Brune’s Spectral Model: Its Impact on Ground Motion Modeling and Interpretation of the Seismic Source

Abstract:

Brune’s seminal paper in 1970 on the spectrum of the earthquake source (cited more than 5600 times) has had a profound effect in seismology. Brune provided a direct method for determining two parameters from the displacement amplitude spectrum of the S wave: seismic moment from the low-frequency asymptote Ω₀ and the corner frequency f_c, from the intersection of the high-frequency and low-frequency asymptotes. Brune related f_c to the apparent source radius and subsequently stress drop. This provided a direct connection between a seismogram and a physical dimension related to an earthquake. With an analytical model for a circular crack (Neuber, 1935; Eshelby, 1957), seismic moment and source radius were connected to Brune’s effective stress (initial stress minus sliding friction stress) for a homogenous rupture. This linkage between the spectrum and effective stress opened the door for a plethora of studies on earthquake scaling, regional and global distributions of stress drop and ground motion. Hanks’ (1979) insightful paper used random vibration theory and the observation that an accelerogram can be represented as bandlimited, Gaussian, white noise to relate root-mean-square acceleration with peak acceleration. Hanks used the coincidence that white noise has a flat acceleration spectrum as does a delta function -- Brune’s acceleration pulse. Boore (1983) took this idea one step further to create near fault ground motion time series which led to countless investigators generating stochastic simulations of ground motion. Boore referred to the stress change as the ‘stress parameter.’ Overlooked in Brune’s paper is that the displacement spectrum of an earthquake that included heterogeneous stress change would have two corners connected by f^-1 decay. This model allows for the ‘stress parameter’ (~10-40 MPa) to produce the high frequency level of the spectrum and a global stress drop (~3-5 MPa) related to the overall rupture duration. Papageorgiou and Aki (1985) developed a 2-corner acceleration spectrum to model data from 5 earthquakes. Their model required a global and local stress drop with a f¹growth between the two corners as had been suggested by Gusev (1983). I will introduce two new 2-corner spectral models (Ji and Archuleta, 2020) that reproduce the mean values of PGA and PGV from the NGA West-2 data for 3.3 ≤ M ≤ 7.7.