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Lattice Representations of Spartan Random Fields

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Random Fields for Spatial Data Modeling

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Abstract

Our discussion of SSRFs has so far assumed that the values of the field are defined on continuum domains \(\mathcal {D} \subset \mathbb {R}^d\). This assumption, which is inherent in the energy functional (7.4), reflects the fact that geophysical and environmental processes take place in a spatial continuum.

The most fruitful areas for the growth of the sciences were those which had been neglected as a no-man’s land between the various established fields.

Norbert Wiener

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Notes

  1. 1.

    Solid state physics focuses on the microscopic properties of materials with periodic crystal lattice structure. However, the latter does not explicitly appear in studies of macroscopic material properties and response to external stimuli for which continuum theories are employed.

  2. 2.

    Alternatively, the centers of the grid cells can be viewed as the field locations.

  3. 3.

    For notational economy we use \({\mathcal {H}}_{0}(\mathbf {x})\) for the discrete energy functional as for the continuum SSRF.

  4. 4.

    The reason for the presence of π is that the reciprocal space is spanned by wavevectors which correspond to cyclical frequencies w = 2πf in the reciprocal time domain.

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Hristopulos, D.T. (2020). Lattice Representations of Spartan Random Fields. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_8

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