 In the previous lecture, we discussed about the earthquake measurement parameters, earthquake measuring equipment and the modification of the earthquake waves due to the soil condition. In earthquake measuring parameters, we discussed about two major earthquake measuring parameters that is the magnitude of earthquake and the intensity of earthquake, both of them denote the size of the earthquake that has occurred. The magnitude of the earthquake express the size of the earthquake in an objective fashion that is it is denoted by some measurement of the ground motion which is then expressed as a magnitude by certain definition whereas the intensity of earthquake is a subjective measure by looking at the damages and destructions that have taken place at different places, we estimate the size of the earthquake. The intensity of earthquake was prevalent mostly in olden days when we did not have very good equipment for measuring the ground motions produced due to earthquake. Therefore, from the visual observation people used to determine the size of earthquake that has taken place and the observations were categorized under different categories and from that one can understand the size of the earthquake that has taken place. Accordingly, a intensity scale was devised and the earthquake used to be measured in that with the help of that particular scale called the intensity scale. In the magnitude of the earthquake the thing that is required is to measure the ground motions or the ground motions giving rise to the particle acceleration velocity and displacements. These measured quantities are utilized in expressing the magnitude of the earthquake and there are certain fixed definition for the magnitude of earthquake that we have discussed in the previous lecture. So, far as the earthquake measurement equipment is concerned we discussed about the seismographs especially the Wood Anderson seismograph that was the old seismograph or rather the oldest seismograph and the principle of working of the seismograph was described. Then we talked about the long period and short period seismograph providing the displacement and the acceleration records directly. After that we discussed about the modification of the ground motions due to the soil condition and mainly we pointed out the kind of modification that takes place for hard soil condition for soft soil condition and what is the kind of energy concentration that takes place when we have a modification of the ground motion due to the presence of the soft soil. In this lecture we will look at the seismic hazard analysis which is very important to earthquake engineers. The seismic hazard of a particular site denotes the hazard potential of the site for an earthquake. By hazard potential what we mean is that the maximum peak ground acceleration that can take place in that particular site or the different kind of damages that can occur due to the earthquake at the site. Mostly the sites for which we perform a seismic hazard analysis are the sites in the vicinity of which we have certain fault lines or earthquake sources especially the presence of earthquake sources which are quite active for those earthquake sources which are surrounding that particular site we perform a seismic hazard analysis of a particular region or the site. There are quite a number of objectives behind the seismic hazard analysis. Firstly seismic hazard analysis enables us to define the maximum value of the peak ground acceleration and for which the structures should be designed in that particular region. Also many a time if we have some systematically recorded earthquake data then from that one can obtain straight away the future response spectrums for which the structures should be designed for future earthquakes. Secondly the seismic hazard analysis is very much useful in obtaining the seismic or in determining the seismic risk of structures in that particular region. In that what is done is that the vulnerability of the structure is first assessed or obtained then after that the vulnerability of the structures for a particular earthquake is then combined with the vulnerability of the region for different sizes of earthquake. So these combined result ultimately provides a seismic reliability analysis of a structure in the region. Also the seismic hazard analysis is important in obtaining the micro donation map of a particular region. This micro donation map is utilized in many ways. One of the most important utilization of the micro donation map one of the major utilization of the micro donation map is in obtaining a systematic planning of a particular region safe against earthquake. Also the micro donation map helps in providing the resource allocation to different parts of the region for earthquake protection. The seismic hazard analysis is a quantitative estimation of the most possible ground shaking at a site. The estimation can be made using deterministic or probabilistic approaches. They require some or all of the following. The firstly the knowledge of earthquake sources, fault activity and fault rupture length. Now the knowledge of earthquake sources mean that the whether the earthquake source is a line source or it is a area source or it is a point source whereas the fault activity means the whether the fault is an active fault in the sense that there is an earthquake that is frequently occurring from that particular fault. The dead faults are the ones in which it is found that for a long period of time there is no earthquake that has been observed coming from that particular fault line. So that is what we understand by fault activity. The other important thing is the fault rupture length that we have discussed before and we have seen that whenever there is an earthquake then along the fault line there is a rupture or the existing fault gets modified because of the earthquake and these fault rupture length is observed from the kind of damages or destruction that takes place at the epicentral region and from there one can assess the fault rupture length. Also these days we have data previous data earthquake data available which basically gives a relationship between or empirical relationship between the rupture length and the energy release of a particular earthquake. So therefore by measuring the ground displacements or ground accelerations at different places one can assess what is the energy release of the earthquake and from that energy release one can indirectly obtain the fault rupture length. So the first part is to assess the earthquake sources that are existing in the vicinity of the site in terms of fault activity nature of the earthquake source and the fault rupture length. Next is the past earthquake data should be available at the particular site so that one can build up a relationship between the rupture length and the magnitude of the earthquake that is what I told you before. The magnitude of the earthquake that takes place in that particular region should be systematically collected and must be available so that one can obtain a probability density function for the magnitude of earthquake coming from a particular fault line and it is very important and we will see later that this probability density function of the magnitude of earthquake is utilized in obtaining the probabilistic seismic hazard analysis. Apart from that the historical and instrumentarily recorded earthquake data or the time history records of the earthquake data must be available and these earthquake data can provide a peak ground acceleration peak ground velocity or peak ground displacement or in other words the time histories of any one of these quantities must be available and if the response spectrum of the earthquake or the previous earthquakes they are available then that can also be usefully used employed for obtaining the seismic hazard analysis. Now as I said before we have two types of seismic hazard analysis one is the deterministic seismic hazard analysis the other is the probabilistic seismic hazard analysis. In the deterministic seismic hazard analysis the computations that are to be carried out are quite simple therefore it is quite often used by practicing engineers for finding out the seismic hazard of a particular region. It is a simple procedure to compute ground motion to be used for safe design of structures in general and for specially these structures in particular specially for the kind of the structures like hospital buildings or very big halls or school buildings for that one should have specifically a maximum peak ground acceleration for that particular site for which those structures must be designed. These deterministic seismic hazard analysis is generally performed for cases where the sufficient earthquake data is not available at the site in the sense that we cannot carry out the probabilistic seismic hazard analysis for the site therefore we take recourse to the deterministic seismic hazard analysis with the limited data that we have. It is also conservative and does not provide likelihood of failure that is the probability of the accidents of certain level of the ground acceleration is not available from the deterministic seismic hazard analysis. In the deterministic seismic hazard analysis one only specifies the maximum earthquake parameter which should be taken into consideration for the design analysis of structures in the region. It can be used specifically for the deterministic design of structures. Probabilistic design of structure is not possible from the results of the deterministic seismic hazard analysis. The deterministic seismic hazard analysis is quite often used for micro donation of large cities for seismic disaster mitigation. In that the entire city is micro zoned into smaller regions and for each region one can specify the maximum level of the ground acceleration for which the structures should be designed in that particular sub zone or sub area. The deterministic seismic hazard analysis consists of the following not 5 steps 4 steps. First one is the identification of sources including their geometry as I told you before these sources could be a point source could be a line source could be an area source. So, one has to identify what kind of sources of earthquake that are existing surrounding the site. Then the evaluation of the shortest epicentral distance or hypocentral distance that means if it is a line source or an earthquake source then one has to find out what is the shortest distance between that line source or the area source to the site in question. So, that is another important thing that is to be carried out for those kinds of sources. Then identification of maximum likely magnitude at each source. So, this is generally obtained from the past earthquake recorded data and from that data one can identify what is the maximum level of the magnitude of earthquake that is expected at the site. And then finally one has to select an appropriate predictive relationship valid for the region and this predictive relationship is one of the predictive relationship is shown over here. This predictive relationship gives or predicts the peak ground acceleration at a particular place given the magnitude of earthquake and the epicentral distance of that particular place. So, this is an empirical equation and this kind of empirical equation is required in the seismic hazard analysis of a particular site. In fact, these predictive relationship constitutes the most important equation that is used in the seismic hazard analysis of a structure. Therefore, one has to find out the appropriate predictive relationship that is valid for the region. If it is not available as such then one has to see that what is the most similar site that is site for which this kind of predictive relationship is existing and then one can use that predictive relationship for this deterministic or probabilistic seismic hazard analysis of that particular site. The method of the deterministic seismic hazard analysis is demonstrated with the help of this example. This is the site in question and surrounding the site we have got three sources of earthquake. These two sources of earthquake are the point sources. This is a line source and the sources or the epicentral distance for the different sources that can be computed from the coordinates that are given over here. For finding out the epicentral distance for the site with respect to this source we simply join these two points. For this source we simply join these two points and get the distances. So far as finding out the epicentral distance for the line source we take the shortest distance of the site from the source and for that we drop a perpendicular from this site to this line and that distance becomes the epicentral distance. We take the shortest distance because if we look at the equation then we find that for the shortest distance would provide the maximum effect on the particular site coming from this source. The maximum magnitudes which are specified for these three sources are 7.5 for source 1, 6.8 for source 2 and for source 3 it is 5. So, they are put in a tabular form over here for source 1, 2, 3 we have the magnitudes recorded as 7.5, 6.8 and 5.0. The epicentral distances that were calculated 60.04, 70.63 for sources 2 and 3 and for source 1 23.70 that is the perpendicular distance of the site to the line source. Then once we have the epicentral distance then we put in the values of the magnitude maximum magnitude of the earthquake and the epicentral distance into this equation and we get the value of peak ground acceleration in gals that is centimeter per second square. So, those values of the peak ground accelerations are computed and expressed in G units and they are shown over here for the source 1 we have a peak ground acceleration of 0.49 g, for source 2 it is 0.1 g and source 3 it is 0.015 g. Thus the maximum peak ground acceleration that is expected at the particular site is 0.490 g and we say that the hazard level for the particular site is represented by a peak ground acceleration of 0.49 g or in other words in designing any structure in that particular site we should take a peak ground acceleration of 0.49 g. Now, for this particular problem even without performing this kind of calculation one can see that it is easily or it is very apparent that line source or the source 1 will have or dictate the hazard level for the site because the distance epicentral distance for this particular source to site is the least amongst them and also the magnitude of earthquake or expected magnitude of earthquake in that particular source is the largest of the three. Therefore, it is quite obvious that this particular source would determine or dictate the hazard level for the site. However, there could be problems in which even if a particular source is very close to the site but the magnitude of earthquake that is expected in that particular source may be small compared to others. In that case one has to perform this kind of calculations and put the quantities in a tabular form in order to identify the maximum hazard level. Now, we come to the probabilistic seismic hazard analysis and this probabilistic seismic hazard analysis is carried out when we have sufficient recorded data for that particular site coming from different sources of earthquake and the data that are recorded will permit to obtain the different quantities of interest that is required in performing the probabilistic seismic hazard analysis. Now, since the probabilistic seismic hazard analysis requires some elementary knowledge of the statistics of random variable, we must have an idea about them. It is presumed that you have already some knowledge or the elementary knowledge of the statistics of random variable. However, for the completeness and for recapitulation of those statistics let me first describe them in connection with the probabilistic seismic hazard analysis. The first is the statistics of the random variable is obtained with the help of certain quantities which are called the statistical quantities of a random variable. Let us consider X to be a continuous random variable means that X can take any value between minus infinity to plus infinity. However, in reality the X can take values only within some finite interval and the X values are also discretized in the sense that the X values are sampled at certain interval. Therefore, making it a quantity which is in fact not a continuous random variable, but a random variable which can take a number of discrete values between the finite interval. So, now let us say we are interested in finding out the probability of X being less than equal to a specified value Xi. Then we can count the number of values of X that are less than or equal to the value of Xi and let that number be n Xi. Then we say that the probability of X being less than equal to Xi is equal to n Xi by n. Although we say that X is a continuous for practical problems we get the values of X as sampled values at an interval that is what I told you before. However, from that sampled values we can obtain the probability of the particular value being less than a certain level in this particular fashion. And then from the plot of them we can obtain a continuous curve by joining the points and express the probability density function or the cumulative distribution function as a continuous curve. So, that is the first part of the random variable X. Now, this random variable X is characterized by certain quantities which we call as the statistics of the random variable. Firstly, we express the random variable by a mean value and the mean value is described by this particular equation that is 1 by n and summation of all the possible values that X can take and that gives the mean value X bar. Next week represent it by a mean square value and the mean square value is defined by this that is 1 by n summation of the square of all the possible values that X can take. Then we take sometimes represent also it by RMS value which is simply a square root of the mean square value. We can define also the variance of the random variable and the variance is defined as 1 by n multiplied by summation of X i minus X bar whole square that is for all the values that X can take from those values we deduct the mean value and then take a square of that and add them and divide by n that gives the variance of the random variable. The standard deviation is simply the square root of the variance. Then we try to describe the cumulative distribution function, probability density function and the moments of the probability density function. So, these are some of the elementary statistics of a random variable that we must know in order to deal with a random variable and carry out the probabilistic seismic hazard analysis. If we talk of the cumulative distribution function then the cumulative distribution function is drawn in this particular fashion. Say there is a sample, a number of sample values of X then we can obtain the probability of X being less than a particular value X 0 or that will be given by n X 0 that is the number of values of X that are below or equal to X 0 and that divided by n and that we put as an ordinate over here. Similarly, for different levels of X 0 one can have such ordinates and if we join the points of those ordinates then we can get a card and this card becomes a continuous card and we say that this denotes the cumulative distribution function for the card. Now if it is possible to give a mathematical expression for this continuous card then we have a mathematical expression for the cumulative distribution function in terms of the variable X and some other statistical quantity that we will see later. Next is the probability density function. The probability density function is defined as the slope of the CDF curve at a particular point and this is denoted by dfx dx. So, therefore if we have a mathematical expression for fx or the cumulative distribution function in terms of X then one can differentiate that with respect to s for every point and then can obtain an expression for the probability density function. However, we can straight away find out the probability density function from the sample values itself the way we have obtained the cumulative distribution function. Say the sample values of X has some maximum value that is the X max and a minimum value which is X minimum. Then we divide the interval between the X max and X mean by a number of discrete value or by a number that gives say interval or the sampled interval delta X. The larger the value of n bar smaller becomes the value of delta X and better becomes the estimate of the probability density function. Now, we find out the number of values which lie within each interval that is we say set X 1 to be is equal to X minimum plus delta X and X 2 as X minimum plus twice delta X then X 3 as X minimum plus 3 delta X in this way we can have the values of X 1, X 2, X 3, X 4 so on till we come to the value of X max. So, the values between X minimum and X max now is represented by a set of values which is varying between X 1 or X minimum to X max and these values are first obtained after we have obtained those values then we count the number of values that are lying between X minimum and X 1 let us say this is equal to n 1 then between X 2 and X 1 let it be n 2 and so on that is X 3 minus X 2 within this interval let the values be n 3. Once we have that then one can obtain these quantities that is probability of the X lying between X 2 and X 1 will be obviously equal to n 2 by n bar where n bar is the total number of values that exist between X mean and X max including X min and X max. Similarly, the value of the X probability of X be equal to X 1 plus X 2 by 2 will be given by n 2 by n. So, the center of the interval that is denoted by X 1 plus X 2 by 2 and we say that what is the probability that X will be equal to these value will be also given by n 2 by n or in other words we assume that within the pocket interval of X 1 and X 2 whatever values we obtain those values are the values or the from those values we construct what is the probability of X being equal to the average ordinate between the pocket interval X 1 and X 2. So, utilizing that one can draw this histogram it starts with X minimum over here and this particular value is equal to n 1 by n bar and we divide it by delta X. Similarly, the next histogram is equal to n 2 by n bar into delta X and next one is n 3 by n or n bar delta X and so on. Now, if we join the centers of these histograms and plot a curve then this curve would give you the probability density function of the random variable X and if we are able to give a mathematical expression to this probability density function then the pdf can be expressed as a mathematical function. So, that is how one obtains the in real situation obtains the cdf and pdf expressions or mathematical expressions for any random values which are sampled. Now, these continuous functions that we or continuous curves that we have obtained for cdf and pdf they are fitted to some standard distribution curves and if we find that those curves that we have obtained that is the pdf and cdf curve are conforming to some standard distribution curves then we say that the probability density function or the cumulative distribution function of the random variable is conforming to a standard distribution and for that we carry out what is known as a goodness of fit test in which we have many kinds of tests that are possible. One of the common kind of type of test that is carried out is the chi-square test and this chi-square test is based on the degree of mean square error and from the degree of the mean square error we denote the goodness of the fit between the target distribution and the distribution that we have obtained from the numerical data. Next is the some common distributions that are utilized in the earthquake engineering and structural engineering and in civil engineering. These distributions, common distributions are the normal distribution, log normal distribution, extreme value distribution type 1 and type 2, viable distribution and a Poisson distribution. Now out of these distributions we find that normal distributions and the log normal distributions they are quite often used in many engineering problems and specially for the cases which are the natural processes and it is generally seen that any natural process if we consider it as a random quantity then its distribution that is the cumulative distribution function or the probability distribution function or density function conform to the normal distribution and log normal distribution also is quite common. The extreme value distributions are the distributions where we talk of the distribution of the maxima that is for example if we wish to find out the distribution of the maximum wind speed in a particular year then for each day one can calculate what is the maximum value of the wind speed and so that way we can have 365 values of the maximum wind speed and if we find out a distribution using those values then that will be called a extreme value distribution. Similarly, maximum early wind speed can be expressed also as a extreme value distribution that is if we have a collected data for 50 years of the maximum wind speed that has taken place at a particular site in a year then from those 50 values one can construct a distribution and that distribution will be called an extreme value distribution. Why will distribution is a specific kind of extreme value distribution? Now, these distributions require two quantities that is the standard deviation of the random variable and the mean value of the random variable and how to find out the mean value and the standard deviation of the random variable that we have discussed before. So, knowing these two quantities one can express or give a mathematical expression to the probability density function and the cumulative distribution function for different kind of distributions. And the data that we collect from the field say in the case of earthquake the past earthquake data in terms of peak ground acceleration or in terms of the magnitude of earthquake those data can be analyzed one can construct a cdf and a pdf from the numerical data. And after we have obtained the cdf and pdf as a continuous curve then we can go for a curve fitting technique to examine whether the distributions conform to normal distribution or log normal distribution. The expressions for the probability density function for the normal variate that is given by the first expression this expression over here. Here sigma x is the value of the standard deviation of the random variable mu is the mean value and fx obviously is an integration of the probability density function because the probability density function is defined as a derivative of the first derivative of the cdf function fx. And in many standard test books in statistics we get a standard normal variate and a value of the probability density function for the standard normal variate. This standard normal variate is defined as x minus mu divided by delta sigma x. Utilizing that standard tables one can find out for the case of normal distribution what is the probability of accidents of certain value or probability of some of the random variable being less than equal to some value. Now when we come to the log normal distribution then the say the value is x becomes is equal to ln of z. z is the values sample values that we have obtained and we take a logarithm natural log of that and that is x. Then we say that the x is a or rather follows a normal distribution designated by a sigma value and a mu value that is the mean value and the standard deviation. So, log of the original variable z becomes normal. So, in that case we say that it is a log normal z is a log normal quantity and the probability density function for the log normal variable is given by this expression. Here this is the z not bar it is a curved line indicating it is not the mean value of the z bar but it is a median of z and there exists a relationship between the mean value and the median. Finally for the case of the earthquake we have another distribution which is very important that is called the Poisson distribution which is associated with the Poisson process. If we consider the distribution of earthquake occurrences with respect to time then it has been found that this temporal distribution follows a Poisson process or in other words we say that temporal distribution of earthquake occurrence for certain magnitude or a peak ground acceleration is given by this expression and this expression is known as the Poisson distribution. What it says is that probability of n being equal to n that means probability of some event n to be is equal to a value n is given by lambda t to the power n e to the power minus lambda t divided by factorial n. Now the lambda basically is known as the average rate of occurrence of that particular event and over the time period t and n basically is the number of events that has taken place. Therefore, the p probability of n being equal to n is given by this expression in which we must know the value of lambda. Now for n is equal to 1 exactly one event occurring in tiers that is one quantity that is of interest. Other is that n greater than equal to 1 means at least one incidence that has taken place in tiers this is another quantity of interest and both of them can be straight away obtained from this particular equation and they take this particular form and out of this one is very important because using this particular expression one can find out the probability of accidents of any seismic parameter y bar. Say y bar is big ground acceleration in a time interval of capital T will be simply given by this expression 1 minus e to the power lambda y bar t that is where lambda y bar is the average occurrence rate of that particular seismic parameter. Now if you know that then in tiers of time what is the probability of accidents of that particular seismic parameter can be walked out. Now with the help of these knowledge of the statistics of random variables let us now look at the different steps that we consider for obtaining the probabilistic seismic hazard analysis. Now it predicts the probability of occurrence of a certain level of ground shaking at a site by considering the uncertainties these uncertainties are size of earthquake, location, rate of occurrence of earthquake and the predictive relationships. We have talked about all these quantities before probabilistic seismic hazard analysis is carried out in 4 steps. The first step is the identification and characterization of the source probabilistically. Here we not only talk of the maximum magnitude of earthquake that can occur at a particular source but we also want to find out or we must know also the probability density function of the magnitude of earthquake. Then it assumes a uniform distribution of the point of earthquake in the source zone that means if we consider a line source or an area source then we assume that in that particular line source or the area source the possibility of the earthquakes occurring at every point is uniform. Then the computation of the distribution of R considering all points of earthquake as a potential source. So, this part we will see how we can obtain from the data that we obtain about the earthquake sources considering every point in the source zone to be a potential point of earthquake. The second step consists of the determination of the hazard rate at which an earthquake of a particular size will be exceeded using Gutenberg recurrence law that is given by equation 1.23a. In 1.23a the lambda m denotes the occurrence of earthquake greater than certain magnitude value or the occurrence or average rate of occurrence of earthquake above certain level of magnitude of earthquake m that is called lambda m is given by this equation 10 to the power a minus bm and this can be also written in the exponential form as exponential alpha minus beta m. Using the above recurrence law and specifying maximum and minimum values of m the following probability density function of m can be obtained that is given by equation 1.26 and here we can see that there is a m0 and mu m0 and mu denotes or the minimum level of the magnitude of earthquake that is considered in the analysis and the maximum magnitude of earthquake that is coming from the source and beta is a parameter that has been explained before. The third step consists of the following we obtain a predictive relationship and these predictive relationship could be of the similar type that I have shown before in data's mystic seismic hazard analysis and the idea of having a predictive relationship is to obtain a peak ground acceleration given a set of magnitude and a value of r form a particular site. Then the uncertainty of the relationship in the predictive relationship is considered by assuming peak ground acceleration to be log normally distributed that is the relationship that the predictive relationship gives that gives a mean value of the variable and the mean value of the variable say is a peak ground acceleration then there will be a standard deviation which will be attached to this particular random variable which is a peak ground acceleration. What it means is that for a given set of magnitude of earthquake and an epicentral distance the peak ground acceleration that is computed at the site is a random variable and this random variable is found to be log normally distributed. The fourth step consists of the following it combines all the uncertainties of location size and predictive relationship by this combination rule which is expressed by equation 1.27. It represents that lambda y bar represents the average rate of occurrence of a seismic parameter say peak ground acceleration being greater than equal to y bar is equal to this summation that is summation over all the sources nu y is the average rate of occurrence of earthquake which is greater than the magnitude of m 0 and that is specified for each source then this double integration within this integrand we have probability of y being greater than y bar conditional upon a set of value of m and r. So, given a m and r value one can obtain a value of the probability of the exceedance of the peak ground acceleration being greater than a value of y bar this we can obtain from the predictive relationship and then we multiply it by the probability density function of the magnitude and the probability density function of the epicentral distance and integrate over the entire values of magnitude and epicentral distances that we have considered for a particular source and some of these integrand weighted by the nu y values for each source to obtain the value of lambda y bar and once we get the value of lambda y bar then one can obtain a seismic hazard curve that is a lambda y bar versus y bar.