In this chapter, we present the basic mathematical framework for the study of input-output models, along with some simple examples that show various approaches to the description and analysis of those models. The mathematical tools are derived from the general field of operational calculus, so for the most part, we are limited to the consideration of linear systems. Linear digital systems are described by linear difference equations, whereas linear analog systems are described by linear differential equations. Fortunately, many cases occur in engineering and science in which the systems either are linear or can be approximated sufficiently closely by a linear representation.
Linear methods have been applied very successfully to the analysis of geophysical systems and the processing of geophysical data. The input and output of a system are related by a difference equation (digital case) or a differential equation (analog case), the solution of which gives the output for a given input. This equation provides a complete description of the system, but often it must be converted to other forms to be useful.
Other modes of description of the system are related to the outputs produced by special types of inputs. Thus, we have
the impulse response of the system, which is the output produced by an impulse input
the frequency response of the system, which relates the outputs produced by sinusoidal inputs
the transfer function (or system function), which is a generalization of the frequency-response function
These modes of description are related to
Figures & Tables
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.