Evaluation of Functional Forms for the Attenuation of Small-to-Moderate-Earthquake Response Spectral Amplitudes in North America

We studied several attenuation forms to evaluate their ability to describe the decay of response spectral amplitudes in Eastern Canada/Northeastern United States, Central United States, Pacific Northwest/British Columbia, Northern California, and Southern California. Linear, bilinear, and trilinear regression forms with different combinations of geometric spreading coefficients were tested to assess their ability to describe spectral amplitude decay from 0.33 to 10 Hz in the distance range from 10 to 400 km. The residual trends and the shapes of the attenuation forms do not show significant differences in goodness-of-fit between these trial forms, making it difficult to see a meaningful distinction between them. Furthermore, statistical measures such as the Akaike information criterion are not particularly helpful in drawing such distinctions. A simple linear model, as well as more complex forms, can fit the data in almost every region. However, this linear model has steeper slope in the west (similar to 1:3) than in the east (similar to 1), and may not extend well over large distances. The linear model effectively smoothes through complications in functional form such as the Moho bounce, or the transition to surface-wave spreading at regional distances. A trilinear model has the best statistical fit in most cases, but the added complications in shape and parameterization do not appear warranted by the marginal improvement in residuals. The bilinear form offers a good compromise between simplicity and the ability to model amplitude decay appropriately at both near and regional distances.