Detection of Missing Repeating Earthquakes Using Recurrence Elements Analysis

Kate Huihsuan Chen, Roland Bürgmann, and Robert M. Nadeau


Stress perturbations influence earthquake recurrence and are of fundamental importance to understanding the earthquake cycle and determining earthquake hazard. The large population of repeating earthquakes on the San Andreas fault at Parkfield provides a unique opportunity to examine the response of the repeating events to the occurrence of moderate earthquakes. We analyze 187 M -0.4 to 1.7 repeating earthquake sequences (RES) from the High Resolution Seismic Network catalog to estimate the impact of M 4-5 events on RES's timing. Here we adopt a recurrence element analysis based on five recurrence elements (Fig. 1a): (1) dt+: the time difference between a major earthquake and the first subsequent recurrence of a repeating event; (2) dt-: the time difference between a major earthquake and the most recent repeating event; (3) Tr_cos: the recurrence interval spanning the major event; that is, the sum ofdt- and dt+; (4) Tr_post: the duration of the first full recurrence interval following the major event, and (5) Tr_pre: last recurrence interval just preceding the potential trigger. These elements are divided by the average 1987-1998 recurrence interval of a given RES to obtain the normalized values of ${\it dt+}^{\star}$, ${\it dt-}^{\star}$, ${\it Tr\_cos}^{\star}$, ${\it Tr\_post}^{\star}$, and ${\it Tr\_pre}^{\star}$. Very short recurrence elements of ${\it dt+}^{\star}$ (i.e., smaller than 10% of the typical cycle) can indicate the immediate triggering due to the major event, whereas longer than 1 ${\it Tr\_cos}^{\star}$, ${\it Tr\_post}^{\star}$, and ${\it Tr\_pre}^{\star}$ reflect a population of missing events.

Recurrence elements associated with M 4-5 events

For each RES, the five recurrence elements associated with every M 4-5 event are calculated (Fig. 1a). In Fig. 1b, the RES within 5 km distance from the major events tend to have a high fraction of short ${\it dt+}^{\star}$ ($<$ 0.1). Fig. 1b-d also show the percentage of events within a given distance range that have a ${\it dt+}^{\star}$ less than the threshold specified. For example, more than 30% of the events within 2 km distance have ${\it dt+}^{\star}$ $<$ 0.1, whereas within distances of greater than 5 km $\sim$10% of the RES exhibit such rapid recurrence. The percentage of short ${\it dt+}^{\star}$ remain unchanged for events beyond 5 km. To confirm that the observed short dt+ population indicates the triggering effect of M 4-5 events, we compare the observed distribution of ${\it dt+}^{\star}$ $< $0.1, 0.1-0.2, and $>$ 0.5 with ${\it dt+}^{\star}$ values generated from randomly generated times of the five M 4-5 events. The 30 sets of 5 randomly generated M 4-5 times (150 runs in total) produce roughly equal percentages of ${\it dt+}^{\star}$ at most distances, as shown by the blue lines in Figure 1. The random behavior of the small ${\it dt+}^{\star}$ population ( ${\it dt+}^{\star}$ $<$ 0.1) is strikingly different from the real population in the near field of the M 4-5 events ($<$ 5 km). Beyond 5 km, however, the observed ${\it dt+}^{\star}$ $<$ 0.1 distribution matches the synthetic ${\it dt+}^{\star}$. Compared to ${\it dt+}^{\star}$ curves, the fraction of short ${\it dt-}^{\star}$ measured over the same range of distances do not reveal systematic change with distance. The percentages of the observed ${\it dt-}^{\star}$ at all distances match the value of ${\it dt-}^{\star}$, as one can expect from random behavior.

Undetected repeating events?

In Figure 2, the histogram of ${\it Tr\_pre}^{\star}$ reveals a somewhat broader distribution with median value of 1.62. The median ${\it Tr\_pre}^{\star}$ is about a half cycle shorter than the median value of 1.14 and 1.29 for ${\it Tr\_cos}^{\star}$ and ${\it Tr\_post}^{\star}$, respectively. This suggests a general pattern of shortened interval at and following the time of M 4-5 events. Note that the small secondary peak in Fig. 2b is about twice the normalizing interval, indicating some missed recurrences that may have occurred during the trigger event. Given that a single skipped event in a sequence leads to a ${\it Tr\_cos}^{\star}$ value of slightly greater than 1, the second peak at ${\it Tr\_cos}^{\star}$ ${\sim}$2 is suggestive a number of unrecognized repeating event. And since the secondary peak near 2 is minor, the undetected repeating events are unlikely to have significant influence on the M 4-5 triggering effect. We also note that ${\it Tr\_post}^{\star}$ is also somewhat reduced compared to pre-event recurrences, indicating the possible role of afterslip or general acceleration of slip in the early 1990s.

Figure 2.4: (a) Schematic illustration of the five recurrence elements, dt+, dt-, Tr_pre, Tr_cos, and Tr_pos. Percentage of (b) ${\it dt+}^{\star}$ $<$ 0.1 (c) 0.1 $<$ ${\it dt+}^{\star}$ $<$ 0.2 (d) ${\it dt+}^{\star}$ $>$ 0.5 as a function of distance from M 4-5 events for real data (red line) and synthetic data (blue lines generated by 30 sets of 5 randomly drawn of M 4-5 event times). Note that the percentage in each distance bin (1 km) is calculated when the ${\it dt+}^{\star}$ number is greater than 3.
\epsfig{file=katechen09_1_1.eps, bb=25 180 575 600...


We illustrate the effect of major events on earthquake cycles of nearby characteristically repeating micro-earthquakes and determine the distance over which triggering can be documented. We find evidence that the five M 4-5 events that occurred at Parkfield triggered small, nearby repeating earthquakes. The triggering effect can only be seen in the near-field ($<$ 5 km) by the measures of rapid recurrence subsequent to the major event. A small population of missing repeating events at the time of a major event is also detected by the longer-than-average intervals spanning and following a major event. In future work, we will consider whether interaction with nearby M $<$ 4 events plays an additional important role in the RES recurrence patterns. We will also explore in detail the response of the RES to the M 6 2004 Parkfield earthquake.

Figure 2.5: Histograms of ${\it Tr\_pre}^{\star}$, ${\it Tr\_cos\star}$, and ${\it Tr\_post}^{\star}$ determined by the average recurrence intervals of full period. Grey lines denote the normalized recurrence intervals of 1 and 2.
\epsfig{file=katechen09_1_2.eps, bb=60 30 500 685, ...

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