Do Aftershock Probabilities Decay With Time?

doi: 10.1785/0220120061

Upon hearing about the devastating 22 February 2011 M 6.1 Christchurch earthquake, I remembered the nearby 4 September 2010 M 7.0 event and thought, “It’s a bit late for this to be an aftershock,” followed quickly by, “Don’t be stupid.” The first thought was based on the modified-Omori law (Utsu, 1969) in which the rate of aftershocks, exceeding a particular magnitude, decays as

where n is the number of earthquakes per unit time as a function of time (t) and K, c, and p are constants. The second thought was triggered by remembering that while the probability of this event being an aftershock on 22 February 2011 was small, it was necessary to consider the total probability over all days that I would consider “late.”

So, do aftershock probabilities decay with time? Consider a thought experiment in which we are at the time of the mainshock and ask how many aftershocks will occur a day, week, month, year, or even a century from now. First we must decide how large a window to use around each point in time. Let’s assume that, as we go further into the future, we are asking a less precise question. Perhaps a day from now means 1 day ± 10% of a day, a week from now means 1 week ± 10% of a week, and so on. If we ignore c because it is a small fraction of a day (e.g., Reasenberg and Jones, 1989, hereafter RJ89), and set p = 1 because it is usually close to 1 (its value in the original Omori law), then the rate of earthquakes (K/t) decays at 1/t. If the length of the windows being considered increases proportionally to t, then the number of earthquakes at any time from now is the same because the rate decrease is canceled by the increase in the window duration. Under these conditions we should never think “It’s a bit late for this to be an aftershock.”

But is this idealized experiment a reasonable approximation of how society should think about aftershock hazard? Three months after the 22 February event, I had the opportunity to meet with a delegation from the Christchurch City Council and leaders of their business community.We discussed the idea that, while the rate of aftershocks was decreasing, they had moved from the immediate response phase into the short-term restoration of services phase and were in the course of planning for the longer-term rebuilding stage; consequently, the risk from aftershocks to each phase of the process would remain significant. While this made intuitive sense to them, let’s explore it quantitatively.

Urban planners divide disaster recovery into a series of overlapping stages: (1) Emergency, consisting of search and rescue, fire fighting, providing emergency shelters, and damage assessment; (2) Restoration, consisting of the restoration of utility services, debris removal, and making temporary repairs to structures to render them usable for the short term; (3) Reconstruction, during which structures are replaced to predisaster levels; (4) Betterment, during which major redevelopment projects improve the community to a new standard (Haas et al., 1977); and to these I add (5), the Long-Term future (note that emergency managers use a different set of stages). The duration of these five phases varies with the scale of a disaster, the success of pre-disaster planning, governmental processes, and local socioeconomic factors. Some documented examples are the 1994 M 6.7 Northridge, the 1995 M 6.8 Kobe earthquakes (Olshansky et al., 2011), and the two Christchurch earthquakes (Anne Wein, personal comm., 2012). To prevent double counting earthquake probabilities in this study, the five stages are broken into non-overlapping windows based on which phase may be dominant at any given time. For that, I apologize to disaster recovery experts who must manage the complexity of overlapping recovery stages that is further complicated by damaging aftershocks.

Based on the studies referenced above, Stage 1, Emergency, is dominant for the first 2 weeks. Almost all survivors are rescued in the first few days but some rescues occur later and other emergency responses last about 2 weeks. Stage 2, Restoration, is dominant from 2 weeks to 1 year. Reconstruction is primarily accomplished within 3 years and so Stage 3 is dominant from 1 to 3 years. Betterment and redevelopment projects go on for several more years, and so Stage 4 may be dominant from 3 to 10 years after the earthquake. Finally, if the duration of the long-term future is the 50-year window frequently used in probabilistic seismic hazard assessment, then Stage 5 is dominant from 10 to 50 years.

For each time period, I compute the probabilities of an aftershock with magnitude M_a ≥ M_m − 1, where M_m is the mainshock magnitude, using the generic model of RJ89 which is valid until a large aftershock creates a secondary sequence. If M_m = 7, then the probabilities are for an M_a≥6 event, which could do substantial damage. These are just example values, and the relative temporal variation in the probabilities, which are the focus of this piece, would remain the same for other choices of M_m and M_a. For comparison, I also compute the probability of an independent event with a rate of 0.01/year, which is reasonable if the mainshock has a rate of 0.001/year.

The results are shown in Table 1. While the probability of an aftershock exceeding M_a is the largest during the initial Emergency period (61%), it remains quite large during the yearlong Restoration period (33%) and substantial for the remaining time periods (11%–4%). The temporal variation of these probabilities is primarily sensitive to the value of p and so it is important to explore a reasonable range of values. In RJ89 p = 1.08, but, if p = 1, then the later time periods have greater probabilities. If p is higher, then the probabilities during the later time periods are relatively lower. This is demonstrated using a three time window model (Felzer et al., 2003) in which p increases from 0.75 at t = 0 to 1.34 for t ≥ 10 days. For all three models, the aftershock probabilities are significant with respect to the independent event probabilities, except for the last two stages if p = 1.34.

The parameters for these models are not based on very long aftershock sequences, so we need to ask if modified- Omori behavior is valid for long time periods. A rare example of a very long data set is the 1891 M8¼ Nobi earthquake. Omori fit its aftershocks to his formula through 1899, and then Utsu showed that they fit the same decay through 1968 (Utsu, 1969). Despite this 77-year example, we have a lot to learn about aftershock behavior over long time horizons.

Table 1 Stages of Disaster Recovery and Aftershock Probabilities
Stage	Dominant Time Period Post- Mainshock	Probability of an Aftershock With Ma ≥ Mm − 1			Probability of an Independent Event With a Rate of 0.01 Events/Year
		RJ89 Generic Model, p = 1.08	RJ89 Generic Model Modified to p = 1	Felzer, 2003, Multi-Window Model, p = 1.34
1—Emergency	0–14 days	61%	61%	35%	0.04%
2—Restoration	14 days–1 year	33%	43%	10%	1%
3—Reconstruction	1–3 years	11%	17%	2%	2%
4—Betterment	3–10 years	11%	19%	1%	7%
5—Long-Term	10–50 years	13%	24%	1%	33%

We have not been conveying the extended duration of aftershock sequences or their societal importance to the public. Many messages simply have not covered the later stages of the recovery process. For example, the STEP forecasts are for 1 day (Gerstenberger et al., 2005) and the routine USGS aftershock message released after M≥5 events in California covers 7 days. The USGS releases on the hazard in Haiti after the 2010 M 7 earthquake limited aftershock discussions to 1 year. But we did not mention them for longer periods, although they were combined with standard long-term models in the calculations. Perhaps a saving grace for long-term hazards assessments is the common use of the Gardner and Knopoff (1974) algorithm to remove aftershocks from a seismicity catalog. Their parameters limit aftershock sequences to less than 3 years, so aftershocks after that are included as independent events in hazards assessments. Treating them as independent events misses the temporal connections between them, but at least assures that they aren’t ignored.

As we move into the era of Operational Earthquake Forecasting (Jordan and Jones, 2010), we need to learn more about earthquake clustering over long time periods, how society can be affected by aftershocks throughout the earthquake recovery process, and how best to convey this information to emergency managers, planners, and the public.    

Andrew J. Michael USGS 345 Middlefield Road Menlo Park, California 94025, U.S.A