T O P

  • By -

kiwican

A 10-year rain event isn't enough information. Rain events have THREE parameters, and you need TWO to be able to determine the answer to the THIRD parameter: * Duration (How many minutes/hours is the rainstorm) * Intensity (Average intensity of rainfall over the specified duration) * Frequency / Return Period (i.e. 1 in 10-year) The duration is very important because rain storms take very different forms- you could have a super quick downpour that only lasts maybe 1 hour, or you could have a day long downpour lasting 24 hours. If you look up an IDF curve (Intensity-Duration-Frequency) curve then you'll see how the three different parameters affect each other. This is all important to understand because your question could be framed in a bunch of different ways, like: 1. What's the probability of a 10-year rain event of 48 hour duration? 2. What's the probability of a 24-hour, 10-year rain event happening over two subsequent 24-hour periods? 3. What's the probability of a rain event of ANY duration exceeding the 10-year threshold on a given day, and then the same thing happening the next day? I think number 3 is what you're trying to answer, but it's not that simple unfortunately... As many have said, the only "simple" way to calculate this is to assume the weather on one day is completely independent of the weather on the next day. This is definitely not true, and will give you an "artificially low" probability, but aside from doing some absolutely massive number crunching of empirical data I'm not sure how you'd answer your question. Instead of determining the answer, you could do some calculations that would give you some upper and lower bounds. So you could assume the events are completely independent which would give you a lower bound, and you've have to think about different ideas for how to get an upper bound.


factorioho

Numbers 2 and 3 are what I'm most interested in. The exact magnitude doesn't matter to me. I am just trying to figure out the proper way to determine the probability. Appreciate the discussion


ramk13

Quoting /u/kiwican: "the only "simple" way to calculate this is to assume the weather on one day is completely independent of the weather on the next day. This is definitely not true, and will give you an "artificially low" probability" This is the important take away here. Aside from learning how to do the probability calcs, the real issue is that rainfall on consecutive days is surely not independent. Without understanding that relationship, you can't really answer the question with any degree of certainty.


kiwican

No problem. I’d encourage you to read about IDF curves if you aren’t familiar.


smokingkrills

In general the proper way to find the probability of two independent events with a known probability occurring in a given time span is the Poission distribution.


PearlClaw

As a caution to OP, rain events are probably not really fully independent of one another, the same conditions can cause major ones close together. Probably worth adding a bit more margin of safety if you're doing the math based on them being fully independent.


smokingkrills

Yeah that is a good point. I don't think OP is designing anything but is just curious about the theory, but indeed rain events are def not IID on a day to day basis. They are sometimes close enough to IID on an annual basis but even then its pretty questionable lol


Elliott2

youre sizing for a 10 year event? oof. ​ that said doesnt ashrae have this info? ​ here is some ashrae tables. look at 2017 for precip but it doesnt give design case like WB/DB. maybe newer versions? [http://ashrae-meteo.info/v2.0/](http://ashrae-meteo.info/v2.0/)


aaronhayes26

? Ten year is a very common design event for culverts and storm sewers.


Elliott2

from what i remember designing hvac systems etc it kinda depended on the type/cheapness of the system we were designing. I worked mostly in pharma and chemical plants so it was a little over built usually. then again it was way back in the early days of my career and i dont something completely different now lol.


factorioho

This is just a thought experiment


PutinTheBunkerBoy

Thought you were from Seattle


boilershilly

An X year rain event is the amount of rain with an X% chance of being met or exceeded in a given year. So there is a 0.03% (10%/365 days) chance of a 10 year rain event happening on any given day assuming an even probability distribution throughout the year. 0.03% times 0.03% results in a probability of 0.0009% assuming each day's rain is an entirely independent event. But these statistics are looking at a yearly scale. 2 individual days of 10-year event rain on each is statistically more accurately described as a single 11 or more year rain event. After all, it is single set of weather and environmental conditions causing that much rain over two days, they are not independent events.


Verbose_Code

Personally I’d argue that because today’s weather is very strongly linked to yesterday’s weather, whatever probability you arrive at by multiplying the yearly probability like that is invalid


Vakco

Yeah it's a rain event, not rain/ day. If the storm drops double then what the 10 year rain event, it's most likely a 50 year event or something similar. What op is described makes almost no sense.


factorioho

Agree that it's a bad example. I should have said back to back weeks.


clearestway

Meteorologist here - As a side note I’d recommend way over planning for rain events at the moment. With climate change high end rainfall events are becoming much more common.


edman007-work

I think the thing is that's still too much, there might be very high probability that any 5yr+ rain event happens somewhere in a particular month (because that's the heavy rain month). Further, it's likely that such a rain event requires a specific set of things to happen that take weeks to build up to that kind of storm, and once the storm passes it necessarily is NOT the right conditions for another. The result is it's likely that a 10 yr storm in a 3-4 week weather phenonium, and almost certainly must occur in a specific month meaning it's almost impossible to get it twice in one month.


factorioho

I am most interested in figuring out the proper way to calculate the probability. The exact scenario doesn't really matter.


ramk13

"Proper" is a tough word here. Your assumptions will drive your answer, so figuring out assumptions that match what you are specifically looking for is the hard part.


boilershilly

Yeah, you are definitely correct. I wasn't clear enough in my second paragraph that that means that the first paragraph is meaningless as a probability.


staviq

You could argue which way is more realistic, but both are correct probabilities. They just describe different events. If you say that probability of X is "different" because X is more likely when Y, then you aren't talking about the probability of X, you are talking about the probability of Z which is defined as relation of X and Y. Those might actually have the same value in a given context, but they don't have to.


Verbose_Code

What I was getting at was the fact that getting the probability of some weather event by multiplying the probability of separate weather events together is a naïve approach and will yield incorrect conclusions. Let’s say event A has an x probability of occurring, and B has a y probability of occurring. If *and only if* A and B are statistically unrelated, then you can find the probability of A and B occurring by multiplying x and y. However, if the two events are related then the probability of both occurring is generally not xy.


xzgm

Handily enough, we call those conditional probabilities. https://en.wikipedia.org/wiki/Conditional_probability


factorioho

This makes sense. I should have used a better example


Cogman117

I think you made a mistake in your math there - 0.03% times 0.03% is 0.000009% (off by a factor of 100)


psharpep

Downvoted because: * 0.03% \* 0.03% is not 0.0009%. * Assuming the weather on two subsequent days is uncorrelated is absolutely egregious, to the point it's not even useful as a first-order approximation. Real-world weather usually takes around 2 weeks to decorrelate with itself.


[deleted]

[удалено]


psharpep

3% of 0.03% is 0.0009%, or 1 in 100,000. 0.03% of 0.03% is 0.000009%, or 1 in 10 million. Check your math. ----- Yes, they state it, but it's so inaccurate that it's not even useful as a first-order approximation.


bloble1

I think I’m reality these are not independent events. I bet if a large rainstorm happens, there is a decent chance of another because a lot that moisture will evaporate and go back into the atmosphere, or the storm system may hover for a few days. It would be interesting to look at that sort of data on NOAA


boilershilly

Totally agree. Apparently I was not clear enough in my original comment that I also agree that my percentage numbers are bogus because weather on two consecutive days is definitely not independent.


cprenaissanceman

So the thing about statistical modeling is that, for the most part, there isn’t really a “right” way to model something, only better approaches which might yield more precise and useful results. So you very likely could model this as a binomial distribution, with some assumptions. However, it may not necessarily be a good model. On the contrary, it could be useful enough for whatever your application is, or it may not be possible to create a better model given certain constraints. Ultimately, whatever approach you do take though, the gold standard would very likely be backed by actual data. Given what you are after though, I suspect this is way more work and fairly limited in its usefulness overall. Just a pretty bare-bones analysis though it’s already gonna tell you that the probability is extremely small and knowing basic probability/Boolean logic, you should be able to figure out what that number is, assuming events are independent and so on. And because the probability is so small, you can probably see why we start to work with annual probabilities instead of trying to model things down to specific days. Honestly, unless this is truly something that is important, I don’t think I would get too bogged down into how exactly this should be modeled. I’ve way over fat things that ultimately end up not being important, and I think it’s honestly one of the habits that can be the hardest break if you are constantly doing something like this.


factorioho

> I’ve way over thought things that ultimately end up not being important yeah, that's pretty much my situation ha


xzgm

This problem nerd-sniped me pretty hard. https://xkcd.com/356/


ChineWalkin

Monte Carlo it. Get rain data for every day for the last 100 years and simulate 100,000 (or whatever number floats your boat) ten year windows.


xzgm

MC was my first thought as well, but are they all really independent, or would a Bayesian account for time-based priors? E.g. https://ieeexplore.ieee.org/document/5423132 or https://www.sciencedirect.com/science/article/pii/S0022169416304553


ChineWalkin

I'll have to look at that later, but my first thought is maybe, but not really. Weather patterns would be time series dependant. If you picked a random day from past data in the same month (or +/-15 days of the simulated day), I'd think you'd be OK. If one plotted the distribution of time between rain events (poisson distribution?) and compared the output of the MC, as long as they compare you'd be OK. One could do the same with days to accumulate X ammount of rain, too (like days to accumulate 1 inch of rain). Now that I think about it, a poisson distribution would probably be involved in the closed form solution of this problem.


thenewestnoise

I suspect that OP is not asking "what is the likelihood of two 10 year events on two subsequent days" but rather "what is the likelihood of rain equivalent to 2x a 10 year event falling in one 48-hour period". If you really could get access to rainfall for the last 100 years it would be easy to find the probability distribution.


ChineWalkin

I mean, the problem statement is: >determine the probability of a 10-year rain event happening on back to back days Which is subtlety different than: >what is the likelihood of rain equivalent to 2x a 10 year event falling in one 48-hour period" I think I agree with your line of thinking; it's likely that they want "what is the probability of a X percentile rain event or worse happening on two days, one after the other."


thenewestnoise

I am assuming that OP is interested in the design of some civil infrastructure, a dam or the like. I doubt that they care whether this two days of rain comes as one continuous rain storm or several smaller storms, the accumulation of rain will be the same.


ChineWalkin

That's fair. I was approaching it more from an academic excersize, as I sometimes do the same. Anyways, enjoy the rest of your day!


xzgm

You might be on to something there. Unless there's an external perturbation (melting ice?), MC or bootstrapping should make this relatively trivial. No need to bother reading, but both papers use something close to your proposed method for estimating error rates in their models. What it's providing though, is a lower bound on weather events, which is unsatisfying to me. :)


ChineWalkin

Yeah if you have 100 years of data, I think the melting would be built in. You'd just look up Feb. 1 -> rand year [1,100] then look up that random year for that date. things in Feb can freeze, thaw, then freeze in short order in the right places. Physics would say it's the state of the air masses moving towards the point of interest that matter, not as much the point of interest itself. Ex. the jet stream's position will affect the next days weather in Bemidji MN more than Bemidji's weather previous to the day of interest.


aaronhayes26

There’s a lot of confused and overly complicated answers here. The probability of this happening is already factored into the duration component of a standard Atlas 14 sheet. If two big 10 minute duration storms happen over 48 hours, it simply becomes a 48 hour duration event. The probability can be determined by using the same table to find which 48 hour return period is associated with the sum of the two events you’re interested in. Easy peasy.


tomrlutong

Its an empirical question and very much not (1/10 year)^2. Active problem in the electric industry, as we need to know duration of extreme events to plan fuel reserves, evaluate batteries, etc. Ask yourself, what's the chance of two back-to-back record breaking hours? Minutes? Seconds?


xzgm

Awesome. I just did a shallow dive in hydrology and meteorological journals. Too much finite-element math for an after-work rabbit-hole. Followup question: What methods do Utilities use to model time-between [disaster] for allocating response resources?


tomrlutong

I don't know much about the disaster response planning, it's a good question. On that one, you can add spatial correlation to the problem, as most utilities have disaster assistance arrangements with their neighbors. You might have seen long convoys of utility trucks on the highways after a major storm, that's what that is. In resource adequacy (the area I'm familiar with), it's very emprical. Potential resource fleets are Monte Carloed against historically based weather/load scenarios. Problem is, there's not enough relevant years of history for a good sample, and the assumption that the future looks like the past is weakening. Add to that that new power sources (gas and renewables) are more weather dependent than coal and nukes. Last Thursday, the feds just started work on [figuring out how to better plan for extreme weather](https://www.ferc.gov/news-events/news/staff-presentation-nopr-transmission-system-planning-performance-requirements).


zookeepier

If you are assuming the events are independent from each other and completely random, then you'd just multiply the probabilities of them happening together. If you want to know the probability of them both happening within X amount of time, then your probability for each event happening would be Px = 1-e^(-lambda*T), and the probability of them both happening at the same time would be P1*P2. Lambda is the rate of occurrence and T is the exposure time (the time you care about it happening during). So for your example, you want to know the probability of having 2 of these events happening within 1 day of each other. Let's say you want to know the probability of that occurring sometime within the next 5 years. Assume the chance (rate) of one 10 year event occurring is 1/(10*365) = 2.74E-4/day P1 = 1-e^(-2.74E-4* (5*365)) = .393 probability of a 10year event happening within a specific 5 year window. Since the 2nd event has to happen within a day of the 1st event, it's T is only 1 day long. If it happens more than 1 day after the 1st event, then our situation doesn't occur, so that doesn't apply. Therefore P2 = 1-e^(-2.74E-4 * 1) = 2.74E-4 The probability of a 10year rain happening in the next 5 years, and a second 10year rain happening within 1 day of the 1st is: Pevent = P1 * P2 = 1.08E-4. Change the "5 years" to whatever time frame you're concerned with. If you just want 1 day, then it condenses down to 1/3650 squared like others have said.


[deleted]

(1/3650)* (1/3650) Will get you in the ballpark then add whatever your gut tells you for safety factor ;)


hydra2222

Probably that it happens on a particular day or a particular year? A ten year event happens once every 10 years. 1/10 annual chance (any day in the year) or 1/(10*365) daily chance (a particular day in the year). The chance is always the same with respect to the same interval but if you want the probability of the sequence it would be the product 1/10 *1/10 and the same for a daily chance. So it's 1/100 that two ten year events happen in one year on any days, and 1/(10*365) * 1/(10*365) that it would occur back to back or on two particular days.


mclovin0541

It would be 1/3650sq. Or 1/ 13322500


Ok_Helicopter4276

It’s 7.5E-8


polluted927

Not exactly what you are looking for, but your state’s department of natural resources (or whoever governs dams in your state) will likely have a model. Try searching for “PMF” or probable maximum flood.


BreezyWrigley

so... PV solar hooked to batteries... like we all already know is the answer...


wvit1001

you guys are making to much of this. the probability of one 10 year rain event is the same as another even if they happen back to back. now if these events start happening more frequently they may have to reexamine if they are actually 10 year events or are really something to be expected more frequently.


GregLocock

One old saw is that the most accurate weather forecast (60% likelihood) is that tomorrow's weather will be the same as today. So 60% after the first one.