## 4 Discussion  
NTA enables the detection of magnitude correlations [4; 51] when comparing the distribution of the order parameter \(\kappa_{1}\) of seismicity of the original EQ catalog with that of randomly shuffled copies of the same catalog. This has led, as already mentioned, to the conclusion that EQs with \(M\geq 7\) are correlated in global scale [52; 113]. This lies behind our selection of \(M_{\lambda}=7.0\) for earthquake nowcasting in Section 2.4. Indeed, the fact that EPS can be described by the Weibull distribution of Equation (8) with an exponent  
Figure 6: Average EPS maps calculated on the date that \(\beta_{160}\) minimizes (see Table 1 and Section 3) for the case of coarse grain radius \(R=200\) km. Panels (**a**–**h**) correspond to the \(M<8.5\) EQs Papua–Indonesia, Solomon Islands, Iquique–Chile, Illapel–Chile, Alaska, Chignik–Sandwich, Chiapas, and Fiji, respectively (see also Table 2). The EQ epicenters are depicted by red circles indicated by arrows for the readers’ convenience. The values of \(\langle\)EPS\(\rangle\) at the grid point closest to each epicenter location vary from 7% to 77%. \((0.862\pm 0.003)\) which is definitely different from unity (Poissonian statistics) provides an independent verification of the presence of these correlations. We now turn to the statistical properties of average EPS maps. We first focus on the statistics of \(\langle\)EPS\(\rangle\) at the lattice points closest to the epicenters of the six EQs with \(M\geq 8.5\) in GCMT, i.e., those depicted in Figure 5 and typed boldface in Table 2. Figure 7 depicts their average value \(\mu(R)\) together with their standard deviation \(\sigma(R)\) versus the coarse grain radius \(R\). We observe the presence of a logarithmic singularity in the fitting function of \(\mu(R)\) in a fashion similar to that identified in Reference [79]. This is reminiscent, as mentioned, of the two-dimensional Green's function for Poisson equation and reflects the fact that the future EQ epicenters (cf. \(\langle\)EPS\(\rangle\) maps were drawn on the date of their preceding \(\beta_{160,min}\), see Table 1) act as'sources' in these maps. This strengthens the potential of earthquake nowcasting to be generalized to forecasting, see also Reference [111]. Let us now consider the \(R\) dependence of the mean value:  
\[m_{n}(R,R^{\prime})=\frac{1}{N_{ij}}\sum_{ij}\langle\text{EPS}\rangle(x_{ij}, y_{ij}), \tag{12}\]  
where the summation is made over all the \(N_{ij}\) lattice points \((x_{ij},y_{ij})\), estimated numerically in an average EPS map drawn with a coarse grain radius \(R\), while \(R^{\prime}\) is a length scale. The average value \(\overline{\langle\text{EPS}\rangle}\) of \(m_{n}(R,R^{\prime})\) estimated for the \(\langle\)EPS\(\rangle\) maps related to the six EQs with \(M\geq 8.5\), i.e., those depicted in Figure 5, versus the coarse grain radius \(R\), can be seen in Figure 8. In Section 3 of Reference [81], it has been shown that \(m_{n}(R,R^{\prime})\) may take the form:  
\[m_{n}(R,R^{\prime})=\left(\frac{R}{R^{\prime}}\right)^{d_{f}}, \tag{13}\]  
where \(d_{f}\) is related to the fractal dimension [122] of EQ epicenters. Figure 8 reveals that \(d_{f}=1.398\pm 0.019\) for the global seismicity, which does not differ much from the value 1.32 \(\pm\) 0.06, estimated by the same method in the Eastern Mediterranean area [81]. Figure 7: The average value \(\mu(R)\) (red bullets) together with the standard deviation \(\sigma(R)\) (green crosses) of the \(\langle\)EPS\(\rangle\) values closest to each epicenter of the six \(M\geq 8.5\) EQs (see Table 2 and Figure 5) versus the coarse grain radius \(R\). The expressions for the fitting function \(\mu(R)\) (blue) and the average value of \(\sigma(R)\) (green horizontal line) are also shown. According to the regional studies of References [79; 81], when drawing an \(\langle\)EPS\(\rangle\) map on the date of a variability minimum of seismicity, we can have an estimation of the epicenter of a future strong EQ.