# Presentation: 722111

## 3DVAR on the JMA-NHMandEnKF on the SPEEDY model

- Takemasa Miyoshi
- Department of Meteorology, University of Maryland
- and
- Numerical Prediction Division, Japan Meteorological Agency

- 6/21/05 EMC/NCEP seminar

## Two topics:

- 3DVAR on the JMA-NHM
- Before coming to UMD, I developed the 3DVAR for nonhydrostatic model (JMA-NHM) at NPD/JMA (Apr. 2002- Jun. 2003)
- Currently, 4DVAR has been developed by Mr. Honda (NPD) and several scientists at MRI/JMA
- EnKF on the SPEEDY primitive equations model
- Doctoral dissertation (advisor: Prof. Eugenia Kalnay)
- Defended on June 7
- Purpose is to develop a path towards operational implementation

## JMA-Nonhydrostatic Model (NHM)

- In operation since Sep. 2004
- Prognostic Variables
- U,V,W,Prs,θ,qv,qc,qr,qs,qci,qg
- Terrain-Following Vertical Coordinates (z*)
- Staggered Grid Structure
- Arakawa C type (Horizontal)
- Lorenz type (Vertical)

## Design of control variables

- Gaussian horizontal correlation
- Recursive filter, Cholesky decomposition
- Full vertical correlation
- Inter-variable correlation:

- Geostrophic Balance (Synoptic Scale)
- Mass Conservation

- Control Variable: Unbalanced Stream Func.

- Unbalanced Vertical Wind

- PB: Hydrostatic Pressure

## Control variables

| Control Variables | Definition |
| --- | --- |
| Unbalanced Stream Function |  |
| Unbalanced Velocity Potential |  |
| Unbalanced Vertical Wind |  |
| Unbalanced Pressure |  |
| Potential Temperature – the bottom level pressure |  |
| Water Vapor Quantity |  |

- Different variables are assumed to be uncorrelated

- Regression coefficients r1-r4 are determined statistically

## Background error statistics

- NMC Method
- Error = 12hrs fcst - 6hrs fcst (at the same valid time)
- Model Setting
- 62cases Statistics in 4/24/2002 - 5/28/2002
- Grid: 122×122×45
- Horizontal Resolution: 5km
- Region:

## A case study

- Aircraft overran at Narita Int’l Airport at around 21:50

- Synoptic chart 21:00JST

- No clear front around NRT in synoptic scale analysis

- Rapid growth of the Low

## Local front around NRT airport

- Very Shallow Structure
- Low Temperature Side
- Weak Northern Wind
- High Temperature Side
- Strong South Wind

- Cross Section of Local Front

## 7-hr fcsts by several initial conditions

- _Valid: 22:00JST_

- _Initial: 15:00JST_

- Hourly 3DVAR performs as well as the routine MSM4DVAR!

- Still, NRT is in the warmer side in all cases.

## Keep updating...

- 04Z

- 05

- 06

- 07

- 08

- 09

- 10

- 3DVAR

- 3DVAR

- 3DVAR

- 3DVAR

- The latest analysis available without hourly analysis

- _Newer analyses by Hourly 3DVAR_

- ・・・

- Observations assimilated:
-  Synoptic conventional observations
-  Wind profiler radar
-  Aircraft observations

## Forecasts valid at the same time (22JST)

- _Hourly New_
- _Initial Values_

- FT=3

- FT=4

- ★

- ★

- ★

- ★

- ★

- ★

- FT=5

- FT=6

- FT=2

- FT=1

- _Different initial conditions_

- _Initial: 19:00JST_

- _Initial: 18:00JST_

## Analysis field by hourly 3DVAR

- ★

- ★

- Strong shear line at the height of 2-300m

- Low Temperature zone and the local front is well analyzed

- NRT is located in the low temperature side of the front

- _Valid: 22:00JST_

## Summary

- I developed 3DVAR on JMA-NHM during 2002-2003. (Currently, 4DVAR has been developed by Mr. Honda and others.)
- 3DVAR shows reasonable analysis performance.
- The hourly updating cycle using 3DVAR shows an advantage in a case study.

## Ensemble Kalman filter experiments with a Primitive-Equation global model

- Dissertation Committee:
- 	Prof. Eugenia Kalnay (chair/advisor)
- 	Prof. James Carton
- 	Prof. Owen Thompson
- 	Prof. Brian Hunt
- 	Dr. Joaquim Ballabrera

## Overview

- _Data assimilation_

- _Ensemble Kalman filtering (EnKF)_

- A way to keep the model state close to the nature using observations

- A sophisticated method of data assimilation

- Several implementations have been proposed but not compared

- _Model imperfections_

- Sophisticated methods are generally more sensitive to model imperfections

- _Perfect model experiments_

- _Imperfect model experiments_

- No model can simulate the Mother Nature perfectly

- Experiments

- Compare EnKF methods to 3DVAR

- Investigate the effect of model imperfections

## Purpose

- To develop a path towards an operational _ensemble forecast-analysis system_.

- _The ultimate goal:_

- _Relevant scientific questions:_

- What are the relative advantages and disadvantages among different implementations of EnKF?
- How do they compare with 3DVAR?
- What is the effect of model imperfections in data assimilation?

- Realized by ensemble Kalman filtering (EnKF)

## Data assimilation (filtering)

- Time

- Data assimilation is an analysis technique in which the observed information is accumulated into the model state by taking advantage of consistency constraints with laws of time evolution and physical properties. – Bouttier and Courtier (1999)

- Obs

- Obs

- Model integration

- Assimilate

- Analysis

- Analysis

- Analysis

- Accumulate past observations

- Obs

## Kalman filtering (KF)

- _Kalman filtering_ (KF) is an optimal weighted mean between forecast and observation.

- Here, the optimal weight (Kalman gain) is given as

- KF (estimation of P) is optimal when the model is linear and perfect.

- Error covariance is estimated by the forecast:

- Error covariance forecast is usually underestimated partly because of model nonlinearity. Thus, covariance inflation is required.

- Inflation parameter

## Ensemble Kalman filtering (EnKF)

- _Ensemble formulation_

- Assumption: Limited number of ensembles can estimate **P**

- To avoid sampling errors, we localize covariance.

- By the ensemble formulation, we can forecast error covariance from ensemble forecasting:

## An advantage of Kalman filtering

- In Kalman filtering, we forecast **P**

- We use constant **P** in 3DVAR

- An example of using the flow-dependent **P**
- There is a cold front in our area...

- Kalman filtering can consider the flow-dependent error structure.

## EnKF approaches

- Serial EnSRF (Whitaker and Hamill 2002)

- ETKF (Bishop et al. 2001)

- EAKF (Anderson 2001)

- LEKF (Ott et al. 2002; 2004)

- Effective in serial treatment of observations

- Simultaneous treatment of observations

- We implement and compare Serial EnSRF and LEKF

- Perturbed observation (PO) method

- Square root filtering (SRF) method

- EnKF

- We do not investigate this

- _Serial EnSRF_ localizes covariance **P** around observation locations.

- _LEKF_ treats local patches independently.

- N local patches cover the entire globe.

- Weighting function

- Local patch

## Results on the Lorenz-96 model

- LEKF with obs. localization

- 0.33

- Serial EnSRF

- Observe 20 points out of 40 total grid points, with error stdev of 1.0

- 3DVAR RMSE: 1.15

- 0.34

- Lorenz-96 model (Lorenz 1996; Lorenz and Emanuel 1998)

- Covariance inflation factor

- Localization length scale

- Localization length scale

## The SPEEDY model (Molteni 2003)

- _Primitive-equation dynamic core_

- _Simplified physics_

- Convection (a simplified mass-flux scheme)

- Vertical diffusion

- Surface fluxes of momentum and energy (bulk aerodynamic formula)

- Long wave radiation (four spectral bands)

- Short-wave radiation (two spectral bands)

- Large-scale condensation, Clouds

- _No diurnal forcing_

- _Prognostic variables_

- u, v, T, q, Ps

- _Resolution_

- T30L7 ( 96 x 48 x 7 )

| Level | Sigma
(p/ps) | Pressure
(hPa) |
| --- | --- | --- |
| 1 | 0.950 | 925 |
| 2 | 0.835 | 850 |
| 3 | 0.685 | 700 |
| 4 | 0.510 | 500 |
| 5 | 0.340 | 300 |
| 6 | 0.200 | 200 |
| 7 | 0.080 | 100 |

- _Computational time_

- 2 seconds for 6-hour forecast on a 2.7GHz Celeron PC

## Data assimilation experiments

- OSSEs (Observation Systems Simulation Experiments)

- Nature run

## Experimental setup

| Variables | Obs. Err. Stdev. |
| --- | --- |
| u [m/s] | 1.0 |
| v [m/s] | 1.0 |
| T [K] | 1.0 |
| q [kg/kg] | 0.0001 |
| Ps [Pa] | 100.0 |

- _Data assimilation systems_

- 3DVAR

- Serial EnSRF

- LEKF

- _Observational network_

- Uniform (mainly used)

- Realistic

- _Perfect model experiments_

- Nature run is generated by the SPEEDY

- _Cov. inflation_

- 4% fixed, multiplicative

## Results: 500hPa height RMSE

| Localization scale | 1 | 2 | 3 | 4 | 5 | 6 |
| --- | --- | --- | --- | --- | --- | --- |
| NBV=10 | 12 | 25 | 40 | D | D | D |
| NBV=20 | 11 | 11 | 11 | D | 22 | D |
| NBV=30 | 15 | 5 | 5 | 8 | 10 | 17 |

| Localization scale | 1 | 2 | 3 | 4 | 5 | 6 |
| --- | --- | --- | --- | --- | --- | --- |
| NBV=10 | 60 | 76 | D | D | D | D |
| NBV=20 | 35 | 20 | 17 | 31 | D | D |
| NBV=30 | 18 | 17 | 9 | 8 | 13 | 15 |

- Average skills in 500hPa height RMSE [meters]

- _3__DVAR_: 31m

- _Serial EnSRF_

- _LEKF_

- D: filter divergence

## Results: time sequence

- 3DVAR (31m)

- LEKF (8m)

- Serial EnSRF (5m)

- 30 ensemble members

## Results: zonal mean RMSE

- Zonal mean RMSE of u-wind

- 3DVAR

- Serial EnSRF

- 3DVAR shows clearer striped pattern corresponding to observational locations.

## Analysis increments

- 3DVAR

- EnSRF

- Lattice-like pattern in the 3DVAR background error field.

- EnSRF analysis increment is better capturing error structures.

- Shades: background error, Contour: analysis increment

## Error growth

- SPREAD

- RMSE

- 3DVAR

## Online estimation of inflation parameter

- _Concepts of the method_

- Kalnay suggested that covariance inflation parameter can be estimated using the statistics:

- At each analysis step, we can have an estimate of

- Assuming this estimate as observation, estimate     online using KF.

## Online estimation of inflation parameter

- _Results for EnSRF with 10 members_

- We confirm this online estimation provides stable performance in a simple system (the Lorenz-96 model).

| Inflation | Analysis RMSE | Ensemble spread |
| --- | --- | --- |
| 4% multiplicative | 2.3 | 4.9 |
| Online estimation | 2.4 | 2.1 |

- On the SPEEDY, the relative amplitude of analysis RMSE and ensemble spread is adjusted, but analysis RMSE is not decreased.

- Energy norm

- With this method, we can avoid tuning the parameter.

## Observational error covariance localization

- Hunt suggested covariance localization by localizing the observational error covariance.

- Weight the observational variance by the inverse of the Gaussian:

- _Concepts of the method_

## Observational error covariance localization

- _Results for LEKF with 10 members_

- With the localization, LEKF is improved.

- With this method, we can fix the patch size as large and tune the localization scale.

- Local patch parameter

## Summary of the perfect model experiments

- Results using the Lorenz-96 model show that with the observation localization, LEKF outperforms EnSRF
- EnKF strongly outperforms 3DVAR on the SPEEDY model
- 2-day forecast “errors of the day” are similar to analysis errors, but not similar among different methods. In ensemble low-dimensional regions, the errors show similar structures
- Serial EnSRF generally outperforms LEKF, but the difference is substantially reduced when we apply observational error covariance localization or increase the ensemble size

## Imperfect model experiments

- _Use NCEP/NCAR reanalysis (NNR) as the nature run_

- Time evolving model of NNR is different from the SPEEDY model.

- (the SPEEDY model error) = (6-hour forecast) – (NNR)

- Initial condition is the NNR

- The statistics gives model bias (time average) and model error EOFs

## Imperfect model experiments

- _Data assimilation experiments_

-  No treatment of the model error (use the same DA system as perfect model cases)

-  Dee and da Silva (1998) model bias estimation scheme

-  Low order bias estimation scheme (Kalnay and Danforth)

## Model bias fields

## Model error EOFs

- EOF1

- EOF2

- Since the SPEEDY model has no diurnal forcing, first 2 EOFs show diurnal signals with phase differences.

## Results without bias correction

- The advantage of EnKF is not as large in the presence of model imperfections.
- (31m : 12m in the perfect model case)

- 3DVAR is more sensitive to observational networks.

## Dee and da Silva’s bias correction

- _Concepts of the method (Dee and da Silva 1998) _

- Apply EnKF for bias estimation

- Observational increment

- Observation of the model error bias

- Persistence for the model bias evolution

## Dee and da Silva’s bias correction

- _Results with 10+10 members_

- With the bias correction, the filter diverges.

- Possibly because low dimensional assumption of EnKF is not good due to the dynamics of the model bias (we assumed identity (persistence) as time-evolving law of the model bias)

- Dee and da Silva bias estimation

## Low order bias estimation

- N

- =

- Nbias

- Nbias

- 1

- 1

- Each column is a model field with the shape of the bias

- Bias amplitude to be estimated

- The same equations are applied to estimate the low dimensional bias variable

- Mean bias field

- EOFs

## Results with low-order bias correction

- Bias correction works better in EnKF.

| Nbias | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| RMSE [m] | 66.6 | 42.8 | 40.6 | 39.2 | 47.6 | 43.9 | 42.5 |

- Mean bias

- Mean bias + EOFs

- 500hPa height RMSE

- No correction

## Summary of imperfect model experiments

- Without the bias correction, the advantage of EnKF over 3DVAR is reduced
- Bias correction improved EnKF substantially
- Full dimensional bias estimation by Dee and da Silva (1998) did not work with 10 members and is computationally expensive
- We obtained substantial improvement by low order bias correction, which is less expensive

## Towards an operational EnKF...

-  LEKF is efficient with a large number of observations and parallel computers, it may be the only possible choice in operations.

-  With observational error covariance localization, the disadvantage of LEKF is reduced.

-  Online estimation of covariance inflation parameter can be used instead of tuning.

- _This research addresses..._

-  EnKF has the potential to outperform 3DVAR in operations.

## Towards an operational EnKF...

- _Future..._

** ****I will work on the development of 4D-LETKF in JMA**

-  Application of 4D-EnKF (Hunt et al. 2004) to treat asynoptic observations

-  Comparison with 4DVAR

-  Implement more efficient LETKF

## Acknowledgements

- I express my deep gratitude to Prof. Kalnay and the Chaos group, especially Profs. Hunt, Ott and Szunyogh, and to Dr. Ballabrera.
- The Japanese Government financially supported my graduate study.
- Drs. Franco Molteni and Fred Kucharski kindly provided the SPEEDY model.
- Many thanks to the committee professors and other faculty and students for insightful discussions.
- Many many thanks to all for help and support.

**Notes:**
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