Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (2012) 188, doi: /j X x Full waveform inversion strategy for density in the frequency domain Woodon Jeong, 1 Ho-Yong Lee 2 and Dong-Joo Min 1 1 Department of Energy Systems Engineering, Seoul National University, 599 Gwanak-ro, Gwanak-gu, Seoul ,Korea 2 Korea National Oil Corporation, , Gwanyang-dong, Dongan-gu, Anyang, Gyeonggi , Korea. cap250@naver.com Accepted 2011 November 22. Received 2011 November 22; in original form 2011 June 28 INTRODUCTION The delineation of subsurface structures and the determination of material properties are necessary in oil and gas exploration. Among several seismic data processing techniques, seismic inversion and migration often play a key role in delineating subsurface structures and inferring material properties (Tarantola 1984; Pratt et al. 1998; Shipp & Singh 2002; Shin & Min 2006). Since the early 1980s, numerous studies have been devoted to developing robust waveform inversion and migration algorithms. In the early stages, full waveform inversion and migration algorithms are mainly based on the acoustic wave equation and consider only P-wave propagation. As multicomponent data have become more common in recent years, seismic migration and inversion based on the elastic wave equations are used to obtain more reliable subsurface information. Elastic full waveform inversion is practically challenging because it has many SUMMARY To interpret subsurface structures properly, elastic wave propagation must be considered. Because elastic media are described by more parameters than acoustic media, elastic waveform inversion is more likely to be affected by local minima than acoustic waveform inversion. In a conventional elastic waveform inversion, P- and S-wave velocities are properly recovered, whereas density is difficult to reconstruct. For this reason, most elastic full-waveform inversion studies assume that density is fixed. Although several algorithms have been developed that attempt to describe density properly, their results are still not satisfactory. In this study, we propose a two-stage elastic waveform inversion strategy to recover density properly. The Lamé constants are first recovered while holding density fixed. While the Lamé constants and density are not correct under this assumption, the velocities obtained using these incorrect Lamé constants and constant density may be reliable. In the second stage, we simultaneously update density and Lamé constants using the wave equations expressed through velocities and density. While density is updated following the conventional method, the Lamé constants are updated using the gradient obtained by applying the chain rule. Among several parameter-selection strategies tested, only this strategy gives reliable solutions for both velocities and density. Our elastic full waveform inversion algorithm is based on the finite-element method and the backpropagation technique in the frequency domain. We demonstrate our inversion strategy for the modified Marmousi-2 model and the SEG/EAGE salt model. Numerical examples show that this new inversion strategy enhances density inversion results. Key words: Inverse theory; Numerical approximation and analysis; Seismic tomography; Computational seismology; Wave propagation. interdependent parameters in comparison to acoustic full waveform inversion. For this reason, elastic full waveform inversion is more likely to become trapped in local minimum. To avoid this problem, waveform inversion has been performed under the assumption that Poisson s ratio and density are fixed over the entire model (Brossier et al. 2009, 2010; Bae et al. 2010; Lee et al. 2010). On the other hand, Connolly (1999) emphasized the importance of elastic impedance in AVO (amplitude versus offset) and rock property analysis. To extract elastic or acoustic impedance, density information is required, which is usually inferred from well data. However, well data are spatially limited. If density, in addition to P- and S-wave velocities, could be correctly estimated through waveform inversion, it would be helpful in seismic data analysis. However, density is difficult to properly recover (Forgues & Lambaré 1997; Choi et al. 2008; Virieux & Operto 2009) when prior information is not of sufficient accuracy. Furthermore, GJI Seismology C 2012 The Authors 1221

2 1222 W. Jeong, H.-Y. Lee and D.-J. Min incorrect estimates of density degrade the velocity results in an elastic waveform inversion. Although several studies on elastic waveform inversion have tried to estimate density in addition to P- and S-wave velocities, their results are unsatisfactory, even for synthetic data. While P-andS-wave velocities were properly recovered, density was not reasonably estimated (Mora 1987; Choi et al. 2008; Virieux & Operto 2009; Köhn et al. 2010). In this study, we propose a strategy for elastic full waveform inversion that properly estimates both velocities and density. In our algorithm, the Lamé constants and density are resolved sequentially over two stages. First, the Lamé constants are recovered with density fixed at an arbitrary value. Because density is assumed incorrectly, the Lamé constants are also incorrect, but the velocities extracted from these incorrect Lamé constants and density may be comparable to true values. As a result, the velocity models, rather than the Lamé constants recovered in the first stage, are used to reestimate the Lamé constants and density in the second stage. To update the Lamé constants in the second stage, we reversely apply the chain rule used by Mora (1987). Our 2-D frequencydomain elastic waveform inversion algorithm uses the finite-element method. In the inversion algorithm, we estimate model parameters and the source wavelet using the gradient and full Newton methods, respectively (Song et al. 1995; Pratt 1999; Shin & Min 2006). The model parameter gradients are computed using the backpropagation technique (Pratt et al. 1998; Shin & Min 2006) and the conjugate-gradient method (Fletcher & Reeves 1964), and they are scaled using the pseudo-hessian matrix (Shin et al. 2001). Our inversion strategy is applied to synthetic data sets for the modified Marmousi-2 model and the SEG/EAGE salt model. Figure 1. The true elastic Marmousi-2 model for (a) P- and(b)s-wave velocities and (c) density.

3 2-D FREQUENCY-DOMAIN ELASTIC WAVE EQUATION According to Hooke s law, the stress-strain relationship can be expressed as σ ij = C ijkl e kl ( i, j, k, l = x, y, z ), (1) where σ and e are stress and strain, respectively, and C ijkl are the elastic constants. Although this linear relationship originally consists of 81 elastic moduli, it only requires 21 elastic moduli due to the symmetry of the stress and strain tensors. For 2-D isotropic Elastic waveform inversion for density 1223 media, the stiffness tensor is given by C 11 C 13 0 C = C 13 C (2) 0 0 C 44 Therefore, eq. (1) can be written in a matrix form as σ xx C 11 C 13 0 e xx σ zz = C 13 C 11 0 e zz. (3) σ xz 0 0 C 44 2e xz Figure 2. The initial models for (a) P-and(b)S-wave velocities and (c) density used to invert the Marmousi-2 model.

4 1224 W. Jeong, H.-Y. Lee and D.-J. Min In heterogeneous isotropic media, 2-D frequency-domain elastic wave equations can be written as ρω 2 ũ = σ xx x ρω 2 ṽ = σ zx x + σ xz z, (4) + σ zz z, (5) where ω is angular frequency, ũ and ṽ are the horizontal and vertical displacements, respectively, and ρ is density. Based on the relationship between strain and the particle displacements, e ij = 1 ( ui + u ) j (i, j = x, y, z), (6) 2 x j x i where u is the particle displacement. Eqs (4) and (5) can then be expressed by ρω 2 ũ = ( ) ũ C 11 x x + C ṽ 13 + [ ( ṽ C 44 z z x + ũ )], z (7) ρω 2 ṽ = [ ( ṽ C 44 x x + ũ )] + ( ũ C 13 z z x + C 11 ṽ z (8) Replacing C 11, C 13 and C 44 with Lamé constants λ + 2μ, λ and μ, respectively, gives ). Figure 3. Reconstructed models at the 800th iteration using Conventional method I for the Marmousi-2 model: (a) P- and(b) S-wave velocities and (c) density.

5 Elastic waveform inversion for density 1225 Figure 4. Depth profiles at distances of (a) 3 km and (b) 6 km showing P-wave velocity (left-hand panel), S-wave velocity (centre panel), and density (right-hand panel) inverted using Conventional method I for the Marmousi-2 model. The solid lines indicate the true model, and the dashed lines indicate the inverted model. Velocities are shown in km s 1, and density is shown in g cm 3.

6 1226 W. Jeong, H.-Y. Lee and D.-J. Min ρω 2 ũ = [ (λ + 2μ) ũ x x + λ ṽ ] + [ ( ṽ μ z z x + ũ )], z ρω 2 ṽ = [ ( ṽ μ x x + ũ )] + [ λ ũ ] ṽ + (λ + 2μ). z z x z (9) (10) Employing the finite-element method, eqs (9) and (10) can be written (Marfurt 1984) as Sũ = f, (11) where S is the complex impedance matrix, and ũ and f are the displacement and source vectors, respectively. By solving eq. (11), we can simulate wave propagation in isotropic elastic media. FREQUENCY-DOMAIN ELASTIC WAVEFORM INVERSION Gradient direction Our inversion process is divided into two parts: updating model parameters and estimating the source wavelet. In general, waveform inversion is performed by building an objective function based on the residuals between modelled and field data. The source wavelet is required to compute the modelled data, but the exact source wavelet Figure 5. Reconstructed models at the 800th iteration using Conventional method II for the Marmousi-2 model: (a) P- and(b) S-wave velocities and (c) density.

7 Elastic waveform inversion for density 1227 is generally unknown. Because the accuracy of the source wavelet affects the accuracy of the inverted model parameters, and vice versa, the source wavelet and model parameters should be inverted together to obtain more reliable subsurface information. The same objective function can be used for waveform inversion and source wavelet estimation, but the gradients are different. We build an objective function based on the l 2 norm of residuals between the modelled and observed data: E = 1 (ũ d )T ( ũ d ), (12) 2 ω s where u and d are the modelled and observed wavefield vectors, Figure 6. Depth profiles at distances of (a) 3 km and (b) 6 km showing P-wave velocity (left-hand panel), S-wave velocity (centre panel), and density (right-hand panel) inverted using Conventional method II for the Marmousi-2 model. The solid lines indicate the true model, and the dashed lines indicate the inverted model. Velocities are shown in km s 1, and density is shown in g cm 3.

8 1228 W. Jeong, H.-Y. Lee and D.-J. Min respectively, and the superscripts T and indicate the transpose and the complex conjugate, respectively. The angular frequency is denoted by ω,ands denotes the source. The model parameters are updated by the gradient method in each iteration using p l+1 = p l α p E, (13) where l denotes the iteration number, α denotes the step length, and p is the model parameter vector. The gradient of the objective function can be derived by taking the partial derivative of the objective function with respect to the kth model parameter E = [ ( ) ũ T (ũ Re d ) ]. (14) p k p ω s k In general, the gradient of the objective function can be computed by either calculating the partial derivative wavefields directly or using the backpropagation technique, which has widely been used in reverse-time migration (Pratt et al. 1998; Shin & Min 2006). We use the backpropagation technique, in which the gradient is computed by a zero-lag cross-correlation of the virtual source and the backpropagated wavefield (Pratt et al. 1998). Consequently, the gradient of the objective function for the kth model parameter can be expressed by the complex impedance matrix and the virtual source vector E = [ (f ) ] v T Re k (S 1 ) T (ũ d) (15) p k ω s with f v k = S p k ũ, (16) where fk v is the virtual source vector with respect to the kth model parameter, and ũ d is the residual vector expanded to the whole dimensions of the model by padding it with zeroes. If we consider the whole-model parameters, then the virtual source vector should be replaced by the virtual source matrix E p = ω Re [ (F v ) T (S 1 ) T (ũ d) ]. (17) s For source wavelet estimation, the modelled wavefield is decomposed into the numerical Green s function and the source wavelet in the objective function u sr = [ G R sr + igi sr] [e + if], (18) where G is the numerical Green s function, and e and f are real and imaginary part of the frequency-domain source wavelet, respectively. The subscripts s and r indicate source and receiver, respectively, and the superscripts R and I indicate the real part and the imaginary part of the Green s function, respectively. By computing the gradient with respect to the real and imaginary parts of the source wavelet, we can update the source wavelet in each iteration. Following several previous studies (Song et al. 1995; Pratt 1999; Shin & Min 2006; Shin et al. 2007), we compute the Figure 7. Reconstructed models at the 500th iteration in the second stage using (a) Conventional method I and (b) Conventional method II for the Marmousi-2 model. The initial density model gradually increases with depth.

9 Elastic waveform inversion for density 1229 Figure 8. Depth profiles at distances of 3 km (left-hand panel) and 6 km (right-hand panel) showing the density model inverted in the second stage of (a) Conventional method I and (b) Conventional method II for the Marmousi-2 model. The solid lines indicate the true model, and the dashed lines indicate the inverted model.

10 1230 W. Jeong, H.-Y. Lee and D.-J. Min gradient using the full Newton method: [ d R sr Gsr R + d sr I G sr] I s r e + e [ (G ) R 2 ( ) ] sr + G I 2 sr s r = [ f d R sr Gsr I d sr I G sr] R. (19) s r f [ (G ) R 2 ( ) ] sr + G I 2 sr s r Therefore, the source wavelet at the (l+1)th iteration can be obtained using ( d R sr Gsr R + d sr I G sr) I s r ( (G ) R 2 ( ) ) el+1 sr + G I 2 sr s r f l+1 = ( d R sr Gsr I d sr I G sr) R. (20) s r ( (G ) R 2 ( ) ) sr + G I 2 sr s r Scaling and optimization The convergence of the waveform inversion can be accelerated by scaling the gradients of the model parameters. The full and approximate Hessian matrices also play a role in scaling the gradient in the full Newton and Gauss Newton methods, respectively. Considering the computational overburden, Shin et al. (2001) proposed using the pseudo-hessian matrix instead of the approximate Hessian matrix. Because the pseudo-hessian matrix is computed with the virtual source for the model parameters, it is useful to apply the pseudo-hessian matrix when using the backpropagation technique to compute the gradients of model parameters. Therefore, we scale the gradient for model parameters by the diagonal of the pseudo-hessian matrix. The gradient is computed using E = ω Re [ (F v ) T (S 1 ) T (ũ d) ] s [ {( diag (Fv ) T (F v ) ) + λ I }], (21) s where λ is the damping factor and I is the identity matrix in the Marquardt Levenberg regularization. The conjugate-gradient method (Fletcher & Reeves 1964) is employed to optimize the gradient direction. At the (l + 1)th iteration, the conjugate gradient direction can be written as β l+1 = ( l+1e) T ( l+1 E), (22) ( l E) T ( l E) c l+1 = l+1 E + β l+1 c l, (23) p l+1 = p l + α c l+1, (24) where c is the conjugate gradient direction. Figure 9. Reconstructed models at the 800th iteration in the first stage of the new strategy for the Marmousi-2 model: (a) P- and(b) S-wave velocities. The density is fixed at 2 g cm 3.

11 Elastic waveform inversion for density 1231 INVERSION STRATEGY FOR DENSITY Conventional waveform inversion Because the elastic wave equations are expressed using the Lamé constants, λ and μ, and density as shown in eqs (9) and (10), we may update the Lamé constants and density by computing the virtual sources for each parameter in an elastic waveform inversion: (f v ) λ = S u, (25) λ (f v ) μ = S u, (26) μ Figure 10. Depth profiles at distances of (a) 3 km and (b) 6 km showing the P- (left-hand panel) and S-wave (right-hand panel) velocity models obtained in the first stage of the new strategy for the Marmousi-2 model. The solid lines indicate the true model, and the dashed lines indicate the inverted model. Velocities are shown in km s 1.

12 1232 W. Jeong, H.-Y. Lee and D.-J. Min (f v ) ρ = S u, (27) and then extract velocity information from the Lamé constants and density. This approach is called Conventional method I. Alternatively, replacing the Lamé constants with P- and S-wave velocities and density gives ρω 2 ũ = [ ρα 2 ũ x + z x + ( ρα 2 2ρβ [ ( ṽ ρβ 2 x + ũ z ] 2) ṽ z )], (28) ρω 2 ṽ = [ ( ṽ ρβ 2 x + z x + ũ )] z [ (ρα 2 2ρβ 2) ũ ṽ + ρα2 x z ]. (29) We can update velocity and density in two ways. We may update them independently, even though velocities are dependent on density: (f v ) α = S u, (30) α Figure 11. Reconstructed models at the 500th iteration in the second stage of the new strategy for the Marmousi-2 model: (a) P- and(b) S-wave velocities and (c) density. The P- ands-wave velocity models shown in Fig. 9 are used as the initial velocity models. The initial density model gradually increases with depth.

13 (f v ) β = S u, (31) β (f v ) ρ = S u. (32) Otherwise, we update velocities using the chain rule (Mora 1987; Köhn et al. 2010), which gives virtual sources as follows [ S (f v λ ) α = λ α + S Elastic waveform inversion for density 1233 μ μ α + S α [ S λ (f v ) β = λ β + S μ μ β + S β [ = S S 4ρβ + λ μ 2ρβ ] [ ] S u = λ 2ρα u, (33) ] u ] u, (34) Figure 12. Depth profiles at distances of (a) 3 km and (b) 6 km showing the P- (left-hand panel) and S-wave velocity (centre panel) and density (right-hand panel) models obtained in the second stage of the new strategy for the Marmousi-2 model. The solid line indicates the true values, and the dashed line indicates the inverted values. Velocities are shown in km s 1, and density is shown in g cm 3.

14 1234 W. Jeong, H.-Y. Lee and D.-J. Min [ S (f v ) ρ = λ = λ + S μ μ + S ] u [ S ( α 2 2β 2) + S λ μ β2 + S ] u. (35) Comparing the final expressions of eqs (33) (35) with those of eqs (30) (32), we note that they are exactly the same as each other. This method of updating velocities and density is called Conventional method II. Köhn et al. (2010) and Köhn (2011) provided numerical inversion results generated using the aforementioned inversion methods for the Cross-Triangle-Square model. In their results, the inverted densities generated using Conventional method I deviate slightly more from the true densities compared to those obtained using Conventional method II, although the model obtained using Conventional method II shows a larger ambiguity. In our experiments, inversion strategies that simultaneously invert for all the model parameters can provide good results for velocities; however, these strategies do not have satisfactory solutions for density when linearly increasing parameter models are used as an initial guess possibly because of the different sensitivities of the objective function to model parameters or because of non-uniqueness problems resulting from the large number of parameters. To overcome this limitation, Tarantola (1986) proposed a parameter-selection strategy based on sensitivity analysis, but he did not provide numerical examples for a complex test problem. Figure 13. The true models for SEG/EAGE salt model: (a) P-and(b)S-wave velocities and (c) density.

15 Strategy for density inversion We apply a parameter-selection strategy to the aforementioned inversion methods because although conventional inversion methods do not correctly estimate density, they properly reconstruct velocities. When we parametrize the elastic wave equations with the Lamé constants and density, we first invert the Lamé constants while holding density fixed and then reestimate the Lamé constants and density using the inversion results obtained in the first stage as initial guesses. When the velocities are updated using the wave equation parametrized by velocities and density, the velocities are first updated with density fixed, and then both the velocities and density are reinverted based on the velocities from the first stage. In the first stages of both methods, density is fixed at an arbitrary value. The second method, which updates velocities, does not have reasonable results because the velocities Elastic waveform inversion for density 1235 are dependent upon density and are severely affected by the arbitrary density. Therefore, we choose the first method, which updates the Lamé constants rather than the velocities. In this case, the inverted Lamé constants are not correct because we fixed density at an arbitrary value. However, the velocities can be properly estimated using these incorrect Lamé constants and density, and are described by defining the incorrect Lamé constants as virtual Lamé constants: λ V + 2μ V μv α =, β =, (36) ρ C ρ C where the subscripts V and C mean virtual and constant, respectively. After obtaining the proper velocity structures in the first stage, the density is estimated in the second stage, where we can also update the velocities or Lamé constants. Here, the Lamé Figure 14. The initial models for (a) P- and(b)s-wave velocities and (c) density used to invert the SEG/EAGE salt model.

16 1236 W. Jeong, H.-Y. Lee and D.-J. Min constants and density are updated simultaneously using the elastic wave equations expressed through velocity and density. For updating the density, we compute the gradient of density using eq. (32), whereas for updating the Lamé constants, we compute the gradient of the Lamé constants based on the chain rule, which is the reverse of that used by Mora (1987): [ ( u S λ = α S 1 α λ + S β β λ + S [ ( u S μ = α S 1 α μ + S β β μ + S Using α λ = 1 2αρ, β λ = 0, λ μ ) ] u, (37) ) ] u. (38) = 0, (39) λ α μ = 1 βρ, β μ = 1 2βρ, = 0, (40) μ Eqs (37) and (38) can be expressed by [ ( ) ] u S λ = 1 S 1 u, (41) α 2αρ [ ( u S μ = S 1 α 1 βρ + S 1 β 2βρ ) ] u. (42) In this case, density is affected only by the velocities, and the Lamé constants are affected by both the velocities and density. Among several inversion strategies tested, this strategy is the only one that gives reliable solutions for both velocities and density. Figure 15. Reconstructed models at the 800th iteration using Conventional method I for the SEG/EAGE salt model: (a) P- and(b) S-wave velocities and (c) density.

17 Elastic waveform inversion for density 1237 NUMERICAL EXAMPLES Elastic Marmousi-2 model Before demonstrating the inversion strategy for density, we first test two conventional waveform inversion methods for synthetic data from the modified elastic Marmousi-2 model (Martin et al. 2002). The original Marmousi-2 model has a water layer and very low S-wave velocities (i.e. high Poisson s ratio), which requires a large number of grid points. To avoid computational overburden, we remove the water layer and the left- and right-side part of the original Marmousi-2 model. The low S-wave velocities are replaced by high S-wave velocities so that Poisson s ratio can be fixed at Fig. 1 shows the true model with dimensions of 9.2 km 3.04 km with a grid interval of 0.02 km. To generate synthetic seismograms that are used as real data, we assume that receivers are placed at all of the nodal points and that there are Figure 16. Depth profiles at distances of (a) 9 km and (b) 10 km showing the P- (left-hand panel) and S-wave velocity (centre panel) and density (right-hand panel) models inverted using Conventional method I for the SEG/EAGE salt model. The solid lines indicate the true model, the dashed lines denote the initial model, and the dotted lines represent the inverted model. Velocities are shown in km s 1, and density is shown in g cm 3.

18 1238 W. Jeong, H.-Y. Lee and D.-J. Min 219 shots at an interval of 40 m. The maximum recording duration is 5 seconds. The assumed source signature is the first derivative of the Gaussian function with a maximum frequency of 10 Hz. At each inversion step, we estimate the source wavelet and model parameters. To obtain more reasonable inversion results, we also apply a frequency-selection strategy. Three types of frequencyselection strategies can be applied: the discretized method (Sirgue & Pratt 2004), the overlap-grouping method (Bunks et al. 1995), and the individual-grouping method (Kim et al. 2011). They all give similarly good results, although the computational efficiency of each is different. Because we do not consider computational efficiency in our study, we choose to apply the overlap-grouping method. We perform waveform inversion, broadening frequency ranges toward higher frequencies over several steps with a frequency interval of 0.2, 0.2 2, 0.2 4, 0.2 6, and Hz. In conventional waveform inversion, the model parameters are updated independently and simultaneously, as mentioned previously. As shown in Fig. 2, the parameter values increase with depth in the initial model. Figs 3 and 4 show the inverted models and depth profiles through them obtained using Conventional method I, which updates the Lamé constants and density based on eqs (9) and (10). In Figs 3 and 4, we observe that although the velocities are reconstructed properly, density is severely distorted. The inverted density model is similar to the true model, but its values deviate from the actual values over the entire model. In Figs 5 and 6, we display the inversion results generated using Conventional method II, which updates the velocities and density based on eqs (28) and (29). Here, the velocities are well recovered, but density is not properly estimated. Fig. 6 shows that inverted density values deviate from the true density model. Considering that the two conventional waveform inversion methods yield reliable velocity models, we re-estimate density in the second stage where the parameter models (the Lamé constants and velocities) obtained in the first stage are used as initial guesses but are not updated. By updating only density in the second stage, we try to enhance the density results. Figs 7 and 8 show the density results from the second stage obtained using the two methods. In Fig. 8, depth profiles of density still show some discrepancies between the inverted results and true data. Now, we apply our parameter-selection strategy to the same model. Figs 9 and 10 show the P- ands-wave velocity models that were inverted while holding density fixed at a constant value of2gcm 3 in the first stage. Figs 11 and 12 show P- ands-wave velocities and the density estimated in the second stage, in which the velocity models obtained in the first stage and a gradually increasing density model are used as the initial models. Comparing Figs 11 and 12 with the earlier results shows that our parameter-selection strategy yields better solutions for density. The inverted density values are closer to the true values than those obtained using the aforementioned techniques, although there are some discrepancies in the deeper part of the model. SEG/EAGE salt model To further demonstrate our parameter-selection strategy, we apply both the conventional waveform inversion method and the new strategy to synthetic data for line AA through the SEG/EAGE salt Figure 17. Reconstructed models at the 800th iteration in the first stage of the new strategy for the SEG/EAGE salt model: (a) P- and(b) S-wave velocities. The density is fixed at 2 g cm 3.

19 Elastic waveform inversion for density 1239 model (Aminzadeh et al. 1997). The dimensions of the model are 15.6 km 4.2 km, with a grid interval of 0.02 km. The S-wave velocity and density models are generated based on the P-wave velocity model. The S-wave velocities are built so that Poisson s ratio is constant at The background density is constructed by using the empirical formula suggested by Gardner et al. (1974). For the main salt body, the density is set at 2.2 g cm 3, following House et al. (2000). The true models are shown in Fig. 13. The P- ands-wave velocities have ranges of km s 1 and km s 1, respectively, and density varies from 1.98 to 2.44 g cm 3. We model 379 shots at a spacing of 40 m, with receivers at all the nodal points on the surface. The source signature is modelled as the first derivative of the Gaussian function with a maximum frequency of 10 Hz. The frequency-selection strategy is Figure 18. Depth profiles at distances of (a) 9 km and (b) 10 km showing the P- (left-hand panel) and S-wave (right-hand panel) velocity models obtained in the first stage of the new strategy for the SEG/EAGE salt model. The solid lines indicate the true model, the dashed lines denote the initial model, and the dotted lines represent the inverted model. Velocities are shown in km s 1.

20 1240 W. Jeong, H.-Y. Lee and D.-J. Min also applied for three frequency ranges: Hz, Hz and Hz with an interval of Hz. In general, salt structures are very difficult to recover through waveform inversion, particularly the low-velocity zone below the salt body. If we use an initial model containing gradually increasing parameter values in the inversion, then the salt body becomes thin, and the velocities below the salt are too high. Because the velocities are not properly recovered in this case, we may fail to reconstruct the density model, even when using the new strategy. Thus, we use the velocities inverted by Chung et al. (2010) in the Laplace domain as an initial guess for the P-wave velocity model (Fig. 14). Laplace-domain waveform inversion can recover long-wavelength velocity structures, which can provide a good choice for an initial model. The initial S-wave velocity model is constructed from the P-wave velocity model obtained in the Laplace-domain waveform inversion, assuming a constant Poisson s ratio of The initial density model gradually increases with depth. Figs 15 and 16 show the inversion results generated using Conventional method I, which updates the Lamé constants and density simultaneously. While the inverted velocities are comparable to the true values, even for the low-velocity zone below the salt body, the densities are severely distorted. When we apply the new strategy to the salt model, we first invert for velocities while holding density fixed at 2 g cm 3 in the first Figure 19. Reconstructed models at the 450th iteration in the second stage of the new strategy for the SEG/EAGE salt model: (a) P- and(b) S-wave velocities and (c) density. The P-andS-wave velocity models shown in Fig. 17 are used as the initial velocity models. The initial density model gradually increases with depth.

21 Elastic waveform inversion for density 1241 stage, as is done for the previous model. Figs 17 and 18 show the inverted P- and S-wave velocities at the 800th iteration. Comparing Fig. 17 with Fig. 15, we can see that velocity models recovered in the first stage of the new strategy are slightly better than those obtained using the conventional method. Figs 19 and 20 show the final inverted velocity and density models in the second stage. Comparing depth profiles showing density obtained using the conventional method and the new strategy, we find that the new strategy provides much better solutions than the conventional method. However, the inversion results for the salt model are not as good as those for the Marmousi-2 model due to the improperly recovered velocities for the salt model. These results demonstrate that the new parameter-selection strategy enhances inverted density models compared with conventional Figure 20. Depth profiles at distances of (a) 9 km and (b) 10 km showing the P- (left-hand panel), and S-wave velocity (centre panel) and density (right-hand panel) models obtained in the second stage of the new strategy for the SEG/EAGE salt model. The solid lines indicate the true model, the dashed lines denote the initial model, and the dotted lines represent the inverted model. Velocities are shown in km s 1, and density is in g cm 3.

22 1242 W. Jeong, H.-Y. Lee and D.-J. Min elastic waveform inversion, and the accuracy of the inverted density models are dependent upon the inverted velocities from the first stage. CONCLUSIONS One of the main problems of elastic full waveform inversion is that it cannot properly describe density. We have developed an inversion strategy to properly recover density through elastic full waveform inversion. Our inversion strategy consists of two stages. In the first stage, Lamé constants are inverted with density fixed, from which we can extract velocity information. In this case, although the Lamé constants and density are incorrect, the velocities can be reasonable. We refer to the unreliable Lamé constants as virtual Lamé constants. In the second stage, both the Lamé constants and density are simultaneously re-inverted based on the velocity information estimated in the first stage. In addition, in the second stage, the Lamé constants and density are updated using the wave equations expressed through velocities and density. To update the Lamé constants, we use a reversed version of the chain rule. Applying the conventional waveform inversion method and the new inversion strategy to synthetic data for the modified version of elastic Marmousi-2 model and the SEG/EAGE salt model, we find that the new inversion strategy recovers more reliable density models than the conventional methods. To obtain accurate density models using the new parameter-selection strategy, accurate velocity models are necessary, which can be obtained by performing elastic waveform inversion with density fixed in the first stage. In this study, we have presented only numerical examples for simplified models with a fixed Poisson s ratio because inverting models with varying Poisson s ratio is still challenging. Further study is needed to verify the new inversion strategy for realistic models with varying Poisson s ratios. ACKNOWLEDGMENTS This work was financially supported by the Brain Korea 21 project of Energy System Engineering, the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( ), the Energy Efficiency & Resources of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 2010T ), and the Korea Ocean Research and Development Institute (PMS198). We would like to thank Prof. Changsoo Shin at Seoul National University for providing computational resources and Laplace-domain inversion results for the SEG/EAGE salt model. 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