## Abstract

The wavefield dependence on a virtual shift in the source location can provide information helpful in velocity estimation and interpolation. However, the second-order partial differential equation (PDE) that relates changes in the wavefield form (or shape) to lateral perturbations in the source location depends explicitly on lateral derivatives of the velocity field. For velocity models that include lateral velocity discontinuities this is problematic as such derivatives in their classical definition do not exist. As a result, I derive perturbation partial differential wave equations that are independent of direct velocity derivatives and thus, provide possibilities for wavefield shape extrapolation in complex media. These PDEs have the same structure as the wave equation with a source function that depends on the background (original source) wavefield. The solutions of the perturbation equations provide the coefficients of a Taylor's series type expansion for the wavefield. The new formulas introduce changes to the background wavefield only in the presence of lateral velocity variation or in general terms velocity variations in the perturbation direction. The accuracy of the representation, as demonstrated on the Marmousi model, is generally good.

Original language | English (US) |
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Pages (from-to) | 627-634 |

Number of pages | 8 |

Journal | Geophysical Prospecting |

Volume | 59 |

Issue number | 4 |

DOIs | |

State | Published - Jul 2011 |

### Bibliographical note

KAUST Repository Item: Exported on 2020-10-01Acknowledgements: I thank Sergey Fomel for many useful discussions on this subject. I thank KAUST for its support.

This publication acknowledges KAUST support, but has no KAUST affiliated authors.

## Keywords

- Laplacian
- Perturbation
- Wavefield

## ASJC Scopus subject areas

- Geochemistry and Petrology
- Geophysics