting. It then goes on to outline the basic aspects of input-output time series modelling, considering both discrete-time and continuous-time ‘transfer function’ models that.

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