Faraday (1831) investigated the water patterns produced under vibration. Transitory ripples disclosed distinct patterns. From the ripples, Faraday discerned an oscillatory condition that proved useful in his subsequent investigations on light. In the same way, geophysicists must investigate each ripple on a seismic wavelet to unravel the deep secrets of the earth.
For the record, all of the digital filters that we consider (unless otherwise stated) fall under the category of linear time-invariant filters. Note that filter is called linear if it satisfies the additive property and the multiplicative property. Let a given input yield a given output. A filter is called time-invariant if the same input delayed (or advanced) by a given amount yields the same output delayed (or advanced) by the same amount (Robinson and Silvia, 1978).
Recall our discussion in Chapter 4,
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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.