QPF/PQPF question

[mappy]

This is from my husband, who has a work thing involving QPF/PQPF. I can't give too many details regarding the project.

From Mr. Mappy

I would like to know if there is a way to establish an expected precipitation quantity (inches) based on the forecasted quantity and its probability.

Based on my limited understanding of statistics I would think that the expected value would be equal to the forecasted value (QPF) multiplied by its probability (PQPF). So for a forecast of 1" with a 50% chance, one would expect 0.5" of precipitation. However, this seems over-simplified and I'm assuming is an incorrect way to attempt to quantify an expected precipitation.

Thank you for any insight and information you can share.

I've tried talking it through, but I don't have the answer for him. Thanks guys! -Mappy

[WinterWxLuvr]

Just now, mappy said:

 

This is from my husband, who has a work thing involving QPF/PQPF. I can't give too many details regarding the project.

From Mr. Mappy

I would like to know if there is a way to establish an expected precipitation quantity (inches) based on the forecasted quantity and its probability.

Based on my limited understanding of statistics I would think that the expected value would be equal to the forecasted value (QPF) multiplied by its probability (PQPF). So for a forecast of 1" with a 50% chance, one would expect 0.5" of precipitation. However, this seems over-simplified and I'm assuming is an incorrect way to attempt to quantify an expected precipitation.

Thank you for any insight and information you can share.

I've tried talking it through, but I don't have the answer for him. Thanks guys! -Mappy

 

I thought the PQPF was simply probability of getting at least that amount.

[mappy]

I thought the PQPF was simply probability of getting at least that amount.

Yes, it is. But he wants to know what the expected value would be, given the pQPF forecast.

So if the forecast was a 50% chance of 1" rain, he would expect .5" of rain.

[winterymix]

This is a devilish combo of "boring" and "challenging".

I don't understand the idea in full but the portion that I cut and pasted below

gives an actual explanation:

 

 

 Probabilistic quantitative precipitation forecasts
(PQPFs) are then generated by using the relative fre-
quency of the event in the analyzed ensemble; for ex-
ample, if three of the four analyzed members at a grid
point had greater than 10 mm of accumulated rain, the
probability of exceeding 10 mm at that grid point was
set to 75%. Sample probability forecasts from the four-
member ensemble are shown in the fourth row in Fig. 3.
Note, however, that the probabilities are retained only
in the region enclosed by the dashed box. Probabilities
in adjacent regions are computed by shifting the search
region one coarse-mesh grid point. A national-scale 32-
km  PQPF  is  generated  by  tiling  together  the  local
PQPFs; further discussion of this is supplied in section
3b(10) below (see Fig. 4). The final step of the process
is to compare the probability forecasts to the analyzed
precipitation, which is shown in the bottom row of Fig. 3.

https://www.esrl.noaa.gov/psd/people/tom.hamill/reforecast_analog_v2.pdf

[cae]

57 minutes ago, mappy said:

Yes, it is. But he wants to know what the expected value would be, given the pQPF forecast.

So if the forecast was a 50% chance of 1" rain, he would expect .5" of rain.

 

In that case, 0.5" would be the minimum expected value, but you can't really tell what the expected value would be. 

If you have a range of qpf / pqpf values, you can estimate the expected value.

For example, consider a situation in which you know that:

• There is a 100% chance of at least 0"
• There is a 40% chance of at least 0.5"
• There is a 0% chance of at least 1"

We can let the variables p1, p2, and p3 be 100%, 40%, and 0%, and the variables q1, q2, and q3 be 0", 0.5", and 1".  An estimate of the expected value is then

(p1 * q2 - p2 * q1)/2 +

(p2 * q3 - p3 * q2)/2

 

That same pattern holds in general.  If you have n different pqpf values, and q1 = 0 and pn = 0, then you can estimate the expected value as

(p1 * q2 - p2 * q1)/2 +

(p2 * q3 - p3 * q2)/2 +

(p3 * q4 - p4 * q3)/2 +

...

(p(n-1) * qn - pn * q(n-1))/2

 

[mappy]

Thanks guys, I passed along the info