Seismic rhythms: Earthquake response to tectonic, hydrological, and tidal forcing in California

Abstract

Seismicity is primarily driven by tectonics, but stress variations of natural or human-made origin can induce detectable modulations, offering insights into earthquake physics. Here, we identify regions in California exhibiting significant seasonal modulations of seismicity rate linked to hydrological surface loading but no significant semidiurnal tidal modulation. The peak seismicity rate lags behind the peak stressing rate by half a month. Assuming instantaneous nucleation substantially overpredicts the response, whereas time-dependent nucleation governed by rate-and-state friction accurately captures both the amplitude and the time delay with friction-stress parameter aσ ~ 1 to 10 kilopascals (kPa) and characteristic relaxation time t_a ~ 0.05 to 1 year. The seismicity response to tidal and seasonal stress perturbations allows us to probe fault mechanical properties, providing a way to improve seismic hazard assessment.

California faults respond to seasonal stress but not tides, offering a natural lens to probe fault conditions and seismic hazards.

INTRODUCTION

Earth’s crust constantly deforms through various processes, such as tectonic strain accumulation (1, 2), stress changes from large earthquakes (3, 4), seasonal changes in groundwater and snowpack (5–8), Earth tides and ocean loading (9), slow-slip events (10), postseismic afterslip and deformation (11, 12), sea level changes (13), climatic processes (14), and anthropogenic activities related to geo-energy production (15). These forcings introduce stress changes on faults, which may promote or discourage the occurrence of seismicity (3, 5, 7). The amplitude and timing of the seismicity response provide insights into the physical quantities governing earthquake nucleation, such as fault friction and the absolute ambient stress on the faults (10, 16–19). A better knowledge of these quantities would revolutionize our ability to forecast seismicity and associated seismic hazard.

A possible approach based on the analysis of aftershock sequences (4, 16) is limited by the necessity of large enough mainshocks and the difficulty to disentangle the effect of the stress changes due to the mainshock from the effect of afterslip, pore pressure variations, dynamic stresses, and earthquake interactions (12, 20, 21). We therefore focus on analyzing the seismicity response to slow, quasistatic stress fluctuations.

Despite small stress changes of a few kilopascals (kPa) from the seasonal addition or removal of up to a few-decimeter-thick layer of terrestrial water or ice in comparison to the lithostatic stress of 100 to 150 MPa, they have been known to modulate observed seismicity rates either through changes of surface loading, poroelasticity, or pore pressure diffusion in various regions, including the Himalayas (18, 22), California (7, 23–26), New Madrid Seismic Zone (27), East African rift (28, 29), Japan (5, 28, 30, 31), Alaska (32), Italy (33), and Taiwan (34, 35). In contrast to hydrological surface loading, semidiurnal solid Earth tides produce stress changes of similar magnitude (9), but tidal modulation of seismicity is limited to certain areas and often with weaker effects (36–39). This suggests that the seismicity response to harmonic stress perturbation is period dependent in nature, as also observed in laboratory experiments and shown in theoretical models in which earthquake nucleation is not instantaneous (40–42). The amplitude and phase of the seismicity response to harmonic stress variations are therefore a potentially powerful approach to test earthquake nucleation models, although collecting such measurements is challenging.

Because of the required large number of events to provide sufficient statistics for the detection of periodic modulations of seismicity rates (43), most studies thus far have considered relatively large regional domains and it remains unclear whether the response is really regional or dominated by certain areas with higher sensitivity. Here, we examine the variations of seismicity rates in space and time in relation to tectonic, hydrological, and tidal loading and explain their causations using a model of earthquake nucleation. The strong seasonal surface loading (6, 7, 25, 44, 45) and high-quality seismicity catalogs (46, 47) make California an ideal case study.

RESULTS

Long-term seismicity driven by tectonic loading

We analyze earthquakes in California between 2006 and 2021 with magnitudes larger than the local magnitude of completeness (fig. S1). We separate the events directly and independently induced by tectonic, hydrological, and tidal stressing, from those occurring in clusters or swarms which were triggered by other earthquakes (16) or short-term transient strain, e.g., the swarms in the Imperial Valley (10, 48). To that effect, we use an enhanced declustering method based on the nearest-neighbor distances (49). Unlike conventional approaches with a fixed cutoff value to distinguish background from clustered events, our method allows the cutoff to vary spatially (fig. S2; Materials and Methods), enabling improved separation between the two modes in the overlapping region (Fig. 1, A and B). As a result, one-third of all events are classified as background and retained for subsequent analyses. The quality of declustering is validated a posteriori, showing that background events exhibit Poissonian behavior as expected (fig. S3). The remaining background seismicity is concentrated along mapped faults, correlating with high tectonic strain rates (Fig. 1, C and D). We observe a linear relationship between background seismicity rate and tectonic strain rate (inset of Fig. 1D). Given that aftershock sequences and postseismic transients have been filtered out, we cannot inform the nucleation process from comparing the temporal variations of seismicity and strain rates. We further analyze the background seismicity for seasonal and tidal modulation.

Fig. 1. Long-term background seismicity driven by tectonic loading.

(A and B) Bimodal distribution of the nearest-neighbor distance (η) for earthquakes in Northern California and Southern California (regions outlined in blue in C). The mode with larger η (red) represents background seismicity, whereas the mode with smaller η (blue) represents clustered seismicity, including aftershocks and swarms. Earthquakes are sourced from the NCSS (47) and HYS (46) catalogs. Events from the Geysers geothermal field are excluded from the histogram. (C) Map of the average background seismicity rate on a decimal logarithmic scale for earthquakes with magnitudes M ≥ 2.5 from 2006 to 2021. (D) Map of geodetically derived tectonic strain rate (second invariant ${\dot{\mathrm{\epsilon }}}_{\mathit{II}}=\sqrt{{\dot{\mathrm{\epsilon }}}_{\mathit{xx}}^2+{\dot{\mathrm{\epsilon }}}_{\mathit{yy}}^2+2{\dot{\mathrm{\epsilon }}}_{\mathit{xy}}^2}$ (1) on a decimal logarithmic scale with the inset that illustrates a positive correlation between background seismicity rate and tectonic strain rate.

Seasonal but no detectable tidal seismicity modulation

To quantify the amplitude of seismicity modulation at any period T, we apply the Schuster test (43, 50), performing a successive walk, in which unit-length steps are taken for each earthquake occurrence time t_i in the direction corresponding to its phase angle ${\mathrm{\theta }}_i=2\mathrm{\pi }{t}_i/T$. The total drift distance between the walk’s origin and endpoint after a number of full cycles of period T determines the Schuster P value, quantifying the probability of a uniform seismicity rate (i.e., no periodic modulation). A larger drift distance corresponds to a smaller P value, indicating a lower likelihood of a uniform seismicity rate. The cutoff P value, below which the seismicity modulation is considered statistically significant (Materials and Methods), depends on the tested period T and the catalog length (it is not simply 0.05 for a 95% confidence level as often incorrectly assumed) (43). Assuming sinusoidal variations of seismicity rates, the Schuster P value directly relates to the modulation amplitude (41), defined as α = ∆R/r, where ∆R is the half peak-to-peak variation and r is the average seismicity rate, according to Eq. 5 (fig. S4; Materials and Methods).

A low P value does not necessarily indicate genuine periodicity, as clustered events, particularly aftershocks, can artificially inflate statistical significance and bias the P value. A Schuster spectrum, which consists of Schuster tests across a range of periods, allows distinguishing true periodic signals (isolated low P value) from aftershock-related signals due to imperfect declustering (drifting P values at periods larger than the decay time of aftershocks) or multiples (43). The spectrum for Northern California exhibits an isolated low P value at a period of 1 year, whereas P values at shorter periods are mostly above the 95% range of expected values for a Poisson process including at tidal periods, confirming a genuine seasonality (Fig. 2C). Drifting low P values at larger periods suggest a small bias due to imperfect declustering (Fig. 2, C and D), emphasizing the importance of constructing Schuster spectra for robust detection rather than relying on the sole P value calculated for a particular period. Low P values also show up at multiples of 1 year, as expected.

Fig. 2. Seismicity rate modulated by seasonal (annual) variations of hydrological surface loads.

(A) Map of amplitudes of seasonal modulation of background seismicity rates calculated using Schuster analysis (43) with 5000 nearest earthquakes within each grid point. Only statistically significant values (≥0.038) are displayed. (B) Map of amplitudes of seasonal Coulomb stress changes at a 5-km depth, induced by hydrological surface loading on fault planes optimally oriented with respect to the tectonic stress field. The surface load model is based on geodetic and gravity measurements (8), and stress calculations use a semianalytical solution in an elastic half-space (56). Sharp spatial transitions in stress amplitudes reflect changes in the tectonic regime. (C and D) Schuster spectra of seismicity showing an isolated low P value at the annual period in Northern California, indicating significant modulation of seismicity rates. Drifting low P values at longer periods suggest the presence of aftershocks, highlighting the importance of constructing the Schuster spectrum to ensure that low P values are isolated and not the result of aftershocks. Tidal modulation is not statistically significant in both regions. (E to H) Schuster walks demonstrating consistent annual modulation of seismicity rates in Northern California, northern San Andreas fault (SAF), and Coso geothermal area, with no clear pattern in Southern California. Dashed circles mark detection threshold (thr) and 95% confidence intervals. Northern California analysis excludes seismicity from the Geysers geothermal area. Southern California analysis excludes seismicity from the Coso geothermal area.

Because the Schuster test can only reveal synchronous modulation of seismicity within the analyzed region, aggregating events over broad areas with spatial variation of fault orientations and tectonic regimes or heterogeneous seasonal loading may dilute or bias any underlying seasonal signal. Therefore, we perform a more localized analysis, considering 5000 nearest events to each grid point. Our analysis identifies multiple regions with strong seasonal modulation of seismicity (Fig. 2A), but no statistically significant semidiurnal tidal modulation (fig. S5). Seasonal modulation is more pronounced in Northern California than in Southern California. To the first order, regions exhibiting strong seasonal modulation align with areas of large hydrological surface loading, driven by groundwater fluctuations and snowpack changes (Fig. 2B), suggesting a causal relationship.

To further examine the temporal consistency of these signals, we perform the Schuster walks for representative regions to assess whether seasonal modulation remains consistent across multiple years in amplitude and phase. In Northern California, excluding the Geysers area where the strong seasonality is driven by geothermal energy production, we observe a consistent steady drift over the study period, with the peak seismicity rate occurring in mid-August (Fig. 2E). In contrast, Southern California exhibits no drift, indicating no statistically significant seasonal modulation (Fig. 2F).

Having established the temporal consistency, we next examine the spatial variations of the amplitude (Fig. 2A) and phase (Fig. 3A) of the seasonal modulation of seismicity. In the northern segment of the San Andreas fault, the modulation amplitude reaches up to 15% of the background seismicity rate, with the peak seismicity rate occurring in July (Fig. 2G). This pattern is consistent with the overall timing of the peak seismicity rate observed across Northern California (Figs. 2E and 3A). A similar trend is also observed for the Eastern Sierra and Long Valley areas (Fig. 2A and fig. S6). In contrast, the Geysers geothermal area exhibits a large seasonality but peaks in February (Figs. 2A and 3A and fig. S6). The discrepancy with the broader pattern observed in Northern California confirms that seasonal seismicity in the Geysers area is driven by geothermal production, which peaks in the winter due to the availability of water for injection at the site (51). A similar pattern is observed in the Coso (Fig. 2H) and Brawley (fig. S6) areas and is also most likely attributed to the geothermal energy operations (52). The Santa Clara area exhibits weaker but significant seasonality (fig. S6). Schuster spectra from all examined regions reveal distinct, isolated low P values at an annual period, with minimal clustering-related bias (fig. S6).

Fig. 3. Time lag between peak seismicity rate and peak stressing rate.

(A) Map of the time of year when peak background seismicity rates occurred, which were between July and August for most regions. (B) Map of the time of year for peak stressing rates caused by changes in hydrological surface load, which were mostly between late June and early July. (C) Map of the time lag between peak stressing rate and subsequent peak seismicity rate, with a median lag of 0.52 months. Insets in all panels include histograms displaying the distribution of the correspond values shown in the maps.

Temporal variations of seismicity sensitivity to seasonal stress

Although a significant annual modulation in the Parkfield area has previously been reported for the 1984 to 2003 period (23), which we also recover when we apply the Schuster analysis to the same time period, we do not see similarly clear evidence of seasonal modulation in the 2006 to 2021 period analyzed here (Fig. 4). It is possible that the 2004 M6.0 Parkfield earthquake may have altered the stress field or the fault zone properties in a way that disrupted the sensitivity of seismicity to seasonal stress variations. The pronounced seasonal modulation before the Parkfield earthquake could reflect a critically stressed state, consistent with laboratory experiments demonstrating stress-dependent sensitivity (53) and observations of increased tidal triggering preceding some large megathrust earthquakes (54, 55).

Fig. 4. Temporal variations of seismicity sensitivity to seasonal stress.

(A) Map of background (red) and clustered seismicity (blue) in the Parkfield area between 1984 and 2021. (B) Rescaled distance versus rescaled time diagram, (C) Schuster spectrum, and (D) Schuster walk (annual period) for seismicity between 1984 and 2003, revealing clear seasonal modulation of seismicity. (E to G) Same as (B) to (D) but for seismicity between 2008 and 2021, showing the disappearance of seasonal modulation of seismicity after the 2004 M6.0 Parkfield earthquake.

Explaining seismicity modulation with a physical model

The observed seasonal modulation of seismicity provides a unique opportunity to explore the physics governing earthquake nucleation. Given its strong spatial correlation with hydrological surface loading (Fig. 2, A and B), we hypothesize that seasonal seismicity variations are primarily driven by direct stress perturbation from surface loading and assess the extent to which their relationship can be described by a physical model.

Seasonal fluctuations in groundwater levels and snowpack across California induce detectable vertical surface displacements of up to 1 cm (6, 7, 25). On the basis of these surface deformation and gravity measurements, an equivalent water thickness variation model can be derived (8). Using a semianalytical solution describing the deformation of an elastic half-space resulting from a vertical point load (56), we calculate Coulomb stress changes induced by hydrological surface loading. The calculation requires an assumption about the orientation of the faults associated with these earthquakes. Given that much of the seasonally modulated seismicity is of too small magnitude to have reliable focal mechanisms, we explore three plausible sets of fault plane orientations: (i) fault planes oriented optimally with respect to the tectonic stress field (fig. S7), (ii) fault planes oriented optimally with respect to the stress changes induced by hydrological loading, and (iii) vertical right-lateral strike-slip faults parallel to the San Andreas fault (Materials and Methods). For each configuration, we fit the Coulomb stress variations with a sinusoidal function and express them in terms of amplitude and phase. We find that the resulting amplitudes and phases are broadly similar for these three assumptions (figs. S8 and S9). We prefer the model with fault planes oriented optimally with respect to the background tectonic stress field because it provides a physically consistent framework that aligns with long-term fault kinematics and regional stress orientations (Figs. 2B and 3B).

We observe a delayed seismicity response to stress changes, with seismicity consistently lagging behind the peak stressing rate in the summer by a median of 0.52 months throughout most of California (Fig. 3C). The phase is notably different at the Geysers and Coso geothermal areas where seasonal seismicity peaks in the winter and is most probably dominated by the cyclic injection and extraction activities associated with geothermal energy production (52). Another exception is the Santa Clara Valley in the Bay Area where the seismicity rate peaks at about the same time as in Northern California whereas the stressing rate is late by about 3 months (Fig. 3). We suspect that the hydrological model (8) may be locally flawed as the conversion of surface deformation to surface load ignores poroelastic effects, which are notable in this region (57), leading to an incorrect result that is out of phase by 180° and hence the apparent negative phase lag (Fig. 3). This issue does not occur in the Central Valley where the GNSS sites with clear poroelastic signals were excluded in the derivation of the hydrological surface load model (8).

Various models have been proposed to describe the seismicity response to stress perturbations. We first test the Coulomb failure model (CFM), assuming that earthquakes nucleate instantaneously once the Coulomb stress exceeds the fault’s frictional strength. In this simple model, the seismicity rate is directly proportional to the stressing rate and only occurs when stress surpasses its previous peak value (5, 41). The CFM prediction therefore depends on whether the annual period is greater or smaller than the critical period $T_{\mathrm{\tau }}=2\mathrm{\pi }\mathrm{\Delta }S/\dot{S_0}$ (Eq. 9; see also Materials and Methods). In both cases, the model substantially overpredicts the seismicity response (Fig. 5A) and fails to explain the observed time lag between the peak seismicity rate and the peak stressing rate (Fig. 3). A similar issue was observed in the case of a swarm driven by a slow-slip event (10) and in the case of earthquakes induced by gas extraction (19). In both of these cases, the observed amplitude and time lag could not be explained by the CFM but was found to be consistent with a nucleation process driven by rate-and-state friction (16). We therefore evaluate this rate-and-state model (RSM) as well.

Fig. 5. Evaluating earthquake nucleation models using seismicity response to seasonal changes in hydrological surface load.

(A) The CFM, which assumes instantaneous earthquake nucleation, substantially overpredicts the seismicity response, regardless of the critical time $T_{\mathrm{\tau }}=2\mathrm{\pi }\mathrm{\Delta }S/\dot{S_0}$ where ΔS is the stress amplitude and $\dot{S_0}$ is the background stressing rate. (B) The RSM incorporates a more realistic nucleation process using a laboratory-derived friction law, providing a better explanation of the observed seismicity response. Response amplitude is controlled by the friction-stress parameter aσ, depicted as varying colors. Dashed lines indicate RSM predictions for cases in which the characteristic relaxation time t_a is >1 year. (C) Same as (B) but the colors depict different characteristic relaxation time t_a, which is controlled by the time delay between peak seismicity rate and peak stressing rate. Each data point corresponds to a spatial grid cell with a 0.05° resolution (Fig. 2, A and B).

The RSM response depends on the friction-stress parameter aσ (product of the rate-and-state friction parameter a and fault effective normal stress σ), which controls primarily its amplitude (larger for smaller aσ), and on the characteristic relaxation time t_a, which controls its duration (fig. S10; Materials and Methods). By examining the relative amplitudes between seismicity modulation and stress (Fig. 2, A and B) alongside the time lag between the peak seismicity rate and the peak stressing rate (Fig. 3), we uniquely invert for the parameters aσ and t_a (Figs. 5, B and C, and 6). The spatial variations of the amplitude and phase of the seasonal seismicity yield values that fall in a relatively narrow range of aσ between 1 and 10 kPa (Figs. 5B and 6A) and t_a between 0.05 and 1 year (Figs. 5C and 6B). Similar results are obtained if Coulomb stress changes are calculated on faults optimally oriented with respect to the stress changes or on faults parallel to the San Andreas fault (fig. S11). Fault segments capable of hosting dynamic slip (earthquakes) generally share similar values of aσ and t_a, although some probably meaningful spatial variations are also visible (Fig. 5).

Fig. 6. Spatial distribution of frictional parameters governing earthquake nucleation.

(A) Map of the frictional-stress parameter aσ derived from the RSM using seismicity responses to seasonal changes in hydrological surface load. The inset shows a histogram of aσ values with a median of 1.5 kPa. (B) Map of the characteristic relaxation time t_a on a decimal logarithmic scale. The inset shows a histogram of t_a values in decimal logarithmic scale with a median of 0.05 years. The response of seismicity to stress perturbations can be used to probe the stress state of the Earth’s crust and to characterize the processes governing earthquake nucleation.

DISCUSSION

The RSM parameters aσ and t_a are linked via the background stressing rate $\dot{S_0}$ through the relation $t_a=a\mathrm{\sigma }/\dot{S_0}$, enabling a consistency check. Using a median aσ of 1.5 kPa (Fig. 6A) and a median t_a of 0.05 years (Fig. 6B), we estimate $\dot{S_0}$ to be ~30 kPa/year. Assuming a shear modulus of 30 GPa, this corresponds to a strain rate of ~1000 nanostrain/year, which is comparable to the peak long-term tectonic strain rate along the San Andreas fault (Fig. 1D). However, the spatial patterns of $\dot{S_0}$ derived from our estimates of aσ and t_a do not match the geodetic strain rate pattern. This discrepancy may reflect the influence of mixed faulting regimes across the study area. In particular, faults that are activated by transient hydrological surface loading may not be optimally oriented with respect to the regional tectonic stress field. As a result, the effective background stressing rate on these faults can differ from what would be expected based solely on the tectonic strain rate. Spatial variations in elastic properties, such as shear modulus, may also help explain the discrepancy.

We now check whether the nucleation process derived from the seasonal seismicity is consistent with the absence of a statistically significant semidiurnal tidal modulation of seismicity (fig. S5). Ocean tides are ignored as the regions exhibiting a strong seasonal modulation of seismicity are primarily inland. Solid Earth tidal stress in California has an amplitude of ~1 kPa, but it comprises multiple superimposed periods. The Schuster test evaluates one period at a time and is sensitive to the amplitude at each period individually. For the M2 tide (period = 12.4 hours), the stress amplitude is ~0.6 kPa (10). Given the lack of statistically significant semidiurnal tidal modulation of seismicity, the amplitude must be below 8.16% of the background seismicity rate (Eq. 5; Materials and Methods). Applying Eq. 11, we infer aσ > 7.5 kPa. Although this value exceeds many of the values derived from the seasonal modulation, the discrepancy remains within the plausible uncertainty of our results, including uncertainty in the time lag estimated from the hydrological model, which has a monthly resolution. In addition, parts of the discrepancy may stem from neglecting finite fault effects in the RSM, which can enhance the response at 1 year compared to the response at tidal periods (41).

Last, we can check whether the characteristic time t_a derived from our analysis is consistent with the estimates derived from aftershock sequences. For each cluster identified from the nearest-neighbor approach, we fit the temporal decay of aftershock rate with Omori’s law and use the fitted parameters to infer t_a (Eq. 12; Materials and Methods). We find t_a values ranging from 0.01 to 1 year for sequences with mainshock magnitude smaller than M5 (fig. S12), consistent with estimates from seasonal modulation analysis (Fig. 6B). In contrast, sequences with larger mainshocks exhibit considerably longer t_a, likely due to afterslip playing a role in driving aftershocks (12, 20) and prolonging their duration, although additional mechanisms such as dynamic stresses, fluid migration, and viscoelastic deformation may also contribute (21). Swarms in Long Valley show shorter t_a, reflecting their burst-like behavior, although their inferred t_a may not be physically meaningful as Omori’s law generally does not fit swarms well. Although some studies have proposed a magnitude-dependent seasonal response and thus a magnitude-dependent t_a (5), we do not observe strong evidence for such behavior in our dataset. The amplitude of seasonal modulation and the phase of peak seismicity rate remain largely invariant across different earthquake magnitudes (fig. S13).

Although hydrological surface loading combined with RSM nucleation explains both the amplitudes of seasonal seismicity modulation and the observed time lag between the peak seismicity rate and the peak stressing rate, other processes may also contribute to the stress changes. In sedimentary basins and aquifers, where rocks are highly compressible (e.g., Central Valley), fluid pressure changes induce notable mechanical deformation through poroelasticity, creating nonnegligible stress changes that are out of phase with the surface loading (45). However, because the timing of the peak seismicity rate remains consistent across aquifers and the broader regions (Fig. 2A), poroelastic stress is not the main driver of seasonal modulation of seismicity in California and can be neglected. Additional seasonal stress perturbations, such as temperature and atmospheric pressure fluctuations, wobble-induced pole tides, solid Earth tides (annual period), and other ocean loading, are estimated to be minor compared to hydrological surface loading (25). Regarding the time lag between the peak seismicity rate and the peak stressing rate, rather than attributing it solely to the noninstantaneous nature of earthquake nucleation in the RSM, an alternative explanation could involve the diffusion time for pore pressure to migrate toward deeper depths, as proposed for the seasonal seismicity observed in southern Alaska (32). If diffusion were the dominant control, the time lag should increase with earthquake depth. However, our results show relatively uniform time lags across California (Fig. 3), dismissing pressure diffusion as a dominant mechanism.

Our analysis suggests a friction-stress parameter aσ of the order of a few kilopascals, in the range of values derived from a slow-slip driven earthquake swarm (10), seasonal seismicity in the Himalayas (18), induced seismicity (19), and aftershock sequences in California (58). A value of a = 0.001 typical of laboratory experiments (59) implies an effective normal stress σ of a few megapascals. At the seismogenic depth of 5 km, the expected lithostatic pressure is ~150 MPa, implying that the parameter a of natural faults may be substantially lower than under laboratory conditions or that the fluid pressure is extremely high, leading to substantially lower effective normal stress, or a combination.

Although the RSM framework provides a tractable and physically motivated approach to infer fault nucleation parameters from the response of seismicity to transient stresses due to hydrological loading and solid Earth tides, it carries important limitations as it does not account for finite fault effects. In the RSM, the slip velocity is monotonically increasing under steady tectonic loading with spatially uniform behavior. In contrast, finite faults exhibit spatially variable slip velocities due to heterogeneous prestress histories shaped by prior seismic events. Numerical models incorporating rate-and-state friction on finite faults reveal that the RSM tends to underestimate the seismicity response to periodic stress perturbations, often leading to an underestimation of friction-stress parameter aσ (41, 60). Despite these limitations, the RSM remains valuable as a semianalytical tool that enables first-order estimates of nucleation parameters across large spatial domains without the requirements for intensive computational resources. Future work incorporating finite fault geometry, rupture propagation, and spatial stress heterogeneity, although computationally expensive, may refine these estimates and improve predictive capabilities.

The observed seasonal modulation of seismicity in California and its relation to hydrology-induced stress variations provides a unique opportunity to refine our understanding of earthquake nucleation and fault friction properties. Our methodology allows us to identify localized zones of strong seasonal seismicity response that would interfere destructively if the entire region were considered as a whole due to their different time lags. The stronger seasonal modulation is observed in Northern California where hydrological variations are larger. The observed time lag between the peak seismicity rate and the peak stressing rate highlights the noninstantaneous nature of earthquake nucleation and offers constraints on the characteristic time t_a associated with the nucleation process. This characteristic time is large enough (larger than 2 weeks) so that the response to semidiurnal solid Earth tides is muted and mostly undetectable given the amplitude of tidal stresses.

Beyond its geophysical implications, these findings are relevant to sustainable energy production and decarbonization efforts. Many technologies, including geothermal energy, carbon sequestration, and enhanced oil recovery, involve perturbations of the subsurface reservoir via fluid injection or extraction, which may induce seismicity (15). In regions where fault stress state and friction properties remain poorly understood, the presence or absence of seismicity response to tidal and seasonal stress perturbations could provide valuable constraints for modeling the seismicity induced by such operations and help mitigate the resulting seismic hazard.

MATERIALS AND METHODS

Removing events below the completeness magnitude (M_c)

In this study, we use the NCSS (47) and HYS (46) catalogs for Northern California and Southern California, respectively. To ensure that our analysis is free from detection bias, we first estimate the completeness magnitude (M_c), the threshold above which all earthquakes are reliably detected and cataloged and remove events smaller than M_c. Given the heterogeneous distribution of seismic stations across California, we adopt a spatially varying M_c rather than a single uniform value for the entire region. At each grid point, if there are at least 50 events within a 10-km radius, we estimate M_c using the maximum curvature method (MAXC) (61), resulting in local values ranging from 0.5 to 2.5 (fig. S1). Although the MAXC is known to underestimate M_c in regions with heterogeneous detection capabilities (62) and a correction factor of 0.2 magnitude units is commonly applied (63), this limitation is mitigated by restricting the analysis to small spatial bins with relatively homogeneous detection. As a result, we do not need to apply any correction factor in our analysis. Each earthquake is evaluated against its corresponding local M_c and events below that threshold are excluded. If no local M_c is available due to fewer than 50 events within a 10-km radius, the event is discarded.

Distinguishing background seismicity from aftershocks and swarms

To focus on hydrologically and tidally modulated earthquakes, we further remove clustered events driven by other factors (aftershocks driven by mainshocks and swarms driven by aseismic slip and fluids) using the nearest-neighbor distance approach (49) modified with a spatially varying mode separator. The declustering analysis includes earthquakes from 1981 to 2021, aiding the identification of aftershocks from older mainshock, but subsequent analysis is limited to 2006 to 2021 to align with the hydrological surface load model (8). Declustering is performed separately for Northern California and Southern California.

For each event j, we identify the preceding event i* most likely to be its parent (mainshock), quantified by the proximity distance η_ij (49)

$${\mathrm{\eta }}_{\mathit{ij}}=t_{\mathit{ij}}{\left(r_{\mathit{ij}}\right)}^{d_f}{10}^{-b(m_i-m_0)}$$ (1)

where t_ij is the time difference, r_ij is the distance between the events, d_f is the fractal dimension, b is the Gutenberg-Richter b value, m_i is the magnitude of event i, and m₀ is a reference magnitude. Because of depth uncertainty, we analyze only epicenters and set d_f = 1.6, b = 1, and m₀ = 1.0. The event i* minimizes η_ij and is referred to as the nearest neighbor.

The distribution of the nearest-neighbor distance ${\mathrm{\eta }}_j={\mathrm{\eta }}_{i^{\ast }j}=\underset{i}{\text{min}}({\mathrm{\eta }}_{\mathit{ij}})$ generally exhibits a bimodal pattern with one mode representing background events and another representing clustered events. The intersection of probability densities from the two modes defines the optimal mode separator η₀, which appears as a diagonal line with slope of −1 in the two-dimensional (2D) visualization using the rescaled time T_j and the rescaled distance R_j (49)

$$\begin{array}{c}{T}_j=t_{i^{\ast }j}{10}^{-\frac{b}{2}(m_{i^{\ast }}-m_0)}\\ R_j={\left(r_{i^{\ast }j}\right)}^{d_f}{10}^{-\frac{b}{2}(m_{i^{\ast }}-m_0)}\end{array}$$ (2)

Because the productivity of the background seismicity varies spatially, the optimal mode separator η₀ must also vary spatially. One approach to determine location-specific η₀ is to use randomized-reshuffled earthquake catalogs excluding most of the clustered events (64). However, this method requires generating and analyzing a sufficiently large ensemble of reshuffled catalogs, which can be computationally intensive and less tractable for large catalogs. Here, we propose an alternative method by applying the nearest-neighbor declustering within multiple smaller spatial bins, where η₀ can reasonably be assumed to be constant within each bin. To achieve this, we apply a quadtree algorithm to recursively subdivide the space into quadrants until each contains fewer than 5000 events. For each quadrant, we subset 2000 nearest events to the center and fit their η_j distribution with a 1D Gaussian mixture model with two Gaussians, one representing background events and the other representing clustered events. The optimal mode separator η₀ is taken to be where the two density functions intersect. If the automated fit is unsatisfactory, we fit the interevent times distribution with a gamma distribution and estimate the proportion of clustered events as 1 − γ, where γ is the reciprocal of the variance of the normalized interevent times (65). We then adjust η₀ so that the proportion of events classified as clustered matches the estimates from the gamma distribution. Once η₀ is determined for all quadrants, we interpolate the values η₀(x, y) from the quadrant centers to construct its full spatial distribution (fig. S2). Any event j for which ${\mathrm{\eta }}_j\ge {\mathrm{\eta }}_0(x_j,y_j)$ is classified as a background event.

To assess declustering quality, we use the coefficient of variation (CoV), defined as the ratio of the standard deviation to the mean of the duration between subsequent earthquakes (66). A CoV of 1 suggests a Poisson process, CoV > 1 suggests clustering, and CoV < 1 suggests periodic occurrence. Regions with aftershocks exhibit high CoV (>>1), whereas a properly declustered catalog should yield CoV ≈ 1. Residual aftershocks from the July 2019 M7.1 Ridgecrest earthquake persist, prompting us to remove events after 2019 in affected areas for this and subsequent analyses. In addition, there are residual swarm sequences present in the Long Valley area, as evidenced by abrupt jumps in the Schuster walk during certain years compared to others. To mitigate this, we manually increase the cutoff η₀ for the affected spatial bins and year with the abrupt jump by an increment of 0.05 until the anomalous jump disappears from the Schuster walk. After correction, most regions achieve CoV ≈ 1 (fig. S3), signifying Poissonian behavior and confirming the effectiveness of the declustering method in identifying background seismicity. The spatial distributions of CoV (fig. S3) and background seismicity rates (Fig. 1C) are derived from earthquakes within 25 km of each gridded point. For the background seismicity rates, only events with M ≥ 2.5 are considered to mitigate bias from varying rates due to changes in M_c.

Although earthquakes occurring during the peak rates of strong, periodically modulated sequences could be misinterpreted as aftershocks due to temporal clustering, this bias is minimal in our study given the modest modulation amplitudes (<20 to 25%) and low event counts per cycle resulting from fine spatial binning. In addition, the nearest-neighbor declustering method incorporates temporal, spatial, and magnitude separations, allowing it to distinguish aftershock-like clusters and transient increases in seismicity rate. As a result, only events exhibiting both temporal and spatial clustering are identified as aftershocks, whereas those associated with periodic modulation are identified as background events.

Identify periodic modulation of seismicity rates with the Schuster spectrum

To examine periodic modulation of seismicity rates, we apply the Schuster test separately to broad regions in Northern California and Southern California and to smaller spatial bins. For a given testing period T, each earthquake occurrence time t_i is converted into a phase angle ${\mathrm{\theta }}_i=2\mathrm{\pi }{t}_i/T$. A 2D successive walk is then performed, with unit-length steps in the direction of the computed phase angles. After N steps from N earthquakes occurring over a number of full cycles of period T, the Schuster P value is calculated on the basis of the total drift distance D from the walk’s origin to its endpoint (50)

$$P=e^{-D^2/N}$$ (3)

The Schuster P value quantifies the probability of the null hypothesis, which assumes a uniform seismicity rate following a random Poisson point process. The null hypothesis is rejected if the P value is smaller than the threshold $P_{\text{threshold}}=T/t_{\text{catalog length}}$. For a traditional 95% confidence level to reject the null hypothesis, the cutoff value becomes $P_{95}=0.05·P_{\text{threshold}}=0.05·T/t_{\text{catalog length}}$, rather than simply 0.05 as the confidence level must be scaled to account for the number of independent trials embedded over the catalog duration (43). Each trial is an independent opportunity for a periodic signal to manifest over one full cycle of the tested period. In addition to the P value, the total drift direction of the Schuster walk corresponds to the phase at which seismicity rate is highest, offering insight into the timing of peak activity.

A Schuster spectrum, composed of multiple Schuster tests, can be constructed to examine a range of periods. The optimal sampling is performed over the frequency space with a step of $t_{\text{catalog length}}/T_{\text{min}}$, which is fine enough to avoid missing periods while preventing excessive computation (43).

Evaluating Schuster P values without visualizing the Schuster walk or constructing a Schuster spectrum is precarious as drift behaviors may be dominated by residual aftershocks and swarms occurring over one cycle instead of persistent periodic modulation over many cycles. A genuine periodicity detection appears as an isolated low P value at a specific period, whereas imperfect declustering manifests as a drifting low P value at periods equal to the characteristic decay time of aftershocks or longer.

For a sinusoidally varying seismicity rate R(t) with period T, the rate can be expressed as

$$\frac{R\left(t\right)}{r}=1+\text{α cos}\left(\frac{2\mathrm{\pi }t}{T}\right)$$ (4)

where r is the average seismicity rate, and α is the modulation amplitude of the seismicity rate. The Schuster P value is related to α as follows (41)

$$\mathrm{\alpha }=\sqrt{-\frac{4(\text{ln}P+1)}{N}}$$ (5)

Because periodicity detection requires a threshold P value, there is a corresponding modulation amplitude threshold α_threshold necessary for statistically significant detection (fig. S4), which also depends on both the catalog length and the number of events analyzed.

The spatial distributions of α for annual (Fig. 2A) and semidiurnal tidal periods (fig. S5) are derived from the 5000 earthquakes nearest to each gridded point. Only grid points with 5000 earthquakes located within 200 km and those with a background seismicity rate of at least 0.0001 M ≥ 2.5 events/km² per year (Fig. 1C) are considered. Note that these two conditions are not equivalent as the background seismicity rates were calculated using only events within 25 km, resulting in different spatial smoothing. With a fixed number of events and catalog length, the resulting α_threshold is constant, which is 0.038 for the annual period and 0.0818, 0.0816, and 0.0815 for K2 (11.967 hours), M2 (12.421 hours), and N2 (12.658 hours) tidal periods, respectively. To preserve localized behaviors, events from Coso and Geysers geothermal areas are excluded for gridded points outside these regions. Because of Coso’s small area (~1000 background events), we limit the number of events used for Schuster analysis in this area to 1000, which changes α_threshold to 0.084 for the annual period and 0.1830, 0.1826, and 0.1834 for K2, M2, and N2 tidal periods. For Ridgecrest region (north of 35°N, east of 118°W within the Southern California polygon), removing events after 2019 alters the catalog length, resulting in an adjusted α_threshold of 0.035 for the annual period and 0.0808, 0.0806, and 0.0805 for K2, M2, and N2 tidal periods. Furthermore, Schuster spectra and Schuster walks are also shown for selected regions (Fig. 2 and fig. S6).

Stress changes due to hydrological surface load

Seasonal changes in water storage from groundwater fluctuations and snowmelt were modeled using surface displacements recorded by Global Positioning System (GPS) stations and gravity measurements recorded by Gravity Recovery and Climate Experiment (GRACE) satellites (8). Changes in the equivalent water thickness Δh induce variations in the vertical load of magnitude ${\mathrm{\rho }}_{\text{water}}g\mathrm{\Delta }h$, which alter the stress state.

To compute stress changes at a seismogenic depth of 5 km where most earthquakes occurred in California, vertical loads are modeled as multiple point forces distributed across a 2.5-km grid on the Earth’s surface. Assuming a homogeneous elastic half-space, the stress change (extension positive) at point (x, y, z) due to a downward point load N acting at the origin is computed using the following semianalytical solutions (56, 67)

$$\begin{array}{c}{\mathrm{\sigma }}_{\mathit{xx}}(x,y,z)=\frac{N}{2\mathrm{\pi }}[\frac{3x^{2}z}{r^5}-\frac{(1-2\mathrm{\nu })(y^2+z^2)}{r^3(r-z)}-\frac{(1-2\mathrm{\nu })z}{r^3}+\frac{(1-2\mathrm{\nu })x^2}{r^2{(r-z)}^2}]\hfill \\ {\mathrm{\sigma }}_{\mathit{yy}}(x,y,z)=\frac{N}{2\mathrm{\pi }}[\frac{3y^{2}z}{r^5}-\frac{(1-2\mathrm{\nu })(x^2+z^2)}{r^3(r-z)}-\frac{(1-2\mathrm{\nu })z}{r^3}+\frac{(1-2\mathrm{\nu })y^2}{r^2{(r-z)}^2}]\hfill \\ {\mathrm{\sigma }}_{\mathit{zz}}(x,y,z)=\frac{N}{2\mathrm{\pi }}\left[\frac{3z^3}{r^5}\right]\hfill \\ {\mathrm{\sigma }}_{\mathit{xy}}(x,y,z)=\frac{N}{2\mathrm{\pi }}[\frac{3\mathit{xyz}}{r^5}+\frac{(1-2\mathrm{\nu })\mathit{xy}(2r-z)}{r^3{(r-z)}^2}]\hfill \\ {\mathrm{\sigma }}_{\mathit{yz}}(x,y,z)=\frac{N}{2\mathrm{\pi }}\left[\frac{3y{z}^2}{r^5}\right]\hfill \\ {\mathrm{\sigma }}_{\mathit{xz}}(x,y,z)=\frac{N}{2\mathrm{\pi }}\left[\frac{3x{z}^2}{r^5}\right]\hfill \end{array}$$ (6)

where r is the distance between the origin and point (x, y, z). In this system, the coordinate axes are defined as x = East, y = North, and z = Up. The total stress changes are obtained by summing the contributions from all individual point loads.

To evaluate whether stress changes promote failure along a given plane, we compute the Coulomb stress change: ΔS = Δτ + μΔσ, where Δτ and Δσ are the shear and normal stress changes (extension positive) on that plane, respectively, and μ is the coefficient of friction, chosen to be 0.4 to approximately account for poroelastic effects (3). Because our analysis includes many smaller-magnitude earthquakes that lack reliable focal mechanisms, there is considerable uncertainty in the orientation of the fault planes along which they slip. To account for this ambiguity, we explore three distinct plausible sets of fault plane orientations.

In the first scenario, we assume that fault planes are oriented optimally with respect to the background tectonic stress field. Although substantial efforts are being made by the Statewide California Earthquake Center (SCEC) to develop community stress models (CSMs) that include the full 3D stress tensor across California, currently available results are limited to Southern California. Existing datasets typically provide only the orientation of the maximum horizontal stress (S_Hmax) and the relative magnitudes of principal stresses, quantified by the Anderson modified shape parameter (A_ϕ). A_ϕ can be used to classify the tectonic regime: Values between 0 and 1 correspond to normal faulting, 1 to 2 to strike-slip faulting, and 2 to 3 to reverse faulting (68). Here, we use S_Hmax orientation from Heidbach and Ziegler (69) based on the World Stress Map database (70) and A_ϕ from Lundstern and Zoback (71) (fig. S7). By further assuming that the vertical stress corresponds to one of the principal stress directions, we can infer the full principal stress orientation (σ₁, σ₂, and σ₃) consistent with the tectonic regime. This allows us to rotate the stress change tensor into these orientations. The optimally oriented fault plane would orient at an angle $\mathrm{\beta }=\frac{1}{2}\text{arctan}(-\mathrm{\mu })$ from the σ₁ axis in the σ₁-σ₃ plane. Although two such optimal planes exist and we cannot distinguish which one is more favorable, both experience identical shear and normal stress changes. These shear and normal stress changes on the plane can be expressed in terms of the maximum principal stress change ΔS₁, the minimum principal stress change ΔS₃, and the angle β, without requiring explicit specification of strike and dip, using the following expressions (3)

$$\begin{array}{c}\mathrm{\Delta }{\mathrm{\tau }}_{\text{optimal}}=\frac{1}{2}(\mathrm{\Delta }{S}_1-\mathrm{\Delta }{S}_3)\text{sin}2\mathrm{\beta }\hfill \\ \mathrm{\Delta }{\mathrm{\sigma }}_{\text{optimal}}=\frac{1}{2}(\mathrm{\Delta }{S}_1+\mathrm{\Delta }{S}_3)-\frac{1}{2}(\mathrm{\Delta }{S}_1-\mathrm{\Delta }{S}_3)\text{cos}2\mathrm{\beta }\hfill \end{array}$$ (7)

Note that Δτ_optimal represents the maximum shear stress along an optimal rake, which is not constrained to any particular slip direction and may evolve over time.

In the second scenario, we assume that fault planes are oriented optimally with respect to the stress changes induced by hydrological loading. This additional scenario is motivated by the observation that not all fault planes are optimally oriented to the background tectonic stress field, including the San Andreas fault (72). Using the stress change tensor, we compute the principal Coulomb stress changes (Δσ₁, Δσ₂, and Δσ₃) and apply Eq. 7 to determine the shear and normal stress changes on fault planes optimally oriented with respect to the hydrologically induced stress perturbation. In addition to the rake direction evolving with time, the optimally oriented plane may also evolve over time, as the orientation of the principal stress changes induced by hydrological loading is itself time dependent.

In the third scenario, we assume that fault planes are vertical and parallel to the San Andreas fault, with a fixed strike, dip, and rake corresponding to right-lateral strike-slip motion (strike = 162°, dip = 90°, and rake = 180°). This assumption is appropriate for the western part of California but becomes less representative eastward toward the Central Valley and Sierra Nevada, where fault geometries and stress regimes diverge.

After computing the Coulomb stress changes, we quantify the amplitude of seasonal variations and identify the timing of the peak stressing rates. This is achieved by detrending the time series of Coulomb stress changes using a moving average with a 1-year window and fitting the residuals with a sinusoidal function (fig. S8).

Modeling seismicity responses to harmonic stress perturbations

The simplest model to explain the response of seismicity to stress perturbations is the CFM. In this model, earthquakes are assumed to nucleate instantaneously once the driving stress exceeds the fault’s strength. As a result, the seismicity rate R(t) is proportional to the Coulomb stressing rate $\dot{S}$ as follows

$$\frac{R\left(t\right)}{r}=\frac{\dot{S}}{\dot{S_0}}$$ (8)

where r is the background seismicity rate occurring at the background stressing rate $\dot{S_0}$. This relation is only valid when the stressing rate is positive. If the Coulomb stress change decreases, seismicity will shut off and only resume when the stress on the fault exceeds its preceding largest value. For a superposition of a sinusoidally varying stress perturbation and the background stressing rate $\dot{S_0}$, the amplitude of seismicity response (α in Eq. 4) depends on whether the period T of the perturbing stress is larger or smaller than the critical period $T_{\mathrm{\tau }}=2\mathrm{\pi }\mathrm{\Delta }S/\dot{S_0}$ (41)

$$\begin{array}{c}\mathrm{\alpha }=\frac{T_{\mathrm{\tau }}}{T}=\frac{2\mathrm{\pi }\mathrm{\Delta }S}{T\dot{S_0}} \text{when} T\ge T_{\mathrm{\tau }}\hfill \\ \mathrm{\alpha }=2\sqrt{\mathrm{\pi }}\sqrt{\frac{T_{\mathrm{\tau }}}{T}}=2\mathrm{\pi }\sqrt{\frac{2\mathrm{\Delta }S}{T\dot{S_0}}} \text{when} T \ll T_{\mathrm{\tau }}\hfill \end{array}$$ (9)

A major limitation of the CFM is that it does not incorporate the intrinsic time delay between a stress perturbation and fault failure. As a result, it cannot explain phenomena such as the time lag between the peak seismicity rate and the peak stressing rate (Fig. 3 and fig. S9) or the temporal decay of aftershock rates following a mainshock (73).

To address the limitations of the CFM, laboratory measurements of friction during the sliding of rock surfaces or gouge layers led to the development of the empirical rate-and-state friction formulation, which accounts for both slip rate-dependent and time-dependent frictional behavior (59, 74). Subsequent analysis of a one-degree-of-freedom spring-slider system obeying rate-and-state friction led to the development of the RSM, which describes the response of the seismicity rate R(t) to Coulomb stress perturbations ΔS(t) (16, 41)

$$\frac{R\left(t\right)}{r}=\frac{e^{\mathrm{\Delta }S\left(t\right)/a\mathrm{\sigma }}}{1+\frac{1}{t_a}{\displaystyle {\int }_0^t}{e}^{\mathrm{\Delta }S\left(x\right)/a\mathrm{\sigma }}\mathit{dx}}$$ (10)

where aσ is the friction-stress parameter, represented as the product of the rate-and-state parameter a and the fault normal stress σ, and $t_a=a\mathrm{\sigma }/\dot{S_0}$ is the characteristic relaxation time of the seismicity following a stress step (i.e., the characteristic duration of an aftershock sequence). The RSM captures the noninstantaneous nature of earthquake nucleation and successfully explains a number of time-dependent behaviors of earthquakes that cannot be explained by the CFM, such as the Omori’s decay of aftershock rates following a mainshock (4), the correlation of earthquakes with periodic loadings (40), slow-slip driven earthquake swarms (10), volcanic earthquakes (17), and induced seismicity (19).

The response of seismicity to sinusoidally varying stress can be used to constrain the RSM. We perform forward modeling using Eq. 10 for a range of parameters aσ and t_a, simulating 10 stress cycles. The initial cycles are used for model spin-up to eliminate transient effects from initialization (75), and the seventh cycle is used to extract the amplitude of seismicity modulation and the time lag between the peak stressing rate and the peak seismicity rate. These outputs form a lookup table for inverting the RSM parameters (aσ and t_a) from the observed seismicity response (fig. S10). The inversion is performed using Coulomb stress changes calculated for three distinct plausible sets of fault plane orientations (fig. S11), allowing for uncertainty in fault geometry.

In the limit where the period of the perturbing stress T ≪ t_a, such as the semidiurnal tidal periods, the seismicity response in the RSM becomes period independent (41) and Eq. 10 reduces to

$$\mathrm{\alpha }=e^{\mathrm{\Delta }S/a\mathrm{\sigma }}-1$$ (11)

where α is the amplitude of seismicity response (Eq. 4).

Using aftershocks to constrain the earthquake nucleation process

The decay of aftershock rates reflects the response of seismicity to stress change following a mainshock, which can be modeled using Omori’s law (73). By approximating the Coulomb stress change from a mainshock as a stress step of magnitude ΔS, the RSM (Eq. 10) naturally simplifies to Omori’s law (16)

$$R\left(t\right)=\frac{r{t}_a}{t_a{e}^{-\mathrm{\Delta }S/a\mathrm{\sigma }}+t}=\frac{k}{c+t}$$ (12)

with k = rt_a and $c=t_a{e}^{-\mathrm{\Delta }S/a\mathrm{\sigma }}$. This formalism links the empirical Omori’s law parameters k and c to the RSM parameters t_a and aσ, offering an independent method to constrain the earthquake nucleation process and providing cross-validation with the parameters determined from seasonal modulation of seismicity.

To extract aftershock sequences, we use the results of nearest-neighbor declustering analysis described earlier. We select events belonging to the clustered mode with ${\mathrm{\eta }}_j<{\mathrm{\eta }}_0(x_j,y_j)$ and organize them into a spanning forest, where each event is linked to its nearest neighbor, which serves as its parent in the hierarchy. The immediate children of each node represent the aftershocks directly triggered by the parent event. Using sequences with more than 100 directly triggered aftershocks and fitting aftershock rates within 20 km of the mainshock with Omori’s law, we constrain the parameters k and c. With the fitted k and the background seismicity rate r, calculated as the average seismicity rate from 2006 up to the mainshock within the same spatial window, the characteristic time t_a can be inferred (fig. S12). Although the friction-stress parameter aσ can also be estimated in principle, it remains difficult to constrain using this method as the magnitude of the Coulomb stress change ΔS is highly sensitive to the slip model of each mainshock and varies considerably across spatial locations. As a result, we do not pursue the estimation of aσ from aftershock sequences in this study.

Supplementary Materials

This PDF file includes:

Figs. S1 to S13