Stochastic Signal Processing > Stochastic wave separation
  • Step 1: Stationarization.
    The probability model works under local stationarity assumption meaning that the raw V S P data set shown in figure a must first be horizontalised according to the wave field to be separated and then stationarized. This is done by normalizing the amplitudes as shown in figure b.

  • Step 2: Spatial analysis.
    The experimental variogram of stationarized V S P data displayed in figure a is computed in the vertical time and horizontal depth directions as shown in figure b. The horizontal variogram computed in the offset direction shows how energy is split between short range structures corresponding to upgoing wave and noise and large range structure associated to down going wave signature. It is modeled using 2D factorized covariance functions that are the product of two 1D covariance functions operating respectively in the horizontal and vertical dimension.

  • Step 3: Factorial Kriging.
    Factorial kriging gives the best estimate of the wave field to be separated as shown in figure a and b for separating downgoing wave from the whole field.
    Notice that the kriging process is not a filtering process as geophysicists understand it, but rather the best linear estimate of a spatial factor of the regionalisation. This is a fundamental difference as a filter refers to the measured data when kriging refers to the probability model.

  • Step 4: wave decomposition.
    Repeating the same workflow for separating upgoing wave from the remaining noise leads to the final wave decomposition of raw V S P data shown in figure a into down going waves in figure b, upgoing wave in figure c and remaining noise in figure d.