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    Fig. 1.

    Illustration of the hybrid ETKF–3DVAR analysis and the ensemble generation cycle for a hypothetical three-member ensemble.

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    Fig. 2.

    The WRF domain and the 6-hourly tracks of Ike during 0000 UTC 7 Sep–1200 UTC 13 Sep 2008 (•) and Gustav during 0000 UTC 27 Aug–1800 UTC 3 Sep 2008 (□). The data are obtained from the best-track data of the National Hurricane Center.

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    Fig. 3.

    A snapshot of the observations assimilated at 0000 UTC 7 Sep 2008: (a) QSCAT, (b) SYNOP, (c) SHIP, (d) METAR, (e) SOUND (including dropsondes), (f) BUOY, (g) PROFILER, (h) PILOT, (i) AIREP, (j) SATEM, and (k) SATOB. The naming conventions of these observations follow the World Meteorological Organization (WMO) standard (see www.nws.noaa.gov/tg/tableb1.html).

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    Fig. 4.

    Observation errors as a function of pressure for the SOUND wind (solid), the PROFILER and PILOT wind (dashed), and the SATEM temperature (dotted) observations.

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    Fig. 5.

    The 6-hourly analyzed tracks of HYBRID-Ens, 3DVAR, and HYBRID-Static for Ike during the data assimilation period. The best-track data are also shown as a reference.

  • View in gallery
    Fig. 6.

    The 6-hourly absolute errors of the analyzed tracks verified against the best-track data for HYBRID-Ens, 3DVAR, and HYBRID-Static for Ike during the data assimilation period.

  • View in gallery
    Fig. 7.

    RMS errors of the track forecasts from HYBRID-Static (dotted), HYBRID-Ens (solid), and 3DVAR (dashed) up to the 72-h lead time for forecasts initialized every 12 h during the assimilation period for Ike.

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    Fig. 8.

    Track forecasts initialized by the analyses at 0000 UTC Sep 9 2008 for HYBRID-Static, HYBRID-Ens, and 3DVAR for Ike. The best-track data and the WRF forecasts initialized by the GFS analysis (denoted as GFS) are shown as references.

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    Fig. 9.

    The 500-hPa geopotential height (m) for the (a) HYBRID-Ens and (b) 3DVAR analyses valid at 0000 UTC 9 Sep 2008 for Ike.

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    Fig. 10.

    The 48-h forecasts of 500-hPa geopotential height (m) initialized at 0000 UTC 9 Sep 2008 from the analyses generated by (a) HYBRID-Ens and (b) 3DVAR.

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    Fig. 11.

    East–west vertical cross section of the wind speed (m s−1) for the analyses at 0000 UTC 9 Sep 2008 for the (a) HYBRID-Ens and (b) 3DVAR analyses. The cross section cuts cross the maximum relative vorticity at 500 hPa.

  • View in gallery
    Fig. 12.

    The 500-hPa geopotential height increments (color shading) from (a) 3DVAR and (b) HYBRID-Ens, and (c) the 500-hPa geopotential height background ensemble spread (color shading) for HYBRID-Ens valid at 0000 UTC 8 Sep 2008 for Ike. The black contours in (a)–(c) are the corresponding background 500-hPa geopotential height fields valid at 0000 UTC 8 Sep 2008.

  • View in gallery
    Fig. 13.

    The 6-hourly analyzed tracks from HYBRID-Ens, 3DVAR, and HYBRID-Static for Gustav during the data assimilation period. The best-track data are also shown as a reference.

  • View in gallery
    Fig. 14.

    The 6-hourly absolute errors of the analyzed tracks verified against the best-track data from HYBRID-Ens, 3DVAR, and HYBRID-Static for Gustav during the data assimilation period.

  • View in gallery
    Fig. 15.

    The RMS errors of the track forecasts from HYBRID-Static (dotted), HYBRID-Ens (solid), and 3DVAR (dashed) up to 72-h lead time for forecasts initialized every 12 h during the assimilation period for Gustav.

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    Fig. 16.

    The 500-hPa geopotential height (m) for the (a) HYBRID-Ens and (b) 3DVAR analyses valid at 0000 UTC 29 Aug 2008 for Gustav.

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    Fig. 17.

    Locations of dropsondes in the vicinity of Gustav during 0000–1800 UTC 29 Aug: • denotes the dropsonde around 0009 UTC 29 Aug and △ denotes the dropsondes around 1800 UTC 29 Aug. Also shown are the Gustav locations in the background forecast from 3DVAR valid at 0009 UTC 29 Aug (○) and at 1800 UTC 29 Aug (▽).

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    Fig. 18.

    The 500-hPa increments (color shading) from the (a) 3DVAR and (b) HYBRID-Ens analyses valid at 0000 UTC 31 Aug 2008. The black contours are the corresponding background forecasts of 500-hPa geopotential height (m).

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Application of the WRF Hybrid ETKF–3DVAR Data Assimilation System for Hurricane Track Forecasts

Xuguang WangSchool of Meteorology, University of Oklahoma, and Center for Analysis and Prediction of Storms, Norman, Oklahoma

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Abstract

A hybrid ensemble transform Kalman filter (ETKF)–three-dimensional variational data assimilation (3DVAR) system developed for the Weather Research and Forecasting Model (WRF) was studied for the forecasts of the tracks of two major hurricanes, Ike and Gustav, in 2008 over the Gulf of Mexico. The impacts of the flow-dependent ensemble covariance generated by the ETKF were revealed by comparing the forecasts, analyses, and analysis increments generated by the hybrid data assimilation method with those generated by the 3DVAR that used the static background covariance. The root-mean-square errors of the track forecasts by the hybrid data assimilation (DA) method were smaller than those by the 3DVAR for both Ike and Gustav. Experiments showed that such improvements were due to the use of the flow-dependent covariance provided by the ETKF ensemble in the hybrid DA system. Detailed diagnostics further revealed that the increments produced by the hybrid and the 3DVAR were different for both the analyses of the hurricane itself and its environment. In particular, it was found that the hybrid, using the flow-dependent covariance that gave the hurricane-specific error covariance estimates, was able to systematically adjust the position of the hurricane during the assimilation whereas the 3DVAR was not. The study served as a pilot study to explore and understand the potential of the hybrid method for hurricane data assimilation and forecasts. Caution needs to be taken to extrapolate the results to operational forecast settings.

Corresponding author address: Dr. Xuguang Wang, School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Norman, OK 73072. E-mail: xuguang.wang@ou.edu

Abstract

A hybrid ensemble transform Kalman filter (ETKF)–three-dimensional variational data assimilation (3DVAR) system developed for the Weather Research and Forecasting Model (WRF) was studied for the forecasts of the tracks of two major hurricanes, Ike and Gustav, in 2008 over the Gulf of Mexico. The impacts of the flow-dependent ensemble covariance generated by the ETKF were revealed by comparing the forecasts, analyses, and analysis increments generated by the hybrid data assimilation method with those generated by the 3DVAR that used the static background covariance. The root-mean-square errors of the track forecasts by the hybrid data assimilation (DA) method were smaller than those by the 3DVAR for both Ike and Gustav. Experiments showed that such improvements were due to the use of the flow-dependent covariance provided by the ETKF ensemble in the hybrid DA system. Detailed diagnostics further revealed that the increments produced by the hybrid and the 3DVAR were different for both the analyses of the hurricane itself and its environment. In particular, it was found that the hybrid, using the flow-dependent covariance that gave the hurricane-specific error covariance estimates, was able to systematically adjust the position of the hurricane during the assimilation whereas the 3DVAR was not. The study served as a pilot study to explore and understand the potential of the hybrid method for hurricane data assimilation and forecasts. Caution needs to be taken to extrapolate the results to operational forecast settings.

Corresponding author address: Dr. Xuguang Wang, School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Norman, OK 73072. E-mail: xuguang.wang@ou.edu

1. Introduction

Hybrid ensemble Kalman filter–variational data assimilation (DA) methods have recently been proposed (e.g., Hamill and Snyder 2000; Lorenc 2003; Etherton and Bishop 2004; Zupanski 2005; Wang et al. 2007b), implemented, and tested with the real numerical weather prediction (NWP) models and real data (e.g., Buehner 2005; Wang et al. 2008b; Buehner et al. 2010a,b). Compared to a typical variational method (VAR), instead of using a static error covariance, the ensemble covariance is incorporated into the VAR framework to provide a flow-dependent estimate of the background error covariance; the ensemble is generated by a version of the ensemble Kalman filters (EnKF).1

Recent studies have suggested that the hybrid DA systems may yield the “best of both worlds” by combining the best aspects of the variational and the EnKF systems. Studies by, for example, Wang et al. (2007a), Wang et al. (2008a,b, 2009), and Buehner et al. (2010a,b) demonstrated that the hybrid method can improve upon a stand-alone VAR system because of the inclusion of the flow-dependent ensemble covariance; these studies also suggested that a hybrid system improved upon the standalone EnKF system for relatively small ensembles. In addition, a study by Campbell et al. (2010) also suggested that the hybrid system may potentially improve the utilization of observations with nonlocal forward operators such as the satellite radiances because the hybrid method uses the model space covariance localization rather than the observation space covariance localization. Another advantage of the hybrid system is that since the hybrid adopts the variational framework, the dynamic constraint can be conveniently added during the data assimilation. The hybrid system is also attractive for the operational NWP centers because it is built on the existing operational variational framework so that the established capabilities in VAR (e.g., variational quality control, dynamical constraint) can be easily adopted. The above advantages of the hybrid system were also discussed in Wang (2010). Therefore, several operational centers have started to develop and test hybrid DA systems (Buehner et al. 2010a,b; Wang 2010; D. Barker 2010, personal communication; Bishop and Hodyss 2011).

Initial tests of the hybrid DA system with real NWP models and data have shown the great potential of the method (e.g., Wang et al. 2008b; Buehner et al. 2010a,b). However, to the author’s best knowledge, to date there have been few published studies on testing the hybrid DA method for hurricane predictions, which is one of the important missions in operational NWP, especially for those countries that are prone to hurricane-induced damage. While the development of a hybrid system based on the operational systems is on going (e.g., Wang 2010), this study served as a pilot study where we used the hybrid ensemble transform Kalman filter (ETKF)–three-dimensional variational data assimilation (3DVAR) system developed for the Weather Research and Forecasting Model (WRF; Wang et al. 2008a) to explore the potential of a hybrid DA method for hurricane track forecasts. The potential was revealed by demonstrating the fundamental differences between the static and ensemble covariances during the assimilation and their impacts on the subsequent forecasts of the hurricanes.

In the hybrid ETKF–3DVAR data assimilation system for WRF, we chose to use the ETKF to generate the ensemble perturbations. In the ETKF, the ensemble perturbations were generated by solving the Kalman filter equations in the ensemble subspace without the covariance localization. Early studies (Wang and Bishop 2003; Wang et al. 2007a, 2008a,b, 2009; Hacker et al. 2011) suggested that while the ETKF can generate the ensemble perturbations in a relatively less computationally expensive fashion compared to, for example, a singular vector method (Palmer et al. 1998) and an EnKF with covariance localization (e.g., Whitaker and Hamill 2002), it can still produce relatively skillful ensembles. In this paper we further examine if the flow-dependent ensemble covariance generated by the ETKF is a useful estimate of the background error covariance for the hurricane data assimilation, which has not been studied before.

One of the limiting factors of the hurricane track forecasts is the initial condition error associated with both the hurricanes and their environments. For the hurricane forecasts, the number of direct in situ observations that sample the hurricanes and their environments, especially when the hurricanes are far over the ocean, is limited. Previous studies [e.g., a hybrid DA experiment in Hamill and Snyder (2000)] suggested that in such a scenario data assimilation methods using a flow-dependent ensemble covariance were more appropriate than those using a static covariance to spread the observational information to correct the background forecast. Before the emergence of the ensemble-based DA techniques such as the EnKF where flow-dependent ensemble covariance was adopted, many modeling studies adopted the vortex relocation and/or bogusing (e.g., Liu et al. 2000; Kurihara et al. 1995; Zou and Xiao 2000) techniques to initialize the hurricane forecast. While such techniques have been shown to improve the hurricane forecast, they are limited by their assumptions. How to maintain the dynamical and thermodynamical coherency of the hurricane and its environment while the vortex is relocated is also nontrivial and needs further studies. Several recent studies employing the EnKF to initialize the hurricane forecasts have shown great promise (e.g., Torn and Hakim 2009; Zhang et al. 2009; Li and Liu 2009; Hamill et al. 2011). In these studies, no covariance localization or bogusing was applied. These studies further motivated this work, where we explore the potential of the hybrid ETKF–3DVAR method for hurricane analyses and forecasts. Note that in the operational data assimilation system, the 3DVAR approach is accompanied by the relocation technique to improve the hurricane analysis generated by the 3DVAR (Liu et al. 2000). However, since this pilot study was designed to help us to understand the fundamental differences of the hybrid method and the 3DVAR method for the hurricane data assimilation and forecast, we did not conduct the vortex relocation in our experiments.

This study is among the first that we are aware of that studies the hybrid ETKF–3DVAR method for hurricane data assimilation. As an initial attempt to explore the potential of the hybrid DA method for hurricane predictions, we focus on the hurricane track forecast. A study using the hybrid data assimilation system for WRF to assimilate the radar data to examine the hurricane intensity forecast is being conducted and will be reported upon in a forthcoming paper (Li et al. 2011). The impacts of the flow-dependent ensemble covariance on the data assimilation and the forecast of the hurricanes were explored by comparing the forecasts and analyses generated by the hybrid method with those of the 3DVAR approach through the diagnostics of two major hurricanes, Ike and Gustav, in 2008 over the Gulf of Mexico. The results of this study motivated an ongoing experiment, where we are developing and testing a hybrid data assimilation system based on the National Oceanic and Atmospheric Administration’ s operational gridpoint statistical interpolation (GSI) data assimilation system for a season’s hurricane forecasts. The results of this experiment with more cases will be reported upon in future papers. For GSI-based hybrid data assimilation algorithm, please refer to Wang (2010).

In section 2, the hybrid ETKF–3DVAR data assimilation system for WRF is briefly described. Section 3 introduces the experiment design. The results of the track analyses and forecasts for Ike and Gustav (2008) are presented in section 4. Section 5 provides the conclusions and a discussion.

2. The hybrid ETKF–3DVAR for WRF

The detailed description of the hybrid ETKF–3DVAR data assimilation system developed for WRF was documented in Wang et al. (2008a). As shown in Fig. 1, the following four steps were repeated for each data assimilation cycle: 1) Update the ensemble mean by the hybrid ETKF–3DVAR method where the flow-dependent ensemble covariance is provided by the ensemble generated by the ETKF. 2) Update the forecast perturbations using the ETKF. 3) Add the updated ensemble perturbations in 2) to the updated ensemble mean in 1) to generate K initial ensemble members. 4) Make K forecasts starting from the K initial ensemble members forward to the next analysis time and repeat from step 1).

Fig. 1.
Fig. 1.

Illustration of the hybrid ETKF–3DVAR analysis and the ensemble generation cycle for a hypothetical three-member ensemble.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

One of the critical elements in the hybrid DA is the method to incorporate the ensemble covariance in the VAR framework (the thick black box in Fig. 1). In step 1), during the variational minimization, the flow-dependent ensemble covariances were incorporated into the VAR framework by extending the control variables. For the mathematical details on this approach, please see Eqs. 1–5 of Wang et al. (2008a). As proved in Wang et al. (2007b), effectively, this approach is equivalent to replacing the background error covariance in a traditional 3DVAR with a weighted average of the static covariance and the ensemble covariance. The relative weights given to the ensemble covariance and the static covariance are determined by a tunable parameter 1/β1. Another tunable parameter S is also set in the system to determine the scale of the covariance localization for the ensemble covariance. The meaning of 1/β1 and S is further explained in the next section where the experiment design is described.

In the hybrid ETKF–3DVAR system, the ETKF was adopted to generate the ensemble (Wang and Bishop 2003; Wang et al. 2004, 2007a, 2008a,b, 2009). In this system, the ETKF was used to generate ensemble perturbations around the updated mean state. The ETKF updated the forecast ensemble perturbations to obtain the analysis ensemble perturbations through a transformation matrix. The transformation matrix was derived within the ensemble perturbation subspace. Assuming the covariance of the raw forecast ensemble perturbations were equal to the true forecast-error covariance, the goal of the ETKF was to choose the transformation matrix so that the outer-product of the transformed perturbations were equal to the true analysis error covariance. Equation (9) in Wang et al. (2008a) and references therein provided the latest formula for the ETKF. There were two parameters in the ETKF that were intended to ameliorate the systematic underestimate of the analysis-error variance because of the limited ensemble size. One was an inflation factor (π) and the other was the factor (ρ) that account for the fraction of the forecast error variance projected onto the ensemble subspace. Both parameters were determined adaptively within the ETKF algorithm. Wang et al. (2007b) provided details on how to estimate these two factors adaptively for each data assimilation cycle using the innovation statistics. Table 1 lists an example of the averaged parameters π and ρ adaptively determined in one set of the Ike and Gustav experiments. Note that to maintain the computational efficiency, the ETKF update was performed in the low dimensional ensemble subspace and no covariance localization was applied. Therefore the ETKF alone has not been applied for data assimilation so far due to the known filter divergence problem (Hamill et al. 2001). So far the ETKF has been mainly applied for targeting observation and for generating ensemble perturbations either for ensemble forecasting purpose or to be used in a hybrid DA system (e.g., Bishop et al. 2001; Majumdar et al. 2002; Wang and Bishop 2003; Wang et al. 2008a,b; Hacker et al. 2011).

Table 1.

The averaged inflation factor (π) and fraction factor (ρ) adaptively determined in the ETKF algorithm in the HYBRID-Ens experiment. The covariance localization scale (S) and the weighting factor 1/β1 adopted in the HYBRID-Ens experiment are also shown.

Table 1.

3. Experiment design

Experiments were performed running version 3.1 of the WRF (Skamarock et al. 2005), with the model domain shown in Fig. 2. The model was configured to have 30-km horizontal grid spacing and 35 vertical levels. The WRF physics components were the WRF single-moment five-class (WSM) microphysics scheme (Hong et al. 2004), the Yonsei University (YSU) boundary layer scheme (Hong et al. 2006), the Kain–Fritsch cumulus parameterization scheme (Kain and Fritsch 1990), the Rapid Radiative Transfer Model (RRTM) longwave radiation scheme (Mlawer et al. 1997), and the Dudhia shortwave scheme (Dudhia 1989).

Fig. 2.
Fig. 2.

The WRF domain and the 6-hourly tracks of Ike during 0000 UTC 7 Sep–1200 UTC 13 Sep 2008 (•) and Gustav during 0000 UTC 27 Aug–1800 UTC 3 Sep 2008 (□). The data are obtained from the best-track data of the National Hurricane Center.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Conventional in situ data and satellite-derived wind and temperature2 data available from the Global Telecommunications System (GTS) and archived in real time at the National Center for Atmospheric Research were assimilated. Figure 3 shows a snapshot of the distribution of each type of observation. Table 2 and Fig. 4 shows the corresponding observation errors adopted. Observations were preprocessed using the observation preprocessor developed for the WRF-VAR system outlined in Barker et al. (2003). Such sources of data and data-processing procedures have been adopted in early published studies (e.g., Xiao et al. 2009a,b). Note that dropsondes near the hurricane and in the remote environment of the hurricane that were archived in real time were also assimilated. (Figure 3e includes dropsondes for Ike at 0000 UTC 7 September. Figure 17 shows an example of the dropsonde data for Gustav.) For all experiments conducted, the same sets of observations were assimilated and the DA was performed every 3 h. For the experiment using the hybrid method, a 32-member ensemble was used. As in Wang et al. (2008b), the initial ensemble at the very beginning of the data assimilation cycles and the (lateral boundary condition, LBC) ensembles during the cycles were generated by adding 32 random perturbations to the National Centers for Environmental Prediction (NCEP) final analyses (FNL; information online at https://dss.ucar.edu/datazone/dsszone/ds083.2). Following Torn et al. (2006), these perturbations were drawn from a normal distribution having the same covariance as the 3DVAR static background-error covariance.

Fig. 3.
Fig. 3.

A snapshot of the observations assimilated at 0000 UTC 7 Sep 2008: (a) QSCAT, (b) SYNOP, (c) SHIP, (d) METAR, (e) SOUND (including dropsondes), (f) BUOY, (g) PROFILER, (h) PILOT, (i) AIREP, (j) SATEM, and (k) SATOB. The naming conventions of these observations follow the World Meteorological Organization (WMO) standard (see www.nws.noaa.gov/tg/tableb1.html).

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Table 2.

Assimilated observation types, variables, and their corresponding observation errors (numbers below). The leftmost column denotes the observation types using the WMO GTS standard (information online at http://www.nws.noaa.gov/tg/tableb1.html). The errors in the wind observations for SOUND, PROFILER, and PILOT, as well as the errors in the temperature observations for SATEM, shown in the table are vertically averaged values. Their corresponding vertical profiles of observation errors are shown in Fig. 4.

Table 2.
Fig. 4.
Fig. 4.

Observation errors as a function of pressure for the SOUND wind (solid), the PROFILER and PILOT wind (dashed), and the SATEM temperature (dotted) observations.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Since the default 3DVAR static covariance available from the WRF release may not be the optimal model of the static background-error covariance suitable for the current data assimilation experiment, we recalculated the static error covariances using the WRF–ETKF background-forecast ensembles following the same method used by Wang et al. (2008b). Specifically, we ran the ETKF ensemble over a 1-week period during the 2005 hurricane season for the same domain as shown in Fig. 2. Like the hurricane season of 2008, the hurricane season of 2005 also featured a number of storms that made landfall over the United States. The following steps were adopted to generate the ensemble perturbations for calculating the static covariance. The ensemble-mean background forecast was updated by the WRF–3DVAR model, using the default static error covariance. The ensemble perturbations were updated by the ETKF. The updated perturbations were then added to the updated ensemble mean to generate an ensemble of analyses and started ensemble forecasts. The procedures were repeated for 1 week and the ensemble perturbations collected were used to calculate the static covariance following Skamarock et al. (2005). Briefly, the calculation of the static covariance was split into several stages including the calculation of the eigenvectors and eigenvalues of the vertical background errors, the calculation of the balance regression statistics, and the calculation of recursive filter length scales. The above steps in calculating the static covariance were also adopted in earlier studies, such as Wang et al. (2008a,b) and Meng and Zhang (2008). After obtaining the newly calculated static covariance, the correlation length scale and the variance of the new static covariance were further varied by increasing and decreasing the scale and the magnitude of the variance relative to their original values. Then, the 3DVAR experiments were rerun with these new static covariances. It was found that the accuracy of the track analyses of the 3DVAR experiments using the tuned static covariances was not significantly different from the default static covariances (not shown). Early studies by Wang et al. (2008b) and Meng and Zhang (2008) also found the newly generated static covariance either performed slightly better than or was comparable to the default static covariance for their analyses and forecasts over the continental U.S. domain. In all of the following experiments, the newly generated static covariance was used.

As discussed earlier, as an initial attempt to reveal the potential of the hybrid method for hurricane data assimilation and prediction, the hybrid method was applied for the prediction of two major hurricanes in 2008, Ike and Gustav. They both reached category 4 status at their peak intensity and made landfall in the United States after passing through the Gulf of Mexico. Both Ike and Gustav caused many deaths and extensive damage along their paths (information online at http://www.nhc.noaa.gov/2008atlan.shtml). Table 3 gives the starting and ending dates of the data assimilation cycling periods for Ike and Gustav in this study. Figure 2 shows the corresponding paths from the best-track data available from the National Hurricane Center (information online at http://www.nhc.noaa.gov/2008atlan.shtml).

Table 3.

Starting and ending dates of the data assimilation cycling periods for Ike and Gustav.

Table 3.

As described in section 2, there were two tunable parameters in the hybrid method. One was the weighting factor, 1/β1, which determined the weight placed on the static covariance; 1/β1 = 1 meant the full weight was placed on the static covariance and 1/β1 = 0 meant the full weight was placed on the ensemble covariance. Another parameter was the covariance localization scale S, which determined the extent that the localization was applied on the ensemble covariance. To reveal the fundamental difference of the hybrid and the 3DVAR, in the following the results of three experiments are shown. The first experiment, denoted as HYBRID-Ens, was run with parameters 1/β1 = 0 and S = 1500 km;3 the second experiment, denoted as 3DVAR, was the traditional 3DVAR method. There were two major differences between HYBRID-Ens and 3DVAR. One was that the former adopted the flow-dependent ensemble covariance as the background covariance, whereas the 3DVAR adopted the static covariance. The second was that the former, as in other ensemble-based data assimilation methods, used the mean or average of the ensemble forecasts as the background forecast, whereas the 3DVAR used a single deterministic forecast as the background forecast. To isolate the impacts of these two differences on the performance of the 3DVAR and the HYBRID-Ens, we included the third experiment, denoted as HYBRID-Static. The only difference between the HYBRID-Static and the 3DVAR simulations was that in the former the ensemble mean was used as the background whereas 3DVAR used the single deterministic forecast as the background. The experiment HYBRID-Static was conducted by simply setting 1/β1 = 1. If the HYBRID-Static and the 3DVAR results were similar to each other and were both different from those of HYBRID-Ens, then it was the flow-dependent ensemble covariance that made the difference between HYBRID-Ens and 3DVAR. In addition to the above three experiments, we also tried several experiments with different combinations of S and 1/β1, such as 1/β1 = 0, 0.5, and 1 and S = 250, 500, 1000, and 1500 km. Our evaluation showed that the performances of these experiments did not significantly improve upon the results of the experiment HYBRID-Ens. Therefore, we focus on the comparison between HYBRID-Ens and 3DVAR in this paper.

As mentioned in section 2, there were two parameters inherent in the ETKF: the inflation factor and the fraction factor. The inflation factor defined the amount of inflation needed in order to make the spread of the ensemble consistent with the forecast errors on average. The fraction factor defined the percentage of the forecast error variance explained by the ensemble subspace. Both were estimated adaptively in the space of observed variables in the ETKF. For details, please refer to Wang et al. (2007a). Table 1 shows the averaged inflation factor and the fraction factor adaptively determined in the HYBRID-Ens experiments for both Ike and Gustav. On average, for the HYBRID-Ens experiments, inflation factors of 6.70 and 7.22 were used for Ike and Gustav, respectively. Since covariance localization was not used in the ETKF approach, the inflation factor for the ETKF method was in general larger than that typically used by EnKF with covariance localization. Fraction factor of 39.96% and 38.94%, which meant the ensemble subspace4 explained 39.96% and 38.94% of the total 3-h forecast error variance in the observation space, were used for Ike and Gustav, respectively. Since the inflation factor and the fraction factor were determined adaptively, their values reflected the quality of the ensemble in representing the forecast errors. The magnitudes of the inflation factor and the fraction factor depend on the configuration of the ensemble (e.g., the ensemble size), the observation network, and the dynamics of the system being forecasted. Examples of such factors in other applications can be found in Wang et al. (2007a, 2008a,b, 2009). In the current real-observation experiments, the inflation factor accounted not only for the ETKF’s systematic underestimation of the error variance owing to the limited ensemble size, but also for other misrepresented error sources such as errors from the model.

To evaluate the skill of the track forecasts, we initialized deterministic forecasts using the analyses generated by different DA methods. The forecasts were initialized every 12 h during the data assimilation cycles. The statistics of the track forecast errors were collected by using these forecasts initialized after 2 days’ worth of data assimilation cycles. The root-mean-square (RMS) track forecast errors up to 72-h forecast lead times were calculated by comparing the forecast against the best-track data. A 2-day spinup period was adopted following Torn (2011) so that the ensemble had little memory of the initial randomly generated ensemble.

4. Results

In this section, we evaluate the skill of the track analyses and forecasts from the HYBRID-Ens, HYBRID-Static, and 3DVAR experiments. Diagnostics were conducted to understand how the differences in the analyses could contribute to the difference in the forecasts and how different the analyses increments were in different experiments. The goal of such diagnostics was to facilitate our understanding of the differences among the data assimilation schemes and how such differences contributed to the differences in their performance. The results for Ike and Gustav are presented in sections 4a and 4b, respectively.

a. Results for Ike

Figures 5 and 6 show, respectively, the analyzed tracks and the absolute errors of the analyzed tracks for Ike every 6 h when using the 3DVAR, HYBRID-Static, and HYBRID-Ens. The analyzed tracks from the three experiments all aligned with the best-track data in general. The root-mean-square errors of the track forecasts (Fig. 7) showed that the track forecasts of the HYBRID-Ens on average were more accurate than both 3DVAR and the HYBRID-Static, especially for forecast lead times longer than 2 days. Figure 7 also shows that the accuracy of the track forecasts from HYBRID-Static was similar to that of 3DVAR. These results suggested that the improvement of the HYBRID-Ens forecast relative to 3DVAR was mainly a result of the use of the flow-dependent ensemble covariance as opposed to a static covariance to estimate the background error covariance, rather than the use of ensemble averaging in place of a single forecast as the background forecast. Further examining the forecasts where 3DVAR was significantly worse than HYBRID-Ens, we found that the 3DVAR track forecasts had a significant westward bias whereas the HYBRID-Ens track forecast aligned with the best-track data much better. Figure 8 showed an example of such forecasts initialized with the analyses at 0000 UTC 9 September 2008. Note that many of the operational forecasts from the National Hurricane Center also showed a westward bias over the western Gulf of Mexico several days before landfall (information online at http://www.nhc.noaa.gov/pdf/TCR-AL092008_Ike_3May10.pdf). The WRF forecast initialized by the analyses of the operational Global Forecast System (GFS) model was similarly biased toward the west and is shown in Fig. 8 as a reference.

Fig. 5.
Fig. 5.

The 6-hourly analyzed tracks of HYBRID-Ens, 3DVAR, and HYBRID-Static for Ike during the data assimilation period. The best-track data are also shown as a reference.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Fig. 6.
Fig. 6.

The 6-hourly absolute errors of the analyzed tracks verified against the best-track data for HYBRID-Ens, 3DVAR, and HYBRID-Static for Ike during the data assimilation period.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Fig. 7.
Fig. 7.

RMS errors of the track forecasts from HYBRID-Static (dotted), HYBRID-Ens (solid), and 3DVAR (dashed) up to the 72-h lead time for forecasts initialized every 12 h during the assimilation period for Ike.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Fig. 8.
Fig. 8.

Track forecasts initialized by the analyses at 0000 UTC Sep 9 2008 for HYBRID-Static, HYBRID-Ens, and 3DVAR for Ike. The best-track data and the WRF forecasts initialized by the GFS analysis (denoted as GFS) are shown as references.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

What were the differences in the analyses at 0000 UTC 9 September 2008 generated by the different DA methods that contributed to their differences in the track forecasts of Ike? Since the accuracy of the hurricane track forecasts depended both on the analysis of the hurricane itself and the environment that the hurricane was embedded in, analyses for both the hurricane itself and its environment were examined. Figure 9 shows the 500-hPa geopotential height analyses at 0000 UTC 9 September 2008 from HYBRID-Ens and 3DVAR. Ike was embedded in the southwest periphery of the subtropical high. The subtropical high in the 3DVAR analysis extended more to the south in the southwest quadrant of Gulf of Mexico than did that in HYBRID-Ens. This difference in the 3DVAR and HYBRID-Ens analyses suggested that in Ike’s environment analyzed by the 3DVAR there was a stronger easterly wind. This stronger easterly could contribute to the westward bias in the subsequent forecasts of 3DVAR. To further confirm this hypothesis, following Chan and Gray (1982), we calculated the averaged hurricane environmental wind 5°–7° from the center of the hurricane at 500 hPa. It was found that the easterly wind around Ike in 3DVAR was 1.23 m s−1 stronger than that in HYBRID-Ens. Figure 10 shows the forecast of the 500-hPa geopotential height at 48-h lead time. The westward bias of the track forecast by 3DVAR was associated with a stronger subtropical high forecast compared to HYBRID-Ens. Consistently, a recent study by Brennan and Majumdar (2011) examining the track forecast errors from the operational model also hypothesized that the westward bias was a result of the ridge to the north of Ike being too strong. In addition to the differences in the analyzed environment, Ike itself was also analyzed differently by 3DVAR and the HYBRID-Ens. Figure 11 showed the vertical cross section of the wind speed of Ike in the 3DVAR and HYBRID-Ens analyses at 0000 UTC 9 September 2008. The Ike analyzed by 3DVAR was smaller in size and less intense than that by examined HYBRID-Ens, which was further confirmed by verifying the sea level pressure and the vertical cross section of the relative vorticity (not shown). This difference in size and intensity led to a weaker beta drift in 3DVAR than in HYBRID-Ens (Smith 1993), which could also contribute to the relative westward track forecast by 3DVAR as compared to HYBRID-Ens.

Fig. 9.
Fig. 9.

The 500-hPa geopotential height (m) for the (a) HYBRID-Ens and (b) 3DVAR analyses valid at 0000 UTC 9 Sep 2008 for Ike.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Fig. 10.
Fig. 10.

The 48-h forecasts of 500-hPa geopotential height (m) initialized at 0000 UTC 9 Sep 2008 from the analyses generated by (a) HYBRID-Ens and (b) 3DVAR.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Fig. 11.
Fig. 11.

East–west vertical cross section of the wind speed (m s−1) for the analyses at 0000 UTC 9 Sep 2008 for the (a) HYBRID-Ens and (b) 3DVAR analyses. The cross section cuts cross the maximum relative vorticity at 500 hPa.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

To further understand why the analyses at 0000 UTC 9 September 2008 from the 3DVAR and HYBRID-Ens simulations were different, we examined the analysis increments before reaching this time. The increments of 3DVAR and HYBRID-Ens were different for both the hurricane and its large-scale environment. Figure 12 shows a representative example at 0000 UTC 8 September 2008 for the 500-hPa geopotential height. For HYBRID-Ens, the absolute increment for Ike was larger than that for the environment (Fig. 12b), whereas for 3DVAR the absolute increment for Ike was less than or comparable to the increment for the environment (Fig. 12a). Figure 12c shows the spread of the background ensemble of the 500-hPa geopotential height in HYBRID-Ens. The spread suggests that the ETKF-estimated forecast uncertainty was larger around Ike than its environment, which was consistent with the larger increment around Ike than its environment in HYBRID-Ens (Fig. 12b). For 3DVAR, the increment for the environment featured an increase in the geopotential height between the Yucatan Peninsula and Cuba and a decrease in the height along Florida, Cuba, and Jamaica. The increment for Ike itself was also different between the 3DVAR and the HYBRID-Ens experiments. The increment for Ike from 3DVAR was much weaker compared to that from HYBRID-Ens. The increment in HYBRID-Ens showed a dipole pattern. This increment pattern suggested that HYBRID-Ens used observations to correct the position of Ike in the background forecast by moving it to the southeast of the forecasted position. This result suggested that HYBRID-Ens, using the flow-dependent ensemble covariance, naturally and systematically corrected the position of the hurricane without using the extra vortex relocation procedure often employed by the operational 3DVAR system. Note that here flow dependent meant the ensemble covariance provided an estimate of the uncertainty of the position of the hurricane. In other words, the hybrid provided a hurricane-specific background error covariance.

Fig. 12.
Fig. 12.

The 500-hPa geopotential height increments (color shading) from (a) 3DVAR and (b) HYBRID-Ens, and (c) the 500-hPa geopotential height background ensemble spread (color shading) for HYBRID-Ens valid at 0000 UTC 8 Sep 2008 for Ike. The black contours in (a)–(c) are the corresponding background 500-hPa geopotential height fields valid at 0000 UTC 8 Sep 2008.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

b. Results for Gustav

Figures 13 and 14 show, respectively, the analyzed tracks and the absolute errors of the analyzed tracks for Gustav every 6 h from the 3DVAR, HYBRID-Static, and HYBRID-Ens simulations. Tracks analyzed by both 3DVAR and HYBRID-Static drifted to the southwest relative to the best-track data starting from 0000 UTC 29 August 2008. The largest deviations in the 3DVAR and HYBRID-Static analyses from the best-track occur at 0000 UTC 30 August 2008, after which Gustav started to turn to the north and caught the best track around 1200 UTC 31 August 2008. While the tracks analyzed by 3DVAR and HYBRID-Static made a spurious loop, the track analyzed by HYBRID-Ens followed the best track more closely.

Fig. 13.
Fig. 13.

The 6-hourly analyzed tracks from HYBRID-Ens, 3DVAR, and HYBRID-Static for Gustav during the data assimilation period. The best-track data are also shown as a reference.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Fig. 14.
Fig. 14.

The 6-hourly absolute errors of the analyzed tracks verified against the best-track data from HYBRID-Ens, 3DVAR, and HYBRID-Static for Gustav during the data assimilation period.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

The root-mean-square errors of the track forecasts (Fig. 15) showed that the track forecast by HYBRID-Ens was more accurate than those of both 3DVAR and HYBRID-Static, with the former gaining 1–2 day of lead time relative to the latter two. The accuracy of the track forecast by HYBRID-Static was similar to that by 3DVAR. As in the Ike case, these results also suggested that the improvement seen in HYBRID-Ens relative to 3DVAR was mainly a result of the use of the flow-dependent ensemble covariance in HYBRID-Ens. Further examining those forecasts, we found that forecasts initialized during 0000 UTC 29 August–1200 UTC 31 August 2008 using the analyses generated by the 3DVAR were significantly worse than those of HYBRID-Ens. Note that as shown in Figs. 13 and 14, during this period of time, the track analyzed by the 3DVAR method was less accurate than that from HYBRID-Ens, which contributed to the poorer subsequent forecasts of the 3DVAR. Therefore, in the rest of this section we show what could contribute to the poorer analyses by the 3DVAR during this time period (0000 UTC 29 August–1200 UTC 31 August 2008).

Fig. 15.
Fig. 15.

The RMS errors of the track forecasts from HYBRID-Static (dotted), HYBRID-Ens (solid), and 3DVAR (dashed) up to 72-h lead time for forecasts initialized every 12 h during the assimilation period for Gustav.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

Figure 16 show the 500-hPa geopotential height analysis at 0000 UTC 29 August 2008. Gustav was positioned in the southeast quadrant of a high pressure system over Gulf of Mexico. This high pressure system was stronger in the 3DVAR analysis than in the HYBRID-Ens analysis. Therefore in the 3DVAR analysis, Gustav’s environment had a stronger northeasterly wind, which was verified by calculating the environmental wind 5°–7° from the center following Chan and Gray (1982). It was found that the easterly (northerly) wind around Gustav in the 3DVAR was 1.03 m s−1 (1.03 m s−1) stronger than that in HYBRID-Ens. The same was found at 1200 UTC 29 August 2008, at which time relative to HYBRID-Ens, 3DVAR had a 2.36 m s−1 (1.67 m s−1) easterly (northerly) environmental wind anomaly around Gustav. The stronger high pressure system in the 3DVAR seen at 0000 UTC 29 August therefore contributed to the subsequent southwest drift of the track starting from 0000 UTC 29 August 2008.

Fig. 16.
Fig. 16.

The 500-hPa geopotential height (m) for the (a) HYBRID-Ens and (b) 3DVAR analyses valid at 0000 UTC 29 Aug 2008 for Gustav.

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

While dropsondes data were available before 0000 UTC 29 August in the archive, these dropsondes were mostly in the vicinity of Gustav rather than its remote environment, such as the Gulf of Mexico (Fig. 17a). Therefore, 3DVAR, with a static covariance, could not use these dropsondes to directly correct the environment over the Gulf of Mexico where a stronger high pressure system was seen at 0000 UTC 29 August (Fig. 16). During the period of 0000 UTC 29 August–0000 UTC 30 August 2008 when the 3DVAR-analyzed Gustav gradually deviated from the best track, 3DVAR using a static covariance could not systematically adjust the position of Gustav as well as a flow-dependent covariance was capable of doing (e.g., the dipole increment for the hurricane seen in HYBRID-Ens). Previous studies (e.g., Hamill and Snyder 2000) have suggested that given the nature of the static covariance, in order to correctly adjust the position of the hurricane during the assimilation, a large amount of data that directly sample the hurricane were needed. Note that during the time period when the analyzed Gustav started to drift (0000 UTC 29 August–1800 UTC 29 August 2008), the number of dropsondes in the vicinity of Gustav that were available from the real-time data archive used was relatively small (Fig. 17b). Also as shown in Fig. 17b, by the time of the dropsondes, Gustav as estimated in the 3DVAR background forecasts had already drifted away from the best track. Therefore, assimilating these limited dropsondes using the static covariance could not bring the drifting Gustav back to the right locations. This result was consistent with early studies, such as Hamill and Snyder (2000), where static covariance was shown to be notably poorer than the flow-dependent covariance in the data-sparse region. In the data archive, dropsondes that sampled the Gulf of Mexico started to be available at 1800 UTC 29 August, by which time Gustav as analyzed by 3DVAR had already drifted away from the best track.

Fig. 17.
Fig. 17.

Locations of dropsondes in the vicinity of Gustav during 0000–1800 UTC 29 Aug: • denotes the dropsonde around 0009 UTC 29 Aug and △ denotes the dropsondes around 1800 UTC 29 Aug. Also shown are the Gustav locations in the background forecast from 3DVAR valid at 0009 UTC 29 Aug (○) and at 1800 UTC 29 Aug (▽).

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

At 0000 UTC 30 August 2008, the 3DVAR-analyzed Gustav reached its southernmost point along the analyzed track (Fig. 13). During the period of 0000 UTC 30 August~1200 UTC 31 August 2008, 3DVAR assimilated observations that sampled Gustav and formed spurious double vortices (e.g., Fig. 18a). While the increments from 3DVAR further enhanced the double vortices, the dipole pattern increments of HYBRID-Ens again showed the systematic adjustment of the position of the hurricane as in the Ike case (Fig. 18b).

Fig. 18.
Fig. 18.

The 500-hPa increments (color shading) from the (a) 3DVAR and (b) HYBRID-Ens analyses valid at 0000 UTC 31 Aug 2008. The black contours are the corresponding background forecasts of 500-hPa geopotential height (m).

Citation: Weather and Forecasting 26, 6; 10.1175/WAF-D-10-05058.1

As mentioned in section 1, the vortex relocation technique is used in the operational data assimilation system to further improve the hurricane analyses by the operational 3DVAR method (Liu et al. (2000)). By accompanying the 3DVAR experiment with a procedure like the vortex relocation, the erroneous behavior of the analyzed track such as the drift of the vortex and the double vortices in the 3DVAR experiment could be reduced. The results of the hybrid on the other hand suggested that adopting the flow-dependent ensemble covariance can correct the location of the hurricane systematically without using the vortex relocation procedure or bogusing. As is also discussed in section 1, vortex relocation relied on several assumptions and further work is still needed to maintain the dynamical and thermodynamical coherency of the hurricane and its environment while the hurricane was relocated (Liu et al. 2000).

5. Conclusions and discussion

A hybrid ETKF–3DVAR data assimilation (DA) system developed for WRF was applied to explore the potential of a hybrid ensemble–variational data assimilation method for hurricane track forecasting. The impacts of the flow-dependent ensemble covariance were revealed by comparing the forecasts, analyses, and analysis increments generated by the hybrid DA method with those generated by 3DVAR, that used the static background error covariance. Two major hurricanes, Ike and Gustav, in 2008 over the Gulf of Mexico were considered in this study.

The root-mean-square errors of the track forecasts initialized by the analyses generated by the hybrid DA method were smaller than those from the 3DVAR simulations for both Ike and Gustav, with the hybrid method gaining 1–2 days of lead time. Such an improvement was shown to be due to the flow-dependent ETKF ensemble covariance used in the hybrid DA method as opposed to a static covariance, as is typically used in 3DVAR. Such a conclusion was consistent with earlier studies that compared the ensemble-based DA methods with the 3DVAR results for other applications (e.g., Wang et al. 2008a,b; Meng and Zhang 2008; Whitaker et al. 2008; Buehner et al. 2010b). Detailed diagnostics further revealed that the increments produced by the hybrid and the 3DVAR approaches were different for both the analyses of the hurricane itself and the environment of the hurricane. In particular, it was found that the hybrid using the flow-dependent ensemble covariance was able to systematically adjust the position of the hurricane during the assimilation whereas 3DVAR was not.

The current study has served as a pilot study for examining the potential of a hybrid DA system for hurricane forecasts and for improving our understanding of the fundamental differences of the flow-dependent ensemble covariance versus a static covariance for hurricane forecasts. Caution needs to be taken to extrapolate the results to the operational system. First, no satellite radiance data were assimilated and not all hurricane surveillance observations were in the data archive. Second, the operational 3DVAR system is accompanied by a vortex relocation technique (Liu et al. 2000), which was not applied in the current study. We also examined and conducted diagnostics for only two major hurricanes. Further experiments with more cases and assimilating more complete observations are needed to assess the generality of the results. Results from experiments with a season’s cases using the hybrid data assimilation system are being developed based on the National Oceanic and Atmospheric Administration’s (NOAA) operational 3DVAR data assimilation system, which is assimilating all operational data including the direct assimilation of satellite radiance data, as will be reported upon in future papers. The impacts of conducting the vortex relocation will also be examined in such experiments. In addition to the track forecasts, experiments using the WRF hybrid data assimilation system to assimilate radar data to explore the potential of the hybrid data assimilation system for the hurricane intensity forecasts are also on going and will be reported upon in future papers (Li et al. 2011).

In addition, as an initial effort to help broaden our understanding of the potential of the hybrid DA system for hurricane forecasts, we compared it with the 3DVAR system. We also recommend direct and thorough comparisons with other data assimilation techniques such as EnKF and 4DVAR so as to understand the relative advantages and disadvantages of different techniques in hurricane forecasts. The hypothesis of the advantages of the hybrid system relative to a stand-alone EnKF and VAR methods, as discussed in section 1 should be further explored within the context of the hurricane forecasts.

Work is still needed to further improve the hybrid system by improving the covariance localization method, such as defining the covariance localization scale based on the scale of the background flow, by applying the ETKF over local regions to reduce the impacts of sampling errors in the ETKF update (Bowler et al. (2009)), and by using more sophisticated ways to represent the model errors in the ensemble.

Acknowledgments

The author was supported by a University of Oklahoma faculty start up award (122-792100), NOAA THORPEX funds (NA08OAR4320904), and a NASA New Investigator Program award (NNX10AQ78G). The experiments were conducted on the supercomputers hosted by the Supercomputing Center for Education and Research at the University of Oklahoma.

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1

In the EnKF, ensemble forecasts are automatically produced and unified with the data assimilation (Hamill 2006).

2

Satellite-derived data were from Geostationary Operational Environmental Satellites (GOES), Meteorological Satellites (METSAT), the Moderate Resolution Imaging Spectroradiometer (MODIS), the Advanced Very High Resolution Radiometer (AVHRR), and NOAA polar-orbiting satellites.

3

Note that a hybrid method means a hybrid of the variational and ensemble Kalman filter frameworks. The weighted average of the static covariance and the ensemble covariance where the weights on the static and ensemble covariance are nonzero is not essential for a hybrid method. Section 1 describes other advantages of the hybrid of the two frameworks in addition to the flexibility of varying the weights on the static and ensemble covariances.

4

The dimension of the ensemble subspace is the number of ensemble members minus 1, which was 31 in the current study. This dimension was smaller than the number of degrees of freedom of the forecast errors. Therefore, only a fraction of the forecast errors were projected or explained by the ensemble subspace (Wang et al. 2007b).

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