# Presentation: 819288

## P1.26                                 Refining the Operation of the Ensemble Transform Analysis Perturbation Scheme

- Assemble from some archive a matrix          of forecast perturbations.
- Obtain a new matrix           by carrying out an ensemble transform
-      using           and        :
- Obtain a matrix          by carrying out a set of constrained optimizations
-      using          and         :
- Scale the eigenvectors             such that
- Create random linear combinations of the eigenvectors            .
-      At this point one has perturbations in * ****u***, ***v***, ***t***  that are quasi-balanced, have realistic spatial correlations, and have collective variance that mimics the tropical forecast error variance deficit defined in      .

_**VI. SYNTHESIZING TROPICAL VARIANCE - PROCEDURE**_
- Compute the ET-ensemble’s tropical forecast error variance deficit matrix,      :
- Seek perturbations       such that                                   The steps are:

**P1.26                                 ****Refining the Operation of the Ensemble Transform Analysis Perturbation Scheme**
**                                                                                ****through Tropical Forecast Variance Augmentation**
- Justin G. McLay 2, 1, Carolyn A. Reynolds1, and Craig H. Bishop1
- 1 Naval Research Laboratory, Monterey, CA, USA, 2 National Research Council, Washington, D.C., USA

- REFERENCES:
- Bishop, C. H., and Z. Toth, 1999:   Ensemble transformation and adaptive observations.  *J. Atmos. Sci.,* **56**, 1748-1765.
- Wang, X., and C. H. Bishop, 2003:  A comparison of breeding and ensemble transform kalman filter ensemble forecast schemes.  *J. Atmos. Sci.,* **60**, 1140-1158.
- Daley, R., and E. Barker, 2001: *NAVDAS source book 2001.*  NRL/PU/7530--01-441, NRL, Monterey, CA, 93943-5502, 161pp.
*Acknowledgements*: This research is sponsored by the Naval Research Laboratory and the Office of Naval Research under program element 0601153N, project number BE-033-03-4M.

_**IX.  **__**SUMMARY**_
- A technique to augment tropical forecast variance is devised for use in conjunction with the ensemble transform analysis perturbation scheme.  The core element of the technique is a synthesis of global wind and temperature perturbations with maximum amplitude in the tropics and with realistic structure and spatial correlation.  The technique concludes with the addition of the synthesized perturbations to existing 24h-leadtime ensemble members.  The addition enables the ensemble transform scheme to generate analysis perturbations that are more consistent with benchmark NAVDAS analysis error variance estimates.  Over some number of analysis/ensemble cycles this greater consistency leads to substantially better agreement between forecast-error variance derived from the ET-based ensemble and observed forecast-error variance.  The improved performance of the ET-based ensemble engendered by the augmentation technique opens the door to use of the ensemble transform scheme in operational environments that involve constraints on ensemble size and imperfect numerical models.  Refinement of the variance augmentation technique is ongoing.

_**ET IMPLEMENTATION – RESULTS**_

_**VII. STRUCTURE OF SYNTHESIZED TROPICAL PERTURBATIONS**_

_**IV.  IMPROVING ET OPERATION - ALTERNATIVES**_

_**VIII.  CYCLING WITH THE AUGMENTED ET – INITIAL RESULTS**_
- The variance augmentation procedure was used with a 32-member ET ensemble in a 24h analysis cycle for the six-week period 0000UTC 31 December 2004 to 0000UTC 11 February 2005.  The  **Q**  matrix was updated after three weeks.  The augmentation has a pronounced effect on the variance signature of both the ET analysis perturbations and the ET forecast perturbations.  The curves for the u- and v-wind perturbations are much flatter, with greatly reduced peaks in the extratropics.  The curves for the analysis temperature perturbations are less affected, but the corresponding curves for the forecast temperature perturbations nevertheless are substantially flattened as well.

**ETKF TARGET S. MAJUMDAR**

- Figure 6, above.  Manifestation of synthesized tropical perturbations in ET analysis perturbation variance.  0000UTC 31 December 2004 zonally and vertically averaged ratio of ET analysis perturbation variance and NAVDAS analysis error variance with (closed-circle white line) and without (open-circle white line) use of synthesized tropical variance for (left to right) u-wind, v-wind, and temperature.  Note the relative increase in tropical variance and corresponding decrease in extratropical analysis perturbation variance following use of the synthesized tropical variance.

- where
**            ****X****a****   =  ***n *x* m*  matrix of analysis perturbations from current cycle              *m* = number of ensemble members
**            ****X****f****   =  ***n* x *m* matrix of forecast perturbations from previous cycle              *n* = dimension of model state vector
**            ****P****a****   =  ***n* x *n*  diagonal matrix of analysis error variance estimates
**            ****T    =  ***m* x *m*  matrix of weighting coefficients

_**V. FLOW-CHART OF TROPICAL VARIANCE AUGMENTATION**_

**f****1**

**f****0**

**a****1**

**f****1**

- Figure 3, left, shows the sequence of operations in the variance augmentation process.  Synthetic tropical variance is added to existing ET-ensemble forecast perturbations  ***f******0***, resulting in modified forecast perturbations  ***f******0******mod*** with enhanced tropical variance (a).  The modified forecast perturbations ***f******0******mod***, when input into the ET scheme (b), will result in analysis perturbations ***a******1*** with increased variance in the tropics relative to that of analysis perturbations ***a******0*** obtained from un-augmented forecast perturbations (c).   Furthermore, because of the global constraint involved in the ET scheme, the analysis perturbations ***a******1****** ***will have relatively less variance outside the tropics.  When the analysis perturbations ***a******1*** are integrated in the NOGAPS model (d), they should yield forecast perturbations  ***f******1*** with more tropical variance and less extratropical variance than the forecast perturbations  ***f******0 ***(e).  Parts (a) - (e) of this process are repeated over the next analysis cycle, starting with the forecast perturbations ***f******1*** in place of the forecast perturbations  ***f******0***.  In this way, the variance curves of the forecast and analysis perturbations  ***f******i***  and  ***a******i***, respectively, can be gradually flattened over a sequence of analysis cycles.

- Figure 4, above, shows the zonally and vertically averaged amplitude of the 1st, 2nd, and 10th eigenvectors (black, blue, and magenta lines, respectively) of the **Q** matrix for 31 December 2004 for (left to right) u-wind, v-wind, and temperature.  Note the successful localization of the u- and v-wind perturbation amplitude within the –300S to 300N latitude band.

- The basic ET scheme was implemented using ensemble forecasts from the Naval Operational Global Atmospheric Prediction System (NOGAPS) model and analyses and analysis error variance estimates from NAVDAS.  32-member ensemble forecasts were generated every 24h at 0000UTC for the period 29 November 2004 to 31 December 2004 using a T119L30 version of NOGAPS.  Figure 1, left, indicates that relative to the NAVDAS analysis error variance estimates the ET-derived analysis perturbations are strongly sub-variant within the tropics and strongly super-variant outside the tropics.  The ET-ensemble 24h forecast perturbations exhibit a similar variance signature relative to observed forecast error variance.  This undesirable variance signature is reflected as poor performance in a number of ensemble diagnostics.

- Figure 1, above.  Top row:  0000UTC 31 December 2004 zonally and vertically averaged ratio (open-circle white line) of ET analysis perturbation variance and NAVDAS analysis error variance for (left to right) u-wind, v-wind, and temperature.  Red solid line highlights a ratio value of 1.  Bottom row:   Zonally and vertically averaged difference (solid black line) between the ET-ensemble 24h forecast error variance and the observed 24h forecast error variance for (left to right) u-wind, v-wind, and temperature.  Red solid line highlights a difference value of 0.  ET-ensemble 24h forecast error variance calculated as the average of the ET-ensemble variance about the 0000UTC 24h control forecast over the 14d period ending 0000UTC 31 December 2004.  Observed 24h forecast error variance calculated using the differences between the ET-ensemble 0000UTC 24h control forecast and the corresponding 0000UTC verifying analysis over the 14d period ending 0000UTC 31 December 2004.

- The principal advantages of the ET are:
- It maintains ensemble variance in as many directions as there are ensemble members, unlike the bred-vector approach (Wang and Bishop 2003).
- It incorporates multiple flow-dependencies into the analysis perturbations:
- The influence of targeted observations can be imparted through the use of NRL Advanced Variational Data Assimilation System (NAVDAS) flow-dependent analysis error variance estimates (Daley and Barker 2001).
- Amplitudes and spatial correlations unique to the current flow’s dynamical makeup are imparted through the use of forecast perturbations generated from the previous analysis cycle.

_**I. INTRODUCTION**_
- The ensemble transform analysis perturbation scheme has theoretical advantages over bred-vector and correlated-random-noise perturbation schemes, and is computationally less expensive for a given ensemble size than the singular-vector scheme.  However, in practice the operation of the ensemble transform is hindered by the relatively small size of current operational ensembles and by these ensembles’ lack of diversity in tropical regions.  This study describes a technique to improve the operation of the ensemble transform through augmentation of ensemble forecasts with synthetic tropical variance.

_**II.  THE ENSEMBLE TRANSFORM: **_
- The ensemble transform (ET) obtains analysis perturbations for the current analysis cycle as weighted linear combinations of short-range forecasts from the previous analysis cycle, subject to a constraint which enforces global consistency with analysis error variance estimates (Bishop and Toth 1999).  The equation set of the ET is:

**f****0****mod**

**(b)**

**(c)**

**(a)**

**(d)**

**(e)**

- Figure 7, right.  Left column:  Zonally and vertically averaged ratio of ET analysis perturbation variance and NAVDAS analysis error variance prior to augmentation (open-circle white line), after three weeks of augmentation (closed-circle white line), and after six weeks of augmentation (open-square green line) for (top to bottom) u-wind, v-wind, and temperature.  Red solid line highlights a ratio value of 1.  Right column:   Zonally and vertically averaged difference between ET-ensemble 24h forecast error variance and observed 24h forecast error variance prior to augmentation (black line), after three weeks of augmentation (magenta line), and after six weeks of augmentation (blue line) for (top to bottom) u-wind, v-wind, and temperature.  Black dotted line highlights a difference value of 0.  Observed 24h forecast error variance and ET-ensemble 24h forecast error variance calculated as described in the caption for Figure 1.

**90S                60S               30S                EQ                30N               60N              90N**

**90S                60S               30S                EQ                30N               60N              90N**

**90S                60S               30S                EQ                30N               60N              90N**

**90S                 60S                 30S                 EQ                 30N                 60N                90N**

**90S                 60S                 30S                 EQ                 30N                 60N                90N**

**90S                 60S                 30S                 EQ                 30N                 60N                90N**

**difference in forecast error variance**
**(ET – observed)**

**difference in forecast error variance**
**(ET – observed)**

**difference in forecast error variance**
**(ET – observed)**

**Ratio of analysis error variance**
**(ET / NAVDAS)**

**Ratio of analysis error variance**
**(ET / NAVDAS)**

**Ratio of analysis error variance**
**(ET / NAVDAS)**

**V-wind**

**V-wind**

**Temperature**

**Temperature**

**U-wind**

**U-wind**

**difference in forecast error variance**
**(ET – observed)**

**difference in forecast error variance**
**(ET – observed)**

**difference in forecast error variance**
**(ET – observed)**

**Ratio of analysis error variance**
**(ET / NAVDAS)**

**Ratio of analysis error variance**
**(ET / NAVDAS)**

**Ratio of analysis error variance**
**(ET / NAVDAS)**

**U-wind**

**V-wind**

**Temperature**

**U-wind**

**V-wind**

**Temperature**

- Figure 5, above.  Examples of perturbation flow fields for the 9th      realization for 0000UTC 31 December 2004.  a) Sigma-level 28 wind vectors (white arrows) and relative vorticity (shaded).  b) Sigma-level 15 streamlines (blue solid lines with arrowheads).  Note the well organized, coupled features in the wind field at both low and high levels.

**a)**

**b)**

- authors’ emails:  mclay@nrlmry.navy.mil
-                             reynolds@nrlmry.navy.mil
-                             bishop@nrlmry.navy.mil