Multimodel Ensemble Forecasts for Weather and Seasonal Climate (2024)

1. Introduction

The notion of multimodel forecasting was evident in the studies of Lorenz (1963), where he examined the initial state uncertainties in a simple nonlinear system. Much progress has been made in multimodel forecasting for the conventional weather prediction problems, using the singular-vector-based perturbations, Molteni et al. (1996) and the use of breeding modes, Toth and Kalnay (1997). Several other formulations have appeared in the current literature, including the simpler Monte Carlo methods, Mullen and Baumhefner (1994). In seasonal climate forecasts, the multimodel forecasts are normally constructed by using initial perturbations from adjacent start dates, LaRow and Krishnamurti (1998).

Our primary interest is not in the evaluation of the ensemble mean and its probability characteristics, but in the evaluation of a statistical combination of various models in a postprocessing procedure (super forecast/analysis). The skill of this technique appears to far exceed a conventional average of model fields. A number of model forecasts, solutions based on initial state perturbations, and the mean all cluster in one region of the system phase space. The errors of these models are sufficiently large generally such that the “observed” field often lies in another part of this phase space. A multimodel analysis tends to lie closer to the observed state compared to all of these other solutions. This single algorithm is used for all of the proposed problems, that is, global weather, hurricane track and intensity forecasts, and seasonal climate simulations.

This notion first emerged from the construction of a low-order spectral model that has been discussed extensively in the literature; see Lorenz (1963) and Palmer (1999). Section 2 of this paper discusses the low-order multimodel forecast. Here we show the time invariance of a statistic that makes it possible to project solutions, during a forecast phase, and exhibit fewer errors than most conventional models. This idea of resilience has been tested further, in this paper, using a variety of datasets on weather forecasts and seasonal to multiseasonal climate simulations. The seasonal simulations are based on AGCMs (a list of acronyms is provided in appendix B). These are multimodel, 10-yr-long integrations based on AMIP datasets, Gates et al. (1999). The global numerical weather prediction applications of this algorithm (appendix A) are based on 1998 datasets from a large number of operational weather centers: ECMWF, UKMO, NCEP, RPN, JMA, NRL (NOGAPS), BMRC, and those developed from the FSU global spectral model forecasts. Also included in this paper are applications of the same algorithm for hurricane tracks and intensity forecasts during the 1998 season.

2. Motivation

The motivation for this paper came from the examination of the solution of a low-order spectral model, following Lorenz (1963), where we constructed models by assigning different values to the parameters of this problem. These parameters provide different values for the implied heat sources and sinks of this system. Assignment of different values for these parameters enabled us to construct an ensemble of models. One of these member models was arbitrarily defined as a nature run (i.e., a proxy for the real atmosphere).

This is a simple numerical system that can be integrated forward in time using the standard leapfrog, time-differencing scheme with an occasional forward differencing (to dampen the computational mode). The model ensemble can be thought of as set of models with different versions of physical parameterization (and/or diffusion) for each model. The different model versions were generated using the values shown in Table 1, which also includes the initial state definition that was altered slightly by random perturbations. The different initial states reflect different analyses from different forecasts systems.

At this point we define the superensemble that is central to this paper. A long-term integration through a time T (of the multimodels) is arbitrarily divided into two time periods, T1 and T2. Here T1 is regarded as a training period and T2 is regarded as the test (or forecast) period. During the training period, the multimodel variables were regressed against the observed (i.e., the nature run) variables X, Y, and Z. This is simply a least square minimization of the differences of the time series of the multimodels and of the nature run. This procedure provides weights from the regression for the individual multimodels.

There is an essential time invariance of the weights during the training and the test period; see Fig. 1. The abscissa in Fig. 1 denotes time. The training period was arbitrarily assigned as 70 time units (nondimensional time following Lorenz), and the test period is between time units 71 and 200. The time histories of weights shown in Fig. 1 were calculated using a cross-validation technique. To accomplish this, the weights for any given date are calculated from all other dates. This time invariance is an important feature, which if confirmed for weather and seasonal climate forecasts, could provide the possibility for a major improvement in the skill of our current prediction capability.

Figure 2 represents the principal variables X, Y, and Z for the Lorenz model runs. The thin, dashed lines represent the individual model runs, while the solid, heavy line shows the solution for the nature run. The heavy, dashed line starting at time 70 shows the solution for the superensemble. It is clear that the best solution, that is, the closest to that of the nature run, is seen for the superensemble, whereas the individual models exhibit large errors since their solutions carry large phase errors. The solution of the superensemble begins to exhibit a slow growth of error after around 190 time units. That type of behavior has also been noted in the application of this procedure for weather and seasonal climate forecasts. However, this growth of error begins to occur so late that one can make a major improvement in the skill of forecasts over periods of interest. That is a goal of this paper. The key element in this proposed technique is the time invariance (resilience) of the statistics displayed in Fig. 1, and that appears to be present if systems are not highly chaotic. The systems studied by Lorenz (1963) and Palmer (1999) were considerably more chaotic (for the choice of their parameters) compared to the examples we have presented here. We feel that our choice of parameters is more relevant for the atmospheric behavior as will be shown from the display of the time invariance of the statistical weights for the weather and seasonal climate forecasts.

The procedure above is defined by training and test phases using a simple multiple regression of the model anomaly forecast fields with respect to the observed fields. The general procedure is provided in appendix A and may be thought of as a forecast postanalysis procedure. During the test phase, the individual model forecasts together with the aforementioned statistics provide a superensemble forecast.

3. Datasets

The following datasets were used for the three components of the modeling studies reported here.

4. Seasonal/multiseasonal precipitation simulations

We shall next present the superensemble forecast procedure based on the simulation datasets of AMIP. Phillips (1994, 1996) described the AMIP models. AMIP consisted of 31 different global models; the eight models selected are shown in Table 2. This choice was somewhat arbitrary. This table shows the resolution of the models and their salient physical parameterizations. These are all atmospheric general circulation models of the late-1980s vintage. This was one of the most complete datasets presently available for carrying out our proposed study on the superensemble. The 10-yr integration from these respective models had a start date around 1 January 1979 but was not all the same. The initial analyses (common to all models) were provided by ECMWF. Other common datasets shared by all these models included the distribution of monthly mean sea surface temperatures and sea ice. A common grid was used for the proposed superensemble studies. Data at all vertical levels and for all basic variables were interpolated to a common 2.5° latitude–longitude grid that was close to the horizontal resolution of most of these models. In the vertical, these datasets were next interpolated to common standard (so-called mandatory) vertical levels (i.e., 1000, 850, 700, 500, 300, and 200 hPa).

Figure 3a illustrates the monthly mean tropical precipitation skill for option 1. The individual model precipitation rms error lie between 3 and 5.5 day⁻¹. All AMIP rms errors displayed are calculated using anomalies from the respective model’s monthly mean “climatology” that is based on the training period. The error of the ensemble mean, around 2.5 mm day⁻¹, is superior to those of the individual models. The error of the superensemble is approximately 2 mm day⁻¹ and is superior to all other measures shown here. Options 1 and 2 have nearly identical results for the errors of the superensemble during each of their respective training and test periods. That is because 6 yr of data were common during their respective training periods.

Several questions naturally arise at this stage: How good is an error of 2 mm day⁻¹? Is there a real skill for the climate simulations? An rms errors of monthly precipitation totals of the order of 1–2 mm day⁻¹ does imply some useful skill, since the range of values of monthly totals lie between 0 and roughly 20 mm day⁻¹. The issue of its usefulness is further addressed in the geographical distribution of these simulations separately. We found that all simulations for different variables did not exhibit any large distinctions between options 1 and 2. Hence, we shall limit our future discussions to one or the other option.

Figure 3c illustrates the skill for the meridional wind over the global Tropics (30°S–30°N) at 850 hPa for option 1. These are rms errors from continuous 10-yr simulations of the models. The individual model errors lie in the range of 1.5–2.5 m s⁻¹, whereas the errors of the superensemble (thick black line) lie around 1 m s⁻¹. Also shown in the illustrations are the errors of the ensemble mean. Those errors are of the order of 1.3–1.4 m s⁻¹; again we note that the errors of the superensemble are distinctively lower than those of the ensemble mean. Also shown in this diagram are the errors of climatology based on observations (17-yr average from 1980 to 1996 based on ECMWF analysis). Similar results are seen in Fig. 3b for option 2. Overall, the proposed superensemble procedure has the least errors compared to all models, the ensemble mean and the observed climatology. Very similar results are found for the meridional wind at 850 hPa over a monsoon domain (Fig. 3d) bounded by 30°S–35°N, 50°–120°E. The results show a clear separation of errors of the superensemble compared to those of the individual models. The monsoon region is of special interest because of the rainfall variability. The rms error of the rainfall over the monsoon region (Fig. 3e) lies roughly between 3 and 6.5 mm day⁻¹ for the individual models and around 2 mm day⁻¹ for the superensemble. The errors for the mean of the models are between 3 and 4 mm day⁻¹. This holds for both the training and test periods.

Next, we analyzed results for the monthly mean over various selected domains. Precipitation simulation rms errors of the monthly are briefly discussed here. The domains of analysis (Figs. 3f,g,h,i,j) include the Northern Hemisphere, Southern Hemisphere, the globe, Europe (between 30°–55°N and 0°–50°E) and North America (between 25°–55°N and 70°–125°W). Observed monthly rainfall estimates were obtained from the GPCP estimates [Arkin and Janowiak (1991)]. The precipitation simulation skill over the globe, Northern Hemisphere, North American domain, and the Tropics from the superensemble technique stand out and are superior to those of the individual model errors. The monthly errors during the entire 2 yr of the test period (January 1979–December 1980) are quite small, that is, of the order of 1–2 mm day⁻¹ for the superensemble, whereas the errors of the ensemble average were around 2.5 mm day⁻¹. The range of individual model rms errors is between 2 and 5 mm day⁻¹ over the various domains. Furthermore, we note that over several periods, the improvement in skill of the superensemble over the best model exceeds the difference in skill between the best and the worst individual model. Among all of the fields we analyzed, we noted that the best results were obtained for the superensemble statistics of the meridional wind errors over the Tropics, where that error of the superensemble was around 1 m s⁻¹ throughout the 10-yr period (training as well as the test period). This is how far the multiple regression pushes the superensemble solutions toward the observed or the analysis state during the training phase and that accuracy is nearly retained during the test period. The tropical results are expected to be better due to the strong boundary forcing from the large oceans, whereas in the extratropics, there are typically large external errors. An ensemble mean does not have access to spatially weighted models. The procedure selects the best weighting at each individual point in space and for each separate variable, creating a collective, localized reduction of errors.

A comparison of the rms errors of the superensemble and the ensemble mean for the global domain are shown in Figs. 3k and 3m for precipitation and 850-hPa zonal wind, respectively. Here the results for option 2 are shown, but again, it is emphasized that the option does not change the representation of the results. In both cases, the rms errors of the superensemble are less than those of the ensemble mean. The reduction in error of the superensemble over those of the ensemble mean is of the order of 40%–100%. These results are very similar to the results obtained for the meridional wind (Figs. 3o and 3p), as well as in the Indian monsoon domain for all variables (Figs. 3l,n,p).

This study raises several questions. Are we seeing a useful skill here, in terms of seasonal and multiseasonal climate simulations? Why is this skill so high? Given these very low rms errors for the superensemble, a mapping of the predicted fields from the superensemble leaves no doubt that they carry much more skill compared to the individual models. The collective, useful information content of multimodels is extracted in the construction of the proposed postprocessing superensemble technique comparing these to the analysis fields during a training period. The removal of biases for each geographical region, at each vertical level and for each variable, appears to make this a very powerful scheme. The AMIP results from the individual models did not seem very impressive. This procedure has been able to extract some very useful information from the results of the individual models from local regions where their skill was higher. As stated earlier, the success of this scheme lies in the resilience of a relationship between the individual models and the analysis field. Further work is needed to understand the significance of this relationship and further improve the performance.

5. Global numerical weather prediction

Two datasets covering the periods JJA 1998 and JFM 1999 were used for the postprocessing superensemble forecasts from various global weather prediction models. Here we show the results for the global winds at 850 hPa during JJA 1998. Table 3 lists the models whose forecasts were available for this period. In total, we had seven models, each making 92 forecasts with a start time at 1200 UTC for each day. The 61 days during June and July were treated as the training period and the 31 3-day “forecasts” were prepared during the entire month of August 1998.

Given the differences in the physics and complexity of weather models, it is not surprising that one notes varying skills in the performance of these models. Figure 6 illustrates the 850-mb rms forecast errors on day 3, for various regions. These include the global belt, Tropics (30°S–30°N); the Asian monsoon region (between 30°S–40°N and 30°–150°E); USA (25°–55°N, 125°–70°W); Europe (30°–55°N, 0°–40°E); and the entire Northern and Southern Hemispheres. Here we include day-3 histograms of skills for the seven models as well as those for the multimodel ensemble (orange). Verification skills are calculated against ECMWF analysis. For the individual models, the rms wind error increases from roughly 2 to 6 m s⁻¹ during the 3-day forecast. The multimodel superensemble has the least error compared to each of the models. Furthermore, we noted that the 3-day skill of the superensemble (rms error 1.6–4 m s⁻¹) is comparable to the day-1 errors for several of the models. We also show the rms errors of the ensemble mean (orange) in these histograms. The superensemble has a lower error compared to that of the ensemble mean in all cases. Over many regions, the ensemble mean has a higher error compared to some of the best models. This is one of the most promising results of the proposed approach.

Figure 7b shows an example of a typical 3-day forecast percent improvement over Europe for the 850-hPa winds. Although the ECMWF and superensemble forecasts are visually similar, there was a 26% area average improvement of the rms wind error for the superensemble over the ECMWF prediction. This is also typical of most forecasts as seen in Figs. 7a and 7c over the monsoon domain and the United States, respectively (Note:NCEP forecast used in 11c). Small discrepancies over most domains account for roughly 20% differences in skill. The superensemble always shows a closer agreement to the analysis field compared to the multimodels by invoking the models’ bias corrections.

The rms error reduction of the zonal wind at 850 hPa over the entire Tropics for August 1998 is shown in Fig. 8a. The different lines show the percent improvement over the performance of respective models for forecast days 1–3. The analysis field of the respective models is used to calculate the rms errors. Overall, we note a 10%–30% improvement in the 850-hPa zonal wind rms errors from the superensemble. The ensemble means percentages lie in the range of −20% to +20%. In all cases, the superensemble exhibits a much larger reduction of error with respect to the multimodel errors and the ensemble mean errors. It is also interesting to note that the percent improvement increases with the number of days of forecast.

To assess how many models are minimally needed to improve the skill of the multimodel superensemble, we examined sequentially the issue using one to seven models. Results for global wind rms errors at 850 mb are shown in Fig. 8b. We sequentially added models with lower and lower skill as we proceeded from one model to seven models. The dashed line shows the error for the ensemble mean and the solid line indicates that of the superensemble. The superensemble skill is higher than that of the ensemble mean for any selection of the number of ensemble members. The skill of the superensemble between 4 and 7 is small, that is, around 3.6 m s⁻¹. The ensembles mean error increases as we add more ensemble members beyond 3. That increase came from the gradual addition of models with low skill. For two and three models, the skill differences are the smallest. That rapid increase of error beyond three models is not seen for the superensemble since it automatically assigns low weights to the model’s low skills. It is also worth noting that half the skill improvement comes from a single model for this procedure. That is roughly a 5% improvement for the 850-mb winds for 3-day forecasts.

The ensemble of one is essentially the procedure for the removal of the bias of a single model. Here the regression utilizes the past history of the model for each geographical location, each vertical level, and for each variable separately. We carried out this exercise for each of the seven models separately. We then performed an ensemble mean of these individual bias removed fields. This was carried out separately for days 1, 2, and 3 of forecasts. The rms errors of the tropical (30°S–30°N) 850-mb winds for this ensemble mean were compared to that of the statistical multimodel superensemble, Fig. 8c. We clearly see that removal of collective bias is superior to doing that separately.

When we remove the bias of each model separately, we find many interesting results. The skill of forecast of each model is increased. The poorer models, however, do not compare as well as the best model after the removal of the individual biases. If we perform an ensemble mean of all the models, after the bias is removed individually, the mean still includes the poorer models. That is because an equal weight is assigned to each model for the ensemble averaging (after the bias removal). That has an effect of degrading the results. This is apparent in Fig. 8b, where as more models are included, the results eventually become inferior to using a single (best) model. The proposed multimodel forecast technique is superior to the “individual bias removed ensemble mean” because the superensemble does not assign an equal weight of 1.0 to all the models. Thus the poorer models are not excluded in this averaging; however, in those geographical locations and vertical levels for certain variables where they may have a higher skill, this inclusion is helpful. This global bias removal (geographically, vertically, and for different variables) is a dominant aspect of the superensemble. For proper application of such a technique, a priori determination of which models perform well and at what locations could greatly enhance the procession time.

6. Hurricane track and intensity forecasts from the superensemble

Over the Atlantic, Caribbean, and the Gulf of Mexico, there were 14 named tropical systems during the 1998 hurricane season. Of these, four became tropical storms and the remaining reached hurricane strength. The 1998 hurricane season was active between 27 July and 1 December. Our interest here is on the superensemble forecasts skill for the tracks and intensity of these storms. Ideally, it would have been desirable to develop the statistics of a control (a training) period from one or more years of past history. It is imperative that no major model changes occur within the multimodels during the course of these “control” and “forecast” periods. A number of models generally provide hurricane forecast information; these include NCEP/NOAA, GFDL, UKMO, NOGAPS/NRL, official forecasts of the NHC and the FSU suite of models. Table 4 provides a short summary of these models. In addition to these forecasts, there exists some in-house prediction models at the NHC that routinely provide hurricane intensity forecasts (SHIPS, SHIFOR) De Maria and Kaplan (1999). It was not possible to acquire uniform datasets, for a string of years, without major model changes. Four of the aforementioned models, NCEP/MRF, FSU, GFDL, and NOGAPS, underwent major resolution changes after the 1997 season. Thus, it was not possible to derive the statistics based on the multimodel performance with the 1997 datasets and use those to forecast during the 1998 hurricane season. It was still possible to carry out forecasts for each and every storm of the 1998 season using a cross-validation approach, which excludes all days of a particular hurricane while computing the weights for its forecast. All other hurricanes of the season are included in the evaluation of the weights for the storm being forecasted. This entails deriving the multimodel statistics from all storms of 1998, sequentially, excluding the specific storm that is being forecasted. This is a robust approach for assessing the validity of the proposed approach for the forecasts of the storms of 1998. Each of the 14 named storms in 1998 lasted over several days. Thus, it was possible to develop a multimodel forecast database for 68 sets of forecasts for this entire season. The methodology for the calculation of regression coefficients is identical to that described in appendix A. The databases are the model forecasts, the observed, and the official forecast estimate of the track (position) and intensity (maximum wind estimate) every 12 h, starting from an initial time and ending at hours 72–144, as dictated by the termination of such forecasts. The weights for the multimodels vary for each forecast period (i.e., hour 12, 24, 36, 48, and 72). We found that this gave better results than providing a single set of multimodel weights for all three days.

First we shall look at the overall statistics of the track and intensity forecasts for the entire year of 1998. Here we show the day 1, day 2, and day 3 skills for these forecasts. The results for the training period are shown in Fig. 9a, and those for the superensemble forecasts (cross validation) are presented in Fig. 9b. During 1998, the best-track forecasts came from the NHC official component. These are in fact subjective forecasts made by the forecasters of the NHC in Miami. They are essentially based on consensus from among the suite of model forecasts available to them. In addition, they also make use of their past experience on subjective hurricane forecasting while arriving at these official forecasts. The histograms on the left panels show, respectively, the track forecasts. The superensemble track forecasts are superior to those of all other models and official forecasts for each of the three days. The superensemble, in the training phase, has position errors of the order of 0.85°, 1.5°, and 1.9° for days 1, 2, and 3 of forecasts, respectively. The corresponding position errors for the superensemble forecasts (Fig. 9b) are 1.25°, 1.9°, and 2.6° latitude for days 1, 2, and 3, respectively. Similar results hold for the intensity forecasts as well. The rms errors of intensity for the training and forecasts from the superensemble are better than those of all other models including the ensemble mean.

In Figs. 10a–f we show several examples of predicted tracks for Atlantic hurricanes such as Alex, Bonnie, Danielle, Georges, and Mitch of 1998. Here the predicted tracks from several models are displayed. These also include three models from FSU as well, which are

1. a control forecast with the global spectral model at the resolution T126 with no physical initialization,

2. a forecast at the resolution T126 that includes physical initialization (Krishnamurti et al. 1991),

3. an ensemble forecast with a high-resolution regional spectral model following Zhang and Krishnamurti (1999).

In these illustrations, the tracks use the following symbols:

• BEST denotes the observed best track;

• FSUC denotes the FSU control forecast that does not include rain-rate initialization;

• FSUP denotes the results for FSU global spectral model that includes physical initialization;

• FSUE denotes the ensemble averaged track for an FSU regional spectral model, co*cke (1998), which is nested within the FSU global spectral model;

• OFCL denotes the official forecast from the NHC;

• GFDL denotes the forecast made from the geophysical fluid dynamic or Multiple-Mesh Model of Princeton (NCEP/NOAA);

• NGPS denotes the U.S. Navy’s NOGAPS model;

• UKMT denotes the UKMO’s global model; and

• SENS denotes the track based on the superensemble forecasts.

In Fig. 10 we note that the tracks of the superensemble are superior to the forecasts made by the multimodels. A higher accuracy of track forecasts results from the improved timing for the locations of the superensemble with respect to the “best track.” In some instances the heading of some of the multimodel storms appears more accurate, but the timing of their locations compared to the best track is less accurate compared to the superensemble.

Several specific examples of intensity forecasts from the superensemble are shown in Figs. 11a–f. A number of these multimodels provided 3-day forecasts; these included FSU (three models) through day 6 of forecasts, NGPS, and GFDL (not all are illustrated in Fig. 11). We have illustrated two other intensity-forecast estimates provided by the NHC, the SHIPS and SHIFOR statistical models; see De Maria and Kaplan (1999). SHIFOR is a statistical hurricane forecast model. It is based on climatology and persistence and applies only to storms over the ocean. SHIPS is another simple climatological scheme of the Hurricane Research Division that makes use of parameters such as maximum possible intensity, current intensity, vertical shear of the tropospheric horizontal wind, persistence of intensity change over the last 12 h, eddy flux convergence of momentum at 200 mb, zonal wind, and temperature within 1000 km of the storm center. The forecasts for days 4, 5, and 6 came from the FSU models (global without physical initialization, global with physical initialization, and an ensemble average intensity). Overall, a skill in the intensity forecasts is clearly evident from the superensemble. The FSU models contribute to the skill of the superensemble during days 4, 5, and 6, providing very useful information for these extended superensemble forecasts of intensity. The intensity forecasts from SHIPS and SHIFOR are reasonable for the first three days of forecasts; however, it is apparent from the overall statistics, presented earlier (Fig. 9), that the superensemble outperforms all other models in the intensity forecast. It is generally recognized that numerical models have rather poor skill in intensity forecasts. This was reflected in the multimodel forecasts of the storms of 1998. In most instances the intensity forecasts of the superensemble are within a “category” (as indicated by the horizontal lines on the plots) of the observed best estimates of intensity. Overall these intensity forecasts are quite impressive through day 6 of forecasts, given that the current state of the prediction is accepted as being quite poor. SHIPS and SHIFOR often underestimate the storm intensity by several categories.

7. Concluding remarks and future outlook

One of the major issues we have addressed here relates to the removal of the collective errors of all models. That approach exhibits different characteristics from an ensemble average of the results. The straight average approach assigns an equal weight of 1.0 to all the models and may include several poor models. The mean of these poor models degrades the overall results, as more of such models are included. Our approach of the superensemble assigns weights to each model, based on its performance, geographically, vertically, and for each variable separately. This would not assign a high weight to the poorer models over selected regions based on their past performance.

A postprocessing algorithm based on multiple regression of multimodel solutions toward observed fields during a training period shows promise for various applications during a subsequent test period. The resulting superensemble reduces the forecast errors below those of the multimodels. Application of the algorithm to the simulation of seasonal climate, global weather, and hurricane track and intensity shows the most promise. Root-mean-square errors of the seasonal superensemble simulations for winds are reduced to less than one-half the errors of the best models. Precipitation forecast errors of the order of a few millimeters per day are around 300% better than the individual models. The results herein clearly show that the superensemble has higher skills. This performance is attributed to the superior statistical combination of model anomaly fields compared to the standard technique.

One reason for the algorithm’s superior performance is its use of rms error for defining the statistical forecast weights. The same metric is used to define rms error skill scores. The individual model anomalies are not as good as the removal of the statistical reconstruction of all the models using the regression technique. The rms error of the superensemble is shown to be considerably less than that a simple reconstruction of models. Initially, these results seemed strange since we are dealing with a linear regression. Upon closer examination, we noted that the computation of the root-mean-square differences in obtaining the statistical weights implies that this is not a simple linear problem.

The seasonal climate simulations using the AMIP dataset show that the superensemble rms errors for the monthly mean fields (such as winds, temperature, and precipitation) are quite small compared to all other models. Due to regional interannual variability in temperature and precipitation, we examined the AMIP and WMO/WGNE seasonal monsoon datasets in the context of the present modeling. We prepared a dataset from 1-yr differences for the models and the observed fields during the control (or training) phase. We created a simulation of differences using the statistics resulting from the multiple regression technique. The rms errors of the difference fields of the superensemble were far less than those of the individual models. Monsoon rainfall differences between 1988 (heavy rainfall year) and 1987 (a weak rainfall year) were very promising, again revealing that the superensemble had less errors compared to the multimodels. The AMIP includes atmospheric general circulation models that utilize prescribed SSTs and sea ice. Therefore, similar work is needed to explore the skill of the proposed methodology for coupled atmospheric–ocean models.

Our analysis on the number of models needed to improve the skill of the superensemble shows that roughly six models produce the lowest rms errors for the superensemble global NWP. The number of models needed is likely to change with changes in the models making up the multimodel set. In the global NWP application, the superensemble stands out, providing roughly a 20% improvement over the best models. While examining local improvements arising from the superensemble, we noted that it extracts the best information from a number of models. An example of that shown in Fig. 7b illustrates that an erroneous forecast over a given region from the best model is corrected from information extracted from another model that carries a higher weighting over that region. Therefore, improvement by collective inclusion of good features makes the superensemble stand out.

Forecasting the Atlantic hurricanes of 1998 required using that season’s dataset for both the control and the forecast phase. This was based on the cross-validation technique where no data for a storm being forecasted was used for the evaluation of the statistical weights. These forecasts showed the best results compared to all other multimodels. There was a major improvement in the position and intensity of each storm compared to most multimodels and in the overall seasonal statistics;the superensemble outperformed all other models.

Further exploration of methods other than the proposed simple linear multiple regression deserves to be explored. We have started to examine the use of nonlinear regression and neural network training methods for possible further improvement in skill. We have also tested seasonally varying weights for climate applications, which prove to be very promising, but the training size is greatly reduced. Further work is needed on detailed analysis of the probabilistic properties and application for the technique and the weights obtained, particularly for climate. This may also lead to a more efficient and superior technique.

Finally, a detailed separate study on the probabilistic estimates is in preparation using the so-called Brier skill scores among others. In that work we will show that the superensemble does have measurably much higher skills for seasonal climate forecasts compared to all individual models, climatology, and the ensemble mean. These results will be published in a separate paper.