Anisotropic elastic waveform inversion with modified ...

Anisotropic elastic waveform inversion with modified total-variation regularization

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Kai Gao∗ , Youzuo Lin, and Lianjie Huang, Los Alamos National Laboratory, Los Alamos, NM 87545; John Queen, Hi-Q Geophysical Inc., Ponca City, OK 74601; Joseph Moore, University of Utah, Salt Lake City, UT 84108; Ernest Majer, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 SUMMARY Waveform inversion is a promising tool to build subsurface medium property models for seismic imaging and reservoir characterization. Fractured reservoirs often exhibit anisotropy because of aligned fractures along one or several dominant directions in space. Previous attempts to include the influence of medium anisotropy in elastic waveform inversion often invert for reference velocity values and the Thomsen parameters. We develop a new anisotropic elastic waveform inversion method with modified total-variation regularization to directly invert elasticity parameters in heterogeneous, anisotropic elastic media. The modified total-variation regularization helps preserve sharp interfaces while reducing inversion artifacts in anisotropic elastic waveform inversion. We validate our new inversion method using synthetic seismic data for a geophysical model built for the Raft River geothermal site.

INTRODUCTION Full-waveform inversion (FWI) is a promising tool to construct accurate high-resolution subsurface medium parameters models. Originally developed for inverting velocity values of acoustic media (e.g., Tarantola, 1984; Pratt et al., 1998), FWI later was developed to invert both P- and S-wave velocities (e.g., Tarantola, 1986; Mora, 1987; Shipp and Singh, 2002; Brossier et al., 2009) as well as attenuation parameter (Brossier, 2011) for elastic media. Elastic waveform inversion (EWI) is generally more difficult to arrive at satisfactory solutions because of increased number of unknowns, which brings increased nonlinearity and non-uniqueness of the inversion problem compared with the acoustic FWI. Geological formations and reservoirs may contain anisotropic elastic media resulting from finely layering (Carcione, 2015) or fracture alignment along one or more preferential directions (e.g., Tsvankin, 2001; Sayers and Kachanov, 1995). The existence of anisotropy in subsurface brings extra difficulties to waveform inversion in two aspects. First, the modeling of wave propagation in anisotropic media is more complicated than that in isotropic media. Second, the waveform inversion becomes more nonlinear as there are more parameters to be inverted compared to waveform inversion in isotropic media. One method to account for anisotropic signatures in seismic data during waveform inversion is to apply acoustic approximation of anisotropic elastic media (Alkhalifah, 2000; Plessix and Cao, 2011; Wang et al., 2012; Warner et al., 2013). These acoustic approximations use the weak anisotropy assumption of elastic media, and describe the anisotropy and invert the subsurface models using the Thomsen parameters and reference velocity values (Thomsen, 1986). However, since the

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Earth medium is intrinsically elastic, the acoustic approximation in FWI can result in inaccurate inversion results of medium parameters. There are also similar attempts to use elastic media with the Thomsen parameters to approximate anisotropic media (Podgornova and Charara, 2011). We develop a new anisotropic elastic waveform inversion (AEWI) method with a modified total-variation (MTV) regularization scheme using the parameterization of elasticity constants in anisotropic media. The method is abbreviated as AEWI-MTV. Our new method is able to handle complicated anisotropy such as transversely isotropy with tilted symmetric axis (TTI) or monoclinic anisotropy in fractured reservoirs. For 2D case, this inversion involves 6 elasticity parameters, making the convergence of the minimization more difficult than that in isotropic media. We employ the modified total-variation (MTV) regularization scheme (Lin and Huang, 2015) to improve anisotropic elastic waveform inversion. The MTV technique enables us to reformulate the original nonlinear inversion into to subproblems, and solves the least-squares minimization problem using an alternating minimization approach, leading to a robust inversion algorithm. In our AEWI-MTV, we also employ the wavefield-energy preconditioning approach that is developed for EWI in isotropic media (Zhang et al., 2012). We validate our new AEWI-MTV method using synthetic seismic data for a geophysical model built for the Raft River geothermal site.

THEORY Forward modeling of anisotropic elastic wave propagation The velocity-stress equations for wave propagation in 2D anisotropic elastic media is given by (e.g., Carcione, 2015) ∂σ = CΛT v, ∂t

∂v = ρ−1 Λσ, ∂t

(1)

where σ = (σ11 , σ33 , σ13 ) is the stress wavefield, v = (v1 , v3 ) is the particle velocity wavefield, ρ is the mass density of the medium, C is the elasticity tensor in Voigt notation defined as   C11 C13 C15 C =  C13 C33 C35  , (2) C15 C35 C55 and Λ is the differential operator defined as ! ∂ ∂ 0 ∂x1 ∂x3 Λ= . ∂ ∂ 0 ∂x3 ∂x1

(3)

To minimize numerical dispersion in wavefield simulation, we use an optimized rotated staggered-grid finite-difference method (Gao and Huang, 2014). The method introduces a few additional

DOI http://dx.doi.org/10.1190/segam2015-5874837.1 Page 5158

off-axis grid points to achieve high-order spatial and temporal accuracy. Anisotropic elastic waveform inversion with modified totalvariation regularization

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We develop an anisotropic elastic waveform inversion method with modified total-variation regularization that solves the following minimization problem (Lin and Huang, 2015) χ = min{||d − f (m)||22 + λ1 ||m − u||22 + λ2 ||u||TV }, (4)

where Ns and Nr are the number of sources and receivers, respectively, and † represents the adjoint wavefield for model m with the source being the data misfit at the receivers, i.e., d − f (m), and εij is the strain wavefield. We use a wavefield reconstruction method similar to Tan and Huang (2014) to remove the necessity of saving the whole wavefield at each time step. We store the particle velocity wavefields at the boundaries and then propagate them in a reverse-time manner to achieve forward wavefield reconstruction.

m,u

where m = (Cij , ρ) are the model parameters, d is the observed data, f (m) is the forward modeling wavefield for model m, and || · ||2 represents the L2 -norm, and λ1 and λ2 are ′ regularization parameters. The variable u = (Cij , ρ′ ) contains the axillary regularization model parameters, the term λ1 ||m − u||22 is the Tikhonov regularization and λ2 ||u||TV is the total-variation (TV) regularization term for interface preserving. The most important feature of AEWI-MTV is that the minimization problem (4) is decomposed into two subproblems and solved with an alternating minimization approach, i.e., in the k-th iteration (Lin and Huang, 2015), m(k) = argmin{||d − f (m)||22 + λ1 ||m − u(k−1) ||22 }, m

(5) u

(k)

= argmin{||m

(k)

−

u||22

+ λ2 ||u||TV },

(6)

u

where the first subproblem (5) is the traditional full-waveform inversion part with the Tikhonov regularization that can be solved using the adjoint method along with an appropriate search direction update and a step size calculation method, and the second subproblem (6) is a L2 -TV minimization problem and can be solved efficiently with the split Bregman method (Goldstein and Osher, 2009). To invert for elasticity parameters Cij , we need to formulate the gradients for Cij explicitly. The gradients can be calculated using the adjoint method. Since we use the first-order elastic wave equation, the gradients are different from those that are based on the second-order form. Following Vigh et al. (2014), the gradients for elasticity parameters Cij can be calculated with the zero time lag cross-correlation between the forward wavefields and adjoint wavefields, X Z T ∂ε11 † ∇C11 χ = − ε dt, (7) ∂t 11 Ns ,Nr 0  X Z T  ∂ε33 † ∂ε11 † ∇C13 χ = − ε11 + ε33 dt, (8) ∂t ∂t Ns ,Nr 0  X Z T  ∂ε13 † ∂ε11 † ∇C15 χ = − 2 ε11 + ε13 dt, (9) ∂t ∂t Ns ,Nr 0 Z T X ∂ε33 † ∇C33 χ = − ε dt, (10) ∂t 33 Ns ,Nr 0  X Z T  ∂ε13 † ∂ε33 † ∇C35 χ = − 2 ε33 + ε13 dt, (11) ∂t ∂t Ns ,Nr 0 Z T X ∂ε13 † ∇C55 χ = − 4 ε dt, (12) ∂t 13 0 N ,N s

We employ a wave-energy-based precondition method to calculate the gradients (Zhang et al., 2012). For brevity, we do not provide the complete expressions for each ∇Cij here, but take only ∇C55 as an example, X Z T  ∂ε13 2 Es,C55 = 4 dt, (13) ∂t Ns ,Nr 0 X Z T  † 2 Er,C55 = 4 ε13 dt, (14) Ns ,Nr

0

and the preconditioned gradient for C55 is therefore ˜ C55 χ = p ∇C55 χ ∇ . Es,C55 Er,C55

(15)

The gradients are then regularized with the first-order Tikhonov term as described in equation (5). We also apply an anisotropic diffusion method (Grasmair and Lenzen, 2010) to remove unwanted artifacts in the gradients. We use the non-linear conjugate-gradient method to determine the search direction for each parameter. We employ the PolakRibi`ere formula (e.g., Norcedal and Wright, 2006) to obtain the search direction in the k-th iteration, h i ˜ (k) χ · ∇ ˜ (k) χ − ∇ ˜ (k−1) χ ∇ C C C ij ij ij (k) (k−1) ˜ (k) χ + γCij = −∇ γCij , Cij ˜ (k−1) χ · ∇ ˜ (k−1) χ ∇ Cij Cij (16) ˜ (k) χ is the preconditioned gradient of the misfit funcwhere ∇ Cij tional χ with respect to Cij in the k-th iteration. For k = 1, (1) ˜ (1) χ. γCij = −∇ Cij We then use a line search method (quadratic fit along with a bisection method) to update the model. The step lengths are different for each parameter Cij and therefore, the model update is a multi-step-length update. We implement the MTV regularization for each of the model parameter Cij . In each model update iteration, the updated model is used as the prior model, and the auxiliary model pa′ rameter Cij is obtained after a limited number of split Bregman iterations. The calculation time of this MTV regularization with the split Bregman iteration is tiny compared with that of gradients calculation and the line search for optimal step size, yet it can effectively remove the difficulties of instability and artifacts in conventional TV regularization. For a detailed procedure of the split Bregman algorithm, please refer to Goldstein and Osher (2009) and Lin and Huang (2015).

r

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Figure 1: True C33 (a) and C55 (b) models. Unit is GPa. 1.2

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Figure 3: Anisotropic elastic waveform inversion results of (a) C11 model (b) C13 , (c) C33 , and (d) C55 . Unit is GPa.

Figure 2: Initial C33 (a) and C55 (b) models for AEWI-MTV inversion. Unit is GPa.

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Figure 5 is the convergence rate of the AEWI-MTV inversion. The relative data misfit is the sum of v1 and v3 wavefield components. The initial data misfit is about 94%, and the final data misfit after 63 iterations is approximately 12%, a reduction of more than 80%.

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Depth (km)

0.7

cal narrow structure, small features like the curved interfaces are much clearer than those in the initial model, particularly in the inverted C55 model. Figure 4(a) shows the model difference between the true C55 model and the initial smoothed C55 model, and Fig. 4(b) displays the model difference between the true C55 and the inverted C55 model obtained using our AEWI-MTV method. These two figures demonstrate that our AEWI-MTV well reconstructs both the isotropic and anisotropic regions in the model.

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Figure 4: (a) Differences between the true C55 model and the initial C55 model; (b) Difference between the true C55 model and the AEWI-MTV inversion result of C55 . Unit is GPa.

0

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Figure 5: The convergence rate of anisotropic elastic waveform inversion of synthetic surface seismic data for a geophysical model of Raft River geothermal site.

NUMERICAL RESULTS We validate our new anisotropic elastic waveform inversion with modified total-variation regularization using synthetic seismic data from an anisotropic model. The model is built using sonic logs and geological information from Raft River geothermal site (Ayling and Moore, 2013). Figure 1(a) and 1(b) are the true C33 and C55 models containing a vertical narrow structure with vertical fractures in the middle-bottom of the model. The vertical narrow zone contains transversely isotropic media with a horizontal symmetry axis, or HTI media. In the vertical narrow zone, C11 = C33 /2, and C13 = 3(C33 − 2C55 )/4. We generate synthetic surface seismic data for 192 shots and 192 multicomponent receivers on the surface of the model. The source wavelet is the first derivative of Gaussian function. The initial Cij models for inversion are derived from the true elasticity parameter models by smoothing the true models with a moving-average filter of approximately two wavelengths of shear waves. Figures 2(a) and 2(b) show the initial C33 and C55 models. The initial C11 and C13 models are similar. We apply our new AEWI-MTV method to the synthetic data. Figures 3(a)–3(d) show the AEWI-MTV inversion results for elasticity parameter models C11 , C13 , C33 and C55 , respectively. The shallow interfaces are clearly reconstructed. This demonstrates the AEWI-MTV preserves well the sharp interfaces in the model. Although the deeper part of the model is not so well resolved, we can still see that in the HTI verti-

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CONCLUSIONS We have developed a new anisotropic elastic waveform inversion method with modified total-variation regularization. The method employs a modified total-variation regularization scheme to preserve sharp interfaces and reduce inversion artifacts. Our numerical example demonstrates the capability of our new inversion method for reconstructing subsurface isotropic and anisotropic elasticity parameters. Future work includes inverting anisotropic properties from isotropic initial models when anisotropic initial models are not available, and applications to field data.

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy through contract DE-AC52-06NA25396 to Los Alamos National Laboratory (LANL). The computation was performed using the super-computer resources of LANL’s Institutional Computing Program.

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EDITED REFERENCES Note: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2015 SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web.

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REFERENCES

Alkhalifah, T., 2000, An acoustic wave equation for anisotropic media: Geophysics, 65, 1239– 1250. http://dx.doi.org/10.1190/1.1444815. Ayling, B., and J. Moore, 2013, Fluid geochemistry at the Raft River geothermal field, Idaho, USA: New data and hydrogeological implications: Geothermics, 47, 116– 126. http://dx.doi.org/10.1016/j.geothermics.2013.02.004. Brossier, R., 2011, Two-dimensional frequency-domain visco-elastic full waveform inversion: Parallel algorithms, optimization and performance: Computers & Geosciences, 37, no. 4, 444– 455. http://dx.doi.org/10.1016/j.cageo.2010.09.013. Brossier, R., S. Operto, and J. Virieux, 2009, Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion: Geophysics, 74, no. 6, WCC105– WCC118. http://dx.doi.org/10.1190/1.3215771. Carcione, J. M., 2015, Wavefields in real media, 3rd ed.: Elsevier. Gao, K., and L. Huang, 2014, Anisotropic elastic-waveform modeling for fracture characterization in EGS reservoirs: Proceedings of the 40th Workshop on Geothermal Reservoir Engineering. Goldstein, T., and S. Osher, 2009, The split Bregman method for L1-regularized problems: SIAM Journal on Imaging Sciences, 2, no. 2, 323–343. http://dx.doi.org/10.1137/080725891. Grasmair, M., and F. Lenzen, 2010, Anisotropic total variation filtering: Applied Mathematics & Optimization, 62, no. 3, 323–339. http://dx.doi.org/10.1007/s00245-010-9105-x. Lin, Y., and L. Huang, 2014, Acoustic- and elastic-waveform inversion using a modified total-variation regularization scheme: Geophysical Journal International, 200, no. 1, 489– 502. http://dx.doi.org/10.1093/gji/ggu393. Mora, P., 1987, Nonlinear two-dimensional elastic inversion of multioffset seismic data: Geophysics, 52, 1211–1228. http://dx.doi.org/10.1190/1.1442384. Norcedal, J., and S. Wright, 2006, Numerical optimization: Springer. Plessix, R.-E., and Q. Cao, 2011, A parametrization study for surface seismic full-waveform inversion in an acoustic vertical transversely isotropic medium: Geophysical Journal International, 185, no. 1, 539–556. http://dx.doi.org/10.1111/j.1365-246X.2011.04957.x. Podgornova, O., and M. Charara, 2011, Multiscale time-domain full-waveform inversion for anisotropic elastic media: Presented at the 81st Annual International Meeting, SEG, Expanded Abstracts, doi: 10.1190/1.3627703. Pratt, G., C. Shin, and G. J. Hicks, 1998, Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion: Geophysical Journal International, 133, no. 2, 341– 362. http://dx.doi.org/10.1046/j.1365-246X.1998.00498.x. Sayers, C. M., and M. Kachanov, 1995, Microcrack-induced elastic wave anisotropy of brittle rocks: Journal of Geophysical Research. Solid Earth, 100, B3, 4149– 4156. http://dx.doi.org/10.1029/94JB03134.

© 2015 SEG SEG New Orleans Annual Meeting

DOI http://dx.doi.org/10.1190/segam2015-5874837.1 Page 5162

Shipp, R. M., and S. C. Singh, 2002, Two-dimensional full wavefield inversion of wide-aperture marine seismic streamer data: Geophysical Journal International, 151, no. 2, 325– 344. http://dx.doi.org/10.1046/j.1365-246X.2002.01645.x. Downloaded 08/21/15 to 174.56.100.41. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/

Tan, S., and L. Huang, 2014, Reducing the computer memory requirement for 3D reverse-time migration with a boundary-wavefield extrapolation method: Geophysics, 79, no. 5, S185– S194. http://dx.doi.org/10.1190/geo2014-0075.1. Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259–1266. http://dx.doi.org/10.1190/1.1441754. Tarantola, A., 1986, A strategy for nonlinear elastic inversion of seismic reflection data: Geophysics, 51, 1893–1903. http://dx.doi.org/10.1190/1.1442046. Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 51, 1954– 1966. http://dx.doi.org/10.1190/1.1442051. Tsvankin, I., 2001, Seismic signatures and analysis of reflection data in anisotropic media: Pergamon. Vigh, D., K. Jiao, D. Watts, and D. Sun, 2014, Elastic full-waveform inversion application using multicomponent measurements of seismic data collection: Geophysics, 79, no. 2, R63– R77. http://dx.doi.org/10.1190/geo2013-0055.1. Wang, C., D. Yingst, R. Bloor, and J. Leveille, 2012, VTI waveform inversion with practical strategies: Application to 3D real data: Presented at the 82nd Annual International Meeting, SEG. Warner, M., A. Ratcliffe, T. Nangoo, J. Morgan, A. Umpleby, N. Shah, V. Vinje, I. Stekl, L. Guasch, C. Win, G. Conroy, and A. Bertrand, 2013, Anisotropic 3D full-waveform inversion: Geophysics, 78, no. 2, R59–R80. http://dx.doi.org/10.1190/geo2012-0338.1. Zhang, Z., L. Huang, and Y. Lin, 2012, A wave-energy-based precondition approach to full-waveform inversion in the time domain: 82nd Annual International Meeting, SEG, Expanded Abstracts, doi: 10.1190/segam2012-1555.1.

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