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Full-waveform Velocity Inversion Based on the Acoustic Wave Equation

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Full-waveform velocity inversion based on the acoustic wave equation in the time domain is investigated in this paper. The inversion is the iterative minimization of the misfit between observed data and synthetic data obtained by a numerical solution of the wave equation. Two inversion algorithms in combination with the CG method and the BFGS method are described respectively. Numerical computations for two models including the benchmark Marmousi model with complex structure are implemented. The inversion results show that the BFGS-based algorithm behaves better in inversion than the CG-based algorithm does. Moreover, the good inversion result for Marmousi model with the BFGS-based algorithm suggests the quasi-Newton methods can provide an important tool for large-scale velocity inversion. More computations demonstrate the correctness and effectives of our inversion algorithms and code.

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The authors declare no conflicts of interest.

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*American Journal of Computational Mathematics*, Vol. 3 No. 3B, 2013, pp. 13-20. doi: 10.4236/ajcm.2013.33B003.

[1] | A. Tarantola, “Inversion of Seismic Reflection Data in the Acoustic Approximation”, Geophysics, Vol. 49, No.8, 1984, pp. 1259-1266. |

[2] | C. Bunks, F. Saleck, S. Zaleski, and G. Chavent “Multiscale Seismic Waveform Inversion”, Geophysics, Vol.60, No.5, 1995, pp.1457-1473. |

[3] | R. G. Pratt, C. Shin, and G. J. Hicks, “Gauss-Newton and Full Newton Methods in Frequency-space Seismic Waveform Inversion”, Ceophysical Journal International, Vol.133, No.2, 1998, pp.341-362. |

[4] | C. Burs and O. Ghattas, “Algorithmic Strategies for Full Waveform Inversion: 1D experiments”, Geophysics, Vol.74, No.6, 2009, pp.WC37-WC46 |

[5] | R. G. Pratt, “Seismic Waveform Inversion in Frequency Domain Part I: Theory and Verification in a Physical Scale Model”, Geophysics, Vol. 64, No.3, 1999, pp.888-901. |

[6] | Z. M. Song, P. R. Williamson, and R. G. Pratt, “FreQuency-domain Acoustic-wave Modeling and Inversion of Cross Hole data: Part 2—Inversion Method, Synthetic Experiments, and Real-data Results”, Geophysics, Vol.60, No.3, 1995, pp.796-809. |

[7] | R. G. Pratt and M. H. Worthington, “The Application of Diffraction Tomography to Cross Hole Data”, Geophysics, Vol.53, No.10, 1988, pp.1284-1294. |

[8] | C. Shin and W. Ha, “A Comparison between the Behavior of Objective Functions for Waveform Inversion in the Frequency and Laplace Domains”, Geophysics, Vol.73, No.5, 2008, pp.VE119-VE133. |

[9] | W. Hu, A. Abubakar and T. M. Habashy, “Simultaneous Multifrequency Inversion of Full-waveform Seismic Data”, Geophysics, Vol.74, No.2, 2009, pp.R1-R14. |

[10] | L. Sirgue and R. G. Pratt, “Efficient Waveform Inversion and Imaging: A Strategy for Selecting Temporal Frequency”, Geophysics, Vol.69, No.1, 2004, pp.231-248. |

[11] | R. Fletcher and C. Reeves, “Function Minimization by Conjugate Gradients”, Com-puter Journal, Vol.7, No.2, 1964, pp.149-154. |

[12] | C. G. Broyden, “The Convergence of a Class of Double- Rank Minimization Algorithms. 2. The New Algorithm”, J. of the Institute of Math. And its Appl., Vol. 6, 1970, pp.222-231. |

[13] | R. Fletcher, “A New Approach to Variable Metric Algorithms”, Computer Journal, Vol.13, No.3, 1970, pp.317-322. |

[14] | D. Goldfarb, “A Family of Variable Metric Methods Derived by Variational Means”, Mathematics of Computation., Vol. 24, No. 109, 1970, pp. 23-26. |

[15] | D. F. Shanno, “Conditioning of Quasi-Newton Methods for Function Minimization”, Math. Comput., Vol.24, No.111, 1970, pp.647-656 . |

[16] | R. Clayton and B. Engquist, “Absorbing Boundary Conditions for Acoustic and Elastic Wave Equations”, Bulletin of the Seismological Society of America, Vol.67, No.6, 1977, pp.1529-1540. |

[17] | J. P. Berenger, “A Perfectly Matched Layer for Absorbing of Electromagnetic Waves”, J. Comput. Phys., Vol. 114, No.2, 1994, pp.185-200. |

[18] | M. R. Hestenes and E. L. Stiefel, “Methods of Conjugate Gradients for Scaling Linear Systems”, J. Res. National Bureau Standards, Vol. 49, No.6, 1952, pp.409-436. |

[19] | E. Polak and Ribiére, “Note Sur la Convergence de Directions Conjugate”, Rev. Francaise Informat Recherche Opertionelle, 3e Année, 16, 1969, pp.35-43. |

[20] | B.T. Polyak, “The Conjugate Gradient Method in Extreme Problems”, USSR Comp. Math. and Math. Phys., Vol.9, No.4, 1969, pp. 94-112. |

[21] | L. Armijo, “Minimization of Functions Having Lipschitz Continuous First Partial Derivatives”, Pacific Journal of Mathematics, Vol.16, No.1, 1966, pp.1-3. |

[22] | P. Wolfe, “Convergence Conditions for Ascent Methods”, SIAM Rev., Vol.11, No.2, 1969, pp.226-235. |

[23] | P. Wolfe, “Convergence Conditions for Ascent Methods II: Some Corrections”, SIAM Rev., Vol.13, No.2, 1971, pp.185-188. |

[24] | J. Nocedal, Y. Yuan, “Analysis of a Self-scaling Quasi-Newton Method”, Math. Program, Vol.61, No.1-3, 1993, pp.19-37. |

[25] | W. Zhang, “Imaging Methods and Com-putations Based on the Wave Equation”, Beijing, Science Press, 2009. |

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