Stochastic Signal Processing > Kriging
  • Step 1: The ALPHA example.
    Kriging with external drift is the estimation operator associated to non-stationary probability models where m of x is a linear combination of a number of shape factors called external drifts, and the residual R of x is stationary.
    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 represents the spatial behavior of the spatial average of ALPHA.
    Figure d displays the correlation between ALPHA samples and the BETA values at the same location and the variogram model associated to the ALPHA residuals.
    Kriging with external drifts computes the scaling factor to be applied to the BETA values and the 16 weighting factors to be applied to the 16 ALPHA samples. The resulting linear combination of the 16 ALPHA 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. 90% of the actual ALPHA values are included in the 2 kriging standard deviation confidence interval around the kriged with external drift map.

  • Step 2: Time to depth conversion example.
    Depth converting a single time interpreted seismic horizon is about building an average velocity model that fits the well depth marker data. When translated into a probability model, the time to depth conversion is a classical estimation problem of mapping well depth markers under control of time and velocity model as shown in figure a.
    The spatial operator is a Kriging of depth marker data with time external drifts. The number and type of time external drifts depends on the velocity model.
    Kriging is a georegression of the well depth data by the external drifts data at the well locations as shown in figure b.
    The added value of kriging is that it provides in the same run the estimated depth map shown in figure d and the map of minimized depth estimation variance shown in figure e that is a confidence interval on the depth map.