New Source Duration Relationships for Mexican Earthquakes: Practical Application to Stochastic Summation Methods

Abstract

From the gathering of 127 source time functions (STFs) of seismic events in Mexico, relationships have been developed between the seismic moment (M₀) and the total source rupture duration (τ_d), for normal-faulting inslab, interplate and strike-slipe mechanisms. This relationship has been explored by different authors like Furomoto and Nakanishi (J Geophys Res 88:2191–2198, 1983), Tanioka and Ruff (Seismol Res Lett 68:386–400, 1997), Singh et al. (Bull Seismol Soc Ame 90:468–482, 2000), among others; the latter authors present a similar general relationship for Mexico, considering 53 STFs and with information until 1998. In this research, complementary databases have been integrated, and the available functions updated base on recent research works about the SCARDEC method are used (Vallée et al. in Geophys J Int 184:338–358, 2011; Vallée and Douet in Phys Earth Planet Interiors 257:149–157, 2016). With the proposed expressions, it is possible to estimate reasonably, the main parameters of interest for algorithms of stochastic summation, such as the corner frequencies or stress drops in order to simulate strong ground motions, commonly used for source-rupture studies, site-amplification analysis, or for earthquake-resistant design. Besides, a new flexible model of stochastic summation is presented, which allows considering several ways to generate more realistic motions, according to the spectral behavior, both for low and for intermediate frequencies, under earthquakes with moderate and high magnitude; this model was compared with results from a two-stage alternative method considering two corner frequencies.

Appendix: Alternative Form for SSM TS-TCFM

In Sect. 5.1, simulations were performed with the proposed approach and additionally, comparison was carried out with the SSM of two-stages defined by two corner frequencies TS-TCFM, based on the principles of Kohrs-Sansorny et al. (2005) and Niño et al. (2018); although the latter authors recently presented this concept, two critical points were not discussed then: (1) the effects on simulations due to seed earthquakes that do not follow the model of two corner frequencies, which, as it has been shown, has a considerable impact depending on the selected record to be scaled, and (2) it is not explained in detail how the total number of time delays is distributed in the first and the second stage; according to Kohrs-Sansorny et al. (2005) and for the considered two-stages method, the N = f_c/F_c ratio must be an integer value, where f_c denotes the corner frequency for the seed and F_c for the event to be simulated. The above mentioned implies that not all combinations of M₀ and Δσ are numerically possible, without rounding or excluding time delays. Below, an alternative way to define the TS-TCFM depending only on corner frequencies and seismic moment is presented.

For the source spectra averaged and overall simulations to reproduce the ω² model defined by TCF, it is necessary that the probability densities |ρ_c(f)| and |ρ_d (f)| follow the next equality:

$$\frac{S{s}_{S}(f)}{S{s}_{R}(f)}={\left[\frac{1+({\eta }_{c}-1)\cdot {|{\rho }_{c}(f)|}^{2} }{{\eta }_{c}}\right]}^{1/2}\cdot {\left[\frac{1+({\eta }_{d}-1){\cdot |{\rho }_{d}(f)| }^{2}}{{\eta }_{d}}\right]}^{1/2},$$

(13)

where Ss_S and Ss_R are the spectra defined by two corner frequencies, according to Eq. (12), and for the seed and simulated earthquake, respectively; η_c is in the first stage, a small number of delays t_c randomly generated with a probability density ρ_c (t), in the second stage, η_d delays t_d are once again generated with a second probability density ρ_d (t); suggested procedures for estimating the time delays can be found for example in Ordaz et al. (1995) and Kohrs-Sansorny et al. (2005).

In this research, the probability densities in the frequency domain are defined by:

$$|{\rho }_{c}(f)| ={\left[\left[\left[{\left[\frac{(1+{\left(f/{F}_{da}\right)}^{2}\cdot (1+{\left(f/{F}_{db}\right)}^{2}}{[(1+{\left(f/{F}_{a}\right)}^{2}\cdot (1+{\left(f/{F}_{b}\right)}^{2}{]}^{1/2}}\right]}^{2}\cdot {\eta }_{c}\right]-1\right]\cdot \frac{1}{{\eta }_{c}-1}\right]}^{1/2}$$

(14)

$$|{\rho }_{d}(f)| ={\left[\left[\left[{\left[\frac{(1+{\left(f/{f}_{a}\right)}^{2}\cdot (1+{\left(f/{f}_{b}\right)}^{2}}{[(1+{\left(f/{F}_{da}\right)}^{2}\cdot (1+{\left(f/{F}_{db}\right)}^{2}{]}^{1/2}}\right]}^{2}\cdot {\eta }_{d}\right]-1\right]\cdot \frac{1}{{\eta }_{d}-1}\right]}^{1/2}$$

(15)

where

$${F}_{da}={F}_{a}\cdot {\eta }_{c}^{1/4}\text{ and }{F}_{db}={F}_{b}\cdot {\eta }_{c}^{1/4}$$

(16)

Also, and according to Boatwright and Choy (1992), Atkinson and Boore (1998) and García et al. (2004), f_a and/or F_a = 1/(2τ_d); there, τ_d can be obtained from the relationships generated in Sect. 2.2, f_c and/or F_c are estimated by fitting an ω² model to the source acceleration spectrum or, alternatively, they can be estimated through the relationships generated in Sect. 3. To obtain f_b and/or F_b, it is required that the high-frequency spectral level from the two-corner source spectrum, f_a·f_b·M_0R (i.e., for the seed case), to be equal to the corresponding level for the ω² source f_c²·M_0R (García et al. 2004; Niño et al. 2018), that is f_b = (f_c/f_a^0.5)².

Regarding the number of time delays t_c and t_d, it is mandatory to point out that in total, N⁴ = η = η_c·η_d; notice the essential condition η = (f_c/F_c)⁴ and that N is the parameter that must be an integer. For this reason, only discrete values of f_c and F_c can be selected (Kohrs-Sansorny et al. 2005; Salichon et al. 2010).

Finally, the information needed to simulate the acceleration time histories is a function that describes the signal which would arrive at the seismic station and the equivalent source time function ESTF R (t). For each simulation, the simulated event S_k (t) (Eq. 18) is given by the convolution of the recorded small event S_R (t) used as empirical Green’s function, and the ESTF R(t), for two-stages, defined by:

$${R}_{k}\left(t\right)=\xi \cdot {\sum }_{j=0}^{{\eta }_{d}-1}\left[{\sum }_{i=0}^{{\eta }_{c}-1}\delta (t-{t}_{c}-{t}_{d})\right],$$

(17)

$${S}_{k}\left(t\right)={R}_{k}\left(t\right)* {S}_{R}\left(t\right).$$

(18)

In Eq. (17), ξ is a scaling parameter obtained through the ω² source relationships (Brune 1970) and described in Eq. (7); note that, unlike the one-stage method, a second stage of random delays is introduced with a second probability function ρ_d (t), which can be interpreted as a sequence of sub-ruptures during the duration time of each of the main ruptures.

1.1 Numerical Test

In two-stage methods, η_c can be selected in different ways. A common way is to take η_c = N and η_d = N³ as was done in the simulations of this research, but it can also be taken as η_c = η_d = N², or on the extreme side η_c = N⁴, equivalent to the one-stage (OS) summation scheme proposed by Ordaz et al. (1995). The influence on the simulations of this constant is such that increasing the number η_c of generated delays in the first stage would result in more similar time histories but, the mean of simulations will tend to be the same.

For numerical test, the seismic moment of the target event is assumed 1000 times larger than those of the small event (difference of magnitude units: M_w8–M_w6 = 2, 1.22 × 10²¹ N m/1.22 × 10¹⁸ N m = 1000). The corner frequencies are estimated by the self-similar scaling (Eq. 3) and then f_c = 0.671 and F_c = 0.0671, in total η = 10⁴ (η_c = 10 and η_d = 1000) small events represented by a Dirac delta function are summed together. Analogously, τ_d is determined by relationships proposed in Sect. 2.2, then f_a = 0.153 and F_a = 0.0153 and therefore, f_b = 2.946 and F_b = 0.2946; with Eqs. (9) and (11) and the previous data, it can be easily constructed the theoretical spectral ratio for a model defined by two corner frequencies (Fig. 8 -right) assuming that the seed and the target event follow the same model (which will be the same for both, the one and two stages methods). For the presented numerical test, 500 simulations were performed showing that, whereas the one-stage summation schemes (η_c = N⁴) produce similar time histories with an excessive concentration of energy around the half-duration of the rupture (Fig. 8-left), the two-stage summation schemes can distribute the energy during the whole rupture duration (Fig. 8-center) and produce different time histories, which can be associated with a multitude of rupture processes (Kohrs-Sansorny et al. 2005).

In Fig. 8 (right), the average over 500 simulations obtained using the OS-TCFM (continuous red line) and TS-TCFM (discontinuous orange line) schemes are compared with the teoritical spectral ratio (thick blue line) (reference or target event/small event); it is important to note that, for both methods, the average of the simulations tends to match correctly to the theoretical reference spectrum, which does not happen in results presented by Niño et al. (2018), where at low frequencies (f < 0.09 Hz) the average moves away from the theoretical reference spectral level. Nevertheless, according to the theoretical basis of the current stochastic summation methods, it should not occur, the reason is unknown, and authors do not explain or address that in detail; however, this probably happens because of some consideration taken by the authors, in cases where N is not an integer.

Fig. 8

Equivalent source time functions obtained using the OS-TCFM (left), the TS-TCFM (center), and the average of spectral ratio over 500 simulations (right) obtained using the OS-TCFM (continuous red line) and TS-TCFM (discontinuous orange line) compared to different theoretical models analyzed in this study