Stochastic Signal Processing > Kriging
  • Step 1: The ALPHA example.
    Co-kriging extends kriging probability models to the multivariable case. The multivariate probability models assume same stationarity order for all variables and make use of cross correlation or co variogram functions that quantifying the spatial cross correlation between 2 variables.
    Figure b shows the location of 16 available samples of ALPHA data set recalled in Figure a .
    Figure c shows the map of a BETA data set that is correlated to the ALPHA data set.
    Figure d displays the correlation between ALPHA samples and the BETA values at the same location, the variogram models associated to ALPHA and BETA and the Cross Variogram ALPHA BETA.
    Co-kriging computes the 16 weighting factors to be applied to the 16 ALPHA samples, the 16 BETA values at ALPHA sample location and the BETA value at the target location. The resulting linear combination of the 16 ALPHA and Beta sample values and the BETA value at the target location 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. To reduce the size of the kriging equation system, only the Beta value at the target point may be considered. The co kriging operator is then called co located co kriging.
    90% of the actual ALPHA values are included in the 2 kriging standard deviation confidence interval around the co kriged map.

  • Step 2: The Multi-layer depth conversion example.
    When several time interpretations must be depth converted simultaneously, the spatial operator turns into a co-kriging or Bayesian co-kriging of well depth markers with time interpretations external drifts.
    Figure a illustrates the multilayer time interpretation and figure b recall the input data to the co kriging system. Figure c displays the depth converted horizons and figure d recalls the results of the co-kriging operator.
    Bayesian co-kriging depth with time related external drifts provides a consistent interval velocity model that takes into account all types of correlations between layers and fits to all well marker data in a single co-kriging run.