The Hilbert transformation > Polarization filter

We propose a synthetic signal having 2 components, V (the vertical component: the geophone having its spindle axis vertical) and H (the horizontal component: the geophone having its spindle axis horizontal), each carrying a different polarisation.
For explanatory purposes, the 2 waves are separated in time so that the transformation can be better understood.

The different stages of separation for using the polarisation filter are as follows:

1) Polarisation estimation of the first wave. The polarisation P is given by a phase shift so as to make the polarisation linear, and this is then rotated so to bring the oscillation onto the vertical component V (figure in the middle). The first wave is now entirely localised on the V component. The vertical component V is chosen as being the signal domain.

2) Once this operation has been carried out , the second wave appears on the two components as its polarisation is not orthogonal to P. The separation can be carried out in 3 steps:

  • The phase of the horizontal component of the second wave must be change to obtain a rectilinear polarisation. We should note that the polarisation of the first wave, which has been entirely projected onto the vertical component, has not changed.
  • The polarisations of the two waves must be orthogonal by amplifying the horizontal component. The amplification factor can be infinite. To avoid this inconvenience, a rotation of a number of degrees from the reference (V, H) was carried out before calculating the amplification factor. Once the amplification factor has been determined, an opposing angle of rotation must be applied to place the reference in its initial position.
  • A rotation must be operate to reposition the first wave on the vertical component.