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.
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