Stochastic Signal Processing > The Theory of regionalized variables

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Example: Regionalized variable ALPHA

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Examples of regionalized variables in geophysics

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Support effect

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Information effect

Step 1: Example: Regionalized variable ALPHA.
Regionalized variables (RV) are properties or measurements with coordinates in space or time domains. Example of a Regionalized variable ALPHA sampled in a 2D space. Each of the 895 ALPHA samples ,color dots, is identified by x and y coordinates ranging from 0 to 30 on each axis and take values from 0 to 930.
Usual statistics do not take into account the spatial distribution of the sampled variable. The theory of regionalized variables, known as Geostatistics, does.
Spatial distribution of ALPHA samples shows local random fluctuations ALPHA RAND values displayed inside more continuous areas with low or high ALPHA STRUCT values displayed. The theory of regionalized variable provides a mathematical framework to consistently model the random and structural continuity of a regionalized variable.

Step 2: Examples of regionalized variables in geophysics.
In seismic reservoir processing, these regionalized variables are usually called seismic data, spatial attributes or properties, as for example, seismic records in time, seismic amplitudes and velocities, geological properties such as velocity, density, porosity or permeability.
Seismic pre stack gathers, seismic velocity fields, amplitude seismic cube, seismic time depth interpretations and results of seismic inversion are examples of such regionalized variables in geophysics.

Step 3: Support effect.
The spatial distribution or histogram of regionalized variable ALPHA depends on the size of the measurement support, 1by1 unit in figure1, 3by3 units in figure 2 and 5by5 units in figure3. The mean or average of the 3 data set does not vary with the size of the measurement support, but the variance or fluctuations around the average decreases when the size of support increases. This is the support effect.
In geophysics, support effect relates to seismic resolution.
The theory of regionalized variable provides a mathematical framework to consistently model the support effect.

Step 4: Information effect.
When looking at the same measurement support 1by1, the spatial distribution or histogram of regionalized variable ALPHA also depends on the density of the data sampling: 900 samples in figure 1, 100 samples in figure 2 and 36 samples in figure 3.
When gridded on the same 900 support 1by1 grids nodes, and whatever the interpolator, the average or mean of the gridded data sets does not vary with the density of the sampling, but the dispersion variance or fluctuations around the mean decreases when the sampling density decreases. This is the information effect.
In geophysics, information effect relates to seismic acquisition folding or coverage. The theory of regionalized variable provides a mathematical framework to consistently model the information effect.