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1 Nonlinear scale separation and misfit configuration of envelope inversion Jingrui Luo, * and Ru-Shan Wu, University of California at Santa Cruz, Xi an Jiaotong University Summary We first show the scale separation property of envelope inversion. The nonlinear envelope extraction provides a nonlinear scale separation of the information contained in the waveform data, which can reduce the local minima in the misfit function and can be used to recover the largescale component of the model. The misfit configuration tests show that the misfit for envelope data has no local minima. Also, we further show the noise resistant property of this method. Numerical examples show that envelope inversion is to some extent resistant to seismic interference noise. Introduction Several approaches have been developed to overcome the problem of local minima and the difficulty of recovery the long wavelength components in FWI (full waveform inversion). Bunks et al. (995) introduced multiscale FWI, where the inversion is done from low frequency to high frequency. Frequency domain inversion is an intrinsic multiscale approach (Pratt, 999; Pratt and Shipp, 999). Brenders and Pratt (7) used complex-valued frequencies to ease the local minima problem. However, the recovery of long wavelength components depends on the availability of low-frequency signal in seismic source. The generation of low-frequency signal below 5Hz is very expensive. Therefore, effort has been focused to the recovery of long wavelength background without very low frequencies. Travel time inversion (Dines and Lytle, 979; Paulsson et al., 985; Luo and Schuter, 99; Woodward, 99) and migration velocity analysis (Liu and Bleistein, 995; Biondi and Sava, 999; Shen and Symes, 8; Xie and Yang, 8) are two traditional methods. In recent years, Shin and Cha (8) developed Laplace domain FWI which can give a smooth background from an inaccurate initial model. Later, they extended this method to Laplace- Fourier domain (Shin and Cha, 9). Liu et at. () proposed the normalized integration method. There are also some methods which combine waveform inversion and some other techniques. For example, Zhou et al. (995) used traveltime and waveform inversion. Biondi and Almomin () and Almomin and Biondi () combined waveform inversion with wave-equation migration velocity analysis. Wang et al. () combined wave equation tomography and full waveform inversion. Last year, we (Wu, et al., 3; Luo and Wu, 3) proposed the envelope inversion method which is very efficient for recovering the large-scale component of the model. In this paper, we further discuss the nonlinear scale separation property and misfit configuration of envelope inversion. We also show the noise resistant of this method to seismic interference noise. Numerical tests showed that envelope inversion can greatly reduce the local minima and to some extent is resistant to seismic interference noise. Review of envelope inversion method In envelope inversion, the misfit function is defined as the envelope misfit (Wu et al., 3): T T σ ( d) = ( ) ( ) d syn dobs dt = e syn eobs dt () T = Edt where d is the envelope-function, which is also represented by e, E is the instant envelope data residual. The envelope of signal f() t is extracted by using Hilbert transform: et ()= f () t H f() t + { } () where e(t) is the envelope of signal f() t, and H{ f (t)} is the Hilbert transform of f() t. Then equation () can be written as: T σ ( d) = { ( ) ( ) ( ) ( ) } H y t + yh t u t u t + dt (3) T = E dt where u and y are the observed and synthetic waveforms respectively, u H and y H are the corresponding Hilbert transform. Consider velocity v as the model parameter. The derivative of σ with respect to v is: σ T yt () = Ey( t) H { Ey ( )} H t dt v () v Introduce two new vectors F and γ, where yt () F =, γ = Ey() t H { EyH () t } (5) v So equation () can be expressed as: σ T = F γ (6) v From equation (6) we can see that envelope inversion can also be realized by using backpropagation method as in the conventional waveform inversion (Tarantola, 98). Nolinear scale separation by envelope operator It is well-known that seismic data has a wavenumber gap in terms of subsurface structure inversion. For surface SEG SEG Denver Annual Meeting DOI Page 6

2 Envelope inversion reflection survey with limited acquisition aperture, the gap is mainly caused by the lack of low-frequencies in source spectra (Baeten, et al., 3). The linear scale separation is usually realized by a multi-scale inversion and the expensive low-frequency source is a critical armament for success. Here we discuss the nonlinear scale separation through envelope operator. To simplify the analysis, we mainly discuss the scale response for vertical structural variations as in previous discussions (Jannane, et al., 989; Baeten, et al., 3). By extracting the envelope, we can separate the large-scale response coded in the envelope from the waveform data which are the band-limited response to the medium perturbation. Figure (a) shows the comparison of waveform spectrum (blue line) and envelope spectrum (red line) for the Marmousi data with a full Ricker source. We also plot the average medium velocity spectrum (thin grey line) and the average velocity perturbation (medium velocity subtracts the background velocity which is a -D linear model) spectrum (dotted line). In order to see the correspondence of medium wavenumber spectra to the data frequency spectra, we perform a z-t transform using the known velocity structure. The medium spectrum is the average over all the vertical profiles. From Figure 5(a), we see that for the waveform data, the low-frequency spectral components are missing so the long-wavelength part of the perturbation spectrum is not covered. Meanwhile we see the complimentary role of envelope data which has strong low-frequency but very weak high-frequency components. This property of the envelope function can reduce the cycle skipping and local minima problems of waveform data and can recover the long-wavelength components of the perturbation structure. In Figure (b) we plot the case using a low-cut Ricker source filtered with a 5 Hz low-cut taper. We see very little energy exists below 5 Hz for the waveform data. However, the spectrum of the envelope data does not show too much difference from the full-band source case. This demonstrates the nonlinear nature of the envelope extraction, which does not have the linear correspondence between the source spectrum and the data spectrum. Misfit function configuration The test models (Figure ) are the full Marmousi model, its top left corner, and its top right corner respectively. The misfit function is a function of velocity model, which varies in the multi-dimensional parameter space. In order to show the behaviors of the misfit, we simplify the multidimensional space into a -dimensional space. We first decompose the models v (x) into a background model v ( x) and a perturbation δ v (x) = v (x) v (x). v ( x) is a linear gradient model varying from.5km/s (Vmin) to 5km/s (Vmax). The two parameters are Vmax and a strength parameterα. By varingα, we get a series perturbations δv() x = αδv () x. We keep Vmin as.5km/s, and change Vmax from.5 to 7.5 km/s, and changeα from % to %. By summing the background and the perturbation, we get a series of trial models. We calculate the waveform and envelope data from the model and the trial models. The misfit for each trial model is then obtained. Figures 3,, 5 show the misfit configurations for the three models. We see that for the waveform misfit, besides the global minimum, there are some local minima, which will lead the inversion to wrong results. While for the envelope misfit, there is only a global minimum. So using envelope inversion can avoid or reduce the local minimum problem. 6 8 Figure Test models for misfit configuration. The full Marmousi model, its top left corner (in the red border) and its top right corner (in the black border). Study on noise resistance of envelope inversion In last year, we have shown that envelope inversion is resistant to white Gaussian noise. In this study, we further investigate the influence of other noises, such as the seismic interference (SI). We invert Marmousi velocity model starting from a -D linear initial model. The source wavelet is the Ricker wavelet with a dominant frequency of Hz. There are shots and 8 receivers equally spaced on the surface. SI usually comes from others sources operating in the same area. To simulate SI, we put another source behind the th shot, on the right side of the surface. Figure 6 shows a shot profile. (a) is the original data, (b), (c), and (d) are the data with SI noise and the SNR (signal to noise ratio) are,, and respectively. Figure 7 shows the envelope inversion results. (a) is the result without noise, we can see the large-scale component clearly; (b), (c), (d) are the results when the SNR is,, respectively. When the SNR is, we see the result is almost the same as (a). When the SNR is, the result is affected a little, however we can still see the large-scale component. When the SNR is, the noise is too strong (same level as the data). We see that the envelope inversion is affected very much, and we can only see some structures in the very shallow part. Figure 8 shows the final EI+FWI SEG SEG Denver Annual Meeting DOI Page 7

3 Envelope inversion results using envelope inversion results in Figure 7 as initial model. We can see when the SNR are and, the final EI+FWI results are as good as the situation without noise. When SNR=, we see that in this case the result has converged to a local minimum. From the above test we can see that envelope inversion is very useful to recover the large-scale component of the model, and is somewhat resistant to SI noise, and has potential to apply to noisy data. Conclusion In this paper, we showed the nonlinear scale separation property and misfit configuration of envelope inversion. Envelope extraction provides a nonlinear scale separation of the information contained in the waveform data, and can be used to reduce the local minima and recover the largescale component of the model. Numerical tests also show that envelope inversion is to some extent resistant to seismic interference noise. Acknowledgement We thank Xiao-bi Xie, Bo Chen and Jinghuai Gao for helpful discussions. The work is supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project at University of California, Santa Cruz. (a) (b) Figure (a) Comparison of waveform spectrum and envelope spectrum for the Marmousi data with a full Ricker source excitation; (b) Same as (a) but with a low-cut source excitation. The low-cut source is obtained by a 5Hz low-frequency taper. Envelope misfit (a) (b) (c) Figure 3 Misfit configuration for the whole Marmousi model. (a) Waveform misfit; (b) enlarged figure of the marked region; (c) envelope misfit. Envelope misfit (a) (b) (c) Figure Misfit configuration for the top left corner of Marmousi model. (a) Waveform misfit; (b) enlarged figure of the marked region; (c) envelope misfit. SEG SEG Denver Annual Meeting DOI Page 8

4 Envelope inversion Envelope misfit (a) (b) (c) Figure 5 Misfit configuration for the top right corner of Marmousi model. (a) Waveform misfit; (b) enlarged figure of the marked region; (c) envelope misfit. (a) (b) (c) (d) Figure 6 A shot profile data with seismic interference noise. (a) is the original data without noise; the SNR in (b), (c), (d) are,, respectively (a) (b) (c) (d) Figure 7 Envelope inversion results. (a), (b), (c), (d) are the results using data set in Figure 6 (a), (b), (c), (d) respectively (a) (b) (c) (d) Figure 8 EI+FWI results. (a), (b), (c), (d) are the results using Figure 7 (a), (b), (c), (d) as initial model respectively. SEG SEG Denver Annual Meeting DOI Page 9

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES Almomin, A., and B. Biondi,, Tomographic full-waveform inversion: Practical and computationally feasible approach: Presented at the 8 nd Annual International Meeting, SEG. Baeten, G., J. W. de Maag, R.-E. Plessix, M. Klaassen, T. Qureshi, M. Kleemeyer, F. ten Kroode, and Z. Rujie, 3, The use of the low frequencies in a full-waveform inversion and impedance inversion land seismic case study: Geophysical Prospecting, 6, no., 7 7, Biondi, B., and A. Almomin,, Tomographic full-waveform inversion (TFWI) by combining fullwaveform inversion with wave-equation migration velocity analysis: 8 nd Annual International Meeting, SEG, Expanded Abstracts, doi:.9/segam-75.. Biondi, B., and P. Sava, 999, Wave-equation migration velocity analysis: 69 th Annual International Meeting, SEG, Expanded Abstracts, Brenders, A. J., and R. G. Pratt, 7, Full-waveform tomography for lithospheric imaging: Results from a blind test in a realistic crustal model: Geophysical Journal International, 68, no., 33 5, Bunks, C., F. M. Saleck, S. Zaleski, and G. Chavent, 995, Multiscale seismic waveform inversion: Geophysics, 6, 57 73, Dines, K., and R. Lytle, 979, Computerized geophysical tomography: Proceedings of the IEEE, 67, no. 7, 65 73, Jannane, M., W. Beydoun, E. Crase, D. Cao, Z. Koren, E. Landa, M. Mendes, A. Pica, M. Noble, G. Roeth, S. Singh, R. Snieder, A. Tarantola, D. Trezeguet, and M. Xie, 989, Wavelength of earth structures that can be resolved from seismic reflection data : Geophysics, 5, 96 9, Lailly, P., 983, The seismic inverse problem as a sequence of before stack migrations: Proceeding of the Conference on Inverse Scattering, Theory and Applications, SIAM, 6. Liu, Z., and N. Bleistein, 995, Migration velocity analysis: Theory and an iterative algorithm: Geophysics, 6, 53, Liu, J., H. Chauris, and H. Calandra,, The normalized integration method An alternative to fullwaveform inversion?: 7 th European Meeting of Environmental and Engineering Geophysics, 7. Luo, J., and R.-S. Wu, 3, Envelope inversion Some application issues: Presented at the 83 rd Annual International Meeting, SEG, Expanded Abstracts, 7. Luo, Y., and G. T. Schuster, 99, Wave-equation traveltime inversion: Geophysics, 56, , Paulsson, B., N. Cook, and T. McEvilly, 985, Elastic wave velocities and attenuation in an underground granitic repository of nuclear waste: Geophysics,, 55 57, SEG SEG Denver Annual Meeting DOI Page

6 Pratt, R. G., 999, Seismic waveform inversion in the frequency domain, Part : Theory and verification in a physical scale model: Geophysics, 6, 888 9, Pratt, R. G., and R. M. Shipp, 999, Seismic waveform inversion in the frequency domain, Part : Fault delineation in sediments using crosshole data: Geophysics, 6, 9 9, Shen, P., and W. Symes, 8, Automatic velocity analysis via shot profile migration: Geophysics, 73, no. 5, VE9 VE59, Shin, C., and Y. H. Cha, 8, Waveform inversion in the Laplace domain: Geophysical Journal International, 73, no. 3, 9 93, Shin, C., and Y. Ho Cha, 9, Waveform inversion in the Laplace-Fourier domain: Geophysical Journal International, 77, no. 3, 67 79, Tarantola, A., 98, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 9, 59 66, Wang, H., S. C. Singh, H. Jian, and H. Calandra,, Integrated inversion of subsurface velocity structures using wave equation tomography and full waveform inversion: Presented at the 8 nd Annual International Meeting, SEG. Woodward, M. J., 99, Wave-equation tomography: Geophysics, 57, 5 6, Wu, R.-S., J. Luo, and B. Wu, 3, Seismic envelope inversion and modulation signal model: Geophysics, 79, no. 3, WA3 WA, doi:.9/geo3-9.. Xie, X., and H. Yang, 8, The finite-frequency sensitivity kernel for migration residual moveout and its applications in migration velocity analysis : Geophysics, 73, no. 6, S S9, Zhou, C., W. Cai, Y. Luo, G. T. Schuster, and S. Hassanzadeh, 995, Acoustic wave-equation traveltime and waveform inversion of crosshole seismic data: Geophysics, 6, , SEG SEG Denver Annual Meeting DOI Page