In a second example, we examine similar issues related to information content, sample size, upscaling, and the distribution of the upscaled variables. The notion of typical sets from information theory is introduced, and a connection between this concept and the concept of a representative (elementary) volume (REV or RV) is made. In this example, we also focus on how the entropy of the data changes with increasing sample size, and how the sample size relates to the distribution of the observed volume fraction. The first examines how information content is changed by upscaling for the observable volume fraction in a heterogeneous porous material with a binary distribution of textures. Two more involved examples are investigated. While the subject matter relates primarily to applications from hydrologic systems, the concepts are general and may be applied to upscaling in any physical system. In this paper, we present a number of introductory examples that help illustrate the fundamental concepts and definitions of information theory.
The concept of using scaling postulates to reduce the dimensionality or amount of information in a problem is a central idea in upscaling. The intent of this paper is to provide both an introduction to the concepts of information theory, and to review how such concepts might be effectively applied in computing information reduction via upscaling methods.
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