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  • View in gallery

    Illustration of (a) the original background ensembles and (b) the VTS-populated background ensembles applying a shifting time interval τ (enclosed by the blue dashed rectangles) being ingested into the 4DEnVar variational update at the three analysis time levels: the beginning (0300 UTC), middle (0600 UTC), and end (0900 UTC) of a 6-h DA window.

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    Spaghetti-contour plots of (a)–(c) the −120-gpm geopotential height at 1000 hPa in Typhoon Usage (2013) and (g)–(i) the 1400-gpm geopotential height at 850 hPa in a midlatitude closed low from the (left) original 80-member background ensemble and (middle) VTSM- and (right) VTSP-populated 240-member background ensemble with applying a shifting time interval τ = 3 h at the 6-h lead time. In (a)–(c) and (g)–(i), the thin blue contours represent the original 6-h 80-member background ensemble. The thin orange and magenta contours denote the 3- and 9-h 80-member background ensemble produced by VTSM and VTSP (see text for the differences), respectively. The thick green contour represents the ECMWF analysis valid at the 6-h lead time. The short dashed, solid, and long dashed red contours denote the 80-member background ensemble mean valid at the 3-, 6- and 9-h lead times, respectively. Histogram plots for (d)–(f) the Typhoon Usage (2013) example and (j)–(l) the midlatitude closed low example by sampling for the geopotential height variable at the grid points enclosed by the corresponding black dashed rectangles in (a)–(c) and (g)–(i), and the averaged spread calculated for the geopotential variables within the corresponding black dashed rectangles is listed in the top-left corner.

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    Horizontal localization length scales (km) as a function of model levels applied in the GSI-based GFS hybrid 4DEnVar system. Note that the horizontal localization length scales are e-folding scales.

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    Globally and temporally averaged RMSE of the 6-h (a) temperature and (d) wind background forecasts in ENS80 (black) and ENS240 (orange) to the rawinsonde observations as a function of pressure levels. PI relative to ENS240 of the 6-h (b),(c) temperature and (e),(f) wind background forecasts in (middle) VTSM240Hτ (solid lines) and (right) VTSP240Hτ (dashed lines) experiments, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue). The asterisks suggest that the RMSE difference from ENS80 in ENS240, VTSM240H2, and VTSP240H3, respectively, is significant at or above the 95% confidence level by applying the paired t test. The percentage number listed in the bottom-left corner of (b),(e) and (c),(f) is the averaged PI over all the pressure levels with the same color indexes applied, corresponding to different shifting time intervals in VTSM240Hτ and VTSP240Hτ experiments.

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    Globally and temporally averaged RMSE difference from ENS80 for the global (a) temperature and (b) wind forecasts in ENS240 against ECMWF analyses as a function of forecast times to 5 days on the horizontal axis and pressure levels on the vertical axis. Blue (red) color indicates the improved (degraded) forecasts from the other experiments relative to ENS80. The asterisks at the corresponding forecast times and pressure levels indicate that the RMSE difference from ENS80 is significant at or above 95% confidence level by applying the paired t test.

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    As in Fig. 5, but for the global (a)–(f) temperature and (g)–(l) wind forecasts against ECMWF analyses in (a)–(c),(g)–(i) VTSM240Hτ and (d–f),(j)–(l) VTSP240Hτ experiments, applying a shifting time interval τ = (left) 1, (middle) 2, and (right) 3 h.

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    The 700-hPa temperature ensemble autocorrelations (color shaded) between the centered grid point (marked by the black dot) and other grid points calculated from the original 6-h background ensemble in (a) ENS240 and (b) ENS80, and the (c) VTSM- and (d) VTSP-populated 6-h background ensembles applying a shifting time interval τ = 3 h. The solid black contours represent the geopotential heights of the 6-h background ensemble mean at 700 hPa.

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    ARCE of ENS80 at 500 hPa calculated from the 6-h background ensemble as a function of bin numbers on the horizontal axis in (a) NH, (b) TR, and (c) SH for the temperature autocorrelations (solid lines) and the cross correlations between the temperature and the zonal wind (dashed lines). Larger bin number on the horizontal axis indicates larger absolute values of the underlying correlations.

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    ARCED from ENS80 in (a)–(c) VTSM and (d)–(f) VTSP experiments, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue) calculated from the 6-h background ensemble at 500 hPa in (left) NH, (middle) TR, and (right) SH for the temperature autocorrelations (solid lines) and the cross correlations (dashed lines) between the temperature and the zonal wind. The horizontal solid black line represents the ARCED with zero magnitude.

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    Vertical profiles for each experiment of the square root of the globally and temporally averaged innovation variance (solid lines) and the predictions of what it should be if the assimilation assumptions are correct (dashed lines). These predicted values are the square root of the observation error variance plus the variance from the original 6-h background ensemble in ENS80 (black) and ENS240 (orange) and the (a),(b) VTSM- and (c),(d) VTSP-populated 6-h background ensembles, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue) for the (left) temperature and (right) wind forecasts at the 6-h lead time. In (c),(d) many of the curves are very similar and have been overplotted by the blue curves, which were plotted last.

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    The E dimension calculated using the temperature and zonal wind perturbations at 500 hPa from the original 6-h background ensemble in ENS80 (black) and ENS240 (orange) and the VTSM- and VTSP-populated (filled with slash lines) 6-h background ensembles with applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue) in (a) NH, (b) TR, and (c) SH.

  • View in gallery

    Best track of the TCs during the experiment period in the (a) Atlantic, (b) east Pacific, and (c) west Pacific basins.

  • View in gallery

    (a) Track forecast errors in ENS80 (solid black), ENS240 (solid orange), VTSM240Hτ (solid), and VTSP240Hτ (dashed) experiments, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue). The circle (asterisk) signs right above the horizontal axis in (a) indicate that the track error difference from ENS80 in VTSM240H3 (VTSP240H1) is significant at or above the 95% confidence level by applying the paired t test at the corresponding forecast time. (b) Percentage of the track forecasts that are more accurate than that in ENS80 with the same line style and color indexes applied in (a). The numbers right above the horizontal axis in (b) denote the sample size at the corresponding forecast time.

  • View in gallery

    (a)–(c) Scatterplots of the 6-h background track errors on the vertical axis against the 6-h background ensemble track spread on the horizontal axis for (left) ENS80, (middle) ENS240, and (right) VTSM240H3. Blue (red) circles in (a)–(c) denote the equally populated samples representing small (large) background ensemble track spread in each experiment. The black dashed line is the diagonal line. (d)–(i) Rank histogram plots of (left) ENS80, (middle) ENS240, and (right) VTSM240H3 created from the samples representing (d)–(f) small background ensemble track spread and (g)–(i) large background ensemble track spread, which correspond to the blue and red circles in (a)–(c), respectively (see detailed descriptions of the rank histogram plots in the text).

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On the Use of Cost-Effective Valid-Time-Shifting (VTS) Method to Increase Ensemble Size in the GFS Hybrid 4DEnVar System

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  • 1 School of Meteorology, University of Oklahoma, Norman, Oklahoma
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Abstract

Valid-time-shifting (VTS) ensembles, either in the form of full ensemble members (VTSM) or ensemble perturbations (VTSP), were investigated as inexpensive means to increase ensemble size in the NCEP Global Forecast System (GFS) hybrid four-dimensional ensemble–variational (4DEnVar) data assimilation system. VTSM is designed to sample timing and/or phase errors, while VTSP can eliminate spurious covariances through temporal smoothing. When applying a shifting time interval (τ = 1, 2, or 3 h), VTSM and VTSP triple the baseline background ensemble size from 80 (ENS80) to 240 (ENS240) in the EnVar variational update, where the overall cost is only increased by 23%–27%, depending on the selected τ. Experiments during a 10-week summer period show the best-performing VTSP with τ = 2 h improves global temperature and wind forecasts out to 5 days over ENS80. This could be attributed to the improved background ensemble distribution, ensemble correlation accuracy, and increased effective rank in the populated background ensemble. VTSM generally degrades global forecasts in the troposphere. Improved global forecasts above 100 hPa by VTSM may benefit from the increased spread that alleviates the underdispersiveness of the original background ensemble at such levels. Both VTSM and VTSP improve tropical cyclone track forecasts over ENS80. Although VTSM and VTSP are much less expensive than directly running a 240-member background ensemble, owing to the improved ensemble covariances, the best-performing VTSP with τ = 1 h performs comparably or only slightly worse than ENS240. The best-performing VTSM with τ = 3 h even shows more accurate track forecasts than ENS240, likely contributed to by its better sampling of timing and/or phase errors for cases with small ensemble track spread.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xuguang Wang, xuguang.wang@ou.edu

Abstract

Valid-time-shifting (VTS) ensembles, either in the form of full ensemble members (VTSM) or ensemble perturbations (VTSP), were investigated as inexpensive means to increase ensemble size in the NCEP Global Forecast System (GFS) hybrid four-dimensional ensemble–variational (4DEnVar) data assimilation system. VTSM is designed to sample timing and/or phase errors, while VTSP can eliminate spurious covariances through temporal smoothing. When applying a shifting time interval (τ = 1, 2, or 3 h), VTSM and VTSP triple the baseline background ensemble size from 80 (ENS80) to 240 (ENS240) in the EnVar variational update, where the overall cost is only increased by 23%–27%, depending on the selected τ. Experiments during a 10-week summer period show the best-performing VTSP with τ = 2 h improves global temperature and wind forecasts out to 5 days over ENS80. This could be attributed to the improved background ensemble distribution, ensemble correlation accuracy, and increased effective rank in the populated background ensemble. VTSM generally degrades global forecasts in the troposphere. Improved global forecasts above 100 hPa by VTSM may benefit from the increased spread that alleviates the underdispersiveness of the original background ensemble at such levels. Both VTSM and VTSP improve tropical cyclone track forecasts over ENS80. Although VTSM and VTSP are much less expensive than directly running a 240-member background ensemble, owing to the improved ensemble covariances, the best-performing VTSP with τ = 1 h performs comparably or only slightly worse than ENS240. The best-performing VTSM with τ = 3 h even shows more accurate track forecasts than ENS240, likely contributed to by its better sampling of timing and/or phase errors for cases with small ensemble track spread.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xuguang Wang, xuguang.wang@ou.edu

1. Introduction

Instead of utilizing the static climatological background error covariances in the traditional variational (Var) systems, the ensemble-based data assimilation (DA) systems are able to simulate the background error covariances in a flow-dependent fashion by using an ensemble of short-range forecasts. One of the best-known forms is the ensemble Kalman filter (EnKF; Evensen 1994). Different variants of EnKF were developed in recent decades for the purpose of efficient implementations (Houtekamer and Mitchell 1998, 2005; Anderson 2001; Bishop et al. 2001; Whitaker and Hamill 2002; Wang and Bishop 2003; Hunt et al. 2007). Recently, the hybrid DA method has shown increasing popularity and has been adopted by many operational numerical weather prediction (NWP) centers. The hybrid DA method incorporates the ensemble background error covariances into the Var framework (Hamill and Snyder 2000; Lorenc 2003; Buehner 2005; Wang et al. 2007b; Wang 2010). Extensive studies have demonstrated that the hybrid DA method outperforms the stand-alone variational or pure ensemble method (e.g., Wang et al. 2007a, 2008a,b, 2009, 2013; Wang 2011; Zhang and Zhang 2012; Buehner et al. 2013; Clayton et al. 2013; Kuhl et al. 2013; Gustafsson et al. 2014; Wang and Lei 2014; Lorenc et al. 2015; Kleist and Ide 2015a; Kutty and Wang 2015; Buehner et al. 2015).

Within the Monte Carlo approximation, a large-sized ensemble is required to accurately sample the forecast errors in the ensemble-based DA methods. This requirement is especially stringent for high-dimensional NWP models. However, the affordable ensemble size is limited to O(100) in many operational centers (Houtekamer and Zhang 2016;their Table 1) due to the computational constraints. Use of a small-sized ensemble causes sampling errors typically characterized by the remote spurious covariances. If not properly treated, the sampling errors, for example, will incur noisy analysis increments and even cause filter divergence (Hamill 2006). Covariance localization is commonly applied in the ensemble-based DA systems to eliminate the spurious covariances (Houtekamer and Mitchell 2001; Bishop and Hodyss 2009). However, covariance localization may incur imbalance in the analysis (Buehner and Charron 2007; Wang et al. 2009; Buehner 2012; Holland and Wang 2013) and probably remove the realistic signals in distant regions (Miyoshi et al. 2014). On the other hand, the benefits from directly increasing the ensemble size in the ensemble-based DA systems were demonstrated in the operational or near-operational settings (Bonavita et al. 2012; Houtekamer et al. 2014; Bowler et al. 2017; Lei and Whitaker 2017). Unfortunately, the computational cost is significantly increased due to running a large-sized ensemble.

Instead of directly increasing the ensemble size, early studies explored alternative ways to populate the background ensemble. One method was to include ensemble forecast members valid at the same time but initialized from different previous cycles (Van den Dool and Rukhovets 1994; Lu et al. 2007; Gustafsson et al. 2014). This approach is hereafter termed the “time-lagged method.” Such sets of ensemble forecast members with different forecast lengths may be able to sample part of the forecast errors (Van den Dool and Rukhovets 1994). However, previous studies have shown limited success of the time-lagged method to improve the analysis and the subsequent forecasts (e.g., Gustafsson et al. 2014).

In addition, Xu et al. (2008), in a convective-scale forecast context, proposed a time-expanded sampling method by taking advantage of the ensemble forecast members that were initialized in the same previous DA cycle but valid at different lead times. This method was inspired to sample the timing and/or phase errors typically seen in the convective-scale background forecasts. In its implementation, the ensemble forecast members, valid around but not at the analysis time, were included to populate the background ensemble at the analysis time. Because this method requires shifting the ensemble forecast members valid at different lead times to the analysis time, this method is hereafter denoted the valid-time-shifting method for ensemble members (VTSM), adapted from the notation used by Gustafsson et al. (2014). The time difference between the analysis time and the valid time of the shifted ensemble forecasts is named as the shifting time interval for brevity. VTSM was shown to be useful for the regional mesoscale EnKF or hybrid-4DVar systems with the assimilation of either simulated or real observations (Xu et al. 2008; Lu et al. 2011; Gustafsson et al. 2014; Zhao et al. 2015).

While the efficacy of VTSM has been demonstrated in previous studies for the meso- and convective-scale DA and forecasts in the regional models, its usefulness for a global modeling system featured with a variety of weather phenomena remains to be investigated. For example, the global model houses weather phenomena with different degrees of predictability and scales (e.g., midlatitude trough and ridge vs tropical storm). The background errors associated with these phenomena can be diverse, ranging from timing to phase to magnitude and structure errors and featured with various growth rates. For the weather systems with low predictability, the original background ensemble is likely not able to comprehensively sample the background errors from different sources. In such cases, the populated ensemble by VTSM may have a better chance of capturing missing sources of background errors. However, as shown in section 2, for the more predictable cases, populating the background ensemble by utilizing the ensemble forecast members valid at different lead times may introduce some members that are irrelevant to the background errors. One objective of the current study is to provide a thorough investigation of VTSM in a global ensemble-based DA system.

In addition, to ameliorate the potential limitations of VTSM while taking advantage of the ensemble forecasts freely available at different lead times to enlarge the ensemble size, a method extended from VTSM is explored in this study. Different from VTSM, the ensemble members valid at different lead times are recentered on the original ensemble mean valid at the analysis time. In other words, rather than shifting the ensemble members in VTSM, the ensemble perturbations calculated as the deviation from its own ensemble mean are shifted. Hereafter, this method is termed the valid-time-shifting method for ensemble perturbations (VTSP). In the ECMWF global 4DVar DA system, Bonavita et al. (2016) formed the background covariances by blending the perturbations generated by VTSP and those drawn from the climatology. It was found that this blending was beneficial for their global forecasts, compared to either using the static climatological background error covariances or the background error covariances estimated from the ensemble perturbations with the same lead time but initialized most recently. In the Met Office’s global hybrid 4D ensemble–variational (4DEnVar) system, Lorenc (2017) combined the VTSP approach and the time-lagged approach to further increase the ensemble size. It was found that the positive impact of the covariances from this populated ensemble was more apparent when proper localization method was implemented.

In this study, VTSM and VTSP were implemented and investigated in the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS) hybrid 4DEnVar DA system (Wang and Lei 2014; Kleist and Ide 2015b). Different from early studies, this study explores and compares the impacts of both VTSM and VTSP. In addition, to examine the impacts of both methods in a global system featured with weather systems of different scales and predictability, their impacts on both the general global forecasts and the tropical cyclone (TC) forecasts are examined. It is expected that the optimal parameters such as the shifting time interval may be dependent on the specific forecast application. Furthermore, various diagnostics are carried out, and experiments are designed to reveal the causes of the impacts of VTSM and VTSP. The paper is organized as follows. Section 2 describes and illustrates VTSM and VTSP. Section 3 describes the experiment design. Sections 4 and 5 discuss the results and various diagnostics with respect to the general global forecasts and tropical cyclone track forecasts, respectively. The computational cost from VTSM and VTSP is compared with the baseline GFS 4DEnVar experiment in section 6. Section 7 presents a summary and discussion.

2. VTSM and VTSP

In VTS, the ensemble forecasts—either in the form of full ensemble members or ensemble perturbations, which are initialized from the same analyses produced by the previous DA cycle but valid at different lead times—are used to populate the background ensemble at the current analysis time. Figure 1 illustrates the original background ensembles and the VTS-populated background ensembles being ingested into the 4DEnVar variational update within a 6-h DA window of 0300–0900 UTC. In the original GFS 4DEnVar system (Fig. 1a), an ensemble of 80-member forecasts out to the 9-h lead time is initialized from the analyses produced in the previous DA cycle. Considering producing a 3-hourly temporal resolution of the ensemble within the 6-h DA window, the analysis increments in the 4DEnVar variational update are produced at three time levels: the beginning (t = 0300 UTC), middle (t = 0600 UTC), and end (t = 0900 UTC) of the DA window. When applying VTS (Fig. 1b), a shifting time interval τ is first selected. Then, the background forecasts initialized from the previous DA cycle are shifted both forward and backward for the time length equal to the shifting time interval τ. Specifically, the ensemble forecasts valid at times tτ and t + τ will be used to supplement the background ensemble at each analysis time t (enclosed by the blue dashed rectangles in Fig. 1b). Since these additional ensemble forecasts are not valid but shifted to the time t, they are termed the shifted ensembles in contrast to the original ensemble valid at time t. As such, for the example given in Fig. 1b, the ensemble size is tripled at each time t. The ensemble size can be further enlarged by selecting multiple different shifting time intervals (Lorenc 2017). For instance, by selecting τ = 1, 2, and 3 h, the populated ensemble size would be 7 times as large as the original ensemble. However, these ensemble forecasts with smaller lead time differences can be strongly correlated and therefore not effectively add additional degrees of freedom or rank. Given that the focus of this study is to reveal the impact differences of VTSM and VTSP, only one single shifting time interval is selected, as described in the next section. The following two subsections describe and illustrate VTSM and VTSP using both a tropical cyclone and a midlatitude closed low as examples.

Fig. 1.
Fig. 1.

Illustration of (a) the original background ensembles and (b) the VTS-populated background ensembles applying a shifting time interval τ (enclosed by the blue dashed rectangles) being ingested into the 4DEnVar variational update at the three analysis time levels: the beginning (0300 UTC), middle (0600 UTC), and end (0900 UTC) of a 6-h DA window.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

a. VTSM

In VTSM, the ensemble at time t is populated by directly including the original ensemble forecast members valid at times tτ and t + τ. The populated ensemble mean is the average of the original ensemble means valid at times tτ, t, and t + τ. The populated ensemble perturbations are calculated as the deviation of each member from this populated ensemble mean and are used to compute the ensemble background error covariances. As shown in Eq. (A1) in the appendix, the background ensemble covariances after applying VTSM are calculated by summing up two components. The first component approximately represents an average of the ensemble covariances from the original and shifted ensembles. The second component is the contribution from the ensemble mean differences between the original and shifted ensembles. As discussed in the next subsection, the first component is approximately equivalent to the VTSP-populated ensemble covariances. In the forthcoming examples featuring systems of different scales and predictabilities, the role of the second component from the VTSM-populated ensemble covariances will be discussed.

Figure 2, first using Typhoon Usage (2013), illustrates that by directly utilizing the ensemble forecast members at different lead times, VTSM can better sample the timing or phase errors, consistent with Xu et al.’s (2008) earlier study. As seen in Fig. 2a, a westward location error is observed from the original 6-h background ensemble mean relative to the verifying ECMWF analysis. The original background ensemble has relatively large spread, indicating the large uncertainty of the forecasts. By shifting the background ensemble forecast members at the 3- and 9-h lead times to the 6-h lead time via VTSM (Fig. 2b), the spread of the ensemble is increased. In the VTSM-populated ensemble, more members enclose the verifying ECMWF analysis, especially due to the shifted 3-h background ensemble members. This result suggests the VTSM-populated ensemble may better sample the location errors than the original 6-h ensemble. As shown in this typhoon example, the ensemble means at the three different lead times still reside within the envelope of the original 6-h background ensemble (e.g., the blue curves in Fig. 2b), and the contribution from the ensemble mean differences is less likely to dominate the VTSM-populated ensemble error covariances, as shown in Eq. (A1).

Fig. 2.
Fig. 2.

Spaghetti-contour plots of (a)–(c) the −120-gpm geopotential height at 1000 hPa in Typhoon Usage (2013) and (g)–(i) the 1400-gpm geopotential height at 850 hPa in a midlatitude closed low from the (left) original 80-member background ensemble and (middle) VTSM- and (right) VTSP-populated 240-member background ensemble with applying a shifting time interval τ = 3 h at the 6-h lead time. In (a)–(c) and (g)–(i), the thin blue contours represent the original 6-h 80-member background ensemble. The thin orange and magenta contours denote the 3- and 9-h 80-member background ensemble produced by VTSM and VTSP (see text for the differences), respectively. The thick green contour represents the ECMWF analysis valid at the 6-h lead time. The short dashed, solid, and long dashed red contours denote the 80-member background ensemble mean valid at the 3-, 6- and 9-h lead times, respectively. Histogram plots for (d)–(f) the Typhoon Usage (2013) example and (j)–(l) the midlatitude closed low example by sampling for the geopotential height variable at the grid points enclosed by the corresponding black dashed rectangles in (a)–(c) and (g)–(i), and the averaged spread calculated for the geopotential variables within the corresponding black dashed rectangles is listed in the top-left corner.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

On the other hand, as discussed in the introduction, VTSM may introduce irrelevant members. For example, Fig. 2g shows a midlatitude closed low example where both the phase and structure are accurately predicted by the 6-h background ensemble mean. The ensemble encloses the verifying ECMWF analysis, and the spread of the ensemble is small. The VTSM-populated ensemble with the 3-h shifting time interval in this example shows three distinct clusters in the eastern section of the midlatitude low (Fig. 2h). The added members fall completely outside the envelope of the original ensemble and the verifying ECMWF analysis, therefore producing irrelevant sampling of the background errors. Consistently, the histogram plot for the geopotential height variable sampled from the grid points within the areas of three distinct clusters (enclosed by black dashed rectangle in Fig. 2h) shows three peaks (Fig. 2k), significantly altering the background distribution of the original ensemble (Fig. 2j). As a consequence, the VTSM-populated ensemble violates the Gaussian assumption typically used in the ensemble-based DA methods. In contrast, for the typhoon example, in spite of the increased spread, VTSM does not significantly alter the distribution of the original background ensemble by comparing Figs. 2d and 2e. Furthermore, different from the typhoon example discussed earlier, in the eastern section of the midlatitude closed low, the ensemble means at the 3- and 9-h lead times are located totally outside the envelope of the original 6-h ensemble. Considering the relatively small spread of the original ensembles, the contribution of the ensemble mean differences in the VTSM-populated ensemble covariances could possibly dominate the VTSM-populated ensemble covariances in this midlatitude closed low example.

b. VTSP

In VTSP, the shifted ensemble members at time t are produced by recentering the original ensemble perturbations at times tτ and t + τ on the original ensemble mean at time t. So the VTSP-populated background ensemble shares the same original background ensemble mean valid at time t. VTSP therefore reduces the possibility that the shifted members sample irrelevant background error space. For example, in the midlatitude low example, different from VTSM, VTSP produces a populated ensemble without distinct clusters (Fig. 2i). Furthermore, the VTSP-populated ensemble (Fig. 2l) follows the Gaussian distribution more than the original ensemble (Fig. 2j). In the typhoon example, compared to the original ensemble, VTSP still increases the chance that the truth is sampled by adding more members enclosing the verifying ECMWF analysis (Fig. 2c), although VTSP does not increase the spread as much as VTSM. In the typhoon example, VTSP (Fig. 2f), like VTSM, does not show significant change of the distribution of the original background ensemble.

As shown in Eq. (A3), VTSP by design functions as averaging the ensemble covariances at three different valid times. Therefore, VTSP produces a temporal smoothing effect on the ensemble covariances. In a chaotic system, temporal smoothing has a similar effect as spatial smoothing (Raynaud et al. 2008). Buehner and Charron (2007) proved that the spatial smoothing of the ensemble correlations in the grid space was equivalent to applying the localization in the spectral space. Raynaud et al. (2008, 2009) applied a spatial averaging technique on the ensemble background variances to reduce the sampling errors that often have smaller scales, compared to the background errors. Therefore, it is expected that the built-in smoothing effect in VTSP can contribute to eliminating the spurious small covariances caused by sampling errors. As discussed in the previous subsection, the VTSP-populated ensemble error covariances are approximately equal to the first component in the VTSM-populated ensemble error covariances. By comparing with VTSP, it assists in isolating and evaluating the impact of the ensemble mean differences in the VTSM-populated ensemble error covariances.

3. Experiment design

The hybrid 4DEnVar system for the GFS model was developed as an extension of the Gridpoint Statistical Interpolation (GSI) 3DEnVar system (Wang 2010; Wang et al. 2013; Wang and Lei 2014; Kleist and Ide 2015a,b). In contrast to 3DEnVar, 4DEnVar is able to account for the temporal evolution of the background error covariances by utilizing the 4D ensemble forecast errors, therefore enhancing the assimilation of 4D observations within a DA window. The GSI hybrid 4DEnVar system was operationally implemented at NCEP beginning in May 2016.

The DA cycling experiments were carried out for a 10-week period from 0000 UTC 25 July to 1800 UTC 30 September 2013. The conventional and satellite observations operationally used in the NCEP Global Data Assimilation System (GDAS) were assimilated every 6 h. Detailed descriptions of different types of observations are available online (http://www.emc.ncep.noaa.gov/mmb/data_processing/prepbufr.doc/table_2.htm and table_18.htm). The same observation quality control and bias correction for the satellite radiances were used as in the operational GDAS system (Zhu et al. 2014).

The baseline 4DEnVar experiment (ENS80 in Table 1) without applying VTSM and VTSP is configured similarly as the operational system, except that a reduced resolution is adopted due to the computational constraints. The dual-resolution configuration is applied with a control or deterministic member running at a relatively high resolution of T670 and an 80-member ensemble running at a relatively low resolution of T254. In the DA step, the control background is updated by adopting the 4DEnVar algorithm, where the extended control variable method is used to ingest the 4D ensemble perturbations. Detailed mathematical formula and implementation of 4DEnVar in the GSI variational minimization can be found in Wang and Lei (2014). The ensemble members are updated using the EnKF (Whitaker and Hamill 2002; Whitaker et al. 2008). Following the two-way coupling method [Fig. 1b in Wang et al. (2013)], the EnKF ensemble analyses are recentered on the control 4DEnVar analyses.

Table 1.

List of DA experiments.

Table 1.

In the 4DEnVar update, the weights of 12.5% and 87.5% are applied on the static climatological and ensemble background error covariances, respectively, as in the operational system. Different from utilizing the hourly ensemble perturbations as in the operational system, 3-hourly ensemble perturbations are ingested in GSI 4DEnVar in the current experiments due to computational constraints. To treat the sampling errors associated with the ensemble-based covariances, the covariance localization is implemented by a spectral filter transform for the horizontal and the recursive filter for the vertical [see details in Wang et al. (2013)]. Following Lei and Whitaker (2017), the level-dependent localization length scales in Fig. 3 are applied in the horizontal direction and a single value of 0.5-scale heights (i.e., the natural log of the pressure) in the vertical direction. The horizontal and vertical localization length scales in the 4DEnVar variational update are the e-folding scales. To alleviate the imbalance issue in the control analysis, a tangent linear normal mode initialization constraint (TLNMC; Kleist et al. 2009) is applied to the analysis increments during the minimization of GSI 4DEnVar as in Wang et al. (2013), Wang and Lei (2014), and Kleist and Ide (2015a,b).

Fig. 3.
Fig. 3.

Horizontal localization length scales (km) as a function of model levels applied in the GSI-based GFS hybrid 4DEnVar system. Note that the horizontal localization length scales are e-folding scales.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

The 4D serial ensemble square root filter (EnSRF; Whitaker and Hamill 2002) is adopted for the EnKF component of the hybrid system, as in the operational system. In the EnKF update, all the observation operators are calculated by GSI. The ensemble background mean at low resolution is used for data selection and quality control so that all the ensemble members are updated by the same set of observations. The same localization parameters used in the 4DEnVar variational update are applied for EnKF. A normalization factor of 0.388 is applied in EnKF to convert the e-folding scales to the distance at which the amplitude of Gaspari–Cohn localization function (Gaspari and Cohn 1999) approaches zero. To account for the spread deficiency of the background ensemble produced by EnKF, the multiplicative inflation (Whitaker and Hamill 2012) is employed by relaxing the posterior ensemble spread to 85% of the prior ensemble spread. Stochastic parameterization schemes (Palmer et al. 2009; Lei and Whitaker 2016, 2017) are applied to further account for the model uncertainty in the ensemble forecasts.

The GFS model (detailed descriptions available online at http://www.emc.ncep.noaa.gov/GFS/doc.php) is configured similarly as the operational system for the control and ensemble forecasts, except for running at the reduced horizontal resolutions as discussed before. Sixty-four vertical levels are used. In addition, the 4D incremental analysis update (4DIAU) is applied for both the control and ensemble forecasts instead of the digital filter (DFI; Lynch and Huang 1992) used in the operational GFS model, given the superiority of 4DIAU in suppressing the spurious high-frequency oscillations, compared with DFI (Lorenc et al. 2015; Lei and Whitaker 2016, 2017). A 4DIAU implementation is planned for the operational GFS 4DEnVar system (R. Mahajan 2017, personal communication).

In addition to the baseline experiment, two sets of experiments—VTSM240Hτ and VTSP240Hτ in Table 1—are designed where VTSM and VTSP are applied to populate the background ensemble before being ingested into the 4DEnVar variational update. VTSM240Hτ and VTSP240Hτ denote experiments where VTSM and VTSP are applied to increase the background ensemble size from 80 to 240 for a given shifting time interval τ. Since the shifted ensembles defined in VTS originate from the ensemble forecasts that are initialized from the same analyses produced in the previous DA cycle, three different shifting time intervals (τ = 1, 2, or 3 h) are experimented within a 6-h DA window of 0300–0900 UTC. Note that in these experiments, EnKF and ensemble forecasts still run with 80 members, as in the baseline ENS80 experiment. Only the number of ensemble members ingested to the 4DEnVar variational update is increased from 80 to 240 by using the VTS methods. Finally, the experiment ENS240 is conducted. ENS240 is the same as the baseline ENS80 experiment, except that the ensemble size is directly increased from 80 to 240 for each component of the hybrid DA system. As discussed in section 6, although 240 members are used in the 4DEnVar variational update for both the ENS240 and VTS experiments, the VTS methods are computationally less costly. ENS240 is therefore used as a reference to reveal to what extent the inexpensive VTS methods can improve or even outperform ENS240. Within similar experiment configurations, Lei and Whitaker (2017) found that the performance of the GFS hybrid 4DEnVar system showed little sensitivity to the localization length scale changes by increasing the ensemble size from 80 to 320. Therefore, our experiments of ENS240, VTSM240Hτ, and VTSP240Hτ apply the same localization length scales as the baseline ENS80 experiment. Detailed experiment descriptions are presented in Table 1.

4. Evaluation of global forecasts

In this section, the performance of VTSM and VTSP on the general global forecasts is evaluated. Various diagnostics are performed to understand the causes of their impacts on the general global forecasts. In section 5, VTSM and VTSP are further evaluated on the tropical cyclone track forecasts.

a. Verification against conventional observations

Root-mean-square errors (RMSEs) of the 6-h temperature and wind forecasts against the rawinsonde observations were calculated at different pressure levels for all the experiments. Samples were collected from the last 8 weeks during the 10-week experiment period to remove the DA spinup period. The paired t test was performed to examine the significance of the RMSE difference between ENS80 and the other experiments. ENS240 consistently significantly improves the 6-h temperature and wind forecasts over ENS80 at all pressure levels at or above the 95% confidence level (Figs. 4a,d), especially for the global wind forecasts.

Fig. 4.
Fig. 4.

Globally and temporally averaged RMSE of the 6-h (a) temperature and (d) wind background forecasts in ENS80 (black) and ENS240 (orange) to the rawinsonde observations as a function of pressure levels. PI relative to ENS240 of the 6-h (b),(c) temperature and (e),(f) wind background forecasts in (middle) VTSM240Hτ (solid lines) and (right) VTSP240Hτ (dashed lines) experiments, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue). The asterisks suggest that the RMSE difference from ENS80 in ENS240, VTSM240H2, and VTSP240H3, respectively, is significant at or above the 95% confidence level by applying the paired t test. The percentage number listed in the bottom-left corner of (b),(e) and (c),(f) is the averaged PI over all the pressure levels with the same color indexes applied, corresponding to different shifting time intervals in VTSM240Hτ and VTSP240Hτ experiments.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

To quantify the RMSE difference of VTSM240Hτ or VTSP240Hτ relative to ENS80 and the extent to which the improvement of ENS240 can be recovered by VTSM240Hτ or VTSP240Hτ, the percentage improvement (PI) of VTSM240Hτ and VTSP240Hτ relative to ENS240 was defined as
e1
where “exp” denotes the experiments of VTSM240Hτ or VTSP240Hτ. Since ENS240 consistently shows smaller RMSE than ENS80, positive PI indicates the improved forecasts of VTSM240Hτ or VTSP240Hτ over ENS80, and vice versa. The averaged PI over all pressure levels was also calculated. VTSP240Hτ with all shifting time intervals shows positive PI for the 6-h temperature and wind forecasts at almost all pressure levels (Figs. 4c,f). The VTSP experiments generally show larger PI with larger shifting time interval. For instance, in terms of averaged PI, compared to VTSP240H1 and VTSP240H2, VTSP240H3 achieves the largest averaged PI of 50.2% and 60.4% for the 6-h temperature and wind forecasts, respectively. In particular, the improved 6-h temperature and wind forecasts in VTSP240H3 over ENS80 are statistically significant at or above the 95% confidence level at all pressure levels. On the other hand, VTSM240H3 shows nearly zero and even negative PI (Figs. 4b,e). With a smaller shifting time interval, VTSM240H1 and VTSM240H2 generally show more instances of positive PI. Specifically, in the VTSM experiments, VTSM240H2 (VTSM240H1) shows the best averaged PI of 36.1% (25.2%) for the 6-h temperature (wind) forecasts. However, this percentage improvement is less than the best-performing VTSP experiment (VTSP240H3). In summary, VTSP240H3 shows the most consistent improvement and therefore recovers the improvement by ENS240 the most for the 6-h temperature and wind forecasts in all VTS experiments.

b. Verification against ECMWF analyses

The global forecasts out to 5-day lead times were further verified against the ECMWF analyses with the resolution of 1° × 1° grid (http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=pl/). The RMSEs between the global forecasts and the ECMWF analyses were calculated every 6 h and averaged temporally and globally at each pressure level. The paired t test was performed to examine the statistical significance of the RMSE difference from ENS80. Figures 5 and 6 show the RMSE difference of ENS240 and VTS experiments relative to ENS80 for the global temperature and wind forecasts as a function of forecast lead time and pressure level. ENS240 significantly improves over ENS80 for both the temperature and wind forecasts out to 5 days (Fig. 5), which is consistent with Lei and Whitaker (2017).

Fig. 5.
Fig. 5.

Globally and temporally averaged RMSE difference from ENS80 for the global (a) temperature and (b) wind forecasts in ENS240 against ECMWF analyses as a function of forecast times to 5 days on the horizontal axis and pressure levels on the vertical axis. Blue (red) color indicates the improved (degraded) forecasts from the other experiments relative to ENS80. The asterisks at the corresponding forecast times and pressure levels indicate that the RMSE difference from ENS80 is significant at or above 95% confidence level by applying the paired t test.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for the global (a)–(f) temperature and (g)–(l) wind forecasts against ECMWF analyses in (a)–(c),(g)–(i) VTSM240Hτ and (d–f),(j)–(l) VTSP240Hτ experiments, applying a shifting time interval τ = (left) 1, (middle) 2, and (right) 3 h.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

VTSP240Hτ significantly improves temperature and wind forecasts in the stratosphere above 200 hPa over the 5-day lead times. In the troposphere, VTSP240H2 overall is able to maintain the statistically significant improvement out to the 5-day lead time (Figs. 6e,k). VTSP240H1 and VTSP240H3 only show significant improvement within the first 3 days, and the differences between these two experiments and ENS80 beyond the 3-day lead time are statistically insignificant. VTSM240Hτ consistently improves the global forecasts in the stratosphere above 100 hPa over the 5-day lead times, except for VTSM240H3, which shows neutral impacts on the temperature forecasts at early lead times. In the troposphere, however, the VTSM experiments show either nearly neutral or negative impacts. Stronger degradation is found with larger shifting time interval in the VTSM experiments. For example, VTSM240H3 degrades the forecasts below 100 hPa for the entire 5-day period (Figs. 6c,i).

In summary, among the experiments of VTSM240Hτ and VTSP240Hτ, VTSP240H2 shows the most consistent improvement for the global temperature and wind forecasts verified against the ECMWF analyses. The improvement of VTSP240H2 bears similar structure to that in ENS240 by comparing with Fig. 5, though the magnitude is generally smaller. To further understand the causes of the different impacts of VTSM and VTSP on the general global forecasts, in the next several subsections, aspects including ensemble correlation and spread, the effective rank in the ensemble-based covariance matrices are further examined in a global context using ENS240 as a referencing truth.

c. Accuracy of ensemble correlations

As discussed in the introduction, sampling errors due to the limited ensemble size manifest themselves with spurious correlations in the ensemble-based covariances. Figure 7 shows the 2D temperature ensemble autocorrelations at the 6-h lead time between the central grid point (marked by the black dot) and other grid points for a midlatitude low case. The ensemble correlations in the referencing ENS240 experiment display flow-dependent structures stretching along the geopotential height contours (Fig. 7a). Compared to ENS240, ENS80 shows three spurious negative correlation areas away from the central grid point (Fig. 7b) that do not appear in ENS240. VTSP240H3 (Fig. 7d) is able to remove the spurious correlations shown in ENS80 and maintain a similar structure to ENS240. These results illustrate the effectiveness of VTSP in alleviating the sampling errors, possibly owing to its smoothing effect as discussed in section 2b. On the other hand, VTSM240H3 shows a largely different correlation structure (Fig. 7c), compared to ENS240, characterized by the expanded negative correlation areas away from the centered grid point. As discussed in section 2, by comparing with VTSP240H3, the deterioration of the estimated correlations in VTSM240H3 could be caused by the inclusion of the ensemble mean differences that may fail to appropriately sample the forecast errors at the 6-h lead time (as discussed in Figs. 2h and 2k).

Fig. 7.
Fig. 7.

The 700-hPa temperature ensemble autocorrelations (color shaded) between the centered grid point (marked by the black dot) and other grid points calculated from the original 6-h background ensemble in (a) ENS240 and (b) ENS80, and the (c) VTSM- and (d) VTSP-populated 6-h background ensembles applying a shifting time interval τ = 3 h. The solid black contours represent the geopotential heights of the 6-h background ensemble mean at 700 hPa.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

To quantify the accuracy of the background ensemble correlations, the ensemble correlation samples were collected after the spinup period. Each correlation sample consists of a 2D box covering an area of 40° × 40°, as used in Fig. 7. The size of the box is slightly larger than doubling the localization length scale. In each cycle, 165 boxes, evenly distributed over the globe, are selected at 850, 500, and 200 hPa. Within each box, the auto- and cross correlations for the temperature and zonal wind variables are calculated between the centered grid point and other grid points. The absolute value of the relative correlation error (ARCE) with ENS240 as the referencing truth is defined as
e2
where “exp” denotes the experiments of ENS80, VTSM240Hτ, and VTSP240Hτ; “Corr” denotes the background ensemble correlations; and “abs” is the absolute sign. Only samples of the absolute correlation values larger than 0.0001 in ENS240 were collected to calculate ARCE. This setting of the threshold intends to reduce the chance of contaminating the averaged ARCE statistics by a limited number of extremely large ARCE produced by a very tiny denominator in Eq. (2). To further quantify the error reduction or increase relative to the baseline ENS80 experiment, another metric, ARCE difference (ARCED), is defined as
e3
Positive ARCED suggests the improved correlation accuracy from VTSM240Hτ or VTSP240Hτ relative to ENS80, and vice versa. To evaluate the accuracy of ensemble correlations as a function of the value of the underlying correlations, 10 bins with an increasing order of the absolute correlation values are first defined using the absolute correlations in ENS240. ARCE and ARCED are then grouped and averaged for each bin.

Figure 8 shows the ARCE of ENS80 for the temperature variable at 500 hPa in the Northern Hemisphere (NH), tropical region (TR), and Southern Hemisphere (SH). Similar results are also found at other pressure levels and for the zonal wind variable (not shown here). In all hemispheres, ARCE decreases as the underlying absolute correlations increase, especially for the small correlations (e.g., the first two bins show sharp decrease of ARCE). In addition, the cross correlations between the temperature and zonal wind variables show larger errors than the temperature autocorrelations. The results are consistent with the expectation that for a given ensemble size, it is more difficult to estimate the small correlations and cross-variable correlations using the ensembles.

Fig. 8.
Fig. 8.

ARCE of ENS80 at 500 hPa calculated from the 6-h background ensemble as a function of bin numbers on the horizontal axis in (a) NH, (b) TR, and (c) SH for the temperature autocorrelations (solid lines) and the cross correlations between the temperature and the zonal wind (dashed lines). Larger bin number on the horizontal axis indicates larger absolute values of the underlying correlations.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

Figure 9 shows the ARCED of VTSM240Hτ and VTSP240Hτ for the temperature variable at 500 hPa in different hemispheres. Similar results are also found at other pressure levels and for the zonal wind variable (not shown here). In all hemispheres, VTSP240Hτ consistently improves the correlation accuracy for all bins except for VTSP240H3, which shows slightly degraded cross correlations in the last three bins in SH. In contrast, VTSM240Hτ degrades the accuracy, especially for the small underlying correlations. With the increase of underlying correlations (increasing bin numbers), the improvement in VTSP240Hτ and the degradation in VTSM240Hτ are reduced. Larger improvement in VTSP240Hτ and larger degradation in VTSM240Hτ are generally found for the cross-variable correlations than for the same-variable autocorrelations. With a larger shifting time interval, VTSP240Hτ generally shows larger improvement, and VTSM240Hτ results in larger degradation. It is speculated that when applying a larger shifting time interval in VTSP240Hτ, additional ensemble perturbations added by VTSP are more independent from the original background ensemble perturbations, which therefore can more effectively enrich the ensemble. Consistently, VTSP240H3 shows the best percentage improvement for the 6-h global temperature and wind forecasts (Fig. 4c,f). On the other hand, by comparing with VTSP240Hτ, the more severely degraded correlation accuracy with larger shifting time interval in VTSM240Hτ could be attributed to the increased amount of the ensemble mean differences that dominate the VTSM-populated ensemble correlations. It is also noted that VTSM240Hτ shows smallest degradation in TR and largest degradation in SH. Synoptic-scale weather systems typically controlled by the barotropic instability in TR may not evolve as rapidly as those typically controlled by the baroclinic instability in NH or SH (Straus and Paolino 2008). During the experiment period when SH experiences wintertime, strong baroclinic instability is expected. As a result, VTSM240Hτ is likely to induce larger ensemble mean differences in SH than that in TR, thus possibly more severely degrading the VTSM-populated ensemble correlation accuracy in SH.

Fig. 9.
Fig. 9.

ARCED from ENS80 in (a)–(c) VTSM and (d)–(f) VTSP experiments, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue) calculated from the 6-h background ensemble at 500 hPa in (left) NH, (middle) TR, and (right) SH for the temperature autocorrelations (solid lines) and the cross correlations (dashed lines) between the temperature and the zonal wind. The horizontal solid black line represents the ARCED with zero magnitude.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

d. Statistical evaluation of ensemble spread

In this subsection, the relation of the 6-h background forecast errors and the 6-h background ensemble spread is evaluated for all the experiments (Houtekamer and Mitchell 2005; Whitaker et al. 2008; Wang et al. 2013). As shown in Fig. 10, in a globally averaged context, the original background ensemble in ENS80 is underdispersive for both the 6-h temperature and wind forecasts in the stratosphere and lower troposphere, but overdispersive especially for the wind forecasts in the midtroposphere. VTSP240Hτ and ENS240 show negligible spread change from ENS80 (Figs. 10c,d), while VTSM240Hτ increases the spread, especially with a larger shifting time interval (e.g., VTSM240H3; Figs. 10a,b). The increased spread in VTSM240Hτ could be contributed to by the ensemble mean differences by comparing with VTSP240Hτ. As a result, VTSM240Hτ is able to alleviate the underdispersiveness of the original background ensemble in the stratosphere, but exacerbates the overdispersiveness of the original background ensemble in the midtroposphere. These different effects may explain the improved global temperature and wind forecasts above 100 hPa but the degraded forecasts in the midtroposphere in the VTSM experiments, as shown in Fig. 6.

Fig. 10.
Fig. 10.

Vertical profiles for each experiment of the square root of the globally and temporally averaged innovation variance (solid lines) and the predictions of what it should be if the assimilation assumptions are correct (dashed lines). These predicted values are the square root of the observation error variance plus the variance from the original 6-h background ensemble in ENS80 (black) and ENS240 (orange) and the (a),(b) VTSM- and (c),(d) VTSP-populated 6-h background ensembles, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue) for the (left) temperature and (right) wind forecasts at the 6-h lead time. In (c),(d) many of the curves are very similar and have been overplotted by the blue curves, which were plotted last.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

In different hemispheres, VTSP240Hτ and ENS240 do not show apparent spread change from ENS80, while the spread increase in VTSM240Hτ in TR is much smaller than that in NH and SH (not shown here). The smaller spread increase in TR in VTSM240Hτ could be associated with fewer ensemble mean differences added to the total variance in TR than in NH and SH due to their different types of instabilities, as discussed in section 4c.

e. Measure of effective rank in ensemble covariance matrices

The sampling errors in the ensemble covariances are also manifested in the form of a small number of independent subspaces sampled or sharp eigenvalue spectra of the ensemble covariances (Wang and Bishop 2003). The E dimension (Patil et al. 2001; Oczkowski et al. 2005; Kuhl et al. 2007) is therefore calculated to further evaluate the effectiveness of VTSM and VTSP in increasing the effective rank of the ensemble covariance matrix.

Detailed procedures of calculating the E dimension were documented in Oczkowski et al. (2005). Specifically, the E dimension was calculated by collecting the temperature and zonal wind perturbations at the 6-h lead time in each box, as in section 4c. A total energy rescaling norm is employed following Eq. (26) in Wang and Bishop (2003). The temperature perturbations are multiplied by a factor of , where Cp is the specific heat at constant pressure, and Tr is the reference temperature with 300 K (Palmer et al. 1998). Figure 11 shows the E dimension at 500 hPa in different hemispheres. Although ensemble size is tripled in ENS240, the E dimension in ENS240 is about 2.3 times as large as that of ENS80 in different hemispheres. VTSM240Hτ and VTSP240Hτ increase E dimension, compared to ENS80. Although they have the same background ensemble size of 240, VTSM240Hτ and VTSP240Hτ have smaller E dimension than ENS240. For a given shifting time interval, VTSP240Hτ shows larger E dimension than VTSM240Hτ, which suggests that the inclusion of ensemble mean differences in the VTSM-populated ensemble error covariances will reduce the effective rank relative to VTSP.

Fig. 11.
Fig. 11.

The E dimension calculated using the temperature and zonal wind perturbations at 500 hPa from the original 6-h background ensemble in ENS80 (black) and ENS240 (orange) and the VTSM- and VTSP-populated (filled with slash lines) 6-h background ensembles with applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue) in (a) NH, (b) TR, and (c) SH.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

On the other hand, larger shifting time interval results in larger E dimension in VTSP240Hτ in different hemispheres. This result is consistent with the expectation that when separated by larger lead time differences, the ensemble perturbations are more independent. However, by applying a larger shifting time interval in VTSM240Hτ, the E dimension is decreased in SH while it is increased in TR. In NH, VTSM240H3 also shows slightly decreased E dimension than VTSM240H1. These results of the VTSM experiments may be attributed to different amounts of contribution from ensemble mean differences added to the total VTSM-populated ensemble covariances in different hemispheres, controlled by different types of instabilities as discussed in sections 4c and 4d. For example, when a larger shifting time interval is applied in VTSM240Hτ, the relatively larger ensemble mean differences in NH and SH may dominate the total VTSM-populated ensemble covariances to a higher degree and cause smaller E dimension as a result. However, in TR, the ensemble mean differences in VTSM240Hτ contribute less to the total covariances; therefore, the VTSM-populated ensemble covariances are mostly contributed to by the original ensemble perturbations at the three different lead times [e.g., the first component in Eq. (A1)], which are expected to have more degrees of independence with larger time separation.

5. Evaluation of tropical cyclone track forecasts

a. Tropical cyclone track forecast verification

As discussed in sections 1 and 2, a global forecast system houses diverse weather phenomena. The impacts of VTSM and VTSP can be highly dependent on the scales and predictability of the weather systems of interest. Tropical cyclone is selected in this section, distinct from the general global forecasts in section 4, to further examine the impacts of VTSM and VTSP.

During the experiment period, a total of 25 named storms occurred at the Atlantic and Pacific basins, 12 of which reached the hurricane or typhoon category (Fig. 12). The NCEP tropical cyclone tracker (Marchok 2002) was used to track the storm locations in the forecasts. The same criteria described in section 4d of Wang and Lei (2014) were used to collect the forecast samples for the purpose of making a homogeneous comparison among different experiments.

Fig. 12.
Fig. 12.

Best track of the TCs during the experiment period in the (a) Atlantic, (b) east Pacific, and (c) west Pacific basins.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

Figure 13a shows the RMSE of the track forecasts verified against the best track data out to 5 days averaged over the 25 storms. A paired t test was conducted to evaluate the significance of the track error difference between ENS80 and the other experiments. ENS240, VTSM240Hτ, and VTSP240Hτ all statistically significantly improve the TC track forecasts, compared to ENS80, at most lead times out to 5 days.

Fig. 13.
Fig. 13.

(a) Track forecast errors in ENS80 (solid black), ENS240 (solid orange), VTSM240Hτ (solid), and VTSP240Hτ (dashed) experiments, applying a shifting time interval τ = 1 (red), 2 (green), and 3 h (blue). The circle (asterisk) signs right above the horizontal axis in (a) indicate that the track error difference from ENS80 in VTSM240H3 (VTSP240H1) is significant at or above the 95% confidence level by applying the paired t test at the corresponding forecast time. (b) Percentage of the track forecasts that are more accurate than that in ENS80 with the same line style and color indexes applied in (a). The numbers right above the horizontal axis in (b) denote the sample size at the corresponding forecast time.

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

VTSM240Hτ produces smaller track errors with larger shifting time interval. The best-performing VTSM experiment, VTSM240H3, is statistically significantly better than ENS80 beyond the 1-day lead time. VTSM240H3 performs the best among all VTS and ENS240 experiments. The VTSP experiments do not show strong sensitivity to the shifting time intervals. All VTSP experiments statistically significantly improve the TC track forecasts over ENS80 at most lead times, with the 1-h shifting interval performing slightly better. Although the VTSM and VTSP experiments are less costly than ENS240 (discussed in section 6), VTSP240H1 only performs slightly worse than ENS240 within the 4-day lead times, and VTSM240H3 even outperforms ENS240 beyond the 2-day lead time.

Following Zapotocny et al. (2008) and Wang and Lei (2014), the percentage of the track forecasts that are more accurate than those in ENS80 was calculated (Fig. 13b). Beyond the 1-day lead time, generally more than 50% of the forecasts in each of the experiments of ENS240, VTSM240Hτ, and VTSP240Hτ provide more accurate TC track forecasts than those in ENS80, and the percentage is generally increased at longer lead times. Compared to ENS240, VTSP240H1 shows a larger percentage of the improved track forecasts at most lead times, especially beyond the 3-day lead time. Consistent with Fig. 13a, VTSM240H3 outperforms ENS240 even more. Specifically, 53.1%–82.0% of the track forecasts in VTSM240H3, as opposed to 50.4%–62.3% in ENS240, are improved over ENS80 beyond the 1-day lead time. The result is that VTSM240H3 even outperforms ENS240, consistent with the previous studies where VTSM was found to improve the meso- and convective-scale weather forecasts where phase and timing errors contribute significantly.

b. Background track forecast error and ensemble track spread

Given the more accurate track forecasts of VTSM240H3 than ENS240, metrics of ARCE and ARCED with ENS240 as the referencing truth, defined in section 4c, are not appropriate to evaluate the ensemble correlation accuracy of VTSM240Hτ and VTSP240Hτ for the TC track forecasts. The E dimension1 for the TC track forecasts (not shown here) is similar to the general global forecasts in the TR region (Fig. 11b). Briefly, both VTSM240Hτ and VTSP240Hτ obtain larger E dimension than ENS80, but smaller than ENS240. The E dimension in VTSM240Hτ and VTSP240Hτ is further increased with larger shifting time intervals applied. For a given shifting time interval, VTSP240Hτ shows larger E dimension than VTSM240Hτ.

The improved TC track forecasts of VTSP240Hτ relative to ENS80 are therefore hypothesized to be related to the improved ensemble covariances, such as the reduced spurious covariances and increased effective rank, as discussed in the general global forecast diagnostics in sections 4c and 4e. On the other hand, the most accurate TC track forecasts in VTSM240H3 are likely contributed to by its capability of capturing background errors from sources that are missing in ENS80, ENS240, and VTSP240Hτ experiments (e.g., model timing or phase errors, as discussed in section 2 for such weather systems featured with relatively small scales and low predictability). This capability of VTSM in sampling background errors from missing sources is illustrated by the increased spread, as discussed in Fig. 2b in section 2a.

To further demonstrate this capability of VTSM, the background ensemble track spread at the 6-h lead time is evaluated against the background track error to reveal if the increased spread in VTSM240H3 is another contributor to its most improved TC track forecasts. In Figs. 14a–c, the scatterplots were created by collecting a total of 290 paired samples of the background ensemble track spread and absolute background track error from all 25 storms in ENS80, ENS240, and VTSM240H3, respectively. Although the background track errors have similar ranges for all three experiments, VTSM240H3 overall displays a wider range of background ensemble track spread, compared to ENS80 and ENS240. Following Wang and Bishop (2003), the spread–skill relationship for the 6-h background forecast in each experiment is further evaluated. Given the relatively small number of samples, instead of following Wang and Bishop (2003), who divided the samples into multiple equally populated bins, two equally populated bins representing the samples with small and large background ensemble track spread in each experiment (denoted by the blue and red circles in Figs. 14a–c, respectively) were formed. In each group, a rank histogram plot was further created. Given that the “distance” is evaluated, the rank histogram is formed slightly differently from the traditional scalar rank histogram (Hamill 2001). Specifically, the ranks are formed by collecting the distances between the predicted background ensemble mean storm location and the storm locations predicted from the individual background ensemble members, which is positive definite. Then, the number of samples for each rank is determined by throwing the corresponding samples of the distance between the predicted storm location from the background ensemble mean and the observed storm location. Different from the traditional rank histogram plot introduced for the scalar variable in Hamill (2001), a left (right) tail suggests the overdispersiveness (underdispersiveness) of an ensemble. For the first group representing small background ensemble track spread, compared to the severe underdispersiveness of the ensemble in ENS80 and ENS240 (e.g., the right-tailed distribution in Figs. 14d,e), the reliability of the ensemble in VTSM240H3 is improved, evidenced by a relatively flat distribution (Fig. 14f). On the other hand, for the second group featuring large background ensemble track spread, compared to ENS80 (Fig. 14g), the ensembles in ENS240 and especially VTSM240H3 show apparent overdispersiveness (Figs. 14h,i). Therefore, we speculate that the improved reliability of the ensemble in VTSM240H3 for the cases with small background ensemble track spread may be another contributor to its overall outperformance over ENS80 and ENS240 in terms of the TC track forecasts.

Fig. 14.
Fig. 14.

(a)–(c) Scatterplots of the 6-h background track errors on the vertical axis against the 6-h background ensemble track spread on the horizontal axis for (left) ENS80, (middle) ENS240, and (right) VTSM240H3. Blue (red) circles in (a)–(c) denote the equally populated samples representing small (large) background ensemble track spread in each experiment. The black dashed line is the diagonal line. (d)–(i) Rank histogram plots of (left) ENS80, (middle) ENS240, and (right) VTSM240H3 created from the samples representing (d)–(f) small background ensemble track spread and (g)–(i) large background ensemble track spread, which correspond to the blue and red circles in (a)–(c), respectively (see detailed descriptions of the rank histogram plots in the text).

Citation: Monthly Weather Review 146, 9; 10.1175/MWR-D-18-0009.1

6. Cost comparison with ENS80

As shown in Table 1, compared to ENS80, the increased computational cost in VTSM240Hτ or VTSP240Hτ is only incurred by ingesting 240 members instead of 80 members during the 4DEnVar variational update and extending the 80-member background ensemble forecasts by additional τ hours. Table 2 shows the cost for each of the four components in a single 4DEnVar DA cycle in each experiment. The cost in each of the four components in each experiment was estimated by the wall clock time from running the same number of cores on the same type of node on the National Oceanic and Atmospheric Administration (NOAA) High-Performance Computing System Jet machine. Compared to ENS80, the cost of ENS240 almost doubles in the 4DEnVar update and triples in the EnKF update and the ensemble background forecasts. In addition to the similar cost increase in the 4DEnVar update as in ENS240, VTSM240Hτ and VTSP240Hτ only increase the cost in the ensemble background forecasts by 9%, 14%, and 20% for 1-, 2-, and 3-h shifting time interval, respectively. Overall, as shown in the last column in Table 2, in contrast to ENS80, the total cost in ENS240 is increased by 160%, while VTSM240Hτ and VTSP240Hτ only increase the cost by 23%, 25%, and 27% for 1-, 2-, and 3-h shifting time interval, respectively. Sections 4 and 5 show that VTSP240H3 improves the 6-h temperature and wind forecasts by more than 50% and 60%, respectively, relative to the improvement in ENS240, and produces TC track forecasts with comparable or only slightly reduced skills, compared to ENS240. VTSM240H3 even shows more accurate TC track forecasts than ENS240. These performance and cost results suggest that the VTS methods provide a cost-effective means to treat sampling errors in the ensemble-based data assimilation system.

Table 2.

Wall clock time (min) for each of the four components in a single 4DEnVar DA cycle. The wall clock time is estimated from running on the same xJet node on the NOAA High-Performance Computing System Jet machine. The same number of 480 cores was used in each component for different experiments.

Table 2.

7. Conclusions and discussion

Instead of directly increasing the ensemble size, VTSM and VTSP are implemented and explored as inexpensive means to populate the background ensemble in the NCEP GFS hybrid 4DEnVar system. With the goal of sampling timing and/or phase errors, VTSM directly takes advantage of the ensemble members at different valid times to populate the background ensemble at the analysis time. By the design of shifting the ensemble perturbations at different valid times to the analysis time, VTSP performs temporal smoothing on the ensemble covariances, therefore eliminating the spurious covariances caused by sampling errors. To study the impacts of VTSM and VTSP in a global modeling system featured with different scales and predictabilities, both are evaluated for the general global forecasts and tropical cyclone forecasts in the GFS hybrid 4DEnVar system. By applying one single shifting time interval (τ = 1, 2, or 3 h), VTSM240Hτ and VTSP240Hτ triple the background ensemble size from 80 (ESN80) to 240 in the 4DEnVar variational update. Directly running 240 members, ENS240 is designed as the reference to evaluate the effectiveness of the inexpensive VTSM240Hτ and VTSP240Hτ experiments.

VTSP240Hτ generally improves the global temperature and wind forecasts to 5 days. Verified against rawinsonde observations, more than 50% and 60% of the improvement from ENS240 is recovered by the best-performing VTSP experiment (VTSP240H3) for the 6-h temperature and wind forecasts, respectively. Verified against the ECMWF analyses, VTSP240H2 produces the most consistent improvement for the temperature and wind forecasts to 5 days. Detailed diagnostics reveal that the improved global forecasts in VTSP240Hτ can be attributed to the populated background ensemble being closer to Gaussian distribution, improved accuracy of ensemble-estimated background error correlations, and increased effective rank (see summary in Table 3). VTSP240H3 overall shows better performance than VTSP240H2 in improving the ensemble correlation accuracy and increasing the effective rank; this is consistent with the more accurate 6-h global forecasts in VTSP240H3 (verified against both the rawinsonde observations and ECMWF analyses). On the other hand, in the global forecast verification against ECMWF analyses, the reduced forecast skills of VTSP240H3 relative to VTSP240H2 at longer lead times suggest that a trade-off in VTSP needs to be taken into account. This trade-off is between the loss of the flow-dependent features (e.g., eliminating the small-scale signals) and the gain of alleviating the sampling errors (e.g., removing the small-scale noises), owing to the smoothing impact of VTSP. VTSP240H2 may achieve a better balance between these two factors. In VTSP240H3, however, the benefits of alleviating the sampling errors may dominate in the short lead times and contribute to its most improved 6-h global forecasts, while the loss of the flow dependency or the small-scale signals may explain its neutral impacts on the global forecasts at longer lead times.

Table 3.

Summary of impacts of VTSM and VTSP on different aspects in a global context compared to ENS80.

Table 3.

In contrast to VTSP240Hτ, VTSM240Hτ shows degraded global forecasts in the troposphere, especially with a larger shifting time interval, VTSM240H3. This degradation may be attributed to degraded ensemble correlation accuracy, increased spread at such levels, and deviation from Gaussianity in the VTSM-populated ensemble (see summary in Table 3). The improved global forecasts in VTSM240Hτ above 100 hPa may be caused by the increased spread that alleviates the underdispersiveness of the original 80-member background ensemble at such levels (see summary in Table 3). By comparing the components of the VTSM- and VTSP-populated background ensemble error covariances, shown in Eqs. (A1) and (A3), the different impacts of VTSM and VTSP on those aspects in Table 3 are caused by the inclusion of the ensemble mean differences between the original and shifted ensembles in the VTSM-populated ensemble error covariances. This also suggests that the ensemble mean differences between the original and shifted ensembles fail to appropriately sample the background errors in a global forecast context.

For the TC track forecasts, experiments of ENS240, VTSM240Hτ, and VTSP240Hτ are all able to improve over ENS80. The performance of VTSP240Hτ does not show strong sensitivity to the shifting time intervals. Although much less costly, VTSP240Hτ produces comparable or slightly less accurate TC track forecasts than ENS240 within the 4-day lead times and even outperforms ENS240 beyond the 4-day lead time. Like ENS240, the improved TC track forecasts in VTSP240Hτ may originate from the improved accuracy of the estimated ensemble covariances. Larger shifting time interval in VTSM240Hτ shows enhanced improvement for the TC track forecasts. Especially, though much less costly, VTSM240H3 even shows more accurate track forecasts than ENS240. Further diagnostics suggest that the best performance of VTSM240H3 among all the experiments may be caused by its superior capability of capturing the errors from the missing sources, which is featured with the increased spread and therefore improves the reliability of the ensemble for the cases with small ensemble track spread.

Compared to ENS80, ENS240 increases the cost by 260%, while the cost in VTSM240Hτ and VTSP240Hτ is only increased by 23%, 25%, and 27% for τ = 1, 2, and 3 h, respectively. Therefore, these results suggest VTSM240Hτ and VTSP240Hτ provide cost-effective ways to improve sampling errors in ensemble-based data assimilation.

As discussed in section 3, Lei and Whitaker (2017) found little sensitivity of forecast performance with further tuned localization scales in a similar 4DEnVar setting. Therefore, our experiments of ENS240, VTSM240Hτ, and VTSP240Hτ apply the same localization length scales as in ENS80. Lorenc (2017), on the other hand, shows that increasing horizontal and vertical localization length scales were beneficial for direct increase of ensemble size and for using the time-lagged and time-shifted perturbation methods to increase ensemble size in the Met Office’s hybrid 4DEnVar system. The different response to the localization length scales for these two hybrid 4DEnVar systems when increasing the ensemble size is likely due to the different EnKF methods and different baseline ensemble sizes used to generate the ensembles. EnSRF with sequential assimilation and running 80 members is implemented in the GFS hybrid 4DEnVar system, while the Met Office’s hybrid 4DEnVar system adopts the ensemble transform Kalman filter (ETKF; Bishop et al. 2001; Wang and Bishop 2003; Wang et al. 2004, 2007a) and runs 44 members.

Overall, VTSP240H2 shows the most consistent improvement for both the global forecasts and storm track forecasts in the current experiment settings. VTSM240H3 shows the best hurricane track forecasts among all the experiments, whereas it generally degrades global forecasts in the troposphere. The impacts of further increasing the shifting time interval by more than 3 h in VTSM on the hurricane track forecasts remains to be investigated in the future work by adapting the time-lagged and time-shifted perturbation methods to use the form of full ensemble members in Lorenc (2017). These results also illustrate a challenge of optimizing DA algorithm in a multiscale data assimilation system. An additional experiment is warranted when experimenting with the methods in operational GFS hybrid 4DEnVar system, where a wider range of scales is resolved with a higher resolution (T1534/T574).

Another set of experiments was also attempted to inexpensively increase the ensemble size within the GFS hybrid 4DEnVar system by collecting the perturbations of the ensemble forecasts that are valid at the same analysis time but initialized from previous different cycles (i.e., the time-lagged approach with using the ensemble perturbations; not shown here). Compared to the baseline experiment ENS80, as discussed in this manuscript, this approach showed minimal or even negative impact on the global forecasts in the troposphere and the hurricane track forecasts. The only significant improvement from this approach was found for the global forecasts in the stratosphere above 100 hPa, as seen in the VTSM experiments. The improved global forecasts in the stratosphere could be attributed to the increased spread by utilizing the ensemble perturbations at longer lead times, which alleviates the underdispersiveness of the original 80-member ensemble in the stratosphere, as shown in Fig. 10. Given the inferior performance of this time-lagged approach, only the time-shifted approach is discussed in the manuscript.

Acknowledgments

The study was supported by NOAA NGGPS Award NA15NWS4680022. Computational resources provided by the NOAA High-Performance Computing System Jet machine and the OU Supercomputing Center for Education and Research at the University of Oklahoma were used for this study. The authors are thankful to many NOAA collaborators, in particular, Rahul Mahajan, Lili Lei, and Jeff Whitaker, for providing needed observations and scripts. Junkyung Kay is also acknowledged for helpful discussion. Three anonymous reviewers helped to significantly improve the manuscript.

APPENDIX

Components of the VTSM- and VTSP-Populated Background Ensemble Error Covariances

In VTSM, the background ensemble members valid at times tτ and t + τ, treated as the shifted background ensembles, are directly shifted to be valid at time t to supplement the original background ensemble at time t. If the original ensemble size is K, VTSM produces a populated background ensemble with size 3K. The covariances from the VTSM-populated background ensemble at time t can be derived as
ea1
where n is the dimension of the model state variables. The terms , , and are the background ensemble matrices of dimension at times tτ, , and t + τ, respectively, and their corresponding background ensemble mean vectors of dimension are denoted by , , and ; is a matrix with all the elements equal to 1, and its dimension is denoted by its subscript (e.g., dimension for the matrix ). Term is the VTSM-populated background ensemble mean equal to the average of the original background ensemble mean at the three different times. Term “diag” functions as converting a vector to a square diagonal matrix with the elements aligned on the diagonal, and superscript T is the matrix transpose sign. In Eq. (A1), the last step is derived by reformulating each matrix–multiplication term in the second row of Eq. (A1). For example, the term corresponding to the time is reformulated by
ea2

Equation (A1) shows that if the original ensemble size is large enough, the VTSM-populated background ensemble covariances can be obtained by summing up two components. One is an approximate average of the original background ensemble covariances at times tτ, , and t + τ [the terms within the first braces in the last row of Eq. (A1)]. The second component is approximately equal to averaging three matrices, obtained by an outer product of the vector representing the difference between the VTSM-populated ensemble mean and the original ensemble mean at the three different times. Since the VTSM-populated background ensemble mean is calculated as the average of the original background ensemble means at three different times, the second component can also be interpreted as the contribution from the background ensemble mean differences between the original and shifted background ensembles [the terms within the second braces in the last row of Eq. (A1)].

In VTSP, the populated background ensemble perturbations at time t are constructed by shifting the original background ensemble perturbations valid at times tτ and t + τ to the time t. Correspondingly, the covariances from the VTSP-populated background ensembles with size of 3K at time t can be expressed asA1
ea3

In Eq. (A3), because the three groups of the original background ensemble perturbations are calculated from its own background ensemble means, 3 degrees of freedom are removed from the VTSP-populated background ensemble. So instead of is supposed to be used as the denominator in Eq. (A3) to obtain the best unbiased covariances of the population from the VTSP-populated background ensemble sample. It shows the VTSP-populated background ensemble error covariances are equal to the average of the original background ensemble covariances at the three different times.

As noted, the calculations of the VTSM- and VTSP-populated background ensemble covariances in Eqs. (A1) and (A3) can be also applied to the scenarios of more than three time levels.

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1

Following section 4e, the samples for calculating the E dimension for the TC track forecasts were taken by collecting the temperature and zonal wind ensemble perturbations at the 6-h lead time in a box of 5° × 5° located around the TC center from all 25 tropical storms.

A1

In the calculations of the VTSP-populated background ensemble covariances of Eq. (A3), a factor of (3K − 3) is supposed to be used as the denominator in Eq. (A3), as explained in the text. In the practical implementation of VTSP, a factor of (3K − 1) was used as the denominator in Eq. (A3). But the misuse of the factor of (3K − 1) only causes a very tiny error by a factor less than 1%.

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