Stochastic Signal Processing > Variograms
  • Step 1: Alpha example.
    Spatial data analysis or variography is about computing statistics and experimental variograms and modelling experimental variograms according to the choice of the probability model. It is first step of any stochastic processing.
    In the case of ALPHA, experimental variograms are computed in 4 main directions for 20 lags and displayed in figure a. They show similar behavior for lags 1 to 10, meaning that the spatial variability of ALPHA inside the 30 *30 units field is isotropic. They reach a constant sill value at lag 10 called the range.
    The variogram model used to fit the experimental variogram is a nested isotropic model that is the sum of a nugget effect and a spherical variogram as shown in figure b. Nugget effect accounts for the random component of ALPHA shown in figure c1 and the spherical variogram for the structured component shown in figure c2.

  • Step 2: Pre stack amplitude example.
    The modelling of experimental variogram computed on a prestack amplitude gather shown in figure a must be consistent with the geophysical processing of the gather. Figure b displays the experimental and modelled variograms in the vertical time and offset directions for the whole gather as the sum of signal and noise.
    Figure c displays the experimental and modelled variogram maps and the difference that highlight minor spatial features that are not taken into account by the current modelling.
    Figure d displays the inverse Fourier transforms of the experimental and modelled variograms that corresponds to the frequency power spectrum. Variography enables to model the power spectrum as the sum of signal and noise spectrum interpreted in the spatial domain.