Evaluation of Return Period and Occurrence Probability of the Maximum Magnitude Earthquakes in Iraq and Surroundings

It has long been clear that earthquake prediction is important from both social and economic perspectives; therefore, the practical objective of today’s earthquake seismology researchers is an effective earthquake prediction program. The purpose of this study is to estimate earthquake probabilities and return periods using an updated earthquake catalogue (1900-2019) for Iraq and its surroundings. Weibull’s formula and inverse Weibull’s formula were employed to calculate the period of return and the occurrence probability of the maximum magnitude earthquake. The return period for earthquakes magnitudes 5 and 7Mw was 1.1 and 10.54 years, respectively, while the occurrence probability was 93.79% and 9.5%, respectively. The greatest magnitude is 7.7, with a 121-year return period and likelihood of approximately 0.82%. The probability of exceedance increased as the time period increased. The return duration was greater for earthquakes of higher magnitudes.


Introduction
Earthquakes that affect humans and their environments are among the worst types of natural catastrophes.It has long been clear that earthquake prediction is important from both social and economic perspectives; therefore, the practical objective of today's earthquake seismology researchers is an effective earthquake prediction program [1].The difficulty of earthquake prediction in seismology has long attracted the interest of both the scientific community and general public [2].Predicting earthquakes has been highly challenging for a long time [3].The difficulty in predicting earthquakes is due to several reasons: (1) it is extremely difficult to predict the time and size of seismic events because of the complex interactions that occur between tectonic plates, faults, and other geological factors [4]; (2) the lack of substantial and long-term data is a major obstacle to earthquake prediction; and (3) large earthquakes frequently occur at long intervals (hundreds to thousands of years) [5], making it challenging to detect trends and patterns over a long period of time [6].In addition, traditional prediction techniques based on empirical (physical or statistical) models frequently oversimplify and are flawed when used for real-world events [7].Earthquake prediction is one of the many scientific subjects that has benefited from the recent rapid progress in artificial intelligence (AI) [2] .
There are three types of earthquake predicting models in seismology [8], the first is a statistical probability forecasting model based on the Gutenberg-Richter (GR) relationship [9].Physical prediction models, which are divided into two categories, constitute the second category.The first is predicated on the intricately observed space-time patterns of earthquake behaviour.The second is predicated on seismic quiescence that occurs before major quakes [10].The third type is a hybrid model that combines statistical probability forecasting models with physical earthquake prediction models [11], [12].Many investigations have been carried out to evolve reliable estimates of the likelihood, magnitude, and return period relationships in light of the widespread occurrence of earthquakes.When applying probabilistic seismic hazard assessment (PSHA) models to forecast earthquakes, there are specific requirements for the geographic data of the area and earthquakes [8].To reliably predict the occurrence of earthquakes in the future, a PSHA model is based on in-depth knowledge of the mechanism of earthquakes [13].The main benefit of PSHA is its ability to integrate all types of seismicity: time, space, and ground motion to produce a cumulative exceedance probability that considers the relative frequency of different quakes and ground motion features [14] .
Although it is difficult to provide an exact date for a predicted earthquake, it is feasible to estimate its likelihood with a certain degree of inaccuracy.Probability distribution functions are important for estimating earthquake hazards [15].A method for applying a straightforward point process approach to various characteristics of recorded seismicity is the statistical modeling of earthquakes [16], [17].It is possible to forecast the long-term process of earthquake production at a specific location by using best-fit statistical models [18].To compute conditional probabilistic time-dependent seismic renewal models for future earthquakes, statistical distributions of earthquakes such as Gumbel, Gaussian, Lognormal, Gamma, and Weibull were used [19], [20].To predict future earthquakes and perform a probabilistic study of seismic hazards, it is crucial to choose a distribution model that best fits the data for a specific location [21].
In Iraq, Al-Abbasi and Fahmi (1985) used the earthquake catalogue for the period 1905-1982 to determine the earthquake maximum magnitude, return period, and occurrence probability using the Gumbel statistical distribution model [35].Ammer et al. (2004) used the earthquake catalogue for the period 1900-2000 to assess the maximum magnitude and recurrence periods of moderate and large earthquakes using the Gumbel statistical distribution model and the Gutenberg-Richter relation [36].
The purpose of this study is to estimate earthquake probabilities and return periods using an updated earthquake catalogue (1900-2019) for Iraq and surroundings .

Seismicity of Iraq
Iraq is located in the north eastern region of the Arabian Plate, close to the convergence of the Eurasian and Arabian plates.While the eastern and northern regions of Iraq, which lie near the convergence borders of the Arabian and Eurasian plates, are subject to significant seismic activity, other regions of the country, which are farther from the plate boundary, are only subject to weak seismic activity [37].The seismicity of Iraq has been studied by many researchers (for example, [38], [39], [35], [40], [41], [42], [43], [44]; [45], [46], [47], [48] [49], [50], [51].The general trend of the BTC-Zagros fold-thrust belt is closely related to the seismicity of Iraq [44]. Iraq is divided tectonically into three regions: the Bitlis-Zagros Fold and Thrust Belt, the Mesopotamia Foredeep, and the Inner (stable) Arabian platform (Figure 1) [52].The Mesopotamia Foredeep is divided into the Al-Jazira Plain and the Mesopotamia Plain.The Bitlis-Zagros Fold and Thrust Belt and the Mesopotamia Foredeep are classified as Outer (unstable) Arabian platform.Frequent earthquake activity is a feature of the Bitlis-Zagros Fold and Thrust Belt of the Alpine Orogeny [53], [54], [37].
The Lower Zab and Diyala River faults are two examples of active NE-SW trending (transverse) faults in the Bitlis-Zagros fold and thrust belt.Listric (longitudinal) faults running parallel to the fold axes are also active in this region [37].The Badra-Amarah fault, which runs along the Iraqi-Iranian border and is thought to be the most seismically active fault in Iraq, the Euphrates fault, the Hummar fault (north of Basra), the Al-Refaee fault, and the Kut fault are all seismically active faults in the Mesopotamian Foredeep [37].A recent study of the seismicity of the Western Desert and its surrounding areas, which represent an important part of the Inner (stable) Arabian Platform, showed that the region was exposed to earthquakes ranging in size from 2 to 3.5, during the period from 1900 to 2017 [55].The epicenters were grouped into five seismic zones.A causal association may exist between the seismic activity in the research region and zones of weakness and/or stress condensation at the fault junctions.While there are faults in the Inner Arabian Platform, they have undergone far less recent deformation and show less evidence of Quaternary activity [37].Local deformation contributes to the seismicity of a stable shelf [56].Based on the information gathered from the International Seismological Center (ISC) by [49], Figure 1 shows the seismicity map of Iraq for 1900-2019 .

Methodology
In the current study, extreme (maximum or minimum) value analysis was used to calculate the probability of occurrence and return period of maximum magnitude earthquakes.The statistical study of unusual events is the focus of extreme value theory, which is concerned with the statistical laws of the extreme values of a random variable.This method requires basic computations and can efficiently characterize the tail properties of the data.Extreme value theory is an essential tool in the study of natural catastrophes [8].Most severe event analyses focus on the yearly distribution of the lowest or largest values at a particular location [14].The periods of return and earthquake occurrence probability were calculated using Weibull's formula [57].It is a simple method and is still employed by the U.S. Geological Survey and other researchers.The average recurrence interval over a long period is represented by the return period of an earthquake, which is a statistical measurement [25].The procedures listed below were used to determine the return period: (1) acquiring information over time on the frequency of earthquakes of a certain magnitude in a particular region, (2) sorting the earthquakes by magnitude in decreasing order, and (3) calculating the return period of a specific magnitude using Weibull's formula : (1) Where T = return period (year) m= event rank (in reverse order) n = number of earthquakes in the earthquake catalog Using Weibull's formula, the yearly probability of exceeding each magnitude is computed as follows : (2) According to Equations 1 and 2, the occurrence probability (P) of an earthquake with a given magnitude is as follows : ( For example, an earthquake with a 20-year recurrence period would have an annual exceedance probability of 1/20, 0.05, or 5%.According to this, there is a 0.05 or 0.05% probability that an event with a magnitude ≤ 20-year event will occur in any particular year.Similarly, the likelihood of an event larger than the 50-year event occurring in any given year is 1/50 = 0.02, or 2%.Although these percentages are the same every year, an earthquake of this magnitude might occur the following year or be far more than it over a period of 50 years [14], [32] .
The following formula can be used to determine the likelihood that an earthquake of a certain magnitude will occur at any time t [28] : Pt is the occurrence likelihood throughout the entire time period t, and P is the occurrence likelihood in any given year .

Earthquakes Data
The International Seismological Center (ISC) earthquake occurrence data were the datasets that were used.The data that were chosen pertain to earthquakes that occurred between latitude 28° and 38° N and longitude 38° and 49° East.The selected data extend from January 1, 1900, to December 31, 2019.The magnitude of the earthquakes ranges from 0.3 7.7Mw.The data for the earthquakes in Iraq and its surroundings, including the year of occurrence, earthquake number, and maximum and minimum magnitudes, are listed in the Appendix [49].

Results and Discussion
The results of the calculations of the maximum magnitude (Mmax), rank (m), annual exceedance probability (P%), and period of return T for the earthquake data are listed in Table 1.The earthquakes were classified into several ranks based on their magnitude.The highest magnitude event takes the first rank (m1) and the subsequent event the second rank (m2), and the lowest magnitude event takes the last rank, equal to the number of years of the earthquake catalog .

Return Period and Annual Exceedance Probability
The results of the calculation of the return period of the annual maximum magnitude of the earthquakes are listed in Table 1.The relationship between the annual maximum magnitude and the return period is shown in Figure 3.The period of recurrence for every earthquake magnitude between 4.7 and 7.7 Mw can be calculated from Figure 2.For example, a magnitude 6 Mw earthquake has a return time of approximately 2.45 years, whereas a magnitude 7 Mw earthquake has a return period of approximately 10.54 years.The results of the current study on the return period of many earthquakes were compared with those of a previous study conducted by [35] using extreme value analysis.They used the earthquake catalogue for the period 1900-1982.They employed the Gumbel distribution model, while the Weibull distribution model was used in the current study.The obtained values of the return period in the current study were compared with those reported by [35] (Table 2) .The difference in the return period values of the earthquake calculated in the current study and in the previous study, especially for large magnitudes, may be due to the difference in the coverage period of the earthquake catalogs used in both studies.The record periods for the earthquake catalogs in the current and previous studies were 120 and 82 years, respectively.According to Weibull formula (equation 1) the record period affects the value of the return period for the same rank (m) of given magnitude.For example, an earthquake of seismic magnitude 7, which rank (m) 2, has a period of 41.5 years if the record period (n) equals to 82 years while the return period equals to 60.5 years if the record period (n) is equal to 120 years.Several studies [33], [8] should use different statistical distribution models (e.g., Gumbel, Weibull, Gamma, etc.) affect the calculation of the return period and the occurrence probability of an earthquake.
The relationship between the yearly maximum magnitude of the earthquake and the yearly exceedance probability is shown in Figure 3.An event-related magnitude and recurrence interval can be derived from this figure.For example, the event magnitude related with the return period of 1.1 years is about 5 Mw.The 1-year earthquake is the name given to this event.The probability that an event this year or any other year will have a magnitude greater than that of the one-year earthquake is 93.79%.Other example, the earthquake magnitude associated with the return period of 10.54 years is about 7 Mw.The 10-year earthquake was the name assigned to this event.The probability that an event this year or any other year will have a magnitude greater than that of the 10-year earthquake is 9.5% .Figure 3. Earthquake magnitude and annual exceedance probability relationship .

Probability during a period of time (Pt)
The Pt value for the time period t (120 years) was calculated for earthquakes of magnitudes 6.5, 7, and 7.4 Mw, using formula 4. The results are shown in Figure 4.The Pt values at 30years for earthquakes of magnitudes 6.5Mw, 7Mw and 7.4Mw were 99.97%, 95%, and 58.39%, respectively.The earthquake probabilities and their magnitudes over a period of 30 years are shown in Figure 5.An earthquake of magnitude 6Mw for example, has a 99.99% probability of occurrence, whereas an earthquake of magnitude 7.7Mw has a probability of 21.89%.The calculated values show that, as the time period increases, the probability of exceedance increases.Additionally, it should be noted that the return period is longer for earthquakes of higher magnitudes.It is necessary to realize that the calculated probability of an earthquake occurring and its return period are statistical predictions derived from a collection of earthquake data for Iraq.Nobody actually knows when or where an earthquake of magnitude M will strike with a probability of 1% or higher [25].

Conclusions
Weibull's formula and its inverse were used to calculate the probability of occurrence and return period, respectively.The return period for the earthquake's magnitudes of 5 and 7Mw was 1.1 and 10.54 years, respectively, while the occurrence probability was 93.79% and 9.5%, respectively.The largest magnitude is 7.7 with a period of return 121 years and likelihood of about 0.82%.The probability during a period of time (Pt) at 30-years for the earthquakes of magnitudes 6.5Mw, 7Mw and 7.4Mw are 99.97%,95% and 58.39% respectively.The probability of an exceedance increases as the time period lengthens.The return duration is greater for earthquakes with higher magnitudes .

Figure 4 .
Figure 4. Earthquake probability for earthquake magnitudes in a time span

Figure 5 .
Figure 5. Earthquake probability and their magnitudes in a time span of 30 years .

Table 1 .
Ranking, maximum magnitude, probability, and return period .Figure 2. The relationship between return period and earthquake magnitude.

Table 2 .
Comparison of the obtained return period values with the previous study results.