Tidal Triggering Of

Analysis of the tidal stress tensor at the time of moderate to large earthquakes fails to confirm an earlier hypothesis that the origin times of shallow dip-slip earthquakes correlate with solid-earth tidal shear stress, Furthermore, no correlation is seen for either tidal shear stress or tidal normal-to-the-fault compressive stress with shallow strike-slip earthquakes or with deep earthquakes. INTRODUCTION In an earlier s tudy (Heaton, 1975), I repor ted on the results of an investigation tha t suggested a correlat ion between the origin t imes of modera te to large shallow dip-slip ear thquakes and tidal shear stress as resolved into a coordinate frame defined by the slip vectors of individual earthquakes. In tha t investigation, I computed the solid ear th tidal shear stress for 107 ear thquakes for which the focal mechanisms were known. Although no significant tidal correlat ion was found for the entire data set, which contained ear thquakes of all depths and mechanisms, a fairly striking correlat ion was seen for those 34 ear thquakes tha t were classified as shallow dip-slip earthquakes. Because of tha t result, I concluded tha t there is good reason to believe tha t larger shallow dip-slip ear thquakes are tidally triggered. In order to test the validity of tha t hypothesis, in this s tudy I calculate tidal stress histories for 222 ear thquakes tha t were not considered in the previous study. Of these, 68 are classified as shallow dip-slip. If the previous hypothesis is to be considered physically meaningful, then a similar correlat ion between tidal shear stress his tory and the origin t imes of ear thquakes should be seen for this new data set. The purpose of this study, then, is to confirm or reject the hypothesis tha t the origin t imes of shallow dip-slip ear thquakes correlate with tidal shear stress. As geophysical problems go, the problem solved in this paper is very well posed. T h a t is, there are very few subjective judgments which must be made. Both the way in which the data are chosen and the me thod of statistical analysis are defined by the earlier study. Having well-defined rules is essential if statistics are to have meaning. Fur thermore , the statistics are invalid if results are examined before deciding whether to play the game. This is a common problem with many ear thquake predict ion statistical studies and one which I seem to have poorly unders tood in my earlier s tudy (Heaton, 1975). However, it seems clear tha t professional casinos would not allow their pat rons to make this same mistake. EARTHQUAKES AND TIDES In this study, a very simple ear thquake tidal triggering mechanism for ear thquakes is tested. T h a t is, do ear thquakes occur preferential ly at t imes when solid ear th tides increase the shear stress on faults? In order to answer this question, the solid body tidal stress is computed as a function of t ime and then ro ta ted into the coordinate frame which is defined by the ear thquake fault plane and the slip vector. In this way, the tidal shear stress tha t can be considered to be sympathet ic to failure can be plot ted as a function of time. The coordinate frame tha t is used in this s tudy is i l lustrated in Figure 1. The procedure used for calculating the solid ear th tidal stress is described by Hea ton (1975) and identical computer codes are used in bo th 2181 2182 THOMAS H. H E A T O N these studies. In addition to calculating the shear stress, the normal-to-the-fault compressive tidal stress, and the hydrostatic stress (Tll + z22 + T33)/3 are also computed for each earthquake. The fault plane and slip vector of each earthquake are defined from published source studies. Due to the symmetry of the stress tensor, either of the two complementary fault planes and slip vectors that are obtained from focal-mechanism studies can be used to specify the coordinate frame into which the tidal stress tensor is rotated. The effects of oceanic tides on crustal stress are ignored in this study. This is not to imply that oceanic tides are unimportant. On the contrary, oceanic tidal stress can cause significant pertubations in the phase and amplitude of crustal tidal stress and in some instances may dominate over the solid earth tidal stress (Beaumont and Berger, 1975). Unfortunately, computation of tidal stress due to oceanic loading is quite difficult and beyond the scope of this study. Because of the statistical nature of this problem, one may argue that inclusion of the contribution of oceanic tides is FIG. 1. Coordinate sy s t em used in this report. Fau l t str ike is defined as clockwise f rom north. Dip is posit ive for a faul t s tr iking no r th and dipping east. Rake is 0 ° for left-lateral and posit ive 90 ° for thrus t . el is al ined parallel to t he slip vector and ea is perpendicular to the fault. not crucial for this test of tidal triggering. Nevertheless, it is clear that modeling the contribution of oceanic loading is desirable. CHOICE OF DATA SET The purpose of this study is to determine whether the origin times of earthquakes depend upon tidal stress. Thus, it seems clear that only earthquakes whose origin time is independent of other obvious factors should be chosen. Therefore, an attempt was made to include only earthquakes that were main shocks and that were not preceded by obvious large foreshock activity. Furthermore, the fault plane and slip vector for each earthquake should be known and well constrained. These parameters were obtained from either observed surface rupture or seismically determined faultplane solutions. Because of the free-surface boundary condition, T31 (the shear stress sympathetic to failure) for dip-slip earthquakes on vertical faults is always nearly zero. The polarity of the computed tidal shear stress for dip-slip earthquakes on nearly vertical faults is thus sensitive to minor errors in the fault-plane solution. Therefore, earthquakes with dip-slip motion on near-vertical faults are excluded. Earthquakes that appear to meet the specifications listed above were rather ranTIDAL TRIGGERING OF EARTHQUAKES 2183 domly chosen from a variety of sources. The earthquakes were then classified according to focal depth and mechanism. They are listed in chronological order in Tables 1 to 3. Entries with an asterisk represent earthquakes that were considered in my earlier study (Heaton, 1975). Earthquakes with slip angles of less than 30 ° from horizontal are classified as strike-slip and earthquakes with depths greater than 30 km are considered as deep. However, since a focal depth of 33 km is often used as a default value in published catalogs, some of the earthquakes that are called deep may have actually been significantly shallower. To protect against systematic error due to my own predjudices (i.e., a desire to repeat my previous results), the decisions were made about earthquake parameters before any tides were computed and the decisions were final. DATA ANALYSIS The data analysis in this study is identical to that in my earlier study (Heaton, 1975). Tidal stresses are plotted as a function of time for each earthquake. A phase is then assigned using a linear scale (with time) from 0 ° to 360 °, where 0 ° and 360 ° are defined by the times of tidal stress maxima immediately before and after the earthquake, respectively (see Figure 2). These phases are then plotted on rose diagrams. If earthquake origin times and tidal stress are independent, then the phases will appear uniformly distributed about the rose diagram. Clustering of the phases on one side of a rose diagram indicates a possible relationship between tides and earthquakes. The statistical significance of clustering is evaluated by a clever and simple method developed by Rayleigh (1919). Consider a random walk in two dimensions (see Figure 3). Let each earthquake phase ~i represent a unit step in the ~i direction. If the magnitude of the vector sum of m unit two-dimensional vectors (1, ~i) is denoted by R, then the probability PR that a random set of m phases will produce a vector sum whose magnitude exceeds R is approximately equal to exp(-R2/m). This approximation is sufficient when m is larger than 10. Thus, the smaller is PR, the greater becomes our confidence in tidal triggering. RESULTS The results of this study are summarized in Tables 1 to 3 and Figures 4 and 5. Phases of the hydrostatic stress, normal-to-the-fault compressive stress, and shear stress are given for each earthquake. Zero degrees phase denotes either maximum tensile or maximum shear stress. Figure 5, a and b, shows no apparent correlation between earthquake origin times and tidal shear stress or normal-to-the-fault compressive stress for the entire data set (328 earthquakes). The same conclusion is reached when only those earthquakes deeper than 30 km are considered (Figure 5, c and d). If the data set is restricted to shallow (depth < 30 km) strike-slip (slip vector < 30 ° from horizontal) earthquakes, then no apparent correlation can be seen for either shear stress or normal-to-the-fault compressive stress (Figure 5, a and b). All of the above results are compatible with my earlier study (Heaton, 1975). In that earlier study, however, there was a rather striking correlation between the origin times and tidal shear stress seen for shallow dip-slip earthquakes (34 events). An additional 68 shallow dip-slip earthquakes are investigated in this study. The phases of the tidal shear stress for those new earthquakes are shown in Figure 5c; no correlation can be seen. The phases of the tidal shear stress for shallow dip-slip earthquakes, and the combined data set are shown in Figure 5d; once again, the correlation is not statistically significant. Therefore, I conclude that the previously 2184 T H O M A S H . H E A T O N