# Appendix J: Exercises

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 10 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store
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—Theodore Roosevelt


1. Explain why deconvolution normally is applied before stack. Can deconvolution be applied to stacked data?

2. Deconvolution is a process that improves the temporal resolution of seismic data by compressing the basic seismic wavelet. What does temporal resolution mean? The ringy character of a reverberating record without deconvolution limits resolution considerably. Describe what we mean by ringy. Why can deconvolution change a section with a blurred, ringy appearance into one that is crisp and finely detailed?

3. What is meant by the observation that deconvolution compresses the basic seismic wavelet? What is the seismic wavelet? Is this wavelet a physical entity within the earth or is it a mathematical construct?

4. Discuss what is meant by the statement: Deconvolution removes “ringyness” on the record and also compresses the waveforms of the more prominent reflections so that they stand out more distinctly on the deconvolved gathers.

5. Deconvolution can do more than just simple wavelet compression; it can remove a significant part of multiple energy from the section. What is meant by the phrases multiple energy, reverberating energy, and compressing the wavelet?

6. To understand deconvolution, we need a model of the recorded seismic trace. Now discuss the validity of the following model: The recorded trace can be represented as the convolution of the earth’s impulse response with the seismic wavelet. This wavelet consists of various components, including the source signature and the receiver impulse response. The earth’s impulse response is its response to a spike input. This impulse response consists of primary reflections (the reflectivity series) plus all possible multiples. Ideally, deconvolution should compress the wavelet as well as eliminate all multiples so that only the earth reflectivity series remains behind.

7. The inverse filter, when convolved with the seismic wavelet, converts the wavelet to a spike. For an input wavelet of length ${\displaystyle N+\alpha }$, the prediction-error filter contracts it to an ${\displaystyle \alpha }$-long wavelet, where ${\displaystyle \alpha }$ is the prediction distance. Examine this statement with the use of two distinct convolutional models, the first based on a minimum-phase input wavelet and the second based on a mixed-phase input wavelet.

8. Consider the two cases:

Case 1. Deconvolution of a trace using a known, mixed-phase source wavelet.

Case 2. Deconvolution of a trace assuming an unknown, mixed-phase source wavelet. Why does spiking deconvolution yield a degraded output for case 2? Does this also happen for case 1, when the wavelet is known?

9. Which statement better describes gap deconvolution?: (1) Gap deconvolution is a general process encompassing spiking deconvolution or (2) Gap deconvolution is a specialized process derived from spiking deconvolution by the use of a special filter.

10. For vertical incidence, the Fresnel reflection coefficient is the ratio of the reflected wave amplitude to the incident wave amplitude. Show that the Fresnel reflection coefficient is the ratio of the change in acoustic impedance to twice the average acoustic impedance. If density is invariant with depth or if density varies much less than velocity, show that the Fresnel reflection coefficient is ${\displaystyle c=\left(V_{2}-V_{1}\right)\left(V_{2}+\ V_{\rm {l}}\right)}$. If ${\displaystyle V_{2}{>}}$${\displaystyle V_{1}}$, show that the reflection coefficient is positive. If ${\displaystyle V_{2}{<}V_{1}}$, show that the reflection coefficient is negative. Are we talking here about a particle velocity or a pressure reflection coefficient?

11. Why is convolution in the time domain equivalent to multiplication in the frequency domain? Why is the amplitude spectrum of the trace equal to the product of the amplitude spectrum of the seismic wavelet and the amplitude spectrum of the reflectivity? The similarity in the overall shape between the amplitude spectrum of the wavelet and the amplitude spectrum of the trace often is quite apparent. In fact, a smoothed version of the trace amplitude spectrum is nearly indistinguishable from the amplitude spectrum of the wavelet. It is generally thought that the rapid fluctuations observed in the amplitude spectrum of a seismogram are a manifestation of the earth’s impulse response, whereas the lower frequency components are primarily associated with the source wavelet.

12. Let ${\displaystyle a\left(n\right)}$ be given by the infinite convolution ${\displaystyle c\left(1\right)*c\left(2\right)*c\left(2\right)*c\left(2\right)*\ldots }$, where ${\displaystyle c\left[1\right]=\left\{{1\ ,\ }-1\right\},c\left[2\right]=\left\{{1\ ,\ 0,\ }-1\right\},c\left[{3}\right]=\left\{{1\ ,\ 0,\ 0,\ }-1\right\},c\left[{4}\right]=\left\{{1\ ,\ 0,\ 0,\ 0,\ }-1\right\}}$, etc. Compute ${\displaystyle a(n)}$ for ${\displaystyle n={0}}$ to 50. Then compute the causal inverses ${\displaystyle h\left(n\right)}$ for n = 0 to 50. Plot the results. This problem first arose in the theory of numbers, because ${\displaystyle h\left(n\right)}$ is equal to the number of unrestricted partitions of the natural number n. For example, the number 5 can be represented as sums made up of the numbers 1, 2, 3, 4, 5 in the following seven ways:

5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1, so ${\displaystyle h\left(5\right)={7}}$. Can you compute ${\displaystyle h\left({200}\right)={3972999029388}}$?

13. Comment on the following discussion dealing with delay and reverberations in music and with equivalent phenomena in seismology: Music generated in an inert studio does not sound natural when compared with music performed inside a concert hall. In a concert hall, the sound waves propagate omnidirectionally and reach a listener from various directions and with various time lags that depend on the source-to-listener distances. The sound wave reaching the listener directly, called the direct wave, arrives first and determines the listener’s perception of the location, strength, and nature of the sound source. A few closely spaced echoes, called early reflections, follow the direct wave. The early reflections, generated by reflections of waves from all sides of the room, reach the listener at irregular times. These echoes provide the listener with subconscious cues about the size of the auditorium. Following these early reflections, more and more densely packed echoes reach the listener. These are caused by multiple reflections; such a densely packed group of echoes is called a reverberation wave train. As a result of attenuation at each reflection, the amplitude of the echoes decays exponentially with time. The period of time during which the reverberation falls by 60 dB is called the reverberation time. Because the absorption characteristics of different building materials are not the same at different frequencies, the reverberation times vary from frequency to frequency. In an inert studio, delay systems with adjustable delay factors are used to create the early reflections artificially. Electronically generated reverberations combined with artificial echo reflections then are added to the raw studio recordings.

14. Compare the structure of speech signals with the structure of seismic signals. A speech signal is formed by exciting the vocal tract and consists of two sound types: voiced and unvoiced. Voiced sounds include the vowels and a number of consonants such as B, D, L, M, N, and R. A voiced sound is excited by the pulsatile airflow resulting from the vibration of the vocal folds. Unvoiced sound is produced downstream in the forward part of the mouth with the vocal cords at rest and includes sounds such as F, S, and SH. For example, A is a slowly varying low-frequency voiced waveform, whereas S is a high-frequency unvoiced fricative waveform. The voiced waveform “A” is quasi-periodic, so it can be modeled by a sum of a finite number of sinusoids. The lowest frequency of oscillation in a speech waveform is called the pitch frequency. The unvoiced sound “S” has no regular waveform structure and is more like noise.

15. Let us now consider the electroencephalogram (EEG) in comparison with the seismogram. The electroencephalogram signal consists of the sum of all electrical activity caused by the random firing of billions of individual neurons in the brain. In multiple EEG recordings, electrodes are placed at various positions on the scalp, with two common electrodes placed on the earlobes. Potential differences between the various electrodes then are recorded. Typical bandwidths for this type of EEG range from 0.5 to ~ 100 Hz, with amplitudes ranging from 2 to ${\displaystyle 100\;{\rm {\mu V}}}$. Both frequency-domain and time-domain analyses of the EEG signal have been used to diagnose epilepsy, sleep disorders, psychiatric malfunctions, etc. To this end, the EEG spectrum is subdivided into the following five bands: (I) the delta range, occupying the band from 0.5 to 4 Hz, (2) the theta range, occupying the band from 4 to 8 Hz, (3) the alpha range, occupying the band from 8 to 13 Hz, (4) the beta range, occupying the band from 13 to 22 Hz, and (5) the gamma range, occupying the band from 22 to 30 Hz. The delta wave is normal in the EEG signals of children and sleeping adults. However, because it is not common in alert adults, its presence indicates certain brain diseases. The theta wave usually is found in children, although it has been observed in alert adults. The alpha wave is common to all normal humans and is more pronounced in a relaxed and awake subject whose eyes are closed. Likewise, beta activity is common in normal adults. The EEG exhibits rapid, low-voltage waves, called rapid-eye-movement (REM) waves, in a subject who dreams during sleep. Otherwise, in a sleeping subject, the EEG contains bursts of alphalike waves called sleep spindles. The EEG of an epileptic patient exhibits various types of abnormalities depending on the type of epilepsy, which is caused by uncontrolled neural discharges. A frequency analysis of a seismogram can be conducted in a way similar to division of the EEG spectrum into different frequency bands. However, division of the seismic spectrum into the frequency bands is not as useful as is seismic wavelet analysis. As a matter of discussion, do you think that EEG signals might also benefit from wavelet analysis?

16. What can you say about the following signals? (a) A ${\displaystyle \delta }$ function at the origin. Its spectrum is a horizontal line. (b) A cosine wave. Because this function is symmetrical, its spectrum consists of a symmetric pair of ${\displaystyle \delta }$ functions each spaced astride the origin by a distance proportional to the reciprocal of the cosine wave’s period. (c) A sine wave. Because this function is antisymmetrical, its spectrum consists of two elementary lines spaced about the origin but with opposite signs. The spacing of the lines on either side of the center is given by the reciprocal of the spacing between the crests of the sine curve. (d) A rectangular pulse. Its spectrum follows a sinc curve. (e) A fine-ruled grating or a comb filter. Its spectrum consists of a similar series of beams. (f) Now assume that each comb tooth has finite width. The rectangular pattern of each line is equivalent to that of example (d). Hence, we see a sinc form modulation envelope. (g) A line or ${\displaystyle \delta }$ function offset from the origin. Its spectrum is a function of constant amplitude and linear phase.

Figure J-1.  Gap filter for prediction distance 4 ms (i.e., ${\displaystyle N=1}$).

17. (a) Let us illustrate predictive deconvolution. In this exercise, we view various aspects of deconvolution, with emphasis on the decisions based on visual inspection of the signals. Much geophysical interpretation depends on the eye and not on the mathematics alone. In Figure J-1, the time spacing (in ms) is ${\displaystyle dt={4}}$. The dimensionless prediction distance is ${\displaystyle \alpha =1}$. Thus, prediction distance (expressed in ms) is ${\displaystyle \alpha \ dt=\left({\rm {l}}\right)\left({4}\right)={4}}$ ms.

The minimum-delay wavelet and the white reflectivity are unknown. Describe how the wavelet and the reflectivity combine to produce the trace. The trace is known. The deconvolved trace, filter, and inverse filter are computed from the trace.

(i) Do the wavelet and the inverse spike filter look alike? Why?

(ii) Do the reflectivity and the deconvolved trace look alike? Why?

(iii) Is the spike filter minimum delay? Is the choice ${\displaystyle \alpha ={4\,{\rm {ms}}}}$ correct? Why?

Figure J-2.  Gap filter for prediction distance 8 ms (i.e., ${\displaystyle N={2}}$).

(b) Illustration of predictive deconvolution (continued). In Figure J-2, the time spacing (in ms) is ${\displaystyle dt={4}}$. The dimensionless prediction distance is ${\displaystyle \alpha ={2}}$. Thus, prediction distance (in ms) ${\displaystyle =\alpha \,dt={8}}$ ms. The minimum-delay wavelet and the white reflectivity are unknown. The trace is known. The gap-deconvolved trace, the gap filter for prediction distance 8 ms, and the inverse gap filter are computed from the trace.

(i)Do the wavelet and the inverse gap filter look the same? Why not?

(ii) Why does the inverse gap filter blow up? Is this good or bad?

(iii) Do the reflectivity and the deconvolved trace look the same? Why?

(iv) Do you expect the gap filter to be minimum delay? Why? Is this choice of ${\displaystyle \alpha }$ correct if the goal is to recover the reflectivity?

Figure J-3.  Gap filter for prediction distance 12 ms (i.e., ${\displaystyle N={3}}$).

(c) Illustration of predictive deconvolution (continued). In Figure J-3, the time spacing (in ms) is ${\displaystyle dt={4}}$. The dimensionless prediction distance is ${\displaystyle \alpha ={3}}$. Thus, ${\displaystyle \alpha }$ (in ms) ${\displaystyle =\left(\alpha \right)\left(dt\right)={12}}$. The minimum-delay wavelet and the white reflectivity are unknown. The trace is known. The gap-deconvolved trace, gap filter, and inverse gap filter are computed from the trace.

(i) Do the reflectivity and the deconvolved trace look the same? How do they differ?

(ii) Do the wavelet and the inverse gap filter look the same? Why not?

(iii) Is the inverse gap filter stable? Is the gap filter minimum-delay?

(iv) What periodicity do you see on the wavelet? On the inverse gap filter?

(v) Is this choice of ${\displaystyle \alpha }$ correct?

Figure J-4.  Gap filter for prediction distance 16 ms (i.e., '${\displaystyle N={4}}$).

(d) Illustration of predictive deconvolution (continued). In Figure J-4, the time spacing (in ms) is ${\displaystyle dt={4}}$. The dimensionless prediction distance is ${\displaystyle \alpha ={4}}$. Thus, ${\displaystyle \alpha }$ (in ms) ${\displaystyle =\left(\alpha \right)\left(dt\right)={16\,{\rm {ms}}}}$. The minimum-delay wavelet and the white reflectivity are unknown. The trace is known. The gap-deconvolved trace, gap filter, and inverse gap filter are computed from the trace.

(i) In what way are the wavelet and the inverse gap filter similar?

(ii) Why is the inverse gap filter stable? What does this indicate?

(iii) How is the deconvolved trace related to the reflectivity?

(iv) Is the gap filter minimum delay?

(v) What periodicity do you see on the wavelet?

(vi) What periodicity do you see on the inverse filter? Is this choice of ${\displaystyle \alpha }$ correct?

Figure J-5.  Gap filter for prediction distance 20 ms (i.e., ${\displaystyle N={5}}$).

(e) Illustration of predictive deconvolution (continued). In Figure J-5, the time spacing (in ms) is ${\displaystyle dt={4}}$. The dimensionless prediction distance is ${\displaystyle \alpha ={5}}$. Thus, ${\displaystyle \alpha }$ (in ms) ${\displaystyle =\left(\alpha \right)\left(dt\right)={20\,{\rm {ms}}}}$. The minimum-delay wavelet and the white reflectivity are unknown. The trace is known. The gap-deconvolved trace, gap filter, and inverse gap filter are computed from the trace.

(i) Do the wavelet and the inverse gap filter look the same? In what way?

(ii) Why is the inverse gap filter stable? What does this indicate?

(iii) Do the reflectivity and the deconvolved trace look the same?

(iv) Is the gap filter minimum delay?

(v) What periodicity do you see on the wavelet?

(vi) What periodicity do you see on the inverse filter?

(vii) Is this choice of ${\displaystyle \alpha }$ correct? Compare with the case of ${\displaystyle \alpha =4}$.

Figure J-6.  Gap filter for prediction distance 24 ms (i.e., ${\displaystyle N={6}}$).

(f) Illustration of predictive deconvolution (continued). In Figure J-6, the time spacing (in ms) is ${\displaystyle dt={4}}$. The dimensionless prediction distance is ${\displaystyle \alpha ={6}}$. Thus, ${\displaystyle \alpha }$ (in ms) ${\displaystyle =\left(\alpha \right)\left(dt\right)={24\,{\rm {ms}}}}$. The minimum-delay wavelet and the white reflectivity are unknown. The trace is known. The gap-deconvolved trace, gap filter, and inverse gap filter are computed from the trace.

(i) Do the reflectivity and the deconvolved trace look the same? If not, how do they differ?

(ii) Do the wavelet and the inverse gap filter look the same? Why not?

(iii) Is the inverse gap filter stable? Is the gap filter minimum delay?

(iv) What periodicity do you see on the wavelet? On the inverse gap filter?

(v) Is this choice of ${\displaystyle \alpha }$ correct? Before deciding, compare with the case of ${\displaystyle \alpha =4}$.