1 Introduction

Statistical earthquake occurrence models have get more importance while amount of the available earthquake data increase. These models allow one to reduce the large data sets of earthquake occurrences to statistical parameters for any region. They can be used to predict earthquake occurrences, recurrence periods, maximum ground motions and earthquake hazard at a given region (Cornell 1968). Several statistical models have been proposed to represent the process of earthquake occurrence. The most common model is Poisson model, which assumes spatial and temporal independence of all earthquakes including great earthquakes: i.e. the occurrence of an earthquake does not affect the likelihood of a similar earthquake at the same location in the next time unit.

Although most of the earthquake generation models currently used for seismic hazard evaluation assume a Poisson (Cornell 1968; Caputo 1974; Shah and Movassate 1975) or other memory-less distributions, studies in the last few decades have concluded that the large earthquakes in many regions are temporally dependent (Bufe et al. 1977; Sykes and Quittmeyer 1981; Papazachos 1989; Stein et al. 1997; Parsons et al. 2000). Two kinds of time-dependent models have been suggested: the slip-predictable model and time-predictable model.

A few other probabilistic models have been used to represent earthquake sequences as strain energy release mechanisms. Hagiwara (1975) has proposed a Markov model to describe an earthquake mechanism simulated by a belt-conveyer model. A Weibull distribution is assumed by Rikitake (1975) for the ultimate strain of the Earth’s crust to estimate the probability of earthquake occurrences. Knopoff and Kagan (1977) have used a stochastic branching process that considers a stationary rate of occurrence of main shocks and a distribution function for the space–time location of foreshocks and aftershocks.

Bağcı (1996) investigated seismic risk of western Anatolia between 36°–41° N and 25°–31° E using the Poisson model for earthquake data (1930–1990). Altınok (1991) evaluated the seismic risk of west Anatolia by the application of Semi-Markov model. Sayıl and Osmanşahin (2004, 2005) and Sayıl (2005) applied time-and magnitude-predictable model to the Marmara Region, west and east Anatolia for long-term earthquake prediction.

The subject of this study is to estimate the probability of earthquake occurrences and recurrence periods by using Poisson model from historical and instrumental data for selected characteristic sub-regions in western Anatolia.

2 Seismotectonics of the region

One of the most seismically active regions in the world is the Alpine-Himalayan Belt which extends from the Azores to Indonesia. Anatolia locates in the most active section of this belt in the eastern Mediterranean and involves several important tectonic structures such as North Anatolian Fault Zone (NAFZ), East Anatolian Fault Zone (EAFZ), North-East Anatolian Fault Zone (NEAFZ) and Bitlis Thrust Belt (BTB) shown in Fig. 1. Previous studies about this section have brought out that the Anatolian (which is a part of Eurasian), Arabian and African plates joined and formed a triple junction structure (Karliova Junction, KJ, in Fig. 1) in the east Anatolia (McKenzie 1972; Ketin 1977; Osmanşahin 1983; Özer 1983; Osmanşahin et al. 1986; Kenar et al. 1996). Western Anatolia is one of the four major neotectonic provinces in Anatolia (Şengör et al. 1985).

Fig. 1
figure 1

Plate tectonics model of Anatolia and surrounding area (The Institute of Mineral Research and Exploration, Ankara, Turkey)

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The seismicity in western Anatolia is high (Fig. 2) and displays swarm-type activity with remarkable clustering of low-magnitude earthquakes in time and space (Üçer et al. 1985). Focal mechanisms of earthquakes in western Anatolia indicate that intra-plate deformations arising from vertical movements are occurring inside of the Aegean-Anatolian block. Most of the fault-plane solutions in western Anatolia represent normal faulting, indicative of crustal extension. Tensional axes for these solutions are nearly horizontal and perpendicular to the general east–west trend of graben structure. The Arabian plate moves northward, and forces the smaller Anatolian plate westward between the North and the East Anatolian Fault Zones as from Karliova triple junction. McKenzie (1972, 1978) showed that this motion is transferred into the Aegean in a southwesterly direction, resulting in the northern Aegean being dominated by dextral strike-slip faulting of northeasterly strike. This faulting type has been seen in the recent strong earthquakes, and confirmed by neotectonic observations.

Fig. 2
figure 2

Epicentral map and selected 13 sub-regions of western Anatolia

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3 Data

Both historical (since BC-496, I0 ≥ 5.0 corresponding to MS ≥ 4.4) and instrumental period (until the end of 2006, MS ≥ 4.0), data obtained from the catalogues and bulletins of international data centres given in Tables 1 and 2 have been used in this study. Homogeneity of the data is very important in the analysis. In order to ensure homogeneity, all of the magnitudes have been taken as surface wave magnitude (MS). These magnitudes have been determined by seismologists who compiled the catalogues either from recordings of long-period seismometers, or through the use of experimental scaling relations. Possible discrepancies between the magnitudes computed by different authors for the same earthquake are small, and do not affect the results much in any case.

Table 1 Some of the historical period earthquake catalogues for Turkey and surrounding area
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Table 2 Instrumental period earthquake catalogues made use to form data
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In this study, the experimental scaling relation between surface (MS) and body wave magnitudes (mb) has been estimated by using 204 earthquakes of MS ≥ 4.0 and mb ≥ 3.0 taken from the dataset in the instrumental period (Fig. 3, Eq. (1)). Likewise, the correlation between intensity and magnitude has been determined from the data of 115 earthquakes (MS ≥ 4.4) occurred in the instrumental period (Fig. 3, Eq. (2)). Computed MS−I0 and MS−mb relations are consistent with those of Ambraseys (2001) and Bath (1966), respectively.

$$ \hbox{M}_{\rm s} =1.40\hbox{m}_{\rm b}-2.33 $$
(1)
$$ \hbox{M}_{\rm s} =0.47\hbox{I}_{\rm o} +2.05 $$
(2)
Fig. 3
figure 3

Correlations of Ms−mb and Ms−I0 used in this study

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An important criterion for the analysis is completeness of the data. Namely, the data must include all of the earthquakes that occurred in a certain seismogenic region during a specific time–period with magnitudes larger than a specific minimum (cut-off) magnitude. According to this statement, the smallest magnitude from which earthquakes were reliably reported in the catalogues (historical and instrumental period) used has been chosen as a minimum magnitude (Mmin = 4.0 in our case for all sub-regions) in each region. Maximum magnitude value (Mmax) is magnitude of the biggest earthquake for each sub-region.

Another important criteria is to select the main shocks from dataset. Namely, the dataset has to be cleared from after- and for-shocks to be able to use it in Gutenberg-Richter relationship. Thus, after- and for-shocks have been rejected from dataset.

4 Definition of the method

Since magnitude–frequency relations are formed as the basis of the earthquake occurrence, it is used for the criterion of earthquake activity as times ago. In the investigation of earthquake occurrence frequencies, it seems that they exhibit usually a linear relation. An equation to represent the relation between the magnitude and earthquake occurrence frequencies has been suggested by Gutenberg and Richter (1954). The general form of this well-known equation is:

$$ \hbox{LogN}\left( \hbox{M} \right)=\hbox{a}-\hbox{bM}, $$
(3)

where N(M) (cumulative frequency), is the number of earthquakes equal or larger than M magnitude. Gutenberg–Richter relation does not become linear for all magnitudes. Therefore, the magnitude interval (M1, M2) in which the logN(M) is linear must be known. So that, the relation is undetermined for the large earthquakes since they are a few number. On the other hand, it must be sure that earthquake array is complete for the small earthquakes. Parameters (a) and (b) in the magnitude–frequency relation are constants. Parameter (a) depends on the observation period, the order of the region interested and the seismic activity, and defines a mean annual seismic activity index. Parameter (b) is related to the physics of the earthquakes and gives slope of the linear relation. According to the analysis of worldwide data, it has been noted that b values considerably change as depending on the geological age of the seismotectonic belt (Miyamura 1962). In general, low b-values are related to high stress-drop, high b-values are related to high heterogeneity of material and crack density (Weeks et al. 1978; Urbancic et al. 1992; Wiemer and Katsumada 1999).

For (M, LogN) dataset, a and b-values are commonly computed by using the linear least square approximation. According to this method, parameters (a) and (b) are found by Eq. (4);

$$ \sum_{\rm i=1}^{\rm n} \hbox{LogN}_{\rm i} =\hbox{an}-\hbox{b}\sum_{\rm i=1}^{\rm n} \hbox{M}_{\rm i} $$
$$ \sum_{\rm i=1}^{\rm n} \hbox{M}_{\rm i} \cdot \hbox{LogN}_{\rm i} =\hbox{a}\sum_{\rm i=1}^{\rm n} \hbox{M}_{\rm i} -\hbox{b}\sum_{\rm i=1}^{\rm n}\hbox{M}_{\rm i}^2, $$
(4)

where n is the number of group. Other parameters are as described in Eq. (3). The earthquake occurrence probability with the specific magnitude in a specific period could be determined by using a and b values obtained from Eq. (4). Equation (5) could be obtained from the relation between cumulative and normal frequency. Equation 6 is obtained by dividing the relation of magnitude and frequency to the specified time-period (T1). Equation (7) is obtained by logarithm of Eq. (6). Equations (8) and (9) are obtained from Eq. (7).

$$ \hbox{a}^\prime=\hbox{a}-\hbox{Log}\left(\hbox{bln10} \right), $$
(5)
$$ \frac{\hbox{N}\left( \hbox{M} \right)}{\hbox{T}_1 }=\frac{10^{\rm a-bM}}{\hbox{T}_1}, $$
(6)
$$ \hbox{Log}\left( {\hbox{N}\left( \hbox{M} \right)\hbox{T}_1 } \right)=\hbox{a}-\hbox{bM}-\hbox{LogT}_1 \Rightarrow \hbox{n}\left( {\hbox{M}\rangle \hbox{M}_1 } \right)=10^{\rm a-bM-LogT_1 }, $$
(7)
$$ \hbox{a}_1^{\prime} =\hbox{a}^{\prime}-\hbox{LogT}_1, $$
(8)
$$ \hbox{n}\left( \hbox{M} \right)=10^{\rm a_1^{\prime} -bM}. $$
(9)

The annual mean number n of earthquakes (M ≥ M1) with specific magnitude equal and larger than M1 value in a specific time can be estimated by using these relations. In any regions, occurrence risk in T years of an earthquake with any magnitude M for observation interval of T1 year is calculated by Eq. (10) and recurrence period of an earthquake is estimated by Eq. (11) (Tabban and Gencoğlu 1975).

$$ \hbox{R}\left( \hbox{M} \right)=1-\hbox{e}^{\rm -n\left( M \right)T} $$
(10)
$$ \hbox{Q}=\frac{1}{\hbox{n}\left( \hbox{M} \right)} $$
(11)

5 Analysis for western Anatolia

5.1 Definition of seismogenic sub-regions

A seismogenic sub-region must include seismically homogenous fault segment where every point is assumed as having the same probability for a future earthquake. Sub-regions are mainly defined by two fundamental characteristics. These are a seismic profile and the tectonic regime of the region. Sub-regions should be defined as characteristic seismic areas which are as homogenous as possible. Marking the boundaries between sub-regions is quite difficult in the seismically complex regions like Anatolia. The boundary between sub-regions of different seismic potential should be located close to the highest concentration around the hard core of the more active ones. In these cases, all the possible characteristics such as the distribution of epicentres, the type of faulting, geomorphological conditions, seismicity and the largest event should be taken account. Under the points of this view, study area has been divided into 13 sub-regions (Fig. 2).

5.2 Computation of seismic risk and recurrence period

In this study, the linear least square method (Eq. (4)) has been applied to obtain a and b parameters in Eq. (3) for each sub-region shown in Fig. 2 using the earthquakes of MS ≥4.0 occurred from BC-496 to the end of 2006. Distribution of the earthquakes with the magnitude increment of 0.5 and cumulative frequency values for each sub-region have been given in Table 3, and the computed a and b values for these sub-regions are in Table 4. Figure 4 shows the magnitude–frequency relations.

Table 3 Cumulative frequencies (Ni) with the magnitude increment of 0.5 for the earthquakes occurred in each sub-region
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Table 4 a and b values with a standard errors estimated by the linear least square method for each sub-region
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Fig. 4
figure 4

Magnitude-frequency relations computed by the linear least square method. Thick line shows the estimated relation. Broken and thin lines show confidence interval band of 95% and prediction interval band. σ and R are standard deviation and correlation coefficient, respectively

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Seismic risk and recurrence period values have been estimated by using a and b parameters given in Table 4. In the computations, magnitudes of MS ≥ 5.0 and increment interval of 0.5 were chosen, and Eqs. 5, 8, 9 and 10 for seismic risk and Eq. (11) for recurrence period are used. Observational time interval (T1 year) has been determined by the completeness condition (Table 4, second column) of each sub-region. Maximum magnitude value (Mmax) has been selected as magnitude of the biggest earthquake for each sub-region. Computations have been done for decades in the next 100 years. The results are given in Table 5. Figure 5 shows the standardized residuals of the predicted seismic risk values.

Table 5 Seismic risk and recurrence period values estimated by using a and b values for the earthquakes of 5.0 ≤ MS ≤ Mmax with the magnitude increment of 0.5 in observation interval (T1 year) of each sub-region. Maximum magnitude value (Mmax) is magnitude of the biggest earthquake for each sub-region
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Fig. 5
figure 5

The standardized residuals versus the predicted seismic risk for each sub-region. σ and R are standard deviation and correlation coefficient, respectively

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6 Discussion and conclusions

In this study, seismicity of western Anatolia has been investigated by means of computations of the magnitude–frequency relation, seismic risk and recurrence period, and the results tried to be interpreted and related with the active tectonic of region. The map in Fig. 2 showing the main faults and epicentre distribution demonstrate quite high seismic activity in the region. At the result of these observations, the study area has been separated to 13 sub-regions (Fig. 2). Magnitude–frequency relations have been determined by the data sets (MS ≥ 4.0) in different observation intervals for each sub-region. Then seismic risk and recurrence periods for the time periods of decades in the next 100 years and magnitude interval of 5.0 ≤ MS ≤7.5 have been estimated from a and b values computed to determine the magnitude–frequency relations.

In twos of the highest and lowest b-values were determined as 0.66 ± 0.07, 0.66 ± 0.07 for sub-regions 3, 12, and as 0.43 ± 0.03, 0.42 ± 0.03 for sub-regions 1, 9, respectively. As it is well-known, the high b value implies that the high seismic activity had rolled in that region.

According to the seismic risk estimations, the highest-earthquake occurrence probability of MS ≥ 7.0 in the next 100 years is 80.6% (σ = 0.20, R = 0.87) for sub-region 9 and 77.8% (σ = 0.17, R = 0.90) for sub-region 1. Recurrence times for the earthquakes with the same magnitude have been found as 61 and 67 years in these sub-regions. The highest-occurrence probability and recurrence time of an earthquake with MS ≥ 7.5 in the next 100 years have been found as 63.2% (σ = 0.20, R = 0.87) and 100 years for sub-region 9, respectively. Sayıl and Osmanşahin (2005) had applied the regional time- and magnitude-predictable model at the same sub-regions of western Anatolia by using earthquakes with the magnitude MS ≥ 5.5 to compute the occurrence probabilities and the recurrence intervals of large earthquakes. The conclusions of present study agree with the results of the regional time- and magnitude-predictable model.

In conclusion, the large earthquakes have occurred in western Anatolia like its other regions and will occur in the future. For this reason, seismicity studies should be continued for minimizing the losses of life and property caused by earthquakes. Therefore, the tectonics features and active faults and activity of the region should be defined carefully and followed continuously.