P032 3D Seismic Diffraction Modeling in Multilayered Media in Terms of Surface Integrals

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1 P032 3D Seisic Diffraction Modeling in Multilayered Media in Ters of Surface Integrals A.M. Aizenberg (Institute of Geophysics SB RAS, M. Ayzenberg* (Norwegian University of Science & Technology, H.B. Helle (Norsk Hydro ASA, O&E Research Centre, K.D. Kle-Musatov (Institute of Geophysics SB RAS, J. Pajchel (Norsk Hydro ASA, O&E Research Centre & B. Ursin (Norwegian University of Science & Technology SUMMARY We present an iproved ultiple tip wave superposition ethod (MTWSM for the 3-D seisic odeling in ultilayered edia in ters of the Kirchhoff type integrals. The ain iproveent consists in correct description of their boundary values by the reflection/transission operators. The high-frequency approxiation of the operators (effective coefficients generalizes the plane-wave coefficients used in the conventional seisics and allows adequate reproduction of ultiple reflections and transissions accounting for the nearcritical effects. To show the potential of the MTWSM we give results of the Green's function odeling for a strong-contrast curvilinear interface in two-layered acoustic ediu.

2 Introduction. One of the widely used analytical ethods for seisic odeling, inversion and igration of 3-D wave fields in ultilayered inhoogeneous edia is the Kirchhoff integral ethod. It is well-known that in any high-frequency realization of the Kirchhoff integral there are two probles: description of the Green s function for inhoogeneous ediu and the boundary values of the wave field for curvilinear interfaces. To overcoe the first proble, the TWSM was introduced as a high-frequency approxiation of the Kirchhoff integral in a for invariant to inhoogeneity of the ediu (Aizenberg et al.996. Regarding the second proble, it is coon to represent the boundary values as ultiplication of incident wavefield and the plane-wave coefficient which have irregularities at critical angles. However such an approach causes artifacts in the for of edge waves diffracted at the contours of critical points on the interface (Kapfann988; Wenzel et al.990; Sen and Frazer, To overcoe this proble, the exact description of the boundary values in ters of integral reflection and transission operators was introduced (Kle-Musatov et al., 2004; Kle-Musatov et al., A high-frequency approxiation of these operators in the for of effective coefficients (Aizenberg et al., 2004; Aizenberg et al., 2005 allows eliination of the diffraction artifacts and accounting for the head waves. Kirchhoff integral propagators. Assue the source wavefield be ties scattered at the syste of interfaces in ultilayered ediu. Let the interfaces, sequentially be intersected by the wave in accordance to the wave code, be nubered S,... S. Note that the sae interface ay be intersected several ties. The scattered wavefield can be expressed by the Kirchhoff integral (Kle-Musatov et al., 2005 ( x,x ( x p b g ds 1 g 1 b ( x = ( x ( x,x ρ ( x ( x S n ρ n, where g ( x,x interfaces (Kle-Musatov et al., 2004 and ρ ( x b ( x and b ( x n S and its derivative. The wavefield ( x is the Green s function for the -th layer with the absolutely absorbing is the density. The boundary values describe the wave field locally reflected/transitted at the point x of the interface p represents superposition of the wave fields coing fro various reflection/transission points. The liiting value of x b x. integral p ( does not in general coincide with the boundary value Integral reflection and transission operators. It was shown in (Kle-Musatov et al., 2005 that the boundary values b ( x and b ( x n at any chosen point x of the ( 1 interface S can be exactly expressed through the traces of the wavefield p ( x and its ( 1 noral derivative p ( x n as the reflection/transission transfor ( x b x ρ x p x b K p K, ( ( 1 ( 1 x = x,x x, = x,x n ρ 1 n using the reflection and transission operators K 1 x,x = F ( x,ζ k ( ζ F ( ζ,x. ( 1 The space Fourier operator F ( ζ,x transfors the traces p ( 1 ( x and ( x fro the curvilinear interface coordinates x ( x 1, x 2 ζ = ( ζ, ζ. Coefficient 1 2 coefficient. ζ p n = into the wave nuber doain k is siilar to the plane-wave reflection or transission

3 Effective reflection and transission coefficients. In the high-frequency approxiation the action of the reflection/transission operators K ( x,x can be substituted by the ultiplication with the corresponding effective coefficients χ ( x (Aizenberg et al., 2004; Aizenberg et al., Therefore the reflection/transission transfor can be represented as b b ( x ( ( 1 ( 1 x χ x p x, χ x n ρ 1 n x ρ x p x The effective coefficients represent a generalization of the plane-wave coefficients to nonplanar waves, arbitrary interface curvature and finite doinant frequency of the wavelet. In the exceptional case of a plane incident wave and plane interface the effective and plane-wave coefficients coincide.. Modeling of the Green s function for two-layered ediu. The Green s function is the basic analytical tool used in various algoriths of solving inverse probles such as paraeter inversion, angle igration etc. (Ursin, It is usually assued that ultilayered overburden could be approxiated by soe sooth inhoogeneous ediu. In the case of strong-contrast internal interfaces and/or steeply dipping interfaces this approach ay lead to decreasing of the seisic iage resolution. To account for such type of interfaces it is natural to use the Green s function for ultilayered ediu instead of the free-space Green s function. Using the Kirchhoff integral propagators together with the reflection/transission operators allows odeling of the Green s function for a ultilayered overburden. In the nuerical exaple given below we deonstrate odeling of the Green s function for twolayered overburden using the single transission approxiation. Nuerical siulation is done with help of the MTWSM algorith. For coparison we show synthetic seisogras obtained both with the effective and the plane-wave transission coefficients. The 2-D section of the two-layered 3-D overburden, source and data window are shown in Figure 1a. Figure 1b displays odulus of the effective transission coefficient (bold line for the doinant linear frequency 30 Hz as a function of the horizontal coordinate. Modulus of the plane-wave transission coefficient (noral line is given on the sae figure. The seisogras (Figures 1c and 1e present the Green s function coputed using the plane-wave and effective coefficients respectively. On the seisogras we highlight the traces corresponding to the offset 2.0 k and show the separately in Figures 1d and 1f respectively. In coparison to the second seisogra (Figure 1e, the first seisogra (Figure 1c is distorted by the artificial diffraction caused by the curves of the critical transission points with radii k, k, k (Figure 1b. These critical curves coincide with the discontinuities of the first derivatives of the plane-wave coefficient. The artifacts are clearly visible on the zero-offset traces. To deonstrate that the aplitudes and fors of the wavelet are also essentially different for the other traces, we give the highlighted traces in Figures 1d and 1f. Fro the figures it is seen that the plane-wave coefficient produces the phase-shift of π 2 and the three-fold increase of the aplitude. These differences can be explained by the fact that the Fresnel zone strongly influencing the data window and belonging to the anticline part of the interface contains the local perturbation of the plane-wave coefficient in coparison to the effective coefficient between k and k. As a result this perturbation is spread over all the seisogra in Figure 1c. Siilar aplitude effect can be expected when odeling of the Green s function using the asyptotic ray theory. The effect can be even stronger due to absence of the diffraction soothing of the wavefield in coparison to the Kirchhoff integral approach.

4 Conclusions. We have presented the iproved MTWSM algorith for the 3-D seisic diffraction odeling in ultilayered edia. On the nuerical exaple we have deonstrated that the algorith provides an adequate representation of the acoustic Green s function for the two-layered ediu. Acknowledgeents. The authors would like to thank the Russian Fund for Basic Research (grants and , the Federal Agency for Science and Innovations of the Russian Federation (grant RI-112/001/252 and Norsk Hydro ASA (Bergen, Norway for support of this study. References Aizenberg, A.M., Aizenberg, М.А., Helle, H.B., Kle-Musatov, K.D. and Pajchel, J. [2004] Modeling of single reflection by tip wave superposition ethod using effective coefficient. Extended Abstracts, 66 th EAGE Conference & Exhibition, Paris, France, P187. Aizenberg, А.М., Aizenberg, М.А., Helle, H.B. and Pajchel, J. [2005] Reflection and transission of acoustic wavefields at culvilinear interface between two inhoogeneous edia. Proceedings of the Lavrentiev s Institute of Hydrodynaics SB RAS, Dynaics of solid ediu, Acoustics of inhoogeneous edia 23, (in Russian. Aizenberg, A.M., Helle, H.B., Kle-Musatov, K.D. and Pajchel, J. [1996] The tip wave superposition ethod based on the refraction transfor. Extended Abstracts, 58 th EAGE Conference & Exhibition, Asterda, Netherlands, Paper C001. Kapfann, W. [1988] A study of diffraction-like events on DECORP 2-S by Kirchhoff theory. J. Geophys., 62, Kle-Musatov, K., Aizenberg, A., Helle, H.B. and Pajchel, J. [2004] Reflection and transission at curvilinear interface in ters of surface integrals. Wave Motion, 39, Kle-Musatov, K., Aizenberg, A., Helle, H.B. and Pajchel, J. [2005] Reflection and transission in ultilayered edia in ters of surface integrals. Wave Motion, 41, 4, Sen, M.K. and Frazer, L.N. [1991] Multifold phase space path integral synthetic seisogras. Geophys. J. Int.04, Ursin, B. [2004] Tutorial: Paraeter inversion and angle igration in anisotropic elastic edia. Geophysics, 69, Wenzel, F., Stenzel, K.-J. and Zierann, U. [1990] Wave propagation in laterally heterogeneous layered edia. Geophys. J. Int.03,

5 faplitude daplitude a 0,0 source b 2,2 2,0 1,8 z (k -0,5 - -1,5 c 1 = 2 k/s ρ 1 = 2 g/c 3 c 2 = 4 k/s ρ 2 = 2 g/c 3 interface Transission coefficients -2,0 data window 0,6 - -0,5 0,0 0,5 1,5 2,0 2,5 3,0 3,5 x (k 0,4 - -0,5 0,0 0,5 1,5 2,0 2,5 3,0 3,5 x (k c 0,7 0,10 0,9 0,05 1,1 Tie (sec 1,3 0,00 1,5-0,05 1,7 0,0 0,5 1,5 2,0 2,5 3,0 Offset (k -0,10 0,9 1,1 1,3 Tie (sec e 0,7 0,10 0,9 0,05 Tie (sec 1,1 1,3 1,5 0,00-0,05 1,7 0,0 0,5 1,5 2,0 2,5 3,0 Offset (k -0,10 0,9 1,1 1,3 Tie (sec Figure 1: Green s function for two-layered ediu.