The study of the seismic response of inhomogeneous soil deposits, underlying rigid bedrock, has traditionally been framed as a one-dimensional wave propagation problem in a linear viscoelastic medium. When it comes to modelling soil inhomogeneity, the current trend to model the continuous variation of soil stiffness with depth employs a ‘generalised parabolic function’. Exact solutions have already been obtained, but these come in terms of Bessel functions, which muddle the interpretation of results and obstruct assessment of the parameters’ influence. This also hinders the definition of an ‘equivalent homogeneous’ soil deposit (a relevant concept utilised in many seismic codes). It is shown herein that the so-called inhomogeneity factor plays a secondary role in determining the response in many cases; thus straightforward guidelines are suggested for simplifying the problem, leading to elementary scaling relations. The scalings provide simple yet meaningful relations that reveal and explain the fundamental traits of the dynamic behaviour of these systems.

中文翻译：

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非均质土的地震谐响应：标度分析

传统上，对刚性基岩下面的非均质土壤沉积物的地震反应的研究通常被认为是线性粘弹性介质中的一维波传播问题。在对土壤非均质性进行建模时，当前对土壤刚度随深度的连续变化建模的趋势采用了“广义抛物线函数”。已经获得了精确的解决方案，但是这些解决方案是根据Bessel函数进行的，该函数混淆了结果的解释并阻碍了对参数影响的评估。这也阻碍了“等效均质”土壤沉积物的定义（在许多地震法规中使用的相关概念）。在此表明，在许多情况下，所谓的不均匀性因素在确定响应中起次要作用。因此，建议使用简单的指南来简化问题，从而建立基本的比例关系。缩放比例提供了简单而有意义的关系，这些关系揭示并解释了这些系统动态行为的基本特征。