## 1. Introduction

Doppler radar has long been a valuable observational tool in meteorology. It is capable of observing the internal structure of storm systems from remote locations at high spatial and temporal resolutions. Timely and accurate analyses of low-level wind conditions at high resolutions are critical for severe weather warning and monitoring. Based on a single radial component of velocity observations by radar, low-level wind conditions and related fine features (such as convergence and rotation) are usually assessed subjectively by forecasters. However, analysis tools for quantitatively estimating the unobserved cross-beam wind component are still lacking.

So far, many methods of single-Doppler radar retrieval have been developed. Lhermitte and Atlas (1961) initially put forward the velocity azimuth display (VAD) method, which can retrieve mean horizontal wind magnitude and direction. Browning and Wexler (1968) showed how resultant deformation and orientation of the axis of dilatation could also be determined from a single Doppler radar using the VAD method. Waldteufel and Corbin (1979) proposed the volume velocity processing (VVP) method, which can make full use of radar velocity data filling a volume. Some studies (Rinehart 1979; Tuttle and Foote 1990) retrieved wind velocity by tracking the motion of small features in reflectivity. “Synthetic” analysis is also used to retrieve wind field information. (Peace et al. 1969; Bluestein and Hazen 1989).

In recent years, assimilation of Doppler radar data for short-term numerical weather forecasting or nowcasting has become the highlight of research. Some assimilation techniques were used to retrieve wind field from single-Doppler radar observations. Doppler radar radial wind observation is usually assimilated in the form of the VAD wind profiles in real-time applications (Lindskog et al. 2002; Benjamin et al. 2004). On the research front, more sophisticated techniques have been applied to the assimilation of high-resolution raw radar observations rather than radar-estimated fields, such as VAD wind profiles. Horn and Schunck (1980) retrieved data by requiring the wind field to satisfy the Lagrangian advection equation of reflectivity. Liou et al.’ (1991) method retrieved data by “nudging” a full numerical model. Gao et al. (1999) developed a variational Doppler radar analysis system to analyze dual-Doppler radar for the Advanced Regional Prediction System (ARPS). Some studies (Qiu and Xu 1992; Sun et al. 1991) used an adjoint technique to retrieve the wind field. Sun and Crook (1997, 1998, 2001) reported a study in which Doppler radar radial velocity and reflectivity observations were assimilated into a cloud-scale numerical model using a four-dimensional (4D) variational Doppler radar assimilation system (VDRAS). Weygandt et al. (2002a,b) used a technique that employs a single-Doppler wind retrieval algorithm in the study to initialize an observed supercell storm.

Recently, some efforts have been contributed in the three-dimensional variational data assimilation (3DVAR) approach to retrieve wind velocity from Doppler radar radial velocity. Xiao et al. (2005) examined the impact of Doppler radar radial velocity on the prediction of a heavy rainfall event using 3DVAR in which a Richardson equation is employed. Xu et al. (2006) constructed background error covariance functions for wind vector analyses using Doppler radar radial velocity observations. Xue et al. (2007) conducted a variational analysis of oversampled dual-Doppler radial velocity data and applied it into the analysis of tornado circulations.

In 3DVAR Doppler radar radial velocity data assimilation used in the mentioned literature, a background error covariance matrix is always needed to determine the spatial spreading of observational information. But it is well known that an analysis field at different locations may have different correlation scales, which are difficult to estimate. So, a traditional 3DVAR using either correlation scales or recursive filters can only correct certain wavelength errors, and the short wavelength error cannot be sufficiently corrected until the long one is corrected (Xie et al. 2005).

To correctly minimize the errors of longwaves and shortwaves in turn, a sequential 3DVAR approach has been proposed by Xie et al. (2005), and it has been implemented using both recursive filter and multigrid techniques at the Global Systems Division (GSD) of the Earth System Research Laboratory (ESRL) of the National Oceanic and Atmospheric Administration (NOAA) in 2004 for a Federal Aviation Agency (FAA) project joined by the research team from the Lincoln Laboratory at the Massachusetts Institute of Technology (MIT). Because this system also uses the temporal observation information in its analysis, it is called a space and time mesoscale analysis system (STMAS). This system treats a data assimilation problem in two steps. The first step is to analyze the information that can be resolved by given observation networks or background, and the second step is to treat the residues between the previously analyzed fields and observations using a statistical analysis, such as the current data assimilation techniques. In other words, STMAS supplements an additional deterministic analysis procedure to retrieve features that can be resolved by observations or background before performing a standard statistical analysis [e.g., 3DVAR or ensemble Kalman filter (EnKF)]. In this study, we assimilate 2D Doppler radar radial velocities in this first assimilation step in STMAS to test how STMAS deals with 2D Doppler radar radial velocity and to what degree the 2D Doppler radar radial velocity can improve the conventional (in situ) observation analysis.

In the following section, the theory of multigrid method applied in the first step of the two-step data assimilation scheme used in STMAS is briefly reviewed. Smoothing and radar radial wind operators in STMAS are discussed in section 3. Two idealized 2D Doppler radar radial velocity data assimilation experiments are described in section 4. Section 5 presents the results of real 2D radar radial velocity data assimilation of a 2001 typhoon case. Summary and conclusions are in section 6.

## 2. Theory of multigrid data assimilation scheme in STMAS

The multigrid method in the first step of the two-step data assimilation scheme used in STMAS (described in section 1) is applied to assimilate the 2D Doppler radar radial velocity resulting from its efficiency comparing to its recursive filter counterpart. The basic idea of this multigrid implementation can be found in Xie et al. (2005) and Li et al. (2008), and it is briefly reviewed here.

**X**is the increment fields from the model background fields, 𝗕 is the background error covariance matrix,

**Y**is the innovation vector between the observations and the interpolated model background at the observation locations, 𝗢 is the observation error covariance matrix, and 𝗛 is a simple bilinear interpolation operator from a model space to an observation space. Superscripts T and −1 represent matrix transpose and inverse, respectively.

*L*represents the longwave, which can be resolved by observation system, and subscript

*S*represents the shortwave, which cannot be resolved by observation system. Superscript

*b*represents the background field, superscript

*a*represents the analysis field, and

**X**

*denotes the increment over longwaves.*

_{L}A function for relative shortwaves can be iteratively formed to extract the resolvable short information of the residual by keeping the relative longwave **X*** _{L}* extracted. A multigrid method will be applied to carry out this basic idea and correct the increments over longwaves and shortwaves sequentially.

*n*represents the

*n*th multigrid grid level. Because the correlation scales are reflected by coarse or fine grids, in the case of no accurate covariance being known, the 𝗕 matrix is simplified to a diagonal matrix with the element of diagonal being error variance of background. If we set the error variance of background as ɛ

_{b}

^{2}and the error variance of observation as ɛ

_{o}

^{2}, a new observational error variance equal to ɛ

_{o}

^{2}/ɛ

_{b}

^{2}can be set to 𝗢 matrix diagonal element and the original 𝗕 can be simplified to identity matrix with the ratio of the background term to the observation term unchanged.

**Y**

^{(1)}=

**Y**

^{obs}− 𝗛

^{(1)}

**X**

*in the first level grid be the difference between the observations and the interpolated model background at the observation locations; then, the observations used in the other grid levels are defined as where*

^{b}**X**

^{(n−1)}is the solution or approximate solution of

*J*

^{(n−1)}.

*n*= 1, with

**X**

^{(1)}computed by projecting

**X**

^{b}onto the coarsest grid level and calculating the innovation of the projected

**X**

^{b}. After solving

*J*

^{(n−1)},

**X**

^{(n−1)}is interpolated onto the finer grid of the

*n*th level; then,

**Y**

^{(n)}can be calculated by (7) and

**X**

^{(n)}can be solved by minimizing

*J*

^{(n)}. This process is repeated until it reaches at the finest grid level. During the procedure of sequential multiple-scale analyses, the error covariance matrix 𝗢

^{(n)}remains the same as the full observation dataset is used through all multigrid levels, because the covariance matrix 𝗢

^{(n)}reflects the observation error covariance and has nothing to do with multigrid levels. In STMAS, the half V cycle (Briggs et al. 2000) property of the multigrid technique is employed, because it continuously refines the resolutions. Then, the final analysis is This implementation of STMAS has been successively applied to surface analysis for FAA boundary detection (Xie et al. 2005) and to the China Sea’s temperature forecasts (Li et al. 2008).

## 3. Smoothing and radar radial wind operator in STMAS

**X**

^{(n)}= (

**U**

^{(n)T},

**V**

^{(n)T})

^{T}, and a smoothing term is introduced in the cost functional to reduce the bilinear interpolation truncation error effect, where subscript

*b*is the background term, subscript

*o*is the observation term, and subscript

*s*is the smoothing term: where

*c*is the conventional observation term and

*r*is the radar radial wind observation term. The smoothing matrixes 𝗦

_{U}and 𝗦

_{V}in the smoothing term are derived from the Laplacian of control variables at grid points. The details of the conventional observation term and the radar radial wind observation term are as follows: where

*M*is the number of radar radial wind observations and

*θ*is the azimuth angle of the radar beam relative to north, with positive clockwise. In this study, we projected the radial wind onto horizontal planes; thus, there are only horizontal wind components in this analysis formulation. A full 3D radial wind analysis is under development at GSD ESRL.

The error variances of radial velocity observation and conventional observation should be determined by the measurement error of instruments. Here, for simplicity, the same error variance is set for each kind of data. However, because the amount of radial velocity data is much larger than that of conventional data, a scaling scheme is used to balance the weights of these two kinds of observations. Thus, the conventional observation can have the same weight as that of radial velocity observation, which may comprise these two kinds of observations to get to a reasonable wind analysis.

The limited-memory Broyden–Fletcher–Goldfarb–Shanno (BFGS) method (Fletcher 1987) to solve the bound constrained optimization problem (Byrd et al. 1995) is used as the minimization method in this study, and 50 iterations are set in the minimization for each grid level for simplicity.

## 4. 2D idealized Doppler radar radial velocity data assimilation experiments

It is known that radial wind observation data from single radar cannot determine a wind analysis. The interest of this study is to combine single radar with a few in situ observations and see how much information the radial wind can provide to resolve a full wind field. Here, we will show that single radar can really improve STMAS analysis by combining some conventional observations, where neither the single radar nor the set of conventional observations can resolve this detailed structure of the wind fields.

### a. Idealized experiment 1

In the first idealized experiment, 2D storm-like analytic fields are constructed, and the wind vector field is plotted in Fig. 1a. The single radar is located at the point (350 km, 150 km), and 2D Doppler radar radial velocity (Fig. 1b) is derived from this stormlike analytic field. Then, 24 randomly distributed conventional (in situ) wind data and 400 randomly distributed radial wind data (Fig. 1b) are generated by using this true wind vector field and corresponding radar radial velocity data, respectively.

Six level grids are employed, ranging from about 500 km × 500 km (*n* = 1) to 15.625 km × 15.625 km (*n* = 6) with the grid ratio being 0.5 [i.e., the grid spacing in the *n*th level grid is half of that in the (*n* − 1)th level grid]. The background is set to zero for simplicity. The observation error covariance matrices, which are assumed to be diagonal, are the same in every grid level.

Figure 2a shows the analysis result by assimilating only the 24 given conventional data; because the conventional data are too sparse, the analysis obtains a roughly large pattern of the wind only, and one big circulation can be seen. Figure 2b shows the analysis result by assimilating only radar data. Because there is no information of tangential wind and no statistical or empirical correlation information between the *U* and *V* components is used in this study, some of the wind vectors in the analysis may have opposite directions against the reference truth. But the radial wind analysis is good and very similar to the reference truth in radial wind directional projection. Therefore, the analysis is simply to match the radial wind without other useful information. However, if a combination of the conventional data and radar radial wind data is assimilated, the analysis is significantly improved, as shown in Fig. 2c. The basic structure of this storm is captured in this analysis, but the analysis shows large difference to the true wind field at the top-left corner, where there is no conventional or radar observation by our random observation location generator. If only one conventional observation is added at the top-left corner, the analysis can be improved remarkably (Fig. 2d).

It would be interesting to examine plots of the analyzed *U* (east) component, *V* (north) component, and radial wind for all of these assimilation results (i.e., conventional only, radar only, and the combined analysis). As shown in Fig. 3, because the conventional observations are too sparse to provide sufficient information to resolve this storm, the wind analysis of only the conventional observations is far away from the true radial wind if it is projected on the radial wind direction (Fig. 3f). Assimilating radar radial wind data only, although the radial wind is good (Fig. 3i), the *U* and *V* components are not good (Figs. 3g,h). However, if a combination of these two datasets is used in a STMAS analysis, the analysis is much better (Figs. 3j,k,l), which shows that these two datasets can compensate each other to resolve the wind well.

With dense conventional observations or the combination of dense conventional observations and radar observations, an almost perfect analysis can be obtained for the given storm, which is shown in Fig. 4. Experiments with two radars are also tested, where the other radar is located at (150 km, 150 km), and almost the same good results as the analysis of the dense conventional observations can be obtained (not shown). The *U* and *V* components root-mean-square (RMS) differences between the true field (Fig. 1a) and STMAS analysis (Fig. 2) by using different observations in this idealized experiment are shown in Table 1, in which the same results can be obtained. Note that this is a simple test function with single scale. A more interesting case would be a multiscale test function.

### b. Idealized experiment 2

Another idealized experiment is to show the radar radial wind impact on a multiscale analytic function. In this experiment, we put radar at the center of the study domain. A total of 1000 randomly distributed radar radial wind data are generated, and 441 conventional data are uniformly distributed.

Six level grids are employed ranging from about 100 km × 100 km (*n* = 1) to 3.125 km × 3.125 km (*n* = 6), with a grid ratio of 0.5. The background is set to zero for simplicity. The observation error covariance matrices, which are assumed to be diagonal, are the same in every grid level. The conventional data will be added gradually to investigate the impact of radar radial wind observations.

Figure 5a is a true field, and Fig. 5b is the analysis assimilating only radar radial wind data. It is evident that the radial wind of the analyzed wind field is so close to the true radial wind field, but the *U* and *V* components are not good, because there is no information of tangential wind. Figure 6a is the analysis assimilating only 25 conventional data points, which also shows an inaccurate analysis. Figure 6b shows the analysis assimilating both the 25 points of conventional data and the radar radial wind data, and it demonstrates significant improvement. By gradually adding the conventional data, the results are shown in Figs. 7 and 8 for examining the sensitivity of the number of conventional observations. The analyses become better and better when more and more conventional data are added, and the results of using only conventional data come closer to the results of using both datasets gradually. In Figs. 8a,b, conventional data are adequate to provide the information of the true field, so the analyses using only conventional data (Fig. 8a) are almost the same as that using both datasets (Fig. 8b). From Figs. 5 –8, it is evident that radar radial wind data have a big impact on STMAS data assimilation analysis, especially in the situation of sparse conventional data. The *U* and *V* components RMS differences between the true field and STMAS analysis by using different observations in this idealized experiment are shown in Table 2. It is obvious that the RMS error by assimilating the combination of conventional data and radar data is smaller than that by assimilating conventional data or radar data each, which indicates that the combination of conventional data and radar data can provide more useful information than each type of data only.

## 5. 2D real Doppler radar radial wind data assimilation experiment

The Central Weather Bureau (CWB) of Taiwan provides these real Doppler radar radial wind data, and the data covered Typhoon Lekima at 0304 UTC 26 September 2001, near Taiwan Island. The Doppler radar is located in Kenting (21.9028°N, 120.8469°E). For this particular dataset, no conventional observation data are available, and the radar radial wind cannot resolve the typhoon successfully, because the tangential wind component is missing. Therefore, we have to find an alternative to generate a rough estimation of the wind before applying a radial wind analysis experiment by STMAS. This alternative is the velocity distance azimuth display (VDAD) or ground-based velocity track display (GBVTD) method (Lee et al. 1999), which can directly analyze the radial wind data and derive a first-order approximation of the wind field. The difference between VAD and VDAD is that the coordinate of VAD is at radar center, whereas the coordinate of VDAD is at typhoon center. The algorithm of VAD retrieves mean horizontal wind and its deformation, but the algorithm of VDAD retrieves the typhoon’s tangential and radial wind. Then, this VDAD method is used to derive an approximation of the wind field. These derived wind data are used in STMAS analysis combining with radial wind data and see whether STMAS can provide additional information beyond the first-order approximation. There are 44 393 points of radar radial wind data and 3713 points of derived data. The cost function of STMAS is normalized so that the radial wind and conventional observations have equal weights, assuming there is no other statistical variance or covariance information available.

Eight level grids are employed, ranging from about 460 km × 460 km (*n* = 1) to 3.6 km × 3.6 km (*n* = 8), with a grid ratio of 0.5. The background is set to zero. Note that in future typhoon analysis a model background field will be included. The observation error covariance matrices, which are assumed to be diagonal, are the same in every grid level.

In Fig. 9 (top–bottom), the analyses by assimilating radar radial wind data and derived data are shown, and the derived data are added gradually for 10, 35, 143, and 920 points. From left to right, they are radial wind, *U* component, *V* component, wind vector, *U* component difference *DU*, and *V* component difference *DV*, where *DU* and *DV* are the differences between the analyses and derived data. From these *DU* and *DV* patterns, it can be seen that these differences have a structural pattern that decreases gradually as we add the derived data gradually. From Fig. 10, it will be shown that the structural pattern cannot vanish completely, even if all of the derived data are added.

In Fig. 10, the last row shows the *U* component (Fig. 10i) and *V* component (Fig. 10j) patterns of derived data, and the second row shows the analyses using the 3713 points of derived data only (Figs. 10e–h). The differences *DU* and *DV* are almost zero in this row, as expected because no other data source is used (Figs. 10g,h). The first row of plots in Fig. 10 shows the analyses assimilating both radar radial wind data and all of these derived data (Figs. 10a–d); the differences *DU* and *DV* still contain the structural pattern (Figs. 10c,d), and the maximum difference is about 15 m s^{−1}. It would be interesting to understand why the structural pattern exists. So, we will see the analyzed radial wind information (Fig. 11) from these two analyses, one obtained by assimilating both radar and derived data and the other by assimilating the derived data only.

Figure 11a is the radar radial wind pattern, and Fig. 11c is the radial wind of the derived data. Figure 11b is the analyzed radial wind by assimilating both radar data and derived data, and Fig. 11d is the analyzed radial wind by assimilating the derived data only. From both the top and bottom rows, it is clear that the analyzed radial wind (Fig. 11b) assimilating both radial wind and derived wind is very close to the observed radial wind (Fig. 11a). The analyzed radial wind (Fig. 11d) assimilating only derived wind is very close to the radial wind of the derived data (Fig. 11c) in the STMAS data assimilation process. From the left column (Figs. 11a,c), it is obvious that the analyzed radar radial wind of derived wind data does not match the observed radial wind at all, which results in the corresponding analyzed radial wind fields (Figs. 11b,d) also not matching each other. So, a conclusion can be drawn that the structural pattern difference is caused by the difference between the observed radar radial wind and the radial wind of derived wind data. The derived wind vector data can only catch some certain wavelength information under a perfect circular typhoon assumption, and the results shown in the right column convince us that radar radial data provide some additional useful information in the STMAS data assimilation process.

So far, it is well known that the strength of a typhoon is difficult to simulate well. Moreover, the initial condition of a wind field is important to simulate. If the current wind field is captured precisely, it will provide a good initial condition for the model to make a better forecast. In this experiment, it is shown that radar radial wind data really provide some additional useful information in the multigrid method data assimilation process. Of course, these results need to be further verified in future studies.

This experiment also suggests a practical approach for data assimilation in an extreme weather condition, such as hurricane or typhoon, where conventional observations are few. It is useful to use the STMAS to assimilate the combined datasets of the derived wind data and radar radial wind data to produce a better analysis that cannot be obtained by assimilating either of these two datasets. So, the process to analyze the wind field in extreme weather conditions, in the case of insufficient conventional data, contains two steps. The first step is to directly analyze the radial wind data and derive a first-order approximation of the wind field, and the second step is to use STMAS to assimilate these two datasets as the final analysis. This approach is ready for extreme weather forecasts in future research.

## 6. Conclusions

In this study, the multigrid method used in STMAS is introduced into radar radial wind data assimilation. Two idealized experiments and one real radar radial wind experiment are performed. The main conclusions can be drawn as follows:

- Because the radar radial wind data can provide additional information, better analyses can be obtained by assimilating both radial wind data and conventional data. Particularly in the case of sparse conventional data, radar radial wind data will provide relatively significant information and improve the analyses considerably.
- In extreme weather conditions, such as hurricanes or typhoons, it is also useful to use the multigrid method to assimilate the combined datasets of the derived wind data and radar radial wind data.

## Acknowledgments

The authors would like to express their gratitude to three reviewers for their helpful comments and suggestions, which contributed to greatly improve the original manuscript. The first author of this paper was jointly supported by grants of the National Basic Research Program of China (2007CB816001), the National Natural Science Foundation of China (40776016, 40906016, 40906015), and the National High-Tech RandD Program of China (2006AA09Z138).

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The *U* and *V* components RMS differences between the true field and STMAS analysis by using different observations in the first idealized experiment. CNVTN: analysis result by assimilating conventional data only; RADAR: analysis result by assimilating radial velocity only; RADAR+CNVTN: analysis result by assimilating the combination of radial velocity data and conventional data; RADAR+CNVTN+1: as in RADAR+CNVTN but adding just one conventional observation at the top-left corner; DENSE DATA or 2RADAR: analysis result by using dense conventional observations or the combination of dense conventional observations and radar observations or two radars.

The *U* and *V* components RMS differences between the true field and STMAS analysis by using different observations in the second idealized experiment. RADAR: analysis result by assimilating radial velocity only; CNVTN_25: the analysis by assimilating 25 points of conventional data only; RADAR+CNVNT_25: the analysis by assimilating both the same 25 points of conventional data and radial velocity data; CNVTN_100: as in CNVTN_25, but for 100 points of conventional data; RADAR+CNVNT_100: as in RADAR+CNVNT_25, but for 100 points of conventional data; CNVTN_441: as in CNVTN_25, but for 441 points of conventional data; RADAR+CNVNT_441: as in RADAR+CNVNT_25, but for 441 points of conventional data.