A Seismic Model for the Source of Long-Period Events and Harmonic Tremor

Abstract

The sharply peaked spectra of long-period events and harmonic tremor can be explained by the resonance of a three-dimensional, fluid-driven crack induced by a pressure transient applied over a small area ΔS of the crack wall. The crack resonance is sustained by a very slow dispersive wave called the crack wave, the phase velocity of which decreases rapidly with increasing wavelength and with increasing values of the crack stiffness, C = (b/μ)(L/d), a dimensionless parameter where b is the bulk modulus of the fluid, μ is the rigidity of the solid, L is the crack length and d the crack thickness. The excitation of modes depends on the position of the pressure transient, the extent of crack surface affected by the transient, the time history of the transient, and the boundary conditions in effect at the crack perimeter. The far-field spectrum depends critically on the crack stiffness and on the impedance contrast between the fluid and solid, Z = (ρ _s α)/(ρ _f α), in which ρ _s , ρ _f are the densities of the solid and fluid, and α and a represent the velocity of the compressional wave in the solid and acoustic wavespeed of the liquid, respectively. Lowering the ratio b/μ, or increasing the ratios α/a and ρ _s /ρ _f increases Z, which increases the signal duration. The duration of resonance is measured by the quality factor Q describing the damping of oscillations of the dominant mode observed in the far field. Values of Q ≃ 10 - 20 that are consistent with those observed for shallow tremor and long-period events can be obtained with Z ≃ 3 - 40, b/μ ≃ 0.5 - 0.01, α/a ≃ 2 - 8, and ρ _s /ρ _f ≃1.5 - 5. A high impedance contrast Z and sustained oscillatory source within the fluid are required to explain Q > 20 observed in some events. The low ratio b/μ associated with the large values of Z can be achieved with void fractions of gas in the liquid ranging up to a few percent. The presence of these bubbles drastically reduces the acoustic wavespeed of the fluid so that resonance at a long period is possible in a crack of small dimensions. The far-field signature is strongly dependent on the spatio-temporal characteristics of the transient driving the crack. These features of the model suggest that the dynamics of the gas phase may play a critical role in the excitation mechanism of both long-period events and tremor.