Numerical calculations and observations indicate that surface waves reverberate within sedimentary basins. A site response approach is inapplicable to the nonlinear attenuation of these waves as the energy passes repeatedly through the shallow subsurface. Computed dynamic stresses obtained by using published ShakeOut calculations (Graves et al., 2008) indicate that nonlinear attenuation is modest but not negligible in the Los Angeles, California, basin for large earthquakes on the San Andreas fault. The dominant surface waves have velocity amplitudes of ∼1.5 m sec-1 and periods ∼3.5 sec. Around 20% of the total elastic strain energy resides above dynamic/lithostatic stress ratios of 0.2 for both Rayleigh and Love waves where prestressed rock is highly likely to fail somewhat in friction.