## Abstract

We show a correlation between the occurrence of shallow thrust earthquakes and the occurrence of the strongest tides. The rate of earthquakes varies from the background rate by a factor of 3 with the tidal stress. The highest correlation is found when we assume a coefficient of friction of μ = 0.4 for the crust, although we see good correlation for μ between 0.2 and 0.6. Our results quantify the effect of applied stress on earthquake triggering, a key factor in understanding earthquake nucleation and cascades whereby one earthquake triggers others.

For more than a century, researchers have sought to detect the effect on the timing of earthquakes of the gravitational perturbations on Earth from the Moon and Sun (*1*). However, the tidal stresses in most locations are small, and usually it is difficult to ascertain the orientation of the fault plane, which is critical when calculating the effect of the stress variations. Earthquake-tide correlations have been observed to be small or nonexistent in normal crust (*2*–*4*); however, correlations have been shown in shallow, possibly hydrothermal or magma-related areas (*5*, *6*). Here, we take advantage of accurate accounting of ocean tides (*7*) and a large data set of earthquake focal mechanisms with fairly well known fault planes (*8*) to look for a correlation.

We used global earthquakes in the Harvard Centroid Moment Tensor (CMT) catalog (*9*). For each event, we calculated a tidal-stress time series that includes the solid Earth tide and an ocean-loading component (*7*, *10*, *11*). Solid-Earth tides induce stresses only up to 5 × 10^{3} Pa (0.05 bar), whereas in ocean basins, water loading builds stresses up to nearly 5 × 10^{4} Pa (0.5 bar). Both components must be accurately determined to fully resolve tidal influences on the initiation of earthquakes globally. We resolved tidal stresses into normal and shear stress acting on each of the two possible fault planes of the CMT earthquake focal mechanism. Shear failure under compressive stress can be described by the Coulomb criterion, in which a fault fails under a combination of shear and normal stress: τ_{c} = τ + μσ_{n}, where τ and σ_{n} are the shear and normal stresses, respectively, and μ is the coefficient of friction. In addition to examining shear and normal stress independently, we tested different values of μ (0.2, 0.4, or 0.6). For each event, we calculated the tidal phase angle (θ) between –180° and 180° (*11*); 0° phase is defined to be at the time of maximum stress that can promote failure, which is extensional for normal stress and in the direction of slip for shear stress (Fig. 1). In addition, we defined the average of the tidal stress amplitudes at the peaks just before and after each earthquake (τ_{pb} and τ_{pa}, respectively) to be the peak tidal stress τ_{p}.

We focused on a subset of shallow thrust earthquakes with depths of 0 to 40 km because these earthquakes are in regions with the largest tidal stresses; the true fault plane is better known for these events; and shallow events are under lower confining pressures, so tidal stress may be proportionally more influential. In addition, thrust and normal faults have relatively larger tidal stress amplitudes than strike-slip faults because of a larger influence of the ocean loading component. As the ocean moves back and forth in an ocean basin, the additional weight of the water acts to clamp and unclamp dipping faults. In subduction zones, ocean loading tends to be largest and low-angle thrust events are most common. The focal mechanisms for shallow thrust faults show a bimodal distribution of fault dips for fault planes 1 and 2 (Fig. 2A). The shallower-dipping fault planes (plane 1) for the 19 events with τ_{p} greater than 0.02 MPa (0.2 bar) are aligned with the local geometry of the subduction zone (Fig. 2B), so for this study we assume that to be the true fault plane.

Correlation with the tides is found for shallow-dipping thrust events, assuming μ = 0.2, 0.4, or 0.6 (Fig. 3). Schuster's test, used to find statistical significance of periodicity, indicates less than a 1% probability that the distribution is random for τ_{p} above about 0.01 MPa (0.1 bar). More events occur during the period of encouraging stress (–90° < θ < 90°) than occur during times of discouraging stress (–180° < θ < –90° or 90° < θ < 180°); the sinusoidal fit to the nonrandom distribution of events peaks at a tidal phase near θ = 0°, which is the phase of tidal stress expected to promote failure (Fig. 4A).

We also performed a binomial test of significance because Schuster's tests may overestimate significance. We estimated the probability of seeing at random the larger fraction of earthquakes we observed to occur during the half of the tidal phase period with encouraging stress, –90° θ < 90°. If we assume that events occur at random, the number of events that occur with a tidal phase between –90° and 90° should be equal to the number that occur in the other half of the tidal phase range. This simple binomial test gives a probability of >99.99% that earthquakes correlate with tides when tidal stress amplitudes are high (>0.01 MPa) (Fig. 3D). The binomial statistic is calculated assuming that just one choice of parameters was tested; the true odds of finding the correlation at random are expected to be somewhat lower than those obtained here. Tidal correlation with earthquake timing at the highest stress levels is apparent for most choices of coefficient of friction, but the most significant correlation is for μ = 0.4 (Table 1 and Fig. 3D). Assuming μ = 0.4, a minimum binomial probability of 0.0027% of fortuitous correlation (with a corresponding *P* value of 7.6 × 10^{–5} is observed for the 255 events that occur at τ_{p} > 0.003 MPa (*11*).

Events | Binomial (%) | P value (%) | N_{ex} (%) | θ_{peak} (degrees) |
---|---|---|---|---|

μ = 0 (shear) | 4.38 | 10.36 | 5.6 | -22.2 |

μ = 0.2 | 0.1439 | 0.6253 | 9.6 | -1.2 |

μ = 0.4 | 0.0032 | 0.0157 | 12.8 | 0.2 |

μ = 0.6 | 0.0942 | 0.3265 | 10.0 | 6.0 |

μ = ∞ (normal) | 0.4688 | 3.677 | 8.4 | 15.8 |

We see fluctuations in the binomial statistics (Fig. 3D). The most highly correlated 45 events are the most highly stressed, and most are shallow, with 41 events at or above 15 km depth. The next 100 events are not as highly correlated and tend to be deeper, with only 45% of the earthquakes at or above 15 km depth. We observe a recovery of the tidal correlation as more events are included. This trend suggests a depth dependence to the tidal correlation, but this trend is not statistically significant. Event depth may influence correlation with the tides because confining pressures increase with depth, so there is a comparatively larger tidal influence at shallow depths. However, tidal stress amplitudes decay with depth, so event depth and tidal stress amplitude are not independent.

On the basis of the earthquake-tide correlation observed, we have tried to estimate the tidal stress amplitude required to trigger an earthquake. We define the percent of excess events, *N*_{ex}, to be the number of additional or excess events in the tidal phase period with encouraging stress, *N*_{ex} = [*N*_{enc} – (*N*_{tot}/2)]/*N*_{tot}, where *N*_{enc} is the number of events with –90° < θ < 90° and *N*_{tot} is the total number of events. For τ_{p} greater than 0.02 MPa (0.2 bar), *N*_{ex} is 24%. This corresponds to 74% of the events (14 of 19 earthquakes) occurring in 50% of the phase time of encouraging stress (–90° < θ < 90°). Successively lower peak tidal stresses show lower tidal correlations, indicating less effective earthquake triggering (Table 2 and Fig. 4B). Standard errors, calculated using data variance, are large because of small numbers of earthquakes in each τ_{p} bin. In addition to the global data set, 27,464 strike-slip events from California were found to have roughly 1 to 2% more events during times of encouraging tidal stress of <0.004 MPa (Fig. 4B).

Data set | τ_{p} (MPa) | N | N_{ex} (%) |
---|---|---|---|

Global thrust | >0.02 | 19 | 23.7 ± 10.10 |

Global thrust | 0.01 to 0.02 | 41 | 13.4 ± 7.52 |

Global thrust | 0.004 to 0.01 | 155 | 8.7 ± 3.95 |

Global thrust | <0.004 | 1,813 | 1.2 ± 1.17 |

California strike-slip | <0.004 | 27,464 | 0.60 ± 0.30 |

A previous study using global data hinted at an earthquake-tide correlation, suggesting that reverse and normal earthquakes correlate either with the shear stress or the trace of the stress tensor J1 (*7*). In addition, a few regional studies (*5*, *12*–*14*) in areas with a large ocean-loading component of tidal stress have observed triggering of earthquakes or volcanic tremor. Most often, triggered events have been shallow, normal-type faulting along axial ridges, such as Juan de Fuca (*5*, *12*), perhaps associated with hydrothermal circulation. A recent study of seismicity in Japan showed earthquake-tide correlation in subregions, with the best correlations found in regions that experienced a large earthquake (*15*). Our study shows a statistically significant increase in triggering of globally distributed large tectonic events (*M* > 5.5) with increasing tidal stress amplitudes (Fig. 4).

Tidal stress amplitudes required to trigger earthquakes are similar to thresholds suggested for static and dynamic triggering of aftershocks of 0.003 MPa (*16*–*18*). A study of triggering after the *M*7.4 Landers event, for example, showed that aftershocks were triggered by stress increases greater than 0.01 MPa (*16*). In addition, the observed trend of increased triggering with higher imposed tidal stress can be well fit to friction theories of rate- and state-dependent friction and stress corrosion (*11*, *19*, *20*).

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/1103961/DC1

Materials and Methods

References