Abstract 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. The concept of using scaling postulates to reduce the dimensionality or amount of information in a problem is a central idea in upscaling. In this paper, we present a number of introductory examples that help illustrate the fundamental concepts and definitions of information theory. 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. Two more involved examples are investigated. 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. 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 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 a second example, we examine similar issues related to information content, sample size, upscaling, and the distribution of the upscaled variables. To add additional concreteness to the example, we also introduce the idea of a simple utility function as a constraint for discerning among different upscaling options. This latter concept is important when the relative costs of upscaling (i.e., the loss of resolution) must be weighted against the benefits (reduction in the number of degrees of freedom). Such analyses may be important when considering upscaling options (model selection) in applications.

中文翻译：

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升级中的信息处理入门

摘要 本文的目的是介绍信息论的概念，并回顾这些概念如何通过升级方法有效地应用于计算信息减少。使用缩放假设来减少问题中的维数或信息量的概念是升级的中心思想。在本文中，我们提供了一些介绍性示例，这些示例有助于说明信息论的基本概念和定义。虽然主题主要涉及水文系统的应用，但这些概念是通用的，可以应用于任何物理系统中的升级。研究了另外两个涉及的例子。第一个研究如何通过放大具有二元纹理分布的异质多孔材料中的可观察体积分数来改变信息内容。在这个例子中，我们还关注数据的熵如何随着样本量的增加而变化，以及样本量如何与观察到的体积分数的分布相关。介绍了信息论中典型集的概念，并建立了该概念与代表性（基本）体积（REV 或 RV）概念之间的联系。在第二个例子中，我们研究了与信息内容、样本大小、放大和放大变量的分布相关的类似问题。为了增加示例的具体性，我们还引入了一个简单的效用函数的想法，作为区分不同升级选项的约束。当升级的相对成本（即分辨率的损失）必须相对于收益（自由度数量的减少）进行加权时，后一个概念很重要。在考虑应用中的升级选项（模型选择）时，此类分析可能很重要。