Stochastic Signal Processing > Variograms
  • Step 1: The spherical model example.
    The variogram model is a mathematical function called gamma of h used for fitting an experimental variogram. It provides the stochastic processing algorithms with consistent variogram values for any distance h.
    The variogram model function must be consistent with the theoretical variogram definition and must ensure the positivity of all types of variances computed by the probability model.
    The example of spherical variogram model is expressed in figure a and displayed in figure b.
    The Spherical variogram model reaches a constant value called the sill for a lag h called the range.
    Figure c displays a simulated regionalization using a the spherical model in figure b.

  • Step 2: Modelling spatial continuity .
    Derivability near lag 0 of the variogram model function represents the spatial continuity of the regionalization. Figure a displays cubic, exponential and spherical variograms models that only differ by their behavior near lag 0. They represent different types of spatial continuity as shown in figure b, from rugosity for exponential and spherical to smoothness for cubic models as displayed in cross section figure c.

  • Step 3: More variogram models .
    Figure a displays a variogram model with no sill that allows for modelling a non-stationary regionalization.
    Figure b displays a nested variograms model which is the sum of a nugget effect or discontinuity at the origin and a cubic variogram.
    Figure c displays an anisotropic variogram showing different ranges in different direction. The larger the range the more elongated the spatial structure in the corresponding direction.

  • Step 4: Variogram and power spectrum.
    When dealing with seismic amplitudes, the variograms model is the Fourier transform of the frequency power spectrum. This is illustrated on a pre stack amplitude gather displayed in figure a.
    The variogram model displayed in figure b is a factorized model that is the product of a variogram model in the offset direction and another variograms model in the time direction. This factorized variograms is the Fourier transform of the power spectrum variograms model displayed in figure b.