Power signals are signals with finite power, and energy signals are signals with finite energy (Anstey, 1958). A wavelet is defined as an energy signal that is concentrated in a certain time interval. As a result, an arrival time, or origin time, can be associated with a wavelet. The arrival time is the reference point of the wavelet and usually is designated as time zero. Thus, a wavelet is a signal characterized by two properties:
1) The stability property: A wavelet has finite energy; that is, it is a transient phenomenon.
2) The arrival-time property: A wavelet has a definite origin (or arrival) time.
We distinguish among three types of wavelets. In seismic processing, we use all three:
1) Causal wavelet: All values of a causal wavelet before its origin time are zero. Other terms are realizable wavelet and one-sided wavelet. All wavelets originating from a physical source in real time must be causal.
2) Noncausal wavelet: Nonzero values of a noncausal wavelet occur both before its origin time and after its origin time. Other terms are nonrealizable wavelet and two-sided wavelet. Only conceptual wavelets can be noncausal. Such wavelets can be created easily on the computer.
3) Purely noncausal wavelet: The value of a purely noncausal wavelet at its origin time is zero, and all values after its origin time are also zero. Another term is purely nonrealizable wavelet.
In 1953, the general feeling of the exploration industry was that the digital computer was at best a research device. As a result, any digital processing method eventually would have to be implemented by a corresponding analog device to be useful on a routine basis. Thus, it was suggested that the Geophysical Analysis Group should not use either noncausal wavelets or seismic attributes. However, with faith in the future of the digital computer, this suggestion was not followed. The use of attributes was natural because much of mathematical statistics is concerned with transforming raw data into more meaningful forms. It was suggested at the time that seismic interpreters needed to use the character of the causal wavelets. It was feared that the use of either noncausal wavelets or seismic attributes would be detrimental to seismic interpretation. Nevertheless, the following years showed the opposite to be true, and the use of noncausal interpreter wavelets and seismic attributes is widespread in the industry today.
- Anstey, N. A., 1958, Why all this interest in the shape of the pulse: Geophysical Prospecting, 6, no. 4, 394–403.
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Also in this chapter
- Time series
- Sinusoidal motion
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises