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The Gtaenbei}g一Richter relation, log n=“一b M, }vhere n is the cumulative number ofearthquakesgrealer than the magnitude, M and a and b tire constants determined einpiricalY, is discussed itsing a mathematical model‘,/ a加ul and the momew magnitude, MjV. A simple power law of,A)=k nI which relaics the earthquake moment, Mo andfirequencY, n was discussed and the meanings qj constants k and I vere derived. The二/atim ofM。二△a (LA) usuig the stress drop, A(T,八itilt Icigth, L and area qfajaidt plaie, A, gives MO=Au L-' and Mo=A a hL2 for two“。。ofA=L2 wid A二Lh,respectivelY, wh。二 the depth‘了the britle-ductile transition laYer, h=2 x 104 in and Aa=5 x 10' M12‘,二‘isitined. The,/,二quenn-magninide relation is obtained considering dif ferent rupture inechanisnis of earthquakes. Deviation o.t the empirical Gulenberg-Richter relation tit‘,large inagnitude is interpreted‘,、the e加 0 ,.Te 。扩the seisinogenic laver in earth interior Theoretical‘,]‘,/、二is from assumption ofdi加 rent fe ,Vtil,二,nechanisms iidicated varied b、alues fi)rsmal and large events, as wel/ (is the d}ferent tyes o/ 'earthquakes. Mainshoc-ks gave b。/1.0 and 0. 75, respective/Y.户rsmal and lat}qe events. The b values ol'O.7-5 atid 0.50 for afiershocks and 0.50‘,,‘/ 0.375户r了oreshocks ivere also obtaified. Some,二、1/is firom both statistic calibration ofearthquake catalogue and experiments ivere hi agreement wilh our predicted values. the finite thickne.,s of' the scisino-cnic la}ci changes the scalilq-, ot Jarge earthquakes (except Sornete. ct al., 1996), the obtained b \alues are in conti-mer,N, (Rundle. 1989. Pacheco et al., 1992): Main and Berton. 1986: Main. 1992: Hamilton and McCloskey, 1997) Diferent intcrprctations are con,istcnt with diferent modelsof larl-c earthquakc., Such:、the 1-model (Scholz, 1982) in }\hich the slip and I'aUlt Icn-th wil keep increasing, Ailh earthquake size and the W-model (Romanowicz, 1992: Rornano),vicz and Rundle, 1993) in \A,'hich both slip and \} idth of J'ault ,aturatc lOllal-2C events. Ulo\A,'ever. the contro\ ersy of‘ /) values 1rorn each evidence support leads to the confusion Il Llndcrs(andim-, the physical insight of the fi-equenc}ma-nitude relation and b valucs. I Airiathernaticjl fault model, moment magnitude,M*as \el as a power law()(、mo=k n1. \,here M() is carthqUake moment and 4 and/are constants, are diSCUsscd in this paper. GRR is derived considerim, the difercm rupture mechanisins ol' smal and ]arge earthqUak-CS (ilClUding, \olcanic eruptions, forcshock.}, mainshocks and aftershocks). The physical incanim, Ol'b values for diflerell t} PC} Of earthquakes is anal)zed. The obtained theoretical b }alt-[CS alC Ll,cd to predict the h \aILICS cither 1}or earthquake CatalOOLIC Or士 01-、 itl 1LlI t'I s11 lated experiments.

Earthquake frequency and moment magnitude relations for mainshocks, foreshocks and aftershocks: Theoretical b values