P/S separation of OBS data by inversion in a homogeneous medium |

The isotropic elastic wave equation relates displacements to stresses via two elastic constants - the Lam parameters and :

where are the particle displacements in each dimension, is the force function and is medium density. An alternate formulation is:

From equation 10, the explicit form for a heterogeneous two dimensional medium can be expressed in a matrix-vector notation as:

The forward elastic propagation operator can then be expressed as:

For a homogeneous medium, and using a Green's function to describe the energy propagation between any two locations and , the equation takes the form:

The forward elastic propagation operator injects sources from a model into some location in the medium, and records the resulting wavefield at some other location:

where is the model of injected sources at location in the medium, and are the recorded displacement fields at location in the medium. is angular frequency and is the shot gather. In vector notation, this is expressed as

(15) |

where is the forward operator. The adjoint operator injects the data from the same recording locations, and records the resulting wavefield at the model injection points:

which in vector notation is

(17) |

where is the adjoint operator.

P/S separation of OBS data by inversion in a homogeneous medium |

2012-05-10