Partial Ruptures Cannot Explain the Long Recurrence Intervals of Repeating Earthquakes

Repeating earthquakes repeatedly rupture the same fault asperities, which are likely loaded to failure by surrounding aseismic slip. However, repeaters occur less often than would be expected if these earthquakes accommodate all of the long‐term slip on the asperities. Here, we assess a possible explanation for this slip discrepancy: partial ruptures. On asperities that are much larger than the nucleation radius, a fraction of the slip could be accommodated by smaller ruptures on the same asperities. We search for partial ruptures of repeating earthquakes in Parkfield using the Northern California earthquakes catalog. We find 3991 individual repeaters which have 4468 partial ruptures. The presence of partial ruptures suggests that the asperities of repeating earthquakes are much larger than the nucleation radius. However, we find that partial ruptures could accommodate only around 25% of the slip on repeating earthquake patches. A 25% increase in the slip budget can explain only a small portion of the long recurrence intervals of repeating earthquakes.


Long Recurrence Intervals of Repeating Earthquakes
Repeating earthquakes rupture the same asperity of a fault time and time again, with surprisingly regular recurrence intervals.These earthquakes are identified by their co-located rupture asperities, equal magnitudes, and waveform similarity (Gao et al., 2021;Uchida & Bürgmann, 2019;Waldhauser & Schaff, 2021).At first glance, repeating earthquakes seem to be a simple phenomenon; these earthquakes represent locked asperities on a fault, which are loaded to failure by the surrounding fault creep (Beeler et al., 2001).In this simple framework, the time between repeating events also seems intuitive; if the asperity is locked between earthquakes, the slip in each earthquake (S) should match the slip rate (V creep ) in the creeping area surrounding the repeater asperity.If the average time between repeating earthquakes is T r , the slip per repeater should be S = V creep T r .
To relate the recurrence interval T r to the moment M 0 of an earthquake, we note that the seismic slip scales with the cube root of the seismic moment: (2) And if the stress drop is magnitude-independent, as often observed (e.g., Allmann & Shearer, 2007), this simple model of repeaters would suggest that the recurrence interval should scale as   ≈  1∕3 0 .However, the observed recurrence intervals of repeating earthquakes are much longer than this calculation would imply, at least given seismological estimates of the stress drop (Abercrombie, 2014;Abercrombie et al., 2020) and geodetic or geological estimates of the regional creep rate (Harris & Segall, 1987; R. M. Nadeau & Johnson, 1998).Repeater recurrence intervals observed globally scale with moment as   ∝  0.17 0 , not   1∕3 0 (K. H. Chen et al., 2007; R. M. Nadeau & Johnson, 1998).One can think of these discrepancies as a slip deficit.The observed seismic slip in the repeating earthquakes is smaller than the long-term slip on the surrounding fault.
Nevertheless, repeating earthquakes are often used as embedded creep meters on faults.Their recurrence times are coupled with the empirical  0 ∝  0.17  scaling to estimate slip rate (e.g., Uchida & Bürgmann, 2019;Waldhauser & Schaff, 2021).However, the difference between the observed and theoretical scaling implies that we still do not fully understand the processes that create repeating earthquakes.Until we can understand the difference between the observed and theoretical scaling, repeater-based creep meters will remain empirical, making it difficult to expand their use or understand their uncertainty.

Proposed Origins of the Missing Slip
Researchers have proposed a range of physical models to explain the long recurrence intervals of repeating earthquakes.One set of models allows stress drop to increase as earthquakes get smaller.To match the geodetically observed slip rate in Parkfield and recover the   ∝  0.17 0 scaling, the stress drop would have to scale as   −1∕4 0 (K. H. Chen et al., 2007).In this case, very small repeating events would require high stress drop (∼2 GPa, Sammis & Rice, 2001).In Parkfield, repeaters are observed to have median stress drops around just 10 MPa (Abercrombie, 2014;Allmann & Shearer, 2007;Imanishi et al., 2004), though these stress drops could be underestimated if earthquakes have heterogeneous slip distributions with highly localized slip (Dreger et al., 2007;Kim et al., 2016).
A second set of models allows spatial variations in creep rate.A locally lower creep rate could be created by a boundary effect along the border between locked and creeping sections of the fault (Sammis & Rice, 2001).However, the common occurrence of repeating earthquakes is hard to reconcile with the geometrical constraints of this model-in Parkfield, 55% of earthquakes are repeating (Nadeau et al., 2004), and it is difficult to place all of these earthquakes along creeping boundaries.Instead, Williams et al. (2019) suggests that the creep rate varies among the strands that compose the fault zone.In this model, repeaters have long recurrence times because the fault strands have lower slip rates than the system they compose.However, there are few observations to support this more recent model.
A final set of models allows slip on the repeater asperity between repeating earthquakes.These models suggest that much of the slip on repeater asperities accumulates via two mechanisms: via aseismic slip (e.g., Beeler et al., 2001;Cattania, 2019;Chen & Lapusta, 2019), or via smaller earthquakes (e.g., Cattania, 2019).The smaller earthquakes rupture rupture parts of the repeater asperity and are referred to as "partial ruptures."Since the rupture area of the repeaters and partial ruptures are likely to overlap, the partial ruptures will take up a part of the asperity's slip budget.The recurrence interval estimate above, which includes only the slip in repeaters, will then underestimate repeaters' recurrence times.
Such inter-repeater slip seems plausible-recent studies have observed "hierarchical rupture distributions" of repeating earthquakes in Japan, whereby the ruptures of small earthquakes are located within the rupture areas of larger repeating events (Chang & Ide, 2021;Ide, 2019;Okuda & Ide, 2018;Uchida et al., 2007).If the ruptures of a given patch followed a Gutenberg-Richter distribution that extended to a maximum magnitude equal to the repeater magnitude, each magnitude 3 repeater should be accompanied by 10-20 observable partial ruptures.Outside of just repeating events, we regularly see partial ruptures of locked faults around the world (e.g., Konca et al., 2008;Qiu et al., 2016;Ruiz et al., 2014). 10.1029/2023JB027870 3 of 14

Modeled Partial Ruptures
In this study, we focus on this last model: where the asperity can slip during smaller earthquakes between the larger characteristic repeating events.In this model, the behavior of the repeating earthquake asperity depends on the asperity radius.Specifically, behavior depends on how big the radius is relative to the "nucleation radius" R nucl : the radius of the smallest asperity that can host a seismic event (e.g., Cattania & Segall, 2019;Chen & Lapusta, 2009, 2019;Ruina, 1983).On repeater asperities that are only slightly larger than the nucleation radius, all ruptures on the asperity will be around the same size.On repeater asperities that are much larger than the nucleation radius, there are also small earthquakes that do not rupture the entire asperity.There are "partial ruptures" between complete repeater ruptures.
As such, with increasing asperity size, we expect to observe a transition from a regime where partial ruptures are absent to a regime where a large portion of the slip budget is made up of partial ruptures.The transition is estimated to occur between R ∼ 4.3 R nucl -6 R nucl (Cattania & Segall, 2019).The presence or absence of partial ruptures could thus allow us to place a constraint on the size of repeating earthquake asperities relative to the nucleation radius.
In this study, we aim to identify and count the partial ruptures of repeating earthquakes in Parkfield, California.We will use our observations to (a) determine if slip in partial ruptures can account for the repeaters' slip deficit and explain the long recurrence intervals of repeating earthquakes and to (b) determine the size of repeater asperities relative to the nucleation radius.We will use this calibration to further tune and assess numerical models of repeating earthquakes' long recurrence intervals.

Finding Repeaters and Partial Ruptures
We begin by searching for repeating earthquakes and partial ruptures in Parkfield, California.We consider two repeating earthquake catalogs.First, we use a simple approach to identify co-located earthquakes from their locations without new waveform correlation.We take advantage of the high-quality earthquake locations already obtained in this area (Waldhauser & Schaff, 2008) and identify co-located earthquakes as earthquakes located within one rupture radius of each other.Second, we use a more sophisticated and extensive repeater catalog created using waveform correlation by Waldhauser and Schaff (2021).

Identifying Repeating Earthquakes
To search for repeaters in the NCSN double-difference relocated catalog (Schaff & Waldhauser, 2005;Waldhauser, 2013;Waldhauser & Schaff, 2008), we first select earthquakes in the 90-km-long area around Parkfield (Figure 1), where over 50% of seismicity occurs in repeating clusters (Nadeau et al., 2004).We analyze events between 1984 and 2021, excluding 10 years after the 28th September 2004 M w 6 Parkfield earthquake; this large-magnitude event could affect the moment and recurrence interval of repeating sequences via direct loading or via changes in the aseismic slip that loads the repeaters (K.H. Chen et al., 2010Chen et al., , 2013;;Uchida et al., 2015).The analyzed catalog contains 7590 events with magnitudes between M w −0.3 and 4.9.
We calculate each event's moment (M 0 ) from the catalog magnitude (M) assuming M 0 = 10 1.2M+10.15(Wyss et al., 2004).We then project each earthquake onto a fault vertical plane with a strike of 135°, and estimate the ruptures' radii.For circular ruptures, the radii R are In our primary analysis, we assume a stress drop Δσ of 10 MPa, as has been inferred for events in the Parkfield region (Abercrombie, 2014;Allmann & Shearer, 2007;Imanishi & Ellsworth, 2006).We also test using a stress drop of 3 MPa (Section 3.4).
To search for repeating earthquakes, we cut the catalog at the magnitude of completeness (M w 1.1) to identify mostly complete sets of repeating earthquakes: without too many missed events.We consider each M w > 1.1 earthquake in the NCSN catalog as a potential repeater and search for co-located events: earthquake pairs whose catalog locations are within one radius of this reference event horizontally along and perpendicular to the fault 10.1029/2023JB027870 4 of 14 as well as vertically.This criterion leads to potential repeater pairs with a minimum of 38% overlapping source area (Gao et al., 2023).These co-located earthquakes are classified as potential repeaters if their magnitudes are within 0.3 magnitude units of each other.However, we remove repeater pairs separated by less than 50 days (as shown in Figure 3), as pairs with short recurrence intervals are likely to be ruptures triggered by a nearby larger mainshock, not "normal" repeating earthquakes loaded by aseismic slip.To calculate the recurrence interval, we also remove pairs of events which span the 10-year gap after the Parkfield earthquake.Our constraint on recurrence intervals is similar to that has been applied to repeaters by Li et al. (2007) and Bohnhoff et al. (2017).
To account for the catalog location error, we allow an 80-m uncertainty on the horizontal location and a 97-m uncertainty on the vertical location.These uncertainties are the 90% confidence limits for relative location errors in the combined relocated and real-time catalogs.This lenient constraint will include separated earthquake pairs, providing an upper bound on the number of repeating earthquakes and partial ruptures.We additionally use the error ellipse reported in the NCSN catalog for each event pair to provide a lower bound on the number of repeating earthquakes and partial ruptures (see Section 3.4).

Identifying Partial Ruptures
Our search of the NCSN catalog reveals 3991 individual repeating earthquakes: 3991 earthquakes plausibly co-located with at least one other earthquake within 0.3 magnitude units.We also have 2976 repeating earthquakes from the Waldhauser and Schaff (2021) catalog, grouped into 612 sequences.We can now search for partial ruptures of each of these earthquakes.We again search the entire catalog for co-located events.Here we do not truncate the catalog at M w 1.1, and there is no constraint on the recurrence interval between a repeating event

Analyzing Repeating Earthquakes and Partial Ruptures
Our earthquake search results in two collections of repeating earthquakes and partial ruptures.In the first collection, made by searching the relocated NCSN catalog, we find 3991 individual repeaters.These events have 4468 partial ruptures.In the second collection, using the Waldhauser and Schaff (2021) catalog, we find 2976 repeaters which have 2463 partial ruptures.Four examples of these repeaters and partial ruptures are illustrated in Figure 2. The repeating earthquakes are colored in blue, and the smaller-magnitude partial ruptures are in orange.Some repeating asperities host numerous partial ruptures (e.g., panel b) while other asperities host mostly similar-magnitude events (e.g., panel c).
An example of a patch that hosts many smaller magnitude events is the M w 6.0 repeating earthquake asperity (Figure S13 in Supporting Information S1).In the 19 years since the last magnitude 6.0 earthquake occurred, TURNER ET AL. 10.1029/2023JB027870 6 of 14 there have been over 400 smaller earthquakes within the same area where the 2004 earthquake happened.The total moment of these smaller partial ruptures is equivalent to half of the moment of the M w 6.0 repeater.However, this analysis does neglect the rupture's complex slip distribution, and there is no opportunity for averaging given only one recurrence interval, so the Mw 6.0 Parkfield repeater is not considered in the rest of the analysis.

Moment-Recurrence Scaling of Repeaters
We now analyze the numbers and timings of the two collections of repeating earthquakes and partial ruptures.We first analyze the repeaters' recurrence intervals.We take each identified repeater and determine the time between that event and the next repeater on its asperity.We plot this recurrence interval against the pair's average moment in Figures 3a and 3c for each collection of repeating earthquakes.Since there is significant scatter in the individual TURNER ET AL. 10.1029/2023JB027870 7 of 14 recurrence intervals, we also bin the pairs by moment and calculate the median recurrence interval in each moment bin.We estimate the uncertainty of these median recurrence intervals using a bootstrapping approach.
In each of the 1,000 bootstrap iterations, we randomly choose 80% of the events and recompute the median recurrence interval in each bin.Finally, we perform a linear regression between the log recurrence interval and the log moment.In this regression, each recurrence interval estimate is down-weighted by the bootstrap-derived standard deviation.
In Figure 3a, the best-fitting line implies that the recurrence interval scaling for repeater pairs in the NCSN collection is   ∝  0.17 0 , with 95% confidence limits placing the exponent between 0.16 and 0.18 (confidence limits plotted in Figure S2 in Supporting Information S1).The scaling is similar to previous estimates in the Parkfield region (R. M. Nadeau & Johnson, 1998) and elsewhere (K.H. Chen et al., 2007).In Figure 3b, the best-fitting moment-recurrence scaling for sequences from the Waldhauser and Schaff (2021) collection is   ∝  0.17 0 , with 95% confidence limits placing the exponent between 0.11 and 0.23.

Summed Moment in Repeating Earthquakes and Partial Ruptures
Next, we analyze the moment released by repeating earthquakes and partial ruptures.For each identified repeater, we calculate the sum of the moment accommodated in similar-magnitude co-located events-the total repeater moment.We also calculate the sum of the moment in all co-located events, including smaller magnitude partial ruptures-the total moment.To create groups of events that include a single repeater and all of its co-located events, it is only necessary for the events to be located with the target repeater.They do not need to be located with each other-we use a single linkage criterion to form these event groups for analysis.In Figures 2b and 2d, we plot the total moment against the total repeater moment.The dots are colored by the median magnitude of the co-located repeaters.Note that there is one dot per repeating earthquake (not per repeating earthquake sequence) since we analyze each repeater and its co-located events separately.Since we plot one dot per repeater but repeaters occur in sequences, we effectively analyze some earthquakes more than once, but that repetition should not influence our interpretation.As expected, the total moments are larger than the repeater moments.Including the partial rupture moment pushes the dots slightly above a one to one line in Figures 2b and 2d.
We are not interested in individual dots, but in the average moment accommodated by partial ruptures and how that moment changes with repeater magnitude.We therefore bin our observations by repeater magnitude.The repeater magnitude bins have a width of 0.43 magnitude units between M w 1 and M w 3.6, but varying the bin size does not strongly influence our analysis (Section 3.4).In each repeater magnitude bin, we average the moments plotted in Figures 2b and 2d to obtain the mean total repeater moment and the mean total moment.The pink dots in Figures 3b and 3d show the mean total moment in each repeater magnitude bin plotted against the mean total repeater moment in that magnitude bin.The mean total moments are only 10%-20% larger than the mean repeater moments in each magnitude bin; the average moment in partial ruptures seems to be small compared to the total seismic moment.
We note, however, that we are likely missing some partial rupture moment.Some small partial ruptures are likely not detected and included in the NCSN catalog.To account for these missing earthquakes, we estimate and then correct for the NCSN catalog's detection bias as a function of magnitude.We compute the magnitude distribution of the NCSN catalog in the Parkfield region and note that it follows a linear Gutenberg-Richter relationship with a b value of 0.97 above the magnitude of completeness of M w 1.1 (Figure S14 in Supporting Information S1).
We hypothesize that this distribution extends to at least M w = −0.5, which is the smallest partial rupture likely to contribute a significant moment.We therefore use the observed Gutenberg-Richter distribution to compute a theoretical cumulative moment.We compute the theoretical moment between M w = −0.5 and some cutoff magni tude M cut , which will represent the maximum magnitude we are considering for each repeater.We also compute the observed cumulative moment: the moment in all observed earthquakes between M w = −0.5 and M w = M cut .The ratio of the observed to the theoretical moment is a detection ratio: the fraction of the moment detected in each magnitude range.These theoretical and observed moment distributions are illustrated in Figure S15 in Supporting Information S1.
We use the detection ratio as a simple correction for the moment in undetected partial ruptures.For a repeater with magnitude M rep , the maximum magnitude partial rupture is (by our definition) M rep − 0.3.We therefore take the detection ratio between M w = −0.5 and M cut = M rep − 0.3, and we estimate the true partial rupture moment for this repeater and its co-located events by dividing the partial rupture moment by the detection ratio.This correction adds on average around ∼15% to the moment observed in partial ruptures.We do not use this simple correction to correct the total repeater moment, as we only use repeaters above the magnitude of completeness.
Now that we have corrected all of the partial rupture moments-and thus the total moment of the events co-located with each repeater, we again average the total moments within various repeater magnitude bins.The pink triangles in Figures 3b and 3d show the mean corrected total moment in each repeater magnitude bin plotted against the mean total repeater moment in each magnitude bin.The median moment in partial ruptures still seems to be small compared to the total seismic moment.
We plot the fraction of the moment in partial ruptures more explicitly in Figure 4.In this figure, we divide the total partial and total repeater moments by the number of repeaters in each group to obtain the mean repeater and the mean partial moments per repeater cycle.Panel a shows the partial rupture moment per cycle as a function of the mean repeater moment, and panel b shows the fraction of the moment in partial ruptures as a function of median repeater moment, with and without the correction for detection bias.The corrected moment in partial ruptures in each cycle increases from 5% to 30% between M w 1 and M w 2 and then decreases back toward 5%-10%.Note, however, that we may still underestimate the moment in partial ruptures for repeaters smaller than M w 2 because the location uncertainty is similar to the size of the asperity.The 90% error bars plotted in Figure 5 are derived from bootstrapping the earthquakes in our analysis (Section 3.1); they cannot account for partial ruptures that are systematically missing because of location error.For the most robust interpretation, one may wish to focus on the results for M w ≥ 2 repeaters in Figure 5 and ignore the results for smaller repeaters.

Corrected Moment-Recurrence Scaling
We were motivated to identify the moment in partial ruptures to assess whether partial ruptures could help explain the surprisingly long recurrence intervals of repeating earthquakes, as the partials could account for part of the slip budget.As such, we consider two ways to illustrate the partial ruptures' role in repeaters' slip budget: (a) by adjusting the total seismic moment and (b) by adjusting the expected slip per repeater.These equivalent representations are presented in Figure 5.
The gray circles in Figure 5a are re-plotted from Figure 3a; they show recurrence interval versus moment for individual repeating earthquakes in the NCSN collection.The larger light blue circles show averages of these values: the median recurrence intervals versus median moment for repeaters in each magnitude bin, again re-plotted from Figure 3a.However, comparing the recurrence interval to the median repeater moment ignores the moment in partial ruptures.We therefore correct these moments to include the observed partial rupture moment in each 10.1029/2023JB027870 9 of 14 magnitude bin.We multiply the repeater moments by our inferred total-to-repeater moment ratios: by 1 plus the values plotted in Figure 4b.These corrected total moments are plotted in orange in Figure 5.The values are very similar to the uncorrected blue dots, and the best-fitting recurrence interval scaling is still   ∝  0.17 0 .The absolute values of the recurrence intervals, and thus the y-axis intercept, also change very little.3a).We multiply the repeater moments by our inferred total-to-repeater moment ratios: by 1 plus the values plotted in Figure 4b.These corrected total moments are plotted in orange.These values are very similar to the uncorrected blue dots, and the new best-fitting recurrence interval scaling is still   ∝  0.17 0 .(b) Slip per repeater versus moment, corrected for slip in partial ruptures.We convert reassurance interval to slip assuming a longterm fault slip rate of 23 mm/year.We divide the repeater slip by our inferred total-to-repeater moment ratios.These corrected slips are plotted in orange.The new best-fitting recurrence interval scaling is   ∝  0.13 0 .
TURNER ET AL. 10.1029/2023JB027870 10 of 14 We also find minimal change in the scaling if we instead correct the recurrence interval for the partial rupture contribution.In Figure 5b, we convert the recurrence interval to a slip per repeating earthquake cycle.As noted in the introduction, the slip on the asperity per cycle should match the long-term slip outside the asperity so that the slip per cycle should be S = V creep T r , or 23 mm/yr times T r in Parkfield (R. M. Nadeau & Johnson, 1998).The gray and blue dots in Figure 5b show the slip per cycle plotted against the repeater moment using this simple mapping from panel a.However, some of the slip per cycle is accommodated by partial ruptures.To account for the slip in partial ruptures, we divide the slip per cycle in each magnitude bin by the ratio of the total to repeater moment in that bin.As expected, the orange dots move down by 5%-20%.The best-fitting weighted slopes increase by 23%.
Results are similar when we carry out the same analysis for the Waldhauser and Schaff (2021) repeater collection (Figure S10 in Supporting Information S1).

Testing for Bias in Analysis
Identifying repeaters can be challenging, and there are no standard identification criteria for reliably selecting true repeaters (Gao et al., 2023).But we aim to choose criteria that reflect the basic physics of repeating earthquakes-that the rupture areas overlap.Finding pairs of events that overlap depends on choices made about the percentage of overlap between events, the stress drop (which is used to calculate a circular rupture area), and the magnitude difference between the events.In this study, we identify repeaters using a stress drop of 10 MPa, a magnitude difference of less than 0.3 M w units, and a rupture overlap such that the separation between events is less than the rupture radius of the largest event (d < r max ).The parameter choices are based on physical observations.In Parkfield, repeaters generally have median stress drops of about 10 MPa (Abercrombie, 2014;Allmann & Shearer, 2007;Imanishi & Ellsworth, 2006).The distribution of magnitude differences between pairs of events that occur in the same location shows that 50% repeaters will have a magnitude difference of less than 0.3 (Figure S16 in Supporting Information S1).
These criteria are similar to the optimum criteria suggested by Gao et al. (2023), which propose that the distance between events should be less than or equal to 80% of the maximum rupture diameter of the larger event, and their magnitude difference should be less than or equal to 0.3.The constraint we use, which allows event separation up to 100% of the rupture diameter, could allow us to include a few extraneous earthquakes: events in the same location that are not true repeaters.However, we suspect that the nature of earthquakes in Parkfield, where 50% of earthquakes are repeaters, means that most M w 3 events within 1 rupture diameter of a M w 3 are on the same asperity.Choosing an area that is too big may not add many extraneous events or significantly affect our results.
To test this hypothesis and the robustness of our partial rupture moment estimates, we repeat our analysis after modifying the earthquake selection criteria.First, we test whether our result changes if we reduce the assumed stress drop.A smaller stress drop leads to a larger rupture radius, and more events are considered repeaters.We find that a stress drop of 3 MPa increases the number of identified repeating pairs by up to 2% and changes the slope of the recurrence-magnitude scaling by up to 20% (Figure S3 in Supporting Information S1).However, the maximum moment in partial ruptures is 20% of the total moment in the sequence, which is not significantly different to the results obtained using a stress drop of 10 MPa.We see similarly modest changes in our results when we modify other criteria: the local magnitude conversion, magnitude threshold, or triggered event cutoff.These factors can modify the slope by up to 30% and increase the number of events identified as repeaters by up to 6%.The main finding of this study is not affected by changes in the assumed stress drop, local magnitude scaling, magnitude cutoff, or threshold for an event to be considered repeating (e.g., Figures S4-S8 in Supporting Information S1).
Next, we test the influence of the binning and down-weighting by the bootstrap-derived standard deviation on the scaling.Changing the bin size and location can influence the number of events in each bin and the standard deviation down-weighting, particularly for larger magnitude events, where bins can include as few as ∼20 events.Different binning can change the slope of the scaling relationship by up to 30%, up to   ∝  0.24 0 , but even with uncertainty never reaches the theoretical scaling of   ∝  1∕3 0 .And in any case, we note that this data set is not intended to determine this scaling relationship accurately but to determine the moment accommodated in partial ruptures; that moment remains a few tens of percent or less.
We further test the influence of the events' location uncertainty with a more sophisticated approach: using the location error ellipse reported in the NCSN catalog for each event pair instead of using cutoffs on horizontal and 11 of 14 vertical distances separately.We compute the maximum distance between the two earthquakes that is allowed given the 95% error ellipse.Repeaters and partial ruptures are only identified if this maximum distance between a pair of events is within one rupture radius, ensuring events are co-located.This more time-consuming approach reduces the number of identified repeaters and partial ruptures by ∼90%.However, the scaling relationship and the ratio of the moment in repeaters to the total moment in each sequence are similar (Figures S8 and S9 in Supporting Information S1).
In this study, we consider two collections of repeaters and partial ruptures (Figure 4; Figure S9 in Supporting Information S1).We find similar results when using both collections of repeaters.That does make sense, as 77% of repeaters in the NCSN collection are in the Waldhauser and Schaff (2021) collection of repeaters, and 63% of the missed events are below the magnitude of completeness (see Figure S12 in Supporting Information S1).Our simple location-based criterion for locating repeating earthquakes appears to be suitable for this application in this region.

Partial Rupture Slip Budget and Repeater Recurrence Intervals
We were motivated to search for partial ruptures to assess whether slip in partial ruptures could account for repeaters' slip deficit and explain why repeating earthquakes occur less often than predicted.Our do observations reveal numerous partial ruptures.On typical repeater asperities, the moment in partial ruptures is 5%-30% of the repeater moment.This is likely to be an upper bound due to the inclusion of non-repeaters in our data set and the simplified assumptions about rupture dynamics-we assume that partial ruptures occur on circular patches that fully overlap with the repeating earthquake.
Even as an upper bound, those small moment fractions imply that partial ruptures could accommodate up to 25% of the slip on repeating earthquake asperities.However, a 25% increase in the slip budget can explain only a factor of 1.25 increase in the recurrence intervals of repeating earthquakes.That is a small portion of the recurrence interval discrepancy that is often observed.M W 2 repeaters, for instance, occur about 5 times less often than one would expect given a 10 MPa stress drop and a 23 mm/year long-term slip rate.
The small amount of slip accommodated by partial ruptures is observed in Parkfield and potentially in other areas with repeating earthquakes.Chang and Ide (2021) studied three repeating patches in Japan that experienced hierarchical ruptures, which we refer to as partial ruptures.The moment in the reported smaller magnitude events that are observed is only 1%-23% of the total sequence moment, which aligns with what we have observed in Parkfield.
The partial rupture moment also appears unable to explain the scaling of repeater recurrence interval T r with the moment.The recurrence does not change when we adjust for the partial rupture moment (Figure 5).Smaller repeating earthquakes still seem to occur particularly less often than one would expect given the long-term slip rate.

How Big Are Repeaters Relative to Their Nucleation Radius R nucl ?
Partial ruptures may do more than accommodate slip.The presence or absence of partial ruptures allows us to place a constraint on the size of repeating earthquake asperities relative to the nucleation radius: the size of the smallest asperity capable of hosting seismic slip (R nucl , e.g.Cattania, 2019;Cattania & Segall, 2019;Chen & Lapusta, 2009;Dieterich, 1992;Rubin & Ampuero, 2005).If repeating earthquake asperities were only slightly larger than the nucleation radius, then all ruptures on a given asperity would be around the same size, and there would be no partial ruptures.Repeaters and partial ruptures can be used as an additional method to estimate the nucleation radius, complementing observations such as a break in self-similarity at small magnitudes, resulting in magnitude-independent duration proportional to the nucleation length or deviations in the Gutenberg-Richter relationship (e.g., Abercrombie, 1996;Imanishi & Ellsworth, 2006;Lin et al., 2016).Recent observations of magnitude-independent duration yields a nucleation length between 45 and 80 m (Cattania, 2023).
Most repeaters in our collection do have partial ruptures.We do not observe a clear transition from no partial ruptures to partial ruptures with magnitude (Figures 3b and 4b).For instance, asperities with M w 2 repeaters accommodate 25% of their moment in partial ruptures, and that percentage stays the same or decreases as repeater magnitude increases to M w 3.Even asperities with M w 1 repeaters accommodate 5% of their moment in partial ruptures, and that partial moment is likely underestimated because of earthquake location uncertainty.The consistent existence of partial ruptures implies at least that most M w > 2 repeaters have R ≫ R nucl .

Tuning a Numerical Model to Match Repeater Recurrence?
As a final use of our partial rupture observations, we assess some models of repeating earthquakes based on crack-like ruptures (e.g., Cattania & Segall, 2019;Chen & Lapusta, 2009, 2019).These models can reproduce the observed   ∼  0.17 0 recurrence interval-moment scaling.But to match observed recurrence intervals and moments, the models are tuned; modelers indirectly specify the nucleation radius (R nucl ), stress drop (Δσ), and the long-term fault creep rate (V pl ) as they attempt to match the observed moment-recurrence scaling.For instance, Cattania and Segall (2019) infer that matching the recurrence interval scaling implies that: where μ′ = μ for antiplane shear and μ/(1 − ν), where ν is the Poisson's ratio for plane strain deformation.Our observations introduce an additional constraint on the tuning: that at least M w > 2 repeaters have R > ∼6R nucl , since partial ruptures only occur in this regime (Cattania & Segall, 2019).This constraint implies that the nucleation length R nucl is a few 10s of meters or less.
This new constraint proves challenging for the models.Assuming a nucleation length of between 1 and 10 m as inferred here, a stress drop of 10 MPa (Abercrombie, 2014), and a long-term creep rate near Parkfield of 23 mm/ yr (R. Nadeau et al., 1994), Equation 4.3 underestimates recurrence intervals by a factor of 20 and 6 respectively (Figure S17 in Supporting Information S1) This tuning failure could indicate that a crack model coupled with rate and state friction is a poor representation of repeating earthquakes.
However, it is also possible that the models are a good representation of repeaters, and one of these observational constraints is incorrect or misinterpreted.Perhaps earthquake stress drops are actually ≥100 MPa, not 10 MPa, and seismic observations underestimate the stress drop because rupture models do not account for heterogeneous slip (Nadeau et al., 2004).Or perhaps the relevant long-term slip rate is much smaller, of order 4.5 mm/yr (as used in the model by Chen & Lapusta, 2009), because the fault zone is composed of several fault strands, and it is the strand's slip rate, not the regional slip rate, that drives repeaters (Williams et al., 2019).Alternatively, observations of partial ruptures may not accurately indicate the size of a repeater asperity relative to its nucleation size.It is also plausible that the nucleation length varies spatially due to heterogeneity in frictional properties and normal stress (e.g., Cattania & Segall, 2021;Schaal & Lapusta, 2019), so that partial ruptures may occur on patches with a local small nucleation length, while a larger, spatially averaged value controls the recurrence interval.
Given these uncertainties, it may be of interest to consider the implications of the crack model when relaxing the assumption of constant stress drop.The crack model predicts that T r ∝ R 1/2 (Cattania & Segall, 2019).For a constant stress drop, M 0 ∝ R 3 so that   ∝  1∕6 0 .More generally, we can write M 0 ∝ SR 2 , with S the coseismic slip, which is at most equal to the slip accumulated interseismically outside the asperity (V pl T r ).We consider a particular scenario: where the fraction of the moment accommodated by inter-repeater slip-by aseismic slip or partial ruptures-remains constant, independent of magnitude.A constant fraction around 20% would match our observations, for instance, though it is not specifically predicted by crack-based thresholds for rupture coupled with rate and state friction given simple frictional properties (Cattania & Segall, 2019;Chen & Lapusta, 2009, 2019).Given such a magnitude-independent fraction, we can write  0 ∝  2 ∝  5  .Therefore, the model predicts that if the inter-repeater moment fraction remains constant, the recurrence interval should scale with the moment as  Tr ∝  1∕5 0 .This scaling is close to the previously observed scaling of Tr ∝ M0 0.17 .Further, the stress drops should decrease with increasing repeater moment, following a  Δ ∝  −1∕5 0 scaling.This magnitude scaling is small enough to be hidden within the current uncertainty of stress drop estimates (Abercrombie et al., 2020).Perhaps it will be observed in future studies.

Conclusion
With this work, we sought to test the hypothesis that small repeating earthquakes have exceptionally long recurrence intervals because small earthquakes accommodate slip on the asperities between repeating earthquakes.We identify numerous partial ruptures by searching for small co-located earthquakes in Parkfield, California, using the NCSN catalog (Waldhauser & Schaff, 2008), employing two collections of repeaters: one based on the relative locations and another created by Waldhauser and Schaff (2021).In both collections of repeaters, we find that partial ruptures accommodate only a small fraction of the moment.These fractions imply that partial ruptures could accommodate up to 25% of the slip on repeating earthquake asperities.This is not enough slip to explain why small repeating earthquakes often occur 5 times less often than one would expect.

Figure 1 .
Figure 1.Seismicity of the Parkfield region.Gray dots show the events in the Northern California seismic network double-difference relocated catalog.Dark blue events are repeating earthquakes identified by Waldhauser and Schaff (2021).Light blue events are repeating earthquakes identified in this study.Faults plotted from the USGS Quaternary faults and folds database.

Figure 2 .
Figure2.Timing and asperity size of examples of groups of co-located earthquakes, including partial ruptures and repeating earthquakes.Repeating earthquakes are defined as similar-magnitude (within 0.3 magnitude units) co-located ruptures and are plotted in blue.Partial ruptures are smaller co-located ruptures and are plotted in orange.The event circled in black is the reference event used to identify the group of co-located events.The median latitude, longitude and magnitude of the repeating earthquakes are printed at the top of each panel.The gray box in the third panel is the 10 years after the 28 September 2004 Mw 6.0 earthquake, which is excluded from this study.

Figure 3 .
Figure 3. (a) Recurrence interval versus moment for each repeater set from the location-based NCSN repeater collection(Waldhauser, 2013).Individual values are plotted as gray circles, and medians for moment bins are plotted as blue circles.The error bars on the medians indicate 95% confidence limits, which were estimated via bootstrapping (details in the text).The best-fitting line is plotted in solid black and has a gradient of 0.17.The dashed line shows the predicted recurrence intervals assuming a stress drop of 10 MPa.(b) The total moment in repeating earthquakes (x-axis) compared to the total moment in each group of co-located events, including repeating earthquakes and partial ruptures (y-axis).Each dot is colored by the median moment of the repeating earthquake group.Light pink dots are the means for various magnitude bins.The dark pink triangles are the binned means corrected for missing small events (see text for more details).(c, d) Are the same as (a, b) using sequences from the Waldhauser and Schaff (2021) repeater collection.

Figure 4 .
Figure 4. (a) Total moment in partial ruptures as a function of repeating earthquake moment.Both values are per cycle; the values are normalized by the number of repeating earthquakes in each sequence.The dark pink triangles show the binned averages, and the black lines show the 5th and 95th percentiles of these binned medians, as derived from bootstrapping.The dark pink triangles show the binned values corrected for detection bias, as described in the text.(b) The y-axis shows the partial-to-repeater moment ratio: the ratio of the partial rupture moment per cycle to the mean repeater moment.The x-axis is as in panel (a) the mean repeater moment.The gray-shaded region in panel (b) highlights events below M w 2 that may have higher uncertainty due to location errors.

Figure 5 .
Figure 5. (a) Recurrence interval versus moment, corrected for moment in partial ruptures.The gray circles are the recurrence interval versus moment for individual repeating earthquakes in the NCSN collection, and medians for moment bins are plotted as blue circles (same as Figure3a).We multiply the repeater moments by our inferred total-to-repeater moment ratios: by 1 plus the values plotted in Figure4b.These corrected total moments are plotted in orange.These values are very similar to the uncorrected blue dots, and the new best-fitting recurrence interval scaling is still   ∝  0.17 0 .(b) Slip per repeater versus moment, corrected for slip in partial ruptures.We convert reassurance interval to slip assuming a longterm fault slip rate of 23 mm/year.We divide the repeater slip by our inferred total-to-repeater moment ratios.These corrected slips are plotted in orange.The new best-fitting recurrence interval scaling is   ∝  0.13 0 .