## 1. Introduction

During 25 January 2000, a major winter storm affected most of the U.S. East Coast. Heavy rain, mixed with snow and freezing rain, fell in North and South Carolina, producing an enormous natural disaster. Further north, in the Washington, D.C., area, the heavy snow was falling for almost 24 h without stopping, blocking most roads, and causing a closure of businesses. To make things worse, the forecast starting from the 1200 UTC 24 January analysis, which served as guidance for weather forecasters at National Weather Service (NWS) offices and elsewhere, did not predict any significant precipitation over those areas. All operational models did produce correctly a major storm off the coast, but failed to create strong convection and precipitation over Georgia and Carolinas. The operational forecast starting from the 0000 UTC 25 January analysis did produce accurate precipitation. By that time, however, the storm had been already producing significant amounts of precipitation in the Carolinas. In order to better understand the reasons for the failure of the operational weather forecast, a large number of reruns of this storm have been performed at the National Centers for Environmental Prediction (NCEP; results can be found on NCEP's Web site: http://sgi62.wwb.noaa.gov:8080/BLIZZFCST).

One of the most advanced data assimilation methods today is the four-dimensional variational (4DVAR) data assimilation. It has been an operational method at the European Centre for Medium Range Weather Forecasts (ECMWF) since December 1997 (Rabier et al. 2000; Mahfouf and Rabier 2000; Klinker et al. 2000), and at Météo France since June 2000 (Gauthier and Thépaut 2001). The 4DVAR method has been under development at many meteorological centers around the world [Canadian Meteorological Centre, U.K. Met Office, NCEP, National Center for Atmospheric Research (NCAR)], as well as at several universities. The 4DVAR has come a long way since its beginnings in late 1980s and early 1990s (Lewis and Derber 1985; Le Dimet and Talagrand 1986; Talagrand and Courtier 1987; Thépaut and Courtier 1991; Navon et al. 1992; Zupanski, M., 1993a; Courtier et al. 1994, etc.). It employs the adjoint models with physics (Zou et al. 1993; Zupanski D., 1993; Zupanski and Mesinger 1995; Vukicevic and Errico 1993; Tsuyuki 1996a,b, 1997; Mahfouf 1999; Janisková et al. 1999; Zou et al. 2001), and it has been used with nonhydrostatic and cloud models (Verlinde and Cotton 1993; Zou et al. 1995; Sun and Crook 1994, 1996, 1997; Kuo et al. 1996; Zou and Xiao 2000), etc. More advanced 4DVAR algorithms use the forecast model as a weak constraint in variational formalism (Bennett et al. 1993, 1996, 1997; Zupanski 1997). The 4DVAR method is computationally very demanding, however, and for practical reasons, the less general three-dimensional variational (3DVAR) method (e.g., Parrish and Derber 1992; Courtier et al. 1998; Cohn et al. 1998; Daley and Barker 2001; Lorenc et al. 2000) is still widely and successfully used in operations and research.

The impact of initial conditions in this snowstorm case is examined, by using 3DVAR and 4DVAR data assimilation algorithms. In particular, NCEP's operational mesoscale 3DVAR (Parrish et al. 1996) and NCEP's mesoscale 4DVAR (Zupanski, M., 1993a,b, 1996; Zupanski, D., 1993; Zupanski and Mesinger 1995; Zupanski 1997; Zupanski et al. 2002) algorithms are used. Special attention is paid in this study to the impact of model error adjustment and assimilation of precipitation observations in 4DVAR data assimilation. These two components are often neglected in variational data assimilation applications.

Details of the 4DVAR algorithm are described in section 2, synoptic overview, forecast model, observations, and experimental setup are explained in section 3, results are presented in section 4, and conclusions are drawn in section 5.

## 2. 4DVAR algorithm

A 4DVAR algorithm, based on the NCEP mesoscale Eta Model and its adjoint, has been developed (Zupanski, M., 1993a,b, 1996; Zupanski, D., 1993; Zupanski and Mesinger 1995; Zupanski and Zupanski 1995; Zupanski 1997; Zupanski et al. 2002). Specific features of NCEP's mesoscale 4DVAR system are:

it allows for model error adjustment (augmented control variable includes initial conditions, model error—random and bias components, and lateral boundary conditions),

only the nonlinear forecast model is used (no tangent linear model),

it is defined for the limited-area gridpoint model,

it has an advanced, case-dependent preconditioning based on a change of variable, coupled with a robust minimization restart procedure,

the background and model error covariances are modeled using the symmetric, positive-semidefinite, space-limited Toeplitz matrix,

it is developed for the message passing (MPI) parallel computing environment, and

it has a capability to assimilate NCEP's Stage IV hourly precipitation observations (Baldwin and Mitchell 1997).

### a. Cost function

*G*is the nonlinear digital filter operator,

*K*is the nonlinear observation operator, and

*M*is the nonlinear forecast model. The variable

*z*= (

*x,*

*r*

_{1},

*r*

_{2}, …,

*r*

_{L}) is the augmented control variable, adjusted during the minimization of the cost function (2.1). The vector

**x**denotes the initial conditions (analysis),

**x**

_{t}is the model forecast at time

*t,*and

**r**is the random model error component, that will be explained later. Note that

*z*

_{t}=

*M*(

*z*) at time

*t.*The control variable includes the model state variables, that is, surface pressure, temperature, winds, and specific humidity, and the random model error components for the same variables. The vector

**y**denotes the observations, the subscript

*b*refers to the background variable, and

*N*defines the number of observation times in assimilation period. The specific components of the cost function are explained below.

#### 1) Observation Term

The observations are taken from the NCEP operational database. The observation error covariance 𝗿 is diagonal (e.g., variance), dependent on the observation density as in the NCEP operational mesoscale 3DVAR algorithm (Parrish et al. 1996). Larger observation density implies larger observation error, thus smaller weight is assigned to such observations.

#### 2) Background term

_{o}is the forecast error covariance, and 𝗾

_{i}is the

*i*th component of the random model error covariance. In practice, the error covariance is defined through the variance 𝗯

_{D}and the correlation 𝗰 as

^{1/2}

_{D}

^{1/2}

_{D}

The correlations are modeled in a simple, but efficient manner. A convolution of one-dimensional (along each spatial axis) positive-definite Toeplitz matrices represents a correlation matrix. Therefore, separability is assumed. For convenience and simplicity reasons, only the horizontal autocorrelations were modeled, neglecting the correlations in the vertical, as well as the cross correlations between different variables. This may be a limitation of the current system, and work is underway to include the vertical correlations as well. The correlations are approximately homogeneous and isotropic, except for the wind which has a different decorrelation length along two orthogonal wind components. In general, the correlations can be anisotropic along each of the spatial axes. It is assumed that distant correlations are zero. The cutoff (decorrelation) length is empirically defined. In the presented experiments the decorrelation lengths for the background (forecast) error covariance are: 1000 km for surface pressure, 700 km for temperature, 300 km for specific humidity, and for winds 800 km in the direction of the wind and 300 km in the orthogonal direction. For model error covariance the decorrelation lengths are about two to three times shorter: 400 km for surface pressure, 300 km for temperature, 100 km for specific humidity, and for winds 300 and 100 km. A compactly supported, space-limited Second Order Auto-Regressive (SOAR) correlation function (Gaspari and Cohn 1999, their Eq. (4.4)) was employed to form a symmetric, positive-semidefinite Toeplitz matrix (e.g., Golub and Van Loan 1989). In general, the structure of the Topelitz matrix allows only one set of one-dimensional correlation values to be stored for each variable. For the change of variable, however, a square root of this matrix is needed. An eigenvalue decomposition (EVD) of each one-dimensional matrix is performed. Then, the elements of the (symmetric) square root matrix are computed and stored. Note that all this is done offline, as a preparation process. The correlation matrices along a particular axis have small dimensions and are inexpensive to calculate. In our implementation, the standard IBM SP mathematical library EVD and matrix-vector product subroutines are employed. In order to avoid the EVD computation, an alternative way is to define the square root correlation matrix directly. This approach is currently under testing.

To illustrate the effect the model and adjoint have in 4DVAR data assimilation, a 4DVAR experiment with a single temperature observation at the end of the assimilation period is performed. The value of a radiosonde temperature observation, located at 36°N and 80°W, at 830 hPa, is 2.5 K higher than the background field. The observational error for this observation is 0.8°, and the background error is 2.2°. The 4DVAR response at the end of the assimilation period is shown for temperature (Fig. 1a) and *u* wind component (Fig. 1b). As in all other 4DVAR experiments presented here, the assimilation period is 12 h. Recall that at the beginning of the assimilation period, only a simple isotropic and homogeneous autocorrelation of temperature was defined, with no cross correlations. Due to the use of the forecast and adjoint models, however, the 4DVAR does change the initial correlations according to the model dynamics, and it produces the cross correlations as well. By inspecting the 850-hPa temperature analysis response (Fig. 1a), one can note the stretching along the coast, a possible indication of the frontal zone in this area (higher correlations along the frontal zone). The 850-hPa *u* wind analysis response (Fig. 1b) further illustrates the complexity of the 4DVAR analysis error covariance, and the impact of the forecast model integration on cross correlations.

#### 3) Gravity-wave penalty term

_{D}, empirically scaled to produce a reasonable value of the gravity wave component. This value is relatively small, producing about a 100 times smaller gravity wave penalty term than the observation term in (2.1). Using the definition of

*z*

_{t}

*z*

_{t}

*M*

_{t}

*z*

*g*

_{gw}

*M*

^{T}

_{t}

*G*

^{T}

^{−1}

*z*

_{t}

*G*

*z*

_{t}

*z*

_{t}are easily obtained by simply saving the forward nonlinear forecast at time

*t.*Similarly, the adjoint integration

*M*

^{T}

_{t}

*t,*then employing the regular 4DVAR adjoint integration. Therefore, the additional computational cost of the gravity wave calculation in 4DVAR is negligible (Gauthier and Thépaut 2001).

### b. Model error

*r*is a random model error component,

*x*represents model state variable,

*ϕ*is a serially correlated model error,

*α*and

*β*are empirical constants, index

*t*refers to time, and

*L*is the number of random model error vectors defined during the assimilation period. Although the control variable includes only the random error component, the relation (2.7) allows for systematic model error adjustment. For

*L*= 1, only the systematic model error is adjusted.

### c. Change of variable and preconditioning

*z*

*z*

^{b}

^{1/2}

^{−1/2}

*ζ,*

*ζ*is the transformed (nondimensional) control variable, 𝗯 is given by (2.2), and 𝗱 represents a positive-definite diagonal matrix. As shown in the appendix, an ideal change of variable should include a nondiagonal matrix 𝗮, instead of the diagonal matrix 𝗱. In practice, the matrix 𝗮 is difficult to calculate and store. For that reason it is often neglected, thus employing only the square root of the background error covariance as a change of variable. In our 4DVAR implementation, however, the matrix 𝗮 is not neglected. Rather, it is approximated by an empirical, case-dependent diagonal matrix, denoted 𝗱, calculated using current observations and the gradient information. The diagonal elements of 𝗱 are defined following Zupanski, M., (1993b, 1996) as

*b,*‖ · ‖ denotes an

*l*

_{2}norm, and the index

*l*corresponds to each control variable component at each model (analysis) vertical layer. The index

*i*denotes an analysis (model) grid point, and

*C*is an empirical constant. The overbar in (2.9) represents an averaged value over all horizontal points. The symbol

*g*denotes the gradient, and

*F*is the initial cost function. The value of

*F*is the sum of the observation cost function in analysis space,

*F*

^{obs}, and the gravity wave penalty cost function,

*F*

^{pen}, for example,

*F*=

*F*

^{pen}+

*F*

^{obs}. The observation cost function in analysis space,

*F*

^{obs}, is defined as:

*g*

^{obs}, is defined using (2.1) as

*g*

^{obs}

_{n}

*K*

^{T}

_{n}

^{−1}

_{n}

*K*

*M*

*z*

*y*_{n}

*n*refers to observation times, and the subscript tot denotes the total value, summed over all variables. The formula (2.10) gives an analysis space equivalent of the observation cost function, using the gradient forcing from observations and the norm defined by 𝗯

_{D}. The limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) quasi-Newton algorithm (Nocedal 1980) is used in minimization, with the restart procedure of Shanno (1985), described also in Zupanski (1996). Although the quasi-Newton minimization is often characterized by large variations of the gradient (e.g., Liu and Nocedal 1989; Gilbert and Lemarechal 1989), the described minimization and preconditioning produce very smooth convergence. This allows the stopping criteria to be defined at any point of minimization, without any adverse effects.

### d. Adjoint model

The adjoint of the Eta Model is developed using the TAMC (Tangent Linear and Adjoint Model Compiler; Giering and Kaminski 1998; Giering 1999). At present, the adjoint model includes dynamics and partial physics: horizontal and vertical diffusion, large-scale and convective processes, prognostic cloud water scheme, surface processes, and turbulence. The radiation package and the soil model are not included in the adjoint model, but are used in the forward (forecast) run. To regularize the minimization problem a smoothing is introduced in cumulus convection (e.g., Zupanski and Mesinger 1995) and K-theory approximation is used in the adjoint of turbulence (instead of prognostic turbulent coefficients as in the forward model).

## 3. Experimental design

### a. Synoptic overview

By 0000 UTC 24 January 2000, a strong upper-level low has formed over the eastern United States. A deep low at upper levels has stretched south to reach the Gulf of Mexico, with peak jet streak values over the Louisiana and Mississippi coasts. At the same time, an exit area of the upper-level jet, associated with a previous system, was located over the Carolinas. At 1200 UTC 24 January a surface low was already formed over northern Florida. During the next 12–24 h, a strong convection formed over Georgia. Combined with an excess of moisture, the convective system deepened and produced excessive amounts of precipitation over Georgia and Carolinas. At 0000 UTC 25 January, the storm moved to the Washington, D.C., area, producing excessive snow during the following 24 h. At 0000 UTC 26 January, the storm has weakened and moved further north, toward the New York City and Boston areas, producing less damaging snow accumulation. The path of the surface low is presented in Fig. 2, in 12-h intervals, from 1200 UTC 24 January until 0000 UTC 26 January 2000. The NCEP global analysis (Derber et al. 1991; Parrish and Derber 1992; Derber and Wu 1998) is used for verification as an independent analysis. Only for the initial analysis (Fig. 2a) is the operational 3DVAR analysis used, due to technical reasons. The surface low, located in northern Florida, was already formed at 1200 UTC 24 January 2000. During the next 36 h the center of the surface low pressure moved over the ocean in a northeast direction, parallel to the coastline.

The operational data assimilation was apparently unable to correctly initiate strong convection over Georgia at 1200 UTC 24 January. In addition, it appears that insufficient moisture was available in this area, consequently causing a failure of the models to produce significant precipitation amounts observed over land. Only the forecast cycle starting 0000 UTC 25 January, when the secondary storm was already formed, was able to make a correct prediction. The forecast cycle starting 1200 UTC 24 January was the most critical cycle,—one that would have given the NWS forecasters enough time to make a snowstorm warning to the public. Most of the model reruns of this snowstorm focused on this forecast cycle, assessing the analysis and observations, model physics (e.g., convection), and model resolution effects (Rogers and Manikin 2001). Overall, some of the forecasts did show some improvement, however most of them are still missing the accurate precipitation amounts over the Carolinas during the early stage of the storm.

### b. Model

The numerical model used for operational forecasting, and in the adjoint calculation, is NCEP's regional Eta Model (Mesinger et al. 1988; Janjić 1990, 1994; Lobocki 1993; Black 1994; Janjić et al. 1995; Zhao et al. 1997; Chen et al. 1997). This is a gridpoint model, defined using the steplike (Eta) vertical coordinate. At the time of the storm the operational resolution was about 32 km in the horizontal, with 45 vertical layers, which is the resolution used in all assimilation experiments. The adjoint model has the same resolution as the forecast model. Note that since September 2000 the operational resolution of the Eta Model was increased to 22 km horizontally and 50 layers.

### c. Observations

The assimilated observations, taken from NCEP's operational database, include rawinsondes, automated weather reports from commercial aircraft (ACARS) and conventional wind reports, cloud-tracked winds from satellites, surface marine and land observations, Special Sensor Microwave Imager (SSM/I) wind speed over water, dropwindsondes, retrieved temperature profiles from Geostationary Operational Environmental Satellite (GOES) and National Oceanic and Atmospheric Administration (NOAA) satellites, profiler winds, and GOES precipitable water. All details are given in Table 1. The 4DVAR observation operator is the same as the one used in the operational 3DVAR assimilation (Parrish et al. 1996). Since the operational 3DVAR method assimilates observations every 3 h during the 12-h assimilation period, the same setup is employed in 4DVAR. Thus, the number of observation times [index *N* in (2.1)] is 4. Note that both the 3DVAR and 4DVAR use the same amount of observations. In some 4DVAR experiments, however, hourly accumulated precipitation observations are assimilated. These observations are a combination of the radar and rain gauge observations over the continental United States (NCEP's Stage IV National Mosaic 4-km database; Baldwin and Mitchell 1997). As for other observations, the precipitation observation error covariance is assumed diagonal and is empirically determined. This implied a constant value for all observations.

### d. Experiments

The assimilation experiments are performed using: (i) the operational mesoscale 3DVAR alghorithm and (ii) 4DVAR with model error adjustment and with precipitation observations assimilation. In order to assess the impact of model error and precipitation observations, additional 4DVAR experiments are performed: (iii) 4DVAR without precipitation assimilation, (iv) 4DVAR without model error adjustment, and (v) 4DVAR without model error adjustment and without precipitation observations assimilation.

In all experiments (including the 3DVAR), the assimilation period is 12 h, from 0000 UTC until 1200 UTC 24 January 2000, with observations available every 3 h. This means that the analysis (initial conditions) is produced for 1200 UTC 24 January 2000, which was the most critical forecast cycle for forecasting this storm. The forecasts were made up to 60 h, with outputs saved every 12 h. The most important forecast period for precipitation, however, was during the first 36 h of the forecast, from 1200 UTC 24 January until 0000 UTC 26 January 2000.

In all experiments presented here NCEP's operational mesoscale 3DVAR analysis from the previous data assimilation cycle is used as an initial guess and the background vector. The background model error vector is set to zero in 4DVAR experiments, although if known, a better estimate of the guess random model error vector may be used instead. The background variance is computed via the so-called National Meteorological Center (NMC) method (Parrish and Derber 1992). The difference between the 36-h and the 24-h forecasts, valid at the same time, is used. About 60 forecast differences were available, from September–October 2000. In order to correct for the sampling error, an arbitrary threshold of 10% of the maximum standard deviation (for the specific variable and the model vertical layer) was introduced. All variances smaller than the threshold value were inflated to that value, thus making the variance fully invertible.

## 4. Results

East Coast snowstorm formation is a well-studied subject (Rogers and Bosart 1986; Uccellini and Kocin 1987; Nuss and Anthes 1987; Uccellini et al. 1987; Cione et al. 1993; Wichansky and Harnack 2000). Generally, there are few factors needed for their formation (e.g., Wichansky and Harnack 2000): (i) cold air damming within the boundary layer, (ii) strong vertical velocity, and (iii) excess of moisture. As Uccellini et al. (1987) pointed out, the interaction between an upper-level jet streak and surface diabatic processes is also critical for the storm formation, and for maintaining the secondary circulation associated with a storm. With this in mind, several fields are examined: precipitation, surface convergence (indicative of vertical motion), precipitable water (indicative of moisture availability), the 850-hPa temperature, and the sea level pressure. Also, the baroclinic potential vorticity at 850 hPa and 250 hPa is examined, indicative of the instability associated with the storm. Some assessment of the low-level jet (LLJ) (wind at 850 hPa) is also performed.

### a. Comparison with operational data assimilation

In the first set of experiments, the 4DVAR data assimilation is compared with the results of the operational (3DVAR) data assimilation. This comparison attempts to illustrate the value of the 4DVAR analysis and ensuing forecast as generally comparable to the operational data assimilation system. Furthermore, since unlike the 4DVAR, the operational data assimilation system and ensuing forecast failed to predict strong precipitation during the first 24 h, a probable cause for the failure is discussed in some details.

#### 1) Precipitation forecast

The 24-h precipitation forecast is probably the most important forecast aspect worth examining in this weather event, since the heavy snow accumulation created a major natural disaster for the east coast of the United States during that time. The 24-h accumulated precipitation observations, derived using only rain gauge observations (Stage IV, 4-km rain gauge database) are commonly used for precipitation verification at NCEP, and these data are used as precipitation verification in this study as well. However, these verifications are available only at 1200 UTC, therefore making possible a comparison with 24-h and 48-h precipitation forecasts only. The 24-h accumulated precipitation observations, valid at 1200 UTC 25 January 2000, are shown in Fig. 3a. The maximum values exceed 50 mm. Note that these data are different from the precipitation observations used in assimilation, since no radar observations are included. They represent 24-h accumulated amounts, and are more accurate than the combined radar and rain gauge observations. There are, however, some areas with data gaps, that should be viewed with caution. For example, in the snowstorm region, there is a data gap (white area) where heavy precipitation amounts are expected. The observed 24-h accumulated precipitation valid at 1200 UTC 26 January (Fig. 3b) indicate smaller precipitation amounts, located over the New York and New England areas, with the maximum amounts somewhat over 20 mm.

The 24-h precipitation forecasts initiated using NCEP's operational mesoscale 3DVAR data assimilation system, and the 4DVAR algorithm are shown in Figs. 4a and 4b, respectively. The 4DVAR algorithm is applied in the most complex setting, including assimilation of precipitation observations and model error adjustment. The 24-h forecast based on the operational 3DVAR system failed to produce heavy precipitation amounts over the Carolinas (Fig. 4a). The 4DVAR experiment provided an accurate precipitation forecast, as seen in Fig. 4b. Both the location and intensity match very well with the observed values. By comparing Figs. 4a and 4b, we conclude that the 4DVAR based forecast produced larger amounts of precipitation over the Carolinas, as well as over the East Coast, in better agreement with the observations.

In order to see the impact of 4DVAR data assimilation on the precipitation forecast in more detail, a difference between the 4DVAR and 3DVAR-produced forecasts is plotted for 12-h accumulated precipitation amounts (Figs. 5a,b,c), during the 36-h forecast interval. The sequence shows that the positive impact of 4DVAR initial conditions is evenly spread over each 12-h forecast interval. During the first 12 h (Fig. 5a), 4DVAR has produced improved precipitation mostly over South Carolina and Georgia. Due to the movement of the storm, in the 12–24-h period the precipitation forecast is improved over South and North Carolina (Fig. 5b). One can also note a negative difference over the Chesapeake Bay area. It is not clear, however, how important this may be, since the precipitation observations (Fig. 3a) do not show any precipitation in this area. During the next 12 h (24–36-h forecast, Fig. 5c), there is also an excess of precipitation in the 4DVAR forecast, mostly located over the Long Island and New England area. One can also note a small negative difference over Delaware and New Jersey, indicative of more precipitation in the 3DVAR forecast. However, the precipitation amount and the area covered by the negative difference are much smaller than the amount and area covered by a positive precipitation difference. For the 36–48-h forecast (not shown), there is no significant difference.

#### 2) Precipitable water

This quantity is used to examine the availability of moisture, effectively representing the amount of water potentially available for precipitation. In Fig. 6 the precipitable water difference between the 4DVAR and 3DVAR-produced initial conditions, valid at 1200 UTC 24 January, is shown. An excess of positive precipitable water difference is seen over Georgia. This indicates that the 4DVAR experiment produced a significantly larger amount of precipitable water than did the 3DVAR data assimilation in the area of interest. However, in order for this to be utilized, a strong upward motion is required.

#### 3) Surface convergence

For strong vertical motion to take place, an excess of surface convergence (negative divergence) is needed. In Fig. 6a, 1000-hPa divergence fields at the initial time of forecast, valid at 1200 UTC 24 January, is shown. In the area of interest, the 3DVAR-produced initial divergence field shows no notable convergence over Georgia, only a weak convergence in northern Florida (Fig. 7a). On the other hand, the 4DVAR initial conditions indicate a strong convergence, not only in northern Florida, but over Georgia as well (Fig. 7b). A difference between the 4DVAR and 3DVAR-produced divergence field (Fig. 7c) shows a large excess of convergence over Georgia, at the same location where the precipitable water differences were largest (e.g., Fig. 6). Combined effect of the excess of surface convergence and precipitable water in initial conditions produced by the 4DVAR data assimilation is the most probable cause for the initial precipitation amounts generated in 4DVAR experiment.

#### 4) Sea level pressure and 850-hPa temperature

An indication of already existing convection over Georgia, noted in 4DVAR initial conditions, can be seen in sea level pressure differences between the 4DVAR and 3DVAR experiments, valid at 1200 UTC 24 January (Fig. 8a). One can note a 1-hPa lower pressure in 4DVAR initial conditions. Although small, this difference is also supported by lower 850-hPa temperatures in the 4DVAR initial conditions (Fig. 8b). The 4DVAR initial conditions have generally 1–2 K lower temperatures in Georgia.

The 24-h sea level pressure forecast difference between the 4DVAR and 3DVAR experiments, valid at 1200 UTC 25 January, is shown in Fig. 9. A 1-hPa lower pressure can be noted in the 3DVAR experiment, in better agreement with the verifying analysis (Fig. 2c). It can be also noted that there is no difference in the position of the surface low in these two experiments. In the 4DVAR forecast however, there is a lower pressure over a large area southwest of the sea level pressure center, a possible indication of a better-defined surface trough. The 36-h sea level pressure forecast did not reveal any notable difference between the 3DVAR and 4DVAR experiments (not shown). This suggests that the sea level pressure differences between 3DVAR and 4DVAR experiments were not very significant.

#### 5) 850-hPa wind field (LLJ)

About 6 h into the forecast, a LLJ is formed in the 4DVAR experiment, as a result of strong convection and early precipitation, in agreement with the results of Uccellini et al. (1987). The 850-hPa wind difference between the 4DVAR and 3DVAR 6-h forecasts, valid at 1800 UTC, 24 January, shows an excess of wind speed in the 4DVAR experiment, located over Georgia and South Carolina (Fig. 10). The maximum excess amount is about 10 m s^{−1}, mostly enhancing the southeast winds (toward the coast) in the 4DVAR experiments.

#### 6) Potential vorticity (PV)

Baroclinic PV is another important quantity that may be used to diagnose the development of this storm. Since the most dramatic difference between the 4DVAR and 3DVAR-initiated forecasts happened during the first 24 h of forecast, the PV difference is shown for the period 1200 UTC 24–1200 UTC 25 January. The 850-hPa PV difference between 4DVAR and 3DVAR is shown in 6-h intervals (Figs. 11a–e). The 500-hPa PV (not shown) is very much correlated with the 850-hPa PV difference. As seen in Figs. 11a–e, there is a short-lived positive 850-hPa PV anomaly in the 4DVAR experiment over the area of interest, starting at the 6-h forecast, and lasting until about 24 h into the forecast, with a strong maximum at 12 and 18 h into the forecast. At upper levels, there was a somewhat similar pattern, however, not as well defined as at 850 hPa. There is no notable PV anomaly over the area of interest until 12 h into the forecast. A 12-h forecast difference of 250-hPa PV between the 4DVAR and 3DVAR experiments (Fig. 12), valid at 0000 UTC 25 January, shows a positive anomaly with a primary maximum over North Carolina, and a secondary maximum located to the south over the ocean. At the 18-h forecast there was a weaker, although still notable, positive 250-hPa PV anomaly (not shown). By the 24-h forecast, there is no positive PV anomaly in the 4DVAR experiment. These results (Figs. 11 and 12) suggest that deep convection responsible for excessive precipitation amounts over the eastern United States during the first 24 h of forecast was formed in the 4DVAR experiment. This storm was short-lived (12–18 h), and confined mostly to the lower troposphere. Only for about 6 h (between 12 and 18 h of the forecast) did the convection deepen throughout the troposphere.

### b. Impact of model error adjustment and precipitation assimilation in 4DVAR data assimilation

In the second set of experiments, various configurations of 4DVAR algorithms are compared. This comparison will show the impact of model error adjustment and the impact of precipitation assimilation in 4DVAR. These two components are often neglected in 4DVAR applications, and it is instructive to see their value in a complex data assimilation setting.

In general, the error in initial conditions (e.g., surface convergence, precipitable water) exists also in 4DVAR experiments with no model error adjustment and/or no precipitation assimilation. Although the error appears to be smaller than in the 3DVAR experiment, it is still causing significant differences in the precipitation forecast. A possible explanation may be that the 4DVAR with model error adjustment and with precipitation assimilation does a better adjustment of the dominant unstable modes. The initial fit may look relatively similar in all 4DVAR experiments, but the growth of errors during the forecast after data assimilation is different. The impact of model error adjustment and precipitation assimilation on the growing errors during the forecast will be investigated in future work.

#### 1) Precipitation forecast

First, the impact of precipitation observations is assessed by comparing the forecast difference of the 12-h accumulated precipitation between the 4DVAR and 4DVAR_NOPCP (4DVAR with no precipitation assimilation) experiments, during the first 36-h forecast period (Figs. 13a–c). A 12-h forecast difference, valid at 1200 UTC 25 January, is shown in Fig. 13a. One can note relatively small differences during the first 12-h, suggesting that even without precipitation observations, it is possible to obtain a satisfactory precipitation forecast. Without precipitation observations, however, this effect does not last long. During the next 12 h of forecast (12–24 h), the difference becomes more pronounced (Fig. 13b). There is a positive difference for the 24–36-h forecast as well (Fig. 13c), indicating a positive effect due to precipitation assimilation even 36 h into the forecast.

The impact of model error on the precipitation forecast is also examined. In Fig. 14, the forecast difference of the 12-h accumulated precipitation between the 4DVAR and 4DVAR_NOERR (4DVAR with no model error adjustment) is shown. There is a notable difference during the first 24 h of forecast (Figs. 14a,b). This suggests that the model error adjustment played an important role in this situation. Also, the positive difference can be seen in Fig. 14c (36-h forecast), indicating a long-lasting positive impact of model error.

Figures 13 and 14 suggest that both the model error adjustment and precipitation observations had a similar, long-lasting positive impact on the precipitation forecast. During the first 12 h of the forecast it appears that the model error adjustment had a more important contribution. The combined effect of model error adjustment and precipitation observations is important to illustrate because the practice in 4DVAR applications is often to neglect these two components. In Figs. 15a–c the forecast difference of the 12-h accumulated precipitation between the 4DVAR and 4DVAR_NOERR_NOPCP (4DVAR with no model error adjustment and no precipitation assimilation) is presented—during the 36-h forecast interval. The combined positive impact of model error adjustment and precipitation assimilation on the precipitation forecast is significant, with clear implications on the use of model error adjustment and precipitation assimilation in 4DVAR data assimilation in this case. An inspection of Figs. 15a–c shows that the combined effect of model error adjustment and precipitation assimilation had a significant, and long-lasting positive effect on the precipitation forecast. In each of the 12-h forecast periods, there is a substantial improvement in precipitation forecast when both the model error adjustment and precipitation assimilation are included.

#### 2) Potential vorticity

A 850-hPa PV forecast difference between the 4DVAR and 4DVAR_NOERR experiments (Fig. 16a) shows a large positive maximum over South Carolina. This suggests that the role of model error adjustment was very significant. On the other hand, a 850-hPa PV forecast difference between the 4DVAR and 4DVAR_NOPCP experiments (Fig. 16b) shows smaller, although notable, amounts suggesting less pronounced impact of precipitation assimilation. It is interesting to note that the combined effect of model error adjustment and precipitation assimilation (Fig. 16c) shows an impact very similar to the impact of the model error adjustment alone (Fig. 16a).

#### 3) 250-hPa wind field

A 250-hPa wind field is used here to illustrate the impact of precipitation assimilation on the 24-h forecast. A positive impact may be an important indication that the precipitation observation information from data assimilation was able to (dynamically) penetrate to upper levels of the troposphere. As earlier, the NCEP global analysis is used for verification. In Fig. 17a, a 250-hPa wind analysis, valid at 1200 UTC 25 January, is shown. A 250-hPa wind difference between the 4DVAR_NOPCP and NCEP global analysis, valid at 1200 UTC 25 January, is shown in Fig. 17b. Among other errors, there is a large error over the southeastern United States, with a maximum over Georgia. Note that this is also near the location of the left quadrant of the jet streak exit region, characterized by a strong divergence at upper levels and associated convergence at the surface (e.g., Uccellini and Kocin 1987). In the 4DVAR experiment, however, the error is significantly reduced (Fig. 17c). It appears that the 4DVAR_NOPCP forecast has produced much stronger winds in that area, effectively dumping the divergence at upper levels and consequently diminishing the convergence and convection at the surface. This suggests that the precipitation observations, essentially related to the convection and surface physics, can have a notable impact on the ensuing 24-h wind forecast at upper levels. On the other hand, the model error did not have a notable impact on the upper-level wind field in this case (not shown).

#### 4) 850-hPa temperature

The impact of model error is illustrated here by examining the 24-h forecast of the 850-hPa temperature field. The global NCEP 850-hPa temperature analysis is shown in Fig. 18a. One can note a strong gradient off the U.S. east coast, an indication of a frontal zone in that area. The difference between the 24-h temperature forecast in 4DVAR_NOERR experiment and the NCEP global analysis (Fig. 18b), valid at 1200 UTC 25 January, shows an error in the temperature forecast along the coast, in the region of this front. In the 4DVAR experiment (Fig. 18c) the error is smaller, and the gradient of the error dipole is weaker. The dipole structure of the difference suggests a dominant error of the frontal position in all experiments. This implies that, although in both 4DVAR and 4DVAR_NOERR experiments there was an incorrect position of the frontal zone, the model error adjustment had a notable positive impact. Unlike the upper-level wind forecast results, the precipitation observations were not very important.

## 5. Concluding remarks

One of the major recent failures of NCEP's operational forecasting system was related to the prediction of the East Coast snowstorm of 25 January 2000. NCEP's operational Eta Model, as well as the models from other centers, had difficulty in predicting the precipitation location and amounts for this storm. For the forecast cycle at 1200 UTC 24 January, the operational forecast did correctly produce the surface low pressure system located off the coast. It failed, however, to create a deep convection over Georgia and the Carolinas. Consequently, the precipitation amounts over land were not predicted correctly until the cycle at 0000 UTC 25 January.

Probably the most important aspect of this storm was related to the precipitation forecast, due to the enormous life and property damage that this storm caused along the east coast of the United States. A comparison between the 4DVAR and operational (3DVAR) data assimilations traced the reason for the operational forecast failure back to the initial conditions produced by the 3DVAR assimilation. Most notably, the operational data assimilation failed to create surface convergence over Georgia, coupled with an insufficient amount of precipitable water in the same area.

Especially significant is the beneficial role of the model error adjustment and precipitation observation assimilation in 4DVAR experiments, considering that these two components are often neglected in variational data assimilation algorithms. In fact, it appears that adjusting the model error, and having the assimilation of precipitation, are two major components in making the 4DVAR algorithm successful for this case.

The 4DVAR data assimilation presented here is obtained using the operational model resolution for all components of the 4DVAR cost function. Although there are ways to reduce the cost of the 4DVAR data assimilation by integrating the adjoint model in coarser resolution (Courtier et al. 1994; Rabier et al. 2000), it is not clear if this approximation may be acceptable when dealing with smaller-scale features, such as convection and precipitation. This aspect of the 4DVAR data assimilation will be examined in more details in future work. The results presented here may be viewed as an encouragement for applying the 4DVAR method in mesoscale data assimilation.

## Acknowledgments

We would like to thank Dr. Surunjana Saha for invaluable help regarding the use of GRADs plotting package. The encouragement and support of Dr. Steve Lord is greatly appreciated. Our thanks also goes to Dr. Jordan Alpert for stimulating discussions. We would also like to thank two anonymous reviewers for careful and thoughtful reviews. This research was fully supported by the NOAA/NCEP/UCAR Visiting Scientist Programs.

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## APPENDIX

### Preconditioning in Variational Assimilation

^{T}

*z*

*z*

^{b}

^{−T}

*ζ*

_{ζ}

^{−1}

^{−T}

^{−1}

^{T}

^{−T}

^{1/2}𝗯

^{T/2}, the Hessian is

^{−T/2}

^{−1/2}

*z*

*z*

^{b}

^{1/2}

^{−T/2}

*ζ.*

NCEP's operational database used in data assimilation experiments. Note that hourly accumulated precipitations are not assimilated in the 3DVAR data assimilation algorithm