USING THE QTM CATALOG
3.4 Testing for Variations of the Background Seismicity Rate
migration of similar amplitude each year. In addition to this clear seasonality, we note a substantially larger gap between 2010 and 2011. The annual loop during that year shows increased seismicity starting in April at approximately the time of the Cucapah-El Mayor earthquake and lasting for a few months.
We perform the same Schuster analysis as detailed above on the HYS catalog as well (Figure 3.6 c-d). Like for the QTM catalog, we observe a drift to low p-values at periods larger than a few months, which suggests non-stationarity of the earthquake rate. Again however, whether this non-stationarity is in the background rate itself or due to aftershocks in the catalog is not obvious. Unlike the QTM catalog, there is no clear seasonal periodicity at one year, and we see no significant migration of the end points in the yearly random walk. For each cycle, the probability that the endpoint location is still the product of a random walk is greater than 1% (Figure 3.6d). The plot however shows considerable variation of the size of the annual loops, suggesting that the non-stationarity of the process is not due to a single transient period of increased seismicity, as was the case for the QTM catalog. Based on the QTM Schuster spectra though, we see two factors contributing to the departure from the hypothesis of a stationary Poisson process, one being the seasonal variations of seismicity and the other being a transient increase of the rate of mainshocks in 2010.
The HYS catalog also suggests apparent variations of the background seismicity rate. However, the test does not show whether the apparent non-stationarity of the background rate in both catalogs is true or due to remaining aftershocks.
occurrence rate, with a mean IET of about𝑡
0 =0.0367 days, while the time period after the earthquake had a mean IET of about𝑡
0 =0.0361 days (Figure 3.7a). The two IET distributions for the two periods are likewise different as they have slightly different Poisson parameters (Figure 3.7b). However, individually both sets of the catalog still follow an exponential distribution and appear to be stationary.
0 2 4 6
Years 0
20 40 60 80
QTM No. of Events (x103 )
(a)
2008-2009 2011-2018
0 0.1 0.2 0.3 0.4 0.5
Interevent Time (Days) 100
102 104
Cumulative Count
(b)
Poisson Model Data
0 5 10 15 20 25
Years 0
2 4 6 8 10
HYS No. of Events (x103 )
(c)
1981-1989 1990-2017
0 5 10 15 20 25
Interevent Time (Days) 100
102
Cumulative Count
(d)
Figure 3.7: IET distributions of the QTM and HYS catalogs split before and after historical changes in the earthquake rate. (a) Cumulative number of events with time and (b) the interevent time distribution from the QTM catalog for events before (Jan. 2008 – Dec. 2009; black line) and after (Jan 2011 – Dec. 2017; gray line) the 2010 El Mayor Cucapah M 7.3 earthquake in Mexico, omitting the year 2010. Dashed lines in (b) are the Poisson models for each subset of the catalog and solid lines are the QTM data. (c) Cumulative number of events with time and (d) the interevent time distribution from the HYS catalog for events before (Jan. 1981 – Dec. 1989; black line) and after (Jan. 1990 – Dec.2017; gray line) 1990, where there is a notable change in the mean earthquake rate.
Dashed lines in (d) are the Poisson models for each subset of the catalog and solid lines are the HYS data.
To further investigate non-stationarity, we also examine earthquake occurrences for each year in the catalog individually, to see if the earthquake rate, or equivalently the mean interevent time, is changing systematically year to year. Figure 3.8 illustrates
(a) the number of events with time for each year in the catalog plotted separately against days in the year, and (b) the IET distributions for each individual year. All of the years, except 2010, have roughly the same mean IET, averaging about 0.0357 days. Only that of 2010, with an IET of 0.0304 days, is significantly different from the others, falling more than two standard deviations lower than the mean IET across all years (Figure 3.8c). This drop in mean interevent time, meaning a higher earthquake occurrence rate, coincides with the El Mayor-Cucapah event, suggesting there are still a number of remaining aftershocks and/or an increase of the background seismicity following the El Mayor-Cucapah earthquake. Likewise, the individual IET distributions of each year more-or-less collapse onto a single exponential distribution and their variation is within that expected for the Poisson model, as discussed below. The only distribution that notably differs is that of 2010 (Figure 3.8b). These results suggest that the slight tail on the IET distribution of the QTM catalog is a signature of temporary non-stationarity due to the presence of remaining aftershocks in the declustered catalog. The lower mean IET of 2010 when included in the whole catalog acts to skew the associated Poisson model to the left, giving the appearance of a fat-tail. Non-stationarity, however, does not appear to be long-lived past the end of the aftershock sequence, as the mean IET returns to its pre-El Mayor-Cucapah value. Note that the large size of the catalog allows for detecting very small changes of the background seismicity rate of only a few percent.
The 2010 event is also present in the HYS catalog, but the other large events that could affect the seismicity rate and IET distribution in this catalog are the Landers 1992 and Hector Mine 1999 earthquakes. However, in the non-declustered catalog, the seismicity rate, and hence the slope of the number of events with time in Figure 3.4d, changes instead at around 1990. To investigate the effect of this change on the IET distribution, we cut the catalog before and after 1990. The catalog subsets then yield relatively straight cumulative events with time curves (Figure 3.7c), possibly suggesting stationarity. However, their corresponding IET distributions (Figure 3.7d) both still exhibit significant fat tails, though with a stronger one on the period from 1981-1989.
Breaking the HYS catalog down into yearly subsets reveals a much more pronounced change in the seismicity rate with time, as was also suggested by the random-walk paths in the Schuster test (Figure 3.6d). There is a systematic shift in the number of events with time over the years of the catalog, from higher rates of earthquake
0 100 200 300 Days
0 2 4 6 8 10 12 14
No. of Events (x103 )
2010 (a)
2008 2009 2010 2011 2012 2013 2014 2015 2016 2017
Year 0.03
0.035 0.04
Mean t 0 (Days)
(c)
t0 mean t
0
1 2
0 0.1 0.2 0.3 0.4 0.5
Interevent Time (Days) 100
102
Cumulative Count
2010 (b)
2008 2010 2012 2014 2016
Figure 3.8: Interevent time results for the QTM catalog split between each consecutive year to examine temporal variations in the earthquake occurrence rate. (a) Total number of events with time within each separate year against days in the year. Shading represents the year. The slope of 2010 notably differs from those of the other years. (b) Interevent time distribution for each of the individual years. Each year other than 2010 collapses onto more or less the same distribution. (c) Change in the mean interevent time with year. 2010 exhibits a significant reduction in mean interevent time outside the 2𝜎bound due to the abundance of events in the aftermath of the El Mayor-Cucapah quake.
occurrence in the 1980s to generally lower rates in the 2000s (Figure 3.9a). This is inversely seen in the general increase in the mean IET over the length of the catalog (Figure 3.9c), which likewise affects the associated IET distribution (Figure 3.9b). When each year is considered individually, the deviation of the data from the exponential curve is not significant, as was also the case for the QTM catalog, but taken together, the mean produced by the entire catalog does not produce a model that is an accurate representation of the earthquake interevent times due to the non-stationarity in the rate.
Unlike the QTM catalog, the change in the earthquake rate for the HYS catalog does not directly correlate with any particular year and actually happens before the Landers and Hector Mine earthquakes, suggesting that non-stationarity may be a more long-lived characteristic of the background seismicity rather than a result of the presence of aftershocks from any one large quake. However, the lower
0 100 200 300 Days
0 0.2 0.4 0.6 0.8
No. of Events (x103 )
(a)
1985 1990 1995 2000 2005 2010 2015
Year 0.5
1 1.5
Mean t 0 (Days)
(c)
t0 mean t0
1 2
0 5 10 15 20
Interevent Time (Days) 100
101 102
Cumulative Count
(b)
1985 1990 1995 2000 2005 2010 2015
Figure 3.9: Interevent time results for the HYS catalog split between each consecutive year to examine temporal variations in the earthquake occurrence rate. (a) Total number of events with time within each separate year against days in the year. Shading represents year. There is a systematic decrease in the seismicity rate over the course of the catalog. (b) Interevent time distribution for each of the individual years. The shift in mean interevent time is again evident in the data. (c) Change in the mean interevent time as a function of year. There is a long-term increase in the mean interevent time of the catalog, pointing to non-stationarity in the seismicity rate.
completeness of this catalog compared to the QTM could create artefacts of non- stationarity, especially considering that the magnitude of completeness has likely decreased with time as the detection capability of the network improved. That effect alone, if we were using a magnitude of completeness lower than the value in the early 1990s, could have resulted in an apparent increase of the seismicity rate. This is actually opposite to what we observe, however.
The overall apparent decrease in the earthquake rate since the early 1990s may in fact be an artifact due to improved location uncertainties resulting from the increased density of seismic stations (Ben-Zion and Zaliapin, 2020). Zaliapin and Ben- Zion (2015) demonstrate that reducing location errors results in statistically smaller distances between parent and offspring events, which means a larger proportion of events will be identified as clustered after the locations are improved, starting in the
early 1990s (Ben-Zion and Zaliapin, 2020). If more events are identified as clustered for the later years in the catalog, then this could artificially result in fewer mainshocks per year, and hence a lower apparent seismicity rate, as seen in the later half of the HYS catalog. If fewer events are identified as aftershocks in the period before 1991 due to location error, then the likelihood that some of those aftershocks will remain in the catalog after declustering is higher. This means a higher proportion of events with shorter interevent times than there would have been had they been identified as aftershocks, which skews the mean interevent time, and equivalently increases the seismicity rate of the catalog, leading to a fat tail. This could explain why the tail of the catalog subset for the period prior to 1990 is substantially larger than that of the later subset of the catalog (Figure 3.7d). As the location error decreases and the number of events in the catalog increases, as in the case of the QTM catalog, the earthquake IET distribution should approach the exponential distribution.