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  • View in gallery

    (a) The SLP (black line), temperature (color shaded), and wind vectors at 850 hPa at 1200 UTC 3 Oct 2007. The locations of the cyclone at various times and its track are marked. (b) The corresponding geopotential heights and wind vectors at 500 hPa are shown. (A full wind barb represents 5 m s−1.)

  • View in gallery

    Snapshot of (a) sounding, (b) QuikSCAT, (c) ships, and (d) SYNOP observations at 1200 UTC 2 Oct available for assimilation.

  • View in gallery

    A typical WRF-4DEnVar data assimilation flowchart. In the chart, WPS means WRF preprocessing system; OBS_Proc means WRF-Var observation processing; Little_R is a data format for observations; IC and BC mean initial and boundary conditions; Eobsi means the perturbed observations for each ensemble member; EXa means 4DEnVar ensemble analyses; Xa is 4DEnVar analysis; and Gen_eib, Gen_pe, and Gen_po are three new components specific for WRF-4DEnVar system.

  • View in gallery

    The background temperature error correlation (color shaded) at the 11th level at (a) 0600 and (b) 1200 UTC 2 Oct 2007, and the corresponding background temperature error variance (color shaded) at (c) 0600 and (d) 1200 UTC. The black contours are the corresponding background forecasts of temperature. The location used for the correlation is marked by the black dot.

  • View in gallery

    The temperature analysis increments (color shaded) at the 11th level by (a) 3DVar, (b) FGAT, (c) En3DVar, and (d) 4DEnVar valid at 1200 UTC 2 Oct 2007. The contours are the corresponding background forecasts of temperature.

  • View in gallery

    The sea level pressure (black line), temperature (color shaded), and wind vectors at 850 hPa at 1200 UTC 2 Oct 2007 from (a) FNL, (b) CTRL, and (c) 4DEnVar.

  • View in gallery

    As in Fig. 6, but for 72-h forecasts from (a) FNL, (b) CTRL, (c) 3DVar, (d) FGAT, (e) En3DVar, and (f) 4DEnVar. The locations of the cyclone are marked with a black dot.

  • View in gallery

    The temperature difference (color shaded) at the 11th level between FNL and (a) 3DVAR, (b) FGAT, (c) En3DVar, and (d) 4DEnVar valid at 1200 UTC 5 Oct 2007. The contours are the corresponding temperature from FNL.

  • View in gallery

    The vertical profiles of domain-averaged RMSEs of (a) u winds, (b) υ winds, (c) temperature, and (d) humidity at 1200 UTC 5 Oct 2007. The results of CTRL, 3DVar, FGAT, En3DVar, and 4DEnVar are denoted by cross-line, dot–dash, dash, dot, and solid lines, respectively.

  • View in gallery

    The variation of domain-averaged RMSE of (a) u winds, (b) υ winds, (c) temperature, and (d) humidity. The line marks are as in Fig. 9.

  • View in gallery

    The variations of the domain-averaged forecast-analysis RMSEs (asterisk-solid line) and the responsive ensemble spread (circle-dash line) without (blue line) and with (red line) the inflation factor for (a) u winds, (b) υ winds, (c) temperature, and (d) humidity.

  • View in gallery

    Locations of the cyclone at various time (1200 UTC 5 Oct, 1200 UTC 6 Oct, 1200 UTC 7 Oct, and 1200 UTC 8 Oct 2007) from FNL, CTRL, 4DEnVar, and FGAT, respectively; their tracks are marked.

  • View in gallery

    Distribution of the CSLPs (hPa) of FNL, CTRL, FGAT, and 4DEnVar at 1200 UTC 2 Oct, 1200 UTC 3 Oct, 1200 UTC 4 Oct, and 1200 UTC 5 Oct 2007, respectively.

  • View in gallery

    The 24-h forecast domain-averaged RMSE of (a) zonal wind u, (b) meridional wind υ, (c) temperature T, and (d) specific humidity q.

  • View in gallery

    As in Fig. 14, but for the 48-h forecast.

  • View in gallery

    As in Fig. 14, but for the 72-h forecast.

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An Ensemble-Based Four-Dimensional Variational Data Assimilation Scheme. Part III: Antarctic Applications with Advanced Research WRF Using Real Data

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  • 1 College of Marine Science, University of South Florida, St. Petersburg, Florida
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Abstract

A four-dimensional ensemble-based variational data assimilation (4DEnVar) algorithm proposed in Part I of the 4DEnVar series (denoted En4DVar in Part I, but here we refer to it as 4DEnVar according to WMO conference recommendation to differentiate it from En4DVar algorithm using adjoint model) uses a flow-dependent background error covariance calculated from ensemble forecasts and performs 4DVar optimization based on an incremental approach and a preconditioning algorithm. In Part II, the authors evaluated 4DEnVar with observing system simulation experiments (OSSEs) using the Advanced Research Weather Research and Forecasting Model (ARW-WRF, hereafter WRF). The current study extends the 4DEnVar to assimilate real observations for a cyclone in the Antarctic and the Southern Ocean in October 2007. The authors performed an intercomparison of four different WRF variational approaches for the case, including three-dimensional variational data assimilation (3DVar), first guess at the appropriate time (FGAT), and ensemble-based three-dimensional (En3DVar) and four-dimensional (4DEnVar) variational data assimilations. It is found that all data assimilation approaches produce positive impacts in this case. Applying the flow-dependent background error covariance in En3DVar and 4DEnVar yields forecast skills superior to those with the homogeneous and isotropic background error covariance in 3DVar and FGAT. In addition, the authors carried out FGAT and 4DEnVar 3-day cycling and 72-h forecasts. The results show that 4DEnVar produces a better performance in the cyclone prediction. The inflation factor on 4DEnVar can effectively improve the 4DEnVar analysis. The authors also conducted a short period (10-day lifetime of the cyclone in the domain) of analysis/forecast intercomparison experiments using 4DEnVar, FGAT, and 3DVar. The 4DEnVar scheme demonstrates overall superior and robust performance.

Corresponding author address: Dr. Chengsi Liu, Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, OK 73072. E-mail: cliu@ou.edu

Abstract

A four-dimensional ensemble-based variational data assimilation (4DEnVar) algorithm proposed in Part I of the 4DEnVar series (denoted En4DVar in Part I, but here we refer to it as 4DEnVar according to WMO conference recommendation to differentiate it from En4DVar algorithm using adjoint model) uses a flow-dependent background error covariance calculated from ensemble forecasts and performs 4DVar optimization based on an incremental approach and a preconditioning algorithm. In Part II, the authors evaluated 4DEnVar with observing system simulation experiments (OSSEs) using the Advanced Research Weather Research and Forecasting Model (ARW-WRF, hereafter WRF). The current study extends the 4DEnVar to assimilate real observations for a cyclone in the Antarctic and the Southern Ocean in October 2007. The authors performed an intercomparison of four different WRF variational approaches for the case, including three-dimensional variational data assimilation (3DVar), first guess at the appropriate time (FGAT), and ensemble-based three-dimensional (En3DVar) and four-dimensional (4DEnVar) variational data assimilations. It is found that all data assimilation approaches produce positive impacts in this case. Applying the flow-dependent background error covariance in En3DVar and 4DEnVar yields forecast skills superior to those with the homogeneous and isotropic background error covariance in 3DVar and FGAT. In addition, the authors carried out FGAT and 4DEnVar 3-day cycling and 72-h forecasts. The results show that 4DEnVar produces a better performance in the cyclone prediction. The inflation factor on 4DEnVar can effectively improve the 4DEnVar analysis. The authors also conducted a short period (10-day lifetime of the cyclone in the domain) of analysis/forecast intercomparison experiments using 4DEnVar, FGAT, and 3DVar. The 4DEnVar scheme demonstrates overall superior and robust performance.

Corresponding author address: Dr. Chengsi Liu, Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, OK 73072. E-mail: cliu@ou.edu

1. Introduction

Since the variational adjoint technique was introduced in 1980s (Lewis and Derber 1985; Le Dimet and Talagrand 1986), four-dimensional variational data assimilation (4DVar) has played an important role in numerical weather prediction (NWP). The 4DVar seeks for initial conditions giving the best fit between forecast and observations within the assimilation window (Daley 1991; Rabier et al. 2000; Lorenc 2003; Gustafsson 2007; Kalnay et al. 2007; Huang et al. 2009). As a retrospective assimilation algorithm, it can derive the optimal time-trajectory fit to observational data, including nonsynoptic data (Xiao et al. 2002; Simmons and Hollingsworth 2002), at closer observation times than three-dimensional variational data assimilation (3DVar). The 4DVar schemes have been implemented in NWP at many operational centers [the European Centre for Medium-Range Weather Forecasts (ECMWF), Météo-France, the Met Office, the Japan Meteorological Agency (JMA), etc.]. However, it is well known in the data assimilation community that developing adjoint models is tedious, labor intensive, and often subject to errors.

Another advanced data assimilation technique, the ensemble Kalman filter (EnKF), has gained attention in recent years. The EnKF can use a flow-dependent background error covariance ( matrix) calculated from ensemble forecasts and can easily be implemented without tangent linear and adjoint models. The advantages and weaknesses of 4DVar and EnKF have been discussed in Lorenc (2003), Gustafsson (2007), and Kalnay et al. (2007); and combining the advantages of the variational and EnKF techniques has attracted a lot of effort (Evensen and van Leeuwen 2000; Hamill and Snyder 2000; Hunt et al. 2004; Zupanski 2005; Wang et al. 2007a,b; Fertig et al. 2007; Wang et al. 2008a,b; Zhang et al. 2009; Wang 2010; Zhang et al. 2011; Zhang and Zhang 2012).

To add our contribution, we proposed a four-dimensional ensemble-based variational (4DEnVar) algorithm (Liu et al. 2008, 2009, hereafter Part I and Part II, respectively). The new data assimilation technique, 4DEnVar, uses the flow-dependent background error covariance matrix ( matrix) calculated from ensemble forecasts and performs 4DVar optimization (Part I; Part II). It adopts the variational approach, but avoids the tangent linear model and its adjoint, the component that is difficult to develop and that makes 4DVar minimization computationally expensive (Xiao et al. 2008).

Since the establishment of the Weather Research and Forecasting Model (WRF) variational data assimilation system (WRF-Var; Barker et al. 2004; Skamarock et al. 2008), we have made a significant effort to implement 4DEnVar in it (Part II). The execution of 4DEnVar can produce both a probabilistic ensemble analysis and a deterministic analysis. Applying the WRF-4DEnVar with the observing system simulation experiment (OSSE) to a midlatitude cyclone case showed a very positive impact of the algorithm on WRF forecasts (Part II).

Recently, Buehner et al. (2010a,b) have applied ensemble-based variational methods using real observations to global numerical weather forecasting, and they have shown that the ensemble-based matrix has a positive impact on a global variational data assimilation system. However, performance of the ensemble-based variational algorithm on the regional and higher-resolution numerical weather model still needs to be explored.

Therefore, for the third part of the 4DEnVar series, we examine the WRF-4DEnVar performance using real observational data for a case of cyclone evolution in the Antarctic and the Southern Ocean on 2–8 October 2007. Over the Antarctic, there is a real-time regional forecasting system, the Antarctic Mesoscale Prediction System (AMPS; Powers et al. 2003). The AMPS system is dedicated to providing regional prediction over Antarctica in support of the U.S. Antarctic Program, Antarctic science, and international Antarctic efforts. Although WRF-3DVar has been applied in AMPS, there are some challenges in the AMPS data assimilation. A major challenge for data assimilation over Antarctica and the Southern Ocean is the lack of traditional meteorological observations. Extracting as much useful information as possible from observational data is critical for Antarctic weather analysis and prediction. Since the background error covariance plays an important role in most data assimilation systems, the flow-dependent background error covariance calculated from an ensemble forecast is expected to improve the performance of traditional variational approaches. Other than assimilating nonsynoptic observations, using more multitime observations in the assimilation window is an easy and straightforward way of increasing the amount of observational information.

An intercomparison of four different WRF variational approaches—3DVar, first guess at the appropriate time (FGAT; Rabier et al. 1998; Lawless 2010), En3DVar, and 4DEnVar—is conducted in this study. The intercomparison experiments will answer the following two questions. First, how does the flow-dependent background error covariance impact the 4DEnVar analysis and forecast? Second, does the increase in the number of observations from different times lead to improved performances of the variational approaches with the flow-dependent background error covariance in WRF-EnVar compared with the assumed homogeneous and isotropic covariance in WRF-Var? In addition, the performance of WRF-4DEnVar cycling is evaluated with the Antarctic cyclone case as a preliminary test for 4DEnVar on ensemble data assimilation and forecast. We carried out WRF-4DEnVar and FGAT 3-day cycling experiments and their corresponding 72-h forecast by using the analysis at the end of cycling window as its initial condition. For prediction of the unusual cyclone, which landed on 5 October in Antarctica and went back to the Southern Ocean on 8 October, comparison of the results from WRF-4DEnVar and FGAT can provide us with some preliminary insight into the newly developed data assimilation approach. To reduce the effect of sampling errors in the analysis of the ensemble-based sequential data assimilation approach, the inflation factor (Anderson and Anderson 1999) has been applied in EnKF. We added the inflation factor to 4DEnVar and tested it in the cycling experiment.

After successful implementation of the WRF-4DEnVar system in real case studies, short-period (10 day) analysis/forecast experiments with different data assimilation methods, including 4DEnVar, FGAT, and 3DVar, are conducted and their performances are evaluated. This is a further step toward the 4DEnVar system in real-time operations. Our results from both the case studies and the short-period executions indicate that 4DEnVar has great potential in future real-time applications.

The organization of the paper is as follows. The 4DEnVar basic algorithm is briefly reviewed in section 2. An overview of the synoptic cyclone formed in the Ross Sea in early October 2007 is given in section 3. The experimental designs, including assimilated observations and configuration of data assimilation and forecast model systems, are introduced in section 4. Section 5 presents the experimental results. The examination of background error covariance is presented in section 5a; the comparison among 3DVar, FGAT, En3DVar, and 4DEnVar is provided in section 5b; the short-term cycling experiments of FGAT and 4DEnVar are discussed in section 5c; and the 10-day experiments of 3DVar, FGAT, and 4DEnVar are examined in section 5d. The last section provides a summary and discussion.

2. Brief description of 4DEnVar

The column of perturbation matrix is the normalized deviations from the ensemble mean and estimated by N ensemble members:
e1
The background error covariance is approximately calculated by
e2
Referring to Part I, the cost function in control variable space is then formulated by
e3
where w is the control variable, I is the total number of time levels on which observations are available, is the tangent linear observation operator, is the tangent linear forecast model, and is the observation error covariance; d refers to innovations at different times (with subscript i).

Similar to 3/4DVar, if all the observations are simultaneous and no forecast model is involved in the formula of (3), the problem is reduced to En3DVar. If observations are from different times and the model is involved in the formula of (3), the problem becomes 4DEnVar.

The advantage of 4DEnVar is that we can reformulate the algorithm to avoid tangent linear and adjoint models in calculating the gradient of cost function. Introducing the perturbation matrix in observation space via
e4
the gradient of the cost function is then calculated by
e5
Using (5), the adjoint model T is elegantly avoided in the calculation of the cost function gradient by transformation of the background error to observation space using (4). Moreover, (4) indicates that 4DEnVar does not need linear approximation in the forward model and observation operators. After minimization iteration, the optimal analysis xa is obtained:
e6

According to (6), the analysis increment is a linear combination of the predicted ensemble perturbations. The coefficients of the linear combination w can be estimated by minimizing the cost function (3).

Like EnKF, the 4DEnVar is subject to the so-called sampling error in ensemble-based data assimilation. To relieve this problem, we applied the Schur operator for localization in the WRF-4DEnVar analysis and demonstrated that the localization technique can effectively alleviate the impacts of sampling errors upon analysis (Part II).

3. Synoptic overview of a cyclone in Antarctica

On 1 October 2007, a baroclinic disturbance was observed in the Ross Sea of the Southern Ocean. The disturbance developed to a synoptic cyclone that was clearly identifiable from mean sea level pressure on 2 October (as shown in Fig. 1a). The cyclone moved quickly along the ice sheet after 3 October with strengthened intensity. From 6 to 7 October, it crossed the West Antarctic ice sheet (WAIS) inland and began to weaken; it redeveloped on the Ronne Ice Shelf and reached the deepest central sea level pressure on 8 October. The cyclone maintained its strength for two days and started to fill after 10 October while it moved out of the domain. We chose this cyclone as our case study because its track is unusual compared to most Antarctic cyclones. It had a long life span and covered most of WAIS at its peak intensity. Moreover, it moved away from the Southern Ocean and across the WAIS. The locations of the cyclone at various times and its track are marked in Fig. 1a. There was strong cold advection present in Victoria Land and the Ross Sea, which pushed the cyclone to move quickly across the Southern Ocean.

Fig. 1.
Fig. 1.

(a) The SLP (black line), temperature (color shaded), and wind vectors at 850 hPa at 1200 UTC 3 Oct 2007. The locations of the cyclone at various times and its track are marked. (b) The corresponding geopotential heights and wind vectors at 500 hPa are shown. (A full wind barb represents 5 m s−1.)

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

4. Experimental design

a. Assimilated observations

Six types of observations [sounding, synoptic observations (SYNOP), pilot reports (PIREPS), ships, aviation routine weather report (METAR), and Quick Scatterometer (QuikSCAT) winds] were tested in this case study. There are some soundings and a fair amount of surface observations over the computational domain (Fig. 2). Table 1 lists the temporal distributions of different observations from 0900 to 1400 UTC 2 October. Most observations were taken at 1200 UTC 2 October. There are some surface observations and PIREPS from other observation times. Although data assimilation over the Antarctic is a challenge because of the limited number of in situ observations, it is expected that the analysis and the forecast can be improved by WRF-4DEnVar in this case study.

Fig. 2.
Fig. 2.

Snapshot of (a) sounding, (b) QuikSCAT, (c) ships, and (d) SYNOP observations at 1200 UTC 2 Oct available for assimilation.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

Table 1.

Observation numbers and types from 0900 to 1400 UTC 2 Oct 2007.

Table 1.

b. Configuration of data assimilation and forecast model systems

The construction of WRF-4DEnVar (Part II) is based on WRF-Var (Barker et al. 2004; Skamarock et al. 2008). Three packages—Gen_eib, Gen_pe, and Gen_po—are added to WRF-Var for implementing 4DEnVar (Fig. 3). The Gen_eib package generates the ensemble initial and boundary conditions, the Gen_pe package produces the ensemble perturbation matrix, and the Gen_po package perturbs the observations. The WRF-4DEnVar can provide a deterministic data assimilation and forecast as well as the corresponding ensemble ones for the same time, rather than only the deterministic data assimilation and forecast provided by the WRF-Var and the WRF. The deterministic data assimilation uses a single deterministic background state and observations to produce a single analysis at each analysis time. The ensemble data assimilation is implemented by running 4DEnVar for each member using each observation perturbed by random noise drawn from the appropriate distribution and a background consisting of one of the ensemble members. The deterministic forecast is updated by observations, whereas ensemble forecasts are updated by perturbed observations and provide the perturbation matrixes for 4DEnVar as well. We chose zonal wind component u, meridional wind component υ, temperature, and humidity as 4DEnVar analysis control variables. Note surface pressure is not selected as a control variable because it is significantly affected by surface boundary conditions and it is unclear if its perturbations are good enough within 6-h forecast.

Fig. 3.
Fig. 3.

A typical WRF-4DEnVar data assimilation flowchart. In the chart, WPS means WRF preprocessing system; OBS_Proc means WRF-Var observation processing; Little_R is a data format for observations; IC and BC mean initial and boundary conditions; Eobsi means the perturbed observations for each ensemble member; EXa means 4DEnVar ensemble analyses; Xa is 4DEnVar analysis; and Gen_eib, Gen_pe, and Gen_po are three new components specific for WRF-4DEnVar system.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

We conducted all experiments over a grid mesh of with grid spacings of 45 km, with the domain shown in Fig. 2. In the vertical direction, there are 44 layers, and the pressure at model top is 10 hPa.

The control run (CTRL) is conducted by integrating the first guess at 1200 UTC 2 October 2007 for 72 h. Because the first guess is a 6-h forecast, no observational information within this 6-h period is assimilated.

Random perturbations are added to the National Centers for Environmental Prediction Final Analyses (NCEP/FNL) on 0600 UTC 2 October to produce 38 ensemble members. These random perturbations are derived from the background error covariance of the WRF-3DVar data assimilation system, similar to the approach of the initial ensemble generating method mentioned in Houtekamer and Mitchell (2005). Therefore, the perturbations are consistent with the background error covariance defined by the WRF-3DVar data assimilation. We also perturbed boundary conditions with random normal perturbations. The ensemble initial conditions are integrated so that the perturbations are flow dependent. The background error covariances at the observation time are obtained from the statistics of perturbations. The FNL provided the first-guess fields and boundary conditions.

To perform localization, we adopt the correlation function operator (Part II). The cutoff distance for the correlation function is 1500 km. The EOF decomposition is applied to the correlation function operator so as to reduce the computational cost. The horizontal and vertical truncation modes are 40 and 10, respectively.

The ARW model has various options when it comes to selecting physical processes. Following AMPS, the physics schemes in all experiments include the Rapid Radiative Transfer Model (RRTM) based on Mlawer et al. (1997) for longwave radiation, the Goddard shortwave radiation scheme (Chou and Suarez 1999), Mellor–Yamada–Janjic turbulent kinetic energy (TKE) boundary layer scheme described by Janjic (1996, 2002), and the Kain–Fritsch (new Eta Model) scheme (Kain 2004).

5. Experimental results

a. Examination of background error covariance

Covariance is a measure of the correlation between two or more sets of random variables. The covariance for two random variables, X and Y, is defined by
e7
where is the correlation of X and Y, and is the variances of random variables. The background error covariance of WRF-3DVar is assumed to be a homogeneous and isotropic structure, which means that its correlation depends only on the distance and its variances are the same everywhere. As mentioned in section 4, the initial perturbations are produced by WRF-3DVar so that the structure of the error covariance calculated from them is consistent with the structure of the background error covariance from WRF-3DVar. Figure 4a shows the horizontal error correlation of temperature in response to a single point perturbation in a larger temperature gradient zone at the 11th model layer, which is approximately at 850 hPa. The temperature error variance at the same level is shown in Fig. 4c. The correlation field is almost isotropic and the variance field is almost homogeneous. However, the background error covariance is neither exactly isotropic nor homogeneous. First, the initial perturbation is generated by adding the normal unbiased perturbation to the streamfunction in WRF-3DVar so that the error variance structures of temperature and streamfunction are not exactly equal. Second, only 38 members are used to calculate the correlation and variance so that sampling errors can contaminate the correlation and variance structures.
Fig. 4.
Fig. 4.

The background temperature error correlation (color shaded) at the 11th level at (a) 0600 and (b) 1200 UTC 2 Oct 2007, and the corresponding background temperature error variance (color shaded) at (c) 0600 and (d) 1200 UTC. The black contours are the corresponding background forecasts of temperature. The location used for the correlation is marked by the black dot.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

Although the error covariance calculated from initial perturbations is almost isotropic and homogeneous, it will gradually become flow dependent after an ensemble forecast. The correlation and variance are calculated again after a 6-h ensemble forecast. Figures 4b,d show the correlation and variance extending along isotherms, which makes the observation information propagate in the flow direction rather than propagating isotropically so that the wrong correction by observations is avoided. The amplitude of the variance in the high temperature gradient region is significantly larger than in other regions because the ensemble spread is likely attributed to the growth of the perturbations due to the baroclinic instability associated with the temperature gradient. The flow-dependent variance makes the analysis rely more on observations than on background in the regions of lesser predictability. Figure 4 indicates that a 6-h ensemble forecast is enough to make the background error covariance depart significantly from the homogeneous and isotropic structure and become flow dependent.

b. Comparison of 3DVar, FGAT, En3DVar, and 4DEnVar

Four different WRF variational data assimilation experiments, including 3DVar, FGAT, En3DVar, and 4DEnVar, are conducted. The homogeneous and isotropic background error covariance is used in 3DVar and FGAT while the flow-dependent one calculated from ensemble forecast members is used in En3DVar and 4DEnVar. The assimilation window of 3DVar and En3DVar is set to 2 h, valid from 1100 to 1300 UTC 2 October. Both FGAT and 4DEnVar have a longer assimilation window, valid from 0900 to 1400 UTC 2 October, so that more observations can be assimilated. It is noted that both FGAT and 4DEnVar innovations are calculated from the background at the appropriate observation time while 3DVar and En3DVar assume all observations originating from the same time, 1200 UTC. Table 2 gives a summary of the differences between 3DVar, FGAT, En3DVar, and 4DEnVar setups.

Table 2.

Summary of the differences among 3DVar, FGAT, En3DVar, and 4DEnVar.

Table 2.

In this study, the FNL is used for validation for the WRF-3DVar, FGAT, En3DVar, and 4DEnVar analyses and WRF forecasts in Antarctic. First we compared the analysis increments of temperature (color shades) on the 11th model layer by the four WRF data assimilation methods at the initial time (Fig. 5). The temperature analysis increment of 3DVar has the smoothest pattern because of its homogeneous and isotropic background error covariance and observations within only a 2-h assimilation window being assimilated. Although the same covariance is applied in FGAT, the amplitude of its analysis increment is obviously larger than that of 3DVar due to a larger number of assimilated observations. Figures 5c,d show that the temperature analysis increments from En3DVar and 4DEnVar are aligned with the isolines of the background temperature field, which is consistent with the variance pattern in Fig. 4d.

Fig. 5.
Fig. 5.

The temperature analysis increments (color shaded) at the 11th level by (a) 3DVar, (b) FGAT, (c) En3DVar, and (d) 4DEnVar valid at 1200 UTC 2 Oct 2007. The contours are the corresponding background forecasts of temperature.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

The analysis increment from either 3DVar or En3DVar near the cyclone is very small because there are no observations in the area (Figs. 5a,c). When the assimilation window is extended, there are some observations in the upstream and downstream of the cyclone so that the analysis increment of FGAT or 4DEnVar is obviously larger in the area. Figures 5b,d indicate that both FGAT and 4DEnVar have positive increment in the upstream and negative increment in the downstream of the cyclone. As can be seen in the figures, the FGAT analysis increment near the cyclone is negative and that of 4DEnVar is positive. It is indicated that the 4DEnVar analysis increment near the cyclone is affected by upstream observations, but the FGAT analysis increment is mainly affected by downstream observations. Since observation information usually propagates downstream along the flow instead of upstream, the 4DEnVar analysis near the cyclone is more reasonable than the FGAT analysis. The flow-dependent matrix in this case gives a positive impact on the cyclone analysis.

Comparing CTRL to FNL reveals that the experiment CTRL produces colder temperatures in Victoria Land and a more intense cyclone (Fig. 6). The central sea level pressure (CSLP) of the storm in CTRL is 5 hPa lower than in FNL. After assimilating observations with 4DEnVar, some positive temperature analysis increments appear in Victoria Land (Fig. 5d), which correct the previous cold bias of CTRL in that area (Figs. 6b,c). Also, Fig. 6 shows the cyclone intensity in 4DEnVar is much closer to FNL than CTRL. Although the analysis increments of 3DVar, FGAT, and En3DVar are somewhat different from that of 4DEnVar as shown in Fig. 5, the patterns of the analysis fields in all experiments are similar, which is not shown separately.

Fig. 6.
Fig. 6.

The sea level pressure (black line), temperature (color shaded), and wind vectors at 850 hPa at 1200 UTC 2 Oct 2007 from (a) FNL, (b) CTRL, and (c) 4DEnVar.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

Even though the patterns of the synoptic analysis given by 3DVar, FGAT, En3DVar, and 4DEnVar are quite similar, the small differences among them can evolve rapidly with time integration so that the synoptic patterns given by subsequent forecasts can be significantly different. As shown by the analysis in Fig. 7a, the cyclone had moved to Amundsen Sea at 1200 UTC 5 October. However, CTRL fails to predict accurate position and intensity of the cyclone during 72-h forecast; the cyclone is predicted to be too strong in CTRL at 1200 UTC 5 October (Fig. 7b). After assimilating the observations through different data assimilation approaches, the CSLP at the initial time is closer to FNL and the 72-h forecast skill of cyclone intensity is improved (Figs. 7c–f). Especially note that the 4DEnVar experiment obtains significant improvement in CSLP prediction, much closer to FNL at 1200 UTC 5 October (Fig. 7f).

Fig. 7.
Fig. 7.

As in Fig. 6, but for 72-h forecasts from (a) FNL, (b) CTRL, (c) 3DVar, (d) FGAT, (e) En3DVar, and (f) 4DEnVar. The locations of the cyclone are marked with a black dot.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

In addition, the cyclone positions in the 72-h forecasts from En3DVar and 4DEnVar are both very close to that of FNL, whereas the cyclones predicted by 3DVar and FGAT seem to move faster than in FNL (as shown in Fig. 7). For temperature forecasts, it is indicated that the CTRL temperature in Antarctica is warmer, with its peak 10° higher than that of FNL. Similar to CTRL, FGAT fails to predict the correct pattern of the temperature in Antarctica as well. The temperature forecasts by other data assimilation approaches, including 3DVar, En3DVar, and 4DEnVar, are close to that of FNL.

Turning now to the differences of temperature analysis among the four assimilation schemes at the 11th level, when we examine the horizontal error distribution after a 72-h forecast in Fig. 8, we find that the major errors in 3DVar and FGAT are in the region of high temperature gradient, especially near the Amundsen Sea, close to the path of the cyclone. However, Figs. 8c,d show that the forecast error in the cyclone zone from En3DVar and 4DEnVar is far less than that from 3DVar and FGAT. As shown in Fig. 5, the major analysis increment of En3DVar and 4DEnVar is in the region of the high temperature gradient, where the background error variance is larger than elsewhere. In addition, the observation information is only propagated along the flow, and inappropriate correction is avoided by the observations out of the flow. Therefore, these features shown in the analysis increment fields indicate that the observation information is assimilated effectively, especially in the cyclone zone. In the forecast from En3DVar, there are still some positive errors aligned with the isolines of the background temperature field in the Amundsen Sea. However, those errors do not exist in the 4DEnVar analysis, which means that 4DEnVar has a more positive impact on the cyclone forecast than En3DVar. It is found that there are negative errors in Antarctica from 3DVar and En3DVar and positive errors from FGAT as well. However, the errors in Antarctica from 4DEnVar are smaller than those from the other experiments.

Fig. 8.
Fig. 8.

The temperature difference (color shaded) at the 11th level between FNL and (a) 3DVAR, (b) FGAT, (c) En3DVar, and (d) 4DEnVar valid at 1200 UTC 5 Oct 2007. The contours are the corresponding temperature from FNL.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

Based on the vertical profiles of domain-averaged 72-h-forecasting RMSEs (Fig. 9) and the variation of domain-averaged RMSEs (Fig. 10), which is calculated from the difference between FNL and the designed experiments, all data assimilation approaches have a positive impact on the forecast, and 4DEnVar has the best performance for the 72-h forecast among the experiments. At the initial time, the analysis RMSEs of En3DVar and 3DVar are similar, with the analysis accuracy of temperature and humidity of En3DVar slightly inferior to those of 3DVar. However, the forecasting errors in the 3DVar experiment increase much more quickly than those in En3DVar. FGAT assimilated more observations than 3DVar but yields an inferior analysis and forecast. Although both FGAT and 4DEnVar have the same observations assimilated, the accuracy of the analysis and forecast of 4DEnVar is far superior to that of FGAT. Since the observations and innovations are the same between FGAT and 4DEnVar, the main difference lies in the fact that FGAT assumes the background error covariance to be homogeneous and isotropic, while 4DEnVar uses the flow-dependent one calculated from ensemble members. Therefore, the observational information is not propagated appropriately by the assumed homogeneous and isotropic background error covariance in FGAT. In addition, FGAT can compute the correct innovation but still assumes that the innovation is valid at the middle of the window, while 4DEnVar considers the proper temporal treatment of the observations. As a consequence, more negative impact is added to the result when more observations are assimilated in FGAT. When the flow-dependent background error covariance is applied, however, the observation information can be extracted more appropriately, which leads to the enhanced performance of En3DVar and 4DEnVar.

Fig. 9.
Fig. 9.

The vertical profiles of domain-averaged RMSEs of (a) u winds, (b) υ winds, (c) temperature, and (d) humidity at 1200 UTC 5 Oct 2007. The results of CTRL, 3DVar, FGAT, En3DVar, and 4DEnVar are denoted by cross-line, dot–dash, dash, dot, and solid lines, respectively.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

Fig. 10.
Fig. 10.

The variation of domain-averaged RMSE of (a) u winds, (b) υ winds, (c) temperature, and (d) humidity. The line marks are as in Fig. 9.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

c. 4DEnVar cycling experiment

As a preliminary evaluation of the performance of 4DEnVar cycling, we carried out 3-day cycling of 4DEnVar for the Antarctic cyclone. Because our case study is based on the analysis and forecast in a limited regional domain, cycling beyond 3 days could cause serious problems because of an unmatched boundary from the FNL boundary conditions with the inner domain. In the cycling experiment, the forecast is updated 2 times a day. The analysis times are 0000 and 1200 UTC, respectively. The assimilation windows are from 2100 to 0300 UTC for the 0000 UTC analysis and from 0900 to 1500 UTC for the 1200 UTC analysis. During cycling, the analyses of ensemble members are also updated by perturbed observations. The ensemble forecast is conducted so that the matrix is produced for the next 4DEnVar analysis.

Because the infinite ensemble can never be obtained, the analysis from the ensemble-based data assimilation approach always contains noise due to the sampling errors. We have applied the Schur operator for localization in the WRF-4DEnVar system so that the impact of sampling errors upon analysis is alleviated (Part II). To reduce the impact of the sampling errors further, we added the inflation factor to 4DEnVar, which can be used to relax the filter divergence problem (Anderson and Anderson 1999). The perturbations in 4DEnVar are multiplied by the inflation factor so that the whole amplitude of background error variance can be changed without changing its structure.

We examined the variations of the domain-averaged forecast-analysis RMSEs as well as the responsive ensemble spread without and with the inflation factor (Fig. 11). It is indicated that the ensemble spread without the inflation factor is about 1.5 times larger than the RMSEs at the beginning time. As ensemble members are integrated, the ensemble spread is decreased to 80% of the RMSEs at the 12-h forecast time. At the end of the cycling period, the ensemble spread is only about 25% of the RMSEs.

Fig. 11.
Fig. 11.

The variations of the domain-averaged forecast-analysis RMSEs (asterisk-solid line) and the responsive ensemble spread (circle-dash line) without (blue line) and with (red line) the inflation factor for (a) u winds, (b) υ winds, (c) temperature, and (d) humidity.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

We apply the inflation factor to 4DEnVar analysis when the ensemble spread begins to be less than the RMSEs at the 12-h forecast time. Different inflation factors are used for different variables according to the reduction of the ensemble spread shown in Fig. 11. The inflation factors of the wind, temperature, and humidity are 1.15, 1.18, and 1.22, respectively. It is found that the ensemble spread is close to the RMSEs after the inflation factor is applied in 4DEnVar. Also, the analysis-forecast RMSEs are decreased by the inflating background covariance (e.g., the temperature analysis-forecast RMSEs at the 72-h leading time are 60% less than those without the inflation factor).

We use the cycling analysis on 1200 UTC 5 October as the initial condition to make a 72-h forecast. As described in section 3, the cyclone landed on 6 October and crossed the WAIS. On 8 October, it redeveloped on the Ronne Ice Shelf and moved into the ocean again. The best track is plotted in Fig. 12 according to the central sea level pressure from the FNL analysis. The track forecast from CTRL has an obvious northward bias. When the cyclone moved back into the Southern Ocean on 1200 UTC 8 October, the simulated cyclone from CTRL moves far slower than the analysis from FNL. The predicted tracks of the cyclone from FGAT and 4DEnVar are much closer to the track from FNL than that from CTRL. Especially, in the period from 1200 UTC 7 October to 1200 UTC 8 October, the predicted tracks of the cyclone based on 4DEnVar analysis are significantly better than those from other experiments.

Fig. 12.
Fig. 12.

Locations of the cyclone at various time (1200 UTC 5 Oct, 1200 UTC 6 Oct, 1200 UTC 7 Oct, and 1200 UTC 8 Oct 2007) from FNL, CTRL, 4DEnVar, and FGAT, respectively; their tracks are marked.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

The improvement of the forecast skill using 4DEnVar is also reflected in the intensity prediction of the cyclone. When the cyclone landed in Antarctica, its strength had not changed significantly. However, after the cyclone left the land, its intensity increased. As shown in Fig. 13, the intensity of the cyclone from CTRL is too strong and its tendency is also incorrectly predicted. Unlike the intensity tendency from the FNL, which has an increasing trend after the 48-h forecast, the forecasted intensity tendency from the CTRL shows a decreasing trend after the 48-h forecast. Although the predicted intensity from FGAT is better than that from the CTRL, its tendency is still not correctly predicted. The intensity prediction from 4DEnVar is much closer to the analysis from FNL, and 4DEnVar also correctly predicts the strengthening of the cyclone on 8 October.

Fig. 13.
Fig. 13.

Distribution of the CSLPs (hPa) of FNL, CTRL, FGAT, and 4DEnVar at 1200 UTC 2 Oct, 1200 UTC 3 Oct, 1200 UTC 4 Oct, and 1200 UTC 5 Oct 2007, respectively.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

d. A short-period (10 day) application of 4DEnVar in the Antarctic

To validate further the performance of 4DEnVar scheme in real data assimilation in the Antarctic, we conducted short-period (10 days, covering the lifetime of the cyclone in the domain) analysis and forecasting experiments with ARW-WRF. The WRF-4DEnVar, FGAT, and 3DVar systems over the same 10-day period, from 0000 UTC 1 October to 1200 UTC 10 October 2007, were also executed in parallel runs and their results compared. The configuration of data assimilation and model is the same as in the previous case study. The analyses from each experiment valid at 0000 and 1200 UTC each day are used to produce 72-h forecasts. The forecast is verified by comparing u winds, υ winds, temperature, and humidity with the FNL. The forecast RMSEs from 24, 48, and 72 h are shown in Figs. 14, 15, and 16, respectively. Since the 10-day period of the experimental application covers the entire cyclone lifetime, the forecast error of the cyclone is one of the major contributions of the RMSEs. The figures show that the RMSEs at 1200 UTC 2 October and 0000 UTC 3 October are obviously larger than those at other times. It explains that the model forecast is more uncertain when the initial condition is on the cyclone genesis stage. The 4DEnVar, in general, consistently has the lowest forecast RMSE at most forecast times during the 10-day period. The amplitude of RMSE for 3DVar, especially in days 3 and 4, is larger than others. The FGAT has a medium performance between the 4DEnVar and 3DVar in 24- and 48-h forecasts, but its performance is similar to 3DVar in the 72-h forecast. Obviously, WRF-4DEnVar obtains the best performance compared with other data assimilation techniques in this study.

Fig. 14.
Fig. 14.

The 24-h forecast domain-averaged RMSE of (a) zonal wind u, (b) meridional wind υ, (c) temperature T, and (d) specific humidity q.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

Fig. 15.
Fig. 15.

As in Fig. 14, but for the 48-h forecast.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

Fig. 16.
Fig. 16.

As in Fig. 14, but for the 72-h forecast.

Citation: Monthly Weather Review 141, 8; 10.1175/MWR-D-12-00130.1

6. Summary and discussion

The lack of traditional observations over Antarctica and the Southern Ocean is a major challenge for data assimilation in the region. Data assimilation in Antarctica and the Southern Ocean is expected to be improved via two ways. One is through the development of an advanced data assimilation technique that allows for full utilization of limited observations. The other is through assimilating more observations, especially nonsynoptic observations.

4DEnVar is formulated by including the flow-dependent background error covariance calculated from the ensemble members into the incremental and preconditioning variational algorithm. After adding some packages and modules into the WRF-Var system, WRF-4DEnVar system has been successfully developed and can be implemented with real observations.

Intercomparisons among four WRF variational approaches, including 3DVar, FGAT, En3DVar, and 4DEnVar, are carried out by examining different impacts of background error covariance and multitime observations on analysis and forecast. This study shows that all variational approaches can have a positive impact on the analysis and the forecast of an Antarctic cyclone, which suggests that the assimilation ability of variational approaches is promising even though few observations are included. When the flow-dependent background error covariance replaces the assumed homogeneous and isotropic background error covariance in WRF-3DVar, the analysis increment becomes flow dependent and the analysis and forecast accuracy are both improved. 4DEnVar, which has the flow-dependent background error covariance and uses multitime observations, leads to significant improvements of forecast quality. It is indicated that the variational approach with flow-dependent background error covariance can make a more promising analysis with more observations included.

By examining the development of background error covariance during the 6-h ensemble forecast, it is shown that a significant flow-dependent structure can be obtained after the 6-h forecast. Because only a short-term ensemble forecast is used, the computational cost of 4DEnVar is not significantly increased compared to WRF-3DVar and is far less than the computational cost of WRF-4DVar.

We conducted a comparison of cycling experiments using FGAT and 4DEnVar data assimilation approaches. Both FGAT and 4DEnVar improved the track and intensity prediction of the cyclone. However, 4DEnVar has better performance on the predicted tendency of intensity and 72-h track than did FGAT. The 4DEnVar cycling experiment also indicated the inflation factor can effectively improve the 4DEnVar performance.

By verifying the forecast of data assimilation experiments against the FNL, it is found that the 4DEnVar outperforms both the 3DVar and FGAT during the 10-day period, and the performance of FGAT is slightly better than that of 3DVar. This indicates that the 4DEnVar benefits from assimilating more observations and from its flow-dependent background error covariance.

Future research regarding WRF-4DEnVar will involve the following aspects. First, more detailed comparisons between WRF-4DVar and WRF-4DEnVar need to be carried out. The flow dependence of the background error covariance in 4DVar is implicitly developed, which made it interesting to investigate the difference between the implicitly flow-dependent error covariance from 4DVar and the explicitly flow-dependent error covariance from 4DEnVar. Second, the impact of 4DEnVar on ensemble forecast needs to be evaluated further. The WRF-4DEnVar provides not only the initial condition for a deterministic forecast but also a set of initial conditions for an ensemble forecast. Because 4DEnVar considers the impact of observations on the initial conditions of each ensemble member, it should be a good initialization technique that provides a set of better initial conditions for an ensemble forecast. In addition, how to conduct 4DEnVar cycling is a good topic for the future research. For the limited-area WRF, lateral boundary could be an issue with 4DEnVar cycling. It is certain that a period of restart should be necessary for the WRF-4DEnVar cycling. We will conduct further research in the area in the future.

Acknowledgments

We appreciate the discussions with Bill Kuo, Chris Snyder, Jordan Powers, Dale Barker, and Hans Huang (NCAR) when we proposed to conduct the experiments with the Antarctic Mesoscale Prediction System (AMPS). We are also grateful to Fuqing Zhang (Penn State) and Xuguang Wang (OU) who provided constructive suggestions of our initial results. This work was supported by NSF/OPP under Grants ANT-0839068. Computations were carried out using the IBM supercomputers at NCAR, and supported by NCAR's Computational and Information Systems Laboratory (CISL).

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