Summary
Theonly way to make an excessively complex velocity model suitable for application of ray-based methods, such as the Gaussian beam or Gaussian packet methods, is to smooth it. We have smoothed the Marmousi model by choosing a coarser grid and by minimizing the second spatial derivatives of the slowness. This was done by minimizing the relevant Sobolev norm of slowness. We show that minimizing the relevant Sobolev norm of slowness is a suitable technique for preparing the optimum models for asymptotic ray theory methods. However, the price we pay for a model suitable for ray tracing is an increase of the difference between the smoothed and original model. Similarly, the estimated error in the travel time also increases due to the difference between the models. In smoothing the Marmousi model, we have found the estimated error of travel times at the verge of acceptability. Due to the low frequencies in the wavefield of the original Marmousi data set, we have found the Gaussian beams and Gaussian packets at the verge of applicability even in models sufficiently smoothed for ray tracing.
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References
ABDULLAEv, S.S.Chaos and Dynamics of Rays in Waveguide Media(Gordon and Breach, Amsterdam 1993).
ADDISON, P.S.Fractals and Chaos: An Illustrated Course(IOP Publishing Ltd, Bristol and Philadelphia 1997).
BRAC, J. and NGUYEN, L.L.Modeling geological objects with splines.In PSI 1990 Annual Report(Institut Francais du Petrole, Rueil Malmaison, France 1990).
BULANT, P. (2001)Sobolev Scalar Products in the Construction of Velocity Models Application to Model Hess and to SEG/EAGE Salt ModelPure appl. geophys. 1591487--1506.
GOLD, N., SHAPIRO, S.A., BOJINSKY, S., and MÜLLER, T.M. (2000)An Approach to Upscaling for Seismic Waves in Statistically Isotropic Heterogenous Elastic Media, Geophysics 65, 1837–1850.
GRUBB, H.J. and WALDEN, A.T. (1995)Smoothing Seismically Derived VelocitiesGeophys. Prosp. 431061–1082.
KEERS, H., DAHLEN, F.A., and NOLET, G. (1997)Chaotic Ray Behaviour in Regional SeismologyGeophys. J. Int. 131361–380.
KLIME§, L. (1984)Expansion of a High frequency Time-harmonic Wavefield Given on an Initial Surface into Gaussian BeamsGeophys. J. R. astr. Soc. 79105–118.
KLIMES, L. (1989)Gaussian Packets in the Computation of Seismic WavefieldsGeophys. J. Int. 99, 421–433.
KLIMES, L.Lyapunov exponents for 2-D ray tracing without interfaces.In Seismic Waves in Complex 3-D Structures,Report 8(Dep. Geophys., Charles Univ., Prague 1999) pp. 83–96.
KLIMEs, L. (2001)Lyapunov Exponents for 2-D Ray Tracing without InterfacesPure apps. geophys. 1591465–1485.
KATOK, S.R.The estimation from above for the topological entropy of a difeomorphism.In Global Theory of Dynamical Systems. Lecture Notes in Mathematics, vol. 819(eds. Netecki, Z. and Robinson, C.) (Springer, Berlin, Heidelberg, New York 1980) pp. 258–264.
LYAPUNOV, A.M.Problème Général de la Stabilité du Mouvement Annals of Mathematical Studies vol. 17 (Princeton University Press 1949).
MCCAULEY, J.L.Chaos Dynamics and Fractals: An Algorithmic Approach to Deterministic Chaos (Cambridge University Press, Cambridge 1993).
MOLLER, T.M. and SHAPIRO, S.A. (2000)Most Probable Seismic Pulses in Single Realizations of Two-and Three-dimensional Random MediaGeophys. J. Int. 14483–95.
OSELEDEC, V.I. (1968)A Multiplicative Ergodic Theorem: Lyapunov Characteristic Numbers for Dynamica SystemsTrans. Moscow Math. Soc. 19179–210 in Russian, 197–231 English translation.
SMITH, K.B., BROWN, M.G., and TAPPERT, F.D. (1992)Ray Chaos in Underwater AcousticsJ. Acoust. Soc. Am. 911939–1949.
TAPPERT, F.D. and TANG, X. (1996)Ray Chaos and EigenraysJ. Acoust. Soc. Am. 99, 185–195.
TARANTOLA, A.Inverse Problem Theory(Elsevier, Amsterdam 1987).
VERSTEEG, R.J. (1993)Sensitivity of Prestack Depth Migration to the Velocity ModelGeophysics 58873–882.
VERSTEEG, R.J.Analysis of the Problem of the Velocity Model Determination for Seismic Imaging(Ph.D. Thesis, University of Paris VII 1991).
VERSTEEG, R.J. and GRAU, G. (eds.)The Marmousi Experience(Eur. Assoc. Explor. Geophysicists, Zeist 1991).
WITTE, O., ROTH, M., and MOLLER, G. (1996)Ray Tracing in Random MediaGeophys. J. Int. 124159–169.
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Žáček, K. (2002). Smoothing the Marmousi Model. In: Pšenčík, I., Červený, V. (eds) Seismic Waves in Laterally Inhomogeneous Media. Pageoph Topical Volumes. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8146-3_7
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DOI: https://doi.org/10.1007/978-3-0348-8146-3_7
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