# Introduction to deconvolution

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

Deconvolution compresses the basic wavelet in the recorded seismogram, attenuates reverberations and short-period multiples, thus increases temporal resolution and yields a representation of subsurface reflectivity. The process normally is applied before stack; however, it also is common to apply deconvolution to stacked data. Figure 2.0-1 shows a stacked section with and without deconvolution. Deconvolution has produced a section with a much higher temporal resolution. The ringy character of the stack without deconvolution limits resolution, considerably.

Figure 2.0-2 shows selected common-midpoint (CMP) gathers from a marine line before and after deconvolution. Note that the prominent reflections stand out more distinctly on the deconvolved gathers. Deconvolution has removed a considerable amount of ringyness, while it has compressed the waveform at each of the prominent reflections. The stacked sections associated with these CMP gathers are shown in Figure 2.0-3. The improvement observed on the deconvolved CMP gathers also are noted on the corresponding stacked section.

Figure 2.0-4 shows some NMO-corrected CMP gathers from a land line with and without deconvolution. Corresponding stacked sections are shown in Figure 2.0-5. Again, note that deconvolution has compressed the wavelet and removed much of the reverberating energy.

Deconvolution sometimes does more than just wavelet compression; it can remove a significant part of the multiple energy from the section. Note that the stacked section in Figure 2.0-6 shows a marked improvement between 2 and 4 s after deconvolution.

To understand deconvolution, first we need to examine the constituent elements of a recorded seismic trace (the convolutional model). The earth is composed of layers of rocks with different lithology and physical properties. Seismically, rock layers are defined by the densities and velocities with which seismic waves propagate through them. The product of density and velocity is called *seismic impedance*. The impedance contrast between adjacent rock layers causes the reflections that are recorded along a surface profile. The recorded seismogram can be modeled as a convolution of the earth’s impulse response with the seismic wavelet. This wavelet has many components, including source signature, recording filter, surface reflections, and receiver-array response. The earth’s impulse response is what would be recorded if the wavelet were just a spike. The impulse response comprises primary reflections (reflectivity series) and all possible multiples.

Ideally, deconvolution should compress the wavelet components and eliminate multiples, leaving only the earth’s reflectivity in the seismic trace. Wavelet compression can be done using an inverse filter as a deconvolution operator. An inverse filter, when convolved with the seismic wavelet, converts it to a spike. When applied to a seismogram, the inverse filter should yield the earth’s impulse response. An accurate inverse filter design is achieved using the least-squares method (inverse filtering).

The fundamental assumption underlying the deconvolution process (with the usual case of unknown source wavelet) is that of minimum phase. This issue is dealt with also in inverse filtering.

The optimum Wiener filter, which has a wide range of applications, is discussed in optimum wiener filters. The Wiener filter converts the seismic wavelet into any desired shape. For example, much like the inverse filter, a Wiener filter can be designed to convert the seismic wavelet into a spike. However, the Wiener filter differs from the inverse filter in that it is optimal in the least-squares sense. Also, the resolution (spikiness) of the output can be controlled by designing a Wiener *prediction error filter* — the basis for predictive deconvolution (optimum wiener filters). Converting the seismic wavelet into a spike is like asking for a perfect resolution. In practice, because of noise in the seismogram and assumptions made about the seismic wavelet and the recorded seismogram, spiking deconvolution is not always desirable. Finally, the prediction error filter can be used to remove periodic components — multiples, from the seismogram. Practical aspects of predictive deconvolution are presented in predictive deconvolution in practice, and field data examples are provided in field data examples. Finally, time-varying aspects of the source waveform — nonstationarity, are discussed in the problem of nonstationarity.

The mathematical treatment of deconvolution is found in Appendix B. However, several numerical examples, which provide the theoretical groundwork from a heuristic viewpoint, are given. Much of the early theoretical work on deconvolution came from the MIT Geophysical Analysis Group, which was formed in the mid-1950s.

## See also

- The convolutional model
- Inverse filtering
- Optimum wiener filters
- Predictive deconvolution in practice
- Field data examples
- The problem of nonstationarity
- Exercises
- Mathematical foundation of deconvolution