Tidal and hydrological seismicity modulations reveal pore fluid diffusion during earthquake nucleation

Abstract

The occurrence of seismic events can be modulated by external periodic stress perturbations, such as daily tidal stress and annual hydrological stress. Such periodic modulations are crucial for understanding earthquake triggering, yet their underlying physical mechanisms are not fully understood. Here, we find that ordinary earthquakes (OEs) and low-frequency earthquakes (LFEs) on the Central San Andreas Fault (CSAF) are more sensitive to the long-period hydrological and the short-period tidal loadings, respectively. These different frequency-dependent modulations suggest pore fluid diffusion during the noninstantaneous earthquake nucleation and confirm different nucleation times of OEs and LFEs. We constrain the depth-varying physical properties of the CSAF and reveal that fluid content distribution and loading conditions fundamentally control slow-to-fast fault slip behaviors. Our study provides an alternative perspective to understand earthquake nucleation by using the information in periodic seismicity modulations, which can be applicable to subduction zones where similar slip behavior transitions occur.

Seismicity responds differently to annual hydrological and daily tidal loading, revealing how fluids affect earthquake triggering.

INTRODUCTION

The response of seismic activity to external stress perturbations contains rich information about the physics of earthquake triggering, nucleation, and rupture (1). Among various types of stress perturbations, periodic loadings, such as tidal loadings with dominant semidiurnal and diurnal periods and hydrological loadings with typical periods of 1 year, are ubiquitous, providing valuable natural experiments for studying earthquake triggering and the underlying mechanisms. Tidal and hydrological loading stresses have similar amplitudes, but their effects on seismicity have been found to be quite different. This phenomenon reveals the frequency-dependent nature of periodic seismicity modulation, shedding light on the earthquake nucleation process (2).

In recent decades, substantial efforts have been made to explore the periodic modulation of both ordinary earthquakes (OEs) and low-frequency earthquakes (LFEs). Studies of the tidal triggering of OEs suggest that they are not or very weakly modulated by tidal loadings (3–5), with the exception of some special situations where the ocean tidal stresses are relatively large (6, 7) or the faults are critically stressed (8, 9). In contrast, the seasonal modulation of OEs due to hydrological loadings has been widely observed in various tectonic settings, encompassing extensional (10), strike-slip (11–13), and convergent environments (14, 15), as well as in tectonically stable intraplate regions (16).

LFEs associated with tectonic tremors are small seismic events whose source properties and response to external loadings are very different from OEs. The clustered occurrence of LFEs and geodetic evidence suggest that they represent the seismic signature of dominantly slow fault slip events (17, 18). Resolving the physical mechanisms of the generation of LFEs and their relationship with OEs is crucial for understanding the wide spectrum of fault slip behaviors and may contribute to the improved assessment of seismic hazards (19). In comparison with OEs, LFEs are characterized by much slower slip velocities, smaller stress drops, and shorter recurrence times (20). Most LFEs are found in the deep crust, downdip of the seismogenic zone hosting OEs and are understood to occur in environments of near-lithostatic fluid pressure (21). Besides their differences in source parameters and settings, LFEs and OEs also have distinctly different responses to tidal and hydrological loadings: Unlike OEs, the occurrence of LFEs is very sensitive to tidal stresses (22–26). On the other hand, annual changes in tremor rates are observed in some cases (18, 27, 28). However, the overall relationship between LFEs and hydrological loading is not clear, in contrast to the case of OEs.

Several unresolved issues remain regarding periodic seismicity modulations. First, the distinctly different modulations of OEs and LFEs are not well understood, as previous studies have focused on different regions and used different statistical methods to estimate the modulation. Second, the classical physical model of earthquake triggering, which assumes a noninstantaneous nucleation process (29, 30), sometimes fails to explain the observed frequency dependence of periodic modulations (1), suggesting that additional important mechanisms during earthquake nucleation have been overlooked. Most importantly, the fault-mechanical implications of different fault slip behaviors inferred from the differences in periodic modulations of OEs and LFEs remain unexplored.

The Central San Andreas Fault (CSAF) near Parkfield hosts a large diversity of fault slip behaviors, including large damaging earthquakes, small repeating earthquakes, LFEs, and aseismic slip (31–34). Dense geodetic and seismological observations have produced a comprehensive dataset for both stress loading histories and earthquake catalogs along the CSAF, allowing for comprehensive study of the in situ periodic modulations of different types of earthquakes by stress loadings with different periods (Fig. 1). Here, we conduct an integrated investigation of the tidal and hydrological modulations of upper-crustal OEs and lower-crustal LFEs along the CSAF based on the same statistical method. We propose a mechanical model that considers the role of pore fluids during earthquake nucleation to explain the observations. Last, we constrain the physical parameters for OEs and LFEs in the framework of the proposed spring-slider rate-and-state model with dilatancy and diffusion (SRM-DD), which provide insights into the factors controlling the transition of fault slip behaviors over depths.

Fig. 1. Maps of earthquakes (OEs and LFEs) and stress loadings (tidal and hydrological) on the CSAF.

(A) Map view and (B) depth profile of the earthquake catalogs on the CSAF. Circles and triangles denote OEs and LFE families, respectively. The colors of the triangles show the number of events in each LFE family. (C) Time series of the hydrological and tidal shear stresses on the fault. The open circles are the original sample points from the hydrological model, while the line is the analytical fitting by multiple frequencies. The colors of the lines denote the phase with reference to the loading stress.

RESULTS

Periodic seismicity modulations on the CSAF

We focus on the ~220-km-long CSAF (from 35.2°N to 36.4°N) along which the fault geometry is relatively uniform (Fig. 1A) (32). OEs and LFEs occur on the shallow and deep parts of the CSAF, respectively, separated by a ~5-km-wide gap at around 15-km depth (Fig. 1B). We investigate declustered catalogs of OEs and LFEs on the CSAF (Materials and Methods and figs. S1 and S2), from 2006 to 2015, a period for which time series of hydrological loading stress on the CSAF are available (35). Only OEs whose magnitudes are larger than the magnitude of completeness (M_c = 1.1) and on-fault OEs whose distances from the CSAF are less than 2 km are selected, yielding 3572 OEs. The declustered LFE catalog contains 107,927 events (Materials and Methods).

We compare the occurrence times of the events with the time series of reference stress loadings and estimate the seismicity modulation in terms of amplitude P_m and phase φ, using the “phase histogram” method (Materials and Methods), which captures the modulation by the actual loading with multiple frequencies (6). The amplitude P_m indicates how strong the modulation is, while the phase φ indicates under which loading stage more events occur. Positive and negative φ mean that the peak of seismicity rate occurs after and before the peak of stress loading, respectively. The reference stress models are the annual loading model, which includes various types of dominantly annual loadings with the largest contribution from hydrological loading (35) (hereafter referred to as hydrological loadings), and the tidal model, which is calculated by the software package Some Programs for Ocean Tide Loading using a shear modulus of 30 GPa and a Poisson ratio of 0.25 at the center of the seismic events region (120.525°W, 39.935°N, 0 km), considering both ocean tides and solid-earth tides (Fig. 1C) (36). The tidal and hydrological stress models are projected into normal- and shear-stress components on the fault plane based on the representative geometry of the CSAF (strike = 320°, dip = 90°, and rake = 180°). Although a small portion of the selected OEs have different focal mechanisms, the overall angles of the maximum horizontal stress orientation relative to the fault plane are 45° ± 15° (32), which leads to little differences in the phase of the projected stresses. Despite the normal-stress cycles being much larger in amplitude than shear stresses (fig. S3), we find that their effects on modulating seismicity are weak for both OEs and LFEs (fig. S4), which is also found in previous studies (11, 24). When using Coulomb stress as the reference stress, the detected modulation is consistently smaller than that of the shear stress (fig. S5). Therefore, we only use the shear stresses as the reference loading stresses in the following analysis.

To assess the significance of the observed modulation, we also calculate the modulations of 2000 random catalogs, each containing the same number of events as the actual catalog. The observed modulation is regarded as significant only if its P_m is larger than the upper bound of the nonoutlier P_m in the random cases, which is defined as 1.5 times the interquartile range above the third quartile, according to the box-plot statistics (Materials and Methods and Fig. 2) (37). Assuming that random P_m follows a Gaussian distribution, this significance threshold corresponds to the 99.65th percentile of the samples, meaning that the probability that P_m of a random catalog exceeds this threshold is only 0.35%.

Fig. 2. Tidal and hydrological modulations of OEs and LFEs on the CSAF.

(A and B) Modulations of OEs by tidal and hydrological loadings, respectively. The histograms show the phase distributions of the observed catalog on the CSAF. The red lines show the periodic fitting of the phase distributions of the observed catalog, with the labels providing the modulation amplitude P_m and phase φ. The gray lines show the periodic fittings of the phase distributions of 2000 random catalogs. The yellow lines show the periodic modulations reproduced by our SRM-DD, using the preferred parameters listed in Table 1. (C) Amplitudes of tidal and hydrological modulations of OEs. The red triangles denote the observed modulation amplitudes on the CSAF, while the gray boxes show the box-plot statistics for the 2000 random modulation results. The top and bottom of the box indicate the 25% (Q1) and 75% (Q3) quartiles of the P_m in the random catalogs, and the line inside the box denotes the median (Q2). The upper and lower bounds of the box plot are Q3 + 1.5*IQR and Q1 − 1.5*IQR, respectively, where IQR = Q3 − Q1 is the interquartile range. The yellow circles denote the modulation amplitudes reproduced by our model. (D to F) Same as (A) to (C) but for LFEs.

Our results clearly demonstrate that OEs along the CSAF are more sensitive to the annual hydrological shear loading, whereas LFEs are more sensitive to the tidal shear loading at semidiurnal and diurnal frequencies. For the upper-crustal OEs, their correlation with tidal loading has a P_m of merely 0.03 ± 0.02, demonstrating insignificant tidal modulations, which is consistent with previous studies (3); but their hydrological modulation is significant, showing ~12% more events occurring during the winter (December and January) when the stress increases (P_m = 0.12 ± 0.03, φ = −24° ± 15°). Such significant hydrological modulation of OEs is also found in previous studies, although different data and statistical methods were used (11–13). Although a modulation of OEs by tidal normal stresses was found at Parkfield during the falling fortnightly tides (38), when considering the entire time series of the tidal loading as reference, there appears to be no significant tidal modulation by either shear- or normal-stress components in their catalog (fig. S6), which is consistent with the findings of this study and Vidale et al. (3).

In contrast, the tidal modulation of lower-crustal LFEs is very strong (P_m = 0.37 ± 0.03), and the maximum seismicity rate coincides with the maximum tidal shear stress (φ = 1° ± 3°). Meanwhile, the hydrological modulation of LFEs is significant but very weak (P_m = 0.05 ± 0.03). The obtained phase of the hydrological modulation of LFEs is −136° ± 26° (peak at August to September), indicating that the peak of seismicity rate occurs when the loading stress discourages failure, which is physically implausible. However, this modulation phase is uncertain, because the estimation of the modulation phase is not very reliable when the modulation amplitude is very small. The reliability of the modulation results is verified by a series of tests, which rule out possible artifacts from randomness, clustering effects, or temporal variations of detectability (Supplementary Text 1 and figs. S7 to S9). Therefore, we conclude that the distinctly different modulations for OEs and LFEs on the CSAF are robust and their implications for underlying earthquake physics merit further explorations.

Spring-slider rate-and-state model with dilatancy and diffusion

The observed frequency-dependent periodic modulations on the CSAF cannot be completely explained by either the Coulomb failure model (CFM), which assumes instantaneous earthquake failure, or by the spring-slider rate-and-state model (SRM), which incorporates a noninstantaneous nucleation process. In the CFM, the seismicity response is directly proportional to the stress loading rate, so short-period loadings always result in larger seismicity modulations (1). The CFM cannot explain the weak correlation between OEs and tides. In the SRM, the seismicity response at short periods is weakened due to the noninstantaneous earthquake nucleation, but the modulation amplitudes still monotonically decrease with increasing loading periods (see also Fig. 3, B and C) (1). Therefore, both the CFM and the SRM fail to explain the stronger response of OEs to the long-period hydrological loading, implying the existence of another critical timescale other than the nucleation time to result in a nonmonotonic response.

Fig. 3. Seismicity response spectra simulated by the SRM-DD.

(A) Illustration of the SRM-DD. (B and C) The amplitudes and phases of modeled periodic modulations over a broad range of periods. The blue and green lines denote the responses simulated with the optimal parameters of OEs and LFEs (Table 1), respectively, and the dots of the same colors show the responses predicted by the analytical solution of the SRM for comparison. The black dashed lines are the nucleation times (T_a) for OEs and LFEs, respectively, while the red dashed line is the characteristic diffusion time (T_f). The shaded period bands mark the ranges of periods for tidal and hydrological loadings. The gray squares are the selected representative periods of OEs. (D to H) Synthetic catalogs and periodic seismicity modulations of OEs at the representative periods marked in (B) and (C). The corresponding modulation amplitudes (P_m) and phases (φ) are provided.

Several mechanisms have been proposed to introduce an additional timescale, including a resonance destabilization effect for a velocity-neutral asperity (39), the response of the surrounding velocity-strengthening portions of the fault (40), and rate-and-state models of faults with finite dimensions (1). These mechanisms require either a critical size of the asperity (1, 39) or a dominant role of creeping faults surrounding the asperities (40). Under these frameworks, if the critical timescales of LFEs and OEs are close to daily tidal and annual hydrological periods, respectively, then the observed variations in periodic modulations may be potentially explained. Here, we attribute the additional timescale to the diffusion of pore pressure inside fault zones, whose characteristic time leads to a nonmonotonic seismicity response over loading periods. To implement this idea, we incorporate the evolution of porosity and fluid pressure (Eq. 4, E and F) into the rate-and-state framework and consider the SRM-DD. These processes have been observed during fault slip in the laboratory (41, 42). We generate synthetic catalogs by simulating a large enough number of sources with properly assigned initial conditions to explore the expected periodic seismicity modulations in the SRM-DD (Materials and Methods and fig. S10).

The simulated seismicity response spectrum under single-frequency shear-stress perturbations with constant amplitude over a broad range of periods reveals different response modes in the SRM-DD, separated by the nucleation time T_a and the fluid diffusion time T_f (Fig. 3, B and C). When the loading periods are much shorter than both T_a and T_f, the response mode is stress controlled, and the fluid diffusion condition is undrained. The modulation amplitudes are smaller than those in the SRM (Fig. 3B), due to the dilatancy strengthening effect that increases the overall effective normal stress. At intermediate periods, the modulation amplitudes gradually increase with increasing loading period (Fig. 3B), while the modulation phases are positive (Fig. 3C). These characteristics are related to the fluid-diffusion process, which compensates the dilatancy effect and induces pore-pressure perturbations with phase delay to the exerted shear-stress perturbations. We refer to this response mode as diffusion-controlled, whose dominant periods are close to the characteristic diffusion time T_f. The maximum modulation amplitude in the diffusion-controlled mode is determined by nondimensional parameters ε/βσ₀ and T_a/T_f, where ε, β, and σ₀ are the dilatancy parameter, combined compressibility, and effective normal stress, respectively (fig. S11). When the loading periods are much longer than both T_a and T_f, the fluid diffusion condition is completely drained, the response mode is stress-rate controlled, the modulation amplitudes decrease with increasing loading periods, and the modulation phases gradually transition to −90°, the same as in the SRM. The nonmonotonic spectrum of modulation amplitudes in the SRM-DD is critical for explaining the observed modulations of OEs on the CSAF, particularly by enabling the hydrological modulation to be larger than the tidal modulation. In addition, the positive modulation phases predicted by the SRM-DD provide a possible scenario for the peak of seismicity to occur after the peak of the loading stress, which has been observed for many of the LFE families near Parkfield (24).

We do not further consider an external normal-stress perturbation in the model for several reasons. First, the applied tidal and hydrological normal-stress perturbations on the CSAF can be largely compensated by the undrained poroelastic response given their long spatial wavelengths (43), resulting in negligible contribution from external normal stress perturbation (fig. S3). Second, even if the external normal-stress perturbation is not largely compensated by pore fluids, then their perturbation amplitudes (<1 kPa) are much smaller than the background effective normal stress (0.1 to 1 MPa, assuming frictional parameter a = 0.001 to 0.01). Under this condition, the effect of an external normal-stress perturbation is exactly the same as an additional shear-stress perturbation divided by a friction coefficient in the form of Coulomb stress (fig. S12) (30). Third, if an external normal-stress perturbation with a phase delay with respect to the external shear-stress perturbation is considered in the SRM without dilatancy and diffusion, then the simulated response spectrum is monotonic (fig. S13), confirming the crucial roles of dilatancy and diffusion in generating the nonmonotonic responses. Overall, the nonmonotonic response over varying loading periods can be realized via dilatancy and diffusion processes with external shear-stress perturbations alone, while it cannot be realized in the absence of dilatancy and diffusion even with external normal-stress perturbations. Therefore, we do not incorporate the effect of external normal-stress perturbations in the model.

Physical properties of the CSAF

We constrain the fault properties of the CSAF by applying the SRM-DD to fit the observed periodic modulations of OEs and LFEs. The realistic tidal and hydrological shear stress loadings on the CSAF are analytically represented by combinations of several harmonic functions (Materials and Methods and table S1). Using these multifrequency analytical loadings as input to the SRM-DD model, we find the optimal model parameters for both OEs and LFEs (Supplementary Text 2, Table 1, and fig. S14). The predicted tidal and hydrological modulations under these sets of parameters well reproduce the observations in both amplitude and phase (Fig. 2).

Table 1. Physical fault properties constrained from periodic seismicity modulations on the CSAF.

a, frictional parameter; ${\mathrm{\sigma }}_0$, effective normal stress; $\mathrm{\epsilon }$, dilatancy parameter; $\mathrm{\beta }$, combined compressibility; ${\dot{\mathrm{\tau }}}_{\mathrm{r}}$, shear stress loading rate; $T_{\mathit{f}}$, diffusion time; $T_{\mathrm{a}}$, nucleation time; T_a = aσ₀/${\dot{\mathrm{\tau }}}_{\mathrm{r}}$.

	aσ0 (kPa)	$\mathrm{\epsilon }/\mathrm{\beta }$ (kPa)	$\dot{{\mathrm{\tau }}_{\mathit{r}}}$ (kPa/year)	Tf (days)	Ta (days)
OEs	1.2	2.8	3.6	50	120
LFEs	0.25	0.5	17.5	50	5

Our estimate of aσ₀ of OEs (1.2 kPa) is on the same order as that constrained by periodic seismicity modulations or earthquake swarms in several other studies (14, 44) but is one order of magnitude smaller than that estimated by other seismicity observations, such as aftershock sequences (45). The reason for the inconsistency of aσ₀ obtained using different observations is not obvious and beyond the scope of this study. As a consequence of the small value of aσ₀, ε/β and ${\dot{\mathrm{\tau }}}_{\mathrm{r}}$ constrained by this study are also much smaller than those measured in laboratory experiments (42) and values estimated from the source properties of repeating earthquakes (31), respectively.

Here, we focus on the remarkable differences of the parameters found for OEs and LFEs (Supplementary Text 2), revealing strong contrasts of fault properties with depth on the CSAF. First, the stress loading rate surrounding LFEs is larger than that surrounding OEs, indicating an increasing average stress loading rate with depth, in agreement with the increasing interseismic slip rates resolved by geodetic observations (33).

Second, the inferred aσ₀ and ε/β indicate that they both decrease with depth, which can be attributed to an increase of fluid content and pressure at depth. We find that aσ₀ of OEs is one order of magnitude higher than that of LFEs at greater depths. Because the frictional parameter a is unlikely to vary by an order of magnitude (46), this contrast suggests a smaller effective normal stress and a correspondingly higher pore pressure in the lower-crustal CSAF, which is consistent with previous studies (40, 47). On the other hand, ε/β is proportional to ε/Φ, where Φ is the porosity. Because ε does not substantially vary with effective normal stress according to laboratory experiments (42), the resolved ε/β suggests a ~6 times larger Φ in the lower-crustal CSAF. Therefore, the contrasting aσ₀ and ε/β values of OEs and LFEs show that the deeper part of the CSAF has substantially larger fluid content, resulting in both larger porosity and higher pore pressure, consistent with geological evidence of abundant fluid-filled cracks in the LFE environment (21). This fluid content distribution suggests the presence of a fluid source beneath it. Magnetotelluric (48) and seismological (49) studies have revealed a deep (~30 km) fluid-rich volume on the western side of the CSAF, which may be supplied by the continuing dehydration of remnant serpentinite in the mantle wedge of the Farallon subduction in the Mesozoic and Paleogene (50). Fluid diffusion from this deep source to the CSAF system might be responsible for the abundant fluids in the lower crust and the corresponding lower values of aσ₀ and ε/β in the deep CSAF (Fig. 4A).

Fig. 4. Fluid distributions and stress loading conditions on the CSAF as controlling factors of slip behaviors at depth.

(A) Cartoon showing depth section of the CSAF. (B) Depth profiles of the physical properties. The dots indicate the, aσ₀ $\mathrm{\epsilon }/\mathrm{\beta }$, and ${\dot{\mathrm{\tau }}}_{\mathrm{r}}$ values constrained at typical depths of OEs (5 km) and LFEs (30 km), while the lines are the extrapolated depth profiles for the respective parameters, assuming simple exponential functions. (C) Depth variations of modulation amplitude by tidal loadings (Tidal P_m) and by hydrological loadings (Hydrological P_m), maximum slip velocity (V_max), stress drop ($\mathrm{\Delta \tau }$), and recurrence times (T_r) simulated by the SRM-DD with the parameter distributions in (B).

Third, the nucleation times of OEs and LFEs are very different. The optimal T_a of OEs (~120 days) is in the lower range of the typical duration of aftershock sequences (51), while that of LFEs is ~5 days. Due to the short nucleation time of LFEs, they are more sensitive to stress perturbations at tidal periods (Fig. 3B). Considering the comparable diffusion time and nucleation time, OEs are more sensitive to perturbations at annual periods (Fig. 3B). As for the diffusion time, T_f of OEs can be constrained to be around 50 days, corresponding to a diffusivity of 10⁻³ m²/s, assuming a diffusion length of 300 m. The diffusivity/permeability may greatly vary with depth, but it cannot be further constrained by this study, because tidal and hydrological periods do not sample the intermediate diffusion-dominant periods for LFEs (figs. S11 and S14). Therefore, we select the T_f of LFEs to be the same as that of OEs (Table 1).

DISCUSSION

Implications for fault slip behaviors

To further explore how the stress loading rate (${\dot{\mathrm{\tau }}}_{\mathrm{r}}$) and fluid content (aσ₀ and ε/β) affect the seismic slip behavior from shallow to deep crustal levels, we extrapolate the parameters constrained at the typical depths of OEs (5 km) and LFEs (25 km) to profiles across the depth range in the crust (1 to 35 km) using an exponential function which produces nonnegative smooth variations (Fig. 4B). These depth profiles show the depth-dependent tidal and hydrological modulations predicted from the SRM-DD. The predicted tidal modulation amplitudes gradually increase with depth (Fig. 4C), consistent with the depth variations of the observed modulations of LFEs (24). While the predicted hydrological modulation amplitude first increases and then decreases with depth, the maximum is found at ~5 km (Fig. 4C). Due to the limited number of events, we are not able to resolve the depth variations of hydrological modulations with high confidence, but it can possibly be validated using enhanced catalogs with a sufficient number of events via template matching or various machine leaning methods (52, 53). In addition to the periodic seismicity modulations, we can also simulate the depth variations of peak slip velocity, stress drop, and earthquake recurrence time based on the depth profiles of stress loading rate and fluid properties (Fig. 4C). The absolute values of these predictions cannot be directly compared with observations because they are affected by the absolute value of aσ₀, whose discrepancy between different studies is not fully resolved. However, the trends of these predicted features all decrease with depth, suggesting a transition of slip behaviors from unstable to stable sliding with depth, manifested as OEs, LFEs, and fully aseismic creeping, respectively. This prediction is overall consistent with the observations, where the apparent gap between OEs and LFEs at ~15 km might be attributed to a change of fault zone rheology and/or frictional parameters at this depth that is not considered in our model (54, 55).

Overall, both the transition of slip behaviors and the variation of periodic seismicity modulations can be simultaneously reproduced by considering the depth-variations of fluid content and loading conditions, highlighting the key roles of fluids in controlling fault slip at different depths of the CSAF. The lateral transition of slip behaviors along the CSAF might also be related to these factors, which can potentially be constrained in future work focused on along-strike variation of periodic seismicity modulation using enhanced earthquake catalogs. Given the similarities in the inferred distribution of fluids and seismicity of the CSAF and global subduction zones (56), we suspect that variable pore fluid properties may also be dominant factors in the ubiquitous variations of fault slip behaviors at subduction zones. Detailed analysis, combining conventional observations (e.g., stress drops, recurrence times, and fault coupling) and frequency-dependent seismicity modulations of OEs and LFEs in subduction zones, will allow for systematically probing variable mechanical properties of subduction megathrusts.

In this study, by an integrated analysis of catalogs of deep LFEs and shallow OEs, we reveal their distinctly different sensitivities to tidal and hydrological loadings, and provide an alternative perspective to elucidate the crucial roles of dilatancy and fluid diffusion processes. Although these processes have previously been proposed as slip-stabilization mechanisms for slow slip events (57, 58), they are not yet widely recognized as universal governing mechanisms in earthquake nucleation. Moreover, the physical properties of the CSAF obtained from the frequency-dependent modulations suggest important roles of fluid content distribution in controlling slow-to-fast earthquakes. Our findings broaden the understanding of earthquake physics and encourage the incorporation of pore fluid behaviors in dynamic simulations to better predict fault slip behaviors and assess seismic hazards.

MATERIALS AND METHODS

Selection of OE and LFE catalogs

The original OE catalog is from the Northern California Earthquake Data Center (NCEDC) (59). First, we select OEs that occurred during 2000 to 2020 and within 20 km from the CSAF (denoted by C1). Spatial and temporal clusters resulting from various types of interactions comprise a large portion of the OE catalog (60), so we need to eliminate their impact on periodic modulation analysis. For tidal modulation, the aftershock durations are generally longer than the tidal periods; thus, the clusters may not bias the analysis of tidal modulation when using a catalog over a long time span (9, 38, 61). We find that the tidal modulations for the original and declustered catalogs are effectively the same [compare Fig. 2 (A and D) and fig. S7 (A and D)]. However, when analyzing periodic modulations over much longer periods (e.g., annual hydrological cycles), aftershocks can create nonuniform distributions over the loading cycles, resulting in some “fake spikes” in the phase histograms, even if the rest of the catalog follows a Poisson process (fig. S7). Therefore, the clustering effect must be properly removed before analyzing the annual periodic modulation. Although declustering does not alter the tidal modulation results (fig. S7), we use the declustered catalogs for both the tidal and hydrological modulation analysis to ensure methodological consistency and reliable detection of both modulations.

We perform the declustering of C1 using the nearest neighbor approach, which defines clusters based on the spatial and temporal distances with other events, and has been widely used in statistical seismology (60, 62–65). For each event i in the catalog, we calculate its nearest-neighbor proximity ${\mathrm{\eta }}_{\mathit{ij}}$ with every earlier event j

$${\mathrm{\eta }}_{\mathit{ij}}=T_{\mathit{ij}} R_{\mathit{ij}}$$ (1A)

$$T_{\mathit{ij}}=t_{\mathit{ij}} {10}^{-\mathit{pb}{m}_j},R_{\mathit{ij}}={r_{\mathit{ij}}}^d {10}^{-(1-p)b{m}_j}$$ (1B)

where $T_{\mathit{ij}}$ and $R_{\mathit{ij}}$ are rescaled time and spatial distances, respectively. $t_{\mathit{ij}}=t_i-t_j$ is the time difference between event pairs, $r_{\mathit{ij}}$ is either their epicentral or hypocentral distance, and $m_j$ is the magnitude of event $j$. $d$ is the fractal dimension, which is in the range of [1, 2] if $r_{\mathit{ij}}$ is the epicentral distances, and [2, 3] if $r_{\mathit{ij}}$ is the hypocentral distances. $b$ is the b value in the Gutenberg-Richter law. $p$ is the parameter describing the contributions of $T_{\mathit{ij}}$ and $R_{\mathit{ij}}$ and is generally set to be 0.5. The minimum ${\mathrm{\eta }}_i=\underset{t_j<t_i}{\text{min}}\left({\mathrm{\eta }}_{\mathit{ij}}\right)$ is the nearest-neighbor distance for event $i$. A critical distance ${\mathrm{\eta }}_0$ can be evaluated on the basis of the distribution of ${\mathrm{\eta }}_i$ using a one-dimensional Gaussian Mixture Model approach (66). Events whose ${\mathrm{\eta }}_i<{\mathrm{\eta }}_0$ are regarded as clustered events and removed in the algorithm. In this study, we use hypocentral distances with $d=2.5$ and $b=1.0$ estimated from the catalog C1. The details of the declustering are shown in fig. S1. We further select events whose distances to the CSAF are smaller than 2 km (representing on-fault events), the evaluated occurrence times are in the range of 2006 to 2015, when both hydrological and tidal loadings are available, and magnitudes are larger than the magnitude of completeness of the catalog (1.1, fig. S8). After these selections, the catalog containing 3572 events (denoted by C2) is used for the analysis in the “Periodic seismicity modulations on the CSAF.”

The original LFE catalog is from that proposed by Shelly (34), which contains millions of LFEs in 88 families from 2001 to 2016. The distributions of interevent times (the elapsed time since the previous event) in each of the LFE families generally show several groups (fig. S2). Events with short interevent times are regarded as clustered events in the family and are removed in our study. Although different families exhibit different features in the interevent time distributions (67), we use a uniform value $T_0$ = 0.1 day as a threshold value for all families for simplicity (fig. S2). The selection of $T_0$ does not substantially affect the modulation results (fig. S9). After declustering, there are 107,927 events remaining in 88 families of the LFEs catalog.

Estimation of the periodic modulation

The correlation between seismic events and periodic loadings can be estimated by the probability of the event distribution. Under a periodic perturbation, the probability density function (PDF) of the seismic events can be expressed as (2, 6)

$$\frac{P\left(\mathrm{\theta }\right)}{P_0\left(\mathrm{\theta }\right)}=1+P_{\mathrm{m}} \text{cos}(\mathrm{\theta }-\mathrm{\phi })$$ (2)

where θ are the phases of the events, which are calculated on the basis of the loading time series. We separate the loading time series into several cycles, each of which spans two local negative troughs and one positive local peak. In each loading cycle, the phases are −180°, 0°, and 180° at the time of the first trough, the peak, and the second trough, respectively. For other times in the time window, the corresponding phases linearly increase from −180° to 0° at the rising part and from 0° to 180° at the falling part. P(θ) is the PDF of the distribution of θ for all events in the catalog. P₀(θ) is the background PDF in which the events are evenly distributed in time. When the durations for positive and negative perturbations are different, P₀(θ) is uneven (68). Therefore, we estimate P₀(θ) for a large enough number of evenly distributed events and use the normalized PDF P(θ)/P₀(θ) to correct for this bias. P_m and φ represent the modulation amplitude and phase due to the periodic loading, respectively. Larger values of P_m indicate larger deviations from the uniform background distribution, indicating a stronger correlation with the periodic perturbation. φ indicates the phase shift between the peak of the seismicity and the peak of the loading time series. Positive φ means more events occur during the falling time of the loading time series, while negative phase shifts mean more events occur during the rising time of the loading time series. The optimal P_m and φ are obtained using the trust-region reflective algorithm, while their uncertainties (one SD) are estimated by bootstrapping for 2000 times.

Fitting tidal and hydrological loading histories with sinusoidal functions

To generate input periodic stresses for the synthetic catalogs that match the observed tidal and hydrological loadings, we use the analytical stresses with the same constituents as the observed loadings. After projecting tidal and hydrological loading to normal and shear stresses on the CSAF, we fit the scattered time series using a combination of several harmonic functions

$$F\left(t\right)={\displaystyle \sum _{i=1}^N}{A}_i\text{sin}(\frac{2\mathrm{\pi }t}{T_i}+{\mathrm{\phi }}_i)$$ (3)

where $F\left(t\right)$ is the analytical stress loading history; $N$ is the number of harmonic functions; $A_i$, $T_i$, and ${\mathrm{\phi }}_i$ denote the amplitude, period, and phase of the ith component, respectively. $A_i$, $T_i$, and ${\mathrm{\phi }}_i$ for normal and shear stresses of tidal and hydrological loadings are resolved by the multifrequential periodogram analysis code (table S1) (13). Because the modulations by normal stresses are much weaker than that of shear stresses (fig. S4), we only consider the shear stresses as the loading stress in the following analysis, whose peak-to-peak amplitudes are ~600 and ~200 Pa for tidal and hydrological loadings, respectively. The fitted analytical stress loadings $F\left(t\right)$ with multiple frequencies are the input stress perturbations to simulate the seismicity response when constraining physical properties on the CSAF.

Significance of the observed modulation

To evaluate the significance of the observed modulation, we compare it with the modulations of random catalogs, whose N event occurrence times are randomly selected during the observation period, where N is the number of events in the observed catalog. We generate 2000 random catalogs and calculate their respective P_m, which are then summarized using box-plot statistics. In this framework, outlier threshold is defined as Q3 + 1.5*IQR (upper bound) and Q1 − 1.5*IQR (lower bound), where Q1 and Q3 are the first (25%) and third (75%) quartiles of the samples, respectively, and IQR = Q3 − Q1 is the interquartile range. Values that are larger than the upper bound or lower than the lower bound are regarded as outliers. If the samples strictly follow a Gaussian distribution N(μ, σ²), Q3 + 1.5*IQR and Q1 − 1.5*IQR are μ + 2.698σ and μ − 2.698σ, corresponding to the 99.65th and 0.35th percentiles, respectively. In our study, the observed P_m is regarded as statistically significant, if it exceeds the upper bound outlier threshold (Q3 + 1.5*IQR) of the random catalog P_m, as the probability for the random cases to have such a large value is merely 0.35%.

Simulation of periodic seismicity modulation based on the SRM-DD

The governing equations for the slip evolutions of the slider in the SRM-DD are (69–71)

$$\dot{v}\left(t\right)=\frac{v\left(t\right)}{a\mathrm{\sigma }\left(t\right)+\mathrm{\eta }v\left(t\right)}\left\{\dot{\mathrm{\tau }}\left(t\right)-\frac{[\mathrm{\tau }\left(t\right)-\mathrm{\eta }v\left(t\right)]}{\mathrm{\sigma }\left(t\right)}\dot{\mathrm{\sigma }}\left(t\right)-\frac{b\mathrm{\sigma }\left(t\right)}{\mathrm{\theta }\left(t\right)}\dot{\mathrm{\theta }}\left(t\right)\right\}$$ (4A)

$$\dot{\mathrm{\theta }}\left(t\right)=1-\frac{v\left(t\right)\mathrm{\theta }\left(t\right)}{d_c}-\frac{\mathrm{\alpha }\mathrm{\theta }\left(t\right)}{b\mathrm{\sigma }\left(t\right)}\dot{\mathrm{\sigma }}\left(t\right)$$ (4B)

$$\dot{\mathrm{\tau }}\left(t\right)=k[v_{\mathrm{L}}\left(t\right)-v\left(t\right)]$$ (4C)

$$\dot{\mathrm{\sigma }}\left(t\right)=-\dot{p}\left(t\right)+∆\dot{\mathrm{\sigma }}\left(t\right)$$ (4D)

$$\dot{\mathrm{\Phi }}\left(t\right)=-\frac{v\left(t\right)}{d_c}\left\{\mathrm{\Phi }\left(t\right)-\text{εln}\left[\frac{v\left(t\right)}{v^{\ast }}\right]\right\}$$ (4E)

$$\dot{p}\left(t\right)=-\frac{1}{\mathrm{\beta }}\dot{\mathrm{\Phi }}\left(t\right)-\frac{p\left(t\right)}{T_{\mathrm{f}}}$$ (4F)

where $v,\mathrm{\theta },\mathrm{\tau },\mathrm{\sigma },\text{and} \mathrm{\Phi }$ are the velocity, state variable, shear stress, effective normal stress, and porosity of the slider, respectively. The initial values of $\mathrm{\sigma }$ and $\mathrm{\tau }$ are the background effective normal stress ${\mathrm{\sigma }}_0$ and background shear stress ${\mathrm{\tau }}_0$, respectively. Equation 4, A to D, are the classical quasidynamic governing equations for the rate-and-state friction (RSF) law, in which $a,b,d_c,\text{and} v^{\ast }$ are RSF parameters (70, 71), $\mathrm{\eta }$ is the radiation damping parameter partly accounting for the inertial effect (72), $k$ is the spring stiffness, $v_{\mathit{L}}\left(t\right)$ is the velocity time series of the spring which loads the slider, and $∆\dot{\mathrm{\sigma }}\left(t\right)$ is the external normal stress perturbation. $\mathrm{\alpha }$ in Eq. 4B is the Linker and Dieterich parameter, describing the change of $\mathrm{\theta }$ in response to the change of $\mathrm{\sigma }$ (73). The default value of $\mathrm{\alpha }$ is 0. Its effect on the seismicity response is discussed in the next section. Equation 4E describes the evolution of porosity in response to the sliding velocity, in which $\mathrm{\epsilon }$ is the dilatancy coefficient (69). The evolution of pore pressure is governed by the dilatancy/compaction of the pore space and the diffusion of pore fluid, described by the first and second terms on the right-hand side of Eq. 4F, respectively. $\mathrm{\beta }$ is the combined compressibility, defined as $\mathrm{\beta }=\mathrm{\Phi }({\mathrm{\beta }}_f+{\mathrm{\beta }}_{\mathrm{\varphi }})$ (69), where ${\mathrm{\beta }}_f \text{and} {\mathrm{\beta }}_{\mathrm{\varphi }}$ are the fluid and pore compressibility, respectively. The diffusion term is modeled by a ‘membrane’ model, assuming the fault is connected to an external reservoir that is sustained at constant pore pressure $p^{\infty }$ (58). $p$ in Eq. 4F denotes the pore pressure difference between the slider and $p^{\infty }$. $T_{\mathrm{f}}$ is the characteristic diffusion time in this model. Given the initial condition for a spring-slider, Eq. 4 (A and F) can be solved by a fourth-order Runge-Kutta method. Specifically, under background loading when $v_{\mathrm{L}}\left(t\right)$ is constant [$v_{\mathrm{L}}\left(t\right)=v_{\mathrm{l}}$], the simulation predicts stick-slip earthquake cycles. The nucleation time in the cycle is defined by $T_{\mathit{a}}=a{\mathrm{\sigma }}_0/{\dot{\mathrm{\tau }}}_{\mathrm{r}}$, where ${\dot{\mathrm{\tau }}}_{\mathrm{r}}=k{v}_{\mathrm{l}}$ is the background stress loading rate. The fixed model parameters are listed in table S3.

To analyze the periodic seismicity modulation in the SRM-DD, we generate synthetic catalogs by simulating the failure times of many sources with properly assigned initial conditions. We use a large enough number of spring-slider sources with identical parameters but different initial conditions to make sure the synthetic catalog covers all loading stages. The rupture time is defined as when the velocity of the source reaches a critical value (10⁻³ m/s). We first simulate an earthquake cycle under background loading [$v_{\mathrm{L}}\left(t\right)$ is constant] and then evenly sample $v,\mathrm{\theta },\mathrm{\tau },\mathrm{\sigma },\mathrm{\Phi },\text{and} p$ in time as the initial conditions for the sources. This kind of assignment of initial conditions ensures that the seismicity rate is constant under background loading. Then, we add the stress perturbations in addition to the background loading. In this study, we only add the shear stress perturbations, while the normal stress perturbations $∆\dot{\mathrm{\sigma }}\left(t\right)$ are assumed zero

$$v_{\mathrm{L}}\left(t\right)=v_{\mathrm{l}}+A\text{cos}\left(\frac{2\mathrm{\pi }t}{T}\right)$$ (5)

In Eq. 5, $v_{\mathrm{l}}$ is the background loading rate, and A and T are the amplitude and period of the periodic loading, respectively. Notably, although we do not incorporate external normal-stress perturbation [$∆\dot{\mathrm{\sigma }}\left(t\right)=0$], the induced pore pressure by dilatancy still affects the seismicity response as an internal normal-stress perturbation (Eq. 4D). We record the rupture times for the sources under this loading condition and generate the synthetic catalog, which is then used for estimating the periodic modulation using the phase histogram method described in the previous section (fig. S10). Furthermore, the seismicity response spectrum is generated by a series of simulations under stress perturbations with same $A$ = 100 Pa but different $T$ (Fig. 3). However, when constraining the physical properties of the CSAF, we use the analytical stress loading with multiple frequencies as the real periodic loadings to properly account for the nonlinear seismicity response to the periodic loadings with multiple frequencies.

Seismicity response spectrum in the SRM-DD

Much of the seismicity response spectrum in the SRM-DD is the same as in the SRM, in which the normalized modulation amplitude $P_{\mathrm{m}}/(∆\mathrm{\tau }/a{\mathrm{\sigma }}_0)$ is uniquely determined by the normalized loading period $T\mathit{/}{T}_{\mathit{a}}$ (1, 30). When the dilatancy and diffusion processes are incorporated, the interactions between pore fluid behaviors and the noninstantaneous earthquake nucleation result in a nonmonotonic spectrum of modulation amplitudes. The nonmonotonic response depends on two nondimensional parameters: $\mathrm{\epsilon }/\mathrm{\beta }{\mathrm{\sigma }}_0$, which quantifies the nondimensional effect of dilatancy, and $T_{\mathit{f}}\mathit{/}{T}_{\mathit{a}}$, which quantifies the nondimensional effect of diffusion (fig. S11). Although an analytical solution as that in the SRM is difficult to derive, we can analyze the response spectrum in some end-member cases: If $\mathrm{\epsilon }/\mathrm{\beta }{\mathrm{\sigma }}_0$ is very small, then the dilatancy effect is very weak, and the induced pore pressure change is very small. Under this condition, the diffusion process can be neglected no matter what the values of $T_{\mathit{f}}\mathit{/}{T}_{\mathit{a}}$ are, and, in this case, the corresponding response spectrum is completely the same as that in the SRM. When $\mathrm{\epsilon }/\mathrm{\beta }{\mathrm{\sigma }}_0$ is larger, the effect of dilatancy increases, resulting in smaller modulation amplitudes at high frequencies ($T\ll T_{\mathrm{a}}$) and larger modulation amplitudes at intermediate frequencies ($T\sim T_{\mathit{a}}$). The overall degree of the nonmonotonic response is stronger for larger values of $\mathrm{\epsilon }/\mathrm{\beta }{\mathrm{\sigma }}_0$ (fig. S11, A and B). However, it should be noted that $\mathrm{\epsilon }/\mathrm{\beta }{\mathrm{\sigma }}_0$ cannot be too large, because larger $\mathrm{\epsilon }/\mathrm{\beta }{\mathrm{\sigma }}_0$ values also tend to stabilize the slip behavior (69) and prevent stick-slip. On the other hand, if $T_{\mathit{f}}\mathit{/}{T}_{\mathit{a}}$ is very small, then the diffusion is very fast, corresponding to a drained condition, the change of pore pressure due to dilatancy will be fully compensated by diffusion, resulting in the same response as in the SRM without dilatancy and diffusion. In contrast, if $T_{\mathit{f}}\mathit{/}{T}_{\mathit{a}}$ is very large, then the system is undrained, and the effect of diffusion can be neglected. The only difference with SRM is the decreased modulation amplitudes at high frequencies due to the dilatancy strengthening effect. Therefore, the nonmonotonic response spectrum does not happen in both drained and undrained end-member conditions and only happens when $T_{\mathit{f}}$ is comparable to $T_{\mathit{a}}$ (fig. S11, C and D).

The Linker and Dieterich parameter $\mathrm{\alpha }$ also affects the seismicity response spectrum. Laboratory measurements of $\mathrm{\alpha }$ show a wide range from 0 to $f^{\ast }$ (i.e., the steady-state frictional parameter when $v=v^{\ast }$) (73, 74). We simulate the seismicity response spectrum under different values of $\mathrm{\alpha }$ within this range. The simulations show that the effect of nonmonotonic response decreases with increasing $\mathrm{\alpha }$. When $\mathrm{\alpha }$ is small, the nonmonotonic response is obvious. When $\mathrm{\alpha }$ increases, the nonmonotonic response becomes weaker, because the pore pressure change induced by dilatancy and diffusion is compensated by the Linker and Dieterich effect as $\mathrm{\alpha }$ approaches $f^{\ast }$ (fig. S15). In this study, we neglect the Linker and Dieterich effect by keeping $\mathrm{\alpha }=0$ (table S3) and constrain the other parameters from observations.

Supplementary Materials

This PDF file includes:

Supplementary Text 1 and 2

Figs. S1 to S15

Tables S1 to S3