[wxqc] Rain Forecast

[Victor Engel]

I was having a conversation with someone recently about rain forecasts.
Something that people (myself included) seem to be confused about is the
probability of rain. What is the probability actually supposed to be
measuring?

I did a lot of searching around the NOAA website, and it looks like their
numbers are not entered by computer, but are entered by people. The numbers
seem to be forecast probabilities for the covered time periods. The time
periods I looked at seemed to be three hours. This seems to be confirmed by
the chart given at the following URL (I really like this chart, by the way):

http://tinyurl.com/jvf7v

I decided to take the data from this chart and compute 24 hour probabilities
as well as probabilities over longer periods, let's call the periods UNTIL
X, where X is some future time.

If the data points represent probabilities that it will rain sometime in the
subsequent three hour period, then the probability that it will rain in the
subsequent 24 hour period is simple to derive.

Let P3(i) be the probability of rain at sample i (a 3 hour period).
P3(i+1) is the probability of rain for the following 3 hour period).

P3 is a number from 0 to 1 where 0 is 0% chance of rain, and 1 is 100%
chance of rain.

Then P24(i), the probability of rain for the 24 hour period, will be:

1 - (product of (1-P3(i)) for i from 1 to 8.

For example, suppose the probabilities of rain for midnight, 3 am, 6am, 9am,
noon, 3pm, 6pm, 9pm are:

0.1, 0.1, 0.2, 0.2, 0.3, 0.1, 0.1, 0.1, then the probability that it will
rain sometime during the whole day is:

1 - (0.9 * 0.9 * 0.8 * 0.8 * 0.7 * 0.9 * 0.9 * 0.9) = 0.735 or about a 74%
chance of rain.

Anyway, in the conversation I had, the person mentioned that it looks like
we'll be without rain until Monday. The link above, run early this morning,
gave probabilities of:

Date/Time Probability
9/14 07:00 0.05
9/15 07:00 0.10
9/17 13:00 0.20
9/18 07:00 0.50
9/18 19:00 0.30
9/19 07:00 0.20
9/20 07:00 0.10

Using the above calculations, I get the following probabilities for the 24
hour periods following those same times:

9/14 07:00 0.34
9/15 07:00 0.57
9/17 13:00 0.93
9/18 07:00 0.98
9/18 19:00 0.90
9/19 07:00 0.83
9/20 07:00 ----

The last line is blank because the projection does not go that far. Now I
show the probabilities that it will rain sometime BY the time listed (these
times are 3 hours later, because the times in the numbers above are for
subsequent periods).

9/14 10:00 0.05
9/15 10:00 0.40
9/17 13:00 0.92
9/18 07:00 0.99
9/18 19:00 1.00
9/19 07:00 1.00
9/20 07:00 1.00

In other words, given the numbers on the site, it's a virtual certainty that
it will rain by 7PM on Monday.

This analysis is very sensitive to error, especially at the low end.
Suppose, for example, I change all the 5% probabilities to 1% instead. Then
we get the following:

9/14 10:00 0.01
9/15 10:00 0.17
9/17 13:00 0.89
9/18 07:00 0.98
9/18 19:00 1.00
9/19 07:00 1.00
9/20 07:00 1.00

Anyone see a problem in my analysis?
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