# Wavelets - Chapter 9

Series | Geophysical References Series |
---|---|

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

Author | Enders A. Robinson and Sven Treitel |

Chapter | 9 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

Sit in reverie and watch the changing color of the waves that break upon the idle seashore of the mind. —Henry Wadsworth Longfellow

What is a wavelet? From a seismic processor’s point of view, a wavelet is one of the basic building blocks used to construct the seismic models on which seismic-processing methods are based. In the various processing steps, the wavelets are removed from the seismic data to yield the final sections. Correct estimation and/or measurement of the wavelets allows the good results that are expected in any exploration program. The wavelet is a basic concept, and good wavelet estimation is fundamentally important in exploration geophysics.

The seismic method is an instrument for remote detection that uses seismic traveling waves to delineate the subsurface structure of the earth. An exploration geophysicist illuminates the earth’s subsurface by an energy source that generates those seismic waves. In a 3D earth, the waves travel in all directions, but to keep the present discussion simple, we consider the case of only vertically upgoing waves and vertically downgoing waves.

Subsurface rock layers transmit and reflect the seismic waves, and seismic theory and practice deal with those traveling seismic waves. The simplest example of a traveling wave is a primary reflection. A primary reflection consists of the downgoing path from the source to the reflection horizon and the returning upgoing path from the reflector to the receiver. A multiple reflection is an event that bounces back and forth among various interfaces as it proceeds on its trip. Directional sensors can be used to record seismic waves, but usually the receiver is either a hydrophone (for exploration at sea) or a conventional geophone (for land exploration).

The purpose of digital seismic processing is to transform raw data into computer-generated images of the subsurface. Each computer-processing method is based on a specific model that is used to explain the propagation of seismic waves. One of the most popular models is the convolutional model, which appears in one form or another in most seismic-processing and interpretation methods. The convolutional model consists of three components: the input signal, the unit-impulse response function, and the output signal. The output signal is equal to the convolution of the input signal with the unit-impulse response function (Szaraniec, 1985^{[1]}).

The signals that appear in the convolutional model usually are seismic wavelets. Therefore, estimation of these wavelets becomes an important aspect of using the convolutional model efficaciously; in fact, such wavelet estimation is crucial to seismic processing. Much of exploration geophysics is concerned with manipulation of wavelets.

What is vertical resolution? Vertical resolution refers to the ability of an interpreter to distinguish reflections from the top and bottom of a thin subsurface rock layer (Yilmaz, 1987^{[2]}). Clearly, it is easier to determine the top and bottom reflections from a 30-m-thick bed than from a 3-m-thick bed (given, of course, that all other things are equal).

The seismic wavelet is an important factor in determining vertical resolution, which depends on the wavelet’s sharpness. We can see a thin layer better with a sharp wavelet than we can with a broad wavelet. Wavelet sharpness depends primarily on its bandwidth — that is, on the effective spread of frequencies in its amplitude spectrum. A wavelet with a wide band of frequencies (a large bandwidth) is sharper than one with a narrow band of frequencies. In fact, a very narrow band of frequencies would correspond to a wavelet with many high side lobes — a broad “ringing wavelet.” In seismic work, we try to use wavelets with large bandwidth. Good bandwidth is achievable with a well-designed source, with use of appropriate spread geometries and recording filters, and with proper use of deconvolution. However, bandwidth limitations are imposed by the filtering effects of the earth’s sedimentary column, over which we have no control. Beyond deconvolution, we can, as a final step, follow up with a procedure known as wavelet processing.

Figure 1 shows some typical seismic wavelets. A field wavelet is causal and hence has nonzero phase (pulse *d*). To bring such wavelets to the zero-phase condition requires a phase-zeroing filter in the wavelet-processing step. Vertical resolution depends on the sharpness of the seismic pulse. The sharper the pulse is, the thinner the layer is that we can see. The sharpness of the pulse depends primarily on its bandwidth. The wider the band of frequencies is, the better the sharpness is (pulses *a* and *b*). Wavelet sharpness also depends on its phase-lag characteristic. The best that can be achieved in this sense is the zero-phase wavelet (pulses *a*, *b*, and *c*). In these examples, the pulses are symmetric about the central peak. The secret of resolution is bandwidth, not merely the presence of high frequencies. Of course, we always can raise the spectral content on a section by filtering out the low frequencies, but that does not necessarily improve resolution. It just tends to make a wavelet leggier so that it loses sharpness because of the profusion of extra cycles (pulses *b* and *c*).

Figure 2 shows the Widess (1973)^{[4]} diagram for a thin soft bed that pinches out in a hard formation. The top and bottom traces correspond, respectively, to the thick and thin edges of a wedge. *T* is the peak-to-peak period of an individual wavelet. In the top trace, the two wavelets are separated by 2*T*. In other words, 2*T* is the distance between the small black circle and the small white circle. In the bottom trace, the separation is only *T*/20.

Figure 3 shows the Widess diagram used to explore the details of a reservoir, here represented by a wedge. The tuning effect we see represents the constructive or destructive interference resulting from two or more closely spaced reflectors. An exceptionally strong event occurs when the reflections from the upper and lower interfaces interfere constructively. This strong event occurs on the trace labeled “Tune” in the figure.

## References

- ↑ Szaraniec, E., 1985, On direct recovery of the impulse response: Geophysical Prospecting,
**33**, 498-502. - ↑ Yilmaz, Ö., 1987, Seismic data processing: SEG Investigations in Geophysics No. 2.
- ↑
^{3.0}^{3.1}^{3.2}Anstey, N. A., 1980, Seismic exploration for sandstone reservoirs: IHRDC. - ↑ Widess, M. B., 1973, How thin is a thin bed?: Geophysics,
**38**, 1176-1254.

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Synthetics | Deconvolution |

## Also in this chapter

- The shaping filter
- Spiking filter
- White convolutional model
- Wavelet processing
- All-pass filter
- Convolutional model
- Nonminimum-delay wavelet
- Signature deconvolution
- Vibroseis
- Dual-sensor wavelet estimation
- Deconvolution: Einstein or predictive?
- Summary
- Appendix I: Exercises