# Piecemeal convolutional model

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

Author | Enders A. Robinson and Sven Treitel |

Chapter | 10 |

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

ISBN | 9781560801481 |

Store | SEG Online Store |

In the most basic model, the seismogram consists of two entities: (1) the pure primary events representing the reflection coefficients of the interfaces and (2) the myriad of multiple reflections. Most of the reflection coefficients for sedimentary layering are small in magnitude. In such cases, the model can be simplified greatly. Specifically, the strongest and most troublesome multiple events can be associated with horizons that have large reflection coefficients. These are called the major interfaces, and the corresponding primaries from these interfaces are called the major reflections.

An observed seismogram can be subdivided into time intervals (called windows). The boundaries of those windows are established by the arrival times of the major primaries. At such boundaries, the dynamic structure of a seismic trace changes. We will show that within any window, the dynamics remain essentially the same. Admittedly, this is just an approximation, but it clarifies how the mathematics of deconvolution works.

We can divide the trace into a series of windows. This division implies a corresponding division of the reflectivity function. In each window, the dynamic structure remains fixed, but the dynamic structure in a given window differs from that of any other window. The constant dynamics within a window imply that the same multiple wavelet is attached to each reflection coefficient in the corresponding reflectivity window. We will see that each multiple wavelet is a minimum-delay wavelet (or minimum-phase wavelet). The linear mathematical process of convolution describes the dynamics. Thus we have the following dynamic model, which holds within each trace window: (1) trace in window # 1 = (reflectivity in window # 1) * (wavelet # 1); (2) trace in window # 2 = (reflectivity in window # 2) * (wavelet # 2) … ; and (N) trace in window #N = (reflectivity in window #N) * (wavelet #N).

The asterisk (*) indicates convolution. Therefore, a different convolutional model holds for each separate window. This representation of the seismic trace is called the *piecemeal convolutional model* (i.e., the *window-by-window convolutional model*). The important point is that this model of a seismic trace is not *continuously* time-varying: The model is *time-invariant* within any given window and varies only from window to window. The window-by-window convolutional model represents an important simplification. It holds when the reflection coefficients are small except at major horizons. In summary, the window-by-window approach assumes that a given convolutional model holds only over specified time windows of the seismogram rather than over the entire seismogram.

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Seismic deconvolution | Time-varying convolutional model |

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Wavelet Processing | Fine Points |

## Also in this chapter

- Model used for deconvolution
- Least-squares prediction and smoothing
- The prediction-error filter
- Spiking deconvolution
- Gap deconvolution
- Tail shaping and head shaping
- Seismic deconvolution
- Time-varying convolutional model
- Random-reflection-coefficient model
- Implementing deconvolution
- Canonical representation
- Appendix J: Exercises