Stochastic Signal Processing > Kriging
  • Step 1: The ALPHA example.
    Factorial kriging probability model is a specific case of co-kriging. The probability model assumes that Z of x is the sum of independent factors Y of x and that the variogram of Z of x is nested model and the sum of Y of x variograms.
    Figure a shows the ALPHA data set. The experimental variogram shown in figure b is modelled as the sum a nugget effect and a spherical model.
    ALPHA regionalization is interpreted as the sum of two independent components of ALPHA, a white noise or random component and a structured component, the variogram of which are a nugget effect and a spherical model.
    Factorial kriging computes the 895 weighting factors to be applied to the 895 ALPHA data. The resulting linear combination of the 895 ALPHA data best estimates the two components of the decomposition , that are the white noise as shown in figure d and the structured component displayed in figure e. Variograms computed on the random and structured components of ALPHA are shown in figure f and g. They confirm the performance of factorial kriging as a spatial filtering operator.

  • Step 2: The Wiener filter example.
    In the example, the measured amplitudes in figure a left contain organized noise in the time direction and correlated from trace to trace and not correlated to the signal .
    Figure b expresses the covariance model of measured amplitudes as the sum of the covariance of the noise and the signal as a modelled equivalent of the experimental autocorrelation function.
    Figure c compares the results of Factorial Kriging to the usual wiener filter in figure d.
    Factorial kriging is variogram model driven whereas the Wiener filter is data driven.
    The added value of factorial kriging is that it enables to optimize the parametrization of the deterministic filter. Its performance is quantified thanks to the minimized kriging estimation variance.