Chances of a Storm

[Faulkner]

Hi all,

I work in drainage design, and storm water drainage in particular. I'd appreciate your thoughts on this maths problem below. As a piece of background information, "storms" are defined by the chance they might occur in a particular year - for example, a 1 in 30yr storm is a storm so big that you'd expect something that big to occur only once in 30yrs. A 1:30yr storm is much less common than 1:2yr storm for example. Here's the question:

I'm designing a drainage network that will only be around for 10yrs before it is dug up and replaced. What is the chance of a 1:100yr (or worse) storm happening during its lifetime?

What is the best approach to this question?


Oll

 

[Deleted member 4993]

Hi all,

I work in drainage design, and storm water drainage in particular. I'd appreciate your thoughts on this maths problem below. As a piece of background information, "storms" are defined by the chance they might occur in a particular year - for example, a 1 in 30yr storm is a storm so big that you'd expect something that big to occur only once in 30yrs. A 1:30yr storm is much less common than 1:2yr storm for example. Here's the question:

I'm designing a drainage network that will only be around for 10yrs before it is dug up and replaced. What is the chance of a 1:100yr (or worse) storm happening during its lifetime?

What is the best approach to this question?


Oll

For this calculation, you need to look at the recorded history of rain-fall in that area, looking as far back as possible.

When did that last 10 yr. rain happen? When did it happen again? When did it happen again?

When did that last 20 yr. rain happen? When did it happen again?

When did that last 50 yr. rain happen? .... and so -on

In other words, need to analyze historical data to get a reliable answer.

 

[Dr.Peterson]

I work in drainage design, and storm water drainage in particular. I'd appreciate your thoughts on this maths problem below. As a piece of background information, "storms" are defined by the chance they might occur in a particular year - for example, a 1 in 30yr storm is a storm so big that you'd expect something that big to occur only once in 30yrs. A 1:30yr storm is much less common than 1:2yr storm for example. Here's the question:

I'm designing a drainage network that will only be around for 10yrs before it is dug up and replaced. What is the chance of a 1:100yr (or worse) storm happening during its lifetime?

What is the best approach to this question?

Given only that a 1:100-year storm means that that level or worse has probability 1/100 of happening in a given year, the probability that it will not happen in a given year is 99/100. The probability that it will not happen in a given 10-year span is (99/100)^10. Therefore, the probability that it will happen is 1 - (99/100)^10 = 0.0956, that is, 9.656%. That's just a little less than you might naively expect, namely 10/100 = 10%.

In my experience, this would just be called a "100-yar storm"; that is confirmed here, along with the calculation I just did: https://en.wikipedia.org/wiki/100-year_flood

 

[Deleted member 4993]

Given only that a 1:100-year storm means that that level or worse has probability 1/100 of happening in a given year, the probability that it will not happen in a given year is 99/100. The probability that it will not happen in a given 10-year span is (99/100)^10. Therefore, the probability that it will happen is 1 - (99/100)^10 = 0.0956, that is, 9.656%. That's just a little less than you might naively expect, namely 10/100 = 10%.

In my experience, this would just be called a "100-yar storm"; that is confirmed here, along with the calculation I just did: https://en.wikipedia.org/wiki/100-year_flood

Recently though those "yearly" designations are becoming very fluid. That is why I was suggesting a deeper dive into recorded data and trend.

 

[Dr.Peterson]

Recently though those "yearly" designations are becoming very fluid. That is why I was suggesting a deeper dive into recorded data and trend.

Which is precisely why I started with "Given only that ...". We are just saying different sides of the same thing. And the link I provided has a lot more to say that is of interest, though I failed to emphasize that.

Yes, reality is likely certain to be different from whatever calculations someone has done to estimate these probabilities. And, perhaps even more to the point, probability tells us nothing at all about what to actually expect.

 

[Faulkner]

Thanks for your responses!

In terms of historical data, the "Design" storms that we work with are based on statistical data produced through either the Flood Studies Report or the Flood Estimation Handbook. These are used to ensure a standardised approach can be made for drainage design across the UK, without the need for localised historical data (whilst the UK has some of the longest running and extensive rainfall data in the world, it's still not perfect!). As such, I think we can look at this purely from a purely mathematical probability perspective.

Looking at the suggestion from Dr Peterson, I ran the same process but for the chances of a 1:100yr storm happening within a 99yr period. I'd expect this to be quite a high probability, close to 1. However, 1 - (99/100)^99 = 0.630, or 63%. This implies that the maths may not be correct? Am I missing something here?

Your replies are really interesting, thanks for responding! Hope the above makes sense!

Oll

 

[Dr.Peterson]

Looking at the suggestion from Dr Peterson, I ran the same process but for the chances of a 1:100yr storm happening within a 99yr period. I'd expect this to be quite a high probability, close to 1. However, 1 - (99/100)^99 = 0.630, or 63%. This implies that the maths may not be correct? Am I missing something here?

Did you not read the Wikipedia link I provided? It says this, even before giving the formula:

A common misunderstanding is that a 100-year flood is likely to occur only once in a 100-year period. In fact, there is approximately a 63.4% chance of one or more 100-year floods occurring in any 100-year period. On the Danube River at Passau, Germany, the actual intervals between 100-year floods during 1501 to 2013 ranged from 37 to 192 years.​

This is how probability works. Intuition often fails.

 

[Faulkner]

I have to admit, I tend not to trust Wikipedia! Particularly when it makes statements such as the above without references. And as I mentioned above, I think trying to relate the observed historical events may only muddy the waters (hah!), so would like to consider this from a purely mathematical perspective.

From the article, it provides the expression for a binomial distribution, but I can't tell if this is for the chance of precisely a 100yr event, or the chance of a 100yr event or worse. Following the link to the article on Frequency of Exceedance gives a similar but slightly different formula, and it's hard to tell immediately if these are actually the same formula but presented differently.

If I were to take assumed data from an infinite amount of time, and plot the frequency of 100yr storms against the time interval since the last storm event, what sort of curve would you expect? Assuming a perfect theoretical 100yr return period storm, not real data. I'd kind of expect to see a bell-curve skewed to the left, but with the peak precisely on 100yrs. But, as you say, intuition and probabilities don't always match!

Thanks,

Oll