Angle-domain illumination gathers by wave-equation-based methods |

Linearized modeling (Born modeling) from a prestack image parameterized as a function of subsurface offset can
be described as follows:

where is the seismic data with a source located at and a receiver located at ; is the angular frequency; is the prestack image located at for a half subsurface offset ; is the sensitivity kernel defined as follows:

(2) |

where is the source signature, and and are the Green's functions connecting the source and receiver, respectively, to the image point .

Reconstruction of the prestack image
can be posed as an inverse problem by minimizing the following objective function
defined in the data space:

(3) |

where is the data residual and is the acquisition mask operator, which contains unity values where we record data and zeros where we do not. The gradient of the objective function reads

where denotes complex conjugation. Equation 4 is similar to the prestack shot-profile migration formula and it produces migrated reflectivity images defined in the subsurface offset domain (Rickett and Sava, 2002).

The Hessian can be obtained by taking the second-order derivatives of
with respect to the model parameters as follows:

(5) |

When and , we obtain the diagonal elements of the Hessian operator

The diagonal of the Hessian is often known as the illumination map of the subsurface, it contains illumination contribution from both sources and receivers for a given acquisition configuration.

Angle-domain illumination gathers by wave-equation-based methods |

2010-05-19