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<oai_dc:dc xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:oai_dc="http://www.openarchives.org/OAI/2.0/oai_dc/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd">
  <dc:contributor>Sean K Ahdi</dc:contributor>
  <dc:contributor>Julie A. Herrick</dc:contributor>
  <dc:contributor>Junko Iwahashi</dc:contributor>
  <dc:contributor>Alexandros Savvaidis</dc:contributor>
  <dc:contributor>Alan Yong</dc:contributor>
  <dc:creator>Utkarsh Mital</dc:creator>
  <dc:date>2021</dc:date>
  <dc:description>&lt;p&gt;&lt;span&gt;The time‐averaged shear‐wave velocity in the upper 30&amp;nbsp;m depth from the ground surface, or&amp;nbsp;&lt;/span&gt;&lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;⁠&lt;/span&gt;&lt;span&gt;, is often used as a predictor to describe local site effects in ground‐motion models. Although &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;is typically determined from in situ measurements, it is not always feasible to obtain such measurements due to project restrictions or site accessibility. This motivates the development and use of proxy‐based &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;predictions that leverage more readily available secondary information such as surface geology, topographic slope, or geomorphic terrain classes to estimate the mean &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;and associated uncertainty. Traditionally, empirical distributions of &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;have been observed to have long right tails, leading to high levels of associated uncertainty. In this study, we present a physical framework that is grounded in fundamental principles of geostatistics and probability to explain the uncertainty and skewness associated with &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;measurements. Specifically, by invoking Lyapunov’s central limit theorem, we hypothesize that the distribution of &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;can be theoretically approximated by a reciprocal–normal distribution. We show that a non‐normal and skewed distribution of &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;is to be expected and is not a sign of measurement error or sampling bias, although sampling bias can exaggerate such skewness. Our framework also enables us to propose the mode as a characteristic value of &lt;span class="inline-formula no-formula-id"&gt;&lt;span id="MathJax-Element-3-Frame" class="MathJax" data-mathml="&lt;math xmlns=&amp;quot;http://www.w3.org/1998/Math/MathML&amp;quot;&gt;&lt;msub xmlns=&amp;quot;&amp;quot;&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;mn&gt;30&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;"&gt;&lt;span id="MathJax-Span-15" class="math"&gt;&lt;span id="MathJax-Span-16" class="mrow"&gt;&lt;span id="MathJax-Span-17" class="msub"&gt;&lt;span id="MathJax-Span-18" class="mi"&gt;V&lt;/span&gt;&lt;sub&gt;&lt;span id="MathJax-Span-19" class="mrow"&gt;&lt;i&gt;&lt;span id="MathJax-Span-20" class="mi"&gt;S&lt;/span&gt;&lt;/i&gt;&lt;span id="MathJax-Span-21" class="mn"&gt;30&lt;/span&gt;&lt;/span&gt;&lt;/sub&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;span&gt;&amp;nbsp;measurements, as opposed to the mean or median, which can overestimate the most probable value.&lt;/span&gt;&lt;/p&gt;</dc:description>
  <dc:format>application/pdf</dc:format>
  <dc:identifier>10.1785/0120200281</dc:identifier>
  <dc:language>en</dc:language>
  <dc:publisher>Seismological Society of America</dc:publisher>
  <dc:title>A probabilistic framework to model distributions of VS30</dc:title>
  <dc:type>article</dc:type>
</oai_dc:dc>