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, - 1) and (1, 1).
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 cw and two-way time tw. Design an inverse filter to suppress the multiples. [This is called the Backus filter .
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:
- 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.
- Introduction to deconvolution
- The convolutional model
- Inverse filtering
- Optimum wiener filters
- Predictive deconvolution in practice
- Field data examples
- The problem of nonstationarity
- Backus, 1959, Backus, M. M., 1959, Water reverberations: Their nature and elimination: Geophysics, 24, 233–261.