Stochastic Signal Processing > Kriging
  • Step 1: The ALPHA example.
    Simple kriging is the estimation operator associated to stationary probability models where m of x is a known constant. It is called Ordinary kriging when the probability model assumes that m of x is an unknown constant.
    Figure b shows the sampling of ALPHA data set recalled in Figure a at 35 locations. Figure c shows the associated variogram model. Simple kriging computes the 35 weighting factors to be applied to the 35 samples. The resulting linear combination of the 35 sample values best estimates ALPHA over the whole field. It gives the kriged result shown in figure c and the minimized standard deviation of the unknown error ALPHA -Kriging in figure d. 90% of the actual ALPHA values are included in the 2 kriging standard deviation confidence interval around the kriged map.

  • Step 2: The stack example.
    The Stack process is about averaging the gathered amplitudes in the offset direction to enhance the signal to noise ratio of the seismic trace as shown in figure a. It is a linear operator that may be translated into it stochastic equivalent, a stationary probability with constant average and a variogram shown in figure b. The probability model enables to reproduce the usual stack by computing the stacked trace and a spatial quality index related to the estimation variance associated to the arithmetical average. The result is shown in figure c for a single gather and in figure d for a whole stacked section.
    The same probability model can be used to optimize the linear combination of prestack amplitudes in order to minimize the estimation variance. This is simple kriging of pre stack amplitudes. The result of the kriged stack is shown in figure e for the whole section and it is visible that the Spatial Quality Index map indicates enhanced signal to noise ratio when compared to the usual stack.