Page 92 - MATLAB Recipes for Earth Sciences

5 Time-Series Analysis

5.1 Introduction

Time-series analysis aims to understand the temporal behavior of one of
several variables y(t). Examples are the investigation of long-term records

of mountain uplift, sea-level fluctuations, orbitally-induced insolation varia-

tions and their influence on the ice-age cycles, millenium-scale variations of
the atmosphere-ocean system, the impact of the El Niño/Southern Oscillation
on tropical rainfall and sedimentation (Fig. 5.1) and tidal infl uences on no-
bel gas emissions of bore holes. The temporal structure of a sequence of
events can be random, clustered, cyclic or chaotic. Time-series analysis pro-
vide various tools to detect these temporal structures. The understanding of
the underlying process that produced the observed data allows us to predict
future values of the variable. We use the Signal Processing Toolbox, which
contains all necessary routines for time-series analysis.
The fi rst section is on signals in general and a technical description how
to generate synthetic signals to be used with time-series analysis tools
(Chapter 5.2). Then, spectral analysis to detect cyclicities in a single time
series (autospectral analysis) and to determine the relationship between two
time series as a function of frequency (crossspectral analysis) is demon-
strated in Chapters 5.3 and 5.4. Since most time series in earth sciences are
not evenly-spaced in time, various interpolation techniques and subsequent
spectral analysis are introduced in Chapter 5.5. In the subsequent Chapter
5.6, the very popular wavelets are introduced having the capability to map
temporal variations in the spectra. The chapter closes with an overview of
nonlinear techniques, in particular the method of recurrence plots, which are
more and more used in earth sciences (Chapter 5.7).

5.2 Generating Signals

A time series is an ordered sequence of values of a variable y(t) at certain