Received 30 March 2016; accepted 28 May 2016; published 31 May 2016
The natural disaster damaged the old man’s life does not specify a period of time, and perhaps the most important “earthquakes” that threatens “human civilization”. The most destructive disaster of nature is a severe earthquake and its destroying effects. If the earthquake occurs in a populated area, it may cause many deaths and injuries and extensive property damage regions. The ultimate goal of seismic hazard assessment and risk evaluation for a particular site or area is to condense seism-tectonic knowledge and experience into parameters used for predicting seismic parameters which in turn can be applied by engineers in design and subsequent earthquake resistant construction.
Statistical surveys support researches on the likelihood of future earthquakes. A primary goal of earthquake research is to increase the reliability of earthquake probability estimates. With a greater understanding of the hazard parameters of earthquakes, we may be able to reduce damage and loss of life from this destructive event. Statistics help us to predict the future events based on previous events.
1.1. Seismicity of the West Coast of Saudi Arabia
We find that in the western region of Saudi Arabia is growing concern on the volcanic activity associated with earthquakes in the Red Sea have been the work of affluent studies to assess the seismic risk level. Western Region of Saudi Arabia is considered to be a moderately active seismic zone as shown in Figure 1  . Seismic events also include a sequence of earthquakes which occurred in 1967 along the Red Sea rift system at a distance of about 150 km to the south west of Jeddah  . In  , a report that a high level of micro-earthquake activity was detected near the border of the Red Sea and near the transition from oceanic to continental crust. Recently, EI-Isa et al.  reported that about 500 local earthquakes with magnitudes less than 4.85, occurred in the Gulf of Aqaba area during the period from January 21 to April 20, 1983.
Figure 1. Red sea, west coast of Saudi Arabia, seismic foci in Saudi Arabia and solutions in all the years until 2015.
Figure 2. Earthquake intensity map for the western region, west coast, return period = 100 years.
1.2. Seismicity of Red Sea (Figure 2)
Red Sea is a body of water located between the western coast of the Arabian Peninsula and Africa. The overlooking countries on the Red Sea are: Saudi Arabia, Egypt, Sudan, Yemen, Eritrea and Djibouti. Its strategic movement of marine transportation as connection of the South Ocean through the Strait of Bab el Mandeb and extending north to reach the Sinai Peninsula, and there are branches off to the Gulf of Aqaba and the Gulf of Suez, which leads to the Suez Canal. The length of this sea 1900 kilometers and currently up in some areas to 300 km. The deepest point in the Red Sea up to 2500 m and the rate of decline is 500 m. Red Sea, an area of 450,000 km2.
2. Literature Review
When looking around us to natural disasters, particularly earthquakes, we find they are warning of a disaster are taken into account in the event of predictable after a study and give warning of potentially harmful earthquakes in enough time to prepare appropriate for this disaster and minimize the loss of life and property  . Many studies have been presented to develop reliable estimates, of probability, magnitude and recurrence relations given the large pattern of earthquake occurrence. The primary advantage of probabilistic seismic hazard analysis (PSHA) is that it integrates over all seismicity: temporal and spatial along with ground motions to calculate a combined probability of exceedence, which incorporates the relative frequencies of occurrence of different earthquakes and ground-motion characteristics. Practically in any earthquake catalogue the quality of different parts (periods and areas) varies significantly with respect to completeness, magnitude reliability, homogeneity and location accuracy  -  .
 El-Quliti in  estimated the frequency, magnitude and recurrence of extreme earthquakes in Medina Area in Saudi Arabia. Statistical theory of extreme values has been used to analyze the observed extremes of any phenomena and to forecast the further extremes based on the appropriate distribution, Gumbel  . In earthquake engineering, this theory has been applied successfully by many researchers in the past few decades (Nordquist  ; Epstein  ; Yegulalp  ; Al-Abbasi  and Jaiswal  . This theory does not require analysis of the complete record of earthquake occurrence, but uses the sequence of earthquakes constructed from the largest values of the magnitude over a set of predetermined intervals.
Abe and Suzuki  analyzed the seismic data from the viewpoint of science of complexity, where one of the main goals of seismology is to predict when and where the next main shock will occur after an earlier main shock. The rate of recurrence of earthquakes on a seismic source can be represented with the Gutenberg-Richter relation  . Kasap and Gürlen  studied the return periods of earthquakes. Ogata  investigated the statistical models for earthquake occurrences. Utsu  applied gamma, log-normal, Weibull and exponential distributions to describe the probability distribution of inter-occurrence time of large earthquakes in Japan. Aktaş et al.  , used Poisson distribution to describe the recurrence times, and estimated the expected value and variance computed for the loss of life and damaged buildings after the change point using the compound Poisson process. Bayrak et al.  evaluated the seismicity and earthquake hazard parameters of Turkey based on maximum regional magnitude. Öztürk et al.  estimated the mean return periods, the most probable magnitude in a time period of t-years, and the probability of earthquake occurrence for a given magnitude during a time span of t-years for different regions in and around Turkey. They also showed that in the specific region, the most probable earthquake magnitude in the next 100 years would be over 7.5  . The goodness of fit is evaluated employing Kolmogorov-Smirnov test  .
3. Extreme Earthquake Analysis
Earthquake prediction can be considered into two types. First is the statistical prediction which is based on previous events; Data are collected from the records. Second is deterministic prediction which is made from the earthquake signs. Table 1 shows the data for earthquakes in Red Sea province and surrounding area representing the range and maximum magnitude.
Most extreme event analysis is concerned with the distribution of annual maximum or minimum values at a given site. These events are given a rank, m, starting with m = 1 for the highest value, m = 2 for the next highest and so on in descending order. Each earthquake magnitude is associated with a rank, m, with m = 1 given to the maximum magnitude over the years of record, m = 2 given to the second highest magnitude, m = 3 given to the third highest one, etc. The smallest earthquake magnitude will receive a rank equal to the number of years over which there is a record, n. Thus, the discharge with the smallest value will have m = n = 100.
There are several formulas for calculating the probability value. The Weibull formula will be used because of its ease of use. The US Geological Survey  , among others, also uses this formula.
According to the Weibull equation  the return period or recurrence interval T (in years) is calculated using the following equation:
Table 1. Data for the earthquakes in red sea province.
where: m = event ranking (in a descending order), and n = number of events in the period of record.
The percentage probability the (annual exceedence probability) for each magnitude is calculated using the inverse of the Weibull equation as follows:
From Equations ((1) and (2)) it is clear that P = 100/T%. For example, an earthquake equal to that of a 10- year one would have an annual exceedence probability of 1/10 = 0.1% or 10%. This would say that in any given year, the probability that an earthquake with a magnitude equal to or greater than that of a 10 year earthquake would be 0.1% or 10%. Similarly, the probability of an earthquake with a magnitude exceeding the 50 year one in any given year would be 1/50 = 0.02, or 2%. Note that such probabilities are the same for every year, but in practice, such an earthquake could occur next year, or be exceeded several times in the next 50 years.
Table 1 shows the Number of Earthquakes, Minimum Magnitude, Range, and Maximum Magnitude for the years 1913 to 2016  .
Table 2 shows the calculations of the rank m, the probability P and the return period T for the data of the yearly maximum magnitude given in Table 1. The return period or recurrence interval T (in years) and the probability (P) are calculated using Equations ((1), (2)).
4. Earthquake Parameters
4.1. Annual Exceedence Probability and Return Period
Return period or Recurrence interval is the average interval of time within which a flood of specified magnitude is expected to be equaled or exceeded at least once. 100 year earthquake is an earthquake that is expected to occur, on the average, once every 100 years, or has a one percent chance of occurring each year.
Figure 3 is a plot of earthquake magnitude and annual exceedence probability relationship (linear scales) with the annual maximum magnitude per year on the Y axis versus the annual exceedence probability on the X axis. The X and Y axes both use linear scales.
A best-fit curve is drawn through the data points. From the best-fit curve, one can determine the earthquake magnitude associated with an earthquake with a recurrence interval of say 10 years, it is about 5.8 on Richter scale. This would be called the 10-year earthquake.
Similarly the recurrence interval associated with an earthquake magnitude of magnitude of 5 on Richter scale is about 17 year.
The annual peak information may also be presented with a logarithmic rather than a linear scale. This is often done to make the curve appear as a straight line and also to avoid a graph that will suggest either a zero or a one-
Table 2. The rank, probability and the return period results.
Figure 3. Earthquake magnitude and probability relationship (linear scales).
hundred percent exceedance probability. Moreover, a straight line curves are more easily allow extrapolation beyond the data extremes.
Figure 4 represents the earthquake magnitude and the annual exceedance probability (log scale) relationship.
Percentage probability is determined by dividing one by the recurrence interval and multiplying by 100. For example, the probability that an earthquake magnitude will exceed the 100-year earthquake this year or any other year would be 1%.
Figure 5 shows the earthquake magnitude and return period relationship on linear scales and Figure 6 shows the earthquake magnitude and return period relationship using a log scale. From the figure it can be noticed that the return period of an earthquake of magnitude 6.1 on Richter scale is about 104 years, and an earthquake of magnitude of 5.8 on Richter scale has a recurrence interval of about 34.6 years.
Sometimes it is suitable to add a second X-axis to represent the return period to the first X-axis representing the annual exceedance probability.
Figure 7 shows the earthquake magnitude and the annual exceedance probability on the first X-axis and the return period on the second X-axis. Both the two X-axes use a variable scale so the relationship appears as a perfect straight line, this will allow for easier findings.
Figure 4. Earthquake magnitude and annual exceedence probability relationship (log scale).
Figure 5. Earthquake magnitude and return period relationship (linear scales).
Figure 6. Earthquake magnitude and return period relationship (loq scale).
Figure 7. Earthquake magnitude probability and return period relationship.
From the fit line, one can determine the magnitude associated with an earthquake of a recurrence interval of say 30 years. This would be called the 30-year earthquake. The magnitude associated with the 30-year earthquake is about 5.8 Richter scale. Similarly the magnitude associated with an earthquake with a recurrence interval of 104 years (the 104-year earthquake) would have a measure of about 6.1 Richter scale.
4.2. The Probability during a Time Period
The probability of a certain-magnitude earthquake occurring during any period t can be calculated using the following equation:
where P is the probability of occurrence over the entire time period, t, and P is the probability of occurrence in any year.
It is worth to apply Equation (3) for earthquakes of highest magnitudes which represent the most dangerous events in the location. The equation is applied for earthquakes of magnitudes 6.1, 5.9 and 5.8 Richter scale of probabilities of 0.96%, 1.92% and 2.88% respectively. The result is depicted in Figure 8 for earthquakes of magnitudes: 6.17 Richter scale (P = 0.96%), 5.9 Richter scale (P = 1.92%) and 5.8 Richter scale (P = 2.88%).
A homeowner considering the costs of reinforcing a house against earthquakes will want to know how the risk varies during an average mortgage span of 30 years.
Figure 9 shows the earthquake probability and earthquake magnitudes in a time span of 30 years. An earthquake of magnitude of 4.5 on Richter scale for example, has a 100% probability of occurrence but, if the earthquake of magnitude 6.1 on Richter scale is chosen, the probability drops to 25%.
Figure 8. Earthquake probability for some earthquake magnitudes in a time span period.
Figure 9. Earthquake probability and earthquake magnitudes in a time span of 30 years.
5. Conclusions and Points for Future Researches
In this study, the statistical frequency analyses are applied to the recorded annual maximum earthquake magnitudes for Rea Sea province and the surrounding area in the westcoast of Saudi Arabia since 1913.
The risk of earthquakes, the return period and the probability of occurrence for a particular magnitude during any given year are calculated based on the analysis of data over a period of 104 years by applying Weibull equations.
The relation between magnitude and frequency and between magnitude and return period is represented as a curve in a linear scale graph and as a straight line on a logarithmic scale and variable scale graphs to facilitate the findings. The results lead to a general conclusion that Red Sea is considered to be located within the existing areas of the seismic belt and the region is exposed to earthquakes with strength ranging about 3.0 or less on the Richter scale with a high probability. The maximum magnitude is 6.1 with a return period of 100 years and probability of about 0.96%.
The conventional approach of hazard estimation based on magnitude frequency relationship is more useful when the dataset is complete for the entire time span and for the magnitude range. With good and complete datasets, the method is more appropriate and accurate for seismic hazard estimation.
5.2. Points for Future Researches
Points for future researches can be summarized as follows:
・ To study in details the influence of missed data like that for the years in the intervals (1913-1921), (1921-1962), (1962-1964), (1965-1967), (1967-1969) and 1980-1982) on the earthquake parameters.
・ There are several methods of comparing the results of evaluation of earthquake parameters and can be applied in other areas in the Kingdom of Saudi Arabia
・ Can assess the risk parameters for a number of potential risks in Saudi Arabia, an example of the risks of landslides; severe storms; floods.
・ Drawing seismic maps of the areas in which they are registered seismic activity in Saudi Arabia each year for use in the evaluation and the likelihood of the occurrence of such activity on a long-term process.
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