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Geophysical Journal International
, Volume 212 (1) – Jan 1, 2018

13 pages

/lp/ou_press/reciprocal-data-acquisition-and-subsequent-waveform-matching-for-zb9dakuKM2

- Publisher
- The Royal Astronomical Society
- Copyright
- © The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.
- ISSN
- 0956-540X
- eISSN
- 1365-246X
- D.O.I.
- 10.1093/gji/ggx374
- Publisher site
- See Article on Publisher Site

SUMMARY In this study, experiments on seismic reciprocity utilizing various acquisition tools, including airguns, vibrators, dynamite, hydrophones and geophones, in combination are conducted on onshore–offshore transects that crossed the coastline. Because airguns, vibrators and dynamite generate different wavelets (i.e. have different source functions) and because hydrophones and geophones observe different physical field quantities, waveform matching processes must be applied to the recorded time-series to perform integrated seismic profiling. For the best integration of data acquired with onshore and offshore instruments, processes which compensate the instrumental differences with no loss of bandwidth should be employed. We present a combined onshore–offshore acquisition system and associated signal processing procedures for obtaining broad-band seismograms that support the comparison of reciprocal data. In the first of two experiments, we considered the data acquired by using two receiver–source pairs, one consisting of an onshore geophone and offshore airgun and the other consisting of an offshore hydrophone and onshore vibrator. The two waveforms were found to be in very good agreement after applying data processing including signature deconvolution. In the second experiment, we succeeded in matching time-series data from a pair consisting of an onshore geophone and offshore airgun and a pair consisting of an offshore hydrophone and an onshore dynamite source, by tracing subsurface dilatation through the use of a buried array of multicomponent geophones and the moving least squares method. Both signature deconvolution and the moving least squares method preserved the bandwidth of each recorded seismogram and were noise-independent. This study confirms that seismic reciprocity for onshore–offshore seismic profiling can be achieved with a combined survey system using all required instrument types and modern data processing preserving recorded bandwidth. Image processing; Time-series analysis, Controlled source seismology, Seismic instruments, Theoretical seismology, Wave propagation 1 INTRODUCTION Seismic profiling is increasingly essential in investigations of crustal-scale structures. The targets of these efforts, such as earthquake source faults or the Moho, reach depths greater than 10 km. Deep-target seismic surveys typically depend on imaging of wide-angle reflections or traveltime tomography, techniques that entail long survey lines that often cross coastlines (Lafond & Levander 1995; Stern et al.2002; Nazareth & Clayton 2003; Okaya et al.2003; Kodaira et al.2005; Stankiewicz et al.2008). Because human activity is highly concentrated near coastlines, precise information about the subsurface in these areas is required for disaster mitigation. However, human activities produce significant noise in seismic surveys and may preclude ideal data sampling at the source and receiver points. The principle of seismic reciprocity, which holds that the Green's function at point A from point B is equal to that at B from A, underlies flexible survey designs and effective wavefield reconstruction across the ocean–land transition without requiring complicated phase compensation and other manipulations. In addition, because wave propagation is significantly affected by conditions in the shallow subsurface, reciprocal data are valuable for the compensation of source and receiver perturbations (Vermeer 1991; van Vossen et al.2006). The investigations of deep targets under the coastal areas require data derived from sources and receivers on both sides of the coastline (Fig. 1). The different sources used—airguns at sea, vibrators and dynamite on land—have different source functions (generating different wavelets), and the different receivers used—hydrophones at sea, geophones on land—evaluate different physical field quantities (i.e. hydrophones evaluate pressure and geophones evaluate particle velocity). Because the acquisition styles are different, integrated processing up to full stacking is required for seismogram waveform matching. Seismic reciprocity provides a useful mechanism to integrate the various types of data, but a question remains whether this has been experimentally confirmed. Figure 1. View largeDownload slide Illustration of a seismic survey across a coastline using an airgun as the marine source and a vibrator truck and dynamite as the land sources. Beneath the diagram are examples of common-shot data from a land and marine shot acquired near the experiment site. Hydrophones are the marine receivers and geophones are the land receivers. The green and blue lines denote the refracted and wide-angle reflected waves, respectively. Studies of deep subsurface features under a coastal area require a long survey line that crosses the coastline. Figure 1. View largeDownload slide Illustration of a seismic survey across a coastline using an airgun as the marine source and a vibrator truck and dynamite as the land sources. Beneath the diagram are examples of common-shot data from a land and marine shot acquired near the experiment site. Hydrophones are the marine receivers and geophones are the land receivers. The green and blue lines denote the refracted and wide-angle reflected waves, respectively. Studies of deep subsurface features under a coastal area require a long survey line that crosses the coastline. Fenati & Rocca (1984) successfully recorded reciprocal data, but because both their sources (dynamite and vibrators) and receivers (geophones) were on land, they did not need to use sophisticated data processing. Analogous experimental results can be found on the Stanford Exploration Project website (http://sepwww.stanford.edu/sep/prof/bei/sg/paper_html/node7.html). Having been verified under various conditions, seismic reciprocity has been theoretically confirmed for the case of acquisition across a coastline (Dellinger & Nolte 1997; Arntsen & Carcione 2000; Eisner & Clayton 2001; Aki & Richards 2002; Wapenaar et al.2004). It remains for a practical data acquisition system and processing method to be established for this case. A transfer function (e.g. Kennett & Fichtner 2012), describing a modified division or cross-correlation between recorded seismograms and a reference record, can be readily applied to data sets recorded by various sources and receivers. However, waveform matching processes using a transfer function are often restricted by the available bandwidth, which is determined by the combination of source and receiver with the narrowest bandwidth. Background noise also reduces the accuracy of the transfer function. If the source function is derived from the data set acquired through a seismic survey, signature deconvolution can be applied to seismograms to preserve the bandwidth of the recorded data (e.g. Yilmaz 2001). Since the signature deconvolution is independently applied to each of onshore and offshore record, we can explicitly confirm whether the reciprocal data have been obtained or not. Ground checking of reciprocity is quite important, both theoretically and practically, for the quality control of instrumental designs and installation protocols. Although reciprocity is commonly accepted implicitly in current practice, few relevant experiments have assessed reciprocity in a combined onshore–offshore acquisition system. This study carried out two experiments on reciprocity in real cross-coastal settings in Japan. The first compared the recorded signal at an onshore geophone propagated from the offshore airgun versus the signal at an offshore hydrophone propagated from an onshore vibrator. The second compared the record from an onshore geophone–offshore airgun pair with the record from an offshore hydrophone–onshore dynamite pair. 2 EXPERIMENT I: ONSHORE VIBRATOR AND OFFSHORE AIRGUN 2.1 Experimental layout This experiment was conducted in Niigata Prefecture facing the Sea of Japan (Fig. 2). We compared the recorded signal at an onshore geophone propagated from an offshore airgun versus the signal at an offshore hydrophone propagated from an onshore vibrator. Our instrumental layout was based on the numerical simulation described in Appendices A and B. A vibrator and a geophone were placed at each of three points, 2100, 2579 and 2762 m from a point where an airgun and a hydrophone were placed. The airgun was 8.0 m below sea level, and the hydrophone was mounted 10.9 m below sea level on an ocean bottom cable; the two points were 5.3 m apart horizontally (Fig. 3). The analogue signals were sampled and digitized at intervals of 4 ms. The detailed instrumental specifications are listed in Table 1 and photographs of the installations are shown in Fig. 3. Figure 2. View largeDownload slide Map of the coast of Niigata Prefecture, Japan, showing the experiment sites. AP, airgun point; VP, vibrator point; HP, hydrophone point; GP, geophone point; DP, dynamite point; MP, multicomponent geophone. Figure 2. View largeDownload slide Map of the coast of Niigata Prefecture, Japan, showing the experiment sites. AP, airgun point; VP, vibrator point; HP, hydrophone point; GP, geophone point; DP, dynamite point; MP, multicomponent geophone. Figure 3. View largeDownload slide Topographic profile (vertically exaggerated) and instrumental scheme for experiment I. Seismic data were acquired for two receiver–source pairs: onshore geophone–offshore airgun (GP–AP) and offshore hydrophone–onshore vibrator (HP–VP). Figure 3. View largeDownload slide Topographic profile (vertically exaggerated) and instrumental scheme for experiment I. Seismic data were acquired for two receiver–source pairs: onshore geophone–offshore airgun (GP–AP) and offshore hydrophone–onshore vibrator (HP–VP). Table 1. Instrumental specifications for onshore–offshore reciprocity experiments. Experiment Instrument Specification I Airgun Number of guns: 14 Total gun volume: 3020 in3 Chamber pressure: 2000 psi Depth: 8 m Vibrator Number of vehicles: 4 Hold-down weight: 18.6 t/vehicle Sweep frequency: 6–40 Hz Sweep length: 20 s Number of sweeps: 12 Hydrophonea Cut-off frequency: 3 Hz Number of sensors: 1 Geophone Natural frequency: 10 Hz Number of sensors: 9b II Airgun Same as experiment I Depth: 16 m Dynamite Volume: 20 kg (10 kg × 2 holes) Depth: 26 m Hydrophone Natural frequency: 10 Hz Number of sensors: 1 Multicomponent geophone Natural frequency: 10 Hz Number of sensors: 1c Experiment Instrument Specification I Airgun Number of guns: 14 Total gun volume: 3020 in3 Chamber pressure: 2000 psi Depth: 8 m Vibrator Number of vehicles: 4 Hold-down weight: 18.6 t/vehicle Sweep frequency: 6–40 Hz Sweep length: 20 s Number of sweeps: 12 Hydrophonea Cut-off frequency: 3 Hz Number of sensors: 1 Geophone Natural frequency: 10 Hz Number of sensors: 9b II Airgun Same as experiment I Depth: 16 m Dynamite Volume: 20 kg (10 kg × 2 holes) Depth: 26 m Hydrophone Natural frequency: 10 Hz Number of sensors: 1 Multicomponent geophone Natural frequency: 10 Hz Number of sensors: 1c aMounted on the ocean-bottom cable. bParallel connection of three groups, each consisting of three sensors connected in series. cFor each ( x, y and z) direction. View Large Table 3. Values of relative and absolute angle corrections. Correction Geophone Azimuth α (°) Dip β (°) Relative angle MP-1 7 3 MP-2 35 5 MP-3 − 25 4 MP-4 − 1 2 MP-5 − 8.5 1 MP-6 − 16.5 4 MP-7 (centre) 0 0 Absolute angle All 1.3 − 15 Correction Geophone Azimuth α (°) Dip β (°) Relative angle MP-1 7 3 MP-2 35 5 MP-3 − 25 4 MP-4 − 1 2 MP-5 − 8.5 1 MP-6 − 16.5 4 MP-7 (centre) 0 0 Absolute angle All 1.3 − 15 View Large In this study, the most important procedure was determining the compensation for the difference in source functions (source signatures). The airgun source signature with the ghost reflection from the sea surface was modelled by the airgun manufacturer, based on the clustered-airguns theory (Laws et al. 1990), where the source signature for a single airgun was experimentally observed. The source function for the vibrator was a linear up-sweep function with a frequency band of 6–40 Hz and a length of 20 s. Because the airgun source signature differed from the vibrator source signature, we applied zero-phase conversion and bandpass filtering to the airgun source signature in order to integrate the two source signatures. After this procedure, the two source signatures were almost identical (Fig. 4). Figure 4. View largeDownload slide Left, source signatures from experiment I: (a) airgun, (b) zero-phased waveform of (a), (c) bandpass filtered (6–40 Hz) waveform of (b), and (d) vibrator. Right, spectra for adjacent waveforms. Solid line, amplitude spectrum; dotted line, unwrapped phase spectrum. Figure 4. View largeDownload slide Left, source signatures from experiment I: (a) airgun, (b) zero-phased waveform of (a), (c) bandpass filtered (6–40 Hz) waveform of (b), and (d) vibrator. Right, spectra for adjacent waveforms. Solid line, amplitude spectrum; dotted line, unwrapped phase spectrum. 2.2 Waveform matching and experimental results In the raw data for the 2100, 2579 and 2762 m offsets, the reciprocal waveforms appeared to coincide moderately well despite the differing source signatures of the airgun and vibrator (top panels of Figs 5a–c). This coincidence may simply reflect the proximity in time (0.0–0.02 s) of the peaks and troughs of the airgun and vibrator signatures (Figs 4a and d). Nevertheless, the two source signatures had certain differences. Figure 5. View largeDownload slide Experiment I results from offsets of (a) 2100, (b) 2579, and (c) 2762 m. Black line, vibrator source data recorded by hydrophone; red line, airgun source data recorded by geophone. Each plot shows the raw data, signature-deconvolved waveforms, filtered signature-deconvolved waveforms, ghost-crossed waveforms, that is, obtained by cross-convolution with the ghosting function, and filtered ghost-crossed waveforms (Left) and spectra (Right). Solid line, amplitude spectrum; dotted line, unwrapped phase spectrum. Outside of vibrator sweep range or the bandpass filter of 6–40 Hz, phase spectra are not plotted due to unstable computation. Figure 5. View largeDownload slide Experiment I results from offsets of (a) 2100, (b) 2579, and (c) 2762 m. Black line, vibrator source data recorded by hydrophone; red line, airgun source data recorded by geophone. Each plot shows the raw data, signature-deconvolved waveforms, filtered signature-deconvolved waveforms, ghost-crossed waveforms, that is, obtained by cross-convolution with the ghosting function, and filtered ghost-crossed waveforms (Left) and spectra (Right). Solid line, amplitude spectrum; dotted line, unwrapped phase spectrum. Outside of vibrator sweep range or the bandpass filter of 6–40 Hz, phase spectra are not plotted due to unstable computation. We computed the amplitude and phase spectra in a time window of 800 ms, of which centre is at a point where the vibrator data observed the maximum amplitude. This time window was chosen since it held a stable signal part in the seismogram. As many notches are seen in the amplitude spectrum of the 2100 m offset data (Fig. 5a), the data are supposed to be composed of not only the direct wave but the reflected waves. Since the notch frequencies for both vibrator and airgun data show a good extent of similarity, we can reasonably expect that the Green's function in the vibrator data would be equal to the function in the airgun data. These notches in the amplitude spectrum, however, become less obvious in the 2762 m offset data (Fig. 5c); this indicates the decay of the signal-to-noise ratio. To mitigate the differences between the source signatures, we carried out signature deconvolution (also called a Wiener or prediction error filter) for the airgun data, converting the phase characteristics to zero-phase characteristics, because vibrator data have zero-phase characteristics. The filter is described as \begin{equation}y = f * x,\end{equation} (1)where x is the observed raw data, y is the data after signature deconvolution, and * denotes convolution. The zero-phase converter (f) is \begin{equation}f = {\left( {{S^T}S} \right)^{ - 1}}{S^T}{S_0},\end{equation} (2)where S is the original source signature (Fig. 4a), S0 is the zero-phase source signature (Fig. 4b), and T denotes the matrix transpose. Further, the zero-phase conversion was applied to the airgun data recorded by geophones. This conversion increased the similarity of the two recorded waveforms (compare the top and middle panels of Figs 5a–c). For more clarity, a 6–40 Hz bandpass filter was applied to both airgun and vibrator waveforms, and suppressed the slight noise seen in the vibrator record of the 2100 m offset and 1.4–1.6 s time window. The frequencies higher than 40 Hz in the airgun records were mostly attenuated with the wave propagation. The zero-phase conversion being applied, the phase spectrum of 6–22 Hz for the airgun data closely overlapped the spectrum for vibrator data, whereas the phase spectrum higher than 22 Hz did not so in case of 2100 and 2579 m-offset data because of the notch frequencies. These notch frequencies of 22.5 and 22.9 Hz were observed in 2100 m offset and 2579 m offset data respectively and produced a huge gap between the unwrapped phase spectra of airgun and vibrator data. Some disagreements (i.e. 40°–80°), detected in the phase spectrum between airgun and vibrator data for all offsets not less than 20 Hz, probably stemmed from a slight error on the vibrator control system, discrepancy between the real and modelled airgun signatures, and effects of the background noise, but we may as well say that the vibrator and airgun were appropriately controlled in practice as we can see the excellent agreement of the seismograms of airgun and vibrator data. The waveforms of the airgun and vibrator data were still slightly different. Considering that this discrepancy may be explained by ghosting arising from the 2.9 m difference between the airgun and hydrophone depths (Fig. 6a), we tried a simple method of compensation (e.g. Soubaras 1996) based on the ghosting function \begin{equation}\delta \left( t \right) + \gamma \delta \left( {t - \Delta t} \right)\!,\end{equation} (3)where δ is the delta function, γ is the reflection coefficient at the sea surface, assumed to be –1, and Δt is the time lag between the upward-moving signal and the downward-moving ghost signal: \begin{equation}\Delta t = \frac{{2H}}{C},\end{equation} (4)where H is the depth of the airgun or hydrophone and C is the wave speed in seawater (1500 m s–1 in this study). The cross convolutions between the waveforms and ghosting functions are given by \begin{equation}{z_1}\left( t \right) = {F^{ - 1}}\left[ {F\left[ {{y_1}\left( t \right)} \right]\left( {1 + {e^{i\omega \Delta {t_2}}}} \right)} \right]\end{equation} (5)and \begin{equation}{z_2}\left( t \right) = {F^{ - 1}}\left[ {F\left[ {{y_2}\left( t \right)} \right]\left( {1 + {e^{i\omega \Delta {t_1}}}} \right)} \right],\end{equation} (6)where y1 and y2 are the signature-deconvolved waveforms (middle panels in Fig. 5) for the offshore airgun and onshore vibrator, respectively, and F and F–1 are the forward and inverse Fourier transforms, respectively. For an airgun depth of 8.0 m and a hydrophone depth of 10.9 m, Δt1 and Δt2 are 10.7 and 14.5 ms, respectively. The ghost reflections of the data emitted from the airgun and the data received by the hydrophone yielded respective amplitude spectra of $$( {1 + {e^{i\omega \Delta {t_1}}}} )$$ and $$( {1 + {e^{i\omega \Delta {t_2}}}} )$$ and notch frequencies of 69 and 93 Hz (Fig. 6b). Although the vibrator sweep range of 6–40 Hz does not include these notch frequencies, some differentials were confirmed in this range, as shown by the double-headed arrows in Fig. 6(b). Figure 6. View largeDownload slide (a) Schematic diagram illustrating ghost reflections for the airgun source and the hydrophone receiver. (b) Amplitude spectra of ghosting functions for the airgun source (dashed line) and the hydrophone receiver (solid line). The double-headed arrows indicate the difference in the reciprocal data caused by the ghost reflections. Figure 6. View largeDownload slide (a) Schematic diagram illustrating ghost reflections for the airgun source and the hydrophone receiver. (b) Amplitude spectra of ghosting functions for the airgun source (dashed line) and the hydrophone receiver (solid line). The double-headed arrows indicate the difference in the reciprocal data caused by the ghost reflections. The waveforms after the ghost-cross method (z1 and z2) showed that in the data from 2100 and 2579 m, the mismatches seen in the signature-deconvolved records were remarkably improved (compare the middle and bottom panels in Figs 5a and b). In addition, comparison of the amplitude spectra before (y1 and y2) and after convolving the ghosting function (z1 and z2) indicates that the ghost-cross method clearly increases the similarity of the two spectra at frequencies less than 20 Hz but not at frequencies greater than 20 Hz, likely because the effect is overpowered by background noise. On the other hand, in the data from 2762 m, the raw data showed greater similarity in their phase characteristics than the signature-deconvolved or ghost-crossed waveforms (Fig. 5c). The mismatches remaining after these waveform matching processes likely stem from the difficulty of attaching the vibrator to the ground, which typically produces a small error in the phase control of the vibrator. A comparison of the reciprocal data shows that matching processes employing signature deconvolution yielded excellent results. Signature deconvolution preserves the bandwidth of each seismogram and is noise-independent. Moreover, convolving ghosting functions is considered effective when the depths of the airgun and hydrophone are different, provided that the ghost-cross method reduces the effective signal near notch frequency. 3 EXPERIMENT II: ONSHORE DYNAMITE AND OFFSHORE AIRGUN 3.1 Experimental layout The site of the second experiment was south of the first experiment (Fig. 2) and included mountains, a setting appropriate for the use of a dynamite source. The dynamite shot point and multicomponent geophones, all buried at a nominal depth of 26.0 m, were 8288 m from the marine airgun shot point and hydrophone receivers, which were both at 16.0 m depth (Fig. 7). The analogue signals were sampled and digitized at intervals of 4 ms. The detailed instrumental specifications are listed in Table 1 and photographs of the installations are shown in Fig. 7. Figure 7. View largeDownload slide Topographic profile (vertically exaggerated) and instrumental scheme for experiment II. AP, airgun point; HP, hydrophone point; DP, dynamite point; MP, multicomponent geophone. The central illustration depicts the buried array of seven multicomponent geophones. Upper right photos show the PVC pipe tools used to insert geophones into boreholes (see the text). Lower right diagram is a map of the buried geophone array showing the weight centre and distance from the weight centre (r). Figure 7. View largeDownload slide Topographic profile (vertically exaggerated) and instrumental scheme for experiment II. AP, airgun point; HP, hydrophone point; DP, dynamite point; MP, multicomponent geophone. The central illustration depicts the buried array of seven multicomponent geophones. Upper right photos show the PVC pipe tools used to insert geophones into boreholes (see the text). Lower right diagram is a map of the buried geophone array showing the weight centre and distance from the weight centre (r). To obtain reciprocal data for the dynamite source, it was necessary to observe the subsurface pressure. Since the subsurface pressure on land cannot be directly observed, we observed subsurface particle velocity and calculated the dilatation which is proportional to the pressure. As an acquisition system suitable for the moving least-squares (MLS) method (Appendix B), we employed a buried array of seven multicomponent geophones MP-1–MP-7 (Fig. 7). The geophones were positioned in their boreholes with the aid of marked PVC sleeves where tick marks on each sleeve were used to record exact depth. A notch on the end of a PVC tool made to hold the geophone, oriented toward magnetic north, was used to determine exact azimuths (Fig. 7). The holding tools and geophones were left in the boreholes after the PVC sleeves were removed. The positions of the seven geophones are listed in Table 2. After the airgun signals were recorded by the geophones, two 10 kg charges of dynamite were simultaneously detonated in boreholes MP-4 and MP-6, and the offshore hydrophone recorded the resulting seismic signal. Table 2. Locations of multicomponent geophones in the buried array relative to the wellhead for geophone MP-7. MP x (east) (m) y (north) (m) z (depth) (m) MP-1 0.6 − 1.2 − 24.8 MP-2 1.0 − 0.45 − 27.05 MP-3 0.75 0.6 − 25.2 MP-4 − 0.5 0.8 − 26.75 MP-5 − 0.9 0.1 − 25.0 MP-6 − 0.65 − 1.1 − 26.6 MP-7 0.0 0.0 − 26.0 MP x (east) (m) y (north) (m) z (depth) (m) MP-1 0.6 − 1.2 − 24.8 MP-2 1.0 − 0.45 − 27.05 MP-3 0.75 0.6 − 25.2 MP-4 − 0.5 0.8 − 26.75 MP-5 − 0.9 0.1 − 25.0 MP-6 − 0.65 − 1.1 − 26.6 MP-7 0.0 0.0 − 26.0 View Large 3.2 Angle corrections for geophones As the boreholes housing the geophones were composed of mudstone, of which hardness was medium, the geophone spike was not expected to penetrate the bottom rock, which means that the geophones were leaning against the sides of the boreholes. We performed a two-step procedure to correct the angle errors resulting from this geophone orientation. First, we applied the maximum correlation method used in vertical seismic profiling surveys (e.g. Zeng & McMechan 2006) to correct the relative angle error, the difference between the central geophone (MP-7) and the other geophones. To determine the correction angles, we found the azimuthal (α) and depth (β) angles for which the cross-correlation between MP-7 and each other geophone reached its maximum value (Table 3). After applying the angle correction, we found that the waveforms of the seven geophones showed high similarity (Fig. 8). The largest correction was an adjustment of 35° in the x direction of geophone MP-2 (Fig. 8a). Figure 8. View largeDownload slide Waveforms and orientation of seven multicomponent geophones (a) before and (b) after relative angle correction. Figure 8. View largeDownload slide Waveforms and orientation of seven multicomponent geophones (a) before and (b) after relative angle correction. Second, we corrected the absolute angles, which are the angle errors from the exact design coordinates x, y, z corresponding to east, north, and depth. The correction angles in the azimuthal directions were obtained from the average of the relative correction angles. The depth angle correction was determined from the hammer test, which imposes a vertical excitation at the top of the central borehole (MP-7) and induces a wavefield with a maximum oscillation amplitude in the vertical direction that can be used to correct the z component of geophone MP-7. The obtained absolute correction angles α and β were 1.3° and –15°, respectively. 3.3 Minimum-phase conversion for airgun data After correcting for the angles of the geophones, we compensated for the difference between the source signatures of the airgun and the dynamite. The source signature of dynamite is unknown, but in practice dynamite is a more powerful and instantaneous source than an airgun. Since the dynamite source signature was often modelled under the shock wave theory (Peet 1960; Ziolkowski 1993; Ziolkowski & Bokhorst 1993), the airgun data recorded by geophones should be tested to convert them to a minimum-phase system using signature deconvolution (i.e. minimum-phase conversion). Even after signature deconvolution was applied, the waveform of an individual geophone is not a truly reciprocal waveform with respect to the dynamite data recorded by a hydrophone. However, we examined the effect of signature matching between the airgun and dynamite sources by utilizing the first-break event. Because the dominant component of the first-break wave is the refracted wave from the deep subsurface, the incidence angle of the first break is close to 90°; thus, the first break in the geophone data has the greatest amplitude in the z component (approximately 3.5 s in Fig. 8). Under the assumption that the first break is approximated by the response of a 1-D elastic body, the dilatation (ε) and the particle displacement in the z direction (uz) have the relationship ε = (uz—uz0)/L, where uz0 is the standard location and L is the volume length. Therefore, ε is proportional to uz. To check the effect of the minimum-phase conversion, we compared the hydrophone data from the dynamite source (black line in Fig. 9) to the z component of the angle-corrected data from geophone MP-7 (red line in Fig. 9). The geophone data must be integrated with respect to time to convert particle velocities to particle displacements. The amplitude of the first-break wave for the geophone data was smaller than that for the hydrophone data, as indicated by the arrow in Fig. 9(a). Figure 9. View largeDownload slide (a) Raw waveforms from the geophone recording an offshore airgun (red line) and from the hydrophone recording an onshore dynamite source (black line) in experiment II. (b) Waveforms after conversion of the geophone data by signature deconvolution to a minimum-phase system. Note that the first-break amplitude of the geophone data (arrows) is now closer to its counterpart in the hydrophone data. Figure 9. View largeDownload slide (a) Raw waveforms from the geophone recording an offshore airgun (red line) and from the hydrophone recording an onshore dynamite source (black line) in experiment II. (b) Waveforms after conversion of the geophone data by signature deconvolution to a minimum-phase system. Note that the first-break amplitude of the geophone data (arrows) is now closer to its counterpart in the hydrophone data. This comparison showed that the first-break event of the airgun data recorded by the z component of the geophone (uz) became closer to the dynamite data recorded by the hydrophone after the minimum-phase conversion, as indicated by the arrows in Fig. 9(b). We concluded that a minimum-phase conversion improves the airgun source signature and should be applied to all of the geophone records to better match the reciprocal waveform in the hydrophone record from the dynamite source data. 3.4 Waveform matching and experimental results We derived the subsurface dilatation from all seven geophone waveforms through the MLS method, applying the weighting function derived in Appendix B (eq. B.6 with n = 7 and r0 = 1.9 m). This dilatation was the final processed waveform of the offshore airgun source as recorded by the geophones (red line in Fig. 10a), which was the reciprocal of the waveform of the onshore dynamite source as recorded by the hydrophone (black line in Fig. 10a). The two waveforms were in very good agreement with each other. The slight mismatches likely stemmed from background noise near the hydrophone (e.g. ship noise). Figure 10. View largeDownload slide Comparison of the dynamite source waveform recorded by the hydrophone (black line) with the airgun source waveform recorded by the geophone array (red line) in experiment II. Data from (a) all geophones, (b) all geophones excluding MP-4, and (c) all geophones excluding MP-6. Figure 10. View largeDownload slide Comparison of the dynamite source waveform recorded by the hydrophone (black line) with the airgun source waveform recorded by the geophone array (red line) in experiment II. Data from (a) all geophones, (b) all geophones excluding MP-4, and (c) all geophones excluding MP-6. Geophones MP-4 and MP-6 were positioned near each other in the x–z plane (Table 2). When the dilatation was obtained from the six geophones other than MP-4 or MP-6 (Figs 10b and c, respectively), the dilatation waveforms were similar to the waveform derived from all seven geophones. Although the result without MP-4 appeared to yield the best result, probably because the deviation of MP-4 was the greatest among the seven geophones (Fig. 7), every dilatation waveform was in good agreement with the hydrophone waveform. 4 DISCUSSION AND CONCLUSIONS We carried out two experiments investigating the seismic reciprocity of onshore and offshore observation points. The first compared the time-series of an offshore airgun (explosive) recorded by an onshore geophone (vertical velocity) and the time-series of an onshore vibrator (vertical force) recorded by an offshore hydrophone (pressure). The second compared the time-series of an offshore airgun (explosive) recorded by an onshore array of multicomponent geophones (dilatation) and the time-series of an onshore dynamite (explosive) source recorded by an offshore hydrophone (pressure). By applying independent data processing to each of onshore and offshore record, we obtained almost identical waveforms in both experiments. Therefore, seismic reciprocity was confirmed for the case of onshore–offshore seismic profiling by a combined survey system. The principle of reciprocity has given flexibility to survey designs for seismic surveys of all kinds, but the ideal processes for waveform matching have not previously been applied to the case of combined onshore–offshore acquisition campaigns. This study demonstrated that waveform matching processes, including signature deconvolution and the MLS method, are efficient because they preserve the bandwidth of each recorded seismogram and are noise-independent. We also showed that the spacing of buried geophones does not need to be exactly regular for success using the MLS method. Although the MLS method yields higher accuracy as more geophones are deployed, it is important for purposes of economy to know the minimum number of geophones needed to ensure data of adequate quality. Our results suggested that dilatations can be computed with sufficient accuracy by the MLS method when six geophones are positioned at random. However, buried geophone array was, in any case, time-consuming and costly system for seismic surveys. As a further consideration, when the data from three source types (an airgun, a vibrator, and dynamite) recorded by two receiver types (a hydrophone and a geophone) are integrated into a full-range shot-gather or receiver-gather between all combinations of source and receiver, the data processing presented in this paper should be chosen for the data integration, rather than the transfer function method, as long as the source function is available. This is because the processes exclusive of the ghost-cross method preserve the recorded bandwidth and are free of noise. The ability to derive broad-band reciprocal data may help improve the geometric continuity and density of survey results by the use of data interpolation using reciprocity, and may also significantly improve subsurface imaging from seismic surveys that cross a coastline. There are several onerous problems in seismic surveys: for instance, it would be obviously costly and time-consuming to bury geophones underground; further, if both a vibrator and dynamite are used on land, then the deviation between surface and subsurface shootings would have to be appropriately corrected. Hence, additional procedures, such as time shift (static interface) and phase shift in the time domain or re-datuming in the depth domain, need to be investigated and developed so as to synthesize the subsurface records from surface records. ACKNOWLEDGEMENTS We thank David Okaya for his encouragement and constructive advice on the manuscript. We also thank Kei Koshigoe for his encouragement and fruitful comments. We are grateful for the support of Japan Petroleum Exploration Co., Ltd. We also thank the crew of JGI, Inc. for the experiment data acquisition. REFERENCES Aki K., Richards P.G., 2002. Quantitative Seismology , 2nd edn, University Science Books. Arntsen B., Carcione J.M., 2000. A new insight into the reciprocity principle, Geophysics , 65( 5), 1604– 1612. Google Scholar CrossRef Search ADS Belytschko T., Lu Y.Y., Gu L., 1994. Element-free Galerkin method, Int. J. Numer. Methods Eng. , 37( 2), 229– 256. Google Scholar CrossRef Search ADS Cerjan C., Kosloff D., Kosloff R., Reshef M., 1985. A nonreflecting boundary condition for discrete acoustic-wave and elastic-wave equations, Geophysics , 50( 4), 705– 708. Google Scholar CrossRef Search ADS Dellinger J., Nolte B., 1997. A crossed-dipole reciprocity ‘paradox’, Leading Edge , 16( 10), 1465– 1471. Google Scholar CrossRef Search ADS Eisner L., Clayton R.W., 2001. A reciprocity method for multiple-source simulations, Bull. seism. Soc. Am. , 91( 3), 553– 560. Google Scholar CrossRef Search ADS Fenati D., Rocca F., 1984. Seismic reciprocity field tests from the Italian Peninsula, Geophysics , 49( 10), 1690– 1700. Google Scholar CrossRef Search ADS Graves R.W., 1996. Simulating seismic wave propagation in 3D elastic media using staggered-grid finite difference, Bull. seism. Soc. Am. , 86( 4), 1091– 1106. Katou M., Matsuoka T., Mikada H., Sanada Y., Ashida Y., 2009. Decomposed element-free Galerkin method compared with finite-difference method, Geophysics , 74( 3), H13– H25. Google Scholar CrossRef Search ADS Kennett B., Fichtner A., 2012. A unified concept for comparison of seismograms using transfer functions, Geophys. J. Int. , 191( 3), 1403– 1416. Kodaira S., Iidaka T., Nakanishi A., Park J.O., Iwasaki T., Kaneda Y., 2005. Onshore-offshore seismic transect from the eastern Nankai Trough to central Japan crossing a zone of the Tokai slow slip event, Earth Planets Space , 57, 943– 959. Google Scholar CrossRef Search ADS Lafond C.F., Levander A., 1995. Migration of wide-aperture onshore-offshore seismic data, central California: seismic images of late stage subduction, J. geophys. Res. , 100( B11), 22 231–22 243. Google Scholar CrossRef Search ADS Laws R.M., Hatton L., Haartsen M., 1990. Computer modelling of clustered airguns, First Break , 8( 9), 331– 338. Levander A.R., 1988. Fourth-order finite-difference P-SV seismograms, Geophysics , 53( 11), 1425– 1436. Google Scholar CrossRef Search ADS Nazareth J.J., Clayton R.W., 2003. Crustal structure of the Borderland-Continent Transition Zone of southern California adjacent to Los Angeles, J. geophys. Res. , 108( B8), 2404, doi:10.1029/2001JB000223. Google Scholar CrossRef Search ADS Okaya D., Stern T., Holbrook S., Avendonk H., Davey F., Henrys S., 2003. Imaging a plate boundary using double-sided onshore-offshore seismic profiling, Leading Edge , 22( 3), 256– 260. Google Scholar CrossRef Search ADS Peet W.E., 1960. A shock wave theory for the generation of the seismic signal around a spherical shot hole, Geophys. Prospect. , 8, 509– 533. Google Scholar CrossRef Search ADS Soubaras R., 1996. Ocean bottom hydrophone and geophone processing, Annual International Meeting, SEG Expanded Abstract , 24– 27. Stankiewicz J., Parsiegla N., Ryberg T., Gohl K., Weckmann U., Trumbull R., Weber M., 2008. Crustal structure of the southern margin of the African continent: Results from geophysical experiments, J. geophys. Res. , 113, B10313, doi:10.10029/2008JB005612. Google Scholar CrossRef Search ADS Stern T., Okaya D., Scherwath M., 2002. Structure and strength of a continental transform from onshore-offshore seismic profiling of South Island, New Zealand, Earth Planets Space , 54, 1011– 1019. Google Scholar CrossRef Search ADS Vermeer G., 1991. Symmetric sampling, Leading Edge , 10( 11), 21– 27. Google Scholar CrossRef Search ADS van Vossen R., Curtis A., Laake A., Trampert J., 2006. Surface-consistent deconvolution using reciprocity and waveform inversion, Geophysics , 71( 2), V19– V30. Google Scholar CrossRef Search ADS Wapenaar K., Slob E., Fokkema J., 2004. Reciprocity and power balance for piecewise continuous media with imperfect interfaces, J. geophys. Res. , 109, B10301, doi:10.1029/2004JB003002. Google Scholar CrossRef Search ADS Yilmaz O., 2001. Seismic Data Analysis, Processing, Inversion, and Interpretation of Seismic Data , Society of Exploration Geophysicists. Google Scholar CrossRef Search ADS Zeng X., McMechan G.A., 2006. Two method for determining geophone orientations from VSP data, Geophysics , 71( 4), V87– V97. Google Scholar CrossRef Search ADS Ziolkowski A., 1993. Determination of the signature of a dynamite source using source scaling, Part 1: Theory, Geophysics , 58( 8), 1174– 1182. Google Scholar CrossRef Search ADS Ziolkowski A., Bokhorst K., 1993. Determination of the signature of a dynamite source using source scaling, Part 2: Experiment, Geophysics , 58( 8), 1183– 1194. Google Scholar CrossRef Search ADS APPENDIX A: FINITE DIFFERENCE SIMULATION We use the staggered-grid finite difference method with fourth-order spatial accuracy by Levander (1988) to introduce the theoretical aspect of seismic reciprocity. The elastic wave equations consist of the stress–strain relationships \begin{equation}\frac{{\partial {\sigma _{xx}}}}{{\partial t}} = \left( {\lambda + 2\mu } \right)\frac{{\partial {v_x}}}{{\partial x}} + \lambda \frac{{\partial {v_z}}}{{\partial z}},\end{equation} (A1) \begin{equation}\frac{{\partial {\sigma _{zz}}}}{{\partial t}} = \lambda \frac{{\partial {v_x}}}{{\partial x}} + \left( {\lambda + 2\mu } \right)\frac{{\partial {v_z}}}{{\partial z}},\end{equation} (A2) \begin{equation}\frac{{\partial {\sigma _{xz}}}}{{\partial t}} = \mu \left( {\frac{{\partial {v_x}}}{{\partial z}} + \frac{{\partial {v_z}}}{{\partial x}}} \right),\end{equation} (A3)and the equations of motion \begin{equation}\frac{{\partial {v_x}}}{{\partial t}} = \frac{1}{\rho }\left( {\frac{{\partial {\sigma _{xx}}}}{{\partial x}} + \frac{{\partial {\sigma _{xz}}}}{{\partial z}}} \right) + {f_x},\end{equation} (A4) \begin{equation}\frac{{\partial {v_z}}}{{\partial t}} = \frac{1}{\rho }\left( {\frac{{\partial {\sigma _{xz}}}}{{\partial x}} + \frac{{\partial {\sigma _{zz}}}}{{\partial z}}} \right) + {f_z},\end{equation} (A5)where λ and μ are Lamé’s parameters, ρ is the density of the medium, σxx and σzz are normal stresses, σxz is shear stress, vx and vz are respectively the x and z components of the particle velocity, and fx and fz are respectively the x and z components of the external acceleration (body force). In the staggered-grid finite difference method, the stress–strain relationships and the equations of motion are updated in alternation. The simulation model is shown in Fig. A1, where the reciprocity of locations A and B is evaluated. The free surface is defined as z = 0, where stresses σzz and σxz are always zero (Levander 1988). The source function is a Ricker wavelet with a peak of 2 Hz, the sampling interval Δt is 0.001 s, the grid size Δx × Δz is 10 m × 10 m, and the total computation space is 60 km (horizontal) × 16 km (vertical) and includes an absorbing boundary layer 0.4 km wide (40 layers) around its periphery except on the free surface. The absorbing boundaries of the damping method of Cerjan et al. (1985) are employed, and the damping factor of the exponential function is 0.012. Figure A1. View largeDownload slide Model used for elastic wave simulations. A and B are onshore and offshore observation points, respectively. Figure A1. View largeDownload slide Model used for elastic wave simulations. A and B are onshore and offshore observation points, respectively. Explosive energy sources (airgun and dynamite) are expressed in terms of additional stress (Graves 1996) as \begin{equation}{\sigma _{ii}}({x_S},{z_S}) = {\sigma _{ii}}({x_S},{z_S}) + M\Delta t,\,\,\,\,i = x,z,\end{equation} (A6)where M is the source function (Pa m–1 s–1), xS and zS are the coordinates of the source position, and the source depth zS is 2Δz = 20 m. Note that the additional σxz is zero. Because the external acceleration fz is located a half-grid spacing below the free surface in the staggered-grid finite difference method, fz cannot strictly express the vibrator source (a vertical force). We then express the stress in terms of the vertical force: \begin{equation}\sigma _{zz}^{}({x_S},0) = \frac{{{F_z}}}{{\Delta x}}.\end{equation} (A7) In the 2-D simulation, the source function fz is expressed in units of N m–1. Fig. A2 shows the vz component for two shot-gathers from an explosive source. This figure shows that P waves are absorbed at the boundaries of the model space. Although the absorbing boundary layers cannot completely remove the artefacts which are the reflections from the sides of the model, such artefacts would not hinder the confirmation of the reciprocity. Significant grid dispersion does not occur in this simulation space since the simulation parameters such as grid spacing are properly set based on our experience (Katou et al.2009) so that a P wave can be accurately computed with distance being equal to or more than 50 wavelengths. Note that a very high frequency artefact is produced from the triple point at (x, z) = (30 km, 2.5 km), and a high-cut filter is applied to remove this noise. Figure A2. View largeDownload slide The vz component of simulated shot-gathers from (a) an onshore explosive source and (b) an offshore explosive source. The P-waves have been adequately absorbed in the boundary layers of the model. Figure A2. View largeDownload slide The vz component of simulated shot-gathers from (a) an onshore explosive source and (b) an offshore explosive source. The P-waves have been adequately absorbed in the boundary layers of the model. APPENDIX B: SIMULATION RESULTS Using the finite difference method described in Appendix A, we performed two types of studies to simulate the field experiments. One of which is the combination of onshore vibrator and offshore airgun, and the other of which is the combination of onshore dynamite and offshore airgun. B1 Onshore vibrator and offshore airgun Because a vibrator system provides a vertical force and an airgun is an isotropic explosive source, these are not considered to be a reciprocal source combination. Furthermore, in typical recording systems, onshore geophones record vertical velocity (or acceleration) and offshore hydrophones record pressure. Therefore, when data from these two source types are recorded across a coastline, the difference between the two source components is counteracted by the difference between the receiver components. Although this interaction between the source (vertical force) and receiver (vertical velocity) was validated by Arntsen & Carcione (2000), here we re-examine this interaction for the case of an onshore–offshore transition. We performed a finite difference simulation and compared two waveforms: the time-series of an onshore vibrator recorded by an offshore hydrophone (A to B) and the time-series of an offshore airgun recorded by an onshore vertical geophone (B to A). We found that the two computed waveforms matched perfectly (Fig. B1). Note that the amplitudes were normalized with respect to their maximum values to compensate for the difference between the source types. Figure B1. View largeDownload slide Simulation results. The solid grey line is the simulated waveform from location A (onshore vibrator source) recorded at location B (offshore hydrophone). The dashed black line is the waveform from location B (offshore airgun source) recorded at location A (onshore vertical-component geophone). Figure B1. View largeDownload slide Simulation results. The solid grey line is the simulated waveform from location A (onshore vibrator source) recorded at location B (offshore hydrophone). The dashed black line is the waveform from location B (offshore airgun source) recorded at location A (onshore vertical-component geophone). B2 Onshore dynamite and offshore airgun Although onshore dynamite and offshore airgun sources are both explosive seismic sources, observation of pressures by an onshore geophone is not typical. Thus, we must propose a special observation system. The relation between pressure P and displacement (ux, uz) is \begin{equation}\varepsilon = \frac{{\partial {u_x}}}{{\partial x}} + \frac{{\partial {u_z}}}{{\partial z}} = \int {\left( {\frac{{\partial {v_x}}}{{\partial x}} + \frac{{\partial {v_z}}}{{\partial z}}} \right){\rm{d}}t} = \frac{P}{\lambda }.\end{equation} (B1)The pressure and dilatation ε are found to satisfy a linear system. In the finite difference simulation, the dilatation can be directly derived from the finite difference operator. Note that displacements are computed by integrating the observed particle velocity with respect to time dt through a rectangular approximation, and the spatial difference between ux and uz due to the staggered-grid arrangement is considered to be negligible. When the computed dilatation waveform at point A from the source at point B (dashed black line of Fig. B2a) is compared to the pressure waveform at point B from the source at point A (grey line of Fig. B2a), these two waveforms match perfectly. Because the waveform of the onshore dynamite recorded by the offshore hydrophone (A to B) and the waveform of the offshore airgun computed by onshore dilatation from geophone records (B to A) match perfectly, applying dilatation to the obtained waveforms is appropriate to ensure reciprocal data observation of a dynamite source. Figure B2. View largeDownload slide Results of simulations when the onshore dilatation is derived from (a) the finite difference operators and (b) MLS inversion. The solid grey line is the simulated waveform from location A (onshore dynamite source) recorded at location B (offshore hydrophone). The dashed black line is the simulated waveform from location B (offshore airgun source) recorded at location A (onshore dilatation). Figure B2. View largeDownload slide Results of simulations when the onshore dilatation is derived from (a) the finite difference operators and (b) MLS inversion. The solid grey line is the simulated waveform from location A (onshore dynamite source) recorded at location B (offshore hydrophone). The dashed black line is the simulated waveform from location B (offshore airgun source) recorded at location A (onshore dilatation). In a real field experiment, it is very difficult to arrange geophones underground with precise spacing to obtain the spatial difference. Therefore, we implemented the MLS method, which can accurately compute elastic wavefields, including dilatation, even if the geophone geometry includes slight irregularities. If five multicomponent geophones are placed around the onshore observation point (xA, zA), then \begin{eqnarray} \left( {{x_1},{z_1}} \right) &=& ({x_{\rm A}}-\Delta x,{z_{\rm A}}-\Delta z),\nonumber\\ \left( {{x_2},{z_2}} \right) &=& ({x_{\rm A}}-\Delta x,{z_{\rm A}}+\Delta z),\nonumber\\ \left( {{x_3},{z_3}} \right) &=& ( {{x_{\rm A}},{z_{\rm A}}}),\nonumber\\ \left( {{x_4},{z_4}} \right) &=& ({x_{\rm A}}+\Delta x,{z_{\rm A}}-\Delta z),\nonumber\\ \left( {{x_5},{z_5}} \right) &=& ({x_{\rm A}}+\Delta x,{z_{\rm A}}+\Delta z). \end{eqnarray} (B2) To apply the MLS method, displacement functions are defined as \begin{equation}{u_x} = {a_1} + {a_2}x + {a_3}z,\end{equation} (B3) \begin{equation}{u_z} = {b_1} + {b_2}x + {b_3}z.\end{equation} (B4) Thus, the dilatation becomes \begin{equation}\varepsilon = {a_2} + {b_3}.\end{equation} (B5) It is important to employ an effective weight function so that the performance of MSL method can be enhanced. Based on the previous work (Katou et al.2009), we adopt the following weight function: \begin{equation}{w_i} = n\frac{{{r_i}}}{{{r_0}}}{\left( {1 - \frac{{{r_i}}}{{{r_0}}}} \right)^{n - 1}} + {\left( {1 - \frac{{{r_i}}}{{{r_0}}}} \right)^n},\end{equation} (B6)where n is natural number, ri is the distance between the location (xi, zi) of the ith geophone and the centre point (xA, zA), and r0 is the radius of the affected region. The value of r0 should be slightly larger than the largest ri (Belytschko et al.1994; Katou et al.2009); hence, we choose r0 = 15 m (=1.5 × Δx). As with r0, n should be set to 4 or more, and n = 7 was thus chosen by a perturbation test. The quantities aj and bj are \begin{equation}{\bf A}\, = {[{{\bf X}^{\rm{T}}}{\bf WX}]^{ - 1}}{{\bf X}^{\rm{T}}}{\bf W}{{\bf U}_x},\end{equation} (B7) \begin{equation}{\bf B}\, = {[{{\bf X}^{\rm{T}}}{\bf WX}]^{ - 1}}{{\bf X}^{\rm{T}}}{\bf W}{{\bf U}_z},\end{equation} (B8)where \begin{equation}{\bf A}\, = {[\begin{array}{c@\quad c@\quad c} {{a_1}}&{{a_2}}&{{a_3}} \end{array}]^{\rm{T}}},\end{equation} (B9) \begin{equation}{\bf B}\, = {[\begin{array}{*{20}{c@\quad c@\quad c}} {{b_1}}&{{b_2}}&{{b_3}} \end{array}]^{\rm{T}}},\end{equation} (B10) \begin{equation}{\bf X}\, = \left[ {\begin{array} {c@\quad c@\quad c} 1&{{x_1}}&{{z_1}}\\ 1&{{x_2}}&{{z_2}}\\ {}& \vdots &{}\\ 1&{{x_5}}&{{z_5}} \end{array}} \right]\,,\end{equation} (B11) \begin{equation}{\rm{diag}}\,[{\bf W}]\, = {[\begin{array}{*{20}{c@\quad c@\quad c@\quad c}} {{w_1}}&{{w_2}}& \cdots &{{w_5}} \end{array}]^{\rm{T}}}.\end{equation} (B12) The operator [ ]T denotes the matrix transpose, and Ux and Uz are the observed displacements for five geophones. Thus, \begin{equation}{{\bf U}_x}\, = {[\begin{array}{*{20}{c@\quad c@\quad c@\quad c}} {{u_{{x_1}}}}&{{u_{{x_2}}}}& \cdots &{{u_{{x_5}}}} \end{array}]^{\rm{T}}}\end{equation} (B13) \begin{equation}{{\bf U}_z}\, = {[\begin{array}{*{20}{c@\quad c@\quad c@\quad c}} {{u_{{z_1}}}}&{{u_{{z_2}}}}& \cdots &{{u_{{z_5}}}} \end{array}]^{\rm{T}}}.\end{equation} (B14) The MLS solution of dilatation, shown by the black dashed line in Fig. B2(b), agrees reasonably well with the reciprocal data, shown by the grey line in Fig. B2(b). The MLS approach is, therefore, reasonable for practical field campaigns as it is very difficult to ensure borehole spacing regular enough to directly calculate the spatial difference. The calculations in Appendices A and B provide the basis for reciprocal data acquisition across a coastline, ensuring that the time-series of an onshore vibrator recorded by an offshore hydrophone is reciprocal to that of an offshore airgun recorded by onshore geophones and that the time-series of an onshore dynamite detonation recorded by offshore hydrophone is reciprocal to that of an offshore airgun recorded by a buried array of multicomponent geophones (Table B1). Table B1. Instruments and physical field quantities required for onshore–offshore reciprocal data acquisition. Experiment Source (parameter) Receiver (parameter) I Vibrator (fz [N]) Hydrophone (P [Pa]) Airgun (P [Pa]) Geophone (vz [m s−1]) II Dynamite (P [Pa]) Hydrophone (P [Pa]) Airgun (P [Pa]) Geophone array (ε [-]) Experiment Source (parameter) Receiver (parameter) I Vibrator (fz [N]) Hydrophone (P [Pa]) Airgun (P [Pa]) Geophone (vz [m s−1]) II Dynamite (P [Pa]) Hydrophone (P [Pa]) Airgun (P [Pa]) Geophone array (ε [-]) View Large © The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society.

Geophysical Journal International – Oxford University Press

**Published: ** Jan 1, 2018

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