Space-time combined correlation integral and earthquake interactions

Scale invariant properties of seismicity argue for the presence of complex triggering mechanisms. We propose a new method, based on the space-time combined generalization of the correlation integral, that leads to a self-consistent visualization and analysis of both spatial and temporal correlations. The analysis was applied on global medium-high seismicity. Results show that earthquakes do interact even on long distances and are correlated in time within defined spatial ranges varying over elapsed time. On that base we redefine the aftershock concept.


Introduction
Seismicity appears to be scale invariant in many of its aspects.Several papers (Kagan, 1994;Bak et al., 2002;Parson, 2002;Marsan and Bean, 2003;Corral, 2004) investigate spa tial and temporal correlations of epicentres, in volving for example the concepts of Omori law and fractal dimension.We think that the com plex phenomenon of seismicity calls for an ap proach capable of analysing spatial localisation Mailing address: Dr. Patrizia Tosi, Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 605, 00143 Roma, Italy; e-mail: tosi@ingv.itand time occurrence in a combined way and without subjective a priori choices.In this pa per we introduce a new method of analysis that leads to a self-consistent analysis and visuali zation of both spatial and temporal correlations based on the definition of correlation integral (Grassberger and Procaccia, 1983).

Method
We define the space-time combined correla tion integral as where Θ is the Heaviside step function ( Θ(x) =0) if x ≤ 0 and Θ(x) =1 if x > 0) and the sum counts all pairs whose spatial distance x i -x j # r and whose time interval t i -t j # x.When applied over all possible values of τ or r, the well-known correlation integral (Grassberger and Procaccia, 1983) is returned.It results that C c(r, τ) is the generalisation of the correlation in tegral for a phenomenon that explicates in di verse dimensions with not comparable measure ment units.When applied on seismicity Cc(r, τ) takes into account the distribution of all time in tervals and epicentral inter-distances between all pairs of events, irrespective of the relationship between the main event and any aftershock.
From the space-time combined correlation integral we define the time correlation dimen sion and the space correlation dimension for sets of events within space-time distances r and τ, respectively as If Cc(r, τ) was a pure power-law in both variables, then Dt and Ds would correspond to the temporal and spatial fractal dimensions, respectively.More generally, the behaviour of Dt and Ds as a function of r and τ will characterise the clustering features of earthquakes in space and in time.This method has been applied to global seismicity to study the space-temporal correlation between earthquakes all over the world.Data come from the catalogue of the National Earthquake Information Center, USGS, in the time period between 1973 and 2002, with magnitudes mb greater than 5.This catalogue selection was conditioned by com pleteness criteria and it presents medium to high magnitude distribution.

Results
The space-time combined correlation inte gral Cc for global seismicity is represented in fig. 1 with black contour lines.The local slopes of this surface in the direction parallel to the time axis is the time correlation dimension Dt, plotted in colours as a function of space and time.The colour coding of each pixel quantifies the time correlations existing between events occurring within a given distance and time in terval.D t ≅ 1 corresponds to the random occur-   1 significantly support the hypothesis that earthquakes are correlated inside some space temporal ranges.In order to check this hypoth esis we applied the same analysis to the global catalogue after a reshuffling procedure.Reshuf fling consists in mixing the time occurrence of each event keeping fixed its epicentre coordi nates.The characteristic of this procedure is that of maintaining the separate statistical prop erties of data.The results show (fig.2) that all patterns vanish, evidencing constant high val ues of Dt at all distances and time intervals.
To delineate a limit of the clustering resulting from fig. 1, all the points with Dt =0.8 were plot ted on a separate figure (fig.3).It is interesting to see that, within the temporal ranges shown, the clustering boundary can well be approximated by a straight line on this log-log plot, thus indicating a power-law behaviour.Using the least squares fitting we obtained logr = − 0.55logτ +3.8.This relation places a strong constraint on time rela tions among events, evidencing how distance plays a dynamic role.In particular, the relation can be read as defining a temporal correlation  reaching long distances that quickly shrinks over time following a power-law.For relatively short spatial ranges (around 100 km) events are time clustered and correlated for long time in tervals (around 3 years).Over longer distances time correlation lasts for a short period (less than 30 days for 1000 km).
The local slopes of the surface of the space time combined correlation integral, in the direc tion parallel to the space axis, correspond to the space correlation dimension Ds, plotted in colours in fig. 4 as a function of space and time.The colour coding of each pixel quantifies the space clustering existing between events occur ring within a given distance and time interval.Ds ≅ 2 identifies a random distribution of earth quakes, Ds ≅ 1 indicates that epicentres tend to dispose along lines and D s < 1 corresponds to space clustering.Even in this plot different do mains are easily recognised.At short distances, a high space correlation dimension domain is clear ly separated from space clustering ( 0 < Ds < 1) that is present at greater distances: both condi tions last for inter-time up to 100 days.The dis appearance of clustering with time leaves room to a general D s ≅ 1, interpreted as the activity of seismicity on plate boundaries.Even in this case we tested the goodness of the results ap plying the same analysis on the reshuffled cata logue.The resulting plot in fig. 5 shows that the single statistical properties of data are not suffi cient to produce the clustering domains appear ing in the combined approach of fig. 4, but a real connection between space and time is needed.
Localisation errors certainly plays an im portant role at short spatial ranges, generating high values of Ds (fig.4), but it appears that the area with high space correlation dimension is evolving with time calling for the presence of a physical process.In particular, plotting in fig.6 the points with Ds = 1, chosen as the limit sepa rating random behaviour from space clustering, it appear that in the shown ranges they follow a straight line.Fitting with least squares we ob tained the relation logr = 0.1logτ +1.2.The sep aration line defines an area around each epicen tre, slowly growing in time, within which seis mic events are randomly distributed.

Discussion
Figures 1 and 4 show that earthquakes are connected with each other in a non trivial way and that a dynamic interaction appears when space and time are analysed together with the combined correlation integral.The results reveal a statistical property of the global seismicity of medium-high magnitude, but interpreting them as an average behaviour of events after the oc currence of each earthquake of magnitude greater than 5, a possible scenario appears.The term 'aftershock' can be redefined on the basis of our findings: aftershocks are all earthquakes con- nected to one reference event preceding them, as revealed by their temporal correlation (low Dt).
In this sense all earthquakes occurred at a dis tance from reference event less than the radius r, defined by the relation logr = − 0.55logτ +3.8. (where τ is the elapsed time, fig.3), are after shocks of that event.This aftershock region reaches long distances from the reference 'main' event, but it quickly shrinks over time.If compared to an homogeneous time distribution, this area of influence can be interpreted as a re gion of modified probability of earthquake time occurrence.The epicentres of these connected events tend to cluster in space, apart from the near field (an area of radius 10-20 km), where earthquakes are randomly placed.Even this near field aftershock region has a dynamic boundary, increasing slowly in size according to the equation logr = 0.1logτ +1.2 (fig.6).This result is in agreement with other authors (Taji ma and Kanamori, 1985;Marsan et al., 2000;Helmstetter, 2003;Huc and Main, 2003) who found a migration of aftershocks, defined with classic methods, away from a main shock.This migration is described in terms of a law r ( ) + t H , where d t d t r ( ) is the mean distance be tween main event and aftershocks occurring af ter time t, with an exponent H < 0.5 correspon ding to a sub-diffusive process.

Conclusions
In summary, we have introduced a new sta tistical tool, the combined space-time correla tion integral, which allows us to perform a si multaneous and self-consistent investigation of the correlation properties of earthquakes.This tool leads to the discovery, visualization and deep analysis of the complex interrelationships existing between the spatial distribution of epi centers and their occurrence in time.The analy sis performed on the worldwide seismicity cat alogue and the corresponding reshuffled cata logue, strongly suggests that earthquakes of medium-high magnitude do interact with each other.This result led to a new definition of af tershocks, as all earthquakes with non-random occurrence with respect to the reference 'main' event, without considering their magnitude.Fi nally the analysis revealed how the aftershock region modifies over elapsed time.

Fig. 1 .
Fig. 1.Space-time combined correlation integral Cc(r,τ) (dark contour lines) and time correlation dimension Dt (coloured shaded contour) for the catalogue of global seismicity.

Fig. 3 .
Fig. 3.The limit separating time clustering from time randomness (fixed to Dt = 0.8) as a function of distance is shown.Points are fitted by the line of equation logr = − 0.55logτ +3.8.

Fig. 4 .
Fig. 4. Space-time combined correlation integral Cc (r,τ) (dark contour lines) and space correlation dimension Ds (coloured shaded contour) for the catalogue of global seismicity.

Fig. 6 .
Fig. 6.The limit separating spatial clustering from spatial randomness (fixed to Ds = 1.0) as a function of time is shown.Points are fitted by the line of equation logr = 0.1logτ +1.2.