a. The sample time period

All experiments were carried out with a 10-member (five pair), 0000 UTC subset of the NCEP global ensemble forecasts. Error statistics were accumulated for a 30-day period from 13 January through 11 February 1999. Though the choice of a contingent time period has the disadvantage of possibly having strong autocorrelations between errors of consecutive days, thus limiting the effective size of the statistical sample, it has the advantage that persistent, slowly varying error patterns can be detected in the ensemble.

This particular 30-day period, overlapping with the 1999 Winter Storm Reconnaissance program, was selected because a large number of diagnostics prepared for an earlier study (Szunyogh et al. 2000) were already available. Here, we show only eddy statistics that are crucial to exploring the relationship between the location of the storm tracks and the geographical distribution of forecast error reductions due to increased horizontal resolution. The eddy quantities are defined by the deviation from the monthly mean and the 500-hPa geopotential height variance, the meridional temperature flux, and the vertical temperature flux are shown in Fig. 1. Overlapping regions of poleward and upward temperature fluxes mark areas of available potential to eddy kinetic energy conversion. The three main regions of baroclinic energy conversions (North Pacific, North Atlantic, and SH midlatitudes) are well distinguishable. The seasonal differences are also well marked by the more intense temperature fluxes in the Northern Hemisphere. The propagation of the baroclinic wave packets in the Northern Hemisphere was also well documented for the sample period by Fig. 4 in Szunyogh et al. (2000).

b. The operational ensemble configuration

Until June 2000, the implementation of the operational global EFS at NCEP consisted of 17 16-day forecasts. All forecasts were made with the NCEP Medium Range Forecast (MRF) model (Derber et al. 1998). In addition to the 10 and 4 perturbed integrations made with a T62 horizontal resolution, 28 vertical levels version of the model at 0000 and 1200 UTC, respectively, 2 control integrations were made at 0000 UTC, and 1 at 1200 UTC. The high-resolution control forecasts were run at T126 (T170 from 15 June through 5 October 1998, and after 24 January 2000) resolution with 28 (42) vertical levels out to day-7 and day-3 (3.5 day) lead times at 0000 and 1200 UTC, respectively, after which the fields got truncated to T62 resolution and the runs were extended to day-16 lead time (Tracton and Kalnay 1993; Toth and Kalnay 1997).

While ECMWF uses leading initial and evolved singular vectors defined by a norm with energy dimension (Molteni et al. 1996; Buizza et al. 1999), NCEP runs breeding cycles (Toth and Kalnay 1993, 1997; Iyengar et al. 1996) to generate the perturbed initial conditions. In a breeding cycle, first half of the difference between the initially oppositely perturbed pairs of 24-h forecasts is taken. This three-dimensional difference field is then rescaled by a two-dimensional regional rescaling factor, r(λ, ϕ, t), which is varying with the geographical location, but is fixed for all model variables at all levels. The rescaling factor is defined by the ratio r(λ, ϕ, t) = mask (λ, ϕ, t)/K(λ, ϕ, t). Here, mask denotes the daily varying average rms difference between two independently run analysis cycles, computed using the rotational kinetic energy at the 500-hPa pressure level as the inner product, and K(λ, ϕ, t) is the square root of the rotational kinetic energy at the 500-hPa pressure level for the difference field to be rescaled. Both fields are smoothed by a Gaussian filter before the ratio is computed, removing most of the variance in the kinetic energy field associated with wavenumbers larger than 8 and 9. The purpose of this rescaling procedure is to ensure that the initial ensemble perturbations are representative of the typical large-scale geographical distribution of the analysis uncertainty. The only reason why the large-scale distribution of the rotational kinetic energy of the initial perturbations can depart from the mask at the 500-hPa level is that the size of the initial perturbations is never increased during rescaling: the value of r is set to 1 at locations where it would be larger otherwise.

The regional rescaling algorithm was designed with the aim of retaining the structure of synoptic and smaller-scale features developed in the 24-h evolved perturbations. Thus we can expect that by increasing the horizontal resolution the initial structure of the bred perturbations will also change since the magnitude and the structure of the 24-h evolved perturbations may be different in the different resolution models and only the largest-scale components of these perturbations are constrained strongly by the regional rescaling.

c. The experimental ensemble configurations

Four sets of ensembles were generated. The first ensemble (referred to as T62 hereafter) was an almost exact replica of the 0000 UTC low-resolution subset of the operational ensemble, while in the other three ensembles (referred to as D1, D3, and T126 hereafter) the horizontal resolution was increased to T126 out to day-1, day-3, and day-15 forecast lead times. To save computer time the D1 (D3) runs were stopped after 4 (7) days of model integration. A control forecast, started from the unperturbed analysis, was also run for each ensemble following the same truncation strategy that was applied to the associated ensemble.

Testing ensemble configurations that changed their resolution after 1 or 3 days was dictated by the operational constraint that an ensemble integrated at resolution T126 up to 16 days would not be affordable at NCEP in the near future. We note, however, that reducing the horizontal resolution of high-resolution MRF control forecasts after the first few days of model integration has been a long-time practice at NCEP. This strategy is based on the experience that increased horizontal resolution for the first few days of model integration has significant positive impact on forecast quality for the entire 16-day forecast range; a reduction of the horizontal resolution after a few days does not degrade the skill scores substantially. The general belief is that this is due to to the better quality of the higher-resolution analysis and the shorter predictability time limit of the smaller-scale weather phenomena. For example, experience shows that an analysis taken from a T126 cycle and truncated to resolution T62 usually leads to a better T62 forecast than the one that was started from an analysis of a T62 cycle (S. Tracton 1993, personal communication). Because of this, the T62 perturbed initial analyses of the operational NCEP EFS have been created around a T126 analysis truncated to resolution T62 (Tracton and Kalnay 1993).

It must be emphasized that reducing resolution after a few days of model integration is more than a simple truncation (spectral filtering) of the meteorological fields. It also means using a different model after the truncation. Most importantly, the reduced-resolution model at the bottom of the atmosphere is forced by boundary conditions and orography that are different from those used in the high-resolution model. Second, the physical parameterization schemes may behave differently in the different horizontal resolution models. Finally, while at T62 resolution the total wavenumber of the smallest retained feature is 62, not all interactions between structures with wavenumbers smaller than 62 are retained (Machenauer 1991; Kadar et al. 1998). In fact, all of those interactions are neglected that would result in entities characterized by wavenumbers larger than 62. Consequently, the interactions between features characterized by total wavenumbers smaller than 62 are better represented, especially for the high-wavenumber components close to the cutoff wavenumber 62, in a T126 version of the model.

The bred perturbations were generated by T62 model runs for the T62 ensemble, and by T126 model runs for the initially T126 resolution ensembles. The high- and low-resolution breeding cycles were run by using the mask that was designed for the operational T62 ensemble. More precisely, the mask for the T126 breeding cycle was prepared by first transforming the mask to the T62 spectral space from the associated Gaussian grid, then filling up the spectral coefficients related to wavenumbers higher than 62 by zeros, and transforming the field to the high-resolution Gaussian grid. Since the mask was the same for the high- and the low-resolution breeding cycles and the Gaussian filter that was used to compute the rescaling factor preserves the areal average of a scalar quantity for the globe, the global mean of the bred perturbations was identical for the two cycles. This means that the differences between the performance of the different resolution ensembles reported in this paper are due to differences in the local magnitude and the structure of the bred perturbations.

The initial conditions for the ensemble members were created by adding the bred perturbations to the operational T126 analysis of NCEP (truncated to T62) in the initially high-resolution (low resolution) ensembles. The breeding cycles were initiated with the operational bred perturbations from 3 January 1999. Since these perturbations had horizontal resolution T62 the spectral coefficients related to higher wavenumbers in the T126 runs were simply set to zero. As was expected, after 3–4 days of running the breeding cycle at resolution T126 no transient behavior could be observed in the initially high-resolution cycles.

After interpolating the forecasts and analyses to a 2.5° × 2.5° grid, all ensemble and control forecast products were verified against the control analysis. We note that the resolution of this verification grid is lower than the resolution of the T62 analysis–forecast fields, which means that only features resolved in both the low- and the high-resolution runs are verified and the effect of high-wavenumber modes in the T126 simulations are taken into account only implicitly.

Verification results are presented for the 500-hPa geopotential height in the Northern Hemisphere (NH) and Southern Hemisphere (SH) midlatitude regions. The NH (SH) verification region is defined by the 30°–70°N (30°–70°S) latitude band.

Some of the verification statistics require the knowledge of climatology at the grid points on a daily basis. In this study, the computation of climatology (denoted by c) is done in two steps: first, the monthly averages of 36 yr of NCEP reanalyses (Kalnay et al. 1996) are taken and then the daily values are computed by a linear time interpolation assuming that the monthly average is representative at the middle of a given month.

a. Condition for optimal smoothing

The most widely used forecast score for the verification of the ensemble mean is the rms error (RMS) defined as the root-mean-square distance between the forecast and the verifying dataset:

View Expanded

Here, f is the forecast and a is the verifying analysis, both given on the 2.5° × 2.5° grid. The angled brackets stand for the mean that will be taken over the 30-day sample period and over all points within the verification domain. When definition of a local error is needed, the mean is taken over the 30-day sample period at each grid point. This time mean is denoted by an overbar; for instance, the local rms error is

View Expanded

The mean-square error (MS), the square of RMS, can be decomposed (Simmons et al. 1995) as

View Expanded

When c is computed on a daily basis and averages are taken for large regions over an extensive time period, 〈(f − c)〉 and 〈(a − c)〉 become negligible. In this sense, the first (second) term of the rhs in Eq. (3) is usually referred to as forecast (analysis) variance. Here, especially when only the time averages are taken, 〈(f − c)〉 (〈(a − c)〉) is not negligible and for clarity the mean-square distance between the forecasts (analyses) and the climatology will be referred to as the mean-square forecast (analyzed) anomaly.

For a good NWP analysis–forecast system, the mean-square analyzed anomaly provides a good estimate of the climate variance for the selected sample period and the mean-square forecast anomaly [first term of rhs. in Eq. (3)] is near the mean-square analyzed anomaly [second term of rhs in Eq. (3)] for all forecast lead times. Since the covariance between the analyzed and the forecast anomalies [third term of rhs in Eq. (3)] goes to zero as the forecast lead time increases, the long-time limit of MS (RMS) is twice (2) as large as the (root-) mean-square analyzed anomaly. When climatology is used as forecast (f = c), the first and third term on the rhs in Eq. (3) are identically zero. This means that the long-time limit of MS (RMS) is reduced to the (root-) mean-square analyzed anomaly, but at the price of significantly increasing the short-term forecast errors. The use of an appropriate ensemble has the benefit of retaining the forecast anomaly (f − c) at short lead times, similarly to a single forecast, and removing all anomalies, like a climatology-based forecast, in the long-time limit.

Operational EFS systems are not perfect. Most obviously, their size is limited and even if they were otherwise perfect the RMS error in the mean of an m-member ensemble would converge to (1 + m⁻¹) times the mean-square analyzed anomaly (Leith 1974). It means that if the model and the 10-member ensemble investigated here were otherwise perfect, the RMS (MS) for a large sample of forecasts would converge with increasing lead time to 1.1 (1.1) times the mean-square analyzed anomaly.

Based on the above discussion, a condition for the optimal smoothing by a finite-sized ensemble can be defined in the long-time limit. The problem is to find a condition that can be used at shorter forecast lead times. To derive a condition of this type it is useful to rescale Eq. (3) by the mean-square analyzed anomaly:

View Expanded

Here, the normalized MS (NMS), the normalized mean-square forecast anomaly (NFA), and the anomaly correlation (AC) are, respectively,

View Expanded

We note that this expression for AC, which can also be found in Wilks (1995), is equal to the correlation between the forecast and analyzed anomalies only if the mean of the forecast and analyzed anomalies (〈(f − c)〉 and 〈(a − c)〉) are negligible.

The value of MS is smaller for a forecast f than for the climatology-based forecast c, or in other words, NMS can be smaller than one, if and only if, SKILL > 0. This inequality is satisfied for a given value of AC whenever 0 < NFA < 2 × AC, and then the largest possible reduction in NMS is achieved when

View Expanded

and in that case

²(9)

Equation (9) was first derived by Murphy and Epstein (1989), but in a somewhat different context. They assumed, as is the case for a good single NWP model forecast, that NFA is equal to one; hence, except for the initial time, SKILL is smaller than AC². Therefore, they argued, the square of the anomaly correlation should be considered as a measure of potential rather than actual skill. Here we make use of the fact that by taking an ensemble mean, NFA gradually decreases with increasing forecast lead time. This means that ensemble forecasting provides a way of turning the potential skill provided by AC² of the ensemble mean to actual skill. It must also be emphasized that ensemble averaging can also increase AC, thus increasing the potential skill itself. In other words, a change in the EFS improves the SKILL if it leads to the retention of more skillful features with more realistic amplitude and/or to the more efficient filtering of unpredictable details. The optimality condition we were searching for is given by Eq. (8).

The above arguments can be extended to give an estimate of the RMS reduction that can be attributed to the reduced forecast anomalies in the ensemble mean compared to the control forecast. Let us assume that AC is equal for the ensemble and the control forecasts, its value is ac, and the value of NFA for the control forecast is nfa. The NMS is smaller for the ensemble mean than for the control forecast if and only if SKILL − (2 × ac × nfa − nfa) > 0. This condition is satisfied if and only if NFA is between (i) 2 × ac − nfa and (ii) nfa, while the ensemble mean has the largest possible advantage over the control if Eq. (8) is satisfied. In most cases, including the one when nfa = 1, condition i provides the lower and condition ii the upper bound, but when AC is high and nfa is smaller than one (the numerical model is unrealistically diffusive), the regular order of the bounds may be reversed. Since the smallest meaningful value of NFA is zero, that should be used as the lower bound whenever condition i gives a negative number. If the anomaly correlation (ac) is high (close to 1), there is only a narrow range of reduced forecast error variance (NFA) that can improve the normalized RMS error (NMS) and a strong smoothing can only degrade the forecast quality (RMS). When the anomaly correlation (ac) is low, however, the similarly low value of condition i provides a wide range of forecast variance reduction (NFA) that can lower the rms error (NMS).

b. Relative RMS error

A relative RMS

View Expanded

is introduced here as the percentage of the RMS in the initially high-resolution runs (RMS_h) compared to that in the T62 run (RMS_T62). For small rms values, RRMS is a more sensitive measure than the difference between the RMS of the two forecasts (RMS_T62 − RMS_h); thus, it can be more efficiently used to analyze the RMS reduction. Throughout this section the relative rms is used in figures to demonstrate changes in the forecast quality due to increased model resolution. Values smaller (larger) than 100 indicate forecast improvement (degradation) due to increased horizontal resolution.

c. RMS and AC for the control forecasts

Figure 2 summarizes the NH RRMS results. All initially high-resolution control runs have lower rms error. Furthermore, the use of reduced model resolution after 1 day has a clear negative effect on the quality of the control forecasts for the day-2–4 forecast range. Reducing the resolution after 3 days, however, has an opposite effect in the day-4–7 time range by improving the scores. The benefit from increased resolution is largest (7.3% error reduction) at day-2 lead time, after which the gain in forecast quality gradually diminishes.

For the SH RRMS (not shown), the initial impact of increased resolution on the control forecast is positive but less significant than for the NH region. For the day-2–4 range the T126 and the D3 runs are superior to the D1 run again, but beyond day-4 the T126 has the largest rms error.

The NH AC results are in Fig. 3, (D1 and D3 are not shown). The evaluation of AC suggests similar conclusions to those drawn based on rms statistics. The AC is higher for the runs that were started at resolution T126 than for the low-resolution control at all forecast lead times. The D3 run produced consistently higher scores again than the T126 run for the day-4–7 forecast range.

SH AC (not shown): In the SH, the T126 run has a superior AC to the others only up to 4 days.

d. RMS and AC for the ensemble means

The NH RRMS results are presented in Fig. 4. The improvement in the skill of the ensemble mean forecasts due to increased resolution is more substantial than that for the control forecasts. Interestingly, the error reduction is largest (10.1%) at the shortest verified lead time (day 1), in contrast to the case of the control forecasts, for which the largest error reduction was observed at day 2. Another important difference is that the T126 forecast remains superior to the D3 run for the day-4–7 forecast range, too.

Regarding SH RRMS (not shown), the ensemble mean shows a behavior similar to that of the control forecast: the maximum benefit of increased resolution is at day-2 lead time, the T126 and the D3 runs are superior to the D1 run for the day-2–4 forecast range, and the T62 forecast outperforms the T126 run after 4 days.

The NH AC results are in Fig. 3. D1 and D3 are not shown because they are almost indistinguishable from those for the T126 forecasts. The initially high-resolution runs performed better than the T62 reference ensemble in terms of AC, too. Note that the advantage of the higher-resolution runs at day 7 for the ensemble mean is about 12 forecast hours compared to only about 2 h for the control. This means that the increased resolution had a more positive impact on the potential skill of the ensemble mean than on that of the control forecast.

For the SH AC (not shown), the initially high-resolution mean forecasts have an advantage in terms of AC for the first four days. At day 5 the AC for the different runs is identical, while beyond that time the T62 run has the highest AC values.

e. Comparison of the ensemble means and controls

The NH NMS results are shown in Fig. 5. In the T62 forecasts, the RMS is 6.5% larger for the mean forecast than for the control at day 1 and only after 4 days does it become smaller. In contrast, for the initially high-resolution runs the ensemble mean RMS (not shown) is larger only at day 1, and only by a small amount (less than 1.5%), while beyond that time the mean has a gradually increasing advantage over the control.

The NMS for the T126 (T62) control reaches the error level of the climatological forecast at day-9 (day 8.5) lead time. The advantage of the ensemble mean over the control at this time is 32.7% (29.9%), which is equivalent to a 18% (16.3%) RMS reduction. At day-15 lead time the NMS is 0.903 for the ensemble mean, an indication that the ensemble mean provides a forecast typically better than that based on climatology even at this extended forecast range. This is consistent with earlier results shown in Zhu et al. (1996) and Toth et al. (1998). In the SH (not shown), the RMS (MS) is lower for the means than for the corresponding control forecasts at all lead times. The error level of the forecast based on climatology is reached at around day-6 lead time by both the T62 and the T126 control forecasts. The advantage of the means at this time is 30.4% (28.1%) for the T126 (T62) ensemble, which is very similar to that observed for the NH region. The controls and the means converge to their asymptotes by day 12, indicating that there are no predictable features in the SH region beyond that time.

The NH AC (shown in Fig. 3) indicates a lower potential skill for both the T62 and the T126 mean than for the associated controls during the first three forecast days. The increased resolution, however, reduced the difference from 0.002 to 0.001 at day 1, and from 0.005 to 0.002 at day 2 and 3. The SH AC scores indicate that the mean forecasts have higher potential skill than their controls.

In summary, the increased horizontal resolution has a greater positive effect on the mean than on the control forecasts in the NH region. This is in part due to the fact that while the forecast quality of the T62 mean is significantly lower than that of the T62 control during the first three days, the T126 control only slightly outperforms the T126 mean during the same period. Truncating the fields after 3 days of model integration has no benefit in the case of the mean forecasts, which is in contrast to the behavior observed for the control forecasts. This indicates that the effects of the unpredictable small-scale features are efficiently filtered by the ensemble average and an additional nonselective spectral filtering of the meteorological fields degrades the forecast quality.

The skills of the ensemble mean and the corresponding control forecasts are more similar in the SH region and there is no obvious advantage of integrating the ensembles at increased horizontal resolution beyond 4 days.

f. Time evolution of NFA

NFA is first evaluated for the control forecasts (not shown). As can be expected from a good NWP model, NFA remains near one for all controls during the entire 15-day forecast range. More precisely, in the NH region there is a slight (less than 0.5%) initial decay of NFA during the first day, but beyond that time the mean-square distance between the forecasts and climatology is practically perfect. In the SH region the initial decay is somewhat more pronounced (still less than 2.5%) and NFA shows a slow growing trend for the longer lead times, which explains why the long-term limit of ms is larger than 2.

Figure 6 presents the time evolution of NFA for the T126 and the T62 ensemble mean forecasts in the NH region. The mean-square forecast anomaly in the NH region is considerably larger for the T62 than for the T126 run at all forecast lead times. The fact that the T62 model produces higher NFA than the T126 model for the ensemble mean is also indicated by the relatively high NFA for the D1 and D3 runs: after the resolution is reduced there is a well-distinguishable sudden increase in NFA (not shown).

The optimal value of NFA, AC² [Eq. (8)], is also shown in Fig. 6 for both ensembles. NFA for the T126 run is optimal at day 1, after which it gradually drifts away from its optimum. This is not surprising if we recall that the long-term limit of NFA would be 0.1 in a 10-member ensemble. At day 15 the asymptotic values are not established yet, but the difference between NFA (0.17) and its possible lowest value (0.1) is small.

The behavior of the T62 ensemble mean is drastically different. The most striking feature is the curious result that NFA is the largest for the T62 ensemble mean among all forecasts including the controls. This indicates that nonlinearity must play an important role in the early evolution of the T62 ensemble perturbations since linearly evolving perturbations would result in identical ensemble mean and corresponding control forecasts. This behavior will be discussed in more detail in section 5.

Apparently, the RMS could be improved by rescaling the forecast anomalies (f − c) such that NFA would become optimal. The RMS reduction that could be achieved by such a rescaling would be 1.2% (2.8% and 3.3%) at day-3 (day 7 and day 10) forecast lead time for the T62 ensemble, while the same number for the T126 ensemble is less than 0.1% (0.7% and 1.1%). The RMS values for the T126 ensemble mean are nearly as good as they can be for a 10-member ensemble. This indicates that structures retained in the ensemble mean usually have realistic magnitude and significant improvements in the mean forecasts can be expected only if AC can be further improved. This also means that for the high-resolution ensemble mean the actual skill is almost equal to the potential skill given by AC².

a. Decomposition of the mean-square error

In this part of the study, all variables are decomposed into two parts. A mean forecast (analyzed) flow, f (a), is defined by time averaging the forecasts (analyses) for each lead time separately, over the 30-day sample period. Eddy quantities, f′ (a′), are then computed by taking the deviation of the forecast (analyzed) values from the forecast (analyzed) mean. The local mean-square error can also be decomposed by partitioning the meteorological fields into mean and eddy components:

View Expanded

The square root of the first term on the rhs of Eq. (11) (the error in the predicted mean) is conventionally called bias, while the second term (the error in the predicted eddy component) is called the forecast error variance. In Figs. 7 and 8 the two error components for the T126 control run are shown at day-1 and day-4 lead times. The localized patterns associated with large bias have negligible contribution to the total error along the storm track regions. Meanwhile, a close relationship between storm tracks and the dominant patterns of short-term forecast error variance is already well established at day 1 and it becomes even more evident at day 4. This means that decomposing MS into bias and variance components provides a good way to define a forecast error component that is dominantly determined by errors in the prediction of high-frequency transients at short lead times.

The global MS can also be decomposed into bias and error variance terms by taking the average of Eq. (11) over the grid points:

View Expanded

In what follows the relative importance of the increased resolution in reducing the bias versus the error variance component of MS is investigated by plotting figures based both on Eqs. (11) and (12). When reduction in the local bias (forecast error variance) is of concern, the difference between the square of the bias (forecast error variance) terms for the low- and the high-resolution runs are mapped. Positive (negative) values in these maps mark locations of reduced (increased) error terms. In case of the space–time-averaged bias (forecast error variance) a relative bias (forecast error variance) is defined by the ratio of the square of the the bias (forecast error variance) terms for the initially high-resolution and the T62 runs. Like in the case of the relative rms shown in earlier figures, the ratio is expressed as a percentage and the bias (forecast error variance) is reduced whenever the relative bias (forecast error variance) is smaller than 100.

b. Time evolution of the forecast bias component

Figures 9 and 10 summarize the NH forecast bias for the control. The strongly localized patterns of large forecast bias are over or near to land at day 1. In particular, at the location where the largest error reduction occurs in the region of the Tian-Shan Mountains (in central Asia), there is an extremely large gradient in surface height. An independent study (H. M. Van den Dool and S. Saha 1999, personal communication) concluded that this is the typical location of the largest short-term forecast bias in the NCEP global forecast of the 500-hPa height. The same authors, using the technique of empirical orthogonal teleconnections (Van den Dool et al. 2000), found that the error at that location is also in a close relationship with the short-term coherent large-scale Northern Hemisphere error patterns in the NCEP global model. By day 4 the strongly localized patterns of improvement disappear and a mixture of modest amplitude improvements and degradations is left behind.

Since the magnitude of the local bias reductions is large the increased resolution has a dramatic positive influence on the control forecast. The spatially averaged error reduction (Fig. 10) is the largest (35%) at the shortest verified lead time and the spectral truncation has an immediate negative impact. Truncating the forecast after day 1 has a clear negative impact, while reducing the resolution after day 3 is less damaging but still clearly negative. In the SH (not shown), the improvements in forecast bias are smaller than in the NH region. The largest error reduction (16%) is at day 3. The bias reduction rapidly decreases beyond day 4 and completely diminishes by day 6.

The NH forecast bias for the ensemble mean is shown in (Figs. 11 and 12). The dominant improvements are concentrated in the same region as for the control forecast at day-1 lead time. Similar, but smaller magnitude, improvement patterns (not shown) can be observed by comparing the T126 and the D1 (D3) runs at day 2 (day 4). The impact of increased resolution is more dramatic than in the case of the control forecast: the error reduction is 56% at day 1 and day 2 and the error reduction is still significant (29%) at day 7.

While the SH forecast bias for the ensemble mean (not shown): shows a similar trend to that observed for the control forecast in the SH region. The only difference is that the error reduction for the short forecast lead times is larger for the ensemble mean than for the control.

c. Time evolution of the forecast error variance

The NH error variance for the control is shown in Figs. 13 and 14. Large errors in the prediction of transient eddies along the midlatitude storm tracks were significantly reduced by the increased resolution. There are also significant error reductions in regions where the T126 control forecast produced no significant error variance but where it was burdened by large forecast bias. The most obvious example for this is the large-bias area in the Tian-Shan Mountains. It means that in the T62 model not only the forecast means but also the forecast transients are in error in the above regions. While truncating the forecasts after day 1 has negative forecast effects in the day-2–4 forecast range, the D3 run is superior to the untruncated T126 run at and beyond the day-4 forecast lead time. The largest error reduction, 15%, is achieved at day 3 by the T126 forecast.

For the SH error variance with the control (not shown), the initially high-resolution forecasts are less efficient in reducing the variance component of the error than in the NH region. Also, the D1 and D3 runs clearly outperform the T126 run in that 1) the error for the T126 beyond day 4 is even larger than that for the T62 run; 2) the largest error reduction, 6%, is achieved by the D1 run at day-2 lead time; and 3) the truncation always has an immediate positive impact on the error variance component.

Figs. 15 and 16 depicts NH error variance for the ensemble mean results. The main areas of error reduction are in the storm track regions and over Europe. The largest spatially averaged error reduction, 12% achieved by the T126 forecast at day 3, is somewhat smaller for the ensemble mean than for the control forecast. On the other hand, for the longer than 5 day forecasts, the error reduction is larger for the mean than for the control and the T126 mean outperforms the truncated D3 mean. Another interesting feature is that the forecast error variance was clearly increased north of 70°N by increasing the model resolution.

The SH error variance for the ensemble means (not shown) indicates that the positive effect of increased model resolution is relatively modest. The largest error reduction, which is only 4.9% at day 3, was realized by the T126 run. The D1 run performs slightly worse in the day-2–4 range, while the D3 run is somewhat better in the day-4–7 range than the T126 run.