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  • View in gallery

    Schematic diagram of the 4DVAR data assimilation algorithm, including forecast and adjoint model runs. Precipitation observations are assimilated every hour. All other observations are assimilated in 3-hourly intervals. Data assimilation period is 12 h

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    The 18-h forecasts of the 250-hPa wind field (kt) valid at 1800 UTC 3 May 1999, initiated using (a) 3DVAR and (b) weak-constraint 4DVAR data assimilation algorithm. (c) The EDAS verification analysis. Contouring interval for wind intensity is 5 kt

  • View in gallery

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

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

  • View in gallery

    The 18-h forecast of the surface CAPE (J kg–1; dots and shading) and vertical wind shear (0.0002 s–1; solid lines) in the 250–700-hPa layer, and the 1000-hPa wind (m s–1; barbs: full barb is 10 m s–1) valid at 1800 UTC 3 May 1999, initiated using (a) 3DVAR and (b) the weak-constraint 4DVAR data assimilation algorithm. (c) The EDAS verification analysis. Contouring interval for wind shear is 5 units

  • View in gallery

    As in Fig. 5 but for the 24-h forecasts and corresponding EDAS verification (valid at 0000 UTC 4 May 1999)

  • View in gallery

    The 18-h forecast of the 850-hPa relative humidity (%; dots and shading), temperature (°C; solid lines), and wind (m s–1; full barb is 10 m s–1) valid at 1800 UTC 3 May 1999, initiated using (a) 3DVAR and (b) the weak-constraint 4DVAR data assimilation algorithm. (c) The EDAS verification analysis. Contouring interval for temperature is 2°C

  • View in gallery

    As in Fig. 7 but for the 24-h forecasts and corresponding EDAS verification (valid at 0000 UTC 4 May 1999)

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    A series of optimized random error terms ri for surface pressure (0.1 × Pa), 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. Contouring interval is 4 units

  • View in gallery

    A series of optimized random error terms ri for wind (0.001 × 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 is 2 units

  • View in gallery

    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 (ERR_4); (c) NCEP stage-IV multisensor 12-h precipitation accumulations valid at the same time; and (d) 3DVAR

  • View in gallery

    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 (ERR_4); (c) NCEP 4-km rain gauge analysis of 24-h accumulated precipitation valid for the same time (RFC analysis); and (d) 3DVAR

  • View in gallery

    The forecast difference for 4DVAR experiments with and without model error adjustment (ERR_4 − NOERR experiment) for the surface CAPE (100 × J kg–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 4 units

  • View in gallery

    The forecast difference (ERR_4 − NOERR experiment) for the vertical wind shear in the 250–700-hPa layer (0.0002 s–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 2 units

  • View in gallery

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

  • View in gallery

    The forecast difference for 4DVAR experiments with and without precipitation assimilation for the surface CAPE (100 × J kg–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 4 units

  • View in gallery

    The forecast difference for the vertical wind shear in the 250–700-hPa layer (0.0002 s–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 2 units

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

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  • 1 NOAA/NCEP/UCAR Visiting Scientist Programs, Camp Springs, Maryland
  • | 2 NOAA/NCEP/EMC, Camp Springs, Maryland
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Abstract

The National Centers for Environmental Prediction fine-resolution four-dimensional variational (4DVAR) data assimilation system is used to study the Great Plains tornado outbreak of 3 May 1999. It was found that the 4DVAR 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 adjustment played a substantial role. The experimental results suggest that the common practice of neglecting the model error in data assimilation systems may not be justified in synoptic situations similar to this one.

Current affiliation: Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Corresponding author address: Dusanka Zupanski, Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO 80523-1375. Email: zupanski@cira.colostate.edu

Abstract

The National Centers for Environmental Prediction fine-resolution four-dimensional variational (4DVAR) data assimilation system is used to study the Great Plains tornado outbreak of 3 May 1999. It was found that the 4DVAR 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 adjustment played a substantial role. The experimental results suggest that the common practice of neglecting the model error in data assimilation systems may not be justified in synoptic situations similar to this one.

Current affiliation: Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

Corresponding author address: Dusanka Zupanski, Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, CO 80523-1375. Email: zupanski@cira.colostate.edu

1. Introduction

One of the most advanced data assimilation methods today, the four-dimensional variational (4DVAR) method (e.g., Lewis and Derber 1985; LeDimet and Talagrand 1986; Courtier and Talagrand 1987; Thépaut and Courtier 1991; Navon et al. 1992; Zupanski 1993a,b, 1997; Zou et al. 1993; Zou and Kuo 1996), has been under very active investigation in the last 10–15 years. It has become the operational data assimilation method at the European Centre for Medium-Range Weather Forecasts (ECMWF; Rabier et al. 1998, 2000; Klinker et al. 2000; Mahfouf and Rabier 2000) and Météo-France (Gauthier and Thépaut 2001). As an operational method, it was rigorously tested and has proven to work well for larger-scale atmospheric problems.

The issue of mesoscale data assimilation has received less attention. In recent years, however, the research in this area has become more intensive (e.g., Sun and Crook 1994, 1997, 1998, 2001; Park and Droegemeier 1997, 2000; Zou and Xiao 2000; Zou et al. 2001; Guo et al. 2000), and it will probably gain even more popularity as computer power increases. It has become obvious that in the context of mesoscale and storm-scale data assimilation a number of problems require additional research. First of all, the method needs to combine in an optimal way the forecast model with numerous high-resolution observations (e.g., radar moist microphysical parameters and velocity, satellite cloud, soil moisture, lightning, GPS moisture information) to produce realistic initial mesoscale features. Observation errors for each data type are required as an input to the data assimilation problem. In cases of high temporal and spatial data resolution (e.g., satellite and radar observations) observation errors may be correlated (Daley 1992). To take into account correlated observations, the observation error covariance matrix should be defined to include off-diagonal components. The validity of the quasigeostrophic balance constraints, successfully used in larger-scale data assimilation problems (e.g., Parrish and Derber 1992; Courtier et al. 1998; Derber and Bouttier 1999), needs to be reexamined. Model error has to be considered. These issues are just some examples of problems encountered in mesoscale data assimilation. More information about mesoscale 4DVAR data assimilation methods can be found in a review paper by Park and Zupanski (2002).

The focus of this study is the tornado outbreak that occurred across Oklahoma and Kansas on 3 May 1999. We apply the 4DVAR method to assimilate observations 36–24 h prior to the event and examine the forecast results after data assimilation. Even though this study makes an attempt to improve our understanding regarding this specific tornado episode, the grid spacing of chosen forecast and adjoint models (32 km/45 layers) provides only a limited capability to examine the tornado itself. Rather, it offers an insight about the nature and benefits of a 4DVAR method in application to a severe-weather event. Indeed, smaller-scale processes (50–100 km) are often mislocated in the forecast results because they are unresolved by the model (scales between meso-α and meso-β are actually resolved). Nevertheless, the method is capable of producing realistic larger-scale precursors of tornadic activity [e.g., low- and upper-level jet streaks, convective available potential energy (CAPE), wind shear], 24–36 h in advance of the event, which may serve as a warning to field forecasters. Benefits of some unique features of the National Centers for Environmental Prediction (NCEP) mesoscale 4DVAR, such as model error and precipitation assimilation, are examined in more detail in this case.

After a short theoretical background review given in section 2, we proceed with describing NCEP's 4DVAR algorithm in section 3. In section 4, experimental results are presented and discussed. In section 5, conclusions are drawn.

2. Theoretical background

Theoretical background for the 4DVAR data assimilation method comes from optimal control theory (Lions 1971; LeDimet and Talagrand 1986; Tarantola 1987) and Kalman filtering (Kalman 1960; Jazwinski 1970). In application to atmospheric data assimilation problems, the 4DVAR method seeks an optimal solution to the atmospheric state by combining the forecast model, observations, and first-guess (e.g., forecast from the previous data assimilation cycle) information (Lorenc 1986). This goal is achieved through an iterative minimization of a cost function, measuring forecast error. The minimization problem is defined over a time period (data assimilation interval), during which all available observations are taken into account at the observation locations and in the form of directly observed quantities. As input, the method requires an estimate of the observation and first-guess errors in the form of error covariance. The observation error covariance matrix is commonly assumed to be diagonal (uncorrelated observations), whereas the background error covariance typically includes off-diagonal elements to account for space and/or time correlations and for cross correlations between different variables.

The forecast model is used as a dynamical constraint [strong or weak, according to Sasaki (1970a,b)], nonlinearly linking different atmospheric variables. The adjoint model (conjugate transpose of a linearized forecast model) uses observations as forcing while running backward in time to provide the gradient of the functional at the beginning of the data assimilation interval. This information is used as a main input to the minimization algorithm (Gill et al. 1981).

In general, the control variable of the 4DVAR problem can be any model input parameter capable of influencing the forecast at the end of the data assimilation interval. It typically includes initial conditions of the forecast model at the beginning of the data assimilation interval. It can also include some other model input parameters, such as empirical constants, as well as model error, boundary conditions, and so on. All these variables are assumed to be independent elements of a large-size control vector (typically on the order of 107–109). Even though the control vector (also referred to as the augmented control variable) includes components of a different nature (such as initial conditions and model error), the variational formalism provides an optimal solution to each component through a single algorithm. The control variable is iteratively adjusted during the course of minimization until the convergence is reached and the minimum solution is obtained.

The basic assumption of the 4DVAR method is that the optimal solution will stay optimal during the forecast time and, therefore, will provide both optimized initial conditions and an improved forecast. This depends on prior information, sometimes poorly known (first-guess forecast, model and observation error covariances, gravity wave amount, errors in the gradient due to linearization and/or discontinuity, etc.), as well as on the predictability of a particular atmospheric process.

A schematic diagram of a data assimilation scheme is given in Fig. 1. The specific example includes assimilation of precipitation observations over a 12-h data assimilation interval, as used in the experiments presented later. Specific components of NCEP's 4DVAR algorithm, used in the experiments, are explained in the next section.

3. NCEP's 4DVAR algorithm

NCEP's fine-resolution 4DVAR algorithm (Zupanski 1993a,b, 1996, 1997; Zupanski and Mesinger 1995; Zupanski and Zupanski 1995) is used in this study. It includes the following components: forecast model, observations, cost function, adjoint model, model error, minimization, and preconditioning. Details of each component of the 4DVAR algorithm are given below.

a. Forecast model

NCEP's 32-km, 45-layer Eta Model (Mesinger et al. 1988; Janjić 1990, 1994; Łobocki 1993; Black 1994; Janjić et al. 1995; Zhao et al. 1997; Chen et al. 1997) is used in this study. At the time the experiments were performed (autumn of 2000), this was the operational model. In September of 2000, the resolution of the Eta Model was increased to 22 km and 50 layers (Rogers et al. 2000). In August of 2001 the nudging of precipitation observations (Lin et al. 1999) was included in the Eta Data Assimilation System (EDAS).

The same forecast model is used in both data assimilation and forecast experiments (except for the fact that the model error is accounted for during the data assimilation interval, as will be explained later in this section). Only the nonlinear forecast model (no tangent-linear assumption) is used to describe forecast evolution. We believe that the nonlinear aspect is an important ingredient of the 4DVAR algorithm in application to cases of mesoscale severe-weather events, because these processes are changing fast and are often highly nonlinear.

b. Observations

The 4DVAR uses the NCEP operational EDAS database, including rawinsondes, aircraft reports, profiler winds, pibal winds, cloud-tracked Geostationary Operational Environmental Satellite (GOES) winds, surface land and marine observations, Special Sensor Microwave Imager (SSM/I) wind speeds, dropwindsondes, National Oceanic and Atmospheric Administration (NOAA) Television and Infrared Observation Satellite Operational Vehicle Sounder (TOVS) temperature retrievals, and GOES and SSM/I precipitable water. In addition, in the 4DVAR experiments, NCEP hourly multisensor (radar and rain gauge) stage-IV, 4-km precipitation data (Baldwin and Mitchell 1997) are used. No quality 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 assimilation, some caution should be taken. Since August of 2001, the multisensor precipitation is used operationally in EDAS via a nudging technique (Lin et al. 1999). The observation types used in 4DVAR data assimilation experiments are listed and briefly explained in Table 1. The observation error covariance is assumed to be diagonal in space and time and to be time independent, depending on the observation type only. This assumption is commonly made in data assimilation studies, although it may not be always justified, such as in the cases of dense and frequent (satellite) observations (Daley 1992).

c. Cost function

The cost function minimized in the 4DVAR algorithm, denoted J, is given by
i1520-0434-17-3-506-e1
The augmented control variable of the 4DVAR problem, denoted z, includes initial conditions, model error, and lateral boundary conditions [as in Zupanski (1997)]. It includes the following state variables: surface pressure, temperature, east–west u and north–south υ wind components, and specific humidity. The vector y denotes observations, the subscript b refers to the background variable, and N defines the number of observation times in the assimilation period; 𝗕 is the background error covariance, including two components—1) forecast error covariance and 2) model error covariance; 𝗣 is the gravity-wave penalty covariance; 𝗥 is the observation error covariance; H is the nonlinear observation operator; and M is the nonlinear forecast model. In addition, F stands for the gravity-wave filter operator (Lynch and Huang 1992; Huang and Lynch 1993) at time t0 (central time of the filter). We choose the time span for the filter to be 2 h, thus making the central time t0 take 1 h after the beginning of data assimilation. During this initial time period of 2 h, a digital filter is applied (as a weak constraint) to smooth out high-frequency time oscillations. The digital filter operator is applied in a similar manner as in Gauthier and Thépaut (2001).

The background error covariance, for both the forecast and the model error, is defined using a compactly supported space-limited analytical function as in Gaspari and Cohn (1999). The only difference between the two components of the covariance matrix is in magnitude (the ratio between model error variance and forecast error variance is defined empirically as 10–4) and in decorrelation lengths (approximately 30%–50%-shorter decorrelation lengths are assigned to model error). The background error covariance is assumed to be univariate, isotropic, and homogeneous at the initial time of the data assimilation interval. From the beginning to the end of the data assimilation interval, the forecast error covariance evolves into a complex flow-dependent multivariate function, thus achieving more realistic forecast error features at the end of assimilation (actual analysis time). This specific aspect of the 4DVAR method is explained in Thépaut et al. (1996).

d. Adjoint model

The adjoint model includes all dynamical subroutines of the Eta Model as well as most of the physics (radiation and soil-hydrology routines are not included in the adjoint, but they are used in the forward forecast model). The adjoint model has the same time and gridpoint resolution as the forecast model (no coarse-resolution approximation is used when calculating the gradient, as is done in the operational ECMWF and Météo-France 4DVAR applications). The fine-resolution gradient is more expensive to calculate, but it can provide more accurate and, perhaps, critically important information in cases of mesoscale severe-weather events.

To regularize the minimization problem, discontinuous on–off switches of the cumulus convection are treated by applying a smooth function as in Zupanski and Mesinger (1995). Also, the K-theory approximation is used in the tangent-linear and adjoint models, instead of using the Mellor–Yamada 2.5-level turbulence closure approach as in the nonlinear model. The adjoint of the Eta Model is developed using the automatic tangent-linear adjoint model compiler (TAMC) of Giering and Kaminski (1998) and Giering (1999).

e. Model error

The random part of the model error (denoted r) is a component of the augmented control variable z in (1). The total model error ϕ is defined as a first-order Markov process variable and includes both random and serially correlated parts, as in Zupanski (1997). It is applied as an additive correction to the model's equations once per time step, according to the following equations:
i1520-0434-17-3-506-e2
where x is the model state variable, r is a random model error vector, ϕ is a serially correlated model error, and α and β are weighing constants for serially correlated and random parts, respectively. Index t refers to model's time step, and Imax is the total number of random model error vectors defined during the assimilation period. Although the control variable includes only the random error component, the prescribed relation (3) allows for systematic model error adjustment. In our experiments, the initial value of ϕ is set to zero. The choice of Imax = 1 is equivalent to adjusting only the systematic model error as in Derber (1989), because the model error asymptotically approaches the value of r as the integration time progresses. In the experiments presented, we chose Imax = 4, thus allowing for time-changing model error [every 3 h a new optimized random error r is inserted into (3) during a 12-h data assimilation interval]. Constants α and β are given empirical values, 0.67 and 0.33, respectively, thus giving considerably more weight to the serially correlated part as compared with the random part.

In our 4DVAR algorithm, the model error (2)–(3) is assumed to include two different components: 1) the error of the forecast model inside the integration domain and 2) the boundary condition error at the domain lateral boundaries. Through this assumption, the lateral boundaries are also considered to be imperfect and are adjusted during the minimization.

f. Minimization and preconditioning

The memoryless quasi-Newton minimization algorithm of Nocedal (1980), with the restart procedure of Shanno (1985), developed by Zupanski (1993b, 1996), is used in the experiments. It includes the following change of variable (preconditioning):
zzb1/2–1/2ζ
where ζ is the transformed (nondimensional) control variable, 𝗕 is a background error covariance, and 𝗗 represents a positive-definite, case-dependent, empirical, diagonal matrix.

4. Experimental results

a. Experimental setup

The observations are assimilated during the 12-h interval from 1200 UTC 2 May to 0000 UTC 3 May 1999, approximately 36–24 h prior to the tornado event. The data insertion frequency is 1 h for precipitation observations and 3 h for all other observations. A schematic diagram of the data assimilation algorithm is given in Fig. 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 observations and use the perfect model assumption (strong constraint), to assess the impact of a specific 4DVAR component in this synoptic case. To evaluate 4DVAR results, we use the NCEP operational EDAS 3DVAR data assimilation method (Parrish et al. 1996; Rogers et al. 1996, 1997) as a control. The 3DVAR analysis is performed every 3 h during the same 12-h data assimilation interval as in the 4DVAR case. The 3DVAR and 4DVAR experiments use the same observations, except for precipitation data, which are used only in 4DVAR. The same first-guess forecast, valid at time t − 12 (Fig. 1), taken from EDAS, is used in both 3DVAR and 4DVAR experiments. In all experiments, the same Eta Model (NCEP's former operational code) is used to produce the 48-h forecast after the data assimilation. During the 48-h forecast, the model error is assumed to be equal to zero in all experiments. For verification, we use EDAS 3DVAR analyses and the stage-IV precipitation observations. For the purpose of having an independent analysis as verification, we also made use of Advanced Regional Prediction System (ARPS) analyses (Brewster et al. 1994; Xue et al. 1995), during the course of the investigation. The ARPS analyses were kindly provided by the Center for Analysis and Prediction of Storms as a part of a research database for this tornado event.

The number of minimization iterations performed in all 4DVAR experiments is 10. The improvements after 10 iterations were typically marginal.

b. Results

1) 4DVAR versus 3DVAR

Figures 2a,b show the 18-h forecast of the 250-hPa wind valid at 1800 UTC 3 May 1999, initiated using the 3DVAR and 4DVAR data assimilation algorithms, respectively. This 4DVAR experiment used both precipitation assimilation and model error adjustment [four different random error terms are adjusted during the 12-h data assimilation interval, i.e., Imax = 4 in (3)]. This experiment was named 4DVAR (ERR_4). In Fig. 2c, the corresponding EDAS analysis, used as verification, is presented. Figures 3a–c show the same fields, but 6 h later (valid at 0000 UTC 4 May 1999). As seen in the 1800 UTC EDAS analysis (Fig. 2c), there was a pronounced jet streak in central Texas, extending into western Oklahoma and Kansas, across the Texas–New Mexico border. Over southern and central Oklahoma, a local minimum of upper-level wind existed. As we compare the analysis 6 h later (0000 UTC 4 May 1999, when the tornado episode had already started), a wind speed minimum, though weaker, is still seen in southern and central Oklahoma. The axis of maximum wind has moved over New Mexico, and a number of additional smaller-scale wind minima and maxima appeared across Texas, Oklahoma, and Kansas. A diffluent flow pattern over eastern New Mexico and western Texas has formed. These features, if present in the forecast, would be a first warning to a forecaster that a severe-weather event is about to happen. For example, M. Branick, the lead forecaster for the National Weather Service (NWS) in Norman, Oklahoma, noticed that the 250-hPa short-wave wind patterns “dig well south into the southwestern states, and move rapidly as they ejected eastward into the central states…. By late in the day, the ingredients for a significant severe weather outbreak began to come together over the southern half of the Plains. The upper-level jet turned east across New Mexico and then northeast across Oklahoma and Kansas by early evening, with a local 75-knot speed maximum near the Kansas–Oklahoma border.” (from the NWS Norman Web site: http://www.srh.noaa.gov/oun/storms/19990503/). A comparison of the forecast results from 3DVAR and 4DVAR experiments (Figs. 2a,b and 3a,b) with the verification (Figs. 2c and 3c) shows that both forecast experiments underpredicted the jet maximum across the New Mexico–Texas border, extending to Oklahoma and Kansas (Fig. 2c). More careful examination of the figures indicate that the 3DVAR experiment produced a stronger wind maximum in this area, which is in better agreement with the verification. The minimum over southern and central Oklahoma is missing in both experiments. 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 experiment over Oklahoma, though the location error is present. Both the 3DVAR and 4DVAR forecast missed the minimum/maximum patterns over Texas and Kansas. By examining the low-level wind field (850 hPa), we also noticed an indication of a short-wave minimum/maximum wind pattern in the 4DVAR experiment, in better agreement with the verification, as compared with the 3DVAR experiment (figures not shown). Similar minimum/maximum patterns were also noticed in the ARPS wind analyses (250 and 850 hPa).

One cannot necessarily assume that these forecasted maximum/minimum wind patterns may be of some significance to this specific event, because there is a substantial mismatch between the model resolution and the scale of the event, resulting in a considerable location error. As explained in the introduction, the scales below 100 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 4DVAR experimental results, without model error adjustment. We refer to this experiment as strong constraint or 4DVAR (NOERR). Figures 4a and 4b show 18- and 24-h forecasts of the 250-hPa wind field, respectively. We notice an indication of a slight short-wave trough in eastern Oklahoma at 1800 UTC, and a minimum/maximum pattern moved to eastern Oklahoma, Arkansas, 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 link exists between the wind pattern and the severe-weather event (which may not necessarily be a tornado event). This link is nicely seen when comparing 3DVAR and 4DVAR (ERR_4) precipitation with the upper-level wind. Again, the minimum/maximum wind pattern is correlated well with intensive precipitation in the 4DVAR (ERR_4) experiment (figures presented in the next section). It is also important to note that the 4DVAR experiment with model error adjustment produced considerably more realistic forecast results than the one without model error adjustment.

Let us now examine some other atmospheric parameters that played an important role as precursors for tornadic activity. As noted in Thompson and Edwards (2000), important atmospheric conditions associated with the Oklahoma–Kansas tornado outbreak of 3 May 1999 were the midtropospheric (4–10 km) vertical wind shear, low-level wind shear, and surface-based CAPE. It was also demonstrated in Hamill and Church (2000) that a probabilistic model based on CAPE and low-level wind shear, as well as on CAPE and helicity, was able to produce a realistic forecast for this tornado event. In the following figures we present and discuss surface-based CAPE and midlevel vertical wind shear as important ambient conditions associated with this severe-weather event. In Figs. 5a,b we present the surface CAPE and midtropospheric (250–700-hPa layer) vertical wind shear, valid at 1800 UTC 3 May 1999, obtained as an 18-h forecast after 3DVAR and 4DVAR (ERR_4) data assimilation, respectively. As an illustration of surface flow, the 1000-hPa winds are also plotted. In Fig. 5c, the EDAS verification valid at the same time is given. Figures 6a–c show the same fields as Figs. 5a–c, respectively, but 6 h 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. 5c and 6c), large CAPE (exceeding 3000 J kg–1) was developing during this 6-h time interval, associated with minimum/maximum patterns in wind shear. At 0000 UTC 4 May 1999, the CAPE maximum split into two parts, one over Texas, and the other over north Oklahoma and Kansas. The comparison of forecast CAPE in the 3DVAR and 4DVAR forecast experiments reveals a stronger severe-weather event in the 4DVAR experiment at the beginning (18 h) of the interval and the CAPE maximum splitting into two maxima at the end of the interval (24 h). For wind shear, the minimum/maximum wave pattern was poorly predicted in both experiments, with only a slight indication of a short wave in this field over Oklahoma. Because of large errors in both experiments, it is hard to determine which one produced more realistic midtropospheric vertical wind shear. It may be that a combination of the two parameters (wind shear and CAPE) was more realistic (dynamically more consistent) in the 4DVAR experiment, because it produced stronger indications of a severe-weather event in other fields (humidity, precipitation, etc.), which were in better agreement with the observations.

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 dryline gradient across western Oklahoma and slightly cooler temperature in the area of interest, again better verified by the EDAS analysis.

Also compared were differences between 3DVAR and 4DVAR analyses at the initial time (0-h forecast). One interesting finding was that the 3DVAR analysis was more humid, and in better agreement with the observations (figures not shown). After only a few hours of forecast time, however, the atmosphere dried out, thus leaving less favorable conditions for supercell development. This is a typical difference between 3DVAR and 4DVAR analyses. The 3DVAR analysis commonly provides a better fit to the observations. The 4DVAR method provides optimal fit to the observations over a period of time, often producing a dynamically more consistent analysis and an improved forecast but also a degraded analysis fit to the observations.

We also compared other forecast fields and parameters (such as convective inhibition, helicity, 500- and 850-hPa wind, and temperature) The 4DVAR results generally gave more indications of a severe-weather event than did the 3DVAR results. Neither experiment, however, was even close to predicting a tornado or a supercell because of a large discrepancy between the model resolution and the tornado scale, which made it difficult to perform a more strict quantitative evaluation. Even so, we found that the 4DVAR method can contribute much to a better understanding of tornado events, especially when some poorly known features, such as the impact of model error and precipitation observations, are studied.

2) The impact of model error

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

General definition of the model error [(2)–(3)] allows for any number of random error terms during the data assimilation interval. The equations indicate that including more random error terms allows for more time variability in the model error (finer timescales). It makes it 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 does not force fast-changing model error components. It just allows for better capturing of fast processes, if any are present in the atmosphere but not resolved by the forecast model.

We performed experiments using only one random error component during the entire data assimilation interval (Imax = 1) and experiments using four random errors (Imax = 4). A decision was made to examine in more detail the results with more random error components, based on the fact that this experiment provided slightly, but consistently, improved forecast results as compared with the experiments with only one random error component. For example, the precipitation pattern over Oklahoma and Arkansas (discussed later) was slightly improved in location and intensity (figure not shown). Both experiments provided considerably better forecast results as compared with the strong-constraint 4DVAR experiment (no model error adjustment), thus giving us more confidence that the optimized model error has some realistic features in it [similar results were obtained in our previous studies, e.g., Zupanski (1997)].

Figures 9a–d show a series of optimized random error terms ri for surface pressure, obtained every 3 h during the 12-h data assimilation interval. The model 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, we notice that the model error term has a very small magnitude (on the order of 1–2 Pa for surface pressure and 0.005–0.01 kt for wind). We also notice that the model error has some organized structures (synoptic and mesoscale), similar in shape and magnitude from one time interval to another. The experimental results with only one random error term (Imax = 1) showed very similar model error shapes, resembling an average model error structure over the data assimilation interval (figures not shown). Note that the 4DVAR problem was posed assuming no time correlation between different random error components (model error covariance allows for space correlations only). Yet, there is a striking similarity between the successive error patterns, thus suggesting that the model error travels throughout the model domain. It can be seen that the error pattern moved approximately the distance of 1500 km during a 12-h period. This translates to an average phase speed of the model error on the order of 35 m s–1. This speed is much greater than the phase speed of any weather system. It is comparable to internal gravity wave speed. This is a new finding regarding the nature of the model error that has never been observed before. At this point it is not clear if this model error feature is related to the specific forecast model (Eta) and specific synoptic case, or has a more general meaning. It is also not known to what extent the particular definition of the model error covariance influenced the optimal solution for model error. These are all very interesting but complicated issues and will be addressed in our future work. Nevertheless, we would argue that the obtained model error features are realistic. The experimental results presented next should support this argument.

Figures 11a,b present 24-h forecasts of 12-h accumulated precipitation, valid at 0000 UTC 4 May 1999, for 4DVAR, strong-constraint (NOERR) and weak-constraint (ERR_4) experiments, respectively. In both experiments, model error was neglected during the forecast time. Figure 11c shows NCEP stage-IV multisensor 12-h precipitation accumulation valid at the same time, used as verification. Caution should be taken when using this verification, because there are data gaps present (white squares). Also, as mentioned earlier, observations might be biased. For reference, we also show the 3DVAR forecast result in Fig. 11d. Obviously, 4DVAR (ERR_4) produced considerably better results than the 4DVAR (NOERR) experiment. The maximum in southern and central Oklahoma is more pronounced and in better agreement with the observations (Fig. 11c), although a location error is present. The excessive precipitation over Arkansas in the 4DVAR (NOERR) experiment (Fig. 11a) was not observed. Let us now recall the issue of maximum/minimum patterns in the 250-hPa wind intensity, discussed when comparing 4DVAR versus 3DVAR [see section 4b(1)]. As we can see, the excessive precipitation pattern is collocated with the minimum wind pattern over Arkansas in Fig. 4b. The 4DVAR (ERR_4) experiment produced a different maximum/minimum pattern, shifted more to the west (Fig. 3b), which was also reflected in more realistic precipitation. Also note that 3DVAR experiments did not produce the minimum/maximum upper-level wind pattern in this area at all (Fig. 3a), and, as a result, weak precipitation was forecast 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 accumulated precipitation, valid at 1200 UTC 4 May 1999, obtained in the 4DVAR (NOERR) and 4DVAR (ERR_4) experiments, respectively. As before, in both experiments the model error is neglected during the forecast after data assimilation. In Fig. 12c the NCEP 4-km rain gauge analysis of 24-h accumulated precipitation valid for the same time [River Forecast Centers (RFC) analysis; Baldwin and Mitchell (1997)] is plotted. Note that this precipitation observation database is of better quality than the hourly multisensor precipitation observations used in previous considerations and in data assimilation. For that reason, one can have more confidence in the verification field presented in Fig. 12c. The data gaps (white squares) are unfortunately present in this dataset as well. For comparison, we also use the precipitation forecast obtained from the 3DVAR experiment, given in Fig. 12d. By comparing Figs. 12a and 12b with Fig. 12c (verification) we notice similar features as before: in both 4DVAR experiments the maximum precipitation amounts are reasonably well predicted, but the precipitation patterns are shifted to the south. Also, the strong-constraint 4DVAR experiment produced too much precipitation over Arkansas. This was considerably improved by adjusting the model error (weak-constraint 4DVAR experiment in Fig. 12b). The 3DVAR 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 paper on mesoscale 4DVAR data assimilation by Park and Zupanski (2002).

To examine further the model error impact we consider some other important fields in this severe-weather event. For example, we would like to find out if the model error had any impact on the surface CAPE, as well as on the midlevel wind shear. In Figs. 13a,b a forecast difference (ERR_4 − NOERR experiment) for the surface CAPE is shown for 18-h and 21-h forecast times, respectively. The largest differences are located in southern Texas and Louisiana at 18 h (Fig. 13a), and later at 21 h (Fig. 13b) the maximum difference is observed in Oklahoma, indicating more CAPE in the ERR_4 experiment. It is therefore confirmed that the model error adjustment had an important (and positive) contribution in forecasting CAPE correctly in the area where the actual tornado episode took place, even though the difference pattern at 18 h does not necessarily have any relation to this event.

The impact of model error on midtropospheric vertical wind shear (250–700-hPa layer) is seen in Figs. 14a,b, which show the 18- and 24-h forecast differences, respectively, for the ERR_4 − NOERR experiment. The shear values in these figures are multiplied by 5000. Again, we notice that something is happening in the area of interest. In this case the differences are not only localized in this region. For the tornado outbreak area, it is especially interesting to notice a dipole pattern in the Oklahoma, Arkansas, and Missouri region. Recall from the previous figures that large differences in the precipitation forecasts were found between the two experiments in this area. This result is just another confirmation that the model error adjustment indeed played an important role in this event.

It is important to point out that the weak-constraint 4DVAR experiments presented here were able to reduce the problem of model error, but the problem was not eliminated completely. Even the best forecast was still far from perfect. This situation is partly because the method was designed to account for the model error only during the data assimilation period, and the model error during the forecast time was completely neglected. It is not obvious, however, how the model error can be predicted, especially if it is progressing as fast as our experiments indicated. In addition, a predictability issue may be an obstacle for further forecast improvements when smaller-scale processes and longer forecast times are considered. These are the issues that need more attention in future research.

In the next section the impact of precipitation assimilation is examined, following similar guidelines as for the model error impact.

3) Impact of precipitation assimilation

The 4DVAR experiments presented in the previous figures were all carried out with inclusion of assimilation of precipitation observations. For comparison, we also performed 4DVAR data assimilation experiments without precipitation assimilation. In Fig. 15a the 24-h forecast of the 12-h accumulated precipitation valid at 0000 UTC 4 May 1999 is plotted (cf. Figs. 11a–c). Figure 15b shows the 36-h forecast of 24-h accumulated precipitation (comparable to Figs. 12a–c). As the figures indicate, the differences between 4DVAR experiments with and without precipitation are small. It is not even clear that the impact was positive in the area of interest. One of the reasons for this is the fact that the precipitation amounts associated with this event were sporadic and small at the time, 36–24 h prior to the event. In other experiments performed using the same NCEP 4DVAR data assimilation system, such as in the case of the East Coast blizzard of 2000, a substantial positive impact from precipitation assimilation was obtained even after 36 h of forecast time. In this case, the conditions were very different: a large area of precipitation associated with the event of interest was observed and assimilated, thus providing both a substantial improvement during data assimilation and improvements in the subsequent forecast. These results will be reported and discussed in a forthcoming paper.

Also examined was the impact of precipitation assimilation on surface CAPE and the midlevel vertical wind-shear forecast. In Figs. 16a,b the surface CAPE forecast is shown (18 and 21 h, respectively, to be compared with Figs. 13a,b for the model error impact). As the figures 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 a later time (Fig. 16b), this precipitation assimilation signal 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 corresponding figures for model error impact (Figs. 14a,b). Considerable impact of precipitation assimilation is seen but is located away from the area of interest (there were other active regions throughout the model domain). In association with the tornadic event, we notice a positive area in the wind shear over Arkansas at 1800 UTC (Fig. 17a), which became more pronounced later, at 2100 UTC (Fig. 17b). Comparison of Fig. 12b (experiment with precipitation assimilation) and Fig. 15b (without precipitation assimilation) indicates a slight difference in the precipitation forecast patterns in this area, thus confirming some impact of precipitation assimilation. Because of large location error in both experiments, it is not obvious whether the impact of precipitation assimilation was positive or negative.

5. Conclusions

NCEP's fine-resolution 4DVAR data assimilation system is used to study the Great Plains tornado outbreak of 3 May 1999. The data from the NCEP operational database, 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 its adjoint are used in the study.

The experimental results indicate that the 4DVAR method is very well suited to initialize the forecast in a synoptic situation similar to this tornado event. It performed very well in capturing the important precursors of the tornadic activity, such as upper- and low-level jet streaks, humidity field, surface CAPE, vertical wind shear, and so on.

The experiments performed showed little forecast sensitivity to precipitation assimilation. The reason for that was the fact that the assimilated precipitation, prior to the tornado episode, was sporadic and light.

On the other hand, it was shown that the model error played a substantial role in this particular case. The 4DVAR results using the forecast model as a weak constraint (allowing the model error to adjust along with the initial conditions) were superior to the strong-constraint (model error neglected) 4DVAR results. The results of this study may be used as a strong argument against the common practice of neglecting the model error in data assimilation algorithms.

Our experimental results indicate that the model error is a fast-moving structure, moving with the internal gravity wave speed. This is a new finding regarding the model error nature, not observed before. It remains to be seen if this finding is generally applicable (e.g., for various models and synoptic situations). These issues will be studied in our future work.

Acknowledgments

This research was fully supported by the NOAA/NCEP/UCAR Visiting Scientist Programs. 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 us during different stages of the study and to Dr. S. Lord, the director of the EMC, for giving us encouraging support. We are also very thankful to Dr. K. Droegemeier, the chair of the National Symposium on the Great Plains Tornado Outbreak of 3 May 1999, who gave us the idea to study this event. Extremely valuable comments from the three anonymous reviewers are highly appreciated.

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Fig. 1.
Fig. 1.

Schematic diagram of the 4DVAR data assimilation algorithm, including forecast and adjoint model runs. Precipitation observations are assimilated every hour. All other observations are assimilated in 3-hourly intervals. Data assimilation period is 12 h

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 2.
Fig. 2.

The 18-h forecasts of the 250-hPa wind field (kt) valid at 1800 UTC 3 May 1999, initiated using (a) 3DVAR and (b) weak-constraint 4DVAR data assimilation algorithm. (c) The EDAS verification analysis. Contouring interval for wind intensity is 5 kt

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 3.
Fig. 3.

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

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 4.
Fig. 4.

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

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 5.
Fig. 5.

The 18-h forecast of the surface CAPE (J kg–1; dots and shading) and vertical wind shear (0.0002 s–1; solid lines) in the 250–700-hPa layer, and the 1000-hPa wind (m s–1; barbs: full barb is 10 m s–1) valid at 1800 UTC 3 May 1999, initiated using (a) 3DVAR and (b) the weak-constraint 4DVAR data assimilation algorithm. (c) The EDAS verification analysis. Contouring interval for wind shear is 5 units

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 6.
Fig. 6.

As in Fig. 5 but for the 24-h forecasts and corresponding EDAS verification (valid at 0000 UTC 4 May 1999)

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 7.
Fig. 7.

The 18-h forecast of the 850-hPa relative humidity (%; dots and shading), temperature (°C; solid lines), and wind (m s–1; full barb is 10 m s–1) valid at 1800 UTC 3 May 1999, initiated using (a) 3DVAR and (b) the weak-constraint 4DVAR data assimilation algorithm. (c) The EDAS verification analysis. Contouring interval for temperature is 2°C

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 8.
Fig. 8.

As in Fig. 7 but for the 24-h forecasts and corresponding EDAS verification (valid at 0000 UTC 4 May 1999)

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 9.
Fig. 9.

A series of optimized random error terms ri for surface pressure (0.1 × Pa), 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. Contouring interval is 4 units

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 10.
Fig. 10.

A series of optimized random error terms ri for wind (0.001 × 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 is 2 units

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 11.
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 (ERR_4); (c) NCEP stage-IV multisensor 12-h precipitation accumulations valid at the same time; and (d) 3DVAR

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 12.
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 (ERR_4); (c) NCEP 4-km rain gauge analysis of 24-h accumulated precipitation valid for the same time (RFC analysis); and (d) 3DVAR

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 13.
Fig. 13.

The forecast difference for 4DVAR experiments with and without model error adjustment (ERR_4 − NOERR experiment) for the surface CAPE (100 × J kg–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 4 units

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 14.
Fig. 14.

The forecast difference (ERR_4 − NOERR experiment) for the vertical wind shear in the 250–700-hPa layer (0.0002 s–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 2 units

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 15.
Fig. 15.

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

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 16.
Fig. 16.

The forecast difference for 4DVAR experiments with and without precipitation assimilation for the surface CAPE (100 × J kg–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 4 units

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Fig. 17.
Fig. 17.

The forecast difference for the vertical wind shear in the 250–700-hPa layer (0.0002 s–1) for (a) 18- and (b) 21-h forecast. Contouring interval is 2 units

Citation: Weather and Forecasting 17, 3; 10.1175/1520-0434(2002)017<0506:FRDAFT>2.0.CO;2

Table 1. 

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

Table 1. 
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