## 5 Space-Time Correlation between AE and Seismic Events  
Along the lines of studies on the space-time correlation among earthquakes based on a generalization of the Grassberger-Procaccia correlation integral [8], we calculate the degree of correlation between the AE time series and the sequence of nearby earthquakes recorded during the same period. No a priori assumptions on causal relationships between AE and seismic activities drive this approach, which is merely statistical. Only the choice of the time window and the region size is arbitrary. The space-time combined correlation integral is defined as follows [7,9]:  
\[C(r,\tau)=\frac{1}{N_{EQ}N_{AE}}\sum_{k=1}^{N_{EQ}}\sum_{j=1}^{N_{AE}}\theta(r- \left|x_{k}-x_{0}\right|)\cdot\theta\big{(}\tau-\left|t_{k}-t_{j}\right|\big{)} \tag{3}\]  
where the index \(j\) runs over all the \(N_{AE}\) event bursts \(\{x_{0},\,t_{j}\}\), with \(x_{0}\) being the coordinate vector of the monitoring site, while \(k\) runs over all the \(N_{EQ}\) seismic events \(\{x_{k},\,t_{k}\}\), with \(x_{k}\) being the epicentral coordinate vector. Since the double-sum in Equation (3) counts pairs formed by an AE event (\(x_{0},\,t_{j}\)) and a seismic event (\(x_{k},\,t_{k}\)) with mutual epicentral distance \(\left|\,x_{0}-x_{k}\,\right|\leq r\) and time intervals \(\left|\,t_{j}-t_{k}\,\right|\leq\tau\) among all \(N_{AE}\) and \(N_{EQ}\) possible pairs, \(C(r,\tau)\) can easily be interpreted as the probability of an AE burst and an earthquake occurring with an inter-distance \(\leq\)\(r\) and a time interval \(\leq\)\(\tau\). If \(C(r,\,\tau)\), exhibits power-law behavior both in space and time variables, time and space fractal dimensions \(D_{t}\) and \(D_{s}\) can be defined similarly to Equation (2):  
\[D_{t}(r,\tau)=\frac{\partial\log C(r,\tau)}{\partial\log\tau} \tag{4}\]  
and  
\[D_{s}(r,\tau)=\frac{\partial\log C(r,\tau)}{\partial\log r} \tag{5}\]  
The time correlation dimension \(D(r,\,\tau)\) characterizes the time-coupling of AE bursts to the earthquakes occurring up to a given distance \(r\) from the AE monitoring site, whereas the space correlation dimension \(D_{s}(r,\,\tau)\) characterizes the spatial distribution of nearby earthquakes with separation in time from AE bursts not exceeding a given \(\tau\). It is worth noting that Equation (5) identifies with Equation (2) if \(\tau\) equals the whole time data span, as all earthquakes would be taken into account in the Grassberger-Procaccia integral \(C(r,\,\tau)\). Actually, the imposed condition \(\left|\,t_{j}-t_{k}\,\right|\leq\tau\) does not specify the chronological order between the two types of event. On the other hand, the AE time series and the sequence of nearby earthquakes are two very closely interrelated sets in the time domain. Therefore, a given AE burst might be either due to structural damage triggered by a seismic event or due to precursive microseismic activity. In spite of intrinsic difficulties in high-frequency propagation across disjointed media (in particular at the ground-building foundation interface), AE bursts apparently indicate widespread crustal stress crises during the preparation of a seismic event [9,12], when part of the related deformation energy stored in the Earth's crust might be transferred to the building foundations. It is interesting to carry out a probabilistic analysis of the available data considering AE events once as preceding an earthquake, _i.e._, as potential seismic precursors, and next as following an earthquake, _i.e._, as structural aftershocks. In this way, the obtained conditioned probability distributions can be compared in order to discover the prevailing trend. This analysis is performed applying the modified correlation integral [9]:  
\[C^{\pm}(r,\tau)=\frac{1}{N_{EQ}N_{AE}}\sum_{k=1}^{N_{EQ}}\sum_{j=1}^{N_{AE}} \theta(r-|x_{k}-x_{0}|)\cdot\theta\big{(}\tau-\big{|}t_{k}-t_{j}|\big{)} \tag{6}\]  
where \"+\" and \"-\" are used to account for AE events, respectively, as seismic precursors and as aftershocks.