Youzuo Lin and Lianjie Huang - PDF Free Download

PROCEEDINGS, Thirty-Ninth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, February 24-26, 2014 SGP-TR-202 Building Subsurface Velocity Models with Sharp Interfaces and Steeply-Dipping Fault Zones Using Elastic-Waveform Inversion with Edge-Guided Regularization Youzuo Lin and Lianjie Huang Los Alamos National Laboratory, Geophysics Group, MS D452, Los Alamos, NM 87545, USA Emails: ylin@lanl.gov (YL); ljh@lanl.gov (LH) Keywords: Edge information, Elastic-waveform inversion, Modified total-variation regularization, Regularization techniques, Velocity estimation. ABSTRACT Imaging correct locations of fault zones is important for geothermal exploration and optimizing enhanced geothermal systems. Elastic reverse-time migration has the potential to directly image steeply-dipping fault zones. However, it requires both accurate compressional and shear wave velocity models. Elastic-waveform inversion is a promising tool for velocity estimation. Because of the ill-posedness and complexity of elastic-waveform inversion, it becomes much more challenging to accurately obtain velocity estimation compared to acoustic-waveform inversion, particularly in the deep regions and fault zones. To improve velocity estimation, we develop an elastic-waveform inversion method with an edge-guided modified total-variation regularization scheme to improve the inversion accuracy and robustness, particularly for steeply-dipping fault zones with widths of much smaller than the seismic wavelength. The new regularization scheme incorporates the edge information into waveform inversion. The edge information of subsurface interfaces is obtained iteratively using migration imaging during elastic-waveform inversion. Our new elastic-waveform inversion takes advantage of the robustness of migration imaging to improve velocity estimation. We validate the improved capability of our new elastic-waveform inversion method using synthetic seismic data for a complex model containing several steeply-dipping fault zones. Our inversion results are much better than those produced without using edge-guided regularization. Elastic-waveform inversion with an edge-guided modified total-variation regularization scheme has the potential to accurately estimate both the compressional- and shear-wave velocities within steeply-dipping fault zones, which would significantly improve direct imaging of these faults. 1. INTRODUCTION Direct imaging steeply-dipping fault zones is important because these fault zones may provide paths for hydrothermal flow or confine the boundaries of geothermal reservoirs. Huang et al. (2011) demonstrated that reverse-time migration has the potential to image steeply-dipping fault zones. The method requires an accurate velocity model for direct imaging of steeply-dipping fault zones. Full-waveform inversion (FWI) is a promising tool for velocity estimation of subsurface structures (Tarantola 1984, Mora 1987, Pratt et al. 1998, Sirgue and Pratt 2004). However, one of the difficulties with FWI is the convergence to the local minima that makes the technique very sensitive to the starting velocity model. Besides the local minima issue, the computational complexity in solving an elastic-waveform inversion problem becomes much more expensive than the acoustic-waveform inversion. Regularization techniques are often used to alleviate the ill-posedness problem of waveform inversion caused by the limited data coverage (Hu et al. 2009, Burstedde and Ghattas 2009, Ramirez and Lewis 2010, Guitton 2011). Tomographic inversion usually gives smoothed image reconstructions. Dewaraja et al. (2010) used the boundary information to improve the reconstruction quality for SPECT imaging. Guo and Yin (2012) encoded the edge information in MRI reconstructions, which also yields better imaging results. Baritaux and Unser (2010) applied the edge information to the Fluorescence Diffuse Optical Tomography, and obtained enhanced reconstructions compared to results yielded without using the edge information. We have recently employed an edge-guided regularization scheme and reverse-time migration (RTM) in acoustic-waveform (or full-waveform) inversion to obtain accurate and high-resolution velocity reconstructions of compressional-wave velocities in complex geophysical models containing sharp interfaces and steeply-dipping fault zones (Lin and Huang 2013). Elastic-waveform inversion is needed for estimating both compressional- and shear-wave velocities using multi-component seismic data. We develop an elastic-waveform inversion (EWI) method with edge-guided regularization to improve the accuracy of reconstructions of both compressional- and shear-wave velocities using multi-component seismic data. The edge-guided regularization can be combined with any regularization techniques. In this paper, we study the performance of elastic-waveform inversion with the edge-guided regularization and a modified total-variation regularization scheme. The method uses the interface information from elastic-wave reverse-time migration to regularize elastic-waveform inversion. Combining with the modified totalvariation regularization scheme, our new elastic-waveform inversion is able to reconstruct velocities for complex geophysical models containing sharp interfaces and steeply-dipping fault zones. We use synthetic multi-component seismic reflection data to validate the improved capability of our new elastic-waveform inversion for estimating both the compressional-wave and shear-wave velocities of a complex geophysical model from the Soda Lake geothermal field. The geophysical model is constructed using a prestack depth migration image at the Soda Lake geothermal field. The model contains several steeply-dipping fault zones with widths of much smaller than the seismic wavelength. Our results demonstrate that our new elastic-waveform inversion method produces much more accurate estimation of both compressional- and shear-wave velocities in the regions with sharp interfaces and steeply-dipping fault zones compared to those obtained without using the edge-guided regularization technique. 1

2. ELASTIC-WAVEFORM INVERSION The elastic-wave equation in the time domain is given by 2 U T ρ ( K U ) ( μ ( U U )) s, (1) 2 t where is the density, K and μ are the elastic moduli, s is the source term and U is the displacement wavefield. The solution to the above elastic wave equation can be written as U f ( K, μ, ρ, s ) (2) where the function of f is the wave-propagation operator. Numerical techniques, such as finite-difference and spectral element methods, can be used to solve for the forward problem. Let m be the model parameter, Eq. (2) becomes U f ( m ). (3) To invert for the model parameter m, we need to solve the minimization problem 2 E ( m ) m in p f ( m ) 2, (4) m where E ( m ) is the misfit function, 2 stands for the L 2 norm, and p represents recorded multi-component waveforms. The resulting model m minimizes the square difference between observed and synthetic waveforms. 3. ELASTIC-WAVEFORM INVERSION WITH EDGE-GUIDED MODIFIED TV REGULARIZATION The cost function with a modified total-variation (TV) regularization scheme is given by Huang et al. (2008) E ( m, u ) m in{ p f ( m ) 2 1 m u u }, (5) T V m, u where 1 and 2 are both positive regularization parameters, u is an auxiliary vector with a dimension of m, and the TV term u for a 2D model is defined as the L 1 norm given by TV u T V u 1 ( x u ) i, j ( z u ) i, j, (6) i, j with ( x u ) i, j u i 1, j u and i, j ( z u ) i, j u i, j 1 u i, j. To incorporate the edge information, we reformulate the TV term given by Eq. (6) as u E T V w u 1 wi, j ( ( x u ) i, j ( z u ) i, j ), (7) i, j where the weighting parameter w controls the amount of regularization among adjacent spatial grid points. We set the weighting value as the following: 0 if p o in t (i, j) is o n th e e d g e w i, j. 1 if p o in t (i, j) is n o t o n th e e d g e (8) We then obtain elastic-waveform inversion with the edge-guided modified TV regularization scheme given by E ( m, u ) m in{ p f ( m ) 2 1 m u w u }. (9) 1 m, u We rewrite elastic-waveform inversion with the edge-guided modified TV regularization in Eq. (9) as E ( m, u ) m in { m in { p f ( m ) 2 1 m u } u }. (10) E T V u m 4. EDGE DETECTION During each iteration step of elastic-waveform inversion, we compute forward-propagation wavefields from sources and backwardpropagation wavefields from receivers. Therefore, we exploit these wavefields to obtain the edges of heterogeneities or interfaces of subsurface structures using elastic reverse-time migration. Consequently, we gain the edge information during elastic-waveform inversion. After the edges are determined, we apply the weighting coefficients according to Eq. (8). 2

5. NUMERICAL RESULTS We use synthetic surface seismic data for a model with the compressional and shear velocities in Figs. 1a and 1b to demonstrate the improvement of our new elastic-waveform inversion method with the edge-guided modified TV regularization scheme for velocity estimation. The compressional and shear models are constructed using geologic features found at the Soda Lake geothermal field. It contains several steeply-dipping fault zones. There are three basalt regions in Fig. 1 with a high velocity value. A total of 165 common-shot gathers of synthetic seismic data with 720 receivers at the top surface of the model are used to invert for velocity values of the model. The shot interval is 20 m and the receiver interval is 5 m. A Ricker wavelet with a center frequency of 25 Hz is used as the source time function. We smooth the original velocity models in Figs. 1a and 1b by averaging the slowness within two wavelengths, resulting in a model in Figs. 1c and 1d. We use these two smoothed models as the starting models for EWI reconstructions. Figure 2 shows EWI reconstructions with no regularization (Figs. 2a and 2b) and with edge-guided modified TV regularization (Figs. 2c and 2d). The EWI results obtained with the edge-guided modified TV regularization scheme in Figs. 2c and 2d shows accurate velocity reconstruction, particularly in the deep regions of the models that are poorly reconstructed without using any regularization in Figs. 2a and 2b. In addition, our new method greatly improves reconstructions of the steeply-dipping fault zones. For comparison of the velocity values estimated using the two EWI methods, we display two profiles: a horizontal profile and a vertical profile. The horizontal profile cuts through two basalt regions and all five fault zones. The vertical profile is along the center of the largest basalt region. Figure 3 displays the horizontal profiles of the two EWI reconstructions along the depth of 0.7 km. Both methods reconstruct the locations of all five fault zones. EWI with edge-guided modified TV regularization significantly improves the reconstructed velocity values of the fault zones compared to EWI without regularization. The true values of the fault zones are 2125 m/s for compressional velocity and 1062 m/s for shear velocity. The estimated velocity values of EWI with edge-guided modified TV regularization are approximately 2180 m/s for the compressional velocity and 1068 m/s for the shear velocity. However the approximated reconstructed compressional and shear velocity values of EWI without regularization are far away from the true values. Figure 4 shows a comparison of the vertical profiles of EWI reconstructions in Fig. 2. EWI without regularization produces oscillated profiles in both compressional and shear velocity results. By contrast, EWI with edge-guided modified TV regularization accurately reconstructs both compressional and shear velocity values and the interfaces. 6. CONCLUSIONS We have developed a novel elastic-waveform inversion method with edge-guided modified total-variation regularization. The method employs the edge information in combination with a modified total-variation regularization scheme. We have validated the capability of our new elastic-waveform inversion method for accurate reconstructions of velocities for steeply-dipping fault zones and the deep regions of the model. Our elastic-waveform inversion results of synthetic seismic data for a Soda Lake geothermal velocity model demonstrate that our new method can accurately reconstruct not only velocity values but also sharp interfaces. Our novel elastic-waveform inversion method with edge-guided modified total-variation regularization could be a powerful tool for accurate velocity estimation, which could lead to significantly improved elastic reverse-time migration for direct imaging of steeply-dipping fault zones. ACKNOWLEDGEMENTS This work was supported by the Geothermal Technologies Program of the U.S. Department of Energy through contract DE-AC52-06NA25396 to Los Alamos National Laboratory (LANL). The computation was performed using super-computers of LANL's Institutional Computing Program. We thank Magma Energy (U.S.) Corp. for providing us with migration images and James Echols for his help in constructing the velocity model for the Soda Lake geothermal field. 3

(a) True compressional-wave velocity model (b) True shear-wave velocity model (c) Initial compressional-wave velocity model (d) Initial shear-wave velocity model Figure 1: Velocity models used for EWI: (a) compressional velocity model and (b) shear velocity model from the Soda Lake geothermal field for generating synthetic seismic data; (c) smoothed compressional velocity model and (d) smoothed shear velocity model used as the starting models for EWI. 4

(a) Reconstruction of compressional velocity (b) Reconstruction of shear velocity (c) Reconstruction of compressional velocity (d) Reconstruction of shear velocity Figure 2: Reconstructions of the velocity model in Fig. 1 produced using EWI without regularization (a, b) and with edgeguided modified TV regularization (c, d). EWI without using regularization yields poor velocity reconstructions. EWI with edge-guided modified TV regularization produces significantly improved velocity reconstructions, particularly in the deep regions of the model and within the steeply-dipping fault zones. 5

(a) Reconstruction of compressional velocity model (b) Reconstruction of shear velocity model (c) Reconstruction of compressional velocity model (d) Reconstruction of shear velocity model Figure 3: Comparison of horizontal profiles of EWI reconstructions in Figs. 2 along the depth of 0.7 km. In each panel, the red line shows the true velocity value, the green line is the initial guess, and the blue line is the EWI reconstruction result. EWI without regularization in (a) and (b) yields oscillated profiles. EWI with edge-guided modified TV regularization in (c) and (d) yields both accurate compressional and shear velocity values. The true compressional velocity value within the fault zones is 2125 m/s and the true shear velocity value is 1062 m/s, while the velocity values of EWI with edge-guided modified TV regularization in (c) and (d) are approximately 2180 m/s and 1068 m/s. By contrast, the reconstructed velocity values in (a) and (b) are far away from the true values. 6

REFERENCES Baritaux, J.-C., and Unser, M.: A priori guided reconstruction for FDOT using mixed norms, Proceedings, IEEE International Symposium on Biomedical Imaging: From Nano to Macro, (2010). Burstedde, C., and Ghattas, O.: Algorithmic strategies for full waveform inversion: 1d experiments, Geophysics, 74, (2009), 37 46. Dewaraja, Y. K., Koral, K. F. and Fessler, J. A.: Regularized reconstruction in quantitative SPECT using CT side information from hybrid imaging, Physics in Medicine and Biology, (2010), 2523 2539. Guitton, A.: A blocky regularization scheme for full waveform inversion, Expanded Abstracts, 81th SEG Annual Meeting, (2011), 2418 2422. Guo,W., and Yin, W. : Edge guided reconstruction for compressive imaging, SIAM Journal on Imaging Sciences, 5, (2012), 809 834. Hu, W., Abubakar, A. and Habashy, T.: Simultaneous multi-frequency inversion of full-waveform seismic data, Geophysics, 74, (2009), 1 14. Huang, L., Kelley, S., Zhang, Z., Rehfeldt, K., Albrecht, M., and Kaufman, G.: Imaging faults with reverse-time migration for geothermal exploration at Jemez Pueblo in New Mexico, Transactions, Geothermal Resources Council, 35, (2011), 833-837. Huang, Y., NG, M. and Wen, Y. : A fast total variation minimization method for image restoration, SIAM Journal on Multiscale Modeling and Simulation, 7, (2008), 774 795. Lin, Y., and Huang, L.: Accurate velocity estimation for steep fault zones using full-waveform inversion with edge-guided regularization, Transactions, Geothermal Resources Council, (2013), 1031 1035. Mora, P.: Nonlinear two-dimensional elastic inversion of multi-offset seismic data, Geophysics, 54, (1987), 1211 1228. Pratt, R. G., Shin, C. and Hicks, G. J.: Gauss-newton and full newton methods in frequency space seismic waveform inversion, Geophysical Journal International, 13, (1998), 341 362. Ramirez, A., and Lewis, W.: Regularization and full-waveform inversion: A two-step approach, Expanded Abstracts, 80th SEG Annual Meeting, (2010), 2773 2778. Sirgue, L., and Pratt, R. G.: Efficient waveform inversion and imaging: A strategy for selecting temporal frequency, Geophysics, 69, (2004), 231 248. Tarantola, A.: Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, (1984), 1259 1266. 8