Improved Skill for the Anomaly Correlation of Geopotential ...inavon/pubs/krish.pdf · Krishnamurti et al. (2000b), by using multiple regres-sions of the model forecasts (applied

1082 VOLUME 131M O N T H L Y W E A T H E R R E V I E W

q 2003 American Meteorological Society

Improved Skill for the Anomaly Correlation of Geopotential Heights at 500 hPa

T. N. KRISHNAMURTI, K. RAJENDRAN, AND T. S. V. VIJAYA KUMAR

Department of Meteorology, The Florida State University, Tallahassee, Florida

STEPHEN LORD, AND ZOLTAN TOTH

Environmental Modeling Center, National Centers for Environmental Prediction, Camp Springs, Maryland

XIAOLEI ZOU, STEVEN COCKE, AND JON E. AHLQUIST

Department of Meteorology, The Florida State University, Tallahassee, Florida

I. MICHAEL NAVON

CSIT and Department of Mathematics, The Florida State University, Tallahassee, Florida

(Manuscript received 7 October 2002, in final form 27 November 2002)

ABSTRACT

This paper addresses the anomaly correlation of the 500-hPa geopotential heights from a suite of globalmultimodels and from a model-weighted ensemble mean called the superensemble. This procedure follows anumber of current studies on weather and seasonal climate forecasting that are being pursued. This study includesa slightly different procedure from that used in other current experimental forecasts for other variables. Here asuperensemble for the ¹2 of the geopotential based on the daily forecasts of the geopotential fields at the 500-hPa level is constructed. The geopotential of the superensemble is recovered from the solution of the Poissonequation. This procedure appears to improve the skill for those scales where the variance of the geopotential islarge and contributes to a marked improvement in the skill of the anomaly correlation. Especially large im-provements over the Southern Hemisphere are noted. Consistent day-6 forecast skill above 0.80 is achieved ona day to day basis. The superensemble skills are higher than those of the best model and the ensemble mean.For days 1–6 the percent improvement in anomaly correlations of the superensemble over the best model are0.3, 0.8, 2.25, 4.75, 8.6, and 14.6, respectively, for the Northern Hemisphere. The corresponding numbers forthe Southern Hemisphere are 1.12, 1.66, 2.69, 4.48, 7.11, and 12.17. Major improvement of anomaly correlationskills is realized by the superensemble at days 5 and 6 of forecasts. The collective regional strengths of themember models, which is reflected in the proposed superensemble, provide a useful consensus product that maybe useful for future operational guidance.

1. Introduction

In several operational numerical weather predictioncenters, the anomaly correlation of 500-hPa forecastshas always been used as a measure of the models’ over-all performance. Professor Fred Sanders from MIT (seeTable 1 for a list of acronyms and their definitions) hasbeen a frequent lecturer at FSU in recent years. One ofhis favorite comments was that the 500-hPa anomalycorrelation (a measure of skill of 500-hPa geopotentialheight forecasts) was not weather. He always wonderedwhy so much was said on that skill parameter whileassessing the relative performance of models when infact what mattered was the rain and the severe weather.

Corresponding author address: T. N. Krishnamurti, Dept. of Me-teorology, The Florida State University, Tallahassee, FL 32306-4520.E-mail: [email protected]

The counter argument he usually received was that ifthe troughs and ridges were not correctly placed, thelikelihood of a good weather forecast were slim. Forwhat it may have been worth, an anomaly correlationindex has been used uniformly over several decades bythe weather services of the world to assess the perfor-mance of their models.

The motivation for this paper emerged from our re-cent examination of anomaly correlations of 500-hPaheights in the context of a multimodel superensemblefollowing several of our recent studies (Krishnamurti etal. 1999, 2000a,b, 2001). The multimodel superensem-ble is best explained by the schematics in Fig. 1. Theword ‘‘superensemble’’ was used by the first author ofthis paper in a series of publications, only to stress thefact that this ensemble does carry the highest skill com-pared to participating member models of the ensemble

JUNE 2003 1083K R I S H N A M U R T I E T A L .

TABLE 1. List of acronyms.

Acronym Definition

AMIP Atmospheric Model Intercomparison ProjectBMRC Bureau of Meteorology Research CenterDMSP Defense Meteorological Satellite ProgramECMWF European Centre for Medium-Range Weather

ForecastsEMC Environmental Modeling CenterEOF Empirical orthogonal functionEPS Ensemble Prediction SystemFSU The Florida State UniversityhPa HectopascalsJMA Japan Meteorological AgencyMIT Massachusetts Institute of TechnologyNCEP National Centers for Environmental PredictionNOGAPS Navy Operational Global Atmospheric Prediction

SystemNRL Naval Research LaboratoryNWP Numerical weather predictionRmse Root-mean-square errorRPN Recherche en Prevision NumeriqueSSM/I Special Sensor Microwave ImagerSVD Single value decompositionTRMM Tropical Rainfall Measuring MissionTSDIS TRMM Science Data and Information SystemUKMET U.K. Met officeUTC Universal time coordinated

FIG. 1. A schematic diagram that illustrates the division of the timeline; prior to day 0 is the training phase that includes 120 experimentsfrom the multimodels whose daily results are regressed against anobserved ‘‘analysis’’ benchmark. To the right of the zero line is thefuture forecasts phase that make use of the statistics from the trainingphase.and also carries skills above those of the bias-removed

ensemble mean representations. Here the multimodelforecasts are divided into two categories: (i) past me-dium-range forecasts and (ii) current real-time medium-range forecasts. The past covers roughly 100 recent pastforecasts made by the multimodels. Given a benchmarkanalysis for this entire period, it is possible to obtain amodel-weighted bias of forecasts of the multimodels ateach geographical location. This is done followingKrishnamurti et al. (2000b), by using multiple regres-sions of the model forecasts (applied on individual en-semble members) against a benchmark analysis. In thisstudy, ECMWF analysis was selected as the benchmark.This superensemble, based on selective weights as-signed to the member models, can be viewed in a prob-abilistic sense as providing information from a numberof models (Stefanova and Krishnamurti 2002).

An anomaly correlation of 0.6 is generally regardedas an indication of a useful forecast. This threshold valuecomes from experience gained watching the forecastcharts. A forecast with a skill greater than 0.6 generallyimplies that troughs and ridges at 500 hPa are beginningto be properly placed in that forecast.

Reviewing the anomaly correlations of U.S. opera-tional forecasts, Kalnay et al. (1991) reported the sum-maries for the decade of the 1980s. During this decade,the anomaly correlations for 5-day forecasts increasedfrom values such as 0.6 to approximately 0.75 over theNorthern hemisphere and from approximately 0.4 to0.625 over the Southern Hemisphere, as seen in Fig. 2a.Here, only the first 12 zonal wavenumbers were in-

cluded in their analysis. If we take the first 12 wave-numbers (from forecasts and the analysis), the currentskill during November 2000 for the best model and theproposed superensemble are approximately 0.85 and0.90. This implies that major improvements for the 5-day forecasts are continually being realized. This pro-gress of the forecast quality was attributed to factorssuch as improved computing power, improved models,data coverage, and assimilation methodologies.

Figure 2b shows the 500-hPa anomaly correlationskill over the European region for October–December1981 carried out with the ECMWF model (Nieminen1983). What is most encouraging here is the steadymaintenance of very high skill in day-3 forecasts wherethe anomaly correlations are around 0.9 for almost theentire 3-month period. This speaks for the high qualityof the modeling and data assimilation of the ECMWFsystem. Variability in skill from one day to the nextincreases with skills at day 5 for forecasts ranging from0.4 to 0.9 and for day 7 for forecasts ranging from 20.4to 0.7. This is nearly the state of the art of these forecastsat 500 hPa from the current best models.

Kalnay et al. (1998) also reported on the anomalycorrelation at the 500-hPa level, covering a more ex-tensive period of nearly 43 yr of forecasts. Here, theforecasts were based on the NCEP–NCAR reanalysisdatasets, described by Kalnay et al. (1996), and the 1998version of the NCEP forecast model runs at T62 reso-


FIG. 2. Historical progress on anomaly correlations of the 500-hPa level. (a) The 5-day forecast 500-hPa height anomaly correlation(seasonally averaged), including zonal wavenumbers 0–12 for both the Northern and Southern Hemispheres (Kalnay et al. 1991). (b) Dailyvariation of the anomaly correlation of error of the 500-hPa height for forecast days 1, 3, 5, and 7 from 1 Oct to 31 Dec 1981 in Europe(Nieminer 1983). (c) Comparison of operational and reanalysis 5-day forecast anomaly correlations for the Northern and Southern Hemispheres(Kalnay et al. 1998). (d) ECMWF operational forecast skill since 1980 as represented by the forecast day on which the 500-hPa geopotentialheight anomaly over the Northern Hemisphere has dropped to 0.6. Heavy line is the 12-month average of the monthly mean values (Brown1987).

TABLE 2. Outline of multimodels used in this study.

ModelVerticallevels

Horizontalresolution

ECMWFUKMETBMRCJMAFSUNCEPNRLRPN

3130294014422428

T2130.83338 lon 3 0.55558 latT239T213T126T170T1590.98 lon 3 0.98 lat

lution for day-5 forecasts. Results for the Southern andNorthern Hemispheres (208–808 latitude) are presentedin Fig. 2c, which shows a slow increase in skill overthe Northern Hemisphere with skills reaching around0.7 and around 0.6 for the Southern Hemisphere. Therecent operational scores (also averaged over yearly pe-riods) are shown by the dashed lines. The latter are basedon higher-resolution (operational) models and revealslightly higher skills.

Brown (1987) has provided a review of the rapidprogress in the improvements of global numericalweather prediction during the 1980s. The anomaly cor-relation at 500 hPa was one of the measures historicallymonitored by the weather services during that decade.Figure 2d from Brown’s paper shows rapid increase in500-hPa skill of the ECMWF forecasts. Here the lengthof forecast period (along the ordinate) for which theforecasts with anomaly correlation greater than 0.6 arereached during different years (along the abscissa) areshown. As the analysis and forecasting system has beenimproved, the useful length of the prediction has in-creased from 5.5 days in 1980 to 6.5 days in 1985, whichis reflected in the 12-month average of the forecastlength where the anomaly correlation of the 500-hPageopotential height over the Northern Hemisphere dropsto a value of 0.6. The monthly mean values indicatedthat the longer forecast skills are realized in the coldermonths.

Anomaly correlations of 500-hPa geopotentialheights from ensemble forecasts at various operational


FIG. 3. Comparison of two graphs of anomaly correlation skill: (a) nonoptimized training and (b) optimizedtraining. The ordinate denotes anomaly correlation and the abscissa denotes forecast days. The dashed verticalline separates the training and the forecast phases of the superensemble.

centers also have shown major improvements in skillin recent years. The ECMWF forecasts now show that3-day skills can be close to 0.9. This implies that onday 3 these forecasts are placing the troughs, ridges,and contours right on top of the observed fields. Thisis an extraordinary accomplishment when we compareresults obtained in the 1970s and 1980s from singlemodels. Recent skills scores for the NCEP EPS andseveral other models also show similar major improve-ments in recent years. It is interesting to note that al-though there are major differences in the horizontal res-olution of these models, somewhat comparable anomalycorrelations are being achieved by these groups simplyfrom overall model improvements.

2. Superensemble methodology

A main tool of this study is a multimodel superen-semble that was recently developed at The Florida StateUniversity (Krishnamurti et al. 1999, 2000a,b, 2001).

The superensemble is developed by using a number offorecasts from a variety of weather and climate models.Along with a benchmark observed (analysis) field, pastforecasts are used to derive statistics on the past be-havior of these models. These statistics, combined withmultimodel forecasts, enable us to construct a super-ensemble forecast.

Given a set of past multimodel forecasts, we used amultiple regression technique (for the multimodels), inwhich the model forecasts were regressed against anobserved (analysis) field. We then used least squaresminimization of the difference between the anomaliesof the model and the analysis fields in order to determinethe weights. We carried out this minimization at all ver-tical levels, at all geographic locations (the grid pointsof the multimodels), and for all model variables. In all,some six million statistical coefficients describe the pastbehavior.

The motivation for this approach came from the con-struction of a multimodel superensemble from a low-


FIG. 4. Distribution of the optimal number of days for the training phase that provides thehighest anomaly correlations at the 500-hPa surface over the North American Region for a day-6 forecast.

order spectral model (Lorenz 1963). In this low-ordermodel, it was shown that it is possible to introducevarious (proxy) versions of cumulus parameterization(or model physics) by simply altering forcing terms(Krishnamurti et al. 1999). Time integration of this mul-timodel system showed that the multiple regression co-efficients of these multimodels (regressed against a na-ture run) carry a marked time invariance. This timeinvariance was a key element for the success of theproposed method.

We have used many models at diverse horizontal andvertical resolutions. Model output was interpolated to acommon grid of the lowest-resolution multimodel.These global models include several different parame-terizations of physical processes; effects of ocean, snow,and ice cover, as well as treatment of orography. Theobserved (or the analysis) fields are used only duringthe control period to determine the weights, and theverification of forecasts in the forecast phase of the su-perensemble. The training period for global weathercomprises about 120 forecast experiments for each ofthe multimodels.

Approximately seven or eight multimodel forecastswere found to be minimally needed to produce veryeffective superensemble forecasts. The effectiveness ofweather and seasonal climate forecasts has been as-sessed from measures of standard skill scores such ascorrelation against observed fields, root-mean-square er-rors, anomaly correlations, and the so-called Brier skillscores for climate forecasts (assessing skills above thoseof climatology). The horizontal resolution of the su-perensemble is around 125 km (a common denominatorof the multimodel resolutions).

In some sense the construction of the proposed su-perensemble is a postprocessing of multimodel fore-casts. This is still a viable forecast product that is beingprepared experimentally on real time at The Florida

State University using 11-member models. Thus it is auseful product for people to see and is currently avail-able online on real-time basis (http://lexxy.met.fsu.edu/rtnwp).

3. Summary of past results

The following is a summary of computations basedon our past publications (Krishnamurti et al. 1999,2000a,b, 2001).

It is consistently noted in our past studies that thesuperensemble forecasts generally have higher skillcompared to all participating multimodels and the en-semble mean. The ensemble mean assigns a weight of1/N to all the member models (N is the number of mod-els) everywhere (and for all variables), including severalpoorer models. As a result, assigning the same weightof 1/N to some poorer models was noted to degrade theskill of the ensemble mean. It is possible to remove thebias of models individually (at all locations and for allvariables) and to perform an ensemble mean of the bias-removed models. This too has somewhat lower skillcompared to the superensemble, which carries selectiveweights distributed in space, multimodels, and variables.A poorer model does not reach the levels of the bestmodels after its bias removal.

Training is a major component of this forecast initia-tive. We have compared training with the best qualitypast ‘‘observed’’ datasets versus training deliberatelywith poorer datasets. This has shown that forecasts areimproved when higher quality training datasets are de-ployed for the evaluation of the multimodel bias statis-tics. It was felt that the skill during ‘‘forecast phase’’could be degraded if the training was executed witheither poorer analyses or poorer forecasts. That was not-ed in our recent work on precipitation forecasts wherewe had shown that the use of poorer rainfall estimates


during the training period affected the superensembleforecasts during the forecast phase (Krishnamurti et al.2001).

In medium-range real-time global weather forecasts,the largest skill improvement is seen for precipitationforecasts both regionally and globally. The overall skillof the superensemble is 40%–120% higher than the pre-cipitation forecast skills of the best global models. Therms error and the equitable threat scores were the skillparameters used in that study. The training datasets forprecipitation came from the daily TSDIS operationalfiles of TRMM microwave radiometer-based rainfall es-timates. These were augmented from the use of the U.S.Air Force polar-orbiting DMSP satellites that providedSSM/I data from a number of current satellites (F11,F13, F14, and F15) in order to extend the global cov-erage. An application of these precipitation forecastsincluded the forecast guidance for some recent floodepisodes.

In real-time global weather forecasts the superensem-ble exhibited major improvements in skill for the di-vergent part of the wind and the temperature distribu-tions. Tropical latitudes show major improvements withthe superensemble for daily weather forecasts. For mostvariables, we have used the operational ECMWF anal-ysis at 0.58 latitude–longitude for the training phase inthese previous studies.

Real-time hurricane track and intensity forecasts areanother major component of superensemble modeling.This approach of carrying out a training phase followedby real-time forecasts has shown improved forecasts forthe tracks and intensity (up to 5 days) for the Atlantichurricanes. Improvements in track forecasts were 25%–35% better than those of the participating member mod-els. The intensity forecasts for hurricanes have been onlymarginally better than the best models. In some recentreal-time tests during 1999, marked skill in the forecastsof difficult storms such as Floyd and Lennie was noted,where the performance of the superensemble was con-siderably better than that of the member models.

The area of seasonal climate simulations has onlybeen addressed recently in the context of atmosphericclimate models where the sea surface temperatures andsea ice were prescribed, such as the AMIP datasets. Inthis context, given a training period of some 8 yr anda training database from the ECMWF the results ex-hibited improved skill compared to the member modelsand the ensemble mean, which were based on seasonaland multiseasonal forecasts of monthly mean precipi-tation, temperatures, winds, and sea level pressure dis-tributions. Further extension of this work is currentlybeing pursued in the area of improved multimodel sea-sonal forecasts using coupled climate models.

4. Computational methodology

A main tool for this study is a multimodel superen-semble that was recently developed at The Florida State

University (Krishnamurti et al. 1999, 2000a,b, 2001).The proposed superensemble is defined by

N

S 5 O 1 a (F 2 F ), (1)O i i ii2i

where S 5 superensemble prediction, 5 time meanOof ‘‘observed’’ state, ai is the weight for model i withi being the model index, N is the number of models,

i is the time mean of prediction by model i, and Fi isFthe prediction by model i. Here the coefficient ai isdetermined from the use of the least square minimizationprocedure. The weights ai are computed at each gridpoint by minimizing the following function:

t2train

2G 5 (S 2 O ) , (2)O t tt50

where O 5 observed state, t 5 time, and t 2 train 5length of training period (100 days in the current casefor NWP).

The variance of the final geopotential field was ob-tained using two different methods: (i) the use of heightfield Z to construct the superensemble and (ii) the useof the ¹2Z field to construct the superensemble. Hereinstead of constructing a superensemble of the geopo-tential height Z at the 500-hPa level, we have first con-structed the superensemble of the ¹2Z field. The reasonbehind this was to extract some extra skill from thegeopotential gradients and its Laplacian. The geopoten-tial is thereafter recovered from a solution of the Poissonequation using the spectral transform:

2 m m m(¹ Z) 5 2n(n 1 1)Z Y ,n n n (3)

where m is the zonal wavenumber and n is the index ofthe associate Legendre function (which denotes the de-gree of its polynomial). As one proceeds to smaller andsmaller scales, n becomes larger and we note then ¹2Zis directly proportional to n2, whereas Z is inversely pro-portional to n2; thus, this property would be reflected inthe two-dimensional spectral distribution of ¹2Z and Z.

This method appears to improve the superensemblesolution for the geopotential compared to a direct con-struction of the superensemble of Z. We are furthermoreable to assess the skills where this method appears tocontribute to the improvement of forecasts.

We have also carried out a comparison of the classicalbias versus that of the proposed superensemble for theforecasts. A simple way of finding the bias for the NWPof a specific model is to take a daily string of NWPforecasts, obtain a monthly average of these, and com-pare those with the analysis (or observed) mean for thatmonth. This procedure has been used by most weatherservices to assess whether the model has a cold, warm,moist, or dry bias, etc. (e.g., Heckley 1985; Sumi andKanamitsu 1984; Kanamitsu 1985). This is the classicalbias of a forecast. The proposed superensemble does notdo quite the same thing. The multiple-regression-based


FIG. 5. Geographical distribution of statistical weights for different member models: (a) NorthernHemisphere and (b) Southern Hemisphere. Color scale of the fractional weights is shown at the bottom.

least squares minimization of errors is different fromthe simple bias correction in the following manner.

The simple classical bias is given by

CB (classical bias) 5 {Z (l, f) 2 Z (l, f)}/N.Fn on

(4)

Here the nth-day forecast bias for a total number of Ndays is considered; Fn is the average forecasted geo-Zpotential height value at 500 hPa for a period of N days

while On is the averaged of the observed (analyzed)Zgeopotential height for that period.

The superensemble-based bias, following Eq. (1), isgiven by

SB (superensemble bias) 5 Z 2 Zsn on

t5N P3Q

5 a (Z 2 Z ). (5)O O i Fi FMit51 1


FIG. 5. (Continued)

Here Sn is the average geopotential height obtainedZfrom superensemble forecasts for the period of N days,ais are the coefficients (weights) for individual membermodels, ZFi is the forecast from model i, and FMi is theZsimple forecast mean of geopotential height from theith member model. The summation is done both on spa-tial (P 3 Q grid points, latitude–longitude) and temporal(days) scales for all member models. If the ai were allset equal to 1/N, then the classical bias and the super-

ensemble-based bias are very close to each other. Theweights take into account the local relative error char-acteristics of each model in the formulation of the su-perensemble.

The anomaly correlation coefficient is computed forindividual model forecasts and the ensemble based onthe method suggested by Brankovic et al. (1990). Anom-aly correlation for forecast variables is defined as thecorrelation between the predicted and analyzed anom-


alies of the variables. Here anomalies are deviationsfrom the mean climatological values. The following ex-

pression is used for computing the anomaly correlationof geopotential height at 500 hPa:

{[(Z 2 Z ) 2 (Z 2 Z )][(Z 2 Z ) 2 (Z 2 Z )]}O F C F C V C V CACC 5 . (6)

2 2Î [(Z 2 Z ) 2 (Z 2 Z )] [(Z 2 Z ) 2 (Z 2 Z )]O OF C F C V C V C

FIG. 6. A typical example of a forecast on day 6 over the Northern Hemisphere valid on 4 Sep 2000:(top left) analysis, (bottom left) the best model, (top right) superensemble, and (bottom right) the modelwith the lowest skill. Contour interval is 100 m.

Here suffix F denotes forecast, suffix C denotes cli-matology, and suffix V stands for verifying analysis. Theoverbar is the area mean and Z is the geopotential heightat 500 hPa.

5. Datasets

The datasets used in this study are identical to thoseused in a recent study on precipitation forecasts

(Krishnamurti et al. 2001). The daily global analysisand forecasts from the following prediction centerswere used: NCEP (Washington, D.C.), RPN (Canada),NOGAPS (NRL, Monterey, California), BMRC (Aus-tralia), UKMET (Reading, United Kingdom), andJMA (Japan). In addition to these, we used six dailyforecasts from the FSU global spectral model thatutilizes different initial analyses. The differences inthe initial analysis are obtained through the physical


FIG. 7. Same as in Fig. 6 but for the Southern Hemisphere.

initialization of observed rain rates (Krishnamurti etal. 1991) using different rain-rate algorithms (Krish-namurti et al. 2001). All of these forecasts com-menced at 1200 UTC and in most cases 6-day forecastdatasets were available for extended periods. In ad-dition to these, we had access to the initial assimilateddatasets from the European Centre for Medium-RangeWeather Forecasts for extended periods. It should benoted that in this study we have used ECMWF da-tasets in the following context. (i) For the trainingphase and forecast verification we have used the an-alyzed fields from ECMWF. (ii) We have not usedECMWF forecast datasets in the training phase. (iii)We had access to the daily anomaly correlation ofgeopotential height at 500 hPa of ECMWF forecastsfrom a Web site provided by NCEP. These are in-cluded in our tabulations of anomaly correlation skillsbut were not a part of the superensemble.

Formal computation of the anomaly correlation for mostof these models was possible on a daily basis. Table 2provides an outline of all the models used in this study.

These models deploy different horizontal and vertical res-olutions, topography, physical parameterizations, andmodes of dynamic computations. The superensemble wasconstructed at an interpolated horizontal resolution ofT126 (triangular truncation at 126 waves around the globe)at the 500-hPa surface for this study.

The Environmental Modeling Center of the U.S. Na-tional Weather Service maintains daily records of the500-hPa anomaly correlation. Some of these datasetsfor the diverse models were extracted from their Website and used in this study. In the computation of anom-aly correlation, it should be noted that the anomaly cor-relations do depend on the choice of the source of cli-matology and also the verifying analysis. In this study,both ECMWF and the NCEP analyses were used sep-arately for the training and forecast validation and inthe evaluation of anomaly correlation skills. The resultsfrom the use of these different analyses for training werequite similar; hence, those based on the use of ECMWFonly are shown here.

Since we carry a daily inventory of rms errors of


FIG. 8. Differences between forecast and analysis from Figs. 6 and 7: (top) superensemble forecasts, (middle)forecast from the best model, and (bottom) forecast from the model with the lowest skill. Units, m; color scale inm shown at the bottom.

forecasts during the training phase, it was possible toidentify days where the forecasts skills were low. Wearbitrarily removed those dates where the rms errors ofthe training superensemble were greater than 120 m forthe 500-hPa geopotential heights. Using 100 days of‘‘higher skill’’ training days, we noted that the skill of

the superensemble was much improved. Figure 3 showsthe results of computations of anomaly correlations withand without this optimization of the training phase, re-spectively. It is clear from this procedure that somefurther improvement of the skill of the superensembleis achievable.


6. Results of computations

In this section we present some of our results on themultimodel forecasts covering several months of fore-casts. This is an ongoing effort, which is being carriedout in real time with the datasets from the models de-scribed in section 5. Unless stated otherwise, most com-putations and results presented here are based on a train-ing period covering 20 March to 15 July 2000 and theforecasts covered the period from 16 July to 17 Sep-tember 2000. This period includes some missing dayswhere data were not available.

a. Optimizing the number of training days

We had noted that the forecast skill degrades some-what having either too few or too many training daysfor the construction of the superensemble. Thus it wasfelt that a certain optimal number of days could be de-termined in order to obtain the best statistics for thetraining phase, which in turn could provide the highestforecast skill. In order to determine this, we workedseparately at each grid point and assessed the optimalnumber of days that provided the highest skill for thesuperensemble. Figure 4 shows the distribution of theoptimal number of days for the best anomaly correla-tions of geopotential height at the 500-hPa level for day6 of the forecasts over the North American region. Theforecasts here were carried out for the entire month ofAugust 2000; the optimal training days preceded that.It is interesting to note that a large-scale distribution ofthe optimal number of days is present in this analysis.This number around North America shows large-scalevariation from oceans to mountains to the Great Plains.This behavior reflects the possible effects of bias errorsof the member models over these different geographicalregions. We have not addressed the issues of seasonalitywith respect to the optimal number of training days; thismay need to be addressed for practical applications.

b. The distribution of weights

The essence of the proposed superensemble lies withthe distribution of multiple regression weights for themember models. The training phase provides theseweights. These weights exhibit a distribution of positiveand negative fractional values. Figures 5a and 5b illus-trate some of these distributions based on the trainingperiod covering the months April–July 2000. We havearbitrarily selected four of the member models, whoseweights for the Northern and Southern Hemisphere aredisplayed in Figs. 5a and 5b, respectively. These weightsdisplay positive and negative fractional distributions.The scales of the centers of maxima and minima of theseweights appear to be smaller over the Northern Hemi-sphere compared to those of the Southern Hemisphere.Some interesting features seen here are, for instance,that the weights for model 1 over North America show

positive fractional weights over the western UnitedStates and negative fractional weights over the easternUnited States. Over the South Pacific and Australia, theweights for model 2 are predominantly positive whereasthey are negative for model 3. Collectively, positive ornegative signs are contributed by the different models.

If one were to contrast these models with the ensem-ble mean, then all these distributions would bear thesame constant value 1/N (where N denotes the numberof ensemble models). This is the major difference be-tween a superensemble and a member mean. The pasthistory of performance of the member models makesthe superensemble superior in skill, which is reflectedby the geographical distribution of the statisticalweights. This exercise of determining the weights isbased on training of the ¹2Z fields and not the geo-potential Z.

c. On the number of models

We noted that differences in the design on the modelsarise from the choice of physics, resolution, air–sea in-teraction, and the definition of orography. Thus thequestion of the optimal minimum number of models isimportant in the construction of the superensemble. Ifmore than two of the models are included, the errors ofthe ensemble mean start to increase (see Krishnamurtiet al. 2000b). This growth in error arises from assigningan equal weight of 1.0 to all models including the mod-els with relatively lower performance levels. However,the error of the superensemble decreases as these ad-ditional models are included, since all models seem tohave something to contribute over different regions. Theerror growth rate starts to decrease as these models areincluded, and beyond the inclusion of six top models,the error reduction almost stops. The reduction of errorfrom six models is substantial and much higher in com-parison to the ensemble mean. The reason for this be-havior of the superensemble arises from its selective useof fractional and negative weights for the member mod-els of the superensemble. Thus, we feel that improved500-hPa anomaly correlations require minimally sixmodels. These results are quite similar to what had al-ready been noted from the datasets of 1998 (Krishna-murti et al. 2000b).

d. Predicted maps on day 6 of forecasts

Figures 6–8 illustrate a typical 6-day forecast. In Fig.6 we show a forecast over the Northern Hemisphere forday 6 of a forecast from the superensemble (panel b),the best model (panel c), and a model with the lowestanomaly correlation skill (panel d). These are to be com-pared to the analysis valid on that date (i.e., 1200 UTC4 September 2000). The anomaly correlations for thesethree respective forecast categories were 0.75, 0.69, and0.44. We can see a slight improvement in the forecastfeatures over the best model and a considerable im-


FIG. 9. Anomaly correlation as a function of forecast days: thin lines show results for member models and thick line to the top shows resultsfor the superensemble. The different panels show forecasts for days 1–6.

provement with respect to the model with the lowestskill. A similar example of the day 6 forecast for theSouthern Hemisphere is shown in Figs. 7a–d. The skillof the superensemble was generally quite high over theSouthern Hemisphere. The respective anomaly corre-lations for the superensemble, the best, and the lowestskill models for this day-6 forecast were 0.81, 0.71, and0.32.

Given that roughly 10%–15% improvement in anom-aly correlation skill is possible from the superensembleover the best model, does it convey any useful synopticinformation? Inspecting numerous 500-hPa forecastsfrom the best model and from the superensemble, wefind that there is generally some more information con-tent in the superensemble forecasts. The superensembleplaces the trough along 1208W more accurately, as com-pared to the best model, which moves it somewhat far-ther east in 6 days. The ridge over North America westof Lake Michigan shows a short-wave trough (ridingthe ridge) that is captured by the superensemble; thebest model fails to capture that very short wave feature.

In general, the model with the lowest skill has muchlower heights over the entire United States. The troughalong 1208E in the analysis is placed too far east, near1408E, by the best model and near 1258E by the su-perensemble. In the highest latitudes, north of 508N, thedifference between the forecasts of the superensembleand the best model does not appear to be large.

The above features are more clearly seen from thedifferences between the forecasts and the analysis fields.Those fields for the Northern and Southern Hemispheresare shown in Figs. 8a–e. The preponderance of darkblue and dark brown coloring shows the large forecasterrors for the model with the lowest skill. This coloringdiminishes somewhat as we proceed to the best modeland the fewest errors are seen for the superensemble.This was an example that was selected randomly. Thereare several other instances where the forecast skills onday 6 were strikingly larger for the superensemble com-pared to the best model where these differences are evensharper than what are shown in Fig. 8.


FIG. 9. (Continued)

e. Anomaly correlations at 500 hPa

Figures 9a–c illustrate the anomaly correlation of thegeopotential height at 500 hPa for the member models(thin lines) and for the superensemble (thick line) for arecent 20-day period (this includes 100 days of trainingand 20 days of forecast). These results include thosemodels that directly participated in providing daily da-tasets, apart from those models whose anomaly corre-lations were available on the NCEP Web site. Althoughthe member models exhibit considerable variation intheir anomaly correlation, varying from 0.1 to 0.8 forthese forecasts, we note more consistent results for thesuperensemble. The ¹2Z-based superensemble doesshow higher skill on almost all occasions for days 1–6of forecasts. Figure 9a shows the results over the South-ern Hemisphere, while Figs. 9b and 9c show the resultsfor the Northern Hemisphere and the whole globe, re-spectively. Through day 3 of forecasts, the anomalycorrelation skill for superensemble is consistently higherthan 0.9. This shows the major improvement in the stateof numerical weather prediction from some of the mem-

ber models and from the superensemble. Overall theresults for the superensemble are quite impressive. Anoteworthy feature of these forecasts is the consistenthigher skills over the Southern Hemisphere that is wellabove those of the last two decades shown earlier. Al-though we present a sample of results here, we havenoted a major uniformity of these results in our contin-ued computation of this algorithm on a real-time basis.

Table 3 provides a summary of these results for theSouthern Hemisphere, Northern Hemisphere, and theglobal belt. Here the entries for the anomaly correlationskills covering a forecast period from 20 August to 17September 2000 are presented. Results for the membermodels, the ensemble mean, and the superensemble areincluded here. Results for days 1–6 of forecasts areprovided in these tables.

The wintertime skills of the anomaly correlation aregenerally higher than those for the summer season. Theoverall member model skills over the Southern Hemi-sphere are quite high. Over the Northern Hemisphereduring this period, the best model’s skill at days 1 and


FIG. 9. (Continued)

6 were 0.992 and 0.653, respectively. The correspondingnumbers for the superensemble were 0.995 and 0.748for days 1 and 6. This shows roughly a 14% improve-ment on day 6 of forecasts. The corresponding figuresfor the Southern Hemisphere are an improvement fordays 1 and 6 from 0.979 and 0.715 (for the best model)to 0.990 to 0.802 (for the superensemble, i.e., roughlya 13% improvement on day 6). Also shown in this tableare the entrees for the ensemble mean, which lie roughlyhalfway in between the best model and the superensem-ble. Thus it appears that a substantial improvement inskill is possible from the use of the proposed super-ensemble. The global results presented in Table 3 ba-sically confirm these same findings. The fall of skillfrom the best to the poorest model on day 6 of forecastsfor the Northern Hemisphere can be seen ranging from0.65 to 0.45 and for the Southern Hemisphere from 0.71to 0.53. We have noted a consistently high skill for thesuperensemble around 0.75–0.8 for day 6 in our ex-perimental runs.

Table 3d describes the global results for the case

where two of the member models exhibiting the highestanomaly correlation skills are entirely excluded. Theentries in this table are to be compared with the entriesin Table 3c where these high skill member models areincluded. It appears as though the addition of the bestmodels contributes to roughly 1%–2% improvement forthe superensemble, while the overall improvement ofthe superensemble over the best (available) model isaround 10%. This improvement in the superensembleis a result of the selective weighting of the availablemodels during the training phase.

f. Reduction of systematic errors

In Fig. 10, we show the systematic errors (predictedminus observed mean) for day 6 of forecasts for theentire month of August 2000. Here the results for amember model with the least systematic error are com-pared with those of the superensemble for the Northernand Southern Hemisphere. In this illustration, the dis-tribution of colors from blue to red displays the negative


TABLE 3. (a) The 500-hPa geopotential height anomaly correlationvalues from the superensemble (bold), ensemble mean (italic), andthe multimodels averaged for the period 20 Aug–17 Sep 2000 forthe Northern Hemisphere for forecasts from day 1 to day 6. (b) Sameas in (a) except for the Southern Hemisphere. (c) Same as (a) exceptfor total globe. (d) Same as in (c) except the two best models areexcluded from the superensemble suite.

Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

Superensemble 0.995 0.981 0.956 0.905 0.843 0.748Ensemble mean 0.989 0.970 0.936 0.880 0.807 0.700Model 1Model 2Model 3Model 4Model 5Model 6

0.9920.9870.9750.9740.9690.978

0.9730.9650.9510.9370.9370.931

0.9350.9290.8920.8870.8730.866

0.8640.8570.8120.7760.7620.740

0.7760.7550.7210.6450.6360.622

0.6530.5380.5260.4990.453

(b)Day 1 Day 2 Day 3 Day 4 Day 5 Day 6


0.9790.9780.9570.9560.9450.921

0.9620.9500.9250.9140.9030.852

0.9310.9290.8880.8560.8370.813

08930.8820.8260.7700.7330.724

0.8300.7990.7230.6930.6150.613

0.7150.6360.5960.5580.535

(c)Day 1 Day 2 Day 3 Day 4 Day 5 Day 6


0.9840.9810.9630.9620.9560.941

0.9670.9570.9300.9250.9180.889

0.9360.9320.8850.8710.8580.846

0.8890.8800.8150.7860.7670.739

0.8240.7960.7060.6970.6650.632

0.7130.6230.5790.5780.549

(d)Day 1 Day 2 Day 3 Day 4 Day 5 Day 6

Superensemble 0.978 0.957 0.927 0.891 0.834 0.740Ensemble mean 0.970 0.948 0.915 0.866 0.798 0.684Model 1Model 2Model 3Model 4

0.9630.9620.9560.941

0.9300.9250.9180.889

0.8850.8710.8580.846

0.8150.7860.7670.739

0.7060.6970.6650.632

0.5790.5780.549

to positive spread of systematic errors. A preponderanceof dark blue color over the Southern Hemisphere reflectslarge positive systematic errors in excess of 40 m forthe best model. It can be clearly seen that the systematicerror of the model with the lowest skill is quite large(as seen by the coloring). Even the best model has ratherlarge systematic errors. The superensemble is not freeof these errors; however, the error is small compared tothat of the member models. Over the Northern Hemi-sphere the preponderance of white, yellow, and lightbrown colors for the superensemble clearly reflect alarge reduction of the systematic error. These same fea-

tures have been monitored on a real-time basis for thelast 2 yr. It is also clear from these computations thatthe superensemble does not simply remove the classicalbias (i.e., forecast mean minus observed mean equal tozero). The proposed superensemble, although not sys-tematic error free, is able to reduce the forecast errorwith obvious practical advantages but that kind of anartificial bias removal is not possible in a truly predictivesense. The bias correction is an after-the-fact correctionand cannot be implemented for forecasts of any practicalutility.

g. Percent improvement in rms errors fromsuperensemble forecasts

Figure 11 shows the percent improvements from thesuperensemble forecasts compared to those of the en-semble mean over the best model. These are for 2- and5-day forecasts of the global geopotential heights at 500hPa. These improvements are related to the respectiverms errors. It is clear that the improvements for thesuperensemble are quite large at day 2 of the forecasts,approaching about 40%, whereas the corresponding im-provement for the ensemble mean is around 28%. Atday 5, the improvements appear significant and larger;here the corresponding numbers are 52% and 46% forthe superensemble and the best model, respectively. Themean rms errors at 48 and 120 h of forecasts for severalof the member models and for the ensemble mean andsuperensemble are shown in Fig. 11b. These are resultsover the tropical belt 308S–308N. The period coveredfor the results shown in Figs. 11a and 11b includes 92days starting from 1 June to 31 August 2000 (averaged).It is clear that the errors of the superensemble are smallerthan the other representations shown here.

h. Improvements in the planetary and synoptic scales

When we examine the zonal harmonics of the geo-potential at the 500-hPa level (Fig. 12), we note thatthe superensemble (based on the use of ¹2Z) carries alarger portion of the variance compared to the individualmodels (assessed in terms of their respective anomalycorrelation) for the first seven zonal wavenumbers.These are the scales where the zonal harmonics of the500-hPa geopotential carries the largest proportion ofthe total variance. The predicted geopotential field forthe worst model is quite flat by day 6 of forecasts com-pared to its variance field, shown in Fig. 12, and itappears that the long waves are somewhat misrepre-sented. The best model exhibits some improvement forthe zonal percent variances while the superensemble,holding the highest anomaly correlation skill, exhibitsa robust structure for the planetary and synoptic scalewaves.

These same features can be viewed using two-di-mensional variances in the triangular truncation space.In Figure 13 we show these variances as a function of


FIG. 10. Systematic errors: forecast minus analysis over 120 days over the 500-hPa level (top left) NorthernHemisphere superensemble, (bottom left) Northern Hemisphere best model, (top right) Southern Hemispheresuperensemble, and (bottom left) Southern Hemisphere best model. Units, m; color scale in m shown at thebottom.

the east–west and north–south wavenumber m and n.Here again we note a very robust structure for the var-iances centered around zonal wavenumbers 2–5 and me-ridional wavenumbers 4–12. The variances for the bestand the worst models are much lower in comparison.When we first started on this exercise we felt that theconstruction of the superensemble would enhance thestructure of the smaller scales (wavenumbers greaterthan 10) since the ¹2Z field exhibits many smaller scalescompared to the Z field. This exercise revealed that largegeopotential gradients on the planetary and synopticscales were much improved by the construction of thesuperensemble of the ¹2Z field.

7. Future outlook

The results on an anomaly correlation at the 500-hPalevel are a part of an ongoing real-time numerical weath-

er prediction exercise that is being pursued at The Flor-ida State University. The total problem includes 11 dif-ferent models where all the variables at 12 vertical levelsare being subjected to the construction of the super-ensemble. For these weather models, some 107 statisticalweights, which are being used, carry the past behavior.The reason for this large volume of statistics has to dowith the model’s performance. Some models appear tohandle local water bodies better; others have greaterskill over orographic regions while others seem to de-scribe the oceanic convection and rainfall better. Thesystematic errors, in detail, vary from one region toanother for these diverse models. Results for all of thevariables, including daily precipitation (Krishnamurti etal. 2001), show a rather similar behavior; that is, thesuperensemble generally exhibits a much higher skillcompared to the ensemble mean and the member mod-els.


FIG. 11. (a) Percent reduction of mean rms errors at the 500-hPa surface over the best model by the superensembleand by the ensemble mean for Jul 2000 over the global tropical belt 308S–308N. (left) Results of 48-h forecast and(right) results at 120 h. (b) Mean rms errors of the respective member models, the ensemble mean, and the superensembleat hours 48 and 120 of forecasts over the tropical belt 308S–308N.

The emphasis on the 500-hPa geopotential is basedon historical and practical reasons. Noting that it is nowpossible to construct a superensemble of the geopoten-tial heights at 500 hPa with an accuracy of 0.95–0.80between days 1–6 of forecasts implies that troughs andridges are very nearly accurately placed close to theircorrect locations for the medium-range forecasts.

Achieving these skills in a consistent manner appearsto be an accomplishment of current NWP systems com-bined with the superensemble postprocessing method.This may be one of the reasons why operational fore-casters should consider the implementation of this sim-ple procedure of construction of superensemble fore-casts. The datasets provided by the superensemble con-


FIG. 12. Meridionally averaged percent variance of the geopotential heights as a function ofzonal wavenumber.

TABLE 4. The 500-hPa geopotential anomaly correlation averagedfor the period from 20 Aug to 17 Sep 2000 for the global belt, theNorthern Hemisphere, and the Southern Hemisphere for forecastsfrom day 1 through day 6. Here the results from two different meth-ods, the conventional superensemble (SENS) and the SVD-based su-perensemble (SE p SVD), are presented.

Day

Global

SENS SEpSVD

NH

SENS SEpSVD

SH

SENS SEpSVD

123456

0.9920.9790.9580.9280.8810.799

0.9940.9830.9600.9520.9110.845

0.9950.9810.9560.9050.8430.748

0.9960.9830.9590.9480.8690.801

0.9900.9780.9560.9330.8890.802

0.9930.9820.9610.9530.9210.849

tain the weighted averages that are carried out inde-pendently at each grid point of the domain of calculationfor each day of forecast separately. The issue of dy-namical consistency of this dataset has been raised fol-lowing our first study (Krishnamurthi et al. 2000b),where we examined the quasi static and quasigeostroph-ic fields (over middle latitudes) of this data and notedthat the balance is quite acceptable. It should be notedin this context that the construction of the superensem-ble is based on Eq. (1) where the regression coefficientsare based on anomalies with respect to a time mean andnot from a direct use of the full geopotential fields. Itshould also be noted that the individual models haveindeed improved considerably over the last three de-cades. Further improvement, no doubt, will occur as themodels improve their data assimilation, resolution, dy-namical representations, and physical parameterizations.From what we have seen in our recent work it wouldseem that further improvements of the superensembleforecast would also follow.

An area of future work would be to explore the use-

fulness of singular value decomposition (SVD) methodto address the removal of cases of possible ill-condi-tioned matrices in the current superensemble. The cur-rent multiple regression procedure in the training phaseinvolves the solution of matrices at each location anduses the Gauss–Jordan method. We have essentially by-passed the difficulty at a few hundred grid points usingthe ensemble mean wherever ill conditioning was en-countered. The SVD method, as well as several othermethods that invoke EOFs, Z transforms, the Kalmanfilter, and cyclostationary EOFs can also be used toremove the degeneracy. An example of anomaly cor-relation improvement from the SVD method is shownin Table 4. These are the results for the same period asthose presented earlier in Table 3. We notice that animprovement of over 5% from the use of SVD over theSouthern and Northern Hemispheres and the global belt.On day 6 of forecasts, an increase of anomaly correlationfrom 0.799 to 0.845 can be seen over the global belt;while it increased from 0.748 to 0.801 and 0.802 and0.849 over the Northern and Southern hemispheres, re-spectively, using the SVD method. We propose to in-corporate these refinements in our future studies andhopefully they would also be useful for operational fore-casts.

It is worth noting that the anomaly correlation skillsover the Southern Hemisphere are beginning to ex-ceed those of the Northern Hemisphere. At first wethought that this might have been peculiar for theperiod we had investigated, but it appears that as themember models are improving, the Southern Hemi-sphere forecast skills have indeed been going up inrecent years.

Currently the multimodel datasets have only beenavailable to us through day 6 of forecasts. It should bepossible to obtain these datasets through day 10 of the


FIG. 13. Variances of the geopotential Z in the two-dimensional triangular truncation space. The ordinate denotes ameridional wavenumber n and the abscissa denotes the zonal wavenumber m. (top) Results for the superensemble.(bottom left) Variances for the best model and (bottom right) those for the model with the lowest skill.

forecasts. It may also be possible to receive the datasetsas ensemble forecasts for the modeling groups. Givensuch datasets we expect further improvements in theforecasts of the 500-hPa anomaly correlations. Havingsuch a forecast on the operational suite of products canprovide useful guidance for the weather especially overthe Tropics and midlatitudes.

Acknowledgments. The research reported here wasfunded by NASA Grants NAG5-9662 and NAG8-1537, NSF Grant ATM-0108741, and FSURF COE.The authors wish to express their thanks to Dr. RicardoCorrea Torres for computing the optimal training pe-riods for this study. We wish to acknowledge the op-erational global modeling community for the datasets


they have provided. Special thanks go to Tony Hol-lingsworth for the ECMWF datasets used for modelverification.

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