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Representation of the Nested Structures

As far as the second-order moments of the random function $Z({/boma x})$ are concerned, these nested structures can be conveniently represented as the sum of a number of variograms (or covariances), each one characterizing the variability at a particular scale.

/begin{displaymath}/gamma({/boma h}) = /gamma_0({/boma h}) + /gamma_1({/boma h}) + /gamma_2({/boma h})
+/ldots+ /gamma_k({/boma h})./end{displaymath}

For example, $/gamma_0({/boma h})$ may be a transition model (spherical or exponential) which very rapidly reaches its sill value $C_0$ for distances ${/boma h}$ that are only slightly larger than the data support. This model thus combines all the micro-variabilities (e.g., measurement errors and petrographic differentiations). $/gamma_1({/boma h})$ may be another transition model with a larger range (e.g., $a_1 = 10 m$) characterizing the lenticular beds and $/gamma_2({/boma h})$ may be a third transition model with a range ($a_2 = 200 m$) representing the alternation of strata or the extent of homogeneous mineralized zones.

At smaller distances ($h < 30 m$), the observed total variability depends on $/gamma_0({/boma h})+ /gamma_1({/boma h})$, cf. Figure 4.1, while for large distances it will depend on all the $/gamma_i({/boma h})$.

Figure 4.1: Nested Structures.
/begin{figure}/begin{center}
/mbox
{/beginpicture
/setcoordinatesystem units <1c...
...3.1 8.0 3.1 /
/plot 1.55 2.05 8.00 2.05 /
/endpicture}
/end{center}/end{figure}


next up previous contents
Next: Example 4.1: Nested Structure Up: Nested Structures Previous: Scale and Support of   Contents
Rudolf Dutter 2003-03-13