Data Assimilation: Formulation and Single-Resolution Experiments with Real Data for NCEP Global Forecast System

SEPTEMBER 2014 W A N G A N D L E I 3303

GSI-Based Four-Dimensional Ensemble–Variational (4DEnsVar) Data Assimilation: Formulation and Single-Resolution Experiments with Real Data for NCEP Global Forecast System

XUGUANG WANG AND TING LEI School of Meteorology, University of Oklahoma, and Center for Analysis and Prediction of Storms, Norman, Oklahoma

(Manuscript received 26 September 2013, in ﬁnal form 15 April 2014)

ABSTRACT

A four-dimensional (4D) ensemble–variational data assimilation (DA) system (4DEnsVar) was developed, building upon the infrastructure of the gridpoint statistical interpolation (GSI)-based hybrid DA system. 4DEnsVar used ensemble perturbations valid at multiple time periods throughout the DA window to estimate 4D error covariances during the variational minimization, avoiding the tangent linear and adjoint of the forecast model. The formulation of its implementation in GSI was described. The performance of the system was investigated by evaluating the global forecasts and hurricane track forecasts produced by the NCEP Global Forecast System (GFS) during the 5-week summer period assimilating operational conventional and satellite data. The newly developed system was used to address a few questions regarding 4DEnsVar. 4DEnsVar in general improved upon its 3D counterpart, 3DEnsVar. At short lead times, the improvement over the Northern Hemisphere extratropics was similar to that over the Southern Hemisphere extratropics. At longer lead times, 4DEnsVar showed more improvement in the Southern Hemisphere than in the Northern Hemisphere. The 4DEnsVar showed less impact over the tropics. The track forecasts of 16 tropical cyclones initialized by 4DEnsVar were more accurate than 3DEnsVar after 1-day forecast lead times. The analysis generated by 4DEnsVar was more balanced than 3DEnsVar. Case studies showed that increments from 4DEnsVar using more frequent ensemble perturbations approximated the increments from direct, nonlinear model propagation better than using less frequent ensemble perturbations. Consistently, the performance of 4DEnsVar including both the forecast accuracy and the balances of analyses was in general degraded when less frequent ensemble perturbations were used. The tangent linear normal mode constraint had positive impact for global forecast but negative impact for TC track forecasts.

1. Introduction algorithms using simple models and simulated observations (e.g., Hamill and Snyder 2000; Lorenc 2003; Data assimilation systems that bridge the gap between Etherton and Bishop 2004; Zupanski 2005; Wang et al. two traditionally parallel variational and ensemble-based 2007a,b, 2009; Liu et al. 2008; Wang 2010; Wang et al. methods have gained increasing interests recently in 2010). More recently, this method has been imple- both the research and operational numerical weather mented and successfully tested for both regional (e.g., prediction (NWP) communities. Instead of using a typi- Wang et al. 2008a,b; Wang 2011; Zhang and Zhang cally static covariance, the background error covari- 2012; Barker et al. 2012; Li et al. 2012) and global NWP ances in the variational system (Var) are estimated flow models (e.g., Buehner 2005; Buehner et al. 2010a,b; dependently from an ensemble of background states. Bishop and Hodyss 2011; Clayton et al. 2013; Wang Such ensemble background states are typically produced et al. 2013; Buehner et al. 2013). These studies suggest by the ensemble Kalman filter (EnKF) or its simplified that the coupled ensemble-variational system can le- variants. Early studies on such coupled data assimila- verage the strengths of both a stand-alone EnKF and tion include proposing, testing, and demonstrating new a stand-alone Var system, producing an analysis that can be better than either system. The potential advan- tages of the coupled ensemble-variational (EnsVar) data Corresponding author address: Dr. Xuguang Wang, School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., assimilation as compared to a stand-alone Var and EnKF Norman, OK 73072. are discussed in Wang (2010). Briefly, compared to a E-mail: [email protected] stand-alone Var, EnsVar can benefit from flow-dependent

DOI: 10.1175/MWR-D-13-00303.1

Ó 2014 American Meteorological Society Unauthenticated | Downloaded 10/06/21 01:01 PM UTC 3304 MONTHLY WEATHER REVIEW VOLUME 142 ensemble covariance as EnKF. Compared to a stand- variational system. The 3DEnsVar hybrid system was alone EnKF, EnsVar can be more robust for small en- implemented operationally for global numerical pre- semble size or large model errors (e.g., Hamill and diction at NCEP beginning in May 2012. Snyder 2000; Etherton and Bishop 2004; Wang et al. The current GSI-based 3DEnsVar and 3DEnsVar 2007a, 2009; Buehner et al. 2010b), benefit from dy- hybrid did not account for the temporal evolution of namic constraints during the variational minimization the error covariance within the assimilation window. A (e.g., Wang et al. 2013), and take advantage of the es- GSI-based 4DVar data assimilation (DA) system where tablished capabilities such as the variational data quality the innovation is propagated in time using a tangent control and outer loops to treat nonlinearity in Var. linear and adjoint (TLA) of the forecast model is being Although in theory EnKF can adopt model space lo- developed. However, efforts are needed to improve the calization, to save computational costs, EnKF often computational efficiency of the TLA model before sys- adopts serial observation or batch observation process- tematic tests can be conducted (Rancic et al. 2012). In ing algorithms (Houtekamer and Mitchell 2001) and the this study, an alternative method to account for the covariance localization is often conducted in observa- temporal evolution of the error covariance within the tion space. In EnsVar, ensemble covariance localization GSI system was implemented. In this method, the en- is often conducted in model space rather than observa- semble perturbations valid at multiple time periods tion space, which may be more appropriate for obser- within the DA window are used during the variational vations without explicit position (e.g., Campbell et al. minimization. Effectively, the four-dimensional (4D) 2010). Motivated by these earlier studies, several oper- background error covariance was estimated by the ational NWP centers in the world have implemented or ensembles, avoiding the need of the TLA of the are implementing the ensemble-variational data assimi- forecast model. Hereafter, the method is referred as lation system operationally (e.g., Buehner et al. 2010a,b; 4D-Ensemble-Var (4DEnsVar). Clayton et al. 2013; Wang et al. 2013; Kuhl et al. 2013). Incorporating the ensemble perturbations spanning Although the ensemble-variational data assimilation the DA window in the variational framework to avoid system documented in these studies share the same the TLA model have been proposed and implemented spirit to incorporate flow-dependent ensemble covari- in different ways in early studies. Qiu et al. (2007), Tian ance into variational systems, the specific implementa- et al. (2008), and Wang et al. (2010) proposed methods tion can be different in several aspects. Such differences to reduce the dimension of the problem by reducing the include if the ensemble covariance is incorporated into ensemble perturbations produced by the Monte Carlo a three-dimensional variational data assimilation system methods or historical samples to a set of base vectors (3DVar) or a four-dimensional variational data assimi- during the variational minimization. Liu et al. (2008, lation system (4DVar); if the background covariance is 2009) implemented the method in a one-dimensional fully or partially replaced by the ensemble covariance shallow-water model by directly ingesting the ensem- and if the tangent linear adjoint of the forecast model ble perturbations and then increased the size of the was used in the four-dimensional variational minimi- ensemble perturbation by applying the covariance lo- zation. Appendix A summarizes various flavors. In this calization matrix outside the variational minimization study the abbreviation of the names of various ensemble- to alleviate the sampling error issue associated with a variational data assimilation experiments follow those limited ensemble size. Buehner et al. (2010a,b) im- defined in appendix A. plemented the method to the Meteorological Service A 3DVar-based ensemble-variational (3DEnsVar) of Canada’s operational data assimilation system where hybrid data assimilation system was recently developed covariance localization was adopted within the varia- based on the gridpoint statistical interpolation (GSI) tional minimization following Buehner (2005), and sys- data assimilation system operational at the National tematically compared it with their EnKF and 4DVar. Centers for Environmental Prediction (NCEP), and was Bishop and Hodyss (2011) implemented 4DEnsVar to first tested for the Global Forecast System (GFS). It the Naval Research Laboratory (NRL) 4DVar system was found that the new 3DEnsVar hybrid system pro- called Atmospheric Variational Data Assimilation System– duced more accurate forecasts than the operational GSI Accelerated Representer (NAVDAS-AR; Xu et al. 3DVar system for both the general global forecasts 2005) and proposed and tested an adaptive covariance (Wang et al. 2013) and the hurricane forecasts (Hamill localization method in the context of 4DEnsVar using et al. 2011). Wang et al. (2013) also found that GSI- a single case study. It is noted that the model-space based 3DEnsVar without inclusion of the static co- 4DEnsVar algorithm is a natural extension of earlier variance outperformed GSI-based EnKF as a result of proposed 3DEnsVar (Lorenc 2003; Buehner 2005; Wang the use of tangent linear normal mode constraint in the et al. 2007b, 2008a; Wang 2010). One critical component

Unauthenticated | Downloaded 10/06/21 01:01 PM UTC SEPTEMBER 2014 W A N G A N D L E I 3305 in these algorithms is to incorporate ensemble covari- the convergence rates of the minimization? These ques- ances in the variational minimization through augment- tions will be addressed in a real data context where op- ing the control variables. In 3DEnsVar, the ensemble erational observations from NCEP are assimilated. perturbation at a single time period (e.g., the center of The resolution of the operational implementation of the assimilation window) is used whereas in 4DEnsVar, the 3DEnsVar hybrid at NCEP is T254 (triangular ensemble perturbations at multiple time periods span- truncation at total wavenumber 254) for the ensemble ning the assimilation window are used. and T574 for the variational analysis. T. Lei and The 4DEnsVar implemented within Meteorological X. Wang (2014, unpublished manuscript) found that with Service of Canada’s 4DVar system (Buehner et al. this dual-resolution configuration, including the static 2010a) takes the model-space-based minimization for- covariance (i.e., 3DEnsVar hybrid defined in Table A1 mula with the variational minimization preconditioned in appendix A) significantly improved the performance upon the square root of the background error covari- compared to without including the static covariance. ance. 4DEnsVar implemented within NAVDAS-AR is Therefore, in the operational implementation, 3DEnsVar based on the observation-space minimization formula hybrid was adopted. Here we present the evaluation (Bishop and Hodyss 2011). Different from these sys- results and address the aforementioned questions using tems, operational GSI minimization is preconditioned experiments conducted at a reduced spectral resolution upon the full background error covariance matrix of T190 for both the ensemble and the variational anal- (Derber and Rosati 1989). Therefore, the formulation yses (hereafter single-resolution experiments). Wang for the implementation of 4DEnsVar where the mini- et al. (2013) compared the GSI-based 3DEnsVar and mization is preconditioned upon the full background 3DEnsVar hybrid using the same single-resolution con- error covariance is described in this paper. The perfor- figuration and an 80-member ensemble. It was found that mance of the newly developed GSI-based 4DEnsVar the inclusion of the static covariance component in the system is evaluated by comparing it with GSI-based background-error covariance did not improve the fore- 3DVar and 3DEnsVar. In addition to examining the cast skills beyond using the full ensemble covariance as performance of the system for general global forecasts, the background-error covariance. Given this result and the performance of the 4DEnsVar system is studied that the current study represents a first step of testing for hurricane track forecasts for the first time. Using the newly extended system using real data, this study the newly developed GSI-based 4DEnsVar system, focuses on the impact of 4D extension of the ensemble a few other questions were investigated. As far as the covariance in a single-resolution configuration and with- authors are aware, these questions have not been docu- out involving the static covariance. This single-resolution mented in previously published studies on 4DEnsVar configuration is different from Buehner et al. (2010b) in real data context. In 4DEnsVar, temporal evolution where the ensemble was run at a reduced resolution as of the error covariance is approximated by the co- compared to the variational analysis (termed as dual- variances of ensemble perturbations at discrete times. resolution experiments). One method to include the How is the performance of 4DEnsVar dependent on static covariance in 4DEnsVar without involving the the temporal resolution of or the number of time pe- TLA is proposed in Buehner et al. (2013). In this method, riods of ensemble perturbations? In 4DEnsVar, the the same static background error covariance is used for temporal propagation through covariance of ensemble all time periods (e.g., Buehner et al. 2013). Investigation perturbations contains linear assumption. How is the of the impact of including the static covariance using such linear approximation compared to the full nonlinear a method in GSI-based 4DEnsVar (i.e., 4DEnsVar hy- model propagation? Will using 4D ensemble covari- brid) is ongoing and will be documented in future papers ances to fit the model trajectory to observations dis- (D. Kleist 2013, personal communication). tributed within a finite assimilation window improve The rest of the paper is organized as followed. Section 2 the balance of the analysis and how is the balance of the and appendix B describe the formulations and im- analysis dependent on the temporal resolution of the plementation of 4DEnsVar within the GSI. Section 3 ensemble perturbations? 4DEnsVar is implemented describes the design of experiments. Section 4 discusses such that the tangent linear normal mode constraint the experiment results and section 5 concludes the paper. (TLNMC; Kleist et al. 2009)withintheGSIisallowed. What is the impact of such balance constraint on the 2. GSI-based 4DEnsVar formulation and 4DEnsVar analysis and forecast and how is that depen- implementation dent on different types of forecasts such as the general global forecast or hurricane track forecasts? How does Specific formulation of implementing 4DEnsVar including multiple time periods of perturbations impact within GSI is given in this section and appendix B.

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Different from other variational systems, the minimi- In the current implementation, each ak, k 5 1, ..., K,is zation in the operational GSI is preconditioned upon a three-dimensional field collocated at the model grid the full background error covariance matrix. Wang points. Each ak varies in both the horizontal and vertical (2010) describes the mathematical details on how the direction so that spatial localization is applied both hor- ensemble covariance is incorporated in the GSI 3DVar izontally and vertically. The same three-dimensional through the use of the augmented control vectors (ACV) fields of ak are applied for all variables and all time using such preconditioning method. As shown in Wang periods. Therefore, in the current implementation of et al. (2007b), effectively, the static covariance in GSI 4DEnsVar, the number of augmented control variables 3DVar was replaced by and linearly combined with the used is the same as that used in 3DEnsVar. For the im- e ensemble covariance in 3DEnsVar and 3DEnsVar plementation of 4DEnsVar with GFS, (xk)t of Eq. (1) on hybrid, respectively. As discussed in the introduction, which ak are applied include ensemble perturbations the current study focuses on the impact of 4D exten- of surface pressure, wind, virtual temperature, relative sion of the ensemble covariance without involving the humidity, cloud water mixing ratio, and ozone mixing static covariance. Therefore, the formula shown in ratio at different time t. The covariance localization in this section and in appendix B excludes the static co- 4DEnsVar follows the same method adopted in GSI- variance. Formulations including the static covariance based 3DEnsVar described in Wang et al. (2013). The will follow similar lines. Below describes the formulas of vertical covariance localization part (Ay) of the locali- 4DEnsVar following the notation of Wang (2010). Fur- zation matrix A is realized through a recursive filter ther mathematical details of implementing 4DEnsVar in transform (Hayden and Purser 1995) with the distance the GSI variational minimization framework are pro- measured either in scaled heights (i.e., natural log of vided in appendix B. the pressure) or in the number of model levels. For 0 In 4DEnsVar, the analysis increment xt at time period GFS, the horizontal localization is realized through t is defined as a spectral filter transform. Specifically, the horizontal localization part (Ah)ofmatrixA is converted into the K A 5 S21A S S 0 5 å + e spectral space by h s ,where represents the xt [ak (xk)t]. (1) k51 transformation from horizontal grid space to spectral 2 space and S 1 is the inverse spectral transformation. e The quantity (xk)t is thepkffiffiffiffiffiffiffiffiffiffiffiffith ensemble perturbation The As is a diagonal matrix containing the spectral at time t normalized by K 2 1,whereK is the en- coefficients corresponding to the horizontal localiza- semble size. The vectors ak, k 5 1, ..., K, denote the tion function predefined in model grid space. The augmented control vectors for each ensemble member. e-folding distances equivalent to 1600 km and 1.1 scaled The symbol ‘‘+’’ denotes the Schur product. The four- height (natural log of pressure is equal to 1.1) cutoff dis- 0 dimensional analysis increment xt is obtained by min- tances in the Gaspari–Cohn (1999) localization function imizing the following cost function: were adopted for the horizontal and vertical localizations, respectively, in the current study. These locali- L 5 1 TA21 1 1 å o0 2 H 0 TR21 o0 2 H 0 zation radii follow those adopted in the 3DEnsVar and J(a) a a (yt txt) t (yt txt). 2 2 t51 3DEnsVar hybrid experiments in Wang et al. (2013) (2) where the same experiment configurations were used. The last term of Eq. (2) is the observational term as in 0 Comparing Eqs. (1) and (2) with the 3DEnsVar formula the traditional 4DVar except that xt is defined by Eq. o0 H in Wang (2010), 4DEnsVar is a natural, temporal ex- (1). In Eq. (2) yt and t are the innovation and linear- tension of its 3D counterpart. In 3DEnsVar, ensemble ized observation operator, respectively, at time period t. perturbations at a single time, the center of the assimilation window was incorporated. In comparison, ensem- 3. Experiment design ble perturbations at multiple time periods t 5 1, ..., L within the assimilation window were incorporated in The data assimilation cycling experiments were con- 4DEnsVar. ducted during a 5-week period, 0000 UTC 15 August– The first term in Eq. (2) is associated with the aug- 1800 UTC 20 September 2010. The operational data mented control vector a, which is formed by concatenat- stream including conventional and satellite data were ing K vectors ak, k 5 1, ..., K. These augmented assimilated every 6 h. A list of types of operational control vectors are constrained by a block-diagonal conventional and satellite data are found on matrix A, which defines the horizontal and vertical the NCEP website (http://www.emc.ncep.noaa.gov/ covariance localization on the ensemble covariance. mmb/data_processing/prepbufr.doc/table_2.htm and

Unauthenticated | Downloaded 10/06/21 01:01 PM UTC SEPTEMBER 2014 W A N G A N D L E I 3307 http://www.emc.ncep.noaa.gov/mmb/data_processing/ TABLE 1. A list of experiments. prepbufr.doc/table_18.htm). The operational NCEP Expt Description Global Data Assimilation System (GDAS) consisted of an ‘‘early’’ and a ‘‘final’’ cycle. During the early cycle, GSI3DVar The GSI 3DVar experiment 4DEnsVar 4D ensemble-variational DA experiment observations assimilated had a short cutoff window. with hourly ensemble perturbations The analyses were then repeated later including the 4DEnsVar-2hr 4D ensemble-variational DA experiment data that had missed the previous early cutoff to pro- with 2-hourly ensemble perturbations vide the final analyses for the 6-h forecast, which was 3DEnsVar 3D ensemble-variational DA experiment used as the first guess of the next early cycle. As a first 4DEnsVar-nbc As in 4DEnsVar-2hr, but without the use of the tangent linear normal mode balance test of the newly developed hybrid system, only obser- constraint (TLNMC) vations from the early cycle were assimilated following Wang et al. (2013). The same observation forward oper- atorsandsatellitebiascorrectionalgorithmsasinthe A few experiments were designed to address the operational GDAS were used. The quality control de- questions proposed in the introduction. Table 1 sum- cisions from the operational GDAS were adopted for marizes all experiments and their acronyms. To inves- all experiments. The GFS model was configured the tigate the sensitivity of the performance of 4DEnsVar same way as the operational GFS except that the hori- to the temporal resolution of ensemble perturbations zontal resolution was reduced to T190 to accommodate spanning the assimilation window [i.e., t, in Eqs. (1) and the sensitivity experiments using limited computing re- (2)], two experiments, one with hourly ensemble per- sources. The model contained 64 vertical levels with the turbations (4DEnsVar) and the other with 2-hourly en- model top layer at 0.25 hPa. An 80-member ensemble semble perturbations (4DEnsVar-2hr) were conducted. was run following the operational configuration. The Specifically, in the 4DEnsVar experiment with L 5 6, if digital filter (Lynch and Huang 1992) was applied during one denotes the time valid at the center of the data as- the GFS model integration following the operational similation window as t 5 0, forecast ensembles valid at configuration. Verification was conducted using data the t 523-, 22-, 21-, 0-, 1-, and 2-h lead times were collected during the last four weeks of the experiment used. In the 4DEnsVar-2hr experiment with L 5 3, period. Verification of general global forecasts against forecast ensembles valid at the t 522-, 0-, and 2-h lead European Centre for Medium-Range Weather Fore- times were used. To study the impact of TLNMC on casts(ECMWF)analysesandinsituobservations the 4DEnsVar analysis and forecast and how the im- were conducted. Statistical significance test using the pacts depend on different types of forecasts such as paired t test (Wilks 2005,p.121)wasconductedfor general global forecasts and hurricane track forecasts, these verifications. A significance level of 95% was experiments withholding the TLNMC (4DEnsVar-nbc) used to define if the differences seen in the compari- were conducted. In addition, studies with single obser- son are statistically significant or not. Hurricane track vation and a case study assimilating full observations at forecasts for cases during the verification period were a particular time were conducted to explore the down- verified against the National Hurricane Center (NHC) stream and upstream impacts of the 4D ensemble co- best track data. variances and how well the linear propagation through Following Fig. 1a of Wang et al. (2013), a one-way ensemble covariances approximates the full nonlinear coupled 4DEnsVar was adopted. The ensemble sup- propagation. plied to 4DEnsVar was initialized by an EnKF. The ensemble square root filter algorithm (EnSRF; Whitaker 4. Results and Hamill 2002; Whitaker et al. 2008)wasadopted. A recent implementation of EnSRF for GFS was de- a. Single-observation experiments scribed more fully in Hamill et al. (2011) and Wang 1) DOWNSTREAM AND UPSTREAM IMPACTS et al. (2013). This EnKF code has been directly inter- OF 4D ENSEMBLE COVARIANCES faced with GSI by using GSI’s observation operators, preprocessing, and quality control for operationally A single-observation experiment was conducted to assimilated data. In the EnKF, to account for sampling illustrate the impact of the temporal evolution of the errors due to the limited ensemble members and mis- background error covariance in the newly developed representation of model errors, covariance localizations, 4DEnsVar. The observations were at the same location multiplicative, and additive inflation were applied. The but valid at three different times: the beginning (t 5 detailed treatments and parameters used follow those 23 h), middle (t 5 0), and end (t 513 h) of the 6-h as- in Wang et al. (2013). similation window. Their increments valid at the analysis

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FIG. 1. (a)–(c) Temperature increments (red contours; units: K) and (d)–(f) geopotential height increments (red contours; units: m) at 700 hPa valid at the middle of a 6-h assimilation window after assimilating a single temperature observation valid at three different times: (a),(d) beginning, (b),(e) middle, and (c),(f) end of the 6-h assimilation window. Black contours are the background temperature in (a)–(c) and height fields in (d)–(f) valid at the analysis time. The observation was located at the same place denoted by the plus sign, but at the three different times. time which was the middle of the assimilation window location, the center of the maximum increment was (t 5 0) were compared. The observed variable was displaced upstream westward of the observation loca- temperature at 700 hPa. The value of the temperature tion as shown in Figs. 1c and 1f. As a comparison, the observation was set to be 18 warmer than the corre- analysis increment from assimilating the single temper- sponding background value and the observation error ature observations valid at three different times was also standard deviation was set to be 18. In the first experi- computed using 3DEnsVar. Because of the absence ment, a single temperature observation at t 523 h was of the temporal evolution of the background error co- assimilated. Figures 1a and 1d show the resulting anal- variance, the analysis increments produced by 3DEnsVar ysis increment of temperature and geopotential height were independent of observation time. The analysis at 700 hPa valid at t 5 0. Relative to the observation increments were exactly equal to that produced by location, the center of the maximum increment was 4DEnsVar when the observation was at t 5 0(Figs. 1b displaced downstream toward the east and northeast. and 1e). This result was consistent with that the analysis time was 2) COMPARISON WITH FULL NONLINEAR MODEL 3 h later than the observation time and the prevailing PROPAGATION background wind was blowing eastward. The second experiment was identical to the first except that the In 4DEnsVar, the temporal propagation of observa- observation time was at t 5 0. The analysis increments tion information within the data assimilation window valid at t 5 0 were plotted in Figs. 1b and 1e. Different is effectively achieved through covariance of ensemble from Figs. 1a and 1d, the center of the maximum in- perturbations at discrete times. Although the ensemble crement was now more closely collocated with the lo- forecasts were generated by full nonlinear model inte- cation of the observation. For the third experiment, the grations, the temporal propagation through covariance observation was at t 5 3 h. Relative to the observation of ensemble perturbations contains a linear assumption.

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FIG. 2. Geopotential height increments (color shades, units: m) at 850 hPa valid at the middle of a 6-h assimilation window after assimilating a single meridional wind observation. Black contours are background height ﬁelds valid at the middle of the assimilation window. The observation was located at the plus sign and valid at the beginning of the assimilation window. Increments are by (a) model integration, (b) 4DEnsVar, (c) 4DEnsVar-2hr, and (d) 3DEnsVar.

Another single-observation experiment was conducted resulting analysis increments of geopotential height at to illustrate how well the linear propagation compared the middle of the assimilation window (t 5 0) at 850 hPa with the full nonlinear model propagation and how were shown in Fig. 2. Figure 2a shows the increment such comparison depended on the number of time pe- by using the full nonlinear model propagation. First, the riods of ensemble perturbations used in 4DEnsVar. single wind observation at t 523 h was assimilated to Figure 2 illustrates such experiment where a tropical update the state valid at t 523 h. Two 3-h forecasts cyclone [Hurricane Daniel (2010)] was contained in the were then launched. These two forecasts were initialized background forecast. A single meridional wind obser- by the states at t 523 h with and without assimilating vation at 850 hPa was assimilated. The value of the wind the single observation, respectively. The difference be- 2 observation was set to be 5 m s 1 stronger than the tween the two forecasts was shown in Fig. 2a. Such dif- background value with an observation error standard ference reﬂected the actual increments valid at t 5 0by 2 deviation of 1 m s 1. The observation was valid at the propagating the increment at t 523 h through nonlinear beginning of the assimilation window (t 523 h). The model integration (Huang et al. 2009) and, therefore,

Unauthenticated | Downloaded 10/06/21 01:01 PM UTC 3310 MONTHLY WEATHER REVIEW VOLUME 142 can be served as the verification of increments generated ensemble perturbations. For example, the negative by 4DEnsVar and 3DEnsVar. The spatial pattern of the increment on the west side of the eye was too strong in increment through nonlinear model propagation con- 4DEnsVar-2hr than in 4DEnsVar. In addition, while sisted of a dipole structure with a negative increment 4DEnsVarcorrectedthevortexlocationbymovingit and a positive increment located to the southwest and to the southwest similar to the nonlinear propagation, northeast side of the hurricane eye, respectively. Such 4DEnsVar-2hr moved vortex more to the west. Quanti- increment suggested the assimilation of the single wind tative and systematic comparisons of 4DEnsVar using observation at t 523 h corrected the position of the hourly and 2-hourly ensemble perturbations are in- tropical cyclone in the background forecast valid at cluded in section 4g. t 5 0 by moving the vortex southwestward. The incre- b. Verification of forecasts against the ECMWF ments from 4DEnsVar using hourly ensemble perturba- analyses tions, 4DEnsVar with 2-hourly ensemble perturbations, and 3DEnsVar are shown in Figs. 2b, 2c, and 2d, re- To evaluate the performance of the 4DEnsVar sys- spectively. The increments from both 4DEnsVar ex- tem, its forecast quality is measured by computing ver- periments better approximated the increment from ification scores against the ECMWF analyses. Figure 3 nonlinear model propagation than 3DEnsVar. While shows the anomaly correlation (AC) of the geopotential the dipole pattern of the increments by 4DEnsVar height, temperature, and wind forecasts over the globe suggested that the position of the tropical cyclone in the verified against the ECMWF analysis. The ECMWF background was shifted to the west or southwest, the analysis data were obtained from the historical The increment from 3DEnsVar was dominated by a nega- Observing System Research and Predictability Experi- tive increment that was nearly centered at the eye ment (THORPEX) Interactive Grand Global Ensemble with a slight positive increment on the west side of the (TIGGE) data archive hosted by the National Center eye. 4DEnsVar, using hourly ensemble perturbations for Atmospheric Research (http://tigge.ucar.edu/home/ (4DEnsVar), approximates the increments from non- home.htm). The anomaly correlation was calculated linear propagation more closely than using 2-hourly following this formula:

[(xF 2 xc) 2 (xF 2 xc)][(xec 2 xc) 2 (xec 2 xc)] AC 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi. (3) [(xF 2 xc) 2 (xF 2 xc)]2 [(xec 2 xc) 2 (xec 2 xc)]2

In Eq. (3), xF , xec, and xc denote the forecast variable, and Southern Hemisphere (SH) extratropics and tropics. the ECMWF analysis variable, and the corresponding The absolute improvement of 4DEnsVar relative to variable from the climatological average. The overbar 3DEnsVar tends to be slightly larger in the SH extra- denotes the areal mean (i.e., average over the domain tropics than in the NH extratropics especially at longer considered). The climatological average was obtained lead times (e.g., Fig. 4). The statistical significance by averaging the NCEP–NCAR reanalysis data over of the differences of the anomaly correlations among 1981–2010 (Kalnay et al. 1996). All data were first bili- different experiments shown in Figs. 3 and 4 are cal- nearly interpolated to a common grid with a 2.58 reso- culated using the paired t test for each forecast lead lution before calculating the AC. Equation (3) was time. The samples were accumulated by pairs of ACs applied to each model level and each forecast during the from forecasts initialized at different times and located verification period. The arithmetic average for all levels at different model levels. The differences between and forecasts is shown in Fig. 3. The verification started 4DEnsVar and GSI3DVar and between 4DEnsVar at the 2-day lead time to reflect that it was more ap- and 3DEnsVar for the lead times considered in Figs. 3 propriate to use the analyses to verify longer forecasts and 4 are all statistically significant (i.e., greater than (Kuhl et al. 2013). Forecasts from 3DEnsVar are more 95% confidence level). skillful than GSI3DVar, consistent with Wang et al. c. Verification of forecasts against in situ observation (2013). 4DEnsVar further improves the skill of the forecasts compared to 3DEnsVar. The improvement of The 4DEnsVar system is also evaluated by comparing 4DEnsVar relative to 3DEnsVar is smaller than the with the radiosonde observations. Figure 5 shows the improvement of 3DEnsVar relative to GSI3DVar. The root-mean-square fit (RMSF) of 6-h forecasts to in AC was also calculated in Northern Hemisphere (NH) situ observations from marine and land surface stations,

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FIG. 3. The globally averaged anomaly correlation of (a) geopotential height, (b) temperature, and (c) zonal and (d) meridional wind forecasts verified against the ECMWF analyses. Solid, dotted, and dashed lines are for GSI3DVar, 3DEnsVar, and 4DEnsVar experiments, respectively. The plus signs right below the upper x axes denote the lead time when the difference between 3DEnsVar and GSI3DVar is statistically significant. The plus signs right above the lower x axes denote the lead time when the difference between 4DEnsVar and 3DEnsVar is statistically significant. rawinsondes, and aircrafts. Statistical significance of the shows consistent improvement relative to 3DEnsVar difference between 4DEnsVar and 3DEnsVar is calcu- for wind forecasts and neutral or slightly positive im- lated using the paired t test for each level. The samples pact for temperature forecast. Over TR, 4DEnsVar were accumulated by pairs of RMSFs from forecasts shows mostly neutral impact compared to 3DEnsVar initialized at different times. A blue cross is marked for both wind and temperature forecasts. at the level where the difference is significant at and Forecasts at longer lead times were also verified above the 95% level. A black cross is marked when against in situ observations (Fig. 6). Same statistical 4DEnsVar was statistically significantly more accurate significance tests as Fig. 5 were conducted. Tempera- than 3DEnsVar averaged for all levels. Wind and ture forecasts from 4DEnsVar show overall positive temperature forecasts from 3DEnsVar and 4DEnsVar impact relative to 3DEnsVar for both NH and SH at experiments are more accurate than GSI3DVar at most the 4-day lead time. 4DEnsVar shows neutral impact levels over NH, SH, and TR. More appreciable im- on wind forecasts over NH and positive impact over SH provement is seen in the wind forecasts than in the at the 4-day lead time. Over TR, 4DEnsVar shows temperature forecasts. Over NH and SH, 4DEnsVar positive impact relative to 3DEnsVar only for wind

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FIG. 4. Averaged anomaly correlation of geopotential height at (a) Northern Hemisphere (NH), (b) Southern Hemisphere (SH), and (c) tropics (TR). Lines and plus signs are as in Fig. 3.

forecasts at low levels. These results are in general 1) REVIEW OF HURRICANE CASES DURING consistent with those found in Buehner et al. (2010b) THE EXPERIMENT PERIOD except that Buehner et al. (2010b) found that the pos- A total of 16 named storms (8 storms from the At- itive impact of 4DEnsVar relative to 3DEnsVar in NH lantic basin and 8 storms from the Pacific basin) during was similar to that in SH at longer lead time. Such the 2010 hurricane season occurred in the verification differences could be because our experiment was con- period. During the experiment verification period, for ducted during NH summer whereas Buehner et al. the Atlantic basin as shown in Fig. 7a, Hurricanes (2010b) conducted the experiments during NH winter Danielle, Earl, Julia, and Igor reached category 4. Igor or because of the differences in numerical models. It was the strongest tropical cyclone of the Atlantic basin could also be because other differences between the during the 2010 season. In addition to the above 4 two data assimilation systems such as the methods hurricanes in the Atlantic basin, 3 storms reached the employed by each system in treating wind–mass im- tropical storm category. In the east Pacific, Frank, a balance during the variational minimization. category-1 hurricane, was close to the southwest coast of Mexico. In the west Pacific, during the experiment ver- d. Verification of hurricane track forecasts ification time, as shown in Fig. 7b, Typhoon Kompasu Early studies have shown that ensemble-based data made landfall at South Korea. Typhoon Fanapi caused assimilation may be particularly helpful with hurricane heavy rainfall in Taiwan and southern China. According initialization because of the use of flow-dependent es- to the hurricane forecast verification reports by the timates of the background error covariances (Torn and NHC (Cangialosi and Franklin 2011) and Joint Typhoon Hakim 2009; Zhang et al. 2009; Wang 2011). Several Warning Center (JTWC; Angove and Falvey 2010), the studies in particular explored the use of 3DEnsVar hy- official hurricane track forecasts were more accurate dur- brid DA in hurricane forecasts (Wang 2011; Hamill et al. ing the 2010 season than the average of previous years. 2011; Li et al. 2012). They have found that deterministic 2) COMPARISON OF TRACK FORECASTS forecasts from the 3DEnsVar hybrid were superior to those initialized from 3DVar. To date, however, no ex- The cyclones in the forecasts were tracked using the periments have been performed with a 4DEnsVar ap- NCEP tropical cyclone tracker (Marchok 2002). To plied for hurricane predictions. As shown in Fig. 2, ensure a head to head comparison among forecasts application of 4DEnsVar for hurricane initialization and initialized by different data assimilation methods, the predictions can be particularly interesting because of the following criteria were followed to include a particular temporal variation of the error covariance associated forecast in the verification sample pool: (i) forecasts with TC structure and location changes within the DA must have been available for all systems involved in window. In this section, the performance of 4DEnsVar comparison; (ii) the cyclone must have been reported for hurricane forecasts was evaluated. Given that the in tropical cyclone vital statistics (TCVITALS; http:// experiments were conducted at the reduced resolution, www.emc.ncep.noaa.gov/mmb/data_processing/tcvitals_ only the hurricane track forecasts were verified. description.htm) at the initial time of the forecast; and

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FIG. 5. The rms fit of the 6-h forecasts to the in situ observations for (a),(c),(e) temperature and (b),(d),(f) wind as a function of pressure for (a),(b) the Northern Hemisphere extratropics; (c),(d) Southern Hemisphere extratropics; and (e),(f) tropics for the GSI3DVar, 3DEnsVar, and 4DEnsVar experiments. Line definition is as in Fig. 3. Blue plus signs indicate the levels where 4DEnsVar is statistically significantly better than 3DEnsVar. Black plus signs and black dashed lines after ‘‘total’’ denote if 4DEnsVar is or is not statistically significantly better than 3DEnsVar, respectively, after averaging over all levels.

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FIG.6.AsinFig. 5, but at the 96-h forecast lead time.

(iii) the observed TC must have been a tropical cyclone Figure 8a shows the root-mean-square error of or a subtropical cyclone at the lead time being evaluated the track forecasts from 4DEnsVar, 3DEnsVar, and following the NHC practice (http://www.nhc.noaa.gov/ GSI3DVar. 3DEnsVar outperforms GSI3DVar, con- veriﬁcation/verify2.shtml). sistent with the results in Hamill et al. (2011).Track

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FIG. 7. Tracks of tropical cyclones during the verification period in the (a) Atlantic and east Pacific and (b) west Pacific basins. forecasts by 4DEnsVar are more accurate than 3DEnsVar calculated using the paired t test for each forecast lead after the 2-day forecast lead time. The statistical sig- time. The samples were accumulated by pairs of track nificance of the differences of the track forecast errors errors from forecasts initialized at different times. among different experiments shown in Fig. 8 are The differences among 3DEnsVar and GSI3DVar are

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FIG. 8. Skill of tropical cyclone track forecasts measured by (a) the rms errors and (b) percentage of forecasts that were better than the reference GSI3DVar forecast for 4DEnsVar (thick solid), 3DEnsVar (dotted), and GSI3DVar (thin solid) experiments. The numbers above the x axis of (a) denote the statistical significant confidence level of the difference between 4DEnsVar and 3DEnsVar. The differences among 3DEnsVar and GSI3DVar are statistically significant for all lead times considered in (a). The numbers above the x axis of (b) denote the number of samples used in the calculation at each lead time. statistically significant for all lead times considered. periods were used during the variational minimization The differences among 4DEnsVar and 3DEnsVar are in 4DEnsVar. To investigate the impact of including statistically significant after 1-day lead time. In addition multiple time periods of perturbations on the conver- to examining the averaged track forecast errors, a sepa- gence of the minimization, the convergence rates of rate measure of the performance of the track forecast 3DEnsVar, and 4DEnsVar were compared. Figure 9 following Zapotocny et al. (2008) was adopted to fur- shows the level of convergence measured by the ratio ther examine the robustness of the difference seen in Fig. 8a. In this measure, the percentage of forecasts from one DA method that was better than forecasts from GSI3DVar was computed. Figure 8b shows the percentage of forecasts from 3DEnsVar and 4DEnsVar that were better than GSI3DVar. A range of 60%– 68% of the forecasts from 3DEnsVar are better than GSI3DVar for the forecast lead times considered. For 4DEnsVar, 68%–80% of the forecasts are better than GSI3DVar. Comparing 3DEnsVar and 4DEnsVar shows that the percentage of better forecasts by 4DEnsVar is larger than that of 3DEnsVar especially after the 1-day lead time. This result is consistent with that in Fig. 8a. e. Impact of 4D ensemble covariances on convergence rate during the variational minimization and discussion on the second outer loop With a similar experiment configuration, Wang et al. (2013) found that 3DEnsVar showed a slightly slower (faster) convergence rate at early (later) iterations than FIG. 9. Averaged ratios of gradient norms as a function of the GSI 3DVar for the first outer loop, and a faster con- number of iterations in the first and second outer loops during the vergence rate for the second outer loop. Compared to variational minimization of 3DEnsVar (solid line) and 4DEnsVar 3DEnsVar, ensemble perturbations at multiple time (dashed line) experiments.

Unauthenticated | Downloaded 10/06/21 01:01 PM UTC SEPTEMBER 2014 W A N G A N D L E I 3317 of the gradient norm relative to the initial gradient t 523 h to form the new ensemble analyses at t 523h. norm during the variational minimization averaged New ensemble forecasts within the DA window will over the experiment period. Following the configura- then be initialized by this set of ensemble analyses. This tion of the operational GSI, two outer loops were used procedure propagates the ensemble covariance fol- during the variational minimization. In the current lowing the trajectory defined by the analysis resultant experiments, the maximum iteration steps were 100 from the first outer loop. However, because of the and 150 for the first and second outer loops for all ex- computational cost of rerunning the ensemble, this step periments. The same numbers were used in the oper- was omitted in this study and the ensemble perturba- ational system. The minimization was terminated at the tions throughout the DA window used for the second maximum iteration step in most cases. Figure 9 also outer loop were the same as the first outer loop. An shows that the iterations were terminated at a similar attempt was made to illustrate the impact of using the value of the ratio of gradient norm for the 3DEnsVar updated trajectory to evolve the ensemble covariance and 4DEnsVar experiments. For the first outer loop, through a single-observation experiment using the same 4DEnsVar shows slightly a slower convergence rate hurricane case as in Fig. 2. The result (not shown) sug- than 3DEnsVar. For the second outer loop, 4DEnsVar gested a slightly improved increment using re-evolved shows faster convergence than 3DEnsVar. For the ex- ensemble perturbations when using the increment from periments conducted in this study, the cost of 4DEnsVar nonlinear propagation as verification. Future work is variational minimization is approximately 1.5 times of needed to systematically explore the impact of the that of 3DEnsVar. Tests comparing the computational second outer loop in 4DEnsVar and the impact of using costs have shown that 4DEnsVar is about one order of re-evolved ensemble perturbations in the second outer magnitude less expensive than the TLA 4DVar being loop. developed (Rancic et al. 2012). f. Impact of 4D ensemble covariances on balance As shown in Fig. 9, in the operational implementation of GSI, two outer loops were adopted to treat the non- Imbalance between variables introduced during data linearity during the assimilation. In GSI 3DVar, the assimilation can degrade the subsequent forecasts. The implementation of the outer loops follows the same mass–wind relationship in the increment associated method in the incremental 4DVar in Courtier et al. with the ensemble-based method was defined by the (1994) and Lawless et al. (2005). The only difference is multivariate covariance inherent in the ensemble per- that in GSI 3DVar, the mapping from the control vari- turbations. Such an inherent relationship can be al- able to the observations does not involve the component tered by the commonly applied covariance localization of the tangent linear model. Compared to the first outer (e.g., Lorenc 2003; Kepert 2009; Holland and Wang loop, in the second outer loop of GSI 3DVar, the in- 2013). Compared to 3D analysis methods, one attrac- novation was updated by using the analysis resultant tive aspect of the analysis produced by a 4D method from the first outer loop as the background and the is the temporal smoothness. In 4DVar, this is achieved reference state for the linearization of the observation through the explicit use of a dynamic model. In 4DEnsVar, operator was changed from the first guess to the analysis instead, the 4D increments were obtained through resultant from the first outer loop. The equivalence be- the Schur product of extended control variables and tween such outer loop implementation and the Gauss– ensemble perturbations valid at discrete times. The bal- Newton method for solving the nonlinear assimilation ance of the analysis produced by 4DEnsVar is investi- problem (Bjorck A 1996) was shown in Lawless et al. gated in this section. The mean absolute tendency of (2005). In incremental 4DVar, the background error surface pressure (Lynch and Huang 1992) is a useful covariance at the beginning of the DA window is the diagnostic metric to show the amount of imbalance for same in the first and second outer loops. However, the an analysis generated by a data assimilation system. reference state upon which the error covariance is The hourly surface pressure tendency averaged over propagating across the DA window is updated by using the experiment period was calculated and summarized the analysis resultant from the first outer loop (Courtier in Table 2. For all hemispheres, the forecasts initialized et al. 1994; Jazwinski 1970, 279–281). Following in- by4DEnsVarareslightlymorebalancedthanthe cremental 4DVar, in the current implementation of 3DEnsVar. Note that for all the experiments, following GSI-based 4DEnsVar, ensemble forecasts should be re- the operational configuration of GFS, the digital fil- run before the second outer loop. Specifically, ensemble ter was applied during the model integration. In this perturbations used in the first outer loop valid at t 523h study, the digital filter was configured with a 4-h fil- should be maintained. These perturbations will then be tering window where the forecast state at the center of added to the analysis from the first outer loop valid at the window was replaced by the weighted average of

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TABLE 2. Averaged hourly absolute surface pressure tendency at t 523 h. Two 6-h forecasts were then launched. 21 (hPa h ) during the experiment period for the Northern Hemi- These two forecasts were initialized by the states at t 5 sphere extratropics (NE), tropics (TR), and Southern Hemisphere 23 h with and without assimilating those observations, extratropics (SE) for the 3DEnsVar, 4DEnsVar, 4DEnsVar-2hr, and 4DEnsVar-nbc experiments, respectively. respectively. Such differences reflected the actual increments valid at t 5 3 h by propagating the increment NE TR SE at t 523 h through nonlinear model integration. In- 3DEnsVar 0.402 0.524 0.643 crements by 4DEnsVar and 4DEnsVar-2hr were eval- 4DEnsVar 0.392 0.516 0.635 uated by computing the spatial correlation of these 4DEnsVar-2hr 0.396 0.519 0.638 increments with the true increments. Figure 10 shows 4DEnsVar-nbc 0.461 0.571 0.719 such correlations for different state variables at various model levels. It is found that increments by 4DEnsVar forecast states spanning the 4-h window. The impact of using hourly ensemble perturbations correlate with the the digital filter on the forecasts started from the sec- true increments more than using the 2-hourly ensemble ond hour of the model integration. The results in Table perturbations for most of the model levels and vari- 2 suggest that the forecasts initialized by 4DEnsVar ables considered. werestillmorebalancedthan3DEnsVarevenwhen Another way to systematically evaluate the perfor- DFI was applied. mance of 4DEnsVar as a function of the number of time periods of ensemble perturbations is to compare g. Quantitative evaluation of the sensitivity the performance of forecasts initialized by 4DEnsVar to the number of time periods of the ensemble using hourly ensemble perturbations versus 4DEnsVar perturbations using 2-hourly ensemble perturbations. Therefore, a sep- In a typical 4DVar, the analyses are obtained via fit- arate data assimilation cycling and forecast experiment ting the model trajectory to observations distributed where the ensemble perturbations were sampled at within a finite assimilation window through the use of three time periods instead of six time periods were the tangent linear and adjoint of the forecast model. In conducted during the whole experiment period. In 4DEnsVar, the 4D analyses are obtained through vari- other words, in this experiment the ensemble pertur- ational cost function minimization within the tempo- bations were sampled every 2 h. This experiment was rally evolved ensemble forecast perturbation space named as 4DEnsVar-2hr. Figure 11 shows that the spanning the assimilation window. Effectively, the four- performance of forecasts initialized by 4DEnsVar was dimensional (4D) background error covariance of a degraded when less frequent ensemble perturbations nonlinear system was approximated by the covariances were used especially at longer forecast lead times. of ensemble perturbations at discrete times. Using a Similar statistical significance tests as Fig. 3 were con- single-observation experiment, section 4a illustrates ducted for the results in Fig. 11.Itwasfoundthatsuch how the 4DEnsVar increments approximate the incre- degradation is statistically significant at the 72- and ments made by the nonlinear model propagation and 96-h lead times for geopotential height and meridional how such approximation depends on the number of time wind forecasts, and at 96-h lead time for the tempera- periods of ensemble perturbations used in 4DEnsVar. ture and zonal wind forecasts. The AC calculated for This section provides further quantitative investigation NH, SH, and TR showed similar results (not shown). on how the performance of 4DEnsVar depends on the The balance of the 4DEnsVar analyses to the tem- number of time periods at which the ensemble pertur- poral resolution of the ensemble perturbations was also bations are sampled. To evaluate the linear approxi- examined. Table 2 shows that using less frequent en- mation quantitatively, the correlation of the increments semble perturbations, the 4DEnsVar analyses became from nonlinear model propagation and 4DEnsVar was less balanced. calculated. Figure 10 shows an example for a case where The hurricane track forecast was also degraded after the center of the assimilation window was at 0600 UTC the 1-day lead time when less frequent ensemble per- 25 August 2010. All operational observations within the turbations were used (Fig. 12). Similar statistical signif- first hour (between t 523 and t 522 h) of the 6-h DA icance tests as Fig. 8 were conducted for the results in window were assimilated. The increments valid at t 5 Fig. 12. The degradation was statistically significant 3 h, the end of the assimilation window, were evaluated. after 1-day lead time. This result is further confirmed Following the same method in Fig. 2a of section 4a,the by calculating the percentage of forecasts with hourly true increment at t 5 3 h was calculated through non- ensemble perturbations that were better than the fore- linear model propagation. First, all observations within casts with 2-hourly ensemble perturbations (Fig. 12b). the first hour were assimilated to update the state valid These results are consistent with the expectation that

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FIG. 10. The correlation of analysis increments by 4DEnsVar using hourly ensemble perturbations (4DEnsVar) and using 2-hourly ensemble perturbations (4DEnsVar-2hr) with the increment by nonlinear model propagation: (a) height, (b) temperature, and c) zonal and (d) meridional wind increments The increments are valid at the end of the 6-h assimilation window by assimilating all observations within the ﬁrst hour of the assimilation window. See text for more details. the temporal evolution of the error covariance with the obtained from a tangent linear version of a general, assimilation window is better approximated with more hydrostatic, adiabatic primitive equation model. The frequent ensemble perturbations. tendency model used for the TLNMC purpose also did not include parameterized physics. More details on h. Impact of TLNMC TLNMC implemented in GSI 3DVar were provided in The TLNMC was implemented in the GSI minimi- Kleist et al. (2009). Kleist et al. (2009) showed that the zation to improve the balance of the initial conditions. impact of TLNMC resulted in substantial improvement The TLNMC operator was applied to the analysis in- in the global forecasts initialized by GSI 3DVar. The crement during the variational minimization. The op- TLNMC was applied on the analysis increments asso- erator contained three steps including calculating the ciated with the ensemble covariances when 3DEnsVar tangent linear tendency model, projecting the tendency and 4DEnsVar were implemented within GSI. Wang onto the gravity modes, and reducing the gravity mode et al. (2013) found that TLNMC improved the global tendencies. For simplicity, the tendency model was forecasts initialized by 3DEnsVar and also concluded

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FIG. 11. As in Fig. 3, but with the solid, dotted, and dashed lines denoting 4DEnsVar-nbc, 4DEnsVar-2hr, and 4DEnsVar experiments, respectively. The plus signs right above the lower x axes denote the lead time when the difference between 4DEnsVar and 4DEnsVar-2hr is statistically significant. The plus signs right below the upper x axes denote the lead time when the difference between 4DEnsVar-2hr and 4DEnsVar-nbc is statistically significant. that the better performance of 3DEnsVar relative to 4DEnsVar-2hr, but without the use of the TLNMC, EnKF was due to the ability of 3DEnsVar in using such were conducted (hereafter the experiment is named as constraint during the variational minimization. 4DEnsVar-nbc). The impact of TLNMC on the balance In 4DEnsVar, the TLNMC operator was applied to of the analyses is first measured following the same 0 the analysis increment at different time periods xt.In method in section 4f. It was found that TLNMC resulted the current implementation, the reference state for the in substantial decrease of surface pressure tendency and tangent linear tendency model was assumed to be time was, therefore, more balanced 4DEnsVar analyses invariant throughout the assimilation window. In addi- (Table 2). The accuracy of the forecasts initialized by tion, as discussed in section 4f, one attractiveness of 4DEnsVar withholding the TLNMC was also evaluated. 4DEnsVar analyses compared to its 3D counterpart is Figure 11 shows that TLNMC yields significant positive the temporal smoothness, which itself can lead to more impacts measured by the global AC for the forecast lead balanced analyses. Given the simplification and as- times considered. A similar statistical significance test as sumptions made in TLNMC and the inherently more in Fig. 3 was conducted. Such a test revealed that the balanced analyses in 4DEnsVar, the impact of further positive impact of TLNMC was statistically significant applying TLNMC within 4DEnsVar was examined in for the lead times and variables considered. Further this study. Experiments configured to be the same as calculating the AC in NH, SH, and TR showed that most

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FIG. 12. As in Fig. 8, but for the reference 4DEnsVar-2hr forecast for 4DEnsVar (thick solid), 4DEnsVar-2hr (thin solid), and 4DEnsVar-nbc (dotted) experiments. The black (blue) numbers above the x axis in (a) denote the statistical significant confidence level of the difference between 4DEnsVar and 4DEnsVar-2hr (between 4DEnsVar-2hr and 4DEnsVar-nbc). of the positive impact of TLNMC was from SH. The 5. Conclusions and discussion TLNMC showed neutral impact over the TR. For NH, A GSI-based four-dimensional ensemble–variational slight positive impacts were found at early lead times data assimilation system (4DEnsVar) was developed. and slight negative impacts were found after the 3-day Different from its 3D counterpart (3DEnsVar), the en- lead time. The slight negative impact over NH at longer semble perturbations valid at multiple time periods lead times could be due to the time-invariant reference throughout the DA window were effectively used to state used when the tendency model was calculated. estimate the 4D background error covariances during This hypothesis was consistent with a neutral or slight positive impact of TLNMC on the 3DEnsVar over NH the variational minimization. The TLA of the fore- (not shown). cast model was conveniently avoided. Different from The TLNMC implemented in GSI does not include 4DEnsVar implemented in other systems, 4DEnsVar the diabatic processes in the tendency model. There- implemented within GSI minimization was precondi- fore, it may not be appropriate for the forecasts asso- tioned upon the full background error covariance ma- ciated with strong moist processes such as the tropical trix. The specific formulations and implementations of cyclone forecasts. In addition, the impacts of TLNMC 4DEnsVar within GSI were introduced first. Using the on the TC forecasts were examined. Figure 12 shows newly developed GSI-based 4DEnsVar system, a few that the TLNMC showed negative impact on TC track questions were investigated. What is the value of using forecasts. Similar statistical significance test as in Fig. 8 the 4D ensemble covariance in 4DEnsVar for general was conducted for results in Fig. 12. It was found that global forecasts and for hurricane track forecasts? In the negative impact of TLNMC on TC track forecasts 4DEnsVar, temporal evolution of the error covariance was statistically significant for all the lead times in is approximated by the covariances of ensemble per- Fig. 12a. This result is further confirmed by calculating turbations at discrete times. How is the performance of the percentage of forecasts without TLNMC, which 4DEnsVar dependent on the temporal resolution of or were better than the forecasts with TLNMC (Fig. 12b). the number of time periods of ensemble perturbations? The TLNMC showed a negative impact on TC track How is this linear approximation compared to the full forecasts initialized by 3DEnsVar also (not show). With- nonlinear model propagation? Will using 4D ensemble holding TLNMC, 4DEnsVar showed an improved TC covariances to fit the model trajectory to observations track forecast than 3DEnsVar even with reduced num- distributed within a finite assimilation window improve ber of levels of perturbations (not shown). These results the balance of the analysis and how is the balance of suggest that further development of the balanced the analysis dependent on the temporal resolution of constraints by considering the moisture processes are the ensemble perturbations? What is the impact of fur- needed for forecasts with strong moisture processes. ther applying the tangent linear normal mode balance

Unauthenticated | Downloaded 10/06/21 01:01 PM UTC 3322 MONTHLY WEATHER REVIEW VOLUME 142 constraint on the 4DEnsVar analysis and forecast and impact for hurricane track forecasts. For the first outer how is that dependent on different types of forecasts loop, 4DEnsVar showed slightly a slower convergence rate such as the general global forecast or hurricane track than 3DEnsVar. For the second outer loop, 4DEnsVar forecasts? How does including multiple time periods showed a slightly faster convergence than 3DEnsVar. of perturbations impact the convergence rates of the As discussed in the introduction, in this study, as a first minimization? step of testing the newly developed 4DEnsVar for The performance of the system and aforementioned GSI, experiments were conducted where the single questions were investigated using NCEP GFS at a re- control forecast and the ensemble were run at the same, duced resolution. The ensemble was supplied by an reduced resolution. Wang et al. (2013) found little EnKF. The experiments were conducted over a summer impact of including the static covariance in the back- month period assimilating NCEP operational conven- ground error covariance when comparing 3DEnsVar tional and satellite data. The findings from these ex- and 3DEnsVar hybrid with similar experiment settings periments in addressing those aforementioned questions as this study. Therefore, in the current study no static are summarized below. A series of single-observation ex- covariance was included. Recent experiments (T. Lei periments revealed that the newly developed 4DEnsVar and X. Wang 2014, unpublished manuscript) found that was able to reflect the temporal evolution of the back- with the dual-resolution configuration where the en- ground error covariance in the DA window. The global semble was run at a reduced resolution compared to forecasts were verified against both the in situ observa- the control forecast and analysis, the static covariance tions and the ECMWF analyses. 4DEnsVar in general showed a significant positive impact. Buehner et al. improved upon 3DEnsVar. At short lead times, the im- (2013) found that when the static covariance was in- provement of 4DEnsVar relative to 3DEnsVar over NH cluded in the Canadian 4DEnsVar system, the 4D en- was similar to that over SH. At longer forecast lead times, semble covariance only resulted in small improvement 4DEnsVar showed more improvement in SH than NH. in forecast quality. Further work is needed to compare The improvement of 4DEnsVar over TR was neutral or GSI-based 4DEnsVar and 3DEnsVar at dual-resolution slightly positive when forecasts were verified against the mode and to explore the impact of the 4D extension of in situ observations. Track forecasts of 16 named trop- the ensemble covariance relative to its 3D counterpart ical cyclones during the verification period were verified when a static covariance is included. against the NHC best track data. The track forecasts ini- In the current study, no temporal covariance locali- tialized by 4DEnsVar were more accurate than 3DEnsVar zation was applied on the ensemble covariance in after the 1-day forecast lead time. A single-observation 4DEnsVar. A temporal localization is being developed case study where Hurricane Daniel (in 2010) was the within GSI-based 4DEnsVar. Preliminary tests showed background and a case study assimilating all opera- a positive impact of the temporal localization on the tional observations at the beginning of the assimilation performance of 4DEnsVar. Future work is needed to window were conducted to reveal how well covariance further explore such an impact. As discussed in section 4e, of ensemble perturbations approximated the propa- in 4DEnsVar currently implemented in GSI, to save the gation using the full nonlinear model both qualitatively computational cost, during the second outer loop, the and quantitatively. It was found that increments from same evolved ensemble perturbations as in the first outer 4DEnsVar using more frequent ensemble perturbations loop were used although the trajectory was updated approximated the increments from direct, nonlinear during the first outer loop. Future work is needed to ex- model propagation better than using less frequent en- plore the impact of the second outer loop and the impact semble perturbations. Consistently, using experiments of recentering the ensemble perturbations on the new over the full experiment period, it was found that when less trajectory during the second outer loop in 4DEnsVar. frequent ensemble perturbations were used in the assimilation window, the performance of the forecasts initialized Acknowledgments. The study was supported by by 4DEnsVar was degraded especially after 2–3-day lead NOAA THORPEX Grant NA08OAR4320904, NASA times for global forecast and after 1-day lead time for hurri- NIP Grant NNX10AQ78G, NOAA HFIP Grant cane track forecasts. Analyses generated by 4DEnsVar NA12NWS4680012, and NA14NWS4830008. Comput- were more balanced than those by 3DEnsVar. The ing resources at the National Science Foundation, 4DEnsVar using more frequent ensemble perturbations XSEDE, the National Institute of Computational Sci- produced analyses that were more balanced than using ence (NICS) at the University for Tennessee, and less frequent ensemble perturbations. TLNMC showed a NOAA High Performance Computing System Jet were positive impact on 4DEnsVar for global forecasts veri- used for this study. Jeff Whitaker and Daryl Kleist are fied using the anomaly correlation metric and a negative acknowledged for helpful discussion.

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TABLE A1. Characteristics of different ﬂavors of coupled ensemble–variational data assimilation and their acronyms.

No. of time periods of ensemble perturbations incorporated in the DA window during Weights on static and Tangent linear and adjoint the variational minimization ensemble covariance of the forecast model 3DEnsVar One, usually valid at the center 0% on static and 100% on ensemble Not needed of the DA window 3DEnsVar hybrid One, usually valid at the center Nonzero on static and ensemble Not needed of the DA window covariances, sum of weights is usually constrained to be equal to 1 4DEnsVar Multiple 0% on static and 100% on ensemble Not needed 4DEnsVar hybrid Multiple Nonzero on static and ensemble Not needed, same static covariance is covariances, sum of weights is used for multiple time periods, usually constrained to be equal to 1 equivalent to assuming a numerical model of identity matrix Ens4DVar One, usually valid at the beginning 0% on static and 100% on ensemble Needed of the DA window Ens4DVar hybrid One, usually valid at the beginning Nonzero on static and ensemble Needed of the DA window covariances, sum of weights is usually constrained to be equal to 1

APPENDIX A The increment in 4DEnsVar can be expressed as

K È É 0 5 å + e 5 e e Acronyms for Coupled Ensemble-Variational xt ak (xk)t diag[(x1)t] diag[(xK)t] a, Data Assimilation k51 (B2) Table A1 lists the acronyms for coupled ensemble- variational data assimilation examined in this paper. where diag is an operator that turns a vector into a diagonal matrix where the nth diagonal element is given APPENDIX B by the nth element of the vector (Wang et al. 2007b). We further denote È É Mathematical Framework for Implementing D 5 e e diag[(x1)t] diag[(xK)t] . 4DEnsVar in GSI Variational Minimization t Different from other variational data assimilation sys- Then the increment becomes tems, operational GSI minimization is preconditioned 0 x 5 D a 5 D x. (B3) upon the full background error covariance matrix. t t t Wang (2010) and Wang et al. (2013) introduced and Denote the new background error covariance as described the formulas of 3DEnsVar hybrid in GSI. In other words, those formulas describe how the en- B 5 A. (B4) semble covariance is implemented in the GSI 3DVar variational minimization through the augmented control As in the original GSI 3DVar and 3DEnsVar, 4DEnsVar vectors (ACV) under such preconditioning. In this sec- is also preconditioned by deﬁning a new variable: tion, we further extend this framework and derive for- 21 21 mulas to show how the 4DEnsVar is implemented within z 5 B x 5 (A a). (B5) GSI. The key of this derivation is that the minimization $ 5 of the new cost function in Eqs. (1) and (2) can be pre- In the rest of the derivation, we will show that zJ B$ conditioned in the same way as shown in Wang (2010).In xJ, and therefore the minimization for the 4DEnsVar other words, the same conjugate gradient minimization cost function can follow the same conjugate gradient procedure from the original GSI-based 3DEnsVar will method used by the original GSI, which then concludes be followed for GSI-based 4DEnsVar. the derivation. The rest of the terms in Eqs. (B1)–(B5) are Denote the new control variable as deﬁned as those in Eqs. (1) and (2). First, we derive the gradient of the hybrid cost func- x 5 a. (B1) tion with respect to x 5 a. The gradients of the new cost

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