Earthquake recurrence models fail when earthquakes fail to reset the stress field



[1] Parkfield's regularly occurring M6 mainshocks, about every 25 years, have over two decades stoked seismologists' hopes to successfully predict an earthquake of significant size. However, with the longest known inter-event time of 38 years, the latest M6 in the series (28 Sep 2004) did not conform to any of the applied forecast models, questioning once more the predictability of earthquakes in general. Our study investigates the spatial pattern of b-values along the Parkfield segment through the seismic cycle and documents a stably stressed structure. The forecasted rate of M6 earthquakes based on Parkfield's microseismicity b-values corresponds well to observed rates. We interpret the observed b-value stability in terms of the evolution of the stress field in that area: the M6 Parkfield earthquakes do not fully unload the stress on the fault, explaining why time recurrent models fail. We present the 1989 M6.9 Loma Prieta earthquake as counter example, which did release a significant portion of the stress along its fault segment and yields a substantial change in b-values.

1. Introduction

[2] The Parkfield segment of the San Andreas Fault (SAF) in California is the transition zone between the creeping section of the SAF in the north and the locked Cholame segment to the south [Bakun and Lindh, 1985]. The segment does not intersect other faults and features abundant mircoseismicity (Figure 1a). Additionally, it ruptures repeatedly in magnitude Mw ∼ 6 events: six times since 1857, on average every 24.7 years, with individual interval lengths of 12–38 years. This ‘characteristic’ behavior early on caught the attention of seismologists to be an excellent natural laboratory for the study of earthquake cycles, a laboratory that could help to develop a better understanding of the ongoing loading processes in the earth's crust [Bakun and Lindh, 1985]. Based on the quasi-periodic occurrence of M6 events, the ‘Parkfield Earthquake Prediction Experiment’ was set up in 1985 [Bakun and Lindh, 1985]. It applied the widely used time-predictable recurrence hypothesis [Shimazaki and Nakata, 1980], in which an earthquake occurs when the long-term loading rate on the fault has recovered the stress relieved in the most recent event, the latest Parkfield rupture of 1966, and forecast the next one to occur before 1993. The most recent ‘Parkfield earthquake’ eventually ruptured in 2004 after the longest known inter-event time of 38 years. Being too late to fulfill the prediction by the time-predictable model [Jackson and Kagan, 2006], it was also of the same size as its priors, M6, invalidating a slip-predictable forecast at the same time, which would have expected an M6.6–6.9 [Murray and Segall, 2002]. In the slip-predictable model, the size of the next earthquake is proportional to the time since the latest event. What, from the historic record, seemed to be an ideal candidate to overcome the ‘unpredictability-status’ of earthquakes, was in the end rejected by all so far applied forecast models [Jackson and Kagan, 2006].

Figure 1.

(a) Schematic setting of the Parkfield and Loma Prieta study areas along the San Andreas Fault, Central California. Insets: local seismicity, red: earthquakes used for the analysis, stars: 2004 Parkfield and 1989 Loma Prieta mainshocks. (b) Parkfield's seismicity M ≥ 1.3 since 1981, black: events inside the asperity volume; orange, red and purple: sampled seismicity volumes for three nodes for which the FMDs are shown in Figure 2a. Open and solid stars: epicenters of the 1966 and 2004 mainshocks, respectively. (c–f) b-value cross-sections for different time periods, the dashed line follows the b = 0.87 contour, indicating the asperity volume. (g) Loma Prieta's seismicityM ≥ 1.3 since 1981, star and ellipses: 1989 epicenter and approximate areas of maximum co-seismic slip [Beroza and Zoback, 1993]. (h–k) b-value cross-sections for different time-periods.

[3] Although the Parkfield experiment failed to forecast the timing, a study from Schorlemmer and Wiemer [2005]could claim to have ‘postcast’ the rupture area of the 2004 earthquake through a high-resolution survey of the microseismicity data from the two decades before the earthquake. They analyzed the spatial variation in the size distribution of earthquakes, the so-called ‘b-value’ of the Gutenberg-Richter law [Gutenberg and Richter, 1944]: log N = a − bM, where ‘N’ is the cumulative number of earthquakes above magnitude ‘M’, ‘a’ describes the productivity, and ‘b’ the size distribution, i.e., the higher the b-value the smaller the fraction of large versus small magnitudes. b-values are known from the laboratory [Scholz, 1968; Amitrano, 2003] and observations [Schorlemmer et al., 2005] to be inversely correlated with applied shear stress, that means high stresses produce low b-values, and vice versa. A number of studies have established that the b-value acts as a crude stress-meter for identifying asperities, highly stressed areas of a fault where future mainshocks are likely to originate [e.g.,Wiemer and Wyss, 1997, 2002; Schorlemmer and Wiemer, 2005]. Several moderate to large earthquakes, e.g., Northridge 1994, Kobe 1995, and Landers 1999 have been documented for spatially heterogeneous and high post-mainshock b-values, which often coincide with areas of high co-seismic slip, i.e., significant stress release [Wiemer and Katsumata, 1999].

[4] The destructive 1989 M6.9 Loma Prieta earthquake ruptured the transition zone between the northern end of the SAF creeping section and the adjacent locked San Francisco 1906 segment further north. The faulting structure in that region is far more complex than in Parkfield, and less well understood; it is debated whether and to what parts the earthquake and its aftershocks broke the main trace of the SAF or rather the close-by Sargent fault [Dietz and Ellsworth, 1990]. Similar to the M6 Parkfield mainshocks, which occur at the northern-most extent of the M8 class 1857 type earthquakes, the M6.9 Loma Prieta mainshock occurred at the southern-most extent of the 1906 rupture zone, and equally marks the transition between an interseismically stably-looked and a freely-creeping fault segment (Figure 1a). Estimated recurrence times for the Loma Prieta event range from 100–150 years [Shaw et al., 1994] to 300–600 years [Valensise and Ward, 1991].

[5] We study both, the Parkfield and Loma Prieta segments, in terms of their b-value distributions. We demonstrate that high-resolution b-value mapping in space AND time is a powerful tool to reveal likely locations, estimate sizes and even resolve temporal variations in earthquake probabilities for future events.

2. Data

[6] We base our analysis on the Advanced National Seismic System (ANSS) catalog between January 1981 and June 2011. For each of the two fault segments, we sample all earthquakes located within a swath centered on the fault (Figure 1a). For Parkfield we consider a swath width of 10 km, which well includes all relevant seismicity; for Loma Prieta we extend the width to 5.5 km either side of the trace to accommodate for the possibility of dip at depth and the uncertainty of which fault segment ruptured. The chosen swath is consistent with the major regional seismicity distribution as judged from the mapview (Figure 1a). We assess the completeness magnitude for both segments, and confirm the previously published Mc = 1.3 for Parkfield [Schorlemmer and Wiemer, 2005]; the same applies to Loma Prieta. The Parkfield catalog amounts to nearly 5500 events, about half of which occurred before the 2004 mainshock. From about 4200 events in Loma Prieta not quite 400 occurred in the pre-mainshock period. More than 2000 events occurred during the first month of the aftershock sequence, and the activity since 1993 is again low with only about 700 events.

3. Method

3.1. Cross-Sectional b-Value Mapping

[7] To map the cross-sectional b-values we project the sampled earthquakes onto a vertical fault plane. At each grid point (1 km spacing), we select all events within a search radius of 5 km, resulting at each grid node in a cylindrical sample volume of 10 km diameter and 10 km width, 11 km width for Loma Prieta (example volumes shown inFigures 1b and 2a). If we find a minimum of 50 events for a grid node, we calculate the maximum likelihood b-value [Aki, 1965]. The formal uncertainty in the single b-value estimates [Shi and Bolt, 1982] is largely driven by the number of sampled events, and the minimum of 50 events corresponds to average uncertainties of about ±20% [Wiemer and Wyss, 2002]. Since for most grid nodes many more events contribute, the individual errors in Parkfield's b-estimates average below ±0.1, for Loma Prieta below ±0.15. Larger errors are rare and only observed at the very edges of the cross sections.

Figure 2.

(a) Example FMDs for three nodes from different b-value ‘regimes’ along the Parkfield cross-section (locations seeFigure 1b), (b) FMDs for the full asperity volume (b = 0.87 contour) for three time periods (Figures 1d–1f). Temporal b-value and probability evolution for Parkfield and Loma Prieta, given in average values for pre-mainshock, aftershock, and post-mainshock time periods (solid lines), and the actual time series with associated standard deviation (shading). (c) b-value evolution for Parkfield. (d) Translation of a- and b-values into daily probabilities for the occurrence of one ore more M6+ events. Black line/grey shading: observed recurrence times, Tr, of M6 earthquakes, 12–38 yrs, mean: 24.7 yrs. (e) b-value evolution for Loma Prieta. (f) Daily probabilities for the occurrence of one ore more M6.9+ events. Grey shading: geologically estimated recurrence times of Loma Prieta type earthquakes, 100–600 yrs.

[8] Since b-values are thought to represent stresses, and mainshocks are thought to change the stress regime along the rupturing segments, we investigate potential changes in the spatial b-value patterns for different time periods. For Parkfield, we repeat the mapping for four periods: 1. the full 30-year time period, 2. before the 2004 mainshock, 3. during the aftershock sequence (first six months), and 4. the roughly 6 years since then (Figures 1c–1f). For Loma Prieta, we calculate cross-sections for: 1. before the 1989 mainshock, 2. during the first month of aftershocks, 3. during three years of aftershock activity following the first month, and 4. the nearly two decades since 1993 (Figures 1h–1k).

3.2. The b-Value Time-Series

[9] To analyze the temporal evolution of the b-value in more detail, we calculate time series of b-values. In Parkfield, some of the structure of the segment is understood, enabling us to concentrate our analysis on the asperity volume itself, which means that we select all earthquakes located inside the b = 0.87 contour line (Figure 1c). We contrast this time series by data from a high b-value patch of the creeping segment NW of the asperity (orange earthquakes,Figure 1b). We use a moving window of 100 events and move it continuously, event by event, through the catalogs, calculating a maximum likelihood b-value [Aki, 1965] and its formal standard deviation according to Shi and Bolt [1982]at each step. To produce a homogeneous sampling density through time, we interpolate the time series on a daily basis and smooth the signal with a 3-month kernel. The technique of using a constant number of earthquakes for each step implies varying lengths of the time windows, they average around 3 years for the asperity data. We show the calculated b-values at the end of their corresponding time windows, which is why the time series starts in 1985 although it uses all events since 1981 (Figure 2c). For Loma Prieta we know less about the fault structure at depth, which is why we calculate a time series from all events located between Q1 and Q2 (Figure 1g), which corresponds to the estimated extent of the 1989 rupture. We use the same time window of 100 events (Figure 2e).

[10] Taking the time series b-values and the corresponding a-values, we extrapolate the Gutenberg-Richter relation to extract the associated rates of target magnitude earthquakes (Mtarg ≥ 6 (Parkfield) and Mtarg ≥ 6.9 (Loma Prieta)), leading to expected recurrence times [Wiemer and Wyss, 1997]. From those we calculate the time series of daily probabilities of target magnitude events and compare these values to the observed or estimated recurrence probabilities (Figures 2d and 2f).

4. Results

4.1. A Stable Asperity Structure in Parkfield

[11] For the cross-sectional analysis of Parkfield, we observe a strikingly stable pattern of the b-values through the seismic cycle (Figures 1c–1f). The low b-value anomaly documented bySchorlemmer and Wiemer [2005]before the 2004 mainshock is, surprisingly, still a prominent feature after the mainshock. As expected, the b-values do rise a little during the aftershock sequence, highlighting the areas of largest afterslip (light blue and green part of formerly blue asperity), however, they return to low values within half a year (Figures 1e, 1f, and 2b). The high b-value structures surrounding the asperity remain equally stable. The fundamental differences between the statistical behavior of earthquakes in different parts of the segment are illustrated by the frequency-magnitude-distributions (FMD) for three selected nodes (Figure 2a), their relative locations are shown in Figure 1b. Comparing the FMDs for the full asperity volume for the three different time periods (Figure 2b), they reinforce the results seen in Figures 1d–1f: the distributions are very much the same for the pre- and post-mainshock periods, the two b-values are only slightly different according to the associated standard variation [Shi and Bolt, 1982]; during the aftershock sequence the rate is strongly (factor ∼20) increased, the b-value a little. Both observations are confirmed by the time series analysis shown inFigure 2c: the differences between the asperity and the high-b-value volume are prominent, while the average values inside the asperity before and after the mainshock are similar and the perturbation through the mainshock does not last long. This translates also into the probability domain, the probabilities for anotherM ≥ 6 event do not reduce due to that mainshock. The order of average probability level for the asperity volume is close to the observed recurrence times (Figure 2d), and the match is even better, if we do not use the overall asperity ‘a’ and ‘b’-values but calculate the minimum recurrence time from the cross-sectional local ‘a’ and ‘b’-values. This yields a 44 year recurrence time estimate. Considering that this M6+ rate estimate is mainly deduced from the properties of M1–M4 earthquakes, it is remarkably close to the maximum observed value of 38 years. As expected, the creeping volume produces probabilities several orders of magnitude less (Figure 2d).

4.2. Substantial Changes in Loma Prieta

[12] Comparing the b-value cross-sections before and after the mainshock reveals a very different story in Loma Prieta than the one observed at Parkfield (Figures 1h–1k). Since the pre-mainshock activity on that segment was rather low, the spatial resolution of b-values is not as good as at Parkfield, but where we can resolve them they are low. The aftershock sequence was highly productive, so that we separated it into the first month (to avoid incompleteness complications in the direct aftermath of the mainshock, we used an increasedMc = 1.6 for this month of data) and the following three years. In both consecutive periods we observe increasing b-values, starting in the areas of greatest co-seismic slip, and growing through all of the segment. During the last two decades the activity along the segment reduced significantly, especially in the areas of greatest co-seismic slip, where we cannot longer resolve b-values. In the remaining parts of the segment the b-values stay significantly increased, though (Figure 1k). As suggested by the cross-sectional view, the time series of b-values and probabilities for a mainshock-repeat shows an opposite behavior compared to Parkfield, with stably increased b-values over the last two decades, combined with low seismicity rates, and consequently very low probabilities, three orders of magnitude below geologically estimated recurrence times (Figures 2e and 2f).

[13] We note that in both cases, the spatial variability in Parkfield and the temporal changes in Loma Prieta, the b-value signals that we interpret are statistically robust and significant, i.e., their overall patterns are largely insensitive to the choice of the free parameters (e.g., swath width,Nmin, sampling radius, and window- and step size); and their amplitudes are several times larger than the individual uncertainties.

5. Discussion

[14] In order to interpret and understand the observations along the Parkfield and Loma Prieta segments, we build on the inverse relationship between b-values and relative applied shear stress. Low b-value patches, as documented for the Parkfield asperity volume, represent high stress across that area relative to the neighboring patch of high b-values, which represent a lower stress level. The high b-value structures likely represent physical barriers to a moderate sized rupture: in the NW, the freely slipping fault surface that presumably has no energy stored for participating in a rupture, and SE of the asperity the distinct high b-value patch which aligns with the en echelon step over, where the surface trace of the fault crosses the Cholame valley. Observing the stable structure through the seismic cycle, we conclude that the stress distribution has been remarkably stable over the last three decades at least, and was not substantially changed by the 2004 Parkfield mainshock. The relative stress release in the mainshock, therefore, seems to have been only a minor fraction of the overall stress on the majority of the fault plane.

[15] We speculate that the physical cause of the stability of Parkfield's stress pattern is rooted in the underlying strength distribution of the fault, which in the Parkfield case is governed by the contrast in material properties across, and frictional behavior and pore pressure differences along the fault. All three have been reported to vary greatly along the segment [Eberhart-Phillips and Michael, 1993; Thurber et al., 2006; Zhao et al., 2010]. Together they seem to have created a stable and robust structure capable of constantly storing energy. This can explain why time-recurrent models fail at Parkfield: if the asperity always has enough energy stored to rupture in another moderate earthquake, the cyclic loading-unloading model underlying the time- and slip predictable models is inapplicable. Based on different aseismic and co-seismic displacement models, the current slip deficit in Parkfield since 1857, which might be released in the next large Cholame-segment earthquake, has been estimated to range from 1–2 m [Toke et al., 2006]. This deficit is even without further loading the equivalent of the cumulative slip of 3–5 more M6 earthquakes in the asperity.

[16] The stability of the spatial pattern of b-values mapped out by the microseismicity supports part of the characteristic earthquake hypothesis [Bakun and Lindh, 1985]: a significant future event that will nucleate within the apparently highly pre-stressed Parkfield asperity is likely to rupture again the same area as observed in past events. Because earthquake size scales with the available rupture area [Wells and Coppersmith, 1994], the size of the asperity as mapped through the b-values prescribes a maximum possible magnitude within that structure ofM ∼ 6.6. The possibility remains, that when the Cholame segment is ready to rupture, a nucleating earthquake in Parkfield might jump the en echelon step over and grow into an M8 class Fort Tejon type rupture of several hundred kilometers length, releasing the energy that has been accumulated in the locked segment for more than 150 years. These gap filling events are likely able to reset the stress field along that fault segment, as it has been documented for the great 1906 San Francisco rupture [Jaumé and Sykes, 1996].

[17] The analysis of spatial b-value patterns through time allows distinguishing ‘Parkfield-type mainshocks’ from those mainshocks that do release a significant portion of the available energy along a fault segment, as has been suggested for the 1989 Loma Prieta M6.9 event [Beroza, 1991]. The setting for that rupture appears much more complex and we lack the knowledge of the governing structures, which is why we cannot differentiate an asperity volume comparable to Parkfield for the Loma Prieta case. Nevertheless, even from the simplified ‘bulk’ analysis along the full rupture length of 1989, we resolve substantial differences for this earthquake. Loma Prieta has changed the local b-value distribution, and the increase still lasts more than two decades after the mainshock, suggesting, together with the low seismicity rate, a relaxed but increasingly locked setting.

6. Conclusion

[18] Based on our analysis we suggest that Parkfield earthquakes are somewhat ‘characteristic’ in the sense that the shape and strength of the asperity are geometrically constrained, and therewith the locations and sizes of these moderate events. Their inter-event times, however, are close to a stationary Poissonian process. This suggestion had been tested as null-hypothesis byKagan [1997] and Jackson and Kagan [2006], who could not reject the simple model for characteristic time-recurrent hypotheses. The Parkfield mainshocks apparently do not require a substantial tectonic re-loading, explaining the variation of more than a factor of three between individual inter-event intervals observed over the last 150 years, and a paleoseismologically deduced range of inter-event times between 8–188 years [Toke et al., 2006]. Recurrence models that are based on tectonic loading cycles as driving force seem not to be applicable in the Parkfield setting.

[19] We expect this type of recurrence models to be applicable in the Loma Prieta setting, though, where the mainshock together with its aftershocks seems to have released a substantial fraction if not all of the energy on that segment, which will therefore not soon be ready to rupture in another large earthquake. Here we expect stress-buildup through loading to be the necessary driving force to prepare the next major rupture. This is consistent with significantly longer recurrence times assumed for Loma Prieta type events.

[20] Studying the time series of b-values and their translation into M6 or M7 recurrence probabilities does not allow to accurately predict the timing of a future mainshock, but we can assess the current likelihood of a fault segment failing. In Parkfield this likelihood behaves in the post-mainshock phase much different from state-of-the-art estimates: these tooth-saw type renewal models, like, e.g., Brownian Passage Time, assume largely decreased probabilities after the mainshock which slowly recover over time, according to the tectonic loading [Gonzales et al., 2006]. The b-value deduced probabilities, however, stay at pre-mainshock probabilities, and the level is consistent with observed inter-event times. The situation is different in the Loma Prieta case. Probabilities before the mainshock were close to the level of geologically deduced recurrence times of 100–600 years, they have within a few years after the mainshock reduced by about three orders of magnitude, and have currently stabilised at this very low probability level, equivalent to recurrence times of many thousand years, suggesting ‘unloaded’ conditions for the time being.


[21] We thank J. Zechar, C. Kreemer, A. Michael, J. Rubinstein, W. Bakun, two anonymous reviewers and the editor A. Newman for their help in improving this paper. Earthquake catalog data was obtained from the Advanced National Seismic System ( The figures were produced with the generic mapping tools ( Part of this study was funded through SNF grant PMPDP2 134174.

[22] The Editor thanks two anonymous reviewers for assisting in the evaluation of this paper.