Sosie method

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Problem 7.6a

Imagine an impulsive source striking the ground at times apart, where is a random number between 10 and 20, and is the sampling interval. Assume reflections with arrival times and amplitudes as follows:

Each new impact generates the reflection sequence.

One wants the reflections for each impact to add constructively, so Sosie processing involves adding the geophone responses for several impacts after each response has been shifted so that is the same for the responses from all the impacts. Add the reflection sequences as would be done in Sosie recording for 10, 20, and 30 impulses to see how the signal builds up as the multiplicity increases.

Table 7.6a. Random impact time intervals.
0, 16, 14, 12, 17, 19, 12, 16, 15, 11,
14, 11, 13, 20, 14, 18, 16, 14, 17, 10;
18, 18, 13, 11, 15; 10, 19, 12, 20, 17.
Figure 7.6a  Illustrating the buildup of reflection signals in the Sosie method. From top to bottom: the signal and sums for the first (i) 10, (ii) 20, and (iii) 30 impacts.

Background

The Sosie method often uses impactors as seismic sources. Each impactor strikes the ground at random intervals 5 to 10 times per second for approximately 3 minutes, a total of 900–1800 impacts, the times between impacts being much shorter than the reflection wavetrain. Each impact generates the reflection sequence. The reflections from each impulse are much smaller than the noise but the Sosie process builds up the reflections systematically while the noise partially cancels.

In digital recording, an analog signal is measured at fixed intervals , called the sampling interval; is usually 1 or 2 ms. Table 6.22a is a table of random numbers.

Solution

We obtained 29 random numbers in the 10–20 range by reading systematically from Table 6.22a, using even or odd values to determine the tens place and every 4th digit to give the units place. We used these numbers as the time intervals between 30 successive impacts. The impact times we used are listed in Table 7.6a.

The time required for 30 impacts separated by about plus the record length means that we have about 500 samples. We have made the impact intervals about 1/3 of the length of the reflection wavetrain.

Table 7.6b lists for the first ten impulses the signal sequences starting at each impulse time, each sequence having length , hence they overlap. is the sum of the signal values occurring at time . Toward the right, subsets of are listed, each being long, each subset constituting noise for the other subsets. Finally the subsets are summed. Figure 7.6a shows the reflection wavetrain and the result of summing the first 10, 20, and 30 impulses. (The data for the impulses 11–30 are not listed.)

Note the result for the two smallest reflections with amplitudes , that is, the 3rd (negative) reflection at and the 5th at ; the 3rd shows up clearly, but the 5th is no larger than the “noise,” which in this exercise is that caused by the overlapping signals. The noise is attenuated relative to the reflections as more sequences are added.

Table 7.6b  Calculations for signal buildup in Sosie recording; no random noise added.
Figure 7.6b  Same as Figure 7.6a except for added random noise.

Problem 7.6b

In part (a), we examined how the reflection sequence builds up. We now examine the effect of random noise by adding random noise values between to the signal. We obtain these from a random-number table by some scheme, such as systematically selecting pairs of numbers and changing the first member of each pair to a minus sign whenever it is odd and to a plus sign whenever it is even. The signal sequence has an rms value of 2.7, so adding noise values within the range (rms value 3.9) gives a signal/noise ratio of about 0.7.

Solution

The first page of our worksheet is shown as Table 7.6c. The first column is the sample number , the 2nd is the signal sequence from Tables 7.6a,b, the 3rd is the added noise, and the fourth the sum of the signal and noise. The signal plus noise subsets, each starting at the time of a source impact, are listed to the right followed by their sum.

The first 45 members of 30 subsets, each starting at one of the impulses, is shown in Table 7.6d. Summations for 10, 20, and 30 of the subsets are shown in Table 7.6e and in Figure 7.6b.

Examination of the numbers in the signal + noise columns in Table 7.6c indicates the difficulty of finding any of the signals. Table 7.6e lists the sums of the first subset, the first two subsets, then the first 4, 10, 20, and 30 subsets, with the times of the signals underlined. The first member of the signal sequence, which considerably exceeds the average noise, becomes distinguishable with the addition of only a few subsets, whereas the two smallest signal members (at times 13 and 33) still do not stand out even after summing 30 subsets. The broad, smeared-out peak between times 14 and 18, which corresponds roughly to the repetition rate of the impulses, would probably be erroneously interpreted as the interference of several reflections.

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