APPARENT STRESS: SEISMIC ESTIMATE OF TAUa

Seismologists (for example, Wyss and Brune, 1968; Savage and Wood, 1971; Wyss and Molnar, 1972) have defined apparent stress as

taua = eta tau-bar, (6)

where eta is the seismic efficiency, defined by

eta = Ea/E (7a)

= taua/tau-bar , (7b)

where equation 7b follows from 7a according to equations 2 through 5; that is, eta is simply the fraction of the total energy release, or the fraction of the average earthquake stress, allocated to seismic radiation.

To estimate taua, seismologists first determine the radiated energy Ea and the seismic moment M0, defined as

M0 = GAu, (8)

where u is the average slip of an earthquake over a fault-surface area A, and G is the modulus of rigidity (Aki, 1966). Equations 1 and 6 through 8 then yield the simple relation

taua = GEa/M0 (9)

A numerical estimate of taua can be obtained from equation 9 with the following commonly used formulas relating earthquake magnitude M to Ea or M0 (Gutenberg and Richter, 1956; Hanks and Kanamori, 1979),

log M0 = 16 + 1.5M (10a)

log Ea = 11.8 + 1.5M, (10b)

where M0 and Ea are in ergs. Substitution of equations 10 in 9 yields

taua = 6.3 x 10-5 G. (11)

With G = 3 x 104 MPa, the value for taua is 2 MPa. Almost without exception, estimates of taua fall within the range 0-5 MPa, with no indication of any systematic dependence on either earthquake size or tectonic environment (Spottiswoode and McGarr, 1975; Fletcher and others, 1983; Boatwright and Choy, 1986). In short, 5 MPa appears to be a conservative upper bound to taua. Thus, the contribution of taua is small, and the average fault stress tau-bar can be large only if the frictional resistance taur is large (eq. 5).

If laboratory "earthquakes" are proper analogs of crustal earthquakes, which may or may not be the case, then data for such events, including those illustrated in figure 10.3, indicate that taua is indeed small, only a tiny fraction of taur. By inducing unstable frictional failure (earthquakes) across a 2-m-long fault between slabs of granite 40 cm thick (Dieterich, 1981), Lockner and Okubo (1983) measured seismic efficiencies eta for numerous events to conclude that eta ~~ 0.05. If this result were true also for natural earthquakes - a big "if" - then for a typical value taua of 2 MPa, the corresponding value of tau-bar, from equation 6, would be 40 MPa, which, as will be seen, is nearly 3 times higher than the limit inferred from an analysis of heat-flow data (Lachenbruch and Sass, 1980).