Hello everyone, let's try to understand what do we mean by return period. So in hydrology return period can be given for two things in general that is for rainfall and for flood. So in case of rainfall it is associated also with the duration of rainfall that is if you are talking about the return period then we also have to mention that what was the duration of this rainfall that is 2 hour or 24 hour and so on and in case of flood it is related to discharge that is what kind of or what is the magnitude of this discharge for which we are trying to find out this return period. So for understanding that I mean these are the two things so let's try to understand first for flood so let's say this is some data we have collected from some stream gauging site so stream gauging side is nothing but a point at which we can measure the discharge in the stream that we call as stream gauging site.
So from the help of this data we get something like this that is if we try to find out what is the maximum discharge in a in any particular year so we can plot that so here it is the data that is in some year 1980 this was the maximum discharge then in 1981 it is here and so on. Now let's talk about this return period. So for return period first of all we have to define a magnitude so here let's assume this magnitude for which we are trying to find out the return period is 5000 thing.
m3 per second. So, this data we have assumed for a time period of 30 years. So, let's see. So, this value 5000 m3 per second, we can see that in 30 years, how many times this value is getting equal or it is getting greater, I mean to say greater than equal to 5000 m3 per second. That is the discharge in this stream when it is becoming greater than equal to 5000 m3 per second.
So, we can count it here that is 1, 2, 3, 4 and 5. So, 5 times we are seeing that is that the value of discharge is greater than equal to 5000 m3 per second. Now, if we try to find out the time interval that is time interval which I mean which is nothing but the time which is required for this value to become greater than equal to 5000 meter cube per second. The value of discharge to become greater than equal to 5000 meter cube per second. So, that is here you can see that If we count exactly also, so 1, 2, 3, 4, 5. So here it is taking let's say 5 years. In 5 years this value is again exceeding.
After that from here 1, 2, 3, 4. So here it is happening in 4 years. After that 1, 2, 3, 4. Again 4 years. After that it is 1, 2. So, this time 2 years.
So, we can notice one thing here that is the time interval is not same. So, we can also say that this time interval is nothing but your return period. So, this return period is not same but if we try to find out the average of these values. Try to find out the average or we can say average time interval.
So that is nothing but what is the total time period that is 30 years divided by this time interval. So how many periods we are getting? So that is one, then this is second, this is the third one and this is the fourth one. So four periods we are getting that is four intervals we are getting.
So 30 by 4 that is 7.5 years is nothing but the average time interval which is required for the discharge to become greater than equal to 5000 meter cube per second and this value this value is nothing but what we know as return period. So in case of flood it is return period of flood in case of rainfall we call it as return period of rainfall. So that is how we find it.
What do we not mean by this return period? So if you again see this plot, will you say that 7.5 years is the time in which the discharge will, I mean every time it will exceed this value. So here you can see it is happening here in five years then four years then here it is happening just in two years.
So I mean it is not happening in 7.5 years. It is just the average value, it does not mean that every time in 7.5 years the discharge will become greater than equal to 5000 meter cube per second. It is just a probable case that it takes on an average 7.5 years for the discharge to become greater than 5000 meter cube per second.
It may take less than 7.5 years. It may take. greater than 7.5 years. If you see this plot here again, so here just in 6 years it is happening twice. So, it is twice it is happening in less than 7.5 years.
So, return period does not mean that in every 7.5 years the discharge will become greater than equal to a particular value. So, that is about your return period. After that this return period is also associated with the probability of occurrence of that particular event.
So, in this case we can talk about this probability of occurrence of this event that is the discharge to be become greater than equal to 5000 meter cubes per second. So, here probability that is and we can say that what is the probability that in any given year any given year the discharge will become greater than equal to 5000 m3 per second. So this probability is given as 1 upon T and T here is your return period.
So that's how you can find out the probability that in any given year what is the probability that the discharge will become greater than equal to this value. So, in case of this probability when we plot this data between discharge and time or if we talk about the rainfall then it will be between precipitation and time. So, here these values of discharge or rainfall what we get in particular year we treat them as independent.
days The discharge in this particular year has nothing to do with the discharge in the next year or the preceding year. So these value I mean the probability of occurrence here we treat it as independent and with replacement. So the best way to model this with some probability distribution to use binomial distribution. So if we want to find out that in n years we want to find out that what is the probability that this discharge becomes I mean the probability of occurrence of this event r times in n years so we can use this binomial discharge.
that is P and R equal to NCR P to the power R and Q to the power N minus R. So P here is nothing but your probability of occurrence that we give as 1 upon T and Q is 1 minus P. So using this you can find out that what will be the probability that if we talk about N successive years then If you want to find out that R times this event happens, R times event happens. And so two more definitions are associated here, that is risk and reliability. So what do we mean by risk?
So, risk is the probability of occurrence, probability of occurrence of an event at least once in once over a period of once in n years we can say. So r is equal to 1 minus q to the power n. You can use this formula also to find out the probability. at least once in n years and then other one is reliability. So reliability is the probability of an event not happening in n years.
So reliability is given as 1 minus risk. If R is the risk then reliability is 1 minus R that is 1 minus 1 minus Q to the power n that is Q to the power n. So we'll see examples after that to understand it better. Now let's see this question to see how we use the return period to find out the probability of an event.
So it says in a city rainfall of depth 280 mm in one day has a return period of 50 years that is t is given as 50 years then it says determine the probability of one day rainfall depth equal to or greater than 280 mm occurring then three different questions are there once in 20 successive years twice in 15 successive years and at least once in 20 successive years. So one more thing to notice here is as I told when we talk about the rainfall. it is always associated with some duration.
So here the duration is given as one day that is 24 hours. So if we say 280 mm in 12 hours, so it would make it a different event, different event. And if you talk about the return period of these two events, so these would also be different. So if it is t1, this would be t2 and t1 would not be equal to t2.
So that's How it is that is here we also talk about the duration when we talk about the rainfall in return period. But you don't have to worry about this at least here. So simply we have to find out the probability. So t is given.
So probability of this event occurring in any particular year would be 1 by t that is 1 by 50. And probability of not happening would be 1 minus P that is 49 by 50. Now simply we can use the binomial distribution. So it is saying once in 20 successive years. So it would be P nR nCr P to the power R Q to the power n minus R. So here once in 20 years would be n is equal to 20 and R is equal to 1 so if you put the values that is 20 C 1 1 by 50 to the power 1 and 49 by 50 to the power 19 and if you solve this answer would be 0.272 this is the required probability.
Then next one is twice in 15 successive years so for that simply just you have to change the data so that would be 15c2 1 by 50 to the to the power 2 and 49 by 50 to the power 13. So if you solve this it comes as 0.0323. This is the required probability and the last one is at least once in 20 successive years. So that would be we can find this one as 1 minus the probability of not happening at all in 20 successive years the probability of not occurring in 20 successive years.
That would be 1 minus, so not happening in 20 successive years, so that is n is equal to 20 and r is equal to 0. So, 1 minus 20C0 1 by 50 to the power 0 and 49 by 50 to the power 20. That is nothing but 1 minus 49 by 50 to the power 20 and in general terms we can write this one as 1 minus q to the power n and that is nothing but your as we earlier studied these definitions this is your risk. So if someone asks you to find out the risk, this is what you will calculate. That is the probability of happening at least once, at least once in n successive years. So that is if you solve this, this comes out as 0.332. So this is your required answer here.