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Weather.com will happily give me temperature predictions for the next ten days, but no indication of how reliable those predictions are. So if I want to plan something for a week from today, or four days from today, I don't really know whether there's much information in the forecasts. My personal, unscientific (and certainly unreliable) experience says there's not, but there's plenty of variation in the forecasted temperatures, even between days 9 and 10, so the weathermen are not simply falling back to the long-term averages for the time of year.

I'm hoping for a rule of thumb here (so my question is different from this theory-heavy discussion from two years ago). Conceivably something as simple as a fixed interval size for each number of days in the future could work, although I could imagine there being low-hanging fruit in, for example, atmospheric pressure or the proximity of the location being forecasted for to water.

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Look at ensemble forecasts. Although they do not exactly give confidence intervals, they do give the same kind of information you would use a confidence interval for.

Weather is chaotic. So are the models. If initial conditions change slightly, outcome changes dramatically.

Therefore, models are typically run a number of times, for example, ten runs. Where the runs coincide, the forecast is rather certain. Where they diverge, the forecast is not. Below is an example of such for Bucharest, Romania, run on 28 January 2014:

Ensemble forecast for Bucharest

As you can see, the 850 hPa temperature agrees fairly well the first couple of days, but by the end, they are “all over the place”. This kind of plot, which may also be on a map, is also referred to as a spaghetti plot. Although they are not difficult to understand, they do tend to focus on “expert” variables and do not directly translate to precipitation amounts or sunshine hours. They are direct model output.

Below is an example of an NCEP run for the US and surroundings. At 0 hours, all the model runs agree fairly well (fortunately):

NCEP ensemble 0h

After 24 hours, the picture still looks fine:

NCEP ensemble 24h

At 240 hours, however, it's pretty much spaghetti. This means the forecast that weather.com gives you is pretty much useless:

NCEP ensemble 240h

Through the NCEP website, you can also look at animations and maps of the standard deviations. There are various sources for such ensemble forecasts, and not all are free. The line graph above is from the German website Wetterzentrale. ZMAW links to meteogrammes for European cities. The maps are from NOAA ESRL PSD. Weather.gov also links to a number of sources. If you search the web for spaghetti diagram or ensemble forecast, you may find a lot more.

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Thank you! Is it reasonable to assume in this context that all the variation is contained in the model? That is, could it be that each Monte Carlo path i at, say two days out gives a forecast f_i of f_i = T + X + e_i, where T is the true temperature (if you could see the future), e_i is an idiosyncratic noise term, and X is a source of error that is shared by all the paths? I guess what I'm really asking is, is var(X) small enough to safely be ignored? –  kuzzooroo Jan 28 at 23:57
    
I'm not sure, I'm not really an expert. The models are prety well-tested so I think that $var(X)$ would be small enough if we had near-perfect knowledge of the initial state. However, if there is an incorrect initialisation of the models — perhaps a satellite picture being miscalibrated or so — that would cause a systematic error to go into each run. As I said, I'm not really an expert so I can't say how often that is the case, but I have heard more expert colleagues refer to wrong initialisation what ordinary people would call the weather forecast was totally wrong. –  gerrit Jan 29 at 0:51

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