P/S separation of OBS data by inversion in a homogeneous medium |
The Helmholtz separation operator is based on the assumption that any isotropic vector field can be described as a combination of a scalar and vector potential fields:
Equations 3 and 4 apply only for an isotropic medium. Dellinger and Etgen (1990) extend these operators for an anisotropic medium.
The Helmholtz separation operator is useful for distinguishing between P and S-wave amplitudes, but it is not reversible. The derivation of a reversible P-wave and S-wave displacement decomposition by Zhang and McMechan (2010) is done in the wavenumber domain. They formulate a linear equation system based on characteristics of P and S particle displacements in an isotropic elastic medium. The solutions to this linear system in a two-dimensional medium are:
and
where are the spatial fourier transforms of the observed displacement fields in direction , are the wavenumbers, and and are the decomposed P and S displacements. It is important to note that the in these equations is normalized by the absolute value of the wavenumber . This decomposition is reversible, since .
P/S separation of OBS data by inversion in a homogeneous medium |