In this study, we propose a multi-parameter full waveform inversion strategy that can simultaneously recover the velocity and density in acoustic media.
We P01 09 A Multi-parameter Full Waveform Inversion Strategy in Acoustic Media J.Z. Yang* (Tongji University), Y.Z. Liu (Tongji University) & L.G. Dong (Tongji University)
SUMMARY Full waveform inversion (FWI) is a challenging data-matching procedure that exploits the full information from the seismic data to obtain high-resolution models of the subsurface. To best fit the observed seismograms, the forward modeling should correctly account for the wave propagation phenomena present in the recorded data, especially for the wide-aperture acquisition geometry. Thus, the mono-parameter acoustic FWI should be extended to multi-parameter FWI, such as P-wave velocity, shear-wave velocity, density, attenuation, anisotropy, or other related parameters. In this study, we propose a multi-parameter full waveform inversion strategy that can simultaneously recover the velocity and density in acoustic media. The strategy consists of two stages and the acoustic wave equation is parameterized by velocity and density. In the first stage, the velocity and density are simultaneously inverted. In this case, the inverted velocity is reasonable while the density profile is deviated from the true one. During the second stage, the recovered velocity model in the first stage is reused as the initial velocity model. Then, the velocity and density are reconstructed at the same time. The final results show that both the velocity and density are reconstructed well. The synthetic numerical examples prove the validity of the method.
76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014
Introduction Multi-parameter full waveform inversion is an ill-posed inverse problem. Kӧhn et al. (2012) showed that, in multi-parameter elastic FWI, the Lamé parameters, seismic velocities and impedances could be well reconstructed. However, the inverted density showed the largest ambiguity by all parameterizations. Based on radiation pattern analysis, Forgues and Lambaré (1997) found that the density was difficult to be revealed. Thus, most authors estimate density using an empirical formula or fixed the density to a constant value in inversion. Jeong et al. (2012) proposed a full waveform inversion strategy for density in the frequency domain. They first inverted the Lamé constants with density fixed at an arbitrary value, and then both the velocities and density were updated simultaneously. In this study, we propose a hierarchical multi-parameter full waveform inversion strategy that can simultaneously recover both velocity and density in acoustic media. The procedure is different from that of Jeong et al. (2012). Our strategy consists of two stages and the acoustic wave equation is parameterized by velocity and density. In the first stage, the velocity and density are simultaneously inverted. In the second stage, the recovered velocity model in the first stage is reused as the initial velocity model. Then, the velocity and density are updated at the same time. The synthetic numerical examples prove the validity of the method. For computational efficiency, the random phase-encoded simultaneous-source technique (Krebs et al., 2009) is employed. Inverse Theory The goal of FWI is to find a subsurface parameter m by minimizing the data residual δ u between the synthetic data u mod and the observed data u obs . Mathematically, such a misfit can be measured by an objective function: 1 E = δ u Tδ u . (1) 2 The objective function can be minimized by updating the model parameters m n using the quasiNewton method: m n +1 = m n − α n H n ∇E ( m ) , (2) Here ∇E ( m ) is the gradient direction of the objection function with respect to the model parameter,
α is the step length, and H n an approximation of the inverse of the Hessian found by the LBFGS method. The 2D frequency domain acoustic wave equation for isotropic heterogeneous media can be written as: 1 ω2 P ( x, ω ) − ∇ − ∇P ( x, ω ) = δ ( x − xs ) S (ω ) , (3) 2 ρ ( x ) VP ( x ) ρ (x)
where ρ and VP are density and velocity respectively, ω is angular frequency, P ( x, ω ) is frequency
domain pressure wavefield, and S (ω ) is the source function at source location xs . The gradient of the objective function with respect to velocity and density can be expressed as: ∂E 2ω 2 ∂E ω2 1 =− Pf Pb , (4a) = − 2 2 Pf Pb + 2 ∇Pf ∇Pb , (4b) 3 ∂VP ∂ρ ρVP ρ VP ρ where Pf is the forward wavefield, and Pb is the backward wavefield of the residual. If we directly use equations (4) to update velocity and density simultaneously in the inversion, we can only obtain reasonable velocity, the density would be much worse. That’s because the wavefield is much more sensitive to velocity than to density (Jeong et al., 2012). So a hierarchical inversion strategy shoule be proposed to obtain both parameters. In general, it is reasonable to invert velocity first and then density since velocity is easier to be recovered than density. However, the velocity 76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014
should not be inverted alone without density. Therefore, we propose a new hierarchical inverse strategy. It is composed of two stages. In the first stage, the velocity and density are jointly inverted upon the given initial velocity and density models using equations (4). After this, the inverted velocity should be close to the true model, and it should be better than the inverted velocity when density is fixed during the inversion. However, the density will be deviated from the true one. In the second stage, the velocity and density are updated at the same time again. The inverted velocity model in the first stage is reused as the initial model for the second stage. Thus, we can obtain reliable inversion results for both velocity and density. Numerical Examples To demonstrate this strategy, we apply it to the modified elastic Marmousi-2 model (Martin et al., 2006). The S-wave velocity is assumed to be zero, and P-wave velocity and density are to be inverted in inversion. The resampled true velocity and density are shown in Figure 1a and Figure 1c. The dimension is 8 km × 3.5 km with the interval of 0.02 km. The receivers are uniformly distributed on the surface, and there are 79 shots with the interval of 0.1 km. The first shot is located at 100 m. The record time is 5.7 seconds with the time interval of 2 ms. The Ricker wavelet with the main frequency of 7Hz is used as the source function. The gauss-smoothed velocity and density, as shown in Figure 1b and Figure 1d, are used as the initial models in the inversion. a) b)
c)
d)
Figure 1 True velocity (a); gauss-smoothed initial velocity (b); true density (c); gauss-smoothed initial density (d).
In the first stage, the velocity and density are simultaneously updated using equations (4). The inverted results are shown in Figure 2. Although the inverted density is deviated from the true one, the recovered velocity shows good convergence to the true model. The vertical profiles of velocity and density, shown in Figure 3, make a clear demonstration. a)
b)
Figure 2 The inverted results in the first stage: velocity (a); density (b). 76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014
a)
b)
Figure 3 Vertical profiles at distances of 3 km (left), 4 km (middle) and 5 km (right) of the models shown in Figure 2 for (a) P-wave velocity and (b) density. The black line indicates the true model, the red line denotes the initial model and the green line denotes the reconstructed model. The units of velocity and density are km·s-1, and g·cm-3, respectively.
Using the inverted velocity in the first stage as the initial velocity, we proceed to recover both velocity and density in the second stage. The gauss-smoothed density is reused as the initial density. The final inverted results are shown in Figure 4 and the vertical profiles are shown in Figure 5. By comparison of the inverted results with the true models, we find that our hierarchical strategy for acoustic FWI yields reliable solutions for both velocity and density. The resolution of the updated velocity in the second stage is much better than that of the inverted velocity in the first stage. a)
b)
Figure 4 The inverted results in the second stage: velocity (a); density (b).
76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014
a)
b)
Figure 5 Vertical profiles at distances of 3 km (left), 4 km (middle) and 5 km (right) of the models shown in Figure 4 for (a) P-wave velocity and (b) density. The black line indicates the true model, the red line denotes the initial model and the green line denotes the reconstructed model. The units of velocity and density are km·s-1, and g·cm-3, respectively.
Conclusions Inverse strategy is important to multi-parameter FWI. In this study, we propose a hierarchical strategy for multi-parameter FWI in acoustic media. The strategy allows us first to obtain a reasonable velocity through simultaneous inversion. Then the inverted velocity is used to estimate density and improve velocity simultaneously in the next stage. This strategy can be easily extended to elastic or anisotropic multi-parameter FWI. Acknowledgements We would like to thank the financial supports of National Natural Science Foundation of China (Grant No: 41274116), National Important and Special Project on Science and Technology (2011ZX05005005-007HZ), and Self-determined Project of State Key Laboratory of Marine Geology in China (MGG2013001). References Forgues, E. and Lambaré, G. [1997] Parameterization study for acoustic and elastic ray+Born inversion. Journal of Seismic Exploration, 6, 253–277. Jeong, W., Lee, H.-Y. and Min, D.-J. [2012] Full waveform inversion strategy for density in the frequency domain. Geophysical Journal International, 188, 1221–1242. Kӧhn, D., De Nil, D., Kurzmann, A., Przebindowska, A. and Bohlen, T. [2012] On the influence of model parametrization in elastic full waveform tomography. Geophysical Journal International, 191, 325-345. Krebs, J., Anderson, J., Hinkley, D., Neelamani, R., Lee, S., Baumstein, A. and Lacasse, M.D. [2009] Fast full-wavefield seismic inversion using encoded sources. Geophysics, 74(6), WCC177–WCC188. Martin, G.S., Wiley, R. and Marfurt, K.J. [2006] Marmousi2: an elastic upgrade for Marmousi. The Leading Edge, 25, 156–166. 76th EAGE Conference & Exhibition 2014 Amsterdam RAI, The Netherlands, 16-19 June 2014
