# Presentation: 524433

## A Stochastic Parameterization Scheme withinNCEP Global Ensemble Forecast SystemDingchen Hou, Zoltan Toth and Yuejian Zhu  Acknowledgements: Mark Iredell, Henry Juang, Richard Wobus, Bo Cui, Cecile Penland and Prashant SardeshmukhWeiyu Yang, James Purser and Mozheng Wei 

**Notes:**
Outline slide

## OUTLINE

**INTRODUCTION**
**A STOCHASTIC PHYSICS SCHEME**
**        ****Formulation**
**           ****Characteristics of the Perturbations**
**RESULTS OF EXPERIMENTS**
**         ****Spaghetti diagrams **
**            ****Outliers**
**            ****Ensemble mean and spread**
**            ****Probability forecasts**
**            ****Comparison with a bias-correction procedure**
**SUMMARY AND ONGOING RESEARCH**

## Introduction 

**Some Shortcomings of the Current GEFS forecasts**
**      **Cold bias and insufficient spread
**Two Major Sources of Forecast Uncertainty**
-       Uncertainty due to INITIAL CONDITIONS
-       Uncertainty related to the MODEL, not considered in the system
**Representation of Model Uncertainties**
**       **Perturbation rescaling (*Toth and Kalnay, 1995*)
-        Multi-model ensemble (e. g. *Houtekamer et al., 1996; Stensrud et al.,2000*)
-        Multi-model + multi-center ensemble (THORPEX, NAEFS)
-        Multi-model version ensembles (*e. g. Hou et al., 2004*)
*       *Stochastic parameterization (e.g., *Buizza et al., 1999*)
**Nature of Stochastic Parameterization (Palmer, 2001, 2003)**
**       **Using stochastic noise to represent unpredictable small-scale variability
**       **Stochastically sampling the probability distribution of the sub-grid tendency, rather than always sampling the mode of the distribution.
**Types of Stochastic Schemes**
**       **Spatio-temporal stochastic forcing (e.g. *Perez-Munuzuri et al, 2003*)
-        Stochastic backscatter (e.g. *Frederisksen and Davies, 1997; Shutts 2004*)
*       *Stochastic physics, ECMWF EPS (*Buizza et al., 1999*)

## A Stochastic Parameterization Scheme: Formulation

- Model equation:

- T for Tendency; S for Stochastic term

- Single Forecast:

- ECMWF: S is independent of other members

- Ensemble Forecast:

- i=1,2,...,N for N members;
- 0 for control

- One can use

- Linear combination of members or mean

- Desired Characteristics of the Stochastic Forcing
- 1. Forcing for all variables
- 2. Approximately balanced
- 3. Random noise +structure
- 4. Flow dependent

## Use combinations of the perturbation tendenciesStochastically determine the combination coefficients

- The combination coefficients        (i=1,N; j=1,N) forms a square matrix  C (N rows and N columns). With matrix notation, the problem is
- We want Q to be stochastic with the following characteristics: (a) Q vectors sum to 0, i.e., centralized perturbations; (b) Re-scaled to a particular size and the rescaling should be regionally based. (c) Q could be an orthogonal set.
- Most importantly, matrix C should be orthonormal (all N C vectors have unit length and are orthogonal to each other) and have certain temporal variation.

- For an ensemble of N forecasts at M grid points, the N perturbation tendency vectors P (M elements in each P) can be randomly combined to get N new vectors Q (M elements each ), with coefficients

## A Stochastic Parameterization Scheme: Formulation

-        ’s represent coefficients used to re-scale the perturbations to a representative size in NH, SH and TR, using 500hPa kinetic energy as the norm, and

- Model equation:

- T for Tendency; S for Stochastic term

- Single Forecast:

- ECMWF: S is independent of other members

- Ensemble Forecast:

- i=1,2,...,N for N members;
- 0 for control

- One can use

- Linear combination of members or mean

- or

- Random match between i and j

- In the current scheme

- Its finite difference form is

- For t=k*6hr, where
- k=1,2,3,... and

- otherwise

## Characteristicsof the Stochastic Forcing

- 1. Forcing for all variables
- 2. Approximately balanced
- 3. Random noise +structure
- 4. Flow dependent
- Shown are examples of stochastic forcing in vorticity equation and the corresponding total kinetic energy added to the model at 18h of forecast, initialized 00Z, Sep. 25, 2004.

- Vorticity Increment

-  Kinetic Energy Increment

## Spaghetti Diagrams, Smaller areas of outlying analysis (green) 

- Without SP

- With SP

## Outliers: Reduced in the experiment with SP Without SP                                              WITH SP             

## Ensemble Mean Forecast and Spread Increased Spread and Reduced Bias (mean error)

## Reduced Mean Absolute Systematic Error (MASE)

- MASE

- MASE

## Comparison with Post-Processing (Cui and Toth)Stochastic Physics                        Bias-CorrectionMASE

## Systematic Error (Mean error Oct. 2004, 8 day fcst.)

## Comparison with Post-Processing (Cui and Toth)RPSS: Improved in both casesStochastic Physics                    Bias-Correction

## Comparison with Post-Processing (Cui and Toth)BSS: improved in both cases, but only reliability in SP  Stochastic Parameterization                    Bias-Correction

## More Verification ScoresSH z500, Error and RPSS                                                 TR t850, BSS

## AC and ROC area, Less impressive improvement

## Understanding the Improvement

- Benefits of Stochastic parameterization Expected (Palmer, 2003) and
-  Realized with the current scheme
-        More Complete representation of model uncertainty
-              Increase spread
-        Reduction in model systematic error (noise-induced drift)
-              Significant reduction in ME and MASE,  less or no in AC
-              Improved in some probabilistic scores, less in others
-        More Accurate estimate of internal climate variability
-              More improvement in week 2 than in week 1 ???

- Mean state without noise

- Mean state with noise

## SUMMARY OF EXPERIMENTS 

-  Summary
-      A stochastic parameterization scheme is proposed and tested for NCEP Global Ensemble Forecast System. The structured stochastic forcing terms are based on the differences in tendencies between ensemble members and the control run.
-        The stochastic forcing added are balanced, flow-dependent and have random noise structure and geographic patterns, changing smoothly in time and space.
-         By including this stochastic scheme, the ensemble forecast can be improved, with fewer outliers and increased spread. The Mean Absolute Systematic Error (MASE) of the ensemble mean forecast is significantly reduced.
-         For probabilistic forecasts, the stochastic parameterization scheme can significantly improves the RPSS and BSS scores. Unlike the post processing procedure, the stochastic
-         parameterization scheme improves BSS mainly by reducing its reliability component.
-         Compared with a post-processing procedure of bias-correction, the effect of this stochastic parameterization scheme is less effective in week 1, but equally or more effective in week2 forecast. This may indicate potential benefit in CFS.

## CURRENT AND SUGGESTED RESEARCH 

- IMMEDIATE TASKS
-    Test the effect of the combination of the stochastic scheme and the post processing algorithm.
-    Evaluate the results for other variables (wind, precipitation) and perform a number of case studies.
- POSSIBLE REFINEMENT OF The SCHEME
-    Refine the scheme by using linear combination of the ensemble perturbations as the stochastic forcing, and tuning its size, to maximize its effects.
-    Refine the scheme by including temporal variation in the match between ensemble members and stochastic forcing perturbations.
-    Refine the scheme by including the difference between high resolution control and low resolution control forecast as stochastic forcing.
-    Further explore the theoretical basis of the scheme for better understanding and further development of the scheme.
- FOR POSSIBLE OPERATIONAL IMPLEMENTATION
-    Test the scheme with the current operational GEFS configuration (higher resolution, upgraded GFS and 6 hour cycling).
-    Test the scheme in various seasons and weather regimes, and for longer period to confirm preliminary results, and compare with other stochastic parameterization schemes.
-     Develop software capacity to synchronize the integration of the ensemble members and control run (ESMF project, Weiyu Yang) so that the scheme can be tested with its refined versions.

## Combination of SP and PP, Ens. Mean and SpreadFurther reduction in ME, negative effect with over-reductiondue to inconsistent data set for bias estimation

- ---OPT        ---SP
- ---OPT+PP ---SP+PP

## Combination of SP and PP, Probabilistic ForecastConsistent improvement with RPSS and BSS for all lead time andROC for 11+ days, even with the negative effect of over-correction of bias

- ---OPT        ---SP
- ---OPT+PP ---SP+PP

## Combination of SP and PP, OutliersSP reduces outliers more effectively than PP

- ---OPT        ---SP
- ---OPT+PP ---SP+PP

- Forecast consistency

## Effect of Combination of SP and PP, MASESP+PP effectively reduces MASE in tropics

## Systematic Error (Mean error Oct. 2004, ensemble mean 10day fcst)

## Use combinations of the perturbation tendenciesStochastically determine the combination coefficients

- The combination coefficients     (i=1,N; j=1,N) forms a square matrix  C (N rows and N columns). With matrix notation, the problem is
- We want Q to be stochastic with the following characteristics: (a) Q vectors sum to 0, i.e., centralized perturbations; (b) Re-scaled to a particular size and the rescaling should be regionally based. (c) Q could be an orthogonal set.
- Most importantly, matrix C should be orthonormal (all N C vectors have unit length and are orthogonal to each other) and have certain temporal variation.

- For an ensemble of N forecasts at M grid points, the N perturbation tendency vectors (P, dimension M) can be randomly combined to get N new vectors (Q, dimension M), with coefficients

## Use combinations of the perturbation tendenciesStochastically determine the combination coefficients

- Expression of Matrix C
- Express the matrix C as a temporally random rotation (in an N dimensional Space) from a “mean” matrix C0, which is constructed by random numbers.
- Independently select the NxN elements of C0  from a specified distribution: uniform in (-0.5, +0.5) or Gaussian centered at 0.0 ???
- Generation of C0
- Manipulate matrix C0 (Gram-Schmidt) to make its vectors orthogonal and normalized. (Suggested by J. Purser)
- Determination of Random Rotation
- The rotation matrix R has stochastic temporal variation and the following characteristics: (1) orthonormal; (2)close to    ; (3) the temporal variation is a random walk.
- R can be generated by manipulating a matrix in the form of
-      where is a random number and A is a matrix with
-      (Suggested by J. Purser)