## 1. Introduction

The ground-based weather Doppler radar has been an important observational tool for deducing kinematic structures of precipitating (Browning and Wexler 1968; Waldteufel and Cobin 1979; Lee et al. 1994), and more recently nonprecipitating (Kollias et al. 2001; Bluestein et al. 2004), cloud systems. The ability to observe a large region with higher spatial and temporal resolutions than those obtained from the rawinsonde and surface station measurements has greatly advanced our understanding of various mesoscale and convective weather systems (Houze 2004). Despite its wide applications in both research and operational communities, the direct interpretation of the three components (*u*, *υ*, *w*) of a wind field from the observed Doppler velocity is not straightforward because a Doppler radar only measures the component of a wind field projected along the radar beam direction. The retrieval of basic physical properties (horizontal velocity, vertical velocity, divergence, deformation, and vorticity) from single-Doppler radar observations can only be accomplished with certain assumptions on the structure of the actual wind field. Previous studies have used a linear assumption (e.g., Lhermitte and Atlas 1961; Browning and Wexler 1968; Waldteufel and Cobin 1979; Srivastava and Matejka 1986; Matejka and Srivastava 1991), circular vortex assumption (Lee et al. 1994, 1999), and steady assumption (Peace et al. 1969; Bluestein and Hazen 1989) to model and retrieve the physical parameters of a real wind field observed by a single-Doppler radar. Except for these Doppler velocity-based algorithms, the horizontal wind vectors have also been deduced by tracking radar reflectivity patterns (Rinehart and Garvey 1978; Tuttle and Foote 1990; Tuttle and Gall 1999). These studies significantly advanced the utilization of single-Doppler radar observations to estimate key characteristics of atmospheric kinematic structure.

Doppler velocity can be expressed in analytic functions on the aforementioned simplified wind fields. Limited physical information of these real simplified wind fields can be obtained by conducting a least squares fitting (LSF) of the observations to the analytic functions. The VAD method (Browning and Wexler 1968) expressed Doppler velocities along “a ring of constant radius centered at the radar site” (hereinafter referred to as a VAD ring) as a Fourier series, a function of azimuth angle. If the wind field is linear, the resulting Fourier coefficients of the LSF can be physically interpreted as the mean divergence, mean horizontal winds, and mean horizontal deformation averaged over an area encircled by the VAD ring. By processing the data of multiple VAD rings on different radii and/or different elevation angles, the VAD vertical wind profile^{1} (e.g., Fig. 1 in Chrisman and Smith 2009) can be deduced and has been widely used in both research and operations (e.g., Davis and Lee 2012; Giammanco et al. 2013; Kingsmill et al. 2013).

The spatial linearity is the fundamental assumption for deducing physical parameters (i.e., the Fourier coefficients) of a wind field from the VAD analysis. However, the departure of actual wind fields from linearity is common in real weather situations (Rabin and Zawadzki 1984). When nonlinear^{2} wind components exist, the Fourier coefficients deduced in the VAD analysis are not representative of the mean kinematic properties in a real wind field (Waldteufel and Cobin 1979; Koscielny et al. 1982; Caya and Zawadzki 1992). Unresolved nonlinear wind components could contaminate the retrieval of linear wind components (Waldteufel and Cobin 1979; Koscielny et al. 1982) and lead to significant errors in retrieved vertical wind profiles (VWPs). In their formulation of the extended VAD (EVAD) method, Matejka and Srivastava (1991) also noticed the necessity to include certain nonlinear wind components to retrieve the mean divergence within a VAD ring. Although real wind fields may not deviate significantly from a linear distribution under certain weather conditions, the intrinsic linear wind hypothesis in the VAD analysis is unduly restrictive in obtaining accurate kinematic properties for both research and operational purposes (Caya and Zawadzki 1992, hereinafter CZ92). Despite the documented detrimental effect of unresolved nonlinear wind components on VAD analysis, there is no simple way to examine whether the wind field is linear. Hence, the existence of nonlinear wind components was mainly neglected in the VAD analysis in practice.

CZ92 extensively examined the relationship between the physical parameters of a nonlinear wind field and the Fourier coefficients from the VAD analysis. It was shown that the Fourier coefficients (e.g., Browning and Wexler 1968) no longer represent those aforementioned kinematic properties for nonlinear wind fields, except for the divergence term. However, the Fourier coefficients become functions of radius at a constant altitude in nonlinear wind fields. By utilizing this range dependence, a second polynomial fit was used to retrieve the horizontal wind vector at the radar site for nonlinear wind fields. The work presented in CZ92 not only clarified the relations between the physical parameters of a wind field and the results of VAD analysis under nonlinear conditions, but also proposed a new method (hereinafter referred to as NVAD) to account for the nonlinearity and obtain physically meaningful results. However, CZ92 only tested the NVAD method in limited cases in which its practical limitations were not fully explored.

Mesoscale weather systems typically contain nonlinear winds. During the Southwest Monsoon Experiment (SoWMEX) and the Terrain-Influenced Monsoon Rainfall Experiment (TiMREX) field campaign (Jou et al. 2011), the NVAD method was implemented to deduce more frequent and accurate VWPs from Doppler velocity observations collected by the National Center for Atmospheric Research (NCAR) S-band dual-polarization Doppler radar (S-Pol) to supplement the rawinsonde VWPs, which were collected 4–8 times per day. After processing a large set of SoWMEX/TiMREX data, it was found that the NVAD method generated inconsistent VWPs in many circumstances. The NVAD method is sensitive to the distribution of data noise and voids in the Doppler velocity observations. The sensitivity is again related to the mathematical formulation and corresponding computation process used in NVAD.

The purpose of this study is to propose an alternative method, the distance velocity azimuth display (DVAD; Lee et al. 2014), to deduce VWPs in nonlinear wind fields. Instead of using Doppler velocity *r*) as a variable. A description of the general use of ^{3} equations. In addition to its graphical intuition and mathematical conciseness, the potential of using DVAD to quantitatively retrieve physical parameters from both linear and nonlinear wind fields is exploited in this study, and the results are compared with the VWPs deduced from the NVAD method. It is shown that DVAD is consistent with NVAD under idealized conditions but is more robust when processing real observations. The concise mathematical expression of DVAD allows the estimation of the degree of nonlinearity of real wind fields in an objective way.

The rest of this paper is organized as follows. In section 2, a brief review of the mathematical formulation of NVAD and DVAD is provided. In section 3, an evaluation of VWPs retrieved from NVAD and DVAD using an idealized dataset is conducted, and the potential limitations of NVAD are discussed. In section 4, field campaign and operational Next Generation Weather Radar (NEXRAD) datasets are used to demonstrate the robust performance of DVAD. In section 5, an objective way of assessing the nonlinearity of actual wind fields is discussed. Discussion and conclusions are given in section 6.

## 2. Review of NVAD and DVAD

### a. NVAD

*u*,

*υ*, and

*w*), terminal fall speed

*α*, elevation angle

*β*, and the slant range

*r*, as illustrated in Armijo (1969):The horizontal components

*u*and

*υ*of a linear wind field can be expressed using their Taylor expansions at a plane of constant altitude by neglecting the vertical velocity

*w*and terminal fall speed of precipitation particles

*α*at a fixed range

*r*, the coefficients of (3) are related to the mean values of kinematic parameters of the real wind field within a VAD ring aswhere

*x*,

*y*of variables

*u*,

*υ*stand for the partial derivatives of the horizontal winds (e.g.,

*α*from 0 to 2

*π*as follows:A comparison of terms on the right-hand sides of (6) and (7) indicates that, except for the mean divergence term, the physical wind parameters in (7) no longer possess the same relations to the Fourier coefficients in (6) as in the linear case shown in (4). The physical wind parameters cannot be solved by the same VAD technique by fitting

*r*. Therefore, these physical quantities at the radar site (i.e.,

*r*= 0) can be evaluated by a second LSF via the Fourier coefficients obtained at different ranges. This two-step LSF procedure that is based on (6) formed the basis for the NVAD analysis proposed by CZ92. Plausible VWPs in clear-air boundary layers were retrieved by NVAD (CZ92).

### b. DVAD

*r*, the quantity

*x*,

*y*, and

*z*) are centered at the radar. The altitude

*z*of a data point is determined using the

*u*,

*υ*, and

*w*at a constant altitude can be represented by their Taylor series with respect to the radar site asOn the spatial scale (~100 km) covered by a Doppler radar, the mean magnitudes of

*w*and

*u*and

*υ*. Moreover, the limited value of the maximum elevation angle (<20°) scanned by an operational ground-based Doppler radar results in a small contribution of

*w*and

*w*and

*w*and

*w*and/or

*u*and

*υ*can be sufficiently approximated by finite terms of their Taylor series, then

*x*and

*y*without the complexity of trigonometrical basis functions. Equation (10) can be written in a more concise form using the summation notation:where

*n*is the highest degree of the 2D polynomial function. Assuming there are

*l*Doppler velocity observations at the same height, the coefficient

*l*items each containing observed

*l*is the number of observations and

*m*is the total number of coefficients of the polynomial function) depending on the distribution of radar observations and the order of the nonlinear wind model,

## 3. Evaluation of NVAD and DVAD using an idealized dataset

The mathematical expressions of NVAD and DVAD are formulated to retrieve the same VWP at the radar, so the same result is expected when applying the two methods to the same dataset. Despite their theoretical equivalence, the different solving processes associated with the different mathematical formulations of NVAD and DVAD could lead to different results when processing real Doppler velocity observations. Since the true VWP above the radar site is generally unknown for real observations, a synthetic dataset with a known true VWP is used to evaluate the accuracy of VWPs produced by NVAD and DVAD, respectively. Common characteristics of Doppler velocity in real observations, in terms of data quality, are simulated on the synthetic dataset by sequentially adding measurement noise and voids. The VWPs deduced from NVAD and DVAD could be dramatically different in these situations and possible causes are investigated.

### a. Comparison of VWP

The synthetic dataset was obtained from an observation simulation experiment (OSE). The original 3D wind field in the OSE was extracted from a numerical simulation of a squall line (Sun and Zhang 2008), and the wind vectors at three different altitudes with three

- E
_{I}, no artifacts; - E
_{N}, random noise drawn from a normalized distribution is added to the ideal data in E_{I}; and - E
_{N+V}, data voids are added to the noisy data in E_{N}to simulate a realistic radar observation.

^{−1}) from the true values when the data are free of artifacts (E

_{I}). The minor deviations are caused by the fact that the original wind field has an infinite number of nonlinear terms, while the results of both methods shown in Fig. 2a (the same for Figs. 2b and 2c) are based on a fourth-order nonlinear wind model. The deviation from the true value is mainly caused by the unresolved higher-order terms.

After adding random noise with a standard deviation of 2 m s^{−1}, the retrieved VWPs from both methods in Fig. 2b show little changes. It is worth pointing out that the random noise is added directly to the resampled Doppler velocities. As shown in (12), the normally distributed noise has been taken into account in the least squares formulation. As a result, adding random noise in evenly distributed data (without data void) will not affect the result.

Nevertheless, the results can be dramatically different when the same noisy data are combined with data voids. The data voids added in E_{N+V} are missing data points taken from a real volume scan collected during the SoMWEX/TiMREX field campaign. Each PPI scan of the resampled _{N+V} are shown in Fig. 3. Figure 2c shows that the VWP derived from NVAD deviates significantly from the true value, especially at upper levels, while the VWP derived from DVAD is generally not influenced by the same data noise and voids. Possible reasons for the distinct results are discussed in the next subsection.

### b. Limitations of NVAD method

The mathematical formulation of NVAD requires two sequential LSFs to solve for the horizontal wind at the radar site. The first step is similar to the traditional VAD analysis with higher-order Fourier series used to fit _{N+V}).

The uncertainty in the first-step VAD analysis of NVAD is closely related to data voids and noise in observed Doppler velocities. As illustrated in Daley (1993, 45–49), fitting observations of uneven density (due to the existence of data voids) and noise is prone to the overfitting problem, especially when higher-order mathematical models are used. Data noise and voids are inevitable in Doppler radar observations as a result of weak signal, ground clutter, nonmeteorological objects, abnormal propagation, beam blockage, and velocity aliasing. Figure 4 and Table 1 present an example of an incorrect VAD analysis caused by the overfitting problem in real observations. Figure 4a shows a PPI plot of Doppler velocities observed at 0.5° elevation angle during the SoWMEX/TiMREX field campaign. The large black circle in Fig. 4a indicates the VAD ring used to perform the VAD analysis. The fitted curves of different orders from this VAD ring are shown in Fig. 4b, and the corresponding values of the first five Fourier coefficients and their standard deviations (SDs) of fitting are listed in Table 1. It is noted that the SD values of fitting decrease monotonically with higher-order VAD analyses, which indicates higher-order wind models fit better to the observed data. The fitted curves in Fig. 4b also demonstrate the same tendency where the curves of higher-order models are closer to the observed data. Nevertheless, a comparison of the two coefficients

The first five Fourier coefficients and SD of fitting from high-order VAD analyses. Under the linear assumption,

The uncertainty in the second-step polynomial analysis of NVAD can be understood by examining the SD values listed in Table 1. Despite the incorrect results of the fourth-order VAD analysis, its SD value of fitting is nevertheless the smallest. NVAD uses the SD values of the first step fitting as weights for the Fourier coefficients used in the second step polynomial fitting. These physically incorrect coefficients will yield higher weights and further contaminate the final results.

The two limitations of NVAD are mutually related and essentially determined by the distribution of data noise and voids. To better illustrate the impact of data noise and voids on the accuracy of NVAD and DVAD retrievals, an azimuthally continuous gap with increasing width is added to E_{N}. The results in Table 2 show that, as the width of the gap and the order of fitting increase, the error of retrieval by NVAD increases accordingly while those of DVAD remain small. The higher-order results of NVAD become unusable when the gap width exceeds 150°. It is noted that when the maximum gap width is greater than 180° (not shown), the errors of both methods become significant and unusable. Table 3 shows that the DVAD result is not sensitive to the azimuthal location of the gap. In real weather observations, data noise and voids are not only inevitable but also variable in both time and space. It is difficult to examine the data quality of each VAD ring of a volume scan used in NVAD and to rule out those that could potentially lead to incorrect results.

Speed errors (m s^{−1}) of DVAD and NVAD subjected to different continuous gap sizes (30°, 60°, 90°, 120°, and 150°).

Speed errors (m s^{−1}) of DVAD subjected to a 180° azimuthally continuous gap placed in different directions.

Within the DVAD framework, linear and nonlinear wind fields at a constant altitude are expressed as a single polynomial function only with different orders. Unlike CZ92, the VWP at the radar is directly resolved with only a one-step 2D LSF whether the wind field is linear or nonlinear. The greater redundancy of data used in the 2D fitting process effectively reduces the overfitting problem and is able to overcome the limitations of NVAD as demonstrated in E_{N+V}. Although the above experiments demonstrated the robust performance of DVAD when processing simulated Doppler velocity observations with noise and voids, they only provided a specific scenario of the distribution of noise and voids, while their distributions could vary greatly in real weather systems. A more comprehensive comparison between NVAD and DVAD as well as the limitations of DVAD, using both SoWMEX/TiMREX field campaign and operational NEXRAD datasets, is presented in the next section.

## 4. Evaluation of NVAD and DVAD using real datasets

### a. VWP during SoWMEX/TiMREX

The SoWMEX/TiMREX field campaign was designed to address the physical processes leading to the heavy precipitation events in southwestern Taiwan influenced by the interactions between the onset and northward advances of the Asian summer monsoon in late spring and Taiwan’s steep topography (Jou et al. 2011). The primary radar data used in this section were collected with the NCAR S-Pol radar from 1800 UTC 2 June to 0630 UTC 3 June 2008. Rawinsondes were released four times daily between 15 May and 30 June 2008, and the frequency increased to eight times daily during selected intensive operation periods (IOPs). These rawinsonde data provided an independent reference for evaluating the VWPs obtained from NVAD and DVAD. Figure 5 shows sample

Figure 7 shows retrieved VWPs using second-order nonlinear wind models to approximate the real wind field. For the purpose of verification, the sounding data from the Pingtung rawinsonde site in southern Taiwan (blue plus sign in Fig. 5) are included for comparison with results from NVAD and DVAD. It is noteworthy that the rawinsonde measures a wind profile along the trajectory of the balloon, while the VWPs obtained from Doppler radar (both NVAD and DVAD) represent compatible winds over the large area covered by the radar. Close agreement between the two types of VWPs is not expected (CZ92). Nevertheless, good agreement between the radar-derived VWP with that of the rawinsonde confirms the representativeness of the VWP obtained from NVAD and/or DVAD.

During the analysis time period, a cold front approached the S-Pol domain from the northwest and passed the S-Pol site at

Figure 8 shows retrieved VWPs using fourth-order nonlinear wind models to approximate the real wind field. When the fourth-order nonlinear model was used, there were significant differences between the results of the two methods. As shown in Fig. 8a, the result of the NVAD method had abnormal and extreme values at different altitudes and times (e.g., 2000, 2230, and 0200 UTC). The missing wind barbs between 2000 and 2100 UTC were outliers where wind speed was greater than 40 m s^{−1}. It is noted that the occurrences of inconsistent and unreasonable results were intermittent, which was probably caused by the temporal variation of data voids. The result of DVAD using a fourth-order nonlinear model was consistent with that of the second-order model and compared favorably to the sounding data. The southwest-to-northwest veering of wind direction over time was still clearly shown by the time series of VWPs in Fig. 8b. Even-higher-order nonlinear models were used to retrieve VWPs at the S-Pol site, and the results (not shown) were similar to those in Fig. 8, where NVAD showed more incorrect results while DVAD continued to show consistent results with the sounding data.

In the calculation of Figs. 7 and 8, all Doppler velocity observations within the maximum range (150 km) of the S-Pol radar were used. It is possible to select only Doppler observations within a certain distance from the radar. Figure 9 shows VWPs retrieved in the same way as those in Fig. 8, but using Doppler velocity observations within 100 km from the radar. It is clear that the retrieved VWPs are consistent with those in Fig. 8, which indicates that neither method is sensitive to the range of data used for calculation. Figures 7–9 demonstrate the improvement of DVAD over NVAD when higher-order models are used; the performance of the different orders of DVAD will be addressed in section 5.

### b. VWP using NEXRAD datasets

The results using the dataset from the SoWMEX/TiMREX field campaign demonstrate that DVAD is generally more robust than NVAD when dealing with real observations contaminated by noise and data voids. In this section, two winter storm cases collected by the operational NEXRADs were used to examine the performance of NVAD and DVAD under different weather regimes and geographic locations.

The dataset of the first winter storm was collected during the fourth IOP of the Profiling of Winter Storms (PLOWS; Rauber et al. 2014) field campaign. It consists of Doppler velocity observations from 0100 to 2000 UTC 8 March 2009 collected by the NEXRAD located at Davenport, Iowa (KDVN), and high-resolution sounding data every 2 h during the same time period compiled by the NCAR Earth Observation Laboratory. Figure 10 shows four sample

The second winter storm was observed on 29 November 2006. The synoptic chart for this day is characterized by two highs in the west and east directions and a north–south-oriented shear line (http://www.hpc.ncep.noaa.gov/dailywxmap/). The NEXRAD dataset during 0800–2000 UTC 29 November 2006 from KDVN is used because there is a Global Telecommunication System (GTS) sounding site near the radar for verification. The sounding site records observations 2 times per day at 0000 and 1200 UTC. Thus, there is only one sounding VWP for verification during the period of radar observation. Figure 12 shows four sample

## 5. Objective estimation of nonlinearity

When extending the single-Doppler analysis from linear to nonlinear wind models, one of the critical questions is how many nonlinear terms of a wind model are necessary to sufficiently represent the real wind field. CZ92 did not specify how to determine the optimal order of NVAD to represent the wind field, and the degree of nonlinearity was empirically estimated before the computation. Previous studies (Waldteufel and Cobin 1979; Koscielny et al. 1982; Boccippio 1995) have shown that the results of single-Doppler analyses can be biased by the underfitting or overfitting problem due to the choice of an improper wind model. One must be able to dynamically and objectively adjust the nonlinear wind model for best results, especially for operational purposes.

The mathematical models for different orders of nonlinear wind fields used in NVAD and DVAD are nested, which means that the coefficients of a lower-order model are strictly a subset of those of a higher-order model. The standard way of determining the optimal model out of such nested models is to perform an *F* test (Wilks 2011). An *F* test basically compares the residual sum of squares (RSS) between higher- and lower-order models to determine if the higher-order model is still necessary. Since two LSFs are used in the NVAD method, and each LSF has its own RSS value, it is not straightforward to determine the goodness of fit of the two steps combined. Furthermore, incorrect Fourier fitting may still have small RSS values (RSS is proportional to SD) because of the overfitting problem shown in previous sections; the small RSS values in NVAD may not truly indicate the goodness of fit. In the DVAD method, the RSS of the entire domain at each analysis altitude can be directly obtained since only a one-step fitting is required, and the RSS is more representative of the goodness of fit of the mathematical model to the true wind field. Moreover, since the calculation of DVAD uses Doppler velocity observations within a finite vertical extent, the differences in nonlinearity at different altitudes can be estimated independently. The independent evaluation of nonlinearity at different altitudes is not possible in NVAD because the vertical variation of the wind field is forced to be the same as those of the horizontal winds a priori. It has been shown that real wind fields have different degrees of nonlinearity at different altitudes (CZ92); therefore, the DVAD method is likely to produce more accurate VWPs throughout the vertical extent of the analysis domain.

*F*test requires that the data samples used in the fitting model are independent. Since spatially adjacent radar observations are essentially dependent, the standard

*F*test is not applicable here. Following the principle of the

*F*test, the coefficient of determination

*F*test. By comparing the values of

Table 4 shows an example of

The

The objective procedure was applied to the same Doppler velocity observations from the SoMWEX/TiMREX field campaign. The retrieved VWPs alongside the objectively determined orders of nonlinear winds are shown in Fig. 14. The retrieved VWPs in Fig. 14a were consistent with Fig. 8b using fourth-order uniformly and compared favorably to the sounding VWPs. The determined orders of the nonlinear wind model shown in Fig. 14b provide some insights into the structure and variation of the wind field being examined. Figure 14b first shows that a higher-order wind model is generally required at lower levels. This is consistent with the increased variabilities of the wind field near the ground. Before 0100 UTC 3 June 2008 when the cold front moved into the radar scope, the nonlinearity of the wind field at lower levels was generally lower. During the passage of the cold front, the wind field with the radar scope consisted of both northwesterly and southwesterly winds, and Fig. 14b shows an increased order of the nonlinear wind field accordingly.

## 6. Summary and conclusions

In this study, the DVAD method was used to quantitatively analyze nonlinear wind fields observed by a single-Doppler radar. DVAD uses

The use of only a one-step LSF in DVAD allows for the direct estimation of the degree of nonlinearity in real wind fields. The same estimation is not feasible in NVAD because two separate LSFs are used. An optimal nonlinear wind model to approximate the real wind field can be objectively determined during the analysis by comparing the

The study shows the potential of DVAD as a quantitative analysis tool for single-Doppler radar observations beyond the qualitative interpretation presented in Lee et al. (2014). Although the case studies presented in this study demonstrate that DVAD is generally more robust than NVAD, they by no means suggest that the performance of DVAD is universally robust. In addition to the operational limitations mentioned in previous sections, DVAD is subject to the same fundamental limitation as that of VAD and NVAD. All of these single-Doppler analysis methods are based on a prescribed wind model, which results from a truncated Taylor expansion. The methods are essentially limited by the extent to which real wind fields can be represented by such a truncated Taylor expansion. In addition, as discussed in Lee et al. (2014), although DVAD simplifies the interpretation of single-Doppler observations, the vorticity still cannot be retrieved regardless of either the linear or nonlinear wind fields. Therefore, the method proposed in this study cannot be used in flow fields where the vorticity dominates (e.g., mature tropical cyclones). Despite its own limitations, the simplicity and robust performance of the DVAD method make it a good candidate for single-Doppler analysis in operational use.

## Acknowledgments

The authors thank Dr. Scott Ellis and Dr. Wei-Yu Chang for their helpful comments on the manuscript and Dr. Juanzhen Sun for providing the model output of the squall-line simulation. Comments and suggestions by three anonymous reviewers greatly improved the manuscript. The first author is grateful for the support by the Graduate Student Visitor Program of the NCAR Advanced Study Program (ASP) and the Earth Observing Laboratory (EOL) during this research. This study is supported by National Fundamental Research 973 Program of China (2015CB452801) and the research fund of the Key Laboratory of Transportation Meteorology (BJG201202).

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^{1}

The VAD vertical wind profile is commonly abbreviated as VWP in operational radar product catalogs. The same abbreviation is also used for the vertical wind profile, and logically the VAD wind profile is just one particular vertical wind profile calculated by the VAD method. Since the vertical wind profiles referred to in this study are not computed by the original VAD method, the abbreviation VWP is saved in later parts of this paper to refer to its original meaning, “vertical wind profile.”

^{2}

The word “nonlinear” is a common nomenclature meaning any variation in time and/or space that is not constant. Since this paper is focusing on the retrieval of horizontal winds at different altitudes, here nonlinear means higher-order variations of winds in a plane, or simply the nonuniformity in the *X*–*Y* plane of a Cartesian coordinate system.

^{3}

There will be three sets of equations characterized by their “orders” in this paper. The order of a wind model means the highest order in a truncated Taylor series used to approximate the real wind field. The order of VAD analysis means the highest order of a truncated Fourier series used to model the observed Doppler velocities. Since single-Doppler analysis methods are always based on a prescribed wind model, a VAD analysis with order *n* (Caya and Zawadzki 1992). The order of DVAD analysis means the highest order of a two-dimensional polynomial function, which is also one order higher than the corresponding wind model.