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  • View in gallery

    Domain partitioning of the cycled banded ET for the choices of Nband = (a) 5, (b) 9, and (c) 17. (d) Domain partitioning of the cycled block ET for the choice of Nblock = 18.

  • View in gallery

    Illustration of the centering of the weighting matrices 𝗧1, 𝗧2, and 𝗧3 in their respective latitude bands, for the choice of Nband = 5.

  • View in gallery

    Expected ratio of the ET analysis perturbation variance (σa2)ENS and the estimated analysis error variance from the NAVDAS data assimilation system (σa2)DA for the cases of (a) zonal wind. (b) meridional wind, (c) temperature, and (d) relative humidity. Results are the average for 10 days in the test period. For clarity, only results for the cycled global ET and cycled five- and nine-banded ETs are shown.

  • View in gallery

    Average eigenvalue spectrum of the (a) T + 24 and (b) T + 120 forecast-error covariance matrices under the dry total energy norm. The eigenvalue spectrum of the forecast-error covariance matrices was determined for each of the 92 forecast ensembles in the period 25 Jun–17 Jul 2005. Each spectrum was normalized by its leading eigenvalue, and the resulting 92 normalized spectra were averaged.

  • View in gallery

    Solid lines indicate ensemble-average dry total energy of the forecast perturbations as a function of lead time, domain averaged over the (a) NH midlatitudes (20°–65°N), (b) SH midlatitudes (65°–20°S), and (c) tropics (20°S–20°N) and averaged over the twenty-three 0000 UTC forecast ensembles generated within the period 25 Jun–17 Jul 2005. The dashed line indicates the dry total energy of the observed errors in the ensemble mean of the 0000 UTC cycled global ET, averaged over the same domain and period. See the text for further details of the error calculation.

  • View in gallery

    Area-weighted rmse of the ensemble-mean 500-hPa geopotential height in the (a) NH midlatitudes (20°–65°N), (b) SH midlatitudes (65°–20°S), and (c) tropics (20°S–20°N) averaged over the 92 forecast ensembles in the period 25 Jun–17 Jul 2005. (d)–(f) As in (a)–(c), respectively, but for 925-hPa zonal wind.

  • View in gallery

    Area-weighted average, in the (a) NH midlatitudes (20°–65°N), (b) SH midlatitudes (65°–20°S), (c) tropics (20°S–20°N), and (d) deep tropics (10°S–10°N), of the Brier score for the event that the 10-m zonal wind magnitude exceeds 10 m s−1.

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A Local Formulation of the Ensemble Transform (ET) Analysis Perturbation Scheme

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Abstract

Following ideas from the local ensemble transform Kalman filter, a local formulation of the ensemble transform (ET) analysis perturbation scheme is developed by partitioning the numerical weather prediction model domain into latitude bands or latitude–longitude blocks. In comparison with analysis perturbations from the original “global” ET formulation, analysis perturbations from the “banded” or “block” ET formulations are much more consistent with estimates of analysis error variance. Banded or block ET forecast ensembles also perform better under a variety of verification metrics than do global ET forecast ensembles. Substantial performance gains are observed for both the midlatitudes and the tropics. A local ET is scheduled to be made operational at the Fleet Numerical Meteorology and Oceanography Center.

Corresponding author address: Justin McLay, Naval Research Laboratory, 7 Grace Hopper Ave., Monterey, CA 93943. Email: justin.mclay@nrlmry.navy.mil

Abstract

Following ideas from the local ensemble transform Kalman filter, a local formulation of the ensemble transform (ET) analysis perturbation scheme is developed by partitioning the numerical weather prediction model domain into latitude bands or latitude–longitude blocks. In comparison with analysis perturbations from the original “global” ET formulation, analysis perturbations from the “banded” or “block” ET formulations are much more consistent with estimates of analysis error variance. Banded or block ET forecast ensembles also perform better under a variety of verification metrics than do global ET forecast ensembles. Substantial performance gains are observed for both the midlatitudes and the tropics. A local ET is scheduled to be made operational at the Fleet Numerical Meteorology and Oceanography Center.

Corresponding author address: Justin McLay, Naval Research Laboratory, 7 Grace Hopper Ave., Monterey, CA 93943. Email: justin.mclay@nrlmry.navy.mil

1. Introduction

Ensemble transform methods include the ensemble transform (ET) analysis perturbation scheme (Wei et al. 2008; McLay et al. 2008) and the ensemble transform Kalman filter (ETKF) data assimilation scheme (e.g., Bishop et al. 2001; Wang and Bishop 2003; Ott et al. 2004; Hunt et al. 2007; Szunyogh et al. 2008; Bowler 2006; Bowler et al. 2009). These methods obtain analysis perturbations for the current analysis time as linear combinations of forecast perturbations initialized at the previous analysis time. The ET analysis perturbation scheme has been operationally implemented at both the National Centers for Environmental Prediction and the Fleet Numerical Meteorology and Oceanography Center. In both operational ET implementations, the transform is a “global” operation, such that a single transformation matrix is applied to the entire numerical weather prediction (NWP) model domain. The ETKF data assimilation scheme has been operationally implemented at the Met Office (Bowler et al. 2009). In this instance, the transform is a local operation. That is, the NWP model domain is partitioned into subdomains and then multiple transformation matrices are obtained, one for each subdomain. This “local” formulation of the ETKF is intended to introduce covariance localization.

The global nature of the present operational implementations of the ET engenders some issues because the ensemble size K is typically very small as compared with the dimension of the entire NWP model state N; that is, K/N ≪ 1. One of the main issues is that the resulting analysis perturbations can be regionally inconsistent with the estimates of analysis error covariance from the data assimilation system (e.g., McLay et al. 2008). The point emphasized here is that a means to mitigate the regional inconsistency exists within the framework of the local ETKF. In particular, one notes that the ensemble size is necessarily larger relative to the state dimension of a given subdomain than it is relative to the state dimension of the entire NWP model domain. It is a property of the ET that, as the ensemble size approaches the state dimension or the rank of the estimated analysis error covariance matrix 𝗣a from the data assimilation system (whichever is smaller), the ET analysis perturbation covariance matrix 𝗣ae will converge to 𝗣a [see property 4 of McLay et al. (2008)]. It follows that the solution of the ET on multiple subdomains rather than on a single, global domain should, in theory, reduce regional inconsistency between the ET analysis perturbation covariance 𝗣ae and the estimated analysis error covariance 𝗣a.

Motivated by the above observations, here two local versions of the ET analysis perturbation scheme are formulated and are compared with the traditional, global version of the ET. These local ET versions are termed the “banded” and the “block” ETs because the NWP model domain is partitioned into latitude bands or latitude–longitude blocks, respectively. The paper is organized as follows. Section 2 describes the method of the banded and block ETs and of the intercomparison experiment. Results and conclusions are presented in sections 3 and 4, respectively.

2. Method

a. The cycled global ET

The fundamental ET equation set is
i1520-0434-25-3-985-e1
where N and K are as defined in section 1, 𝗭f (𝗭a) is an N × K matrix of forecast (analysis) perturbations, 𝗧 is a K × K matrix of weighting coefficients, 𝗣a is an N × N matrix of analysis error covariance estimates, and 𝗜 is the N × N identity matrix. The constraint 𝗭aT(𝗣a)−1 𝗭a = N𝗜 imparts a magnitude to the perturbations that is globally consistent with estimates of analysis error covariance and ensures that the perturbations are quasi orthogonal. The mean of the basis vectors in 𝗭f is required to be 0 to ensure that the mean of the analysis perturbations is 0. Thus, only K − 1 basis vectors are linearly independent. Also, since KN, 𝗭f should be thought of as a (K − 1)-member approximate basis for the N-dimensional NWP-model vector space.

b. The cycled banded ET

In the banded ET the global NWP domain is partitioned into Nband nonoverlapping latitude bands. Each band covers the full horizontal and vertical extent of the NWP domain within the specified latitude bounds. Three banded configurations were tested for this study, corresponding to choices of Nband = 5, 9, and 17 (Figs. 1a–c). For a given banded configuration, the banded ET procedure proceeds in three steps.

In the first step, Nband different K × K weighting matrices 𝗧j (j = 1, … , Nband) are obtained, one for each of the Nband latitude bands. We define nj, 𝗭fj, and 𝗣aj to be the number of elements of the state vector within band j, the nj × K forecast perturbation matrix for band j, and the nj × nj diagonal analysis-error covariance matrix for band j, respectively. Then, 𝗧j is obtained by an eigendecomposition of the symmetric matrix
i1520-0434-25-3-985-e2
in accordance with Eqs. (4)–(7) of McLay et al. (2008). Once the 𝗧j are obtained, they are “centered” in their respective latitude bands, as illustrated by Fig. 2 for the choice of Nband = 5. The centers of the 𝗧j are analogous to the “localization centres” of Bowler et al. (2009).

In the second step, we let M equal the number of Gaussian latitudes of the NWP model. Then, for each Gaussian latitude m (m = 1, … , M), a K × K weighting matrix is obtained through linear interpolation of the centered weighting matrices 𝗧j (j = 1, … , Nband). In the five-banded configuration, for example, for each Gaussian latitude between 40°N and 0° is obtained by linear interpolation between 𝗧2 (which is centered at 40°N) and 𝗧3 (which is centered at 0°).

Third, we define Nm to be the number of elements of the state vector within the vertical slice of domain through Gaussian latitude m. Then, for each Gaussian latitude m (m = 1, … , M), the ensemble transform equation
i1520-0434-25-3-985-e3
is solved, where 𝗭fm(𝗭am) is the Nm × K forecast (analysis) perturbation matrix for Gaussian latitude m.

c. The cycled block ET

In the block ET, the nonpolar region of the global NWP domain is partitioned into Nblock nonoverlapping latitude–longitude blocks. Each block covers the full horizontal and vertical extent of the NWP domain within the specified latitude–longitude bounds. The two polar regions are encompassed by latitude bands, as in the banded ET. A single configuration of the block ET is tested here, corresponding to a choice of Nblock = 18. This is derived from the five-banded ET configuration (Fig. 1a), such that each of the three nonpolar bands is partitioned into six latitude–longitude blocks, with each block spanning 60° of longitude (Fig. 1d). With regard to procedure, the block ET is much the same as the banded ET:

First, Nblock + 2 different K × K weighting matrices 𝗧j (j = 1, … , Nblock + 2) are obtained, one for each of the Nblock + 2 partitions (i.e., Nblock latitude–longitude blocks and two polar latitude bands). Then, 𝗧j are obtained by solving Eq. (2), and the 𝗧j are centered in their respective partitions.

Second, we let P equal the number of Gaussian grid points on a vertical level of the NWP model. Then, for each Gaussian grid point p (p = 1, … , P), a K × K weighting matrix is obtained through bilinear interpolation of the centered weighting matrices 𝗧j (j = 1, … , Nblock + 2).

Third, we define Np to be the number of elements of the state vector within the vertical column through Gaussian grid point p. Then, for each Gaussian grid point p (p = 1, … , P), the ensemble transform equation
i1520-0434-25-3-985-e4
is solved, where 𝗭fp(𝗭ap) is the Np × K forecast (analysis) perturbation matrix for Gaussian grid point p.

d. Miscellaneous discussion for the cycled banded and block ET

Both the banded and block ET formulations require an insignificant increase in operation count relative to the global ET. Also, the template for these formulations is the local ETKF of Bowler et al. (2009). One distinction between the formulations is that Bowler et al. use overlapping domain partitions, whereas the local ET formulations use nonoverlapping domain partitions for expediency. Another distinction relates to the fact that if the local ET is applied in conjunction with a diagonal 𝗣a (as is likely the case in practice) then there is an implicit limit on how small the domain partitions can be. This can be understood as follows: If the size nj of domain-partition j decreases, then Knj and by property 4 of McLay et al. (2008) (𝗣ae)j → 𝗣aj. However, if the off-diagonal elements of 𝗣aj are zero then as (𝗣ae)j → 𝗣aj the analysis perturbations associated with (𝗣ae)j must exhibit progressively less spatial and multivariate correlation. Some reduction in correlation may be beneficial; it implies a randomization of the analysis perturbations, and various types of random analysis perturbations have proven to be effective in ensemble forecasting (e.g., Anderson 1997; Magnusson et al. 2009; McLay and Reynolds 2009; Zhang 2005; Zhang et al. 2007). On the other hand, too great of a reduction in correlation may adversely affect the balance and growth conditioning of the analysis perturbations. The risk of excessive reduction in correlation means that very small domain partitions must be avoided. The local ETKF formulation of Bowler et al. does not have this restriction because it makes use of full error covariance matrices.

e. The not-cycled five-banded ET

McLay and Reynolds (2009) formed an N × K approximate basis matrix 𝗬 that is independent of the analysis cycle by drawing randomly generated forecast perturbations from a historical archive and found that the resulting “not cycled” global ET was highly competitive in comparison with the original, cycled global ET. Thus, for comparison, a not-cycled five-banded ET is also examined here, using the same method as in McLay and Reynolds (2009), as follows: An archive of 1888 forecast perturbations was derived from a set of 32-member Navy Operational Global Atmospheric Prediction System (NOGAPS) ensembles initialized at 0000 UTC each day of the period 1 January–28 February 2006. For each of the fifty-nine 32-member ensembles in the period, 72-h forecast perturbations were obtained by subtracting the 72-h ensemble-mean forecast from each of the 72-h ensemble members. The initial perturbations for each ensemble were generated using a random-sampling method related to that described in Errico and Baumhefner (1987). As pointed out in McLay et al. (2007), the archived forecast perturbations are very small, even though they are technically valid at the 72-h lead time. This is because the random initial perturbations from which the archived forecast perturbations are integrated undergo significant decay during the first 24 h of the 72-h integration. To obtain the not-cycled basis, perturbations are randomly selected from the archive and the mean of the perturbations is removed. The random selection takes place each time an ensemble transform is to be performed. Note also that the choice of a winter-season perturbation archive was dictated by constraints on computational resources; the archive was taken “off the shelf,” and no attempts were made to optimize the results by adjusting its composition.

f. Numerical model and ensemble configuration

The cycled global ET, the cycled banded and block ETs, and the not-cycled five-banded ET were each used to generate analysis perturbations for a global forecast ensemble. The analysis perturbations were integrated to the 168-h lead time using NOGAPS, a primitive equation, fully parameterized spectral model (Peng et al. 2004). The experiments employ NOGAPS at horizontal resolution T119 and with 30 vertical levels.

All ET operations were carried out with a diagonal 𝗣a obtained directly from the Naval Research Laboratory Atmospheric Variational Data Assimilation System (NAVDAS), an operational three-dimensional variational assimilation system [for details see McLay et al. (2008) and Daley and Barker (2001)]. The ensemble size was chosen to be 16, and each ensemble member consisted of perturbations in wind, temperature, specific humidity, and terrain pressure. The perturbations in wind and temperature were calculated for all vertical levels of the numerical model, whereas the perturbations in specific humidity were calculated for those vertical levels between the surface and roughly 300 hPa. The perturbations were generated at 0000, 0600, 1200, and 1800 UTC of each day during the period 25 June–17 July 2005.

3. Results

a. Consistency between analysis perturbation variance and NAVDAS analysis error variance

Figure 3 displays the expected ratio between the ET analysis perturbation variance and the NAVDAS analysis error variance, averaged over the zonal and vertical directions and over 10 dates in the test period. In the ideal case, the expected ratio would be 1. Ratios less (greater) than 1 indicate that the ensemble variance is less (greater) than the NAVDAS analysis error variance. For the wind components (Figs. 3a,b), the expected ratio for the two cycled banded ET variants is clearly much improved in comparison with the expected ratio for the cycled global ET. In particular, the expected ratio for the nine-banded ET is close to 1 at most latitudes. For temperature, the results are mixed (Fig. 3c). The cycled banded ET variants have an expected ratio that is much closer to 1 in the tropics but that isfarther from 1 in the extratropics. Considering relative humidity (Fig. 3d), the expected ratio for the cycled banded ET variants is clearly closer to 1, in general, but the improvement is not as pronounced as that for the wind components. One also finds that the expected ratio for the cycled banded ET variants is consistently less than 1 for temperature and greater than 1 for relative humidity. This result could be a reflection of systematic differences between the physical balance relationships in the NAVDAS models of background and observation error and those in the NOGAPS forecast. That is, the NAVDAS error models are formulated based only upon geostrophic balance and certain relationships involving the rotational and divergent wind components, whereas the NOGAPS forecast incorporates all of the dynamic and thermodynamic relationships of a moist, primitive equation system. Nevertheless, the general result is that the local ET provides substantially improved consistency with the NAVDAS analysis error variance, as was hoped for.

b. Eigenvalue spectrum of the forecast-error covariance matrix

The eigenvalue spectrum of the ensemble forecast-error covariance matrix 𝗣fe indicates the extent to which the ensemble maintains variance in different directions during integration. Figure 4 shows the eigenvalue spectrum of 𝗣fe under the dry total energy norm at two different lead times for all of the forecast ensembles. The spectrum for the cycled banded and block ETs is substantially flatter than that for the cycled global ET at T + 24 (Fig. 4a) and remains discernibly flatter even at time T + 120 (Fig. 4b). The spectrum for the not-cycled banded ET, meanwhile, is slightly flatter than the spectrum for the cycled banded and block ETs at both T + 24 and T + 120. The notable whitening associated with the local ET variants is likely a result in some part of using a diagonal 𝗣a, as discussed in section 2d. The reason the not-cycled five-banded ET has the whitest spectrum may be that these perturbations originated from uncorrelated random perturbations, and that further randomization was imparted when the not-cycled perturbations were randomly drawn from the archive.

c. Variance of the forecast ensemble

In Fig. 5, perturbation variance and amplification are diagnosed using the 0000 UTC ensemble-average, domain-average perturbation dry total energy, averaged over the 23 days in the test period. For reference, each plot in Fig. 5 also presents the dry total energy of the observed errors in the ensemble-mean forecast of the cycled global ET. The observed-error total energy for the T + 0 h lead time is derived from the twenty-three 0000 UTC NAVDAS analysis error variance estimates. The observed-error total energy for all lead times following the T + 0 h lead time is estimated from the 23-day sample variance of error in the ensemble-mean forecast of the cycled global ET, where the error is defined as the forecast minus the verifying analysis.1 One hopes that the ensemble-average perturbation energy and the error energy are roughly equivalent. When considering Figs. 5a and 5b, the variance of the cycled banded and block ETs is found to be much closer to the estimated observed error variance at lead times prior to T + 48 h in the midlatitudes (Figs. 5a,b). Also, the variance of the cycled banded and block ETs is considerably greater at T + 24 h and beyond in the tropics (Fig. 5c). The global ET actually has a more agreeable variance for lead times beyond T + 48 h in the midlatitudes; however, this is a result of the excessive variance of the global ET at initialization time rather than superior perturbation growth. Both the global and local ETs exhibit too little variance at long lead times, regardless of region. This is likely due to the forecast error having a model-error component that is not accounted for in the ensemble design but may also reflect other factors such as poor estimates of analysis-error covariance.

The variance of the not-cycled five-banded ET is not remarkably different from that of the cycled banded and block ETs, except that in the Northern Hemisphere (NH; summer season) midlatitudes it grows more rapidly prior to T + 48 h (Fig. 5a). This gives the not-cycled five-banded ET considerably more variance than the cycled banded and block ETs in the NH midlatitudes at longer lead times. There is no obvious explanation for this circumstance. It may be an indication that the not-cycled perturbations have too much baroclinicity, since these perturbations are drawn from a winter-season archive and are subsequently integrated upon a summer-season basic state. On the other hand, since the not-cycled perturbations are randomized, their strong growth may reflect an ability of random perturbations to initiate moist convection (Zhang 2005; Zhang et al. 2007). Also, since the not-cycled perturbations are independent of the analysis cycle, they may be able to sample important directions of growth (perhaps related to model error) that are outside the cycling subspace. Another possibility relates to the fact that in the midlatitudes the cycled initial perturbations have a greater proportion of energy at the smaller scales than do the not-cycled initial perturbations (not shown). Given this fact, one might expect the cycled perturbations to be at a growth disadvantage if the initial perturbation amplitude specified by the NAVDAS analysis error variance is at or near the small-scale saturation amplitude of NOGAPS. This idea is supported by results in McLay and Reynolds (2009) that indicate that cycled ET initial perturbations grow more rapidly as the proportion of their energy that is in the larger scales increases.

Bin-mean scatterplots of variance versus squared error also were examined [not shown, but similar scatterplots are presented in McLay et al. (2008) and McLay and Reynolds (2009)]. The variance of both the cycled and not-cycled local ETs was found to be better related than the variance of the global ET with squared error at lead times up to T + 72 h.

d. Root-mean-square error (rmse) of the forecast-ensemble mean

Figure 6 presents the rmse of the ensemble-mean 500-hPa geopotential height and 925-hPa zonal wind as a function of region and lead time. For each variable, the mean of the cycled banded and block ETs has considerably less error than the mean of the cycled global ET at most lead times in both the NH midlatitudes and tropics (Figs. 6a, 6c, 6d, and 6f). In the Southern Hemisphere (SH) midlatitudes the mean of the cycled banded and block ETs has less advantage (Figs. 6b,e). Similar results are found for the anomaly correlation (not shown). The best performance for the cycled banded ET is obtained by the 9- and 17-banded configurations. For the not-cycled five-banded ET, its mean 500-hPa geopotential height is appreciably better than that of the cycled five-banded ET in both the NH midlatitudes and tropics (Figs. 6a,c). For mean 925-hPa zonal wind, there is not much difference between the not-cycled and cycled five-banded ETs (Figs. 6d–f).

e. Brier score

The Brier score for an event can be interpreted as the mean-square error of the ensemble-derived probability of the event (Wilks 2006). Figure 7 presents the Brier score for the event that the 10-m zonal-wind magnitude exceeds 10 m s−1, as a function of domain, lead time, and ensemble. Results for this particular event are presented because high winds at the surface are consequential to a great variety of naval operations. To obtain the domain-averaged Brier score for each lead time in Figs. 7a–d, the Brier score was first calculated at each grid point in the domain as the temporal mean (over the 92 dates between 0000 UTC 25 June and 1800 UTC 17 July 2005) of the square error of the raw ensemble-derived event probability. Then, the gridded Brier scores were spatially averaged with area weighting. One sees that the cycled local ET scores notably better than the cycled global ET in the NH midlatitudes and the tropics and deep tropics at many lead times (Figs. 7a, 7c, and 7d). If one considers only the cycled banded ET, the nine-banded configuration obtains the best Brier scores in the tropics and deep tropics at long lead times. The not-cycled five-banded ET scores slightly better than the cycled local ET at longer lead times in the midlatitudes (Figs. 7a,b) and at most lead times in the tropics and deep tropics (Figs. 7c,d).

4. Conclusions

Concepts from the local ETKF are applied to the ET analysis perturbation scheme to remedy the regional disparity between ET analysis-perturbation variance and estimates of the analysis-error variance provided by the data assimilation system. The resulting local formulations of the ET are termed the banded and the block ETs. Results show that the local ET is effective at improving the consistency between ET analysis-perturbation variance and estimates of the analysis-error variance, in particular with respect to the wind components. Furthermore, the local ET provides meaningful gains in forecast performance relative to the original global ET in a majority of verification metrics. These gains are achieved without a significant increase in computational expense. Furthermore, they are particularly notable in the tropics. The advantages of the local ET may owe simply to its more consistent analysis-perturbation variance, or they may also arise from its whiter forecast-error covariance eigenvalue spectrum.

Considering the midlatitudes, the gains of the local ET are more notable in the Northern (summer) Hemisphere than in the Southern (winter) Hemisphere. This discrepancy may reflect the possibility that the baroclinic error dynamics of the winter midlatitudes have much lower dimension than the moist-convective error dynamics of the summer midlatitudes (e.g., Oczkowski et al. 2005). In such a circumstance, the relatively low effective dimension of the global ET forecast ensemble may already be sufficient to describe the important dynamical dimensions in the winter midlatitudes and the greater effective dimension of the local ET forecast ensemble may not add any value.

A not-cycled form of the banded ET generally performs slightly better than the corresponding cycled form of the banded ET in the warm regions of the globe (the tropics and the NH extratropics); however, the performance gap between the not-cycled and cycled banded ETs is much less than the performance gap between the not-cycled and cycled global ETs that was observed by McLay and Reynolds (2009). The not-cycled initial perturbations are randomized and flow independent. Thus, as speculated in McLay and Reynolds (2009), the good performance of the not-cycled ET may reflect an ability of randomized initial perturbations to initiate moist convection, or an ability of flow-independent perturbations to sample directions related to model error that are outside the cycling subspace.

A cycled banded ET is scheduled to be made operational at the U.S. Navy Fleet Numerical Meteorology and Oceanography Center. The cycled formulation was chosen over the noncycled formulation to ensure that data assimilation applications have flow-dependent perturbations at very short lead times and to avoid having to maintain a perturbation archive.

Acknowledgments

This research was sponsored by the Naval Research Laboratory and the Office of Naval Research under Program Element 0601153N, Project BE-033-03-04M. The DoD High Performance Computing program at NAVO MSRC provided the computing resources. The manuscript benefited considerably from the comments of three anonymous reviewers.

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Fig. 1.
Fig. 1.

Domain partitioning of the cycled banded ET for the choices of Nband = (a) 5, (b) 9, and (c) 17. (d) Domain partitioning of the cycled block ET for the choice of Nblock = 18.

Citation: Weather and Forecasting 25, 3; 10.1175/2010WAF2222359.1

Fig. 2.
Fig. 2.

Illustration of the centering of the weighting matrices 𝗧1, 𝗧2, and 𝗧3 in their respective latitude bands, for the choice of Nband = 5.

Citation: Weather and Forecasting 25, 3; 10.1175/2010WAF2222359.1

Fig. 3.
Fig. 3.

Expected ratio of the ET analysis perturbation variance (σa2)ENS and the estimated analysis error variance from the NAVDAS data assimilation system (σa2)DA for the cases of (a) zonal wind. (b) meridional wind, (c) temperature, and (d) relative humidity. Results are the average for 10 days in the test period. For clarity, only results for the cycled global ET and cycled five- and nine-banded ETs are shown.

Citation: Weather and Forecasting 25, 3; 10.1175/2010WAF2222359.1

Fig. 4.
Fig. 4.

Average eigenvalue spectrum of the (a) T + 24 and (b) T + 120 forecast-error covariance matrices under the dry total energy norm. The eigenvalue spectrum of the forecast-error covariance matrices was determined for each of the 92 forecast ensembles in the period 25 Jun–17 Jul 2005. Each spectrum was normalized by its leading eigenvalue, and the resulting 92 normalized spectra were averaged.

Citation: Weather and Forecasting 25, 3; 10.1175/2010WAF2222359.1

Fig. 5.
Fig. 5.

Solid lines indicate ensemble-average dry total energy of the forecast perturbations as a function of lead time, domain averaged over the (a) NH midlatitudes (20°–65°N), (b) SH midlatitudes (65°–20°S), and (c) tropics (20°S–20°N) and averaged over the twenty-three 0000 UTC forecast ensembles generated within the period 25 Jun–17 Jul 2005. The dashed line indicates the dry total energy of the observed errors in the ensemble mean of the 0000 UTC cycled global ET, averaged over the same domain and period. See the text for further details of the error calculation.

Citation: Weather and Forecasting 25, 3; 10.1175/2010WAF2222359.1

Fig. 6.
Fig. 6.

Area-weighted rmse of the ensemble-mean 500-hPa geopotential height in the (a) NH midlatitudes (20°–65°N), (b) SH midlatitudes (65°–20°S), and (c) tropics (20°S–20°N) averaged over the 92 forecast ensembles in the period 25 Jun–17 Jul 2005. (d)–(f) As in (a)–(c), respectively, but for 925-hPa zonal wind.

Citation: Weather and Forecasting 25, 3; 10.1175/2010WAF2222359.1

Fig. 7.
Fig. 7.

Area-weighted average, in the (a) NH midlatitudes (20°–65°N), (b) SH midlatitudes (65°–20°S), (c) tropics (20°S–20°N), and (d) deep tropics (10°S–10°N), of the Brier score for the event that the 10-m zonal wind magnitude exceeds 10 m s−1.

Citation: Weather and Forecasting 25, 3; 10.1175/2010WAF2222359.1

1

Note that verification against the analysis may underestimate the forecast error at short lead times.

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