understanding and predicting subseasonal variations · understanding and predicting subseasonal...

2001 CDC Science Review 27

CHAPTER 3

Understanding and PredictingSubseasonal Variations

3.1 Modeling and understanding thestatistics of weekly averages

The principal mechanism of tropical-extratropical interaction is through dia-batically forced Rossby waves. On sea-sonal and longer scales, tropical diabaticheating is strongly linked to tropicalSST; hence one speaks of an “SST-forced” global response as in Chapter 2.On the subseasonal scales of interesthere, the SST variability is relativelyweak, and its coupling to the heatingvariability is much less rigid. The heatingvariability itself is considerable, how-ever, and has a significant extratropicalimpact. Some of this variability (espe-cially that associated with the MJO) is

predictable, and raises the hope that atleast some aspects of subseasonal extrat-ropical variability may therefore also bepredictable. Unfortunately, for variousreasons the simulation and predictabilityof subseasonal tropical heating variationshas thus far proved difficult in generalcirculation models. This has been amajor stumbling block in capitalizing onthis source of subseasonal extratropicalpredictability.

Inspired by the success in Figs 2.1–2.4 ofsimple empirical predictions of seasonaltropical SST variations and their globalimpact, we have recently constructed alinear inverse model (LIM) suitable forstudies of atmospheric variability and

Given that the details of daily weather are unpredictable beyond about a week, the questions of what aspects of thecirculation remain predictable and what useful information can be extracted from predicting them present interest-ing challenges. The forecast problem is particularly difficult for Week 2, because boundary conditions have begun tobecome important but initial conditions have not yet completely lost their influence; at the same time, the chaos fromunpredictable nonlinear interactions has nearly saturated. This is mainly why prediction efforts have traditionallyfocused on shorter (synoptic) and longer (seasonal to interannual) time scales. And yet there is much to said forshifting some of the focus to the subseasonal scale, if only because variability on this scale accounts for a largefraction of the total atmospheric variability from synoptic to decadal scales. Also, episodes of springtime floods,summertime droughts, and prolonged wet or dry spells are phenomena with obvious societal consequences.

CDC scientists are addressing these issues by focusing on the variability and predictability of weekly averages,through both modeling and diagnosis of the observed statistics, and through detailed investigations of NCEP's oper-ational forecast ensembles for Week 2. A significant recent accomplishment was the construction of a low-dimen-sional 37-component linear empirical-dynamical model that not only successfully represents the statistics of weeklyanomalies but also has comparable forecast skill in Week 2 to that of NCEP's operational ensemble. There is evi-dence that much of this model's skill arises from processes not well represented in the NCEP or other numericalmodels, such as subseasonal variations of tropical convection. On the other hand, much of the skill of the numericalmodels is likely due to processes not well represented in the empirical model, such as nonlinear baroclinic cyclogen-esis or blocking development in Week 1. It is therefore possible that an intelligent combination of the empirical andnumerical model forecasts will yield a Week 2 forecast that is superior to either in isolation. Constructing such acombination is now one of our primary efforts. This effort will benefit from and build on our recent success inimproving both statistical and numerical forecast products for this time scale.

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CHAPTER 3 Understanding and Predicting Subseasonal Variations

28 2001 CDC Science Review

predictability on weekly time scalesusing global observations of the past 30years. Notably, it includes tropical dia-batic heating as an evolving model vari-able rather than as an externally specifiedforcing. It also includes, in effect, thefeedback of the extratropical weathersystems on the more slowly varying cir-culation. We have found both of thesefeatures to be important contributors tothe model's realism.

The model is concerned with the behav-ior of 7-day running mean anomalies ofextratropical streamfunction and col-umn-averaged tropical diabatic heating.It assumes that atmospheric states sepa-rated by time lags τ are related as x(t+τ)= G(τ) x(t) + ε, where G is a linear oper-ator and ε is noise. This implies that thezero-lag and time-lag-covariance matri-ces of x are related as C(τ) = G(τ)C(0).We use this relationship at a particularlag, say τ = 5 days, to obtain G(5) fromobservational estimates of C(5) and

C(0). We then make another assumptionthat is at the heart of the LIM formalism,that distinguishes it from other empiricalmodels, and that enables one to makedynamically meaningful diagnoses ofdirect relevance to modelers. This is thatG(τ) satisfies the relation G(τ) =exp(Lτ), where L is a constant linearoperator. We use this to obtain L fromG(5), and having done so, use it again toobtain G for all other lags. We are finallyin a position to make forecasts for alllags as x(t+τ) = G(τ) x(t). Crucially, hav-ing obtained L, we can also diagnose therelative importance of its elements asso-ciated with tropical-extratropical andinternal extratropical interactions. Forexample, we can use L to estimate whatthe statistics of extratropical variabilitywould be without diabatic forcing fromthe tropics.

Figure 3.1 demonstrates the success ofthis model in reproducing the observedvariance and 21-day lag covariance of 7-

Observed________ Linear Inverse Model (LIM)

Statistics of weekly 250 mb streamfunction anomalies

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Variance

LIM without tropical heating______________________

21 day lag covariance

Fig. 3.1: Observed and modeled (using the full LIM and a version of the LIM in which the effects of tropical heat-ing are removed) statistics of weekly 250 hPa streamfunction anomalies.

Observed Linear Inverse Model (LIM) LIM without tropical heating

Understanding and Predicting Subseasonal Variations CHAPTER 3


day running-mean anomalies of 250 mbstreamfunction during northern winter.Note again that we are effectively usingthe observed 5-day lag covariances topredict the 21-day lag covariances here.The comparison of the observed and pre-dicted covariances is clearly encourag-ing. The right column shows that theeffect of tropical heating is relativelysmall on the variance but relatively largeon the 21-day lag covariance. This isconsistent with our finding that althoughtropical heating contributes a relativelysmall portion of the extratropical vari-ability, it contributes a large portion ofthe predictable variability.

Forecast skill is an important test of anymodel. The LIM is better at forecastingWeek 2 anomalies than a dynamicalmodel based on the linearized baroclinicequations of motion (with many morethan the LIM’s 37 degrees of freedom)that is forced with observed tropicalheating throughout the forecast. Indeedat Week 2 the LIM’s skill is competitivewith NCEP's MRF model with nominallyO(106) degrees of freedom. The upperpanel of Fig. 3.2 shows such a compari-son of Week 3 forecast skill during thewinters of 1985/86–1988/89. Otherexperiments show that this encouragingforecast performance is not limited toyears of El Niño or La Niña episodes.

The LIM assumes that the dynamics ofextratropical low-frequency variabilityare linear, stable, and stochasticallyforced. The approximate validity of theseassumptions has been demonstratedthrough several tests. A potentially limit-ing aspect of such a stable linear model

with decaying eigenmodes concerns itsability to predict anomaly growth. Wehave nevertheless found, through a sin-gular vector analysis of the model’s prop-agator G, that predictable anomalygrowth can and does occur in thisdynamical system through constructivemodal interference. Examination of theinitial structures associated with optimalanomaly growth further confirms theimportance of tropical heating anoma-lies associated with El Niño and La Niñaas well as Madden-Julian oscillation epi-sodes in the predictable dynamics of theextratropical circulation.

The LIM formalism also allows one toestimate predictability limits in astraightforward manner. Indeed it allowsone to estimate the expected skill of anyindividual forecast from the strength ofits predicted signal. Given that in manycases the predictable signal is associatedwith tropical forcing, one can quantifythe effect of that forcing on extratropicalpredictability. Our general conclusion isthat without tropical forcing, extratropi-cal weekly averages may be predictableonly about two weeks ahead, but withtropical forcing, they may be predictableas far as seven weeks ahead. This differ-ence is highlighted in the lower panel ofFig. 3.2. This suggests that accurate pre-diction of tropical diabatic heating, ratherthan of tropical sea surface temperaturesper se, is key to enhancing extratropicalpredictability on these time scales.

As mentioned earlier, most currentGCMs have difficulty in representing andpredicting heating variations on thesescales. This is especially true of the



NCEP MRF model. We have docu-mented significant deficiencies in the“reanalysis version” of that model inmaintaining and propagating MJO-related heating and circulation anoma-lies. Figure 3.3 shows that forecasts ini-tialized when the MJO is active over theIndian ocean are unable to represent thesubsequent eastward propagation of 850-mb zonal wind anomalies; indeed theydo not predict propagation at all but a

rapid decay. This has been demonstratedto have a negative impact on extratropi-cal forecasts.

Figure 3.4 shows that the LIM's forecastskill over the PNA region is comparableto that of the operational MRF ensemblemean, especially in summer. The MRFcan represent some phenomena that theLIM cannot, such as nonlinear barocliniccyclogenesis and blocking. To the extent

(a) LIM Week 3 Forecast SkillDJF 1985/86 - 1988/89

(b) GCM Week 3 Forecast SkillDJF 1985/86 - 1988/89

(c) Potential Predictabilitywith Tropical Heating

(d) Potential Predictabilitywithout Tropical Heating

Predictability of Weekly Averages during Winter

-0.3 -0.15 0 0.15 0.3 0.45 0.6 0.75

7 days 14 days 21 days 28 days 35 days 42 days

Fig. 3.2: Forecast skill and predictability of weekly averages during winter. Top: Correlation of observed andWeek 3 forecasts of upper tropospheric streamfunction anomalies averaged over 52 forecast cases in the wintersof 1985/86–1988/89 for (a) LIM and (b) the NCEP MRF. Bottom: Potential predictability limit: forecast lead atwhich skill (i.e., the correlation of observed and predicted anomalies) drops below 0.5. (c) Determined from thefull LIM. (d) Determined from a version of the LIM in which the effects of tropical forcing are removed.



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that these phenomena are predictable, theMRF should have an advantage. This isindeed the case in Week 1. By Week 2,these phenomena become unpredictable;even so, their role in exciting larger scale,slowly evolving structures such as thePNA pattern in Week 1 can contribute tomaintaining forecast skill in Week 2. Onthe other hand, the LIM is much better atpredicting subseasonal variations of trop-ical convection than the MRF, and beingan anomaly model, also does not sufferfrom climate drift by construction.Therefore, it seems likely that the com-parable skill of the LIM and the opera-tional MRF models is not arising entirelyfrom the same sources. This is in contrastto the seasonal prediction problem dis-cussed in Chapter 2, in which the compa-rable skill of GCMs and simple statisticalmodels arises from essentially the samesource. To the extent that the sources of

Week 2 forecast skill in the statistical anddynamical models are distinct, combin-ing the two forecasts should, in principle,yield forecasts that are superior to eitherin isolation. Constructing such a combi-nation is currently one of our main prior-ities.

3.2 Subseasonal variations in tropicalconvection and predictability of Cali-fornia rainfall.

Figure 3.2 demonstrates that much of theLIM skill in the extratropics arises fromits ability to predict tropical heating vari-ations. Operational models are notori-ously poor at this. This has implicationsfor predictions of extratropical rainfall:

Fig. 3.4: Week two forecast skill (as measured by pat-tern anomaly correlation over the PNA regions) for theoperational NCEP MRF ensemble mean (blue curve)and the LIM (red curve) for four winter and summerseasons.

Fig. 3.3 Composite anomalies of 850 mb zonalwind averaged between 5°N–15°S relative to themaximum of the first EOF of subseasonal tropicalOLR anomalies, when MJO activity is maximumover the east Indian ocean. The upper panel is forobserved anomalies from Days –14 to +14, whereDay 0 refers to the time of maximum EOF coeffi-cient. The lower panel is for the anomalies pre-dicted by the NCEP MRF model, with the meanmodel error removed.



for example, rainfall along the west coastof North America is known to be influ-enced by subseasonal tropical heating.This suggests that operational precipita-tion forecasts over North America couldbe improved, particularly in Week 2–3range, by using statistical methods toaugment the numerical product. CDCscientists have obtained a conservativelower bound on the potential improve-ment through a statistical predictionmodel of weekly precipitation over west-ern North America in winter.

The model is based on Canonical Corre-lation Analysis (CCA), with tropicalOutgoing Longwave Radiation (OLR)anomalies as the predictor and NCEPReanalysis precipitation over the easternPacific and western North America as thepredictand. A single CCA modeaccounts for most of the predictable sig-nal. The rank correlation of this modeand observed rainfall anomalies overSouthern California over a 25-winterperiod is 0.2 for a two-week lag, which iscomparable to the correlation between aweekly ENSO index and weekly rainfallin this region. Figure 3.5 shows that thiscorresponds to a 50% increase above theclimatological risk (33%) of above-nor-mal rainfall in California when the pro-jection of tropical OLR on the leadingCCA mode two weeks earlier is high, i.e.in the upper quintile of its distribution.

The leading CCA mode represents sup-pressed convection over the equatorialIndian Ocean and enhanced convectioneast of the dateline (Fig 3.6). Associatedwith this canonical tropical OLR anom-aly pattern is the development of uppertropospheric westerly wind anomalies

near 30°N in the eastern Pacific (notshown). Synoptic-scale weather systemsare steered farther east toward Californiaby these enhanced westerlies.

An analysis of four years of operationalWeek 2 ensemble forecasts indicates thatthe skill of this statistical model is com-parable to that of the operational ensem-ble mean, just like the LIM forecastsdiscussed earlier (see Fig. 3.4). Since byWeek 2 the operational forecast modelhas lost its ability to represent subsea-sonal tropical heating variability, the sta-tistical model provides essentiallyindependent guidance to the forecaster.The fact that the skill of the two modelsis comparable suggests a significantpotential for improvement of the opera-tional Week 2 precipitation forecasts. Weare investigating ways of optimally com-bining the numerical and statistical fore-casts. This requires estimating thecovariances of the ensemble mean fore-

Fig. 3.5: Probability that rainfall will be in the uppertercile of its distribution when the projection of thetropical OLR on the leading CCA mode two weeksprior is in the upper quintile of its distribution.



cast errors and observed tropical heatingvariability. Since the tropical convectiveevents associated with predictability onthis scale occur only about once or twicea season, a long (20+ year) record ofnumerical forecasts with a frozen modelis needed to estimate the required fore-cast error statistics. Work is underway atCDC to create such a retrospective fore-cast database.

3.3 The role of ENSO-related tropicalheating on operational weather fore-casts

The crucial role of tropical heating in theevolution of at least some extratropicalweather events is evident in a study ofthe effect of the 1997–98 El Niño on theoperational 1–14 day ensemble forecasts.This study was motivated by the hypo-thetical question: What would happen tothe medium-range (up to 14-day) fore-cast if the anomalous tropical SSTs werereplaced by climatological values? Thedifference of such forecasts from thosemade with the actual SSTs— “the ENSOsignal”—could then be used to diagnosethe influence of El Niño or La Niña onevolving midlatitude storm systems.Because of the uncertainty inherent inweather forecasting, we used the opera-

tional MRF ensemble to look at the aver-age of many forecasts of a given storm.The ensemble also provided us with arigorous way of assessing the statisticalsignificance of our results. It is perhapsworth mentioning that the ensemble withclimatological SSTs was run in-house atCDC in real time. Results from the com-parison with NCEP's operational ensem-ble with actual SSTs were madeavailable to public on the web, also inreal time.

This study provided the first demonstra-tion of a direct impact of El Niño SSTanomalies on individual extratropicalweather systems. Perhaps the most inter-esting case was the devastating ice stormthat hit Canada in early January 1998.Figure 3.7 shows the predicted 500 mbensemble-mean height anomaly patternswith and without the El Niño SST forc-ing, as well as the observed verification.The forecast was a lot closer to theobserved with the El Niño forcingincluded, showing that it played animportant role in the evolution of thisstorm. The area of unusually warm mid-level air associated with the productionof freezing rain is indicated by the redarrow. The operational runs (with ElNiño SSTs) show a wavetrain aligned

Fig. 3.6: OLR regressed on leading canonical predictor vector, scaled for one standard deviation of the canonicalpredictor variable. Contour interval is 1.5 W m-2. Positive contours are thicker, the zero line is omitted and shad-ing indicates statistical significance at the 95% level.



along the Atlantic Coast of the UnitedStates that is nearly absent in the runswithout the El Niño forcing. Thiswavetrain appears to have been a productof convective forcing in the eastern tropi-cal Pacific interacting with a deep mid-latitude/subtropical trough over Mexico:a previously unnoticed mechanism for ElNiño teleconnections.

The much-anticipated “El Niño rains” inCalifornia provide another example. Cal-

ifornia rainfall is episodic, even duringEl Niño years. Rain also falls duringnon-El Niño years, making attribution ofan individual storm to El Niño nearlyimpossible using historical data alone.However, our use of a dynamical modelallowed us to make the attributiondirectly. In Fig. 3.8 the runs with andwithout the El Niño forcing differ sub-stantially. The “El Niño effect” is a cleareastward extension of the rainfall intoCalifornia.

We conducted a similar study for the fol-lowing winter, during which there was asubstantial La Niña event. However, theresults were less conclusive, partlybecause of model changes and partly dueto the weaker SST forcing. Nonetheless,

OBSERVED

WITHOUT EL NIÑO (6-10 DAY FCST)

WITH EL NIÑO (6-10 DAY FCST)

Fig. 3.7: Ensemble mean 6–10d average 500 hPaheight anomalies for the ensemble with observed trop-ical SSTs (upper panel) and the ensemble with clima-tological tropical SSTs (middle panel) for forecastsverifying the first week of January 1997. The lowerpanel shows the verifying analysis, and the red arrowsindicate where unusually warm air at mid-levels con-tributed to the development of freezing rain at the sur-face.

Precipitation during the first week

OBSERVEDEL NIÑO EFFECT

WITHOUT EL NIÑO

in February: Week 2 forecastsWITH EL NIÑO

Fig. 3.8: Precipitation accumulated over the secondweek of the forecast for the ensemble with observedtropical SSTs (upper left), climatological SSTs (upperright), and the difference between the two (lower left)for forecasts verifying the first week of February 1998.



over the course of these two winters thisexperiment provided evidence of a largeinfluence of tropical convection onmedium and extended-range forecasts inmidlatitudes, especially in Week 2. Fig-ure 3.9 shows that the tropical influenceon forecast skill became significant afterWeek 1 in this experiment, and wasresponsible for almost all of the skill byday 14. This study also underscored thegreat difficulty, but also the greatrewards, of using an operational forecastmodel in research mode. As an addedbenefit, we were able to assist NCEP dur-ing their fire-related computer outage byrunning the operational ensemble fore-casts at CDC in real time.

3.4 The relationship between spreadand skill in the operational NCEPensemble forecasts

The studies discussed in sections 3.1 and3.2 show that statistical models can haveskill comparable to NWP models in theWeek 2 to Week 3 range because subsea-sonal variations of tropical convection,which the NWP models do not simulatewell, provide significant predictive infor-mation. However, the NWP models,unlike the statistical models, can provideinformation on day-to-day variations ofboth the signal (the amplitude of the pre-dictable component of the forecast) andnoise (the amplitude of the unpredictablecomponent of the forecast). The statisti-cal models assume the noise to be sta-tionary, i.e. to not vary from forecast toforecast. An NWP ensemble can be usedto estimate the noise in each forecastcase, and hence a case-dependent esti-mate of the RMS error of the ensemble-mean forecast.

The simplest measure of forecast noise isthe width, or spread, of the forecast prob-ability distribution for any quantity ofinterest. CDC scientists have investigatedthe relationship between spread and skillin the operational NCEP forecast ensem-bles using an archive of operational fore-casts maintained at CDC since 1995.Simple statistical considerations showthat such a measure is most useful whenthe case-to-case variability of the ensem-ble spread is large. This was shown to betrue in two winters of operational ensem-ble predictions. However, the short datarecord precluded a detailed analysis of

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Fig. 3.9: Week 2 500 hPa height anomaly correlationskill over the PNA region for the ensemble mean with(blue) and without (red) El Niño SST anomalies in thetropics. The black curve shows the RMS 500 hPaheight El Niño Week 2 forecast “signal”, defined as thedifference in the ensemble mean forecasts with andwithout tropical SST anomalies.



the dynamical mechanisms of the spreadvariability.

To get around this limitation, a five-levellinear quasi-geostrophic (QG) model,linearized about three-day segments ofthe observed flow for 21 years, was usedto model the spread variability. The fun-damental assumption was that day-to-day variations of spread are due prima-rily to day-to-day variations in thegrowth rate of small perturbations duringthe forecast period, and that day-to-dayvariations in the initial error, i.e. in thespread of the analysis-error distribution,are either unimportant or not well sam-

pled. The five-level model was able toreproduce the main results (not shown)of the shorter 2-winter study mentionedabove. When run for 21 years, the QGmodel showed the largest spread vari-ability of 3-day forecasts over the easternPacific and eastern Atlantic oceans,which was associated with modulationsof the local jets by the PNA and NAOmodes of low-frequency variability (Fig.3.10). To the extent that such modula-tions are predictable, the results from thisstudy suggest that skill should also bemost predicable in these regions.

Fig. 3.10: Upper panels: 21 winter mean 300 hPa streamfunction spread (S) and standard deviation of lnS (=β),estimated from 3-day integrations of the five-level linear QG model. S is normalized by the mean amplitude of theinitial perturbations used in the ensemble integrations. Contour interval for β is 0.01, with values greater than0.28 shaded. Contour interval for normalized S is 0.25, with values greater than 4 shaded. Lower Panels: Map ofcorrelations between time series of lnS at points indicated by the black rectangles and three-day averaged 300 hPastreamfunction. Contour interval is 0.1, negative values are dashed, and the zero line is thick solid.



3.5 Experimental week 2 forecasts ofextreme events using the operationalNCEP ensemble

The existence of a significant spread-skillrelationship (at relatively short forecastranges) means that changes of both themean and width of the forecast probabil-ity distribution from their climatologicalvalues can be used to estimate the proba-bility that the verification will lie in thetails of the climatological probability dis-tribution (see Fig. 2.6). The statisticalmodels discussed earlier assume that thespread is constant, and that only shifts ofthe mean are important in altering theprobability that the verification will be an“extreme event”.

Unfortunately, this advantage of ensem-ble forecasts, which is modest but signif-icant in Week 1, is lost by the middle ofWeek 2. The main reason is that by Week2 the forecast ensemble spread nearlysaturates to its climatological meanvalue, so that there are no significantspread variations from case to case. Inother words, most of the predictable vari-ation of forecast skill in Week 2 is associ-ated with predictable variations of thesignal, not of noise. For several yearsCDC has exploited this fact in producingan experimental real-time Week 2 fore-cast product based on the NCEP ensem-ble (http://www.cdc.noaa.gov/~jsw/week2/). Tercile probability forecasts of500 mb height, 850 mb temperature, 250mb zonal wind, sea-level pressure andprecipitation are provided. Only the sig-nal, not noise, is used to construct theseprobability forecasts. The procedureinvolves converting maps of the pre-

dicted standardized anomalies into mapsof extreme quantile (in this case, tercile)probabilities. This calibration is doneempirically, using the available historicalrecord of ensemble forecasts and verify-ing analyses. The procedure is as fol-lows: 1) for a positive standardizedforecast anomaly α, all instances inwhich a forecast exceeded this value inthe data record are found, and the proba-bility β that the verifying analysis fell inthe upper tercile of the climatologicaldistribution is computed, 2) the standard-ized anomaly contour α is relabeled as aprobability of above-normal equal to β. Ifα is negative, the probability that the ver-ifying analysis fell into the lower tercileis computed, and the contour is relabeled“probability of below-normal”. If themodel has systematic errors, these proba-bilities need not be symmetric, i.e. theprobability of below-normal for a nega-tive α need not be the same as the proba-bility of above-normal for a positive α.Our calibration thus provides one simpleway of accounting for model error inprobabilistic predictions.

Figure 3.11 shows an example of such aprobability forecast. Note that the inter-pretation of this map is slightly differentfrom that for a conventional probabilityforecast. If all the points on the mapinside the yellow contour (as opposed tothose inside the yellow band) are countedover a large sample of forecasts, 50–60%of these points will verify in the uppertercile of the climatological distribution.Similarly, for points falling in the darkestred regions on the map, over 90% willverify in the upper tercile. The conven-tional interpretation would be that points



in the yellow band would have a 50–60%chance of verifying in the upper tercile.Such a calibration would require a lotmore forecasts to compute reliably, sincethere are far fewer points inside the yel-low band than there are inside the yellowcontour.

Since we assume that the signal, not thenoise, contains all of the useful predic-tive information, the useful subspace ofthe ensemble can be isolated through an

EOF analysis of the correlation matrix ofthe ensemble-mean predictions. (Theidea here is similar to that in Fig 2.8).The right panels of Fig. 3.12 show thethree leading EOFs thus obtained. Forcomparison, the three leading EOFs ofthe correlation matrix of observed 7-dayaverages is also shown, in the left panels.There are two notable aspects to Fig.3.12: 1) the signal and observed EOFpatterns are similar, and 2) the three lead-ing EOFs explain considerably morevariance of the ensemble-mean forecaststhan they do of the observed variability(36% vs. 22%). To understand this better,note that the total forecast covariance canbe decomposed into a part due to the pre-dictable signal (Csignal) and a part due tounpredictable noise (Cnoise). If the fore-cast model is unbiased and the noise isuncorrelated with the signal, theobserved variance (Cobs) is approxi-mately the sum of the two. This relation-ship is exact for the LIM discussed insection 3.1. The fact that the signal varia-tion occurs in a lower dimensional sub-space than the observational 7-dayaverages then simply means that the vari-ance contained in the noise is non-trivial.The similarity of the observed and signalEOF patterns has a subtler interpretation:it implies that the noise component of thecovariance is nearly white, and that theensemble-mean does indeed capturemost of the extractable signal with coher-ent spatial structure.

The product shown in Fig. 3.11 has beenquite popular with operational forecast-ers. A similar method has been adoptedin operations by NCEP/CPC. A detailedanalysis of the performance of this

Fig. 3.11: Example of an experimental week 2 forecastverifying the last week of October 1998.



scheme, and its implications for Week 2predictability, is underway.

3.6 Unifying ensemble forecasting anddata assimilation.

The fundamental goal in subseasonalprediction, just as in seasonal prediction,

is to predict the forecast probability dis-tribution function (PDF) accurately. Inthe previous sections, we have discussedresearch efforts at CDC toward this goal.It is hoped that statistical methods likethe LIM or CCA, when combined withan NWP ensemble, will improve themean of the forecast PDF. The spread-

Observed 7-d averages 59-94 Week2 Forecasts: 3 winters

500 mb DJF Rotated EOF analysis (correlation matrix)

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8.1%Fig. 3.12: Rotated EOFs of weekly average 500 mb height computed using the correlation matrix for DJF 1958-1994 (left panels) and the correlation matrix of week 2 operational ensemble mean forecasts for DJF 1995/96 to1997/98.



skill relationship discussed in section 3.4shows that useful information can beextracted at short forecast ranges fromthe second moment of the NWP ensem-ble. One obvious way to improve theaccuracy of the forecast PDF is toimprove the accuracy of the initial PDF.Currently, all operational centers con-struct an ensemble of initial conditionsby perturbing a single control analysis,obtained from a three-dimensional (as atNCEP) or a simplified four-dimensional(as at ECMWF) data assimilation sys-tem. The methods used to generate theperturbations to the control analysis,breeding vectors at NCEP and singularvectors at ECMWF, are fundamentallyad-hoc and not representative of analysisuncertainty. CDC scientists have beeninvestigating new ways of coupling theensemble forecast and data assimilationsteps, in order to improve both the initialand forecast PDFs.

The coupling of ensemble forecastingand data assimilation is natural. Theessence of data assimilation is statistical,in that it amounts to blending “firstguess” forecasts with new observationsusing weights determined by theirrespective error statistics. Carefully con-structed forecast ensembles can providesuch statistics. Currently, operationalmethods make rather simplistic assump-tions about the error statistics, assuming,for example, that the correlation of fore-cast errors at two locations depends onlyon the distance between them and not onthe location or whether the atmospherehas recently been quiescent or stormy(Fig 3.13a). Results from simple modelexperiments using sophisticated “Ensem-ble Kalman Filter” techniques suggest

that the quality of initial conditions canbe dramatically improved by using fore-cast error statistics estimated from a spe-cially constructed ensemble. Forexample, error statistics from the ensem-ble permits a single observation at a fixedlocation to make very different correc-tions to the first-guess depending on the

Fig. 3.13: Examination of the structure of the “analy-sis increment” (the initial condition minus the prior“first guess” forecast) for the traditional method ofdoing data assimilation, where error statistics do notchange from location to location or day-to-day. In thisexperiment, an observation that is 1 K warmer than theprior forecast is found at the location denoted by thedot. (a) Analysis increments using the “3D-Var” dataassimilation methodology. The “one size fits all”increments are a simple decreasing function ofincreasing distance from the observation location. (b)Analysis increments using the new ensemble dataassimilation methodology. Changes to the prior fore-cast are now stretched out along the frontal zone, sothat the entire position of the warm front is changed bythe one observation.



flow of the day (Fig 3.13b). By estimat-ing the analysis increment to the firstguess in this flow-dependent manner,ensembles of initial conditions can bedramatically improved, perhaps even tothe point that they are more accurate thananalyses based on four-dimensional vari-ational methods.

Recent CDC efforts in this area havefocused on algorithmic details of ensem-ble-based data assimilation experiments.We have sought to understand how thestatistics of forecast errors estimatedfrom an ensemble depend on the size ofthe ensemble, and how one might extractuseful information from smaller ensem-bles—an important issue, since largerensembles make heavier demands oncomputational resources. This researchhas demonstrated that with an accuratespecification of forecast error statistics,new problems can be tackled in a theoret-ically justifiable manner, including prob-lems such as determining wheresupplementary observations would bemost beneficial for reducing analysis orforecast error (the problem of “targeting”observations). In addition, since ensem-ble-based data assimilation techniquesare particularly useful when observationsare sparse, CDC scientists are planningto adapt such techniques to extend theNCEP reanalysis back into the pre-radio-sonde era (pre-1948).

EPILOGUE

The problem of how to make useful fore-casts at lead times between a week and amonth is a challenging and oftenneglected one. Forecast information onthese time scales is in great demand fromusers. This is an area that NOAA has tra-ditionally not focused on in the past.CDC researchers have been addressingthe problem on two fronts; 1) by trying toextract the maximum information fromensemble NWP model forecasts, and 2)by investigating statistical forecast meth-ods that complement the NWP ensem-bles by exploiting predictable signals notwell represented in current models. Ourresearch thus far suggests that the NWPand statistical approaches are comple-mentary, and provide information that isindependent to some degree. The chal-lenge is to combine the two in an optimalmanner, yielding forecasts that are supe-rior to either individually. Due to thelow-frequency nature of the phenomenaat these forecast ranges, determining theoptimal combination would require gen-erating a long (20+ year) dataset ofensemble forecasts with a fixed model toestimate the forecast error statistics withthe necessary accuracy. Work is currentlyunderway at CDC to create such adataset, which will also be useful in sev-eral other applications not discussedhere.

Contributed by: J. Barsugli, T. Hamill,H. Hendon, B. Liebmann, M. Newman, P.Sardeshmukh, and J. Whitaker.

understanding and predicting subseasonal variations · understanding and predicting subseasonal...

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