On Strong Positive Frequency Dependencies of Quality Factors in Local-Earthquake Seismic Studies

Many observations of seismic waves from local earthquakes are interpreted in terms of the frequency-dependent quality factor $Q\left(f\right)=Q_0{f}^{\eta }$, where η is often close to or exceeds one. However, such steep positive frequency dependencies of Q require careful analysis with regard to their physical consistency. In particular, the case of η = 1 corresponds to frequency-independent (elastic) amplitude decays with time and consequently requires no Q-type attenuation mechanisms. For η > 1, several problems with physical meanings of such Q-factors occur. First, contrary to the key premise of seismic attenuation, high-frequency parts of the wavefield are enhanced with increasing propagation times relative to the low-frequency ones. Second, such attenuation cannot be implemented by mechanical models of wave-propagating media. Third, with η > 1, the velocity dispersion associated with such Q(f) occurs over unrealistically short frequency range and has an unexpected oscillatory shape. Cases  η = 1 and  η > 1 are usually attributed to scattering; however, this scattering must exhibit fortuitous tuning into the observation frequency band, which appears unlikely. The reason for the above problems is that the inferred Q values are affected by the conventional single-station measurement procedure. Both parameters Q₀ and are apparent, i.e., dependent on the selected parameterization and inversion method, and they should not be directly attributed to the subsurface. For η  ≈ 1, parameter Q₀ actually describes the frequency-independent amplitude decay in access of some assumed geometric spreading t^−α, where α is usually taken equal one. The case η > 1 is not allowed physically and could serve as an indicator of problematic interpretations. Although the case $0<\eta <1$ is possible, its parameters Q₀ and may also be biased by the measurement procedure. To avoid such difficulties of Q-based approaches, we recommend measuring and interpreting the amplitude-decay rates (such as parameter α) directly.