Recursive integral time extrapolation of elastic waves using low-rank symbol approximation |

where is the wavefield, is the spatial location, is time, and is the a matrix operator containing material parameters and spatial derivative operators. Equation 1 can also be expressed using the first-order system

where . The solution of equation 2 can be formulated using the definition of the matrix exponential:

Defining , the eigenvalue decomposition of can be written as (Du et al., 2014)

(4) |

where

(5) | |||

The solution to the first-order system 3 can now be written as

(6) |

To simplify the system, we can define the analytical wavefield

(7) |

The solution to equation 1 finally takes the form

Selecting the first of the two decoupled solutions of equation 8 leads to a time extrapolation operator

where .

For acoustic isotropic constant-density wave equations for pressure waves, where is velocity and is the Laplacian operator. This corresponds to the one-step extrapolation method proposed by (Zhang and Zhang, 2009).

The wavefield is an analytical signal, with its imaginary part being the Hilbert transform of its real part (Zhang and Zhang, 2009). To see this, we can perform Hilbert transform to the real-valued wavefield in the frequency domain, and use the dispersion relation and derivative property of Fourier Transform

where and denotes forward and inverse Fourier transform in time. The output corresponds to the imaginary part of analytical wavefield .

Recursive integral time extrapolation of elastic waves using low-rank symbol approximation |

2018-11-16