A new method for determining small earthquake source parameters using short-period P waves

We developed a new technique of inverting short-period (0.5–2 Hz) P waveforms for determining small earthquake (M <3.5) focal mechanisms and moments, where magnitude ~4 events with known source mechanisms are used to calibrate the "unmodeled" structural effect. The calibration is based on a waveform cluster analysis, where we show that clustered events of different sizes, for example, M ~4 versus M ~2, display similar signals in the short-period (SP, 0.5–2 Hz) frequency band, implying propagational stability. Since both M ~4 and M ~2 events have corner frequencies higher than 2 Hz, they can be treated as point sources, and the "unmodeled" structural effect on the SP P waves can be derived from the magnitude 4 events with known source mechanisms. Similarly, well-determined magnitude 2’s can provide calibration for studying even smaller events at higher frequencies, for example, 2–8 Hz. In particular, we find that the "unmodeled" structural effect on SP P waves is mainly an amplitude discrepancy between data and 1D synthetics. The simple function of "amplitude amplification factor" (AAF) defined as the amplitude ratio between data and synthetics provides useful calibration, in that the AAFs derived from different clustered events appear consistent, hence stable and mechanism independent. We take a grid-search approach to determine source mechanisms by minimizing the misfit error between corrected data and synthetics of SP P waves. The validation tests with calibration events demonstrate the importance and usefulness of the AAF corrections in recovering reliable results. We introduce the method with the 2003 Big Bear sequence. However, it applies equally well to other source regions in southern California, because we have shown that the mechanism independence and stability of the AAFs for source regions of 10 km by 10 km are typical. By definition, the AAFs contain the effects from the station site, the path, and crustal scattering. Although isolating their contributions proves difficult, the mechanism independence and stability of the AAFs suggest that they are mainly controlled by the near-receiver structure. Moreover, the ratios between the AAFs for the vertical and radial components from various events at different locations appear consistent, suggesting that these AAF(v)/AAF(r) ratios might be simple functions of site conditions. In this study, we obtained the focal mechanisms and moments for 92 Big Bear events with M_L down to 2.0. The focal planes correlate well with the seismicity patterns, while containing abundant finer-scale fault complexity. We find a linear relationship between log(M_0) and M_L, that is, log(M_0) = 1.12M_L + 17.29, which explains all the data points spanning three orders of magnitude (2.0 < M_L < 5.5)

A new identity for SL(2,C)-characters of the once punctured torus group

We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.Comment: 9 page