A Probabilistic Framework to Model Distributions of VS30

The time-averaged shear-wave velocity in the upper 30 m depth from the ground surface, or V-S30, is often used as a predictor to describe local site effects in ground-motion models. Although V-S30 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 V-S30 predictions that leverage more readily available secondary information such as surface geology, topographic slope, or geomorphic terrain classes to estimate the mean V-S30 and associated uncertainty. Traditionally, empirical distributions of V-S30 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 V-S30 measurements. Specifically, by invoking Lyapunov's central limit theorem, we hypothesize that the distribution of V-S30 can be theoretically approximated by a reciprocal-normal distribution. We show that a non-normal and skewed distribution of V-S30 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 V-S30 measurements, as opposed to the mean or median, which can overestimate the most probable value.