I know this is uncharitable and I am wrong but I am having trouble coming up with concrete scenarios where you die with 2 days notice but survive with 3. I am nonethless a believer that more accurate forecasting has value.
Could you imagine a scenario where from warning to complete evacuation takes more than two days? Evacuating a whole area is a hard task, particularly once you start looking at more complex problems (elderly, prisons, hospitals).
Hurricane Maria went from Cat 2 to Cat 5 in less than 24 hours, and turned making a direct hit to Dominica in 2017.
It's not uncharitable, because the system doesn't actually claim to give you an extra day of notice. It would be more accurate to say: "the model can reach a given level of forecast accuracy roughly a day farther in advance". So your question becomes: are there scenarios where I am 80% sure this is a Cat 5 hurricane 3 days before, where I would die if I was only 65% sure it was Cat 5 on that same day? The answer is - probably not, because even in the example used in the PR article, the NHC was already issuing strong early guidance 5 days before landfall.
The 2 vs 3 days makes less of an impact on personal decision making but has massive benefits for decision making at the country wide response level.
I don't think the amount of notice is as important as "this totally is for sure going to wreck you" is. We generally know something is going to hit somewhere at sometime. Which isn't specific enough for everyone to act on
You live on a chain of small islands and travel by boat.
Hurricane blast radius is very small
Imagine you've got to evacuate a hundred thousand people. That extra day is incredibly valuable.
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You can walk 50km in those extra 24 hours which will save your life if you’re in a storm surge area without other means of transportation, which is easily the case when everyone else is evacuating alongside you.