## 3 Results
Such a method, however, required detailed seismological catalogs for a large geographical region such as those provided by the Japan Meteorological Agency. Unfortunately, in the case of the eastern Mediterranean region studied here, such a catalog does not yet exist and, hence, this method could not be applied at present. To partially overcome this lack, we employed the following procedure: We applied the EQ nowcasting to circular areas of radius \(R\) with centers located at the points of a grid of \(0.25\lx@math@degree\times 0.25\lx@math@degree\) and estimated for each center an EPS value based on the Weibull approximation of the CDF depicted in Figure 3a. A self-consistent way to obtain an estimate of the epicenter location of the impending EQ would then be to average the aforementioned calculated EPS values within a circular area of radius \(R\) around the point P and assign this average EPS, \(\langle\)EPS\(\rangle\), to it. In this way, we could obtain for \(R=250\) km the \(\langle\)EPS\(\rangle\) maps shown in Figure 4, which were calculated at the date of the minimum \(\beta_{W,min}\) (see the first column of Table 1) preceding the \(M\geq 7.1\) EQs whose epicenters were depicted by the (black) open squares. Comparing the \(\langle\)EPS\(\rangle\) maps with these epicenters, we saw a reasonable agreement. A model to understand the process of self-consistently averaging EPS is shown in Figure 5; to keep this model simple, we assumed that all EQs occurred practically in the center O of the yellow-colored circular region \"Y\" of radius \(R\). After a long enough period without a large EQ, the EPS took the value of unity for all points inside \"Y\" and, since we assumed that no other EQ took place, the EPS would be zero outside it. Hence, the value of \(\langle\)EPS\(\rangle\) at a point P lying at a distance \(d\) away from O simplified to:  
\[\langle\)EPS\(\rangle=\frac{A(d,R)}{\pi R^{2}}, \tag{7}\]  
where \(A(d,R)\) is the (red) line-shaded area of the two intersecting circles shown in Figure 5, and equaled [140]:  
\[A(d,R)=2R^{2}\arccos\biggl{(}\frac{d}{2R}\biggr{)}-d\sqrt{R^{2}-\biggl{(} \frac{d}{2}\biggr{)}^{2}}. \tag{8}\]  
Hence, the value of \(\langle\)EPS\(\rangle\) in polar coordinates \((\rho,\theta)\) with center at O equaled to:  
\[f(\rho)=\frac{2}{\pi}\arccos\Bigl{(}\frac{\rho}{2R}\Bigr{)}-\frac{\rho}{\pi R }\sqrt{1-\biggl{(}\frac{\rho}{2R}\biggr{)}^{2}} \tag{9}\]  
for \(\rho<2R\) and zero otherwise. If we now calculate the mean value of \(\langle\)EPS\(\rangle\): \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\beta_{W,min}\)**Date** & **EQ Date** & **M** & **Epicenter Location** \\ \hline
10 July 1981 & 19 December 1981 & 7.2 & Lesvos, Greece \\
7 March 1986 & 30 August 1986 & 7.2 & Vrancea, Romania \\
24 August 1995 & 22 November 1995 & 7.2 & Aqaba, Jordan \\
6 April 1999 & 17 August 1999 & 7.6 & Izmit, Turkey \\
4 October 2011 & 23 October 2011 & 7.1 & Van, Turkey \\
8 September 1999 & 12 November 1999 & 7.2 & Düzce, Turkey \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The dates of the five almost simultaneously (within 3 months) observed minima in the limiting geographical areas \(A_{5}\) and \(A_{6}\) of the 5-core and the 6-core of the area \(\mathrm{N}_{25}^{50}\mathrm{E}_{5}^{46}\), respectively, together with the strong EQs that they had preceded. \({}^{a}\) They are marked with (red) open circles in Figure 2. In the last line, a local minimum that was also observed before Düzce EQ was also added; see Figure 2.  
\[m(R,R^{\prime})=\frac{\int_{0}^{R^{\prime}}\int_{0}^{2\pi}f(\rho)\rho d\theta d\rho} {\pi R^{\prime 2}}=\frac{2\int_{0}^{R^{\prime}}f(\rho)\rho d\rho}{R^{\prime 2}} \tag{10}\]  
in a circular region of radius \(R^{\prime}>2R\), we simply obtain that:  
\[m(R,R^{\prime})=\left(\frac{R}{R^{\prime}}\right)^{2} \tag{11}\]  
by virtue of the integrals \(\int_{0}^{2R}\arccos(\frac{\rho}{2R})\rho d\rho=\pi R^{2}/2\) and \(\int_{0}^{2R}\rho^{2}\sqrt{1-\left(\frac{\rho}{2R}\right)^{2}}d\rho=\pi R^{3 }/2\).