Amplitude Attenuation Function

The constrained linear least-squares perturbations to the $\log{A_{o}}$ function were found to be very stable and well represented by a sixth order Chebyshev polynomial at hypocentral distances from 8 km to 500 km. At shorter distances, it is approximated by a line with a slope close to 2. In this study, we use hypocentral distance, rather than epicentral distance as originally used by Richter (1935), to accurately represent variation in the $\log{A_{o}}$ attenuation function at close distances. This $\log{A_{o}}$ form was adopted and a CISN $\log{A_{o}}$ algorithm was developed and used in all subsequent inversions for the $dM_{L}$ SNCL adjustments. A plot of the CISN $\log{A_{o}}$ function is shown in Figure 2.26.

Figure 2.26: Comparison of the new CISN $-log{A_{o}}$ attenuation function (solid line) with those used in Northern California (short dashed line, Richter, 1935) and in Southern California (long dashed line, Kanamori et al, 1999). Note that distance is given from the hypocenter and not from the epicenter.