  The Multifractal Stress-Activated (MSA) model is a statistical model of
triggered seismicity based on mechanical and thermodynamic principles. It
predicts that, above a triggering magnitude cut-off $M_0$, the exponent $p$ of
the Omori law for the seismic decay of aftershocks is a linear increasing
function $p(M) =a M+b$ of the main shock magnitude $M$. We previously reported
empirical support for this prediction, using the Southern California SCEC
catalog. Here, we confirm this law using an updated, longer version of the same
catalog, as well as new methods to estimate $p$. One of this methods is the
newly defined Scaling Function Analysis, adapted from the wavelet transform.
This method is able to measure a singularity ($p$-value), erasing the possible
regular part of a time series. The Scaling Function Analysis also proves
particularly efficient to reveal the coexistence of several types of relaxation
laws (typical Omori sequences and short-lived swarms sequences) which can be
mixed within the same catalog. The same methods are used on data from the
worlwide Harvard CMT and show results compatible with those of Southern
California. For the Japanese JMA catalog, we still observe a linear dependence
of $p$ on $M$, yet with a smaller slope. The scaling function analysis shows
however that results for this catalog may be biased by numerous swarm
sequences, despite our efforts to remove them before the analysis.
