1. Introduction

Since the invention of weather Doppler radar, the Doppler velocity V_d displayed in a plan position indicator (PPI) or a range–height indicator (RHI) are commonly used to interpret the wind flow structure of meteorological phenomena in both operations and research. Doppler velocity is the projection of the three-dimensional (3D) target motion along the radar beam direction. As a result, the full 3D air motion cannot be measured directly and can only be deduced if there are simultaneous observations from two or more Doppler radars on the same target plus application of the continuity equation (e.g., Armijo 1969). Otherwise, only characteristics of certain simple meteorological flow patterns, for example, straight-line wind, rotational wind, and converging/diverging wind, can be qualitatively identified via their Doppler velocity patterns (e.g., Lhermitte and Atlas 1961; Donaldson 1970; Wood and Brown 1992). Quantitative description of the wind structures of the aforementioned specific flow patterns and their derived quantities (e.g., convergence, deformation, and/or vorticity) can be estimated by single-Doppler wind retrieval algorithms such as the velocity–azimuth display (VAD; Browning and Wexler 1968; Caya and Zawadzki 1992; Donaldson 1991; Matejka and Srivastava 1991), volume velocity processing (VVP; Waldteufel and Corbin 1979; Koscielny et al. 1982; Boccippio 1995), and velocity track display (e.g., VTD; Lee et al. 1994; Roux and Marks 1996; Lee et al. 1999). Although these algorithms use the same PPI or constant altitude PPI (CAPPI) data, the underlying flow models are either centered at the radar (e.g., VAD) or at a user-selected location (e.g., VVP and VTD).

While manipulating the Doppler velocity equation to formulate the generalized VTD (GVTD) technique, Jou et al. (2008) showed that the atmospheric vortex signatures displayed in rV_d (V_d scaled by the distance from the radar to the gate, r) had several advantages over those displayed in V_d and simplified the governing equations. First, the signature of a constant wind (Fig. 1a) possesses a set of parallel lines (Fig. 1c) compared to a starburst pattern displayed in V_d (Fig. 1b). It has been shown in Jou et al. (2008) that the gradient of these parallel lines of rV_d is the constant wind vector. Second, the V_d pattern of an axisymmetric atmospheric vortex (Fig. 1d) possesses geometric distortion due to the scanning characteristics of a ground-based Doppler radar; for example, the Doppler velocity dipole is skewed toward the radar as a function of the distance from the radar to the vortex center accompanied by nonparallel constant V_d lines inside the radius of maximum wind (RMW; the dashed circle in Fig. 1e). The aforementioned geometric distortion for an axisymmetric vortex is no longer a factor displayed in rV_d/R (i.e., Fig. 1f), where R is the distance between the radar and the vortex center. Further examining of the properties of rV_d suggests that rV_d can be expressed in a Cartesian coordinate and have a wider application beyond atmospheric vortices documented in Jou et al. (2008).

View Full Size

(a) The flow field of a constant easterly wind; (b) the Doppler velocity, V_d, of the wind field in (a) sampled by a Doppler radar located at (0, 0), the bottom-left corner; (c) the corresponding rV_d display of the same background flow; (d) the flow field of a Rankine-combined vortex; (e) V_d as in (b), but for the vortex in (d); and (f) rV_d/R as in (c), but for the vortex in (d), where R is the distance from the Doppler radar to the vortex center. The units of V_d, rV_d, and rV_d/R are m s⁻¹, m² s⁻¹, and m s⁻¹, respectively.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

The use of rV_d and its accompanied polynomial form in the Cartesian coordinate to represent linear and nonlinear wind fields were first presented in Peace et al. (1969). These authors demonstrated that a convective cell moving in space observed by a Doppler radar at two different times is equivalent to observing the same cell with two Doppler radars from two vantage points, if storm evolution can be ignored. Therefore, the 2D horizontal wind fields within a convective cell can be deduced using two different volumes of data from the same radar. This methodology was later termed “synthetic dual-Doppler analysis” (e.g., Bluestein and Hazen 1989; Bluestein et al. 1994). When the wind field is stationary in the moving frame, Gal-Chen (1982) recognized that the second derivative of rV_d with respect to time is invariant. Hence, rV_d can be used to determine the optimal advection speed of an atmospheric phenomenon. These authors recognized the potential advantages in working with rV_d but they did not focus on its single-Doppler radar applications.

One of main difficulties of utilizing the single-Doppler radial velocity information in interpreting flow patterns is its vectorlike property that lacks a general relationship to the corresponding 3D Cartesian wind vectors. The purpose of this paper is to revisit this nearly forgotten quantity, rV_d, and illustrate the advantages of using rV_d as a new framework to display, examine, and interpret single-Doppler velocities on linear and nonlinear atmospheric flow patterns beyond those documented in Peace et al. (1969) and Gal-Chen (1982). It will be shown that the polynomial representation of rV_d of linear wind fields is mathematically equivalent to the VAD. Therefore, qualitative information of a linear wind field can be deduced by a 2D least squares fit of rV_d. Since the rV_d display of a linear wind field is a quadratic curve, the linear terms except for the vorticity can be graphically estimated from the shape, contour values, and orientation of the corresponding quadratic curve without actually performing a series of VAD analyses. In the presence of nonlinear wind components, rV_d has a more concise mathematical representation over the nonlinear VAD analysis (e.g., Caya and Zawadzki 1992) in deducing kinematic properties. Because of the mathematical similarity to the VAD, the proposed methodology is coined distance velocity–azimuthal display (DVAD).

This paper is organized as follows. Section 2 presents the rationale and the mathematic framework of the DVAD. Section 3 illustrates characteristics of DVAD on several simple, analytical flow fields. Section 4 demonstrates how to quantitatively deduce flow characteristics via least squares fitting and taking straight derivatives of the rV_d field. Graphical interpretation of a real wind field using DVAD is illustrated in section 5. Summary and future work are presented in section 6.

2. The DVAD methodology

a. Mathematic expression and properties of DVAD

For a radar located at the origin, the Doppler velocity V_d at any point P (x, y, z) in space can be expressed in terms of the three-dimensional Cartesian velocities, u, υ, and, w, and the spherical coordinate parameters: mathematic angle θ (defined as 0° pointing east and increasing positively counterclockwise), elevation angle ϕ, and range r, from the radar to each gate, as illustrated in Armijo (1969):

View Expanded

Hence, by multiplying r on both sides, we obtain the following:

View Expanded

Here we ignore the contribution from the terminal fall velocity of the particle, as well as the effects of atmospheric refraction and earth curvature on radar beam height. The characteristics between rV_d and V_d differ in many aspects. First, by examining Eq. (1), rV_d can be expressed exactly by Cartesian coordinate quantities while V_d involves a spherical coordinate quantity r. In general, coordinate transformation changes the form of the expressions but will not add more information. This point will be illustrated in section 2b. Second, taking the gradient of Eq. (1), we obtain the following:

View Expanded

where V = ui + υj + wk is the 3D Cartesian velocity vector of a target at point P. As a result, rV_d possesses a property similar to a type of velocity potential (a scalar) in fluid mechanics where its gradient is the three-dimensional velocity vector V, plus the first derivative of u, υ, and w scaled by their corresponding Cartesian distance x, y, and z. In contrast, V_d behaves like a vector and does not possess similar mathematic and physical characteristics as rV_d. These first-order derivative terms in Eq. (2) prevent the direct computation of the velocity vector V other than at the origin (i.e., the radar). When the last three terms on the right-hand side of Eq. (2) are small, ∇(rV_d) is a proxy of the 3D velocity vector V. However, without the knowledge of the spatial gradient of the velocity field, the use of “∇(rV_d) as a proxy of the 3D velocity vector V” is mostly valid in a region close to the radar.

b. Expressions of DVAD for linear and nonlinear wind fields

To investigate the properties of DVAD of linear and nonlinear wind fields, u, υ, and w in Eq. (1) can be expanded in Taylor series with respect to the origin (x₀, y₀, z₀) in space, then Eq. (1) will take a form of trivariate polynomial as follows:

View Expanded

where u₀, υ₀, and w₀ are the three-dimensional velocities at the point (x₀, y₀, z₀). The right-hand side of Eq. (3) is a polynomial expressed in a Cartesian coordinate system and its highest order is one above the highest order of the underlying linear or nonlinear flow fields. Although the Taylor series can be expanded with respect to any point other than the radar at the origin, there are no advantages because Eq. (3) becomes unnecessarily complicated and the full wind field still cannot be deduced other than at the radar [e.g., Eq. (2)] as presented in many previous single-Doppler radar wind retrieval works (e.g., Browning and Wexler 1968; Peace et al. 1969; Waldteufel and Corbin 1979; Koscielny et al. 1982). Therefore, Eq. (3) can be simplified when expanding the Taylor series with respect to the radar (i.e., x₀ = y₀ = z₀ = 0). For the convenience of illustration, we will focus on the 2D form of Eq. (3) by setting Δz = 0 and Δx and Δy become x and y. Then Eq. (3) becomes

View Expanded

where Eq. (4) is in a form of a standard polynomial where the coefficient of each term is a combination of physical quantities of a given wind field.¹ Note that Eq. (4) should be processed on CAPPI data. Since the geometric characteristics of polynomials are easy to recognize visually (especially for the first- and second-order polynomials), displaying and processing rV_d rather than V_d make the subsequent interpretation and computation of the gross wind field properties much more intuitive.

c. Characteristics of DVAD for linear, nonrotational wind fields

When considering only linear wind fields, the second-order derivatives are zero and Eq. (4) reduces to a bivariate quadratic equation (e.g., Korn and Korn 1961, 39–41):

View Expanded

where

View Expanded

Equation (5) is a bivariate quadratic equation, representing conic sections.² Different linear wind fields [i.e., combinations of u_x, (u_y + υ_x), υ_y, u₀, and υ₀ in Eq. (5)] yield different types of conic sections, including three types of nondegenerate quadratic curves (i.e., ellipse, parabola, and hyperbola). The geometric property of the quadratic equation is determined by the sign of the discriminant, δ = (u_y + υ_x)²/4 − u_xυ_y for Eq. (5), as follows (e.g., Taylor 1959, 297–298):

1. δ < 0, a set of ellipse [If u_x = υ_y ≠ 0 and u_y+ υ_x = 0, Eq. (11) represents a circle, which is a special case of ellipse];

2. δ = 0, a set of parabola;

3. δ > 0, a set of hyperbola.

Physically, δ involves the square of shearing deformation and the product of two components of the divergence. Because the square of shearing deformation is always greater or equal to zero, the only case that rV_d pattern holds an ellipse is for u_xυ_y > (u_y + υ_x)²/4, which implies u_xυ_y > 0 is a necessary but not a sufficient condition. Therefore, a given linear wind field (u_x, υ_y, u_y, and υ_x) yields a rV_d pattern governed by Eq. (5). Meteorologically, divergence, stretching deformation, and shearing deformation, control the rV_d pattern. However, a given rV_d pattern is only unique to u_x, υ_y, and (u_y + υ_x) while the vorticity (υ_x − u_y) cannot be resolved. When the wind field is linear, DVAD and VAD are mathematically identical where the mean wind, divergence, and deformation can be deduced.

The geometric properties of the rV_d patterns for linear wind fields can be further examined via Eq. (5). The presence of a mean wind (u₀, υ₀) is equivalent to translating the rV_d conic sections to a new origin (x₀, y₀) as follows (e.g., Korn and Korn 1961, 39–41):

View Expanded

View Expanded

The magnitude and sign (i.e., direction) of the rV_d pattern translation depend on the values of the u₀ and ₀ and the linear wind field specified in Eqs. (6) and (7). In the VAD framework, the analysis is performed on rings centered at the radar. Hence, the linear wind fields have their centers³ at the radar (see Fig. 2 in Browning and Wexler 1968) and u₀ and ₀ are interpreted as “translation speed.”

Geometrically, Eq. (5) represents a general form of conic sections with an arbitrary orientation [if (u_y + _x) ≠ 0] that can be rotated to realign the primary axes with the x and the y axes. Mathematically, this is equivalent to performing a coordinate transformation by rotating a positive acute angle α as follows (e.g., Taylor 1959, 294–297):

View Expanded

and Eq. (5) can be reduced to the following form:

View Expanded

in the rotated coordinate system. It can be shown that (e.g., Korn and Korn 1961, 39–41)

View Expanded

It can be concluded that the existence of shearing deformation (u_y + υ_x ≠ 0) rotates the major axes of the conic sections of the rV_d pattern at an acute angle from the x and the y axes. The amount of rotation is a function of the stretching deformation and shearing deformation. While the resultant deformation (defined as the square root of the sum of the square of shearing and stretching deformation; Bluestein 1992, p. 100) is invariant, the shearing or stretching deformation is a property that is dependent on the coordinate system and can be made to disappear by selecting a proper coordinate system (e.g., axis of dilatation; Bluestein 1992, p. 99). This property of deformation is clearly illustrated using Eqs. (5), (8), and (9) while the total divergence/convergence is invariant according to Eq. (10).

It has been demonstrated mathematically that the presence of a linear, nonrotational wind field in DVAD framework yields a simple and concise bivariate quadratic polynomial in a Cartesian coordinate. The physical properties are intuitive to identify and interpret based on the straight-forward and well-known geometric relations between conic sections and quadratic equations. It is instrumental to examine basic patterns between V_d and rV_d on linear and low-order nonlinear wind fields and illustrate the fundamental differences between these two quantities.

3. Graphical illustration of V_d and rV_d on low-order polynomial flows

In this section, the V_d and rV_d patterns of basic flow fields from constant, linear to nonlinear winds are examined to illustrate the characteristics and differences of these two approaches. To the authors’ knowledge, the V_d patterns of linear wind fields have not appeared in literature and textbooks. The parameters of those cases illustrated in this section are summarized in Table 1, and descriptive names are given in Table 2.

Table 1.

Parameters of the cases illustrated in section 3.

View Table

Table 2.

Descriptive names for each case listed in Table 1.

View Table

a. Constant flow

When the velocity field is a constant [i.e., u(x, y) = C₁, υ(x, y) = C₂, and V = C₁i + C₂j], Eq. (4) becomes

View Expanded

which represents a set of parallel lines as illustrated in Jou et al. (2008). Then Eq. (2) reduces to

View Expanded

The same result can be directly deduced from Eq. (2). An example of a uniform southwesterly wind (u₀ = υ₀ = 10 m s⁻¹, case A) is illustrated in Fig. 2a1. The corresponding V_d holds a diverging (starburst) pattern centered at the radar (Fig. 2a2) while the rV_d pattern possesses a set of parallel lines oriented from northwest to southeast (Fig. 2a3), perpendicular to the prescribed uniform southwesterly wind (Fig. 2a1). Note that the wind direction is pointing from negative to positive contours according to Eq. (12). Although experienced radar scientists can identify the wind direction and speed from Fig. 2a2, the rV_d pattern is clearly simpler than the traditional V_d display.

View Full Size

Four basic linear flow fields: (a1) u₀ = 10.0, u_x = 0, u_y = 0; υ₀ = 10.0, υ_x = 0, υ_y = 0; (b1) u₀ = 0, u_x = 2 × 10⁻⁴, u_y = 0; υ₀ = 0, υ_x = 0, υ_y = 1 × 10⁻⁴; (c1) u₀ = 0, u_x = 2 × 10⁻⁴, u_y = 0; υ₀ = 0, υ_x = 0, υ_y = −1 × 10⁻⁴; and (d1) u₀ = 0, u_x = 0, u_y = 1 × 10⁻⁴; υ₀ = 0, υ_x = 1 × 10⁻⁴, υ_y = 0. The corresponding Doppler velocities, V_d, sampled by a radar at (0, 0) are shown in (a2)–(d2). The corresponding DVAD patterns, rV_d, are shown in (a3)–(d3). The unit for the velocity is m s⁻¹ and the unit for the first-order derivative is s⁻¹. The circle (○) and plus (+) represent the radar location and the center for the prescribed flow field, respectively. The units of V_d and rV_d are m s⁻¹ and m² s⁻¹, respectively.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

b. Linear wind fields

In this subsection, it is shown that the linear wind fields can be represented by three basic quadratic curves. It is noted that shearing deformation is coordinate dependent (section 2c), all patterns illustrated in this subsection can be obtained by rotating the zero shearing deformation flow. The amount of rotation in the mathematic coordinate system of a given DVAD pattern indicates the relative magnitude of shearing deformation. Therefore, the zero shearing deformation flow is discussed as a building block in this subsection.

1) Zero shearing deformation flow (u_y + υ_x = 0)

When u_y + υ_x= 0, the corresponding rV_d patterns can be one of three nondegenerate quadratic curves, ellipse, hyperbola, and straight lines (a special case of parabola) depending on the sign and magnitude of u_x and υ_y. When u_x and υ_y are both positive (e.g., u_x = 2υ_y = 2 × 10⁻⁴ s⁻¹, case B1), the wind vectors are diverging from the singular point at the radar (Fig. 2b1). The corresponding rV_d patterns (Fig. 2b3) are concentric ellipses with all positive contours according to Eq. (5). In this example, the u gradient along the x axis is twice the υ gradient along the y axis, so the major (minor) axis of the ellipse is along the y axis (x axis). Note that the V_d pattern (Fig. 2b2) appears elliptical but it cannot be shown mathematically as in rV_d. Therefore, it will be referred to as ellipselike curves hereafter. When u_x and υ_y are both negative (u_x < 0 and υ_y < 0; u_x = 2υ_y = −2 × 10⁻⁴ s⁻¹), the wind vectors in Fig. 2b1 are reversed in direction (i.e., pointing toward the radar at the origin) and the patterns remain the same as a set of ellipses but the contour values in Figs. 2b2 and 2b3 are all negative (not shown). Note that the major axis of the ellipse is aligned with either the y (|u_x| > |υ_y|) or x (|u_x| < |υ_y|) axis. If u_x = υ_y, both V_d and rV_d contours of the aforementioned wind fields are concentric circles (scaled differently) with positive (negative) contours representing a pure diverging (converging) flow (not shown).

When u_x and υ_y are opposite signs (e.g., u_xυ_y < 0, u_x = −2υ_y = 2 × 10⁻⁴ s⁻¹, case C), the corresponding vector field is illustrated in Fig. 2c1 while V_d and rV_d patterns are a set of hyperbola-like curves and hyperbola (Figs. 2c2 and 2c3). Note that u_x > 0 (υ_y > 0) represents a set of “east–west (north–south) opening hyperbola” with positive (negative) V_d and rV_d contours along the “opening” of the hyperbola while u_x < 0 or υ_y < 0 represents conjugate hyperbolas (not shown).

If either u_x or υ_y vanishes in the zero shearing deformation flow, the resulting rV_d pattern is a set of parallel lines (i.e., a special case of parabola when δ = 0). When u_x = 0 (υ = 0), the lines parallel the x (y) axis (not shown). Note that in zero shearing deformation flow cases, the major and minor axes are always aligned along either the x or the y axis.

2) Pure shearing deformation flow (u_x = υ_y = 0 and u_y + υ_x ≠ 0)

When u_x = υ_y= 0 and u_y + υ_x ≠ 0, the rV_d pattern possesses a set of rectangular hyperbola with horizontal (x axis) and vertical (y axis) asymptotes and V_d possesses a set of hyperbola-like curves similar to rV_d. For example, the wind vectors of shearing deformation (e.g., u_y + υ_x > 0, u_y = υ_x = 10⁻⁴ s⁻¹, u_x = υ_y = 0, case D) and the corresponding V_d and rV_d patterns are illustrated in Figs. 2d1, 2d2, and 2d3, respectively. Note that the major axes of the rectangular hyperbola or the hyperbola-like curves now are rotated 45° from either x or y axis compared to the zero shearing deformation flow cases. When u_y + υ_x < 0, the V_d and rV_d patterns are conjugate of the pattern illustrated in Figs. 2d2 and 2d3 (not shown).

Here, the rectangular hyperbola depends only on the magnitude and the sign of u_y + υ_x, not on the individual magnitude and/or signs of u_y and υ_x as in the zero shearing deformation flow fields presented in section 3b(1). It is apparent that for a given u_y + υ_x (i.e., shearing deformation), different combinations of u_y and υ_x yield the same V_d and rV_d patterns, respectively. As a result, it is not possible to separate u_y and υ_x for a given wind field so unambiguously deducing vorticity is not possible. Hence, the full linear wind field cannot be retrieved from an observed rV_d pattern even though u_x and υ_y can be uniquely distinguished as illustrated in subsection 3b(1). The result is consistent with the traditional V_d-based VAD formulation where the divergence (u_x + υ_y), stretching deformation (u_x − υ_y), and shearing deformation (u_y + υ_x) can be deduced, but vorticity (u_y − υ_x) remains elusive so the full wind field cannot be completely recovered. Therefore, a simple coordinate transformation from V_d (VAD) to rV_d (DVAD) simplifies the interpretation of the observed flow pattern but does not resolve new terms (i.e., provides new information).

3) Mixed linear wind fields

In sections 3b(1) and 3b(2), the rV_d properties and graphical characteristics for polynomials of several simple cases were illustrated. These aforementioned basic rV_d patterns form the four basic building blocks (Figs. 2a–d) for interpreting other more complicated linear flow fields [i.e., any combinations of u_x, u_y + υ_x, and υ_y in Eq. (11)]. Based on Eq. (5), the combined rV_d pattern (ellipse, parabola, or hyperbola) is determined by the sign of δ (i.e., the relative magnitude between shearing deformation, stretching deformation, and divergence). The following two examples illustrate the flow fields by combining zero and pure shearing deformation flows using those “building blocks.”

The first flow field (Fig. 3a1, case BD) is constructed by superimposing the zero shearing deformation flow (Fig. 2b1, u_x = 2υ_y = 2 × 10⁻⁴ s⁻¹, case B) and the pure shearing deformation flow (Fig. 2d1, u_y = υ_x= 10⁻⁴ s⁻¹, case D) with the resulting V_d and rV_d patterns portrayed in Figs. 3a2 and 3a3. As predicted from Eqs. (5) and (8), the primary axes of the rV_d pattern illustrated in Fig. 3a3 rotates counterclockwise at an acute angle of 31.7°. The V_d pattern (Fig. 3a2) rotates the same magnitude as the rV_d pattern but the shape is a distorted ellipselike curve. The rV_d pattern is able to maintain the elliptical shape because the determinant δ of Eq. (5) is less than zero in this case. If the flow matches the condition of u_yυ_x = (u_y + υ_x)²/4, then the rV_d pattern is a set of straight lines (degenerate parabola with two identical real solutions, not shown). If the combination of zero and pure shearing deformation flows makes δ > 0, then the rV_d pattern becomes hyperbola (not shown).

View Full Size

As in Fig. 2, but for two flow fields including shearing deformation. (a1) u₀ = 0, u_x = 2 × 10⁻⁴, u_y = 1 × 10⁻⁴; υ₀ = 0, υ_x = 1 × 10⁻⁴, υ_y = 1 × 10⁻⁴; and (b1) u₀ = 0, u_x = 2 × 10⁻⁴, u_y = 1 × 10⁻⁴; υ₀ = 0, υ_x = 1 × 10⁻⁴, υ_y = −1 × 10⁻⁴. The unit for the velocity is m s⁻¹ and the unit for the first-order derivative is s⁻¹. The circle (○) and plus (+) represent the radar location and the center for the prescribed flow field, respectively. The units of V_d and rV_d are m s⁻¹ and m² s⁻¹, respectively.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

The second flow field (Fig. 3b1, case CD) is constructed by superimposing the zero shearing deformation flow (Fig. 2c1, u_x = −2υ_y = 2 × 10⁻⁴ s⁻¹, case C) and the pure shearing deformation flow (Fig. 2d1, u_y = υ_x = 10⁻⁴ s⁻¹, case D) with the resulting V_d and rV_d patterns portrayed in Figs. 3b2 and 3b3. This is the case where δ is always greater than 0 when u_x and υ_y have opposite signs. The characteristics of this set of V_d and rV_d patterns are very similar to the previous example in Fig. 3a with a set of hyperbola and hyperbola-like curves rotated 16.8° counterclockwise as predicted in Eq. (8). These examples demonstrate that the rV_d signature of pure linear wind field is cleaner than the V_d signature, an advantage of displaying Doppler velocity in the rV_d framework.

4) Superimposed linear wind fields on constant background wind

The presence of background constant winds, u₀ and υ₀, geometrically translates the center of basic conic sections displayed in rV_d from (0, 0) to (x₀, y₀) where the magnitude and sign (i.e., direction) of the rV_d pattern translation depend on the characteristics of the background flow and the linear wind field specified in Eqs. (6) and (7). The following three examples illustrate the wind fields and the characteristics of V_d and rV_d of a linear wind field (portrayed in Figs. 2b–d) superimposed on a constant southwesterly background wind (Fig. 2a, u₀ = υ₀ = 10 m s⁻¹).

Superimposing a zero shearing deformation flow when u_xυ_y > 0 (u_x > 0 and υ_y > 0; u_x = 2υ_y = 2 × 10⁻⁴ s⁻¹, Fig. 2b1, case B) onto the southeasterly background mean flow (Fig. 2a1, case A) shifts the flow center toward the southwest into the wind (Fig. 4a1, case AB) compared to Fig. 2b1. It has been illustrated before that the V_d and rV_d patterns of a zero shearing deformation flow are similar as concentric ellipse-like curves and ellipses (Figs. 2b2 and 2b3). However, when a constant background flow presents, rV_d patterns conserve their original concentric ellipses whose center is shifted to (−25 km, −50 km) as predicted in Eqs. (6) and (7). Although the V_d display shifts to the same center as in rV_d,⁴ it has lost its original shape with only a hint of the ellipselike pattern remaining.

View Full Size

As in Fig. 2, but for flow fields including translation. (a1) u₀ = 10, u_x = 2 × 10⁻⁴, u_y = 0; υ₀ = 10, υ_x = 0, υ_y = 1 × 10⁻⁴; (b1) u₀ = 10, u_x = 2 × 10⁻⁴, u_y = 0; υ₀ = 10, υ_x = 0, υ_y = −1 × 10⁻⁴; and (c1) u₀ = 10, u_x = 0, u_y = 1 × 10⁻⁴; υ₀ = 10, υ_x = 1 × 10⁻⁴, υ_y = 0. The unit for the velocity is m s⁻¹ and the unit for the first-order derivative is s⁻¹. The circle (○) and plus (+) represent the radar location and the center for the prescribed flow field, respectively. The units of V_d and rV_d are m s⁻¹ and m² s⁻¹, respectively.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

If the zero shearing deformation flow of u_xυ_y < 0 (u_x > 0 and υ_y < 0; u_x = −2υ_y = 2 × 10⁻⁴ s⁻¹, Fig. 2c1, case C) is superimposed onto the southwesterly background flow (Fig. 2a1, case A), the center of the wind pattern shifts to the northwest (Fig. 4b1, case AC). The rV_d pattern of this wind field (Fig. 4b3) remains a set of hyperbola with a new center at (−25 km, 50 km) according to Eqs. (6) and (7). The corresponding V_d pattern (Fig. 4b2) is a set of distorted hyperbola-like curves.

When the constant southwesterly wind (Fig. 2a1, case A) is superimposed on the pure shearing deformation field (u_y + υ_x > 0, u_y = υ_x = 10⁻⁴ s⁻¹, u_x = υ_y = 0, Fig. 2d1, case D) the rV_d pattern maintains its hyperbola characteristics (Fig. 4c3, case AD) and shifts to a new center at (−50 km, −50 km) according to Eqs. (6) and (7) while the V_d display is a set of distorted hyperbola-like curves (Fig. 4c2).

The following two examples illustrate the mixed linear wind fields (Fig. 3a1, case BC and Fig. 3b1, case BD) superimposed onto the constant background flow (Fig. 2a1). In the resulting wind fields (Fig. 5a1, case ABD, and Fig. 5b1, case ACD), the general characteristics of the V_d (Figs. 5a2 and 5b2) and rV_d patterns (Figs. 5a3 and 5b3) are very similar to those shown in simple linear wind fields (Figs. 4a–c) where the basic patterns of the rV_d (Figs. 5a3–5b3) are conserved with a translation while distortion in the V_d patterns (Figs. 5a2–5b2) is apparent. The centers in rV_d patterns were shifted to (0 km, −50 km) and (−33.3 km, 16.7 km) according to Eqs. (6) and (7), respectively. Note that in all five cases above, the zero V_d and rV_d contours are invariant according to the definition, and not affected by the variable transformation. Also one of the zero contours always passes through the radar at (0, 0) by definition.

View Full Size

As in Fig. 2, but for flow fields including mean wind, diverging/converging, and shearing deformation. (a1) u₀ = 10, u_x = 2 × 10⁻⁴, u_y = 1 × 10⁻⁴; υ₀ = 10, υ_x = 1 × 10⁻⁴, υ_y = 1 × 10⁻⁴; and (b1) u₀ = 10, u_x = 2 × 10⁻⁴, u_y = 1 × 10⁻⁴; υ₀ = 10, υ_x = 1 × 10⁻⁴, υ_y = −1 × 10⁻⁴. The unit for the velocity is m s⁻¹ and the unit for the first-order derivative is s⁻¹. The circle (○) and plus (+) represent the radar location and the center for the prescribed flow field, respectively. The units of V_d and rV_d are m s⁻¹, and m² s⁻¹, respectively.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

Recall that the signatures of V_d and rV_d without the constant background flow are quite similar. From the above examples, there is a clear advantage of displaying the Doppler velocity in rV_d rather than V_d for interpretation and operational purposes because rV_d patterns are conserved in the presence of a constant background flow and the flow characteristics can be more easily deduced than with the conventional V_d display.

c. Nonlinear wind fields

The nonlinear wind fields discussed below are constructed by including the second order derivatives in the velocity fields as shown in Eq. (4). The second-order nonlinear wind field therefore possesses a cubic polynomial in the rV_d framework. The graphical expression of a cubic polynomial is complicated and the resulting rV_d patterns are not as straightforward to recognize as those of the quadratic equation, except for a few simple flow patterns. It can be seen, just like the quadratic equation, that several second-order derivatives are grouped together and cannot be individually determined unambiguously.

Examples of both weak and strong simple nonlinear wind fields (with only u_xx ≠ 0 in the nonlinear terms, case E1, not shown) superimposed onto the same linear wind field illustrated in Fig. 5a1 (case ABD) are portrayed in Fig. 6. For the case of a weak nonlinear wind field (Fig. 6a1, case ABDE1), the resulting V_d (Fig. 6a2) and rV_d patterns (Fig. 6a3) are still similar to that of the linear case (Figs. 5a2 and 5a3). For the case of a strong nonlinear wind field (10 times the weak nonlinear case, Fig. 6b1, case E2, not shown), the V_d and rV_d patterns (Figs. 6b2 and 6c2, case ABDE2) possess hardly any signs of the original linear patterns in V_d and rV_d (Figs. 5a2 and 5a3). The general characteristics of these linear wind fields mixed with other second-order nonlinear wind fields (not shown) are similar to the examples shown in Fig. 6.

View Full Size

As in Fig. 2, but for flow fields including linear and nonlinear (weak upper, strong bottom) terms. (a1) u₀ = 10, u_x = 2 × 10⁻⁴, u_y = 1 × 10⁻⁴; υ₀ = 10, υ_x = 1 × 10⁻⁴, υ_y = 1 × 10⁻⁴, u_xx = 3 × 10⁻⁷; and (b1) u₀ = 10, u_x = 2 × 10⁻⁴, u_y = 1 × 10⁻⁴; υ₀ = 10, υ_x = 1 × 10⁻⁴, υ_y = 1 × 10⁻⁴, u_xx = 3 × 10⁻⁶. The unit for the velocity is m s⁻¹, the unit for the first-order derivative is s⁻¹, and the unit for the second-order derivative is m⁻¹ s⁻¹. The circle (○) represents the radar location. The units of V_d and rV_d are m s⁻¹ and m² s⁻¹, respectively.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

The Taylor series expansion can be conducted to third and higher orders, but the graphical characteristics possess only limited applications in practice. Therefore, no further graphical examples of the higher-order nonlinear wind fields will be provided and discussed. Nevertheless, the rV_d display can be used to determine the degree of linearity of the underlying wind field, a valuable tool to assess the validity of the properties deduced by the VAD for both research and operational purposes.

4. Retrieval of quantitative flow information

It has been illustrated in previous sections that linear and nonlinear wind fields can be represented as polynomials in the DVAD framework whose coefficients link to the flow characteristics and/or their combinations. It is also clear that for a given rV_d pattern, only a subset of those flow characteristics, not the full wind field, can be estimated qualitatively. Quantitative information of the wind field (i.e., coefficients of the polynomial) can be obtained by 1) the least squares method or 2) the derivative method. Both methods can be performed on nonlinear wind fields. In this section, we will only illustrate the DVAD analysis using a linear wind field (mixture of a background mean wind, diverging flow and deformation flow, case ABD). To test the sensitivity of the two methods to random noise in the observations, a 2 m s⁻¹ random noise is added to the analytic wind field (Fig. 7a1) that represents ~10% uncertainty. The corresponding V_d and rV_d displays are respectively illustrated in Figs. 7a2,a3 that resemble their perfect fields in Figs. 5a2 and 5a3.

View Full Size

Results of derivative method on a linear wind field (wind field parameters are listed in the following table, with 2 m s⁻¹ of noise). (a1) u, v vector; (a2) contour; (a3) rV_d contour; (b1) ∂(rV_d)/∂x; (b2) ∂²(rV_d)/∂x²; (b3) ∂²(rV_d)/∂x∂y; (c1) ∂(rV_d)/∂y; (c2) ∂²(rV_d)/∂y∂x; and (c3) ∂²(rV_d)/∂y². The circle (○) and plus (+) represent the radar location and the center for the prescribed flow field, respectively. The units of V_d and rV_d are m s⁻¹ and m² s⁻¹, respectively.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

The least squares fit method is a standard approach and has been widely used in many single-Doppler wind retrieval algorithms, such as VAD (e.g., Browning and Wexler 1968), VVP (e.g., Waldteufel and Corbin 1979), and VTD (e.g., Lee et al. 1994). Therefore, the details will not be repeated here. The derivative method is described in detail in this section.

The quantitative DVAD analysis can also be performed via successive differentiation of Eq. (5) with respect to x and y to deduce the coefficients. Each differentiation will eliminate the lowest-order terms in the previous set of equations and eventually the highest-order terms will emerge when the lower-order derivatives vanish. This property is another advantage of DVAD over those V_d-based single-Doppler wind retrieval algorithms operating in the polar coordinate system.

This property can be illustrated by using a bivariate quadratic polynomial in Eq. (5). By taking the derivative of Eq. (5) with respect to x and y, we obtain the following:

View Expanded

View Expanded

Evaluating Eqs. (13) and (14) at the origin (x = y = 0),⁵ we obtain u₀ and υ₀ as shown in Eq. (2). By taking the derivative of Eqs. (13) and (14) with respect to x and y, we obtain three independent equations:

View Expanded

View Expanded

View Expanded

In this linear wind field, Eqs. (15)–(17) should be constant by definition. Hence u_x, (u_y + υ_x) and υ_y can be obtained. Therefore, u₀ and υ₀ can be obtained within the entire domain by evaluating Eqs. (13) and (14) using Eqs. (15)–(17). If the wind field is nonlinear, then Eqs. (15)–(17) will not be constant.

Coefficients of a higher-order polynomial can be obtained theoretically in a similar manner. While taking derivatives of a field is straightforward compared with a 2D least squares curve fit, the derivative method amplifies local noise with each differentiation. Therefore, when applying the derivative method to real data, the 2D field of rV_d and their subsequent derivatives need to be filtered/smoothed before each differentiation.

The results of quantitative DVAD analysis of the prescribed linear wind field (case ABD) using the least squares fit and the derivative methods are illustrated in Table 3. It can be concluded from Table 3 that both approaches yield almost identical results. The retrieved u₀ and υ₀ are within 10% of the specified analytical field while the three linear terms are nearly identical to the true values. The least squares fit method is not sensitive to the random noise. The graphical representation of the deduction in Eqs. (13)–(17) is shown in Figs. 7b1–b3,c1–c3. According to Eqs. (13)–(14), for a linear wind field, the first derivative of rV_d with respect to x and y should be two sets of parallel lines as illustrated in Figs. 7b1,c1 while the second derivative [Eqs. (15)–(17)] should be constant as illustrated in Fig. 7. It is evident from Fig. 7 that the derivative method is still applicable when noise is presented in data, provided proper filtering is applied to the intermediate results.

Table 3.

The true coefficients of this linear wind field and the retrieved coefficient from two different approaches.

View Table

5. Graphical interpretation of real wind fields using rV_d

The concept and mathematic framework of DVAD has been derived and illustrated using analytical wind fields on a constant altitude in previous sections. The application of DVAD to graphically interpret the real, near-surface, flow pattern from low elevation angle single-Doppler radar PPI⁶ observations is portrayed by an example involving two mesoscale convective systems (MCSs). On 7 June 2003, a squall line approached southern Taiwan and was observed by an S-band Doppler radar (RCKT) operated by the Central Weather Bureau of Taiwan. The RCKT 0.5° reflectivity PPI overlaid with V_d and rV_d contours are illustrated in Figs. 8a and 8b, respectively. The target squall line is located ~100 km west of the RCKT, and another MCS was located ~50 km east of RCKT. Note the wedge between 300° and 30° azimuth was blocked by terrain in southern Taiwan.

View Full Size

The Doppler radar observations of a squall line approaching RCKT (at the origin indicated by a circle). (a) Reflectivity (dBZ) in color overlaid with Doppler velocity contours (m s⁻¹), (b) reflectivity in color overlaid with rV_d contours (×100 m² s⁻¹), (c) Doppler velocity in color and contours, and (d) rV_d in color and contours. The red arrows indicate wind directions estimated from the gradient of rV_d.

Citation: Monthly Weather Review 142, 2; 10.1175/MWR-D-13-00196.1

• Download Figure
• Download figure as PowerPoint slide

An experienced radar meteorologist can identify a general southwesterly wind ahead of the squall line and a general westerly jet behind the squall line based on the distribution of V_d (Figs. 8a,c). The convergence along the southern part of the squall line is indicated by a weak Doppler velocity contour followed by stronger Doppler velocities behind the line. These flow characteristics are more intuitive in the rV_d display (Figs. 8b,d) where the general flow directions are indicated by the red arrows. Relatively uniform flow fields are represented by localized near parallel straight contours of rV_d, such as to the southeast of RCKT. For example, the southwesterly winds southwest of the RCKT turned into west-southwesterly winds to the east of the RCKT where the winds approached the back side of the MCS. The location of the squall line (i.e., high reflectivity) is accompanied by curved rV_d contours, indicating a nonuniform wind shift across the squall line. The rV_d contours were also curved within that MCS but the curvature was not as steep as that within the squall line. This region of curved rV_d contours along the squall line is a region of convergence between northwesterly (behind the line) and southerly (ahead of the line).

6. Summary and future work

This paper presents the concept of the distance velocity–azimuth display (DVAD) methodology and basic applications of DVAD to linear and nonlinear wind fields on a constant altitude PPI plane. DVAD is an alternative approach to express and interpret single-Doppler velocities that use a nearly forgotten form of rV_d. The Doppler velocity V_d is a vector field in the radar-centric spherical coordinate that has been used by radar meteorologists in the past six decades. In contrast, the DVAD framework allows rV_d to be expressed in the form of a polynomial in a Cartesian coordinate system, linking rV_d directly to x, y, and z, and the 3D velocities (u, υ, and w). The gradient of rV_d is mathematically related to the true 3D wind vectors and their first-order derivatives at each grid point. Therefore, rV_d possesses a property similar to a type of velocity potential in fluid mechanics rather than a vector property possessed by V_d. Although rV_d has been introduced in Peace et al. (1969) and Gal-Chen (1982) on synthetic dual-Doppler radar applications and determining the optimal moving frame of reference, its advantages in simplifying Doppler velocity signatures, especially on linear wind fields, have not previously been explored.

The major merit of representing the single-Doppler radar velocity in the rV_d framework is its direct link to the well-known polynomial functions. A Taylor series expansion of a 3D wind field with respect to the radar site expressed in the rV_d framework corresponds to a polynomial with one order higher than the order of the prescribed wind field. This relationship is especially useful when the actual wind field is linear (or quasi linear). The resulting mathematic form of rV_d is a bivariate quadratic polynomial representing three conic sections: ellipse, parabola, and hyperbola, depending on the magnitudes and signs of those first-order derivative terms of the wind field.

When illustrated in 2D on a CAPPI, the rV_d patterns of zero shearing deformation flow can be either ellipse or hyperbola with primary axes aligned with the x or y axis while the rV_d patterns of pure shearing deformation flow form rectangular hyperbola with its primary axis rotating 45° from their diverging/converging flow counterparts. The corresponding V_d display possesses a similar pattern as its rV_d counterpart. When mixing linear wind fields, the resulting rV_d patterns remain as one of the conic sections but rotate at an angle determined by the sign and magnitudes of these linear terms. However, the distortion is apparent in the corresponding V_d display. The presence of a constant background flow for the rV_d pattern of a linear wind field is equivalent to translating the center of the rV_d pattern from the radar to a new location determined by the background winds and the linear terms. In contrast, the corresponding V_d display loses its original shape. Although VAD and DVAD deduce the same information when the wind field is linear, in practice, it is feasible to graphically estimate the characteristics of the actual wind field from rV_d without performing a VAD analysis.

The quantitative DVAD analysis can be processed via either a 2D bivariate least squares fitting or a derivative method. The Cartesian-based DVAD analysis can be used to derive the information of linear wind fields by the same way as that of VAD and VVP, but a clear advantage appears when dealing with data with asymmetric distribution and/or large sectors of missing data. The concise formula of DVAD also makes it straightforward to extend the method to analyze single-Doppler radar observations subject to nonlinearity.

For nonlinear wind fields, the differences between processing the same observed Doppler velocity field in VAD and in DVAD are significant because flow characteristics derived from the VAD Fourier coefficients are only valid under linear wind fields (Caya and Zawadzki 1992) while DVAD coefficients can be extended to include nonlinearity in a straightforward way. In comparison, the DVAD framework does not have this limitation where the mean wind can be deduced even for nonlinear wind fields, thus it is more accurate than those obtained from the VAD. Applying DVAD to the nonlinear wind field and examining the vertical wind profiles between DVAD and Caya and Zawadzki (1992) methodologies using analytical data and real cases are currently under way and will be documented in a future paper.