# Presentation: 871394

## Probabilistic Forecasting in Practice

**J. Paul Dallavalle**
**AMS Short Course on Probabilistic Forecasting**
**San Diego, CA**
**January 9, 2005**

## Operational Probabilities

**BALTIMORE WASHINGTON INTERNATIONAL          **
** ****KBWI   GFS MOS GUIDANCE   11/19/2004  1200 UTC                      **
** ****DT /NOV  19/NOV  20                /NOV  21                /NOV  22 **
** ****HR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00 06 12 **
** ****N/X                    49          58          48          64    42 **
** ****TMP  58 57 54 52 52 52 52 54 56 56 54 53 53 52 51 58 62 61 54 48 44 **
** ****DPT  51 51 51 50 51 52 52 52 52 52 53 52 51 50 49 50 49 47 47 40 38 **
** ****CLD  OV OV OV OV OV OV OV OV OV OV OV OV OV BK BK BK BK BK SC FW BK **
** ****WDR  36 06 09 09 08 09 09 11 13 13 17 00 28 29 29 31 30 30 30 31 31 **
** ****WSP  01 02 01 01 02 03 04 03 02 02 01 00 02 02 04 07 09 07 04 05 05 **
** ****P06        44    57    48    34    38     4     6     2     1  1  5 **
** ****P12                    63          40          10           2     5**** **
** ****Q06         1     1     1     1     1     0     0     0     0  0  0 **
** ****Q12                     1           0           0           0     0 **
** ****T06      2/ 8  5/ 0  2/ 0  0/ 0  0/13  0/ 0  0/ 0  0/ 0  1/14  0/ 0 **
** ****T12            5/ 8        2/ 0        1/14        0/ 0     1/15    **
** ****POZ   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 **
** ****POS   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 **
** ****TYP   R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R **
** ****SNW                     0                       0                 0 **
** ****CIG   7  6  6  5  3  3  3  3  3  3  4  4  5  6  8  6  6  7  8  8  8 **
** ****VIS   6  6  6  5  5  3  3  4  5  5  5  5  5  5  2  7  7  7  7  7  7 **
** ****OBV   N  N  N BR BR BR BR BR BR BR BR BR BR FG FG  N  N  N  N  N  N**

## Covert Probabilities

**BALTIMORE WASHINGTON INTERNATIONAL          **
** ****KBWI   GFS MOS GUIDANCE   11/19/2004  1200 UTC                      **
** ****DT /NOV  19/NOV  20                /NOV  21                /NOV  22 **
** ****HR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00 06 12 **
** ****N/X                    49          58          48          64    42 **
** ****TMP  58 57 54 52 52 52 52 54 56 56 54 53 53 52 51 58 62 61 54 48 44 **
** ****DPT  51 51 51 50 51 52 52 52 52 52 53 52 51 50 49 50 49 47 47 40 38 **
** ****CLD  OV OV OV OV OV OV OV OV OV OV OV OV OV BK BK BK BK BK SC FW BK**** **
** ****WDR  36 06 09 09 08 09 09 11 13 13 17 00 28 29 29 31 30 30 30 31 31 **
** ****WSP  01 02 01 01 02 03 04 03 02 02 01 00 02 02 04 07 09 07 04 05 05 **
** ****P06        44    57    48    34    38     4     6     2     1  1  5 **
** ****P12                    63          40          10           2     5**** **
** ****Q06         1     1     1     1     1     0     0     0     0  0  0 **
** ****Q12                     1           0           0           0     0**** **
** ****T06      2/ 8  5/ 0  2/ 0  0/ 0  0/13  0/ 0  0/ 0  0/ 0  1/14  0/ 0 **
** ****T12            5/ 8        2/ 0        1/14        0/ 0     1/15    **
** ****POZ   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 **
** ****POS   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 **
** ****TYP   R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R **
** ****SNW                     0                       0                 0 **
** ****CIG   7  6  6  5  3  3  3  3  3  3  4  4  5  6  8  6  6  7  8  8  8 **
** ****VIS   6  6  6  5  5  3  3  4  5  5  5  5  5  5  2  7  7  7  7  7  7 **
** ****OBV   N  N  N BR BR BR BR BR BR BR BR BR BR FG FG  N  N  N  N  N  N**

## Outline

** ****Model Output Statistics (MOS)**
***Definitions***
***Properties ***
***Regression Estimation of Event Probabilities (REEP)***
** ****Predictand definitions**
** ****Appropriate predictors**
** ****Developmental considerations **
** ****Application in an operational environment**
** ****Sample forecasts**
** ****Subjective probabilities**

## A SIMPLE STATISTICAL MODEL

**Relative frequency of 12-24 h precipitation occurrence as a function of forecast relative humidity**

**FCST MEAN RELATIVE HUMIDITY (%)**

**OBSERVED REL. FREQUENCY**

**0**

**10**

**20**

**30**

**40**

**50**

**60**

**70**

**80**

**90**

**100**

**0.0**

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**0.6**

**0.7**

**0.8**

**0.9**

**1.0**

**3-YR SAMPLE;  200 STATIONS**

**1987-1990 COOL SEASON**

**47%**

## Model Output Statistics (MOS)

**Relates observations of the weather element to be predicted (predictand) to appropriate variables (predictors) via a statistical method**
**Predictors include:**
**	****NWP model output**
**	****Prior observations **
**	****Geoclimatic data – terrain, normals, etc.**
**Current statistical method:  multiple linear regression (forward selection)**

## Properties of MOS Development

** ****Mathematically simple, yet powerful**
** ****Models non-linearity through NWP variables and predictor transformations**
** ****Uses record of observations at forecast points**
** ****Applies equations to future run of similar forecast model**
** ****Produces probability forecasts from a single run of the underlying NWP model**
** ****Can use other mathematical methods such as logistic regression or neural networks**

## Regression Estimation of Event Probabilities (REEP)

** ****Define the meteorological variable (predictand)**
** ****Define the event in terms of categories of the predictand**
** ****Transform predictand to a 0 (event does not occur) or a 1 (event occurs)**
** ****Develop regression model relating predictand to predictors**
** ****Interpret regression fit in terms of estimated relative frequencies, i.e., probabilities**

## Event Definition

**12-24 H FCST 1000 - 500 MB RH**

**12-24 H PRECIPITATION ≥ .01"**

**10**

**20**

**30**

**40**

**50**

**60**

**70**

**80**

**90**

**100**

**0**

**1**

## Simple Linear Regression Fit

**12-24 H FCST 1000 - 500 MB RH**

**12-24 H PRECIPITATION ≥ .01"**

**10**

**20**

**30**

**40**

**50**

**60**

**70**

**80**

**90**

**100**

**0**

**1**

- RV=36.5%

## Regression Estimation of Event Probabilities

** ****12-24 H FCST 1000 - 500 MB RH**

**12-24 H PRECIPITATION ≥ 01"**

**10**

**20**

**30**

**40**

**50**

**60**

**70**

**80**

**90**

**100**

**0**

**1**

- 3 Events

- 7 Events

**RF= 30%**

**P = 30%**

## Properties of MOS Probabilities

** ****Unbiased –average of probabilities over a period of time equals long-term relative frequency of the event**
** ****Reliable – conditionally (“piece-wise”) unbiased over the range of probabilities**
** ****Reflects predictability of event – range of probabilities narrows and approaches relative frequency of event as predictability decreases, with increasing projections or rare events**

## Probability Considerations

**So, if the MOS probability represents the long-term relative frequency of the ****event****, what is the ****event**** definition ???**
**	****Meteorological element**
**	****Characteristics of the observing system**
**	****Point or areal extent**
**	****Temporal resolution**
**	****Categorical breakpoints**
**	****Conditional on another ****event**
**	**

## Predictand Definitions

## Suitable observations?

**Appropriate Sensor?**

**Good siting?**

**Real ?**

**Photo Courtesy W. Shaffer**

## Snowfall GuidanceDealing with Observational Systems

**PORT HOPE, MI**
** **
** ****KP58   GFSX MOS GUIDANCE   1/09/2004  0000 UTC**
** ****FHR  24| 36  48| 60  72| 84  96|108 120|132 144|156 168|180 192**
** ****FRI  09| SAT 10| SUN 11| MON 12| TUE 13| WED 14| THU 15| FRI 16 CLIMO**
** ****X/N  10|  5  17| 14  33| 26  29| 13  15|  3   6|  2  11|  8  17 12 28**
** ****TMP   8|  8  16| 20  31| 27  23| 15  12|  5   6|  4  10| 10  16**
** ****DPT  -2|  1   3| 19  24| 22  15|  8   4|  1  -3| -6   1|  7   6**
** ****CLD  OV| OV  OV| OV  OV| OV  OV| OV  PC| OV  OV| OV  OV| OV  OV**
** ****WND  13|  8   7| 11  13| 13  12| 15  14| 13  12| 11  12| 12  11**
** ****P12  12| 12  18| 49  75| 60  38| 46  28| 17  21| 28  25| 20  23999999**
** ****P24    |     18|     75|     83|     51|     32|     38|     29   999**
** ****Q12   0|  0   0|  1   1|  1   0|  1   0|  0   0|  0    |**
** ****Q24    |      0|      1|      2|      1|      0|       |**
** ****T12   2|  1   0|  1   1|  2   0|  0   0|  0   0|  0   0|  0   0**
** ****T24    |  2    |  1    |  9    |  1    |  1    |  2    |  1**
** ****PZP   0|  0   0|  0   0|  0   2|  0   0|  0   0|  0   1|  1   2**
** ****PSN 100|100 100|100  99| 88  90|100 100| 99 100| 98  96| 96  95**
** ****PRS   0|  0   0|  0   0| 12   9|  1   0|  1   1|  2   3|  2   1**
** ****TYP   S|  S   S|  S   S|  S   S|  S   S|  S   S|  S   S|  S   S**
** ****SNW    |      1|      2|      2|      2|       |       |**

## MOS Snowfall GuidanceObservations from Co-op Observer Network

## Challenges of the Co-op Network

**Station Selection:**
**	*****Some sites don’t report snowfall amount***
***	******Some sites open (or close) during the sample period***
***	******Some sites “move” either horizontally or vertically during the sample period***
***	******Some sites report neither accurately nor reliably***

## More Challenges of the Co-op Network 

**Station Reporting Time:**
**	*****Sites report once in a 24-hr period***
***	******Reporting times are site-specific and are in local time***
***	******Other hydrological elements are sometimes reported separately***
***	******Reporting times change at some sites during the sample period ***

## Strategy
Arrive at one “official” reporting time per site
Discard any site with multiple reporting times or multiple locations

***Arrive at one “official” reporting time per site***
***Discard any site with multiple reporting times or multiple locations***

**Two distinct “windows”**
-           ***12Z: 11 – 17Z, 00Z: 21 – 3Z***
**Discard any site outside the two windows**

## MOS Thunderstorm Probabilities -Dealing with remote sensors

## Creating Predictand EventsLightning strikes are summed over the “appropriate” time period and assigned to the center of “appropriate” grid boxes

| ××××  |  |  |  |  |
| --- | --- | --- | --- | --- |
|  |  |  × |   × | ××× |
| × |  |  |  | ×× |
|  |  | ×× | ×
 |  |

|    |  |  |  |  |
| --- | --- | --- | --- | --- |
|  |  |   |   |   |
|   |  |  |  |   |
|  |  |   |   |  |

- = thunderstorm

- = no thunderstorm

## What is “appropriate” for thunderstorms?

**Time period?**
**	***1 hour*
*	**2 hours*
*	**3 hours*
*	**6 hours*
*	**12 hours*
**Grid size?**
**	***5 km*
*	**10 km*
*	**20 km*
*	**40 km*
*	**120 km**	*

## MOS Thunderstorm Probabilities -12-h period, 40-km grid

## Categorical Breakpoints Define the event “Y” = 1 when ...

| Weather Element | Breakpoint |
| --- | --- |
| Prob. of Precip. |  .01 inches of liquid-equivalent |
| Thunderstorms |  1 strike |
| Visibility (less than 1⁄2 mile) | < 1⁄2 mile |
| Sky cover (clear) | CLR/SKC |

## More Categorical Breakpoints

**Ceiling Height**
**Category 1:  < 200 ft**
**Category 2:  200 – 400 ft**
**Category 3:  500 – 900 ft**
**Category 4:  1000 – 1900 ft**
**Category 5:  2000 – 3000 ft**
**Category 6:  3100 – 6500 ft**
**Category 7:  6600 – 12000 ft**
**Category 8:  > 12000 ft**

**Visibility**
**Category 1:  < 0.5 mi**
**Category 2:  0.5 mi - < 1 mi**
**Category 3:  1 - < 2 mi**
**Category 4:  2 - < 3 mi**
**Category 5:  3 – 5 mi**
**Category 6:  6 mi**
**Category 7:  > 6 mi**

**KLNS   GFS MOS GUIDANCE   11/29/2004  1200 UTC                      **
**DT /NOV  29/NOV  30                /DEC   1                /DEC   2 **
**HR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00 06 12 **
**... **
**CLD  CL BK BK BK OV OV OV OV OV OV OV OV OV OV OV OV OV BK CL CL CL **
**... **
**CIG   8  8  8  8  7  7  7  8  8  7  7  7  4  2  3  3  6  7  8  8  8 **
**VIS   7  7  7  7  7  7  7  7  7  7  7  7  5  5  4  2  6  7  7  7  7**
**... **

## Conditional Probabilities

**If event B is conditioned upon A occurring, then:**
**   ****Prob(B/A)=Prob(A********B)/Prob(A)**
**or: Prob(B/A)=Prob(C)/Prob(A)**

**U**

**A**

**B**

**C**

## Conditional Event Probabilities

**If event B is conditioned upon A occurring, then:**
**   ****Prob(B/A)=Prob(A********B)/Prob(A)**
**or: Prob(B/A)=Prob(B)/Prob(A)**
**Examples:**
**If precipitation occurs, what is the probability that freezing rain will occur?**
**If precipitation occurs, what is the probability that 0.25 inches or more will occur? **

**U**

**B**

**U**

**A**

**B**

## Conditional Predictands

**Precipitation type**** (condition:  precipitation occurrence)**
**	****freezing (FZDZ,FZRA,PL,mixtures)**
**	****snow (SN or SG)**
**	****liquid (RA,DZ,mixtures)**
**  ****NOTE:  exclusive and exhaustive**
**Precipitation amount**** (condition:  **_**> **_**0.01 inches of precip.)**
**	**_**>**_** 0.10 inches**
**	**_**>**_** 0.25 inches**
**	**_**>**_** 0.50 inches**
**	**_**>**_** 1.00 inches**
**	**_**>**_** 2.00 inches**
**  ****NOTE:  not exclusive, but exhaustive**

## Appropriate Predictors

## Predictor Considerations

-  **Describe physical processes associated with event**
***thunderstorms:  CAPE, K-index, vertical velocity, etc.***
** ****Avoid irrelevant variables**
***thunderstorms:  1000-500 hPa thickness***
** ****Use event relative frequencies or high-resolution geophysical variables (terrain) that contribute to local forcing of event**
** ****Mimic forecaster thought process**
***thunderstorms:  interact relative frequency and K-index***
** ****Provide non-linear transforms of predictors**

## Relative Frequency Predictor

## High-Resolution Terrain

## Transform - Point Binary Predictor

**FCST 12-24 H MEAN RH         PREDICTOR CUTOFF = 70%****INTERPOLATE;  STATION RH ≥ 70% ,   SET BINARY = 1;**** BINARY = 0,  OTHERWISE*****  ***

- 96

- 86

- 89

- 94

- 87

- 73

- 76

- 90

- 76

- 60

- 69

- 92

- 64

- 54

- 68

- 93

**RH ≥70%  ;  BINARY AT KCMH = 1**

- •

**KCMH**

**(71%)**

## Linear Regression – Point Binary Predictor Only

**12-24 H FCST 1000 - 500 MB RH**

**12-24 H PRECIPITATION ≥ .01"**

**10**

**20**

**30**

**40**

**50**

**60**

**70**

**80**

**90**

**100**

**0**

**1**

- RV=36.5%

- RV=42.4%

## Linear Regression – Cont. & Point Binary Predictors

**12-24 H FCST 1000 - 500 MB RH**

**12-24 H PRECIPITATION ≥ .01"**

**10**

**20**

**30**

**40**

**50**

**60**

**70**

**80**

**90**

**100**

**0**

**1**

- RV=36.5%

- RV=42.4%

**RV=44.9%**

**POP = -0.234 + (0.007********RH) +**

**(0.478********BINARY RH (70%))**

## Transform - Grid Binary Predictor

**FCST 12-24 H MEAN RH         PREDICTOR CUTOFF = 70%****WHERE RH > 70%, SET GRIDPOINT VALUE = 1; = 0, OTHERWISE;**
**INTERPOLATE TO STATION**

- 1

- 1

- 1

- 1

- 1

- 1

- 1

- 1

- 1

- 0

- 0

- 1

- 0

- 0

- 0

- 1

**0 ≤ VALUE AT KCMH ≤ 1**

- •

**KCMH**

**(.21 )**

## Transform – Logit Fit

**Transform – Logit Fit**

- KPIA (Peoria, IL)   0000 UTC; 18-h projection

## Developmental Considerations

***	***
**	**

## Development in the Real World

**Selection and quality control of the observational dataset**
**Precise definitions of predictand and conditioning (if any) events**
**Simultaneous development for related predictands**
**	**_**precipitation type:**_
**	*****freezing***
***	******snow ***
***	******liquid ***
***  ******	******Note******1******:  exclusive and exhaustive***
***	******Note******2******: sum of 3 probabilities should = 1***
**	**

## More Developmental Considerations

**Choice of appropriate predictors**
**Number of terms in the equation – selection criteria**
**	*****Multi-collinearity***
***	******Overfit***
**Stratification of developmental sample**
**	*****Forecast cycle***
***	******Projection***
***	******Season (cool, warm; winter; spring, summer, cool)***
**	**

## Issues with the Developmental Sample

**Size of sample**
**Representativeness of sample**
**Stability of sample**
**Frequency of event**
**	*****- is event rare?***
**Pooling of data**
**	*****- regionalized equations***
***	******- station specificity***

## Regionalized Probability Equations

**BALTIMORE WASHINGTON INTERNATIONAL          **
** ****KBWI   GFS MOS GUIDANCE   11/19/2004  1200 UTC                      **
** ****DT /NOV  19/NOV  20                /NOV  21                /NOV  22 **
** ****HR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00 06 12 **
** ****N/X                    49          58          48          64    42 **
** ****TMP  58 57 54 52 52 52 52 54 56 56 54 53 53 52 51 58 62 61 54 48 44 **
** ****DPT  51 51 51 50 51 52 52 52 52 52 53 52 51 50 49 50 49 47 47 40 38 **
** ****CLD  OV OV OV OV OV OV OV OV OV OV OV OV OV BK BK BK BK BK SC FW BK**** **
** ****WDR  36 06 09 09 08 09 09 11 13 13 17 00 28 29 29 31 30 30 30 31 31 **
** ****WSP  01 02 01 01 02 03 04 03 02 02 01 00 02 02 04 07 09 07 04 05 05 **
** ****P06        44    57    48    34    38     4     6     2     1  1  5 **
** ****P12                    63          40          10           2     5 **
** ****Q06         1     1     1     1     1     0     0     0     0  0  0 **
** ****Q12                     1           0           0           0     0 **
** ****T06      2/ 8  5/ 0  2/ 0  0/ 0  0/13  0/ 0  0/ 0  0/ 0  1/14  0/ 0 **
** ****T12            5/ 8        2/ 0        1/14        0/ 0     1/15    **
** ****POZ   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 **
** ****POS   0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 **
** ****TYP   R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R  R **
** ****SNW                     0                       0                 0 **
** ****CIG   7  6  6  5  3  3  3  3  3  3  4  4  5  6  8  6  6  7  8  8  8 **
** ****VIS   6  6  6  5  5  3  3  4  5  5  5  5  5  5  2  7  7  7  7  7  7 **
** ****OBV   N  N  N BR BR BR BR BR BR BR BR BR BR FG FG  N  N  N  N  N  N**

## Regionalized Development – Snowfall Guidance

## Operational 12-h PoP EquationMid-Atlantic, cool season, valid 12-24h after 00Z

**Probability = C****0**** + C****1********X****1**** + C****2********X****2**** + ... **
**All variables are grid binaries; mean RH is from 1000-700 hPa**
**Equation based on 11286 cases**
**Reduction of variance =.617**

| Term | Value | Var. | Proj. |
| --- | --- | --- | --- |
| C0 | .234 |  |  |
| C1 | .680 | 12h Prcp. | 24 |
| C2 | .221 | Mean RH | 12 |
| C3 | -.230 | 500 VV | 18 |
| C4 | .210 | Mean RH | 18 |
| C5 | -.109 | K Index | 18 |

## Sample 12-h PoP EquationBWI, cool season, valid 12-24h after 00Z

**Probability = C****0**** + C****1********X****1**** + C****2********X****2**** + ... **
**Variable is a grid binary; breakpoint of 0.05 inches**
**Equation based on 361 cases; 84 precip. events**
**Reduction of variance =.639**

| Term | Value | Var. | Proj. |
| --- | --- | --- | --- |
| C0 | .014 |  |  |
| C1 | .987 | 12h Prcp. | 24 |

## Operational Precip. Type EquationsMid-Atlantic, valid 18h after 00Z

**Probability = C****0**** + C****1********X****1**** + C****2********X****2**** + ...**
**Note:  ****Trans. Thk.**** Is a function of 1000-850 hPa thk.**
**Thk.**** Is a grid binary of the 1000-925 hPa thk.**
**  **

| Term | FZ | SN | RN | Var. | Proj. |
| --- | --- | --- | --- | --- | --- |
| C0 | .003 | .252 | .746 |  |  |
| C1 | -.032 | .823 | -.790 | Trans.
Thk. | 18 |
| C2 | .169 | -.087 | -.082 | Vert.
ZR | 15 |
| C3 | .643 | -.080 | -.563 | ZR
RFreq | 18 |
| C4 | .104 | -.152 | .048 | Vert.
ZR | 21 |
| C5 | -.009 | -.248 | .257 | Thk. | 18 |

## Application in an Operational Environment

***	***
**	**

## Truncating Probabilities

**0 **_**<**_** Prob (A) **_**<**_** 1.0**
**Applied to PoP’s and thunderstorm probabilities**
**If Prob(A) < 0, Prob****adj**** (A)=0 **
**If Prob(A) > 1, Prob****adj**** (A)=1.**

## Normalizing Categorical Probabilities

**Sum of probabilities for exclusive and exhaustive categories must equal 1.0**
**If Prob (A) < 0, then sum of Prob (B) and Prob (C) = D, and is > 1.0.**
**Set:  Prob****adj**** (A) = 0,**
**Prob****adj**** (B) = Prob (B) / D,**
**Prob****adj**** (C) = Prob (C) / D**

## Monotonic Categorical Probabilities

**If event B is a subset of event A, then:**
**   ****Prob (B) should be **_**<**_** Prob (A).**
**Example:  B is **_**>**_** 0.25 in; A is      **_**>**_** 0.10 in**
**Then, if Prob (B) > Prob (A)**
**set Prob****adj**** (B) = Prob (A).**
**Now, if event C is a subset of event B, e.g., C is **_**>**_** 0.50 in, and if Prob (C) > Prob (B),**
**set Prob****adj**** (C) = Prob (B)**

## Unconditional Probabilities from Conditional

**If event B is conditioned upon A occurring:**
** ****Prob(B/A)=Prob(B)/Prob(A) **
** ****Prob(B) = Prob(A) × Prob(B/A) **
**Example:  **
**Let A = event of **_**>**_** .01 in., and B = event of **_**>**_** .25 in., then if:**
**Prob (A) = .70, and **
**Prob (B/A) = .35, then **
**Prob (B) = .70 × .35 = .245**

**U**

**A**

**B**

## Temporal Coherence of Probabilities

**Event A is **_**>**_** 0.01 in. occurring from 12Z-18Z**
**Event B is **_**>**_** 0.01 in. occurring from 18Z-00Z**
** ****A ********B is **_**>**_** 0.01 in. occurring from 12Z-00Z**
**Then P(A********B) = P(A) + P(B) – P(A********B)**
**Thus, P(A********B) should be:**
_**<**_** P(A) + P(B) and**
_**>**_** maximum of P(A), P(B)**

**A**

**B**

**C**

## Temporal Coherence - Partially Enforced

**Thus, P(A********B) should be:**
_**<**_** P(A) + P(B) – ****coherence not checked**
_**>**_** maximum of P(A), P(B) – ****coherence checked**
**SAN DIEGO           **
** ****KMYF   GFS MOS GUIDANCE   12/28/2004  1200 UTC                      **
** ****DT /DEC  28/DEC  29                /DEC  30                /DEC  31 **
** ****HR   18 21 00 03 06 09 12 15 18 21 00 03 06 09 12 15 18 21 00 06 12**
** ... **
** ****P06        79    ****71****   ****100 ****   ****68****     ****5****     6    14     9    16 21 28 **
** ****P12                   ****100          68****          19          25    32 **
** ****Q06         4     3     5     2     0     0     0     0     0  0  1 **
** ****Q12                     5           2           0           0     0 **
** ****T06      ****9****/ 0 ****30****/ 2 22/ 4  9/ 0  0/ 0  0/ 0  0/ 0  0/ 0  1/ 0  0/ 0 **
** ****T12           ****47****/ 3       29/ 4        0/ 0        0/ 0     3/ 0****    **
** **

## Other Possible Post-Processing

** ****Compute the expected value**
**used for estimating precipitation amount**
** ****Fit probabilities with a distribution**
**Weibull distribution used to estimate median or other percentiles of precipitation amount**
** ****Calculate time-interval probability from probabilities for two or more subintervals**
** ****Estimate “best” category forecast**
**definition of “best” depends on the user**
** ****Reconcile meteorological inconsistencies**
**difficult to do**
**inconsistencies minimized somewhat by use of NWP model in developmental and operational processes**

## MOS BEST CATEGORY SELECTION

**MOS BEST CATEGORY SELECTION**

*Using QPF Probabilities as an Example*
*** ***

- NO

**PROBABILITY (%)**

- NO

- NO

- NO

- 4

- 1

- 6

- 3

- 2

- 5

- 0

- YES

- YES

**THRESHOLD**

**EXCEEDED?**

**TO MOS GUIDANCE MESSAGES**

## Meteorological Consistency

**BALTIMORE WASHINGTON INTERNATIONAL          **
** ****KBWI   GFSX MOS GUIDANCE  12/29/2004  0000 UTC                       **
** ****FHR  24| 36  48| 60  72| 84  96|108 120|132 144|156 168|180 192      **
** ****WED  29| THU 30| FRI 31| SAT 01| SUN 02| MON 03| TUE 04| WED 05 CLIMO**
** ****X/N  53| 34  50| 36  57| 41  62| 37  56| 41  54| 34  ****46****| 33  51 25 41**
** ****TMP  45| 38  43| 40  51| 44  50| 40  49| 44  45| 36  39| 37  44      **
** ****DPT  33| 31  36| 38  44| 36  38| 37  42| 39  35| 29  29| 32  35      **
** ****CLD  PC| PC  PC| OV  OV| CL  CL| OV  OV| OV  OV| OV  OV| OV  OV      **
** ...      **
** ****P12   3|  4   1| 20  28| 12   3|  9  36| 34  25| 25  23| 30  26 22 24**
** ****P24    |     10|     31|     12|     36|     38|     36|     30    34**
** ****Q12   0|  0   0|  0   0|  0   0|  0   1|  1   0|  0    |             **
** ****Q24    |      0|      0|      0|      0|      1|       |             **
** ...**
** ****PZP   4| 10  13|  9   7|  9   8| 10   8| 10   8| 21  ****22****| 18  13      **
** ****PSN   0|  3   0|  0   0|  0   0|  0   0|  0   0|  0  11| 13   8      **
** ****PRS  12|  8   5|  0   0|  0   2|  0   0|  0   5|  5   6|  0   0      **
** ****TYP   R|  R   R|  R   R|  R   R|  R   R|  R   R|  R   Z|  R   R      **
** ****SNW    |      0|      0|      0|      0|       |       |**** **

## Some Sample Forecasts

**See http://****www.nws.noaa.gov/tdl/synop/products.shtml**

## 12-h PoPs, valid Dec. 23 & 24, 2004

## 12-h PoPs, valid Dec. 24 & 25, 2004

## 12-h PoPs, valid Dec. 30 & 31, 2004

## 12-h PoPs, valid New Year’s Eve 2004

## Range in Probability 

**KSAN   GFSX MOS GUIDANCE  12/31/2004  0000 UTC                       **
** ****FHR  24| 36  48| 60  72| 84  96|108 120|132 144|156 168|180 192      **
** ****FRI  31| SAT 01| SUN 02| MON 03| TUE 04| WED 05| THU 06| FRI 07 CLIMO**
** ****X/N  58| 50  61| 50  61| 50  59| 47  58| 48  58| 50  59| 47  64 48 66**
** ****TMP  55| 51  59| 52  58| 52  56| 49  56| 50  55| 52  57| 49  61      **
** ****DPT  51| 46  50| 47  48| 46  46| 42  44| 42  50| 46  46| 40  45      **
** ****CLD  OV| OV  PC| PC  OV| OV  OV| PC  OV| OV  OV| OV  OV| PC  PC      **
** ****WND  14| 11   9|  7  12| 11  14| 12  11| 12  14| 12  12|  9  10      **
** ****P12  64| 13   9| 16  46| 93  61| 19  43| 47  58| 65  52| 32  15 15 15**
** ****P24    |     18|     53|     94|     52|     65|     77|     32    23**
** ****Q12   2|  0   0|  0   1|  3   3|  0   1|  1   3|  3    |             **
** ****Q24    |      0|      1|      4|      1|      3|       |             **
** ****T12   2|  0   0|  1   1|  6  14|  0   1|  8   8| 10  10|  4   1      **
** ****T24    |  2    |  1    |  6    | 14    |  9    | 17    | 10**

## Grid Resolution Makes A Difference

**21-24h forecast**
**40-km tstm prob.**

**21-24h forecast**
**20-km tstm prob.**

## Subjective Probabilities

## Objective vs. Subjective PoP ForecastsCool Season(00/12Z  cycles combined)