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Book Chapter

Wavelets

Published:
January 01, 2008

Abstract

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.

What is a wavelet? A wavelet is a signal that has finite energy (Robinson, 1962, 1964a, 1964b). In other words, a wavelet is a waveform with the bulk of its energy confined to a finite interval on the time scale. A wavelet can be written as 
formula
where bn is the coefficient at discrete time n. The present time instant n = 0 represents the critical point in the classification of a wavelet. Past times would be all the instants n 0 before the present time, and future times would be all the instants n > 0 after the present time.

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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Contents

Society of Exploration Geophysicists Geophysical References Series

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Enders A. Robinson
Enders A. Robinson
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Sven Treitel
Sven Treitel
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Society of Exploration Geophysicists
Volume
15
ISBN electronic:
9781560801610
Publication date:
January 01, 2008

GeoRef

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