There’s an urban myth that several years ago, the American Society of Civil Engineers (ASCE) decided to hold its annual conference in Las Vegas. 10,000 engineers were to descend on the city for four days of doing what engineers do in Las Vegas, talk about engineering. You can imagine that these things have to be booked years in advance and the story goes that several months after the booking was taken, the Las Vegas Chamber of Commerce wrote to the ASCE to tell them they were cancelling the booking. The reason given was that they’d done analysis which showed that the engineers would not spend enough money in the casinos. We’re just too risk averse.
I was thinking about risk and chance recently when I read this article.
Source: The Guardian, Feb 2019
Here's a link to the video if you want to see it in action.
Look at the sub-headline: Reservoir’s flood drain was expected to be used once every 50 years, but has seen action twice in last two.
The word that caught my attention was 'but'. Although the article doesn’t clarify, is it saying that if something is designed to operate once every 50 years, it cannot operate twice in two consecutive years?
I think there are three possible reasons why it has operated as it has:
Into which of these three categories a design ‘failure’ falls is often the subject of debate and in many cases legal action. It is the fear of many designers to explain to a householder why their property has flooded twice recently although it is actually protected.
We often use the term ‘return period’ to try and describe the possibility of events happening. So in this case we could say that the reservoir is designed to a 1:50 year return period, sometimes expressed as a 2% or 0.02 chance. This means that it should not fail during an event which occurs on average once every 50 years. Does that mean that we could get two of these events in consecutive years? Does it mean we could go 100 years without an event like this occurring? Yes, both these things are possible, the question is, how likely are they?
An analogy would be a bag containing 50 tokens. 49 of the tokens are black and one is red. You pull a token out at random. After you’ve pulled it out, you put it back and the tokens are thoroughly mixed. If you pull a black token, you get £10. If you pull the red token you have to pay £500. Do you want to play? How many black tokens would there have to be for you to play? How much would the fine for pulling the red token have to be reduced for you to play? How would you spend your time in Las Vegas?
To predict how likely it is that rare events will occur within a defined time period, we can use a binomial or Poisson distribution, defined here.
Using a binomial distribution, we can produce the following table:
The table shows that there is a 36.4% chance that a scheme would not experience a 1:50 year event in 50 years. There is a 37.2% chance it will experience one of these events and a 18.6% chance it will experience two of these events.
What does the table look like if we reduce the time period from 50 years, to 2 years? In other words what is the distribution for a 1:50 year event over two years?
In other words, there is a 96% chance that the reservoir will not experience a 1:50 year event in any two consecutive year period. There is only a 0.04% chance that it would experience two events in this time. Put another way, if we looked at 10,000 reservoirs, around 4 of them would have operated like this one. It gets more complicated when we think that there have been 62 consecutive two-year periods since 1957.
The article also talks about how, as well as the two recent events, there were also events in 2006 and 1996 (as well as implying there was another event in 1997, when a swimmer died.) What are the chances of that?
This time we extend the time period to 62 years (the time since the reservoir was built)
So there was a 2.77% chance that four events would occur during this time. That's not unbelievable, in fact, it's barely even newsworthy.
You’ll notice during this article, we’ve talked about “event” rather than “storm”. This is where things really start to get complicated. Firstly, there isn’t just one 1:50 year storm! In the UK, if 35mm of rainfall lands on a catchment in one hour, that is roughly a 1:50 year storm. However, 60mm landing in 4 hours is also a 1:50 year storm. Notice that although the storm duration has increased by a factor of four, the rainfall depth has not even doubled. So as rainfall events get longer, they also get bigger (more depth) but not necessarily by that much.
A short, intense event would have very little impact on this reservoir, whereas a long event would. We then need to consider a combination of storms. What are the chances of a 1:30 year storm happening, which nearly fills the reservoir, followed closely by a 1:2 year storm which causes it to spill? We then need to consider outside influences. What are the chances of the discharge from the reservoir being reduced (perhaps they’re building a bridge downstream) and then a large storm hitting the catchment? What are the chances of the rainfall landing when the catchment is very dry and therefore all the rainfall simply soaks in to the soil? These are all reasons why we use complex hydraulic models, such as InfoWorks ICM to test all these scenarios.
The headline could have been “reservoir was designed to a 1:50 year return period and has been exceeded twice in the last two, which is quite unlucky”.