The dataset is comprised of earthquakes recorded during the period mentioned. The sum of data values was y₁ = data recorded in the chosen year level D ≥ 500 km for the depths in the region X (A, B, C,D,E)

∑ n 0 n n Y n = y n 0 + y n 1 + ⋯ + y n n (1)

where n = 1987, ∙∙∙, 2017, and range n’ is 30 years.

The mean is defined as

Mean = M = ∑ n 0 n n Y n 30 (2)

Datavalue − mean = Y ′ n (3)

If Y ′ n > 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 Y ′ n < 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.

Calculating variance from annual averages will help to define and explain the locations where ultra-deep earthquakes (UDQ) occur. Table 1 lists the boundary coordinates of each region studied. Our new step for UDQ is to consider the following parameters: depths of ≥500 km, magnitudes M ≥ 2.5, and subduction locations defined by coordinates. Two of these locations are in the Northern Hemisphere, and the three others are in the Southern Hemisphere. They are North Pacific and Philippines for the Northern Hemisphere, South Pacific (Indonesia), South America, and Fiji for the Southern Hemisphere.

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.

Table 2 establishes the seasons worldwide.

With Table 1 and Table 2, is constructed the mathematical model for the five different regions by season. The data are from catalogs [5] and [6] for thirty years. Now it is possible to work with our mathematical model to construct tables for each region, studied around the circum Pacific, following the requests as M ≥ 2.5, depths of ≥500 km, and the recorded variations of earthquakes in each location defined in Table 1. It is important to remind ourselves that we are searching for data varying monthly during the period. Reliability in statistical 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.

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.

The next two tables, Table 5 and Table 6, refer to the Philippines. The first table refers to the average data as the former one, listed monthly. The second table represents the Philippines’ seasons.

Table 5 created a percentage plot of monthly variations in the Philippines. 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, Table 6, presented Summer in the Philippines as responsible

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 shows the monthly average distribution of earthquakes within the initial

conditions formerly defined. South Hemisphere, presented months with a lack of earthquakes as indicated with zero, in higher number than Northern, within the average indicated as one, and enhancement months, when only the values above 1.5 were considered for results. We found anomalies higher than those at the North Hemisphere, which means several years presented one or two months with common occurrences above than normal. In Indonesia, it was highest in 1996 (July), in 2004 (January and June), and in 2015 (February). Those anomalies are not involved in our calculations as particularly crucial for the seasonality of the location and had little influence on the estimates.

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.

The deep depth earthquakes in South America usually do not have many events in a year. If you observe Table 9, there are inumerous gaps, with zero occurrences. It displayed a very small number of occasions as 1990 when almost all months presented zero events. However, another explanation it would be the events from South America are not recorded as in other regions around the globe. Tough, with fewer occurrences was possible to construct a solid table in the period we choose.

There are anomalies in Table 9 occurring mostly during the two years, in November and December of 2015, and the following year in January, October, and November. Statistically, the year 2015 will introduce instability in the period researched and cannot be considered in our study. The discrepancy in November could be associated with other sources besides seasonality. Figure 7 displays the South America monthly variation without the anomaly number, as the entire year of 2015 was taken out. The increases occur in March (Fall) and June (Winter).

Table 10 shows the seasons enhancements and the yearly percentage. The entire year presented for the period shows an increase of 40% in November (Spring) and Fall, and Winter shows a smaller evolution rate. This abnormal activity in November is due to the sudden incongruity in 2015 perhaps caused by a reason other than seasons.

Figure 8 shows the percentages of events by season, apparently during Fall and Winter the events increase in South America. During Spring, it is the lowest number of earthquakes in the region.

Table 11 illustrates the enhancement of earthquake events in this region showed, during the first seven months of the year, there is only 8% of activity and after July it increases to 9% completing the last five months; therefore, increasing of seasonality in the area will be between August to December. Those months perform 45% of all activity.

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

months with the same number of average earthquakes. The seasons did not affect enhancements or decreases during the entire year. Although some variances could be found when we considered earthquakes above magnitude M6 this present search, showed no distinctions among the seasons.

The result is comparable with other areas, such as South America; however, in South America, the number of occurrences is the lowest at the Southern Hemisphere and a few months during several years had a lack of records in the area, such as for April (Fall), August (Winter), and September (Spring).

Yet, it seems the east side of Circum Pacific had unusual activity. In the Northern Hemisphere there were no deep-depth occurrences, in the Southern Hemisphere it happens in traplate as it happens in South America, and, further, around Fiji and Vanuatu. It seems most of the activity is in the West Pacific, in both hemispheres. Seasonality appears to be in the cases studied in this paper, restricted to the western side of the Pacific, at subduction zones.