International Journal of Natural and Engineering Sciences 2 (2): 81-86, 2008
    ISSN: 1307-1149, www.nobelonline.net
    
    A Statistical Approach for Estimating The Probability of Occurrence of Earthquake on The Northern Anatolian Fault Zone
    Veysel Yilmaz, H. Eray Çelik* Department of Statistics, University of Eskişehir Osmangazi, Eskişehir-Turkey Corresponding Author e-mail: ecelik@ogu.edu.tr
    *
    
    Received: January 27, 2008 Accepted: March 09, 2008
    
    Abstract
    Probabilistic methods are useful for characterizing earthquake prediction because they are, for all practical purposes, random phenomena. Earthquake occurrence probabilities can be predicted by using different probability distributions. In this study eleven distributions are examined for determining the best one that represents earthquake data from Turkey during the last 104 years. In such a determination, Kolmogorov-Smirnov, Chi-square, Anderson-darling test statistics are used independently for treating Turkish earthquake data. It is concluded that the Weibull probability distribution is the most appropriate than any other distribution with the data at hand. This result is valid for the earthquake risk evaluation in the area between longitudes 40o 50’ – 41o 50’ N and latitudes 30o 00’ – 40o 00’ E which includes the Northern Anatolian Fault Zone (NAF) as the most active fault in Turkey. Key words: Earthquake, prediction, probability, Turkey, Weibull Distribution.
    
    INTRODUCTION
    It has been one quarter century since Utsu [1,2], Rikitake [3] and [4] proposed probabilistic approaches for predicting the magnitude of the next earthquake on a specific fault. Poisson distribution is commonly applied to estimate the occurrence of earthquakes in seismic hazard studies [5,7]. A number of probability distributions have been proposed and used for computation of conditional probability of future earthquake occurrences, including the double exponential, Gaussian, Weibull, Log-normal [8], Gamma [9], and Pareto distributions [10]. The difficulty lies in determining the suitable distribution given the available earthquake data for a certain active fault zone. Nishenko and Buland [1987] obtained a reasonably good fit to the data using a log-normal distribution. McNally and Minster [11] argued that the Weibull distribution would be more appropriate [22]. Past studies on seismicity and seismic hazard in Turkey includes the use of geostatistical methods without consideration of probability distribution functions and regional seismic hazard estimations [12]. It is necessary to make several assumptions and establish definitions in modeling of earthquake occurrences on a fault system. According to the Poisson model, earthquake occurrences have no memory and they occur independent of each other. The random variable is defined as either the magnitude of the earthquake in modeling and/or as inter-event time between two earthquake incidences in the continuous distributions [13]. A fault system usually creates earthquakes of many different magnitudes, not just one magnitude and inter-event time. The magnitude of the earthquake is the most important parameter used in the modeling studies. Although the devastation caused by the earthquake is directly related to the magnitude of the earthquake when it occurs near urban areas, site conditions and quality of the buildings are also important parameters. For example, while an earthquake of magnitude of
    
    over 7.5 causes no devastation in Japan, 20.000 people died in the earthquake occurred in 1999 in Turkey [14]. In countries where ‘sound’ building codes either do not exist or do exist but are not followed earthquakes with magnitude larger than 5 can be devastating events. For this reason, earthquakes with magnitude equal to or larger than 5 are taken into consideration in this study. In this study, the set of data presented in Table 1 is used, which consists of earthquakes whose magnitudes are equal to or larger than 5 between the years 1900 and 2004 in the area within longitudes 40o 50’ – 41o 50’ N and latitudes 30o 00’ – 40o 00’ E [23]. This area (see Figure 1) includes the north Anatolian fault zone, which is the most active earthquake zone in Turkey. It is obvious from table 1 that the number of the earthquakes with magnitude of 5 or over is 63 during the last 101 years. On the other hand, it is worthy to notice that 3 earthquakes have magnitudes more than 7. The temporal magnitude randomness is obvious and therefore it is important to try and search the most suitable probability distribution model for prediction in this region. The objective of this study is to determine which probability distribution best fits the earthquake data considered. Four probability distributions are commonly used in the literature Poisson, Semi-Markov, Gumbel and Weibull. The comparisons are made by the help of Chi-Square, Kolmogorov-Smirnov and Anderson- Darling test statistics. For the parameter estimations of these distributions three estimation methods, namely, least squares, maximum likelihood and moment methods are considered. Earthquake prediction probabilities based on these three estimation methods have been calculated for different time periods.
    
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    Table 1 (Cont.) The earthquakes at the magnitude of 5 or over between the years 1900-2004 in the region between (40.50 - 41.50) north and (30.00 – 40.00) east coordinates Date 21.09.1957 23.06.1967 22.07.1967 22.07.1967 22.07.1967 30.07.1967 22.12.1969 05.10.1977 14.08.1996 14.08.1996 11.01.1997 13.09.1999 11.11.1999 12.11.1999 12.11.1999 12.11.1999 16.11.1999 14.02.2000 06.06.2000 23.08.2000 26.08.2001 22.10.1905 21.08.1907 21.06.1908 25.06.1910 09.08.1918 29.08.1918 09.06.1919 09.06.1919 29.05.1923 16.08.1923 24.01.1928 21.09.1936 18.11.1936 21.11.1942 02.12.1942 11.12.1942 20.12.1942 20.06.1943 20.06.1943 26.11.1943 07.12.1943 02.01.1944 01.02.1944 02.02.1944 10.02.1944 05.04.1944 30.09.1944 18.10.1944 02.03.1945 21.01.1946 19.12.1947 13.05.1949 13.08.1951 14.08.1951 07.09.1953 19.08.1954 06.01.1956 26.05.1957 26.05.1957 26.05.1957 27.05.1957 01.06.1957 Time 20:16 10:06 16:56 17:14 18:09 01:31 04:47 05:34 01:55 02:59 06:42 11:55 14:41 16:57 17:17 17:29 17:51 06:56 02:41 13:41 00:41 03:42 03:55 19:26 00:39 06:39 07:13 15:47 11:34 03:52 07:36 11:41 15:50 14:01 19:04 02:39 14:03 15:32 16:47 22:20 01:19 10:59 06:08 03:33 12:05 04:40 04:13 12:54 10:39 11:25 17:31 20:14 18:33 18:46 03:59 21:03 14:52 06:33 08:54 09:36 11:01 05:26 Latitude 40,75 40,85 40,67 40,70 40,72 40,72 40,60 41,02 40,75 40,78 40,55 40,75 40,75 40,81 40,75 40,70 40,73 41,02 40,69 40,68 40,95 41,00 40,70 40,60 41,00 40,89 40,58 41,16 40,68 41,00 41,02 40,99 41,21 41,25 40,82 41,04 40,76 40,70 40,85 40,84 41,05 41,00 41,00 40,70 40,74 41,00 40,84 41,11 40,89 41,20 41,05 40,71 40,74 40,88 41,08 41,09 41,21 41,00 40,67 40,60 40,76 40,73 40,75 Longitude 34,02 33,65 30,69 30,80 30,51 30,52 34,20 33,57 35,30 35,31 35,25 30,08 30,25 31,19 31,08 31,47 31,59 31,76 32,99 30,72 31,57 31,00 30,10 35,90 34,00 33,41 35,16 33,20 33,89 30,00 34,41 30,86 33,53 33,33 34,44 34,88 34,83 36,80 30,51 30,73 33,72 35,60 33,70 31,27 31,44 32,30 31,12 34,87 33,47 33,40 33,48 32,82 32,71 32,87 33,18 33,01 36,41 30,20 31,00 30,74 30,81 30,95 30,86 6 35 18 33 10 12 3 10 10,4 7,5 10,4 27,8 11 5 10 10 15 7 27 15 0 0 10 10 10 10 25 40 10 20 10 80 20 40 16 10 10 10 0 0 10 40 10 10 10 10 10 60 10 20 10 40 40 30 10 10 40 10 50 50 Depth (km) 40 Magnitude 5.1 5.1 6.0 5.2 5.1 5.4 5.1 5.3 5.3 5.3 6.0 5.6 5.4 6.2 5.2 5.0 5.0 5.0 6.1 5.8 5.4 5.2 5.5 5.2 6.2 5.8 5.3 5.7 5.0 5.5 5.2 5.3 5.1 5.2 5.5 5.4 6.1 7.0 6.5 5.5 7.2 5.6 5.0 5.0 5.1 5.3 5.5 5.5 5.2 5.6 5.0 5.1 5.1 6.9 5.1 6.4 5.0 5.0 7.1 5.4 5.9 5.8 5.0
    
    V. Yılmaz and H. E. Çelik / IJNES, 2 (2): 81-86, 2008
    
    83
    
    In the equation (1); t is the time passing between the events, β is the shape parameter, and η is the scale parameter. Weibull Cumulative Distribution Function is as follows;
    ⎛t⎞ −⎜ ⎟ ⎜η ⎟ F(t) = 1 − e ⎝ ⎠ β
    
    (2)
    
    The expectation of the earthquake in period t is calculated with the expected value, as shown in equation (3).
    Figure1. The Area of investigation (NAF: 40.50 - 41.50 north and 30.00 – 40.00 east) [http://www.iris.iris.edu/volume2000no1/page-14-16.htm]
    
    E(t) = ∫ t.f (t).dt
    R
    
    STATISTICAL ANALYSES
    In determination of the distribution which fits the earthquake data best, the test statistics, summarized briefly below, have been used. The results of the test statistics used in determination of the effective distribution are given in Table 2. It is seen that the test statistics have minimum values for Weibull Distribution. For this reason, it has been concluded that the distribution that best represents the data set is the Weibull Distribution. In terms of compliance, Pearson VI, lognormal and log-logistic follow the Weibull distribution.
    Table 2 Goodness of fit results Distributions Chi-Square Kolmogorov Smirnov 0.2350 0.3240 0.3240 0.0981 0.0821 0.0901 Anderson Darling 4.11 17 17 0.644 0.53 0.535
    
    ⎛ t ⎞ ⎜ ⎟ ∞ β ⎛ t ⎞β−1 −⎜ η ⎟β ⎛ ⎞ ⎜ ⎟ E(t) = ∫ t. ⎜ ⎟ e ⎝ ⎠ .dt = ηΓ ⎜1 + 1 ⎟ ⎝ β⎠ η⎝ η ⎠ 0
    
    (3)
    
    MLE, LSM and MOM are the most common estimation methods that are used in order to estimate the shape and scale parameters of Weibull Distribution [18,19,20].
    
    DISCUSSION
    Estimations of Weibull distribution parameters are obtained with the three methods mentioned above and the results are presented in Table 3.
    Table 3 Estimated values of shape and scale parameters of Weibull Distribution obtained using three different calculation methods Parameters Method Least Squares Method (LSM) Maximum Likelihood Method (MLE) Method of Moments (MOM)
    
    Value(df) 16.5 (4) Beta 28.4 (4) Erlang 28.4 (4) Exponential 1.43 (4) Gamma 2.38 (4) Log-Logistic 1.06 (4) Lognormal
    
    ˆ β
    0.514 0.507 0.765
    
    ˆ η
    329.26 335 785.066
    
    Pearson V
    Pearson VI
    
    15.8 (4)
    1.43 (4) 110 (4) 140 (4)
    
    0.2120
    0.0821 0.5970 0.7070
    
    3.42
    0.386 67.8 101
    
    Tringular Uniform Weibull
    
    When Table 3 is studied, it is seen that parameter estimations obtained by LSM and MLE methods are close to each other. Especially, , estimated using these two methods are close to each other but different from that obtained by MOM. The probabilities are calculated using equation (2) for magnitudes equal to or greater than 5 in the area of investigation. The substitution of parameters estimated by LSM, MLE and MOM into equation (2) leads to the following probability statements.
    F(t) −[ t ] = 1 − e 329.26 0.514
    
    1.06 (4)
    
    0.0697
    
    0.277
    
    In making predictions by probability distributions, the first step is the estimation of distribution parameters. The three methods mostly used in estimation of parameters are Least Squares Method (LSM), Maximum Likelihood Method (MLE) and Method of Moments (MOM)[15,16]. In this study, fitting of Weibull Distribution for calculation of earthquake prediction probabilities in the mentioned region is introduced and then the mathematical process in the estimation of MLE, LSM and MOM and as well as Weibull Distributions is explained. Weibull Distribution Weibull Distribution with two parameters is given below [17];
    ⎛ t ⎞ ⎜ ⎟ ⎛ t ⎞β−1 − ⎜ ⎟ β η f(t) = ⎜ ⎟ e ⎝ ⎠β η⎜ η ⎟ ⎝ ⎠
    
    LSM
    
    (4)
    
    F(t)
    
    MLE
    
    −[ t ] = 1 − e 335 0.507
    
    (5)
    
    and
    F(t) t −[ ] = 1 − e 785.066 0.765
    
    MOM
    
    (6)
    
    β > 0, η > 0
    
    (1)
    
    Accordingly when ti= 100, 365, 730, 1825, 3650, 5475, 7300 days are substituted in these equations then the probability results appear as in Table 4.
    
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    Table 4. Estimating the probability of occurrence with different prediction periods for the earthquakes at the magnitude of 5 or larger Method LSM MLE MOM Probability p p p Prediction period (days) 730 1825 3650 0.778 0.910 0.968 0.773 0.612 0.906 0.851 0.965 0.961
    
    100 0.418 0.418 0.187
    
    365 0.652 0.648 0.427
    
    5475 0.986 0.984 0.988
    
    7300 0.993 0.992 0.996
    
    The probabilities of an earthquake occurrence with a magnitude of 5 or higher are 0.418, 0.418 and 0.187 according to LSM, MLE, MOM methods, respectively within 100 days after an earthquake of magnitude 5 or larger has already occurred. These probabilities successively increase to 0.652, 0.648 and 0.427 in one year, to approximately 0.90 in 5 years and to 0.99 in 20 years. It can be deduced from the results that there is a very high probability of occurrence of an earthquake with a magnitude of 5 or higher within 20 years after an earthquake with a magnitude of 5 or higher has already occurred in the area of investigation. If Table 4 is studied in terms of parameter estimation methods, MOM gives lowest probability in shorter time periods when compared with LSM and MLE. However, when the time period increases, it is seen that all the three estimation methods give similar results. In order to decide on a proper method to make a better estimation on shape and scale parameters, a simulation study was carried out with samplings composed of 10, 30, 50, 100 and 120 units. At the end of this simulation study on 250 comparisons, 122 comparisons revealed that MLE was the best estimation method, while 104 of the 250 showed the LSM, and the other 24 of 250 showed the MOM as the best estimation method. It was observed that in the simulation studies with 10 units the MLE was the best estimation method while in studies with 30 and 50 units the MLE and LSM, in studies with 100 units the LSM, and in the simulation studies with 120 units the MLE performed best. In general, the MLE appeared to be the best estimation method [21]. For the size of the data set used in this study, it is seen that estimated values of the parameters obtained by using LSM and MLE methods are close each other, while those obtained with MOM show differences. If sampling set is large enough however, it is seen that the results obtained from the three methods are closer to each other. The probability of an earthquake occurrence at a magnitude of 5 or higher for different time periods in the area of investigation is given in the following figure. It is seen in Fig. 2 that as the inter-event time increases, the probability of earthquake occurrence approaches to 1. For example, the probability of earthquake occurrence is 0.87 in 1400 days and 0.91 in 2000 days. The prediction of an earthquake can be defined as the time, place and magnitude parameters of a possible future earthquake and the prediction of uncertainties depend on these parameters. Because of the divergences that earthquakes show in terms of time, place and magnitude, prediction of the seismic hazard with probabilistic methods provides the best results. Statistical approaches based on probability and random processes are the most appropriate methods to use in the analyses of seismic hazard due to the divergences seen in time and location of earthquakes.
    
    The probability of earthquake occurrence of magnitude 5 or larger
    
    Time (day)
    
    Figure 2. The probability of earthquake occurrence of magnitude 5 or larger in the area of investigation
    
    CONCLUSION
    It is known that the probability of occurrence of earthquakes in seismic zones with a certain size and tectonic regime can be estimated using Weibull probability distribution. Hagiwara (1974) and Rikitake (1974) used Weibull distribution to estimate probability of earthquake occurrence. The basic feature of the Weibull distribution is the time between events. Only the occurrence probabilities of the magnitudes that are equal to or greater than Mmin (the smallest magnitude defined), can be computed with the Weibull distribution unlike other probability models (Gumbel, Poisson, Semi-Markov) used in earthquake probability analyses. Contrary to other approaches, the basic principle of the Weibull distribution is to take into consideration the distributions of the events in specific time intervals. In this paper, the authors attempted to find a probability distribution that best represents the set of earthquake data from Turkey. KS, AD and Chi-Square statistical tests have been used in determining that the most representative probability model is the Weibull distribution. Also, the time intervals between the successive earthquake events in days are considered for the probability calculations instead of classifying the set of data according to the years as in other general studies. Shape and scale parameter estimations regarding the Weibull distribution are obtained by using LSM, MLE and MOM methods; and the probability of earthquakes of magnitude 5 or higher is calculated for different time periods by these methods. The study is carried out for a region that includes the most active fault zone in Turkey,
    
    V. Yılmaz and H. E. Çelik / IJNES, 2 (2): 81-86, 2008
    the north Anatolian fault. The probability of an earthquake with a magnitude of 5 or larger in the mentioned region is 0.96 in 10 years; and the repetition period of an earthquake with a magnitude of 5 or higher is 651.8 days. The study area is the most densely populated region of Turkey in terms of both population and industry. In this region, investigation and control on compliance with building codes, new building construction directives, and building rehabilitation studies, which have been neglected until today, must be carried out and completed as soon as possible. The estimations in this study depend just on the magnitudes of the earthquakes with a specific magnitude. For that reason, the results found in this study should be assessed as statistical findings that emerge from the estimation of earthquake probabilities. Statistical results obtained in the study are intended to help promote preventive precautions to minimize losses due to earthquakes. In conclusion, this study aims to act as a preliminary work that would be a base for further interdisciplinary studies to be carried out that are necessary to mitigate earthquake risk in the region. ⎛ n β⎞ ˆ ⎜ ∑ ti ⎟ ˆ η = ⎜ i =1 ⎟ ⎜ n ⎟ ⎜ ⎟ ⎝ ⎠
    1 ˆ β
    
    85
    
    (11)
    
    Least Squares Method (LSM) If we take the natural logarithm of the both sides of equation (2); ⎛t⎞ −⎜ ⎟ ⎜η⎟ β F(t) = 1 − e ⎝ ⎠
    
    ⎛ t ⎞β − Ln(1 − F(t)) = ⎜ ⎟ ⎝ η⎠
    is obtained. When the natural logarithm is taken again; Ln { Ln (1 − F(t) )}= β Lnt − β Lnη − ax b Y (12)
    
    is obtained. When Y = Ln { Ln (1 − F(t) )} is defined with − ax = âLnt and b = âLn; Y = ax+b; the Least Squares Method is obtained. By the help of the ⎧ ⎡ ⎤⎫ ⎥⎪ 1 n ⎪ ⎢ 1 ⎪ ⎢ ⎪ ⎥⎬ t = ∑ ln ⎨ln i ⎞ ⎥⎪ n i =1 ⎪ ⎢ ⎛ 1− ⎪ ⎢⎜ n + 1⎟ ⎥⎪ ⎠ ⎦⎭ ⎩ ⎣⎝ and
    
    APPENDIX
    Maximum Likelihood Method Likelihood function for Weibull Distribution having two parameters is as follows; L(η, β) = ∏ f (t i ; θ)
    i =1 n
    
    (13)
    
    β−1 −⎛ t ⎞ ⎜ ⎟ n β⎛ t ⎞ ⎜ η ⎟β L(η, β) = ∏ ⎜ ⎟ e ⎝ ⎠ η⎝ η⎠ i =1
    
    (7)
    
    Y=
    
    If the natural logarithm of equation (7) is taken, the following formula is obtained; InL(η, β) = ∑ Inβ − β Inη + (β − 1)Int i − η−β t iβ
    i =1 n
    
    1 n ∑ ln t n i =1
    
    (14)
    
    definitions, β and η are estimated with equations (15) and (16).
    ⎫ ⎧ ⎛ ⎡ ⎛ ⎧ ⎡ ⎤⎞ ⎤ ⎫ ⎞⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎢ 1 ⎥ ⎪ ⎟⎪ ⎪ n ⎜ ⎢ 1 ⎥ ⎟ n ⎪ n ⎪ ⎟ ∑ ln t i ⎬ ⎨n ∑ ln t i ⎜ ln ⎨ln ⎢ ⎛ ⎬ ⎟ ⎬ − ⎨ ∑ ln ⎜ ln ⎢ ⎥ ⎞⎥ ⎜ ⎪ ⎢ ⎜1− i ⎟ ⎥ ⎪ ⎟ ⎪ ⎪i =1 ⎜ ⎢1 − i ⎥ ⎟ i =1 ⎪ i =1 ⎪ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎪ ⎝ ⎩ ⎣ ⎝ n +1 ⎠ ⎦ ⎭ ⎠ ⎭ ⎩ ⎪ ⎝ ⎣ n +1⎦⎠ ⎭ ˆ ⎩ β= 2 2⎫ ⎧ n n ⎧ ⎫ ⎨n ∑ ln t i ⎬ − ⎨n ∑ ln t i ⎬ ⎭ ⎩ i =1 ⎭ ⎩ i =1
    
    InL(η, β) = nInβ − nβ Inη + (β − 1)∑ Int i − η
    i =1
    
    n
    
    −β
    
    ∑t
    i =1
    
    n
    
    ( )
    
    β i
    
    (15)
    
    The derivates of the likelihood function is taken according to the parameters and equaled to zero.
    n ∂InL(η, β) = − nβη−1 + βη− (β+1) ∑ t iβ = 0 ∂η i =1 n ∂ ln L(η, β) t = nβ−1 − n ln η + ∑ ln t i − ∑ (ln i )t βη−β i ∂β η i =1
    
    ( )
    
    ( )
    
    (8)
    
    t ˆ η = exp(Y − ) ˆ β Method of Moments (MOM)
    
    (16)
    
    (9)
    
    When η is eliminated in equation (9), equation (10) is obtained.
    
    To estimate the parameters of two-parameter Weibull Distribution using the Method of Moments parameters, the first and second moments. The k. Moment of the Weibull Distribution having to parameters around zero is given equation (17). k E(t k ) = ηk Γ( + 1) β (17)
    
    ∑ ln t
    i =1 n i =1
    
    n
    
    ˆ β i
    
    ln t i
    ˆ β i
    
    ∑t
    
    1 − = ˆ β
    
    ∑ ln t
    i =1
    
    n
    
    i
    
    n
    
    (10)
    
    The first moment around zero is obtained by writing k=1 and k= 2 in equation (17) [21]. 1 E(t) = η Γ( + 1) β (18)
    
    Equation (10) can be numerically solved with Newton ˆ ˆ method and β is obtained from here. Then β obtained is used ˆ equation (11) and η is estimated.
    
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    2 Ε(τ2 ) = η2Γ( + 1) β
    
    V. Yılmaz and H. E. Çelik / IJNES, 2 (2): 81-86, 2008
    (19) the Mexican Subduction Zone. Institute of Geophysics (UNAM), Ciudad Universitaria, Mexico City, Mexico Geofísica International. 42(1): 69-81. [11] McNally KC, Minster C.B. 1981. Non uni-form seismic slip rates along the Middle American Trench. Journal of Geophysical Research. . 86: 4949-4959. [12] Şen Z, Al-Suba’I K. 2001. Seismic hazard assessment in the Tihamat Asir region, southwestern Saudi Arabia. Mathematical Geology. . 33 (8): 967-990. [13] Yılmaz V, Çelik H.E. 2004. The estimation of earthquake risk in Eskişehir, Turkey. Journal of Science and Technology, Anadolu University. 5(2): 279-283. [14] Yilmaz V. 2004. The statistical analyses of the psychological effects of the earthquake in the Marmara region of Turkey on 17 August 1999. Social Behavior and Personality. 32(6): 551-558. [15] Härtler G. 1994. “Best linear unbiased estimation for the Weibull process. Microelectronics and Reliability. 34(7): 1253-1260. [16] Qiao H, Tsokos C.P.1994. Parameter estimation of the Weibull probability distribution. Mathematics and Computers in Simulation. 37(1) :47-55. [17] Weibull W. 1967. Estimation Of Distribution Parameters By A Combination Of The Best Linear Order Statistics Method And Maximum Likelihood. USAF Report, AFML-TR-67-105/ AF. 61(052):522. [18] Chen Z.1997. Statistical inference about the shape parameter of the Weibull distribution. Statistics and Probability Letters. 36: 85-90. [19] Kappenman R.F.1985. Estimation for the threeparameter Weibull, lognormal, and gamma distributions. Computational Statistics and Data Analysis. 3(1):11-23. [20] Mahdi S, Ashkarb F. 2004. Exploring generalized probability weighted moments, generalized moments and maximum likelihood estimating methods in twoparameter Weibull model. Journal of Hydrology. .285: 62–75. [21] Yılmaz V, Erisoglu M. 2003. The use of statistical parameter estimation methods in the calculation of the parameters Weibull distribution and the application of Weibull distribution to earthquake data. Journal of Statistical Research, Turkey. 2(2): 203-217. [22] Workshop on Earthquake Recurrence. 1999. State of the Art and Directions for the Future: Istituto Nazionale De Geofisica. Rome, Italy. 22-25. [23] Data Bank of Seismology Department of General Directorate of Earthquakes and Disaster Affairs, Ankara, Turkey, 2004.
    
    When the square of the equation (18) is taken and divided ˆ equation (19), equation (20) only depended on β parameter is obtained.
    {∑ t i }2 1 {Γ( + 1)}2 β i =1 = n 2 Γ( + 1) n ∑ t i2 β i =1
    n
    
    (20)
    
    Equation (20) is solved by the help of the standard iterative ˆ methods and the β parameter is obtained. η parameter is found ˆ ˆ with equation (21) by substituting β in equation (18). ˆ η= t 1 Γ( + 1) ˆ β (21)
    
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