| Summary: | It is often case that ground-motion records for a given area of interest are available in relative abundance for small (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 Mw>=5, especially at high frequencies, but the magnitude-scaling of PGA and PSA is nonlinear when a broader magnitude range is considered (a cubic function fits quite well). This shows why extrapolation of GMPEs beyond their range of applicability is likely to lead to under- or over-prediction unless constraints are applied to the magnitude scaling. Simple stochastic models are a good basis for such constraints. Finally, we demonstrate that variations in k are much more important for small shocks than at large magnitudes and could be a contributing factor to the often observed magnitude-dependency of the standard deviations of ...
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