A numerical tour of wave propagation |

Time domain FWI was proposed by Tarantola (1984), and developed in Pica et al. (1990); Tarantola (1986). Later, frequency domain FWI was proposed by Pratt et al. (1998). Actually, many authors call it full waveform tomography. (tomography=fwi, imaging=migration) Here, we mainly follow two well-documented paper Pratt et al. (1998) and Virieux and Operto (2009). We define the misfit vector by the differences at the receiver positions between the recorded seismic data and the modelled seismic data for each source-receiver pair of the seismic survey. Here, in the simplest acoustic velocity inversion, corresponds to the velocity model to be determined. The objective function taking the least-squares norm of the misfit vector is given by

where and are the number of sources and geophones, denotes the adjoint and the complex conjugate, while indicates the forward modeling of the wave propagation. The recorded seismic data is only a small subset of the whole wavefield.

The minimum of the misfit function is sought in the vicinity of the starting model . FWI is essentially a local optimization. In the framework of the Born approximation, we assume that the updated model of dimension can be written as the sum of the starting model plus a perturbation model : . In the following, we assume that is real valued.

A second-order Taylor-Lagrange development of the misfit function in the vicinity of gives the expression

(65) |

Taking the derivative with respect to the model parameter results in

(66) |

Briefly speaking, it is

(67) |

Thus,

where

(69) |

and

(70) |

and are the gradient vector and the Hessian matrix, respectively.

- The Newton, Gauss-Newton, and steepest-descent methods
- Conjugate gradient (CG) implementation
- Fréchet derivative
- Gradient computation
- Numerical results

A numerical tour of wave propagation |

2021-08-31