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1 Downloaded 11/08/14 to Reditribution ubject to SEG licene or copyright; ee Term of Ue at Gauian beam baed enitivity kernel calculation and it application in turning wave tomography Yu Geng*, Modeling and Imaging Lab, IGPP, Univerity of California, Santa Cruz, viiting from Intitute of Wave and Information, Xi an Jiaotong Univerity. Xiao-Bi Xie, Modeling and Imaging Lab, IGPP, Univerity of California, Santa Cruz. Summary We introduce a Gauian beam (GB) baed method to calculate finite-frequency enitivity kernel and tet it application in turning wave tomography. The GB ummation method ha the advantage of high computation efficiency and no any angle limitation. Thu it i uitable in generating enitivity kernel for high-frequency, long propagation ditance and wide-angle wave including the turning wave. We firt validate the GB kernel by comparing them with thoe calculated uing analytical and finite-difference (FD) olution. Then, baed on thee enitivity kernel, we build an inverion ytem for turning-wave travel-time tomography. The propoed method incorporate the wave phenomena into the inverion thu avoid certain difficultie encountered by the high-frequency aymptotic method. On the other hand, thi method till keep the implicity of the ray-baed tomography method. The preliminary reult from numerical tet reveal the potential application of thi method in building velocity model uing turning wave. Introduction Although velocity tomography and migration velocity updating are traditionally dominated by the ray-baed technique, wave equation baed method gradually gain their ground and are reported to provide better reult (Luo and Schuter, 1991; Woodward, 199; Sava and Biondi, 004; Spetzler and Snieder, 004; de Hoop, van der Hilt and Shen, 006; Jocker et al., 006; Fliedner and Bevc, 008). Finite-frequency enitivity kernel are introduced in velocity tomography to overcome the diadvantage of ray-baed method which aume an infinitely high frequency. Several method can be ued to calculate finitefrequency enitivity kernel in heterogeneou model (Woodward, 199; Marquering, Nolet and Dahlen, 1998; Marquering, Dahlen and Nolet, 1999; Dahlen, Hung and Nolet, 000; Hung, Dahlen and Nolet, 000; Tian et al., 007; Xie and Yang, 008a; Liu et al., 009). Turning wave data are widely ued to determine the uburface velocitie. Technique ued for thi purpoe include turning-ray tomography baed on the firt arrival information (Zhu, Sixta and Angtman, 199; Stefani, 1995; Bai et al., 003; Simmon, 008; Zhu et al., 008). Recently, following the pioneer work by Lailly (1983) and Tarantola (1984), full-waveform inverion (Pratt and Goulty, 1991; Pratt, 1999; Brender and Pratt, 007; Virieux and Operto, 009) uing the entire recorded eimic data, ha been increaingly applied in velocity tomography. The full-waveform inverion method uually tart with an initial velocity model which can be etimated from travel-time tomography. At high-frequencie, wavefield can be decribed by ummation of paraxial GB (Popov, 198; Cerveny, 1985), and it ha been widely ued to contruct Green function for eimic modeling and imaging (Nowack and Aki, 1984; Hill, 1990; 001; Gray, 005; Nowack, 008; Popov et al., 010). The GB method overcome certain critical difficultie uch a the two-point boundary value problem and the cautic problem, and can handle wide-angle wave including turning wave. Thi make it epecially uitable for tranmitted wave tomography. The diadvantage of the GB method i that it naturally introduce mooth effect to the obtained wavefield. For very complex model, velocity moothing i often required, which may limit the application of the GB method. Xie (011) teted uing the GB method to calculate finitefrequency enitivity kernel. In thi paper, we continue to dicu the GB baed enitivity kernel and tet it application in turning wave tomography. Compared to the ray tomography, the kernel baed method incorporate both wave phenomena and ray concept; while compared to the full-wave equation method, the GB baed method i more efficient. In the ret part of thi paper, we firt dicu how to build finite-frequency enitivity kernel from GB ummation and validate thee kernel by comparing them with analytic and FD olution. We then compare the kernel predicted travel-time delay with thoe directly meaured value. Finally, we create an inverion ytem uing thee enitivity kernel and tet turning wave tomography uing ynthetic data et. Senitivity Kernel Calculation and Calibration The finite-frequency enitivity kernel for tranmitted wave have been dicued by many author (e.g., Woodward, 199; Spetzler and Snieder, 004; Jocker et al., 006). The frequency domain travel time enitivity kernel K can be expreed a F r; rs, Gr; rg, Gr ; r, G KF r, rs, rg, imag k0 (1) G S where k v r i the background wavenumber, i 0 0 the frequency, v0 r i the background velocity, G i the 1 656

2 Downloaded 11/08/14 to Reditribution ubject to SEG licene or copyright; ee Term of Ue at Green function, r i the pace location, r S and r are G the ource and receiver location, and taking imaginary part. imag denote The broadband enitivity kernel can be obtained by tacking ingle-frequency kernel. W( ) KB r, rs, rg KF r, rs, r G, d, () where P( ) W ( ) (3) P( ) d i the weighting function. P( ) S( ) S *( ), and S( ) i the ource pectrum. The travel-time difference caued by velocity perturbation can be calculated a v t( rs, rg ) KB r, rs, r G dv. (4) v0 Equation (4) alo form an integral equation which can be ued to invert velocity perturbation vv0 from travel time reidual t. To validate the GB Green' function and enitivity kernel, we firt compare the analytical, FD and the GB Green function. In a D homogeneou medium, the Green function can be calculated uing the zerothorder firt-kind Hankel function H ( kr ) 0 (1) i (1) G( rr, S; ) H0 ( kr), (5) 4 where r rr. The aymptotic Green function can be written a the ummation of Gauian beam u ( rr,, ) over ray parameter i Q G r r u r r d. (6) 0 (, S; ) GB(, S, ) V0 GB Here Gauian beam u ( rr,, ) can be written in the raycentered coordinate ytem a u(, n, ) (Cerveny, Klime and Pencik, 1984; Nowack and Aki, 1984; Hill, 1990) 1 v i P u, n, exp i n, (7) Q Q where i the coordinate along the ray path, n i the coordinate perpendicular to the ray, i the travel time along the ray, and P and Q are complex calar function, which can be obtained by olving the dynamic ray tracing equation with initial value P0 i V0 and Q0 rw0 V0. r, w0 and V 0 tand for reference frequency, the initial beam width and the velocity at the ource end of the ray, repectively. GB Figure 1 compare the 1 Hz Green function calculated uing analytical, FD and GB method, where, except at cloe to the ource, all three method give almot the ame accuracy. We then compare the enitivity kernel calculated uing different method in a contant velocity model v0 3.5 km/. A 10 Hz Ricker wavelet i ued a the ource time function and the ource-receiver ditance i 5 km. The enitivity kernel calculated uing different method are hown in Figure. Fifty beam are ued to form the GB Green function, and the initial beam width i 0.58 km. The profile along the two dahed line in Figure are hown in Figure 3. We ee different method give conitent reult. Shown in Figure 4 are comparion between the FD and GB kernel in a pixel by pixel way, where the vertical and horizontal coordinate are kernel value obtained by uing GB and FD method, repectively. We ee mot point fall on the unit lope with ome cattered point coming from area cloe to the ource and receiver. Figure 1: Comarion of 1 Hz Green function calculated uing GB, FD and analytical method, with (a) real part and (b) imaginary part. Next, we compare the travel-time difference predicted uing the enitivity kernel with thoe directly meaured. We ue a contant background velocity and a Gauian haped circular velocity perturbation patch with a radiu of 300 m and centered at x 4 km and z 3.1 km. The maximum perturbation at the center of the patch are ranging between 30% and 180%. The predicted travel time difference i calculated uing equation (4) with the enitivity kernel calculated in the background velocity model. The actual travel-time difference i meaured from the cro-correlation between ynthetic eimogram calculated by FD in the background velocity model and the velocity model including the perturbation. In Figure 5, we plot both value along horizontal and vertical axe. For perturbation between -30% and 60%, the predicted and CPS/SEG Beijing 014 International Geophyical Conference 657

3 Downloaded 11/08/14 to Reditribution ubject to SEG licene or copyright; ee Term of Ue at meaured t are cloe to the unit lope. For very large velocity perturbation, the predicted t how apparent deviation due to the nonlinear effect. However, the predicted t from kernel calculated uing different method are conitent. Figure : Senitivity kernel calculated from (a) FD, (b) GB and (c) analytical olution. Figure 3: Selected profile for kernel in Figure, (a) along the vertical dahed line, and (b) along the horizontal dahed line. Figure 4: Comparion between the enitivity kernel caculated uing the GB and FD method. Figure 5: Comparion between the kernel predicted travel time difference δt and directly meaured δt. Inverion Sytem To build an inverion ytem, we dicretize equation (4) to obtain t( rs, rg ) m( ri ) KB ri, rs, r G, (8) i where both unknown velocity perturbation m( r i ) and kernel K have been partitioned following Xie and Yang (008b). B Numerical Example Reolution tet uing a checkerboard model. We firt tet the inverion reolution under the turning wave geometry. A example, hown in Figure 6 are three broadband turning wave enitivity kernel calculated in the background model, of which velocitie change linearly from km / at the urface to 3.5 km / at 3 km depth. The true velocity model (Figure 7a) ha a checkerboard haped perturbation overlapped on the background. The checkerboard ha a grid ize of 1 km 1 km and the perturbation vary between 5% (Figure 7b). We ue a 10 Hz Ricker wavelet a the ource time function. To generate large offet data for inverion, we extend the velocity model on both ide, and ue total of 51 hot, each with 8 right-hand ide receiver, to illuminate the model. Both the hot and receiver interval are 0.5 km and the maximum offet i 14 km. The background model alo erve a the initial model. The travel-time difference are calculated from ynthetic data et generated uing the true velocity model and the background velocity. Shown in Figure 7c i the inverion reult. Comparing Figure 7c with 7b, the turning wave tomography provide reaonable reolution for the uburface tructure. CPS/SEG Beijing 014 International Geophyical Conference 658

4 Downloaded 11/08/14 to Reditribution ubject to SEG licene or copyright; ee Term of Ue at Figure 6: Senitivity kernel calculated in a model with linear vertical velocity gradient, with offet 1.5 km, 5.0 km and 8.5 km. Black line are correponding geometrical ray. Figure 7: Reolution tet uing the checkerboard model: (a) true velocity model, (b) velocity perturbation and (c) retrieved perturbation. Turning wave tomography in a model with high-velocity incluion. In thi example, the true velocity model ha a high-velocity patch added to the background velocity, a hown in Figure 9a. The background velocity i imilar to that ued in the previou example and the patch ha a maximum perturbation of 8% (hown in Figure 9b). 10 Hz Ricker wavelet i ued to generate the ynthetic data. For inverion, we ue 15 hot and 8 fixed receiver. Both of them are located on the urface between 0.5 km to14.5 km, with 1 km hot interval and 0.5 km receiver interval. The background model alo erve a the initial model. In Figure 8, we compare t value in the data with thoe predicted by kernel and velocity perturbation. The conitency between two et of t value pave the bai for velocity inverion. Shown in Figure 9c i the inverion reult. Comparing Figure 9c with 9b, the inverted velocity perturbation reproduce the perturbation pattern with reaonable accuracy. A a preliminary tet, iteration are not ued in the inverion. Concluion We propoed a method to calculate finite-frequency enitivity kernel by umming up Gauian beam. To validate the reulted kernel, we compared the kernel with Figure 8: Comparion between the kernel predicted travel time difference and directly meaured δt from ynthetic data. Figure 9: Turning wave tomography reult for a model with high-velocity incluion: (a) true velocity model, (b) velocity perturbation and (c) inverted perturbation. FD olution and checked the kernel predicted and actually meaured travel time reidual. The GB method ha the advantage of high computation efficiency and no angle limitation, thu uitable for generating turning wave kernel. We teted the GB kernel in the turning wave tomography and achieved reaonable reolution and accuracy. The velocity tomography baed on finite-frequency enitivity kernel combine the wave phenomena into the inverion but till keep the implicity of the ray-baed tomography. It i expected that the propoed method will be ueful in building velocity model. Acknowledgment Thi reearch i upported by the WTOPI Reearch Conortium at the Univerity of California, Santa Cruz. We alo thank CWP for ray tracing algorithm in SU. CPS/SEG Beijing 014 International Geophyical Conference 659

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