On the Formulation of a Displacement-Based Model for Representing Anelastic Attenuation in Wave Propagation Simulations

Abstract:

Energy losses in the form of anelastic attenuation due to material internal friction plays a major role in wave propagation problems and earthquake ground motion simulation. These attenuation effects are typically represented through the characterization of the quality factor, Q. There have been several studies in which Q is modeled using viscoelastic devices, where the effects of internal friction are represented by springs and dashpots. A recently introduced model, called the BKT model (after authors Bielak, Karaoglu and Taborda), proposed the use of two Maxwell elements (each made of a spring and a dashpot connected in series) in combination with a Voigt element (consisting of a spring and a dashpot connected in parallel). The BKT model showed very good adherence to intended values of constant Q = Q_o. It, however, depended on a set of parameters precomputed arbitrarily for different values of Q_o, and was limited to problems under the assumption of frequency independent attenuation. In this work we show that the internal parameters used in the BKT model can be set using exact expressions formulated based on a numerical optimization of the model's fit with Q_o. This formulation holds for any value of Q_o. In addition, we show that using three Maxwell elements substantially improves the accuracy of the model and increases its flexibility, allowing one to model problems with frequency-dependent Q = Q_o(f). This latter extension of the BKT model is of critical significance for deterministic physics-based earthquake simulations done at frequencies of engineering interest (f > 1 Hz).