W024
Full Waveform Inversion for Reflected Seismic Data
S. Xu* (CGGVeritas), D. Wang (Statoil), F. Chen (CGGVeritas), Y. Zhang (CGGVeritas) & G. Lambare (CGGVeritas)
SUMMARY
Full waveform inversion has been successful in building high resolution velocity models for shallow layers. To achieve this, it requires refracted waves or low frequencies in the reflection/refraction data. To relax the dependence on low frequency reflections, we revisit full waveform inversion. We propo a new approach allowing the updating of long wavelength components of the velocity model affecting the reflected arrivals. Our approach is bad on a non-linear iterative relaxation approach where short and long wavelength components of the velocity model are updated alternatively. We study theoretically the associated Fréchet derivatives and gradients and discusd how and why such a strategy improves the resolution that we can expect from full waveform inversion. Finally we prent a first 2D application to a 2D Gulf of Mexico conventional streamer datat.惊喜英文单词
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Introduction
Full waveform inversion (FWI) has been an important method to build velocity models for ismic imaging (Tarantola, 1984; Sirgue and Pratt, 2004; Virieux and Operto, 2009). Classical FWI involves the minimization of a square misfit function between the calculated and obrved data. Non-linear gradient bad optimizations are ud (Pratt et al., 1998; Ravaut et al., 2004; Sirgue and Pratt, 2004; Choi et al., 2008; Ma and Hale, 2011) with complex strategies for regularizing the process (filtering, weighting and mute of the data, etc ...). The strategies mitigate non-linearity but cannot recover the features that are not covered by the intrinsic resolution of the method.
The resolution of FWI is the resolution of a migration operator. The recovered wavelengths in the velocity model correspond to the time recorded wavelengths stretched to depth according to the local velocity and angular aperture. For the transmissions and refractions, the stretching due to the angle aperture allows one to recover the long wavelengths of the velocity model (Gauthier et al., 198
6); while for the reflections, only short wavelengths can be recovered by FWI due to the narrow range of reflection angle apertures. This explains why FWI recovers long wavelength components of the velocity model only in shallow areas and why its resolution improves when lower frequencies and longer offt data are available (Ravaut et al., 2004; Sirgue et al., 2010). Unfortunately for conventional streamer data, low frequencies are not available due to the existence of source and receiver ghost (Lindy, 1960), and the maximum offt is usually limited within 8km. It remains a challenge how to apply FWI to streamer data (Plessix et al., 2010a) to obtain good velocity resolution. We propo here a new approach for FWI. We keep the least square misfit function of FWI, but split the velocity model into long and short wavelength components, i.e. the reference model and the perturbation model. The two parts of the velocity model are updated jointly with an iterative relaxation method. At each loop, first, the perturbation model is obtained from the initial reference model by a true amplitude migration. It is then fixed and the reference model is updated by a local optimization scheme. We show in the prent paper the expression of the associated Fréchet derivatives and the gradients, and discusd how and why such a strategy greatly improves the resolution that we can expect from FWI. Finally we prent a first application of our algorithm to a 2D Gulf of Mexico streamer datat, showing that the approach can update long wavelength components of the velocity model.
Fréchet derivatives and Gradient for FWI
Let’s start from the scalar acoustic wave equation. In the frequency domain the associated Green’s function G(x,ω;s)(where x is the position, ωis the angular time frequency, and s is the shot position) satisfies the equation,
−ω2mG−∆G=δ(x−s)(1) where m(x)=1/v2(x)is the model to be estimated, i.e. the squared slowness. In a classical full waveform inversion we ek for m(x) minimizing the square misfit cost function
∁(m)=12∭ds drdω‖G obs(r,ω;s)−G cal(m)(r,ω;s)‖2(2) where G obs and G cal denote the obrved and calculated Green function, respectively, and s and r the t of source and receiver positions, respectively. Let’s split the model intoabreast
m=m0+δm, (3) where m0contains the long wavelength components of the velocity model (explaining the transmission behavior of the model), and δm contains the short wavelength components (explaining the reflection behavior of the model). Accordingly the Green functions can be split into
G(m0+δm)=G0(m0)+δG(m0,δm).(4) An exact expression of the perturbation of the Green function depending on m0 and δm is given by the Fredholm integral equation from the cond kind
δG(m0,δm)(r,ω;s)=ω2∫dx G0(r,ω;x)G(x,ω;s)δm(x). (5) It is interesting now to study the Fréchet derivatives of δG for δm and m0.
The expression of the Fréchet derivative of δG with respect to δm(fixing m0) is the conventional kernel of the Born operator
达内软件培训ðδG ðδm �m 0(m 0)(r ,ω,s ;x )≈ ω2G 0(r ,ω;x )G 0(x ,ω;s ). (6)
It depends on m 0 but not on δm , and corresponds to the Fréchet derivative of the conventional full waveform inversion problem. As such it is also the kernel of a “normal” migration operator (Lailly 1983; Tarantola 1984).
The gradient of the cost function (2) can be easily computed from the Fréchet derivative as ð∁ðδm �m 0(x )=−∬dsdr ∫dωðδG ðδm �m 0∗(r ,ω,s ;x )R (r ,ω;s ), (7) where ∗ denotes the conjugated, a
nd R are the residuals defined by R (r ,ω;s )=G obs (r ,ω;s )−G cal (r ,ω;s ). (8) The contribution to the gradient of the direct, diving and refracted waves is along the wave paths (Figure 1a), while the contribution of the reflected waves is the “normal” migration respon of their residuals (Figure 1b). We e that if FWI can potentially recover the long wavelength components of the velocity model in the shallow area (in fact along the diving wave paths) it cannot in the deeper area if sufficiently low frequencies are not available in the data (Plessix et al., 2010b). Figure 1 Contribution to the gradients of classical FWI and SRFWI (expressions (7) and (10)) of a
trace with the source location at (2.5, 0.0) km, and receiver location at (12.5, 0.0) km. The source wavelet is a Ricker with a peak frequency at 6. Hz; and the background velocity is 2. km/s. Two signals are considered as residuals: a direct wave and a reflected wave for a reflector at 5km depth. a) Contribution of the direct wave to the gradient of FWI; b) Contribution of the reflected wave to the gradient of FWI; c) Source-reflector contribution of reflected wave to our new gradient (expression
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(13); d) Source-reflector-receiver contribution of reflected wave to our new gradient.
Let’s look now at the Fréchet derivative of δG (5) with respect to m 0 (fixing δm ). From expression
(5) we can write
ðδG ðm 0�
δm (m 0,δm )(r ,ω,s ;x )=ω2∫dy �ðG 0ðm 0(r ,ω;y )G (y ,ω;s )+G 0(r ,ω;y )ðG ðm 0(y ,ω;s )�δm (y ) (9) where we can introduce Born expressions for ∂G 0/∂m 0, and ∂G/∂m 0, i.e. ðG 0ðm 0(x )(r ,ω,y )=ω2G 0(r ,ω;x )G 0(x ,ω;y ) and ðG ðm 0(x )(y ,ω,s )=ω2G (y ,ω;x )G (x ,ω;s ). (10) We obtain
ðδG ðm 0(r ,ω,s ;x )=ω4∫dy �G 0(r ,ω;x )G 0(x ,ω;y )G
(y ,ω;s )+G 0(r ,ω;y )G (y ,ω;x )G (x ,ω;s )�δm (y ) (11)
And finally using equation (5)
ðδGðm
newsletter0(x)�δm(m0,δm)(r,ω,s;x)=ω2�G0(r,ω;x)δG(x,ω;s)+δG(r,ω;x)G0(x,ω;s)� (12) and we e that it d
epends both on m0 and δm (through term δG). The gradient of the cost function with respect to m0 (fixing δm) is
ð∁ðm
0�δm(x)=−∬dsdr∫dωðδGðm0(x)�δm∗(r,ω,s;x)R(r,ω;s), (13) and as shown in Figure 1c and 1d, the contribution to the gradient of the reflected waves is built up along the wave propagation paths of the reflections. Figure 1c shows the contribution corresponding to the wave path from the source to the reflector (first term on the right hand side of equation (12)) while Figure 1d shows the total contribution, source-reflector-receiver. We can have now some hope that a local optimization of C for m0(fixing δm at each non-linear iteration) may be suitable for recovering the long wavelength components in depth of the velocity model available in the reflected arrivals.
Seismic Reflection Full Waveform Inversion
We propo an approach where m0is iteratively updated through a non linear local optimization scheme. Starting from an initial velocity model m0 in expression (9) the term δG is approximated by a wave equation modelling using the model and the reflectivity derived from the true amplitude migration (Zhang et al., 2007) of the previous iteration residuals. Each iteration consists of the followi
ng steps:
1)True amplitude prestack depth migration of the reflected wavefield for the initial velocity
model m0init;
2)From the true amplitude migration result and the initial velocity model m0init simulation of the
perturbed wave field, δG;
3)Computation of the residual data;
4)Computation of the gradient of the cost function;
5)Update of the velocity model to m0updated.
In order to focus on the reflected wavefield the direct and refracted waves are muted in δG. Our propod inversion method is named Seismic Reflection Full Waveform Inversion (SRFWI). It fully corresponds to the migration-bad travel time (MBTT) waveform inversion approach propod by Chavent et al. (1994) which also as some similarity in terms of resolution with the differential mblance optimization (DSO) method propod by Symes and Carazzone (1991).
Figure 2: An application of SRFWI to a 2D Gulf of Mexico datat. left: migration with initial model; Middle: the migration image with inverted velocity model; right: velocity perturbation.
Application
We have developed a 2D SRFWI algorithm and we apply it to a Gulf of Mexico datat. A 2D line was pulled out from a 3D narrow azimuth streamer datat. The target is the velocity anomaly corresponding to a gas cloud. To ensure the efficiency, we decimated the shot numbers to one third. The inversion was performed with traces low pasd filtered to 8 Hz. Figure 2 illustrates the migrated images with the initial velocity (left) and with the inverted velocity (middle) from SRFWI, and the corresponding velocity perturbation (right). Although the acquired maximum offt is about 8km and
the water bottom is about 3km deep, we e clearly long wavelength update from SRFWI in the target area. The migrated image is also improved with the updated velocity.
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Conclusion
We investigated the gradient operator of full waveform inversion and propod a new method to take advantages of the reflected ismic waves. An analysis of the Fréchet derivatives and gradient operator of our new FWI shows that it can u the reflections to reconstruct the long wave length components of the model. A first application to a 2D real data example demonstrates this capability with conventional streamer data.
Acknowledgements
We thank CGGVeritas for the permission to prent this work, and particularly Jerry Young for his encouragements. We thank Patrice Guillaume and Vetle Vinje for reviewing the abstract. References
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欧盟英文缩写
Lindy, J. P., 1960, Elimination of ismic ghost reflections by means of a linear filter, Geophysics, 25 , 1, 130-140.
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结汇英文
Pratt, R., C. Shin, G. Hicks, 1998, Gauss-Newton and full Newton methods in frequency-space ismic waveform inversion: Geophysical Journal, International, 13, 341–362.
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Sirgue, L., O. I. Barkved, J. Dellinger, J. Etgen, U. Albertin, J. H. Kommedal, 2010, Full-waveform inversion: the next leap forward in imaging at Valhall: First Break, 28, 65–70.
Symes, W. W., J. J. Carazzone, 1991. Velocity inversion by differential mblance optimization Geophysics 56, 654-663.
情迷酒吧Tarantola, A., 1984, Inversion of ismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266.
Virieux, J., S. Operto, 2009. An overview of full waveform inversion in exploration geophysics, Geophysics, 74(6), WCC127-WCC152.
Zhang, Y., S. Xu , N. Bleistein and G. Zhang, 2007. True-amplitude, angle-domain, common-image gathers from one-way wave-equation migrations, Geophysics, 72, 1, S49-S58.
