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Random Fields for Spatial Data Modeling pp 127–171Cite as

Additional Topics of Random Field Modeling

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Abstract

We now turn our attention to specialized topics of random field modeling that include ergodicity, the concept of isotropy, the definition of different types of anisotropy, and the description of the joint dependence of random fields at more than two points. Ergodicity, isotropy and anisotropy are properties that have significant practical interest for the modeling of spatial data. On the other hand, the joint N-point dependence is a more advanced topic, primarily of modeling importance for non-Gaussian random fields. In the case of Gaussian random fields the N-point moments can be expressed in terms of the first and second-order moments.

Since we cannot change reality, let us change the eyes which see reality.

Nikos Kazantzakis

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Notes

  1. 1.

    Exceptions are spatial data sets that exhibit slowly changing, quasi-stationary patterns.

  2. 2.

    Expansion of the domain in all orthogonal directions is necessary in cases of stationary but anisotropic processes.

  3. 3.

    On this matter see also the comments by J. Kent in the Discussion of the paper [203].

  4. 4.

    In [487] this equation is expressed in terms of the volume V d of the unit sphere which is related to the surface area by means of \(\mathcal {S}_{d} = d\, V_{d}\).

  5. 5.

    For simplicity we talk about isolevel contours, but in three dimensional problems these actually refer to surfaces.

  6. 6.

    A matrix B is orthogonal if B ⊤ = B −1.

  7. 7.

    Ergodic conditions imply both translation invariance of the hydraulic conductivity correlations and infinite domain size (practically, much larger domain length in each orthogonal direction than the respective correlation length).

  8. 8.

    If the exponent is a functional of the random.

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  1. Technical University of Crete, Chania, Greece

    Dionissios T. Hristopulos

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Hristopulos, D.T. (2020). Additional Topics of Random Field Modeling. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_4

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