Can observations of earthquake scaling constrain slip weakening?

We use observations of earthquake source parameters over a wide magnitude range (M-w similar to 0-7) to place constraints on constitutive fault weakening. The data suggest a scale dependence of apparent stress and stress drop; both may increase slightly with earthquake size. We show that this scale dependence need not imply any difference in fault zone properties for different sized earthquakes. We select 30 earthquakes well-recorded at 2.5 km depth at Cajon Pass, California. We use individual and empirical Green's function spectral analysis to improve the resolution of source parameters, including static stress drop (Delta sigma) and total slip (S). We also measure radiated energy E-S. We compare the Cajon Pass results with those from larger California earthquakes including aftershocks of the 1994 Northridge earthquake and confirm the results of Abercrombie (1995): mu E-S/M-0 << Delta sigma (where mu = rigidity) and both E-S/M-0 and Delta sigma increase as M-0 (and S) increases. Uncertainties remain large due to model assumptions and variations between possible models, and earthquake scale independence is possible within the resolution. Assuming that the average trends are real, we define a quantity G' = (A sigma - 2 mu E-S/M-0)S/2 which is the total energy dissipation in friction and fracture minus sigma S-1, where sigma(1) is the final static stress. If sigma(1) = sigma(d), the dynamic shear strength during the last increments of seismic slip, then G' = G, the fracture energy in a slip-weakening interpretation of dissipation. We find that G' increases with S, from similar to 10(3) J m(-2) at S = 1 mm (M1 earthquakes) to 10(6)- 10(7) J m(-2) at S = 1 m (M6). We tentatively interpret these results within slip-weakening theory, assuming G' approximate to G. We consider the common assumption of a linear decrease of strength from the yield stress (sigma(p)) with slip (s), up to a slip D-c. In this case, if either D-c, or more generally (sigma(p) - sigma(d)) D-c, increases with the final slip S we can match the observations, but this implies the unlikely result that the early weakening behaviour of the fault depends on the ultimate slip that the fault will sustain. We also find that a single slip-weakening function sigma(F)(s) is able to match the observations, requiring no such correlation. Fitting G' over S = 0.2 mm to 0.2 m with G' proportional to S1+n, we find n similar to 0.3, implying a strength drop from peak sigma(p) - sigma(F)(S) proportional to S-n. This model also implies that slip weakening continues beyond the final slip S of typical earthquakes smaller than similar to M6, and that the total strength drop sigma(p) - sigma(d) for large earthquakes is typically >20 MPa, larger than Delta sigma. The latter suggests that on average a fault is initially stressed below the peak strength, requiring stress concentration at the rupture front to propagate slipping.