Probabilistic quantitative precipitation forecasts (PQPF) and quantitative snowfall (PSnow) provide our best estimate of the chance that any given location will receive an amount of rain/snow that 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/melted snow that equals or exceeds 0.01 inches. The PQPF/PSnow is similar, except it is computed for the probability to exceed higher amounts (e.g., 1.00 inches or rain or 6.0 inches of snow)
The PQPF/PSnow is derived from the probability of precipitation (PoP) forecasts and our quantitative precipitation/snowfall forecast (QPF/SnowAmt). For the purpose of the calculations, the standard QPF/SnowAmt, which is an unconditional value, is converted to a conditional value by dividing it by the PoP. The resulting QPF/SnowAmt is then an amount that is conditional upon the occurrence of rain/snow at any specific location. Although this seems to be a subtle difference, it is very important.
The PQPF/PSnow is based on the climatological distribution of precipitation, which very closely matches a linear combination of low order gamma distributions. Generally speaking, this combined distribution indicates that the probability of receiving larger rainfall/snowfall amounts decreases nearly exponentially as the amounts get larger.
The probabilty density function is:
f(x,a,b) = C(a=1) • (1/b) • e-x/b + C(a=2) • (x/b2) • e-x/b + C(a=3) • (x2/2b3) • e-x/b
where b=µ/a, μ is the conditional QPF/SnowAmt, or mean expected rainfall/snowfall amount given that precipitation occurs at the specified location, a is the gamma order and C(PoP,a) is a weighting function .
Each term of this equation can be integrated from any rainfall/snowfall threshold value x, to infinity to determine the probability to exceed that value x and recombined using a PoP based weighting fnction. After integrating, the conditional probability to exceed an amount x is given by:
cPOE(x) = C(a=1) • e-x/μ + C(a=2) • (2x/µ + 1) • e-2x/µ + C(a=3) • 1/2 • (9x2/µ2 + 6x/µ + 2) • e-3x/µ
C(PoP,a) = max(1-abs(2 + tanh(pi/60 • (PoP-60)) - a), 0)
However, it is more useful to provide the unconditional probability to exceed the specified amounts. 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)
Rainfall Example: Assume the forecast QPF (unconditional) is 0.80 inches and the PoP is 70% and we are interested in the probability of exceeding 1.00 inch. The conditional QPF is (0.80)/(0.70) or approximately 1.14 which is now the value μ. The result is:
cPOE(1) = 0.50 = 50%.
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)• (0.50) = 0.35 = 35.0%.
Snowfall Example: Assume the forecast SnowAmt(unconditional) is 3.7 inches and the PoP is 80% and we are interested in the probability of exceeding 6.0 inches. The conditional SnowAmt is (3.7)/(0.80) or approximately 4.62 which is now the value μ. The result is:
cPOE(6) = 0.256 = 25.6%.
Since there is only a 80% chance of snow, the final, unconditional chance to exceed six inches of snow at a location is:
POE(6) = (0.80)• (0.256) = 0.205 = 20.5%.
The probability distribution can also be used to calculate a minimum and maximum value for the specified time periods. The minimum value is defined to be the 15th percentile (85% chance of exceeding this value - which also corresponds to the threshold for "categorical" precipitation - PoP >= 80%). The maximum value is defined to be the 95th percentile (5% chance of exceeding this value).
This work is an extension of the methods employed by the National Weather Service Forecast Office is Tulsa, OK. Many thanks to Steve Amburn - Science and Operations Officer - WFO Tulsa.
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 Techiques Development Laboratory, Silver Spring, MD., March 1969