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Introduction

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Book cover Random Fields for Spatial Data Modeling

Part of the book series: Advances in Geographic Information Science ((AGIS))

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Abstract

This chapter introduces various definitions and concepts that are useful in spatial data modeling: random fields, trends, fluctuations, spatial domain types, different spatial models, disorder and heterogeneity, noise and errors, inductive and empirical modeling, sampling and prediction, are among the topics discussed herein. There are also brief discussions of the connections between statistical mechanics and random fields as well as on stochastic versus nonlinear systems approaches.

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Notes

  1. 1.

    I do not like using “and/or”; herein, “A or B” shall mean that A, or B, or both A and B are valid.

  2. 2.

    In technical jargon, different colors may be used for different types of noise. For example, pink noise, also known as flicker, has a power-law spectral density , where ∥k∥ is the wavenumber and 0 < α < 2. The spectral density of colored fluctuations does not necessarily follow a power law.

  3. 3.

    The maximum integral range should be the reference scale, if the random field is anisotropic.

  4. 4.

    Note that X(s; ω) may represent a linear superposition of random fields with different properties.

  5. 5.

    It is not presumed that readers will be able to solve the groundwater flow problem without training. This example serves more as an illustration and motivation for further reading.

  6. 6.

    It will be useful to allow for dynamic changes of the noise; hence, in addition to space we include a time label for the following discussion.

  7. 7.

    To be more precise, for spatial noise case we should refer to wavenumbers instead of frequencies.

  8. 8.

    We use the letter “W” to denote the Wiener process as is commonly done in the literature.

  9. 9.

    For a precise definition of this statement see Sect. 5.1.

  10. 10.

    See the glossary for the definition of lattice used in this book.

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  1. Technical University of Crete, Chania, Greece

    Dionissios T. Hristopulos

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Hristopulos, D.T. (2020). Introduction. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_1

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