|Roll a 100-sided die once. That is what looking|
for a 100 Year storm at a single rain gauge
in a single year is like.
CBC has reported on this: link
While we are all concerned about flooding, the question on large storm frequency is "So What?". Or more specifically, from a statistical, mathematical, logical point of view, is more than five 100 Year storms over a 14 year period (2005 to 2018) rare and unexpected, or does this have a high probability of occurring? As we know the Insurance Bureau of Canada does not always rely on proper statistics to support statements on extreme weather, confusing theoretical shifts in probabilities of extreme events with real data (see IBC Telling the Weather Story where IBC ignores Environment and Climate Change Canada's Engineering Climate Datasets).
Let's do some math to see if over five 100 Year storms is rare or not.
First, consider that a 100 Year storm has a probability of occurring of 1/100 = 1 percent per year.
Second, count up the number of rain gauges that have been proliferating across the GTA to support inflow in infiltration studies for wastewater studies and to support operational needs. Here are some counts with various sources:
i) City of Toronto (https://www.toronto.ca/city-government/data-research-maps/open-data/open-data-catalogue/water/#09dee024-b840-174f-7270-29c1a1773d14) - 46 rain gauges
ii) Region of York (https://www.york.ca/wps/wcm/connect/yorkpublic/b22ae2f3-5140-48f2-869e-a803d2552893/2017+Inflow+and+Infiltration+Reduction+Strategy+Annual+Report.pdf?MOD=AJPERES) - 71 rain gauges
iii) Peel Region (https://www.peelregion.ca/council/agendas/pdf/ipac-20110811/4b.pdf) - 6 rain gauges (correction July 25, 2019 - Peel has 28 rain gauges ... probabilities in this blog post will go up a bit)
iv) Halton Region (https://www.peelregion.ca/budget/2018/pdf/conservation-halton.pdf) - 14 rain gauges
v) Toronto and Region Conservation Authority (http://126.96.36.199/xcreports/Precipitation/precipitationOverview.aspx) - 14 rain gauges
Total number of gauges = 151. A good first estimate - certainly there are more. (correction July 25, 2019 - as Peel has 28 rain gauges the total is 173 stations)
Third, assuming each rain gauge observes rainfall events independently year to year, what is the chance of getting at least one 100 Year event at a single gauge in 14 years?
Probability = 1 - (1-1/100)^14 = 13.1% chance of a 100 Year storm storm at a single gauge. That seems pretty big.
The number of 'trials' or samples equivalent to 14 rolls of a 100-sided die, meaning 14 independent observations or 'samples' from the statistical population of events.
It is reasonable to assume that a single rain gauge can record a 100 Year event but not surrounding gauges? Yes indeed. The August 2018 storm in Toronto only exceeded 100 Year rainfall totals at one gauge. So it is reasonable for smaller, spatially isolated rainfall events that do occur.
Fourth, assuming all rain gauges observe rain independently what is the chance of getting more than one 100 Year events across all 151 gauge in 14 years?
(Additional comment: we know that storms exceeding 100 Year volumes can cover large areas such that observations are adjacent gauges are not completely independent, especially if they are spatially very close - so this fourth scenario is considered an upper bound on sensitivity analysis considering gauge independence - below, another bound is evaluated assuming less independence).
What about more than five 100 Year storms over 14 years? We have to then consider combinations of events (we do not care which of the 2144 samples has the events) and approach this by subtracting the probability of 1, 2, 3, and 4 events. This summarizes the approach (thanks so much FP!):
The probability of 5 or more 100 Year events is again over 99.9% (see cell F22), showing that when there are many, many trials, the probability of a multiple rare event is very high.
Let's consider over five 100 Year storms again. A keen reader has shown that the probability is 41.6% for this, as shown in cell L22 in the spreadsheet image above. Again,pretty high chance of getting 5 or more events when gauges do not observe extremes independently, but rather in clusters.
For more on this analysis, and the probability of 5 or more occurrences in 423 observations the probabilities considered in deriving the probability are as follows:
- 4 occurrences in 423 observations (P = 0.195038119)
- 3 occurrences in 423 observations (P = 0.183893083)
- 2 occurrences in 423 observations (P = 0.1297298)
- 1 occurrences in 423 observations (P = 0.060868484)
- 0 occurrences in 423 observations (P = 0.014245815)
- Sum = 0.583775302
|Probability of 5 or more 100 Year Storms at Independent Rain Gauges (151 gauges x 14 years = 2114 'trials')|
|Probability of 5 or more 100 Year Storms at Clusters of Rain Gauges With Dependent Observations (30.2 gauge clusters x 14 years = 422.8, say 423, 'trials')|
|Probability of 5 or more 100 Year Storms at Large Clusters of Rain Gauges With Dependent Observations (15.1 gauge clusters x 14 years = 211.4, say 211, 'trials')|
- May 12, 2000 - 1 rain gauge over 100 Year (see slide 9)
- August 19, 2005 - 12 rain gauges over 100 Year (see slide 11)
- July 13, 2018 - 6 rain gauges over 100 Year (see slide 19)
Conclusion - is it not rare to get more than five 100 Year rainfall observations at over 151 GTA gauges, over 14 years. The chances range from near certainty (over 99.9%) for independent events at each rain gauge to relatively high probability (over 40%) if gauges are independent clusters of 5 or more.
As noted in my recent Financial Post OpEd, making a big deal about irrelevant risk facts distracts us from addressing the root cause of flood problems. The City of Toronto should try to not get distracted. And Councilor Mike Layton is probably in the running for a Milli Vanilli "Blame it on the Rain" award this year :)
Terence Corcoran covers this all very well in today's column, referencing analysis on this blog.
Note: probabilities for 5 or more events corrected/updated April 1, 2019. Thanks to keen readers for helping define the probabilities of combination events and for the nostalgic references to University of Toronto's Professor Emeritus Dr. Barry Adams' CIV340 course notes that outline the analysis approach.
What are the probabilities considering the updated number of stations (i.e., more in Peel), meaning a total of 173 stations? That is, 2422 trials if stations are independent and 484 trials if stations are clustered in clusters of 5.
For 5 or more 100-year storms in 14 years, the probability is 99.9% - 53.2% for independent and clustered gauges, respectively.
For 6 or more storms the probability is 99.9% - 35.6% for independent and clustered gauges, respectively.