The envelopes of acceleration time histories

We derive an analytical model for the envelope of the time history of the acceleration radiated by a dynamic rupture process. The critical element of the model is the stochastic assumption that the acceleration pulses radiated by different fault areas or by different sections of the rupture front arrive incoherently at any observer. The model for the resulting acceleration envelope depends on the square root of the line integral, evaluated over the isochrone, of the square of the product of the dynamic stress drop and a high-frequency radiation pattern that incorporates both directivity and diffraction. By evaluating the maximum of this envelope for simple rupture geometries, we predict the variation of peak acceleration with source size and recording bandwidth for earthquakes from ML ≈ −1 to 6. We also test the analytic model for the envelope using numerical sources that exhibit both smooth and rough rupture growth, fitting the acceleration envelopes of the rough rupture models slightly better than the envelopes of the smooth rupture models. Finally, we propose and test a deconvolutional technique that fits the squared acceleration envelope from a large earthquake as the sum of squared acceleration envelopes recorded from a small earthquake.