# Deconvolution exercises

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 |

## Exercises

**Exercise 2-1**. Write the *z*-transform of wavelet Design a three-term inverse filter and apply it to the original. Hint: The *z*-transform of the wavelet can be written as a product of two doublets, (1, - 12) and (1, 12).

**Exercise 2-2**. Consider the following set of wavelets:

- Wavelet A : (3, -2, 1)
- Wavelet B : (1, -2, 3)

- Plot the percent of cumulative energy as a function of time for each wavelet. Use Robinson’s energy delay theorem to determine the minimum- and maximum-phase wavelet.
- Set up matrix equation (
**31**) for each wavelet, compute the spiking deconvolution operators, then apply them. - Let the desired output be (0, 0, 1, 0). Set up matrix equation (
**30**) for each wavelet, compute the shaping filters, and apply them. Find that the error for wavelet B with the delayed spike is smaller.

**Exercise 2-3**. Consider wavelet A in Exercise 2-2. Set up matrix equation (**32**), where *ε* = 0.01, 0.1. Note that *ε* = 0 already is assigned in Exercise 2-2. As the percent prewhitening increases, the spikiness of the deconvolution output decreases.

**Exercise 2-4**. Consider a multiple series associated with a water bottom with a reflection coefficient *c _{w}* and two-way time

*t*. Design an inverse filter to suppress the multiples. [This is called the Backus filter

_{w}^{[1]}.

**Exercise 2-5**. Consider an earth model that comprises a water-bottom reflector and a deep reflector at two-way times of 500 and 750 ms, respectively. What prediction lag and operator length should you choose to suppress (a) water-bottom multiples, and (b) peg-leg multiples?

**Exercise 2-6**. Refer to Figure 2.6-9. Consider the following three bandwidths — low (*F _{L}*), medium (

*F*) and high (

_{M}*F*), for TVSW application:

_{H}*F*: 10 to 30 Hz_{L}*F*: 30 to 50 Hz_{M}*F*: 50 to 70 Hz_{H}

What kind of slopes should you assign to each bandwidth so that the output trace has an amplitude spectrum that is unity over the 10-to-70-Hz bandwidth?

**Exercise 2-7**. If the signal character down the trace changes rapidly (strong nonstationarity), should you consider narrow or broad bandwidths for the filters used in TVSW?

**Exercise 2-8**. Consider a minimum-phase wavelet and the following two processes applied to it:

- Spiking deconvolution followed by 10-to-50-Hz zero-phase band-pass filtering.
- Shaping filter to convert the minimum-phase wavelet to a 10-to-50-Hz zero-phase wavelet.

What is the difference between the two outputs?

**Exercise 2-9**. How would you design a minimum-phase band-pass filter operator?

**Exercise 2-10**. Consider (a) convolving a minimum-phase wavelet with a zero-phase wavelet, (b) convolving a minimum-phase wavelet with a minimum-phase wavelet, and (c) adding two minimum-phase wavelets. Are the resulting wavelets minimum-phase?

**Exercise 2-11**. Consider the sinusoid shown in Figure 1-1 (frame 1) as input to spiking deconvolution. What is the output?

**Exercise 2-12**. Order the panels in Figure 2.E-1 with increasing prediction lag.

## Figures and equations

**(**)

**(**)

**Figure 2.1-1**(a) A segment of a measured sonic log, (b) the reflection coefficient series derived from (a), (c) the series in (b) after converting the depth axis to two-way time axis, (d) the impulse response that includes the primaries (c) and multiples, (e) the synthetic seismogram derived from (d) convolved with the source wavelet in Figure 2.1-4. One-dimensional seismic modeling means getting (e) from (a). Deconvolution yields (d) from (e), while 1-D inversion means getting (a) from (d). Identify the event on (a) and (b) that corresponds to the big spike at 0.5 s in (c). Impulse response (d) is a composite of the primaries (c) and all types of multiples.

## See also

- Introduction to deconvolution
- The convolutional model
- Inverse filtering
- Optimum wiener filters
- Predictive deconvolution in practice
- Field data examples
- The problem of nonstationarity

## References

- ↑ Backus, 1959, Backus, M. M., 1959, Water reverberations: Their nature and elimination: Geophysics, 24, 233–261.