1. Introduction
Over the last decade, numerous operational atmospheric data assimilation systems relying on ensemble approaches to estimate the flow-dependent background error covariances have been implemented in various national forecasting centers for both deterministic (e.g., Clayton et al. 2013; Wang et al. 2013; Buehner et al. 2015; Caron et al. 2015) and ensemble forecasting (e.g., Houtekamer and Mitchell 2005; Hamill et al. 2011; Schraff et al. 2016) applications. One of the fundamental components that allowed the implementation of these systems is the use of spatial covariance localization methods [see, e.g., section 3e of Houtekamer and Zhang (2016)] needed to alleviate the large number of sampling errors in ensemble-derived background error covariances estimated from ensemble sizes of O(10) to O(100) members—the maximum sizes allowed, given the computational and time constraints for operational applications at present time. As a comparison, Kondo and Miyoshi (2016) were able to obtain good results without covariance localization in a global application at a relatively low resolution when using an ensemble Kalman filter (EnKF) with 10 240 members.
In most ensemble-based data assimilation schemes, the spatial localization method consists of using a simple prescribed homogeneous and isotropic function [usually the fifth-order piecewise rational function of Gaspari and Cohn (1999)] to gradually damp distant covariances toward zero, often reaching exactly zero at a given distance. The estimated optimal distance (or length scale) varies greatly as a function of ensemble size, size of the spatial domain, ensemble resolution, and observational network density [see, e.g., Buehner and Shlyaeva (2015, hereafter BS15) and references therein] and is usually determined through manually tuning the length-scale parameter by performing a series of complete data assimilation experiments [though Ménétrier et al. (2015) recently proposed a practical objective approach to estimate the optimal localization length scales]. However, more advanced localization methods have recently been proposed and tested in experimental contexts.
Scale-dependent localization (SDL; Buehner 2012) is one such advanced localization method and consists of applying appropriate (i.e., different) localization length scales to different ranges of background error covariance (spatial) scales while simultaneously assimilating all of the observations. In short, more severe localization is applied to small scales and less localization to large scales within the same single analysis procedure, at the expense of an increase in the computational cost. SDL aims to facilitate efficient data assimilation over a wide range of scales while avoiding the need to perform the analysis using multiple-step strategies like in Zhang et al. (2009) or Miyoshi and Kondo (2013) in the context of the EnKF. On the other hand, SDL was first introduced in the context of spectral localization (Buehner 2012), and therefore, the assumption was made that the covariance between the scales is zero (which has an impact that is equivalent to a spatial smoothing of covariances; Buehner and Charron 2007). With SDL, the use of prescribed homogeneous and isotropic localization functions is retained, as opposed to the “adaptive” localization approach of Bishop and Hodyss (2009, 2011) or the “empirical localization functions” methods of Lei and Anderson (2014) and Flowerdew (2015). The SDL method, in combination with spectral localization, was successfully shown to have positive impact on global NWP forecast accuracy by Buehner (2012) and Lorenc (2017) in global ensemble–variational (EnVar; e.g., Buehner et al. 2013; Lorenc et al. 2015) data assimilation schemes.
BS15 recently proposed a new SDL formulation that avoids the complete removal of the between-scale covariances imposed in the initial formulation of Buehner (2012). In the context of idealized two-dimensional univariate data assimilation experiments, BS15 showed that their SDL formulation leads to a reduction of analysis error over all spatial scales. The goal of the present study is to evaluate the potential benefits of this new approach in the EnVar-based global deterministic weather prediction system (GDPS; Buehner et al. 2015) of Environment and Climate Change Canada (ECCC), thus representing the first evaluation of this new method in a three-dimensional (3D) multivariate analysis system. A comparison with the original SDL formulation of Buehner (2012), as in BS15, is outside the scope of this study. Although the SDL approach can be applied, in theory, to both horizontal and vertical scales, this paper, as in previous studies, only focuses on horizontal-scale-dependent horizontal covariance localization.
The SDL formulation of BS15 is described in section 2, while section 3 presents the separation technique used to decompose ensemble-derived background error covariances in terms of overlapping horizontal wavenumber bands (hereafter referred to as “wave bands”). Section 4 reveals the impact of the SDL method on the horizontal distribution of the analysis increments through the assimilation of a single observation in two different flow regimes, while section 5 presents the impact of using SDL on the accuracy of 7-day forecasts in the ECCC GDPS. Finally, section 6 presents an examination of the impact of the SDL formulation on the dynamical balance of the analysis increments, and section 7 summarizes our findings and explores some potential future applications for the SDL method.
2. SDL formulation
3. Horizontal wave bands decomposition
For the evaluation of the SDL formulation presented in section 2, the approach was implemented in this study using three wave bands [compared to six in Buehner (2012) and four in Lorenc (2017), though for a different SDL formulation]. This arbitrary choice was motivated primarily to keep the total computational cost of the analysis step at a level that would not prevent a potential future implementation in the ECCC operational data assimilation systems, since using three wave bands already nearly triples the total cost.
The spectral coefficients used to decompose the global covariances are presented in Fig. 1. Their design is a rough attempt to isolate synoptic, subsynoptic, and mesoscales [see, e.g., Fig. 1.1 in Bluestein (1992)]. The filter response function to isolate the large scale is equal to 1 from wavenumber 0 (constant value over the globe) to wavenumber 4 (wavelength of 10 000 km) and then decays (following the square of a cosine) to 0 at wavenumber 20 (wavelength of 2000 km). For the small scale, the response function is 0 up to wavenumber 20 (wavelength of 2000 km) and reaches a plateau of 1 starting from wavenumber 80 (wavelength of 500 km). As for the medium scale, the response function is simply equal to the differences between a value of 1 and the sum of the two previous response functions, ensuring that the three overlapping response functions sum to 1, which is mandatory in this SDL formulation to minimize any changes to the overall background error variance.
Spectral filter coefficients used to separate the background error covariances into three horizontal wave bands (large scale in green, medium scale in blue, and small scale in red) expressed as a function of (a) wavelength and (b) spherical harmonic total wavenumber.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
The flow-dependent background error covariances used in this paper come from the ECCC 256-member global EnKF, using a grid spacing of 0.45° (~50 km) [described by Houtekamer et al. (2014), with subsequent modifications described by Gagnon et al. (2014, 2015)], the configuration that was operational in 2017 but executed on a retrospective time period in 2014. Figure 2 presents an example of ensemble perturbations for temperature on a native model level near ~700 hPa for a given ensemble member before (top panel) and after the scale decomposition (lower three panels). These three scale-decomposed perturbations were, as in the rest of this paper, simply obtained by first transforming the original ensemble member into spectral (i.e., spherical harmonic) space, then multiplying the resulting spectral coefficients by the filter coefficients shown in Fig. 1, and finally transforming the results back into gridpoint space. The large differences in scales in the three sets of scale-decomposed perturbations are obvious, and the amplitude of the perturbations is roughly similar in each of the wave bands (see again Fig. 2)—two indicators suggesting that the design of our bandpass filters is a reasonable first attempt for testing the approach.
Six-hour ensemble forecast perturbations for temperature (K) on a model vertical level near ~700 hPa from the EnKF valid at 1200 UTC 17 Oct 2014. (top panel) The full perturbation (ensemble member 1 minus the ensemble mean), and (bottom three panels) the scale-decomposed version of the same perturbation obtained after applying the three bandpass filters shown in Fig. 1.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
To obtain a more objective view of the resulting scale-decomposed ensemble perturbations, we computed vertical profiles of homogeneous and isotropic length scales using the definition given by Daley (1991, p. 110; describes the curvature of the correlation functions near their origin) as well as ratios of total variance (the variance in each wave band divided by the variance of the full perturbation) for each of the 3D analysis variables: zonal and meridional winds, temperature, and the natural logarithm of specific humidity. The length-scale profiles (Fig. 3) reveal again the large differences in scale from the three wave bands. It can also be seen that the vertical variation in length scales (the general increase with height) for zonal wind (Fig. 3a) and temperature (Fig. 3b) is much less pronounced in the wave bands than in the full perturbations. This can be explained by looking at the ratio of variances (Fig. 4), which reveals that the relative amplitude of the perturbations for the three wave bands changes considerably with height for these two variables. A clear increase in the contribution of the large-scale wave band is shown, especially in the stratosphere. It is these changes in the relative amplitude of the scales that influence the profile of length scales of the full perturbations. Taking another look at Fig. 4 also reveals that the relative contribution of the three wave bands is considerably different among the variables. For example, the temperature perturbation variance is dominated by the large-scale wave band at all levels in the troposphere (Fig. 4b), whereas humidity variances are mainly generated by both small and medium scales (Fig. 4c), except near the surface, where the impact of the large scale increases.1 From the above observations regarding Figs. 3 and 4, we can state that an explicit horizontal-scale-dependent horizontal localization approach implicitly leads, on average, to 1) vertical-level-dependent horizontal localization and 2) variable-dependent horizontal localization.
Vertical profiles of homogeneous and isotropic horizontal correlation length scales obtained from 6-h ensemble forecast perturbations from the EnKF valid at 1200 UTC 17 Oct 2014. Length scales were computed for (a) zonal wind, (b) temperature, and (c) the natural logarithm of specific humidity for the full-ensemble perturbations (black lines) and for the scale-decomposed ensemble perturbations obtained after applying the three bandpass filters shown in Fig. 1 (large scale in green, medium scale in blue, and small scale in red).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
Vertical profiles of the ratio of the variance in each of the scale-decomposed ensemble perturbations (large scale in green, medium scale in blue, and small scale in red) with respect to the variance in the full ensemble perturbations. Statistics computed from 6-h ensemble forecast perturbations from the EnKF valid at 1200 UTC 17 Oct 2014.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
Finally, it is important to stress that the length-scale values reported above for each of the wave bands should not be directly compared with the localization length scales reported in the rest of the paper, as the relationship between the two is not straightforward.
4. Impact on the analysis
To illustrate the impact of different localization approaches, we conducted a series of single temperature observation data assimilation experiments in two very different flow regimes present in the background of the GDPS analysis, valid at 1200 UTC 17 October 2014. The first observation is located in the center of Hurricane Gonzalo (point A in the Atlantic Ocean in Fig. 5), where small-scale background error covariance should be important. The second observation is at the center of a high pressure area (point B in Fig. 5; located northeast of Gonzalo, near Saint John’s in Newfoundland), where large-scale background error covariances should predominate. Note that unlike in the operational 4D-EnVar system at ECCC, no hybridization of the covariances was employed here (i.e., no blending with a climatological 
Mean sea level pressure analysis (contour interval every 4 hPa) valid at 1200 UTC 17 Oct 2014 from the ECCC GDPS.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
We first computed analysis increments without any spatial localization. Although the results are contaminated by large amounts of sampling error (i.e., a noisy pattern extending many thousands of kilometers away from the observation locations), it is relatively clear that the observation placed at the center of the hurricane (Fig. 6a) generates a much more compact distribution of analysis increments than the observation located at the center of the high pressure area (Fig. 7a). Using horizontal covariance localization based on the fifth-order piecewise rational function of Gaspari and Cohn (1999) to gradually force the correlation toward 0 at a distance of 2800 km (L28; the setup used in both 4D-EnVar and EnKF operational systems at ECCC) leads to relatively compact analysis increments in both scenarios (Figs. 6b, 7b).
Normalized temperature analysis increments resulting from the assimilation of a single temperature observation at the center of Hurricane Gonzalo (point A in Fig. 5) obtained from four different configurations of 3D-EnVar (a) without spatial localization, (b) using L28, (c) using SDL1, and (d) using SDL2. The values are exactly equal to 1 at the grid point nearest to the observation location, and the area in white represents values between −0.03 and 0.03. Data valid at 1200 UTC 17 Oct 2014.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
As in Fig. 6, but for an observation at the center of the high pressure system (point B in Fig. 5) over Newfoundland.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
The SDL approach using the wave band separation configuration presented in the previous section was tested with two different sets of localization length scales for each of the three wave bands. The first set (SDL1) forces the correlations to zero at a distance of 10 000, 4000, and 1500 km for, respectively, the large-scale, the medium-scale, and the small-scale wave bands. These ad hoc values were inspired by the large differences in homogeneous and isotropic length scales between the wave bands shown in Fig. 3, where both medium and large scales exhibit longer length scale, compared to the original perturbations at most of the vertical levels. As for the second set (SDL2), shorter distances of 3300 and 2400 km were instead used for the large- and medium-scale wave bands, while the same distance of 1500 km is used for the small-scale wave band. This more conservative configuration was inspired by the values used in Buehner (2012).
The impact of SDL1 on the hurricane case (Fig. 6c) leads to moderate changes with respect to L28 (Fig. 6b), with broader and smoother large-scale signal surrounding a narrow core of high-amplitude normalized increments that remains relatively unchanged. However, for the high pressure cases, the changes brought by SDL1 are quite drastic (cf. Figs. 7b,c), as the normalized increments now extend many thousands of kilometers away from the observation location, but without the high level of noise observed without spatial localization (cf. Figs. 7c,a). The large-scale pattern depicted by SDL1 exhibits similarities with the normalized analysis increments that result from using our climatological 
5. Impact on the forecasts
The impact of the SDL approach on the GDPS forecast performance was evaluated in data assimilation experiments using a 3D-EnVar scheme instead of the operational 4D-EnVar configuration. Time-dependent background error covariances were removed, as we aim to evaluate in a separate study the impact of the SDL approach on temporal localization in combination with the inclusion of the Lagrangian advection of the localization, as in Desroziers et al. (2016). Also, our climatological 
Summary of experiments.
The forecast accuracy was evaluated by comparing the forecasts with the set of independent atmospheric analyses from ERA-Interim (Dee et al. 2011). This comparison is done after first using a spatial averaging procedure for both the forecasts and the ERA-Interim analyses to interpolate them onto a global 1.5° latitude–longitude grid. Consequently, this evaluation does not include scales that cannot be resolved on this relatively low-resolution grid. The RMS error (RMSE) was computed for forecast lead times between 24 and 168 h, every 24 h, for five variables in the extratropical regions (temperature at 850 hPa, relative humidity at 700 hPa, geopotential height at 500 hPa, and zonal wind at 250 and 50 hPa) and three variables in the tropics (zonal wind at 850, 250, and 50 hPa).3 The results presented in Figs. 8–10 are expressed in terms of relative changes in RMSE with respect to the L28 (control) experiment multiplied by −1 to simulate a change in a quality index. Therefore, positive (negative) values represent improved (degraded) forecasts with respect to ERA-Interim. Statistical confidence levels were obtained using the permutation technique described in Efron and Tibshirani (1993, see their chapter 15). The changes in this ad hoc quality index were also averaged over all variables and lead times in order to get a single value for each of the three domains considered (see Table 2). The evaluation approach adopted here is similar to that of Lorenc (2017).
Changes in the forecast quality index (defined as the relative change in RMSE times −1) with respect to control experiment L28 and measured using ERA-Interim for the SDL experiments (a) SDL1 and (b) SDL2 over a period of 1.5 months (from 0000 UTC 16 Jun to 1200 UTC 31 Jul 2014). The grid of triangles shows different forecast lengths and several fields at different pressure levels for the northern extratropics, tropics, and southern extratropics. Upward-pointing (downward pointing) red (blue) triangles indicate a reduction (increase) of the RMSE. Filled (empty) triangles indicate that the confidence level is greater (lower) than 90%.
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
As in Fig. 8, but over an extended total period of 2.5 months (i.e., from 0000 UTC 16 Jun to 1200 UTC 31 Aug 2014).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
As in Fig. 8, but using a constant horizontal localization length of (a) 2400 km (experiment L24) and (b) 3300 km (experiment L33).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
Summary of changes in total forecast quality index (in %) with respect to experiment L28 (see text for the details on the computation of this metric). Values greater than 0.5% are highlighted in bold.
The verification scores of the forecasts from SDL1 reveal a statistically significant degradation of the scores for the stratospheric zonal winds in both tropical and northern extratropical regions that peaks at 24 h and diminishes with forecast lead time (Fig. 8a), whereas the rest of the scores are a relatively even mix of statistically nonsignificant improvements and degradations. The overall impact of SDL1 is negative in two out of the three domains (Table 2). However, when using the SDL2 configuration that uses a much less aggressive approach, as compared to SDL1, in the variation of the localization amount between the wave bands (see again Table 1), a clear positive impact can be observed almost everywhere (Fig. 8b). The largest improvements are observed in the Southern Hemisphere extratropics, with changes in the quality index peaking at days 5 and 6. In the Northern Hemisphere, the improvements at short lead times are similar to the other hemisphere, but the positive signal gradually decays beyond 72 h, whereas it increases for longer lead times in the Southern Hemisphere. Further experiments would be necessary to determine if these differences between the hemispheres are related to the differences in seasons or in observing networks. As for the tropics, the improvements are somewhat smaller in maximum amplitude but are remarkably constant with lead time. The above results lead to an overall improvement of the score greater than 0.5% in every region, with maximum improvement in the Southern Hemisphere (Table 2). Since only a few of the positive scores found in SDL2 exceed the 90% confidence level, the SDL2 and the corresponding control experiments (L28) were both extended by an additional month. The verification scores over a 2.5-month period are very similar to the verification over the previous shorter period of 1.5 months (compare Figs. 9, 8b), except that more scores now exceed the 90% confidence level, mainly in the Southern Hemisphere. The averaged positive impacts are even slightly greater in every domain (Table 2).
Two additional 1.5-month experiments were carried out to evaluate the impact of using the length scales adopted for the medium- and the large-scale wave bands in SDL2 in the traditional “one size fits all” localization approach. Experiment L24 (L33) forces the covariances to zero at a distance of 2400 (3300) km (see again Table 1). When compared to L28, the scores resulting from L24 are rather mixed, except in the tropics, where the improvements are similar to SDL2 (see Fig. 10a and compare with Fig. 8b). As for L33, it is obvious that reducing the amount of localization in our EnVar scheme leads to a general degradation of the scores. Taking a final look at Table 2 indicates that SDL2 outperforms every experiment where the same amount of localization is applied to all the scales, except in the tropics, where similar improvements can be obtained by simply increasing the amount of localization.
Note that verification statistics against radiosonde observations also led to similar conclusions about the relative performance in the different experiments reported in this section (not shown). We also remark that in all the experiments with traditional localization (i.e., L24, L28, and L33), the horizontal localization amount was kept constant with height since the introduction of height-dependent horizontal localization had previously been found to have no significant beneficial impact in our EnVar system.
6. Impact on the dynamical balance
It is well known that spatial localization is detrimental to the dynamical balance of the resulting analysis increments in both the EnKF and EnVar schemes (Lorenc 2003; Houtekamer and Mitchell 2005; Kepert 2009; Caron and Fillion 2010). In this section, we investigate if the SDL method causes changes in the dynamical balance. The degree of balance between the mass and the rotational wind components in experiments SDL2 and L28 was measured using the same method employed in Caron et al. (2007) and Caron and Fillion (2010), and the following description in the next two paragraphs is derived from Caron and Fillion (2010), with minor modifications.
The top row of Fig. 11 indicates that the correlations between mass and wind field terms in the analysis increments from SDL2 are increased, compared to L28 in both hemispheres, particularly in the troposphere. However, in terms of ratios between the wind and the mass component amplitudes (wind RMS divided by mass RMS; middle row of Fig. 11), the deviations from a ratio of 1 are greater in the analysis increments from SDL2 than from L28 at most vertical levels in the Southern Hemisphere (Fig. 11d) and, to a much lesser extent, in the Northern Hemisphere (Fig. 11c). Nevertheless, the overall measure of imbalance (U), depicted in the bottom row of Fig. 11, indicates that the analysis increments from SDL2 are generally closer to balance than the analysis increments from L28 in the troposphere of both hemispheres.
Vertical profiles of the degree of nonlinear balance in the analysis increments valid at 0000 UTC 1 Jul 2014 resulting from L28 (blue lines) and SDL2 (red lines) averaged over the extratropics of the (left) Northern Hemisphere and (right) Southern Hemisphere. (a),(b) Correlations between mass and wind components; (c),(d) ratios of average horizontal RMS value of mass and wind components (wind RMS divided by mass RMS); and (e),(f) normalized deviation from nonlinear balance computed using (8).
Citation: Monthly Weather Review 146, 5; 10.1175/MWR-D-17-0369.1
Despite the small overall improvements in the degree of rotational balance obtained with the SDL method, the gain appears negligible, compared to the detrimental impact of using horizontal covariance localization, as shown in Caron and Fillion (2010, see their section 3c). Nonetheless, it appears from the results presented in this section that the SDL approach does not further degrade the rotational balance when compared with the current localization approach.
7. Summary and discussion
The SDL method recently proposed by BS15 for the EnVar data assimilation scheme, which avoids the complete removal of the between-scale covariances imposed in the initial formulation of Buehner (2012), was tested in a 3D-EnVar version of the ECCC GDPS, thus representing the first evaluation of this new method in a 3D multivariate analysis system. Although the SDL approach can be applied to both horizontal and vertical scales, this paper only focused on horizontal-scale-dependent horizontal covariance localization, though, ideally, both horizontal and vertical scales should be modified accordingly based on normal mode theory (e.g., Fillion and Roch 1992).
When compared with using the same amount of localization for all scales, the application of the SDL method during the boreal summer was shown to result in more accurate forecasts in the extratropical part of both the Northern and Southern Hemispheres. In the Northern Hemisphere, the improvements were largest at short lead time and lasted about 5 days, whereas in the Southern Hemisphere, the improvements were larger, lasted over the whole 7 days of the forecasts, and peaked at days 5 and 6. In the tropics, the SDL approach was also able to improve the forecasts’ accuracy, as compared to the usual localization approach used in the operational 4D-EnVar and EnKF systems at ECCC, but led to results similar to those obtained when using increased localization with the “one size fits all” approach. Further examination is thus recommended to understand why the SDL technique and configuration used here was not found to be beneficial in the tropics. We remark that the overall performance gain from using SDL (around 1% on average, as shown in Table 2) is roughly similar to the improvements typically observed (in the range of 1% to 2%) for upgrades of the global operational system at ECCC.
It was shown that a horizontal-scale-dependent horizontal localization leads to implicit vertical-level-dependent, variable-dependent, and location-dependent horizontal localization. It is unclear at this point what the relative contributions of these three changes are on the forecast improvements reported here.
Examination of the dynamical balance of the analysis increments revealed that the SDL approach seems to slightly improve the correlation between the mass and the rotational wind field, compared to the traditional localization approach, but, on the other hand, amplifies somewhat the underestimation of the wind field, compared to the mass field. Nevertheless, an overall measure of imbalance showed small general improvements in the degree of rotational balance when using the SDL method, but this difference appears negligible, compared to the detrimental impact of using any type of horizontal localization.
For various reasons, we choose not to include in this study a comparison of the SDL method proposed by BS15 with the formulation of Buehner (2012). The main reason is that the latter does not appear suitable for a relatively large ensemble size like the ECCC 256-member EnKF since it completely discards the between-scale covariances. This between-scale localization is equivalent to applying a spatial smoothing of the covariances (Buehner and Charron 2007), an approach that is expected to be beneficial only when used in combination with very small ensemble size [i.e., O(10) members; e.g., see Berre and Desroziers (2010) and references therein]. The formulation of Buehner (2012) is also not simpler to implement or computationally cheaper.
It is important to stress that finding the optimal SDL configuration is not trivial and was done in an ad hoc manner in this paper. Therefore, we cannot conclude that the results presented here depict all of the benefits that could be obtained with this method in our global EnVar system. First, since the SDL approach increases the computational cost of the analysis step linearly with the number of wave bands, a three-wave-band decomposition was adopted here to keep the computational cost at a level that would not prevent a potential future implementation in our operational data assimilation systems. It is unclear how to determine both the optimal number of wave bands and the optimal filtering response function design. Second, finding an appropriate amount of localization for each of the wave bands is not straightforward. Two attempts were needed here to obtain improvements, but there is no guarantee that the amounts of localization employed in the so-called SDL2 configuration are optimal. The recently proposed objective approach by Ménétrier et al. (2015) to determine the amount of localization does not appear appropriate to find the optimal amount of localization in each of the wave bands when using the SDL formulation of BS15, since the latter retains the correlations between the wave bands, while the former assumes that the (scale decomposed) ensemble members are independent from each other. Its application to our three-wave-band decomposition led to localization distance (not shown) similar to the SDL1 configuration, which was shown to deteriorate the forecast accuracy. More research is thus recommended on this topic.
Despite the encouraging results reported here, the investigation of the impact of SDL in a 4D-EnVar scheme using hybrid background error covariances will need to be assessed before considering its implementation in an operational context. In such tests with time-varying background error covariances, we also envision combining the SDL approach together with the inclusion of Lagrangian advection of the spatial localization, as in Desroziers et al. (2016).
Though the focus of all the studies so far on SDL have been on horizontal-scale-dependent horizontal covariance localization, this approach could also be used to improve vertical localization. A first step could be to impose a different amount of vertical localization for each range of horizontal scales. A more appropriate implementation would probably consist of combining decomposition into ranges of both horizontal and vertical scales, allowing, for example, more (less) horizontal covariance localization and less (more) vertical localization to be applied on narrow (large) and deep (shallow) weather phenomena like thunderstorms (stratocumulus). This will be the subject of our future research on the SDL approach in the context of limited-area NWP at the convective scale and also a comparison with the original SDL formulation proposed by Buehner (2012).
Acknowledgments
The authors thank Yann Michel (CNRM/Météo-France) for providing code that helped to plot the scorecards shown in Figs. 8–10, as well as Stéphane Laroche for developing and maintaining the forecast verification package used in this study. The authors also thank Luc Fillion, whose comments helped to improve an earlier version of the paper.
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This could be explained by the continental (dry)–ocean (moist) contrast projecting on the largest scales.
In short, the weight given on our climatological 
Tropical region is defined here as the area between 30°S and 30°N.
