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THE RISK ANALYSIS OF SEISMIC ACTIVITY INCIDENCE IN ROMANIA
I. M. Dragan, Al. Isaic-Maniu
Academy of Economic Studies, Bucharest, Romania e-mail: [email protected], [email protected]
ABSTRACT
In Romania there is one of most powerful seismic activity region from Europe, known as Vrancea. In the past 300 years, a single major seismic event occurred with an epicenter outside this area (1916). This paper starts from going over all major seismic events, with a magnitude of over 6 degrees on Richter's scale, which were documented. Was tested the most plausible statistic behavioral model and was determined the probabilities for future large scale earthquakes, by different time horizons.
Key words: seismic risk, modeling, validation, prediction, statistic distribution
1 INTRODUCTIVE FEATURES
Europe, from the geologically point of view, confirms high seismic risk areas such as Italy, Turkey, Iceland, Serbia, Bulgaria, Greece and Romania, as in figure 1. The seismic intensity zones are marked by color code. So in Romania stand out the Eastern region of the country (Figure 2) the Carpathian Mountains. Agglomerations of black dots on the map represent earthquakes frequencies. One can remark a region of high concentration of earthquakes, which is known as the Vrancea area.
In general, is recognized that the occurrence of major seismic phenomena is a "rare event" from a statistical point of view. Due to the very large time horizon that can be taken into observation as against to registering events in artificial systems, as well as the non-periodicity of these events, there is the possibility of interpretation and statistical modeling of these seismic phenomena. In Romanian: Dragomir (2009), Lungu (1999), Lungu and Arion (2000), Radulescu (2004).
The statistical studies regarding the earthquakes usually start from the fact that rare events are best described using the exponential law - if considering the succession of time intervals between events, or Poisson's law - if it is intended to model the frequency of earthquakes (Sacuiu & Zorilescu, 1978; Johnson, Kotz & Balakrishnan, 1994; Evans, Hasting & Peacock, 2000).
The easiness of using these two distribution laws, distinct in nature, consists of the fact that they are defined by the same parameter, characterizing the same phenomenon - the behavior of a system in time, from both continuous and discrete points of view. A previous study made on seismic phenomena in Romania (Voda & Isaic-Maniu, 1983) covering the time period 1400-1977, has failed to confirm the hypothesis of an exponential behavior, the confirmed model being the bi-parametric Weibull model.
In the followings, we shall extend the area of investigation starting with the year 1100, with some additions to the identified supplementary information, as well as to the earthquake in 1977, the last one taken into account in the previous study.
We considered major seismic events those with a level of over 6 degrees on Richter's scale. Obviously, historical assessments are somewhat subjective, as the intensity was evaluated indirectly, since Mercalli (1931) and Richter's (1956) scales are more recent. The chronicles used to register that: "the earth had been shaken and the bells were ringing by themselves in Golia's tower"
(n.n. Iasi - Romania), which indicates that an important seismic event took place. We used information in the profile literature (Constantinescu & Marza, 1980) as well as other official sources as those of the National Institute for the Physics of Earth (www.infp.ro).
PEAK GROUND ACCELERATION <m/ss> 10% PROBABILITY OF EXCEEDANCE IN 50YEARS. 47Svear return period
H
0 c >.2 0.4 0 LOW .8 1.6 2 MODERATE .4 3.2 4 HIGH .0 4.8 VERY HIGH
HAZARD HAZARD HAZARD HAZARD
Source http://geology.about.com/
Figure 1. The hazard of seismic activity in Europe
. « x,
' -r «H Y. K^ll - >"v ' ' V
» * , . * 1 '
s ' • - - jCt.' \
r ? * 9 \
- V g V t V* * T 1
• - * r '•v.gf.'v'i.iitL. .
» il a
• ' v V V'
«V Y!,'^ •
•V* ■ » ' ■ *
y * «
Source Geoscience Interactive Databases - Cornell University/INSTOC Figure 2. The seismic activity in Romania
2 THE RECORD OF MAJOR SEISMIC ACTIVITY
The main seismic events which occurred in Romania, and their characteristics, as they were recorded at the time in documents, or in modern and official registrations, were as follows in table 1.
Table 1. The main indicators of risk and reliability
November 5th, 1107 August 8th, 1126 April 1st, 1170 February 13 th May 10th, 1230 year 1276 year 1327 October 10th, 1446 August 29th, 1471 November 24th, 1516 July 19th, 1545 October 16th, 1550
November 2n< August 17th,
, 1558 1569
May 10th, 1590 August 10th, 1590 August 4th, 1599 May 3rd, 1604 November 24th 1605 January 13th, 1606 October 8th, 1620 August 9th, 1679 August 8th, 1681 June 12th, 1701 October 11th, 1711
May 31st, 1738 December 7th, 1746 year 1750
6.2 degrees
6.2 degrees 7.0 degrees
7.0 degrees
7.1 degrees 6.5 degrees
7.0 degrees
7.3 degrees
7.1 degrees
7.2 degrees 6.7 degrees 7.2 degrees 6.1 degrees 6.7 degrees 6.5 degrees 6.1 degrees 6.1 degrees 6.7 degrees
6.7 degrees
6.4 degrees 7.9 degrees
6.8 degrees 6.7 degrees 7.1 degrees 6.1 degrees 7.0 degrees
6.5 degrees 6.0 degrees
Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter Richter
January 18th, 1778 March 18th, 1784 April 6th, 1790 December 8th, 1793 October 26th, 1802 March 5th, 1812 January 5th, 1823 November 26th October 15 th.
January 23r October 15t October 17t April 27th,
1829 1834 , 1838 , 1847 , 1859 1865
November 13th, 1868
November 23rd, November 26th
1868 1868
October 10th, 1879 August 31st, 1894 September 13 th, 1903 October 6th, 1908 May 25th, 1912 January 26th, 1916 March 29th, 1934 November 10th, 1940 March 4th, 1977 August 30th, 1986 May 30th, 1990 October 27th, 2004
6.1 degrees Richter
5.8 degrees Richter 7-8 degrees Richter
6.1 degrees Richter
7.9 degrees Richter 6.5 degrees Richter 6.0 degrees Richter 7.5 degrees Richter, 6.0 degrees Richter 7.5 degrees Richter
6.2 degrees Richter 6.0 degrees Richter
6.4 degrees Richter
6.0 degrees Richter
6.5 degrees Richter
6.1 degrees Richter
6.2 degrees Richter 7.1 degrees Richter
6.3 degrees Richter 7.1 degrees Richter
6.3 degrees Richter
6.4 degrees Richter 6.9 degrees Richter 7.7 degrees Richter 7.4 degrees Richter 7.1 degrees Richter 6.9 degrees Richter 6.0 degrees Richter
In the area of Vrancea (analyses of the area in Ivan, 2007; Ivan, 2011; Ardelean, 1999) there are registered almost daily earthquakes under 3 degrees.
Figure 3. The distribution of major seismic events in 100 years intervals in Romania
I6n 14 12 10 8 6 4 2 0
c»' f»' f»' f»' <£,' c»' c»' c»
<1? ^ k«3 ^ <V° ^ ^
years
3 THE STATISTIC REPRESENTATION OF MAJOR SEISMIC ACTIVITY
The registered data were processed firs of all, statistically descriptive. The results as distribution series are presented in Table 2, the grouping being done in intervals of 100 years.
Table 2 -
he distribution of major seismic events in 100 years intervals
No. Interval(years) Number of major seismic events
1 1100 - 1200 4
2 1200 - 1300 2
3 1300 - 1400 1
4 1400 - 1500 2
5 1500 - 1600 7
6 1600 - 1700 7
7 1700 - 1800 9
8 1800 - 1900 15
9 1900 - 2000 11
10 2000 - 1
TOTAL n = 59
The series (Table 2 and Figure 3) seems to suggest an acceleration of events in the last 250 years: in the first decade Di one earthquake was registered; Q2 (Me) = 5.5 earthquakes, and in D9 -14.6 . This could be the effect of an energetic acceleration in the intensity of the activity of the terrestrial crust, but most probably it is the result of information inconsistencies in the medieval period which seem to suggest this seismic intensification. The maximum value in an interval of 100 years is 15 major seismic events (1800 - 1900). The total number of major earthquakes is 59. The average in a 100 year interval is 5.9, with a standard deviation of o = 4.75 and a variation coefficient of CV = 0.805 which suggests a strong heterogeneity of the observation series. Standard error = 0.502.
Table 3 - Descriptive statistics
Statistic Value
Sample Size 10
Range 14
Mean 5,9
Variance 22,544
Std. Deviation 4,7481
Coef. of Variation 0,80476
Std. Error 1,5015
Skewness 0,72385
Excess Kurtosis -0,36372
' ( s2 )
The shape of the series is completed (table 3) with the values of the Skewness coefficient:
M 2
= 0.724 (where m3 is the centered moment of rank 3, and s - the centered moment of
rank 2), and respectively (f2 - Kurtosis coefficient: (f2 = =-0.364 (where is the centered
(s2)
moment of rank 4). The minimum value in a 100 year interval was 1, and none of intervals frequencies were zero. The value of the first quartile was Qi = 1.75 and the third was Q3 = 9.5 respectively.
4 THE STATISTIC MODEL OF THE SEISMIC INCIDENCE ACTIVITY
In order to analyse the process of earthquake occurrence, we tested several distribution laws, obviously starting with "the law of rare events" - Poisson, continuing with the exponential law (Evans, 2000) and Weibul (Isaic-Maniu, 1983). For the series of 50 years interval (Dragan & Isaic-Maniu, 2011), the best results were obtained for the log-logistic statistic model (Johnson, Kotz & Balakrishnan, 1995; Evans & Hastings, 2000; Stephens, 1979; Paiva, 1984; Ahmad, Sinclair & Werritty, 1988) by filtering three different selection tests. Probability Density Function (PDF)
<*._ ß r* - 7
f <* )=q
ßr r \ßI 2 r * - 7 I
al a
1 i"
v v a ' y
Cumulative Distribution Function (CDF):
F <* ) =
r f V I-1
a
1 +
v
l 7 y
(1)
(2)
For the distribution of 100 years interval, the best results were obtained for the Beta statistic model. The general formula for the probability density function of the beta distribution is:
( v n )"1-1 ( h x )"2-1
f (X ) = ^--^--r, n < x < h;a1,a2 > 0 (3)
V ' ui \iu \a1 ' 1 2 K J
B (a1,a2)(h-n)
where a1 and a2 are the shape parameters, n and h are the lower and upper bounds, respectively, of the distribution, and B (a1,a2) is the beta function. The beta function has the formula
1
B (a, p) = \ ta1 (1 -t)P-1 dt (4)
0
The case where n = 0 and h = 1 is called the standard beta distribution. The equation for the standard beta distribution is
, , xa1-1 (1- x)a2-1
f (X) = —¿i-V",0 < x < 1;a1,a2 > 0 (5)
B (a1,a2)
Typically we define the general form of a distribution in terms of location and scale parameters. The beta is different in that we define the general distribution in terms of the lower and upper bounds. However, the location and scale parameters can be defined in terms of the lower and upper limits as follows: location = n; scale = h - n.
Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.
The following (figure 4) is the plot of the beta probability density function for four different values of the shape parameters
Figure 4. The beta probability density function
The formula for the cumulative distribution function of the beta distribution is also called the incomplete beta function ratio (commonly denoted by Ix) and is defined as
f > -1 (1 -1)a -1 dt
F ( x) = Ix (a,a2 ) = -——f-^-,0 < x < 1;a,a2 > 0
B (a1, a2)
where B is the beta function defined above.
The formulas below are for the case where the lower limit is zero and the upper limit is one. Mean «
(6)
Mode
a1 + a2
a1 -1 i
-1-,a,a2 > 1
a1 + a2 - 2
Range 0 to 1
Standard Deviation
^a 2
y a +a2 ) (a +a2 +1)
Coefficient of Variation
i
a
Skewness
a1 (a1 +a2 +1)
2(a2 -a1 )^Ja1 + a2 +1 (a1 + a2 + 2 ),Ja]a2
First consider the case where a and b are assumed to be known. For this case, the method of moments estimates are
f-
x (1 - x )
V -1
(7)
a2 = 11 -
(i -x)
^ x (l - x)
^ -1
(8)
where x is the sample mean and s2 is the sample variance. If a and b are not 0 and 1, respectively,
then replace x with X—a and s2 with —S—- in the above equations. b — a (b - a)2
For the case when a and b are known, the maximum likelihood estimates can be obtained by solving the following set of equations
1 n f Y — a I
) —0(a, + «2 ) = - Z lg H-I (9)
n i=1 V b a J
1 n f b — y I
¿(«2 )—^(«1 + «2 ) = - Z lg f I (10)
n i=l V b a J
The maximum likelihood equations for the case when a and b are not known are given in pages 221-235 of Volume II of Johnson, Kotz & Balakrishnan (1994).
5 FITTING THE DISTRIBUTION
In order to test the statistic nature of the distribution, we used the Kolmogorov-Smirnov, Anderson-Darling and Pearson-Fisher tests (Stephans, 1979; www.mathwave.com; www.vosesoftware.com).
Kolmogorov-Smirnov
The test is defined for the hypothesis
H0: the distribution of earthquakes is Beta
Hj: the distribution of earthquakes is not Beta.
We compute the empirical distribution function F(x):
1 n
^M^Ilx, (11)
" i=1
where IXis the indicator function, equal to 1 if Xi < x and equal to 0 otherwise.
The Kolmogorov-Smirnov statistic for a given cumulative distribution function F (x) is
Dn = sup IF ( x)- F ( x) (12)
x
and F(x) the theoretical values of distribution.
The Dn computed value is compared to the maximum admitted equivalent. The statistic computed value for the presented case resulted in Dn = 0,19999 is inferior to the critical level 0.48893 for a significance level of a = 0,01, respectively inferior to value 0.40925 for 95%. The Beta distribution hypothesis is not rejected even for a = 0,2 .
The Anderson-Darling test - is also a distance test, proposed by Wilbur Anderson and Donald A. Darling in 1952. The statistic of the test is
A2 = - N - S (13)
where:
S = 1 [ln F ( Xt) + ln (1 - F (X„+1-.))] (14)
in which F is the cumulative distribution function. For a significance level a, we validate one of the two hypotheses H0 and H1. The critical values for various specified distributions are computed by Stephens (1979).
The value of the statistics of the test: 4.3274 refute the Beta distribution fora = 0,01, with critical value 3.9074, respectively 2.5018 for a = 0,05
Pearson-Fisher Statistic
Chi Square or Pearson-Fisher (%2) test was proposed as a measure of random departure between observation and the theoretical model by Karl Pearson (Pearson 1900). The test was later corrected by Ronald Fisher trough decrease of the degrees of freedom by a unit (decrease duet of the existence of the equality relationship between the sum of observed frequencies and the sum of theoretical frequencies, (Fisher 1922), and by the number of 692 unknown parameters of the theoretical distribution when they come as estimated from measures of central tendency (Fisher 1924).
The chi-square test is used to test if a sample of data came from a population with a specific distribution. An attractive feature of the chi-square goodness-of-fit test is that it can be applied to any uni-variate distribution for which you can calculate the cumulative distribution function. The chi-square goodness-of-fit test is applied to binned data (i.e., data put into classes).
The test is defined for the hypothesis
H0: The data follow a specific distribution
H1: The data do not follow the specific distribution
The statistic is calculated as (in original):
= sji^W =i (15)
[ m J ¿=1 Ei
where O. is the observed frequency for bin i and E. is the expected frequency for bin i and is calculated by
E. = N(F(Yu)-F(Y )) (16)
where F is the cumulative distribution function and Yu and Y. are the upper and lower limits for class i.
The test statistic follows, approximately, a chi-square distribution with (k - c) degrees of freedom where k is number of non-empty cells and c - the number of estimated parameters for the distribution +1.
Therefore, the hypothesis that data are from a population with the specified distribution is
rejected if
2 2 X ^ Xa,k-c
where %2ak-c is the chi-square percent point function with k - c degrees of freedom and a significance level of a.
The computations lead to a value of the xC = 7,3289E-8 statistic inferior to the critical value
xl01 = 6,6349, so that the H0 hypothesis is accepted with a probability of 99%. Either for different
values of a (0.02; 0.05; 0.1) respectively 0.2 (critical value 1.6424) the Beta distribution hypothesis is not rejected.
Considering the three applied tests (Kolmogorov-Smirnov, Anderson-Darling and Pearson-Fisher), two of them confirm with a high confidence degree the Beta distribution, by parameters:
a1 = 0.34225 a2 = 0.63562 a = 1.0 b = 15.0
The Probability Density Function (pdf) for the estimated values of the parameters is presented in Figure 5 and The Hazard Function in Figure 6.
Probability Density Function
1,2 1,1 1
0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1
10
14
Beta (0,34225; 0,63562; 1; 15)
Hazard Function
1
)
Figure 5. The Probability Density Function
Beta (0,34225; 0,63562; 1; 15)
Figure 6. The Hazard Function
1, ..
Table 4 presents the values of the main indicators of the Beta distribution for a number of x , 15 events.
Table 4. The Values for pdf, CDF, h(x) §i S(x)
Statistic Values computed for x (earthquakes) equal to:
Functions 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
pdf- probability density function 0.115 0.075 0.059 0.051 0.045 0.042 0.040 0.039 0.038 0.039 0.040 0.044 0.054
CDF -
cumulative
distribution
function 0.000 0.328 0.419 0.485 0.539 0.587 0.631 0.672 0.711 0.749 0.788 0.827 0.869 0.918 1.000
h(x) - hazard
function - 0.171 0.129 0.115 0.110 0.110 0.114 0.122 0.134 0.153 0.183 0.234 0.338 0.655 -
S(x) -distribution 1.000 0.672 0.581 0.515 0.461 0.413 0.369 0.328 0.289 0.251 0.212 0.173 0.131 0.082 0.000
0
2
4
6
2
4
6
6 CONCLUSIONS
In the followings, through simulation operations for the values of the Beta distribution, we formulate various hypotheses on the occurrence of seismic events, for the confirmed statistic model.
Thus, if we limit, for a 100 years interval, the number of major seismic events between x = 2 and x2 = 10 respectively, we have: P (x < xj) = 32.51 % P (x > X1) = 67.19 % P (x1 < x < X2) = 42.14 % P (x < x2) = 74.95 %
P (x > X2) = 25.05 %
It is an optimistic variant that the chances for less than two major seismic events to occur in a 100 year interval are around 33%, and for more than 10 major seismic events is reduced to 25%. If we modify the limits to x1 = 1 and x2 = 3 major seismic events, then:
P (x < xx) = 0 % P (x > x0 = 99.99 % P (x1 < x < x2) = 41.89 % P (x < x2) = 41.89 % P (x > x2) = 58.11 %
So, there is a small probability that in Romania, less than 1 earthquake will occur, and slim chances that more than 3 earthquakes will occur. There is a probability of approximately 42% that in an interval of 100 years, between 1 and 3 events could occur.
If we modify the limits to x = 4 and x2 = 12 major seismic events, then:
P (x < x0 = 48.5 % P (x > x0 = 51.5 % P (x1 < x < x2) = 34.3 % P (x < x2) = 82.7 % P (x > x2) = 17.3 %
Romania represents an unique case in the world, from a seismic point of view: earthquakes of over 7 degrees Richter in magnitude which originate from Vrancea affect approximately 50% of the territory and approximately 60% of the population, including the capital, Bucharest. Nonetheless, the earthquake in 1977 was not the most powerful. It was only the fourth in magnitude among the earthquakes in the last 200 years. In Romania, there were 6 earthquakes of over 7 degrees Richter in the last 200 years. More technical details on the area Vrancea can be found in Ivan (2007, 2011).
In the case of Romania, the warning period for an earthquake is 25-30 second, which is relatively short in comparison to Mexico City - 60 seconds. However, it is enough to interrupt dangerous activities: nuclear reactors, heavy water production, chemical industry, gases, electricity and water. For trains and subways, stopping the electrical power is enough to stop the carriages.
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