Fine-Resolution 4DVAR Data Assimilation for the Great Plains

506 VOLUME 17W E A T H E R A N D F O R E C A S T I N G

q 2002 American Meteorological Society

Fine-Resolution 4DVAR Data Assimilation for the Great Plains Tornado Outbreakof 3 May 1999

DUSANKA ZUPANSKI* AND MILIJA ZUPANSKI*

NOAA/NCEP/UCAR Visiting Scientist Programs, Camp Springs, Maryland

ERIC ROGERS, DAVID F. PARRISH, AND GEOFFREY J. DIMEGO

NOAA/NCEP/EMC, Camp Springs, Maryland

(Manuscript received 1 March 2001, in final form 26 October 2001)

ABSTRACT

The National Centers for Environmental Prediction fine-resolution four-dimensional variational (4DVAR) dataassimilation system is used to study the Great Plains tornado outbreak of 3 May 1999. It was found that the4DVAR method was able to capture very well the important precursors for the tornadic activity, such as upper-and low-level jet streaks, wind shear, humidity field, surface CAPE, and so on. It was also demonstrated that,in this particular synoptic case, characterized by fast-changing mesoscale systems, the model error adjustmentplayed a substantial role. The experimental results suggest that the common practice of neglecting the modelerror in data assimilation systems may not be justified in synoptic situations similar to this one.

1. Introduction

One of the most advanced data assimilation methodstoday, the four-dimensional variational (4DVAR) meth-od (e.g., Lewis and Derber 1985; LeDimet and Tala-grand 1986; Courtier and Talagrand 1987; Thepaut andCourtier 1991; Navon et al. 1992; Zupanski 1993a,b,1997; Zou et al. 1993; Zou and Kuo 1996), has beenunder very active investigation in the last 10–15 years.It has become the operational data assimilation methodat the European Centre for Medium-Range WeatherForecasts (ECMWF; Rabier et al. 1998, 2000; Klinkeret al. 2000; Mahfouf and Rabier 2000) and Meteo-France (Gauthier and Thepaut 2001). As an operationalmethod, it was rigorously tested and has proven to workwell for larger-scale atmospheric problems.

The issue of mesoscale data assimilation has receivedless attention. In recent years, however, the research inthis area has become more intensive (e.g., Sun andCrook 1994, 1997, 1998, 2001; Park and Droegemeier1997, 2000; Zou and Xiao 2000; Zou et al. 2001; Guoet al. 2000), and it will probably gain even more pop-ularity as computer power increases. It has become ob-

* Current affiliation: Cooperative Institute for Research in the At-mosphere, Colorado State University, Fort Collins, Colorado.

Corresponding author address: Dusanka Zupanski, CooperativeInstitute for Research in the Atmosphere, Colorado State University,Fort Collins, CO 80523-1375.E-mail: [email protected]

vious that in the context of mesoscale and storm-scaledata assimilation a number of problems require addi-tional research. First of all, the method needs to combinein an optimal way the forecast model with numeroushigh-resolution observations (e.g., radar moist micro-physical parameters and velocity, satellite cloud, soilmoisture, lightning, GPS moisture information) to pro-duce realistic initial mesoscale features. Observation er-rors for each data type are required as an input to thedata assimilation problem. In cases of high temporaland spatial data resolution (e.g., satellite and radar ob-servations) observation errors may be correlated (Daley1992). To take into account correlated observations, theobservation error covariance matrix should be definedto include off-diagonal components. The validity of thequasigeostrophic balance constraints, successfully usedin larger-scale data assimilation problems (e.g., Parrishand Derber 1992; Courtier et al. 1998; Derber and Bout-tier 1999), needs to be reexamined. Model error has tobe considered. These issues are just some examples ofproblems encountered in mesoscale data assimilation.More information about mesoscale 4DVAR data assim-ilation methods can be found in a review paper by Parkand Zupanski (2002).

The focus of this study is the tornado outbreak thatoccurred across Oklahoma and Kansas on 3 May 1999.We apply the 4DVAR method to assimilate observations36–24 h prior to the event and examine the forecastresults after data assimilation. Even though this studymakes an attempt to improve our understanding re-

JUNE 2002 507Z U P A N S K I E T A L .

FIG. 1. Schematic diagram of the 4DVAR data assimilation algo-rithm, including forecast and adjoint model runs. Precipitation ob-servations are assimilated every hour. All other observations are as-similated in 3-hourly intervals. Data assimilation period is 12 h.

garding this specific tornado episode, the grid spacingof chosen forecast and adjoint models (32 km/45 layers)provides only a limited capability to examine the tor-nado itself. Rather, it offers an insight about the natureand benefits of a 4DVAR method in application to asevere-weather event. Indeed, smaller-scale processes(50–100 km) are often mislocated in the forecast resultsbecause they are unresolved by the model (scales be-tween meso-a and meso-b are actually resolved). Nev-ertheless, the method is capable of producing realisticlarger-scale precursors of tornadic activity [e.g., low-and upper-level jet streaks, convective available poten-tial energy (CAPE), wind shear], 24–36 h in advanceof the event, which may serve as a warning to fieldforecasters. Benefits of some unique features of the Na-tional Centers for Environmental Prediction (NCEP)mesoscale 4DVAR, such as model error and precipi-tation assimilation, are examined in more detail in thiscase.

After a short theoretical background review given insection 2, we proceed with describing NCEP’s 4DVARalgorithm in section 3. In section 4, experimental resultsare presented and discussed. In section 5, conclusionsare drawn.

2. Theoretical background

Theoretical background for the 4DVAR data assim-ilation method comes from optimal control theory (Li-ons 1971; LeDimet and Talagrand 1986; Tarantola 1987)and Kalman filtering (Kalman 1960; Jazwinski 1970).In application to atmospheric data assimilation prob-lems, the 4DVAR method seeks an optimal solution tothe atmospheric state by combining the forecast model,observations, and first-guess (e.g., forecast from the pre-vious data assimilation cycle) information (Lorenc1986). This goal is achieved through an iterative min-imization of a cost function, measuring forecast error.The minimization problem is defined over a time period(data assimilation interval), during which all availableobservations are taken into account at the observationlocations and in the form of directly observed quantities.As input, the method requires an estimate of the ob-servation and first-guess errors in the form of error co-variance. The observation error covariance matrix iscommonly assumed to be diagonal (uncorrelated ob-servations), whereas the background error covariancetypically includes off-diagonal elements to account forspace and/or time correlations and for cross correlationsbetween different variables.

The forecast model is used as a dynamical constraint[strong or weak, according to Sasaki (1970a,b)], non-linearly linking different atmospheric variables. The ad-joint model (conjugate transpose of a linearized forecastmodel) uses observations as forcing while running back-ward in time to provide the gradient of the functionalat the beginning of the data assimilation interval. This

information is used as a main input to the minimizationalgorithm (Gill et al. 1981).

In general, the control variable of the 4DVAR prob-lem can be any model input parameter capable of in-fluencing the forecast at the end of the data assimilationinterval. It typically includes initial conditions of theforecast model at the beginning of the data assimilationinterval. It can also include some other model inputparameters, such as empirical constants, as well as mod-el error, boundary conditions, and so on. All these var-iables are assumed to be independent elements of alarge-size control vector (typically on the order of 107–109). Even though the control vector (also referred toas the augmented control variable) includes componentsof a different nature (such as initial conditions and mod-el error), the variational formalism provides an optimalsolution to each component through a single algorithm.The control variable is iteratively adjusted during thecourse of minimization until the convergence is reachedand the minimum solution is obtained.

The basic assumption of the 4DVAR method is thatthe optimal solution will stay optimal during the forecasttime and, therefore, will provide both optimized initialconditions and an improved forecast. This depends onprior information, sometimes poorly known (first-guessforecast, model and observation error covariances, grav-ity wave amount, errors in the gradient due to lineari-zation and/or discontinuity, etc.), as well as on the pre-dictability of a particular atmospheric process.

A schematic diagram of a data assimilation schemeis given in Fig. 1. The specific example includes assim-


TABLE 1. The observations assimilated by the 4DVAR and 3DVAR algorithms. Note that precipitation observations are used only in the4DVAR algorithm.

Observation types Comments

Rawinsondes Winds, temperature, specific humidity, surface pressure, heightAircraft Communications and Reporting System and con-

ventional aircraft reports Winds, temperatureProfiler windsPilot balloon windsCloud-tracked winds from GOESSurface land and marine observations Winds, temperature, specific humidity, surface pressureSSM/I wind speeds Wind direction from guessDropwindsondes Winds, temperature, surface pressure, specific humidityNOAA TOVS temperature retrievals Over water onlyHurricane bogus windsGOES and SSM/1 precipitable water SSMI: over water only

GOES: over land and waterStage-IV, 4-km multisensor hourly precipitation Used in 4DVAR experiments only

ilation of precipitation observations over a 12-h dataassimilation interval, as used in the experiments pre-sented later. Specific components of NCEP’s 4DVARalgorithm, used in the experiments, are explained in thenext section.

3. NCEP’s 4DVAR algorithm

NCEP’s fine-resolution 4DVAR algorithm (Zupanski1993a,b, 1996, 1997; Zupanski and Mesinger 1995; Zu-panski and Zupanski 1995) is used in this study. It in-cludes the following components: forecast model, ob-servations, cost function, adjoint model, model error,minimization, and preconditioning. Details of each com-ponent of the 4DVAR algorithm are given below.

a. Forecast model

NCEP’s 32-km, 45-layer Eta Model (Mesinger et al.1988; Janjic 1990, 1994; Łobocki 1993; Black 1994;Janjic et al. 1995; Zhao et al. 1997; Chen et al. 1997)is used in this study. At the time the experiments wereperformed (autumn of 2000), this was the operationalmodel. In September of 2000, the resolution of the EtaModel was increased to 22 km and 50 layers (Rogerset al. 2000). In August of 2001 the nudging of precip-itation observations (Lin et al. 1999) was included inthe Eta Data Assimilation System (EDAS).

The same forecast model is used in both data assim-ilation and forecast experiments (except for the fact thatthe model error is accounted for during the data assim-ilation interval, as will be explained later in this section).Only the nonlinear forecast model (no tangent-linearassumption) is used to describe forecast evolution. Webelieve that the nonlinear aspect is an important ingre-dient of the 4DVAR algorithm in application to casesof mesoscale severe-weather events, because these pro-cesses are changing fast and are often highly nonlinear.

b. Observations

The 4DVAR uses the NCEP operational EDAS da-tabase, including rawinsondes, aircraft reports, profilerwinds, pibal winds, cloud-tracked Geostationary Op-erational Environmental Satellite (GOES) winds, sur-face land and marine observations, Special Sensor Mi-crowave Imager (SSM/I) wind speeds, dropwindsondes,National Oceanic and Atmospheric Administration(NOAA) Television and Infrared Observation SatelliteOperational Vehicle Sounder (TOVS) temperature re-trievals, and GOES and SSM/I precipitable water. Inaddition, in the 4DVAR experiments, NCEP hourly mul-tisensor (radar and rain gauge) stage-IV, 4-km precip-itation data (Baldwin and Mitchell 1997) are used. Noquality control of these precipitation data is performed;thus, some biased observations may be included (K.Mitchell 2000, personal communication). Therefore,when examining the impact of precipitation assimila-tion, some caution should be taken. Since August of2001, the multisensor precipitation is used operationallyin EDAS via a nudging technique (Lin et al. 1999). Theobservation types used in 4DVAR data assimilation ex-periments are listed and briefly explained in Table 1.The observation error covariance is assumed to be di-agonal in space and time and to be time independent,depending on the observation type only. This assump-tion is commonly made in data assimilation studies,although it may not be always justified, such as in thecases of dense and frequent (satellite) observations (Dal-ey 1992).

c. Cost function

The cost function minimized in the 4DVAR algo-rithm, denoted J, is given by


FIG. 2. The 18-h forecasts of the 250-hPa wind field (kt) valid at1800 UTC 3 May 1999, initiated using (a) 3DVAR and (b) weak-constraint 4DVAR data assimilation algorithm. (c) The EDAS veri-fication analysis. Contouring interval for wind intensity is 5 kt.

1b T b21J 5 (z 2 z ) B (z 2 z )

2

1T 211 [z 2 F(z )] P [z 2 F(z )]t t t t0 0 0 02

N1T 211 {H[M(z)] 2 y} R {H[M(z)] 2 y} .O n n2 n51

(1)

The augmented control variable of the 4DVAR problem,denoted z, includes initial conditions, model error, andlateral boundary conditions [as in Zupanski (1997)]. Itincludes the following state variables: surface pressure,temperature, east–west u and north–south y wind com-ponents, and specific humidity. The vector y denotesobservations, the subscript b refers to the backgroundvariable, and N defines the number of observation timesin the assimilation period; B is the background errorcovariance, including two components—1) forecast er-ror covariance and 2) model error covariance; P is thegravity-wave penalty covariance; R is the observationerror covariance; H is the nonlinear observation oper-ator; and M is the nonlinear forecast model. In addition,F stands for the gravity-wave filter operator (Lynch andHuang 1992; Huang and Lynch 1993) at time t0 (centraltime of the filter). We choose the time span for the filterto be 2 h, thus making the central time t0 take 1 h afterthe beginning of data assimilation. During this initialtime period of 2 h, a digital filter is applied (as a weakconstraint) to smooth out high-frequency time oscilla-tions. The digital filter operator is applied in a similarmanner as in Gauthier and Thepaut (2001).

The background error covariance, for both the fore-cast and the model error, is defined using a compactlysupported space-limited analytical function as in Gas-pari and Cohn (1999). The only difference between thetwo components of the covariance matrix is in magni-tude (the ratio between model error variance and fore-cast error variance is defined empirically as 1024) andin decorrelation lengths (approximately 30%–50%-shorter decorrelation lengths are assigned to model er-ror). The background error covariance is assumed to beunivariate, isotropic, and homogeneous at the initialtime of the data assimilation interval. From the begin-ning to the end of the data assimilation interval, theforecast error covariance evolves into a complex flow-dependent multivariate function, thus achieving morerealistic forecast error features at the end of assimilation(actual analysis time). This specific aspect of the4DVAR method is explained in Thepaut et al. (1996).

d. Adjoint model

The adjoint model includes all dynamical subroutinesof the Eta Model as well as most of the physics (radi-ation and soil-hydrology routines are not included inthe adjoint, but they are used in the forward forecast


FIG. 3. As in Fig. 2 but for the 24-h forecasts and correspondingEDAS verification (valid at 0000 UTC 4 May 1999).

model). The adjoint model has the same time and grid-point resolution as the forecast model (no coarse-res-olution approximation is used when calculating the gra-dient, as is done in the operational ECMWF and Meteo-France 4DVAR applications). The fine-resolution gra-dient is more expensive to calculate, but it can providemore accurate and, perhaps, critically important infor-mation in cases of mesoscale severe-weather events.

To regularize the minimization problem, discontinu-ous on–off switches of the cumulus convection are treat-ed by applying a smooth function as in Zupanski andMesinger (1995). Also, the K-theory approximation isused in the tangent-linear and adjoint models, insteadof using the Mellor–Yamada 2.5-level turbulence clo-sure approach as in the nonlinear model. The adjoint ofthe Eta Model is developed using the automatic tangent-linear adjoint model compiler (TAMC) of Giering andKaminski (1998) and Giering (1999).

e. Model error

The random part of the model error (denoted r) is acomponent of the augmented control variable z in (1).The total model error f is defined as a first-order Mar-kov process variable and includes both random and se-rially correlated parts, as in Zupanski (1997). It is ap-plied as an additive correction to the model’s equationsonce per time step, according to the following equations:

x(t) 5 M[x(t 2 1)] 1 f(t), (2)

f(t) 5 af(t 2 1) 1 br , (i 5 1, . . . , I ),i max

a 1 b 5 1, (3)

where x is the model state variable, r is a random modelerror vector, f is a serially correlated model error, anda and b are weighing constants for serially correlatedand random parts, respectively. Index t refers to model’stime step, and Imax is the total number of random modelerror vectors defined during the assimilation period. Al-though the control variable includes only the randomerror component, the prescribed relation (3) allows forsystematic model error adjustment. In our experiments,the initial value of f is set to zero. The choice of Imax

5 1 is equivalent to adjusting only the systematic modelerror as in Derber (1989), because the model error as-ymptotically approaches the value of r as the integrationtime progresses. In the experiments presented, we choseImax 5 4, thus allowing for time-changing model error[every 3 h a new optimized random error r is insertedinto (3) during a 12-h data assimilation interval]. Con-stants a and b are given empirical values, 0.67 and 0.33,respectively, thus giving considerably more weight to theserially correlated part as compared with the random part.

In our 4DVAR algorithm, the model error (2)–(3) isassumed to include two different components: 1) theerror of the forecast model inside the integration domainand 2) the boundary condition error at the domain lateral


FIG. 4. The 250-hPa wind field (kt) forecast, initiated using the strong-constraint 4DVAR data assimilation algorithm: (a) 18- and (b) 24-hforecast. Contouring interval for wind intensity is 5 kt.

boundaries. Through this assumption, the lateral bound-aries are also considered to be imperfect and are adjustedduring the minimization.

f. Minimization and preconditioning

The memoryless quasi-Newton minimization algo-rithm of Nocedal (1980), with the restart procedure ofShanno (1985), developed by Zupanski (1993b, 1996),is used in the experiments. It includes the followingchange of variable (preconditioning):

b 21/21/2z 2 z 5 B (I 1 D) z, (4)

where z is the transformed (nondimensional) controlvariable, B is a background error covariance, and Drepresents a positive-definite, case-dependent, empiri-cal, diagonal matrix.

4. Experimental results

a. Experimental setup

The observations are assimilated during the 12-h in-terval from 1200 UTC 2 May to 0000 UTC 3 May 1999,approximately 36–24 h prior to the tornado event. Thedata insertion frequency is 1 h for precipitation obser-vations and 3 h for all other observations. A schematicdiagram of the data assimilation algorithm is given inFig. 1.

A number of 4DVAR data assimilation experiments

are carried out, varying from full-blown experiments(including all observations and model error adjustment)to experiments that exclude precipitation observationsand use the perfect model assumption (strong con-straint), to assess the impact of a specific 4DVAR com-ponent in this synoptic case. To evaluate 4DVAR results,we use the NCEP operational EDAS 3DVAR data as-similation method (Parrish et al. 1996; Rogers et al.1996, 1997) as a control. The 3DVAR analysis is per-formed every 3 h during the same 12-h data assimilationinterval as in the 4DVAR case. The 3DVAR and4DVAR experiments use the same observations, exceptfor precipitation data, which are used only in 4DVAR.The same first-guess forecast, valid at time t 2 12 (Fig.1), taken from EDAS, is used in both 3DVAR and4DVAR experiments. In all experiments, the same EtaModel (NCEP’s former operational code) is used to pro-duce the 48-h forecast after the data assimilation. Duringthe 48-h forecast, the model error is assumed to be equalto zero in all experiments. For verification, we useEDAS 3DVAR analyses and the stage-IV precipitationobservations. For the purpose of having an independentanalysis as verification, we also made use of AdvancedRegional Prediction System (ARPS) analyses (Brewsteret al. 1994; Xue et al. 1995), during the course of theinvestigation. The ARPS analyses were kindly providedby the Center for Analysis and Prediction of Storms asa part of a research database for this tornado event.

The number of minimization iterations performed in


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FIG. 5. The 18-h forecast of the surface CAPE (J kg21; dots andshading) and vertical wind shear (0.0002 s21; solid lines) in the 250–700-hPa layer, and the 1000-hPa wind (m s21; barbs: full barb is 10m s21) valid at 1800 UTC 3 May 1999, initiated using (a) 3DVARand (b) the weak-constraint 4DVAR data assimilation algorithm. (c)The EDAS verification analysis. Contouring interval for wind shearis 5 units.

all 4DVAR experiments is 10. The improvements after10 iterations were typically marginal.

b. Results

1) 4DVAR VERSUS 3DVAR

Figures 2a,b show the 18-h forecast of the 250-hPawind valid at 1800 UTC 3 May 1999, initiated usingthe 3DVAR and 4DVAR data assimilation algorithms,respectively. This 4DVAR experiment used both pre-cipitation assimilation and model error adjustment [fourdifferent random error terms are adjusted during the 12-h data assimilation interval, i.e., Imax 5 4 in (3)]. Thisexperiment was named 4DVAR (ERRp4). In Fig. 2c, thecorresponding EDAS analysis, used as verification, ispresented. Figures 3a–c show the same fields, but 6 hlater (valid at 0000 UTC 4 May 1999). As seen in the1800 UTC EDAS analysis (Fig. 2c), there was a pro-nounced jet streak in central Texas, extending into west-ern Oklahoma and Kansas, across the Texas–New Mex-ico border. Over southern and central Oklahoma, a localminimum of upper-level wind existed. As we comparethe analysis 6 h later (0000 UTC 4 May 1999, whenthe tornado episode had already started), a wind speedminimum, though weaker, is still seen in southern andcentral Oklahoma. The axis of maximum wind hasmoved over New Mexico, and a number of additionalsmaller-scale wind minima and maxima appeared acrossTexas, Oklahoma, and Kansas. A diffluent flow patternover eastern New Mexico and western Texas hasformed. These features, if present in the forecast, wouldbe a first warning to a forecaster that a severe-weatherevent is about to happen. For example, M. Branick, thelead forecaster for the National Weather Service (NWS)in Norman, Oklahoma, noticed that the 250-hPa short-wave wind patterns ‘‘dig well south into the southwesternstates, and move rapidly as they ejected eastward intothe central states. . .. By late in the day, the ingredientsfor a significant severe weather outbreak began to cometogether over the southern half of the Plains. The upper-level jet turned east across New Mexico and then north-east across Oklahoma and Kansas by early evening, witha local 75-knot speed maximum near the Kansas–Oklahoma border.’’ (from the NWS Norman Web site:http://www.srh.noaa.gov/oun/storms/19990503/). Acomparison of the forecast results from 3DVAR and4DVAR experiments (Figs. 2a,b and 3a,b) with the ver-ification (Figs. 2c and 3c) shows that both forecast ex-



periments underpredicted the jet maximum across theNew Mexico–Texas border, extending to Oklahoma andKansas (Fig. 2c). More careful examination of the fig-ures indicate that the 3DVAR experiment produced astronger wind maximum in this area, which is in betteragreement with the verification. The minimum oversouthern and central Oklahoma is missing in both ex-periments. By comparing the figures 6 h later (Figs. 3a–c), we see that an indication of the wind minimum/maximum pattern appeared in the 4DVAR experimentover Oklahoma, though the location error is present.Both the 3DVAR and 4DVAR forecast missed the min-imum/maximum patterns over Texas and Kansas. Byexamining the low-level wind field (850 hPa), we alsonoticed an indication of a short-wave minimum/maxi-mum wind pattern in the 4DVAR experiment, in betteragreement with the verification, as compared with the3DVAR experiment (figures not shown). Similar min-imum/maximum patterns were also noticed in the ARPSwind analyses (250 and 850 hPa).

One cannot necessarily assume that these forecastedmaximum/minimum wind patterns may be of some sig-nificance to this specific event, because there is a sub-stantial mismatch between the model resolution and thescale of the event, resulting in a considerable locationerror. As explained in the introduction, the scales below100 km are not resolved by the model. We would argue,however, that these patterns do indicate that a severe-weather event may happen.

To make this argument, we also present the 4DVARexperimental results, without model error adjustment.We refer to this experiment as strong constraint or4DVAR (NOERR). Figures 4a and 4b show 18- and24-h forecasts of the 250-hPa wind field, respectively.We notice an indication of a slight short-wave troughin eastern Oklahoma at 1800 UTC, and a minimum/maximum pattern moved to eastern Oklahoma, Arkan-sas, and Missouri at 0000 UTC. As will be shown later,when the effect of model error adjustment is examined,this pattern coincides with excessive (and unrealistic)precipitation over this area, thus proving that a linkexists between the wind pattern and the severe-weatherevent (which may not necessarily be a tornado event).This link is nicely seen when comparing 3DVAR and4DVAR (ERRp4) precipitation with the upper-levelwind. Again, the minimum/maximum wind pattern iscorrelated well with intensive precipitation in the4DVAR (ERRp4) experiment (figures presented in thenext section). It is also important to note that the4DVAR experiment with model error adjustment pro-duced considerably more realistic forecast results thanthe one without model error adjustment.

Let us now examine some other atmospheric param-eters that played an important role as precursors for tor-nadic activity. As noted in Thompson and Edwards(2000), important atmospheric conditions associated withthe Oklahoma–Kansas tornado outbreak of 3 May 1999were the midtropospheric (4–10 km) vertical wind shear,


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FIG. 7. The 18-h forecast of the 850-hPa relative humidity (%; dotsand shading), temperature (8C; solid lines), and wind (m s21; fullbarb is 10 m s21) valid at 1800 UTC 3 May 1999, initiated using (a)3DVAR and (b) the weak-constraint 4DVAR data assimilation al-gorithm. (c) The EDAS verification analysis. Contouring interval fortemperature is 28C.

low-level wind shear, and surface-based CAPE. It wasalso demonstrated in Hamill and Church (2000) that aprobabilistic model based on CAPE and low-level windshear, as well as on CAPE and helicity, was able to pro-duce a realistic forecast for this tornado event. In thefollowing figures we present and discuss surface-basedCAPE and midlevel vertical wind shear as important am-bient conditions associated with this severe-weatherevent. In Figs. 5a,b we present the surface CAPE andmidtropospheric (250–700-hPa layer) vertical wind shear,valid at 1800 UTC 3 May 1999, obtained as an 18-hforecast after 3DVAR and 4DVAR (ERRp4) data assim-ilation, respectively. As an illustration of surface flow,the 1000-hPa winds are also plotted. In Fig. 5c, the EDASverification valid at the same time is given. Figures 6a–c show the same fields as Figs. 5a–c, respectively, but 6h later (valid at 0000 UTC 4 May 1999). In all figures,the wind shear is scaled by a factor of 5000.

As seen in the EDAS verification analyses (Figs. 5cand 6c), large CAPE (exceeding 3000 J kg21) was de-veloping during this 6-h time interval, associated withminimum/maximum patterns in wind shear. At 0000UTC 4 May 1999, the CAPE maximum split into twoparts, one over Texas, and the other over northOklahoma and Kansas. The comparison of forecastCAPE in the 3DVAR and 4DVAR forecast experimentsreveals a stronger severe-weather event in the 4DVARexperiment at the beginning (18 h) of the interval andthe CAPE maximum splitting into two maxima at theend of the interval (24 h). For wind shear, the minimum/maximum wave pattern was poorly predicted in bothexperiments, with only a slight indication of a shortwave in this field over Oklahoma. Because of large er-rors in both experiments, it is hard to determine whichone produced more realistic midtropospheric verticalwind shear. It may be that a combination of the twoparameters (wind shear and CAPE) was more realistic(dynamically more consistent) in the 4DVAR experi-ment, because it produced stronger indications of a se-vere-weather event in other fields (humidity, precipi-tation, etc.), which were in better agreement with theobservations.

Indeed, comparison of 850-hPa relative humidity(RH) fields, in Figs. 7a–c (18-h forecast) and 8a–c (24-h forecast) shows higher RH in the 4DVAR experiment,in better agreement with the EDAS verification analyses.Also, the 4DVAR experiment shows a stronger drylinegradient across western Oklahoma and slightly coolertemperature in the area of interest, again better verifiedby the EDAS analysis.



Also compared were differences between 3DVARand 4DVAR analyses at the initial time (0-h forecast).One interesting finding was that the 3DVAR analysiswas more humid, and in better agreement with the ob-servations (figures not shown). After only a few hoursof forecast time, however, the atmosphere dried out, thusleaving less favorable conditions for supercell devel-opment. This is a typical difference between 3DVARand 4DVAR analyses. The 3DVAR analysis commonlyprovides a better fit to the observations. The 4DVARmethod provides optimal fit to the observations over aperiod of time, often producing a dynamically moreconsistent analysis and an improved forecast but also adegraded analysis fit to the observations.

We also compared other forecast fields and parame-ters (such as convective inhibition, helicity, 500- and850-hPa wind, and temperature) The 4DVAR resultsgenerally gave more indications of a severe-weatherevent than did the 3DVAR results. Neither experiment,however, was even close to predicting a tornado or asupercell because of a large discrepancy between themodel resolution and the tornado scale, which made itdifficult to perform a more strict quantitative evaluation.Even so, we found that the 4DVAR method can con-tribute much to a better understanding of tornado events,especially when some poorly known features, such asthe impact of model error and precipitation observa-tions, are studied.

2) THE IMPACT OF MODEL ERROR

One of the unique features of the NCEP 4DVAR dataassimilation system is the capability to optimize (adjust)the model error along with adjusting the initial condi-tions. In this section we examine the impact of modelerror adjustments in 4DVAR, during the data assimi-lation and the subsequent forecast.

General definition of the model error [(2)–(3)] allowsfor any number of random error terms during the dataassimilation interval. The equations indicate that in-cluding more random error terms allows for more timevariability in the model error (finer timescales). It makesit possible for the model error better to capture fast-changing processes. It is important to note, however,that allowing finer timescales in the random error doesnot force fast-changing model error components. It justallows for better capturing of fast processes, if any arepresent in the atmosphere but not resolved by the fore-cast model.

We performed experiments using only one random er-ror component during the entire data assimilation interval(Imax 5 1) and experiments using four random errors (Imax

5 4). A decision was made to examine in more detailthe results with more random error components, basedon the fact that this experiment provided slightly, butconsistently, improved forecast results as compared withthe experiments with only one random error component.For example, the precipitation pattern over Oklahoma and


FIG. 9. A series of optimized random error terms ri for surface pressure (0.1 3 Pa), obtained every 3 h during the 12-h data assimilationinterval: valid 1200 UTC 2 May plus (a) 3, (b) 6, (c) 9, and (d) 12 h. Contouring interval is 4 units.

Arkansas (discussed later) was slightly improved in lo-cation and intensity (figure not shown). Both experimentsprovided considerably better forecast results as comparedwith the strong-constraint 4DVAR experiment (no modelerror adjustment), thus giving us more confidence thatthe optimized model error has some realistic features init [similar results were obtained in our previous studies,e.g., Zupanski (1997)].

Figures 9a–d show a series of optimized randomerror terms r i for surface pressure, obtained every 3h during the 12-h data assimilation interval. The mod-el error terms are scaled by 10 in the figures presented.Model error wind vectors (intensity in shades) are

plotted in Figs. 10a–d. By examining the figures, wenotice that the model error term has a very smallmagnitude (on the order of 1–2 Pa for surface pressureand 0.005–0.01 kt for wind). We also notice that themodel error has some organized structures (synopticand mesoscale), similar in shape and magnitude fromone time interval to another. The experimental resultswith only one random error term ( Imax 5 1) showedvery similar model error shapes, resembling an av-erage model error structure over the data assimilationinterval (figures not shown). Note that the 4DVARproblem was posed assuming no time correlation be-tween different random error components (model er-


FIG. 10. A series of optimized random error terms ri for wind (0.001 3 kt), obtained every 3 h during the 12-h data assimilation interval:valid 1200 UTC 2 May plus (a) 3, (b) 6, (c) 9, and (d) 12 h. Wind barbs and wind intensity in contours and shades. Contouring interval is2 units.

ror covariance allows for space correlations only).Yet, there is a striking similarity between the succes-sive error patterns, thus suggesting that the modelerror travels throughout the model domain. It can beseen that the error pattern moved approximately thedistance of 1500 km during a 12-h period. This trans-lates to an average phase speed of the model error onthe order of 35 m s 21 . This speed is much greaterthan the phase speed of any weather system. It iscomparable to internal gravity wave speed. This is anew finding regarding the nature of the model errorthat has never been observed before. At this point it

is not clear if this model error feature is related to thespecific forecast model (Eta) and specific synopticcase, or has a more general meaning. It is also notknown to what extent the particular definition of themodel error covariance influenced the optimal solu-tion for model error. These are all very interestingbut complicated issues and will be addressed in ourfuture work. Nevertheless, we would argue that theobtained model error features are realistic. The ex-perimental results presented next should support thisargument.

Figures 11a,b present 24-h forecasts of 12-h accu-


FIG. 11. The 24-h forecast of 12-h accumulated precipitation (mm), valid at 0000 UTC 4 May 1999, for (a) 4DVAR, strong constraint(NOERR); (b) 4DVAR, weak constraint (ERRp4); (c) NCEP stage-IV multisensor 12-h precipitation accumulations valid at the same time;and (d) 3DVAR.

mulated precipitation, valid at 0000 UTC 4 May 1999,for 4DVAR, strong-constraint (NOERR) and weak-constraint (ERRp4) experiments, respectively. In bothexperiments, model error was neglected during the

forecast time. Figure 11c shows NCEP stage-IV mul-tisensor 12-h precipitation accumulation valid at thesame time, used as verification. Caution should be tak-en when using this verification, because there are data


FIG. 12. The 36-h forecast of the 24-h accumulated precipitation (mm), valid at 1200 UTC 4 May 1999, for (a) 4DVAR, strong constraint(NOERR); (b) 4DVAR, weak constraint (ERRp4); (c) NCEP 4-km rain gauge analysis of 24-h accumulated precipitation valid for the sametime (RFC analysis); and (d) 3DVAR.

gaps present (white squares). Also, as mentioned ear-lier, observations might be biased. For reference, wealso show the 3DVAR forecast result in Fig. 11d. Ob-viously, 4DVAR (ERRp4) produced considerably bet-

ter results than the 4DVAR (NOERR) experiment. Themaximum in southern and central Oklahoma is morepronounced and in better agreement with the obser-vations (Fig. 11c), although a location error is present.


FIG. 13. The forecast difference for 4DVAR experiments with and without model error adjustment (ERRp4 2 NOERR experiment) forthe surface CAPE (100 3 J kg21) for (a) 18- and (b) 21-h forecast. Contouring interval is 4 units.

FIG. 14. The forecast difference (ERRp4 2 NOERR experiment) for the vertical wind shear in the 250–700-hPa layer (0.0002 s 21) for (a)18- and (b) 21-h forecast. Contouring interval is 2 units.

The excessive precipitation over Arkansas in the4DVAR (NOERR) experiment (Fig. 11a) was not ob-served. Let us now recall the issue of maximum/min-imum patterns in the 250-hPa wind intensity, discussedwhen comparing 4DVAR versus 3DVAR [see section4b(1)]. As we can see, the excessive precipitation pat-

tern is collocated with the minimum wind pattern overArkansas in Fig. 4b. The 4DVAR (ERRp4) experimentproduced a different maximum/minimum pattern,shifted more to the west (Fig. 3b), which was alsoreflected in more realistic precipitation. Also note that3DVAR experiments did not produce the minimum/


FIG. 15. (a) The 24-h forecast of 12-h accumulated precipitation (mm), valid at 0000 UTC 4 May 1999, for the 4DVAR experiment withoutprecipitation assimilation. (b) The 36-h forecast of 24-h accumulated precipitation (mm), valid at 1200 UTC 4 May 1999, for the 4DVARexperiment without precipitation assimilation.

FIG. 16. The forecast difference for 4DVAR experiments with and without precipitation assimilation for the surface CAPE (100 3 Jkg21) for (a) 18- and (b) 21-h forecast. Contouring interval is 4 units.


FIG. 17. The forecast difference for the vertical wind shear in the 250–700-hPa layer (0.0002 s21) for (a) 18- and (b) 21-h forecast.Contouring interval is 2 units.

maximum upper-level wind pattern in this area at all(Fig. 3a), and, as a result, weak precipitation was fore-cast in this area (Fig. 11d).

Let us now examine the 36-h precipitation forecast.Figures 12a,b show the 36-h forecasts of the 24-h ac-cumulated precipitation, valid at 1200 UTC 4 May 1999,obtained in the 4DVAR (NOERR) and 4DVAR (ERRp4)experiments, respectively. As before, in both experi-ments the model error is neglected during the forecastafter data assimilation. In Fig. 12c the NCEP 4-km raingauge analysis of 24-h accumulated precipitation validfor the same time [River Forecast Centers (RFC) anal-ysis; Baldwin and Mitchell (1997)] is plotted. Note thatthis precipitation observation database is of better qual-ity than the hourly multisensor precipitation observa-tions used in previous considerations and in data assim-ilation. For that reason, one can have more confidencein the verification field presented in Fig. 12c. The datagaps (white squares) are unfortunately present in thisdataset as well. For comparison, we also use the pre-cipitation forecast obtained from the 3DVAR experi-ment, given in Fig. 12d. By comparing Figs. 12a and12b with Fig. 12c (verification) we notice similar fea-tures as before: in both 4DVAR experiments the max-imum precipitation amounts are reasonably well pre-dicted, but the precipitation patterns are shifted to thesouth. Also, the strong-constraint 4DVAR experimentproduced too much precipitation over Arkansas. Thiswas considerably improved by adjusting the model error(weak-constraint 4DVAR experiment in Fig. 12b). The3DVAR experiment suffered from both sources of error:a too-weak precipitation maximum and the fact that the

precipitation pattern is shifted to the south (Fig. 12d).These results are cited and discussed in the review paperon mesoscale 4DVAR data assimilation by Park andZupanski (2002).

To examine further the model error impact we con-sider some other important fields in this severe-weatherevent. For example, we would like to find out if themodel error had any impact on the surface CAPE, aswell as on the midlevel wind shear. In Figs. 13a,b aforecast difference (ERRp4 2 NOERR experiment) forthe surface CAPE is shown for 18-h and 21-h forecasttimes, respectively. The largest differences are locatedin southern Texas and Louisiana at 18 h (Fig. 13a), andlater at 21 h (Fig. 13b) the maximum difference is ob-served in Oklahoma, indicating more CAPE in theERRp4 experiment. It is therefore confirmed that themodel error adjustment had an important (and positive)contribution in forecasting CAPE correctly in the areawhere the actual tornado episode took place, eventhough the difference pattern at 18 h does not necessarilyhave any relation to this event.

The impact of model error on midtropospheric ver-tical wind shear (250–700-hPa layer) is seen in Figs.14a,b, which show the 18- and 24-h forecast differences,respectively, for the ERRp4 2 NOERR experiment. Theshear values in these figures are multiplied by 5000.Again, we notice that something is happening in thearea of interest. In this case the differences are not onlylocalized in this region. For the tornado outbreak area,it is especially interesting to notice a dipole pattern inthe Oklahoma, Arkansas, and Missouri region. Recallfrom the previous figures that large differences in the


precipitation forecasts were found between the two ex-periments in this area. This result is just another con-firmation that the model error adjustment indeed playedan important role in this event.

It is important to point out that the weak-constraint4DVAR experiments presented here were able to reducethe problem of model error, but the problem was noteliminated completely. Even the best forecast was stillfar from perfect. This situation is partly because themethod was designed to account for the model erroronly during the data assimilation period, and the modelerror during the forecast time was completely neglected.It is not obvious, however, how the model error can bepredicted, especially if it is progressing as fast as ourexperiments indicated. In addition, a predictability issuemay be an obstacle for further forecast improvementswhen smaller-scale processes and longer forecast timesare considered. These are the issues that need more at-tention in future research.

In the next section the impact of precipitation assim-ilation is examined, following similar guidelines as forthe model error impact.

3) IMPACT OF PRECIPITATION ASSIMILATION

The 4DVAR experiments presented in the previousfigures were all carried out with inclusion of assimila-tion of precipitation observations. For comparison, wealso performed 4DVAR data assimilation experimentswithout precipitation assimilation. In Fig. 15a the 24-hforecast of the 12-h accumulated precipitation valid at0000 UTC 4 May 1999 is plotted (cf. Figs. 11a–c).Figure 15b shows the 36-h forecast of 24-h accumulatedprecipitation (comparable to Figs. 12a–c). As the figuresindicate, the differences between 4DVAR experimentswith and without precipitation are small. It is not evenclear that the impact was positive in the area of interest.One of the reasons for this is the fact that the precipi-tation amounts associated with this event were sporadicand small at the time, 36–24 h prior to the event. Inother experiments performed using the same NCEP4DVAR data assimilation system, such as in the caseof the East Coast blizzard of 2000, a substantial positiveimpact from precipitation assimilation was obtainedeven after 36 h of forecast time. In this case, the con-ditions were very different: a large area of precipitationassociated with the event of interest was observed andassimilated, thus providing both a substantial improve-ment during data assimilation and improvements in thesubsequent forecast. These results will be reported anddiscussed in a forthcoming paper.

Also examined was the impact of precipitation assim-ilation on surface CAPE and the midlevel vertical wind-shear forecast. In Figs. 16a,b the surface CAPE forecastis shown (18 and 21 h, respectively, to be comparedwith Figs. 13a,b for the model error impact). As thefigures indicate, there is a similar signal in the 18-h

field, as compared with model error effect (Fig. 13a),thus indicating a ‘‘move in the right direction.’’ At alater time (Fig. 16b), this precipitation assimilation sig-nal is less pronounced than the model error impact (Fig.13b) in the area of interest.

For the impact on vertical wind shear (Figs. 17a,b),we compare the forecast differences with the corre-sponding figures for model error impact (Figs. 14a,b).Considerable impact of precipitation assimilation is seenbut is located away from the area of interest (there wereother active regions throughout the model domain). Inassociation with the tornadic event, we notice a positivearea in the wind shear over Arkansas at 1800 UTC (Fig.17a), which became more pronounced later, at 2100UTC (Fig. 17b). Comparison of Fig. 12b (experimentwith precipitation assimilation) and Fig. 15b (withoutprecipitation assimilation) indicates a slight differencein the precipitation forecast patterns in this area, thusconfirming some impact of precipitation assimilation.Because of large location error in both experiments, itis not obvious whether the impact of precipitation as-similation was positive or negative.

5. Conclusions

NCEP’s fine-resolution 4DVAR data assimilation sys-tem is used to study the Great Plains tornado outbreakof 3 May 1999. The data from the NCEP operationaldatabase, including hourly precipitation observations,are assimilated during a 12-h data assimilation interval,36–24 h prior to the event. The operational Eta Model(at the time the experiments were performed) and itsadjoint are used in the study.

The experimental results indicate that the 4DVARmethod is very well suited to initialize the forecast ina synoptic situation similar to this tornado event. It per-formed very well in capturing the important precursorsof the tornadic activity, such as upper- and low-level jetstreaks, humidity field, surface CAPE, vertical windshear, and so on.

The experiments performed showed little forecastsensitivity to precipitation assimilation. The reason forthat was the fact that the assimilated precipitation, priorto the tornado episode, was sporadic and light.

On the other hand, it was shown that the model errorplayed a substantial role in this particular case. The4DVAR results using the forecast model as a weak con-straint (allowing the model error to adjust along withthe initial conditions) were superior to the strong-con-straint (model error neglected) 4DVAR results. The re-sults of this study may be used as a strong argumentagainst the common practice of neglecting the modelerror in data assimilation algorithms.

Our experimental results indicate that the model erroris a fast-moving structure, moving with the internalgravity wave speed. This is a new finding regarding themodel error nature, not observed before. It remains tobe seen if this finding is generally applicable (e.g., for


various models and synoptic situations). These issueswill be studied in our future work.

Acknowledgments. This research was fully supportedby the NOAA/NCEP/UCAR Visiting Scientist Pro-grams. Thanks are given to the many people of NCEP/EMC, including Y. Lin, S. Saha, J. Alpert, P. Kaplan,and T. Black, just to mention a few, who helped usduring different stages of the study and to Dr. S. Lord,the director of the EMC, for giving us encouraging sup-port. We are also very thankful to Dr. K. Droegemeier,the chair of the National Symposium on the Great PlainsTornado Outbreak of 3 May 1999, who gave us the ideato study this event. Extremely valuable comments fromthe three anonymous reviewers are highly appreciated.

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Fine-Resolution 4DVAR Data Assimilation for the Great Plains