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Seasonality Variability and Periodicities for Ultra-Deep Earthquakes Worldwide

Our last study found that deepest-depth earthquakes with larger magnitudes (M6 or above) showed seasonality, which is dependent on the area searched. The main results indicated strong evidence that the causes for the delays in enhancements along the period investigated were due to the tectonics also, not only the season. Therefore, if the inquiry was about an area in Northern Hemisphere, the season in which the increases occurred is different than the season in the Southern Hemisphere. Also, higher latitudes in the Northern Hemisphere or around the Equator, displayed seasonality similarly where the tremors appear to increase during the Spring and Summer. This did not happen to the Southern Hemisphere where disturbances and anomalies occurred without showing much connection to the seasons in the analyzed period. However, some of the regions presented periodicities independent from the seasons.


Introduction
In the first study [1], we examined deep and ultra-deep earthquakes (DQ and UDQ) worldwide, during the period 1996-2017. This study showed that UDQ in South America happened within the continent instead of the ocean, as often happens in other areas. It is notable that these DQ and UDQ were disturbed with periods of intense enhancements during some of the years observed. It could be attributed to the seasons, or not, and is the focus of our next investigations. The second paper [2] analyzed the period 1999-2017 while considering four different locations and only UDQs. The outcomes identified a slight rise during Spring or Fall. Philippines enhancements only occur during the Summer.
The Southern Hemisphere exhibited increases also during the Spring and Fall; however, South America did not display a reliable variability. Our conclusions were that it is necessary to isolate more regions with deepest depths occurrences during a larger period to have a better idea of the history.
We also decided to verify a former paper [3] to confirm if only greater magnitudes show seasonality in those subduction zones. The last study [4] followed the initial conditions of five different areas, two in the Northern Hemisphere, and three in the Southern Hemisphere, earthquakes of magnitude M ≥ 6, and at depths of ≥500 km. It is of paramount importance to isolate these depths to understand if the disturbances are due by seasons or if any other parameter could relate to variations within the Earth. Evidence was found for earthquakes in the Northern Hemisphere occurring in the Summer and in the Southern Hemisphere they happened in the Spring. Nevertheless, these results are specific and could not be generalized, for example, with magnitudes M ≥ 2.5. Therefore, the last step is considered in this paper and will take all the possible magnitudes in the deepest depths to check if it would change the formerly completed evaluation. The initial conditions will be depths of ≥500 km, the same five locations formerly studied, magnitudes of M ≥ 2.5, during the period 1987-2017, which is a reasonable historic range for these events. The magnitude parameter starting at the lower values will allow for a better possibility of determining any connection between the subduction zones considered and the seasons.
The results around the Circum Pacific area showed that Northern Hemisphere regions had similar reactions for the seasons, as expected. The Southern Hemisphere also had feedback, but the enhancements dependencies appeared to be beyond the seasons.
The next sections will describe our statistical, mathematical model and how it is used to find the relationship between seasons and enhancement of events in the five chosen areas.

Construction of Plots and Tables
The dataset is comprised of earthquakes recorded during the period mentioned.
The sum of data values was y 1 = data recorded in the chosen year level D ≥ 500 km for the depths in the region X (A, B, C,D,E) where n = 1987, •••, 2017, and range n' is 30 years.
The mean is defined as If n Y M ′ > , the value is higher than one during that year and it means the number of earthquakes exceeds the average. On the other hand, if n Y M ′ < , the number will be less than one during that year, indicating that the number of earthquakes was below the average. It is also possible to have a result of zero (was not common, most of the dataset), which would mean there was no available data.

Coordinates and Seasons for the Regions Examined
Calculating variance from annual averages will help to define and explain the locations where ultra-deep earthquakes (UDQ) occur. The coordinates to construct our mathematical models are displayed in Table 1.
The coordinates make a subdivision of areas where UDQ happen. Although it is clearly defined at the Northern and Southern Hemisphere, it is not possible to comprise all the locations with deepest earthquakes within these coordinates.
Regardless, the five places give a fair idea of a possible seasonality.  studies is based on the history of data, but also depending on how precise the data in the records is. Next, we divided the data into two parts, two areas located at the Northern Hemisphere, and the three others in the Southern Hemisphere.

North Pacific Variations Monthly and Seasonal
The next Table 3 and Table 4, are the results at the North Pacific area. Table 3 is constructed with the data from the catalog [5].
Using Table 3, we constructed Figure 1, which shows the monthly development of earthquakes. From the figure, we can notice the enhancement of events happened in May (end of Spring), July, and August (Summer), meaning 33% of the performance in the area. Table 4 and Figure 2 are constructed by dividing the past results into the four seasons. We collected the highest monthly average for each season, as displayed in Table 4, to build Figure 2. Figure 2 demonstrates May (Spring) and July (Summer) with 60% of events. Therefore, the outcome at the North Pacific shows most of the enhancement of earthquakes start at the end of Spring, covering the Summer, and are fewer in the Winter and Fall.

Philippines Variations Monthly and Seasonal
The next two tables, Table 5 and  Using Figure 3, we observe that during May (Spring) and July (Summer) in this region the number of events activities is enhanced by 24% while other months had smoother occurrences below 10%. Therefore, the enhancement of earthquakes happens to the same seasons as in the North Pacific.
The next table,     for the 31% of total activity for the year, followed closely by Spring. Table 6 is plotted in Figure 4. Figure 4 shows that May (Spring) and July (Summer) are responsible for 56% of the earthquake events per year; therefore, we conclude that the area at the North Circum Pacific presented the highest seasonality enhancement during the Spring and Summer.
Next, we analyze the Southern Hemisphere and some of the chosen regions, such as those identified in Table 2.       Table 7. The average of earthquakes in Indonesia is calculated by month; zero means during that month and year no events were recorded (it is more frequent than in Northern). The number one means there was on average amount, and only those values above 1.5 are considered an enhancement. There were also anomalies in several years; the biggest occurred in 1996, with nine times the average in July.

South Pacific (Indonesia) Region
Year Consulting Table 7, only in1996 were July events were followed with an increase in August. It is interesting that in Indonesia six months have an increase in earthquakes independent of the season. These months were March, May (Fall), July (Winter), September, October (Spring) and December (Summer), as we see in Figure 5.
Thus, South Pacific (Indonesia area) results showed that UDQ independent of the magnitude value did not display seasonality such as that seen in the North Hemisphere. Table 8 and Figure 6 also presented a smoother difference between seasons although Winter has a percentage just 1% higher than the other seasons.
This could be easily explained by the anomalies observed during the three months of the Winter (June through August), in several years examined.

South America Region
The deep depth earthquakes in South America usually do not have many events in a year. If you observe Table 9,  There are anomalies in Table 9 Research   Table 11. The last region analyzed is Fiji, and the table is constructed in the same statistical manner as the previous tables. This location showed a continue number of earthquakes for all months, with small gaps as zero.

Fiji and Vanuatu Region
Year    It is possible to indicate that the activity on the area has a slight increase after August, which is during Winter. Table 12 shows the Fiji seasonality during the year.
Figure 9 was constructed from Table 12 all seasons, but Summer had two

Discussion of Results
The results concluded that the areas examined at the Northern Hemisphere had is also a little closer to the equator than the areas considered on the east side.
Consequently, UDQ seasonality did not look to be ubiquitous; instead, it applies more to the higher latitudes or most locations nearest to the continents.

Conclusions
The final conclusions for this UDQ worldwide study showed, for the initial conditions with magnitudes of M ≥ 2.5, there is evidence for seasonality at the Northwest Pacific, less evidence for the Southwest Pacific, and none for the Southeast Pacific. Such results cannot be considered final or generalized for earthquake seasonality around the globe. At this point, there are many restrictions in our studies.
It is essential to make other parameters more flexible in our calculations to enable a better statistical study. For example, considering different depths will