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

P/S separation by inversion

If the seismic data were sampled vertically as well as horizontally (i.e. - if receivers were placed at multiple depths as well as on a surface), then the spatial derivatives of the wavefield could be calculated, and the separation between P and S-waves would be trivial. Robertsson and Muyzert (1999) propose doing volumetric recordings of the wavefield using tetrahedral receiver groups, for the purpose of calculating all spatial derivatives and thus separating P from S.

The P/S separation method presented here is a (nearly) medium-independent data-space inversion. The model it attempts to invert for is the wavefield, emitted by a virtual source array at some depth level, which after propagating through a homogeneous medium matches the recorded wavefield at the receiver level.

The basic premise of the separation method is: if there is a good match to the recorded data at the receivers, then there is also a good match to the ``true'' receiver wavefield at a close proximity to the receivers. The method effectively reconstructs the wavefield at the receiver's vicinity, as it was in the field experiment. Since we now have a volumetric spatial sampling of the wavefield near the receivers, separation operators based on three-dimensional spatial derivatives (i.e. - divergence and curl) are applied to separate P-waves from S-waves.

The synthetic seismic data I use in the following examples are displacement data, which represent geophone acquisition. The elements of the inversion are:

1. Observed data: The original displacement components recorded by the geophones in the field.
2. Model: A virtual source gather, consisting of displacement source functions, injected at some location into a homogeneous elastic medium.
3. Calculated data: The recorded displacements at the receiver level, as a result of the injection of the virtual sources.
4. Desired model: The virtual sources which generate recorded data equal to the observed data.
5. What we actually want: The wavefield displacement values both at the receiver level AND one depth level below them, so that we can apply spatial derivative operators to separate P from S.

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