[S-CG55] Dynamics in mobile belts

[SCG55-16] A theoretical basis of the moment magnitude scale

*Mitsuhiro Matsuura¹ (1.University of Tokyo)

Keywords:Radiated seismic energy, seismic moment, scaling law, moment magnitude, stress drop, rupture velocity

The magnitude of seismic events is measured by the amount of radiated seismic energy E_R. The scale of earthquake sources is measured by the seismic moment M₀ define by the product of rigidity μ, average fault slip D, and final rupture area S. The radiated seismic energy and the seismic moment are essentially different quantities; the former depends on the dynamic rupture process, but the latter does not. Kanamori (1977) supposed that the radiated seismic energy is given by the difference between the elastic strain energy released by shear faulting and the dissipated energy by frictional sliding, and obtained a scaling relation E_R = (Δσ/2μ)M₀, where Δσ denotes the stress drop, in an extremely simplified case. Taking the scale-independence of the stress drop into consideration, Hanks and Kanamori (1979) reduced the above scaling relation to E_R = 5×10^-5M₀, and defined the well-known moment magnitude scale M_w as log M₀ = 1.5M_w + 9.1. The moment magnitude scale works well except for some abnormal earthquakes. However, as pointed out by Matsu'ura (2024), the indirect estimation of radiated seismic energy mentioned above contradicts its direct evaluation, E_R = (60πρ)^-1(2V_P^-5+3V_S^-5)× ∫₀^ΔT[∂_t²M₀(t')]²dt', based on the analytical solution of displacement fields for a point dislocation source. By the way, the development of seismographic networks in the 21st century enabled us to directly evaluate the amount of radiated seismic energy. For example, Kanamori et al. (2020) directly evaluated the ratio e_R = E_R/M₀, which is called the scaled energy, from the KiK-net downhole records, and obtained the scaling relation E_R ≃ 3×10^-5M₀ for Japanese inland earthquakes with a magnitude range from M_w = 5.6 to 7.0. Their result is generally consistent with the Hanks–Kanamori's scaling relation, and assures the validity of the moment magnitude scale in a practical sense. Then, what is the theoretical basis of the moment magnitude scale? To answer this question, we consider a classical self-similar crack model, in which a shear crack concentrically expands from r = 0 to R with a constant stress drop Δσ at a rupture velocity v_r (the rupture duration ΔT = R/v_r). In such a case, the instantaneous fault slip distribution is written as D(r,t) = (24/7π)(Δσ/2μ)[(v_rt)²–r²]^1/2H(t–r/v_r), and so the corresponding moment function and its ultimate value (namely seismic moment) are given by M₀(t) = (16/7)Δσ(v_rt)³ and M₀= (16/7)Δσv_r³ΔT³, respectively. Substituting this expression of moment function into the formula for directly evaluating E_R, we obtain an interesting theoretical relation, E_R ≃ (Δσ/2μ)(v_r/V_S)³M₀, which can be regarded as a modified version of the Kanamori's scaling relation. The modified scaling relation clearly indicates the rupture-velocity dependence of the scaled energy e_R; for instance, under the assumption of Δσ/2μ = 5×10^-5, the scaled energy becomes 1.7, 2.6, and 3.6×10^-5 for v_r/V_S = 0.7, 0.8, and 0.9, respectively. So, the scaled energy 3×10^-5 estimated by Kanamori et al. (2020) implies that the average rupture velocity of normal earthquakes is within the range of 0.8—0.9×V_S.

REFERENCES
[1] Kanamori, H. (1977), The energy release in great earthquakes, J. Geophys Res., 82, 2981-2987.
[2] Hanks, T.C., Kanamori, H. (1979). A moment magnitude scale, J. Geophys. Res., 84, 2348-1249.
[3] Kanamori, H., Ross, Z.E., Rivera, L. (2020), Estimation of radiated energy using the KiK-net downhole records—old method for modern data, Geophys. J. Int., 221, 1029-1042.
[4] Matsu'ura, M. (2024), Reconsideration of the energy balance in earthquake faulting, Prog. Earth Planet. Sci., 11:2, https://doi.org/10.1186/s40645-023-00602-x.