 In the last lecture, we discussed about the seismic hazard analysis. There are two types of seismic hazard analysis. One is deterministic seismic hazard analysis called DSHA. The other one is the probabilistic seismic hazard analysis called PSHA. We described the DSHA with the help of an example and then outlined different steps with the help of which we perform PSHA. Now, the way the PSHA is calculated will be explained with the help of this example. Before I go into the example, let me repeat some of the basic formulas that are used for calculating the PSHA which we discussed in the last lecture. The average rate of accidents of earthquake of certain magnitude is given by Gutenberg relationship and this is lambda m is equal to 10 to the power a minus b m is equal to exponential of alpha minus beta m, where m is the magnitude of earthquake which is being exceeded. Now, using this rate average rate of accidents, one can find out the probability density function of the magnitude of earthquake occurring from a particular source of earthquake and this is given by minus beta m minus m 0 divided by 1 minus minus beta m mu minus m 0, where m u is the upper limit of the magnitude of earthquake or the maximum magnitude of earthquake that occurs in that particular source and m 0 is the minimum earthquake or the minimum level of earthquake that we consider for the analysis and beta is the constant for this particular equation. Now, apart from that we used another equation that is the about the arrival of the earthquake, the arrival of the earthquake is modeled as a Poisson model, the temporal distribution of the occurrence of rather the recurrence of an earthquake for some magnitude is given by the recurrence relationship of the PMF that is lambda t to the power n into e to the power minus lambda t divided by n factorial, where the P n is equal to n means the probability mass function that is the number of the event occurring is equal to n. Now, from this one can show that P n is equal to 1 will be equal to e to the power minus lambda t and probability of exceeding at least one event is given by this equation, where e to the power minus lambda t is the probability of not exceeding one event and probability of exceedance is given by 1 minus e to the power minus lambda t. So, this particular equation where used to find out the what is the probability of exceedance of a certain pre ground acceleration in a given period of time t. Now, we will use this equations in the solution of this problem, the problem is the for the site which is shown in the figure 1.20, we have to perform a typical calculation for PSHA and for that the attenuation relationship that we used for the case of DSHA that is the equation this equation ln PGA is equal to 6.74 plus 0.0859 m minus 1.80 ln of r plus 25, this attenuation relationship is will be used here with a standard deviation specified as sigma is equal to 0.57. As I told you these attenuation relationship provides you the average value of the PGA and to this equation a standard deviation is specified because it is assumed that the pre ground acceleration and for a given magnitude of earthquake varies log normally. The sources and the sites are shown in the figure the this is the site and source 1 is a line source, source 2 is a area source and source 3 is a point source. The recurrence law of the Gutenberg which gives the average rate of exceedance of certain magnitude of earthquake is given by this and the mind you this log is a of best 10. So, this is the average rate of the exceedance of the earthquake of magnitude m. So, this is equal to 4 minus m for source 1 for source 2 it is 4.51 minus 1.2 m and for source 3 it is 3 minus 0.8 m. The minimum value of the magnitude of earthquake is considered as 4. The maximum value of the magnitude of earthquake for each source is given over here for source 1 it is 7.7 for source 2 it is 5 and for the source 3 it is 7.3. Now, for each source we can calculate the epicentral distance assuming that for the line source and the area source the earthquake has a significant equal probability of occurrence at every point the or in other words we assume the it is a uniformly distributed that means distribution of the occurrence of earthquake over the line source or in the area source is assumed to be uniformly distributed. Now first we take up the first uncertainty that is the location uncertainty for the first source we calculate the this will be the not minimum this will be the maximum and this will be not max this will be minimum. The maximum epicentral distance calculated is 90.12 that means if I join these two points the distance will be 90.12 and r minimum will be equal to if I draw a perpendicular from this side to this source then that perpendicular distance is 23.72. Now we divide this maximum r and the minimum r into a 10 equal divisions so that there is a pocket interval of certain value. Now the epicentral distance is 23.72 would vary and if we assume that the line is divided into 1000 segments then for each segment we can assume the center of the segment to be a source of earthquake and thus we will get 1000 values of the r. Now these 1000 values of the r would lie between 23.72 to 90.12 so we find out what is the probability of occurrence of the earthquake with the r in a certain interval that interval we obtained by dividing the maximum value minus the minimum value by 10. So counting the total number of radial distances lying within each interval we can find out the probability of occurrence of the earthquake or the epicentral distance lying within certain interval and the middle point of the interval is taken as the radial distance. So here for example for the first source the probability of occurrence of earthquake with epicentral distance of 27.04 is about 0.336. Similarly, for a radial distance of 33.68 the probability of occurrence is about 0.18. So in this way one can calculate the probability of occurrence of an earthquake for a what we call specified epicentral distance. The same kind of calculation is performed for source to here again we find out the maximum distance and this will be the minimum distance. So this maximum and minimum distance this interval is again divided into 10 equal division and assuming that the area is divided into 2500 parts of size 2 into 1.2 meter and assuming that the center of each one of this sub areas of 2 into 1.2 is a point of earthquake then we can have 2500 epicentral distances for the source to and these 2500 epicentral distances can be now categorized into 10 divisions that we have obtained and we can find out the probability of occurrence of an earthquake with epicentral distance as specified for the middle of the what you call middle of the interval. So here say the epicentral distance 47.67 has a probability of occurrence nearly as 0.02 so and 59.24 has a probability of occurrence of about 0.03. In the same way the third probability of occurring earthquake with a certain epicentral distance for the third source can be computed but it is seen here that since it is a point source then it involves only one epicentral distance. So the epicentral distance is since it is 1 so the probability of its occurrence is 1. Next we find out the size uncertainty the occurrence rate of the earthquake is given by this formula that is this formula. So here we get the value of A as 4 and B is equal to 1 for this value of A is equal to 4.51 and the value of B is equal to 1.2 and for this the value of A is 3 and value of B is equal to 0.8 for the equation that I have written on the top. So, using this equation one can find out the value of lambda 1 or rather new 1, new 2 and new 3 this is same as lambda 1, lambda 2 and lambda 3. So assuming the maximum probability of the magnitude of earthquake and the minimum magnitude of earthquake as 4 one can calculate the what is the accidents rate of the minimum magnitude of earthquake for different sources. So they are 1.501 and 0.631 respectively. Note that here the magnitude of the earthquake that we put into this equation is the minimum magnitude of earthquake that we have specified because the accidents of the earthquake of certain magnitude of earthquakes means here that at least the accidents would be considered for the minimum magnitude of earthquake. Now for each source zone again we can calculate the probability of occurrence of certain magnitude of earthquake between 2 intervals that is m 1 and m 2 and this can be determined by integrating the probability density function f m which is shown again here. This probability density function can be computed using this equation and integrating this equation from m 1 to m 2 and this integration leads to this particular equation that is the magnitude of earthquake occurring between an interval of m 1 and m 2. So for source zone 1 m max and m minimum are divided again into 10 divisions and center of the division is considered as one magnitude that way we get 10 such magnitudes and one can calculate the probability of occurrence of certain magnitude of earthquake and that magnitude of earthquake will be referred to the magnitude which lies at the center of the division. So probability of occurrence of that magnitude of earthquake can be obtained for source 1 source 2 and source 3. For all the three sources one can have this probability of occurrence of the magnitude of certain earthquake. Now once they are calculated then let us say we are interested in finding out the probability of accidents of 0.01 g. So what we do is that for the probability of accidents of 0.01 g for a magnitude of earthquake level 4.19 and an epicentral distance of 27.04 or for source zone 1 can be calculated from the histograms that I have shown. For example, for a magnitude of earthquake of 4.19 that means the center of the first interval of the magnitude of earthquake that we obtained for source zone 1 is equal to 4.19. So for that the probability of occurrence is equal to 0.551. Similarly, the probability of occurrence of the epicentral distance of 27.04 again 27.04 is the center of the first interval of the epicentral distance that we computed for source zone 1 and that was computed as 0.336. For example, here one can see the so this is the center of the division for the for this center the probability of occurrence is about 0.336. Now once these two probabilities are obtained then we find out the probability of peak ground acceleration. Being greater than 0.01 g condition upon magnitude of earthquake is equal to 4.19 and epicentral distance is equal to 27.04 that we obtained over here that is obtained from this relationship that is 1 minus f z z. Now this f z z is the normalized distribution of the peak ground acceleration normalized distribution means the this distribution refers to the any peak ground acceleration minus say any peak ground acceleration p g a minus is average value divided by its standard division this is the quantity z. Now the one can find out this normalized quantity and since it is assumed that the peak ground acceleration log normally varies therefore, this normalized quantity also varies log normally. And there are standard tables from where one can find out the this probability function that is f z z for a given value of the z. Now if we calculate the value of z that is 0.01 g then this 0.01 g is first converted into centimeter per second square unit because the attenuation relationship that we have used in that the p g a value is calculated with a unit of centimeter per second square. So therefore, converting this into a centimeter per second square and then finding out the its probability one can get the z value as minus 1.65 and from the table one can get the value of f z z and this leads to a probability of accidents of 0.01 g as 0.951. So we see that the calculation follows like this first we find out the probability of occurrence of certain magnitude of earthquake from the equation that is given for source one. Then probability of occurrence of certain epicentral distance again from by considering that the earthquake can happen at any point within the line source or in the area source and from there we can calculate the probability of occurrence of certain epicentral distance and they are plotted as histograms. And for those combination of m and r one can find out the p ground acceleration exceeding a value of 0.01 g conditioned upon these two quantities and for obtaining the this accidents value we assume that the p g a is varying log normally. So from standard log normal tables one can get the value of f z z and one can calculate this quantity. Now, if you wish to find out the value of the rate of accidents of certain magnitude of a p g a level of 0.01 g then we use the summation equation that we have shown before that is the summation equation that is shown here we integrate or sum up in this case integration will be replaced by summation. So, we will be summing up the product of these quantities that is probability of accidents of certain level of p ground acceleration conditioned upon a given set of value of m and r and then multiply it by the probability of occurrence of this particular m and probability of occurrence of this particular r. So, these quantities are multiplied and then we sum up for all the events. So, here first we make this multiplication and this multiplication turns out to be 0.176 and in this the rate of occurrence of earthquake greater than magnitude m is equal to r that is new one that also we have calculated. So, we multiply this entire thing with new one and the result is 0.176. So, the annual rate of accidents of a p ground acceleration of 0.01 g turns out to be 0.176 for source one given a pair of value of magnitude 4.19 and r is equal to 27.04. Now, since we have made 10 intervals for the magnitudes and 10 intervals for the epicentral distance these intervals were obtained by deducting the minimum value from the maximum value and dividing it by 10. We have 10 intervals of magnitude of earthquake or 10 combinations of magnitude of earthquake and 10 combinations of epicentral distance. So, we have got total 100 such combination. So, out of that we have just taken only one. So, lambda 0.01 g for other 99 combinations of m and r can be obtained for source one and they can be summed up for source zone 2 and 3 similar calculations can be found out. Finally, we get the value of magnitude of the rate of accidents of p g a 0.01 g equal to that what we will get for source one, source 2 and source 3 they will be summed together and we will get finally, the value of this quantity lambda 0.01 g. Now, we can consider next a big ground acceleration level of 0.02 g then 0.03 g then 0.04 g and so on. So, for different levels of big ground acceleration we can find out the lambda value and this lambda value then can be plotted against different levels of the big ground acceleration and this will give a curve and that curve is called the seismic hazard curve for big ground acceleration for the region. Now, with this particular quantity known that with the seismic hazard curve for big ground acceleration or for any other earthquake measurement parameter if we have a seismic hazard curve then for a given level of the seismic measurement parameter one can find out what is the annual probability accidents of that level of the parameter. Then one can consider also what is the probability of accidents in t years of time by using the equation this equation that we have obtained. So, this equation provides the accidents rate of say for example, accidents of a big ground acceleration level what is the probability of accidents of a big ground acceleration level of y bar can be obtained using this equation where the lambda is equal to lambda y bar that is the lambda value that we calculated and is shown in the form of the seismic hazard curve. So, using this temporal relationship of the occurrence of earthquake one can calculate what is the probability of accidents of certain level of say big ground acceleration in t years of time for a region. So, this is about the PSHA and therefore, we can see that by drawing some what to call the histograms showing the probability of occurrence of magnitude of earthquake and probability of occurrence of certain epicentral distance and using the pertinent equations one can obtain the seismic hazard curve for a particular region drawn for a seismic measurement parameter the seismic measurement parameter could be a big ground acceleration or big ground displacement, big ground velocity or for that matter any other earthquake measurement parameter. Now, let us look into the next example which is a very simple state forward example in order to clarify the temporal distribution of the earthquake which is assumed to be a Poisson model. So, here the problem is the seismic hazard curve for a region shows that annual rate of accidents of an acceleration 0.25 g due to earthquake is 0.02 that is lambda 0.25 g is given as 0.02. So, from the hazard curve that is plotted for a region we get particular this value. So, this is given now what you have to find out is that what is the probability that exactly one such event that means, only one event where the 0.25 g of the big ground acceleration will take place in 30 years and the other one is that what is the probability that at least one such event that is one such event means at least once this big ground level will be exceeded in 30 years of time. Also find the value of lambda that has a probability 10 percent probability of accidents in 50 years of time that means, one has to calculate what is the value of this lambda for a 10 percent accidents in 50 years of time. So, this the answer to the first is the probability of occurrence of only one event such event is given by the equation that we have written this is the equation probability of occurrence of only one event. And we put the appropriate value the value of lambda is given as j 0.02 the time period is 30 years and then e to the power minus lambda t. So, that gives a result of 33 percent. So, the answer to the first question that is the probability of occurrence of one event is 33 percent. The probability of occurrence of one event of at least once accidents is given by again this equation and here we put the values that is lambda value as 0.02 and the time as 30 years and we get it as 45.2 percent. Now, this equation can be rewritten in the form of e to the power minus lambda t is equal to 1 minus p n greater than equal to 1. So, l n lambda t is equal to 1. So, l n lambda will be equal to l n of 1 minus p n greater than equal to 1. Here we ignore the sign because the negative value for the occurrence rate that really does not make any meaning. So, therefore and this l n will be equal to lambda t from there we can get the value of lambda is equal to this divided by t. So, that is what has been used here the l n the occurrence accidents occurrence rate lambda will be equal to l n of 1 minus p n greater than equal to 1 divided by t and this is given as 0.1 that is the fine lambda that has a probability of accidents of 10 percent. So, 10 percent and the time is 50 years. So, that is what we have done. So, 10 percent is 0.1 and time is 50 years and the lambda is calculated as 0.0021. So, therefore, given the seismic hazard curve one can find out the probability of occurrence of just one event and accidents of at least one event and given the probability of accidents for or the earthquake for a given period. One can find out what is the average rate of accidents the value of lambda that we have calculated in the second problem. Now, let us come to the seismic risk at a site seismic risk at a site is similar to that of seismic hazard determination. Now, the seismic risk means that again what is the probability of accidents of certain earthquake measurement parameter. Now, based on that one can assess the seismic risk for a site and one can also bring in the period into picture that is one can say what is the probability of accidents of certain level of earthquake measurement parameter say peak ground acceleration in 50 years of time or 100 years of time. So, in that case the previous example that we have done that kind of example would be useful. So, by definition the seismic risk is that probability of certain parameter say this is peak ground acceleration exceeding a particular level during certain period and generally we take that period as one year. So, that we can say that it is the annual probability of accidents and the inverse of this annual probability of accidents or inverse of this risk is called the return period. The study of seismic risk requires source mechanism parameters that is the focal depth orientation of fault etcetera, recurrence relationship which is used to find out the probability density function of the magnitude of earthquake that is what we have seen before and the attenuation relationships again we have used this attenuation relationship in connection with PSHA. Now, using this information one can obtain the seismic risk either using a calculation like probabilistic seismic hazard calculation or using a Cornell's or Milne and Devonport's approach. So, there were certain methods which were the classical methods and they were proposed by Cornell and Devonport. Using their concept many empirical equations were obtained with the help of data information for a particular region. For a particular region these empirical equations have been developed if we wish to use them for some other region then the constants of the equation or the values of the different parameters that have been used that values must be accordingly adjusted. Now I am giving you some equations over here, but many more equations are available of this nature and they are all given in the book. The first equation talks of the annual occurrence of a particular event that is the number of the say magnitude of earthquake or peak ground acceleration of earthquake exceeding a value of y s. So, y s in generally indicates a earthquake measurement parameter. So, let us say it is peak ground acceleration for our problem. So, then what is the annual rate of occurrence of an event specified by the exceedance of certain level of peak ground acceleration. So, that number is given by this equation y s divided by c bar to the power minus p bar. So, here c bar and p bar they are the constants and these constants are obtained from the regional data and y s say here is a level of the peak ground acceleration. The next equation shows the probability of exceedance of a particular event. Of a maximum magnitude of earthquake say maximum magnitude of earthquake specified by M 1 then the probability of exceedance of that magnitude maximum magnitude of earthquake is given again by a equation like this where alpha 1 beta are the constants and alpha 1 is related to the period of observation t 0 and another constant alpha. This equation shows the probability of exceedance of the intensity of an earthquake. So, probability of exceedance of an intensity of earthquake for a level small i 1 is given by an equation of the constant alpha 1 like this again this is an empirical equation all of them are all in empirical equation empirical equation. So, here these values 1.54 and 47 they these values are obtained for a particular region say if the data is from United States. So, for some region in USA these particular values are valid for other regions one has to calculate the appropriate values of these parameters. This equation shows the probability of the magnitude of earthquake. So, the magnitude of earthquake lying between an upper limit and a lower limit the probability of the magnitude being less than equal to M 1. So, that is given by an equation like this where M 1 is the magnitude in question the probability of which you are trying to find out that is what is the probability of the magnitude of earthquake being less than M 1. So, that is what we are trying to find out. So, this is M 1 and M u refers to the maximum magnitude of earthquake and M 0 is the minimum magnitude of earthquake that you consider for the analysis. And if we know the what is the probability of the magnitude of earthquake being less than equal to certain value then one can calculate what is the probability of accidents of the magnitude of earthquake for the same value of the magnitude that will be 1 minus this quantity. Now, the using this equation if I in this equation if we put the value of M u to be very large then this turns out to be almost equal to 1 this one and as a result of that one can simplify this equation to this particular form. That means probability of the magnitude of earthquake being greater than equal to M 1 is equal to 1 minus e to the power minus beta M 1 minus M 0. So, that comes from this particular equation. So, that way we have many such empirical equation which talks of the probability of occurrence or probability of accidents of certain level of the earthquake parameters in a particular region and they are called the seismic risk for the region. Next we come to a very important topic the microzonation using the hazard analysis. Now, the microzonation is a very useful concept for mitigating the earthquake disaster in a particular region and for that one should have a map for the region showing the relative vulnerability of the different sub regions in that particular area. The concept basically is like this say we take a city or a metropolitan area and divide it into a number of sub areas and the center of the sub areas are the points in question so here we at these points we attach some value which indicates the vulnerability of certain parameter that we will now discuss. So, the microzonation say we are wanting to find out with respect to the probability density or the sorry population density. So, what we do that in this region we calculate the population density and that way for all these places we calculate the population density of the region and then what we do the points which are having the similar popular density they are grouped together and from this grouping one can come out with areas showing the population density that means in this area has a one kind of population density this area is having another kind of population density and so on. So, this will be called a microzonation map of the city with respect to population density similarly one can construct a microzonation map of a city for construction. So, a city can be divided into the type of construction that is a bad construction good construction or medium type of construction and one can have a microzonation map similarly for the old structure new structure relatively old structure we can have this kinds of categories and then one can have a microzonation map for the type of the structures that is existing in the city. So, now this concept can be extended to the microzonation of a region with respect to peak ground acceleration and the amplification factor of the ground motion due to soil condition. So, that is these two parameters are very important because for design purposes one should know what is in each sub region what is the peak ground acceleration for which the structure should be designed in this region and what is a likely amplification factor for that region because of the soil condition. So, if these two informations are there then one can and go ahead with a safe design of structures for those regions. So, that is what is done in a microzonation. So, now let us try to formally define the microzonation. Microzonation is delineation of a region of a big city into different parts with respect to seismic hazard potential. Various parameters indicating hazard potential are used to microzone the area like local soil characteristics, earthquake source properties, epicentral distance, topography, population density, types of construction etcetera with respect to each parameter a map may be prepared. They are then combined by giving weightages to each parameter to arrive at a hazard index. So, for each one of the sub area one can calculate and hazard index by giving weightages to each one of this parameter. Although each parameter has its own importance soil amplification, earthquake source properties, epicentral distance are considered very important parameters to denote seismic risk or hazard of a particular region. DSHA or PSHA combined with soil amplification are quite often used to prepare a microzonation map. The steps include divide the region into a number of bits considering variation of soil properties as I have shown here. Then at the center of each grid find PGA either by DSHA or PSHA procedure giving the probability of accidents in the case of PSHA. And then for each one of the site that is the center of the grid one can find out the PGA amplification by performing a one dimensional, two dimensional, three dimensional wave propagation analysis that is what we discussed before. And multiply the PGS that we have obtained and for the from the DSHA or PSHA analysis by the amplification factor and accordingly can prepare a microzonation map. So, a deterministic microzonation map for a peak ground acceleration is shown over here in this figure. You can see that this particular region is controlled by this PGA, this region is controlled by this PGA and so on. And therefore, if you have to construct a building then the building should be designed with a PGA of 0.4 g. This one shows the probabilistic microzonation here the numbers over here associated with each sub region shows the probability of accidents of 0.25 g level of peak ground acceleration is 10 percent. So, we take 10 percent probability of accidents that is 10 percent risk and based on that one can design the buildings with this peak ground acceleration. So, let me summarize at this stage what we discussed today we have discussed a problem in which we have shown how probabilistic seismic hazard analysis can be carried out. And it uses some of the equations that probability equations and a attenuation law and by drawing certain histograms and using certain summation over all the sources one can find out what is known as a seismic hazard curve for the region showing the annual probability of accidents of certain earthquake measurement parameter with respect to the level of that earthquake measurement parameter. Then we have discussed about the seismic risk expressions which are available in the literature for different quantities say magnitude of earthquake, intensity of earthquake, up peak ground acceleration. And you will get in books and in the web sites also a host of such empirical equations. If we do not have any particular data systematic data available for a particular site then one can use one of these empirical equations expressing the seismic hazard or seismic risk at a particular site. Then we discussed about the microzonation of a particular region what microzonation means and how a microzonation of a region or a city can be obtained for a given level of different levels of peak ground acceleration or the probability of occurrence of the peak ground acceleration. And also in that we include the amplification effect due to soil condition. Thank you.