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									             Bulletin of the Seismological Society of America. Vol. 67, No. 3, pp. 849-861. June 1977

                                            I HL
                                        BY MC E E CAPUTO


       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
                                    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.


   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
   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.
 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

                                          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

               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


                                           ~(O)           Po                            (3)
                               e(O) = ½sin2 O -- f~ cos2 O.

  In the time T the number of earthquakes occurring on each fault of direction 0 is

                                       f~ -- po~(O)"

  Considering in first approximation      that the faults have a circular area, the formula
(Keilis Borok, 1959)

                                          E-          ~kl~P2                            (5)

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
   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

                      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
                             (~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
                          2f~R ~-1 Jh/R U(u) du                                          (s)
                                                  %/~ poU3/2
                                         24/3--v 3--v.+ (
                                                 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

                                     ~(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)

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)

then [1 = ll, and for E > E1 the lower limit assumes the value

                                                   R    - -                _             °

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)
                                                                                  f,           U(u) du

                                                        TABLE          1
                    (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

                                                                 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)

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.

  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~
         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         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






                                    •       $                ••|•

                                •   |   •   e   e    •       •

                                I                    [                    ,        I            1
                          5,0                       6,0                      7.0               |.0   9.0

         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,
   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 --
   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

                               TDeM-(~+,;)~ f         [¢(0)](t-~)18   d,Y.        (23)
                              3fa(4-")/~C(~'1)i3 J        sin ,~

  Over the interval defined by
                                               _l        -
                                                     13(79 -- po)
                                                      c sin al

                                       Mo2 -         l~3f~

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)

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
                          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

                                                         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,
                                              - 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)

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

                             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.

  The author wishes to thank Prof. A. Gangi for helpful discussion of the paper.

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                 A MODEL FOR ESTIMATING STATISTICS OF EARTHQUAKES                             861

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  Manuscript received September 15, 1976

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