# Presentation: 194579

## Defining the Quantitative Precipitation Forecast

- Steve Amburn, SOO
- WFO Tulsa, OK

## Current Definition for QPF(NWSI 10-503)

**      ****NWSI 10-503:  *****WFO Public Weather Forecast Products Specification*****      **
**              “****QPF 12HR. - This parameter, quantitative precipitation forecast (QPF) represents**
**	****the total amount of liquid precipitation, in inches, expected during a 12-hour**
**	****period ending at 6:00 a.m., or 6:00 p.m. local time at any point in the forecast**
**	****area. The QPF is presented in locally defined ranges, (e.g., .10-.24), or single**
**	****values.”**
**     ****NWSI 10-506: *****Digital Data Products/Services***
**	****“****Quantitative Precipitation Forecast (QPF) **- the total amount of
- 	expected liquid precipitation (in hundredths of inches). A QPF will be specified
- 	 when a measurable (0.01") precipitation type is forecast for any hour during
- 	 a QPF valid period.
**       ****Problems: **
**	****1.  What does “expected” really mean?  Greater than 50% chance? **
**	****2.  The definition is for a point or grid forecast, not a “basin average” or **
**                     ****areal average.**
**	****3.  How do you forecast QPF when the PoP (actual areal coverage) is, say, 40%?**
- 	**4.  Is this an areal average, basin average, average over time?  **

## QPF Definition

- Can we forecast QPF for a point?
- When you know it will rain somewhere, how do you make a QPF when most of the area will not receive rain?

## Maximum QPF(NWSI 10-503)

**“****MAX QPF**** (optional). The value for MAX QPF is the estimated maximum**
**	****amount of precipitation, in inches, occurring at any point within the **
**	****forecast area during the12-hour period ending at 6:00 a.m. or 6:00 p.m. **
**	****local time. This amount is presented as either a single value or a range, **
**	****and is based upon a 75% confidence level of the QPF forecaster(s).”**
- A potentially valuable forecast. But...how can we
- arrive at that number?

## QPF Solutions

- Acknowledge the definition is inadequate.
- Adopt a specific and scientifically valid definition
- Use the new definition to help us
- Convey more information
- Develop new/better products

## Defining anything

- “When you can measure what you are speaking about and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager, unsatisfactory kind.”  Lord Kelvin (1824-1907)

## Proposed QPF Definition

**Quantitative Precipitation Forecast (QPF6)** – The unconditional **mean of the expected distribution** of liquid precipitation that a forecaster would expect (in hundredths of inches) accumulated over a six-hour period beginning at 00 UTC and 12 UTC, at a point (or NDFD grid box) from a large number of similar events.

##  “QPF6:  The unconditional mean of the expected distribution of liquid precipitation that a forecaster would expect...”-----Why this change?

**We cannot accurately forecast rainfall at a point.**** **** **
**The NWS QPF should, however, describe the **_**expected event.**_** **
**Most frequently, rainfall events can be described as **_**exponential distributions**_**,  usually starting at zero. (Wilks, 1995)**
** **
**Every distribution has a mean**
**So, if the NWS QPF is defined as the mean of the distribution, and if we forecast that mean, then we have effectively forecast the expected distribution.**
**Most WFO meteorologists forecast that “mean” now!**

**Notes:**
This slide shows the logical path connecting any given rainfall event to its rainfall frequency distribution, its mean, and finally to that the mean can be used to describe and define the event.  This assumes the selection of a correct mathematical distribution, usually the exponential distribution (Wilks, “Statistical Methods in the Atmospheric Sciences”, 1995. 

## 10-year Distribution Tulsa

**Notes:**
The shows that the long term precipitation distribution is exponential. This chart was nearly identical for Tulsa, Fort Smith, McAlester and Fayetteville over the same 10-year period. 

## 10-year Distribution Fort Smith

**Notes:**
Fort Smith distribution ... Exponential. 

- Precipitation at a point, over time, has an _exponential distribution and a mean._
- (See Jorgensen and Klein, 1969, for over 100 examples.)

**Notes:**
Distribution at a point (FYV) over a 10 year period. 
Point data
Distributed exponentially
Jorgensen, Klein and Roberts, ESSA tech memo has examples of exceedance probabilities implying exponential distributions for more that 100 cities across the CONUS. 

- Individual events (typically) also have an _exponential distributions _and therefore means._ _

**Notes:**
86 different 12-hour rainfall events. 
Distributed exponentially (except one) 

## Two Seasons of Precipitation

**Notes:**
More events...

## Three Seasons of Precipitation

**Notes:**
More events but for autumn. 

## Two Seasons of Precipitation

**Notes:**
Winter events are distributed exponentially for the most part, but some events are not.  However, this generally has very small effects in computing the exceedance probabilities with the equation.  Examples follow...

## Distributions and Exceedance Probabilities

**Notes:**
  Compare the actual distribution (left) to the exceedance probabilities on the right.  The dark blue line is calculated from the chart data on the left.  The cyan line is computed from the probability of exceedance (POE) formula, shown later in this presentation. 
  The data comes from the ABRFC who compute the rainfall amounts (QPE) for 4745 HRAP grids across the TSA CWFA.  

## Distributions and Exceedance Probabilities

**Notes:**
Another example.  Note that the distribution of rainfall amounts is not exponentially distributed, but the exceedance probabilities are still very close to what actually occurred. It appears the uPOE is a good approximation because the actual distribution beyond 0.25 inches is very close to exponentially distributed.  

## Distributions and Exceedance Probabilities

## Distributions and Exceedance Probabilities

## Distributions and Exceedance Probabilities

## Distributions and Exceedance Probabilities

**Notes:**
The POE is not as good a fit, but the distribution is only loosely exponential.  Still, the POE has value if we forecast the correct mean and PoP. 

## Distributions and Exceedance Probabilities

**Notes:**
This is an rare example where the POE equation did not do well, although, it is clear the actual distribution of rainfall was not even close to exponential.  (Integrating the gamma distribution with an alpha parameter of 3 would have worked better.)  The formula computed probability to exceed 0.50 inches is near 60% while the actual data showed it should have been around 90%.  Still, such a forecast (even as poor as this one) seems to provide much better information to the customer than what we do now. 

## Same as previous slide, but using a gamma distribution, alpha = 3

- If a forecaster could select the non-exponential distribution, then the uPOE could be more accurate.
-    (exponential distribution = gamma distribution, alpha= 1)

## Distributions and Exceedance Probabilities

**Notes:**
Just another more typical example. 

## QPF similar to PoP

- Precipitation distributions can be considered as:
- A combination of events at a point (**point mean**), or
- A combination of points over time (**areal mean**).
- Speaking on PoP verification, “...as long as there are no local effects...the result from any gage and result for all gages averaged together would be the same in the long run.” – L. Hughes, Probability Forecasting – Reasons, Procedures, Problems, NOAA Tech Memo NWS FCST 24

**Notes:**
The point here is that the mean at a point over time will very closely match the mean over an area for a single event, given that the events being averaged are similar.  

## A Series of Similar but Different Events

- The following are three events with:
- Same mean
- Same areal coverage
- Same frequency distribution of rainfall
- _Different_ spatial distribution

## A

- 30% coverage
- Mean QPE = .06

- 0.01”

- 0.25”

- 0.50”

- 1.00”

**A**

**B**

**C**

**Notes:**
Points A and C receive no rain.  Only point B records rain.  

## A

- 30% coverage
- Mean QPE = .06

- 0.01”

- 0.25”

- 0.50”

- 1.00”

**A**

**B**

**C**

**Notes:**
This time, B and C receive no rain. 

## A

- 30% coverage
- Mean QPE = .06

- 0.01”

- 0.25

- 0.50

- 1.00

**A**

**B**

**C**

**Notes:**
Again B and C receive no rain.  

## QPF similar to PoP

- For each of the previous events,
- Areal coverage (PoP) = 30%
- Areal Mean = 0.06”
- For a sufficient number of similar but random events, each point will have:
- the same precipitation frequency
- the same mean as the areal mean.

**Notes:**
Bullet #2 is generally well accepted.  Also, it can be seen that the chance of rain at any point should be 30%.  And, after a sufficient number of similar events, the mean at each point will be 0.06 inches. 

## See the following example of an actual event...

**Notes:**
This slide shows the logical path connecting any given rainfall event to its rainfall frequency distribution, its mean, and finally to that the mean can be used to describe and define the event.  This assumes the selection of a correct mathematical distribution, usually the exponential distribution (Wilks, “Statistical Methods in the Atmospheric Sciences”, 1995. 

## Actual distribution of rainfall for the event

- Conditional Mean

-   Areal coverage = 74%

**Notes:**
September 14, 2005 example:
The unconditional mean is 0.37 inches.  How well does it describe what actually occurred?  Poorly.
Does 0.37” indicate that 26% of the area (time) had no rain?   No. 
Does 0.37” show the possible range of precipitation amounts?  No. 

## Exceedance probabilities defined by the mean (QPF)

-   Areal coverage = 74%

- Conditional Mean

**Notes:**
QPF that defines a distribution:
An accurate mean for the exponential distribution is a good fit of the actual distribution. 
Multiplied by the PoP, the distribution indicates rain will not fall everywhere (or every time). 
Exceedance probabilities can be computed using the forecast distribution. 
The forecast distribution is derived from the QPF.  Forecasters must know their QPF represents the distribution, hence the addition to the definition. 

## More examples of actual distributions and distributions computed from QPF

## Conditional Mean

**Conditional Mean**

- Conditional Mean

**Notes:**
POE from formula for the previous slide showing the distribution. 

**Notes:**
These are mislabeled.  They should be “Actual POE) and “Fcst POE”.  These are not distributions. 

**Notes:**
These are POEs, not distributions....

**Notes:**
POEs, not distributions. 

**Notes:**
POEs, not distributions. 

## QPF = Mean of expected distribution So what?

- QPF defines the expected distribution, so
- We will have a precise definition based on science and math.
- We can develop more precise training.
- Probabilistic QPFs can be computed.
- Probabilities can be computed for significant threshold amounts (user defined).
- Probabilistic River Forecasts may be possible.

**Notes:**
This slide shows the logical path connecting any given rainfall event to its rainfall frequency distribution, its mean, and finally to that the mean can be used to describe and define the event.  This assumes the selection of a correct mathematical distribution, usually the exponential distribution (Wilks, “Statistical Methods in the Atmospheric Sciences”, 1995. 

## Where does the forecaster add value?

- Meteorologists will:
- Identify synoptic conditions
- Assess precipitation type
- Anticipate expected rainfall intensities
- Determine expected areal coverage
- GFE tools are available to help.
- Mets can forecast _conditional amounts_ and convert them, if they wish.

**Notes:**
This slide shows the logical path connecting any given rainfall event to its rainfall frequency distribution, its mean, and finally to that the mean can be used to describe and define the event.  This assumes the selection of a correct mathematical distribution, usually the exponential distribution (Wilks, “Statistical Methods in the Atmospheric Sciences”, 1995. 

## Slide 43

**Notes:**
Example of different areas, different PoPs and different QPFs. 
This is where a forecaster will add value. 
Each area has a different QPF (expected mean of a distribution), therefore, each area has different chances to exceed certain threshold rainfall amounts. 

## What about forecasting a maximum QPF (MQPF)?  

- Why is MQPF6 important ?
-      The MQPF6 grid is defined by the NWS and might provide added information to our customers, such as Emergency Managers, about the potential extreme rainfall during the up-coming event.
-      During summer-time convection, scattered thunderstorms can produce a wide range of rainfall, including a significant threat of heavy rain.

**MQPF6 Definition (from ISST)**:
- ...   MQPF6 is defined to represent the 90th percentile of the conditional QPF distribution; ...

## MQPF6

- If QPF is defined as the mean of the exponential distribution, the MQPF6 (10% chance of exceedance) can be computed.

**The unconditional probability to exceed an amount x is given by: **
**	****	****uPOE(x) = PoP **●**(e****-x/μ** **)**
- 			,where μ is _conditional_ QPF, and
- 			  x is the rainfall amount
- Then,
- 		**MQPF6(.10) =** **-μ ln(.10/PoP) **

**Notes:**
It will be left to the “student” to show the derivation of the lower formula from the upper one.  (My weak attempt at humor!)

## Summary

- PQPF:  Probabilistic quantitative precipitation forecast.
- POE: Probability of Exceedance, or the method of communicating probabilistic QPF.
- Can include a Maximum QPF6 (MQPF6) grid, if desired.
- Gridded verification of QPF is highly recommended so individual can see the accuracy of their forecasts.

## End