Time Series Analysis I
Roy E. Plotnick, Karen L. Prestegaard, 1995. "Time Series Analysis I", Nonlinear Dynamics and Fractals: New Numerical Techniques for Sedimentary Data, Gerard V. Middleton, Roy E. Plotnick, David M. Rubin
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As discussed in Chapter 1, time series whose power spectra exhibit 1/∫ dependency (1/∫, flicker, or pink noise) have long been known to be ubiquitous in natural and experimental systems (Mandelbrot and Wallis, 1968, 1969a, b; Schroeder, 1991). For example, this behavior has been recognized in turbulent velocity fluctuation as measured in laboratory flumes (Nordin et al., 1972; Mollo-Christensen, 1973; Nowell, 1978). Mandelbrot and Wallis (1968, 1969a, b) demonstrated that fractional Brownian motions and discrete fractional Gaussian noises provide mathematical models for these 1/∫ noises (see Chapter 1). In addition, the last few years have witnessed renewed attempts to produce general physical models, such as self-organized criticality (SOC), to explain this behavior (Bak and Chen, 1989; Chapter 2, this volume). In addition to spectral analysis, other techniques, such as rescaled range, autocorrelation, and geostatistics have been applied to the analysis of fractal series. In this chapter we describe these methods and illustrate their use in the analysis of real and synthetic data sets with fractal structure.
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The intention of these notes is to provide sedimentary geologists with an introduction to the new techniques for analyzing experimental and observational data provided by the rapid development of those disciplines generally known as Fractals and Nonlinear Dynamics (chaos theory). A general introduction to a minimum of theory is given, but most of the space is devoted to show how these ideas are useful for interpreting sedimentary data. The main applications are likely to be time series or spatial profiles or two-dimensional maps or images. Sedimentary geologists deal every day with actual time series, such as measurements of current velocity or suspended concentration at a station, or with virtual time series, such as stratigraphic sections, well logs, or topographic profiles yet few geologists know much about the new numerical techniques available to analyze such data.