aa r X i v : . [ phy s i c s . g e o - ph ] A ug Statistics of seismic cluster durations
A. Saichev ∗ and D. SornetteETH ZurichDepartment of Management, Technology and EconomicsScheuchzerstrasse 7, 8092, Zurich, [email protected] ∗ deceased 8 June 2013August 2, 2018 Abstract
Using the standard ETAS model of triggered seismicity, we presenta rigorous theoretical analysis of the main statistical properties of tem-poral clusters, defined as the group of events triggered by a given mainshock of fixed magnitude m that occurred at the origin of time, attimes larger than some present time t . Using the technology of gen-erating probability function (GPF), we derive the explicit expressionsfor the GPF of the number of future offsprings in a given temporalseismic cluster, defining, in particular, the statistics of the cluster’s du-ration and the cluster’s offsprings maximal magnitudes. We find theremarkable result that the magnitude difference between the largestand second largest event in the future temporal cluster is distributedaccording to the regular Gutenberg-Richer law that controls the un-conditional distribution of earthquake magnitudes. For earthquakesobeying the Omori-Utsu law for the distribution of waiting times be-tween triggering and triggered events, we show that the distributionof the durations of temporal clusters of events of magnitudes abovesome detection threshold ν has a power law tail that is fatter in thenon-critical regime n < n = 1. This para-doxical behavior can be rationalised from the fact that generations ofall orders cascade very fast in the critical regime and accelerate thetemporal decay of the cluster dynamics. Introduction
The present article is the last in a series [9,10,18–26,30,31,33,34] devotedto the analysis of various statistical properties of the aftershocks triggeringprocess of the ETAS model [13, 14, 16, 17, 29]), which serves as a standardbenchmark in statistical seismology. Here, we present derivations and re-sults concerning statistical properties of the temporal seismic clusters, madeof future events triggered after the current time t by some main shock ofmagnitude m that occurred as some previous instant taken as the origin oftime.We mostly investigate the statistical properties of the durations of tem-poral seismic clusters and of the maximal magnitude over all events in theclusters. Specifically, we derive the dependence of the probability densityfunction (pdf) of the duration of seismic clusters as a function of the mainshock magnitude m , of a magnitude threshold ν of earthquake detection andof the branching ratio n . In addition, we study the characteristic propertiesof the maximal magnitudes of future offsprings triggered after the currenttime t .The results obtained here have a broader domain of application than sta-tistical seismology, and can be used for any system that can be described bythe class of self-excited conditional Poisson process of the Hawkes family [5–7]and its extension and more generally to branching processes. For instance,some of the stochastic processes in financial markets can be well representedby this class of models, for which the triggering and branching processes cap-ture the herding nature of market participants. The Hawkes process has beensuccessfully involved in issues as diverse as estimating the volatility at thelevel of transaction data, estimating the market stability [3,4], accounting forsystemic risk contagion, devising optimal execution strategies or capturingthe dynamics of the full order book [1].The article is organised as follows. Section 2 presents the formulationof the version of the Hawkes model, known at the ETAS model of triggeredseismicity, with particular emphasis on statistics of temporal seismic clus-ters statistics. Section 2 also introduces the fractional exponential model asa convenient parameterisation of the distribution of waiting times betweentriggering and triggered events (also known as the bare Omori law). Thefractional exponential model provides a rather accurate approximation ofthe well-known modified Omori-Utsu law. Section 3 derives the explicit andapproximate expressions for the generating probability function (GPF) of thenumber of future offsprings in a given temporal seismic cluster, defining, inparticular, the statistics of the cluster’s duration and the cluster’s offspringsmaximal magnitudes. Section 4 presents a detailed statistical analysis of2he seismic cluster’s duration statistics and the statistics of the maximal off-springs magnitude. It is shown in particular that, in the subcritical case,one may use, without significant error, the so-called one-daughter approxi-mation in which each event can trigger not more than one first-generationaftershock. All proofs of our four main results, presented in the form of fourpropositions, are given in appendices. Section 5 conclude. Let us consider a main shock occurring at time t = 0 with magnitude m . To make precise our investigation of the statistical properties of theaftershocks triggered by the main shock (consisting of the main shock’s directaftershocks, the direct aftershocks of the first generation aftershocks and soon), we use the ETAS model [13, 14, 16], whose main assumption is thatall earthquakes obey the same laws governing the generation of triggeredearthquakes. Each earthquake is thus potentially the “mother” of triggeredevents, which themselves can trigger their own “daughters” and so on. In theETAS model, there are two categories of earthquakes: (i) the main shocksthat are supposed to be “immigrants”, i.e. they are not triggered by previousearthquakes, and (ii) all the other earthquakes that are triggered by someprevious event, be it a main shock or one of the event it has triggered eitherdirectly or indirectly through a cascade. The cornerstones of the ETAS model are based on three well-known sta-tistical laws that govern the process of earthquake triggering. The follow-ing subsections 2.1.1-2.1.4 enunciate the four fundamental definitions of theETAS model, which describe the properties shared by all earthquakes.
The well-known
Gutenberg-Richter law (GR) states that earthquakes oc-cur with magnitudes distributed according to the complementary cumulativedistribution function p ( m ) := Pr { m ′ > m } = 10 − bm , (1)where the b -value is often found empirically close to 1 [15].3n the ETAS model, the GR law is assumed to apply both for the mainshocks and for their aftershocks, as well as for subsequent aftershocks of af-tershocks over all generations. Moreover, the ETAS model posits that themagnitudes are independent random variables, i.e. there is no (uncondi-tional) dependence between the magnitudes of any earthquake in a givenseismic catalog. The magnitudes are thus i.i.d. random variables distributedaccording to (1). The fertility law states that the mean number R d ( m ) of direct (first gen-eration) aftershocks triggered by a given earthquake of magnitude m is ex-ponentially large in the magnitude of the mother earthquake [8]: R d ( m ) = κµ, µ := 10 αm , (2)where α is in general found to be smaller than b , with typical values closeto 0 .
8. The parameters κ as well as α may depend on regional properties ofseismicity.For simplicity of notations, we assume that all magnitudes are positiveand all events with a positive magnitude has the ability to trigger futureevents. In the standard ETAS model, one introduces a characteristic cut-off magnitude m , below which events are sterile, i.e. do not trigger otherevents. This cut-off magnitude is needed to ensure that the ETAS modelis well-defined, otherwise, the swarms of arbitrary small earthquakes makethe seismic activity divergent and ill-defined [31]. Our parameterisation thusamounts to take m = 0, which is nothing but a translation in the magnitudescale that has no impact on the calculations and results. The modified Omori-Utsu law specifies the distribution f ( t ) of waitingtimes { T k } between a mother earthquake and its direct (first-generation)offsprings [35]: f ( t ) = θc θ ( c + t ) θ +1 , θ ∈ (0 , , c > . (3)By construction, it gives the dependence of the rate of first-generation after-shocks as a function of time t counted since the mother earthquake.The ETAS model assumes further that the modified Omori-Utsu law ap-plies for all earthquakes, whatever their rank in the generation ordering.Thus, all earthquakes have the potential to trigger their aftershocks withdelays given by expression (3). 4 .1.4 Poisson statistics in the ETAS model Combining the assumptions stated in subsections 2.1.1-2.1.3 that all earth-quakes are treated equally in the sense that they all possess the same propen-sity for triggering earthquakes with the same time dependence given by theOmori-Utsu law and with i.i.d. magnitudes, it derives that the total number R d ( m ) of first-generation daughters triggered by the mainshock of magnitude m obeys the Poisson statistics. This means that the probability q d ( r | m ) thatthe number R d ( m ) of first-generation daughters takes the value r is given by q d ( r | m ) := Pr { R d ( m ) = r } = ( κµ ) r r ! e − κµ . (4)Accordingly, the Generating Probability Function (GPF) of the total numberof first-generation daughters is given by G d ( z | m ) := E (cid:2) z R d ( m ) (cid:3) = ∞ X r =0 q d ( r | m ) z r = e κµ ( z − . (5)As a consequence of the number R d ( m ) of first-generation daughters obey-ing Poissonian statistics (5), and given that their occurrence times { T k } arestatistical independent, we can state that Proposition 1
Given a fixed observation time t , the random numbers offirst-generation daughters of the main shock that occurred in the past (before t ) and that will come in the future (after t ) are statistically independent andobey to the Poissonian statistics.The proof of Proposition 1 is given in the Appendix. For the known and fixed main shock magnitude m , the GPF of the numberof its first-generation daughters is given by expression (5). In contrast, theGPF of the number of first-generation events of an arbitrary first-generationdaughter (i.e. the number of grand-daughters of the main shock via the fil-iation of one of its daughters) is given by the average of (5) over the GRdistribution of magnitudes (1), since the first-generation daughters have ran-dom iid magnitudes: G d ( z ) = − Z ∞ G d ( z | m ) dp ( m ) . (6)5his yields G d ( z ) = γκ γ (1 − z ) γ Γ( − γ, κ (1 − z )) , γ = b (cid:14) α , (7)where Γ( a, z ) is the incomplete gamma function.For real aftershock sequences, the parameter γ belongs to the interval: γ ∈ (1 ,
2) (see more detailed discussion in [18]). In particular, for ( b = 1 , α =0 . γ = 1 .
25. For such a value, for the convenience of futureanalytical derivations, we may replace the exact expression (7) by the firstthree power terms of its Taylor expansion in the variable (1 − z ): G d ( z ) := 1 − n (1 − z ) + ρ (1 − z ) γ , < γ < , (8)where n := κγγ − , ρ = κ γ γ Γ( − γ ) . (9)Parameter n is the so-called “branching ratio”, which quantifies how manyfirst-generation daughters are triggered per mother. A value close to 1 corre-sponds to the approach to the critical regime [9]. n has also the meaning ofbeing the average fraction of triggered events in the whole population [11].Parameter n will play a crucial role in the following analysis of the statisticalproperties of seismic clusters, in particular in the two fundamental regimes,the subcritical ( n <
1) and critical ( n = 1) regimes, which are relevant toreal seismic activity.Plots of the GPF G d ( z ) (7) and its algebraic approximation G d ( z ) (8) forthe typical values ( b = 1 , α = 0 .
8) and for κ = 0 . γ = 1 .
25 and n = 1) are depicted in figure 1.Parameter γ controls the asymptotic decay for r → ∞ of the probability q d ( r ) = 1 r ! d r G d ( z ) dz r (cid:12)(cid:12)(cid:12)(cid:12) z =0 (10)that an arbitrary daughter has r first-generation offsprings. One can show(see, for instance, [18]), using either the exact G d ( z ) (7) and its algebraicapproximation G d ( z ) (8), that q d ( r ) has the following asymptotic tail q d ( r ) ≃ ρ Γ( − γ ) · r γ , r ≫ . (11)Since 1 ≤ γ <
2, the mean number of descendants of first-generation exists,but not its variance. 6 z G o ( z ) , Θ o ( z ) Figure 1: The lower solid line shows the exact GPF G d ( z ) (7) as a function ofthe variable z . The upper solid line is its algebraic approximation G d ( z ) (8).The dashed line is the ratio of the former to the later, which quantifies theaccuracy of the algebraic approximation. The parameters are b = 1, α = 0 . κ = 0 .
2, corresponding to γ = 1 .
25 and n = 1. One can notice thatthe algebraic approximation becomes excellent as 1 − z →
0, and the twoexpressions are undistinguishable for 1 − z < . .3 The modified Omori-Utsu law and its fractionalexponential approximation The modified Omori-Utsu law (3) used in the ETAS specification fol-lows Ogata’s formulation of the ETAS model [16]. Earlier, Kagan andKnopoff [13, 14] introduced a version of ETAS (under a different name),which had essentially all its ingredients, except for the expression of thefunction f ( t ) controlling the distribution of waiting times between a motherearthquake and its first-generation offsprings. Kagan and Knopoff [13, 14]used a pure power law f ( t ) ∼ /t θ truncated to zero for t < c for somepositive characteristic time c . The difference between Ogata’s and Kaganand Knopoff’s specifications of the memory kernel f ( t ) amounts to a changein “ultraviolet” cut-off, which was shown in Ref. [28] to have no significantimpact on the dynamics and overall generating process.Here, we propose to use another ultra-violet cut-off, which has significantadvantages for the analytical computations that we develop below, withoutsignificant impact on the main characteristic of the model, namely its heavytail power law asymptotics f ( t ) ∼ t − θ − , t → ∞ . (12)We thus propose to use the so-called fractional exponential distribution , whichhas the same power law asymptotics (12), as the modified Omori-Utsu law.To motivate the fractional exponential distribution, let us consider Laplacetransform ˆ f ( u ) of the probability density function (pdf) f ( t ):ˆ f ( u ) := Z ∞ f ( t ) e − ut dt . (13)It is well-known that the two following power law asymptotics are equivalent: f ( t ) ≃ ǫ Γ( − θ ) · t − θ − , t → ∞ ⇔ − ˆ f ( u ) ≃ ǫ · u θ , u → , for θ ∈ (0 , . (14) Definition 1
The fractional exponential distribution, denoted f θ ( t ), is de-fined by its Laplace transformˆ f θ ( u ) := 11 + u θ . (15) Remark 1
The fractional exponential distribution, with Laplace image(15), corresponds to the canonical case where the factor ǫ in the above asymp-totics (14) is equal to 1. The modified Omori-Utsu law (3) gives the sameasymptotic power law with the correspondence c θ = Γ − (1 − θ ) ⇒ c = Γ − /θ (1 − θ ) . (16)8ne can show that the fractional exponential distribution, defined by theLaplace image (15), is given by f θ ( t ) = t θ − · E θ,θ ( − t θ ) , θ ∈ (0 , , (17)where E α,β ( z ) is the generalized Mittag-Leffler function E α,β ( z ) := ∞ X j =0 z j Γ( αj + β ) . (18)The fractional exponential distribution is characterised by two power lawasymptotics, one for short times and the other for long times: f θ ( t ) ≃ − t − θ − Γ( − θ ) , t ≫ ,t θ − Γ( θ ) , t ≪ , θ ∈ (0 , . (19)By construction, it has the same power law asymptotics as the modifiedOmori-Utsu law (3) at long times, with the matching (16) of the prefactorsas already mentioned.Figure 2 compares in log-log scales the modified Omori-Utsu law (3) andthe corresponding fractional exponential distribution f θ ( t ) (17), for four dif-ferent values of the exponent θ . These plots illustrate the closeness of thesetwo pdf’s at large times, both embodying the long memory property of theaftershocks triggering process. One can also observe the transition from theslope − − θ at large times to the slope − θ at small times predicting by(19). In contrast with Kagan and Knopoff who assumed that no daughterscan be triggered between times 0 and c after the mother event occurred, orwith Ogata who assumed an asymptotic constant rate of triggering at smalltimes after the mother event occurred, the fractional exponential distributionamounts to assuming a diverging triggering rate as one looks closer and closerto the main event. But, given the value of the exponent − θ > −
1, thetotal number of daughters triggered at short is finite (the pdf is integrable).From the fractional exponential pdf, we define the survival functionΦ θ ( t ) = Z ∞ t f θ ( t ′ ) dt ′ . (20)Its Laplace image and explicit expression are given byˆΦ θ ( u ) = u θ − u θ ⇔ Φ θ ( t ) = E θ (cid:0) − t θ (cid:1) , (21)9 −2 −5 τ f ( τ ) −2 −5 τ f ( τ ) −2 −5 τ f ( τ ) −2 −5 τ f ( τ ) θ = 0 . θ = 0 . θ = 0 . θ = 0 . Figure 2: Plots in double logarithm scales, for four different θ values, of themodified Omori-Utsu law pdf f ( t ) (3) (solid lines) and the correspondingfractional exponential distributions f θ ( t ) (17) (dashed lines). To make thetwo pdf’s comparable, we use the scale parameter c given by (16). For θ = 0 .
4, one can observe best the transition from the slope − − θ at largetimes to the slope − θ at small times.10here E α ( z ) = E α, ( z ) is the Mittag-Leffler function obtained as the specialcase β = 1 of the generalized Mittag-Leffler function (18). The followingproperties holdˆΦ θ ( u ) ≃ ( u θ − , u ≪ ,u − , u ≫ . ⇔ Φ θ ( t ) ≃ − t θ Γ(1 + θ ) , τ ≪ ,t − θ Γ(1 − θ ) , t ≫ . (22)In the limiting case θ = 1, the fractional exponential distribution reducesto the pure exponential pdf: f ( t ) = Φ ( t ) = e − t ⇔ ˆ f ( u ) = ˆΦ ( u ) = 11 + u . (23)Below, we will analyze in details the statistics of seismic clusters bothfor the cases when the pdf of waiting times is described by the fractionalexponential distribution f θ ( t ) (0 < θ <
1) and by the exponential case θ = 1(23). Consider a mother event (an “immigrant” in the language of branchingprocesses) occurring at time 0. At the current time τ >
0, we distinguish thetriggered events of first-generation (direct daughters) represented as emptyand full circles in figure 3 and the triggered events of second and higher gen-erations (grand-daughters, grand-grand-daughters, and so on) representedby empty and full squares in figure 3. Moreover, we separate the events inthe past (empty symbols) from the events that will occur in the future (fullsymbols).Let us defined the GPF Ω( z ; t ) of the number R ( t ) of the future eventsof all generations. By Proposition 1, Ω( z ; t ) can be represented as a productof two GPFs: Ω( z ; t ) := E (cid:2) z R ( t ) (cid:3) = Ω out ( z ; t ) · Ω in ( z ; t ) , (24)where Ω out ( z ; t ) is the GPF of the number of future events triggered after thecurrent instant t by the past daughters of first-generation, and Ω in ( z ; t ) is theGPF of the total number of future daughters and all their higher-generationaftershocks. In other words, the seismic activity in the future (at times afterthe present time t ) can be decomposed as due to two sources: (i) the set of all11 = 0 t = τ Figure 3: Given a fixed current time τ , we give an illustration of the def-initions of past and future direct and indirect aftershocks triggered by amain shock that occurred at time 0, depicted by the bold vertical arrowat the origin of time. The circles represent directly triggered events, i.e.first-generation daughters. The squares represent triggered events of sec-ond and higher generations (grand-daughters, grand-grand-daughters, andso on). Empty (resp. full) symbols correspond to events in the past, i.e.that occurred at times t < τ (resp. in the future, i.e that occurred at times t > τ ).the first-generation daughters that were born up to the current time t , whichis represented by Ω out ( z ; t ); (ii) the mother event that occurred at time 0,which continues to trigger direct daughters and all their grand-daughters andhigher generation events in the future, and which is represented by Ω in ( z ; t ).Using proposition 1, Ω out ( z ; t ) and Ω in ( z ; t ) are given byΩ out ( z ; t ) = exp (cid:18) − κµ Z t [1 − G ( z ; t − s )] f ( s ) ds (cid:19) , Ω in ( z ; t ) = exp (cid:18) − κµ [1 − zG ( z )] Z ∞ t f ( s ) ds (cid:19) , (25)where G ( z ) = G ( z ; t = 0), and G ( z ; t ) is the GPF of the number of after-shocks of all generations that are triggered by an event of arbitrary magnitudethat occurred at time 0. 12 roposition 2 The GPF G ( z ; t ) is determined by the two coupled equa-tions G ( z ; t ) = Q [ H ( z ; t )] ,H ( z ; t ) = 1 − zG ( z )Φ( t ) − G ( z ; t ) ⊗ f ( t ) . (26)where the symbol ⊗ represents the convolution operation, Φ( t ) is the survivalfunction Φ( t ) = Z ∞ τ f ( t ′ ) dt ′ (27)corresponding to the pdf f ( t ) of waiting times, and the auxiliary function Q ( y ) is defined by Q ( y ) = G d (1 − y ) . (28)with G d defined by (5) and (7) and given approximately by (8).The proof of Proposition 2 is given in the Appendix.It follows from the equations (26) that the GPF G ( z ), of the total numberof aftershock of all generations triggered by some event, satisfies the well-known transcendent equation G ( z ) = G d [ zG ( z )] . (29) Remark 2
For the feasibility of analytical calculations, we replace the exactexpression (7) of the GPF G d ( z ) of the number of direct aftershocks by itsalgebraic expansion G d ( z ) (8). After substitution in (28), we obtain Q ( y ) = G d (1 − y ) = 1 − ny + ρy γ . (30)In the following, we will use this expression for the function Q ( y ) in all ourcalculations, offering when needed quantitative assessment of the quality ofthe approximation provided by this expansion. Remark 3
To make explicit the dependence on the magnitude m of themain shock and the importance of the branching ratio n , in the following, wereplace the notation H ( z ; t ) in equation (26) by H ( z ; t, n ) and the functionΩ( z ; t ) (24) by Ω( z ; t, n, m ). Proposition 3
The GPF Ω( z ; t, n, m ) (24) of the number of future off-springs of all generations triggered by the mainshock of magnitude m isgiven by Ω( z ; t, n, m ) = e − κµ · H [ zG ( z ); t,n,m ] , (31)where H ( z ; t, n ) satisfies to the nonlinear integral equation H ( z ; t, n ) − n · H ( z ; t, n ) ⊗ f ( t ) + ρ · H γ ( z ; t, n ) ⊗ f ( t ) = (1 − z )Φ( t ) . (32)13he proof of Proposition 3 is given in the Appendix.For t = 0, the following relation derives from (31) and (32):Ω( z ; n, m ) := Ω( z ; t = 0 , n, m ) = e κµ · ( zG ( z ) − , (33)where G ( z ) is the solution of the transcendent equation (29). As mentioned in section 2.1.2, the fertility law (2) holds for all earthquakeswith non-negative magnitudes. To account for the fact that real seismicityis only detected above a magnitude threshold determined by instrumentalsensitivity and motivated by the fact that one may be interested only inearthquakes of large magnitudes, we introduce the threshold magnitude ν >
0. We can thus count the subset of triggered events whose magnitudes { m j } satisfy the inequality m j > ν >
0. The magnitude ν can thus be consideredto be an observational magnitude threshold, such as only earthquakes withmagnitudes larger than ν are observed. The existence of such a threshold canbe shown to renormalise the parameters (branching ratio n and backgroundseismicity rate or immigrant rate) of the ETAS model when applied to orcalibrated on the observed earthquakes [22, 32]. Definition 2
We denote ν -cluster the set of offsprings of all generationswhose magnitudes exceed the given magnitude threshold ν . We call future ν -cluster the subset of the ν -cluster of future offsprings, i.e. which occur afterthe current time t . Figure 4 illustrates the notion of the future ν -cluster. Proposition 4
Given the GPF Ω( z ; t, n, m ) of the number of future off-springs of any positive magnitude of a mother event of fixed magnitude m ,the GPF Ω( z ; t, n, m, ν ) of the number of future offsprings of magnitude largerthan ν is given by the following relation:Ω( z ; t, n, m, ν ) = Ω (1 + p ( ν )( z − t, n, m ) . (34)where p ( m ) is of the GR law (1).The proof of Proposition 4 is given in the Appendix.14 t0 m ν Figure 4: Schematic representation of the offsprings of a given immigrantmother. Along the vertical axis is the offsprings magnitudes. The horizontaldashed line corresponds to the threshold magnitude ν . The current time t isindicated by the vertical dashed line. The dotted circles show the offspringswhose magnitudes are below ν and thus do not belong to the ν -cluster. Thehollow solid circles are the offsprings that belong to the ν -cluster, but donot belong to the future ν -cluster. The bold solid circles are the offsprings,which belong to the future ν -cluster.15 .6 Probability of absence of offsprings, cluster dura-tions, maximum magnitude and function P ( t, n, m, ν ) Let us introduce the new function P ( t, n, m, ν ) := Ω( z = 0; t, n, m, ν ) = Ω (1 − p ( ν ); t, n, m ) . (35)Our motivation for proposing this function P ( t, n, m, ν ) is that it has threeinteresting probabilistic interpretations. P ( t, n, m, ν )It follows from the first equality (35) and from the statistical meaning ofthe GPF Ω( z ; t, n, m, ν ) that the function P ( t, n, m, ν ) (35) is equal to theprobability that the future ν -cluster is empty, i.e. that the number R ν ( t ) offuture offsprings is equal to zero: P ( t, n, m, ν ) = Pr { R ν ( t ) = 0 } . (36)Defining R ν := R ν ( t = 0), the function P ( n, m, ν ) := Pr { R ν = 0 } = Ω( z ; t = 0 , n, m, ν ) (37)is the probability that the mainshock of magnitude m does not trigger off-springs of magnitudes larger than ν . Below, it will be convenient to definethe complementary probability P ( n, m, ν ) := 1 − P ( n, m, ν ) , (38)that the mainshock triggers at least one observable offspring of magnitudelarger than ν . Figure 5 shows the dependence of the probabilities P ( n, m, ν )as a function of the threshold magnitude ν , for different values of the branch-ing ratio n , for a main shock magnitude equal to m = 9. P ( t, n, m, ν )Let us introduce T ( ν ) as the random duration of the ν -cluster: T ( ν ) := max { T j ( ν ) } , (39)where { T j ( ν ) } is the set of occurrence times of all the events that makeup the ν -cluster. Then, the second probabilistic meaning of the function P ( t, n, m, ν ) is expressed by P ( t, n, m, ν ) = Pr { T ( ν ) < t } , (40)16 ν ¯ P ( n , m , ν ) Figure 5: Dependence of the probability P ( n, m, ν ) that the main shocktriggers at least one offspring of magnitude larger than ν as a function of ν .For all curves, m = 9, b = 1 and α = 0 . γ = 1 . n = 1 , . , . , ..., . , .
1. 17.e., it is the probability that the total duration of the ν -cluster be less than t . Note that this relation (40) does not exclude the possibility that the mainshock does not trigger any offspring.Considering only the ν -clusters that contain at least one offspring, relation(40) has to be replaced by a conditional one, expressing the condition that the ν -cluster contains at least one offspring. Using the law of total probability,the corresponding conditional counterpart of relation (40) readsPr { T ( ν ) < t | R ν > } = [ P ( t, n, m, ν ) − P ( n, m, ν )] (cid:14) P ( n, m, ν ) , (41)where R ν := R ν ( t = 0) is the total number of offsprings in the ν -cluster.Denoting ϕ ( t ; n, m ) as the pdf of the total duration of the ν -clusters thatcontain at least one offspring, expression (41) yields ϕ ( t ; n, m, ν ) = C · ∂ P ( t, n, m, ν ) ∂t , C := 1 (cid:14) P ( n, m, ν ) . (42) P ( t, n, m, ν )Let us define M ( t, n, m ) as the largest magnitude among all those offuture offsprings triggered after time t by a given main shock of magnitude m that occurred at time 0: M ( t, n, m ) := max { m j ( t ) } , (43)where { m j ( t ) } is the set of the magnitudes of all future offsprings triggeredafter time t .Analogous to relation (41), we havePr { M ( t, n, m ) < ν | R ( t ) > } = P ( t, n, m, ν ) − P ( t, n, m ) P ( t, n, m ) , (44)where R ( t ) = R ν =0 ( t ) is the total number of future offsprings, P ( t, n, m ) :=Pr { R ( t ) = 0 } = Ω( z = 0; t, n, m ) and P ( t, n, m ) defined by (38) is the prob-ability that there is at least one triggered future event.Let ψ ( ν ; t, n, m ) be the pdf of the maximal magnitude over all futureoffsprings. It follows from relation (44) that ψ ( ν ; t, n, m ) = 1 P ( t, n, m ) · ∂ P ( t, n, m, ν ) ∂ν . (45)18 .6.4 Properties of the second largest offspring Equalities (44) and (45) are simple consequences of the theory of orderstatistics applied to the offsprings magnitudes. Here, we provide some addi-tional order statistics relations that are relevant to the understanding of the ν -cluster’s statistics. These relations derive by using elementary facts of thetheory of order statistics (see for instance Ref. [2]).Let M ( t, n, m ) be the second largest magnitude in the set of future off-springs. By definition, it is smaller than the largest magnitude M ( M < M ),but is larger than the magnitudes of all other future offsprings. Then, thepdf ψ ( ν ; t, n, m ) of the second largest magnitude M ( t, n, m ) is given by ψ ( ν ; t, n, m ) = 12 · dp ( ν ) dν · Q ( t, n, m, ν ) P ( t, n, m ) , (46)where Q ( t, n, m, ν ) = ∂ Ω( z ; t, n, m ) ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =1 − p ( ν ) , (47)with the GPF Ω( z ; t, n, m ) of the total number of future offsprings numbergiven by expression (31). In addition, we have defined P ( t, n, m ) := Pr { R ( t ) > } = 1 − P ( t, n, m ) (48)as the probability that the number R ( t ) of all future offsprings be larger thanone. One can then show that P ( t, n, m ) = P ( t, n, m ) + ∂ Ω( z ; t, n, m ) ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 . (49)There is another remarkable fact that derives from the structure of theGR law (1) and the ETAS model, which can be called the extremal GR law .The random difference δM = M ( t, n, m ) − M ( t, n, m ) (50)between the largest and second-largest magnitudes is distributed accordingto the same regular GR law (1), for any n and m . H ( z ; t, n ) defining thestatistics of the number of future offsprings In this section, we provide an exact expression for the function H ( z ; t, n )that obeys the nonlinear integral equation (32), and which determines the19PF Ω( z ; t, n, m ) (31) of the number of future offsprings, in the case of theexponential pdf f ( t ) (23). Using the insights obtained from this exact solu-tion together with the properties of the function H ( z ; t, n ), we then formulatea conjecture for its general structure for an arbitrary pdf f ( t ). H ( z ; τ, n ) Let us first consider the case where the pdf of waiting times is the expo-nential function f ( τ ) (23). In this case, the nonlinear integral equation (32)reduces to an initial value problem for the ordinary differential equation: dH ( z ; t, n ) dt + (1 − n ) H ( z ; t, n ) + ρH γ ( z ; t, n ) = 0 ,H ( z ; 0 , n ) = 1 − z. (51)Its solution is H ( z ; t, n ) = D ( z ; φ ) · H ( z ; φ, ρ, n ) , (52)where D ( z ; φ ) := (1 − z ) · φ, H ( z ; φ, ρ, n ) := (cid:18) ρ − n · (cid:0) − φ γ − (cid:1) · (1 − z ) γ − (cid:19) − γ , (53)and φ := Φ (cid:0) t (cid:14) t (cid:1) = e ( n − t , t := 1 (cid:14) (1 − n ) . (54)Two important properties of the function H ( z ; t, n ) can be derived fromthis solution (52). Property 1
The function H ( z ; t, n ) (52) is the product of two factors, whichhave a clear physical meaning. The function D ( z ; φ ) corresponds to the one-daughter approximation , where each offspring triggers not more than onefirst-generation aftershock. In contrast, the function H ( z ; φ, ρ, n ) takes intoaccount that each offspring can trigger more than one first-generation after-shock. Mathematically, this is responsible for the factor ρ in the power lawasymptotics (11) of the probability of first-generation aftershock numbers. If ρ = 0, i.e. if each offspring triggers no more than one first-generation after-shock, then the second factor in the right-hand side of (52) becomes H ≡ H ( z ; t, n ) reduces to D ( z ; φ ). Property 2
Both functions D and H in expression (52) depend on thecurrent time t only via the function φ (54).20 .2 Conjecture for the structure of function H ( z ; t, n ) Based on the physical meaning of the decomposition (52) of the function H ( z ; t, n ) (52), we propose the following overall structure of the function H ( z ; t, n ) for arbitrary waiting time distributions f ( t ). Conjecture 1
We suggest that the structure of the function H ( z ; t, n ) takesthe form of a product of D and H as given by expression (52), independentlyof the waiting time distribution f ( t ), where D ( z ; φ ) corresponds to the one-daughter approximation and H takes into account that each offspring cantrigger more than one first-generation aftershock. Thus, in order to get thegeneral form of the function H ( z ; t, n ), one just needs to find the generalisa-tion of the time-dependent function φ = Φ( t, n ), contained in the functions D ( z ; φ ) and H . In practice, φ can be determined from the calculation of D .From a mathematical point of view, the one-daughter approximation cor-responds to neglecting the parameter ρ in the nonlinear integral equation(32), which amounts to linearise it. The function φ = Φ( t, n ) can be ob-tained from H ( z ; t, n ) − n · H ( z ; t, n ) ⊗ f ( t ) = (1 − z ) · Φ( t ) , (55)which derives by linearising the nonlinear integral equation (32).To solve (55), we apply the Laplace transform with respect to t to equation(55) term by term. The corresponding algebraic equation for the Laplaceimage ˆ H ( z ; u, n ) := Z ∞ H ( z ; t, n ) e − ut dt (56)reads ˆ H ( z ; u, n ) − n ˆ H ( z ; u, n ) ˆ f ( u ) = (1 − z ) ˆΦ( u ) , (57)where ˆ f ( u ) is the Laplace image of the pdf f ( t ), and ˆΦ( u ) is the Laplaceimage of the corresponding survival function Φ( t ) (20). Using relation (20)and the standard properties of the Laplace transform, we obtainˆΦ( u ) = 1 u (cid:16) − ˆ f ( u ) (cid:17) . (58)Using (57) and (58) yieldsˆ H ( z ; u, n ) = (1 − z ) · ˆΦ( u, n ) , ˆΦ( u, n ) := 1 u − ˆ f ( u )1 − n ˆ f ( u ) . (59)21ccordingly, the solution of equation (55) is described by the first equation of(53), where now φ = Φ( t, n ), where Φ( t, n ) is the inverse Laplace transformof the function ˆΦ( u, n ) (59):Φ( t ; n ) = 12 πi Z i ∞− i ∞ e ut − ˆ f ( u )1 − n ˆ f ( u ) du. (60)According to our conjecture, the correct function H ( z ; t, n ) for an arbitrarypdf f ( t ) is obtained by substituting the function φ = Φ( t, n ) (60) into theexpressions (52), (53). Remark 4
We will show below that, in the subcritical case n . . ν & ν -clusters is rather well described by the one-daughter approximation,i.e. by replacing the exact equality (52) by the approximation H ( z ; t, n ) ≃ D ( z ; φ ) = (1 − z ) · Φ( t, n ) . (61)This may be interpreted by the fact that the one-daughter approximationgives accurate expressions for the ν -cluster’s statistics, not only for the ex-ponential pdf f ( t ), but also for arbitrary pdf f ( t ). In the case where f ( t ) is the fractional exponential distribution f θ ( t ) (17),we substitute its Laplace image ˆ f θ ( u ) (15) in expression (59) to obtainˆΦ θ ( u, n ) = u θ − − n + u θ . (62)Obviously, the inverse Laplace transform of ˆΦ θ ( u, n ) is equal to the fractionalexponential survival function itself up to a rescaling of time: φ = Φ θ ( t, n ) := Φ θ (cid:0) t (cid:14) t θ (cid:1) , t θ = (1 − n ) − /θ , (63)where Φ θ ( t ) (21) is the fractional exponential survival function.Using conjecture 1 and relations (63), (52), the function H θ ( z ; t, n ) in thecase where f ( t ) is the fractional exponential distribution (17) is obtained byreplacing φ by Φ θ ( t, n ) in the right-hand side of expressions (52) and (53): H θ ( z ; t, n ) = (1 − z ) · Φ θ ( t, n ) · H [ z ; Φ θ ( t, n ) , ρ, n ] . (64)In the particular case θ = 1, where the fractional exponential survival func-tion Φ θ ( t ) (21) reduces to the exponential one Φ ( t ) = e − t , expression (64)becomes the exact solution of the initial value problem (51).22 Statistical properties of ν -clusters In section 2, we have derived the general relations (42), (45) and (46)describing the statistics of the duration of ν -clusters and of the magnitudes ofthe largest events in the ν -cluster. In section 3, we have obtained the explicitformulas needed to calculate the statistical characteristics of ν -clusters, andtheir dependence on the branching ratio n , magnitude m of the main shockand observation magnitude threshold ν . In the present section, we exploitthese formulas to present detailed results on the statistical properties of ν -clusters. We perform our analysis for the fractional exponential case, forwhich the waiting time distribution of triggered aftershocks is described bythe fractional exponential pdf f θ ( t ) (17), which is asymptotically equivalent(for large t ≫
1) to the modified Omori-Utsu law (3). η ( t, n, ζ ) Consider expression (35) for the probability P ( t, n, m, ν ) of absence offuture offsprings (i.e. the probability that the future ν -cluster is empty).According to the relations (31), (52), (53), we can write P ( t, n, m, ν ) as P ( t, n, m, ν ) = e − χ · φ · η ( φ,n,ζ ) , φ = Φ θ ( t, n ) , (65)where η ( φ, n, ζ ) := (cid:0) S · (cid:2) − φ γ − (cid:3)(cid:1) − γ , (66)and χ := κ · µ · ζ , S := ρ · ζ γ − − n , ρ = κ γ γ Γ( − γ ) ,κ = n ( γ − (cid:14) γ, ζ := 1 − (1 − p ) · G (1 − p ) . (67)It is useful to study the properties of the function η ( t, n, ζ ) (66), for oursubsequent analysis of the ν -clusters statistics. The dependence of η ( t, n, ζ )as a function of the argument φ changes qualitatively depending on whetherthe factor S (67) is small or large. Figure 6 shows the dependence of S asa function of the magnitude threshold ν . For large ν (most events cannotbe observed), S becomes smaller than 1, while for small ν (most events areobservable), S is of the order of 1 or larger. Large values of S are alsoobtained near criticality, i.e. for n → S (67) small ( S ≪ φ , the function η ( φ, n, ζ ), whichquantifies the contribution of the multiple generations of offsprings, is almostconstant: S ≪ ⇒ η ( φ, n, ζ ) ≃ ∀ φ ∈ (0 ,
1) (68)23 ν S Figure 6: Dependence of the factor S (67) as the function of the thresholdmagnitude ν , for ( b = 1 , α = 0 .
8) ( γ = 1 . n = 0 . , . , . , . , . , . , . S &
1, the dependence of η ( φ, n, ζ ) (66) as a functionof φ is strongly nonlinear. Figure 7 shows η ( φ, n, ζ ) as a function of φ forseveral values of n , and thus S . One can see that, for n . .
5, the linearapproximation (68) holds for any threshold magnitude ν .Let us study separately the critical case n = 1. Taking into accountrelations (67), (63) and the asymptotics (22) for the survival function Φ θ ( t ),one get: lim n → Φ θ ( t, n ) ≡ , lim n → S · (cid:2) − Φ γ − θ ( t, n ) (cid:3) = λ · t θ ,λ := ρ · ζ γ − · ( γ − θ ) . (69)As a result, we obtain η ( t, ν ) := lim n → η [Φ θ ( t, n ) , n, ζ ] = (cid:0) λ · t θ (cid:1) − γ . (70)In the critical case ( n = 1), the function η ( t, ζ ) has thus the power lawasymptotics η ( t, ζ ) ≃ λ − γ · t − ̺ , ̺ := θγ − , λ · t θ ≫ , (71)24 φ η Figure 7: Factor η ( φ, n, ζ ) (66) as a function of φ for γ = 1 .
25 and ν = 6.From top to the bottom: n = 0 . , . , . , . , .
9. The corresponding valuesof S (67) are: 0 .
14; 0 .
23; 0 .
39; 0 .
72; 1 . ̺ that renormalises the waiting time distribution kernelvia the exponent γ quantifying the relative importance of different magnituderanges in the generation of offsprings. ν -clusters We are now armed to obtain the statistical distribution of the durationsof future ν -clusters, whose contributing events have magnitudes larger thethreshold ν . After substituting equalities (65), (66) into relation (42), weobtain the following explicit expression for the pdf ϕ θ ( t ; n, m, ν ) of the dura-tions of ν -clusters: ϕ θ ( t ; n, m, ν ) = C · χ · (1 + S ) · f θ ( t, n ) · η γ ( φ, n, ζ ) · e − χ · φ · η ( φ,n,ζ ) , (72)where φ = Φ θ ( t, n ) , f θ ( t, n ) := d Φ θ ( t, n ) dt = 1 t θ f θ (cid:18) tt θ (cid:19) . (73)Three limiting cases of the pdf (72) of the ν -cluster durations are worthdiscussing. 25 −1 −8 −6 −4 −2 t ϕ ( t ; n , m , ν ) ∼ t − n = 0 . n = 0 . n = 0 . n = 0 . Figure 8: Log-log plots of the exact pdf ϕ ( t ; n, m, ν ) given by expression(72) of the durations of ν -clusters in the pure exponential case θ = 1 f ( t ) = e − t for m = 9, α = 0 . b = 1 ( γ = 1 .
25) and ν = 6. The differentcurves correspond to n = 0 .
99; 0 .
9; 0 .
7; 0 . ϕ ( t ; m, ν ) (76), obtained for the criticalregime n = 1. The dashed lines show the pdf ϕ ( t ; n, m, ν ) (74) in the one-daughter approximation, for the following values of the branching ratio: n =0 .
9; 0 .
7; 0 .
5. One can observe the validity of the one-daughter approximation.In the critical regime n = 1, expression (71) predicts a tail exponent ̺ := θγ − = 4 for θ = 1 and γ = 1 .
25, which explain the asymtotics 1 /t ̺ = 1 /t shown in the figure. 26. One-daughter limit
S → and η →
1: then, expression (72) reducesto ϕ θ ( t ; n, m, ν ) = C · χ · f θ ( t, n ) · e − χ · Φ θ ( t,n ) . (74)2. Large time limit t → ∞ for which φ →
0: then, the asymptotics ofthe pdf ϕ θ (72) is defined by the asymptotics (19) of the original pdf f θ ( t ) (17). Remembering that, for θ = 1, the pdf f ( t ) (23) reduces tothe pure exponential, we have ϕ θ ( t ; n, m, ν ) ≃ C · (1 + S ) − γ · (1 − n ) ×× − t − θ − Γ( − θ ) , θ ∈ (0 , ,e ( n − t , θ = 1 , t ≫ (1 − n ) − /θ . (75)3. Critical case n = 1: In this case, using the second limit in (69), weobtain the following expression for the pdf of the ν -cluster’s durations: ϕ θ ( t ; m, ν ) = C · χ · θ ( γ − · t · λ · t θ (1 + λ · t θ ) γγ − · e − χ · ( λ · t θ ) − γ , (76)which has the following power law asymptotics ϕ θ ( t ; m, ν ) ≃ ξ · t − ̺ − , ξ := C · χ · ̺ · λ − γ , t → ∞ , (77)where the exponent ̺ is defined in (71).The following figures illustrate how the pdf ϕ θ ( t ; n, m, ν ) (72) changes itsshape upon variations of the main shock magnitude m , magnitude threshold ν , branching ratio n . The figures also provide a check on the validity of theabove asymptotic relations (74)–(77).Figure 8 shows ϕ ( t ; n, m, ν ) as a function of duration t for a great mainshock ( m = 9), a significant magnitude threshold( ν = 6) for the pure expo-nential case ( θ = 1) and several values of the branching ratio.Figure 9 shows ϕ ( t ; n, m, ν ) as a function of duration t in the pure ex-ponential case θ = 1 f ( t ) = e − t for n = 0 . ν . Note that, for all n < ϕ ( t ; n, m, ν ) tends to zero exponen-tially fast at large time t ≫
1. One can observe that ν substantially influencesthe shape of ϕ ( t ; n, m, ν ) only at small times t .
1. For large enough ν ’s, theexact pdf of the durations of the ν -cluster’s approaches closely at all times t the corresponding one-daughter limit pdf (74).27 −6 −4 −2 t ϕ ( t ; n , m , ν ) ∼ t − ∼ e − . · t ν = 5 ν = 3 ν = 0 ν = 7 Figure 9: Solid lines: plots of the pdf ϕ ( t ; n, m, ν ) of the durations of ν -clusters in the pure exponential case θ = 1, for m = 9, α = 0 . b = 1 ( γ =1 . n = 0 . ν = 0; 3; 5; 7. The dotted line correspondsto the limiting pdf ϕ ( t ; m, ν ) (76), obtained for the critical regime n = 1and ν = 0. The dashed lines are the one-daughter approximations for thepdf of the durations of the ν -clusters, for θ = 1, n = 0 . ν = 5; 7.28 −15 −10 −5 t ϕ θ ( t ; n , m , ν ) ν = 5 ν = 3 ∼ t − θ − ∼ t − θ − ν = 7 ν = 0 Figure 10: Solid lines: plots of the pdf ϕ θ ( t ; n, m, ν ) of the durations of ν -clusters in the fractional exponential case θ = 0 .
3, for m = 9, α = 0 . b = 1( γ = 1 . n = 0 . ν = 0; 3; 5; 7. The dotted line is the pdf ϕ θ ( t ; n, m, ν ) of the durations of ν -clusters in the critical case n = 1 andfor ν = 0. The dashed lines show the pdf ϕ θ ( t ; n, m, ν ) of the durations of ν -clusters in the one-daughter limit for θ = 0 . n = 0 . ν = 5; 7.29igure 10 shows the pdf ϕ θ ( t ; n, m, ν ) of the durations of ν -clusters in thefractional exponential case θ = 0 .
3. One can observe that the key propertiesand shapes of ϕ θ ( t ; n, m, ν ) differ dramatically from those of ϕ ( t ; n, m, ν ).In the later exponential case θ = 1, and for n < ν > ϕ tends tozero exponentially fast for t ≫
1. It is only in the critical case n = 1, that ϕ ( t ; n, m, ν ) develops a power law tail, albeit with a rather large exponent ̺ (71). In the fractional exponential case ( θ < ϕ θ is significantly slower than in the critical case n = 1 at large t . For n < ν > ϕ θ tends to zero, at t ≫ ϕ θ ∼ t − θ − (75), which decays to zero muchmore slowly than the dependence in the critical regime given by ϕ θ ∼ t − ̺ − (where ̺ > θ is given by (71)). Remark 5
The pdf ϕ θ ( t ; n, m, ν ) of the durations of ν -clusters for the frac-tional exponential and the pdf ϕ ( t ; n, m, ν ) for the exponential case, bothdetermined from expression (72), share one important property. In the sub-critical case (for n . .
9) and for sufficiently large magnitude thresholds ν (for figures 8, 9 and 10, for ν & ϕ θ ( t ; n, m, ν ) and ϕ ( t ; n, m, ν ) (72) are both very well approximated by their correspondingone-daughter limit (74). Since the solution of the integral equation (32)in the one-daughter limit ( ρ = 0) is exact, we conjecture that this providesthe almost exact expression for the pdf of the durations of ν -clusters for all θ ∈ (0 ,
1] in these regimes n . . ν & ν -clusters In this section, we study in detail the statistics of the maximal magnitudeof future offsprings of a main shock of fixed magnitude m that occurred attime 0. The pdf ψ ( ν ; t, n, m ) of the maximal magnitude ν is given by ex-pression (45). As a result of the equalities (65)-(67), ψ ( ν ; t, n, m ) depends ontime t only through the function Φ θ ( t, n ). For definiteness, we take the func-tion Φ( t, n ) to be the fractional exponential, which includes as a special casethe exponential function: θ = 1 Φ( t, n ) = e ( n − t . Thus, for convenience,we rename ψ as the function of the argument φ : ψ = ψ ( ν ; φ, n, m ) and willdiscuss its time dependence via the auxiliary argument φ . If one wish torecover the explicit time dependence of the pdf of the maximum magnitudeof future offsprings, one just has to solve the equation φ = Φ θ ( t, n ) for thetime t . In the pure exponential case θ = 1, this correspondence has a simple It is equal to the inverse Laplace transform, with respect to argument u , of the ex-pression (59) t = ln( φ ) (cid:14) ( n − φ ∈ [0 ,
1] ontothe time axis t ∈ (0 , ∞ ).Using relations (45), (65)-(67), we obtain the explicit expression ψ ( ν ; φ, n, m ) = κµφp ′ ( ν ) e − κµφη ( φ,n ) − · ζ ′ [ p ( ν )] · η γ ( φ, n, ζ ) · e − κµ · ζ · φ · η ( φ,n,ζ ) , (78)where η ( φ, n ) := η [ φ, n,
1) = (cid:20) ρ − n (cid:0) − φ γ − (cid:1)(cid:21) − γ ,ζ ′ ( p ) := dζdp , p ′ ( ν ) := dp ( ν ) dν , p = 10 − bν , (79)and ζ = ζ ( p ) is defined in (67).Below, we will compare the exact expression (78) for ψ ( ν ; φ, n, m ) withits one-daughter approximation ψ ( ν ; φ, n, m ) = κµφe − κµφ − · ζ ′ [ p ( ν )] · p ′ ( ν ) · e − κµ · ζ · φ · , (80)where ζ and ζ ′ are given by the following relations: ζ ( p ) = p − n (1 − p ) , ζ ′ ( p ) = 1 − n (1 − n (1 − p )) . (81)Figure 11 shows the pdf ψ ( ν ; φ, n, m ) (78) of the maximal magnitude offuture offspring for different values of the effective time φ . As time increases,the pdf shifts to the left, indicating a decrease of the typical magnitude ofthe largest future offspring.Figure 12 compares the exact expression (78) with its one-daughter ap-proximation (80) of the pdf ψ ( ν ; n, m ) := ψ ( ν ; φ = 1 , n, m ) at the effective“time” φ = 1 as a function of the maximal magnitude ν among all offspringstriggered by the mainshock for different values of the branching ratio n . Onecan observe that the one-daughter expression (80) provides a good approx-imation to the exact expression (78) , the better the approximation, thesmaller the branching ratio.Let M ( t, n, m ) be the maximal magnitude of all future offsprings triggeredby a main shock of magnitude m that occurred at time 0, as defined by (43).Its mean value at the current time t = t ( φ ) reads M ( φ, n, m ) := Z ∞ νψ ( ν ; φ, n, m ) dν . (82)At the specific time t = t ( φ = 1), this reduces to M ( n, m ) := M ( φ = 1 , n, m ) = Z ∞ νψ ( ν ; n, m ) dν . (83)31 ν ψ ( ν ; φ , n , m ) Figure 11: Plots of the pdf ψ ( ν ; φ, n, m ) (78) of the maximal magnitude offuture offsprings for m = 8, γ = 1 .
25 and for n = 0 .
9. From left to right,the effective time is φ = 0 . , . , . , . ,
1, which corresponds to a real timeincreasing from right to left.We use this expression to construct figure 13, which shows the differencebetween the main shock magnitude m and the average magnitude M ( n, m )of its largest offspring: ∆( n, m ) := m − M ( n, m ) (84)for different values of the branching ratio n . One can observe that ∆( n, m )monotonically increases with the main shock magnitude m . The dependenceas a function of m and n is sufficiently slow and smooth that the so-calledB˚ath law, represented in figure 13 by the dashed straight line, if not correct,provides a rough estimate of ∆( n, m ) [12, 20].Another property of interest is that the shapes of the pdf’s ψ (78) arealmost identical for a wide range of main shock magnitude m and values ofthe branching ratio n . By centering the pdf’s according to˜ ψ ( δ ; n, m ) := ψ (cid:0) δ + M ; n, m (cid:1) (85)where the distance from the mean is δ = M ( n, m ) − M ( n, m ) , (86)32 .5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 900.10.20.30.40.50.60.70.8 ν ψ ( ν ; , n , m ) Figure 12: Plots of the pdf ψ ( ν ; φ = 1 , n, m ) of the maximal magnitudeover all offsprings triggered by a main shock of magnitude m = 8, with α = 0 . b = 1 ( γ = 1 . n = 0 . , . , . . . . . , .
1. 33 m ∆ ( n , m ) Figure 13: Plot of the difference ∆( n, m ) (84) between the main shock mag-nitude m and the average magnitude M ( n, m ) of its largest offspring as afunction of the mainshock magnitude m , for α = 0 . b = 1 ( γ = 1 . n = 0 . , . . . , . , . , .
9. The dashed straight linecorresponds to the B˚ath law: ∆( n, m ) ≃ . δ ˜ ψ ( δ ; n , m ) Figure 14: Plots of the pdf ˜ ψ ( δ ; n, m ) (85) of the distance from the mean de-fined by expression (86) of the maximum magnitude of the future offprings ofa main shock of magnitude m = 8, for γ = 1 .
25 and n = 0 . , . , . , . , . n .In figures 12 and 14, one can observe that the pdf’s exhibit rather sharpdecay to the left, so that one can define a low quantile maximum magnitude M q=0.05 ( t, n, m ) of the maximum magnitude of the offsprings in a ν -cluster,such that Pr { M ( t, n, m ) > M q=0.05 ( t, n, m ) } = 0 . , (87)where M ( t, n, m ) is defined by equality (43). This definition (87) means that,typically in only one in twenty clusters, the maximum magnitude among alloffsprings is smaller than M q=0.05 ( t, n, m ). Figure 15 shows the dependenceof M q=0.05 ( t, n, m ) as a function of the effective increasing time 1 − φ (since φ is a decreasing function of time). M q=0.05 ( t, n, m ) is a decreasing functionof time, as the triggering activity decays progressively. Using the standard ETAS model of triggered seismicity, we have presenteda rigorous theoretical analysis of the main statistical properties of temporal35 − φ M q = . ( t , n , m ) Figure 15: Plots of the low quantile maximum magnitude M q=0.05 ( t, n, m )as a function of 1 − φ , for m = 8 and γ = 1 .
25. From top to bottom, n =0 . , . , . , . , . , .
4. The curves shows that M q=0.05 ( t, n, m ) decreaseswith increasing of time (that is, with decreasing φ ).clusters, defined as the group of events triggered by a given main shock offixed magnitude m that occurred at the origin of time, at times larger thansome present time t . The most general and powerful tool to derive analyti-cally rigorously the statistical properties of the numbers of events triggeredby some main shock as a function of time is the technology of generatingprobability function (GPF), of which we have recalled the main propertiesand that we have applied to our problem. We have derived the explicit andapproximate expressions for the GPF of the number of future offsprings ina given temporal seismic cluster, defining, in particular, the statistics of thecluster’s duration and the cluster’s offsprings maximal magnitudes. Our mainresults have been presented in the form of four propositions, whose proofshave been given in appendices.We introduced the probability P ( t, n, m, ν ) that the future cluster ofevents of magnitudes above some detection threshold ν is empty. Thisprobability becomes the workhorse for the derivation of our main results. P ( t, n, m, ν ) can also be interpreted in its time dependence as the probabil-ity that the total duration of the cluster of triggered events is less than t .A third interesting interpretation relates the derivative of P ( t, n, m, ν ) withrespect to ν to the probability density function of the maximal magnitude36ver all events within the temporal cluster. We used P ( t, n, m, ν ) to derivethe remarkable result that the magnitude difference between the largest andsecond largest event in the future temporal cluster is distributed accordingto the regular Gutenberg-Richer law that controls the unconditional distri-bution of earthquake magnitudes.The distribution ϕ θ ( t ; n, m, ν ) of the durations of temporal clusters ofevents of magnitudes above some detection threshold ν was obtained in exactanalytical form, and investigated in three limits: (i) the one-daughter limitfor n < n is equal to its critical value 1. For earthquakes obeyingthe Omori-Utsu law for the distribution of waiting times between triggeringand triggered events, we show that ϕ θ ( t ; n, m, ν ) has a power law tail that isfatter in the non-critical regime n < n = 1. Thisparadoxical behavior is similar to the one explained in Ref. [27], and resultsfrom the fact that generations of all orders cascade very fast in the criticalregime and accelerate the temporal decay of the cluster dynamics. We alsoderive the detailed shape of the distribution of the maximal magnitude overall events in the future cluster triggered by some main shock. We show thatthe so-called B˚ath law, stating that the difference between the main shockmagnitude and the average magnitude of its largest offspring is equal to 1 . Acknowledgements : The second author dedicates this article to thefirst author, who was his cherished friend and long-time collaborator. AS wasextraordinary young in his mind, exceptionally creative and with a freshnessand enthousiasm for research rarely found even in beginning scientists. Thisarticle was almost finalised before the time when AS left us and DS vowedto bring it to completion and have it published to honor his memory. DSregrets that many other commitments has delayed this important endeavor.
A Proofs of propositions 1-4
A.1 Proof of Proposition 1
Let { T k } be the occurrence times of the triggered tdaughters, while R d ( m )is the total number of daughters. Let us introduce the number R d ([ t , t ]; m )37f daughters triggered within a time interval[ t , t ] , ∀ ( t , t ) : 0 < t < t < ∞ . One may represent R d ([ t , t ]; m ) in the form: R d ([ t , t ]; m ) = R d ( m ) X k =1 Π ( T k , [ t , t ]) , (88)where Π ( t, [ t , t ]) := ( , t ∈ [ t , t ] , , t / ∈ [ t , t ] , is the indicator of the time interval [ t , t ].Consider the GPF of the random number R d ([ t , t ]; m ) G d ( z ; [ t , t ]) := E (cid:2) z R d ([ t ,t ]; m ) (cid:3) . (89)Taking into account the identity z Π( t, [ t ,t ]) ≡ z − t, [ t , t ]) (90)and keeping in mind that { T k } are iid random variables, we can rewrite theGPF (89) in the form: G d ( z ; [ t , t ]) = E h (1 + ( z − E [Π]) R d ( m ) i . (91)We have used here the short notation Π = Π ( T, [ t , t ]). The outer expecta-tion E [ · · · ] at (91) represents the statistical average over the total number R d ( m ) of daughters total number. The inner expectation E [Π] correspondsto averaging over the random instant T , distributed according to the pdf f ( t ). The inner expectation is equal to E [Π] = E [Π ( T, [ t , t ])] = Z t t f ( t ) dt . (92)Using this last relation and the Poissonian statistics (4) of the random num-ber R d ( m ), the equality (91) transforms into the Poissonian GPF: G d ( z ; [ t , t ]) = exp (cid:18) κµ · ( z − Z t t f ( t ) dt (cid:19) . (93)In particular, if t < t , t > t , then one may rewrite (93) as G d ( z ; [ t , t ]) = G d ( z ; [ t , t ]) · G d ( z ; [ t, t ]) , (94)which is equivalent to the proposition.38 .2 Proof of Proposition 2 Before deriving equations (26), it is useful to recall some properties ofthe total number of aftershocks that are triggered by some shock, in theframework of the theory of (unmarked) branching processes. Each eventtriggers daughters (its first generation aftershocks), whose total number R d is described statistically by the following GPF, G d ( z ) := E (cid:2) z R d (cid:3) = ∞ X r =0 q d ( r ) · z r , (95)where the { q d ( r ) } are the probabilities that the number R d of first-generationaftershocks is equal to a given integer r . In the framework of branching pro-cesses, all daughters trigger, independently of each other, their own daugh-ters, whose numbers are iid random integers, possessing the same GPF G d ( z ),and so on.Let G k ( z ) be the GPF of the number R k of the aftershocks of the first k generations. Given the iid property of all numbers of any aftershock’sdaughters, we have G k +1 ( z ) = G d [ zG k ( z )] , k = 1 , , . . . (96)The product zG k ( z ) means that each daughter of the initial event triggers in-dependently aftershocks belonging to the first k generations, whose numbersare described by the same GPF G k ( z ).A well-known result in the theory of branching processes states that, for n ∈ (0 ,
1] where n is the branching ratio defined as the average number ofdaughters of first-generation per mother, n := E [ R d ] = dG d ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) z =1 , (97)then the following limit existslim k →∞ G k ( z ) = G ( z ) , (98)where G ( z ) is the GPF of the total number of aftershocks of all generationsthat are triggered by the initial shock. Using the recurrent relation (96) andthe limit (98), G ( z ) is solution of the transcendent equation: G ( z ) = G d [ zG ( z )] . (99)We can now derive the equation analogous to (99), which determines theGPF G ( z ; t ) of the number R ( t ) of future aftershocks of all generations. By39uture, we recall that this refers to aftershocks that occur after the currenttime t , where the origin of time is the time of occurrence of the main initialshock. Recall that the time intervals between a given event and any of itsdirectly triggered daughter are iid random variables with the same pdf f ( t ).We start with the derivation, similar to (96), of the recurrent equation forthe GPF G k ( z ; t ) of the number R k ( t ) of triggered aftershocks of the first k generations. Let us discuss first the simplest case, where the shock has onlyone daughter (that is, G d ( z ) = z ), which is triggered at the random time T . Consider the conditional GPF G k +1 ( z ; t | T ) under the condition that T isequal to the some given value. In this case, the following relation holds G k +1 ( z ; t | T ) = ( zG k ( z ) , T > t,G k ( z ; t − T ) , T < t. (100)Similarly to the right-hand side of equality (96), the first line means that theGPF of the number of aftershocks of the first k + 1 generations, includingthe shock’s daughter and its aftershocks of the first k generations, is equal to zG k ( z ), This is because, if T > t , then the daughter and all its aftershocksare in the future (i.e. after t ). In contrast, the second line of (100) meansthat, if T < t , then the shock’s daughter is not a future offspring, while weshould only consider the future aftershocks of the daughter.Let rewrite relation (100) in the more convenient form for future analyticalcalculations: G k +1 ( z ; t | T ) = zG d ( z ) · ( T − t ) + G k ( z ; t − T ) · ( t − T ) . (101)Averaging both sides of this equality with respect to the statistics of therandom time T with pdf f ( t ), we obtain G k +1 ( z ; t ) = zG k ( z )Φ( t ) + Z t f ( τ ) G k ( z ; t − τ ) dτ . (102)Let us now get the sought recurrent equation in the general case where theGPF G d ( z ) of the number of daughters is arbitrary. Since the time durations { T k } between any shock and its first-generation daughters are iid variables,in order to obtain the recurrent equation, one needs to replace in (96) theGPF G k +1 ( z ) by G k +1 ( z ; t ), and zG k ( z ) by the right-hand side of the equality(102), that is to say G k +1 ( z ) = G d [ zG k ( z ) · Φ( t ) + f ( t ) ⊗ G k ( z ; t )] , k = 1 , , . . . (103)For n ∈ (0 , k →∞ G k ( z ; t ) = G ( z ; t ) . (104)40n this limit, we obtain from (103) the sought equation for the GPF G ( z ; t )of the number of future aftershocks of all generations: G ( z ; t ) = G d [ zG ( z )Φ( t ) + f ( t ) ⊗ G ( z ; t )] . (105)It is easy to check that this equation (105) is equivalent to (26). A.3 Proof of proposition 3
After substitution relations (25) into the right-hand side of equality (24),we obtain expression (31) for the GPF Ω( z ; t, n, m ) of the number of fu-ture aftershocks of all generations. In turn, taking into account the secondequation in (26) and equality (30), we obtain equation (32). A.4 Proof of proposition 4
By definition, the GPF Ω( z ; t, n ) of the number of future aftershocks ofall generations is equal toΩ( z ; t, n, m ) := E (cid:2) z R ( t ) (cid:3) = ∞ X r =0 q ( r ; t, n, m ) · z r , (106)where R ( t ) is the random number of the future aftershocks of all generations,while { q ( r ; t, n, m ) } are the probabilities that the random number R ( t ) isequal to the given integer r .Let R ν ( t ) be the random number of future offsprings whose magnitudesexceed the threshold ν , R ( t ; ν ) = R ( t ) X j =1 ( m j − ν ) , (107)where { m j } are the magnitudes of the future offsprings.Using the law of total probability, we can represent the GPF of the ran-dom number R ν ( t ) in analogy with equality (106) under the form:Ω( z ; t, n, m, ν ) := E (cid:2) z R ( t ; ν ) (cid:3) = ∞ X r =0 q ( r ; t, n, m ) · E (cid:2) z R ν ( t ) | r (cid:3) , (108)where E [ · · · | r ] is the conditioned expectation, under the condition that thenumber of future offsprings is equal to the given integer r : R ( t ) = r .41aking into account that, in the framework of the ETAS model, all off-springs have iid random magnitudes that are statistically independent of thenumber R ( t ) of future aftershocks, we obtain E (cid:2) z R ν ( t ) | r (cid:3) = Λ r ( z, ν ) , Λ( z, ν ) := E h z ( m ′ − ν ) i , (109)where m ′ is the random magnitude of some offspring distributed accordingto the GR law (1). Using the identity z ( m − ν ) ≡ z − · ( m − ν ) , (110)similar to (90), we obtainΛ( z, ν ) = 1 + ( z − · p ( ν ) ⇒ E (cid:2) z R ν ( t ) | r (cid:3) = [1 + ( z − · p ( ν )] r , (111)After substitution the last relation into (108), and after performed the sum-mation of the series, we obtain the sought relation (34).42 eferenceseferences