| Summary: | It is often case that ground-motion records for a given area of interest are available in relative abundance for small (Mw<5) earthquakes but are practically non-existent for larger earthquakes, which have the potential to cause damage to structures. This is a direct consequence of the almost universally observed Gutenberg-Richter relation that a unit increase in magnitude decreases the number of earthquakes observed by a factor of ten. The productive use of data from small earthquakes for seismic hazard assessments relies on knowledge of how earthquake ground motions scale with magnitude. The majority of ground-motion prediction equations (GMPEs) are derived, usually by regression analysis, for the prediction of shaking from earthquakes with Mw>=5 but there is often little consideration given to how they extrapolate to small magnitudes, which are sometimes considered within the hazard integral of probabilistic seismic hazard assessments (PSHAs) or for the testing of GMPEs against observations, especially for regions of low-to-moderate seismicity. These reasons for needing to extrapolate GMPEs pose a significant challenge because the use of GMPEs beyond (or even close to the edges of) the magnitude range for which they were derived can lead to significant under- or over-estimation of ground motions, even if the functional form includes nonlinear magnitude-scaling terms. Here we show that simple stochastic models comprised of a Brune source spectrum and a constant stress (drop) parameter Delta sigma coupled with a site attenuation modeled by k leads to a nonlinear magnitude scaling of peak ground acceleration (PGA) and response pseudo-spectral acceleration (PSA) at 1s that matches the observed dependency over the whole range of magnitudes from Mw 1 to 7. Higher magnitude dependency for models derived using only data from small earthquakes compared to that found using data from large earthquakes is physically-realistic. Assuming a linear magnitude dependence of the logarithm of PGA and PSA is reasonable for ...
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