1. Introduction
Since Doppler radar has been applied in atmospheric wind sounding and weather analysis (Potvin et al. 2012; Liu et al. 2003, 2013; Frehlich 2013; Thompson et al. 2012), many efforts have been made on wind field retrieval by using measured the wind radial velocity. In the early 1960s, Lhermitte and Atlas (1961) developed the velocity–azimuth display (VAD) method, which is based on a uniform wind field assumption. However, the VAD method is not suitable for complex wind retrievals. To obtain more wind field information, many methods were proposed based on a simple wind model assumption, such as velocity area display (VARD; Easterbrook 1975), extended VAD (EVAD; Srivastava et al. 1986; Matejka and Srivastava 1991), concurrent extended VAD (CEVAD; Matejka 1993), airborne VAD (AVAD; Leon and Vali 1998), gradient VAD (GVAD; Gao et al. 2004), volume velocity processing (VVP; Waldteufel and Corbin 1979), tracking radar echo by correlation (TREC; Tuttle and Gall 1999), typhoon circulation TREC (T-TREC; Wang et al. 2010), velocity–azimuth process (VAP; Tao 1992), integrating VAP (IVAP; Liang 2007), velocity plan processing (VPP; Lang et al. 2001), extended VPP (EVPP; Fang et al. 2007), vorticity–divergence method (Jiang and Ge 1997), and the reflectivity–radial velocity co-constraint method (Han et al. 2013).
More complex methods such as 3D or 4D variational analysis methods take successive observations from multi-Doppler radar as input data together with other observation data as the background field and apply dynamic or diagnostic equations as constraints. Using these methods, the structural details of the weather process can be retrieved on a storm scale; and unobserved terms also can be obtained simultaneously (Gao et al. 1999, 2006; Huang et al. 2010; Protat and Zawadzki 1999; Qiu and Xu 1992; Rihan et al. 2005; Shapiro et al. 1995, 2003, 2009; Sun and Crook 1997, 1998; Shun et al. 2005; Weygandt et al. 2002). However, selecting the proper weighting coefficients in a cost function is still a challenge because of uncertainties in observations and equation errors cannot be evaluated exactly (Shapiro et al. 2009). If asumptions such as velocity stationary, frozen turbulence, and pattern translation are not valid, then the analysis becomes significantly degraded. Early studies of single-Doppler velocity retrieval techniques focused mainly on fitting radial velocity measurements to a simple wind model. Because of the simple spatial dependence assumed for the wind field, some retrieval techniques were valid when the wind field is not complex (Shapiro et al. 1995). In addition, simple model results can be used as initial conditions to estimate background wind in 3D or 4D variational analysis. However in some cases using complex retrieval techniques, the real-time computation time must also be considered (Caya et al. 2002; Gao et al. 2006). Since the large distances between radars hinders multiple-Doppler wind synthesis, previous works have had to focus on using single-Doppler radar observations. Therefore, traditional retrieval techniques using simple models are feasible tools and offer valuable insights for wind field analysis by way of real-time observation.
Because of the differences in wind models, the scope of application for each retrieval method differs. Most retrieval algorithms behave well for 2D wind field retrievals, but have difficulty in accurately retrieving the 3D wind field. The VVP method, which is based on a linear wind field assumption, can give a better description of the wind field and obtain more wind field information compared to other simple retrieval algorithms. The ill-conditioned coefficient matrix (CM), which would lead to large errors in retrieval, is impossible to directly solve. Research on reducing the number of retrieval parameters was discussed (Xin and Reuter 1998; Holleman 2005). Waldteufel and Corbin (1979) removed the smaller parameters in the wind model according to the magnitude of the parameters in an attempt to simplify the CM. Boccippio (1995) selected six parameters for retrieval by applying singular value decomposition in the solution; but asserted that potential errors would be introduced by the neglected parameters. Wei (1998) decreased the instability and difficulty in a solution by a matrix balance method that could reduce the CM's condition number. Li et al. (2007) proposed a stepped method that could obtain retrieval results of uniform wind field first and then obtain other parameters according to pre-assumptions of the parameters. In the studies on the causes of the ill-conditioned CM and the difficulties of solution in the VVP method, Koscielny et al. (1982) calculated the correlation between parameters in different volume size analyses and concluded that retrieval results would be inaccurate when the parameters were linearly correlated. Furthermore, some parameters would be unobtainable due to the missing information caused by high linear correlation. Boccippio (1995) noted that the collinearity between matrix vectors might cause numerical instability and variance inflation, and increase sensitivity to bias from neglected factors. The effects mentioned above, especially when dealing with small azimuthal sectors of data, would also hinder the application of the VVP method. Caya et al. (2002) improved the VVP technique by applying it to a moving reference frame with a constant background wind field velocity. Holleman (2005) compared the retrieval result of the VVP and VAD methods and found that nonorthogonal basis functions in the CM may cause fluctuations of the retrieved parameters; the retrieval results of the VVP method were better when using fewer parameters, although fewer parameters were not fit to the model assumptions.
In the current work, the sensitivity of the ill-conditioned CM and the size of the analysis volume on the retrieval accuracy in VVP method are analyzed. A brief review of the VVP method is presented in section 2. In section 3, we estimate the upper-limit error norm of the retrieval result and discuss the errors caused by the neglected parameters. In section 4, we briefly address the features of the condition number and the determinant of the CM. The influence of computer accuracy on retrieval, of the analysis volume size on observation and calculation errors, and a test of uniform wind field are discussed in section 5. A summary of the conclusions follows in section 6.
2. A brief introduction of VVP technique
The analysis volume in the VVP technique is a 3D space containing several PPI observations. The hypothesis adopted for the analysis volume is that the wind field varies linearly and remains constant during radar scanning.
3. Sensitivity analysis
The errors that affect retrieval accuracy can be classified as 1) model errors caused by the inaccuracy of the mathematic description of the wind field model; 2) observation errors that come from sounding or radial velocity measurements; and 3) computation errors caused by the calculation method and computer precision (Wang et al. 2011).
Because of the limitations of a computer word length, the order of the CM and the magnitude of the condition number have a large influence on calculation errors (Ward 1977). Thus, for the VVP method, the accuracy of the retrieval result is mainly determined by the scale and features of the CM, the method used in solving it, and computer precision. It thus can be inferred that a complex CM could lead to large calculation errors.
The influence of computer accuracy on retrieval will be presented in section 6.
a. The upper limit and the estimation of retrieval error
It can be shown that 
Although neglected unimportant factors in the wind field model that will be discussed in the following sections, introduce potential errors in retrieval, choosing the appropriate CM not only improves the analysis’s accuracy but also reduces the difficulties in the solution process. Mathematical skill can be used to modify the coefficient matrix 
b. Errors caused by neglected parameters
Although small parameters could be neglected to decrease the difficulty of the solution, retrieval errors would increase when the neglected parameters are large. Also, the size of the matrix elements varies with the analysis volume’s position and size. Therefore, neglecting partial parameters for increasing the accuracy of the retrieval is not efficient.
4. The condition number and the determinant of coefficient matrix
In this section, we use synthetic wind field data hat has eight layers. The interval between any two adjacent elevation angles is 0.5°, while the number of azimuths is 360 and the interval between any two adjacent azimuths is 1°. There are 600 volume gates in each radial and the gate width is 250 m.
a. The condition number of coefficient matrix
According to (6), 12 fitted parameters need to be retrieved for the full model retrieval. To compare the condition number of CMs containing different retrieval parameters, we sorted these parameters as shown in Table 1 and calculated the 2-norm condition numbers of the CMs (see Fig. 1). As mentioned in the estimation of the upper-limit retrieval error using (11), radial velocity errors of the CM analyzed were not considered.
Groups of fitted parameters.
The variation of the CMS’s condition numbers (colors) for (a)–(l) the different retrieved parameters shown in Table 1. The size of the analysis volume is specified as a sector width of 10° and 20 gates for each radial. The elevation is 2.5°.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00190.1
The condition number is a measure of computing complexity and sensitivity to error. In the VVP method, the large condition number indicates that the solution matrix would be seriously ill conditioned, and the robustness of the retrieval would be decreased (Boccippio 1995; Caya et al. 2002).
As shown in Fig. 1, the distribution of the condition number shows that 1) the condition number varies exponentially with an increase in the number of fitted parameters and 2) the difference of the condition numbers is large even if the number of fitted parameters is the same. For example, an increase of condition number is likely to result if vorticity terms are included. The condition numbers of the 4P2, 5P2, and 6P2 groups, including 
b. The correlation among coefficient matrix vectors
Koscielny et al. (1982) calculated the correlation among CM elements, and found the correlation coefficient could reach 0.9. Because of the limitation of computer accuracy, the values of correlation coefficients of the matrix vectors would be too close to compare when the CM was seriously ill conditioned. Thus, the determinant of the CM is used as the standard to evaluate the correlation of matrix vectors in this study.
As shown in Fig. 2, the variation of the determinant suggests that the correlation among matrix vectors increases gradually when new parameters are added. Moreover, comparing Fig. 2k with Figs. 2i and 2j, it is seen that the difference among the determinants can reach dozens in value; this indicates a strong correlation among matrix vectors 
As in Fig. 1, but for the value of the determinants of the CMSs.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00190.1
For studying further the variation of condition number, we calculated as an example the correlation coefficient among matrix vectors for the 4P1 group (see Fig. 3). The correlation of matrix vectors corresponding to 
The correlation coefficients of the matrix vectors corresponding to 
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00190.1
5. The influence of the analysis volume’s size on retrieval
For the CMs in the 4P1 group (along with the change of sector width), the variations of the matrix elements: (top) (left) a1, (middle) b1, and (right) c1; (middle) as in (top), but for a2, b2, and c2; and (bottom) condition numbers (left to right) a3, b3, and c3 vs azimuth. The elevation is 2.5°. The distance to the radar is 75 km and the volume gates are fixed. The quantities 
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00190.1
Comparing Figs. 4 and 5, the increase of elements in the CM is significantly different when the volume size increases. Specifically, the variation of matrix elements is larger with the change of the width of the volume sector rather than with the number of volume gates; and the decreasing trend of the condition number is obvious with the increasing number of volume gates compared to the increase of the sector width. For the matrix elements 
As in Fig. 4, but different volume sector widths and numbers of gates.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00190.1
Generally, a large analysis volume can offer more wind field information and the strong linear correlation among CM vectors is decreased. However, it should be noted that the condition number is much less sensitive to domain size than to azimuth and model bias (see Fig. 1); and increasing the retrieval domain size will increase the influence of nonlinearities in the true wind field when wind field model assumptions are no longer suitable. Besides, we consider that random errors of calculation and observation can also be smoothed in a large analysis volume, which improves retrievals. This point was confirmed by a test of uniform wind field shown in section 5b.
Because a large difference among matrix elements would generate calculation errors, the volume size cannot be increased arbitrarily. Although the correlation between matrix elements increases, the condition number becomes more optimal when correlations between matrix vectors decrease. Therefore, there are two methods to directly reduce the value of the condition number: one is to increase the volume sector width when the analysis volume is relatively small, and the other is to increase the number of volume gates when the analysis volume is relatively large.
a. The influence of computer accuracy on retrieval
b. The influence of the analysis volume’s size on random error
This infers that at the beginning of retrieval, once the error spread is prevented, the accuracy of the retrieval results improves. Moreover, the larger-sized analysis volume not only introduces more information of the wind field for retrieval, but it also optimizes the condition number and mitigates random errors in calculations and radar observations.
c. A test of uniform wind field
A uniform wind model contains only three parameters u, Ï…, and w, which can be retrieved without model bias by using the parameter groups as shown in Table 1. If a complex wind model is adopted, such as one with a linear wind field that needs more fitted parameters to describe the wind model in order to prevent model bias, then this will always result in an ill-conditioned CM with a large condition number.
In the test, the size of the analysis volume is enlarged in two ways: one is to increase the number of volume gates and the other is to increase the sector width. The number of data points in the analysis volume is the same. Retrieval results are presented in Fig. 6. Using the two approaches mentioned above, the errors of wind velocity and direction both decrease with the enlargement of the analysis volume. This conclusion indicates that enlarging the analysis volume can improve the accuracy by decreasing calculation errors. When the size of the analysis volume is small, the difference of the retrieval errors obtained by the two approaches is slight, but the RRE of the wind velocity with an increase of volume gates (see Fig. 6a) decreases faster than that with an increase of volume sector width (see Fig. 6b). Therefore, increasing the size of the analysis volume can reduce the influence of computational errors and increase the accuracy of the retrieval; this confirms the conclusions in section 5b.
The RREs of horizontal wind speed and direction for: (a) a volume sector width fixed at 20° with the number of volume gates changing from 30 to 120; and (b) the number of volume gates fixed at 40 and the volume sector width changing from 15° to 60°.
Citation: Journal of Atmospheric and Oceanic Technology 31, 6; 10.1175/JTECH-D-13-00190.1
6. Conclusions
In the VVP method, errors in retrieval result come from: computational accuracy, intrinsic features of the wind field model, and observation error. Through the analysis of CM and the sensitivity of retrieval errors to different factors, we draw the following conclusions:
When more fitted parameters are chosen, the mathematical description of the real wind field will be more accurate. But the CM would be more ill conditioned, and the retrieval would be unsolvable. In the case of a certain number of fitted parameters, the condition numbers of the CM can be obviously different depending on the choice of wind models. By analyzing the upper limit of the error norm, choosing properly fitted parameters can reduce the solution difficulty and the calculation error. Even though less fitted parameters might introduce large model errors, less fitted parameters could help to obtain a better retrieval result rather than more fitted parameters.
The linear correlation between matrix vectors is a reason for ill-conditioned CMs. Because a CM is a function of position, the degree of correlation changes notably for the different locations. Retrieval results from regions with large condition numbers should be carefully examined. The different degrees of correlation among matrix vectors also indicate that the choice of wind model is vital retrieval results. When more fitted parameters are chosen, a large condition number and an ill-conditioned CM would mainly be the cause of the strong correlation among matrix vectors.
Because of the limitation of computing accuracy, it is not appropriate to carry out a retrieval when the condition number of a CM is too large and the computational accuracy does not meet the requirements of calculation. The analysis of error spread in calculations shows that the increase in the analysis volume’s size and an improvement in calculation accuracy can mitigate random errors. The large analysis volume could reduce the influence of observation and calculation errors when these errors are from a uniform or normal distribution. Moreover, the retrieval test of a uniform wind field indicates that the decrease of condition numbers and the increase of matrix elements is clearly due to increasing the number of volume gates.
Acknowledgments
This research was supported by the Graduate Students’ Scientific Research Innovation Program of Jiangsu Higher Education Institutions of China (CXZZ_0501, CXLX12_0501), the National Basic Research Program of China (the 973 Program) (2013CB430102), the China Meteorological Administration Special Public Welfare Research Fund (GYHY201206038, GYHY201306040), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (10KJA170030), the National High Technology Research and Development Program of China (the 863 Program) (2007AA061901), the State Key Laboratory Program (2013LASW-B16), the Nanjing Weather Radar Open Laboratory Foundation (BJG201208), the Aero-Science Fund (201320R2001), the key technology projects of the China Meteorological Bureau (CMAGJ2014M21), and the Industry Fund by the Ministry of Water Resources (201201063).
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