Discrete-time signals are sequences of numbers, with each number being identified by a fixed time instant. Such a series of data in a time sequence is called a time series. In other words, a time series xn is a series of data, with each data value xn being associated with a discrete, equally spaced time index n. The time index n is taken to be a whole number, or integer.
Time series occur in all branches of science (Wold, 1938; Kolmogorov, 1941; Wiener, 1942). Economic data always appear in the form of numerical time series. Some meteorologic data, such as daily temperatures, are numerical time series; other meteorologic data, such as continuous barographic records, are continuous-time signals. Continuous-time signals appear in the engineering, biological, and physical sciences. Such continuous-time signals can be read (or measured, observed, or sampled) at equal intervals of time, thereby generating time series (Robinson and Silvia, 1979, 1980).
Because a time series represents only the sampled values of a continuous-time signal, it provides only a limited description of the signal. By taking the sampling instants close enough together, the amount of information that is lost by replacing a well-behaved continuous-time function by a time series can be made small. A time spacing that is too gross would mean substantial information loss in the sampling process. At the other extreme, a time spacing that is too fine would mean substantial redundancy in information produced by the sampling
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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing (SEG Geophysical References Series No. 15), covers the basic ideas and methods used in seismic processing, concentrating on the fundamentals of seismic imaging and deconvolution. Most chapters are followed by problem sets. Some exercises supplement textual material; others are meant to stimulate classroom discussions. Text and exercises deal mostly with simple examples that can be solved with nothing more than pencil and paper. The book covers wave motion; digital imaging; digital filtering; various visualization aspects of the seismic reflection method; sampling theory; the frequency spectrum; synthetic seismograms; wavelets and wavelet processing; deconvolution; the need for continuing interaction between the seismic interpreter and the computer; seismic attributes; phase rotation; and seismic attenuation. The last of the 15 chapters gives a detailed mathematical overview. Digital Imaging and Deconvolution, nominated for the Association of Earth Science Editors award for best geoscience publication of 2008–2009, will interest professional geophysicists, graduate students, and upper-level undergraduates in geophysics. The book also will be helpful to scientists and engineers in other disciplines who use digital signal processing to analyze and image wave-motion data in remote-detection applications. The methods described are important in optical imaging, video imaging, medical and biological imaging, acoustical analysis, radar, and sonar.