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Bulletin of the Seismological Society of America. Vol. 67, No. 3, pp. 849-861. June 1977 A MECHANICAL MODEL F O R T H E STATISTICS OF EARTHQUAKES, MAGNITUDE, MOMENT, AND FAULT D I S T R I B U T I O N I HL BY MC E E CAPUTO ABSTRACT A mechanical model is presented to explain the Ishimoto lida empirical law for earthquake statistics log n(M) = & - bM where n(M) dM is the number of earthquakes in the interval of magnitude M, M dM. The model fits properly the statistics of earthquakes of all of the regions of the world for which exist reliable catalogs of earthquakes (Ranalli, 1974) although the mechanism of the model considered is specifically that of stick slip. The same model explains the empirical law for the statistics of the seismic moment Mo log no(Mo) = &o -- bo log Mo and explains also the scatter of the values in the empirical relation between Mo and M. Finally if one knows the rate at which the elastic energy is stored in a region of the Earth's crust, and if a catalog of earthquakes of the region is available, this model gives an estimate of the number and size of faults of the region, and also of the maximum magnitude and seismic moment possible in the region. INTRODUCTION The problem of developing stochastic models for earthquake occurrence has raised strong interest since the late 1950's. The unsatisfactory conclusion drawn from these studies is that statistical models, based on different principles, lead to similar observed effects because they depend on too large a number of parameters and allow the fitting of almost any data. In spite of the fact that there are plenty of reasons to consider earthquakes as a deterministic phenomenon, less interest was raised in the problem of developing mechanical models; we had to wait until 1967 to have an interesting approach with a paper by Burridge and Knopoff. At present, to proceed further it is necessary to take a deeper look at the physical aspect of the various types of process and therefore to the so-called genetic models. Interesting studies on the mechanism of stick slip have been made in the laboratory (Brune, 1973; Archuleta and Brune, 1975) measuring particle velocities and stress drop. A first attempt to provide a genetic statistical model for the stick-slip mechanism was made by Caputo (1976) and led to the conclusion that the Ishimoto Iida (I.I.) (1939) empirical law can be satisfactory only in a limited magnitude range, and that even in this range, in some cases, it could be satisfactory only in first approximation, in the same paper evidence was also given that the b coefficient of the I.I. law, in the intermediate magnitude range in Europe and in the ~editerranean region, could be decreasing with increasing magnitude. 849 850 MICHELE CAPUTO Recently the problem of explaining the above-mentioned empirical relations be- came more important because of practical implications mostly for engineering pur- poses. Usually one needs to know: 1. whether these relations can actually be linear; 2. in which range they are straight lines; 3. what happens outside of this range; for instance, whether b is decreasing or increasing for small values of M; 4. the maximum possible values of M and Mo in a given region. In this note these problems are considered introducing a more correct formulation of the stress field than in the other note (Caputo, 1976). This leads to different al- gebra and results. The asymptotic behavior of the derived statistical law is studied and the model is appled to estimate the number and dimensions of the faults in the central Apennines. Also, the minimum and maximum stress drop resulting from the model are in agree- ment with the observed values (Brune and Allen, 1967; Wyss and Brune, 1968; Chinnery, 1969). Finally, the model is used to determine the frequency-moment relation, which is a linear function of log Mo in a wide range of Mo ; it is shown that the frequency-magni- tude and frequency-moment relations are strictly related and the relations between their coefficients are established. A quantitative explanation and theoretical formula- tions are given for the empirical relations between the following pairs of parameters: area of the faults and magnitude, area of the faults and moment, magnitude and moment. THE MODEL To obtain the model let us consider that a given portion of the Earth's crust is crossed by a system of faults of area S, of linear dimension 1 = S 1/2, and direction 0 ~dth respect to a dominant tectonic stress field which we assume to be linearly in- creasing with time F = d. In this model it is also assumed that the relaxation of the elastic energy in the volume Ia around a fault has a limited effect on the stress distribution in the region; this is true only if the number and size of the faults is such that the total volume affected by the energy relaxation is much smaller than the total volume of the region considered. This should be checked in each region where this model is applied. Let fc and f~ be the friction coefficient and the cohesive force acting between the ~ faces of the fault, the force acting to lock the fault of area S = 1 after the time t is (£ + fo~t cos20)S (1) while the force acting in the direction to unlock it i~ Ste sin ~.cos 0 (2) The friction coefficient is rather independent of the confining pressures (Byerlee, 1968) at least in the laboratory experiments; i~ the Earth we assume it is constant for crustal slip. The forces (1) and (2) will be equal at failure and therefore the elastic energy will be released when t---- fo A MODEL FOR ESTIMATING STATISTICS OF EARTHQUAKES 851 After each earthquake the stress in the volume around the earthquake may or may not drop to zero; reversed movement may also result when coseismie movement is underdamped and the sides of the fault overshoot the equilibrium position (Fitch and Scholz, 1971). As a first approximation for a generic system of faults we assume that after each earthquake the stress drops to a constant value po ; in this case the return period and the stress drop of the faults of direction 0 are f~ po and fa ~(O) Po (3) where e(O) = ½sin2 O -- f~ cos2 O. In the time T the number of earthquakes occurring on each fault of direction 0 is therefore (4) f~ -- po~(O)" Considering in first approximation that the faults have a circular area, the formula (Keilis Borok, 1959) E- ~kl~P2 (5) 2# gives the elastic energy radiated by the faults of linear dimension l, in a medium of rigidity ~ for an event of stress-drop p (~ is the seismic efficiency and k is a geometric factor). The elastic energy radiated by the faults of direction 0 and linear dimension l is therefore, taking into account (3), ~13 r l, 2 E = ~ [Ja - [po~(O)] 2] (6) To provide a genetic statistical model of earthquake occurrence we need to know the relation for the number of faults in the ranges l, I q- dl and 0, O q- dO. There is an indication that for very large faults this number is given by a log- normal distribution (Ranalli, 1976); in the range of our interest, to simplify the algebra, we may reasonably approximate the log-normal distribution with Dl -~ dl dO where R and u are positive arbitrary constants. The results of this research will be valid to the extent that the law Dl -~ represents the actual distribution of faults. The number of earthquakes which are in that range and which occurred in the interval T is therefore TD~ 1 L - po~,(o) dZ d o ~ v ~ . ~ (7) 852 MICHELE CAPUTO To estimate the number n ( M ) dM of earthquakes in the interval M, M + dM let us introduce the empirical relation between M and E (e.g., see Bath, 1973) E = 10 ~+~M and change the coordinates 1 or 0 to M, we obtain from (6) Expressing (7) in terms of dM dl or dM dO we obtain [, DTe7 In 10 n(M) = --J 2l~f~2 d~p (f~ + po~(O) ) dl dO TDey In 10 [~(8) ](5-2~)/3 (1 - P°Z)) dO n(M) = f 3.2(~-~)~R~-1 LT:-~ J 1 -- p o ~ Ra _ Ett ~?/c ' Z = R -113" (~l~) 1p02) --~ To compute the integrals let us set 11 and /2 as the minimum and maximum linear dimensions of the faults, and introduce the new variable u = 1/R, we obtain TDe7 In 10 rz2/R n(M) 2f~R ~-1 Jh/R U(u) du (s) %/~ poU3/2 24/3--v 3--v.+ ( uj~l 1 + 2po2u3] U(u) = 1 + 2(po ~ -- 4f~2)u 3 -- 4%/'2f~f~ u3/2(1 + 2po2u3)u2 TDe'r In 10 f ' ~ n ( M ) = 3R~-lf~1/3(5--~i 1 [~(O)] u3(5-2~) dO. (9) We must also take into account that 9(0) and l(O) are functions of O symmetric with respect to O, = ½ tan -1 ( - fc)-l; the integrals are therefore computed over half of the path and then multiplied by 2. For O = O~, the stress accumulated has a mini- mum and 1, given by (6), has a maximum; also 1 ~(O~) -- ~ (~/~ + f~2 - f~). The limits of the integral in formula (9) are established by taking into account that M is constant ~long the path of integration and therefore using formula (6) which gives the path of integration. A MODEL FOR ESTIMATING STATISTICS OF EARTHQUAKES 853 The energy E is subject to the limitation imposed by 12, the maximum size of the faults, and by the maximum stress accumulation ~, then E <_ 137 ]~t=2 - 2 tp Po 2) 2tt The E limit sets a limit also for a~, which is obtained from = A (lO) ½ sin 2ai - - f~ cos2 ~1 As we shall see later there is another value of E that is relevant to the path of in- // U /22 = t a n - ' ft. 1 FIG. 1. Paths for the integration of (8) and (9). tegration; it is 1 (fo~ _ (po~,(o~))~) ,kz2 ~ - E2 = 10 ~+~M~. (11) 2 ~[~(os)P The upper limit of the integral in du is given by the point of intersection of the path of integration defined by (6) (Figure 1) with M 2 M ~ or M2M,~. The lower limit is established as follows: If E < E1 with EI(M~) = 113kv(P~ -- po2) (12) 2tt then [1 = ll, and for E > E1 the lower limit assumes the value R - - _ ° (13) 854 M I C F I E L E CAPUTO To study (9) we must consider t h a t since dl/dO > 0 when 0 < O < ~r/2, for the integration of (8) and (9), we may have four differents paths indicated in Figure 1 and in Table 1. Let E~(M2) and EI(Mx) be the energies of the curves (6) passing through M2 and M~, and defined b y (11) and (12), (see Figure 1) ; for the computation of the integral we must separate the cases E2 ~ E~ ; the path of integration for the different values of E(M) is given in the Table 2. We are able now to discuss the log n(M) ; in the zone I we have d log n(M) (1 - ,) v v R (~4) dM f, U(u) du TABLE 1 LIMITS OF THE INTERVAL OF INTEGRATION OF (8) AND (9) (a~ and ~2 are the smallest positive roots of the equations) ll = R(½ sin 2a - fo cos2 ~)m l~ = R({ sin 2q - f , c o s ~ a)~/a Limits of the Interval of Integration Zone From a Point of To a point of in dl in dO L\~/- 11 12 II MmM1 M~MM -~ -~ o~l as III M1MM M~Mm L~J ( fo ~% po~J m, o, L\;~7) / - IV MIMM M2MM L~_ po---~j ~ m ~ depending on the values of the parameters v, k, f~, f , , v, fl, v, when M is sufficiently small, this derivative could be positive, but it becomes negative for the larger values of M of the pertinent interval. In zone II we have d log n(M) (1 - v) "/ 3' ~ - ~ U dM - -3 + 5 ~/~ (15) f, U(u) du which is negative for all values Of M of the pertinent interval. In zone I I I we have dlog~(M) .(1_~) ~ --b (16) dM ~= A MODEL FOR ESTIMATING STATISTICS OF EARTHQUAKES 855 this implies that log n(M) is a straight line for M1 < M < Ms. Finally in zone IV we obtain d log n(M) - (1-- v)'y/3 "yl2U 3R fl/~ U(u) du (17) dM that is negative for all the values of M of the pertinent interval. In zone II, when 11 is sufficiently small, we have d ~ log n ( M ) / d M s < 0 which im- plies that the b coefficient of the I.I. law is a decreasing function of M in this interval. In the zones I and IV we have d ~ (log n ( M ) ) / d M ~ < 0 and the curve log n(M) is concave downward. The minimum and maximum values of M permitted by the geometrical and physical parameters of the system of faults are obtained from (5) associating maximum size of faults with maximum stress drop and minimum size of faults with minimum stress TABLE 2 SELECTION OF THE P A T H S OF INTEGRATION OF (8) AND (9) Value of M M <M~ ~ M < M < 3//i ~ M <M Path of integration I II IV Value of M M < Ma M1 < M < M2 M2 < M Path of integration I III IV drop as shown below M~ = 1(log123(792-- Po2)Ev Mm -- 1 log yllk(fa -- (po¢(O.)) s) 13 (18) 2u[~(o.)] ~ • For M in the central part of this range (i.e., in the zone III) log n(M) is exactly a straight line. The deviation of log n(M) from the straight line are extremely pertinent to the discussion of the geophysical parameters of the system of faults. DETERMINATION OF THE PARAMETERS OF THE MODEL The value of fc is known from laboratory experiments (Byerlee, 1968), for vk we assume the approximate value of ½, from the analysis of the data of a given region we may determine M.~, Ms, M1, MM ; then we may compute ll, 12, ~5, fc from (11), (18) and (12) as shown in (19) \ 3 log 12 + log [~2 _ po2] = ~'MM + Z, Z = log 2t~10~ tic 3log/1 = M m ~ / + Z - - Z1, 3 1 o g / l + l o g [ 1 5 2 - po2] = ~, MI + Z 3 1 o g l 2 = M~'y + Z - Zl, fa ~2 Z l = l o g (\-~(~)] -- Po. S (19) 856 MICHELE CAPUTO o o P" o m qr ~ Log n FIG. 2. Frequency-magnitude relationships. Apennines; an arbitrary constant is added to log n. Considering the logarithms unknown we can see t h a t the matrix of the coefficients of the unknown has characteristic 3, which allows to determine three of the four unknown /1, 12, f~ and ~. Therefore, we need to determine one of t h e m from other sources. 1031 1029 t 1027 I "i • $ ••|• 1025 • | • e e • • I [ , I 1 5,0 6,0 7.0 |.0 9.0 Magnitude{Ms) FIG. 3. Compilation of 87 estimates of M• after Chinnery and North (1975). A MODEL FOR ESTIMATING STATISTICS OF EARTHQUAKES 857 If we have enough data for the small values of M we can add to system (•9) the equation for the maximum of log n ( M ) - where/~ = R(/]~) a n d / ] I is the value of M corresponding to the maximum of the function log n ( M ) ; in this case we should have predetermined the value of M from a best fit to the data. Alternatively from the depth of the foci and the thickness of the crust in a given area we m a y estimate/2 and then compute/1, i~ and f~ using system (19). The best fit of log n ( M ) to the data is made by considering as best-fit parameters A = ( T D e / f ~ ) and v. T is known and so isle; A therefore determines eD. If we know a value for e, then R can be obtained to find the number of faults of dimension I in the range l, 1 -~ dl and 0, 0 + dO. TABLE 3 DISTRIBUTION OF FAVLTS F O R THE APENNINES L e n g t h of F a u l t s (m) 66 100 500 1000 5000 10000 50000 N u m b e r of f a u l t s 56160 61340 4390 2180 150 80 APPLICATION TO THE APENNINES Considering the data of the Apennines (Caputo, Postpischl, 1974; Console, Gas- parini, in preparation) we may assume in first approximation that M1 = 2, M2 = 6, M m = O , M ~ = 8, po = O. Actually in the Apennines there are earthquakes with M < 0; by assuming Mm = 0 we change the curve log n ( M ) only in the range of small values of M, while a large portion of the linear part of the curve (including its slope) and its portion for larger values of M remain unchanged. The value of 12is assumed to be 50 km because of the thickness of the crust (Caputo et al., 1976) and the depth of the foci in the Apennines, fc, is assumed 0.6 (Byerlee, 1968). The values of B and % after Bath (1973), are 12.24 and 1.44. For M1 < M < M2 we assume (Caputo, Postpischl, 1974) b = 0.7, and then by using (16) and (19) we obtain v = 2.45, 11 = 103's2,/~ = 107"~°,J~¢ = 105"61;by fitting the model to the data we find A = 101~'53which in turn gives De = 1015'3. A crude estimate of e can be made by assuming that the crust in the Apennines is subject to a strain of 10-7 year -~, which in turn gives E = 104.4o dyne cm -~ year -~ and D -- 1010"s. The distribution of the faults participating in the earthquake sequence of the catalog is listed in Table 3; it seems acceptable. The total volume of the crust involved in the release of the elastic energy around the faults of the region is 2.10 4 km 8. The volume of the crust in the seismic region, considering that the epicenters have depth to 50 km, is at least ten times larger then the volume where the release of the elastic energy occurs. The values obtained for the minimum ~tress drop f a / ~ ( O , ) - po and for the maxi- 858 MICHELE CAPUTO mum stress drop i~ - po are in agreement with the observed values (Brune and Allen, 1967; Wyss and Brune, 1968; Chinnery, 1969). T H E MODEL FOR THE SEISMIC MOMENT ~) To estimate the seismic moment Mo of a given earthquake (l, v we need to know the average displacement along the fault <s>. Since it is reasonable to assume (s> proportional to the stress drop and to I/c, we may write (s> c~ sin ~ ~ - p° where c is a factor dependent on the geometry. The seismic moment is therefore Mo- f~ - ff~ (22) c sin ~ ~ -- p° c~b(0) According to the theory developed in the preceding sections for the statistics of the magnitudes we obtain that the number of earthquakes with seismic moment in the range Mo , Mo "4- dM is dMo TDeM-(~+,;)~ f [¢(0)](t-~)18 d,Y. (23) 3fa(4-")/~C(~'1)i3 J sin ,~ Over the interval defined by _l - 13(79 -- po) Mol c sin al Mo2 - l~3f~ c~(,~,) the integral in (23) does not depend on Mo, therefore in this interval the number no of earthquakes with seismic moment in the range Mo, Mo + dMo satisfies the relation log no = go 2 -}- u log Mo = do - bo log Mo. (25) 3 For the Apennines we find Mol = 1019"4, Mo2 = lO 26'3 and log no = 11.8 - 1.49 log Mo. For the worldwide data Mo~ is estimated 1027"5. A formula similar to (25) was empirically obtained by Chinnery and North (1975) for worldwide data. The numerical value of the exponent of Mo for the Apennines is A MODEL FOR ESTIMATING STATISTICS OF EARTHQUAKES 859 surprisingly close to that of the empirical formula for the worldwide data (1.61). This in turn implies that the distribution of the size of the faults in the Apennines is not too far from that implied by the empirical formula for worldwide data; we should also take into account that the two ranges of magnitude are different; they are from 4 to 7 in the Apennines and from 5.2 to 8.6 for the worldwide data. This theory can also be checked by the worldwide data. In fact Gutenberg and Richter (1954) give for the empirical frequency-magnitude relation for worldwide data. log n = d -- b M = 7.99 -- 0.93M (26) and Chinnery and North (1975) give for the empirical frequency-moment relation for worldwide data log no = 17.27 -- 1.61 log M o . (27) By comparing (26), (25) and (16) we obtain (5/7) + 1 = bo (28) which gives bo = 1.64, in agreement with the value of the empirical formula. The theory outlined here provides a relationship between a and ao ; from (26) and (9) we find d~ f [~(~)](~-~)/~ ] d - - do = log \ 1 0 ~ ] 71n10 ThE RELATION BETWEEN M AND Mo From formulas (22) and (6) we find that the ratio between the radiated energy and the seismic moment for an earthquake of given (1, 0) is E _ ~kcsin~( fa ) (29) Mo ~, -j(~ ) po or, in terms of magnitude, fa - log e(~---~ - po al < ~ < a 2 (30) log Mo = 12.24 + 1.44M + log c ~ sin ~ ; which fits very well the compilations of the observational data of Chinnery and North and of Kanamori and Anderson (1975): The presence of v~ in formula (30) could explain all or part of the scatter of the data of the above-mentioned compilations which is in average 2 orders of magnitude as is the range of the function of 0 in (30). Kanamori and Anderson (1975) also plot the area of the fault S versus Mo and versus M; in both cases there is a wide scatter of data which is explained by formulas (22) and (6). Formula (30) allows to compute 0 for a ~iven fault when M and Mo of an event or 860 MICHELE CAPUTO the fault are known. From ~, if the fault is known, one may also estimate the direction of the tectonic force; also, if e is known, by means of (3), one may estimate the return period of the events associated to the fault; in other words, one may attempt time prediction of the earthquakes of the fault. The estimate of e can be tentatively made by measuring the strain at the surface. THE MAXIMUM VALUES OF M , ]~ro AND /2 The maximum and minimum seismic moments possible in this model are 3 ffi 12 ( p -- po) _ M o ~ c sin ~i ll~f~ - Morn. (31) c¢(O~) In case po = O, from (24) and (31) and (19) we obtain, eliminating f~, 3 log l~ = l o g [cM°2¢~(~)]2 'YM2 -- Z [~(~.)]2 MM = log ~(V~) Z 1 - - M2 log M o u = log [~(0,)]2 sin 61 -- Z -- 7 M 2 @[M°2~b(~)]~ (32) which give the maximum magnitude, moment, and size of the faults by means of the observable quantities M 2 , Mo2, ~. We should note that 12does not depend on ~5. ACKNOWLEDGMENT The author wishes to thank Prof. A. Gangi for helpful discussion of the paper. ~EFERENCES Arehuleta, R. J. and J. N. Brune (1975). Surface strong motion associated with a stick-slip event in a foam rubber model of earthquakes, Bull. Seism. Soc. Am. 65, 1053-1071. B~th, M. (1973). Introduction to Seismology, Birkhauser-Verlag, Basel. Brune, J. N. (1973). Earthquake modeling by stick-slip along preeut surfaces in stressed foam rubber, Bull. Seism. Soc. Am. 63, 2105-2119. Brune, J. N. and C. R. Allen (1967). A low-stress-drop, low-magnitude earthquake with surface faulting: The Imperial, California, earthquake of March 4, 1966, Bull. Seism. Soc. Am. 57, 501-514. Burridge, R. and L. Knopoff (1967). Model and theoretical seismicity, Bull. Seism. Soe. Am. 57, 341-371. Byerlee, J. D. (1968). Brittle ductile transition in rocks, J. Geophys. Res. 73, 4741-4750. Caputo, M. (1976). Mechanical models of earthquakes and their statistics. Proc. E.S.C. Sym- posium on Earthquake Risk for Nuclear Power Plant (1975), Roy. Neth. Meteorol. Inst. Publ. 158. Caputo, M. and D. Postpischl (1974). Contour mapping of seismic areas by numerical filtering and geological implications, Ann. Geofis., (Rome) 27. Caputo, M., L. Knopoff, E. Mantovani, S. Mueller, and G. Panza (1976). Rayleigh wave phase velocities and upper mantle structure in the Apennines, Ann. Geofis. (Rome) 27, (in press). A MODEL FOR ESTIMATING STATISTICS OF EARTHQUAKES 861 Chinnery, M. (1969). The strength of the earth's crust under horizontal shear stress, J. Geophys. Res. 59, 2085-2089. Chinnery, M. A. and R. G. North (1975). The frequence of very large earthquakes, Science 1197- 1198. Console, R. and C. Gasparini (1977). I1 periodo sismico di Cerreto di Spoleto (dicembre 1974) (in preparation). Fitch, T. J. and C. H. Scholz (1971). Mechanism of underthrusting in South West Japan: A model of convergent plate interactions, J. Geophys. Res. 75, 7260-7292. Gutenberg, B. and C. F. Richter (1954). Seismicity of the Earth and Associated Phenomena, Prince- ton Univ. Press, Princeton, N.Y. Kanamori, H. and D. L. Anderson (1975). Theoretical basis of some empirical relations in Seis- mology, Bull. Seism. Soc. Am. 55, 1073-1095. Keilis-Borok, V. I. (1959). On estimation of the displacement in an earthquake source and source dimensions, Ann. Geofis. (Rome) 12, 205-214. Kolmogorov, A. N. (1941). (~ber die logarithmish normale Verteilungs gesetz der Dimensionen der Teilchen bei Zerstuckelung, Proc. Sc. USSR, 31, 1-99. Ishimoto, M. and K. Iida (1939). Bull. Earthquake Res. Inst., Tokyo Univ. 17,443-478. Ranalli, G. (1974). A test of lognormal distribution of earthquake magnitude, Presented to the International Symposium on Seismology and Physics of the Solid Earth, Jena, April 1974. Ranalli, G. (1976). Length distribution of strike-slip faults and the process of breakage in conti- nental crust, J. Canad. Terre 13, 704-707. Wyss, M. and J. N. Brune (1968). Seismic moment, stress, and source dimensions for earthquakes in the California-Nevada region, J. Geophys. Res. 73, 4681-4694. ISTITUTO DI GEOFISICA UNIVERSITA BOLOGNA~ ITALY ISTITUTO NAZIONALE nI GEOFISICA ROMA, ITALY Manuscript received September 15, 1976