Determination of the signature of a dynamite source using source scaling, Part 2: Experiment

Transcription

1 GEOPHYSICS, VOL. 58, NO. 8 (AUGUST 1993); P , 12 FIGS. Determination of the signature of a dynamite source using source scaling, Part 2: Experiment Anton Ziolkowski* and Karel Bokhorst ABSTRACT In April 1990 we performed an experiment in the Netherlands to test the theory of the determination of the signature of a dynamite source using the scaling law. The theory says that the source signature may be determined from the recorded seismic data using two shots of different charge size at the shotpoint; we used 125 g and 500 g charges. The theory was put at risk with a 250-g test charge at each shotpoint. According to the theory, the test record should be different from the other two and, apart from the noise, should be predictable from them. This experiment was repeated 95 times at approximately 50 m shotpoint intervals, using a 240-channel recording system. The results corroborate the theory within an acceptable error. The second-derivative of the volume injection function was extracted as the source signature; it varied slightly from shot to shot and was minimum phase. This new method of seismic data acquisition allows the signature of the dynamite source to be obtained from the data, uncontaminated by the earth, and avoids the assumptions that must be made in statistical wavelet estimation methods. If there is good shot-to-shot repeatability, the second shot is only needed occasionally for calibration. INTRODUCTION We demonstrate a method of data acquisition that allows the signature of the dynamite source to be extracted from the data in which, for practical reasons, we define the signature as the second derivative of the volume injection function of the source. This method was proposed in Ziolkowski et al. (1980) and was based on the source scaling-law as proposed in Ziolkowski and Lerwill (1979). The idea is to fire two shots of different charge size at the shotpoint, instead of one. At each geophone, the two shots give two different seismograms: the earth impulse response is the same for both seismograms, but the source volume injection functions are different. The two signatures are scaled versions of each other with the amplitude scale factor and the time scale factor both equal to the cube-root of the ratio of the charge masses. The two signatures and the earth impulse response can be derived from the two seismograms and the scaling law along the lines as described in Ziolkowski et al. (1980). Ziolkowski and Lerwill (1979) did not derive the scaling law; it was more of a conjecture based on energy arguments and it was applied to the far-field particle velocity function. Source scaling has been known for a long time, both in exploration geophysics (Gaskell, 1956; O Brien, 1957) and in nuclear explosion seismology (Rodean, 1971). The scaling law used in the present paper is derived using energy arguments in Part 1 (Ziolkowski, 1993) in which it is pointed out that the convolutional model does not apply for the far field particle velocity function: it applies for the volume injection function of the source, and its time derivatives. We explain our computational scheme, and present the results of a field experiment in which the theory was put at risk using an idea proposed by Ziolkowski (1982). The experiment was carried out in the Netherlands in It consisted of a seismic line of 96 shotpoints at approximately 50 m intervals and 240 channels of data per shot. At each shotpoint three different charge sizes were used: 125, 250, and 500 g. The signatures were derived from the 125 and 500 g data. The 250 g seismograms were then estimated from both the 125 g and the 500 g data and the scaling laws and compared with the measured data. The statistics of the error between the measured and estimated data indicate that the theory was corroborated within an acceptable error, and that the source signature may be estimated accurately using this method. Manuscript received by the Editor December 26, 1991, revised manuscript received January 28, *Dept of Geology and Geophysics, University of Edinburgh, Grant Institute, West Mains Road, Edinburgh EH9 3JW, United Kingdom. Geco-Prakla, Poortweg 4, P. 0. Box 148, 2600 AC Delft, The Netherlands Society of Exploration Geophysicists. All rights reserved. 1183

2 1184 Ziolkowski and Bokhorst THE BASIC EQUATIONS If two dynamite shots are fired in more or less the same source position-for example, at the same depth in two holes very close together at the same shotpoint-the received signals are = + (1) = + (2) where is the received signal, is the volume injection function of the source, h,,(t) is the response of the earth at the receiver position to an impulse in volume injection at the source position, and n,(t) is the noise present at the receiver. The subscripts s and r denote the dependence of these functions on the source and receiver positions, and the subscripts 1 and 2 refer to the first and second charges. Following the hypothesis of Ziolkowski and Lerwill (1979) and Ziolkowski (1993), the two volume injection functions are related as where = (3). (4) If the noise is negligible, equations (l), (2), (3), and (4) can be solved for the unknowns and h,,(t), as discussed below. In practice the noise on each trace is not negligible, and this reduces the accuracy of the estimates of these unknowns. However, with multichannel commonshotpoint data it is possible to improve the signal-to-noise ratio and obtain accurate estimates of the source signatures and as we describe below. SOLUTION OF THE EQUATIONS The equations we need to solve are (l), (2), and (3). A practical point in the solution is that it is particle velocity that is measured with geophones that are generally in the far field. Since the far-field particle velocity function of the incident field is proportional to the second derivative of the volume injection function (See, for example, Ziolkowski 1993), it is sensible to recognize the band-limiting effect of this method of measurement on our received bandwidth and extract the second derivative of the volume injection function, rather than the volume injection function itself. We may write equations (l), (2), and (3) as follows in which 1 + (5) + (6) = (7) An extensive discussion of the solution of identical equations is given in Ziolkowski et al. (1980). The main points are as follows, in which it is assumed that is greater than 1. In l (8) the frequency domain and ignoring the noise, equations (5), (6), and (7) become (9) (10) = (11) Division of equation (10) by equation (9) removes the influence of the geology; substitution from equation (11) into the result yields the following equation which can be rearranged as follows = (12) ( s,l =. (13) In this equation (w) and are known from measurements. If the value of were known at one frequency, say, then equation (13) would yield the (complex) value S s, l This could then be substituted into the right-hand side of equation (13) and thus yield S ), and so on. This recursive scheme forms the basis of our algorithm. There are four problems to solve: (1) the initial value must be known; (2) the values at negative frequencies have to be defined; (3) some sort of interpolation must be used to recover values at regularly-spaced samples in the frequency domain to permit the transformation back to the time domain; (4) the algorithm does not handle the noise. All these problems were addressed in Ziolkowski et al. (1980), where the following solutions were proposed. First, must be guessed. The guess is in error by an amplitude factor and a phase factor that are the same for all values of (w) extracted using the recursion. The amplitude error is not important for the shape of the extracted wavelet. The phase error is, and it must be found. The source signature is real and causal, and these properties are used to solve for the phase error. Since the signature is real, the values at negative frequencies are the complex-conjugates of the values at positive frequencies; this solves the second problem. Transforming the frequency spectrum of the signature back to time will yield a noncausal signature if the phase error is nonzero. The phase error can be estimated by trial-and-error: a correction between 0 and 180 degrees is applied until the anticausal part of the signal is minimized. Then the signal is causal, but may have the wrong polarity: the initial value of s(t) must be positive since the initial motion is outwards. The interpolation problem is probably best discussed using an example. Suppose is 2, is chosen to be 1 Hz, and we wish to determine the source spectrum at all frequencies with regular sample intervals = (1.024) Hz up to a Nyquist frequency of 500 Hz. From the initial guess at = 1 Hz and the recursion formula, we obtain (complex) values of at the frequencies etc. (that is, at 2, 4, 8,..., 256 Hz). We recognize that our maximum frequency of 256 Hz is determined by our choice of. We need to know (w) at the frequencies etc. These desired regular frequency values are mapped into the

3 Dynamite Signature Scaling: Experiment 1185 range to by dividing them by the appropriate power of For example, the frequency value maps into this range when divided by = 8, and it then has the value 2 = Hz. Using the known values of at the frequencies 1, 2, 4, 8 Hz, etc., we interpolate to find = (14) We now use the recursion again to obtain values, at = = = = 6Af and finally at = = as required. The procedure is the same for all frequencies within the chosen bandwidth l-256 Hz. The only remaining problem is the noise. In practice, there are usually many receivers for each shot. Therefore the ratio spectrum R,(o) can be estimated as many times for each shotpoint as there are receivers. The noise, being different on each receiver and for each shot, is uncorrelated, and the ratio spectrum can therefore be estimated even in the presence of noise. Thus, from two shots in the same place the second derivative of the volume injection function of the smaller shot is derived. Using equation (7), (t) can be derived. Then estimates of, can be found from equations (5) and (6) by deconvolution. AN EXPERIMENT TO TEST THE THEORY Following a discussion with Dr. G. C. P. King in 1978, an experiment in which this theory could be put at risk was proposed in Ziolkowski (1982). The experiment consists of a third shot of mass M 3 placed close to the first two. From the scaling law the second derivative of the volume injection function of this third shot is where = (15) (16) An estimate of the third seismogram can be made by convolving s (t) with This estimate can then be compared with the true seismogram. If there is not a close resemblance between the true seismogram and the estimate, the theory is refuted. In practice, of course (Gorski, 1983, private communication; Ziolkowski and Holtslag, 1984), it is not necessary to obtain g,,,(t) to predict the seismogram. Consider two filters, (t) and, defined as follows: = (17) = l (18) The source signatures of the first two shots are determined as described above; the source signature of the third shot is then found from equation (15). The filters in equations (17) and (18) can be found easily by least-squares using the Wiener-Levinson approach (Levinson, 1947). Application of to the first seismogram (t) and off, to the second seismogram (t) yields and = (19) (20) Thus two independent predictions of the third seismogram may be made from the first two seismograms. EXPERIMENTAL RESULTS As part of a larger project, an experiment to test this theory was conducted in April 1990 near Almelo in the eastern part of the Netherlands. Data were obtained with a split-spread configuration using the following parameters: Number of channels : 240 Sampling interval : 2 ms Record length :4s Group interval : 10m Group length : 20 m Number of geophones per group : 12 Geophone resonance : 10 Hz Shotpoint interval : 50 m Number of shots : 96 Shot depth : 10 m Shot sizes : 1 detonator, 125 g, 250 g, 500 g at the corners of a 3 m square, centered on the shotpoint. For each shotpoint, the 125 g shot was taken as shot 1, the 500 g shot was taken as shot 2, and the 250 g shot as shot 3. The detonator shot was intended for use in calibration. Each dynamite charge includes a detonator, and the cube-root scaling is therefore not exactly correct: the energy of the detonator should also be included. From our data we found that the energy in the detonator seismograms was negligible compared with that in the smallest dynamite charge seismograms and that this correction was therefore negligible. Figure 1 shows the three shot records of the 125 g, 250 g, and 500 g charges at one shot record. Corresponding spectra for a geophone group at 100 m offset are shown in Figure 2. The convolutional model discussed above applies only for a point source and point receiver. We had geophone groups of 20 m length, which cannot be regarded as point receivers. The response of each group varies with the slowness of each arrival. The response to surface waves in particular is attenuated relative to the body waves. Nevertheless the contribution of the surface waves is still large. We therefore further attenuated the surface waves by velocity-filtering in the frequency-wavenumber domain. The velocity-filtered shot records are shown in Figure 3, in which the surface waves are still evident on the near-offset traces.

4 1186 Ziolkowski and Bokhorst We then calculated the spectral ratio for each of the 240 channels, using shot 1 and shot 2 as defined in equation (13), and formed the arithmetic mean. The result is shown in Figure 4. Inspection of the raw data, Figure 2, shows that the bandwidth is confined to the range Hz. This frequency band was used to perform the wavelet extraction using the scheme described above. Outside the bandwidth the amplitude spectrum of the source signature was set to zero. The extracted source signature was the second derivative of the volume injection function of the 125 g charge and is shown in Figure 5. The energy in the noncausal part of the trace was 0.2 percent of the total energy, which is a very low noise level. The noise comes from a number of sources, including the following: additive noise in the data; Gibbs s phenomena caused by the truncation at low and high frequencies in the wavelet extraction process; and numerical errors caused by our cubic-spline interpolation. We judged that the true signal exists only up to the third zero-crossing at about 0.03 s; beyond that we regard everything as noise and truncate it. The truncated signature and its amplitude and phase spectra are shown in Figure 6. It is minimum phase. Using the scaling law we may now construct the corresponding signatures for the 250 g and 500 g charges. All three signatures are shown in Figure 7. We may now construct filters to calculate the 250 g signature from the 125 g and 500 g signatures, as defined in equations (17) and (18). These filters are shown in Figure 8. Application of these filters to the 125 g and 500 g data yields two estimates of the 250 g data, as described by equations (19) and (20). The results are shown in Figure 9, where we have plotted only every fifth trace and 2 s of data to enable the seismograms to be seen more clearly. In Figure 10 we show the energy in the difference between the true seismogram and the estimate as a percentage of the energy in the true seismogram for every trace. It is clear that the estimate is better with increasing offset, partly because the surface waves were not completely attenuated on the near-offset traces by the velocity filter. There is also a bias: the estimate tends to be better for the traces to the right than for those to the left. We attribute the first error to our use of groups, rather than single geophones. And we attribute the second error to our use of the square shotpoint configuration. The offset varies by 3 m between the three shots, which FIG. 1. Shot record with (a) a 125 g dynamite charge, (b) a 250 g dynamite charge, and (c) a 500 g dynamite charge. The distance between traces is 10 m; the timing lines are at 1 s intervals; the shot was at trace 120.

5 Dynamite Signature Scaling: Experiment 1187 introduces time differences of up to 2 ms, or one sample interval, especially for the refracted arrivals. We were very pleased to see how well the reflections were predicted. To check that this result was not a fluke, we extracted signatures for every fifth shotpoint. The result is shown in Figure 11, where we see good shot-to-shot repeatability if we take into account the occasional polarity reversal. CONCLUSIONS AND RECOMMENDATIONS We have demonstrated that the source scaling law may be used to solve the deconvolution problem for the dynamite source. In our experiment, we used two shots of masses = 125 g and = 500 g at every shotpoint. At every shotpoint the two shots gave two different seismograms containing the two source signatures convolved with the same earth impulse response, plus noise. The spectral ratio of the two seismograms was therefore independent of the geology and approximately equal to the spectral ratio of the source signatures, the error being caused by the noise. The error was reduced by averaging the ratio using multichannel recording. The spectral ratio and the source scaling law are two independent equations that were used to determine the two unknown signatures. For practical reasons, we recovered the second derivative of the volume injection function, rather than the volume injection function itself. We found that this extracted signature was minimum phase. It did not have the same amplitude spectrum as the seismic reflection data as shown in Figure 12. We put the theory at risk using a third shot of mass M 3 = 250 g at every one of our 96 shotpoints, whose corresponding seismogram was predictable from the other two. We performed this experiment within a production environment and have shown that the error between the measured and predicted seismograms is acceptably low, especially at large offsets. Our results were repeatable. We recognize that we do not have the most sophisticated method for solving the equations. An alternative approach could be to formulate the search for a solution as an optimization problem and then apply all the usual optimization algorithms. We believe that there were two principal sources of error in our experiment. The first was the use of groups of geophones, rather than a single geophone at each geophone FIG. 1. (continued)

6 1188 Ziolkowski and Bokhorst station. Groups were used because surface waves are a problem in this area; we also wanted to keep the conventional seismic data acquisition method as little changed as possible. The second was the use of a square configuration for our shots: we should have arranged our shots to have been cross-line; this would have minimized timing errors between the different shots, and it would have been no more difficult to do in practice. A practical point is that the theory assumes the source to be embedded in a homogeneous isotropic medium. This is never the case. There are always inhomogeneities and anisotropic effects present that probably cause the source to generate shear waves as well as pure compressional waves. If this is the case, we would not expect the surface waves to scale. This is a point for further investigation with three-component geophones. In conventional seismic data acquisition with dynamite, using groups and attenuating the surface waves with velocity filters, we would expect this method to be just as successful as it was here. If there is good shot-to-shot repeatability, the second shot is not needed at every shotpoint. It would be needed only for calibration whenever the conditions at the shot changed. This method could therefore be implemented in practice with a very modest increase in data acquisition costs. C) ACKNOWLEDGMENTS We dedicate this paper to Geoff King, who in 1978 recognized that the theory could be put at risk in an experiment therefore qualifying it to be considered a scientific theory, according to Sir Karl Popper s demarcation criterion (Popper, 1959). The experiment was performed to the highest standards by Prakla-Seismos AG. We especially thank the following people: Messrs. E. Tiegs and N. Rossmanek of Prakla-Seismos for their help and cooperation; Klaas-Jan Koster of Amoco Production Research Center, formerly of the section of Applied Geophysics in the Faculty of Mining and Petroleum Engineering, who helped plan the experiment and gave enthusiastic day-to-day attention to the seismic data acquisition; and Mike Bailey of Exploration Consultants Limited, who carried out the quality control on behalf of the project. We thank Neil Goulty and Guy Drijkoningen for their constructive comments on the manuscript. The paper was written mostly on a portable Macintosh computer provided by a $25,000 grant from Schlumberger-Doll Research, Ridgefield, Connecticut. The experiment presented here was undertaken as an integral part of a three-year research project entitled Inversion of (text continued on p. 1194) FIG. 1. (continued)

7 FIG. 2. Amplitude spectra of the raw field data for the (a) 125 g, (b) 250 g, and (c) 500 g charges at trace 100. The amplitude scale is the same for all plots. FIG. 3. Shot record showing data after velocity-filtering in the frequency-wavenumber domain with (a) a 125 g dynamite charge, (b) a 250 g dynamite charge, and (c) a 500 g dynamite charge. The distance between traces is 10 m; the timing lines are at 1 s intervals; the shot was at trace 120.

8 1190 Ziolkowski and Bokhorst FIG. 3. (continued)

9 FIG. 3. (continued) FIG. 5. Extracted source signature for the 125 g shot using a bandwidth of Hz. The dimension of the amplitude scale is volume per unit time per unit time (e.g., the true amplitude is not obtained with this method. FIG. 4. (a) Amplitude and (b) phase spectrum of the spectral ratio averaged over 240 traces. The amplitude scale is dimensionless.

10 1192 Ziolkowski and Bokhorst EXTRACTED SIGNATURE FIG. 6. Extracted source signature: (a) time signal following removal of the noise after the third zero-crossing, (b) amplitude spectrum, and (c) phase spectrum for the 125 g shot. The dimension of the amplitude scale is volume per unit time per unit time (e.g., the true amplitude is not obtained with this method. FIG. 7. Scaled source signatures for the (a) 125 g, (b) 250 g, and (c) 500 g data calculated from the extracted 125 g signature using the scaling law. The amplitudes of the individual plots have been normalized, but the amplitudes relative to the scales are correct. FIG. 8. Wiener filters at s sampling intervals that shape the (a) 125 g and the (b) 500 g data into the 250 g estimated data. The amplitude scale is dimensionless.

11 FIG. 9. (a) Original 250 g velocity-filtered shot record, (b) estimate (from the 125 g record) of the 250 g velocity-filtered shot record, and (c) difference between the original 250 g shot record and the estimated 250 g (every fifth trace plotted). The distance between the plotted traces is 50 m; the timing lines are at 1 s intervals.

12 1194 Ziolkowski and Bokhorst FIG. 10. Energy in the difference between the true shot record and the estimate as a percentage of the energy in the true shot record, for every trace, for (a) the prediction from the 125 g data, and (b) the prediction from the 500 g data. The shot was at trace 120. FIG. 12. Amplitude spectrum of the earth impulse response after deconvolution for source over the chosen Hz bandwidth. The dimensions of the amplitude scale are time per unit area (e.g., s). The true amplitude cannot be determined with this method. FIG. 11. Extracted signatures for the 125 g data for every fifth shot position showing good shot-to-shot repeatability, plotted on the same scale. The timing lines are at 7.5 ms intervals. Land Seismic Data, funded by the European Community under contract number TH /88. REFERENCES Gaskell, T. F., 1956, The relation between size of charge and amplitude of refracted wave: Geophys. Prosp., 4, no. 2, Levinson, N., 1947, The Wiener RMS (root mean square) error criterion in filter design and prediction, Appendix B of Wiener, N., 1949, Time series: The MIT Press. O Brien, P. N. S., 1957, The relationship between seismic amplitude and weight of charge: Geophys. Prosp., 5, no. 3, Popper, K. R., 1959, The logic of scientific discovery: Hutchinson. Rodean, H. C., 1971, Nuclear explosion seismology,: U.S. Atomic Energy Commission, Division of Technical Information; available as TID from National Technical Information Service, U.S. Dept. of Commerce. Ziolkowski, A. M., 1982, Further thoughts on Popperian geophysics-the example of deconvolution: Geophys. Prosp., 30, , Determination of the signature of a dynamite source using source scaling; Part 1: Theory: Geophysics, 58, 1174-l 182. Ziolkowski, A. M., and Holtslag, J. A., 1984, The Delft air-gun experiment: II. Results of the scaling law method: 54th Ann. Internat. Mtg., Soc. Explo. Geophys., Expanded Abstracts, Ziolkowski, A. M., and Lerwill, W. E., 1979, A simple approach to high-resolution seismic profiling for coal: Geophys. Prosp., 27, 2, Ziolkowski, A. M., Lerwill, W. E., March, D. W., and Peardon, L. G., 1980, Wavelet deconvolution using a source scaling law: Geophys. Prosp., 28,