In this paper, we performed data analysis on earthquake data downloaded from the global seismic records website [http://earthquake.usgs.gov/earthquakes/eqarchives/epic/] for the period 1973–2010. Figure 1 shows the distribution of these earthquakes with a 1 × 1° pixel resolution. The color represents the number of earthquakes for each pixel.

Fig. 1 The distribution of earthquakes with magnitudes greater than 2.5 during 1973–2010. Pixel resolution is 1° by 1° and the color indicates the number of earthquakes for each pixel Full size image

The phase folding analysis is carried out using three selected cycles of the Sun, Earth, and Moon according to their dictates of the rotational relationship. These include the Earth tropical year cycle, the rotation of the Earth and the orbital period of the Moon around the Earth. The specific cycle value and the divided interval for each cycle are shown in Table 1. The phase t 0 is set to 0:00:00 on January 1, 1973 and the time t i is the time of the i-th earthquake occurrence. In Eq. (1), f is the reciprocal of the cycle value listed in Table 1, which is taken to be constant, so the first and second order of the derivative respective to the time are both 0.

Table 1 Period selection and interval divisions Full size table

Global phase folding analysis of earthquake distribution

We performed global seismic phase folding analysis for each of the three cycles for different earthquake magnitudes, and gave each a significance value for the phase analysis. From Fig. 2, depicting the phase distribution for the Earth’s tropical year cycle, we see that earthquakes of magnitude 2.5–3.0 occur more frequently around the time of the autumnal equinox than the vernal equinox. In contrast, earthquakes of magnitude 3.0–5.0 take on opposite characteristics to those of earthquakes of magnitude 2.5–3.0 and they tend to be distributed more frequently near the vernal equinox than the autumnal equinox. The more destructive earthquakes of magnitude 5.0–6.0 appear to be significantly impacted by solar activity. The phase of the seismic frequency enhancement is concentrated on or near the winter solstice, and reaches its minimum during the summer solstice. From Fig. 2, we can see that earthquakes with a magnitude greater than 6.0 are not affected by the solar year, with very small χ2/Ndf (Ndf indicates the number of degrees of freedom in seismic cycle distribution) values reflecting the characteristics of cycle distribution.

Fig. 2 The phase-folding distribution of earthquakes with magnitudes greater than 2.5 for the Earth tropical year cycle, because the abscissa covering the 365 days in 1 year is related to the Earth tropical year cycle. The four red dashed lines fromleft to right in the plots denote the position of vernal equinox, summer solstice, autumnal equinox, and winter solstice, respectively Full size image

Figure 3 shows the phase folding analysis associated with the lunar cycle. The χ2/Ndf values for all earthquakes are all <5 suggesting that the lunar cycle plays only a minor role in the timing of major earthquakes.

Fig. 3 The phase-folding distribution of earthquakes with magnitudes greater than 2.5 for the lunar cycle, because the abscissa covering the 30 days in 1 month is related to the lunar cycle Full size image

Figure 4 shows that earthquakes with magnitudes of less than 5.0 are more likely to be affected by the Earth’s rotation. Earthquakes with magnitudes of 2.5–3 have inverse distribution characteristics to earthquakes with magnitudes of 3.0–5.0. Earthquakes of magnitude 2.5–3.0 occur more frequently during the day with maxima at noon and midnight, local time. Earthquakes with magnitudes between 3.0 and 5.0 occur more frequently at night, reaching a maximum at 01:00 local time. The timing of earthquakes with magnitudes greater than 5.0 is not obviously affected by the rotation of the Earth.

Fig. 4 The phase-folding distribution of earthquakes with magnitudes greater than 2.5 for the rotation of the Earth, because the abscissa covering the 24 h in 1 day is related to the rotation of the Earth Full size image

In Figs. 2, 3, 4, the earthquakes of different magnitude from statistical results take on obviously non-uniform timing distribution. Because the final behavior of lithosphere movement are determined by many kinds of factors together, including the lithosphere structure, gravity, the effect of the magnetic field, solar activity, and so on. Those factors will provide different effects to the earthquakes with different magnitudes. So it is possible that the rotation of the Sun plays different role in impacting on seismic occurrence times for the earthquakes with different magnitude, although the precise reason still need to be obtained by further study.

Because of their destructive power, people always pay more attention to earthquakes that have magnitudes above 5.0. According to the above figures that compare the phase folding analysis for each of the earthquakes with magnitudes greater than 5.0, there appears to be a significant seismological non-uniformity in the Earth’s tropical year cycle. The corresponding χ2/Ndf values are up to 23.35 for earthquakes with magnitudes between 5 and 6. This indicates that the Earth’s rotation around the Sun may, in some way, affect the occurrence time of earthquakes, making them accumulate in the end of each year. In total, there are 43,889 earthquakes categorized as magnitude 5.0–6.0, of which the distribution of 5,236 seem to be modulated by the rotation of the Sun. As a result, the rotation of the Sun plays a role in impacting the timing of approximately 12 % of earthquakes with magnitudes between 5.0 and 6.0. On the other hand, the timing of seismic events associated with the lunar cycle and the Earth’s rotation follow a more uniform distribution.

Local phase folding analysis of earthquakes

The local properties of earthquakes and the internal structure of the Earth play a crucial role in the occurrence of earthquakes. So, in the following section, we performed a local phase folding analysis of earthquakes with magnitudes greater than 5.0 because these have the greatest destructive power. Longitude is divided into 18 sections, each with 20°, and latitude is divided into four intervals, each with 45°, forming a grid of 72 regions of 20 by 45°. The phase analysis is carried out for a different cycle length for each individual region. In accordance with the division intervals listed in Table 1, we performed phase folding analysis on each of the 72 regions. Each of the resulting χ2/Ndf values for these regions are shown in Tables 2, 3, 4.

Table 2 χ2/Ndf values for each region corresponding to the solar cycle Full size table

Table 3 χ2/Ndf values for each region corresponding to the rotation of the Earth Full size table

Table 4 χ2/Ndf values for each region corresponding to the lunar orbit Full size table

In order to study the phase distribution characteristics with values of χ2/Ndf exceeding 5, we list the phase distribution of these regions for different cycles in Figs. 5, 6, 7.

Fig. 5 The phase distribution of earthquakes with magnitudes greater than 5 with corresponding χ2/Ndf value exceeding 5 for the Earth’s tropical year cycle. The corresponding region in each diagram is as follows: a 45°N–90°N, 40°E–60°E, b 45°N–90°N, 140°E–160°E, c 45°N–90°N, 160°E–180°, d 45°N–90°N, 160°W–180°, e 0°–45°N, 80°E–100°E, f 0°–45°N, 140°E–160°E, g 0°–45°S, 100°E–120°E, h 0°–45°S, 140°E–160°E, i 0°–45°S, 160°E–180°, j 0°–45°S, 160°W–180°, k 0°–45°S, 60°W–80°W Full size image

Fig. 6 The phase distribution of earthquakes with magnitudes greater than 5 with corresponding χ2/Ndf values also exceeding 5 for the lunar cycle. The corresponding region in each diagram is as follows: a 45°N–90°N, 60°E–80°E, b 0°–45°S, 120°W–140°W Full size image

Fig. 7 The phase distribution of earthquakes with magnitudes greater than 5.0 with corresponding χ2/Ndf values exceeding 5 for the Earth’s rotation. The corresponding region in each diagram is as follows: a 0°–45°N, 80°E–100°E, b 0°–45°S, 60°W–80°W Full size image

From Table 2 and Fig. 5, for the 72 regions analyzed, we can see that there are 11 regions with a significant non-uniform phase distribution (χ2/Ndf exceeding 5) for the timing of earthquakes on the Earth’s tropical year cycle. Selecting a specific 4-month (October to January) time interval, 8 regions have a significant non-uniform phase distribution within this time interval. Assuming that the timing of earthquakes is evenly distributed, the probability that regions with a significant non-uniform phase distribution have earthquakes that occur within this time interval should be 1/3 (4/12), but instead the probability value is 5.59 × 10−3 C 11 8 C 3 1 C 2 1 ( C 3 1 ) 11 . This suggests that, in addition to the random seismological mechanisms operating in the Earth’s tropical year cycle, there must also be another mechanism related to the Earth’s rotation around the Sun. This mechanism must exist in order to trigger earthquakes and make them occur specifically during the winter (October to January). In addition, these areas seem to be geographically isolated, with 7 of the 11 regions located on the western side of the central Pacific Plate.

As can be seen from Table 3 and Fig. 7, the non-uniform phase distribution of earthquakes which are affected by the rotation of the Earth lies in two regions (0°–45°S, 120°–140°W), (45°–90°N, 60°–80°E), where the ratio of χ2/Ndf is much more than 5 and the strong non-uniform feature is concentrated in the afternoon and before dawn for a 24-h period. Meanwhile, it can be seen from Fig. 6 and Table 4, for the lunar cycle, there is also a non-uniform phase distribution with a ratio of χ2/Ndf exceeding 5 in two regions: (0°–45°N, 80°–100°E), (0°–45°S, 60°–80°W). The significantly uneven distribution of these earthquakes cycles implies that the underlying mechanism may be related to the rotational dynamics of the Sun, Earth, and Moon, which may play different roles in each of the geographic fault structures within the Earth.