Tu P7 15 First-arrival Traveltime Tomography with Modified Total Variation Regularization

Transcription

1 T P7 15 First-arrival Traveltime Tomography with Modified Total Variation Reglarization W. Jiang* (University of Science and Technology of China) & J. Zhang (University of Science and Technology of China) SUMMARY The first-arrival traveltime tomography is a robst tool for near-srface velocity estimation. A common approach to stabilize the ill-posed inverse problem is to apply Tihonov reglarization to the inversion. However, the Tihonov reglarization method recovers smooth local strctres while blrring the sharp featres in the model soltion. We present a first-arrival traveltime tomography method with modified total variation reglarization to preserve sharp velocity contrasts and improve the accracy of velocity inversion. In order to solve the minimization problem of the new traveltime inversion method, we decople the original optimization problem into two sbproblems: a standard traveltime inversion problem with the traditional Tihonov reglarization and a L2 total variation problem. We se conjgate gradient method and split-bregman iterative method to solve these two sbproblems, respectively. Or synthetic and field data examples show that the new method prodces higher resoltion models than the Tihonov method for the models with sharp interfaces.

2 Introdction In seismic exploration, accrate near-srface soltions are essential for mapping low-relief strctres and stratigraphic traps. The first-arrival traveltime tomography has been widely applied today for nearsrface imaging (Zhang and Tosöz, 1998). Tihonov reglarization is often incorporated in the tomographic objective fnction to mitigate the ill-posedness of the geophysical inverse problem (Tihonov and Arsenin, 1977). However, tomographic inversion with the Tihonov reglarization tends to prodce smooth model soltions. There are many efforts to develop reglarization schemes for prodcing sparse and blocy model reslts. Most of these reglarization methods can be separated into two categories. The first category defines the nonsmoothing reglarization as the L2 norm of the model parameters (Last and Kbi, 1983; Portniagine and Zhadanov, 1999; Zh and Harris, 2015). The second category of the reglarization methods defines a nonsmoothing reglarization throgh L1 norm of the model parameters. A particlar version of L1 norm reglarization schemes is nown as total variation (TV) (Bertete-Agirre et al., 2002; Anagaw and Sacchi, 2012; Loris and Verhoeven, 2012). However, TV reglarization scheme with the approximation often prodces nwanted inversion artifacts and the convergence rate is highly sensitive to the smoothing parameter (Lin and Hang, 2014). Lin and Hang (2014) propose an acostic and elastic waveform inversion method sing a modified total variation (MTV) reglarization scheme to preserve sharp interfaces in piecewise-constant strctres and improve the accracy of P- and S-wave velocity inversion. The new MTV reglarization scheme avoid adding a smoothing parameter in TV terms, which significantly alleviate instability of the inversion algorithm. In this stdy, we apply the MTV reglarization scheme to the first-arrival traveltime inversion. The objective fnction of traveltime tomography with the MTV reglarization is categorized into two sbproblems: a traveltime inversion problem with the Tihonov reglarization and a standard L2-TV problem. First-arrival traveltime tomography with MTV reglarization The objective fnction of traveltime tomography with the MTV reglarization is defined as: Φ ( m, ) = min { d G( m) + λ (1) 1 ( m ) + λ2 } m, TV where d is the observed traveltime data, G(m) is the calclated traveltime data for the crrent model m, λ 1 and λ 2 are reglarization parameters to balance the weights between the data misfit term and the reglarization terms, is an axiliary variable. The objective fnction (1) can be eqivalently written as: Φ ( m, ) = min{ min { d G ( m) } } 2 + λ 2 (2) 1 m + λ 2 m TV Following Lin and Hang (2014), we employ an alternating minimization algorithm to solve the doble minimization problem in eqation (2). Split the L1 and L2 components of this fnction, we can minimize eqation (2) efficiently by iteratively minimizing m and separately. Using an initial model 0 = m 0, the two sbproblems are as follows: 2 1 m = arg min Φ m = arg min d G( m) + λ m 2 (3) Sbproblem 1: ( ) Sbproblem 2: arg min ( ) m { 1 } { 2 1 2} m 2 { 2 } arg min{ 2 2 TV } = Φ m = m +λ (4) where is the iteration nmber. These two sbproblems have different physical meanings. The first sbproblem minimizes the data misfit term with the constraint of a Tihonov reglarization term. The model 1 is introdced as an apriori model. The second sbproblem sees a reference model with a sharp interface with the help of TV reglarization term. By solving these two sbproblems,

3 respectively, we establish traveltime tomography with MTV reglarization leading to an inversion that not only enhances the sharpness of interfaces, bt also improves the accracy of velocity soltion. Synthetic Examples We apply the traveltime inversion with Tihonov, TV and MTV reglarization schemes to a 2D model. The tre model contains piecewise-constant strctres (Figre 1(a)). The velocity model consist of cells with a niform spacing of 5 m. The acqisition geometry consists of 51 shots with a 40 m interval. A total of 199 receivers are placed along 4 m line at a 40 m interval. The synthetic traveltimes are calclated from the tre model. Gassian random noise of a magnitde at 5% is applied to the synthetic traveltime data. We se a homogeneos model of 3000 m/s as an inpt for inversion. (a) Tre model (b) Tihonov reglarization (c) TV reglarization (d) MTV reglarization Figre 1 Inversion reslts of synthetic example. (a) Tre model with homogenos velocity in the first layer. (b) The reslt calclated with traveltime tomography sing Tihonov reglarization. (c) The reslt calclated with traveltime tomography sing TV reglarization. (d) The reslt calclated with traveltime tomography sing MTV reglarization. The interfaces in the inversion soltions are not accrately resolved in the early iterations since the starting model is set to be homogeneos. We will obtain a reference model with wrong sharpened interfaces when solving the second sbproblem. Hence, we rn traveltime inversion with Tihonov reglarization for the first eight iterations to obtain a model with relatively accrate interfaces. Then we can apply MTV reglarization method to refine the model. (a) Figre 2 Velocity profiles extracted from the reslts of synthetic example. (a) The horizontal profile extracted from z=130 m. (b) The vertical profile extracted from x=2000 m. (b)

4 The traveltime tomography with Tihonov reglarization scheme prodces the smoothest soltion compared with the reslts of other reglarization methods (Figre 1(b)). Traveltime inversion with the TV and MTV reglarization improves the sharpness of interface in the inversion reslts, as shown in Figre 1(c) and Figre 1(d). Figre 2(a) and 2(b) show the horizontal profile at the position z=130 m and vertical profile at the position x=2000 m from the inversion reslts. Velocity profiles of traveltime tomography with Tihonov reglarization are smoothed across the interfaces. The horizontal and vertical profiles show that the traveltime inversion with TV reglarization improves the accracy of velocity, bt it also contains significant flctation. While the velocity profiles of MTV reglarization method are closer to the tre model and contain less flctation. To qantify the accracy of these methods, we calclate the data misfit and model misfit for every iteration. Figre 3(a) and 3(b) show convergence crves of the normalized data misfits and normalized model misfits of the three inversion methods: traveltime inversion with Tihonov, TV and MTV reglarization schemes. All three methods show a comparable converging rate and the data misfit decreases to the same level (Figre 3(a)). The model misfit crves in Figre 3(b) demonstrate that the TV reglarization method converges mch faster than the Tihonov reglarization methods. The MTV reglarization method yields a model soltion with the smallest model misfit among the three reglarization methods. Ths, the reslt of the MTV reglarization method is closer to the tre model than Tihonov and TV reglarization methods. (a) Figre 3 Convergence crves of (a) the normalized data misfits and (b) the normalized model misfit of traveltime inversion with the Tihonov (in ble), TV (in green) and MTV (in red) reglarization schemes. Real data application We apply the first-arrival traveltime tomography with MTV reglarization to a 2-D real dataset collected in the Middle East. This dataset incldes 558 shots with shot interval of 60 m, and approximately 240 receivers for each shot with receiver interval of 30 m. We pic the first-arrival traveltimes from the field records. The average reciprocal error associated with the pics is abot 6 ms. The near srface area incldes moderate topography variations, and the first arrival data indicates complex near srface strctres in the area. We condcted the tomographic stdies with a model consisting of cells and a niform grid spacing of 20 m. The initial model is created by generalized linear inversion (GLI) method (Figre 4(a)). The inversion shold stop when the data misfit decreases to the level of reciprocal errors of piced traveltimes. Figre 4(b) shows that the Tihonov reglarization yields the smoothest inversion reslt. The reslts of TV (Figre 4(c)) and MTV (Figre 4(d)) reglarization exhibit the similar strctres in the first layer, bt the reslt of MTV reglarization method provides more velocity details in deeper part. Both TV and MTV reglarization schemes can yield a model with a sharp and continos edge for the first layer compared with the Tihonov reglarization method. The high velocity bodies in the second layer (b)

5 between the initial model and the inversion soltions are different becase we assme a three-layer model for the GLI method. For all of the reglarization methods, the traveltime tomography can image the high velocity body, which is different from the initial model. (a) Initial model (b) Tihonov reglarization (c) TV reglarization (d) MTV reglarization Figre 4 Inversion the reslts of field data example. (a) Initial model bild from GLI inversion method. (b) The reslt calclated with traveltime tomography sing Tihonov reglarization. (c) The reslt calclated with TV traveltime tomography. (d) The reslt calclated with MTV traveltime tomography. Conclsions We apply the modified total variation reglarization scheme to traveltime tomography and compare the performance of MTV reglarization method with Tihonov and TV reglarization methods. We decople the original traveltime inversion problem into two sbproblems, and employ the conjgate gradient method and split-bregman iterative method to solve these two sbproblems. We nmerically validate the improvement of or new method sing synthetic data and real data. However, or method needs to solve an L2-TV problem compared with the Tihonov reglarization method, which wold increase additional comptational cost. Acnowledgements We appreciate the spport from GeoTomo allowing s to se the software pacage and perform this research. References Anagaw, A. Y. and Sacchi, M. D. [2012] Edge-preserving seismic imaging sing the total variation method. Jornal of Geophysics and Engineering, 9(2), 138. Bertete-Agirre, H., E. Cheraev. and Oristaglio, M. [2002] Non-smooth gravity problem with total variation penalization fnctional. Geophysical Jornal International, 149(2), Last, B. J. and Kbi, K. [1983] Compact gravity inversion. Geophysics, 48(6), Lin, Y. and Hang, L. [2014] Acostic-and elastic-waveform inversion sing a modified total-variation reglarization scheme. Geophysical Jornal International, 200(1), Loris, I. and Verhoeven, C. [2012] Iterative algorithms for total variation-lie reconstrctions in seismic tomography. GEM-International Jornal on Geomathematics, 3(2), Portniagine, O. and Zhadanov, M. S. [1999] Focsing geophysical inversion images. Geophysics, 64(3), Tihonov, A. N. and Arsenin, V. I. [1977] Soltions of ill-posed problems. Vh. Winston. Zhang, J. and Tosoz, M. N. [1998] Nonlinear refraction traveltime tomography. Geophysics, 63(5), Zh, T. and Harris, J. M. [2015] Applications of bondary-preserving seismic tomography for delineating reservoir bondaries and zones of CO 2 satration. Geophysics, 80(2), M33-M41.