Th P10 13 Alternative Misfit Functions for FWI Applied to Surface Waves

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1 Th P0 3 Alternative Misfit Functions for FWI Alied to Surface Waves I. Masoni* Total E&P, Joseh Fourier University), R. Brossier Joseh Fourier University), J. Virieux Joseh Fourier University) & J.L. Boelle Total E&P) SUMMARY The aim of this study is to determine the advantages and limitations of different misfit functions for an alication of Full Waveform Inversion FWI) to surface waves. The difference-based L norm, classically used in FWI and sensitive to both amlitude and hase information, suffers from cycle-skiing and local minima. For slow surface waves roagating in the low velocity near surface, the roblem of cycleskiing is even greater due to their small wavelengths. In the absence of low frequencies, convergence may not be ossible when starting from a smooth initial mode. Alternative misfit functions alied in various data domains are therefore investigated with the aim of overcoming this issue. Taking the difference-based L norm as a basis for comarison, simle synthetic tests are conducted to evaluate a weighted cross-correlation and a singular value decomosition aroach as alternative misfit functions, as well as investigating the effect of calculating the residual in different data domains such as the omega-k), tau-) and omega-) domains. 75 th EAGE Conference & Exhibition incororating SPE EUROPEC 3 London, UK, 0-3 June 3

2 Introduction The construction of subsurface velocity models is a central roblem for oil & gas exloration. As well known rolific hydrocarbon basins have been exlored and roduced, it is now necessary to investigate more comlex regions, such as on-shore foothills. In such cases, the heterogeneous near-surface has a great imact on the comlexity of seismic roagation and can obstruct the imaging of deeer targets. Although conventionally viewed as coherent noise or "ground roll", surface waves samle the near subsurface and can be used for imaging. In civil engineering, disersion curve inversion allows imaging the first tens of meters. However this aroach relies on a D assumtion and only smooth lateral heterogeneities are tolerated. An alternative method may be Full Waveform Inversion FWI), that extends beyond D limitations and avoids time icking or disersion analysis. Classical FWI FWI is a high resolution technique used to derive quantitative models of the subsurface by matching the full observed seismogram with a corresonding synthetic seismogram calculated from a velocity model, and solving a local otimization roblem. The L norm of the difference is conventionally used to calculate the misfit Tarantola, 984), fitting both the amlitude and the hase of the waveforms: C dif f = t x dobs t,x) d cal t,x) ), ) where d obs t,x) is the measured data and d cal t,x) is the calculated data recorded at time t and offset x. As the misfit function is minimized in a least-squares sense, the model is iteratively udated with a gradient-based descent method until a minimum is reached Virieux and Oerto, 09). By exloiting the full data content and using a strict data-matching aroach, this method aears to be very sensitive and may not be robust. Non-linearities, such as cycle skiing, can reduce the convexity of the misfit function Bunks et al., 995; Mulder and Plessix, 08) and the minimization may get stuck in a local minimum. In the absence of very low-frequency data, the initial velocity model needs to exlain the data to within half a wavelength, so that it lies within the small basin of attraction of the global minimum and can converge. For slow surface waves roagating in the low velocity near surface, the roblem of cycle-skiing is even greater due to their small wavelengths. Synthetic datasets were created using a discrete wavenumber summation method Bouchon and Aki, 977), for horizontally layered media with a free surface, and simulating 3D elastic wave roagation with a Ricker wavelet source of 0 Hz eak frequency. Figure a shows the two-layer model used to create the "observed" dataset to which random Gaussian noise is added Figure b). A grid analysis is erformed on the S-velocity and the deth of the first layer to investigate the accuracy required for the initial model. Even for this simle framework and only small shifts in the model arameters examle in Figure c), the grid analysis result for the classical difference-based L norm aroach Figure a) contains many local minima due to the high amlitude of the surface waves that dominate the misfit. In this study, alternative, more robust, misfit functions alied in various data domains are investigated to imrove the convexity of the valley of attraction and reduce the resence of local minima. To evaluate the misfit functions, grid analysis results for the same synthetic test are comared. Alternative misfit functions Current solutions to calculating the misfit more robustly are based on other norms such as the hybrid L /L or Huber norm Brossier et al., 0; Guitton and Symes, 03) or on zero-lag cross-correlation Routh et al., ), but these also suffer from cycle-skiing in the absence of low frequencies. A weighted cross-correlation roosed by Van Leeuwen and Mulder 08) is investigated here as a more robust alternative to the classical difference-based L norm. The misfit is given by a cross correlation on the time axis of the observed and calculated data where events are searated by arrival times. C Wi = Δt x W i Δt) t d obs t + Δt,x)d cal t,x)). ) 75 th EAGE Conference & Exhibition incororating SPE EUROPEC 3 London, UK, 0-3 June 3

3 The weighting W i Δt) is alied to each time samle. Two weightings are tested in this study. The first W Δt)=Δt/Δt max, linearly enalizes with distance away from zero lag. The second W Δt)=e αδt is a Gaussian weighting to maximize zero-lag energy and with a width controlled by the α arameter, giving a misfit function whose negative is minimized. An aroriate width needs to be chosen to at least be in the order of the length of the wavelet, since it can greatly influence the convexity of the misfit function Van Leeuwen and Mulder, 08). The grid analysis result for the enalized version of the cross-correlation Figure b) illustrates how it is highly sensitive to noise at large lags and not robust enough. Instead the sensitivity of the Gaussian weighted cross-correlation can be better tuned to obtain a convex result with no local minima Figure c) allowing convergence from an initial model further away from the true one. The weighting alied to the cross-correlation is therefore critical for a stable misfit function. The weighted cross-correlation is however not sensitive to the frequency and hase rotation of an event. Due to the disersive roerty of surface waves, the frequency may contain key information on the deth of the signal and may need to be identified. Furthermore "cross-talk" may occur for multile arrivals. Therefore couling this misfit function with a strategy to searate arrivals, such as comaring data in a different domain, needs to be considered. For comarison, a recently roosed misfit function based on a singular-value decomosition SVD) aroach Moghaddam and Mulder, ) is also tested. An SVD is alied to data matrix A), of size number of receivers by number of sources, for each frequency so that A obs ) =U obs S obs V H obs. When the calculated data is equal to the observed data then the matrix S)=U H obs A cal)v obs, will be diagonal. A weighting W ij is alied to the misfit function to linearly enalize the off-diagonal values: C SV D = i j Wij S ij ) ). 3) The grid analysis result Figure d) shows a larger basin of attraction than the classical difference-based L norm, but sensitivity is lacking to ensure convergence for high S-velocities and small layer deths. Alternative data domains The data domain in which observed and calculated datasets are comared also affects the sensitivity of the misfit function. Perez Solano et al. ) rooses the, k) domain, to reduce the resence of local minima for the difference-based L norm Eq 4). A weighted cross-correlation of the modulus of the,k) data is alied on the wavenumber k-axis Eq 5): C Wi = C dif f = Δk k dobs,k) d cal,k) ), 4) W i Δk) k d obs,k + Δk) d cal,k) ). 5) The τ, ) domain or "slant-stack" is also investigated. In this domain, data have undergone a linear move-out LMO) correction, and are summed over the offset axis. This is done for a range of slowness values. Searating events by their slowness may reduce the cycle-skiing roblem, and stacking may also make the misfit function more robust in the resence of noise. Both the difference-based L norm Eq 6) and a weighted cross-correlation alied on the slowness axis Eq 7) are considered: C Wi = τ C dif f = τ Δ dobs τ,) d cal τ,) ), 6). W i Δ)d obs τ, + Δ)d cal τ,)) 7) 75 th EAGE Conference & Exhibition incororating SPE EUROPEC 3 London, UK, 0-3 June 3

4 Finally, the, ) domain, equivalent to the,k) domain but with a different samling, may also hel searate events by their slowness as well as identify their frequency, which can be helful to use the disersive roerty of surface waves. Again the difference-based L norm Eq 8) and the weighted cross-correlation Eq 9) misfit functions are tested: C Wi = C dif f = Δ dobs,) d cal,) ), 8) W i Δ) d obs, + Δ) d cal,) ). 9) The difference-based L norm becomes more convex in all alternative data domains tested as shown by Figures e,h,k). Convergence is esecially successful in the,k) and, ) domains. The domain aears to efficiently mitigate non-linearities related to disersive effects. Where local minima are no longer resent, it may be ossible to start with an initial model far from the true one. On the other hand, the enalized cross-correlation is not very successful. Noise dominates the misfit in the,k) domain Figure f), and the global minimum is no longer at the true S-velocity and deth of the layer for the other tested domains Figures i, l). Yet when a gaussian weighting is alied, the global minimum is correctly centered and the convexity can be tuned by the α arameter Figures j, m). Only in the,k) domain Figure g), some local minima remain resent even with an otimized weighting, limiting convergence. Conclusions We have used numerical tests to comare alternative FWI misfit functions for surface wave alications. Both reviously roosed SVD Moghaddam and Mulder, ), and difference-based, k) domain Perez Solano et al., ) misfit functions are validated as imrovements to classical difference-based FWI. Furthermore the difference-based, ) domain aroach as well as a Gaussian weighted crosscorrelation in the t,x), τ, ) and, ) domains are also shown to be romising alternatives. Crosscorrelations on other axis or double cross-correlations Van Leeuwen and Mulder, 08) could also be tested. Our future work will investigate the robustness of these misfit functions on laterally varying models. Acknowledgements The authors would like to thank TOTAL E&P for ermission to show these results. This work was roduced using the CIMENT high-erformance comuting facilities Université Joseh Fourier, Grenoble). References Bouchon, M. and Aki, K. [977] Discrete wave-number reresentation of seismic-source wave fields. Bulletin of the Seismological Society of America, 67), Brossier, R., Oerto, S. and Virieux, J. [0] Which data residual norm for robust elastic frequency-domain full waveform inversion? Geohysics, 753), R37 R46, doi:0.90/ Bunks, C., Salek, F.M., Zaleski, S. and Chavent, G. [995] Multiscale seismic waveform inversion. Geohysics, 605), Guitton, A. and Symes, W.W. [03] Robust inversion of seismic data using the Huber norm. Geohysics, 684), 30 39, doi:0.90/.984. Moghaddam, P. and Mulder, W. [] The Diagonalator, an alternative cost functional for wave-equation inversion. Exanded Abstracts, 74 th Annual meeting, EAGE, W0. Mulder, W. and Plessix, R.E. [08] Exloring some issues in acoustic full waveform inversion. Geohysical Prosecting, 566), Perez Solano, C., Donno, D. and Chauris, H. [] Alternative objective function for inversion of surface waves in D media. Exanded Abstracts, 8 th Euroean Meeting of Environmental and Engineering Geohysics,Near Surface Geoscience, A. Routh, P. et al. [] Encoded simultaneous source full-wavefield inversion for sectrally shaed marine streamer data. SEG Technical Program Exanded Abstracts, 30), , doi:0.90/ Tarantola, A. [984] Inversion of seismic reflection data in the acoustic aroximation. Geohysics, 498), Van Leeuwen, T. and Mulder, W. [08] Velocity analysis based on data correlation. Geohysical Prosecting, 566), , ISSN , doi:0./j x. Virieux, J. and Oerto, S. [09] An overview of full waveform inversion in exloration geohysics. Geohysics, 746), WCC7 WCC. 75 th EAGE Conference & Exhibition incororating SPE EUROPEC 3 London, UK, 0-3 June 3

5 a) b) 0 offset m) c) 0 offset m) time s) 3 4 time s) Observed dataset t,x) 6 Calculated dataset t,x) Figure Two-layer model a) used to create the observed dataset with added random Gaussian noise b), and an examle of the calculated dataset for a layer S-velocity at 480 m/s and deth of m c). a) b) c) layer S-velocity m/s) layer S-velocity m/s) layer S-velocity m/s) layer S-velocity m/s) d) Difference-based L norm t,x) Penalized cross-correlation t,x) Gaussian weighted cross-correlation t,x) SVD aroach omega,x) e) f) g) h) Difference-based L norm omega,k) i) Penalized cross-correlation omega,k) Gaussian weighted cross-correlation omega,k) j) k) Difference-based L norm tau,) l) Penalized cross-correlation tau,) Gaussian weighted cross-correlation tau,) m) Difference-based L norm omega,) Penalized cross-correlation omega,) Gaussian weighted cross-correlation omega,) Figure Two-arameter grid analysis for all misfit functions tested in this study a-m), for an observed dataset with the true global minimum at 450 m/s layer S-velocity and m layer deth. 75 th EAGE Conference & Exhibition incororating SPE EUROPEC 3 London, UK, 0-3 June 3