FEBRUARY 1992 S A H A 345Response of the NMC MRF Model to Systematic-Error Correction within Integration SURANJANA SAHADevelopment Division, National Meteorological Center, Washington, D.C.(Manuscript received 20 March 1991, in final form 17 July 1991) ABSTRACT We describe an extensive nudging (within-integration correction) experiment with a large and sophisticatedatmospheric model. The model is an R30 version of the National Meteorological Center (NMC) TS0 operationalglobal medium-range forecast model. The purpose is to combat the systematic-error growth right from the startof the integration process by adding artificial sources and sinks (the corrections) of heat, momentum, and mass.The corrections derived from 30 antecedent 24-h integrations (by subtracting the forecasts from their verifyinginitial conditions) are applied to 30 subsequent independent 5-day forecasts from 1 July 1988 to 30 July 1988.Verification statistics over these 30 5-day forecasts are computed for the control cases, the nudged cases, andfor forecasts corrected after the fact. The main results show that the nudging process, when carefully designed, does not lead to any technicalproblems and the model accepts the applied corrections quite faithfully. Both nudging and after-the-fact correctedforecasts have greatly reduced systematic errors. In terms of forecast accuracy, nudging is, on the whole, notbetter than after-the-fact correction. However, for forecast lead times beyond 10 days, where after-the-factcorrections are currently not possible, nudging is an attractive alternative. The physical processes most affectedby the nudging process is precipitation. In the nudged model atmosphere without the traditional "cold bias,"both large-scale and convective precipitation is reduced detrimentally relative to the control runs, possibly dueto tuning of the model.1. Introduction Operational forecasting centers, such as the NationalMeteorological Center (NMC) in the United States andthe European Centre for Medium Range WeatherForecasts (ECMWF) in Europe, have been making 110-day integrations daily with their comprehensivenumerical weather prediction (NWP) models for thepast decade. Many improvements have been made inmedium-range forecasting over the last decade or so,enabling these and other centers to carry out 10-dayintegrations with some modicum of success. However,in spite of these improvements, present state-of-theart medium-range forecast (MRF) models have largeerrors in their forecasts, some of which are systematic.Three methods may be considered to correct these systematic errors. The first, obviously, would be to continue improvingthe initial conditions and the physics and numerics inthe model through better parameterization of clouds,radiation, boundary-layer processes, representation ofthe stratosphere, etc. Optimal resolution in the model,especially in the vertical, may also help to reduce thesystematic error. Many of the resources in the field ofNWP are already devoted to these efforts. Experiencetells us that almost every change made to the model Corresponding author address.' Dr. Suranjana Sara, DevelopmentDivision, National Meteorological Center, Room 202, WWB, 5200Auth Rd., Washington, D.C. 20233.leads to a change (often, but not always, an improvement) in the systematic error. The second approach is to minimize the errors bythe use of statistical postprocessing methods. Theknown systematic error that has occurred in a set ofthe most recent verified forecasts can be subtractedfrom the latest forecast as a postprocessing tool beforerelease to the users. The correction thus applied is afunction of forecast lead time, with small correctionearly on, and more substantial correction at I0 days.This approach has something in common with a practice called Model Output Statistics (e.g., Klein andGlahn 1974; Lorenz 1977), which is used to generatethe best possible local weather forecast from a modelforecast using regression techniques. The postprocessorapproach has been shown to be a success (Sara andAlpert 1988) and is now used operationally by NMC(Alpert and Saha 1989) to correct the 1-10-day forecasts. Other relevant papers describing the nature ofsystematic errors are White (1988), Arpe and Klinker(1986), and Thiebaux et al. (1988). The third approach, the primary focus of this paper,is to repeatedly correct the model during integration,a process generally called nudging. The purpose is tocombat the growth of systematic error right from thestart of integration, in the hope that by creating a betterlarge-scale environment in the model by nudging, therewill also be a reduction in the growth of the randomerror. Correction of systematic errors within integrationhas been carded out in simple models of the atmoc 1992 American Meteorological Society346 MONTHLY WEATHER REVIEW VOLUME 17.0sphere by Johansson and Saha (1989), Schemm andFaller(1986), and Johansson and Baer (1987), andin more complex models by Saha and Kanamitsu(1988) and Sausen and Ponater (1989a,b). Sausen andPonater (1989b) have shown that in addition to reducing the initial drift in the model, nudging also enlarges the internal variability of the model and enhancesthe bandpass-fiRered transient meridional transport ofzonal momentum. An important potential advantageof nudging over postprocessing is that it may reducethe random error. Johansson and Saha (1989) haveshown that systematic errors in barotropic forecastscan largely be removed and, more importantly, thattheir removal reduces the growth of random errors inthe barotropic model. However, it remains to be seenwhether this is still true for a large complex state-ofthe-art general circulation model. In the nudging approach, one would ideally like toknow the systematic error in the very first time step ofthe model before interactions and nonlinearity comeinto play. This systematic error can then, with an opposite sign, be applied to all the prognostic equationsof the model as an extra tendency term. The first timestep approach has some unsurmountable practicalproblems, however. The first is the daily cycle; the second is the spinup of the model. Therefore, we estimatethe nudging correction from the systematic error overthe first 24 h in a set of 30 1-day integrations fromconsecutive initial states, details of which will be presented later. This mean 24~h error was then dividedby the number of seconds in a day, and then added(with opposite sign) as a constant nudging term to themodel tendencies at each time step until the end ofindependent integrations out to 5 days. The nudgedforecasts were then verified over a 30-day verificationperiod. Comparisons will be made between control,nudged, and after-the-fact corrected forecasts. The experiments outlined above were applied to anR30 version (rhomboidal truncation at 30 waves inthe zonal direction) of the operational MRF model inuse at NMC in 1988. This provides an experimentalsetting very close to operational forecast models, whichare far more complicated than those used in Johanssonand Saha (1989) and Schemm and Failer (1986).There are apparently many unpublished efforts tonudge complicated models. Many of these attemptsfailed for reasons not fully understood. The obviousreason was technical, in the sense that the models became unstable and eventually blew up. While mostmodels accepted corrections in the temperature field(Tsuyuki and Kanamitsu 1987), the technical problems became worse by correcting momentum, moisture, and surface pressure fields. In this sense, our attempt can be termed a success, because by carefullydesigning the procedure to correct the model duringits integration, there were absolutely no technical obstacles with any of the experiments. Our approach to nudging is only a very simple firstiteration in the process of eliminating the systematicerror altogether. To force a time-varying multivariateflow to produce a desired time-mean state is antremely complicated problem. Even if we have onlyone variable (streamfunction, barotropic model), tl~teproblem has not been resolved of how one forces thetime mean of the model to be the same as climatology(see Fig. 2 in Johansson and Saha 1979, for instance).In the framework of stationary flow in a shallow-waterequation model (i.e., three variables), Murphree(1989) has proposed an iteration scheme in which theerror in each variable is corrected separately and successively in a number of cycles, thus allowing for theirproper interdependence until convergence is obtained.This method, though elegant, is prohibitively expensivein computer resources if one uses large complex generalcirculation models. Moreover, it may not converge fortime-varying flow. We realize from the outset that ideally corrections made to one variable should have animpact on the correction applied to other variables.This aspect, however, is far from perfect in our currentapproach to nudging. Our nudging approach to reduce the systematic errorof a model should be distinguished from nudging techniques used in data assimilation (Haltiner and Williams1980; Hoke and Anthes 1976; Davies and Turner1977). In data assimilation, a term is added to theequations representing a relaxation of the model variable toward the observation to be assimilated, the timeconstant of the relaxation representing our faith in thereliability of the observation relative to the model'sguess. In the approach followed here, a term is addedto the equations that is held constant throughout theintegration and does not depend either on the modelvariables or on the state toward which we want thesystem to be nudged. One has to remember that theremoval of systematic error is not a goal in itself. Strongrelaxation toward the climate would achieve that goal,but would also destroy the time variation we are tryingto predict. In section 2 we describe the model and data used aswell as the forecast experiments. Section 3 deals withthe verification statistics used in the comparison of therelative skill of control, nudging, and after-the-factcorrected forecasts. The impact of nudging on variousphysical fields, in particular the rainfall and divergentcirculation patterns, will be presented in section 4. Thepurpose of this section is also to exploit nudging as anindirect diagnostic tool to find possible sources/causesof systematic errors in medium-range forecast models.Conclusions and discussions are given in section 5.2. Model, data, and experimental procedure An R30 version of the T80 MRF model ( 1988 version) used operationally at NMC was utilized in allthe experiments. A sophisticated diagnostic version ofthe model has been developed by M. Kanamitsu and1992 S A H A 347R. Kistler at NMC, which greatly facilitated thelaunching of the large series of experiments involvedin this research, as well as in the graphics necessary forthe interpretation of results. The model equations are integrated on sigma surfaces with 18 levels in the vertical. The prognostic variables are vorticity, divergence, virtual temperature,specific humidity, and surface pressure. The time stepused was 16 min and all integrations were performedon a CYBER 205 supercomputer. The following is abrief description of some of the physics used in themodel: - short- and longwave radiation with simple diurnalvariation; - zonally averaged climatological clouds; - large-scale condensation; - Kuo convection; - shallow convection; - stability-dependent vertical diffusion; - a Laplacian-type (del-4) horizontal diffusion; - similarity theory surface layer; - gravity-wave drag; - predicted soil moisture; - a fixed observed sea surface temperature; - climatological albedo, sea ice, roughness, etc.; - silhouette orography. Sixty consecutive 5-day control integrations frominitial states 1 June 1988 through 30 July 1988 at 0000UTC were first carried out. Correction of the systematicerror present in forecasts originating from the secondhalf of the above period ( 1 July 1988-30 July 1988)was then done in the following two ways.a. After-the-fact corrections The true systematic error is not known to any greataccuracy, and is thus replaced by the time-mean errorover the N most recent forecasts, where N = 30 is ourso-calle~l "training period." In doing so, some flow dependence of the mean error may in fact be accountedfor by our corrections. Figure 1 illustrates how such amoving training period was used for correction (Epstein1985). As an example, consider the forecast heightfields (for forecast lead times of 1-5 days) over theverification period from I July 1988 to 30 July 1988.The 5-day forecast originating from 5 July (full inclinedline) and verifying on 10 July (point A ) was correctedby subtracting the time-mean error calculated over theprevious 30 5-day forecasts (verifying from 6 June to5 Julymfull lines) from it. (It must be noted here thatthe error statistics over the period 6-10 July cannot begenerated in an operational setting, as the verifyinganalyses would be unavailable.) Similarly, the 5-dayforecast originating from 6 July (dashed line) and verifying on 11 July (point B) was corrected using thetime-mean error from the 30 5-day forecasts verifyingon 7 June to 6 July (dashed lines), and so on. Notethat the after-the-fact correction is different for eachlead time. For short lead time the correction behavessomewhat like a function linear in the forecast leadtime. Several different methods were tested before selectingthe final choice of correcting only waves within a T6truncation. This was done because the systematic errorusually manifests itself in the long waves of the atmosphere, especially in the mass field (White 1988;Saha and Alpert 1989).b. Within-integration correction (nudging) As discussed earlier, 24-h errors derived from antecedent forecasts (and verification) were used to correct the model during integration. Using the same verification period outlined in section 2a, 30 nudged integrations were performed from these initial states ( 1July 1988-30 July 1988). The nudging terms were calculated in the following manner. Using the same ex I JuneI NIT. 1988COND.DAY 1DAY 2DAY 3DAY 4DAY 5 -REAL TIME6June 7June 30June l July 5July 6July 7July 8July 9July 10July 1t Julyerror statistics for nudgingtO COrreCt A 30 days of error statistics \I to co~ct A A BI II 30 days of error statistics~.~ 4/ ..... ~_/._~! t~'correct ~'F1G. 1. The correction method used as explained in section 2.30 July 31 July lAug 2Aug 3Aug 4Aug348 MONTHLY WEATHER REVIEW VOLUME 120ample from section 2a and referring again to Fig. 1, tocorrect the forecast originating from 5 July 1988, weestimated from the control runs the mean 24-h errorover the antecedent 30-day period (verifying from 6June 1988 to 5 July 1988--1ine with asterisks) for themodel variables to be corrected (virtual temperature,divergence, vorticity, and surface pressure), both inthe horizontal (961 complex coefficients correspondingto a R30 truncation) and in the vertical ( 18 sigma layers). Upon dividing the resulting 24-h error (E24) bythe total number of seconds in a day, we obtained thecorrection terms as constant tendencies that were added(with opposite sign) to the model equations at eachtime step until the end of the 5-day integration. In anoninteractive model the difference between controland nudged run would equal one (two) times E24 at24 (48) h, etc. Many test experiments were initially made to determine exactly how we were going to implement the correction so as to get the best results. Trial 5-day runswere made in which we experimented with: (i) the number of spectral modes to be corrected,that is, the first six meridional modes for zonal wavenumber 0, triangular six waves, all R30 waves, and soon; (ii) combinations of one or more of the followingmodel variables to be corrected (virtual temperature,vorticity, divergence, and surface pressure); (iii) nudging only the first six meridional modes forzonal wavenumber 0 in the temperature field towardtheir initial state values (using the nudging-relaxationapproach (Haltiner and Martin 1980); (iv) initialization of the 30-day mean initial statesand 30-day mean 24-h forecasts before obtaining thedifference error correction field; (v) results of forecasts using correction fields truncated in sigma coordinates were compared to thoseusing correction fields truncated in pressure coordinates, to see if any differences occurred near orography. The final choice for correcting model integrationswas to nudge all time-dependent variables of the modelexcept specific humidity. The correction fields in thesevariables (divergence, vorticity, virtual temperature,and surface pressure) were obtained by truncating insigma coordinates, the antecedent 30-day mean initialcondition field, and the 30-day mean 24-h controlforecast field, triangularly at zonal wavenumber 6 (T6).These two fields were then initialized separately beforetaking the difference. The correction so derived, a fieldvarying in all spatial dimensions, but not in time, wasthen applied at all 18 sigma layers of the model.3. Verification statistics We used the following verification skill scores forthe 30 l-5-day integrations, originating from 1 July1988 to 30 July 1988, for control, nudged, and afterthe-fact correction: (a) root-mean-square total error -r = Z (FN- V~) (b) mean error N~30 1 -~1 =76 2 (r~.-v~) N=I (c) root-mean-~uare random e~or ~n = [(gr)2 - (~)2] (d) anom~y co,elation N=30 N=I AC = ~30 ~=30 [ Z (F~-Cn)~ Z (V~--C~r)2] N=I N=Iwhere F~ is the forecast verifying at time N, V~ is thevefificmion initial condition m time N, and C~ is theclimatology obtained from a 7-yr climatology ( 19791985) prepared by the Climate Analysis CenWr, Washingon, D.C. The monthly climatology is inte~olatedfrom the two nearest calendar months to the date Nin question. [Implidt in the definitions (a)-(d) is asummation over space, either sp~tral or grid space.] The anomaly co,elations presented in this paperwere calculated according to the method used at NMC,which differs somewhat from the above definffion.. Thezon~ mean v~ues of (F~ - C~) and (V~ - C~.) ~eremoved from the computations of anomaly co,elations at NMC. Since the remov~ of systematic e~orshows up fi~t and foremost in the large males, thispractice rendem the NMC's definition of anomaly correlation rather insensitive to the removal of the systematic e~or. The above mores were computed for the folloMngv~ablm ofgeopotenfi~ heist field, temperature field,zonal wind field, mefidional ~nd field, and relativehumidity field. They were ~so computed as a functionoff (i) forecast lead time ( 1-5 days); (ii) ve~ical pre~ure level (at 12 standard pressurelevels at 1000, 850, 700, 500, 400, 300, 250, 200, 150,100, 70, and 50 hPa); (iii) zon~ wavenumber; (iv) total wavenumber; (v) geographical location. Due to the large amount of verification statisticsgenerated, we have selected results in ~aphical forepe~ining mainly to 500-hPa geopotential fields. Figures 2-5 summ~ze the verification scores (a)total root-mean-~uare e~or, (b) mean e~o[ (c) ranFEBRUARY 1992 S A H A 3491008O2O 0 0I 2 3 FORECAST TIME IN DAYSb-35I 2 3 ~ 5 E~RECRSI TIHE INC 11o loo 9O Oo- : 70~ ~o I0 0 2 3 ~1FORECAST TIME IN Dt~Y$1.0.9.7.B.5.3dI t I I 2 3 ~1 FORECRST TI~ IN DRYS F]o. 2. Verification scores for the Northern Hemisphere averaged over latitudes 20--80-N and temporally over 30 forecasts originatingfrom I July 1988 to 30 July 1988 for 500-hPa geopotential field (in geopotential meters), for a rhomboidal truncation of 30 waves in thezonal direction as computed in section 3 for (a) total root-mean-square error as a function of forecast lead time, (b) mean error as a functionof forecast lead time, (c) random root-mean-square error as a function of forecast lead time, and (d) anomaly correlation as a function offorecast lead time. Control runs are represented by curve A, nudged runs by curve B, and after-the-fact corrections by curve C,350 MONTHLY WEATHER REVIEW VOLUME I20dom root-mean-square error, and (d) anomaly correlation) for the 500-hPa geopotential .field computedfor the full ,spectrum of rhomboidal 30 waves (R30)as a function of 1-5-day forecasts originating from 1July 1988 through 30 July 1988 for control runs (curveA), nudged runs (curv,e B), and after-the-fact correctedforecasts (curve C). Figure 2 is for the Northern Hemisphere from 20- to 80-N, Fig. 3 is for the tropical beltfrom 20-N tO 20-S, Fig. 4 is for the Southern Hemisphere from 20- to 80-S, and Fig. 5 is for the globe asa whole. Figure 2~.~hows that nudging reduces 24.5% of thetotal error (in squared units) after 5 days of integrationwhile after-the-fact correction reduces the total errorby 21.6% in the Northern Hemisphere from 20- to80-N. There is a substantial decrease in the mean error(Fig.'2b) by both methods, from a negative bias of.n. early 26 m to a negative bias of only about 2 m after5 days of integration, with after-the-fact correctionsdoing a little better than n/~dging out to 4 days. Therandom error (Fig. 2cj shows a small reduction ofabo, ut 8:5% (in squared"uhits) for the nudging runsand about 5% for after-the-fact corrections after 5 daysofintegratioh. It shou!.d be pointed out here that, if theafter-the-fact correction were the ~ame every day, thenthe random 'error would reinain unchanged. The reduction of the random error-proves the utility of amoving training period relative to a static training period (Epstein 1985) t.o account for the flow dependenceof the Systematic error. The reduction of the randomd/'ror .in t. he. nudged run is modest evidence of the basicpremise of~he nudgihg ~tpproach: that the random errorwilt .decrease giver an environment with less systematicerror. The decrease in the total, mean, and randomerror stfirts figh.t from the beginning of integration andcontinues all the way to the end of the 5-day forecast.The anomaly correlation (Fig. 2d) does not show anyperceptible change until 3 days of integration, afterwhich a slight improvement can be seen for the nudgedruns. Figures 3~-d show the scores for the tropics from20-N to 20-S. In Fig. 3a we see that the total error isreduced drastically by about 78% (in squared units)'by nudging and by an even more impressive 87% byafter-the-fact correction after 5 days of integration.Figure 3b shows a near elimination of the negative biasof about 39 m by after-the-fact correction, and a moremodest reduction to a negative bias of only 11 m bynudging, The results of correction in the tropics isstriking because it is the only area where the meanerror is larger than the random error, The reductionin the random error is shown in Fig. 3c with after-thefact corrections doing a slightly better job than nudgingout to 4 days of integration, with reductions of 9.5%(in squared units) for nudging and 17.4% for after-thefact corrections after 5 days of forecast. The anomalycorrelation (Fig. 3d) shows an increase in predictionskill from 2.2 days at the 0.6 level for the comxol runs,to. near!y 2..6 for nudging and to 3.6 days (nearly a dayand a half is gained) for after-the-fact corrections. Turning' our attention now to the Southern Hemisphere from 20- to 80-S, Fig. 4a shows a small reduction in the total error, by about 9.5% (in squared units)by after-the-fact ~orrection and by 11.3% by nudging.The mean errbr (Fig. 4b) reduces from a negative biasof 30 m for.the control runs to a negative bias of 9.5m by.nudging and to a negative bias of only 2 m byafter-the-fact correction. Figure 4c shows a marginalincrease in the random error for both nudging and after'the--act corrections out to 4 days of integration,after which nudging reduces the random error slightlywhile after-the-fact correction keeps on increasing therandom error. The anomaly correlation (Fig. 4d) showsthe same results with nudging improving the correlation very slightly after 4 days of integration, while afterthe-fact corrections decrease the correlation after 5days. Summarizing the effects of correction for the globeas a whole, we see from Fig. 5a that the total errorreduces by about 22.6% (in squared units) by bothmethods after 5 days of integration, with after-the-.factdoing slightly better than nudging to 4 days. The rneanerror (Fig. 5b) shows a reduction from a large negativebias of nearly 31 m for the control runs to a greatlyreduced negative bias of 7.5 m by the nudging process,and a near elimination of the bias occurs with afterthe-fact correction. Figure 5c shows a very smal~ decrease in random error by both methods of correction,with nudging doing a slightly better job after 4 days ofintegration. The anomaly correlation (Fig. 5d) showsno differences with the control runs up to 3 days ofintegration, after which nudging increases and afterthe-fact correction decreases the correlations. Figure 6 shows verification statistics ofgeopotentialheight as a function of pressure. In this case the calculations were made in spherical harmonics and areaverages over the globe for all R30 waves. The statisticswere compiled for the 30 5-day control, nudged, andafter-the-fact corrected forecasts originating from 1 July1988 through 30 July 1988. We can see that the meanerror for the control runs (curve D) constitutes a largepercentage of the total error (curve A), especially above500 mb. Both nudging (curve B) and' after-the-factcorrections (curve C) reduce the total error of the control runs (curve A) at all levels with truly impressivereductions occurring in the stratosphere. This is alsoreflected in the greater reduction of the mean error athigher levels by nudging (curve E) and more so byafter-the-fact correction (curve F) when compared tothe mean error of the control runs (curve D). Figure 7 shows a spectral decomposition of verification scores with respect to the total wavenumber (n)for the 500-hPa geopotential field for all 30 5-day forecasts originating from I July 1988 to 30 July 1988.FEBRUARY 1992 S ^ H A 3511oo80 2O 0 0F~IRECRST TIHE IN DRYS-5-10-15-35-401oo~0802O1ooI 2 3 ~. FORECQST iI~ IN ORYS1.0.8.?.6.S.3 I I I I 0 I 2 3 4 F~RECRST TIME IN ORYSFIG. 3. Same as in Fig. 2, but for the tropics, averaged over latitudes 20-N-20-S.The mean error of the control runs (curve D) dominates the spectrum of the total error for the controlruns (curve A) for the longer waves (total wavenumber0-4), while the random error constitutes much of thetotal error for the smaller-scale waves (beyond totalwavenumber 4). The global-mean value (i.e., totalwavenumber 0) is the single largest mode present inboth the total and mean error fields. Both correctionmethods reduce the total error of the control runs(curve A), with after-the-fact correction (curve C)352 MONTHLY WEATHER REVIEW VOLUME 120 2 3 qFeRECRST TIME IN DRYSbI 2 3 ~J 5 FIaRECRST TIME IN ORYS C80 0 f I l 0 ! 2 3 FOREC, RST TIH[ IN ORYS d.6.S,~ I I I I I 2 3 q F~RECRST TIME IN ORYSFIG. 4. Same as in Fig. 2, but for the Southern Hemisphere, averaged over latitudes 20--80-S.doing better than nudging (curve B) to total wavenumber 3. However, beyond wavenumber 3, after-thefact correction proves to be detrimental and actuallyincreases the total error. The mean error of the controlruns (curve D) is reduced by nudging (curve E) all theway to wavcnumber 12 or so. After-the-fact correction(curve F) reduces the mean error much more for totalwavenumbers 0, I, and 2, beyond which it does notdo as well as nudging up to wavenumber 6. It shouldbe recalled that correction by both methods were ap~EBRUARY 1992SAHA~7o~ so~ so 3O 2O 10 0-5-10-L5-3C353 I I I I0 I 2 3 4 FfRECFIST TIHE IN D~IYS~to Ctoo;90 8o~ soI 2 3 q FORECRST TIHE IN DRYSdI 2 3 ~ F~RECRST TIHE IN DRYS~G. 5. Same as in Fig. 2, but for the globe as a whole.plied to a truncation of triangular 6 waves (T6) only.While there is obviously no difference for curves D andF beyond total wavenumber 6, curve E for nudgingfeatures a modest improvement over curve D as a resultof nonlinear interactions with scales n < 6 when corrections to the model are applied within the integrationprocess. Such indirect improvements, as well as a reduction of the random error, are primary reasons forconsidering nudging an alternative to postprocessing. Figure 8 shows the verification scores for the zonal354 MONTHLY WEATHER REVIEW VOLUME 120~ SO0 ?00" 900 1000 ' 0 20 ~[0 60 SO 1DO 120 le~0 160 180 200 ERRgR IG P HETERS FIG. 6. Total root-mean-square error (curves A, B, and C) andabsolute value of the mean error (curves D, E, and F) for 500-hPageopotential field (in geopotential meters) for 30 5-day control,nudged, and after-the-fact corrected forecasts, respectively (originatingfrom I July 1988 to 30 July 1988), as a function of vertical pressurelevel. (The scores were computed in spherical harmonics and averagedover the globe for a rhomboidal truncation of 30 waves in the zonaldirection.) 3 I I 2 3 ~ Flo. 8. Total root-mean-square error for 250-hPa zonal wind field(m s-~) zonally averaged over the Northern Hemispheric latitudesof 20-80-N for 30 5-day control (curve A), nudged (curve B), andafter-the-fact corrected (curve C) forecasts (originating from 1 July1988 to 30 July 1988), as a function of forecast lead time. (Thescores were computed for a rhomboidal truncation of 30 waves inthe zonal direction.) 28 26 22 0o FiG. 7. Spectral decomposition of the total root-mean-square error(curves A, B, and C) and absolute value of the mean error (curvesD, E, and F) for 500-hPa geopotential field (in geopotential meters)for 30 5-day control, nudged, and a~ter-the-fact corrected forecasts,respectively (originating from I July 1988 to 30 July 1988), as afunction of the total (or two-dimensional) wavenumber averagedover the globe.wind at 250 hPa for total root-mean-square error as afunction of forecast lead time. The scores have beencomputed for the full spectrum of rhomboidal 30 wavesand spatially averaged over the Northern Hemisphericlatitudes from 20- to 80-N. It can be seen that nudging(curve B) reduces the total error of the control runs(curve A) starting from day 1 all the way to the endof integration at 5 days, where the reduction is about3.7% (in squared units). However, there is no changeto the total error up to day 4 by after-the-fact corrections (curve C), after which it actually starts to increasethe error. On other domains (not shown), curves A,B, and C remain equally dose. We therefore concludethat little is gained in reducing error in the motion fieldeither by nudging or by after-the-fact correction. Thismay be due to the fact that the systematic error in thewind field is a very small part of the total error to beginwith.4. Response of the model to nudging We now turn our attention away from verificationscores and focus instead on the response of the modelto the extra sources and sinks ofheat, momentum, andmass that were added to the model during integrationby the nudging process, without regard to whether theyactually improved the forecasts or not. For example,if the nudging was intended to accelerate the westerliesFEBRUARY 1992 S A H A 355(in areas where the forecast zonal wind was underestimated in the 30-day training periods), did the westerlies in the nudged runs indeed show greater speedsthan in the control runs? Questions about the responseof a model to fixed sources of heat and momentumhave previously been addressed mostly with linearmodels (Kok et al. 1987). Figures 9-13 compare the effects of (a) the 24-hmean correction obtained from the 30 30-day movingtraining periods to (b) the actual response of the modelafter 5 days obtained as the difference between the 30day mean nudged runs and the 30-day mean controlruns for 5-day forecasts originating from 1 July 1988through 30 July 1988 for various parameters. Figure 9is for the zonally averaged virtual temperature field asa function of height, Fig. 10 is for the zonally averagedzonal wind field as a function of height, Fig. 11 is for'the zonally averaged meridional wind field as a functionof height, and Figs. 12a and 12b are maps showing,respectively, the mean correction applied to, and the5-day response in, the surface pressure field. Similarly,Figs. 13a and 13b are maps showing, respectively, thecorrection applied to, and the 5-day response in, thezonal wind field at sigma level 14 (nearly 200 mb). It can be seen from Fig. 9a that the heating requiredto offset the cold bias in the model is of the order of akelvin per day in many areas. The model accepted thisheating (Fig. 9b) to a first-order approximation, thenudged runs being warmer than the control runs byseveral kelvins after 5 days of integration. Also, thecooling of the upper-tropospheric tropical atmosphereis realized as was intended with the correction applied.Note that in a noninteracting model, Fig. 9b shouldequal Fig. 9a times 5. Since the magnitude ratio is only1 or 2, considerable interaction is evident, even in thezonal mean temperature response (the most successfulcorrection of all). Near the lower boundary between45- and 65- of both hemispheres the systematic erroris made worse, probably showing how hard it is tonudge several interacting parameterizations (boundarylayer processes, horizontal-vertical diffusion, radiation)at the same time. The correction to the motion field was applied to vorticity and divergence fields during integration, but we discuss the applied correction and model response in terms of the zonal and meridional wind fields. We can see that for the zonally averaged zonal wind field (Fig. 10a), the intention of the correction applied was to slow down the upper-level tropospheric westerly jets in both hemispheres, and the response of the model after 5 days (Fig. 10b) shows that it accepted that change. As in Fig. 9, there is correspondence even in several of the details. The intended (Fig. 1 la) and ac tual (Fig. 1 lb) changes in the Hadley circulation are shown for the zonally averaged meridional wind field, and we can see that the corrections imposed implied the need to accelerate the vertical overturning in the 15Fnl~ 1~le1716I110997 90 70 50 30 I0 -I0 -50 -70 LATI?U~- CONTOUR INTERVAL- 0.20 18 15~t~Fnl3 1217II10907 90 70 50 30 lO -tO =30 -50 -70 =90 L~T I'rUOE FIG. 9. (a) Mean 24-h correction (verification minus forecast)applied to the virtual temperature field in the model (K) during the30 nudged integrations originating from 1 July 1988 to 30 July 1988,and obtained from the 30 30-day moving training periods. The correction field is shown as a function of zonally averaged latitudes fromthe North Pole (90-) to the South Pole (-90-) and height, wheresigma I is at the bottom and sigma 18 is at the top of the atmosphere.Negative values are indicated by the shaded areas. (b) Mean 5~dayresponse of the model to correction applied in the virtual temperaturefield (K) shown as the difference between the 30 5-day nudged and30 5-day control integrations originating from 1 July 1988 to 30 July1988. The response is also shown as a function of latitude and height,as explained in (a). Negative values are indicated by the shaded areas.Hadley cell (its rising branch is north of the equatorin Northern Hemisphere summer). As was the casefor the virtual temperature and zonal wind field, themeridional wind component is quite responsive to thecorrection applied, and the actual changes after 5 daysof integration quite closely match those imposed. Theresponse in the meridional wind field seems to haveshorter vertical scales than was intended. Also, themagnitude of the response in the meridional wind relative to the correction is much smaller than the response in the zonal wind or virtual temperature field. The last of the four fields that was directly impactedby the nudging process was the surface pressure field.The model requires correction to In(Psi (where Ps issurface pressure), and hence, Fig. 12a shows the naturallogarithm of the ratio ofps in the verifying initial conditions to the Ps in the 24-h forecasts averaged over the30 30-day training periods. Positive/negative values inthe figure imply that mass ought to be supplemented/356 MONTHLY WEATHER REVIEW VOLUME 120......... ;;~:?oqL ~o ~o ~o -~o t?FIG. 10. (a) Same as in Fig. 9a, but for the zonal wind field (m s-i). (b) Same as in Fig. 9b, but for the zonal wind field (mdestroyed in the nudging experiments. The responseof the model to this mass forcing, shown in Fig. 12b,is very much as intended, particularly in the zonalmean modes. Assuming that ln(/Ys) = ln(ps), wherethe overbar is a time mean, we can express the corrections and response in millibars for easier interpretation.A value of 1 x 10-3 in Figs. 12a and 12b con'espondsto the correction and response, respectively, of nearlyI mb. Certainly, the correction applied and the 5-dayresponse is clearly nontrivial. The forcing and response of the mass field are probably one of the causes of the insufficient accelerationof the Hadley cell, as noted in Fig. I 1. The nudging ofthe zonal-mean meridional velocity implied an increase~01601~01~(~0~ ~o~CONTOUR INTERVRL~ 0.20 FtlCTOR= I,E+O~tS' FIG. 11. (a) Same as in Fig. 9a, but for the meridional wind field(m s-t). (b) Same as in Fig. 9b, but for the meridional wind field(m90180~301EI30~60.<~ ~' ~ ' 7:?o~ ~,:~,0.,~, .:.,.: ,~ , - .. CONTOUR ]NTERVRL= 1.00 FRCIOR- l.E+03 FIG. 12. (a) Global map showing the mean 24-h correction mppliedto the tendency of the natural logarithm of surface pressure in themodel during the 30 nudged integrations originating from I July1988 to 30 July 1988, in terms of the natural logarithm of a ratio.The ratio is the 30-day mean verification initial condition surfacepressure field to the 30-day mean 24-h forecast surface pressme fieldobtained from the 30 30-day moving training periods. Negative values(shaded areas) show regions where the model will be corrected so asto lose mass, and positive values indicate regions where the modelwill be nudged to gain mass. (All values have been scaled by 1000.0.)(b) Global map showing the mean 5-day response of the model tocorrection applied to the tendency of the natural logarithm of surfacepressure during the 30 nudged integrations originating from 1 July1988 to 30 July 1988, in terms of the natural logarithm of a ratio.The ratio is the 30-day mean 5-day nudged surface pressure field tothe 30-day mean 5-day control surface pressure field. Negative values(shaded areas) show regions where the nudged runs lose mass, andpositive values indicate regions where the nudged runs gain massrelative to the control runs. (All values have been scaled by 1000.0.)FEBRUARY 1992 S A H A 357 b901 ~601 !30~~0180.~ ~0, 90E 180ECONTOUR INTERVRL- 2.00 FIG. 13. (a) Global map showing the mean 24-h correction (verification minus forecast) as applied (indirectly) to the zonal windfield (m s-~) at sigma 14 (about 200 hPa) during the 30 nudgedintegrations originating from I July 1988 to 30 July 1988, and obtained from the 30 30-day mean moving training periods. Negativevalues are indicated by the shaded areas. (b) Global map showingthe mean 5-day response of the model to correction applied (indirectly) to the zonal wind field (ms-l) at sigma 14, shown as thedifference between the 30 5-day nudged and 30 5-day control integrations originating from 1 July 1988 to 30 July 1988. Negative valuesare indicated by the shaded areas.in the strength of the Hadley cell. However, by increas- 4'ing simultaneously the pressure in the equatorial re /gions by 1-3 mb, it becomes difficult for the lower ~branch of the Hadley cell (the trades) to strengthen. ,~Likewise, warming the lower tropical troposphere (Fig. [9b) reduces convection, and thus reduces the intensityof the Hadley cell. In this sense the corrections, si- .~multaneously applied to four variables, may be some- ~what inconsistent. ~ To supplement the zonal cross sections, we also show ~global maps of the correction applied (Fig. 13a) to the ~zonal wind field at sigma level 14 (nearly 200 mb) and Ithe actual difference (Fig. 13b) between the 30-daymean nudged and control 5-day forecasts. Here wecannot see any correspondence. Huge differences showup in Fig. 13b that have no clear-cut relationship tothe forcing applied by the correction. We suspectedthat the dipole in the zonal wind over eastern SouthAmerica (the -11, +9 m s-l pair) was a consequenceof unintended changes in tropical convection, evidenceof potent interaction between the nudged variables involving the convection scheme. This was indeed confirmed when we examined the response in the diabaticheating field (not shown), where there were many"bull's-eyes" of considerable magnitude, much largerthan the intended large-scale correction in the heatingfields. Upon prescribing such bull's-eyes in heating,linear models do indeed show strong response in zonalwind (Kok et al. 1987), not unlike Fig. 13. The above results lead us to discuss the response ofthe model to within-integration corrections for aspectsthat are not directly forced by the corrections and, admittedly, are not always intended. The most strikingexample is the response of global precipitation (Fig.14) to the corrections applied as a function of forecastlead time, where the total rainfall in the 30-day meannudged forecasts (curve NG) is much less than in the30-day mean control forecasts (curve CG) originatingfrom 1 July 1988 through 30 July 1988. This is trueover land (curves NL and CL) and particularly overthe sea (curves NS and CS), in the tropics as well asin middle and high latitudes of both hemispheres (notshown). The reduction in the tropics is particularlyremarkable because the Hadley cell is stronger in thenudged runs than in the control runs (Fig. 1 lb) as aresult of mechanical nudging. Why is this so? Three reasons come to mind. The first is that thecorrection applied warms the atmosphere successfully,which, however, reduces the relative humidity, thusreducing large-scale precipitation. The second is thatthe applied correction in the temperature field (warming throughout most of the troposphere and cooling at3.13.02.92.82.62. s ~ '-.._~. ...... " / "--, '--,.., h/~- .........2.21 ~.0 TIME IN DRYS FIG. 14. Mean global precipitation values (mm day-~) averagedover the 30 nudged runs (dashed curves) and 30 control runs (fullcurves) originating from I July 1988 to 30 July 1988. Precipitationover land areas is indicated by curves NL and CL for nudged andcontrol runs, respectively, precipitation over sea areas by NS and CS,and over the globe as a whole by NG and CG.358 MONTHLY WEATHER REVIEW VOLUME 12018171611106 TIME IN DAYSCONTOUR INTERVRL= 1.0018 17 16 15 11 10 TIME IN DAYS ! 7o 16 ilq - ~ ~.o~-_ ~3 ~ 11 ,7 TIME IN DRY5 FIe. 15. (a) Mean difference in the globally averaged relative humidity field (%) between the 30 nudged and 30 control integrationsoriginating from 1 July 1988 to 30 July 1988, as a function of forecastlead time and height, Where sigma I is at the bottom and sigma 18is at the top of the atmosphere. Negative values are indicated by theshaded areas. (b) Same as in (a),'but for the specific-humidity field(g kg-l). (c) Same as in (a), but for the total diabatic heating field( 10-6 K s-~).the lowest sigma level) tends to stabilize the atmosphere, thus reducing convective precipitation. A further complication over the oceans arises from the factthat the sea surface temperature is kept constant duringintegration, making it virtually impossible to counteract the drastic effects of reduced evaporation over theoceans (not shown). While the rainfall values in thecontrol runs (global mean of nearly 3 mm day-~) areprobably more realistic than those in the nudged runs(global mean of nearly 2.3 mm per day),, the controlvalues may be better for the wrong reasons. Both thelarge-scale and convective rainfall parameterizationschemes contain adjustable parameters that may havebeen tuned to obtain realistic precipitation values :forthe overly cold model atmosphere. By nudging, eventhough we provide more realistic temperature fieldsfor the model, these parameterizations may be out oftune. A third reason is the nature of the dynamical balancebetween heating and vertical motion in the tropics. Bynudging both the heating (directly) and vertical motionfields (indirectly), that balance may have been changedfor the worse. We now examine the model response in terms ofthe difference in globally averaged relative humidity(Fig. 15a), moisture (Fig. 15b), and total diabaticheating (Fig. 15c) fields for the 30-day mean nudgedforecasts and 30-day mean control forecasts (originating from 1 July 1988 through 30 July 1988) as a function of height and forecast lead time. While the relativehumidity decreases almost everywhere as a result ofthe general warming of the troposphere (Fig. 15a), thespecific humidity (Fig. 15b) changes very little in thefree atmosphere with an increase in the planetaryboundary layer (sigma levels 1-5) where moisturesupplied by evaporation remains untapped by the decreased convectiye processes in the nudged runs. Thevertical diffusion of moisture (not shown) also decreases in the nudged runs. As a result, the total diabaticheating in the atmosphere as shown in Fig. 15c (notincluding the nudging heat source) is much reducedin the nudged runs, thereby compensating the heatingrepresented by 'the artificial heat sources by at least50% after 5 days of integration. Other complicationsand compensations (as seen in the layered structure inFig. 15c) may be related to changes in the clouds. Onthe whole, the model responds with great stability toimposed sources and sinks during integration by creating mechanisms to counterbalance these imposedchanges. Of course there are many unintended changes 'thatresult from the prescribed sources and sinks. This isnot typical for nudging only. If one compares a controlrun with an anomaly-perturbation run of any kind,there is always a signal-to-noise problem. So while weare relatively satisfied to see the signal (i.e., the responselooks like the correction applied in zonal mean fields),FEBRUARY 1992 S A H A 359there is a lot of noise generated (unintended changesin the response field), as seen, for instance, in Fig. 13b.5. Conclusions Using a reduced resolution version of NMC's operational MRF model, two sets of 30 5-day forecastswere made for initial conditions in July 1988. The purpose was to compare the control forecasts to nudgedforecasts corrected during integration by added constant sources of heat, momentum, and mass. A threeway comparison is made by also considering forecastscorrected in a postprocessing mode.The conclusions are as follows: 1 ) Nudging a complicated general circulation modelin a quasi-operational setting is technically possible (themodel does not blow i~p). 2) The model was responsive in that the systematicdifference between nudged and control runs had a goodsimilarity to the applied corrections, primarily in thezonal mean temperature and wind. 3) From the point of view of forecast accuracy,nudging was a success in that it sharply reduced thesystematic error of forecasts of the mass field out to 5days. The reduction of the systematic error in the motion field is small or absent, mostly because there isvery little systematic error in these fields to begin with. 4) The reduction of the systematic error throughnudging for total wavenumbers less than 7 (n < 7) has,through nonlinear interaction, a small beneficial impacton the (mostly random) error in scales with total wavenumbers greater than 7. However, for n < 7, after-thefact correction is better, particularly in the tropics andfor forecast lead times of less than 4 days or so. 5) A postprocessor statistical error reductionmethod (after-the-fact correction) produced as muchof a reduction of error as nudging. Producing a postprocessed forecast out to 10 days is much less involvedthan nudging, and can be achieved almost for free inan operational environment where forecasts out to 10days are made routinely every day. 6) Nudging is a viable method in reducing errorsfor longer forecast lead times, especially climate runs.Since no after-the-fact corrections can be applied operationally beyond 10 days, it is the only method presently available. All it would take is a series of 24-hforecasts with the model, to obtain the correction fieldsto be used in runs beyond 10 days.6. Discussion Although nudging was a partial success in that themodel did accept the applied corrections quite faithfully, the systematic error of the forecasts was not always reduced. This is because it is virtually impossibleto know a priori exactly what the systematic error is.The time-mean errors as calculated in our experimentare not free from sampling errors; therefore, there areproblems when we equate them to be the true systematic errors of the model. Both after-the-fact and nudgingapproaches would obviously benefit from a more stabledefinition of the systematic error. The nudging experiment as carded out here can alsobe used as a tool to investigate the possible causes ofsystematic errors in the model. Since the rainfall (bothlarge-scale and convective) shows great sensitivity tothe tropospheric temperature, it seems likely thatphysical processes causing precipitation and heatingrequire attention. For instance, why is there a decreasein the rainfall when there is a better temperature profilein the model? As found in Johansson and Saha(1989), a reducedsystematic error in a global barotropic model also resuits in a reduced random error. One of our hopes wasto reproduce this result in a quasi-operational setting.There is indeed a small improvement in scales n > 7caused by nudging scales n < 7, but this payoffis smallfor the large amount of extra work to be done fornudging. In order to make the experiments more economical,we used a R30 version of the operational T80 model.It is indeed ironic that over the same verification period( 1 July 1988-30 July 1988), the R30 version had lesssystematic error (-25.7 m from Fig. 2b) than the operational T80 model ( -31.5 m from o~cial sources atNMC) for reasons not fully understood. We have found that in terms of forecast improvement, nudging is, on the whole, not much better orworse than after-the-fact corrections, at least out to 4days of integration. There could be two reasons forthis. The first is that the NMC medium-range forecastmodel reacts rather noninteractively to the appliedcorrections. A second, and more likely, explanation isthe internal inconsistency of the applied corrections.We found, for instance, that the much-needed mechanical acceleration of the Hadley cell was renderedalmost impossible by the correction to the surfacepressure and tropospheric temperature fields. Ideally,one would want to correct the variables one after theother, making the corrections dependent on each other(Murphree 1989). Not only is such a procedure prohibitively expensive, but there is no guarantee that aconvergent solution is possible. Studies extending the nudging process beyond 5 daysof integration (up to 30 days are already underway)will enable us to study the impact of within-integrationcorrections for longer forecast lead times, which maybe useful for dynamically extended-range forecasts(DERF) or other climate-oriented integrations. Acknowledgments. I am deeply indebted to Dr.H. M. van den Dool for his ideas and discussions toward the making of this paper. Valuable assistance andguidance from Drs. M. Kanamitsu, E. Kalnay, J. A1360 MONTHLY WEATHER REVIEW VOLUME 120pert, ~. Johansson, H. L.-Pan, T. Murphree, F. Baer,and P. Caplan are also gratefully acknowledged. Dr.Ming Cai was helpful in constructing Fig. 1.REFERENCESAlpert, J. C., and S. Saha, 1989: Operational systematic error cor rection for the NMC operational medium-range forecast model. Office Note 360, National Meteorological Center, NWS/NOAA, U.S. Department of Commerce, Washington, D.C.Arpe, K., and E. Klinker, 1986: Systematic errors of the ECMWF operational forecasting model in the mid-latitudes. Quart. J. Roy. Meteor. Soc., 112, 181-202.Davis, H. E., and R. E. Turner, 1977: Updating prediction models by dynamic relaxation: An examination of the technique. Quart. J. Roy. Meteor. Soc., 103, 225-245.Epstein, E., 1985: Procedure for the statistical corrections of medium range spectral forecasts. 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