A single-channel seismic reflection method for quantifying lateral ...

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32T.M. McGee / Marine Geology 164 (2000) 29–35Fig. 3. Subset of short-offset traces from a profile offshore Costa Rica. This profile has been analyzed using full-waveform inversion by Pecher etal. (1998). The sea floor reflection, the BSR and the first water-layer multiple are indicated.the lower is 449 ms. Reflection coefficients used in thecalculations are 1.0 at the top of the upper layer (tosimulate an air–water interface), 0.2 between thelayers (to simulate a water–sediment interface) and0.5 at the bottom of the lower layer (to simulate aBSR). The magnitudes of the coefficients have beendefined a bit larger than what would usually be realisticin order that the internal multiple between the twointerfaces be clearly visible. Theoretical values of thetrue amplitudes of the synthetic primary reflectionsare calculated to be 0.20 for the sea-floor reflectionand 0.48 for the BSR.The traces of Fig. 1(b)–(e) were created byconvolving Fig. 1(a) with a real seismic source signaturedigitized at sampling intervals of 1, 2, 3, and4 ms, respectively. It may be observed that the shapeof the signature becomes progressively moredegraded as the sampling becomes coarser. Thetrace of Fig. 1(d) is particularly noteworthy in thisregard. The maximum (negative) excursion of thesea-floor reflection appears truncated because no digitalsample happens to coincide with it. The maximum(positive) excursion of the BSR is also distorted, butin a different way because the time between the BSR
T.M. McGee / Marine Geology 164 (2000) 29–35 33excursions are shifted to erroneous times and theiramplitudes are as much as 20% in error.4. Application to real dataFig. 4. Enlarged portion of Fig. 3 plotted as wiggle traces.and the sea-floor reflection is not an integral multipleof the sampling interval. Such differences betweendigital waveforms degrade their coherence and constitutea source of processing error that is entirely due tothe digitizing rate being inadequate.The traces in Fig. 2 display the result of applyingthe spectral division of Eq. (3) and inverse transformationto the traces in Fig. 1. Amplitudes on thetrace of Fig. 2(a) occur at the correct times and arewithin 0.1% of the theoretical values except that theyare negative due to the effect of the water surface.Amplitudes on traces of Fig. 2(b)–(e) describe waveletsthat are approximately symmetrical about theirmaximum excursions. The symmetry results fromboth primary and multiple wavelets having the samestructure, i.e. that of the source signature, and thequotient in Eq. (3) being equivalent to a deconvolutionof that structure. The symmetry degrades atcoarser sampling rates due to reduced coherencebetween numerator and denominator of Eq. (3).When the degradation is negligible, i.e. Fig. 2(b),the maximum excursions occur near the correctonset times of the reflected wavelets and the amplitudesare within a few percent of the theoreticalvalues. When the degradation is substantial, as it isin Fig. 2(d) and (e), the times of the maximumIn order to compare results obtained with real data,the single-channel method described above wasapplied to short-offset traces from a multi-channeldata set previously analyzed by Pecher et al. (1998)using full-waveform inversion.The data were acquired offshore Costa Rica inabout 3 s of water at a location where a BSR occursabout 0.5 s below the sea floor. The source was atuned airgun array and the received signals weredigitized at a 4 ms sampling interval over a totalrecording time of 8 s.A gray-scale image of 57 short-offset traces fromthe data set is shown in Fig. 3. The sea floor reflectionoccurs slightly below 3 s, the BSR at about 3.6 s andthe first water-layer multiple slightly below 6 s.The primary sequence extends from 3 to 6 s,approximately, and the first multiple sequencewould extend from slightly below 6 s to more than9 s had the recording time been that long. Sincecalculation of the spectral quotient of Eq. (3) is greatlysimplified when the numerator and denominator are ofthe same length, one second of zero values wasappended to the data. Undoubtedly, this is a sourceof numerical noise and introduces some error into theresults. In an effort to reduce such noise, data values attimes greater than 7.8 s were linearly tapered tosmooth the transition from non-zero to zero values.An enlarged portion of Fig. 3 is plotted in Fig. 4 aswiggle-traces to illustrate the waveforms thatcomprise the sea floor and BSR reflections. It can beseen that the dominant period of the signal is about25 m. This indicates a dominant frequency of about40 Hz.The data in Fig. 3 were corrected for sphericalspreading (at 1500 m/s which is close to the speedof propagation in water) and the spectral quotient ofEq. (3) computed after appending zeros to each trace.An enlarged portion of the output, Fig. 5, illustratesthe waveforms from which true amplitudes weremeasured. The dominant period at the sea floor andthe BSR remains 25 ms, indicating that the deconvolutionwas not able to collapse the waveform to any
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