Dynamical scaling and generalized Omori law


  • E. Lippiello,

    1. Department of Information Engineering, Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Second University of Naples, Aversa, Italy
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  • M. Bottiglieri,

    1. Department of Environmental Sciences, Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Second University of Naples, Caserta, Italy
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  • C. Godano,

    1. Department of Environmental Sciences, Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Second University of Naples, Caserta, Italy
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  • L. de Arcangelis

    1. Department of Information Engineering, Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, Second University of Naples, Aversa, Italy
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[1] The power law decay of the aftershocks rate is observed only after a characteristic time scale c. The dependence of c on the mainshock magnitude MM and on the lower cut-off magnitude MI is well established. By considering ten sequences recorded in the California Catalog we show that the aftershock number distribution becomes independent of both MM and MI if time is rescaled by an appropriate time scale fixed by the difference MMMI. This result is interpreted within a more general dynamical scaling hypothesis recently formulated, relating time differences to magnitude differences. The above hypothesis gives predictions in good agreement with the recent findings by Peng et al. (2007).

1. Introduction

[2] Statistical models for seismic hazard evaluation are generally based on two well established empirical laws: the Gutenberg-Richter (GR) law [Gutenberg and Richter, 1944] and the Omori law [Omori, 1894]. The GR law states that the distribution P(M) of magnitude M follows an exponential behaviour P(M) ∼ 10bM, where usually b ≃ 1. The Omori law predicts that the number of earthquakes, after the occurrence of a large seismic event, decays in time as a power law, n(t) ∼ (t + c)p, where p ≃ 1. A third scaling relation is the productivity law stating that the number of aftershocks of an MM mainshock is proportional to image with αb [Felzer et al., 2004; Helmstetter et al., 2005]. Combining the above three laws one has that the probability to have an aftershock with magnitude M > MI at the time t after a MM mainshock occurred at the time tM is given by

equation image

[3] A further relation is the Båth's law [Båth, 1965] stating that the magnitude difference between the main shock and the maximum detected aftershock, ΔM ≃ 1.2, is approximately a constant independent of the main shock magnitude.

[4] A topic of wide interest is whether the parameter c has a physical meaning or it is an artifact of instrumental deficiency [Narteau et al., 2002; Shcherbakov et al., 2004; Kagan, 2004; Lise et al., 2004; Lolli and Gasperini, 2006; Peng et al., 2006, 2007]. Shcherbakov et al. [2004, 2005a] show that c scales with the lower magnitude cut off MI. More precisely, combining the GR, the Omori and the Båth laws, they obtain for p > 1

equation image

where β is a constant and c* represents a lower cut off for c's. Using the above dependence of c on MI and MM, they obtain the so-called generalized Omori law that fits very well experimental data. This law, inserted in a non-homogenous Poisson process, reproduces the experimental scaling behavior of interoccurrence time distribution [Shcherbakov et al., 2005a, 2005b, 2006].

[5] A result similar to equation (2) was obtained by Kagan [2004] who investigated the magnitude and temporal distribution during the early stage of aftershock sequences for several strong earthquakes. He observed that the effective sequence starting time c depended on the magnitudes of the mainshock and aftershocks as

equation image

where M1 = 4, δ = 1 are experimental constants and time is measured in days. Kagan proposed a Short Term Aftershocks Incompleteness (STAI) hypothesis stating that many small aftershocks are not reported in the earthquake catalogues during the first days after the main event. According to this hypothesis the dependence of c on MI and MM is only an artifact and catalogs result complete for MMI only for times larger than c. This implies a threshold magnitude MC depending on the time from the main-shock, by setting ttM = c in equation (3)

equation image

[6] For M > MC, the GR law is satisfied whereas for M < MC an almost flat behavior of P(M) is expected. This is consistent with experimental data [Kagan, 2004; Shcherbakov et al., 2006; Enescu et al., 2007]. According to equation (4) for a MM = 7 mainshock, the catalog can be considered complete above MI > 2 only after 10 days from the main event. A direct inspection [Helmstetter et al., 2005] of the time-magnitude distribution for several main-aftershock sequences gives a time dependent threshold magnitude that follows equation (4) but with δ = 0.75 and M1 = 4.5. Notice that equations (2) and (3) are equivalent if δ = equation image and M1 = ΔMδlog10(c*). Furthermore, the dependence of c on the main magnitude has also been observed by Lise et al. [2004] in their analysis of aftershock sequences.

[7] Recently, Lippiello et al. [2007] have considered subsets of the Southern California Catalog, each containing 125 earthquakes with magnitude M ≥ 3, and have computed the average magnitude inside each subset. The averaging procedure reduces statistical fluctuations and allows to detect a significant non-null magnitude correlation not related to STAI. They propose a dynamical scaling (DS) hypothesis relating magnitude to time differences to explain the observed magnitude correlations. This idea originates from a critical dynamics approach in which energy and time do not behave as independent quantities, but one observes a self-similar behaviour if energy is rescaled by an appropriate power of time. Lippiello et al. are also able to obtain the GR and the Omori law and to generate synthetic catalogues reproducing the statistical features of experimental ones.

[8] In this paper we apply the DS hypothesis to the analysis of aftershock sequences and discuss the physical meaning of the constant c. In particular we show that its dependence on MM and MI can be a direct consequence of the DS hypothesis. In the Discussions we compare our prediction with the recent findings of Peng et al. [2007].

2. Dynamical Scaling Hypothesis

[9] The DS hypothesis [Lippiello et al., 2007] proposes that the magnitude difference MiMj fixes a characteristic time scale

equation image

where τ0 is a constant measured in seconds. As a consequence, the conditional probability density p(Mi, tiMj, tj) to have an earthquake of magnitude Mi at time ti given an earthquake Mj at time tj, is magnitude independent when time is rescaled by τij

equation image

where F(z) is a normalizable function. It is possible to show that many statistical features of earthquake occurrence can be recovered using different forms for F(z) [Lippiello et al., 2007]. In order to have an easier comparison with the STAI hypothesis here we consider the choice

equation image

where A is fixed by normalization. From equations (6, 7) one immediately observes that p(Mi, tMM, tM) = A independently of M for zp = τ0p (ttM)p 10b(MMM) ≪ 1, whereas the GR behaviour p(Mi, tMM, tM) ∝ 10bM is obtained in the opposite limit zp ≫ 1. Hence, the condition τ0p(ttM)p image = 1 fixes a crossover magnitude Mc that separates flat from GR behavior. MC is in agreement with the empirical equation (4) with δ = p/b and M1 = −p/b(log τ0).

[10] Next we evaluate the aftershock probability PAS(ttM, MIMM) = equation imagedM p(M, tMM, tM).

[11] From equations (5, 6, and 7) one obtains

equation image

with K = image For ttMτ0K, equation (8) coincides with equation (1) in the regime tc with α = b. As a consequence it is quite natural to identify τ0K with the time scale c fixing the onset of the power law behaviour in the Omori law. Therefore, the DS hypothesis implies a cτ0K in agreement with equation (2), but also constraints the exponents β, b, p, taken as independent by Shcherbakov et al. [2004], to satisfy the relation β = b(2 − 1/p). The experimental values fitted by Shcherbakov et al. [2005a, 2006] fulfill the above relation. In other words the only assumption of equation (6) allows to obtain the Omori law and the scaling of c with MMMI. The important consequence of the above result is that the aftershock number decay is independent of the mainshock magnitude MM and of the threshold magnitude MI when time is rescaled by K.

[12] To investigate this point we have considered ten main-aftershocks Californian sequences in the years 1975 and 2005 with MM ∈ [5.6, 7.3] and different values of MI. We always consider as aftershocks all the events occurred in the box of size 1° × 1° centered in the main-shock epicenter. Data are provided by the Northern California Earthquake Data Center (http://www.ncedc.org). In Figure 1 we show the PAS(ttM, MIMM) for four different sequences (the MM = 7.3 June 28, 1992 Landers earthquake; the MM = 6.7 January 17, 1994 Northridge earthquake; the MM = 7.1 October 16, 1999 Hector Mine earthquake and the MM = 6.5 December 22, 2003 San Simeon earthquake). Curves corresponding to different magnitude thresholds exhibit different c values, ranging from few days for MI = 2.0, to approximately zero when MI = 3.5. This result is in good agreement with Shcherbakov et al. [2004]. Next, following the relation (8), we have rescaled time by K and plotted the aftershock rate as function of (ttM)/K. Figure 2 shows that for all the sequences, curves with different MI collapse onto the same master curve. The same behaviour is observed for six more sequences not reported here. The only tunable parameter is the ratio b/p, whose values, reported in Table 1 for the different sequences, are determined as the ones providing the best collapse.

Figure 1.

Probability of aftershocks versus ttM for four different sequences. The time decay obeys the Omori law for different values of MI.

Figure 2.

Probability of aftershocks versus (ttM)/K for the same sequences as in Figure 1. Rescaling time by K defined in equation (11) leads to the collapse of the different curves, for different MI, onto the same master curve.

Table 1. Values of b/p for the ten different analysed sequences
Imperial Valley6.41979/10/150.7
North Palm Spring5.61986/07/081.1
Superstition Hills6.21987/11/240.8
Hector Mine7.11999/10/161.1
San Simeon6.52003/12/221.1

[13] We observe curve collapse also if we consider different sequences, corresponding to different MM (Figures 3 and 4) and the same MI = 2.5: as expected c increases with MM, from ≈3 hours for San Simeon earthquake (MM = 6.5) to ≈3.5 days for Landers earthquake (MM = 7.3) (Figure 3). The curves, again, collapse onto the same master curve when time is rescaled by K (Figure 4). In Figure 4 we also compare numerical data with the Omori law equation (1) for c = τ0K, and with the predictions of equation (8) using p = 1.1 and τ0 = 0.5 sec as best fit parameters. Equation (8) fits numerical data for ttM > K whereas predicts a lack of recorded events at early times from the mainshock.

Figure 3.

PAS(ttM, MIMM) versus ttM for four different sequences with MI = 2.5.

Figure 4.

PAS(ttM, MIMM) versus rescaled time (ttM)/K for the same sequences as in Figure 3. The curves collapse on the same master curve. The dashed violet curve represents the prediction of equation (8), whereas the dot-dashed pink curve is the Omori law with c given by equation (3). In the inset, the average aftershock probability from equations (9) and (8) (circles) is compared with the power law behaviors, (ttM)−0.58 at short times and (ttM)−0.92 at long times, fitted by Peng et al. [2007], represented as solid blue lines.

3. Discussions

[14] The previous analysis shows that PAS(t, MIMM) does not depend on MI and MM when time is rescaled by K. Within the DS hypothesis, the dependence of c on MI and ML has a physical origin related to a “critical” behaviour of the earthquake occurrence. Although the mechanical details of this criticality are not yet clear, Shcherbakov et al. [2006] suggest a mechanism based on a cascade of stress from large scales to small ones. Within the STAI approach, conversely, the collapse of the rescaled curves is only an artifact due to catalog incompleteness. Therefore, in principle, if one was able to detect all occurred events, would find the Omori power law behaviour extending down to very short times. This implies a basic difference with the STAI hypothesis in the behaviour of PAS(t, MIMM) for times ttM < K. Equation (8), indeed, predicts a logarithmic behaviour at short times different than the Omori law (ttM)p predicted by the STAI hypothesis. This different behaviour implies that the number of events in the early part of the sequence, within the DS approach, is significantly smaller than the one expected according to STAI. To this extent an interesting result has been recently obtained by Peng et al. [2007]. They use a careful high-frequency filtering procedure to seismic signals of shallow earthquakes (3 ≤ MM ≤ 5) recorded in Japan. Focusing on the time interval ttM ≤ 200 sec they find a number of missing events, with M ≥ 0, about five times larger than the number of events previously recorded in the Japan catalog. This number, however, it is not sufficient to match the behaviour (ttM)p fitted at large times. More precisely, they measure the average equation imageAS(ttM) obtained by summing the contribution of all the analysed sequences and they find equation imageAS(ttM) ∼ (ttM)q with q = 0.58 for t < 900 sec, to be compared with the Omori behaviour equation imageAS(ttM) ∼ (ttM)p with p = 0.92 observed for t > 900 sec. It is possible to show that the result of Peng et al., that excludes an Omori law extending to small times, is consistent with the DS approach. From the above definition one has

equation image

with PAS(ttM, MIMM) given in equation (8). We use the parameters fitted by Peng et al., p = 0.92 and P(MM) = image τ0 = 0.5 sec obtained from Figure 4 and MI = 1. The result is presented in the inset of Figure 4. The comparison with the two power laws, obtained by Peng et al. as best fit, show very good agreement between the DS prediction and experimental data. This indicates that the logarithmic behaviour for the aftershock number distribution (equation (10)) is more appropriate to characterize the early part of the sequences.


[15] This work is part of the project “Terremoti probabili in Italia nel trentennio 2005–2035” of the Italian Civil Protection Department, convention ProCiv-INGV 2004-06.