Probabilistic QPF Detailed Definition
Probabilistic quantitative precipitation forecasts (PQPFs) provide our best estimate of the chance that any given location will receive an amount of rain that equals or exceeds a certain threshold value. Our regular “probability of precipitation” (PoP) forecast is the unconditional probability that a location will receive an amount of rain that equals or exceeds 0.01 inches of precipitation. The PQPF is similar, except it is computed for the probability to equal or exceed a higher rainfall amount, such as 0.10, 0.50, 1.00 or 2.00 inches, or any other arbitrary value.
The PQPF is derived from the probability of precipitation (PoP) forecasts and our quantitative precipitation forecast (QPF). For the purpose of the calculations, the standard QPF, which is an unconditional QPF, is converted to a conditional value by dividing it by the PoP. The resulting QPF is then an amount that is conditional upon the occurrence of rain at any specific location. Although this seems to be a subtle difference, it is very important.
The PQPF is based on the climatological distribution of precipitation, which very closely matches the special gamma distribution called the exponential distribution. This distribution indicates that the probability of receiving larger rainfall amounts decreases exponentially as the rainfall amounts get larger. The density function for the exponential distribution is:
Equation 1: f(x) = (1/µ) • e-x/µ
This equation can be integrated from any rainfall threshold value x, to infinity to determine the probability to exceed that value x. The term µ is the conditional QPF, or average expected rainfall amount given that rain occurs at the specified location. After integrating, the conditional probability to exceed an amount x is given by:
Equation 2: cPOE(x) = e-x/µ
We felt it would be more useful to provide the unconditional probability to exceed the specified rainfall amounts. This is easily accomplished. The cPOE(x) is simply multiplied by the probability of precipitation (PoP) at any location to determine the unconditional probability to exceed the amount x. For simplicity, the unconditional probability of exceedance will be denoted by “POE.”
Equation 3: POE(x) = (PoP) • cPOE(x)
Example: Assume the forecast QPF (unconditional) is 0.80 inches and the PoP is 70%. The conditional QPF is then (0.80)/(0.70) or approximately 1.14 which is now the value for µ, in equation 2. The result is:
cPOE(1) = e-1/µ = e-1/(1.14) = 0.41 = 41%.
Since there is only a 70% chance of rain, the final, unconditional chance to exceed one inch of rain at a location is:
POE(1) = (0.70) • (0.41) = 0.287 = 28.7%.
Special thanks to Steve Amburn at the National Weather Service in Tulsa for providing the above information.