The relation between M_E, M_L and M_w in theory and numerical simulations for small to moderate earthquakes

Abstract

This article addresses the question of how moment magnitude, M_w, local magnitude, M_L, and energy magnitude, M_E, of small to moderate earthquakes would scale with each other in an ideal elastic medium and why they scale differently in reality. It turns out that M_L, in the way it is commonly determined, is a poor and inconsistent measure of earthquake size. For moderate to large events, the situation could be improved with a new magnitude, M_A, that is not biased by the Wood-Anderson response. M_w is robust and can in principle be determined in a consistent way over the entire imaginable magnitude range, but it accounts only for the static component of the earthquake source. Could M_E be an alternative? A simple conceptual analysis reveals the relation between radiated seismic energy, seismic moment, static stress drop, apparent stress, radiation efficiency and rupture velocity. Numerical simulations based on an extended-source model show that, to estimate the radiated seismic energy (and thus also M_E), one needs to take into account the effects of rupture directivity and radiation pattern. Anelastic attenuation and the bandwidth of the recording instrument determine the magnitude limit below which pulse widths and corner frequencies remain nearly constant. Below this limit, all information about the dynamics of the source is lost and attempts to correct for this in the presence of noise and aleatory signal variability, even with well-calibrated attenuation models, are probably futile. We thus have to accept the fact that M_E (just as M_L) is a different measure of earthquake size for small earthquakes than for moderate to large earthquakes.

Appendix

1.1 Seismic energy, radiation efficiency and apparent stress

It can be shown (e.g. Aki and Richards 1980; Kostrov and Das 1988; Scholz 2002; Kanamori and Rivera 2006) that the total strain energy and thus the potential energy change, ΔW, associated with the earthquake process can be expressed as

$$ {\Delta} W = \bar{\sigma} \bar{D} F, $$

(64)

where F is the fault area, $\bar {D}$ the average dislocation, $\bar {\sigma } = (\sigma _{1} + \sigma _{2})/2$ the mean stress, and σ₁ and σ₂ are the shear stress before and after the earthquake. Essentially, (64) describes the total work done during the dislocation process at the earthquake source (Wyss and Brune 1968). During the earthquake rupture process, the strain energy is converted into radiated seismic energy, E_R, and into the so-called fracture energy, E_G, as well as frictional energy dissipated along the fault as heat, E_H. Thus, energy conservation requires that

$$ {\Delta} W = E_{R} + E_{G} + E_{H}. $$

(65)

Figure 6 in Venkataraman and Kanamori (2004b) is an instructive graphical representation of the relation between these different energy forms. As noted by these authors, it is probably more appropriate to interpret the fracture energy of earthquake as the energy which is mechanically dissipated in a finite volume near the damaged zone and outside of the main fault surface.

Equation 64 can be rewritten as

$$ {\Delta} W = \frac{\sigma_{1} - \sigma_{2}}{2} \bar{D} F + \sigma_{2} \bar{D} F. $$

(66)

If one assumes that at the end of the rupture process the stress σ₂ on the fault is equal to the dynamic friction, σ_f (neither under- nor overshoot), then the second term in Eq. 66 is equal to the frictional energy E_H. Under this assumption, with Δσ_s = σ₁ −σ₂ and with the definition of seismic moment, $M_{0} = \mu \bar {D} F$, the sum of radiated energy and fracture energy is

$$ E_{R} + E_{G} = \frac{1}{2} \frac{{\Delta}\sigma_{s}}{\mu} M_{0}. $$

(67)

With the definition of the radiation efficiency (e.g. Husseini and Randall 1976; Venkataraman and Kanamori 2004b; Bormann and Di Giacomo 2011),

$$ \eta_{R} = E_{R} / (E_{R} + E_{G}), $$

(68)

and with Eq. 67 this becomes

$$ \eta_{R} = 2 \frac{\mu}{{\Delta} \sigma_{s}} \frac{E_{R}}{M_{0}}. $$

(69)

This relation is equivalent to Equation 1 of Venkataraman and Kanamori (2004b), which in turn can be solved for E_R:

$$ E_{R} = \eta_{R} \frac{1}{2} \frac{{\Delta}\sigma_{s}}{\mu} M_{0}. $$

(70)

From Eq. 68, we see that η_R = 0 when E_R = 0 and that η_R = 1 when E_G = 0. With μ = 3.3 × 10⁴ MPa and Δσ_s = 3.3 MPa and assuming η_R = 1, we obtain the ratio E_R/M₀ = 5 × 10^− 5, which Kanamori (1977) used to estimate the energy radiated from large earthquakes (see also Vassiliou and Kanamori 1982; Bormann and Di Giacomo 2011).

In going from Eq. 66 to Eq. 67 and then on to Eq. 69, it was necessary to explicitly assume that σ₂ = σ_f. Thus, contrary to what is stated in Beeler et al. (2003), the definition of η_R given by Eq. 69 is valid only for the case of no under- or overshoot, whereas (68) is always valid. Modifications to Eq. 69 for the cases of undershoot (partial stress drop) and overshoot (stress drops to a value below the dynamic friction) are derived in Kanamori and Rivera (2006).

The radiation efficiency, η_R, should not be confused with the seismic efficiency, η, which is the fraction of the total strain energy that is radiated as seismic energy,

$$ \eta = \frac{E_{R}}{{\Delta} W}. $$

(71)

According to McGarr (1999), η ≤ 0.06. Making use of this definition of η and appealing once again to the definition of M₀, given above, (64) can be rewritten in terms of E_R as

$$ E_{R} = \eta \bar{\sigma} \frac{M_{0}}{\mu}. $$

(72)

This relation is already implicit in Eqs. 4 and 5 of Wyss and Brune (1968). Now, with the definition of apparent stress as $\sigma _{a} = \eta \bar {\sigma }$ (Wyss 1970), from Eq. 72 one obtains

$$ \sigma_{a} = \mu \frac{E_{R}}{M_{0}}, $$

(73)

and the ratio between apparent stress and static stress drop becomes

$$ \frac{\sigma_{a}}{{\Delta} \sigma_{s}} = \frac{\mu}{{\Delta} \sigma_{s}}\frac{E_{R}}{M_{0}}. $$

(74)

Combining (69) and (74), we obtain the important result

$$ \frac{\sigma_{a}}{{\Delta} \sigma_{s}} = \frac{1}{2} \eta_{R}. $$

(75)

Beeler et al. (2003) refer to this stress ratio as the Savage-Wood efficiency. In fact, this result was already derived by Savage and Wood (1971), who, in their Eq. 10 denote η_R by ξ. Savage and Wood (1971) concluded that apparent stress and static stress drop should satisfy the following inequality:

$$ \sigma_{a} \leq {\Delta} \sigma_{s} / 2, $$

(76)

where the equal sign applies to cases in which η_R = 1, which is equivalent to neglecting the contribution of E_G in Eq. 65. Note again that Eqs. 73 and 74 are always valid, whereas (70) and (75), being based on Eq. 69, are valid only for the case of no under- or overshoot.

1.2 Seismic energy radiated by the Brune source model

The amplitude spectrum of the moment-rate function for the source model of Brune (1970, 1971) with seismic moment M₀ and corner frequency f_c is

$$ |\dot{M}(f)| = \frac{M_{0}}{1 + (\frac{f}{f_{c}})^{2}}. $$

(77)

If we apply (36) to this spectrum, we obtain

$$ E_{\beta} = \frac{\pi^{2}}{5} \frac{{M_{0}^{2}} {f_{c}^{3}}}{\mu \beta^{3}} = 1.974 \frac{{M_{0}^{2}} {f_{c}^{3}}}{\mu \beta^{3}}. $$

(78)

Note that Eq. 78 implies that $E_{\beta } \propto {M_{0}^{2}} {f_{c}^{3}}$, which is equivalent to $E_{\beta } \propto {M_{0}^{2}} / T^{3}$ implicit in Eq. 15.

According to Brune (1971), the corner frequency of the spectrum in Eq. 77 is given by

$$ f_{c} = \frac{2.34}{2\pi} \frac{\beta}{L}, $$

(79)

where β is the shear-wave velocity in the source region and L is the radius of the fault patch that ruptured (assumed to be circular). Solving (22) for L, and inserting the result in Eq. 79, the corner frequency becomes

$$ f_{c} = (\frac{2.34}{2\pi}) (\frac{16}{7})^{(1/3)} \beta (\frac{{\Delta} \sigma_{s}}{M_{0}})^{(1/3)} . $$

(80)

The cube of this equation is ${f_{c}^{3}} = 0.118 \beta ^{3} {\Delta }\sigma _{s} / M_{0}$, which, inserted in Eq. 78, gives the analytical expression for the S-wave energy radiated by the Brune source model as

$$ E_{\beta} = 0.233 \frac{M_{0} {\Delta}\sigma_{s}}{\mu}. $$

(81)

This result is identical to the result implicit in Eq. 20 of Singh and Ordaz (1994). Although the Brune model includes a source dimension (L), it is essentially a point source that accounts neither for the orientation of the fault nor for the rupture directivity associated with the rays leaving the source. So we can use the theoretical point-source ratio (q = 1/23.4) of the P- and S-wave energies (Boatwright and Fletcher 1984) to compute the total radiated seismic energy of an earthquake according to the Brune model, which then is

$$ E_{R} = 0.243 \frac{M_{0} {\Delta}\sigma_{s}}{\mu}. $$

(82)

With reference to Eq. 30, we see that for the Brune model

$$ \frac{\sigma_{a}}{{\Delta}\sigma_{s}} = 0.243, $$

(83)

and if we compare (82) with Eq. 70, we see that 0.243 is equivalent to half the radiation efficiency. Thus, in the light of this interpretation, we must conclude that the radiation efficiency of the Brune model is η_R = 0.486. It is noteworthy that this conclusion is at odds with the basic assumption underlying the Brune model, namely that the stress drop occurs instantaneously, which would imply that η_R = 1.