A probabilistic model of seismicity: Kamchatka earthquakes

ISSN 0742�0463, Journal of Volcanology and Seismology, 2010, Vol. 4, No. 6, pp. 412–422. © Pleiades Publishing, Ltd., 2010.Original Russian Text © V.V. Bogdanov, A.V. Pavlov, A.L. Polyukhova, 2010, published in Vulkanologiya i Seismologiya, 2010, No. 6, pp. 52–64.

412

INTRODUCTION

The most dangerous natural disasters, when theiruncertainties as to size, location, and time of occurrenceare added to their danger, are catastrophic earthquakes(second in the number of lives lost per year [Golitsynet al., 1999]). Mankind exerts itself investigating both theearthquake phenomenon itself and accompanying phe�nomena that might serve as precursors of an imminentdisaster. A great diversity of methods are used in earth�quake science; various models have been developed todescribe their physical, mechanical, thermodynamic, andother processes, as well as different earthquake predictiontechniques that have been successfully applied in practice[Dobrovol’skii, 1991; Keilis�Borok and Kosobokov,1986; Matvienko, 1998; Pisarenko et al., 1984; Fedotovet al., 1987]. There is, however, a property common to allof these, namely, that seismic occurrences are stochasticin nature.

The general (simplified) scheme that describes theoccurrence of seismic events can be thought of as a grad�ual continuous process of elastic stress (σ) buildup in theEarth’s interior. Conditions may arise at a randommoment of time t and in a random volume S giving rise to

exceedance of rock strength. A discrete stress drop thenoccurs. A random event is characterized by energy E andhypocenter coordinates S(ϕ, λ, h). Obviously, the distri�bution of continuous stresses over the coordinates andtime carries information on the probabilities of occur�rence of seismic events [Boldyrev, 2002; Riznichenko,1985]. In other words, knowledge of the space–time dis�tribution σ(ϕ, λ, h, t) provides, at least in principle, anidea of the probability distributions of multivariate seismicevents as functions of random continuous variablesξ= (ϕ, λ, h, k, t) (the coordinates, energy, or energy classk = logE, and time). However, the converse is also obvi�ous: the probability distribution carries information onthe stress distribution σ(ϕ, λ, h, t). Any stochastic processcan in turn be specified by a distribution, which is giveneither as a distribution function F(ϕ, λ, h, k, t) or as a den�sity f(ϕ, λ, h, k, t) that define the probabilities of randomevents P. Consequently, considered in formal terms,when the distribution in the five�dimensional space for aspecific region is known, the associated seismicity is com�pletely described. The following should be remembered.When performing statistical treatment and using distribu�tions of random continuous variables, it is more logical to

A Probabilistic Model of Seismicity: Kamchatka EarthquakesV. V. Bogdanov, A. V. Pavlov, and A. L. Polyukhova

Institute of Space Physics Research and Radio Wave Propagation, Far East Division, Russian Academy of Sciences, Kamchatskii Krai, Paratunka, 684034 Russia

e�mail: [email protected] December 15, 2008

Abstract—The catalog of Kamchatka earthquakes is represented as a probability space of three objects {Ω,

P}. Each earthquake is treated as an outcome ωi in the space of elementary events Ω whose cardinality forthe period under consideration is given by the number of events. In turn, ωi is characterized by a system ofrandom variables, viz., energy class ki, latitude ϕi, longitude λi, and depth hi. The time of an outcome has

been eliminated from this system in this study. The random variables make up subsets in the set and are

defined by multivariate distributions, either by the distribution function (ϕ, λ, h, k) or by the probabilitydensity f(ϕ, λ, h, k) based on the earthquake catalog in hand. The probabilities P are treated in the frequencyinterpretation. Taking the example of a recurrence relation (RR) written down in the form of a power law forprobability density f(k), where the initial value of the distribution function f(k0) is the basic data [Bogdanov,2006] rather than the seismic activity A0, we proceed to show that for different intervals of coordinates andtime the distribution felim(k) of an earthquake catalog with the aftershocks eliminated is identical to the dis�tribution ffull(k), which corresponds to the full catalog. It follows from our calculations that f0(k) takes onnearly identical numeral values for different initial values of energy class k0 (8 ≤ k0 ≤ 12) f(k0). The differencedecreases with an increasing number of events. We put forward the hypothesis that the values of f(k0) tend tocluster around the value 2/3 as the number of events increases. The Kolmogorov test is used to test the hypothesisthat statistical recurrence laws are consistent with the analytical form of the probabilistic RR based on a dis�tribution function with the initial value f(k0) = 2/3. We discuss statistical distributions of earthquake hypocentersover depth and the epicenters over various areas for several periods

DOI: 10.1134/S0742046310060059

F̃,

F̃

F̃

JOURNAL OF VOLCANOLOGY AND SEISMOLOGY Vol. 4 No. 6 2010

A PROBABILISTIC MODEL OF SEISMICITY: KAMCHATKA EARTHQUAKES 413

start by calculating statistical distribution series. The latterdescribe the frequency with which groups of events fall inspecified intervals of the random variables Δj. For this rea�son the simplest procedure is to find, in numerical form,statistical series of distribution densities f(Δϕ, Δλ, Δh, Δk,Δt) or distributions of probabilities P(Δϕ, Δλ, Δh, Δk, Δt),which can be represented as tables or the correspondinghistograms. These series characterize the possibilities ofrandom events falling in the respective intervals Δj of therandom variables (ϕ, λ, h, k, t). Starting from the fre�quency distributions one can construct a multivariate stepfunction F(Δϕ, Δλ, Δh, Δk, Δt) that specifies the descrip�tion of seismicity in space and time, with increasingly bet�

ter accuracy as becomes smaller and the number ofevents under consideration n greater. An earthquake cata�log for a region is the starting point for statistical calcula�tions. In dealing with a probability theoretic description,it is reasonable to represent the catalog as the probabilityspace of three mathematical objects, viz., Ω the space of

elementary events, the set of subsets of random events,and P the probability [Kolmogorov, 1974]. Each earth�quake is viewed as an outcome ωi in space Ω whose cardi�nality for the period of study is given by the number ofevents in the catalog. Each outcome ωi is in turn specifiedby the following system of random continuous variables:latitude ϕi, longitude λi, depth hi, energy class ki = logE,and time ti. One can construct subsets of random events in

by combining several random variables and fixing theothers in specified limits. We assume that the space andenergy characteristics of Kamchatka earthquakes for theperiod 1962–2009 provide an average representation ofKamchatka’s seismicity [Boldyrev, 2002]. The Guten�berg–Richter relation was treated in probabilistic terms[Bogdanov, 2006] to demonstrate the advantages andpromises offered by the stochastic seismicity model wepropose here.

GENERAL CONCEPTS

Calculations of the probabilities. We assume [Sobolev,1993] that a set of conditions Ф [Kolmogorov, 1974] existsin the Earth’s interior upon the background of externaldisturbances that controls the evolution of nonlinear pro�cesses in a rupture zone with subsequent passage to anunstable state and fracture [Bogdanov et al., 2006]. Anoutcome ωi is accompanied by the recording of five ran�dom variables: energy class k, three coordinates (latitudeϕ, longitude λ, and depth h), and time t. The presentstudy is only concerned with the spatial and energy distri�butions in a fixed time interval, hence the time of an eventas a random variable will be excluded from subsequentanalysis of our model. An elementary event ωi thendefines a vector ξi(ωi) given by four random continuousvariables ξ = (ϕi, λi, hi, ki). The distribution of events for a

F̃

F̃

F̃.

period Ti in a volume that is specified by the ranges ofcoordinates Δϕf, Δλf, and Δhf will be represented as a dis�tribution of n points corresponding to the endpoint of thevector ξi in four�dimensional space.

Random events can form an arbitrary combinationconsisting of a variable number of random quantities (theothers being fixed) and give rise to some subsets A, B, C,

etc. in . We give an example by defining subsets of ran�dom events in this set.

Event A: The occurrence of seismic events in differentintervals of energy class Δk in a specified volume V (theRR in the probabilistic representation).

Event B: The occurrence of earthquake hypocentersin fixed intervals of depth Δhf in a specified volume V.

Event C: The epicenters of seismic events that haveoccurred in a specified volume V fall in fixed intervals Δϕf,Δλf.

The volume V is given by the maximum and minimumvalues of the coordinates ϕ, λ, and depth h, which may bevaried. Defining random events in this manner, it is natu�ral to find the corresponding probability distributionfunctions based on the earthquake catalog in hand. Thelist of random events can be enlarged.

Consider, e.g., the random event С. Suppose the eventoccurred in mϕ,λ cases, that is, only mϕ,λ of the n events fallin the fixed intervals Δϕf and Δλf. In that case the “relativefrequency” ν = mϕ,λ/n characterizes the objective possi�bility of an event C given the set of conditions Φ. The fre�quency serves as a measure of the existing relationshipbetween the conditions in the Earth’s interior and theevent C; it shows how frequently these conditions produceour event. Practical application of probability theorymethods and statistics to various areas of human knowl�edge dealing with mass random events revealed a remark�able pattern related to frequency stability. As the numberof samples increases, the frequencies fluctuate aroundcertain numbers. It is therefore natural to relate thesenumbers to all individual events that result from randomexperiments [Venttsel’, 1962]. Below, when we say “therelative frequency of an event” we mean its mathematicalanalogue P, which characterizes the probability of theevent when the set of conditions Φ emerges. Conse�quently, in our case we set

(1)

The probabilities of the other events will be calculatedsimilarly.

For the set of seismic events that occurred during aperiod of time, the probability space is a set of three

objects {Ω, P}. Since the random variables ϕi, λi, hi

and ki are confined within the corresponding intervals

F̃

P C( ) mϕ λ, /n.=

F̃,

414


BOGDANOV et al.

between the maximum and the minimum values, the fol�lowing relation is true for the space of elementary events:

(2)

Relation (2) covers all admissible events in our idealexperiment. The maximum and minimum values of therandom variables that (2) involves are controlled by thegeometry of an actual seismic region under study, as wellas by its internal properties, which control the energy ofthe event. Depending on the problem at hand, one canvary the limits of the intervals Δj in (2), diminishing orincreasing their endpoints, modifying the total number ofevents in the set that form a complete group. In order to

pass from the idealized probability space {Ω, P} to anactual experiment, we apply a statistical treatment tothe earthquake catalog to define the boundary of thespace of elementary events Ω and the probabilitiesthemselves P. The statistical treatment is applied in thisstudy to the full catalog of Kamchatka earthquakes forthe period January 1, 1962 to December 31, 1999, to thesame catalog with aftershocks eliminated, and to thefull catalog for January 1, 1962 to June 30, 2009.

Distributions for a system of random variables. If thedistribution for a system of random variables is specified inanalytical form by its distribution function F(ϕ, λ, h, k, t)or the probability density f(ϕ, λ, h, k, t), then the system isfully specified from the probabilistic point of view. For�mulas are available from which to find the distributions ofindividual variables [Venttsel’, 1962; Kolmogorov, 1974].However, the case under consideration requires the con�verse formulation, viz., given the distributions of randomvariables, find the distribution of the system. To do this weneed to know the relationships that connect the individ�ual random variables that constitute the system, that is, weneed to know the conditional distributions.

The probability density for the continuous variablesthat define a seismic event can also be represented as ahigher derivative of F(ϕ, λ, h, k, t) and as a product of con�ditional and unconditional functions f(ϕ, λ, h, k, t) by thefollowing relation:

(3)

The notation in (3) is as follows: f(ϕ) is the probabilitydensity in the distribution of seismic events as a functionof ϕ; f(λ|ϕ) is the probability density in the distribution ofseismic events over λ, given that the latitudes are equal toϕ; f(h|ϕ, λ) is the probability density in the distribution ofseismic events over h, given that their latitudes and longi�tudes are equal to ϕ and λ, respectively; f(k|λ, ϕ, h) is theprobability density in the distribution of seismic eventsover k, given that their longitudes, latitudes, and depthsare equal to λ, ϕ, and h, respectively. Knowing theanalytical form of the density in (3), one can find theprobability of a seismic event falling in fixed intervals:

Ω ω : ϕmin ϕ ϕmax; λmin λ λmax;≤ ≤ ≤ ≤{=

hmin h hmax; kmin k kmax≤ ≤≤ ≤ }.

F̃,

f ϕ λ h k t, , , ,( ) ∂4F ϕ λ h k, , ,( )/∂ϕ∂λ∂h∂k=

= f ϕ( )f λ ϕ( )f h ϕ λ,( )f k λ ϕ h, ,( ).

latitude Δϕi = ϕi – ϕi – 1, longitude Δλj = λj – λj – 1,depth Δhm = hm – hm – 1, and energy class Δk = kn –kn – 1:

(4)

where i, j, m, and n are subscripts that denote therespective intervals of random variables. The secondequality in (4) is written according to the definition ofF(ϕ, λ, h, k). Expressions (3)–(4), being functions offour variables, cannot be represented graphically,although numerical values of P(Δϕi, Δλj, Δhm, Δkn) caneasily be found in the frequency representation from acatalog of seismic events. In a similar manner one canalso find the unconditional distributions for all ran�dom variables k, ϕ, λ, h, as well as various combina�tions for the conditional distributions for these vari�ables. A statistical treatment of the catalog using (4)can provide information on the mean probability of aseismic event in a fixed interval of geographic coordi�nates, depth, and energy class, as well as yielding numer�ical values of the step distribution function F(Δϕ, Δλ,Δh, Δk). As the number of events n increases and theinterval Δ decreases, the relative frequency νapproaches its mathematical analogue P and F(Δϕ,Δλ, Δh, Δk) tends to a stable continuous distributionF(ϕ, λ, h, k, t). Calculating F for a fixed time intervalT in a specified volume V, we average the sets of con�ditions that produce the occurrence of a randomevent; consequently, F provides a mean description ofseismicity behavior. In other words, the function F forthe period of interest determines, on average, thepotential of seismicity behavior in the volume of inter�est in terms of probability theory. With this approach,the earthquake catalog for the entire period of instru�mental observation Tinstr, when represented as a math�

ematical object of three elements {Ω, P} can betreated as the basic model; upon this background onecan detect changes in seismicity, that is, can recordvariations in the probability distribution for local areasof the seismic region under study due to changes in theactivity of some or other volume and for various timeintervals Т(Т < Тinstr). Consequently, prediction whenstated within the framework of this approach, imply�ing an answer to the questions “where” and with“what energy” a seismic event ωi is expected to occur,is the prediction of the probability of its occurrence Pin the space of elementary events Ω.

P Δϕi Δλj Δhm Δkn, , ,( ) dϕ dλ dh

h1

h2

∫λ1

λ2

∫ϕ1

ϕ2

∫=

× f k ϕ λ h, , ,( ) kd

k1

k2

∫ F ϕi λj hm kn, , ,( )=

– F ϕi 1– λj 1– hm 1– kn 1–, , ,( ) P Δϕi( )P Δλj Δϕi( )=

× P Δhm Δλj Δϕi,( )P Δkn Δhm Δλj Δϕi, ,( ),

F̃,



EXAMPLES: CALCULATING PROBABILITY DISTRIBUTIONS FOR RANDOM

SEISMIC EVENTS

Stakhovskii [2002] uses the term “seismic field” or“seismic measure” for the bivariate probability distribu�tion that characterizes the occurrence of epicenters in aspecified box of a grid. In this study we have considerablyexpanded the class of random seismic events that form

subsets of with probability measures P. In our case any

random event E(E ∈ ) that has the measure PE charac�terizes seismicity in terms of chance.

The earthquake data for the Kamchatka region werekindly supplied by the Kamchatka Branch of the Geo�physical Service of the Russian Academy of Sciences,Petropavlovsk�Kamchatskii (abbr. KB GS RAS). Thesedata include: (1) a complete catalog of seismic events forthe period 1962–1999 (period T1) with epicenters situ�ated in the area delimited by the coordinates Δϕ = 50°–59°N and Δλ = 154°–168°E (area S1); the maximumdepth of the events is H = 599 km and (2) an earthquakecatalog for period T1 with epicenters in area S1 with after�shocks eliminated (using an algorithm created byG.M. Molchan and O.E. Dmitrieva and a program cre�ated by V.B. Smirnov [Molchan and Dmitrieva, 1993;Smirnov, 1997]; the data were supplied by V.A. Saltykov,KB GS RAS). We used the Kolmogorov test for thehypothesis of whether the distributions in the recurrencerelation based on the earthquake catalog are identical withthe theoretical distribution by also considering the seismicevents for the intervals January 1, 1962 to June 30, 2009(period Т4) and between January 1, 2000 and June 30,2009 (Т5).

The lowest completely reported energy class along theeastern coast is k = 9 [Smirnov, 1997; Sobolev, 1999;Fedotov et al., 1998]; further calculation was done for thisor higher values. Our treatment was concerned with theevents for the time intervals between January 1, 1962 andDecember 31, 1999 (period T1), from July 18, 1977 toDecember 31, 1999 (T2), from January 1, 1995 toDecember 31, 1999 (T3), and from January 1, 1962 toJune 30, 2009 (T4). The epicenters are situated in theareas delimited by the coordinates Δϕ = 50°–59°N andΔλ = 154°–168°E (area S1), Δϕ = 51°–56°N and Δλ =156°–163°E (S2), Δϕ = 52°–53°N and Δλ = 159°–160°E(S3). The events that fall in a specified area during a spec�ified time interval form a complete group. The period T2

was chosen because the hypocentral depths were deter�mined to within 1 km from July 18, 1977. The period T3

was chosen arbitrarily, provided it is less than T1 and T2.

We defined random events above as events that aregrouped in corresponding intervals of random variablesΔj. In that case we do not provide individual characteris�

F̃

F̃

tics of the events, but merely indicate the numbers of sam�ple values that fall in specified class intervals.

We now proceed to calculate (in the frequency repre�sentation) the probability distribution for the randomevents A, B, … as defined above. The calculation was doneusing a program that filters, following a given algorithm,the catalog data according to (4) for each event A, B, etc.[Bogdanov and Pavlov, 2007].

Event A. We treat the class interval of energy class Δkas the base of a rectangle with height mk/(n ⋅ Δk) = Pk/Δk,where mk and Pk are the number of seismic events in thisinterval and the probability of that event, respectively. Thearea of the rectangle is equal to the probability of the cor�responding group. As n increases, the area will tend to theprobability of the recorded value of k to fall in the corre�sponding interval Δk. The probability is approximatelyequal to the integral of the probability density f(k) takenover this interval. Consequently (Fig. 1a), the smoothed(adjusted) histogram (the broken line that connects theupper sides of the rectangles in their respective intervalsΔk) is the statistical analogue of the probability densityf(k), which was constructed, as an example, for the seis�mic events whose epicenters are situated in area S1 for theperiod T1(k0 = kmin = 9.5). In Fig. 1b the plot of f(k) isshown in log–log scale and corresponds to the recurrencerelation in the probabilistic representation. The interval12.5 ± 0.5 contains the break in this plot, which divides therecurrence relation into two parts with different values ofthe slope γ. Bogdanov [2006] showed that a broken linethat is a smoothed histogram can be represented analyti�cally for completely reported earthquakes

(5a)

or in the logarithmic form

(5b)

where 9.5 ≤ k1 < 12.5 with γ1 ( = 9.5) and 12.5 ≤ k2 <

16 with γ2 ( = 12.5). Bogdanov [2006] remarks that(5a, 5b) differ from the familiar equation of the recur�rence relation in that the former do not involve a depen�dence on seismic activity A0, which is the number of seis�mic events that have occurred per unit time per unit area inthe energy class interval k0 ± 0.5 and is a variable [Gaiskii,1970; Riznichenko, 1985]. The basic quantity in (5) is thedensity f(k0) = mk(k0)/(n ⋅ Δk) = Pk(k0)/Δk, where mk(k0)is the number of events falling in the energy interval k0 ±0.5.

Figures 2a and 2c show plots of the probability densityf(k) for the epicenters in areas S1, S2, and S3, whichoccurred during the time interval T2, for catalogs with andwithout aftershocks, as well as tables of the correspondingvalues of f(k). These plots are a probability interpretationof the recurrence relation for seismic events in energyclass intervals with Δk = 1 (k = ki ± 0.5). The initial (low�

f1 2, k( ) f k01 2,

( ) 10γ1 2, k1 2, k0

1 2,–( )–×=

f1 2,log k( ) f k01 2,

( )log γ1 2, k1 2, k01 2,–( ),–=

k01

k02

416


BOGDANOV et al.

est) values of the energy class k0 were equal to 10.5 and 9.5.When Δk = 1, the probabilities P(k) are numerically equalto f(k). (The reason that concrete values of f(k) are givenalong with the plots will be clear from the subsequent dis�cussion.) Figures 2b and 2d show distribution functionsF(k) with initial k0 equal to 10.5 and 9.5. An analysis of theresults shown in Figs. 2a and 2c reveals the interesting sta�bility of the respective frequencies. For specified intervalsΔki one notes nearly identical numerical values of f(k)irrespective of the area considered and of whether after�shocks have or have not been eliminated. The greatest dif�ferences arise for the recurrence plot with the least area(S3), where the number of seismic events is more than anorder smaller compared with S1 and S2. From Fig. 2 it fol�lows that all statistical plots are nearly identical irrespec�tive of the area considered; the greater the number ofevents n, the more similar the plots are.

Figure 3a presents smoothed plots of the recurrencerelation when averaged over the areas S1, S2, and S3 forcatalogs with and without aftershocks for the period T1

with two values of k0 equal to 9.5 and 10.5, as well as tablesof f(k). It can be seen from these tables that the respectivevalues of f(k) are similar. Figure 3b shows these plots onthe log–log scale. Appreciable scatter is observed for val�ues of k in the intervals k = 14.5 and 15.5, which is rather

obvious, considering that events are rare in these energyranges and that their number naturally decreases down tocomplete absence as the area considered is diminished. Itcan be concluded from Figs. 2, 3a, and 3b that the RR inthe probabilistic representation is invariant under achange of area, both for a full catalog and for one with theaftershocks eliminated. The respective probabilities arenearly identical and, according to the limit theorems inprobability theory, the agreement will improve as thenumber of events in the time series increases. Hence, onecan also conclude that the RR holds for aftershocks aswell. This can be seen as follows. If Pfull(ki) ≈ Pelim(ki)(the equality is approximate, because the observationalseries is finite; Pfull and Pelim are the probabilities for thefull catalog and for that with aftershocks eliminated,respectively), then

(6)

where Δn(ki) is the number of aftershocks that fall inthe ith energy interval only, ΔN(Σ) is the total number

Pfull ki( )nfull ki( )

Nfull Σ( )��

nelim ki( ) Δn ki( )+Nelim Σ( ) ΔN Σ( )+��= =

= nelim ki( )

Nelim Σ( )��

1 Δn ki( )/nelim ki( )+[ ]

1 ΔN Σ( )/Nelim Σ( )+[ ]��

= Pelim ki( )1 Δn ki( )/nelim ki( )+[ ]

1 ΔN Σ( )/Nelim Σ( )+[ ]��,

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1615141312110

10

–3.0

–0.5

–1.0

–1.5

–2.0

–2.5

–3.5

169 151413121110

–4.0

0

9

f(k)

S1, T1, Nfull = 20927

k

(а) (b)

log

f(k)

k

S1, T1, Nfull = 20927

Fig. 1. An example of the probabilistic representation of the recurrence relation. The histogram and a smoothed plot of probabilitydensity f(k) for the epicenters of seismic events in the full catalog (no aftershock elimination) during time Т1 in area S1. Numeralson the horizontal axis mark the midpoints of the intervals ( (ki ± 0.5), k0 = 9.5 (a). An example of a smoothed plot of probabilitydensity f(k) on a log–log scale (b). Nfull is the number of events in the full catalog.



0.7

0.6

0.5

0.4

0.3

0.2

0.1

161514131211100

f (k)

к

1.0

0.9

0.8

0.7

0.6

161514131211100.5

F (

k)

к

S1, Nelim = 2374

S2, Nelim = 1534

S3, Nelim = 177

S1, Nfull = 4251

S2, Nfull = 2970

S3, Nfull = 259

0.7

0.6

0.5

0.4

0.3

0.2

0.1

16151413121190

f (k)

k

S1, Nelim = 7471

S2, Nelim = 4887

S3, Nelim = 602

S1, Nfull = 12344

S2, Nfull = 8631

S3, Nfull = 813

10

S1, Nelim = 2374

S2, Nelim = 1534

S3, Nelim = 177

S1, Nfull = 4251

S2, Nfull = 2970

S3, Nfull = 259

S1, Nelim = 7471

S2, Nelim = 4887

S3, Nelim = 602

S1, Nfull = 12344

S2, Nfull = 8631

S3, Nfull = 813

1.0

0.9

0.8

0.7

0.6

16151413121190.5

F (

k)

k10

elim full

k S1 S2 S3 S1 S2 S310.511.5

12.513.5

14.5

15.5

elim full

k S1 S2 S3 S1 S2 S3

10.511.5

12.5

13.5

14.5

15.5

9.5

(a)(b)

(c)(d)

0.6887

0.2182

0.06820.0198

0.00460.0004

0.676

0.2216

0.0782

0.0189

0.0046

0.0007

0.6384

0.2316

0.11860.0113

0

0

0.65870.2308

0.0875

0.0193

0.0035

0.0002

0.6411

0.2367

0.09830.0205

0.003

0.0003

0.63320.2124

0.1236

0.0232

0.00770

0.68220.21880.0693

0.0217

0.00630.00150.0001

0.6861

0.21220.06960.0246

0.0059

0.00140.0002

0.7060

0.18770.06810.0849

0.0033

00

0.6556

0.22880.0795

0.0301

0.0066

0.00120.0001

0.6559

0.2206

0.0815

0.0338

0.0071

0.0010

0.0001

0.6814

0.20170.0677

0.03940.0074

0.00250

Fig. 2. The probabilistic representation of the earthquake recurrence relation for earthquakes whose epicenters are in areas S1,S2, and S3. Probability densities f(k) for the epicenters of seismic events in the catalog with aftershocks eliminated and in the fullcatalog for values of k0 equal to 10.5 (a) and 9.5 (c). The integral distribution function F(k) of the recurrence relation for the epi�centers of seismic events in the catalog with aftershocks eliminated and in the full catalog for values of k0 equal to 10.5 (b) and9.5 (d). Nelim is the number of events in the catalog with aftershocks eliminated (abbreviated as “elim”) and Nfull is that in the fullcatalog.

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1514131211109

elim fullk

10.511.512.513.514.515.5

9.5

016

elim full–1.5–2.0–2.5–3.0–3.5–4.0–4.5

90

16

–1.0–0.5

151413121110

0

–1

–2

–3

14.513.5

12.511.5

10.59.5 15.5

mean (S1, S2, S3), full. k > 9

0.6914

0.20620.0690

0.02700.00520.0010

0.000113

mean (S1, S2, S3)

k > 9 k > 10

0.66430.21640.0762

0.03440.00700.0016

0.00007

0.6677

0.22380.0884

0.01670.00310.0004

0.64430.22660.1031

0.02100.00480.0002

mean (S1, S2, S3), full. k > 10

mean (S1, S2, S3), elim. k > 9

mean (S1, S2, S3),elim. k > 10

mean (S1, S2, S3), full. k > 9

mean (S1, S2, S3), full. k > 10

mean (S1, S2, S3), elim. k > 9

mean (S1, S2, S3), elim. k > 10

mean (S1, S2, S3), elim. full. k > 9

mean (S1, S2, S3), elim.full. k > 10

log f1(k) = –1.6185 – 0.617(k–k0) = 6.094 – 0.617k,125 < k < 15.5; log f1(k0) = –1.6185; γ1 = –0.617

log f1(k) = –0.192 – 0.456(k–k0) = 4.14 – 0.456k,95 < k < 12.5; log f1(k0) = –0.192; γ1 = –0.456

k

f(k)

log

f(k)

(a) (b) (c)

log

f(k)

kk

Fig. 3. Averaged recurrence relations for f(k) in areas S1, S2, and S3. Averaged probability densities for the catalog with aftershockseliminated and for the full catalog with initial values of k0 equal to 9.5 and 10.5 (a). Averaged probability densities on a log–logscale. Broken dashed lines show averaged distributions f(k) with k0 equal to 9.5 and 10.5 for the catalog with aftershocks elimi�nated and for the full catalog (b). An analytical relation for f(k) on a log–log scale derived by least squares from the averaged dis�tributions of the catalog with aftershocks eliminated (abbreviated as “elim”) and of the full catalog (c), k0 = 9.5 [Bogdanov, 2006].

of aftershocks that fall in all energy intervals used,from the lowest completely reported to the highest,nfull (ki) and Nfull(Σ) are the numbers of events whichfall in the interval ki and in all energy intervals of the

full catalog, respectively; the events in the catalog withaftershocks eliminated are denoted similarly fornelim(ki) and Nelim (Σ). From (6) we derive to within theapproximate equality Pfull ≈ Pelim:

418


BOGDANOV et al.

(7)

From (7) it follows that aftershocks are distributed inenergy class intervals in the probabilistic representa�tion according to the recurrence relation. We used themethod of least squares for the averaged functions with

= 10.5 to calculate the respective values of

logf( ) from (5b) and the slopes γ1,2 (see [Bogdanov,2006]). These linear relations with their respective val�

ues of logf( ) and γ1,2 are shown in Fig. 3c (10.5 ≤

k1 < 12.5 with γ1 ( = 10.5) and 12.5 ≤ k2 < 16 with γ2

( = 12.5)). The numerical values of γ1,2 are given inFig. 3c.

It is of interest to compare the evolution of the distri�butions f(Si, k) for different time intervals T1, T2, and T3 ina given area, e.g., S2, with distributions for areas S1 and S3

for the longest period T1. Figures 4a and 4b presentsmoothed plots of these distributions for the full catalogand tables of their numerical values. It appears fromFigs. 4a and 4b that the most conspicuous differencebetween the plots occurs for area S2 and the shortest timeperiod T3 with the smallest number of events Nfull = 1306

Δn ki( )

ΔN Σ( )��

nelim ki( )

Nelim Σ( )��≈ Pelim ki( ) felim ki( )Δk.= =

k01

k01 2,

k01 2,

k01

k02

(k0 = 10.5). It follows that the increase in the number ofevents for a given area Si due to increased observationperiod T1 > T2 > T3 is identical with the increase in thenumber of events for a given time interval Ti due to theincrease in area S1 > S2 > S3.

In addition, direct calculation [Bogdanov, 2006]showed that the mean values of f(k0) based on the catalogwith aftershocks eliminated and the full catalog for areasS1, S2, S3 for the period Т1 with different initial values of k0

(Δk = 1) are equal to, respectively, (k0 = 8.5) = 0.648,

(k0 = 9.5) = 0.662, (k0 = 10.5) = 0.674,

and (k0 = 11.5), = 0.678. They are nearly identical

and close to the value (2/3) ≈ 0.667. It goes without sayingthat the conditions of earthquake generation are differentin local seismic areas of a seismic region. However, thePoisson theorem [Venttsel’, 1962], which states that thefrequencies of a random event Х (probabilities Pi) underchanged experimental conditions are stable, gives theresult that the frequency of that event tends to the arith�metic mean of the Pi in probability as the number ofevents n increases. It can thus be hypothesized that themean probability of a seismic event with different initialvalues of energy class k0 (at least in the interval here con�

fS3

fS1 S2 S3, , fS1 S2 S3, ,

fS1 S2 S3, ,

0.7

0.6

0.5

0.4

0.3

0.2

0.1

161514131211100

f (k)

k16151413121110

k

S1, T1, Nfull = 20927

0.7

0.6

0.5

0.4

0.3

0.2

0.1

161514131211100

f (k)

k16151413121110

k

k S1 S2 S3

10.511.512.513.5

14.515.5

k

S1 S2 S3

10.5

11.512.5

13.514.5

15.5

9.5

(a) (b)

(c) (d)

10

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

f (k)

0.7

0.6

0.5

0.4

0.3

0.2

0.10

f (k)

T1 T1 01.01.62 – 17.07.77 T2 T3 T1

0.650.23

0.08

0.03

0.010

0

0.65560317

0.22550657

0.080833760.03086644

0.006391170.00072627

0.0000726

0.6551

0.23370.0798

0.0259

0.00530.0002

0

0.6559

0.2206

0.08150.0338

0.0071

0.001

0.0001

0.6093

0.233

0.099

0.0488

0.0084

0.0012

0.0003

0.66

0.23

0.080.04

0.01

0

0

S3, T1, Nfull = 1360

S2, T1, Nfull = 13709S2, 01.01.82–17.07.77, Nfull = 5138

S2, T2, Nfull = 8631S2, T3, Nfull = 3343

S1, T1, Nfull = 7318

S3, T1, Nfull = 469

S2, T1, Nfull = 4742S2, 01.01.82–17.07.77, Nfull = 1772

S2, T2, Nfull = 2970

S2, T3, Nfull = 1306

S1, T, Nfull = 1182

S2, T5, Nfull = 814

S3, T5, Nfull = 104

theoretical distributiontheoretical distribution

S1, T4, Nfull = 8500

S2, T4, Nfull = 5556

S3, T4, Nfull = 573

theor. k S1 S2 S3

10.5

11.512.513.5

14.515.5

theor.

k

S1 S2 S3

T1 T1 01.01.62 – 17.07.77 T2 T3 T1

10.5

11.512.5

13.514.5

15.5

0.6679

0.23190.0797

0.0174

0.0026

0.0005

0.6548

0.2347

0.08960.0186

0.0021

0.0002

0.6778

0.23140.0751

0.0152

0.0006

0

0.6411

0.2367

0.09830.0205

0.003

0.0003

0.5965

0.25340.1248

0.02140.0031

0.0008

0.6525

0.22390.1023

0.0171

0.0043

0

0.66690.2321

0.08070.0172

0.00260.0005

0.6539

0.2360.0893

0.01890.00180.0002

0.64050.21990.11870.0175

0.00350

0.66670.23330.0816

0.01970.0048

0.0012

0.66070.2335

0.0871

0.01610.0025

0

0.6486

0.24320.08720.0209

00

0.5865

0.20190.1923

0.0192

00

0.6667

0.23330.0816

0.01970.0048

0.0012

Fig. 4. Comparing the probability densities f(k) for the full catalog. Comparison for areas S1 and S3 for the period T1 for distribu�tions over time for area S2 for periods T1, T2, and T3 and for the period January 1, 1962 to July 17, 1977 (a, b). Comparing theprobability densities f(k) of the full catalog for areas S1, S2, and S3 for the period T4 (January 1, 1962 to June 30, 2009) and Т5(January 1, 2000 to June 30, 2009) with the theoretical distribution (c, d).



sidered, 8 ≤ k0 ≤ 12 with Δk = 1) is little different from thevalue (2/3) ≈ 0.667. The above argument suggests that theRR in the probabilistic representation, as calculated fromour earthquake catalog, tends to group itself around thetheoretical distribution with the initial value f(k0) = 2/3 asthe number of earthquakes completely reportedincreases:

(8)

where 9.5 ≤ k1 < 12.5 with γ1 ( = 9.5)and 12.5 ≤ k2 < 16

with γ2 ( = 12.5).

We proceed to test the above statistical hypothesisusing the Kolmogorov test [Venttsel’, 1962]. ConsiderFigs. 4c and 4d, which show the RR calculated values

for areas S1, S2, and S3 for the periods from January1, 1962 to June 30, 2009 (period Т4) and from January 1,2000 to June 30, 2009 (Т5) based on the full catalog with

= 9.5. The tables give for comparison purposes, apartfrom calculated values based on the corresponding inter�vals Δk, also theoretical distributions calculated from (8).The numerical values of γ1,2 are shown in Fig. 3c. Accord�ing to the procedure of the Kolmogorov test, one shouldfind the maximum absolute values of the difference

between the statistical distribution function (k) in thecorresponding interval (ki ± Δk/2) and the theoreticalfunction (8)

for all three distributions corresponding to areas S1, S2,and S3. The next step is to find the quantity λ = D(ki ±

Δk/2) where n(Δki) is the number of eventsthat fall in the energy class interval (ki ± Δk/2); thisequals NΣP(Δki), where NΣ is the total number ofevents that fall in the interval between k0 and kmax.A.N. Kolmogorov showed that whatever the distribu�

f1,2 k( ) 2/3( ) 10γ1 2, k1 2, k0

1 2,–( )–× ,=

k01

k02

f̃ k( )

k01

f̃

D ki Δk/2±( ) max f̃ ki Δk/2±( ) f ki Δk/2±( )–=

n Δki( ),

tion function f(k) of a continuous random variable, the

probability of the inequality D(ki ± Δk/2) ≥ λ

tends to the limit P(λ) = 1 – as

the number of events indefinitely increases. The valuesof P(λ) are found from the formula given above andhave been tabulated [Venttsel’, 1962]. The probabilityP(λ) tells us that the difference of the maximum devi�

ation between (k) and f(k) due to random factors willnot be less than the observed deviation. If the probabil�ity P(λ) is small, the hypothesis should be rejected asunlikely. Large values of the probability indicate thatthe hypothesis is consistent with the observations.

The table gives the calculated results, from which itfollows that P(λ) assumes values equal or close to 1, indi�cating some consistency between the theoretical relationand the observations. However, the theoretical relationinvolves the parameters γ1,2, which were found statisti�cally, and this inflates the probabilities. At the same time,formula (8) has a fixed value of f(k0), which follows fromthe independent assumption about the initial values beinggrouped around 2/3. Clearly, any other value would entailconsiderable discrepancies between the theoretical valuesof f(k) corresponding to the current values of ki > k0, when

the γ1,2 and the statistical distribution function (k) arefixed. Consequently, one can say that the RR in the prob�abilistic representation can describe seismic events bymeans of the single equation (8), irrespective of the area(volume) concerned.

Event B. The unconditional probabilities of seismicevents over depth down to 100 km were calculated at astep Δ = 5 km for periods T1, T2, and T3 and areas S1, S2,and S3 (Fig. 5a). The calculation for these same areas andthe periods T2 and T3 down to 50 km depth was carriedout with step Δ = 1 km (see Fig. 5b). In Figs. 5a and 5b weuse different kinds of lines to indicate the probability dis�tributions over depth for S1, S2, and S3 corresponding to

n ki( )

1–( )αe 2α

2λ

2–.

α ∞–=

∞

∑

f̃

f̃

Results of calculations of parameters required to test compatibility of a statistical hypothesis with experimental data using theKolmogorov test (for explanations see main text)

Period T4(January 1, 1962 to June 30, 2009)

Period T5(January 1, 2000 to June 30, 2009)

S1(NΣ = 8501) S2(NΣ = 5555) S3 (NΣ = 573) S1(NΣ = 1183) S2(NΣ = 814) S3(NΣ = 104)

D 0.002546 Δki = 12.5 ± 0.5

0.01266 Δki = 10.5 ± 0.5

0.037029 Δki= 12.5 ± 0.5

0.00564 Δki = 10.5 ± 0.5

0.01802 Δki = 10.5 ± 0.5

0.08933 Δki = 12.5 ± 0.5

n(Δki) 146 3633 68 782 528 20

12.08 60.27 8.25 27.96 22.98 4.47

l 0.0307 0.763 0.305 0.158 0.414 0.495

P(l) 1 0.605 1 1 0.997 0.98

n Δki( )

420


BOGDANOV et al.

the time period T3. The short observation period (T3 <T2 < T1) leads to considerable fluctuations in the probabil�ity distribution compared with similar distributions overthe same areas but for longer time periods. This is obvi�ous, since when the observation period is short (a smallnumber of events), the random factor in the probabilitydistribution is not negligible. Over longer time periods therandom effects are smoothed out, owing to the largernumbers of events. In Figs. 5a and 5b one notes a similar�ity in the distribution of probabilities of events over depthirrespective of area, which is related to some features inthe hypocenter location program now in use for Kam�chatka earthquakes.

There is a local zone of increased earthquake occur�rence probability in the depth range 22–26 km for theperiod T3 in area S3 (Fig. 5b). It should be noted that thepeak of activity that occurs at depths of 22–26 km for areaS3 (period T3) is absent in the distribution for a largerperiod (T2). The appearance of this peak upon the back�ground of the distribution for a longer time period charac�terizes the generation of a zone of increased seismic activ�ity at these depths during the period T3. This can be shownas follows. Three earthquakes with k = 13.0 (h = 34 km),13.2 (h = 5 km), and 14.2 (h = 3 km) occurred onMarch 8, 1999 and their epicenters were in S3. As well,three more events occurred in the same area for the sameperiod: April 1, 1995 with k = 13.4 (h = 31 km), May 27,

1998 with k = 13.4 (h = 12 km), and May 28, 1998 withk = 13.1 (h = 5 km). One notes one characteristic fea�ture, namely, the hypocenters of these events enclose thevolume of increased seismicity for this time period, sincethey occurred above and below the range 22–26 km.

Event C. The distribution of the probabilities that theepicenters of seismic events fall in a given interval (Δϕ andΔλ) with arbitrary values of depth h and energy class k canbe found from the formula for a bivariate distribution(compare with (4))

where the unconditional probabilities P(λ), P(ϕ) andthe conditional probabilities P(λ|ϕ), P(ϕ|λ) are calcu�lated in the frequency representation from the catalog ofevents. The calculation for the coordinates was done atsteps of Δ = 10. The probability values characterize thedegree of seismic activity shown by individual elemen�tary (10 × 10) boxes. The results of this calculation forarea S2 are shown in Fig. 6. The numerator is therounded probability (in percent) of the epicenters ofseismic events to fall in 10 × 10 boxes from energy classk ≥ 9 upward; the denominator is the same quantitywith k ≥ 12. The same figure shows areas S1, S2, and S3

with the outline of the Kamchatka Peninsula.

P ϕ λ,( ) P λ( )P ϕ λ( ) P ϕ( )P λ ϕ( ),= =

0.20

0.15

0.10

0.05

0.16

0.12

0.18

100806040200

0.20

0.14

0.10

0.08

50403020100

0.06

0.04

0.02

S1, T1f (

h)

h, km

S2, T1

S3, T1S1, T2

S2, T2S3, T2

S1, T3

S2, T3

S3, T3

S1, T2

S2, T2

S3, T2S1, T3

S2, T3

S3, T3

(а) (b)

h, km

f (h)

Fig. 5. Smoothed plots of probability distributions for hypocenters falling in specified depth ranges for different areas and timeperiods: (a) probability distributions over depth down to 100 km at steps of Δh = 5 km for periods T1, T2, and T3 and for areas S1,S2, and S3; (b) probability distributions over depth down to 50 km at steps of Δh = 1 km for periods T2 and T3 and for areas S1,S2, and S3.



CONCLUSIONS

This study is an attempt to approach seismic phenom�ena using standard methods available in the theory ofprobability and statistics, which are based on the notionsof multivariate distributions, conditional and uncondi�tional probabilities, statistical series, etc. [Gaiskii, 1970;Kolmogorov, 1974]. Statistics rests on probability theoryas its only basis, but there is a feedback too, as buildingprobabilistic models and making decisions on whichmodel is preferable requires statistical data. In our case thedata consist of a catalog of Kamchatka events; this is themain issue in this study. Also, probability theory andmathematical statistics study fluctuations and patterns inphenomena that involve random factors, hence the genu�ine changes in seismicity that may occur in a local areaduring a certain period of time produce changes in thefrequency of earthquake occurrence, rearrangementsover depth, power spectra, etc. All this is responsible forvariations in the distributions of random quantities that

occur upon the background of averaged distributions,

which the basic model of the catalog {Ω, P} involvesduring the entire period of observation. In turn, the spaceof elementary events Ω is always expanding, owing to thenatural increase in the period of instrumental observation.Comparison of seismicity patterns with the basic modelenables estimation of earthquake risk connected with thesubsequent evolution of the seismic process. The presentapproach can furnish a basis for transition to the next stepin the development of a mathematical model for Kam�chatka seismicity in terms of chance based on the possi�bility theory, which estimates the potential occurrence ofa seismic event [Pyt’ev, 2000].

We state our main conclusions.

(1) A model has been developed for Kamchatka seis�micity which is represented as a probability space of

three objects {Ω, P}, which enables us to study dif�

F̃,

F̃,

59°N

58°N

57°N

56°N

55°N

54°N

53°N

52°N

51°N

50°N

154°E 156°E 158°E 160°E 162°E 164°E 166°E 168°E

0% 0.10% 0.05% 0.34% 1.33% 1.32% 8.91%

0% 0% 0% 0.19% 0.57% 0.76% 4.19%

0.02% 0.02% 0.13% 0.42% 0.69% 9.52% 10.3%

1.52% 5.53% 7.44%0% 0% 0%

0%

0%

0%

0% 0%

0% 0%

0%

0% 0%

0%

0%

0.14% 0.07% 3.68% 10.9% 7.72% 2.48%

7.06% 10.6% 8.20% 2.48%

1.00% 1.03% 9.87% 11.3% 0.98% 1.25%

11.0% 13.1% 1.33%

0.06% 1.14% 4.37% 8.95% 3.05% 0.42% 0.04%

1.14% 3.81% 18.1% 3.24% 0.57%

Fig. 6. A map showing the positions of areas S1, S2, and S3 relative to the outline of the Kamchatka Peninsula. Probability distri�bution for earthquakes falling in the corresponding boxes of geographic coordinates for area S2 and period T1 with k ≥ 9 (numer�ator) and with k ≥ 12 (denominator).

422


BOGDANOV et al.

ferent aspects of the seismic process in terms of proba�bility theory.

(2) Taking the example of an RR represented as theprobability density f(k) for completely reported events ofenergy class kmin ≥ 9, we showed that this quantity for thecatalog of Kamchatka earthquakes with aftershocks elim�inated in different intervals of coordinates Si(Δϕi, Δλi,Δhi) and time intervals Ti is identical with the probabilitydensity for the full catalog of seismic events. Hence, it fol�lows that the RR is also valid for the aftershock distribu�tion. The RR in the probabilistic representation for anarbitrary area is described by the relations (5a or 5b); eachof these is based on the value of f(k0) of the k0 = kmin com�pletely reported class rather than on the so�called seismicactivity A0, which is a variable quantity.

(3) A statistical treatment of the catalog showed for thefunction f(k) that the increase in the number of events fora given area due to an increased time interval of observa�tion is identical with an increased number of events for agiven time interval due to increased area. As the numberof events increases, the probabilities and statistical distri�butions f(k) tend to stable values.

(4) It is hypothesized that, for different initial k0 of acompletely reported earthquake energy class, the initialvalues of f(k0) (at least in the interval of our calculation8 ≤ k0 ≤ 12) tend to group themselves around the value 2/3as the number of events increases. The Kolmogorov testwas used to test the hypothesis that the analytical form ofthe probabilistic RR based on the distribution functionwith the initial value f(k0) = 2/3 is consistent with the sta�tistical RR’s obtained theoretically.

The authors hope that the probabilistic approach willfacilitate the study of seismicity and the structure of theEarth’s interior within the framework of geodynamic pro�cesses based on comparisons between these processes fordifferent periods of time [Sadovskii et al., 1984].

ACKNOWLEDGMENTSThis work was supported by the Russian Foundation

for Basic Research, project no. 03�05�65302 and by theFar East Division of the Russian Academy of Sciences,regional project 2002.

REFERENCESBogdanov, V.V., A Probabilistic Interpretation of the Earth�quake Recurrence Relation: The Kamchatka Region, Dokl.RAN, 2006, vol. 408, no. 3, pp. 393–397.Bogdanov, V.V., Geppener, V.V., and Mandrikova, O.V., Mod�elirovanie nestatsionarnykh vremennykh ryadov geofizicheskikhparametrov so slozhnoi strukturoi (Modeling of NonstationaryTime Series of Complex�Structured Geophysical Parameters),St. Petersburg: LETI, 2006.Bogdanov, V.V. and Pavlov, A.V., A Program for Calculating theProbabilities of Seismic Events, Moscow: VNTITs, 2007(Inventor’s Certificate no. 50200702092, Certificate of Indus�try Development no. 9093 as of October 10, 2007).Boldyrev, S.A., The Reflection of Lithosphere Structure andProperties in the Seismic Field of the Kamchatka Region, Fiz�ika Zemli, 2002, no. 6, pp. 5–28.

Venttsel’, E.S., Teoriya veroyatnosti (Probability Theory),Moscow: GIFML, 1962.Gaiskii, V.N., Statisticheskie issledovaniya seismicheskogorezhima (Statistical Studies in Seismicity), Moscow: Nauka,1970.Golitsyn, G.S., Pisarenko, V.F., Rodkin, M.V., and Yaro�shevich, M.I., The Statistical Characteristics of TropicalCyclones and Risk Assessment, Izv. RAN, Fizika Atmosfery iOkeana, 1999, vol. 35, no. 6, pp. 734–741.Dobrovol’skii, I.P., Teoriya podgotovki tektonicheskogo zemletr�yaseniya (A Theory of the Preparation of a Tectonic Earth�quake), Moscow: IFZ RAN, 1991.Keilis�Borok, V.I. and Kosobokov, V.G., Periods of HighProbability of Occurrence of the World’s Strongest Earth�quakes, in Mathematical Methods in Seismology and Geody�namics, New York: Allerton Press Inc., 1987, pp. 45–53.Kolmogorov, A.N., Osnovnye ponyatiya teorii veroyatnostei(The Basic Concepts of Probability Theory), Moscow: Nauka,1974.Matvienko, Yu.D., The Application of the M8 Method toKamchatka: A Successful Advance Forecast of theDecember 5, 1997 Earthquake, Vulkanol. Seismol., 1998,no. 6, pp. 27–36.Molchan, G.M. and Dmitrieva, O.E., The Objective FunctionApproach to Aftershock Identification, in Seismichnost’ i seis�micheskoe raionirovanie Severnoi Evrazii (The Seismicity andSeismic Zonation of North Eurasia), Moscow: IFZ RAN,1993, pp. 62–69.V.F. Pisarenko, F.M. Pruchkina, and M.G. Shnirman, Sto�chastic Model of Evolution of a System with Defects, inVychislitel’naya seismologiya (Computational Seismology, vol.17), Mathematical Modeling and Interpretation of GeophysicalData, II, 1984, pp. 24–27 (Translated by Allerton Press in1985).Pyt’ev, Yu.A., Vozmozhnost’. Elementy teorii i primenenie (Pos�sibility. Principles of the Theory and Applications), Moscow:Editorial URSS, 2000.Riznichenko, Yu.V., Problemy seismologii (Problems in Seis�mology), Moscow: Nauka, 1985.Sadovskii, M.A., Golubeva, T.V., Pisarenko, V.F., and Shnir�man, M.G., Typical Rock Sizes and the Hierarchical Proper�ties of Seismicity, Fizika Zemli, 1984, no. 2, pp. 2–15.Smirnov, V.B., Earthquake Catalogs: Evaluation of Data Com�pleteness, Vulkanol. Seismol., 1997, no. 4, pp. 93–105 [Volca�nology and Seismology, 1998, vol. 19, no. 4, pp. 497–510].Sobolev, G.A., Osnovy prognoza zemletryasenii (Principles ofEarthquake Prediction), Moscow: Nauka, 1993.Sobolev, G.A., Precursory Periods of Large Kamchatka Earth�quakes, Vulkanol. Seismol., 1999, no. 4/5, pp. 63–72.Stakhovskii, I.R., The Expansion of the f(α�Spectra of SeismicFields in the Precursory Volumes of Large Earthquakes, FizikaZemli, 2002, no. 2, pp. 74–78.Fedotov, S.A., Chernyshev, S.D., Matvienko, Yu.D., andZharinov, N.A., The Forecast of the December 5, 1997, Mag�nitude 7.8–7.9 Kronotskii Earthquake, Kamchatka, and ItsM ≥ 6 Aftershocks, Vulkanol. Seismol., 1998, no. 6, pp. 3–16[Volcanology and Seismology, Gordon and Breach, cover�to�cover translation, 1999, vol. 20, no. 6, pp. 597–613).Fedotov, S.A., Shumilina, L.S., and Chernysheva, G.V., TheSeismicity of Kamchatka and the Commander IslandsBased on Data of Detailed Investigations, Vulkanol. Seis�mol., 1987, no. 6, pp. 29–60.

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A probabilistic model of seismicity: Kamchatka earthquakes