The Structure of Background-error Covariance in a Four-dimensional Variational Data Assimilation System: Single-point Experiment

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1 ADVANCES IN ATMOSPHERIC SCIENCES, VOL. 27, NO. 6, 2010, The Structure of Background-error Covariance in a Four-dimensional Variational Data Assimilation System: Single-point Experiment LIU Juanjuan 1 ( ), WANG Bin 1 ( ), and WANG Shudong 1,2 ( ) 1 State Key Laboratory of Numerical Modeling for Atmospheric Sciences & Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing Graduate University of the Chinese Academy of Sciences, Beijing (Received 19 March 2009 revised 29 November 2009) ABSTRACT A four dimensional variational data assimilation (4DVar) based on a dimension-reduced projection (DRP- 4DVar) has been developed as a hybrid of the 4DVar and Ensemble Kalman filter (EnKF) concepts. Its good flow-dependent features are demonstrated in single-point experiments through comparisons with adjointbased 4DVar and three-dimensional variational data (3DVar) assimilations using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model (MM5). The results reveal that DRP-4DVar can reasonably generate a background error covariance matrix (simply B-matrix) during the assimilation window from an initial estimation using a number of initial condition dependent historical forecast samples. In contrast, flow-dependence in the B-matrix of MM5 4DVar is barely detectable. It is argued that use of diagonal estimation in the B-matrix of the MM5 4DVar method at the initial time leads to this failure. The experiments also show that the increments produced by DRP-4DVar are anisotropic and no longer symmetric with respect to observation location due to the effects of the weather trends captured in its B-matrix. This differs from the MM5 3DVar which does not consider the influence of heterogeneous forcing on the correlation structure of the B-matrix, a condition that is realistic for many situations. Thus, the MM5 3DVar assimilation could only present an isotropic and homogeneous structure in its increments. Key words: DRP-4DVar, data assimilation, flow dependence, single-point experiment Citation: Liu, J. J., B. Wang, and S. D. Wang, 2010: The structure of background-error covariance in a four-dimensional variational data assimilation system: single-point experiment. Adv. Atmos. Sci., 27(6), , doi: /s Introduction With the development of the global observing system, as well as advances in our computational ability, data assimilation has enjoyed rapid progress in recent years. Considerable attention has been paid to four-dimensional variational data assimilation (4DVar; Courtier et al., 1994; Wang et al., 2000; Xiao et al., 2000) and the Ensemble Kalman Filter (EnKF; Evensen, 1994) because both methods try to address the background error covariance (hereafter B-matrix) through an implicit (4DVar) or an explicit (EnKF) description of flow-dependent forecast error structures (Gustafsson, 2007). There is no doubt that characterization of the B-matrix is the key to the success in data assimilation. There are many approaches for evaluating the B-matrix. For example, the NMC (National Meteorological Center) method (Parrish and Derber, 1992), which is usually used in variational data assimilation, has a known disadvantage in that this method is constrained to have homogeneous and isotropic correlations among variables (Daley, 1991). Some approaches have been explored to improve the influence of the B-matrix, with the EnKF (Evensen, 1994) using Corresponding author: WANG Bin, wab@lasg.iap.ac.cn China National Committee for International Association of Meteorology and Atmospheric Sciences (IAMAS), Institute of Atmospheric Physics (IAP) and Science Press and Springer-Verlag Berlin Heidelberg 2010

2 1304 THE BACKGROUND-ERROR COVARIANCE IN DRP-4DVAR VOL. 27 ensemble forecast statistics to produce flow-dependent B-matrices. By defining the analysis variables in grid space and using recursive filters to define the background error, Derber et al. (2003) obtained inhomogeneous anisotropic background errors. Hamill and Snyder (2000) presented a prototype hybrid 3DVar ensemble Kalman filter. Just as mentioned above, the B-matrix plays an important role in data assimilation methods, but some approaches have various pros and cons 4DVar can assimilate a growing number of various asynchronous observations, but requires tremendous amounts of programming to obtain the adjoint model and limits the B-matrix to flow-dependency over the assimilation window; the EnKF enables the B-matrix to be flowdependent globally, and requires less effort in terms of coding, but has not been available for some types of observations, and incorporated information gradually. Recently, several researchers have proposed a variety of hybrid methods (Lorenc, 2003; Hunt et al., 2004; Liu et al., 2008, 2009; Zhang et al., 2009; Wang et al., 2010). The main thrust behind the hybrid methods is that the ensemble-based background error covariance statistics are provided to the variational data assimilation. Despite the many differences between the various ensemble-based algorithms, all are comprised of a finite number of samples and short-range forecast cycles (Hamill, 2000). The DRP-4DVar approach (Wang et al., 2010) is one of these various hybrid schemes, and its B-matrix is modeled using the ensemble of forecasts, these ensembles are produced by the historical time series of model forecasts, so ensure it to flow dependency globally. Thépaut et al. (1993) showed how complex a forecast error covariance could be when it evolved according to the primitive equations. It is not surprising that the construction of the B-matrix has attracted a lot of attention, with considerable effort being put into designing and coding (Fisher, 2001). This paper will study the B-matrix character within DRP-4DVar. However, with the matrices either having very complicated structure or being too large to fit into computer memory, some simplifications had to be made. Thépaut et al. (1996) presented a study of the structure functions implied in 4DVar with baroclinic development in the Pacific Ocean. Lorenc (2003) discussed how 4DVar summarizes the B-matrix within a short time-window using a linear perturbation model and its adjoint. Huang et al. (2009) presented results from standard single observation test experiments to illustrate the flow-dependent nature of 4D-Var analysis increments. Following the methodology of Thépaut et al. (1996), this study exhibits the flow dependence of the B-matrix in DRP-4DVar through singlepoint experiments under a realistic meteorological situation with the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model (MM5; Grell et al., 1994). A brief introduction to the methodology is presented in section 2. Single-observation experiments are addressed in section 3, which also shows the flowdependent structure through assimilation of various types of observations and a comparison of the results with those from MM5 4DVar and MM5 3DVar. Finally, a summary and discussion are provided in section Methodology The construction of the B-matrix is extremely important to the performance of an assimilation method. Many approaches have been proposed and evaluated in the literature. For example, MM5 3DVar used the NMC method (Parrish and Derber, 1992); EnKF used an ensemble method to ensure the B-matrix s global flow-dependence. The main idea of DRP-4DVar is to combine 4DVar and EnKF to produce a superior hybrid approach. In order to benefit from using the statedependent uncertainty provided by EnKF, Wang et al. (2010) used initial perturbation samples to estimate the B-matrix as in EnKF: B b b T 1 b = (x 1 x, x 2 x,, x m x ) m 1. x = 1 m (x 1 + x x m) (1) Here, the initial perturbation samples x i can be the 72-h forecasts available with 24 and 48 hour lead time prior to analysis time using MM5; thus, the dimension of the B-matrix is enormous, typically This size renders it prohibitive to generate complete information about the temporal evolution and spatial distribution of the B-matrix. To allow it to work, we are forced to simplify the matrix to a more compact form. Single point observation experiments will be used in the following methodology. Following Lorenc (1986) and Thépaut et al. (1993), in 4DVar, the analysis increment of a single point experiment can be written as: M (x a (t 0 ) x b (t 0 )) = x a (t i ) x b (t i ). (2) This denotes the increment trajectory, which is the difference between the analysis trajectory and the background trajectory. According to Fisher (2001), if the forecast model is linear and perfect, the covariance

3 NO. 6 LIU ET AL matrix for background errors at time t i is ε b (t i )(ε b (t i )) T = M t0 t i ε b (t 0 )(ε b (t 0 )) T M T t 0 t i = M t0 t i BM T t 0 t i. (3) investigate how the B-matrix in 4DVar evolves with respect to atmospheric state through the increment trajectories in single-point assimilation experiments. 3. Single-point assimilation experiments A group of single-point assimilation experiments are designed to examine the implicit evolution of the B-matrix in DRP-4DVar. The 6-h accumulated rainfall observation is chosen. These kinds of observations are difficult to assimilate directly by sequential methods, such as 3DVar and EnKF. In addition, in order to compare the influence of the B-matrix on the analysis increment both in DRP-4DVar and 3DVar, we design another group of experiments for assimilating temperature observations directly. All assimilation experiments are conducted using the MM5 as the forecast model, which contains a full set of sub-grid scale physical parameterizations [see Grell et al. (1994) for details]. In this study, the model has 30-km horizontal resolution over a gridpoint domain which covers most of China and has 24 vertical layers from σ=0 to σ=1. The physics packages used are Dudhia s simple ice scheme, the Anthes- Kuo cumulus parameterizations scheme, the Hong- Pan planetary boundary layer, and a Cloud-Radiation scheme [see Grell et al. (1994) for details]. We apply a 6-h assimilation window from 0000 to 0600 UTC 14 June The accumulated rainfall observation at the end of the assimilation window is located at (24.4 N, E), and the temperature observation on σ = is from the same location. The background field is obtained from National Centers for Environmental Prediction (NCEP) Final (FNL) Global Tropospheric Analyses. More information on the two groups of experiments is given in Table 1. Figure 1 shows a significant descent of the cost function and the norm of the gradient in experiment Adjoint R, which demonstrates convergence in the minimization process and successful fulfillment of the adjoint-based 4Dvar process. Meanwhile, in the counterpart experiment DRP R the observational cost function shows a steady decrease from 3021 to 970, which is about a 75% reduction, and the gradient is equal to zero exactly in the reduced space. Note that the adjoint-based 4DVar reduces its cost function much more than the DRP-4DVar does, which means Fig. 1. (a) Cost function J and (b) norm of the gradient G (logarithmic coordinates with base of 10 with respect to the number of iteration in Adjoint R. the analysis in the adjoint-based 4DVar includes very little information from the background. This may be due to the overestimation of the B-matrix by the incomplete and diagonal B-matrix in the MM5 4DVar system. It deleterious to the performance that the Adjoint R diagonal B-matrix is used at the initial time. As discussed in section 2, the increment trajectories from a single-point 4DVar experiment can represent the flow dependence of the B-matrix. Therefore, we show the increments of geopotential height at 500 hpa at three different times in the assimilation window in Figs. 2 and 3 to demonstrate the trajectories of DRP R and Adjoint R, respectively. For both experiments, the increments are anisotropic and are no longer symmetric with respect to the observation location. They are larger in DRP R than in Adjoint R. An easily apparent feature is that the increment trajectories match well to the evolution of the background in experiment DRP R (see Fig. 2). The increments remain concentrated around a trough in the 500-hPa height field, which corresponds to a primary rainfall area in Fig. 2d, and then there is movement eastward following the trough. These features clearly denote the flow dependence of the B-matrix in the DRP-4Dvar

4 1306 THE BACKGROUND-ERROR COVARIANCE IN DRP-4DVAR VOL. 27 Table 1. The setup of the two groups of experiments. Name Assimilation system Observation data Group 1 DRP R DRP 4DVar 6-h accumulated rainfall Adjoint R Adjoint based 4DVar 6-h accumulated rainfall Group 2 DRP T DRP 4DVar Temperature Adjoint T Adjoint-based 4DVar Temperature 3DVar T 3DVar Temperature Fig. 2. The dashed line represent analysis increments (analysis-background) of experiment DRP R for geopotential height at 500 hpa at (a) 0000, (b) 0300, and (c) 0600 UTC 14 June 2002, during which a single rainfall observation was taken at the location (24.4 N, E). Contours are for the background geopotential height. (d) 6-h accumulated rainfall. The contour interval is 10 hpa in (a), (b), and (c) and 5 mm in (d). The symbol + marks the location of the observation. case. In experiment Adjoint R presented above, however, the characteristics of the flow dependence of the B-matrix in MM5 4DVar can be hardly be detected. As we know, 4DVar can propagate B-matrix information during model integration; if this is the case, what causes the failure of MM5 4DVar in terms of the expected propagation? The initial structure of the B- matrix at the initial time might be the main reason. The MM5 4DVar uses a diagonal B-matrix that only considers the variance. This very simple structure of the B-matrix at the initial time leads to a very small possibility of development of covariance along the non-diagonals during the short time of the assimilation. Given that a forecast is sensitive to the initial conditions, it is likely that the initial structure functions are an important element in the evolution of the B-matrix. Some single-point WRF (Weather Research and Forecasting model) 4DVar experiments by Huang et al. (2009) showed that 4DVar is able to implicitly specify flow-dependent covariance functions when a 3DVar B-matrix with statistical information in both the diagonal and some non-diagonals is used in the experiments. The success of DRP-4DVar in describing the flow dependence is owed to the ensemble statistics of historical prediction samples contained in its B-matrix at the initial time, according to Houtekamer et al. (1996) and Fisher (1999). Because the selection of these historical prediction samples is closely related to the background states, which are often 72-h predictions started 24-h and 48-h before the initial time of the assimilation, DRP-4DVar can even continuously propagate the B-matrix from one assimilation window to the next, while other available 4DVar systems generally use the same, fixed B-matrix at the

5 NO. 6 LIU ET AL N 33N (a) N 29N 27N 25N N 100E 104E 108E 112E 116E 120E 35N (b) 33N + 31N 29N 27N N 23N 100E 104E 108E 112E 116E 120E 35N (c) 33N 31N N N 25N N E 104E 108E 112E 116E 120E Fig. 3. Same as in Figs. 2a c, except for experiment Adjoint R. The symbol + marks the location of the observation. initial time for all assimilation windows. This is clearly an advantage that other available 4DVar systems do not have, although the covariance calculated in DRP- 4DVar is still imperfect and suffers from correlations between the samples. Further improvement of B will be achieved by inclusion of an analog prediction sample in which the corresponding simulated observation increment is highly correlated with the real observation increment. The analysis increment from the assimilation of a single observation provides a partial view of the background error covariance. To compare the covariance from the NMC method (Parrish and Derber, 1992) Fig. 4. Analysis increments (analysis-background) of temperature (K; shaded): (a) DRP T, (b) 3DVar T, corresponding to a temperature observation located at the 500- hpa level (24.4 N, E) at the end of the assimilation window. The background temperature is shown in dark contours, with a contour interval of 1 K. with that estimated from historical forecast samples, we include only one temperature observation located at the σ=0.525 level (24.4 N, E) at the end of the assimilation window in the second group of experiments. Figures 4 through 6 depict the increments at 500 hpa computed from assimilating a single temperature of 1.9 K greater than the background temperature. From Figs. 4b, 5b, and 6b, it is evident that when the background error covariance is generated using the NMC method, the temperature increments are nearly isotropic with respect to the observation location, and the wind increments are smaller north of 24 N and nearly zero at the location of the temperature observation. The results are clearly unaffected by the local meteorological conditions. For these reasons, the NMC method in 3DVar T (Parrish and Derber, 1992) is constrained to have homogeneous and isotropic correlations among the variables. Although it can easily be checked for past periods, this assumption can also be justified to some extent on a theoretical basis. However, there is evidence that such correlations are not always appropriate, according to Buehner (2005). Obviously, the influence of heterogeneous forcing on the correlation structure of background errors that are realistic for many situations ca-

6 1308 THE BACKGROUND-ERROR COVARIANCE IN DRP-4DVAR Fig. 5. Same as in Fig. 4, except for zonal wind. The background zonal wind is shown in dark contours, with a contour interval of 2 m s 1. trend can then be captured (Figs. 4a, 5a, and 6a). The increments are now anisotropic and no longer symmetric with respect to the observation location. The temperature increment is trisected and elongated along the temperature contour. In addition, the single temperature observation leads to balanced temperature and wind analysis increments. However, the temperature-wind balance at t0 is the result of using a background covariance B based on a balanced relationship, which is assumed to be uncorrelated between variables, and with the correlations between the fullanalysis variables modeled with a linear balance operator in the 3DVar. This results in the spatial propagation of the information within DRP T and Adjoint T (not shown) being significantly different from that in 3DVar since the influence of the data is extended in nonisotropic patterns in 4DVar, as mentioned above. At 500 hpa, positive temperature increments (Fig. 6a) follow the pattern of the trough fairly closely, and negative responses follow the pattern of the ridge, which shows a dynamic consistency. This is the advantage when the model constraint replaces the simple balance relationship. 4. Fig. 6. Same as in Fig. 4, except for meridional wind. The background meridional wind is shown in dark contours, with a contour interval of 2 m s 1. nnot be illustrated by using the NMC method. To avoid these constraints, a different approach is to estimate the background error covariance by an ensemble method, whereby the effects of the weather VOL. 27 Summary and discussion In this study, we exhibit the structure of the Bmatrix for DRP-4DVar using a series of single-point experiments. DRP-4DVar has an important characteristic: the increment trajectory is well matched to the evolution of the background state. It means the B-matrix of DRP-4DVar is implicitly flow-dependent. But in experiment Adjoint R, the character of the flow dependence of the B-matrix is hardly detectable. The initial structure of the B-matrix at the start time of the assimilation window might be the main reason for the difference, as the MM5 4DVar only uses a diagonal B-matrix for describing the variance. This very simple structure of the B-matrix at the initial time leads to a very small possibility for development of covariance along the non-diagonals in the assimilation during the short time window. Given that a forecast is sensitive to the initial conditions, it is likely that the initial structure functions are an important element in the evolution of the B-matrix. So, there is no doubt that an ensemble could be used to provide flow-dependent background error covariance in 4DVar with no extra computational cost, which will be the subject of future research (Buehner, 2005; Zhang and Snyder, 2007). It is common to consider B from the NMC method Parrish and Derber (1992) to be static in 3DVar, allowing it to describe only the average characteristics. While this assumption can be justified to some ex-

7 NO. 6 LIU ET AL tent on a theoretical basis, there is evidence that such correlations are not always appropriate. Obviously, the influence of heterogeneous forcing on the correlation structure of background errors that are realistic for many situations cannot be illustrated. Our study shows that DRP-4DVar produces much more complex increments than 3DVar and that the influence of the data results in nonisotropic patterns in DRP- 4DVar, which should be considered as an improvement. Since the effects of weather trends can be captured by DRP 4DVar, the increments are anisotropic and no longer symmetric with respect to the observation location. The temperature increment is trisected and elongated along the temperature contour. The main reason for these differences is that the B-matrix in DRP 4DVar is estimated by the ensemble sample. Since the B-matrix is obtained from statistical ensemble that is based on a number of initial conditiondependent historical forecast samples, the background information can be taken as being represented in a continuous propagation from one analysis cycle to the next, although the covariances calculated here are imperfect, suffering from the correlation of samples. Further improvement of the B-matrix will be achieved in the future. Acknowledgements. We acknowledge the Ministry of Science and Technology of China (Grant No. 2006BAC03B01), and the Ministry of Science and Technology of China for funding the 973 project (Grant No. 2005CB321703). REFERENCES Buehner, M., 2005: Ensemble-derived stationary and flow-dependent background error covariances: Evaluation in a quasi-operational setting for NWP setting. Quart. J. Roy. Meteor. Soc., 131, Courtier, P., J.-N. Thépaut, and A. Hollingsworth, 1994: A strategy for operational implementation of 4D-Var using an incremental approach. Quart. J. Roy. Meteor. Soc., 120, Daley R., 1991: Atmospheric Data Analysis. Cambridge University Press, UK., 472pp. Derber, J. C., R. J. Purser, W.-S. Wu, R. Treadon, M. Pondeca, D. Parrish, and D. Kleist, 2003: Flow dependent Jb in a global grid-point 3D-var. Proc. Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, Reading, United Kingdom, ECMWF, Evensen, G., 1994: Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res., 99(C5), , doi: /94JC Fisher, M., 1999: Background error statistics derived from an ensemble of analyses. ECMWF Research Department Technical Memorandum, No. 79, London, 12pp. Fisher, M., 2001: Assimilation techniques (3): 3dVar, April European Centre for Medium-Range Weather Forecasts, London, 11pp. Grell, G. A., J. Dudhia, and D. R. Stauffer, 1994: A description of the fifth-generation Penn State/NCAR Mesoscale Model (MM5). Tech Note NCAR/TN- 3981STR, Boulder, 122pp. Gustafsson, N. 2007: Discussion on 4D-Var or EnKF?. Tellus, 59A, Hamill, T. M., and C. Snyder, 2000: A hybrid ensemble Kalman filter 3D variational analysis scheme. Mon. Wea. Rev., 128, Hamill, T. M., 2002: Ensemble-based data assimilation. Proc. Workshop on Predictability, Reading, United Kingdom, ECMWF, Houtekamer, P. L., L. Lefaivre, J. Derome, H. Ritchie, and H. L. Mitchell, 1996: A system Simulation Approach to Ensemble Prediction. Mon. Wea. Rev., 124, Huang, X.-Y., and Coauthors, 2009: Four-dimensional variational data assimilation for WRF: Formulation and preliminary results. Mon. Wea. Rev., 137, Hunt, B. R., and Coauthors, 2004: Four-dimensional ensemble Kalman filtering. Tellus, 56A, Liu, C. S., Q. N. Xiao, and B. Wang, 2009: An Ensemblebased four-dimensional variational data assimilation scheme: Part II: Observing system simulation experiments with the Advanced Research WRF (ARW). Mon. Wea. Rev., 137, Liu, C. S., Q. N. Xiao, and B. Wang, 2008: An Ensemblebased four-dimensional variational data assimilation scheme: Part I: Technical formulation and preliminary test. Mon. Wea. Rev., 136, Lorenc, A. C., 1986: Analysis methods for numerical weather prediction. Quart. J. Roy. Meteor. Soc., 112, Lorenc, A. C., 2003: Modeling of error covariances by 4D-Var variational data assimilation. Quart. J. Roy. Meteor. Soc., 129, Parrish, D. F., and J. C. Derber, 1992: The National Meteorological Center s spectral statistical interpolation analysis system. Mon. Wea. Rev., 120, Thépaut, J.-N., R. Hoffman, and P. Courtier, 1993: Interaction of dynamics and observations in a four dimensional variational assimilation. Mon. Wea. Rev., 121, Thépaut, J.-N., P. Courtier, G. Belaud, and G. Lemaitre, 1996: Dynamical structure functions in a fourdimensional variational assimilation: A case study. Quart. J. Roy. Meteor. Soc., 122, Wang, B., X. Zou, and J. Zhu, 2000: Data assimilation and its application. Proceedings of the National Academy of Sciences, USA, 97, Wang, B., J.-J. Liu, S. D. Wang, W. Cheng, J. Liu, C. S. Liu, Q. N. Xiao, and Y.-H. Kuo, 2010: An economical approach to four-dimensional variational

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