Suppress Parameter Cross-talk for Elastic Full-waveform Inversion: Parameterization and Acquisition Geometry

Transcription

1 Suress Paraeter Cross-talk for Elastic Full-wavefor Inversion: Paraeterization and Acquisition Geoetry Wenyong Pan and Kris Innanen CREWES Project, Deartent of Geoscience, University of Calgary Suary Full-wavefor inversion (FWI) roises to rovide high-resolution estiates of subsurface roerties but reains exosed to a range of challenges. Involving ultile hysical araeters in FWI increases the non-linearity of the inversion rocess. The araeter cross-talk arising fro the couling effects between different hysical araeters akes the inverse roble ore underined. Proer araeterization and acquisitiion geoetry should be deterined to anage the araeter cross-talk. In this research, we analyze the araeter cross-talk associated with different araeterizations for elastic FWI. We also discuss the influence of acquisition geoetry in anaging the araeter cross-talk. We resent nuerical exales to show that iendace araeterization is ost efficient for elastic FWI in itigating araeter cross-talk. Introduction Full-wavefor inversion (FWI) allows to reconstruct high-resolution velocity odels of the subsurface through the extraction of the full inforation content of the seisic data (Lailly, 198; Tarantola, 1984; Virieux and Oerto, 2009). FWI iteratively iniizes a L-2 nor isfit function, which easures the difference between the odelled data and observed data (Pratt et al., 1998; Pan et al., 2014b, 2015a, 2015b). Shorter-ter FWI is eloyed to rovide high-resolution velocity odel for seisic iaging. Longer-ter FWI is exected to inverse ore elastic araeters for reservoir characterization in exloration geohysics. Coared to ono-araeter FWI, ulti-araeter FWI is a ore challenging task. Siultaneous inversion of ultile araeters increases the non-linearity of the inversion rocess. The couling effects (or araeter cross-talk) between different hysical araeters ake the inverse roble ore undeterined. The overla of the artial derivative wavefields (or scattering atterns) associated with different hysical araeters give rise to the araeter cross-talk difficulty. The ulti-araeter Hessian in FWI is a square and syetric atrix with a block structure. It carries ore inforation than a ono-araeter Hessian. Within the ulti-araeter Gauss-Newton Hessian aroxiation, the off-diagonal blocks easure the correlation of Fréchet derivative wavefields with resect to different hysical araeters, and they act to itigate the couling effects between these araeters. In this research, the scattering atterns of the elastic constants in general anisotroic edia are given for araeter cross-talk analysis. It is verified that the ulti-araeter Gauss-Newton Hessian and its araeter-tye aroxiation can reduce the araeter cross-talk in ulti-araeter FWI (Pan et al., 2015b, 2016). We also coare different araeterizations for effective inversion in elastic FWI. The nuerical exerients show that the iedance araeterization is ore effective for inversion and avoiding araeter cross-talk, coared to velocity and Laé constants araeterizations. GeoConvention 2016: Otiizing Resources 1

2 Review of non-linear least-squares inverse roble As a non-linear least-squares otiization roble, FWI seeks to estiate the subsurface araeters by iteratively iniizing the difference between the synthetic data and observed data. The isfit function is forulated in a least-squares for: 1 2 ( ) d(x r,x s, ), (1) 2 x s x r where x r and x s indicate the locations of receivers and sources, d is the data residual vector and is the angular frequency. To iniize the quadratic aroxiation of the isfit function, the udated odel at k+1 th iteration can be written as the su of the odel at the kth iteration and the search direction: 1 k k k k, (2) where k is the ste length, a scalar constant calculated through a line search ethod satisfying the weak Wolfe condition. The gradient can be constructed by cross-correlating the forward odelled wavefield and back-roagated wavefield. The search direction is the solution of the Newton linear syste: (H + A) g, () k k k where g k is the gradient, H k is the Gauss-Newton Hessian aroxiation, and A is the daing ter. The Gauss-Newton Hessian is the correlation of two artial derivative wavefields: The ulti-araeter Hessian. (4) The ulti-araeter Hessian has a block structure. Considering the 2D subsurface odel with Nx Nz nodes and N hysical araeters are assigned to describe the roerties of each node. The ultiaraeter Hessian is a NxNzN NxNzN square and syetric atrix with N diagonal blocks and N( N 1) off-diagonal blocks. Each block is a NxNz NxNz square atrix. For isotroic and elastic edia, three different hysical araeters 1, 2, can be used to describe the roerties of the edia. Thus, the Newton linear syste can be written as:. (5) The off-diagonal blocks in the ulti-araeter Hessian indicate the correlation of artial derivative wavefields with resect to different hysical araeters and they redict the couling effects of these hysical araeters. Thus, alying its inverse to the gradient can itigate the araeter cross-talk difficulty. Paraeterization and acquisition geoetry issue for elastic FWI Coared to ono-araeter FWI, ulti-araeter FWI is a ore challenging task. Siultaneous inversion of ultile araeters increases the non-linearity of the inversion rocess. The couling effects between different hysical araeters ake the inverse roble ore undeterined. The overla of the artial derivative wavefields associated with different hysical araeters give rise to the araeter crosstalk difficulty. Proer araeterzation and acquisition geoetry should be deterined for anaging the araeter cross-talk. A good choice of araeterization will gives three different scattering atterns which are as different as ossible, to allow easy identification of these araeters. GeoConvention 2016: Otiizing Resources 2

3 In this research, we define three different araeter classes for elastic FWI, which are velcotiy araeterization:,,, Laé constants araeterization:,, and iedance araeterization: I, I,. Figure 1 shows the P-P scattering atterns for different araeter classes (Tarantola, 1986). s Theoretically, the iedance araeterization should give best inversion result (Tarantola, 1986). We also design a reflection survey and a transission survey for coarison. Figure 1. (a), (b) and (c) show the P-P scattering atterns for,,. (d), (e) and (f) show the P-P scattering atterns for,,. (g), (h) and (i) show the P-P scattering atterns for I, I,. Exales In this seciton, we illustrate with nuerical exales to coare the elastic FWI inversion results associated with different araeterizations. We also coare the inversion result by reflection survey with that by transission survey. Figure 2 shows the true odel erturbations for different hysical araeters 1, 2,. For velocity araeterization, 1, 2, reresent,,. For Laé constants and iedance araeterizations, 1, 2, indicate,, and I, Is,. The initial odels are hoogeneous. s Figure 2. (a), (b) and (c) show the true odel erturbations for,, 1 2. The red stars reresent the source locations. The black and green triangles indicate the receiver locations for reflection and transission surveys. We erfored reflection survey and transission survey for inversion resectively and Gauss-Newton ethod is used. Figure a, b and c show the Gauss-Newton Hessian of velocity araeterization, Laé constants araeterization and iedance araeterization for reflection survey. Figure d, e and f show the Gauss-Newton Hessian of different araeterizations for transission survey. Figure 4 shows the velocity araeterization inversion results with reflection survey and transission survey. As we can see, the inverted P-wave velocity is still suffered fro araeter cross-talk. Figure 5 shows the Laé constants araeterization inversion results with reflection survey and transission survey. We can see GeoConvention 2016: Otiizing Resources

4 that the araeters and are difficult to be recovered. Figure 6 show the inversion results for iedance araeterization. We observe that different hysical araeters are be inverted very well without araeter cross-talk artifacts. Figure. (a), (b) and (c) show the Gauss-Newton Hessian of velocity araeterization, Laé constants araeterization and iedance araeterization for reflection survey. (d), (e) and (f) show the Gauss- Newton Hessian for transission survey. Figure 4. Velocity araeterization inversion results with reflection survey and transission survey. Figure 5. Laé constants araeterization inversion results with reflection survey and transission survey. Figure 6. Iedance araeterization inversion results with reflection survey and transission survey. Conclusions In this research, we discuss the araeter cross-talk difficulty for elastic FWI. We also consider the erforances of different araeter classes in anaging araeter cross-talk. It is concluded that the iedance araeterization works better for elastic FWI than other araeterizations. GeoConvention 2016: Otiizing Resources 4

5 References Lailly, P., 198, The seisic inverse roble as a sequence of before stack igration: Conference on Inverse Scattering, Theory and Alications, SIAM, Exanded Abstracts, Mora, P., 1987, Nonlinear two-diensional elastic inversion of ultioffset seisic data: Geohysics, 5, Nash, S. G., 1985, Preconditioning of truncated-newton ethods: SIAM J. Sci. Statist. Cout, 6, Pratt, R. G., C. Shin, and G. J. Hicks, 1998, Gauss-Newton and full Newton ethods in frequency-sace seisic wavefor inversion: Geohysical Journal International, 1, Pan, W., K. A. Innanen, and G. F. Margrave, 2014a, A coarison of different scaling ethods for least-squares igration/inversion: EAGE Exanded Abstracts, We G Pan, W., K. A. Innanen, G. F. Margrave, and D. Cao, 2015a, Efficient seudo-gauss-newton full-wavefor inversion in the t- doain: Geohysics, 80, no. 5, R225-R14. Pan, W., K. A. Innanen, G. F. Margrave, M. C. Fehler, X. Fang, and J. Li, 2015b, Estiation of elastic constants in HTI edia using Gauss-Newton and Full-Newton ulti-araeter full wavefor inversion: SEG Technical Progra Exanded Abstracts, Pan, W., G. F. Margrave, and K. A. Innanen, 2014b, Iterative odeling igration and inversion (IMMI): Cobining full wavefor inversion with standard inversion ethodology: SEG Technical Progra Exanded Abstracts, Pan, W., K. A. Innanen, and W. Liao, 2016, Estiation of elastic constants for HTI edia using Gauss-Newton and full-newton ulti-araeter full-wavefor inversion: Geohysics, subitted. Tarantola, A., 1984, Inversion of seisic reflection data in the acoustic aroxiation: Geohysics, 49, Tarantola, A., 1986, A strategy for nonlinear elastic inversion of seisic reflection data: Geohysics, 51, Virieux, A. and S. Oerto, 2009, An overview of full-wavefor inversion in exloration geohysics: Geohysics, 74, no. 6, WCC1- WCC26. GeoConvention 2016: Otiizing Resources 5