Secondary Aftershocks and Their Importance for Aftershock Forecasting by Karen R

Bulletin of the Seismological Society of America, Vol. 93, No. 4, pp. 1433–1448, August 2003

Secondary Aftershocks and Their Importance for Aftershock Forecasting by Karen R. Felzer, Rachel E. Abercrombie, and Go¨ran Ekstro¨m

Abstract The potential locations of aftershocks, which can be large and damag- ing, are often forecast by calculating where the mainshock increased stress. We find, however, that the mainshock-induced stress field is often rapidly altered by aftershock-induced stresses. We find that the percentage of aftershocks that are secondary aftershocks, or aftershocks triggered by previous aftershocks, increases with time after the mainshock. If we only consider aftershock sequences in which all aftershocks are smaller than the mainshock, the percentage of aftershocks that are secondary also increases with mainshock magnitude. Using the California earthquake catalog and Monte Carlo trials we estimate that on average more than 50% of aftershocks produced 8 or more days after M Ն5 mainshocks, and more than 50% of all aftershocks produced by M Ն7 mainshocks that have aftershock sequences lasting at least 15 days, are triggered by previous aftershocks. These results suggest that previous aftershock times and locations may be important predictors for new aftershocks. We find that for four large aftershock sequences in California, an updated forecast method using previous aftershock data (and neglecting mainshock-induced stress changes) can outperform forecasts made by calculating the static Coulomb stress change induced solely by the mainshock.

Introduction It has long been observed that aftershocks may produce sequence. We also show that while the exact factors that their own aftershocks, commonly known as secondary af- determine aftershock location may be many and complex, tershocks (Richter, 1958). The essential deﬁnition of sec- using only the locations and times of previous aftershocks ondary aftershocks is that they are primarily or entirely trig- can produce a more precise forecast of aftershock locations gered by a previous aftershock and show pronounced spatial than using the calculations of the static Coulomb stress and temporal clustering around this aftershock. The fraction changes induced solely by the mainshock. of aftershocks that are secondary is often assumed to be small. There is good reason to question this assumption, Evidence for Secondary Aftershocks however, with important implications for aftershock forecasting. If secondary aftershock production is indeed low, The Aftershock Production Rate of Small Earthquakes then the best way to forecast aftershock locations should be Most aftershocks are small. Therefore secondary after- by calculating the stress changes produced by the mainshock shocks may be common only if small earthquakes, as a (e.g., Rybicki, 1973; Das and Scholz, 1981; Harris and group, are capable of producing a substantial number of af- Simpson, 1992; King et al., 1994; Stein et al., 1997; Taylor tershocks. It has been recognized that the volumetric static et al., 1998; Zeng, 2001; Kilb, 2003). If secondary after- stress change produced by small aftershocks (when inte- shocks are common, however, many aftershocks might not grated over a ﬁnite volume) is essentially negligible in com- be predictable without using the locations and times of pre- parison to that produced by the mainshock (e.g., Hardebeck, vious aftershocks. Indeed, for general seismicity, many au- 1998). The relative ability of two earthquakes to induce thors have successfully used the locations of previous earth- volumetric static stress change is not, however, equal to quakes to predict the distribution of future ones (e.g., Kagan their relative ability to produce aftershocks. The Gutenberg– and Jackson, 1991; Cao et al., 1996; Woo, 1996; Jackson Richter relationship states that the total number of earth- and Kagan, 1999; Kafka and Levin, 2000; Kagan and Jack- quakes in a population that are larger than or equal to some ;bM (Ishimoto and Iida, 1939מson, 2000; Wiemer, 2000; Kafka, 2002). magnitude M varies as 10 In this article we use statistical tests to demonstrate that Gutenberg and Richter, 1944). Volumetric static stress secondary aftershocks are a common phenomenon and then change varies with earthquake moment, which varies with use Monte Carlo modeling to quantify how many after- earthquake magnitude, M,as101.5M (Kanamori, 1977). In shocks are secondary in the average California aftershock contrast, aftershock productivity varies much more slowly

1433 1434 K. R. Felzer, R. E. Abercrombie, and G. Ekstro¨m with mainshock magnitude, as 10bM (Reasenberg and Jones, of them should occur in small sequences that constitute small 1989; Yamanaka and Shimazaki, 1990; Kagan, 1991; Felzer and frequent temporal peaks that blend into one another. et al., 2002), where b is the parameter from the Gutenberg– That it is possible for an aftershock sequence adhering Richter relationship for the aftershock population, and is typ- well overall to Omori’s law to in fact be made up of many ically found to be close to 1. The physical reason why af- different overlapping aftershock sequences, each individu- tershock production varies as 10bM rather than with volu- ally adhering to Omori’s law, has been demonstrated theo- metric static stress change is not presently known, although retically (Sornette and Sornette, 1999; Helmstetter and Sor- we note that other seismic properties that vary as 10M are nette, 2002). In order to test these results for individual dynamic displacement, velocity, and acceleration in the far aftershock sequences we generate synthetic sequences using field (Richter, 1958) and mainshock faulting area, if we as- the statistical aftershock production model of Felzer et al. sume a constant stress drop (Kanamori and Anderson, 1975). (2002). In this model earthquakes are generated to either To solve for the relative aftershock productivity of small follow a modified Omori decay rate initiating at the time of and large earthquakes we need to multiply the expression the mainshock (direct aftershocks) or initiating at the time giving the number of aftershocks as a function of mainshock of a previous aftershock (secondary aftershocks). The other magnitude by the number of potential mainshocks in each rules of the model are that aftershock magnitudes are se- magnitude range. The latter is given by the Gutenberg– lected randomly from the Gutenberg–Richter distribution, Richter relationship. We assume that the b-value for after- aftershock productivity varies as 10bM, and aftershocks pro- shocks is the same as the b-value for the general population duce aftershocks at the same rate that other earthquakes do (Ranalli, 1969). We note that the assumption of identical b- (for rationalization of the assumptions, see Felzer et al. values is additionally supported by the mathematical dem- [2002]). onstration of Woo (1996) that subpopulations of earthquakes This general method for the statistical generation of syn- cannot have different b-values if the population as a whole thetic aftershocks is similar to that used by Ogata (1988), adheres to the Gutenberg–Richter relationship. Assuming Kagan (1991), Console and Murru (2001), and Helmstetter equal b-values, the product of the aftershock production re- and Sornette (2002). Models of this type have been found lationship and Gutenberg–Richter relationship is given by to produce synthetic aftershock sequences that are in excel- ;a constant. That is, the total number of lent agreement with real aftershock time series (Ogata, 1988 ס 10bM ן bMמ10 aftershocks produced by each magnitude range is the same, Guo and Ogata, 1997; Felzer et al., 2002). We generate a regardless of the magnitude range being considered (Michael trial sequence with this model, using a mainshock of M 7 and Jones, 1998; Felzer et al., 2002). Thus the total number and the average California modified Omori parameters (see of aftershocks produced by small earthquakes, and hence the the Variation of Secondary Aftershocks with Time section). total number of secondary aftershocks produced by small Forty-seven percent of the aftershocks in our trial synthetic aftershocks, is expected to be significant. sequence are secondary. Yet the time series of the sequence shows no excursion beyond the 95% confidence intervals Secondary Aftershocks and Omori’s Law for Poissonian variation around the modified Omori law Utsu (1962) and Page (1968) have argued that few af- (Fig. 1). Thus by counterexample we have proven that the tershocks are secondary because most aftershock sequences absence of significant deviation from the modified Omori decay smoothly with time after the mainshock in accordance law does not imply a rarity of secondary aftershocks. with the modified Omori law (Utsu, 1961). This law is given by Testing for the Existence of Secondary Aftershocks We propose that a better test for the existence of sec- p (1) ondary aftershocks is to search for temporal correlationsמ,(c ם A(t ס R between aftershocks on a local scale. We test this idea on where R is the aftershock rate, t is time after the mainshock, a sample of nine large California aftershock sequences and A, p, and c are constants. It is usually found that p is (Fig. 2). The epicentral information for the mainshocks and slightly larger than 1 and c is on the order of a fraction of a the spatial limits used for the aftershock zones, which are day (e.g., Reasenberg and Jones, 1989). The smooth adher- rectangles selected by visual inspection, are given in Table ence of most aftershock sequences to the modified Omori 1. The data are taken from the Council of the National Seis- law is taken by Utsu (1962) and Page (1968) to indicate a mic System catalog. All of the mainshocks are M Ͼ6, except paucity of secondary aftershocks because secondary after- for the Oceanside and Palm Springs earthquakes (M 5.4 and shocks should cluster close in time to the aftershock that M 5.6, respectively), which we observe had unusually large triggered them, creating sharp peaks in the aftershock decay aftershock sequences for their magnitudes. The half-widths time series. Utsu (1962) and Page (1968) were correct that of the aftershock zones are between one and two times the such temporal peaks should occur, but they erred in their fault length of the mainshock. We use all M Ն2 aftershocks implicit assumption that the peaks should rise higher than that occurred from the time of the mainshock through De- the average noise level. In fact, if secondary aftershocks are cember 2001, with the exception of the Joshua Tree after- continuously being triggered by small aftershocks, then most shock sequence, which we use only until the time of the Secondary Aftershocks and Their Importance for Aftershock Forecasting 1435

4 10 38N model simulation San Francisco best fit modified Omori MH 3 37N LP 10 95% Poissonian confidence interval 36N C 2 10 2 aftershocks/day 35N L HM 1 10 N 34N Los Angeles JT 100 NPS 0 10 33N O

Number of M 0 1 2 10 10 10 122W 121W 120W 119W 118W 117W 116W days Figure 2. California aftershock sequences used in Figure 1. Time series of a simulated aftershock this study. The aftershock zone boundaries were de- sequence of an M 7 California mainshock (dots) (us- termined using all aftershocks recorded in each se- Ն ing average California parameters; see Table 3), fit quence, but only M 4 aftershocks are plotted on the with the modified Omori law (solid line) and plotted figure. The mainshocks are C, Coalinga; MH, Morgan with the 95% confidence intervals for expected Pois- Hill; NPS, North Palm Springs; O, Oceanside; LP, sonian random fluctuation (dashed lines). Although Loma Prieta; JT, Joshua Tree; L, Landers; N, North- none of the data points lie outside of the 95% confi- ridge; and HM, Hector Mine. The cities of Los An- dence intervals, 47% of the aftershocks are secondary. geles and San Francisco are also labeled for reference. We conclude from this counterexample that the absence of significant deviation from the modified Omori law need not indicate that few aftershocks are shock temporal clustering can be seen. To draw the influence secondary. Note that each data point represents the total number of aftershocks to occur on the day in zones, mainshock fault length is estimated from the equa- question, rather than the instantaneous aftershock rate tions of Kanamori and Anderson (1975), assuming a circular at a particular time (for example, the first data point fault and a constant stress drop of 30 bars, and using the ס gives the total of all aftershocks that occurred in the earthquake moment that is calculated from M0 Kanamori, 1977). The resulting fault length is) 10.73םfirst 24 hr of the sequence.) This accounting system 101.5M in combination with the small c parameter value of the total aftershock sequence (c Ϸ 0.05) causes the similar to that estimated empirically by Wells and Copper- graph to curve upward at small times on the log–log smith (1994). We draw the boxes for the influence zones in plot. For aftershock sequences in which c is larger map view and do not use depth in our calculation. This is than about 0.2, or aftershock recording on the first done for simplicity and because depths are more poorly con- day is incomplete, upward concavity on plots of this strained than epicenters. type should be negligible. To test for the presence of temporal clustering within the influence zones we measure the percentage of all after- Landers mainshock, and the Landers aftershock sequence, shocks that occur less than 24 hr after and within the influ- which we use only through the time of the Hector Mine ence zone of a previous aftershock. We call this percentage mainshock. We draw “influence zone” boxes around each the spatiotemporal correlation index. To test whether the aftershock epicenter with the half-widths of the influence value of the spatiotemporal correlation index indicates actual zones equal to two times the estimated fault length. The interaction between aftershocks rather than just coincidental influence zones are not expected to contain all of the sec- correlation, we remeasure the index after shuffling the af- ondary aftershocks of a given aftershock, since there is pres- tershock times and locations. In this process all of the after- ently no known physical limit on how far away a secondary shock times and locations are preserved, but they are ran- aftershock can be. Nor are the earthquakes in a given influ- domly recombined with one another. We then test the null ence zone expected to be exclusively secondary aftershocks hypothesis that the real and recombined catalogs come from of the target aftershock, as secondary aftershock zones un- a base population with the same spatiotemporal correlation doubtedly intersect and direct aftershocks may be scattered index (i.e., there is no real correlation between the after- throughout. The influence zones are chosen to encompass shocks). If multiple data samples are taken from a single an area that is large enough such that we expect to capture large population, the differences between the percentages of many secondary aftershocks if they exist, while at the same events in each sample satisfying a given criterion should be time small enough such that any signal of secondary after- normally distributed. We thus compute z statistics to deter- 1436 K. R. Felzer, R. E. Abercrombie, and G. Ekstro¨m

Table 1 Epicentral Information and the Aftershock Zones Used for Mainshocks in This Study

Date Aftershock Zone, Aftershock Zone, Earthquake (mm/dd/yy) Magnitude Epicenter N–S E–W

119.90מ/120.58מ 35.91/36.47 120.32מ ,Coalinga 05/02/83 6.7 36.23 121.38מ/121.88מ 37.02/37.54 121.68מ ,Morgan Hill 04/24/84 6.2 37.31 116.51מ/116.73מ 33.89/34.1 116.61מ ,N. Palm Springs 07/08/86 5.6 34.0 117.63מ/117.93מ 32.82/33.12 117.87מ ,Oceanside 07/13/86 5.4 32.97 121.37מ/122.35מ 36.68/37.44 121.88מ ,Loma Prieta 10/18/89 7.0 37.04 115.98מ/116.50מ 33.74/34.28 116.32מ ,Joshua Tree 04/23/92 6.1 33.96 116.00מ/117.01מ 33.79/35.00 116.44מ ,Landers 06/28/92 7.3 34.20 118.28מ/118.97מ 34.06/34.56 118.54מ ,Northridge 01/17/94 6.7 34.21 116.01מ/116.45מ 34.28/35.07 116.27מ ,Hector Mine 10/16/99 7.1 34.59 mine whether the real and recombined catalogs might come To solve for the fraction of cumulative aftershocks oc- from the same statistical population (null hypothesis). Using curing from the time of the mainshock until some end time, this procedure we find that for every sequence the null hy- t, that are secondary, we first calculate the cumulative num- pothesis can be rejected at a confidence level of over ber of direct aftershocks, CD, and total aftershocks, CT,by 99.99%, with the higher spatiotemporal correlation occur- integrating the modified Omori law (equation 1) over time ring in the real catalog. This indicates that there is real tem- from 0 to t: poral-spatial clustering between the aftershocks. Therefore,

מ מ in the absence of any plausible alternative explanation for AD p 1 מpDD 1 ם ס ΂΃(t c ) c מthe correlation, we can infer that secondary aftershocks are CDDD 1 pD (3a) present in all of the aftershock sequences. and Secondary Aftershock Rates in the Average מ מ AT p 1 מpTT 1 ם ס California Aftershock Sequence , ΂΃(t c ) c מCTTT 1 pT ϶ ϶ In the previous section we demonstrated with high like- pDT1, p 1. (3b) lihood that secondary aftershocks are routinely and significantly present in aftershock sequences, but we have not The fraction of cumulative aftershocks that are sec- quantified how many there are. The synthetic aftershock ondary, Sc(t), is then given by generation model described earlier can be used to solve for the percentage of aftershocks that are secondary after partic- (C /C . (4 מ 1 ס (S (t ular mainshock magnitudes and elapsed times. It is also pos- cDT sible to write several equations such that the number of after- To find the average parameters for these equations for Cali- shocks that are secondary can be easily estimated for any fornia, we count the cumulative number of M Ն4.8 after- time and mainshock magnitude. In this section we will de- shocks that occurred at intervals from 0.02 to 180 days after rive these equations and the parameters necessary to solve M Ն5.0 California mainshocks. The cutoff of M 4.8 is used them. because we determine using the Gutenberg–Richter relation- Variation of Secondary Aftershocks with Time ship that this is the smallest magnitude to which all of the data are complete (in particular there is potentially very low Both direct and total aftershock sequences follow the reporting of small aftershocks immediately after M Ͼ7 modified Omori law, where the total aftershock sequence is mainshocks in the data set [K. Hutton, personal comm., the sum of the direct and secondary sequences. Therefore, 1999]). Using a cutoff below the completeness magnitude the general equation for the fraction, S0(t), of ongoing after- would reduce the statistical accuracy of the results. We use shocks at time t that are secondary is given by all mainshocks that occurred between 1975 and 2000, except for mainshocks in the Mammoth Lakes volcanic area, be- מ pD cause volcanic and tectonic earthquakes may be governed ם ( ADD(t c מ ס by different processes and have different parameters. We (2) , מ S0(t) 1 c ) pT ם A (t TT also omit (1) mainshocks that occurred as such early aftershocks of previous earthquakes that their own aftershocks where AD, cd, pD, AT, cT, and pT are the modified Omori could not be reliably determined and (2) mainshocks that parameters for the direct and total aftershock sequences, re- had aftershocks larger than themselves before 180 days had spectively. elapsed. This leaves a total of 73 M Ն5 earthquakes. We Secondary Aftershocks and Their Importance for Aftershock Forecasting 1437 aftershocks, over 180 days. The parameters 1.2 ס determine the aftershock region of each earthquake by vi- shocks sually selecting the area with highly clustered earthquakes that we solve for from the average sequence should then give after the mainshock. In the vast majority of cases, the half- us the fraction of aftershocks that are secondary in the av- lengths of these regions are between one and two times the erage aftershock sequence that follows an M 6.04 mainshock fault length of the mainshock. After defining the regions we and contains no aftershocks larger than the mainshock. find a total of 80 recorded M Ն4.8 aftershocks, for all of the mainshocks, over 180 days. Solving for the Direct Sequence Omori’s Law Parameters.

Since some of the individual aftershock sequences con- We solve for the direct Omori parameters AD, cD, and pD for tain few earthquakes we stack the aftershock data to form a the average California aftershock sequence by using the sta- single average sequence, from which we solve for the mod- tistical aftershock-generating model described earlier to per- ified Omori parameters. In stacking the data we follow the form a grid search. In this process, a range of parameter premise of Dieterich (1994) and assume that aftershock se- values are tested (0.087–0.11 for AD, 0.01–0.15 for cD, and quences triggered by mainshocks of different magnitudes 1.05–1.4 for pD), then the averaged results of multiple model have the same duration. In order to normalize the parameters runs are compared with the data. We search for the set of from the stacked sequence we first divide the number of parameters that minimizes the least-squares error between aftershocks in the stack by the total number of mainshocks the synthetic sequences and the data. A set b-value of 1.0 is and then determine the magnitude of the equivalent main- also used as input to the aftershock-generating model. The shock that would produce a sequence of this size. Finding model also requires one more parameter, Mmin, the magni- the magnitude of the equivalent mainshock is done using the tude of the smallest existing earthquake. We chose to set ס magnitudes of the mainshocks of the individual sequences. Mmin 0 based on the results of Abercrombie (1995), that First we note that since aftershock production varies with earthquakes at least as small as M 0 have stress drops com- mainshock magnitude as 10bM, and since the aftershocks fol- parable to those of other earthquakes, and on the results of low the Gutenberg–Richter relationship, we expect that at Richardson and Jordan (2002), that M 0 may be the mini- any given time the accumulated number of aftershocks mum magnitude for traditional shear-failure earthquakes. If (mcמ(C(tמb(M should be equal to10 , where C(t) is a value that the true Mmin is less than 0, then more aftershocks are sec- decreases with time and mc is the smallest magnitude after- ondary than the parameters we solve for would predict; con- Ͼ shock counted. We find that an appropriate b-value for Cali- versely, if Mmin 0, fewer aftershocks are secondary. The ע fornia is 1.0 0.01, using a least-squares algorithm in effect of a particular change in Mmin depends on the mag- which each data point is weighted by the number of earth- nitude of the mainshock (if we only consider aftershocks מ quakes it represents (Bender, 1983). To solve for this b-value smaller than the mainshock) and is proportional to (Mmin מ we use a magnitude completeness cutoff of M 2.6 for south- DMmin)/(M Mmin), where M is the magnitude of the main- ern California and M 2.2 for northern California and elimi- shock. That is, since each magnitude range produces an nate data in the Mammoth Lakes region and north of the equal number of aftershocks, the change in secondary after- triple junction, both places where we find the completeness shocks due to a change in Mmin is proportional to the frac- magnitudes to be larger than M 3. We solve for the cutoff tional change of magnitude ranges. magnitudes by using a v2 test to find the lowest magnitude Our resulting best-fit direct sequence parameters for ס ס 1מp ס for which the Gutenberg–Richter function still adequately California are AD 0.0921 days , pD 1.37, and cD describes the data at the 95% confidence level. For compar- 0.085 days (Table 3). This value of AD specifically gives the ison, other authors have found b-values of 0.99 (Wiemer and number of M Ն4.8 aftershocks produced by an M 6.04 main- Wyss, 2002) and 1.0 (Stein and Hanks, 1998) for southern shock. The value may be adjusted for some other aftershock mcמ4.8 California. Using a b-value of 1.0 we then combine all of magnitude cutoff mc by multiplying it by10 and ad- the mainshock magnitudes to solve for the equivalent main- justed to describe an aftershock sequence initiated by an- shock magnitude, me, other mainshock with magnitude M by multiplying it by .6.04מ10M 73סi 4.8מCמ(m(i 4.8מCמ m .͚ 10 Solving for the Total Sequence Omori’s Law Parameters(1/73) ס e 10 1סi We use a nonlinear least-squares fit to solve for the total sequence modified Omori law parameters directly from our 73סi (i) ס mme 10 (1/73)͚ 10 (5) average California aftershock sequence data. Sornette and 1סi Sornette (1999) demonstrated that stacking direct and sec- ס me 6.04 , ondary aftershock sequences actually causes the total aftershock sequence to have modified Omori law parameters that ם where the m(i)s are the magnitudes of the individual main- vary with time. Specifically, if pD is equal to 1.0 h, then ם מ shocks. This means that we expect the average M 6.04 Cali- pT should vary smoothly with time from 1.0 h to 1.0 Ն fornia mainshock to produce the average number of M 4.8 h, where h is a value between 0 and 1. AT should also in- aftershocks in our stack, which is 80 aftershocks/73 main- crease with time. This temporal parameter variation is the 1438 K. R. Felzer, R. E. Abercrombie, and G. Ekstro¨m

Table 2 Comparison of Probability Map and Coulomb Stress Change Predictions of M Ն5 Aftershocks

Aftershock Date Aftershock Real-time 24-hr Coulomb Mainshock (mm/dd/yy) Magnitude Prob. Map Prob. Map Map

Joshua Tree 05/18/92 5.0 5 6 2 Joshua Tree 06/28/92 7.3 1 378 108 Landers 06/28/92 5.1 576 NA 213 Landers 06/28/92 5.0 576 NA 229 Landers 06/29/92 5.0 19 NA 57 Landers 06/29/92 5.5 6 NA 66 Landers 07/01/92 5.3 89 15 18 Landers 07/05/92 5.4 2090 2090 1337 Landers 07/24/92 5.0 107 23 393 Landers 08/17/92 5.0 78 88 182 Landers 09/15/92 5.1 50 290 103 Landers 11/27/92 5.4 1 217 471 Landers 12/04/92 5.2 5 301 840 Landers 08/21/93 5.0 72 102 138 Landers 06/16/94 5.0 161 141 1965 Landers 03/18/97 5.1 1 288 413 Landers 10/16/99 7.1 1 291 1262 Hector Mine 10/21/99 5.0 241 27 72 Hector Mine 10/21/99 5.0 230 21 92 Hector Mine 10/22/99 5.0 3 265 92

The first three columns give the mainshock and the date and magnitude of the aftershock. The next three columns give the ranking of the cell that the aftershock occurred in on the real-time probability map, the 24-hr probability map, and the Coulomb static stress change map, respectively. A number 1 ranking indicates that the cell was predicted to be the most likely one to contain an aftershock. “NA” indicates that the aftershock occurred too early to be eligible for a 24-hr probability map prediction. likely explanation for the observations of Mogi (1962) and Gross and Kisslinger (1994) that the later parts of many af- 1.1 tershock sequences are not well fit with the same parameters 1 that describe the initial activity. For simplicity, however, we 4.8 0.9 first solve for a single set of Omori parameters that best fits 0.8 the entire sequence (Table 3; Fig. 3). Our best-fit pT parameter is 1.08, the same average pT that was found by Reasen- 0.7 berg and Jones (1989) in a study of 62 California aftershock 0.6 sequences. Our best-fit A parameter is lower than that found data T 0.5 1 parameter 0.4 set fit 0.3 3 parameter Table 3 0.2 set fit Average Modified Omori Law Parameters for California 0.1

Cumulative aftershocks M aftershocks Cumulative - Aftershock Sequences 10 2 10-1 100 101 102 103 p pc, days daysמParameter Type Days A, days1 Direct sequence All 0.0921 1.37 0.085 Total sequence All 0.116 1.08 0.014 Figure 3. Cumulative number of M Ն4.8 aftershocks recorded in the average aftershock sequence 5מTotal sequence 0–1 0.151 0.75 10 Total sequence 2–10 0.191 1.27 0.076 of an M 6.04 California mainshock, as calculated from the aftershock sequences of 73 M Ն5 California 5מTotal sequence Ͼ10 0.222 1.34 10 mainshocks that occurred between 1976 and 2000. The parameters are solved for an M 6.04 mainshock and aftershocks M The data are fit with a single modified Omori law Ն4.8. The first row gives parameters for direct aftershock sequences. The (dashed line) and a modified Omori law that under- remaining rows give parameters for total aftershock sequences (made up goes parameter changes at days 2 and 10 (solid line). of direct and secondary aftershocks). Total sequence parameters are given The change in Omori parameters with time is caused as a single parameter set that averages over all times and as separated by the presence of secondary aftershocks. Parameters parameter sets that give more accurate solutions for shorter time periods. are given in Table 3. Secondary Aftershocks and Their Importance for Aftershock Forecasting 1439 by Reasenberg and Jones (1989) (0.1163 versus 0.2873). 100 The AT parameter solved for by Reasenberg and Jones 90 (A) (1989) is now known to have been biased upward by about 80 a factor of 2 because they solved for parameters from indi- 70 vidual aftershock sequences instead of a stack and thus only 60 used the more active sequences, for which parameters could 50 be solved for most robustly (L. M. Jones, personal comm., 2002). 40 Using the best-fit direct and total parameters and equa- 30 tion (4), we solve for the fraction of cumulative aftershocks 20 that are secondary. We obtain reasonable agreement between 10 these results and the predictions made by Monte Carlo trials 0 with our synthetic aftershock model (Fig. 4A). Our agree- 0 100 200 300 400 500 600 ment with the simulation results deteriorates, however, when 100 we use the same parameters to predict what fraction of ongoing aftershocks should be secondary (equation 2; Fig. 4B). 90 (B) This is because equation (2) is particularly sensitive to the 80 temporal changes of the total Omori parameters. 70 To answer this difficulty we approximate the smooth 60 change in total sequence Omori parameters by fitting the 50 total sequence with three different sets of modified Omori % of aftershocks that are secondary% of aftershocks 40 parameters, with the changeover between parameter sets oc- 30 curring at days 2 and 10 of the sequence. These changeover days give the best least-squares fit to the data (Fig. 3). We 20 find that the least-squares error between the modified Omori 10 law function and the data decreases by 14% when we use 0 0 20 40 60 80 100 two parameter sets rather than one and decreases by 57% days when we use three parameter sets rather than one. An alternative mathematical solution to breaking the sequence into Figure 4. (A) The percentage of cumulative after- three parts is derived by Helmstetter and Sornette (2002). shocks of an average M 6.04 California mainshock that should be secondary, as a function of time, as calculated Even with the three sets of parameters the predictions from equation (4) using the single best-fit modified Omori of equation (2) and the simulation results agree poorly for law parameter set (dashed line) and best-fit triple set of durations less than about 0.3 days. This is because the c- total sequence modified Omori law parameters (solid line). value, which is important at short times, is quite different Both fits produce similar results. (B) The percentage of ongoing aftershocks of an average M 6.04 California 5מ ס ס for the two sequences, with cD 0.085 days and cT 10 c mainshock that should be secondary as a function of time, days. This value for T holds even if we only look at the first as calculated by equation (2) using the best-fit modified few data points in the sequence. The problem may be intrin- Omori law single parameter set (dashed line) and the best- sic with the c parameter itself. The parameter is used to fit modified Omori law triple parameter set (solid line). In correct for the fact that the power law becomes infinite as t this case, the two solutions diverge by nearly 20% after goes to zero; but is not really a true fix since the modified 100 days. The calculation using the triple parameter set is much closer to results from model simulations (black Omori law with a nonzero c parameter has a nonzero after- dots). Each model data point represents the average of ס shock rate even at t 0. This implies that the mainshock 1000 Monte Carlo trials. Error bars on the black dots in- and aftershock should happen at precisely the same moment, dicate the 95% range of values observed with the Monte which, under laws of cause and effect, is clearly nonphysical. Carlo trials. One potential solution is to introduce a Heaviside function such that there are simply no aftershocks before some time Variation of the Fraction of Aftershocks that are t0 and a pure power law afterward (Kagan and Knopoff, 1981). The c-parameter problem should not be important Secondary with Mainshock Magnitude over time periods longer than about half a day. In all of our secondary aftershock calculations we are In general our results indicate that the fraction of after- concerned only with aftershock sequences in which no af- shocks that are secondary after an M 6.04 mainshock is not tershock is larger than the initial mainshock. This restriction insignificant, especially for ongoing aftershocks occurring at is made because if an aftershock larger than the mainshock least several days after the mainshock. For additional insight occurs, it is obvious that most aftershocks will be secondary. into the solution we also plot separate time series of the Using this restriction, however, causes secondary aftershock direct, secondary, and total aftershock rates (Fig. 5). production to vary with mainshock magnitude. 1440 K. R. Felzer, R. E. Abercrombie, and G. Ekstro¨m

0.4 100 0.35 90 ongoing all aftershocks 80 0.3 direct aftershocks cumulative 70 secondary aftershocks 0.25 60 0.2 50 40 4.8 aftershocks/day 0.15 30 0.1 20 0.05 10

Rate of M 0 0 that are secondary% of aftershocks 0 1 2 3 4 5 3 4 5 6 7 8 days mainshock magnitude

Figure 5. Direct, secondary, and total aftershock Figure 6. The average percentage of cumulative rates for the average M 6.04 California mainshock, and ongoing aftershocks that are secondary as a func- calculated using equation (2) and the best-fit triple set tion of mainshock magnitude, after 10 years, for Cali- of modified Omori parameters. Initially the vast ma- fornia aftershock sequences in which no aftershock is jority of aftershocks are direct; but the direct after- larger than the mainshock. shock rate decays quickly and is overtaken by the secondary aftershock rate within 2 days. The graph is started at 0.3 days since we find that equation (2) can- 50% of aftershocks produced after the eighth day of the se- not be used accurately for earlier times. quence are secondary for M Ն5 mainshocks and over 50% of all aftershocks are secondary for M Ն7.0 mainshocks that As noted earlier, each unit magnitude range of earth- have aftershock sequences lasting for at least 15 days quakes contributes equally to aftershock production, so each (Fig. 6). magnitude unit of direct aftershocks is expected to generate equal fractions of the secondary aftershock population. Thus if we eliminate all aftershocks larger than the mainshock, Predicting Aftershock Locations and their progeny, we will eliminate few direct aftershocks, but the fraction of secondary aftershocks eliminated will be We have shown that many aftershocks are secondary, equal to the fraction of magnitude ranges eliminated. That especially following large mainshocks. It does not automat- is, the fraction of total aftershocks that are secondary will be ically follow, however, that the common practice of calcu- proportional to the number of magnitude units that are lating only mainshock-induced stress changes to predict fu- smaller than the mainshock. Therefore if we have a mini- ture aftershock locations (e.g., Papadimitriou and Sykes, mum magnitude Mmin and a mainshock magnitude M, the 2001) could be significantly improved by including after- average fraction of aftershocks that are secondary if we only shock-induced stress changes. This is because the after- consider sequences in which all aftershocks are smaller than shocks might primarily reinforce, rather than modify, the M is given by mainshock-induced stress change pattern. One way to determine whether including aftershock- מ M Mmin induced stress changes is important would be to model stressןס (S(6.04), (6 מ (S(M 6.04 Mmin changes produced by a mainshock and its suite of aftershocks and then see if the areas of stress increase changed where S(M) is the fraction of aftershocks that are secondary significantly as the sequence progresses. It would be ex- after a mainshock of magnitude M and S(6.04) is the average tremely difficult to do such a calculation, however, because fraction of earthquakes that are secondary in the aftershock many aftershocks are missing from catalogs, especially near sequence of an M 6.04 mainshock, which we have already the beginning of sequences; focal mechanisms are often un- calculated. (Any other reference mainshock magnitude may certain and the correct fault plane difficult or impossible to also be used for which the fraction of secondary aftershocks select; and aftershock rupture inversions may be quite dif- is known.) Using this equation with Mmin set to 0, we solve ficult to calculate. Alternatively, we might carry out the cal- for the average fraction of aftershocks that are secondary for culation with synthetic mainshock and aftershocks, but such average California sequences 10 years after the mainshock a model would also be extremely difficult to build. There as a function of mainshock magnitude. We find that over are too many modeling unknowns, one of the most important Secondary Aftershocks and Their Importance for Aftershock Forecasting 1441 being how to set up a realistic fault network with numerous al. (2002) are done at 7 km depth and resolved onto vertical, small faults that is kinematically stable. optimally oriented strike-slip fault planes, which are calcu- It is possible to approach the problem from a different lated using regional N7ЊE maximum compression. angle. Namely, if we can demonstrate that a prediction scheme, any prediction scheme, that neglects mainshock- Aftershock Probability Maps induced stress change and uses only information from pre- The probability map method is based on the premise vious aftershocks performs significantly better than a stan- that the location of future aftershocks is significantly influ- dard prediction scheme using only mainshock-induced stress enced by the timing, concentration, and magnitude of pre- changes, then it is clear that the presence of secondary af- vious aftershocks. The method uses no information about tershocks is of practical importance for the purpose of fore- the fault or slip parameters of the mainshock. We find that casting. Or at least it is clear that the tendency of aftershocks the probability map method works best for sequences with to cluster with each other in time and space gives quite useful at least 50 recorded aftershocks. forecasting information. The essence of the probability map method is that each Toward this end we compare the predictive abilities of point in the aftershock zone is assigned to one of several mainshock-induced static Coulomb stress change calcula- predefined regions based on the point’s proximity in time tions and a time-updated method of our own design, called and space to previous aftershocks. The specific probability the probability map method, that uses no information about of a particular point hosting a future aftershock is then de- the mainshock but calculates the probability of new after- termined based on the percentage of the preceding group of shocks occurring in different locations based on the timing, N aftershocks that located in the same region. This purely location, and magnitudes of previous aftershocks. These two empirical method of solving for probabilities is not common methods do not, of course, exhaust the possibilities of what with other techniques (e.g., Jackson and Kagan, 1999; Wie- we could compare. For the stress change induced directly by mer, 2000) but has the significant advantage that unknown the mainshock, we could also, for example, use dynamic factors such as unrecorded aftershocks and the degree of stress (e.g., Kilb, 2003), viscoelastic stress (e.g., Zeng, 2001; location and magnitude assignment error are taken into ac- Freed and Lin, 2002) or pore pressure change (e.g., Bosl and count automatically as long as the new set of aftershocks is Nur, 1997). We chose static Coulomb stress change because affected by these factors to a similar degree as the preceding it is commonly used and easy to calculate. In addition, static N set of aftershocks was. stress changes can often be used as an estimate for dynamic To define the different aftershock regions we first draw and viscoelastic stress changes. Alternative schemes for pre- square influence zones around each aftershock epicenter, as dicting aftershock locations based on previous aftershock described earlier. Region 1 is defined as the map area that parameters include methods proposed by Wiemer (2000) is not contained within any influence zone. The other regions and Jackson and Kagan (1999). Both of these methods are are within the influence zones and are based on the timing more complex and refined than our probability maps and and concentration of previous aftershocks. We take into ac- require a number of additional calculations. In addition, the count whether an influence zone is older or younger than a method of Wiemer (2000) is based on the idea that the cutoff age t (where the age of an influence zone is defined Gutenberg–Richter b-value and modified Omori p-value as the time elapsed since the aftershock that created it) and vary spatially over the aftershock zone, which is not a uni- whether the number of influence zones overlapping at a par- versally accepted concept; the idea of b-value variation in ticular point are above or below a cutoff number L. This particular runs counter to the findings of Woo (1997). So for gives us four additional mutually distinct regions. In total simplicity we do not include the Wiemer (2000) and Jackson the five regions cover the entire aftershock zone. A given and Kagan (1999) methods at this time. point in the aftershock zone is in region 1 if it is not in any ,influence zones 1 מ The results of our comparison will be specific to our influence zone; region 2 if it is in 1 to L study area of California since the quality of aftershock catalog all of which are more than t days old; region 3 if it is in 1 influence zones, at least one of which is less than 1 מ and mainshock slip inversion data varies from place to place. to L We do our comparison for the North Palm Springs, Joshua t days old; region 4 if it is in L or more influence zones, all Tree, Landers, and Hector Mine sequences, for which we have of which are more than t days old; and region 5 if it is inside gridded static Coulomb stress change data from Stein et al. of L or more influence zones, at least one of which is less (2003). For all of the aftershock sequences we use the after- than t days old. shock zone boundaries given in Table 1 and consider as We limit our number of regions to five in order to main- aftershocks all earthquakes occuring within these boundaries tain statistical robustness in the empirical measurement of through the end of December 2001, except for Joshua Tree the probabilities. The boundaries of the regions will vary aftershocks, which can no longer be independently identified with time, and the regions are noncontiguous. As mentioned and are thus truncated at the time of the Landers mainshock, previously, at forecast time the probability of a new after- and Landers aftershocks, which are similarily truncated at shock occurring in any region is set equal to the percentage the time of the Hector Mine mainshock. of the preceding N aftershocks that occurred in the same The Coulomb stress change calculations from Stein et region. It is important that N not be too large, because the 1442 K. R. Felzer, R. E. Abercrombie, and G. Ekstro¨m distribution of aftershocks between the different regions may in each cell to get the total probability of the next aftershock change with time. By definition, for example, there are no occurring within it. aftershocks in regions 2 or 4 before time t has elapsed, so One disadvantage of our method is that since the re- there is a significant readjustment after time t. In addition, gional boundaries change with the occurrence of each new after a large aftershock there is a sharp, transitory increase aftershock, new forecasts have to be issued each time an in the probability of aftershocks occurring in regions 3 or 5. aftershock occurs. In practice, it could be difficult to calcu- N must be small enough, therefore, to capture these temporal late aftershock parameters quickly enough to do this in real changes, while at the same time be large enough to produce time, especially during the early, active part of an aftershock sequence. Thus we also make and test probability maps that ס reliable statistics. Through trial and error we find that N 50 is optimal for the nine sequences listed in Table 1. For are only updated at midnight and used to forecast the next smaller aftershock sequences smaller N values may be nec- 24 hr of activity. We refer to these as “24-hr probability essary, but we find that N values smaller than about 25 maps” and to the maps that are continuosly updated as “real- should not be used. time probability maps.” Similarily to N, the values of L and t must be chosen to strike a balance. If L is too large then the very densest clusters will be clearly highlighted, but the medium-size and Evaluating the Aftershock Probability Maps and smallest clusters will all be combined together in the same Coulomb Stress Change region, so the higher probability associated with the medium- size clusters will not be differentiated. Likewise, making L We next compare how well the probability map and the too small means the combination of large, medium, and static Coulomb stress change map methods predict after- small clusters in a single region, causing the larger proba- shock locations. We use the Wilcoxon signed-rank test for bility associated with both the large and medium clusters to correlated samples (Weiss and Hassett, 1982; Kilb and be missed. The choice of t is subject to similar constraints. Rubin, 2002) to determine whether one method is signifi- Through trial and error with the nine aftershock sequences cantly better than the other. To use this test, we first rank listed in Table 1, we find that the best balance can be the cells on the probability and Coulomb stress change maps days. For individual se- with a rank of 1 assigned to the cell with the highest assigned 0.5 ס and t 5 ס achieved with L quences with fewer aftershocks than average, we find that probability or the largest positive stress change, and so on. even better results may be acheived with L set equal to 3 or If two or more cells on a given map have the same proba- 4. The best L and t values for a particular aftershock se- bility or stress change we assign them the same rank, which quence cannot really be determined until the sequence is is equal to the rank the lowest cell would have had if they over, however, making the values irrelevant for forecasting were unequal. The rank of the cell that an aftershock occurs purposes. Thus we feel that the most appropriate way to test in then tells us how well the aftershock was predicted. If an the forecasting power of the probability maps against the aftershock occurs in the cell ranked 1 or 2, for example, then Coulomb stress change maps is to use uniform, preset values the prediction map did an excellent job at pinpointing the of L and t for all of the sequences. most likely aftershock location. Once we define the regions and estimate the probability For each aftershock in our sample we record the rank of each region containing aftershocks, we can make a prob- of the cell that it occurred in on the probability map and on ability map for the location of future aftershocks. This is the static Coulomb stress change map. Our sample of after- done by first covering the entire aftershock zone with a dense shocks excludes the first 50 aftershocks of each sequence grid. The smallest earthquake that we use to make our maps because these are needed to initialize the probability map, is M 2, which has an influence zone half-length of about but it includes every other M Ն2.5 aftershock of the North 0.11 km, so we use a grid spacing of 0.1 km. The probability Palm Springs, Joshua Tree, and Hector Mine sequences and of the next aftershock occurring at each grid point is then every other M Ն3.5 aftershock of the considerably larger set equal to the probability of an aftershock occurring in the Landers sequence. A histogram of the static Coulomb stress region that the grid point is in, divided by the total number change map ranks minus the real-time probability map ranks of grid points in that region. We choose M 2 as our magni- (Fig. 7) is made for each sequence. Note that more after- tude cutoff for making the maps after finding that smaller shocks on the right-hand side indicates that the probability earthquakes have such small influence zones in comparison map is performing better, and vice versa. It can be observed to their location error that the very marginal information that all of the histograms contain more aftershocks on the gained from using them is not worth the extra calculation right-hand side, and the Wilcoxon signed-rank test confirms time. with over 99.9% confidence that for each aftershock se- The grid points are so closely spaced that making in- quence the real-time probability map makes more precise dividual forecasts for each grid point is practically meaning- predictions than the static Coulomb stress change map. That less given location error on the order of 2 km. Therefore we is, for each case the null hypothesis that differences in cell next cover the map with square cells with dimensions of 2.5 ranks are due to simple random fluctuations rather than sys- by 2.5 km. A summation is done over all of the grid points tematic differences can be confidently rejected. Secondary Aftershocks and Their Importance for Aftershock Forecasting 1443

50 We also use the Wilcoxon signed-rank test to compare North Palm Springs the predictive ability of the Coulomb stress change map with 40 the 24-hr updated probability map. The contrast between the 30 predictive ability of these two maps is on the order of 25% smaller, but we can still say with over 99.9% confidence that 20 the 24-hr probability map predicts more precisely than the static Coulomb stress change map for each of the sequences. 10 An additional way to compare the two prediction meth- 0 ods is to plot the total number of cells that need to be iden- -80 -60 -40 -20 0 20 40 60 80 100 tified by each method in order to forecast the locations of a 250 given percentage of the aftershocks. As an example, consider an aftershock zone that is covered by 100 cells. At each 200 Joshua Tree forecast time we want to identify a set of cells such that 50% of all of the aftershocks in our forecast interval will occur 150 within some cell that we have identified, but such that the 100 number of extraneous cells included in our set is kept to a minimum. The smaller the number of cells that we can iden- 50 tify and still capture 50% of the seismicity, the better our 0 forecasting method is performing. -400 -200 0 200 400 The number of cells that are needed to locate different percentages of the aftershocks with the different methods 150 (static Coulomb stress change map, real-time probability Landers map, and 24-hr probability map) are given in Figure 8. For 100 reference, each figure also contains a line that indicates the best prediction that could have been made and a line that number of aftershocks indicates the average prediction that would have resulted 50 from random guessing. The best-prediction-possible line is calculated from the completed aftershock sequence by retro- 0 spectively ranking each cell according to the number of fore- -2000 -1000 0 1000 2000 shocks it ended up containing. The random-guess line and corresponding 95% confidence intervals are calculated from 250 1000 Monte Carlo trials. 200 Hector Mine Figure 8 indicates that in general both the static Cou- lomb stress change maps and probability maps provide pre- 150 dictions significantly better than random guessing, and the general ability of both methods to identify the most densely 100 populated cells, where about 20%–40% of the aftershocks 50 will occur, is comparable. The probability maps, however, are uniformly better than the Coulomb stress change maps 0 -600 -400 -200 0 200 400 600 at selecting the cells in which most of the remaining after- difference in cell rank shocks will occur. The only exception is the most scattered 1%–10% of the aftershocks, which neither method shows Figure 7. Comparing the aftershock location prediction precision of the static Coulomb stress change map and prob- any significant ability to anticipate (Fig. 8). We also note ability map. To do the comparison the cells on each map are that the shapes of the probability map forecast curves in ranked according to their predicted likelihood of containing Figures 8 and 13 are quite similar to curves that describe an aftershock, with a rank of 1 corresponding to the highest how well the spatial distribution of the small earthquakes in probability. When an aftershock occurs, the rank of the cell a catalog measured over a limited time period can predict that it occurred in on the probability map is subtracted from the rank of the same cell on the static Coulomb stress change the locations of large earthquakes for years afterward (Kafka map. A positive answer means that the probability map pre- and Levin, 2000; Kafka, 2002). dicted the location of the aftershock more accurately, while One surprising result is that while the static Coulomb a negative answer means that the Coulomb stress change stress change map performs better than random guessing for map made the better prediction. The right-handed skewness nearly all of the aftershocks in the Landers and Hector Mine of all of the histograms indicates that the probability map is performing better overall. Peaks on the right-hand side of sequences, for some of the Joshua Tree and North Palm the graphs for the North Palm Springs and Joshua Tree se- Springs aftershocks the static Coulomb stress change map quences indicate where large secondary aftershock clusters actually performs worse than random guessing (Fig. 8, left- caused concentrated deviations from the Coulomb stress hand side). This is probably due to the near-fault presence change predictions. 1444 K. R. Felzer, R. E. Abercrombie, and G. Ekstro¨m

100 90 Figure 8. Comparison of aftershock loca- Palm Springs 90 Palm Springs 80 tion forecasts made by the probability maps 80 real time prob 70 and by Coulomb stress change calculations. On 24 hr prob 70 60 the left are calculations done for the entire af- 60 Coulomb retrospective 50 tershock sequence; on the right are calculations 50 random 40 40 guess done for only those aftershocks located away 30 30 from the main fault plane. For the Joshua Tree, 20 20 Landers, and Hector Mine earthquakes this is 10 10 done by eliminating all cells experiencing 0 0 more than an absolute value of 8 bars of stress 0 10 20 30 40 50 60 70 80 90100 0 10 20 30 40 50 60 70 80 90 100 change from the mainshock; for the smaller 400 400 North Palm Springs earthquake we eliminate 350 350 Joshua Tree Joshua Tree all cells experiencing more than 5 bars of stress 300 300 change. The percentage of aftershocks are plot- 250 250 ted versus the number of (2.5 by 2.5 km) cells 200 200 that need to be identiﬁed by each forecasting method in order to have this percentage of af- 150 150 100 tershocks contained. Squares give results for 100 the real-time probability map forecasts, stars 50 50 for the 24-hr probability map forecasts, and tri- 0 0 angles for the Coulomb stress change calcula- 0 10 20 30 40 50 60 70 80 90100 0 10 20 30 40 50 60 70 80 90 100 tions. For reference, the ﬁlled circles give the 2000 best prediction that would have been possible, 2000 1800 1800 Landers 1600 Landers measured from the minimum number of cells 1600 1400 needed to cover different percentages of after- 1400 1200 shocks in the completed sequence, and the

minimum number of cells minimum number of cells 1200 1000 empty diamonds give the results that would be 1000 800 800 obtained if the likelihood of each cell having 600 600 aftershocks were guessed randomly. The ran- 400 400 dom-guess results and accompanying 95% 200 200 confidence intervals are calculated from 1000 0 0 0 102030405060708090100 0 102030405060708090100 Monte Carlo trials with each aftershock sequence. For each sequence, the overall best 600 400 forecasting is provided by the real-time prob- 500 Hector Mine 350 Hector Mine ability maps, with the next best forecasts pro- 300 vided by the 24-hr probability maps. 400 250 300 200 200 150 100 100 50 0 0 0 10 20 30 40 50 60 70 80 90100 0 10 20 30 40 50 60 70 80 90 100 percentage of aftershocks percentage of aftershocks of some cells with highly negative static Coulomb stress quake. We find that the Coulomb map predictions indeed changes. These cells will be ranked as the least likely to improve when the near-fault area is eliminated, but remain contain aftershocks, but since they are in the near-fault re- worse than the real-time and 24-hr probability map predic- gion they may contain many secondary aftershocks. The as- tions (Fig. 8), with the exception of the 24-hr prediction for signments of negative stress change to these cells may also the Hector Mine aftershocks, for which the two predictions be in error due to inaccuracies in determining the mainshock appear comparable. This may be because an unusually large slip distribution; calculated near-fault stress changes are ex- proportion (2/3) of aftershocks in the Hector Mine sequence tremely sensitive to the slip distribution and background occured in near-fault cells and were eliminated. We also in- stress orientation used (e.g., Kilb et al., 1997). Thus we vestigate whether the relative predictive ability of the static make new graphs for only those aftershocks occurring in Coulomb stress change and probability maps varies with af- cells experiencing less than an absolute value of 8 bars of tershock location and aftershock magnitude. We find that stress change for the Joshua Tree, Landers, and Hector Mine location is important; even with the near-fault stress change earthquakes and less than 5 bars of stress change for the calculation errors, the static Coulomb stress change map has North Palm Springs mainshock (Fig. 8; right-hand side). a definite advantage in the near-fault region (Fig. 9). This is This eliminates most of the near-fault aftershocks of each not surprising since the mainshock fault location is a primary mainshock. Toda et al. (1998) similarily used an 8-bar cutoff input to the Coulomb stress change calculation, while it is to eliminate near-fault aftershocks of the 1995 Kobe earth- neglected in the probability map calculation. Secondary Aftershocks and Their Importance for Aftershock Forecasting 1445

35N Landers 34.2N 34.8N

34.6N 34.1N

34.4N 34.0N 34.2N 33.9N 34N

117W 116.6W 116.2W 33.8N 116.5W 116.3W 116.1W 34.9N Figure 10. Comparison of the 5% of cells iden- Hector Mine tified to be the most likely to contain the next Joshua 34.8N Tree aftershock by the mainshock-induced static Cou- lomb stress change calculation (black dots) and prob- 34.7N ability map method (gray squares) right before the Landers earthquake occurred. (Note that some of the 34.6N circles and squares overlap.) Small and large stars de- note the epicenters of the Joshua Tree and Landers 34.5N earthquakes, respectively. 34.4N 2002). Specific comparisons of the two methods for all 34.3N M Ն5 aftershocks are given in Table 2. 116.6W 116.4W 116.2W 116W One caveat for our results is that we may not have used the best method to rank cells on the static Coulomb stress Figure 9. Black dots indicate the 50 aftershocks of the Landers and Hector Mine earthquakes best pre- change map. We rank the cells simply according to the dicted by the Coulomb stress change calculations; amount of positive Coulomb static stress change they ex- open squares indicate the 50 aftershocks best pre- perience, which is in keeping with how static Coulomb stress dicted by the probability map calculations. There is change is often applied to predict aftershock locations. When some overlap between the two data sets, but, as ex- Coulomb stress change is applied in conjunction with rate pected, the Coulomb stress change is better at predicting aftershocks lying near the mainshock fault and state friction theory, however, a stress step changes the plane (presumably direct aftershocks) while the prob- rate of ongoing seismicity by a given multiple, so the probability map is better at predicting the more distributed, ability of having an aftershock in a given cell actually be- (and presumably secondary), aftershocks. The epicen- comes a combined function of the stress change and the pre- ters of the Landers and Hector Mine earthquakes are existing seismicity rate. Yet we note that static Coulomb denoted by a large star and circle, respectively. stress change studies that incorporate the effect of pre- existing seismicity rates do not appear to show any large The relative ability of the two methods to predict an predictive improvement over studies that use static Coulomb aftershock does not vary with aftershock magnitude at the stress changes by themselves (compare Hardebeck et al. 95% confidence level. Just as the probability map method [1998] and Toda et al. [1998]). So we feel justified for now predicts aftershock locations more precisely than the static in simply ranking the cells according to the amount of posi- Coulomb stress change map in general, it also more precisely tive static Coulomb stress change they experience, with the predicts the locations of the largest aftershocks. Specifically, idea that this might be refined in future studies. it is interesting to note that the real-time probability maps In addition to the better performance we find here, there do a much better job than the Coulomb stress change map are also other benefits to be gained from using a probability- at pinpointing the locations of the Landers (Fig. 10) and map-type method for aftershock prediction rather than, or in Hector Mine epicenters, primarily a result of the fact that addition to, static Coulomb stress change maps. For one, the both of these earthquakes nucleated at some distance from probability map method is potentially faster; it can be used their respective mainshock faults (hence low Coulomb stress as soon as 50 aftershocks have been located—there is no change) but had active foreshock sequences (hence the need to wait for a mainshock slip inversion. For another, the strong identification by the probability map method.) We probability maps not only identify which cells are most note that the actual static stress change at the Hector Mine likely to contain an aftershock but also assign a specific af- epicenter does remain controversial (Harris and Simpson, tershock probability to each cell. We evaluate how well these 1446 K. R. Felzer, R. E. Abercrombie, and G. Ekstro¨m

25 100 90 Coalinga 80 Morgan Hill 20 Oceanside 70 Loma Prieta 15 60 Northridge 50 random 40 10 30 number of trials 20 5 10 (A) minimum percentage of cells 0 0 0 10 20 30 40 50 60 70 80 90 1 -5% 0 5% 10% 15% 20% percentage of aftershocks probability prediction error 100 90 Figure 11. Error distribution of how well the probability maps predict the actual probability that a 80 given set of cells will contain the next aftershock. For 70 each aftershock sequence in Table 1 we measure what 60 percentage of aftershocks occurs in the sets of probability map cells that add up to 5%, 10%, 20%, 30%, 50 40%, 50%, 60%, 70%, 80%, 90%, and 95% of the 40 total probability. The sets are specifically chosen such 30 that the target probability is obtained with the smallest possible total number of cells. The histogram gives 20 the differences between the measured percentages and 10 (B) predicted probabilities. The distribution is skewed to minimum percentage of cells 0 the right, meaning that the cells assigned the highest 0 10 20 30 40 50 60 70 80 90 100 probability of having an aftershock actually contain percentage of aftershocks slightly more aftershocks than expected and cells assigned the lowest probabilities have somewhat fewer Figure 12. Results of using the probability map aftershocks than expected. For nearly 90% of the to forecast the aftershocks of the Coalinga, Morgan trials the difference between the predicted probability Hill, Oceanside, Loma Prieta, and Northridge earth- and percentage of aftershocks measured is less than quakes. The percentage of aftershocks is plotted 10%. against the percentage of the cells covering each aftershock zone that need to be identified by the probability map method in order to forecast them. Results are given for (A) real-time probability map forecasts specific probabilities are estimated by measuring what per- and (B) 24-hr probability map forecasts. Forecasts for centage of the aftershocks occur in the sets of cells that add all five of the aftershock sequences show similar suc- up to 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, cess rates, suggesting that the probability map method 90%, and 95% of the total predicted probability. The sets of provides forecasts of a consistent quality. cells are chosen such that each set contains as few cells as possible. We calculate the differences between the predicted probabilities and the actual aftershock percentages (Fig. 11). affected by the aftershocks. Estimates of these properties can We find that the error distribution is skewed to the right, best be made by combining a local Omori’s law with sec- meaning that the cells with the highest probabilities tend to ondary activity accounted for with the Gutenberg–Richter contain a slightly larger percentage of aftershocks than pre- relationship and ground-motion calculations. An example of dicted, and vice versa. Overall nearly 90% of the probability such a forecasting method is given by Gerstenberger et al. map probability forecasts are correct to within 10%. We also (2001). find that the probability map method produces predictions that are consistent in quality for different aftershock se- Conclusions quences (Fig. 12). Finally it is important to note that both the Coulomb We find that many triggered earthquakes are actually stress change calculations and the probability map method secondary aftershocks—aftershocks of previous after- provide information only on the most likely location of af- shocks—and are thus not constrained to occur where the tershock epicenters. They provide no information about how mainshock increased stress. The percentage of aftershocks many aftershocks are likely to occur, about the magnitude in a sequence that are secondary increases with time and of those aftershocks, or about the extent of area that may be mainshock magnitude. For the average California aftershock Secondary Aftershocks and Their Importance for Aftershock Forecasting 1447 sequence of an M 6 mainshock with no aftershocks larger Freed, A. M., and J. Lin (2002). Accelerated stress buildup on the southern than the mainshock, we find that over 35% of all aftershocks San Andreas Fault and surrounding regions caused by Mojave Desert earthquakes, Geology 30, 3007–3010. and over 50% of ongoing aftershocks are secondary 10 days Gerstenberger, M. C., S. Wiemer, L. M. Jones, and D. Giardini (2001). after the mainshock. For the average M 7 mainshock over Testing and implementation off real time probabilistic aftershock haz- 40% of the total and nearly 70% of ongoing aftershocks are ard mapping (abstract), EOS 82 no. 47 (Fall Meet. Suppl.), S41C-06. secondary over the same time period. Gomberg, J. (2001). The failure of earthquake failure models, J. Geophys. Because secondary aftershocks are common we should Res. 106, 16,253–16,263. expect many aftershocks to occur near previous, recent af- Gross, S. J., and C. Kisslinger (1994). Test of models of the aftershock rate decay, Bull. Seism. Soc. Am. 84, no. 5, 1571–1579. tershocks. 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