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A 3.5-dimensional variational method is developed for Doppler radar data assimilation. In this method, incremental analyses are performed in three steps to update the model state upon the background state provided by the model prediction. First, radar radial-velocity observations from three consecutive volume scans are analyzed on the model grid. The analyzed radial-velocity fields are then used in step 2 to produce incremental analyses for the vector velocity fields at two time levels between the three volume scans. The analyzed vector velocity fields are used in step 3 to produce incremental analyses for the thermodynamic fields at the central time level accompanied by the adjustments in water vapor and hydrometeor mixing ratios based on radar reflectivity observations. The finite element B-spline representations and recursive filter are used to reduce the dimension of the analysis space and enhance the computational efficiency. The method is applied to a squall line case observed by the phased-array radar with rapid volume scans at the National Weather Radar Testbed and is shown to be effective in assimilating the phased-array radar observations and improve the prediction of the subsequent evolution of the squall line.

Because the radar network provides only single-Doppler scanning over most areas in the U.S., research efforts have been undertaken to develop various methods for meteorological parameter retrievals from single-Doppler observations (Sun et al. [

Equations in an NWP model can be expressed symbolically in the following vector form:_{k}_{k}_{0} at the initial time (_{0} at the initial time (_{0}. A prior estimate of _{m}_{m}

Assume that _{k}_{k}_{m}_{k}_{k}_{m}_{k} = _{m} denotes the summation over _{k} denotes the summation over

The cost function in (_{k} = 0 in (^{−1} and ^{6} observations. To assimilate high-resolution radar observations over a mesoscale area, the model state vector _{k} often needs to contain at least as many as 10^{6} gridded variables. In this case, the P4dVar is still too expensive and unpractical for real-time radar data assimilation.

A further simplification can be made by setting not only _{k}_{k}^{3} domain). To solve the above problems, the simple-model 4dVar (Xu et al. [

As shown by the flowchart in Figure _{k}_{k}_{m}

Flowchart for the three steps of the method. Step

The quality control procedure checks for and corrects two types of errors in raw level II radial velocity data. First, it corrects the possible velocity alias error (if any) caused by the finite range of radar velocity measurements limited by the Nyquest velocity

Interpolate

Take the integer

If the absolute value of the adjusted innovation is less than 0.5

For the second type of correction, the data quality control removes the error caused by the precipitation terminal velocity. The terminal velocity,

The radial velocity analysis is performed by minimizing the following cost function:

To facilitate the formulation and computation of the background term, _{1}, _{2}, and _{3} are the univariate error covariance matrices for the background fields

The background error covariances for (

The observation term has the following general form (for step 1 and step 2)_{m}

As in (

The vector velocity analysis is performed by minimizing the following cost function:

The background term in (

The observation term

The third term in (

The last term in (

The time derivative term

The radial momentum equation in (

As in step 1, by using (

The estimated fields of

As explained in Section

After

The cost function in (

The analyzed total perturbation pressure and potential temperature are given by

After the perturbation pressure and potential temperature fields are updated by the above incremental analyses, the relative humidity is altered and thus needs to be recovered by adjusting the water vapor mixing ratio

Interpolate the radar observed reflectivity

Check each grid point for the conditions of

Check each grid point for the conditions of

In companion with the

The 3.5dVar is applied to the phased array radar radial velocity and reflectivity observations collected during the period of 2100–2200 UTC 2 June 2004 when a four-quadrant electronic scan strategy was tested. During this period, a squall line moved southeastward through the central Oklahoma area in the radial range (140 km) of the phased array radar scans (Figure

Phased-array radar observed reflectivity (a), radial velocity (b) at

The COAMPS is used to produce the background fields. The model is configured with three nested domains centered over the state of Oklahoma with resolutions of 54, 18, and 6 km for the coarse, medium, and fine grids, respectively, and 30 levels in the vertical. The three nested domains are shown in Figure

COAMPS domains nested with 54, 18, and 6 km grids.

By using the 3.5dVar, three consecutive volume scans of the dealiased radial velocity and reflectivity data are assimilated through a single cycle around 2108 UTC, and then a test run, called Test 1, is launched for 58-minute forecast to 2200 UTC. Another test run, called Test 3, is also performed by assimilating the first nine volume scans of the dealiased radial velocity and reflectivity data in three cycles from 2108 to 2118 UTC. The velocity and reflectivity fields at

Velocity (arrows for the horizontal wind and green contours for the vertical velocity) and reflectivity (color field) at ^{−1}. The arrow (=30 m s^{−1}) at the lower-left corner is the vector scale for the horizontal velocity.

The 5-minute forecasts of velocity and reflectivity fields valid at 2118 UTC from the second cycle (at 2113 UTC) in Test 3 are given Figure

As in Figure

Figure

As in Figure

It is necessary to mention that the incremental potential temperature produced by the thermodynamic analysis is relatively small (within ±

The velocity and reflectivity forecasts from the control run and Test 1 valid at 2200 UTC are plotted against the observed reflectivity at

As in Figure

(a) Correlation coefficient between the predicted and observed reflectivity fields. (b) RMS differences between the predicted and observed radial velocities. The red, blue, and green curves are for the results obtained from the control run, Test 1 and Test 3, respectively.

To verify the three-dimensional winds, the model-produced (analyzed and predicted) radial velocity fields are interpolated (using the observational operator) back to the phased-array radar observation points and compared with their respective observed values. The RMS differences between the predicted and observed radial velocities are plotted as functions of time for the control run and the two test runs in Figure

This paper presents a variational approach for Doppler radar data assimilation. This method analyzes four-dimensional radar observations (three consecutive volume scans) but updates the model state only in three-dimensional space at the central time level in each assimilation cycle, and therefore we call it three-and-half-dimensional variational (3.5dVar) method. The method can be considered as an upgraded combination of the previous retrieval methods (Qiu and Xu [

The 3.5dVar method has also been tested and applied to many other cases, either as a stand-alone package (Gu et al. [

Because simplifications are made to the general variational formulation in each of the three steps in the method to achieve the desired computational efficiency for real-time applications, the resulting analyses are suboptimal (relative to the desired maximum likelihood estimates). This implies that the method can be improved by reducing the involved simplifications and refining the error covariance estimation and representation. Such an improvement is made in the current 3.5dVar over the early version (Gu et al. [

The mass continuity equation in (

Assume that

Note that

Denote by

For mesoscale flows, we may assume that

The authors are thankful to Carl Hane, Jidong Gao, and the anonymous reviewer for their comments and suggestions that improved the presentation of the paper and to Douglas Forsyth, Kurt Hondl, Richard Adams, Pengfei Zhang and Kang Nai for their help in obtaining and processing the phased-array radar data. The research work was supported by the NOAA HPCC program, the FAA contract IA# DTFA03-01-X-9007 to NSSL, the ONR Grants N000140410312 to the University of Oklahoma, and the NOAA-University of Oklahoma Cooperative Agreement no. NA17RJ1227.