Magnitude-dependent epidemic-type aftershock sequences model for earthquakes

We propose a version of the pure temporal epidemic type aftershock sequences (ETAS) model: the ETAS model with correlated magnitudes. As for the standard case, we assume the Gutenberg-Richter law to be the probability density for the magnitudes of the background events. Instead, the magnitude of the triggered shocks is assumed to be probabilistically dependent on that of the relative mother events. This probabilistic dependence is motivated by some recent works in the literature and by the results of a statistical analysis made on some seismic catalogs [Spassiani and Sebastiani, J. Geophys. Res. 121, 903 (2016)]. On the basis of the experimental evidences obtained in the latter paper for the real catalogs, we theoretically derive the probability density function for the magnitudes of the triggered shocks proposed in Spassiani and Sebastiani and there used for the analysis of two simulated catalogs. To this aim, we impose a fundamental condition: averaging over all the magnitudes of the mother events, we must obtain again the Gutenberg-Richter law. This ensures the validity of this law at any event's generation when ignoring past seismicity. The ETAS model with correlated magnitudes is then theoretically analyzed here. In particular, we use the tool of the probability generating function and the Palm theory, in order to derive an approximation of the probability of zero events in a small time interval and to interpret the results in terms of the interevent time between consecutive shocks, the latter being a very useful random variable in the assessment of seismic hazard.

Figure 1

Functions $f\left(x\right)=Hp\left(x\right)+Cp\left(x\right)(1-2\frac{p\left(x\right)}{\beta })H(1-H)$ and $g\left(x\right)=Lp\left(x\right)$, where $p\left(x\right)=\beta exp\{-\beta (x-m_0)\}$ and $(\beta ,a,C,m_0,m_c)=(2.0493,1.832,0.8,1,1.8)$.

Figure 2

Interevent time density $F_{\mathrm{inter}}\left(\tau \right)$ in the case of the further approximations $a(\tau -t)\approx a\left(\tau \right)$ for small $\tau$ and $A\left(\tau \right)\approx \tau a\left(\tau \right)$. The three curves are obtained by setting $m_c=2.5$ and $m_0=0.5$, and by varying the parameter $C$ (and then $K$). The interevent time density is plotted here in a loglog scale and as a function of the dimensionless interevent time $\tau$, obtained by multiplying the original interevent time by the average rate of the total observable events $\overline{\lambda }$.

Figure 3

Function $q\left(x\right)=C(1-2exp\{-0.4648(x-1.8\left)\right\})$ when $C$ varies. The value at which the function becomes zero is $m^{*}=3.2913$.

Figure 4

Function $f(m^{\prime },x)=0.8(1-2exp\{-0.4648(m^{\prime }-1.8)\})(1-2exp\{-1.9648(x-1.8)\})$ for different values of the parameter $C$ and with $m^{\prime }=2.7$ and $m^{\prime }=7$ in panels (a) and (b), respectively.

Figure 5

Functions $f(m^{\prime },x)=0.8(1-2exp\{-0.4648(m^{\prime }-1.8)\})(1-2exp\{-1.9648(x-1.8)\})$ and $f(x,m^{\prime \prime })=0.8(1-2exp\{-0.4648(x-1.8)\})(1-2exp\{-1.9648(m^{\prime \prime }-1.8)\})$, respectively in (a) and (b). In panel (a), the varying value is the magnitude of the triggering events. Instead, in panel (b), the varying value is the magnitude of the triggered events.

Figure 6

Transition probability density function $p\left(x\right|m^{\prime })$. In subplot (a), the magnitude of the triggering events varies and $C=0.8$. Subplots (b) and (c) are instead obtained when $C$ varies and $m^{\prime }=2.7,m^{\prime }=7$, respectively.