# Presentation: 045927

## A comparison of hybrid ensemble transform Kalman filter(ETKF)-3DVAR and ensemble square root filter (EnSRF) analysis schemes

**Xuguang Wang**
**NOAA/ESRL/PSD, Boulder, CO**
- Thomas M. Hamill
- Jeffrey S. Whitaker
- NOAA/ESRL/PSD, Boulder, CO
- Craig H. Bishop
- NRL, Monterey, CA

## Motivation for the hybrid

- The hybrid ensemble (inexpensive yet skillful, e.g., ETKF)-3DVAR method is expected to be less expensive than ensemble data assimilation, and can also benefit from ensemble-estimated error statistics.
- The hybrid may be more robust for small ensemble size.
- The hybrid is convenient to be adapted to the existing variational framework.
- The risk of the hybrid is low as it can perform no worse than 3DVAR.

## Hybrid ETKF-3DVAR

- member 1 forecast

- member 2 forecast

- member 3 forecast

- ETKF-3DVAR update mean

- ETKF update perturbations

- member 1 analysis

- member 2 analysis

- member 3 analysis

- member 1 forecast

- member 2 forecast

- member 3 forecast

**data assimilation**

**forecast**

## Hybrid ETKF-3DVAR updates mean

-  Background error covariance is approximated by a linear combination of the sample covariance matrix of the ETKF forecast ensemble and the static covariance matrix.

-  Can be conveniently adapted into the operational 3DVAR through augmentation of control variables.

## ETKF updates perturbations

- ETKF transforms forecast perturbations      into analysis perturbations      by
-        where    is chosen by trying to solve the Kalman filter error covariance update equation, with forecast error covariance approximated by ensemble covariance.
- Latest formula for       (Wang et al. 2004;2005, MWR)
- Computationally inexpensive for ensemble size of o(100).

## Experiment design

-  Observations
- 362 Interface and surface Exner functions taken at equally spaced locations
- Observation values are T31 truth plus random noise drawn from normal distribution
- Assimilated every 24h

-  Numerical model
- Dry 2-layer spectral PE model run at T31;
- Model state consists of vorticity, divergence   and layer thickness of Exner function
- Error doubling time is 3.78 days at T31
- Perfect model assumption

- (Hamill and Whitaker MWR, 2005)

## Experiment design

- In this experiment, the hybrid updates the mean using

- whose solution is equivalent to that if solved variationally under our experiment design.

- The static error covariance** **model is constructed iteratively from a large sample of 24h fcst. errors.** **

-  Hybrid solver and static error covariance model

## RMS analysis errors: 50 member

-  Covariance localization applied on the ETKF ensemble when updating the mean (but *not* applied when updating the perturbations) improves the analyses of the hybrid.
-  Improved accuracy of EnSRF over 3DVAR can be mostly achieved by the hybrid.

## RMS analysis errors: 20 member

-  Both 20mem hybrid and EnSRF worse than 50mem; still better than 3DVAR
-  Hybrid similarly accurate compared to EnSRF

## RMS analysis errors: 10 member

-  EnSRF experienced filter divergence for all localization scales tried.
-  Hybrid is still more accurate than the 3DVAR.
-  Hybrid is robust in the presence of small ensemble size.

## Flow dependent background error covariance models

- (Hybrid 50 mem. result)

- (EnSRF 50 mem. result)

## Spread-skill relationships

- (50 mem. result)

-  Overall average of the spread is approximately equal to overall average of rms error for both EnSRF and Hybrid.
-  Abilities to distinguish an analysis with large error variance from that with a small error variance are similar for EnSRF and Hybrid.

## Summary

- A hybrid ETKF-3DVAR DA scheme was proposed and the skill was compared with EnSRF with a PE two layer model and simulated observations under perfect model assumption.
- The hybrid analyses can achieve similar improved accuracy of the EnSRF over 3DVAR.
- The hybrid is more robust when ensemble size is small.
- The ETKF ensemble variance is as skillful as the EnSRF.
- The hybrid can be conveniently adapted into the existing operational 3DVAR framework.
- The hybrid is expected to be less expensive than the EnSRF.

## Future work

- Preliminary results of OSSE within imperfect model environment, e.g., model with resolution error, also suggest hybrid can be as accurate as EnSRF.
- Incorporate to WRF 3DVAR and GFS GSI frameworks.
- Extend the idea of the hybrid ETKF-3DVAR to the 4DVAR framework.

## Preliminary results with resolution model error 

- Preliminary results with resolution model error

-  T127 run as truth; imperfect model run at T31
-  Additive noise for EnSRF drawn randomly from the truncation error samples
-  Additive noise for hybrid drawn randomly from **B **
- 15000km localization tried on ETKF ensemble when updating the mean of  the hybrid
** **comparable rms errors for hybrid and EnSRF

## Statistics of iteratively constructed B

- Statistics of iteratively constructed **B**

- rms analysis errors

- imbalance

## Single observation analysis increment for 3DVAR

## Initial-condition balance

- (50 mem. result)

-  Analysis is more imbalanced with more severe localization.
-  Hybrid analysis is more balanced than EnSRF.