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Statistical Post Processing

Hong Guan Bo Cui and Yuejian Zhu

EMC/NCEP/NWS/NOAA

Presents for NWP Forecast Training ClassMarch 31, 2015, Fuzhou, Fujian, China

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1. Background

North American Ensemble Forecast System (NAEFS)

International project to produce operational multi-center ensemble products

Bias correction and combines global ensemble forecasts from Canada &

USA

Generates products for:Weather forecasters

Specialized usersEnd users

Operational outlet for THORPEX research using TIGGE archive

StatementThe North American Ensemble Forecast System (NAEFS) combines state of the art

weather forecast tools, called ensemble forecasts, developed at the US National Weather Service (NWS) and the Meteorological Service of Canada (MSC). When combined, these tools (a) provide weather forecast guidance for the 1-14 day period that is of higher quality than the currently available operational guidance based on either of the two sets of tools separately; and (b) make a set of forecasts that are seamless across the national boundaries over North America, between Mexico and the US, and between the US and Canada. As a first step in the development of the NAEFS system, the two ensemble generating centers, the National Centers for Environmental Prediction (NCEP) of NWS and the Canadian Meteorological Center (CMC) of MSC started exchanging their ensemble forecast data on the operational basis in September 2004. First NAEFS probabilistic products have been implemented at NCEP in February 2006. The enhanced weather forecast products are generated based on the joint ensemble which has been undergone a statistical post-processing to reduce their systematic errors.

Summary of 6th NAEFS workshop1-3 May, 2012 Monterey, CA

6th NAEFS workshop was held in Monterey, CA during 1-3 May 2012. There were about 50 scientists to attend this workshop whose are from Meteorological Service of Canada, Mexico Meteorological Service, UKMet, NAVY, AFWA and NOAA.

Following topics have been presented and discussed during workshop:• Review the current status of the contribution of each

NWP center to NAEFS• For each NWP center, present plans for future model and

product updates, for both the base models and ensemble system (including regional ensembles)

• Decide on coordination of plans for the overall future NAEFS ensemble and products (added variables, data transfer for increased resolution grids, FNMOC ensemble added to NAEFS, especially for mesoscale ensemble-NAEFS-LAM)

• Learn about current operational uses of ensemble forecast guidance, including military and civilian applications.

7th NAEFS Workshop in Montreal, Canada

• Time: 17-19 June 2014• Locations:

– 17-18 June – Biosphere, Montreal, Canada– 19 June – CMC, Dorval, Canada

• Co-chairs: Andre Methot and Yuejian Zhu• Topics (or sessions)

– Status and plan of Global ensemble forecast systems;– Operational data management and distribution; – Ensemble verification and validation metrics; – Reforecast, bias correction and post process; – Regional ensemble and data exchange; – Wave ensembles; – Integration of ensemble in forecasts: user feedback and recommendation; – Products – hazard weather, high impact weather and diagnostic variables; – Open discussion of the NAEFS research, development, implementation and operation

plan

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NCEP CMC NAEFSModel GFS GEM NCEP+CMC

Initial uncertainty ETR EnKF ETR + EnKF

Model uncertainty/Stochasti

c

Yes (Stochastic Pert) Yes (multi-physicsand stochastic)

Yes

Tropical storm Relocation None

Daily frequency 00,06,12 and 18UTC 00 and 12UTC 00 and 12UTC

Resolution T254L42 (d0-d8)~55kmT190L42 (d8-16)~70km

About 50kmL72

1*1 degree

Control Yes Yes Yes (2)

Ensemble members 20 for each cycle 20 for each cycle 40 for each cycle

Forecast length 16 days (384 hours) 16 days (384 hours) 16 days

Post-process Bias correction(same bias for all

members)

Bias correction for each member

Yes

Last implementation February 14th 2012 November 18th 2014

NAEFS Current StatusUpdated: November 18th 2014

Milestones• Implementations

– First NAEFS implementation – bias correction Version 1.00 - May 30 2006– NAEFS follow up implementation – CONUS downscaling Version 2.00 - December 4 2007– Alaska implementation – Alaska downscaling Version 3.00 - December 7 2010– Implementation for CONUS/Alaska expansion Version 4.00 - April 8 2014 – Implementation 2.5km/3km NDGD products for CONUS/Alaska Version 5.00 – August 2015

• Applications:– NCEP/GEFS and NAEFS – at NWS– CMC/GEFS and NAEFS – at MSC– FNMOC/GEFS – at NAVY– NCEP/SREF – at NWS

• Publications (or references):– Cui, B., Z. Toth, Y. Zhu, and D. Hou, D. Unger, and S. Beauregard, 2004: “ The

Trade-off in Bias Correction between Using the Latest Analysis/Modeling System with a Short, versus an Older System with a Long Archive” The First THORPEX International Science Symposium. December 6-10, 2004, Montréal, Canada, World Meteorological Organization, P281-284.

– Zhu, Y., and B. Cui, 2006: “GFS bias correction” [Document is available online]– Zhu, Y., B. Cui, and Z. Toth, 2007: “December 2007 upgrade of the NCEP Global Ensemble Forecast System (NAEFS)” [Document is

available online]– Cui, B., Z. Toth, Y. Zhu and D. Hou, 2012: "Bias Correction For Global Ensemble Forecast" Weather and Forecasting, Vol. 27 396-410 – Cui, B., Y. Zhu , Z. Toth and D. Hou, 2013: "Development of Statistical Post-processor for NAEFS"

Weather and Forecasting (In process)– Zhu, Y., and B. Cui, 2007: “December 2007 upgrade of the NCEP Global Ensemble Forecast System (NAEFS)” [Document is available

online]– Zhu, Y, and Y. Luo, 2014: “Precipitation Calibration Based on Frequency Matching Method (FMM)”. Weather and Forecasting (doi:

http://dx.doi.org/10.1175/WAF-D-13-00049.1)– Glahn, B., 2013: “A Comparison of Two Methods of Bias Correcting MOS Temperature and Dewpoint Forecasts” MDL office note, 13-

1– Guan, H, B. Cui and Y. Zhu, 2014: “Improvement of Statistical Post-processing Using GEFS Reforecast Information”, Weather and

Forecasting (in process)

Bias correction: • Bias corrected NCEP/CMC GEFS and NCEP/GFS forecast (up to 180 hrs)• Combine bias corrected NCEP/GFS and NCEP/GEFS ensemble forecasts• Dual resolution ensemble approach for short lead time• NCEP/GFS has higher weights at short lead time

NAEFS products (global) and downstream applications• Combine NCEP/GEFS (20m) and CMC/GEFS (20m) • Produce Ensemble mean, spread, mode, 10% 50%(median) and 90% probability forecast at 1*1

degree resolution• Climate anomaly (percentile) forecasts• Wave ensemble forecast system• Hydrological ensemble forecast system

Statistical downscaling • Use RTMA as reference - NDGD resolution (5km/6km), CONUS and Alaska• Generate mean, mode, 10%, 50%(median) and 90% probability forecasts

NAEFS Statistical Post-Processing System

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Variables pgrba_bc file Total 51

GHT 10, 50, 100, 200, 250, 500, 700, 850, 925, 1000hPa 10

TMP 2m, 2mMax, 2mMin, 10, 50, 100, 200, 250, 500, 700, 850, 925, 1000hPa

13

UGRD 10m, 10, 50, 100, 200, 250, 500, 700, 850, 925, 1000hPa 11

VGRD 10m, 10, 50, 100, 200, 250, 500, 700, 850, 925, 1000hPa 11

VVEL 850hPa 1

PRES Surface, PRMSL 2

FLUX (top) ULWRF (toa - OLR) 1

Td and RH 2m 2

Notes CMC and FNMOC do not apply last upgrade yet

NAEFS bias corrected variables

Last upgrade: April 8th 2014 - (bias correction)

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Variables Domains Resolutions Total 10/10

Surface Pressure CONUS/Alaska 5km/6km 1/1

2-m temperature CONUS/Alaska 5km/6km 1/1

10-m U component CONUS/Alaska 5km/6km 1/1

10-m V component CONUS/Alaska 5km/6km 1/1

2-m maximum T CONUS/Alaska 5km/6km 1/1

2-m minimum T CONUS/Alaska 5km/6km 1/1

10-m wind speed CONUS/Alaska 5km/6km 1/1

10-m wind direction CONUS/Alaska 5km/6km 1/1

2-m dew-point T CONUS/Alaska 5km/6km 1/1

2-m relative humidity CONUS/Alaska 5km/6km 1/1

NAEFS downscaling parameters and productsLast Upgrade: April 8 2014 (NDGD resolution)

All downscaled products are generated from 1*1 degree bias corrected fcst. globally Products include ensemble mean, spread, 10%, 50%, 90% and mode

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2. NAEFS bias correction

NAEFS bias correction

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eaf

eaeaafN

afNt

t

1

)(1

For any given forecast f , it could express as :

Where a is truth (or replaced as best analysis), e is systematic error and ℇ is random error

Therefore, systematic error (or accumulated bias) could be a time average difference of forecast and truth:

In fact, an accumulated bias is changed from time to time which means e is not exactly systematic error, and ℇ is not exactly random error based on the time window for average.

NAEFS Bias Correction

)()()( 0,,, tatftb jijiji

2). Decaying Average (or Kalman Filter method): Average bias will be updated by considering prior period bias and current bias by using decaying average (or Kalman Filter method ) with weight coefficient (w).

)()1()1()( ,,, tbwtBwtB jijiji

1). Bias Estimation: We assume a bias (b) for each lead-time (t) (6-hour interval up to 384 hours), each grid point (i, j) is defined as the different of best analysis (a) and forecast (f) at the same valid time (t0) which is up on latest available analysis.

NAEFS Bias Correction 3). Decaying Weight: Through many experiments for different weights (w = 0.01, 0.02, 0.05, 0.1 and etc…), and different parameters, and different lead times, overall, w equals to 0.02 has been used for GEFS bias correction which is mainly using past 50-60 days information (see figure).

NAEFS Bias Correction

4). Bias corrected forecast: The new (or bias corrected) forecast (F) will be generated by applying decaying average bias (B) to current raw forecast (f) for each lead time, at each grid point, and each parameter.

)()()( ,,, tBtftF jijiji

Simple Accumulated Bias

Assumption: Forecast and analysis (or observation) is fully correlated

NAEFS Bias Correction 5). Performance: The performance is estimated by applying NAEFS bias correction method. The bias is calculated at each grid point for raw forecast (f) and bias corrected forecast (F), then using decaying average method (w=0.02) to get current average bias, taking absolute bias for each grid point, each lead time to generate domain average absolute error (bias) which smaller value is better (see figure: example for Northern Hemisphere 2 meter temperature, decaying average (w=0.2) about 2 months period ended by April 27, 2007).

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500hPa height: 120 hours forecast (ini: 2006043000)Shaded: left – raw bias right – bias after correction

Forecast becomes worse after bias

correction

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2 meter temperature: 120 hours forecast (ini: 2006043000)Shaded: left – raw bias right – bias after correction

Positive biasNegative bias

Comparison of raw and bias corrected T2m for Summer and Fall 2011

RMS and Spread (NH – Fall)RMS and Spread

(NH – Summer)

ME and ABSE (NH – Fall)

ME and ABSE (NH – Summer)

Get worse from bias correction

Raw forecast is bias free

No carry on bias from summer

NAEFS Bias Correction

• Several questions left behind– The correlation of prior joint sample

• Assume samples are fully correlated

– The weight to calculate decaying average• Optimum weights are functions of geographic and forecast

lead times (should be)• Currently, w (weight) is fixed (w=0.02)

– Systematic error for seasonal• Current method is lagged for seasonal information

– 2nd moment adjustment for current method• Slightly adjust for CMC’s ensembles• N/A for single model ensemble

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3. Using ensemble reforecast

GEFS Reforecast

Configurations (Hamill et al, 2013)

• Model version– GFS v9.01 – last implement – May 2011– GEFS v9.0 – last implement – Feb. 2012

• Resolutions– Horizontal – T254 (0-192hrs– 55km) T190 (192-384hrs – 70km) – Vertical – L42 hybrid levels

• Initial conditions– CFS reanalysis– ETR for initial perturbations

• Memberships – 00UTC - 10 perturbations and 1 control

• Output frequency and resolutions– Every 6-hrs, out to 16 days– Most variables with 1*1 degree

• Data is available – 1985 - current

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Using Reforecast Data

• Bias over 24 years (24X1=24) 25 years (25x1=25)• Bias over 25 years and 31day window (25x31)• Bias over recent 2, 5, 10, and 25 years within a window of 31day (2x31, 5x31, 10x31, 25x31)• Bias over 25 years with a sample interval of 7days within a window of 31days and 61days (~25x4 and ~25x8)

.

.

.

1985 1986

20102009

dayday-15 day+15

Winter 2010

Spring 2010

Using 25-year reforecast bias (1985-2009) to calibrate 2010 forecast

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Fall 2010

Summer 2010Solid line – RMSEDash line - Spread

2% decaying is best for all lead time

Winter 2009

Spring 2009

Summer 2009 Fall 2009

Decaying averages are not good except for day 1-2

Decaying average is equal good as reforecast, except for week-2

forecast

Decaying averages are not good except for day 1-3

Using 24-year reforecast bias (1985-2008) to calibrate 2009 forecast

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Another year

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Reforecast bias (2-m temperature)

Perfect bias

warm bias

cold biascold bias

Long training period (10 or 25 years) is necessary to help avoid a large impact to bias correction from a extreme year case and keep a broader diversity of weather scenario!!

Skill for 25y31d’s running mean is the best. 25y31d’s thinning mean (every 7 days) is very similar to 25y31d’s running mean. 25y31d’s thinning mean can be a candidate to reduce computational expense and keep a broader diversity of weather scenario!!!

T2m calibration for different reforecast sample sizes

sensitivity on the number of training years (2, 5, 10, and 25 years)

sensitivity on the interval of training sample (1 day and 7 days)

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0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r2, NH, 2010

Winter

Spring

Summer

Fall

Day

Bias corrected forecast: The new (or bias corrected) forecast (F) will be generated by applying decaying average bias (B) and reforecast bias (b) to current raw forecast (f) for each lead time, at each grid point, and each parameter.

r could be estimated by linear regression from joint samples, the joint sample mean could be generated from decaying average (Kalman Filter average) for easy forward.

Using reforecast to improve current bias corrected product

Additional term (bias from reforecast) is added if r (correlation coefficient) is not equal one. This adjustment is expected to benefit for longer lead time forecast

mjijiji

mji

mji DRBrbrfF

jiji )1()1( ,,

2,

2,, ,,

Using reforecast to improve current bias corrected product (24-hr forecast, 2010 )

Spring Summer

FallWinter

Perfect bias

Using reforecast to improve current bias corrected product (240-hr forecast, 2010 )

Perfect biasWinter Fall

SummerSpring

500hPa height

Winter 2010

Spring 2010

Using 25-year reforecast bias (1985-2009) to calibrate 2010 forecast

Very difficult to improve the skills

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4. 2nd moment adjustment

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Make Up This Deficient ?Improving surface perturbations

Or Post processing

Bias correction does not change the spreads

2nd moment adjustment

N

t

tatfN

E1

2))()((1

N

t

M

m

m tftfMN

S1 1

2))()((1

11

E

SR ))1()1(( tftfD mm R=1 if E=0

Ensemble skill

Ensemble spread

mji

mji

mji DRFF )1(* ,,,

1st moment adjusted forecast 2nd moment adj.

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Estimated by decaying averaging

Almost Perfect!!!

Winter Spring

FallSummer

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After 2nd momentadjustment

Surface Temperature (T2m) for Winter (Dec. 2009 – Feb. 2010), 120-hr Forecast

25% spread increased

7% RMSE reduced

SPREAD

RMSE

SPREAD/RMSE

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0

0.2

0.4

0.6

0.8

1120-hr forecast NA

Spre

ad R

MSE

ratio

SPP2

Winter Spring Summer Autumn

0

0.2

0.4

0.6

0.8

1240-hr forecast NA

Spre

ad R

MSE

ratio

Winter Spring Summer Autumn

improving under-dispersion for all seasons with a maximum benefit in summer.

Improving Under-dispersion for North America (Dec. 2009 – Nov. 2010)

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SPP2

RAW

RAW

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5. Multi-model ensemble

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ts

jtsk

jk z

nw

,,,ˆ

1

Bayesian Model Average

Weights and standard deviations for each model (k - ensemble member) at step j

Finally, the BMA predictive variance is

2

1 1 1

2,,,,,,,,1, )

~~()

~,...,

~|( k

K

k

K

i

K

kktsiitskktsKtsts wfwfwffyVar

ts

jtsk

tstskts

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jk

z

fyz

,,,

,

2,,,,,

2

ˆ

)~

(ˆ

Between-forecast variance Within-forecast variance

Sum of (s,t) represents the numbers of obs.

It is good for perfect bias corrected forecast,Or bias-free ensemble forecast, but we do not

40Courtesy of Dr. Veenhuis

Model 1

Model 2

Model 3

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Bias freeEns forecasts

Observation orBest analysis

Variance Weights

BMA, Decaying process, and adjustment

New weights

New variance

Prior weights

Prior variance

Err and sprd

New Err and sprd

Prior Err and sprd

Flow Chart of Recursive Bayesian Model Process (RBMP)

2nd moment adjustment Adjusted PDF

(We thanks to Dr. Veenhuis for allowing us to adopt his BMA codes).

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T2m for summer and Fall 2014

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The results demonstrate:1. BMA could improve 3 ensemble’s mean, but

spread could be over if original spread is larger2. RBMP could keep similar BMA average future,

but 2nd moment will be adjusted internally3. All important time average quantities are

decaying average (or recursive – save storage)

NUOPCIBC – simple combine three bias corrected ensembles

DCBMA – decaying based Bayesian Model Average

RBMP – Recursive Bayesian Model Process (built in decaying average and internal 2nd-moment adjustment)

Solid line – RMS errorDash line - Spread Over-dispersion

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Summer 2013

ROC are better for all leads

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The results demonstrate:1. BMA could improve 3 ensemble’s mean, but

spread could be over if original spread is larger2. RBMP could keep similar BMA average future,

but 2nd moment will be adjusted internally3. All important time average quantities are

decaying average (or recursive – save storage)

NUOPCIBC – simple combine three bias corrected ensembles

DCBMA – decaying based Bayesian Model Average

RBMP – Recursive Bayesian Model Process (built in decaying average and internal 2nd-moment adjustment)

Solid line – RMS errorDash line - Spread

Over-dispersion

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Fall 2013

Don’t understand this over-dispersion

Slightly degradation

ROC are better for all leads

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U10m for summer and Fall 2014

Assume V10m has the similar statistics as U10m

The difference of error/skills is very smaller, it is very hard to identify the improvement.

Need to consider to have additional plots for difference of NUOPC and DCBMA and NUOPC and RBMA

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RBMP is perfect for 1st moment and 2nd moment adjustment, especially for

2nd moment.

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Summer 2013

Southern hemisphere

Tropical

NH CRPS NH ROC

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The same conclusion as summerPerfect for 2nd moment adjustment

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NH CRPS NH ROC

Southern hemisphere

Tropical

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6. Statistical downscaling

Statistical downscaling for NAEFS forecast

• Proxy for truth– RTMA at 5km resolution– Variables (surface pressure, 2-m temperature, and 10-meter

wind)

• Downscaling vector– Interpolate GDAS analysis to 5km resolution– Compare difference between interpolated GDAS and RTMA– Apply decaying weight to accumulate this difference –

downscaling vector

• Downscaled forecast– Interpolate bias corrected 1*1 degree NAEFS to 5km resolution – Add the downscaling vector to interpolated NAEFS forecast

• Application– Ensemble mean, mode, 10%, 50%(median) and 90% forecasts

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12hr 2m temperature forecast Mean Absolute Error (MAE)

w.r.t RTMA for CONUSaverage for September 2007

GEFS raw forecast

NAEFS forecast

GEFS bias-corr. & down scaling fcst.

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NCEP/GEFS raw forecast

NAEFS final products

4+ days gain from NAEFS

From Bias correction (NCEP, CMC) Dual-resolution (NCEP

only) Combination of NCEP and

CMC Down-scaling (NCEP, CMC)

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From: Bias correction (NCEP, CMC) Dual-resolution (NCEP only) Combination of NCEP and

CMC Down-scaling (NCEP, CMC)

NAEFS final products

NCEP/GEFS raw forecast

8+ days gain

GMOS forecast

NAEFS final productsFrom :Bias correction (NCEP, CMC)Dual-resolution (NCEP only)Down-scaling (NCEP, CMC)Combination of NCEP and CMC

From Valery Dagostaro (MDL)

CONUS 2m Temperature

For September 2007

Verify against RTMA

Verify against observation

BACK

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6. Statistical downscaling

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Bias correction for each

ensemble member

+

High resolution deterministic

forecast

Mixed Multi- Model

Ensembles(MMME)

Probabilistic products at 1*1 (and/or) .5*.5

degree globally

Down-scaling (based on RTMA)

Probabilistic products at NSGD

resolution(e.g. 2.5km – CONUS)

NCEP

Others

CMC

Reforecast

RBMPFor

blenderVaried

decaying weights

Auto-adjustment

of 2nd moment

Smartinitialization

Future NAEFS Statistical Post-Processing System