I'm trying to understand what weather forecasts mean more precisely. As I understand it from reading Wikipedia, blogs, etc., the percentage value for rain/precipitation that you see in a forecast is technically called the "probability of precipitation". To quote the National Weather Service webpage:
"Mathematically, PoP is defined as follows: PoP = C x A where "C" = the confidence that precipitation will occur somewhere in the forecast area, and where "A" = the percent of the area that will receive measureable precipitation, if it occurs at all. So... in the case of the forecast above, if the forecaster knows precipitation is sure to occur ( confidence is 100% ), he/she is expressing how much of the area will receive measurable rain. ( PoP = "C" x "A" or "1" times ".4" which equals .4 or 40%.)"
This definition does not seem well stated to me, for the reason that "confidence" is (presumably) not uniform across an area. For instance, the phrase "how much of the area will receive measurable rain" seems odd, since a forecast would (presumably) only be able to give a probabilistic estimate for this area.
Let's cook up an example. Consider a town (the forecast area) consisting of two parts of equal area (called north side and south side). Each point in the north side will be rained on with 100% probability tomorrow, and each point in the south side will be rained on with 50% probability (at every point) tomorrow. What is the PoP in this example? At face value, the definition could be interpreted as implying that the PoP is 100%, since precipitation will occur somewhere in the forecast area. However, this value seems intuitively unsatisfying, since some people might not get any rain.
Here's what I might expect a more precise definition to be. If $A$ is the area and $C(x)$ is the "pointwise confidence function" depending on a point (location) $x$, then define $$PoP = \frac{1}{area(A)}\int_{A} C(x)\, dx.$$ In words, this is just the expected value of $C(x)$, or the probability that a randomly located person would see rain over the specified time interval. In practice, of course the integral would be estimated based on actual measurement sites. If such a formula is indeed an accurate definition, then I'd be satisfied. In the above example, the PoP would be 75%. (The official definition could in essence be viewed as a shorthand that is more useful for those without any background in calculus.) If this definition is not correct, then some explanation would be helpful.
I've read web articles with statements like the following: "As a student and observer of meteorology, it constantly bums me out that people do not understand what it means when someone says there’s an “X% chance of rain” tomorrow. A 50 percent chance of rain does not mean there’s a 1-in-2 chance that you’re going to get wet."
It's not clear to me why "A 50 percent chance of rain means there’s a 1-in-2 chance that you’re going to get wet" would be an inaccurate interpretation of PoP. If the definition I suggested above is correct, then it is perfectly correct to say that a stationary observer placed at a random location in this scenario would have a 50% chance of getting wet. Am I missing something, or is this author being careless?
I don't have any background in meteorology, and in particular I don't have much of a sense of how PoP is actually computed in practice.