Main Menu. Summary. Introduction. stands for general space-time coordinates. γ

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1 Yu Geng*, ILab, IGPP, University of California, Santa Cruz, visiting from Institute of Wave and Information, Xi an Jiaotong University, Ru-Shan Wu, ILab, IGPP, University of California, Santa Cruz, Jinghuai Gao, Institute of Wave and Information, Xi an Jiaotong University. Summary Time-dependent Gaussian Packets are high-frequency asymptotic space-time particle-like solutions of the wave equation. A Gaussian Packet is a Gaussian shape timelocalized Gaussian Beam. We study the dependence of Packet shape to the initial values and discuss the evolution of Gaussian Packet in smooth media and strong inhomogeneous media. Through comparison between paraxial Gaussian Packets and the results from finite difference simulations in strongly inhomogeneous media, such as the SEG/EAGE salt model, we see the severe deviation of paraxial Gaussian Packet from the accurate solution at large propagation time and near the salt boundaries. The application to strongly heterogeneous media needs to be further studied. Introduction Previous study has shown how to use Gaussian Beams summation in computation of high frequency seismic wavefields in smoothly varying inhomogeneous media (Cerveny, Popov et al. 198; Popov 198; Nowack and Aki 198; Cerveny 1985; Babich and Popov 1989; Nowack ; Cerveny, Klimes et al. 7). ore recently overcomplete frame-based Gaussian Beam summations have been developed (Lugara, Letrou et al. ) and successfully used to depth migration (Hill 199; Hale 199; Hill 1). This Beam form solutions are usually formulated in the frequency domain and it can decompose the wavefield into beam fields that are localized both in position and direction. Time localization for seismic imaging has been discussed by Wu (8; 9) using a Dreamlet decomposition and propagating method. In this paper, we are interested in the particle-like high-frequency asymptotic solution which is localized around a neighborhood of a point in space with a Gaussian envelope and propagates along a ray. This solution is also called quasiphotons (Babich and Ulin 1981), space-time Gaussian Beam (Raslton 198), Gaussian Packet (Klimes 1989; Klimes ) or coherent states (Combescure, Ralston et al. 1999). In this study we will refer this kind of solution as Gaussian Packets. Similar solution is also discussed by Perel (Kiselev and Perel 1999; Perel and Sidorenko 7) in the homogeneous media as Gaussian wave Packets, which is an exact solution and has also been proven to be a mother wavelet with a fixed time parameter. Zacek has discussed the decomposition of the seismic data using optimized Gaussian Packets (Klimes 1989; Zacek 6; Zacek 6) and related migration in common-shot domain (Zacek ; Zacek 5; Bucha 8; Bucha 9). In this paper, we study the evolution of Gaussian Packet in inhomogeneous media in different complexities and discuss the limitation of high-frequency asymptotic approximation. We first review the formulation of Gaussian Packet evolution in smooth media. Then the Gaussian Packets dependence of the initial parameters will be discussed. Numerical result of the evolutions of Gaussian Packets in homogeneous media, gradient media and smoothed complex media are shown. Evolutions using Finite Difference code are also presented to show the validity of the Gaussian Packet method and its deviation from the accurate solution in some regions. In the paper, the vectors and matrices are expressed by means of their components or boldface letters. The capital letter indices take the values of KL=, 1,; the lower case indices take the values kl=, 1,,; the Greek indices take the values α, β = 1,,,. y α stands for the points of the space-time ray, while x α stands for general space-time coordinates. γ l stands for the ray parameters ( γ stands for travel time along the ray). Overview of Gaussian Packet Assume that in D case, a wave process ux ( α ) is described by the scalar wave equation. 1 ux ( α ) ux ( ) α = (1) V ( x i ) x Where V( x i ) is the velocity at point x i. A Gaussian Packet is an approximate space-time high frequency asymptotic solution of the corresponding hyperbolic wave equation. It can be expressed by means of the complex-valued vectorial amplitude A and the scalar phase functionτ gp u ( xα ) = A exp( iωτ ) () where ω is an arbitrary positive parameter, and it can be understood as the center frequency of the wave Packet. Both the phase-function and amplitude are complex-valued function of space coordinates x i and time x. The phasefunction must satisfy the space-time eikonal equation τ τ V τ τ = () x x And Im[ τ ( x α )], herev stands for propagation velocity, and it is real-valued. 1 SEG SEG Denver 1 Annual eeting 97

2 Using the method of characteristics, the eikonal equation can be solved by given space-time slowness vector N α = τ, α and Hamiltonian H = H( xα, Nβ ). Unlike Gaussian Beam, the phase function here is not τ ( xα ) simply in the form ofτ( xα ) = τ( x i ) x, thus. x Choosing N = 1 along the central ray, time y will be equivalent to the corresponding travel time γ and the space-time ray tracing effectively reduced to the usual τ τ space ray tracing, = V. The space-time paraxial approximation of a Gaussian Packet centered at point y α with x = y( γ ) is gp 1 u ( xκ ) = A exp( iω[ Ni( xi yi) + Nij( xi yj)( xi yj)]) () τ where Nij is the spatial part of Nαβ = Nβα =. xα xβ In order to distinguish from the exact Gaussian Packet, this form of Gaussian Packet is called paraxial Gaussian Packet. In ray-centered coordinates system ( q1, q, q ) with q s = ds stands for the arclength measured along the s central ray, the second space-time derivatives matrix αβ of the phase function can be written as τ τ τ ij =, i =, = (5) qi qj qi x x x Denoting =, =, 1 ( q) 1 ( q) V 1 =, v = ( q) (6) V and paraxial-ray propagator matrix by Q1 Q P1 P where V ( q) is the velocity derivative along the ray. m The complex-valued scalar amplitude evolves along the central ray will be, Babich et al(babich and Ulin 1981) A= A V V det( Q1+ Q ) The calculation of N α can done in ray-centered coordinates system, which can be expressed as follow, 1 ( q) = V V v (8) ( q ) (7) = V [ V ] (9) = V (1) 1 The derivative of Ricatti equations in ray-centered coordinate system can be referred to Babich et al(babich and Ulin 1981), Norris et al(norris, White et al. 1987), Klimes(Klimes ). The sub-matrix (Cerveny, Klimes et al. 7) can be calculated in the same way as Gaussian Beams and m can be solved analytically (Babich and Ulin 1981). It is obviously that if m =, the Gaussian Packet will be reduced to the corresponding Gaussian Beam. It can be find out that, will have the following expressions(babich and Ulin 1981; Klimes ), 1 = [( Q1+ Q ) ] T (11) T 1 = ( Q1 + Q ) Q (1) Which means, we can calculate and easily from the formulation (11) (1), and then obtain other quantities through the relations (8), (9), (1). Gaussian Packet evolution in homogeneous media and the initial parameters Imaginary part of determines the initial Gaussian Packet width along the plane perpendicular to the ray, and real part of determines the initial curvature of the phase front, which are the same as in the case of Gaussian Beam. We will discuss the initial value of a Gaussian Packet mainly in D case ( x = and q = ). A good choice of the initial value of can be referred to Hill (Hill 199) 11 = i ωrw (1) P = i V, Q = ωrw V (1) where w = πva ωr is the packet initial width at reference frequency ω r. As shown in Figure 1, in a homogeneous media with velocity equals to km/s, 7 rays are shot at x = km, 5 km, 8km from the surface individually, only the packet with ωr = πrad / s, 1 = and = i propagating along 19 th ray is shown. If purely imaginary number is chosen as initial value of 11 for the evolution of a single Gaussian Packet, the packet has zero curvature at the beginning and spreads less, while complex initial value represents a curve front of packet at the surface and the packet may focused on a certain depth according to the value of Re( 11) during the evolution (Nowack 8), as shown in Figure 1. 1 SEG SEG Denver 1 Annual eeting 98

3 b) Figure 1. Snapshot at t = γ =. s, 1.s for a Gaussian Packet with different initial parameters: from left to right are Re <, Re =, Re >, respectively c) and 1 must be chosen so that the quadratic 1 term Im( N ( x y )( x y )) is positive defined. Using ij i j i j the same velocity model and ray parameter, bigger will produce shorter Gaussian Packet along the central ray as shown in Figure. Also, if 1, the Gaussian Packet shows unsymmetry with respect to the central ray as shown in Figure. With Re( ), the real part of controls the decaying of the packlet along the central ray. Thus, as pure positive imaginary and 1 = will be good choice for Gaussian Packet s initial values. Figure. Snapshot for Gaussian Packets on t = γ =.7s with different initial. From left to right, equals to.1i, 1i, i, respectively. White line stands for corresponding central ray. a) Figure. Snapshot for a Gaussian Packet on t = γ =. s, 1.s with different initial value 1 and. From left to right: a), 1 = 1i, 1 = 1i when = i ; b) 1 = 1, 1 = 1, when = i ; c) = + i, = + i, when 1 =. Gaussian Packets evolution in gradient media The evolution of a Gaussian Packet in a gradient media is similar to the case in a homogeneous medium. Figure a shows the evolution of Gaussian Packets of ωr = π rad / s with different propagating direction and given parameters = i ωrw, = 1i and 1 = in a gradient media of dv / dz =.. White lines are the 7 rays traced in the whole media, and red lines are the central ray for corresponding Gaussian Packets. For comparison, the evolution results using finite difference code (Xie and Yao 1988) are shown in Figure b. The paraxial Gaussian Packet evolution itself can provide fairly accurate phase and amplitude information around the vicinity of the central point in this case. However, the phase begins to be inaccurate when it is too far from the central point. a) 1 SEG SEG Denver 1 Annual eeting 99

4 b) c) d) Figure. Evolution of Gaussian Packet through a linearly gradient medium. a) using paraxial Gaussian Packet method; b) using Finite Difference method. Gaussian Packets evolution in complex media Now we examine the evolution of Gaussian Packet in strongly inhomogeneous media. As an example, we show the numerical results using a smoothed SEG velocity model in Figure 5. We know that the Gaussian Packet solution expressed by () - (1) is a paraxial high-frequency asymptotic approximation, and is only accurate in the vicinity along a central ray. The validity region around the ray and the deviation from the exact solution when shifting away from the central ray depend on the medium velocity variation. For strong heterogeneous media, such as the irregular salt structures, the validity region may be very limited. This is the reason why Gaussian Beam method has some difficulty in applying to subsalt imaging. The Gaussian Packet method may face similar difficulties. However, Gaussian Packet has different diffraction behaviors than the Gaussian Beam depending on the pulse duration with respect to the frequency. Through the comparison of the evolutions of two wave packets in Figure 5e computed by FD simulations to the corresponding Gaussian Packets in Figure 5b and 5c (paraxial approximation), we see that at large propagation distance in strongly varying medium, especially near the salt boundaries, the paraxial solution deviates severely from the accurate solution. Therefore, further study is warranted in order to apply the Gaussian Packet method to imaging in strongly heterogeneous media. a) b) e) Figure 5. Evolution of Gaussian Packets through a smoothed SEG salt model. a) smoothed velocity model; b) to d), ray tracing (white lines) from different shots locations at km, 8km and 1km respectively. Corresponding evolutions of Gaussian Packets (paraxial approximate solution) are shown along a few central rays (black lines) at t = γ = (.5 n) s, n= 1,,... ; e) Snapshot for two packets evolution using Finite Difference method to compare with the paraxial Gaussian Packet solution in b), c), white lines are the corresponding central rays of the packets. Conclusions Initial values of Gaussian Packets will determine the shape and behavior of Gaussian Packet evolution. Therefore, the selection of initial parameters is important for modeling and imaging using Gaussian Packets. Through comparison between paraxial Gaussian Packets and the results from finite difference simulations in strongly inhomogeneous media, such as the SEG/EAGE salt model, we see the severe deviation of paraxial Gaussian Packet from the accurate solution in some areas for large propagation time. The application in those regions needs further study. Acknowledgments The author would like to thank Prof. Ludek Klimes for helpful discussion on the Gaussian Packet. We thank also Yingcai Zheng, Xiao-Bi Xie and Yaofeng He for help and discussions. This work is supported by WTOPI (Wavelet Transform On Propagation and Imaging for seismic exploration) Project at University of California, Santa Cruz. The author Yu Geng would also like to thank the support of China Scholarship Council. 1 SEG SEG Denver 1 Annual eeting 1

5 EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 1 SEG Technical Program Expanded Abstracts have been copy edited 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. REFERENCES Babich, V.., and.. Popov, 1989, The ethod of Gaussian Beams Summing: Izvestiya Vysshikh Uchebnykh Zavedenii Radiofizika,, no. 1, Babich, V.., and V. V. Ulin, 1981, Complex space-time ray method and quasiphotons: Zap. Nauchn. Sem. Leningr. Oth. at. Instit. 117, 5 1. Bucha, V., 8, Gaussian packet prestack depth migration, Part : Optimized Gaussian packets: Seismic Waves in Complex -D Structures Report 18: Bucha, V., 9, Gaussian packet prestack depth migration, Part : Simple -D models: Seismic Waves in Complex -D Structures Report 19: 9-5. Cerveny, V., 1985, Gaussian-Beam Synthetic Seismograms: Journal of Geophysics-Zeitschrift Fur Geophysik, 58, no. 1-, 7. Cerveny, V., L. Klimes, and I. Pšencík, 7, Seismic ray method: Recent developments: Advances in Geophysics, 8, no. 8, 1 16, doi:1.116/s )81-8. Cerveny, V., and.. Popov, et al., 198, Computation of Wave Fields in Inhomogeneous-edia Gaussian-Beam Approach: Geophysical Journal of the Royal Astronomical Society, 7, no. 1, Combescure,., J. Ralston, and D. Robert, 1999, A proof of the Gutzwiller semiclassical trace formula using coherent states decomposition: Communications in athematical Physics,, no., 6 8, doi:1.17/s5591. Hale, D., 199, igration by the Kirchhoff, slant stack and Gaussian Beam methods: CWD-11, Center for Wave Phenomena, Colorado School of ines. Hill, N. R., 199, Gaussian-Beam igration: Geophysics, 55, , doi:1.119/ Hill, N. R., 1, Prestack Gaussian-beam depth migration: Geophysics, 66, 1 15, doi:1.119/ Kiselev, A. P., and. V. Perel, 1999, Gaussian wave packets: Optics and Spectroscopy, 86, no., 7 9. Klimes, L., 1989, Gaussian Packets in the Computation of Seismic Wavefields : Geophysical Journal International, 99, no., 1, doi:1.1111/j.165-6x.1989.tb1699.x. Klimes, L., and I. Pšencík, 1989, Optimization of the Shape of Gaussian Beams of a Fixed Length: Studia Geophysica et Geodaetica,, no., 16 16, doi:1.17/bf Klimes, L.,, Gaussian packets in smooth isotropic media: Seismic Waves in Complex -D Structures Report 1: -5. Lugara, D., and C. Letrou, A. Shlivinski, E. Heyman, and A. Boag,, Frame-based Gaussian beam summation method: Theory and applications : Radio Science, 8, no., 86, doi:1.19/1rs59. 1 SEG SEG Denver 1 Annual eeting 11

6 Norris, A. N., and B. S. White, 1987, Gaussian Wave-Packets in Inhomogeneous-edia with Curved Interfaces: Proceedings of the Royal Society of London Series a-athematical Physical and Engineering Sciences 118): 9-1. Nowack, R. and K. Aki, 198, The Two-Dimensional Gaussian-Beam Synthetic ethod - Testing and Application: Journal of Geophysical Research 89Nb9): Nowack, R. L.,, Calculation of synthetic seismograms with Gaussian beams: Pure and Applied Geophysics, 16, no., 87 57, doi:1.17/pl157. Nowack, R. L., 8, Focused Gaussian Beams for Seismic Imaging. Expanded abstracts, SEG 78th Annual eeting: Perel,. V. and. S. Sidorenko 7, New physical wavelet 'Gaussian wave packet'. Journal of Physics a-athematical and Theoretical 1): Popov,.., 198, A New ethod of Computation of Wave Fields Using Gaussian Beams: Wave otion,, no. 1, 85 97, doi:1.116/ ) Raslton, J., 198, Gaussian Beams and the propagation of singularities, in W. Littman ed., Studies in Partial Differential Equations : AA Studies in athematics : 6-8. Wu, R. S., B. Y. Wu, and Y. Geng, 8, Seismic wave propagation and imaging using time-space wavelets. Expanded Abstracts: 78th Annual International eeting, SEG, Wu, R. S., B. Y. Wu, and Y. Geng, 9, Imaging in compressed domain using Dreamlets: Annual International eeting, SEG, Expanded Abstracts:. Xie, X. B., and Z. X. Yao, 1988, P-SV wave responses for a point source in two-dimensional heterogeneous media: Finite-difference method: Chinese Journal of Geophysics-Chinese Edition, 1, 7 9. Zacek, K.,, Gaussian packet pre-stack depth migration: 7th Annual International eeting, SEG, Expanded Abstracts, Zacek, K., 5, Gaussian packet pre-stack depth migration of the armousi data set: 75th Annual International eeting, SEG, Expanded Abstracts, Zacek, K., 6, Decomposition of the wave field into optimized Gaussian packets: Studia Geophysica et Geodaetica, 5, no., 67 8, doi:1.17/s11-6--y. Zacek, K., 6, Optimization of the shape of Gaussian beams: Studia Geophysica et Geodaetica, 5, no., 9 66, doi:1.17/s11-6--z. 1 SEG SEG Denver 1 Annual eeting 1