Assemble from some archive a matrix of forecast perturbations. Obtain a new matrix by carrying out an

1. To understand the results described in Section III, recognize that the ET can be thought of as a vector-space operation that aims to construct vectors of a certain form. Linear algebra theory tells us that to be sure of producing these vectors we need a basis for the vector space. Considering that the dimension of the NOGAPS vector space is O(106) and the dimension of the forecast ensemble is O(101), it is almost certain that the forecast ensemble does not span the NOGAPS vector space. Thus, the ET is operating without a strict basis, and this fact is manifested in the ET-derived analysis perturbation vectors not having the desired form. Figure 2, below. Zonally and vertically averaged ratio of ET-ensemble v-wind analysis perturbation variance and NAVDAS v-wind analysis error variance when the ET-ensemble has 32 members (open-circle white line), 256 members (closed-circle white line), and 768 members (open-square green line). 1.6 3.5 3.5 1.4 3.0 3.0 1.2 2.5 2.5 1.0 2.0 2.0 0.8 1.5 1.5 0.6 1.0 1.0 0.4 0.5 0.5 0.2 0.0 0.0 0.0 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N • There are several possible ways to surmount the problem: • Apply a spatially varying scaling mask to the analysis perturbations. • This action is simple but inadvisable, as it distorts the perturbations’ spatial correlations in ways that may be detrimental. • Increase the number of forecast perturbations • This action improves the basis that the ET operates with by increasing the variance, and hence the span, of the forecast ensemble. It is effective, as illustrated in Fig. 2, but computationally prohibitive. • Augment the variance of the existing forecast perturbations in regions where it is deficient (i.e. the tropics). • This action also improves the basis that the ET operates with, but can be carried out at affordable computational cost. A variance augmentation procedure is specified in Section V. V-wind 3.5 3.0 2.5 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 2.0 Ratio of analysis error variance (ET / NAVDAS) NOGAPS 1.5 1.0 f0 0.5 with constraint 0.0 90S 60S 30S EQ 30N 60N 90N ET super- variant super- variant sub- variant f0mod a0 + a1 tropical noise 3.5 3.0 V-wind U-wind Temperature 2.5 2.0 1.5 perturbation amplitude perturbation amplitude perturbation amplitude 1.0 0.5 0.0 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 3.5 3.5 3.0 3.0 2.5 2.5 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 20N 1.6 1.6 EQ 1.4 1.4 1.2 1.2 20S 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 3.5 U-wind V-wind Temperature 3.0 2.5 2.0 Ratio of analysis error variance (ET / NAVDAS) Ratio of analysis error variance (ET / NAVDAS) Ratio of analysis error variance (ET / NAVDAS) 1.5 1.0 0.5 0.0 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 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, 2National Research Council, Washington, D.C., USA authors’ emails: mclay@nrlmry.navy.mil reynolds@nrlmry.navy.mil bishop@nrlmry.navy.mil • 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. IV. IMPROVING ET OPERATION - ALTERNATIVES • ET IMPLEMENTATION – RESULTS 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. U-wind V-wind Temperature 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 • 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: difference in forecast error variance (ET – observed) difference in forecast error variance (ET – observed) difference in forecast error variance (ET – observed) 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. where Xa = n x m matrix of analysis perturbations from current cycle m = number of ensemble members Xf = n x m matrix of forecast perturbations from previous cycle n = dimension of model state vector Pa = 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 Figure 3, left, shows the sequence of operations in the variance augmentation process. Synthetic tropical variance is added to existing ET-ensemble forecast perturbations f0, resulting in modified forecast perturbations f0mod with enhanced tropical variance (a). The modified forecast perturbations f0mod, when input into the ET scheme (b), will result in analysis perturbations a1 with increased variance in the tropics relative to that of analysis perturbations a0 obtained from un-augmented forecast perturbations (c). Furthermore, because of the global constraint involved in the ET scheme, the analysis perturbations a1will have relatively less variance outside the tropics. When the analysis perturbations a1 are integrated in the NOGAPS model (d), they should yield forecast perturbations f1 with more tropical variance and less extratropical variance than the forecast perturbations f0 (e). Parts (a) - (e) of this process are repeated over the next analysis cycle, starting with the forecast perturbations f1 in place of the forecast perturbations f0. In this way, the variance curves of the forecast and analysis perturbations fi and ai, respectively, can be gradually flattened over a sequence of analysis cycles. • 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. f0 f0mod f1 a1 f1 (a) (b) (c) (d) (e) VII. STRUCTURE OF SYNTHESIZED TROPICAL PERTURBATIONS 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. U-wind U-wind • VI. SYNTHESIZING TROPICAL VARIANCE - PROCEDURE • Compute the ET-ensemble’s tropical forecast error variance deficit matrix, : • Seek perturbations such that The steps are: Ratio of analysis error variance (ET / NAVDAS) difference in forecast error variance (ET – observed) 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N 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. V-wind V-wind Ratio of analysis error variance (ET / NAVDAS) difference in forecast error variance (ET – observed) 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. • 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 . 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N Temperature Temperature ETKF TARGET S. MAJUMDAR difference in forecast error variance (ET – observed) Ratio of analysis error variance (ET / NAVDAS) a) b) 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. 90S 60S 30S EQ 30N 60N 90N 90S 60S 30S EQ 30N 60N 90N IX. SUMMARY A technique to augment tropical forecast variance is devised for use in conjunction with the ensemble transform analysis perturbationscheme. 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. 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. 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.