Multimode surface-wave analysis of near-surface...
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Multimode surface-wave analysis of near-surface data. Michel Cara, Zacharie Duputel, Luis A. Rivera and Georges Herquel Institut de Physique du Globe de Strasbourg, UMR 7516 CNRS and UdS/EOST, Strasbourg, France. email: [email protected] 1 Abstract The near-surface shear-velocity profile can be determined from the inversion of surface-wave dispersion data. Sev- eral methods of analysis have been developed in the past, mostly using the phase information of seismic signals ob- served on linear arrays of sensors. One drawback of these techniques is a possible misidentification between modes and secondary lobes resulting from the limited appearture of the array. Another limitation comes from the contami- nation by random and/or signal generating noise since the whole wavefield is taken into account. In this work, we pro- pose to overcome these limitations by adding a group-delay time information in the analysis. The wavefield is mapped into the group-velocity/phase-velocity (U-c) domain which enables us to improve the dispersion measurements. The observed multimode dispersion data are then inverted in terms of a 1D velocity profile. We show both on synthet- ics and field data that the U-c diagrams greatly facilitate the identification of each modes and that our inversion proce- dure quickly converge to the expected models. 0 5 10 15 0 5 10 15 20 25 30 depth (m) |K V S |,|K V P | (%) 0 5 10 15 0 5 10 15 20 25 30 depth (m) |K V S |,|K V P | (%) 0 5 10 15 0 5 10 15 20 25 30 depth (m) |K ρ |,|K e | (%) 0 5 10 15 0 5 10 15 20 25 30 depth (m) |K ρ |,|K e | (%) ν = 0.1 ν = 0.25 ν = 0.4 a) b) c) d) Figure 1: Kernel sensitivity of phase-velocities at 28Hz computed for model A (Figure 5a-b). K V P , K e (grey lines), K V S and K ρ (dark lines) are displayed for (a) the fundamen- tal mode and (b) the first overtone for different Poisson ra- tios ν ranging from 0.1 to 0.4. 40 50 60 70 80 90 100 110 120 130 0.0 0.5 1.0 Offset (m) Time (s) 40 50 60 70 80 90 100 110 120 130 0.0 0.5 1.0 Offset (m) Time (s) a) b) Figure 2: Synthetic seismograms computed by summing the first six Rayleigh modes (Herrmann, 2006). (a) and (b) correspond to models A and B shown on Figure 5. -50 -10 c (m/s) U (m/s) 0 1 2 543 412 281 775 382 251 120 c (m/s) U (m/s) 0 1 1114 672 451 1074 632 411 190 -50 -10 0 20 40 60 200 400 600 800 1000 1200 c (m/s) Freq. (Hz) 0 20 40 60 100 200 300 400 500 600 700 c (m/s) Freq. (Hz) 800 r (k) ^ r (k) ^ a) b) Figure 3: f-c Diagrams (left) and U-c Diagrams (right) corre- sponding to the synthetics on Figure 2. Dotted black lines: frequencies used to calculate the U-c diagrams. Black dots: theoretical values. Crosses: location of peak maxima. 2 Dispersion measurements : U-c Diagrams Multi-mode signals recorded by an array of N sensors are stacked in order to reinforce the individual modes by con- structive interference : G ω 0 (k,ω )= N −1 n=0 w n S ω 0 (x n ,ω ) exp[−iK (k,ω )x n ], where S ω 0 (x n ,ω ) is the time Fourier transform of a record observed at epicentral distance x n and filtered around the circular frequency ω 0 . The last term in this expression is the phase-shift filter with K (k,ω ) defined as : K (k,ω )= k + ω −ω 0 U c where k is the wavenumber and U c is the central group- velocity of the multi-mode wave packet. We call “U-c diagram” the modulus of the inverse Fourier transform g ω 0 (k,t) of G ω 0 (k,ω ). It will exhibit peaks at group group-velocity U m (ω 0 ) and phase-velocity c m (ω 0 ), related to each surface-wave mode m (Cara 1976, Duputel et al. 2009). The series of diagrams plotted at different frequencies ω 0 then allows the analyst to retrieve the fundamental and the higher-mode dispersion curves. 3 Inversion for 1-D velocity profile We consider a thin layered model of the subsurface where we invert for V S only since V P and ρ have a small influence on the phase-velocities (Figure 1). This is a non-linear problem which is solved using a two-step inversion scheme (Duputel et al., 2009). The first step is a pre-inversion step providing the a priori model which is necessary to run the second step based on a quasi-Newton algorithm. Two al- ternative approaches were followed to choose the a pri- ori information on the model parameters: 1) An empirical conversion of the fundamental mode phase velocity into V s and 2) extraction of an optimum a priori information from a model-library made of 5000 models. 4 Synthetic tests The fundamental mode and the first higher mode are clearly visible on the U-c diagrams (Figure 3). On Figure 4, the fundamental mode phase-velocity measurements fit well with the theoretical values, even for frequencies lower than 15Hz where the classical f-k method fails. On Figure 4a, we note a phase-velocity misfit which is greater for the first overtone than for the fundamental mode. This is probably due to the low amplitude of the first overtone which is clearly visible on the U-c diagram (Figure 3a(right)), but merely detectable on the f-c diagram (Figure 3a(left)). Figure 5 depicts the inversion results corresponding to the dispersion data measured on U-c Diagrams. c (m/s) Freq. (Hz) 0 20 40 60 200 400 600 800 0 20 40 60 200 400 600 800 U (m/s) Freq. (Hz) c (m/s) Freq. (Hz) 0 20 40 60 100 200 300 400 500 0 20 40 60 100 200 300 400 500 U (m/s) Freq. (Hz) RMS = 37 RMS = 14 RMS = 19 RMS = 28 RMS = 8 RMS = 16 a) b) Figure 4: Measured and Theoretical phase-velocities c and group-velocities U for model A (a) and model B (b) (Fig- ure 5). Amplitude contourmap of f-c diagrams are dis- played. Solid lines: theoretical values, black crosses: val- ues measured on the U-c diagrams, black dots: phase- velocities measured on the f-c diagrams. 400 600 800 0 5 10 15 20 25 30 Depth (m) V S (m/s) 246810 2 4 6 8 10 Layer Layers 0 5 0 5 10 Misfit (%) Iterations 400 600 800 0 5 10 15 20 25 30 Depth (m) V S (m/s) 2 4 6 81012 2 4 6 8 10 12 layer Layers 0 2 4 0 5 Misfit (%) Iterations 200 400 600 0 5 10 15 20 25 30 Depth (m) V S (m/s) 2 4 6 81012 2 4 6 8 10 12 layer Layers 0 5 0 5 10 Misfit (%) Iterations 200 400 600 0 5 10 15 20 25 30 Depth (m) V S (m/s) 246810 2 4 6 8 10 layer Layer 0 0.2 0.4 0 5 0 10 20 Misfit (%) Iterations a) b) c) d) Figure 5: Inversion results from measurements on synthetic U-c diagrams for the model A (a-b) and the model B (c-d). Resolution matrices and data misfits are shown. (a) and (c): a priori empirical conversion of c into V s . (b) and (d): use of a model-library. Plain black line: exact model. Doted black line: a priori model. Grey line: inverted model. 5 Application to field data Dispersion curves related to the records displayed in Fig- ure 6 are estimated by using the U-c diagram technique. Note on Figure 7 that only the fundamental mode and the first overtone are well excited. Even if the two initial models differ, the final inverted mod- els are remarkably similar (Figure 8) and a good fit is ob- served between observed and predicted phase-velocities (Figure 9). This emphasizes the robustness of this solution that is confirmed by the resolution matrices which show a good overall resolution. Three sets of layers can clearly be distinguished on Fig- ure 8. They correspond to the local soil structure which is well known from previous seismic refraction investigations (e.g. Bano et al., 2002). A geological interpretion of is given in Table 1. 10 15 20 25 30 35 40 45 50 55 60 0 0.1 0.2 0.3 0.4 0.5 Offset (m) Time (s ) Figure 6: The shot gather obtained in Riedseltz, Alsace (France), by using a sledgehammer as the energy source. e(m) 〈V s 〉 (m/s) lithology 2.20 178 Undifferentiated loess (Wurm age) 14.14 330 Pliocene sands ∞ 507 Oligocene marls Table 1: Interpretation of the inverted shear velocity profiles displayed on Figure 8. e: layer thickness, 〈V s 〉: averaged shear-velocity in each layer. The lithology can be directly observed on a quarry which is next to the seismic profile. c (m/s) U (m/s) 808 432 244 150 802 332 144 50 c (m/s) Freq. (Hz) -25 -15 r (k) 0 1 0 20 40 60 80 100 200 300 400 500 600 700 800 900 1000 ^ b) a) Figure 7: f-c Diagram (left) and U-c Diagram (right) ob- tained by stacking the records shown in Figure 6. Dot- ted black line: frequency used to calculate the U-c diagram (20Hz). See caption of Figure 3. 200 400 600 800 0 5 10 15 20 Depth (m) V S (m/s) 2 4 6 8 10 12 2 4 6 8 10 12 Layer Layer 2 4 6 8 2 4 6 8 Layer Layer 0 5 0 5 10 Misfit (%) Iterations 0 5 0 20 40 Misfit (%) Iterations 0 0.2 0.4 0 0.2 0.4 a) b) d) c) e) Figure 8: Inversion results from the phase-velocities measured on synthetic U-c diagrams. In (a), plain lines: inverted models, dotted lines: the a priori models. Black lines correspond to inversions based on the a priori conversion of c into V s associed with the resolution matrix in (b) and the data missfit in (c). Grey lines correspond to the use of a model-library (the same as for Figure 5). The corresponding resolution matrix is shown in (d) and data missfit in (e). 6 Conclusions We show that the discrimination between the fundamental and the first higher mode is clearly made easier by using U-c diagrams instead of classical f-k analysis. The high quality dispersion data thus obtained can be processed to infer the near-surface shear velocity structure with a better depth resolution than when using fundamental mode dis- persion only. Our simple inversion procedure converges rapidly to ac- ceptable solutions both for synthetic and actual data. The fact that the inverted V s profiles depend weakly on the a priori model gives us confidence in the robustness of the solutions. 0 20 40 60 200 300 400 500 c (m/s) Freq. (Hz) 0 20 40 60 200 300 400 500 c (m/s) Freq. (Hz) a) b) Figure 9: Plain black line: measured phase-velocities. Doted black line: data predicted from the a priori model. Grey line: data predicted from the inverted model. (a) A priori conversion of c into V s . (b), Use of the model-library. The source-receiver configuration and the test area which we consider in this paper corresponds to a cheap, classical experiment in near-surface prospecting. This limits the higher-mode content in the observed surface-wave signals. A simple but more expensive configuration to increase the resolution of the method at depth could, for example, consist of using a buried instead of a surficical source. Acknowledgement We thank Jean-Jacques L´ ev` eque and Alessia Maggi for helpful comments and the students of the EOST school of engineers for the data acquisition. We wish to acknowledge Robert B. Herrmann to provide programs for the modal superposition of surface waves. We are also grateful to Jean-Claude Gress, St´ ephanie Guerbet, Julien Bois, Olivier Loeffler and Philippe Gorsy which initiated this project. References Bano, M., J. Edel, G. Herquel, and EPGS class 2001-2002, Geophysical investigation of a recent shallow fault, The Leading Edge, 21(7), 648–650, 2002. Cara, M., Observation d’ondes S a de type SH , Pure and Applied Geophysics, 114, 141–157, 1976. Duputel, Z., M. Cara, L. Rivera, and G. Herquel, Improving the analysis and inversion of multimode Rayleigh-wave dis- persion by using group-delay time information observed on arrays of high frequency sensors, Accepted in Geophysics, 2009. Herrmann, R. B., Computer Programs in Seismology, Saint Louis University., 2006.
Transcript of Multimode surface-wave analysis of near-surface...
Multimode surface-wave analysis of near-surface data.Michel Cara, Zacharie Duputel, Luis A. Rivera and Georges Herquel
Institut de Physique du Globe de Strasbourg, UMR 7516 CNRS and UdS/EOST, Strasbourg, France.
email: [email protected]
1 AbstractThe near-surface shear-velocity profile can be determinedfrom the inversion of surface-wave dispersion data. Sev-eral methods of analysis have been developed in the past,mostly using the phase information of seismic signals ob-served on linear arrays of sensors. One drawback of thesetechniques is a possible misidentification between modesand secondary lobes resulting from the limited appeartureof the array. Another limitation comes from the contami-nation by random and/or signal generating noise since thewhole wavefield is taken into account. In this work, we pro-pose to overcome these limitations by adding a group-delaytime information in the analysis. The wavefield is mappedinto the group-velocity/phase-velocity (U-c) domain whichenables us to improve the dispersion measurements. Theobserved multimode dispersion data are then inverted interms of a 1D velocity profile. We show both on synthet-ics and field data that the U-c diagrams greatly facilitate theidentification of each modes and that our inversion proce-dure quickly converge to the expected models.
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Figure 1: Kernel sensitivity of phase-velocities at 28Hzcomputed for model A (Figure 5a-b). KVP
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Figure 2: Synthetic seismograms computed by summingthe first six Rayleigh modes (Herrmann, 2006). (a) and (b)correspond to models A and B shown on Figure 5.
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2 Dispersion measurements : U-c DiagramsMulti-mode signals recorded by an array of N sensors arestacked in order to reinforce the individual modes by con-structive interference :
Gω0(k, ω) =
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wnSω0(xn, ω) exp[−iK(k, ω)xn],
where Sω0(xn, ω) is the time Fourier transform of a record
observed at epicentral distance xn and filtered around thecircular frequency ω0. The last term in this expression is thephase-shift filter with K(k, ω) defined as : K(k, ω) = k+ ω−ω0
Uc
where k is the wavenumber and Uc is the central group-velocity of the multi-mode wave packet.We call “U-c diagram” the modulus of the inverse Fouriertransform gω0
(k, t) of Gω0(k, ω). It will exhibit peaks at group
group-velocity Um(ω0) and phase-velocity cm(ω0), related toeach surface-wave mode m (Cara 1976, Duputel et al.2009).The series of diagrams plotted at different frequencies ω0
then allows the analyst to retrieve the fundamental and thehigher-mode dispersion curves.
3 Inversion for 1-D velocity profileWe consider a thin layered model of the subsurface wherewe invert for VS only since VP and ρ have a small influenceon the phase-velocities (Figure 1 ). This is a non-linearproblem which is solved using a two-step inversion scheme(Duputel et al., 2009). The first step is a pre-inversion stepproviding the a priori model which is necessary to run thesecond step based on a quasi-Newton algorithm. Two al-ternative approaches were followed to choose the a pri-ori information on the model parameters: 1) An empiricalconversion of the fundamental mode phase velocity into Vs
and 2) extraction of an optimum a priori information from amodel-library made of 5000 models.
4 Synthetic testsThe fundamental mode and the first higher mode are clearly visible on the U-c diagrams (Figure 3 ). On Figure 4 , thefundamental mode phase-velocity measurements fit well with the theoretical values, even for frequencies lower than15Hz where the classical f-k method fails. On Figure 4 a, we note a phase-velocity misfit which is greater for the firstovertone than for the fundamental mode. This is probably due to the low amplitude of the first overtone which is clearlyvisible on the U-c diagram (Figure 3 a(right)), but merely detectable on the f-c diagram (Figure 3 a(left)).Figure 5 depicts the inversion results corresponding to the dispersion data measured on U-c Diagrams.
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Figure 5: Inversion results from measurements on syntheticU-c diagrams for the model A (a-b) and the model B (c-d).Resolution matrices and data misfits are shown. (a) and(c): a priori empirical conversion of c into Vs. (b) and (d):use of a model-library. Plain black line: exact model. Dotedblack line: a priori model. Grey line: inverted model.
5 Application to field data
Dispersion curves related to the records displayed in Fig-ure 6 are estimated by using the U-c diagram technique.Note on Figure 7 that only the fundamental mode and thefirst overtone are well excited.Even if the two initial models differ, the final inverted mod-els are remarkably similar (Figure 8 ) and a good fit is ob-served between observed and predicted phase-velocities(Figure 9 ). This emphasizes the robustness of this solutionthat is confirmed by the resolution matrices which show agood overall resolution.Three sets of layers can clearly be distinguished on Fig-ure 8 . They correspond to the local soil structure which iswell known from previous seismic refraction investigations(e.g. Bano et al., 2002). A geological interpretion of is givenin Table 1.
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Figure 6: The shot gather obtained in Riedseltz, Alsace(France), by using a sledgehammer as the energy source.
e(m) 〈Vs〉 (m/s) lithology2.20 178 Undifferentiated loess (Wurm age)14.14 330 Pliocene sands∞ 507 Oligocene marls
Table 1: Interpretation of the inverted shear velocity profilesdisplayed on Figure 8. e: layer thickness, 〈Vs〉: averagedshear-velocity in each layer. The lithology can be directlyobserved on a quarry which is next to the seismic profile.
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Figure 8: Inversion results from the phase-velocities measured on synthetic U-c diagrams. In (a), plain lines: invertedmodels, dotted lines: the a priori models. Black lines correspond to inversions based on the a priori conversion of c intoVs associed with the resolution matrix in (b) and the data missfit in (c). Grey lines correspond to the use of a model-library(the same as for Figure 5). The corresponding resolution matrix is shown in (d) and data missfit in (e).
6 ConclusionsWe show that the discrimination between the fundamentaland the first higher mode is clearly made easier by usingU-c diagrams instead of classical f-k analysis. The highquality dispersion data thus obtained can be processed toinfer the near-surface shear velocity structure with a betterdepth resolution than when using fundamental mode dis-persion only.Our simple inversion procedure converges rapidly to ac-ceptable solutions both for synthetic and actual data. Thefact that the inverted Vs profiles depend weakly on the apriori model gives us confidence in the robustness of thesolutions.
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Figure 9: Plain black line: measured phase-velocities.Doted black line: data predicted from the a priori model.Grey line: data predicted from the inverted model. (a) Apriori conversion of c into Vs. (b), Use of the model-library.
The source-receiver configuration and the test area which we consider in this paper corresponds to a cheap, classicalexperiment in near-surface prospecting. This limits the higher-mode content in the observed surface-wave signals. Asimple but more expensive configuration to increase the resolution of the method at depth could, for example, consist ofusing a buried instead of a surficical source.
AcknowledgementWe thank Jean-Jacques Leveque and Alessia Maggi for helpful comments and the students of the EOST school ofengineers for the data acquisition. We wish to acknowledge Robert B. Herrmann to provide programs for the modalsuperposition of surface waves. We are also grateful to Jean-Claude Gress, Stephanie Guerbet, Julien Bois, OlivierLoeffler and Philippe Gorsy which initiated this project.
ReferencesBano, M., J. Edel, G. Herquel, and EPGS class 2001-2002, Geophysical investigation of a recent shallow fault, The
Leading Edge, 21(7), 648–650, 2002.
Cara, M., Observation d’ondes Sa de type SH, Pure and Applied Geophysics, 114, 141–157, 1976.
Duputel, Z., M. Cara, L. Rivera, and G. Herquel, Improving the analysis and inversion of multimode Rayleigh-wave dis-persion by using group-delay time information observed on arrays of high frequency sensors, Accepted in Geophysics,2009.
Herrmann, R. B., Computer Programs in Seismology, Saint Louis University., 2006.
