Abstract
 Top of page
 Abstract
 1. Forecasterror growth and predictability limits of a numerical weather prediction system
 2. Methodology
 3. Impact of resolution on forecasterror growth
 4. Discussion and conclusions
 Appendix: the ForecastError Growth Model
 Acknowledgments
 References
The impact of horizontal resolution increases from spectral truncation T95 to T799 on the error growth of ECMWF forecasts is analysed. Attention is focused on instantaneous, synopticscale features represented by the 500 and 1000 hPa geopotential height and the 850 hPa temperature. Error growth is investigated by applying a threeparameter model, and improvements in forecast skill are assessed by computing the time limits when fractions of the forecasterror asymptotic value are reached. Forecasts are assessed both in a realistic framework against T799 analyses, and in a perfectmodel framework against T799 forecasts.
A strong sensitivity to model resolution of the skill of instantaneous forecasts has been found in the short forecast range (say up to about forecast day 3). But sensitivity has shown to become weaker in the medium range (say around forecast day 7) and undetectable in the long forecast range. Considering the predictability of ECMWF operational, highresolution T799 forecasts of the 500 hPa geopotential height verified in the realistic framework over the Northern Hemisphere (NH), the longrange time limit τ(95%) is 15.2 days, a value that is one day shorter than the limit computed in the perfectmodel framework. Considering the 850 hPa temperature verified in the realistic framework, the time limit τ(95%) is 16.6 days for forecasts verified in the realistic framework over the NH (cold season), 14.1 days over the SH (warm season) and 20.6 days over the Tropics.
Although past resolution increases have been providing continuously better forecasts especially in the short forecast range, this investigation suggests that in the future, although further increases in resolution are expected to improve the forecast skill in the short and medium forecast range, simple resolution increases without model improvements would bring only very limited improvements in the long forecast range. Copyright © 2010 Royal Meteorological Society
1. Forecasterror growth and predictability limits of a numerical weather prediction system
 Top of page
 Abstract
 1. Forecasterror growth and predictability limits of a numerical weather prediction system
 2. Methodology
 3. Impact of resolution on forecasterror growth
 4. Discussion and conclusions
 Appendix: the ForecastError Growth Model
 Acknowledgments
 References
The first estimate of forecasterror growth was given by Charney et al(1966), who, using a simple model, concluded that a reasonable estimate of the forecasterror doubling time was 5 days. A few years later, Smagorinsky (1969), using a more refined, ninelevel primitive equation model that contained also moist processes, gave a lower estimate of the forecasterror doubling time of 3 days. Lorenz (1982), analysing 500 hPa geopotential height forecasts from the operational model used at the European Centre for MediumRange Weather Forecasts (ECMWF) at that time, a 15level primitive equation model with moist processes and orography, further reduced the estimate to 1.85 days for the Northern Hemisphere (NH) winter. This estimate was obtained assuming that forecast error grows following a quadratic equation that was introduced earlier by Lorenz (1969), which assumes that the shortterm errors grow linearly, while the longterm growth is dominated by a (negative) quadratic term that brings the longrange forecast error to an asymptotic value. Subsequently, Leith (1982) included in Lorenz's (1969) forecasterror growth model a constant term designed to simulate the effect of analysis errors on the shortrange forecast error. Leith's (1982) forecasterror growth model was further modified by Dalcher and Kalnay (1987), who investigated both the chaotic and systematic component of model error, and modified the model to simulate the effects of nonlinear error saturation. Dalcher and Kalnay (1987) applied the newly modified model to the data analysed by Lorenz (1982), and introduced two measures of predictability, the time limits τ(95%) and τ(50%), that are the times when the forecast error reaches 95% or 50% of the forecasterror asymptotic value. They suggested that the forecasterror doubling time that was used up to that time to assess predictability, was not a very good measure of error growth, but that better measures were the forecasterror growth rate, and time limits such as τ(95%) and τ(50%). Using the same dataset of Lorenz (1982), they estimated that the forecasterror growth rate was about 0.43 d^{−1} (which corresponds to an error doubling time of about 1.7 days for the NH winter, a shorter value than the 1.85 days found by Lorenz (1982), longer for the long waves and shorter for the short waves. They concluded that, for the NH winter, the time limit τ(95%) was 12 days, and the time limit τ(50%) was 5.5 days.
Savijärvi (1995) applied the Dalcher and Kalnay (1987) model to assess the error growth of forecasts issued by the National Meteorological Centre (NMC) of Washington between 1988 and 1993. Savijärvi (1995) introduced a third time limit, τ(71%), the time when the forecast error exceeds of the saturation value (see Savijärvi (1995) for more details on the definition of this threshold, which corresponds to the level of climatic variability), and named it the deterministic predictability time limit. He estimated that for the 500 hPa geopotential height during the 2year period 1991–1993, the NMC forecasts reached this level at about day 9 for the large scales (i.e. waves with total wave number up to 3), and about day 3 for the small scales (i.e. waves with total wave number between 3 and 20). He indicated that if the model error and the initial condition error could be halved, this limit would be reached at day 10 for the large scales and at day 6.6 for the short scales. Savijärvi (1995) also estimated that the time limit τ(95%) for the NMC system in 1991–1993 was longer than what was given by Dalcher and Kalnay (1987), about 20 days for the large scales and 7 days for the short scales, and suggested that this limit will remain at these levels even if the model error and the initial condition error could be halved. Reynolds et al(1994) also used the Dalcher and Kalnay (1987) model for forecasts to investigate the threedimensional structure of random error growth in the NMC system for the same period, and concluded that both the model and analysis uncertainties contributed to forecasterror growth.
More recently, Simmons and Hollingsworth (2002) used the Dalcher and Kalnay (1987) model to study forecast skill improvements of the ECMWF single, highresolution forecasts. They showed that between winter 1996/1997 and winter 2000/2001, the forecasterror doubling times during the NH winter decreased by ∼30% in the short forecast range (say at forecast days 1–4). At days 1 and 2, for example, they quoted doubling times in winter 2000/2001 of 1.14 and 1.42 days, values that are ∼50% shorter than the ones quoted by Lorenz (1982). The shortening of the doubling times at forecast day 1 and 2 can be related to the more realistic model activity at all scales, due to the increased resolution of the forecast model and to the inclusion of more realistic schemes that simulate physical processes. From Fig. 11 of Simmons and Hollingsworth (2002), it is possible to estimate that for the ECMWF system in winter 2000/2001, the time limits τ(71%) and τ(95%) were about 9 and 13 days, respectively.
This work presents an updated assessment of the forecasterror growth of synopticscale features represented by instantaneous (as opposed to averaged over a period of time, e.g. 1 day) ECMWF forecasts of the 500 and 1000 hPa geopotential heights and the 850 hPa temperatures. As in Reynolds et al(1994) and Simmons and Hollingsworth (2002), the Dalcher and Kalnay (1987) model of forecasterror growth has been used. Through the years, increases in resolution, especially in horizontal resolution, have brought improvements in forecast accuracy. This work provides a clean (clean since forecasts have been produced using different horizontal resolutions but the same model version for the same period) assessment of the impact of horizontal resolution on the short and the longrange forecast skill. In particular, the predictability times when the forecast rootmeansquare error reaches different levels of the asymptotic limit (25%, 50%, 71% and 95%) are computed. To better understand the role of model uncertainty on forecast skill, error growth is studied not only in a realistic framework, with forecasts verified against analyses, but also in a perfectmodel framework, using highresolution forecasts as verification. The key questions addressed in this work are: how much do the time limits change when resolution is increased from T95 to T799? And how do the past estimates of the socalled predictability time limits compare with these more recent estimates based on winter 2007/2008 forecasts?
After this introduction, section 2 describes the methodology used in this study, section 3 discusses the impact of resolution on forecasterror growth, and finally some conclusions are drawn in section 4.
3. Impact of resolution on forecasterror growth
 Top of page
 Abstract
 1. Forecasterror growth and predictability limits of a numerical weather prediction system
 2. Methodology
 3. Impact of resolution on forecasterror growth
 4. Discussion and conclusions
 Appendix: the ForecastError Growth Model
 Acknowledgments
 References
In the first part of this section, the impact of a resolution increase from T95 to T399 on forecast error during the first 15 days is discussed, both in the perfectmodel and in the realistic frameworks. This analysis is based on 90day average r.m.s.e. computed directly from the experimental data. Then, in the second part of this section, the forecasterror growth model is fitted to the experimental data, and the coefficients (α, β, γ) are computed for all resolutions and in both frameworks. Finally, the forecasterror growth curves are then used to assess the impact of resolution on the medium and longrange predictability.
3.1. Forecast error computed from the experimental data
Figure 2 shows the average r.m.s.e. for the control, the ensemblemean (defined as the average of the control and the four perturbed members) and the perturbed members, with errors computed in the perfectmodel and realistic frameworks for Z500 over NH. In the perfectmodel framework, there is a clear indication of the benefit of increasing the resolution for the whole 15day forecast range, with stronger benefits detected in the case of the control and the ensemble mean. The difference between the r.m.s.e. of the perturbed members run at different resolutions is smaller than the difference between the r.m.s.e. of the corresponding control forecasts because they all have similar initial condition errors (forecasts run at higher resolution start closer to the T799 analysis, see Table II), while the control forecasts run with higher resolution has much smaller initial errors. For each resolution, the r.m.s.e. of the ensemble mean is the lowest due to the filtering effect of the averaging operator. The results obtained in the realistic framework indicate a clear benefit of increasing resolution from T95 to T255, but a negligible impact of any further increase. As was the case for the results obtained in the perfectmodel framework, the impact is more evident for the control and ensemblemean forecasts.
Table II. Average initial error of the control and perturbed members for Z500 over NH.Configuration  95  159  255  319  399 

Initial error control  1.15  0.54  0.27  0.20  0.17 
Initial error perturbed member  3.18  3.01  2.98  2.97  2.96 
For any two ensemble configuration with resolution Tx and Ty, the ranksum Mann–Whitney–Wilcoxon nonparametric test RMW(Tx,Ty) has been used to assess the statistical significance of the difference between the distributions of the 90 forecast errors for each forecast step. The test (see, for example, Wilks (1995) for a definition) gives the probability, herein expressed as a percentage, that any two error distributions, of which Figure 2 shows the 90day average, are samples of the same underlying overall distribution. For any two distributions, the test has been computed using a bootstrapping technique randomly resampling the two distributions 5000 times. Being nonparametric, the test has the advantage of not relying on any assumption on the distribution of the scores. Note that in the computation of the test it has been assumed that there is no correlation between the gridpoint average rootmeansquare errors of consecutive days.
Generally speaking, the test confirms the average results shown in Figure 2, i.e. that the greater the difference in resolution, the more statistically significant the difference between the two average forecasterror curves is, and that shortrange differences are more statistically significant than longrange differences. As an example, Figure 3 shows the test value RMW(T799,Ty) computed between the forecasterror distribution function of the T799 perturbed forecasts and the distributions of the perturbed forecasts (for reason of space, the test is shown only for the perturbed forecasts since most of the results discussed in this paper refer to the perturbed forecast errors) of experiments T399, T319, T255, T159 and T95, in the realistic and perfectmodel frameworks. In the perfectmodel framework, the test RMW(T799,Ty) is less than 10% for up to forecast day 12 for all Ty, while in the realistic framework the test is less than 10% only for some resolutions Ty in the short and mediumrange. This difference between the perfectmodel and the realistic test values reflects the difference between the corresponding average forecast error curves shown in Figure 2.
Table III lists the forecasterror doubling times at forecast days 1–3 for two resolutions, T95 and T399. Before discussing the results, recall that the forecast error has been computed not against observations but against the analysis, which has been generated using the ECMWF 4DVar assimilation system and thus, most likely, projects more strongly onto the model manifold than the observations. It is interesting to note that for both resolutions the doubling times computed in the perfectmodel and the realistic frameworks are very similar, indicating that the model error component not simulated in the perfectmodel framework has a negligible impact on the forecasterror growth in the short forecast range. Table III also shows that both in the perfectmodel and the realistic frameworks, the doubling times are shorter for the higherresolution, more active T399 model, which is more penalized than the lowerresolution ones. The fact that the T399 model is more active can be detected by comparing the spread of the two systems: at forecast days 2 and 4: the T95 spread is already about 10% and 16% lower than the spread of the T399 ensemble (not shown). Toth et al(2002) found a similar effect of increasing the resolution of single forecasts as discussed further in section 4.
Table IIIa. Doubling times of the control forecast error for Z500 over NH.Doubling time control forecast  T95  T399 

 Perfectmodel  Realistic  Perfectmodel  Realistic 


Day 1  0.73  0.72  0.67  0.62 
Day 2  1.23  1.27  0.97  1.06 
Day 3  1.64  1.72  1.15  1.38 
Table IIIb. Doubling times of the control forecast error for Z500 over NH.Doubling time control forecast  T95  T399 

 Perfectmodel  Realistic  Perfectmodel  Realistic 


Day 1  0.86  0.84  0.91  0.84 
Day 2  1.38  1.38  1.31  1.28 
Day 3  1.82  1.83  1.61  1.61 
Figure 4 shows the average r.m.s.e. for the control, the ensemble mean and the perturbed members for T850 computed over NH in the perfectmodel and the realistic frameworks, and Figures 5 and 6 show the corresponding results for the Southern Hemisphere (SH) and the Tropics. Generally speaking, results for T850 confirm the conclusions drawn from the Z500 results: the impact of resolution is very clear in the perfectmodel framework, while there is only a small positive impact when increasing resolution from T95 to T255 in the realistic framework. The T850 forecasterror asymptotic level depends on the region. The comparison of the NH and SH curves show that the asymptotic level is higher during the cold season over NH than during the warm season over SH, and that the asymptotic level is reached earlier during the warm season over SH.
3.2. Estimation of the forecasterror growth model from the experimental data
The coefficients (α, β, γ) of the forecasterror growth model have been computed for all forecasts in both frameworks using the discretized form of the forecasterror growth model (see appendix for more details). The fit of the forecasterror curve to the experimental data is, in general, rather good as indicated by two diagnostics. Figure 7 shows that the scatter plots of the change in forecast error as a function of the average forecast error computed from the data and from the model are close, although not perfect. The poor fit at very short forecast ranges (say up to forecast day 1) is partly due to superexponential error growth, as was pointed out by Bengtsson et al(2008). The differences between the data and the model scatter plots indicate that although the forecasterror growth model captures the key features of forecasterror growth, it is not capable of properly describing it at all forecast ranges. As a further measure of model fit, the correlation coefficient between the forecasterror data and the model values have been computed. Results indicate that in both the perfectmodel and the realistic frameworks, the correlation coefficients are above 90% for the control forecasts and above 95% for the perturbed forecasts for all resolutions (not shown).
3.3. Resolution impact on medium and longrange predictability
Figures 8 and 9 show the forecasterror growth model curves for the control and perturbed members, with the model coefficients estimated from the data, and the curves normalized by the forecast error limit E_{∞},
 (2)
(See the appendix for a deduction of the error growth solution and for the relationship between the parameters in Eq. (2) and the coefficients (α, β, γ) of the forecasterror growth model Eq. (1).)
The normalized curves shown in Figures 8–9 reflect the r.m.s.e. data shown in Figure 7: errors are smaller in the perfectmodel than in the realistic framework, the control errors are smaller than the perturbed member errors, and the sensitivity to resolution is larger for the control forecasts, since the highresolution curves start closer to the analysis, as was pointed out earlier (see also Table II).
To highlight the impact of resolution on forecasterror growth at different forecast ranges, the forecast times τ(25%), τ(71%) and τ(95%) have been computed from the normalized curves (recall that e.g. τ(25%) is the time at which η(t) = 0.25). The top panels of Figure 10 show that when resolution is increased by about a factor of 4 from T95 to T399, the control forecast τ(25%) increases by 3.0 days (from 4.1 to 7.1 d) in the perfectmodel framework (by construction, τ(100%) is infinite for T799 since in the perfectmodel case the T799 control forecast is used as verification). By contrast, it increases only by 0.9 d (from 3.5 to 4.4 d) in the realistic framework. For the perturbed members, τ(25%) increases by ∼0.8 d in the perfectmodel case, and by 0.5 d in the realistic framework. It is not surprising that the increase in predictability of the control in the perfectmodel framework is the largest, almost three times longer than the one in the realistic framework, since not only the model but also the initial condition accuracy improves (see Table II) as the resolution increases. By contrast, the increase in predictability of the perturbed members in the perfectmodel and the realistic case differ by a smaller amount; this is because the initial condition accuracies are more similar (Table II). The middle and bottom panels of Figure 10 show the forecast times τ(50%) and τ(71%); these panels show that as the forecast length increases, the positive impact of resolution decreases and eventually disappears. Results show that τ(50%) for the control forecast increases by 2.5 d (0.4 d) in the perfectmodel (realistic) frameworks, and τ(50%) for the perturbed forecasts increases by 0.5 d (0.2 d) in the perfectmodel (realistic) frameworks. The impact of resolution is much smaller, and in some cases negative, on τ(71%). Table IV lists the impact of the resolution increase of about a factor of 4, from T95 to T399, on τ(25%), τ(50%), τ(71%) and τ(95%) for the perturbed members.
Table IV. Impact of resolution increase from T95 to T399 on the predictability times τ.  Perfectmodel  Realistic 

 T95  T399  Difference  T95  T399  difference 


τ(25%)  3.4  4.2  0.8  2.9  3.4  0.5 
τ(50%)  6.7  7.2  0.5  5.9  6.1  0.2 
τ(71%)  9.9  9.9  0  9.0  8.7  − 0.3 
τ(95%)  18.1  16.4  − 1.7  16.8  15.1  − 1.7 
The fact that the impact of resolution decreases with forecast time is more evident in Figure 11, which compares the difference between the forecast error of the T95, T159, T255, T319 and T399 forecasts and the T799 forecast for all forecast ranges, computed in both frameworks. Figure 11 shows that, in both verification frameworks, a resolution increase has the largest positive impact in the short forecast range, with a peak at around forecast day 1, with the benefit becoming smaller after about forecast day 9 in the perfectmodel framework and day 7 in the realistic framework. The fact that there is a 2day difference between the perfectmodel and the realistic results indicate that a reduction of model error (i.e. the elimination of the model error component that affects the realistic and not the perfectmodel experiments), could increase the benefit of current, and future, resolution increases. The fact that differences are negative for T95 after forecast day 8 is an indication of the limitations of the threeparameter forecasterror growth model to fully describe forecasterror growth.
Figures 12–16 summarize the main results of this work. Figures 12, 13 and 14 show the time limits τ(25%) and τ(71%) computed using the normalized, error growth curves for the perturbed members for Z500, Z1000 and T850 over NH (the time limit τ(95%) has not been shown because, as pointed out above, already τ(71%) shows a very weak sensitivity to resolution). The time limits computed using Z1000 (Figure 13) and T850 (Figure 14) confirm the two key conclusions drawn above, precisely that a resolution increase has a larger impact on the shorter forecast range and that the impact of resolution decreases as resolution increases.
Figures 15 and 16 show the time limits τ(25%) and τ(71%) computed using the normalized, error growth curves for the perturbed members for T850 over SH and the Tropics. These figures support the two conclusions drawn above, although the time limits are quantitatively different. The comparison of the T850 results for NH (Figure 14) and SH (Figure 15) show that the predictability limits are shorter during the warm season over the SH than during the cold season over the NH: for example, τ(71%) for T799 in the realistic case is 9 days over NH and 7.5 days over SH. For the Tropics, the shortrange limit τ(25%) is similar to the NH one, but the longrange limit τ(71%) is longer, 10.5 days for the T799 forecasts verified over the Tropics, compared to 9 days for forecasts verified over NH and 7.5 days for forecasts verified over SH.
4. Discussion and conclusions
 Top of page
 Abstract
 1. Forecasterror growth and predictability limits of a numerical weather prediction system
 2. Methodology
 3. Impact of resolution on forecasterror growth
 4. Discussion and conclusions
 Appendix: the ForecastError Growth Model
 Acknowledgments
 References
The error growth of instantaneous forecasts has been analysed, and the sensitivity of predictability to horizontal resolution has been assessed using ECMWF forecasts. More precisely, the impact of horizontal resolution on short, medium and longrange predictability time limits, defined as the times when the forecast rootmeansquare error reaches different levels of the forecasterror asymptotic limit (25%, 50%, 71% and 95%) has been discussed. Attention has been focused on the 500 and 1000 hPa geopotential heights and the 850 hPa temperature forecasts during winter 2007/08 (December 2007, January and February 2008, 90 cases) verified over the Northern Hemisphere, the Southern Hemisphere and the Tropics. The Dalcher and Kalnay (1987) model of forecasterror growth has been used to extrapolate the forecasterror growth beyond the 15 forecast days for which forecast data have been generated, and to normalize the forecasterror curves of forecasts run with different resolutions using the appropriate asymptotic limit given by the error growth model. Forecasterror growth has been studied both in a realistic framework, with forecasts verified against analyses, and in a perfectmodel framework, using highresolution forecasts as verification. Results obtained in the perfectmodel framework gave an upper bound on the skill gains that could be expected in reality.
Before summarizing the key results, it is important to mention two important caveats. Firstly, analyses have been used as verification in the realistic framework instead of observations. This has the benefit of providing a better coverage of the whole globe for all variables, but has the disadvantage that the shortrange forecast error might be underestimated. Although this leads to an underestimation of the shortrange forecast error (say by up to 25% for forecast lengths shorter than 2 days), it is difficult to gauge whether this could lead to an over or an underestimation of the impact of horizontal resolution on predictability. Secondly, since the same analysis is used in all experiments (albeit interpolated from the operational T799 resolution to the experiment resolution), this study does not take into account any potential benefit that increases in the horizontal resolution used in data assimilation could bring. This might have reduced the sensitivity to horizontal resolution of the shortrange predictability limits, but it is doubtful whether it could have any impact on the medium and longrange predictability limits.
The key results of this work on six issues are summarized hereafter.
4.1. Predictability time limits estimated in the perfectmodel and the realistic frameworks
Results obtained in the two frameworks lead to the same overall conclusions, although quantitatively the predictability times obtained in the perfectmodel framework are longer than the corresponding times obtained in the realistic framework. This is particularly true in the medium range for the control forecasts. The perfectmodel values give an upper bound on the predictability that could be achieved by current, stateoftheart numerical weather prediction models, such as the one used at ECMWF. Considering the impact of a fourfold increase of resolution from T95 to T399 on the error of perturbed forecasts over NH, perfectmodel results listed in Table IV (left columns) indicate gains in the shortrange skill that are about 60% longer than what the realistic results indicate (e.g. 0.8 instead of 0.5 days for τ(25%)). Differences become larger in the medium range, with skill gains up to 100% longer than the realistic results indicate (e.g. 0.5 instead of 0.2 days for τ(50%)). These differences indicate that simple resolution increases without model developments would bring only small improvements in the medium and long forecast ranges.
4.2. Usefulness of simple, parametric forecasterror growth models
The simple forecasterror growth model first proposed by Lorenz (1969, 1982), modified by Dalcher and Kalnay (1987) and used, among others, by Savijärvi (1995), Simmons and Hollingsworth (2002) and Bengtsson et al(2008) has proven again to be a useful tool to investigate forecasterror growth beyond the forecast range spanned by available forecast data. In the case of comparison of error growth of forecasts characterized by different asymptotic limits, it provides an easy way to normalize the different forecasterror curves. Despite this, our investigation also confirmed Bengtsson et al(2008) indication that this forecasterror growth model has difficulties in describing the error growth at very short forecast ranges.
4.3. Impact of resolution on predictability time limits over NH winter
Results obtained in both the perfectmodel and the realistic frameworks indicate that increasing resolution leads to longer time limits in the short forecast range, but has a small impact in the long forecast range (see Table IV for a summary of the NH results). These results are in line with the impact of resolution increases reported in the following published works:
Simmons and Hollingsworth (2002), who reported improvement in the shortrange forecast skill by about 1 day per decade (see their Fig. 4), due to a combination of a resolution increase of about a factor of 2 (from T213 to T511 between 1992 and 2000), model changes and improvements in data assimilation. Our results indicate an increase of τ(25%) for the control forecast in the realistic framework of a resolution increase from T159 to T399 of 0.5 days (from 3.9 to 4.4 days). The gain computed in this work in the realistic framework is lower, possibly because forecasts did not benefit from model or dataassimilation improvements (forecasts have been performed using the same model cycle and starting from the same initial conditions).
Tracton and Kalnay (1993) and Toth and Kalnay (1993), who discussed the implementation of the National Centers for Environmental Prediction (NCEP) ensemble prediction in 1992 with a variable resolution, higher up to day 5 and then lower, because they found that beyond 5 days horizontal resolution did not improve the skill of their system. Toth et al(2002) also reported that for single forecasts the use of higher resolution can reduce the shortrange forecast error, but could have a small or even detrimental effect in the long range.
Buizza et al(2003), who reported gains in the predictability of single control forecasts of about 0.5 days at around forecast day 5 when the ensemble resolution was increased from T159 to T255, and the analysis resolution was increased from T319 to T511 (see their Fig. 4). Our results indicate an increase of τ(50%) for the control forecast in the realistic framework of a resolution increase from T159 to T255 of 0.3 days (from 6.8 to 7.1 days): our gain is lower because Buizza et al(2003) T255 forecasts also benefited from the increase of the analysis resolution.
4.4. Short, medium and longrange predictability limits over NH, SH and the Tropics
The shortrange predictability time limit τ(25%) has shown a strong sensitivity to resolution: for T850 in the realistic case, for example, it has increased by about 50% when horizontal resolution has increased from T95 to T799: from 2 to 3 days over NH, from 1.5 to 2.2 days over SH and from 2.3 to 3.2 days over the Tropics. Results indicate that also a horizontal resolution increase from T399 to T799 leads to small but detectable improvements. By contrast, the mediumrange limit τ(71%) has not shown any sensitivity to resolution. Similar conclusions can be drawn by considering the perfectmodel results. The longrange time limit τ(95%) computed in this work for T255 Z500 forecasts verified over NH is 16.6 days in the perfectmodel framework and 15.3 days in the realistic framework. These values are about 2 days longer than the 13 days estimated by Simmons and Hollingsworth (2002) for the T255 system that was operational at ECMWF at the time of their investigation. The corresponding predictability limits for the T799 system are practically the same as the T255 ones (16.6 and 15.2 days, respectively, in the perfectmodel and the realistic frameworks). Considering T799 forecasts of the 850 hPa temperature (T850) verified in the realistic framework, the time limit τ(95%) is 16.6 days for NH (cold season), 14.1 days for SH (warm season) and 20.6 days over the Tropics.
4.5. Forecast error doubling times over NH
Consistent with the positive impact of a resolution increase, our results have indicated that the forecasterror doubling time decreases when resolution increases, especially in the short range (day 1–4), in agreement with the findings of Simmons and Hollingsworth (2002). For example, they reported a ∼10% decrease in the doubling time at forecast day 2, from 1.59 days in 1996/1997, when the ECMWF model had a T213 resolution, to 1.42 day in 2000/2001, when the resolution was T511 (see their Table I). Our results for the control forecast in the realistic framework indicate a ∼12% reduction of the doubling time at forecast day 2 (from 1.21 days to 1.06 days) when resolution was increased from T255 to T399. For the perturbed forecasts in the realistic framework, our results indicate a ∼5% reduction (from 1.34 days to 1.28 days). As mentioned above, the fact that the decrease is larger for the control forecast is because higherresolution control forecasts start closer to the analysis.
4.6. Predictability limit sensitivity to horizontal resolution and model improvements
To conclude this discussion, let us consider the question that was posed in the introduction: what is the sensitivity of the short and longrange forecast skill to the model resolution? Our results have indicated a strong sensitivity of forecast skill to model resolution in the short range, but no sensitivity in the long range. If, as in Dalcher and Kalnay (1987), the predictability limit is defined as τ(95%), i.e. the forecast time when the forecast error reaches 95% of the saturation level, our estimates indicate that for the prediction of instantaneous, synopticscale features as represented by Z500, our estimates are that the limit is 16.6 and 15.2 days, respectively, for T799 perturbed forecasts verified in the perfectmodel and the realistic frameworks. These values are 3 to 4 days longer than the 12 days quoted by Dalcher and Kalnay (1987), which was obtained applying the forecasterror growth model to the ECMWF Z500 forecasts used by Lorenz (1982). Although a direct comparison of these estimates is not possible, since they used forecasts issued by weather models of different complexity for different periods, and used different versions of the forecasterror growth model, the comparison suggests that we have not yet converged to a number that measures the real predictability limit of the atmosphere, and future work might further lengthen this estimate. Our results suggest that rather than resolution, it is model improvements that might lead to better predictions and longer predictability limits.
Appendix: the ForecastError Growth Model
 Top of page
 Abstract
 1. Forecasterror growth and predictability limits of a numerical weather prediction system
 2. Methodology
 3. Impact of resolution on forecasterror growth
 4. Discussion and conclusions
 Appendix: the ForecastError Growth Model
 Acknowledgments
 References
The forecasterror growth model used to estimate the predictability limits was first proposed by Lorenz (1982), then used and slightly modified by Dalcher and Kalnay (1987) and by Simmons et al(1995). Accordingly to this model, the time evolution of the forecast error E is given by the following equation:
 (A1)
In this study, the forecast error E(t) of Eq. (A1) is the average rootmeansquare error of a forecast, verified either against a T799 analysis in the realistic framework or against a T799 forecast in the perfectmodel framework. The average forecast error has been computed by using gridpoint weights w_{k} = cos(lat_{k}) to take into account the spherical geometry of the Earth:
 (A2)
where in Eq. (A2) f_{j, k}(j, t) is the forecast at time t started on day j at gridpoint k, and a_{j, k}(j + t) is the corresponding verifying analysis.
Equation (A1) can also be written as:
 (A3)
with:
 (A4)
In Eq. (A3), a is the rate of growth of the forecast error, S simulates the effect of model error deficiencies on the error growth, and E_{∞} is the asymptotic value (see Dalcher and Kalnay (1987) for more details). Equation (A3) has solution:
 (A5)
with
 (A6)
where E(0) is the error at initial time. Note that Eq. (A5) can be used to compute the forecasterror doubling time at each forecast step t.
Equation (A1) can be written in a discretized form for the forecast step j:
 (A7)
where for each forecast step j = 1, …, N, Δt is equal to 12 hours, ΔE_{j} = (E_{j+1} − E_{j}) is the forecasterror increase between steps j and (j + 1), and Ē_{j} = 0.5(E_{j} + E_{j+1}) is the average forecast error between the two steps. As in Lorenz (1982) and Simmons et al(1995), the three parameters (α, β, γ) of Eq. (A1) can be estimated by a leastsquare fit of the rootmeansquare differences between consecutive forecast errors ΔE_{j} for j = 1,17, with t_{1} = 0 and t_{17} = 360 hours. Once (α, β, γ) have been determined, the coefficients (a, S, E_{∞}) and (C_{1}, C_{2}) can be computed using the relationships in Eqs (A4) and (A6).
As in Dalcher and Kalnay (1987), predictability time limits τ(η) can be defined as the forecast times when the forecast error equals a fraction η of the asymptotic valueE_{∞}. By using Eq. (A5), it is straightforward to compute the predictability limit as a function of the ratio η(t):
 (A8)
In this work, four time limits will be computed, as the time when η(t) equals 0.25, 0.5, (0.71) and 0.95.