Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (2013) 195, Advance Access publication 2013 July 5 doi: /gji/ggt233 Waveform acoustic impedance inversion with spectral shaping R.-É. Plessix 1 and Y. Li 1,2 1 Shell Global Solutions International, Kesslerpark 1, 2288 GS Rijswijk, the Netherlands. reneedouard.plessix@shell.com 2 Department of Geophysics, Stanford University, 397 Panama Mall, Stanford, CA 94035, USA Accepted 2013 June 7. Received 2013 April 24; in original form 2012 December 13 SUMMARY Full waveform inversion applied to surface seismic data containing only reflection data generally gives an impedance map, the background velocity being assumed known. The first iteration update does not have a spectrum close to the Earth impedance spectrum because of source wavelet and wave propagation effects. To improve the convergence, these effects can be compensated by designing a spectral shaping filter that produces a gradient of the misfit function with a spectrum similar to the Earth spectrum. Based on an asymptotic analysis of the gradient of the misfit function, we rederive the theoretical spectral shaping filter. When the observed source wavelet is known or can be estimated from the data, we retrieve that, after source deconvolution/whitening of the data, the theoretical spectral shaping is in ω β/2 for the data or in k β for the gradient with ω the angular frequency and k the wavenumber. β is an exponent depending on acquisition and equal to 1 with areal (so-called 3-D) acquisition and to 2 with line (so-called 2-D) acquisition. Under acoustic assumption, this leads to a waveform acoustic impedance inversion approach. We test this approach with a small synthetic example and with a real data set. Since we did not use aprioriimpedance information to derive the spectral shaping, we validate the approach by comparing the spectrum of the inverted impedances with the one of the Earth impedance computed from well-log measurements. The results illustrate the relevance of the spectral shaping to improve the convergence of the waveform inversion of reflection data. Key words: Image processing; Fourier analysis; Inverse theory; Numerical approximations and analysis. GJI Marine geosciences and applied geophysics 1 INTRODUCTION Full waveform inversion (FWI) aims at directly retrieving the elastic Earth parameters by minimizing the misfit between observed and computed data (Tarantola 1987). Over the last 30 yr, this approach has been applied to different types of data, surface, vertical seismic profile or crosswell data (Bamberger et al. 1982; Gauthier et al. 1986; Mora 1987; Song et al. 1995; Pratt 1999; Ravaut et al. 2004; Delprat-Jannaud & Lailly 2005; Plessix & Perkins 2010; Sirgue et al. 2010; Vigh et al. 2010; Routh et al. 2011; Baeten et al. 2013). Although a bit arbitrary, we can classify most of the FWI applications into two groups: in a first group, FWI is applied to reflection data and the dynamic aspect of the reflection waves plays the dominant role; in a second group, FWI is applied to transmission data and the kinematics aspect of the transmission waves plays the dominant role. This classification can be explained by looking at the gradient of the misfit function, assuming the background (initial) velocity is in the domain of attraction of the global minimum. With reflection data, the gradient is similar to a migration, the main contributions are at the specular reflection points (Lailly 1983; Tarantola 1984). With transmission data, the gradient is close to a traveltime/tomography inversion gradient, the contributions are spread along the wave paths (Pratt et al. 1996; Virieux & Operto 2009). This classification can also be explained by the shapes of the acoustic velocity and density radiation/diffraction patterns (Wu & Aki 1985; Tarantola 1986). Whereas the acoustic velocity radiation pattern is isotropic, the density radiation pattern is concentrated around the small reflection angles. With reflection data, the least-squares inversion is mainly sensitive to impedance contrasts (Tarantola 1986). This separation between the tomography mode and the migration mode of FWI has been explained in Gauthier et al. (1986), Mora (1989), Pratt et al. (1996) and Mulder & Plessix (2008). The band-limited frequency acquisition of the surface seismic data, especially the lack of low frequencies, makes FWI ill-posed because the seismic data are not really sensitive to intermediate wavenumbers (Claerbout 1976; Jannane et al. 1989). To mitigate this difficulty, one can acquire low-frequency and long-offset data. Acquiring long offsets and low frequencies helps FWI in the tomography mode when we want to retrieve the background velocity, especially with a frequency continuation approach (Pratt et al. 1996; C The Authors Published by Oxford University Press on behalf of The Royal Astronomical Society. 301

2 302 R.-É. Plessix and Y. Li Virieux & Operto 2009; Baeten et al. 2013). With reflection data, the low frequencies significantly improve the impedance inversion results (Soubaras & Dowle 2010; Reiser et al. 2012; Baeten et al. 2013). The frequency continuation also helps FWI with reflection data (Bunks et al. 1995). With reflection data only, we cannot rely on the long offsets to bring intermediate wavenumber information. Inverting a very sparse set of frequencies as it is done with transmission data when FWI works in migration mode may lead to artefacts (Mulder & Plessix 2004). Therefore, in this case, we generally run FWI with the full frequency band of the data sets and we assume that the initial guess for the Earth parameters accurately models the kinematics of the reflections. This gives the linear (with Born modelling) or non-linear least-squares migration (Chavent & Plessix 1999; Nemeth et al. 1999; Kuehl & Sacchi 2001). The objectives of the least-squares migration are to reduce the acquisition imprint, to improve the structural image and to improve the recovery of the amplitudes of the reflection coefficients. When we solve the full-wave equation (at finite frequencies with a finite-difference scheme for instance), the misfit functional is not parametrized with reflection coefficients, but with the elastic parameters or their perturbations. With only reflection data, FWI, in the migration mode, returns an impedance map since, at short/intermediate offsets, FWI is mainly sensitive to impedance. In this work, we focus on this approach and called it waveform impedance inversion. Due to the computational cost of solving the wave equation, the misfit function is minimized with a gradient optimization. With reflection data, the gradient is equivalent to a migration of the data residuals. The gradient, hence the impedance perturbations at the first iteration, contains the source wavelet squared, more precisely the observed source wavelet times the modelled one. The iterative minimization should correct for the illumination effects and remove the effects of the source wavelet squared. A data pre-conditioning that removes the source wavelet should help speeding up the convergence. The wave propagation introduces a frequency dependency that makes the spectrum of the gradient different from the spectrum of the Earth impedance. Correcting this effect via a frequency data pre-conditioning (Lutz & Strijbos 2009) also improves the convergence (Lazaratos et al. 2011). In this work, we follow this line of thought. Assuming known the observed source wavelet, we derive a spectral shaping filter from an asymptotic analysis. We apply it on the data and modelled wavelet. We could also shape the gradient with a post-processing that corresponds to a non-diagonal approximation of the inverse of the Hessian. This derivation is independent of any aprioriknowledge of the impedance spectrum and is not based on the convolution model as in Lazaratos et al. (2011). It is relatively similar to ones used to obtain ray-based inversion formulas (Beylkin 1985; Bleistein 1987; Jin et al. 1992). The outline of the paper is the following. In the next section, we expose the main aspects of the derivation of the spectral filter, some of the details being given in the appendixes. Then, in the example section, we illustrate the approach with a small synthetic data set and with a real data set acquired over a hydrocarbon field in the North Sea. 2 WAVEFORM IMPEDANCE INVERSION FORMULATION In this section, we derive the formula for the spectral shaping filter. We focus on reflection data and assume an acoustic propagation. This is a limitation because the reflection amplitudes depend on elastic parameters. The approach could aprioribe extended to elastic propagation. Since we focus on the reflection, short-offset, data, there is a wavenumber gap (Claerbout 1976; Jannane et al. 1989). We assume that the initial guess of the Earth parameters, namely the background velocity, accurately model the kinematics of the reflections. This means that it contains the correct low wavenumbers. An inaccurate background velocity will introduce some biases in the result because the stack over the diffraction curve will be partially destructive, similarly to migration. After recalling the FWI approach, we reparametrize the misfit function with the impedance perturbation and relate the FWI gradient with the one obtained with Born approximation. This shall lead us, after writing the gradient in term of Green s functions, to an asymptotic formulation. This asymptotic expression of the gradient gives us an approximation of the gradient spectrum we use to determine the spectral filter. We present this theoretical derivation in the frequency domain. However, all the numerical examples are obtained with finite-difference time-domain solutions of the wave equation and with a time-domain FWI implementation, namely the misfit function is computed from time-series. These derivations are apriorinot new and follow the same approach that the ones used to derive ray-based asymptotic inversion formula (Beylkin 1985; Bleistein 1987; Jin et al. 1992). However, we did not explicitly find them in the literature in this context. This is why we also give some details in the appendixes. 2.1 Full waveform inversion We parametrize the Earth with the acoustic impedance, I, and the acoustic velocity, V, because FWI mainly updates the short wavelengths with reflection data (Tarantola 1986). The acoustic velocity is assumed known. The second-order wave equation reads in the frequency domain V (x)i (x) p(x s, x,ω) V (x) I (x) p(x s, x,ω) = w(ω)δ(x x s ), (1) with ω the angular frequency, p the pressure field, w the source wavelet, x s a source location and x a subsurface point. The misfit function reads, with d(x s, x r,ω) the measured data set and x r a receiver location, J(I ) = 1 dx s dx r dω c(x s, x r,ω) f d (ω)d(x s, x r,ω) 2. (2) 2 We introduce the spectral shaping filter, f d. This filter, that needs to be determined, shall reshape the data spectrum. Data weights depending on source and receiver positions could be added. The goal of the waveform inversion is to retrieve the impedance by minimizing the misfit function. The cost of the simulation of the synthetics forces us to use a local optimization that relies on the gradient of the misfit function. This gradient can be efficiently determined by the adjoint-state method (Plessix 2006). First, we compute the backpropagated field q by solving the adjoint-state equation ω2 V (x)i (x) q (x s, x,ω) V (x) I (x) q (x s, x,ω) = dx r δ(x x r )e (x s, x r,ω) (3) with the conjugation and e = c(x s, x r,ω) f d (ω)d(x s, x r,ω)the residuals.

3 Waveform acoustic impedance inversion 303 Secondly, we compute the gradient by cross-correlating the incident field p with the backpropagated field q. The gradient with respect to the impedance, g, is equal to (with the dot product) g(x) = [ dx s dω V (x)i 2 (x) q (x s, x,ω)p(x s, x,ω) V (x) ] I 2 (x) q (x s, x,ω) p(x s, x,ω). (4) We did not write it, but only the real part of the right expression is the gradient. In our implementation, we optimize the misfit with a quasi- Newton method. At the iteration k of the minimization, the impedance is updated by I k+1 (x) = I k (x) α k d y H k (x, y)g( y) (5) with I k the values of the impedance at iteration k, α k the step length determined during a line search, and H k the pseudo-inverse of the Hessian. In our implementation, the pseudo-hessian is approximated by a diagonal matrix (Plessix & Mulder 2004). 2.2 Interpretation with Born approximation When the data mainly contain reflections (this is the case in this study), the gradient of the misfit function, that gives an estimate of the perturbations, corresponds to a migration of the residuals. To analyse the gradient, we reparametrize the inversion with the impedance perturbation, r r = I I b (6) I b with I b a background impedance. r corresponds to the normalized relative impedance. The gradient is kinematically equivalent to a migration (Lailly 1983; Tarantola 1984). The gradient with respect to this perturbation is easily obtained from g. Atr = 0, we have by change of variables g r (x) = [ dx s dω V (x)i b (x) q (x s, x,ω)p(x s, x,ω) V (x) ] I b (x) q (x s, x,ω) p(x s, x,ω). (7) When we assume that the data only contain primary reflections, this is also the gradient under Born approximation. We can then re-interpret the different components of the gradient. Under Born approximation, the incident field p satisfies the wave equation in the background model V (x)i b (x) p(x s, x,ω) V (x) I b (x) p(x s, x,ω) = w(ω)δ(x x s ), (8) and the backpropagated field q satisfies the adjoint-state equation in the background model V (x)i b (x) q (x s, x,ω) V (x) I b (x) q (x s, x,ω) = dx r δ(x x r )e (x s, x r,ω). (9) The scattered field, p 1,isgivenby V (x)i b (x) p 1(x s, x,ω) V (x) I b (x) p 1(x s, x,ω) = V (x)i b (x) r(x)p(x s, x,ω) V (x) I b (x) r(x) p(x s, x,ω), (10) and, ignoring the direct waves, the synthetics by c(x s, x r,ω) = p 1 (x s, x r,ω). (11) The perturbation r is also called reflectivity in the Born approximation context. However, this Born reflectivity has a different meaning that the one used in the classical impedance literature (Becquey et al. 1979; Lancaster & Whitcombe 2009; Lazaratos & David 2009), which can cause some confusions. In Appendix A, we try to illustrate the difference. 2.3 Green s function formulation To go further in the analysis, we now write the pressure fields and gradients in term of Green s functions. With G the Green s functions (see Appendix B), eqs (8) and (10) for the incident and scattered fields give (we do not write the subscript b to simplify the notations) p(x s, x,ω) = w(ω)g(x s, x,ω); p 1 (x s, x r,ω) = ω dx w(ω) 2 r(x)g(x V (x)i (x) s, x,ω)g(x r, x,ω) + dx w(ω) V (x) r(x) G(x I (x) r, x,ω) G(x s, x,ω). (12) In the second term of the right-hand side of the scattered field equation, we remove the spatial derivatives of the impedance perturbation with an integration by part. From eq. (9), we obtain the backpropagated field in term of Green s functions q 0 (x s, x,ω) = dx r e (x s, x r,ω)g (x, x r,ω). (13) We shall evaluate the gradient at r = 0. Although not necessary, by simplicity, we assume that the background Earth does not generate reflections, meaning that the residuals are equal to minus the scattered data (e = f d d). The scattered data, up to a noise that we neglect, should satisfy our modelling assumption, eq. (12). Namely, there exists an impedance perturbation, r, such that d(x s, x r,ω) = dx w d (ω) V (x)i (x) r(x)g(x s, x,ω)g(x r, x,ω) + dx w d (ω) V (x) I (x) r(x) G(x r, x,ω) G(x s, x,ω), (14) with w d the true (observed) wavelet. After substituting eqs (12) (14) into eq (7), the gradient reads in terms of Green s functions: g(x) = dx s dx r d y dωw(ω)w d (ω) f d (ω) V (x)i (x) V ( y)i ( y) r( y)g(x s, x,ω)g(x, x r,ω)g (x s, y,ω)g ( y, x r,ω) V ( y) + dx s dx r d ydωw(ω)w d (ω) f d (ω) r( y) V (x)i (x) I ( y)

4 304 R.-É. Plessix and Y. Li G(x s, x,ω)g(x, x r,ω) G (x s, y,ω) G ( y, x r,ω) + dx s dx r d y dωw(ω)w d (ω) f d (ω) V (x) r( y) I (x) V ( y)i ( y) G(x s, x,ω) G(x, x r,ω)g (x s, y,ω)g ( y, x r,ω) dx s dx r d y dωw(ω)w d (ω) f d (ω) V (x) I (x) G(x s, x,ω) G(x, x r,ω) V ( y) r( y) I ( y) G (x s, y,ω) G ( y, x r,ω). (15) 2.4 Filter definition To define a spectral filter, we estimate the gradient spectrum under the asymptotic approximation. Ignoring multipathing, the asymptotic 3-D Green s function is equal to G(x, y,ω) = A(x, y)exp [ ıωτ(x, y) ] (16) with A(x, y) the amplitude and τ(x, y) the traveltime between the points x and y. With asymptotic Green s functions, eq. (15) for the gradient reads g(x) = dk dx s dx r dx dνν 4 k 5 w(νk)w d (νk) f d (νk)r h (x )ˆr v (k) with A g (x s, x r, x, x )exp { ık[νφ g (x s, x r, x, x ) z ] } (17) φ g (x s, x r, x, x ) = τ(x s, x ) + τ(x r, x ) τ(x s, x) τ(x r, x). Here, A g is an amplitude factor independent of k, andν is a stretching parameter relating wavenumber and angular frequency, ω = νk. We decomposed the perturbation into a vertical and horizontal components, r v and r h, and substituted the vertical component by its Fourier expression (the hat corresponds to quantities in the spatial or frequency Fourier domain). More details can be found in Appendix B. We obtain the asymptotic expression of the gradient, eq. (17), by applying the stationary phase theorem. The derivation is detailed in Appendix B and we found the following expression for the Fourier representation of the gradient, see eq. (B4) ĝ(x, k) = k β w(νk)w d (νk) f d (νk)b(x) r h (x) ˆr v (k), (18) with B an amplitude factor independent of k. The exponent β depends on the acquisition. Assuming infinite aperture, we have β = 3 (n s + n r )/2 with n s and n r the dimensions of the source and receiver sets. For three-dimension acquisition, β is theoretically 1 and for two-dimension acquisition is 2. In practice, this value should however be adapted to account for the specific acquisition. Moreover, this derivation is based on an asymptotic analysis that has some limitations (for instance, the source, receiver, frequency coverage is never infinite in practice). In Appendix C, we describe two possible numerical approaches to determine β. We observe that the spectrum of the gradient, ĝ(x, k), depends on three terms: (i) the spectrum of the perturbation ˆr v ; (ii) the spectrum of the (observed) wavelet w d and the spectrum of the synthetic wavelet w; (iii) a factor k β that is due to the physics of the wave propagation. To have a more efficient optimization, we would like the gradient spectrum close to the perturbation spectrum. This can be achieved by defining the data spectral shaping filter f d. We in fact want w(ω)w d (ω) f d (ω)ω β = 1. In the least-squares inverse problem, the synthetics data should have the same wavelet that the filtered data, namely w(ω) = w d (ω) f d (ω). This completely defines the spectral filter f d f d (ω) = ω β/2 w d (ω). (19) And the synthetic wavelet is equal to w(ω) = ω β/2. (20) This spectral filter contains the deconvolution of the observed source wavelet and a frequency shaping in ω β/2 to compensate the propagation effect. This result, obtained without the convolution model assumption, is similar to the one derived in Lazaratos & David (2009). Here, we however build the spectral filter, f d, without the aprioriknowledge of an average Earth impedance spectrum, but rely on the knowledge of the observed source wavelet spectrum (see Appendix C for further details). This shaping of the data should help to recover a perturbation that has the same spectrum that the Earth impedance. We could also directly apply a post-processing to the gradient, that is directly shape the gradient wavenumber spectrum. Indeed, if we consider flat spectrum wavelets, that is w(ω) = w d (ω) = f d (ω) = 1, eq. (18) becomes ĝ(x, k) r h (x)ˆr v (k). k β If we now define h by its Fourier representation ĥ(k) = 1,the k β gradient shaping in the space domain consists in dz h(z z )g(x, z ). This means a pre-multiplication of the gradient by the operator H[(x, z), (x, z )] = h(z z )δ(x x ). H is an approximation of the inverse of the Hessian. It is a not diagonal approximation. We did not write the geometrical effects that are also compensated by the inverse of the Hessian (Plessix & Mulder 2004). We notice that the ray-based inverse formula for migration can also be seen as a pre-multiplication of the gradient by an inverse of the Hessian that shapes the gradient spectrum (Beylkin 1985; Bleistein 1987; Jin et al. 1992). The approach discussed in this paper follows the same type of derivations. 3 EXAMPLES We now illustrate the advantages and limitations of the approach with a synthetic and a real data set. We assume that the data have been pre-processed to remove the multiples. To compute the synthetics we solve the full two-way wave equation with absorbing boundary conditions. The inversion is then a non-linear inversion since we do not use Born approximation. We perform the waveform impedance inversion in time domain. To plot the spectrum of the impedances, we transform the wavenumber axis into a frequency

5 Waveform acoustic impedance inversion 305 axis using the formula ω = Vk, with ω the angular frequency, k the 2 wavenumber and V a velocity equal to 2000 m s Small synthetic example To build a small synthetic data set, we consider a receiver patch of 800 m 800 m with a receiver spacing of 10 m in both lateral directions. The sources are located in a 200 m 200 m square in the middle of the receiver patch with a spacing of 50 m in both lateral directions. The synthetic data are generated with the acoustic time-domain scheme used in the inversion with absorbing boundary conditions. The source spectrum is between 2 and 40 Hz and the source wavelet is a Ricker wavelet with a 20 Hz pick frequency. The model is a flat layered Earth with 1 km of water. The true impedance below 1 km is plotted in red in Fig. 3. The velocity is 1500 m s 1 in the water and 2000 m s 1 in the Earth. We carried out two inversions. In the first one, we apply no wavelet deconvolution (whitening) and no spectral shaping and in the second one, we apply a wavelet deconvolution and a spectral shaping of ω 3/4. We choose β = 3/2 because the sources have a quite small areal geometry. The initial impedance is homogeneous in the Earth. We first compare the waveform impedance inverted trace in the middle of the domain with the true normalized relative impedance filtered between 2 and 40 Hz, the frequency band used to generate the seismic data set. The results are plotted in Fig. 1 after the first iteration and in Fig. 2 after the fifth iteration. The inversion with whitening and spectral shaping gives an impedance spectrum close to the true spectrum from the first iteration. After five iterations, the inversion has more or less converged within the seismic frequency band. This is not the case when we do not pre-condition the data with a source deconvolution and a spectral shaping. To further analyse the results, we plot the waveform impedance inversion results versus the unfiltered true impedance in Fig. 3. While the spectral shaping helps to recover the impedance within the seismic frequency spectrum, it obviously does not add frequencies. The low frequencies, below 2 Hz, have not been recovered. Acquiring the low frequencies would help to improve the recovery of the impedance. The waveform impedance inversion returns the relative impedance. 3.2 Real data set We now invert a narrow azimuth real data line recorded over a North Sea oil field. The data contains 1345 shots with a spacing of about 35 m. Each shot contains 16 streamer lines of 320 receivers. The receiver spacing is 12.5 m with a maximum offset of about 4 km. The swath width is 230 m. The acquisition geometry is between a 2-D and a 3-D acquisition. We carry out a 3-D inversion. The background velocity and the initial impedance are plotted in Fig. 4. We constructed the initial impedance by simply multiplying the velocity with a density of 2000 kg m 3 in the Earth. A source wavelet has been extracted from the recorded direct wave, that travels only in the water, and the sea-bottom reflections. This wavelet has been used to whiten the data. After looking at the data frequency spectrum, we decided to invert the frequencies between 4 and 20 Hz. The data in the frequency band 4 7 Hz are relatively noisy. However, as seen in the synthetics, it is important to value as much as possible the whole frequency spectrum. We consider five inversions: (i) In the first inversion, we invert the data set (re)convolved with a source wavelet with a spectrum in ω 1/2 in the 4 20 Hz frequency band. This mimics the data set before source wavelet deconvolution. We call this run without whitening and spectral shaping. The results are plotted in Fig. 5. The normalized relative impedance exhibits some high-frequency variations due to the squaring of the source wavelet. After five iterations, some of the high-frequency variations have been attenuated and the image has a stronger low-frequency content. Figure 1. Comparison between the waveform impedance inversion results at iteration 1 (blue line) and the true impedance filtered between 2 and 40 Hz (red line).

6 306 R.-É. Plessix and Y. Li Figure 2. Comparison between the waveform impedance inversion results at iteration 5 (blue line) and the true impedance filtered between 2 and 40 Hz (red line). Figure 3. Comparison between the waveform impedance inversion results at iteration 5 (blue line) and the true impedance. Figure 4. Background velocity and background (initial) impedance.

7 Waveform acoustic impedance inversion 307 Figure 5. Normalized relative impedance at iterations 1 and 5 with the data set without whitening and spectral shaping. Figure 6. Normalized relative impedance at iterations 1 and 5 with the data set with whitening only. Figure 7. Normalized relative impedance at iterations 1 and 5 with the data set with whitening and a spectral shaping with ω 1/2, corresponding to β = 1. (ii) In the second inversion, we invert the whitened data set, namely after source wavelet deconvolution. We call this run with whitening only. The results are plotted in Fig. 6. Comparing with the previous inversion, the first iteration has a lower frequency content, since we do not have the effect of squaring the source wavelet. During the iterations, the low frequencies are slightly boosted. (iii) In the third inversion, we invert the whitened data set and apply a spectral shaping of ω 1/2 ; this corresponds to β = 1, the theoretical value for 3-D acquisition. We call this run with whitening and spectral shaping with ω 1/2. The results are plotted in Fig. 7. The iterations slightly modify the amplitudes, correcting for the illumination effects. (iv) In the fourth inversion, we invert the whitened data set and apply a spectral shaping of ω 3/4 ; this corresponds to β = 3/2, the theoretical value for a line source and an areal receiver acquisition. We call this run with whitening and spectral shaping with ω 3/4. The results are plotted in Fig. 8. The iterations slightly modify the amplitudes, correcting for the illumination effects.

8 308 R.-E. Plessix and Y. Li Figure 8. Normalized relative impedance at iterations 1 and 5 with the data set with whitening and a spectral shaping with ω 3/4, corresponding to β = 3/2. Figure 9. Normalized relative impedance at iterations 1 and 5 with the data set with whitening and a spectral shaping with ω 1, corresponding to β = 2. (v) In the fifth inversion, we invert the whitened data set and apply a spectral shaping of ω 1 ; this corresponds to β = 2, the theoretical value for 2-D acquisition. We call this run with whitening and spectral shaping with ω 1. The results are plotted in Fig. 9. The iterations slightly modify the amplitudes, correcting for the illumination effects. To verify whether the spectral shaping helps to retrieve an impedance with a spectrum close to the one measured by well logs, we compute an average impedance spectrum from nine wells situated within 5 km of the seismic line. The average log impedance is plotted in Fig. 10 together with the log impedance of each well to show the spread. We observe that some differences between the spectra of the nine wells and the average one. In Fig. 11, we compare the average spectra of the retrieved relative impedances of the four last runs with the average log impedance. We do not plot the results obtained with the run without whitening and spectral shaping, since the high frequencies are clearly overemphasized at the early iteration in this case. The first update with the data set with whitening, Fig. 11(a), has a too strong high-frequency content. The results with the three different spectral filters, Figs 11(b) (d), produce an average impedance close to the well-log impedance. The result with the ω 3/4 spectral filter seems to give the best fit. However, we have some noticeable discrepancies between the well logs (Fig. 10). Since one of the wells is almost on the seismic line (at x = 67.5 km), we now compare the impedance spectrum of this well log with the corresponding trace spectra of the FWI results after the first iteration in Fig. 12, and after the fifth iteration in Fig. 13. After the five iterations, the differences between the welllog impedance and the trace extracted from the run with whitening only, Fig. 13(a), is still significant. Nevertheless, we notice that the low frequencies have been boosted during the iterations to better match the Earth impedance. This shows that a spectral shaping that accounts for the wave propagation effects can speed up the convergence. In Figs 12(b) (d), we observe that the spectral filters really help to recover the Earth impedance spectrum. The spectral shaping with ω 3/4 gives the best fit at the first iteration. After five iterations, Fig. 13(b) (d), the three runs with spectral shaping lead to relatively similar results, the spectral shaping with ω 3/4 being still the best one. In Fig. 14, we finally plot the absolute impedances retrieved after five iterations of the four last runs. The layers with a low impedances (dark blue, if we obviously exclude the water layer) correspond to layers with gas. The absolute impedances are in fact the results of the waveform impedance inversion since we solve the full wave equation. However, intermediate wavenumbers can still be missing, as shown in the synthetic example. 4 C O N C LU S I O N S A N D D I S C U S S I O N We have applied waveform inversion to retrieve relative impedance from reflection data. We followed the idea that the convergence

9 Waveform acoustic impedance inversion 309 Figure 10. In blue, the log impedances measure in nine different wells and in red, the average impedance. Figure 11. Comparison between the spectrum of the average log impedance and the average relative impedance retrieved at iteration 1 of the four FWI runs. of the iterative process can be improved by shaping the data in order to obtain a spectrum of the gradient of the misfit close to the Earth impedance spectrum. Thanks to an asymptotic analysis of the gradient of the misfit function, we derive a theoretical spectral shaping filter in order to remove some of the frequency dependencies due to the wave propagation. The data filter has the form of ω β/2, with ω the angular frequency and β an exponent notably depending on the acquisition geometry. It corresponds to a shaping of k β of the gradient, with k the wavenumber. With real applications and surface seismic data, we found that with an ideal line (2-D) acquisition β is equal to 2, and with an ideal areal (3-D) acquisition β equal to 1. With most of the real applications, this gives an upper and lower limit for β. Assuming that we derive the source wavelet directly from the data or measure it, the waveform impedance inversion consists to first whiten/deconvolve the data from the source wavelet, secondly to apply the spectral shaping and to define the synthetic source wavelet accordingly, and finally to iteratively minimize the data misfit function. We tested this approach on a small synthetic example to illustrate the method. This allows us to point out the role of the low frequencies. Indeed, the waveform impedance inversion retrieves the relative impedance. The long wavelengths of the impedance

10 310 R.-É. Plessix and Y. Li Figure 12. Comparison between the spectrum of the log impedance of the well situated at x = 67.5 km and the spectrum of the relative impedance retrieved at iteration 1 at x = 67.5 km of the four FWI runs. Figure 13. Comparison between the spectrum of the log impedance of the well situated at x = 67.5 km and the spectrum of the relative impedance retrieved at iteration 5 at x = 67.5 km of the four FWI runs. should be known apriori. This is why in all our tests we assume the background velocity known. Finally, we invert a real data set recorded over a hydrocarbon field in the North Sea. We compare the results with and without source deconvolution of the data, and with or without spectral shaping filter. Since the filter we derived is independent of the well measurements, we could compare the waveform impedance inversion results with the well-log impedances to evaluate our results. The comparisons confirm the relevance of the spectral shaping to improve the convergence of the inversion iterative process. The theoretical values for the exponent, β, of the spectral shaping filter give good indicative values. The optimal choice for β may be sometimes difficult to estimate because the theoretical values were found with an asymptotic analysis that assumes infinite receiver and source coverage. In practice, the coverage is often incomplete, which may make β depth dependent. However, the value of β should aprioribe between 1 and 2 with classic surface seismic acquisition. In this real example, the results obtained with β equal 3/2 are quite satisfactory.

11 Waveform acoustic impedance inversion 311 Figure 14. Absolute impedance at iteration 5. AC K N OW L E D G E M E N T S The authors thank Shell Upstream International Europe for providing the data and Shell Global Solutions International for allowing the publication of this work. We are also grateful to our colleagues, Sophie Michelet for providing the initial models, Greg Hester and Isabel White for pre-processing the real data set, and Henning Kuehl and Alexandre Stopin for discussions and help with some of the processing tools. We also thank the reviewers whose comments help us to clarify some points of the manuscript. REFERENCES Baeten, G., de Maag, J.W., Plessix, R.-E., Klaassen, M., Qureshi, T., Kleemeyer, M., ten Kroode, F. & Zhang, R., The use of the low frequencies in a full waveform inversion and impedance inversion land seismic case study, Geophys. Prospect., 61, Bamberger, A., Chavent, G., Hemon, C. & Lailly, P., Inversion of normal incidence seismograms, Geophysics, 47, Becquey, M., Lavergne, M. & Willm, C., Acoustic impedance logs computed from seismic traces, Geophysics, 44, Beylkin, G., Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, J. Math Phys., 26, Bleistein, N., On the imaging of reflectors in the Earth, Geophysics, 52, Bunks, C., Salek, F.M., Zaleski, S. & Chavent, G., Multiscale seismic waveform inversion, Geophysics, 60, Chavent, G. & Plessix, R.-E., An optimal true-amplitude least-squares prestack depth-migration operator, Geophysics, 64, Claerbout, J., Fundamentals of Geophysical Data Processing, McGraw-Hill. Delprat-Jannaud, F. & Lailly, P., A fundamental limitation for the reconstruction of impedance profiles from seismic data, Geophysics, 70, R1 R14. Gauthier, O., Virieux, J. & Tarantola, A., Two-dimensional nonlinear inversion of seismic waveform: numerical results, Geophysics, 51, Jannane, M. et al., Wavelengths of Earth structures that can be resolved from seismic reflection data, Geophysics, 54, Jin, S., Madariaga, R., Virieux, J. & Lambare, G., Two-dimensional asymptotic iterative elastic inversion, Geophys. J. Int., 108, Kuehl, H. & Sacchi, D.M., Generalized least-sqaures DSR migration using a common angle imaging condition, in Proceedings of the 71th Annual Meeting, MIG 5.6, pp , SEG. Lailly, P., The seismic problem as a sequence of before-stack migrations, in Conference on Inverse Scattering: Theory and Applications, pp , ed. Bednar, J., SIAM.

12 312 R.-É. Plessix and Y. Li Lancaster, S. & Whitcombe, D., Fast track coloured inversion, in Proceedings of the 70th Annual Meeting, pp , SEG. Lazaratos, S. & David, R.L., Inversion by pre-migration spectral shaping, in 79th Annual Meeting, pp , SEG. Lazaratos, S., Chikichev, I. & Wang, K., Improving the convergence rate of full wavefield inversion using spectral shaping, in 81st Annual Meeting, pp , SEG. Lutz, J.B.M. & Strijbos, F., High definition processing, in Proceedings of the 79th Annual Meeting, p. T001, EAGE. Mora, P., Nonlinear two-dimensional elastic inversion of multioffset seismic data, Geophysics, 52, Mora, P., Inversion = migration + tomography, Geophysics, 54, Mulder, W. & Plessix, R.-E., How to choose a subset of frequencies in frequency-domain finite-difference migration, Geophys. J. Int., 158, Mulder, W. & Plessix, R.-E., Exploring some issues in acoustic full waveform inversion, Geophys. 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Wu, R.S. & Aki, K., Scattering characteristics of elastic waves by an elastic heterogeneity, Geophysics, 50, APPENDIX A: REFLECTIVITY DEFINITION In this appendix, we illustrate some of the differences between the Born reflectivity defined as the impedance perturbation in eq. (6) and the reflectivity used in acoustic impedance inversion and the convolution model (Becquey et al. 1979; Lazaratos & David 2009). We assume a layered Earth with the impedance equal to N I (z) = i 0 + (i n i n 1 )H e (z z n ) n=1 with H e the Heaviside function, z n the depth of the reflector and i n the impedance of the layer n. In a Kirchhoff sense, the migration estimates the reflection coefficients,r. In terms of impedance, at zero-offset incident angle, the reflection coefficient at the depth z can be written as follows: R(z) = I (z + z) I (z z) I (z + z) + I (z z) with z a small depth perturbation. When z is small enough, we obtain R(z) = z I (z) ln[i (z)] = z. I (z) z z In a layered Earth, this gives N i n i n 1 R(z) = (2 z) δ(z z n ) i n + i n+1 n=1 with δ the Dirac distribution that corresponds to the derivative of the Heaviside function. At z n, we estimated the impedance by the average. This is the basis of seismic acoustic impedance inversion (Becquey et al. 1979). Indeed, the migration, m, if we do not mention the amplitudes, is of the form m(z) = dz w m (z z )R(z )dz, with w m a migration wavelet, that depends on the observed source wavelet and on the wavelet used in the computation of the incident field. After spike deconvolution, for instance, we retrieve R(z) andwe obtain the impedance by depth integration. Under Born approximation, we define the Born reflectivity by r(z) = I (z) I b(z). I b (z) Let us consider that the background impedance is obtained by averaging the impedance I as follows: I b (z) = I (z + z) + 2I (z) + I (z z). 4 By noticing that 2 I (z) z 2 I b (z) = I (z) + z 2 2 I (z) z 2. This gives r(z) = 1 I (z) z2 2 I (z) z 2. I (z+dz) 2I (z)+i (z dz),wehave z 2

13 Waveform acoustic impedance inversion 313 Replacing the impedance by its expression gives r(z) = (2 z 2 ) N n=1 i n i n 1 i n + i n+1 δ (z z n ) = z R (z). The presence of the second derivative in z instead of the first one in the formula of r means that we have the derivative of the Dirac instead of the Dirac distribution. There is a π/2 phase difference between the reflection coefficient and the Born reflectivity. With reverse time migration it is usually compensated by pre-processing the data since we have after integration by parts dz w m (z z )R (z ) = dz w m (z z )R(z ). Without the flat layered assumption, the spatial derivatives should be taken along the normal to the reflectors. APPENDIX B: ASYMPTOTIC DERIVATION In this appendix, we derive the asymptotic expression of the gradient. We again assume a flat layered Earth for simplicity. However, this is not needed. The 3-D Green s function is equal to G(x, y,ω) = A(x, y)exp [ ıωτ(x, y) ] with A the amplitude and τ the traveltime, and by reciprocity, we have A(x, y) = A( y, x) andτ(x, y) = τ( y, x). We shall also assume that A and τ are smooth. This leads to y G(x, y,ω) ıω y τ(x, y)g(x, y,ω). We call l the vector defined by y τ(x, y) = l(x, y) V b ( y). Substituting the Green s functions in eq. (15) by their asymptotic expressions gives g(x) = dx s dx r dx dωω 4 (ω)w d (ω) f d (ω)r(x ) A g (x s, x r, x, x )exp [ ıωφ g (x s, x r, x, x ) ], (B1) with A g (x s, x r, x, x ) = A(xs,x)A(xr,x)A(xs,x )A(x r,x ) V (x)i (x)v (x )I (x ) [ 1 l(xs, x) l(x r, x) l(x s, x ) l(x r, x ) +l(x s, x) l(x r, x)l(x s, x ) l(x r, x ) ] ; φ g (x s, x r, x, x ) = τ(x s, x ) + τ(x r, x ) τ(x s, x) τ(x r, x). To determine the high-frequency approximation, that is the approximation for large k, we shall use the stationary phase theorem (Treves 1980) that states that a function of k v(k) = dx u(x, k) exp[ıkφ(x)] with x a n-dimension variable, asymptotically behaves, when u and φ are smooth functions, as ( ) 2π n/2 v(k) a(x c ) u(x c, k) exp[ıkφ(x c )] k with x c, the stationary point defined by φ (x c ) = 0 when φ (x c ) 0. φ is the first derivative (gradient) of φ with respect to x c and φ is the second derivative (Hessian). The term a represents an amplitude and phase term depending on the sign and determinant of φ (x c ). This term is independent of k. Since we are only interested in determining the relative spectrum of the gradient, we don t give its exact expression. When there are several stationary points, the approximation of v contains a summation over the stationary points. To apply the stationary theorem, we represent the impedance perturbation by its Fourier expression in the direction of the discontinuities, with a flat layered Earth in the depth direction r(x) = r(x, z) = r h (x) dk ˆr v (k)exp( ıkz). We have split the perturbation into a smooth horizontal component r h and a discontinuous one r v with ˆr v its Fourier representation. Here, k is the vertical wavenumber and x represents the horizontal coordinates. Injecting this Fourier representation into eq. (B1) gives g(x) = dk g(x, k) (B2) with g(x, k) = dx s dx r dx dνν 4 k 5 (νk)w d (νk) f d (νk)r h (x )ˆr v (k) A(x s, x r, x, x )exp [ ık(νφ g (x s, x r, x, x ) z ) ]. Here, we have applied the change of variables ω = kν. The stationary points satisfy τ(x s, x) + τ(x r, x) = τ(x s, x ) + τ(x r, x ); xs τ(x s, x) = xs τ(x s, x ); xr τ(x r, x) = xr τ(x r, x ); x τ(x s, x ) = x τ(x r, x ); 1 = ν zτ(x s, x ) + z τ(x r, x ). In a smooth background without multipathing, the five first equalities tell us that x = x and that x should be a specular point on a reflector. The last equation defines ν. It corresponds to a (wavelet) stretching parameter (Tygel et al. 1996). With θ the reflection angle at the specular point, we have 1 ν = 2cos(θ). V (x) At a specular point, the asymptotic expression of g reads g(x, k) = k β w(νk)w d (νk) f d (νk)b(x)r h (x)ˆr v (k)exp( ıkz) (B3) with { β = 5 n/2; B(x) = (2π) n/2 ν 4 a(x c )A(x s (x), x r (x), x, x). The dimension coefficient n depends on the acquisition since we have an integral over sources and receivers in eq. (B1). In threedimension space we can write β = 3 (n s + n r )/2 with n s and n r the dimension of the source and receiver acquisition sets. When the sources and receivers have an areal coverage (so-called 3-D acquisition), we have n = 8(n s = n r = 2) and then β = 1. When we just have a line acquisition (so-called 2-D acquisition), we have n = 6(n s = n r = 1) and then β = 2. With actual surface acquisition,

14 314 R.-É. Plessix and Y. Li β is aprioribetween 1 and 2. However, β may be larger than 2 for small acquisitions, for instance when we have only one receiver or one source. We can now write from eqs (B2) and (B3) g(x, k) = dk ĝ(x, k) exp( ıkz) with ĝ the Fourier representation of the gradient given by ĝ(x, k) = k β w(νk)w d (νk) f d (νk)b(x)r h (x)ˆr v (k). (B4) In this calculus based on asymptotic considerations, β does not depend on the Earth parameters. It just depends on the acquisition. In practice, we don t use an asymptotic approach in the inversion and the acquisition has finite aperture. β is used to pre-condition the data and we probably only need an approximate estimation because we still carry out some iterations. The results obtained with the real data seem to confirm this hypothesis. They also suggest that a clearly wrong estimate of β does not help to reduce the number of iterations in the iterative minimization algorithm. APPENDIX C: FINDING β NUMERICALLY To find numerically the exponent β of the spectral filter, eq. (19), one can compute the gradient with a white synthetic source wavelet, that is w(ω) = 1, and with an observed data set after source deconvolution but without spectral shaping, that is w d (ω) = 1andf d (ω) = 1. Eq. (B4) gives after summation over the horizontal coordinates, with ĝ ave the average gradient spectrum: ĝ ave (k) k β ˆr v (k). (C1) ( means proportional to.) If an average impedance spectrum from well logs is known, β can then be found by dividing the gradient and well-log impedance spectrum. Without well-log information, we can compare the data and gradient spectrum. Let us first estimate the data spectrum with the asymptotic analysis. Substituting the asymptotic Green s function in eq. (14) and using the Fourier representation of the perturbation assuming one reflector/contrast, that is r h (z) = δ(z z 0 ) with z 0 the reflector depth, we have d(x s, x r,ω) = dx dz dσ ω 3 w d (ω)a d (x s, x r, x)r h (x)ˆr v (σω) exp [ ıω(φ d (x s, x r, x) σ (z z 0 )) ]. where σ relates frequency and wavenumber k = σω and with { 1 Ad (x s, x r, x) = A(x V (x)i ((x) s, x)a(x r, x)[1 + l(x s, x) l(x r, x)]; φ a (x s, x r, x) = τ(x s, x) + τ(x r, x). The stationary conditions are z = z 0 ; x τ(x s, x) = x τ(x r, x); σ = z τ(x s, x) + z τ(x r, x). The first two equations show that the stationary conditions are satisfied at the specular points on the reflector. The third equation gives the stretching σ = 2cos(θ) with θ the reflection angle at the V (x) specular point. The asymptotic expression for d is then d(x s, x r,ω) = ω α B d (x s, x r )r v (x)w d (ω)exp ( ıω(τ(x s, x) +τ(x r, x)) = ω α B d (x s, x r )r v (x)ˆr h (σω)w d (ω)exp( ( ı(ω(τ(x s, x) +τ(x r, x) σ z 0 )) ). since r h (k) = exp(ıkz 0 ) with r v (z) = δ(z z 0 ). Here B d is an amplitude factor independent of ω that we do not detail. With m the dimensional of the integral, we have α = 3 m/2. The actual acquisitions are always in three-dimension space, therefore m = 4andα = 1. With many reflectors, we would sum the different contributions assuming no multiscattering and we obtain with j indexing the contrasts d(x s, x r,ω) = j ω α B d, j (x s, x r )r v (x j )ˆr h (σ j ω)w d (ω) exp ( ı(ω(τ(x s, x) + τ(x r, x) σ j z j )) ). At short offsets, σ j 1, the arguments of the exponentials are almost 0. We then get the spectrum of the data, d ave after stacking over sources and receivers ( ) 2ω d ave (ω) ˆr h ω α w d (ω). (C2) V Since k = 2ω, we obtain with eq. (B4) after averaging over the V horizontal coordinates g ave (ω) d ave (ω) ωβ α. Since with real data α = 1, computing the ratio between the gradient spectrum and the data spectrum gives an estimate from β. The second approach to derive β is based on the convolution model since to obtain eq. (C2), we (re)derived the convolution model. Finally we notice that, assuming the convolution model, the ratio of the data spectrum by the impedance spectrum multiplied by ω α gives the observed source wavelet spectrum. This leads to the data spectral filter proposed by Lazaratos et al. (2011).