What are the time and frequency domains? A filter's action can be described by its impulse response as well as by its frequency spectrum. The filter's impulse response is in the time domain, as is the input signal itself. The filter's frequency spectrum is in the frequency domain. Both modes of expression are functions of each other – that is, if one is known, the other can be derived from it.
In digital filtering, either domain can be employed, but generally, both the seismic signal and the characteristics of the filter must be converted into the same form. For example, we need for the operation to be in the time domain, but only the frequency spectrum of the filter is specified. In such a case, the frequency spectrum must be transformed into an impulse response (in the time domain) so that operation can be carried out in the time domain. The Fourier transform and the inverse Fourier transform provide the physical basis for such conversions from one domain to the other.
What is a Fourier transform? The Fourier transform converts a function of time (the signal) into the corresponding function of frequency (the temporal frequency spectrum). The inverse Fourier transform works in the reverse direction. Specifically, the inverse Fourier transform converts a function of frequency (the temporal frequency spectrum) into the corresponding function of time (the signal).
The Fourier transform need not apply only to frequency in cycles per second and time in seconds. The Fourier transform also can be
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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
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.