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**Probabilistic Quantitative Precipitation Forecasting**

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. These QPFs can also be produced for snowfall amounts by converting the anticipated precipitation from liquid to the snow equivalent, using the appropriate snow to water ratio, such as 10 inches snow for 1 inch water.

The PQPF is similar to the regular PoP, except PQPF is computed for the probability to equal or exceed higher rainfall amounts, such as 0.10, 0.25, 0.50 or 1.00 inch, or any other meaningful value (user feedback can help determine which thresholds should be provided). 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:

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:

cPOE(x) = e^{-x/μ}

We believe it is more useful to provide the unconditional probability to exceed 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."

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 μ. So the equation for cPOE for determining the chance of exceedance of 1.00 inch produces the following:

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%.

Where a weather event is expected to cause widespread rainfall and the probability of rain is 90% or greater, it has been found that the frequency distribution of rainfall amounts will more closely fit the gamma distribution where the "alpha (α) term" equals three. In those high probability events (typically high areal coverage), the highest frequency of rainfall amounts will be none-zero. Therefore, the computation of exceedance probabilities using the gamma distribution where the alpha term equals 3 provides better results as shown in comparisons with the exponential distribution. The resulting equation for the high probability events is given by the following:

POE(x) =PoP * (0.5) * (e^{-x/β}) * (x^{2}/β2 + 2x/β + 2) ,
where β(beta) = μ/α = (QPF/3) , μ = conditional QPF

These results are very similar to those of Donald L. Jorgensen, William H. Klein, and Charles F. Roberts, Conditional Probabilities of Precipitation Amounts in the Conterminous United States, ESSA Technical Memorandum WBTM TDL 18, Weather Bureau Office of Systems Development Techniques Development Laboratory, Silver Spring, MD., March 1969 Questions regarding this product or how it is computed may be directed to Steve Amburn (Steve.Amburn@noaa.gov) at the National Weather Service in Tulsa.

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