Stochastic Signal Processing > Simulations
  • Step 1: Kriging or simulation.
    The ALPHA regionalization is shown in Figure a. 35 sample values on support 1*1 are available as shown in Figure b, and the variogram model is shown figure c.
    Due to the information effect, the variogram computed on a kriged map is not the same as the one input to the kriging process as kriging operator cannot reproduce local spatial high and lows that have not been sampled. As they are accurate, the kriged maps are smoother than the actual ALPHA data set as shown in figure d.
    When equipped with simulation algorithms, the same probability model may also produce alternative simulations of ALPHA as shown in Figure e, that look like ALPHA data set but are not locally accurate.

  • Step 2: Why simulations.
    Simulating a number of ALPHA maps conditioned by the 35 samples enables to count the number of simulated ALPHA values above 300 on each simulation, leading to 300 possible solutions. The expectation curve displayed in figure a corresponds to the distribution of the simulated values. It shows that 50 % of simulations contain more than 304 alpha values, when the actual number of alpha values computed from the actual ALPHA data set is 300. The P50 value read from the expectation curve is a good estimator of the number of ALPHA values above 300.
    Notice that computing this number from the kriged map would lead to an unrealistic 189 number of ALPHA values above 300, this is called a bias.

  • Step 3: Simulation algorithms.
    Simulation algorithms have been developed to produce stationary and non-stationary numerical data sets according a given variogram model. Turning bands, sequential gaussian, Truncated Gaussian are examples of such algorithms. An example is given of how to simulate depth structural maps using a non-stationary probability model with bayesian constraints.
    Computing the depth realizations includes the following steps:
    • Figure a shows depth conditioning data that all simulations must honor
    • Figure b and c shows how to draw the external drift coefficients from the distribution controlled by the bayesian constraints and to build the simulated depth trend.
    • Figure d shows how to add simulated depth residuals computed by the simulation algorithm and reproducing the variogram of depth residuals.
    • Figure e shows how to condition the simulated depth residuals to depth conditioning data using a Bayesian kriging operator
    • Figure f displays 3 alternative simulated depth maps built according the above computing workflow