# Deconvolution exercises

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Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

## Exercises

Exercise 2-1. Write the z-transform of wavelet ${\displaystyle \left(1,\ 0,\ -{\frac {1}{4}}\right).}$ 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)
1. 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.
2. Set up matrix equation (31) for each wavelet, compute the spiking deconvolution operators, then apply them.
3. 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 cw and two-way time tw. Design an inverse filter to suppress the multiples. [This is called the Backus filter [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 (FL), medium (FM) and high (FH), for TVSW application:

FL : 10 to 30 Hz
FM : 30 to 50 Hz
FH : 50 to 70 Hz

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:

1. Spiking deconvolution followed by 10-to-50-Hz zero-phase band-pass filtering.
2. 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

 ${\displaystyle {\begin{pmatrix}r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}g_{0}\\g_{1}\\g_{2}\\\vdots \\g_{n-1}\end{pmatrix}}}$ (30)

 ${\displaystyle {\begin{pmatrix}\beta r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&\beta r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&\beta r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &\beta r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}1\\0\\0\\\vdots \\0\end{pmatrix}},}$ (32)