## 1. Introduction

The use of airborne Doppler radars has significantly advanced our understanding of meteorological phenomena by providing wind structure information that details the dynamics of an evolving system. Airborne platforms have been particularly important for observing phenomena that occur in remote areas, such as tropical cyclones over the open ocean. Lhermitte (1971) first discussed the idea of using airborne Doppler radars for obtaining three-dimensional wind structures. A single Doppler radar beam measures the along-beam velocity component of precipitation particles within that beam. To retrieve all three components of the wind field, two (or more) Doppler radar beams must scan an area with a sufficient separation angle between the beams (e.g., Armijo 1969; Klimowski and Marwitz 1992). Airborne radars must therefore employ certain scanning techniques that provide multiple views of the wind from sufficiently different angles in order to map the wind structure of the precipitation phenomena.

In one of the first airborne Doppler studies, Marks and Houze (1984) utilized a scanning technique for successful mapping of the three-dimensional wind field. They used data collected by the X-band Doppler radar on board the National Oceanic and Atmospheric Administration (NOAA) WP-3D (P3) aircraft. Located in the tail of the aircraft, the radar antenna pointed orthogonally to the aircraft track and scanned circularly through all elevation angles around a horizontal axis [for more on this radar, see Jorgensen (1984)]. Multiple viewing angles of the same domain were obtained by flying the aircraft at different track angles. Another tail radar was later installed on the second NOAA P3 aircraft, allowing for simultaneous Doppler observations when both aircraft were flown together (Gamache at al. 1995). Both tail radars soon implemented the fore/aft scanning technique (FAST; Jorgensen and DuGranrut 1991), in which the antenna alternately points ~20° to the fore and aft of the aircraft while circularly sweeping around a horizontal axis. With this technique, multiple along-beam velocity measurements from the same domain are obtained along a single flight track by the different fore and aft angles. The National Center for Atmospheric Research Electra Doppler Radar (ELDORA) operates with the same scanning geometry but utilizes two antennas that rotate at a faster rate, allowing for higher-resolution observations (Hildebrand et al. 1996).

Multiple techniques for retrieving the three-dimensional wind field have been developed for the NOAA P3 tail radar and the ELDORA, which both scan around a horizontal axis. One such method is a local solver that interpolates radial velocities from each viewing angle onto a Cartesian grid and solves for the corresponding velocities in the horizontal plane. These horizontal velocities from different viewing angles are then used to calculate two orthogonal horizontal wind components (e.g., Jorgensen et al. 1983; Marks and Houze 1984). The vertical wind component is calculated by integrating the anelastic mass continuity equation using appropriate boundary conditions. This technique is simple and computationally inexpensive, but errors can accumulate in the wind component along the direction of integration (Gao et al. 1999).

A second method is a global optimization approach that minimizes a cost function containing the differences between the radar-measured and retrieved velocity components. This cost function also includes constraints such as the anelastic mass continuity equation and vertical velocity boundary conditions (Gamache 1997; Bousquet and Chong 1998; Reasor et al. 2009). With the avoidance of explicit integration, this variational technique reduces errors in the vertical velocity for the aforementioned scanning geometry (Gao et al. 1999). Since all retrieval strategies are limited by the geometry of the scanning technique, no individual retrieval method is perfect; however, utilizing multiple methods adds to the reliability of the scientific interpretations of the retrieved wind fields.

The High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP), recently developed at NASA’s Goddard Space Flight Center, is a Doppler radar system that employs a different scanning strategy from the previously mentioned airborne Doppler radars (Li et al. 2011). It operates with two beams that point downward at fixed angles (30° and 40° away from nadir) with each beam scanning conically around a vertical axis. HIWRAP flew for the first time in 2010 on the NASA Global Hawk unmanned aircraft during the Genesis and Rapid Intensification Processes (GRIP) field experiment (Braun et al. 2013).

Recent studies have begun exploring how established retrieval methods can be applied to the scanning geometry of HIWRAP. Tian et al. [2013; here and below see also Tian (2014, manuscript submitted to *J. Appl. Meteor. Climatol.*)] applied the velocity azimuth display (VAD) technique (Lhermitte and Atlas 1961; Browning and Wexler 1968) to HIWRAP data to obtain the mean vertical profile of the horizontal wind along the flight track. Under the assumptions that the wind field is linear and the vertical velocity is constant across the scan circle, this method fits the measured radial winds at each altitude to a sinusoidal curve as a function of azimuth. Guimond et al. (2014) implemented the global optimization technique to obtain the three-dimensional wind field in the HIWRAP scanning domain. The cost function for this variational scheme included a modified weighting parameter that was better suited for the different scanning geometry.

In this paper, we extend the application of established retrieval techniques to the HIWRAP geometry by focusing on a simple interpolation and integration approach. The vertical integration scheme used for the P3 tail radars cannot be applied in the HIWRAP case since the scanning geometry does not align sufficiently with the horizontal wind in order to avoid large projection errors introduced by the vertical wind. A better alternative is the coplane method described by Armijo (1969) and Miller and Strauch (1974). The coplane method uses a cylindrical coordinate system in which two components of the wind are readily derived from the observations. The third wind component is completely unobserved by the radar and must be retrieved by explicitly integrating the mass continuity equation with specified boundary conditions. As a local solver, the solution of the coplane method at a certain grid point has a transparent relationship to the local radar observations, whereas in a global solver observations across the radar domain have an impact on the solution at an individual grid point. Without such interference, possible errors in the radar measurements or retrieved winds are more easily identifiable and traceable. With a natural coordinate system, the coplane method is particularly useful for understanding the advantages and disadvantages of the HIWRAP scanning technique. This understanding is necessary for interpreting any Doppler analysis method used on the HIWRAP radar geometry. In this study, we apply the coplane technique to simulated radar data and actual radar data to demonstrate its effectiveness. We also compare this technique to the global optimization solutions and investigate their differences.

Sections 2 and 3 describe the coplane method and its application to the HIWRAP geometry and observations. Section 4 examines the boundary conditions necessary for the coplane method. Section 5 analyzes the coplane retrieval of simulated radar data and section 6 analyzes the coplane retrieval of real HIWRAP data. Section 7 presents the conclusions of this study.

## 2. Coplane method and HIWRAP geometry

### a. Description of HIWRAP

HIWRAP is a dual-beam, dual-frequency (Ka and Ku band) radar system designed to fly on the high-altitude NASA Global Hawk unmanned aircraft system. Rather than scanning around a horizontal axis like the tail radars on the P3 aircraft, the antenna beams of HIWRAP point downward and scan around a vertical axis to obtain multiple angled looks of the tropospheric winds. Figure 1 illustrates this scanning geometry. The two beams point at nominal tilt angles τ of 30° and 40° away from nadir, while the antenna rotates at a typical rate of 100° s^{−1}, such that one complete revolution takes about 3.5 s. The radar beams, each with a range resolution of 150 m, sweep out spiral paths over the ground as the aircraft flies with an ideal level position along a straight flight track. For a typical aircraft speed of 160 m s^{−1} and altitude of 18.5 km, the along-track sampling and swath width are 560 m and ~30 km, respectively. The outer beam operates simultaneously at 13.5 and 33.7 GHz and the inner beam operates simultaneously at 13.9 and 35.6 GHz. HIWRAP employs dual-pulse repetition frequency sampling that can yield an extended unambiguous velocity of ~110 m s^{−1}. A more detailed description of HIWRAP can be found in Li et al. (2011).

### b. Description of the coplane method

The coplane dual-Doppler technique was developed to retrieve the three-dimensional winds with two or more ground radars (Armijo 1969; Miller and Strauch 1974) and later applied to airborne tail radars employing the FAST (Chong and Testud 1996). This technique is implemented in a cylindrical coordinate system whose central axis is the line between the location points where the radar (or radars) provides two different looks of a single point in the domain. For aircraft observations, the ideal situation for the coplane method would have a straight flight track and constant flight altitude across the analysis domain. The two looks of the wind field, obtained with fore- and aft-pointing beams, are considered independent and, for the purpose of this study, instantaneous. For the typical Global Hawk speed and altitude, the largest time gap between the observations is 200 s. These two measurements can then be readily converted into two orthogonal wind components. Recovery of the third wind component at every point in the domain requires well-posed data (i.e., data exist at every point). To describe the application of the coplane method to the downward-pointing conically scanning HIWRAP geometry, we follow the discussions from Tian et al. (2014, manuscript submitted to *J. Appl. Meteor. Climatol.*) and Guimond et al. (2014).

*ρ*,

*α*, and

*Y*. The flight track serves as the main axis

*Y*where the origin is some arbitrary point along

*Y*. The variable

*ρ*is the radial distance from the central axis, and

*α*is the coplane angle beginning at 0° for the nadir plane and increasing to the right of the flight track. For every rotation angle

*θ*(0° points in +

*Y*direction), range

*r*, and current track position

*D*,

*P*,

*R*, and

*τ*are the drift, pitch, roll, and tilt angles, respectively. Equations (1) and (2) are similar to those in Guimond et al. (2014) and are derived for the current scanning geometry following Lee et al. (1994). The cylindrical coordinates of the observations are then calculated by

Cylindrical coordinate system used for the coplane dual-Doppler retrieval from HIWRAP observations. The position of the target in the cylindrical coordinate system is *P*, defined by the radius *ρ*, coplane angle *α*, and flight track distance *Y*. The red dashed lines and red arrows represent line segments and vectors, respectively, along the radar beam. The black dashed line and black arrows within the coplane represent a line segment and vectors in the cylindrical coordinate system. The symbols *r*_{1} and *r*_{2} are the ranges of the target from the radar positions at *Y*_{1} and *Y*_{2}. Angles *β*_{1} and *β*_{2} are the positive angles between the ranges and the radius *ρ*. Components *V*_{r1} and *V*_{r2} are the radial Doppler velocities at point *P*, and *U*_{ρ} and *U*_{Y} are the corresponding cylindrical coordinate velocity components in the coplane. The axes *X*_{t}, *Y*, and *Z*_{t} define the corresponding track-following Cartesian coordinate framework, where *Y*_{t} is coincident with *Y*. [Figure adapted from Tian et al. (2014, manuscript submitted to *J. Appl. Meteor. Climatol.*).]

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Cylindrical coordinate system used for the coplane dual-Doppler retrieval from HIWRAP observations. The position of the target in the cylindrical coordinate system is *P*, defined by the radius *ρ*, coplane angle *α*, and flight track distance *Y*. The red dashed lines and red arrows represent line segments and vectors, respectively, along the radar beam. The black dashed line and black arrows within the coplane represent a line segment and vectors in the cylindrical coordinate system. The symbols *r*_{1} and *r*_{2} are the ranges of the target from the radar positions at *Y*_{1} and *Y*_{2}. Angles *β*_{1} and *β*_{2} are the positive angles between the ranges and the radius *ρ*. Components *V*_{r1} and *V*_{r2} are the radial Doppler velocities at point *P*, and *U*_{ρ} and *U*_{Y} are the corresponding cylindrical coordinate velocity components in the coplane. The axes *X*_{t}, *Y*, and *Z*_{t} define the corresponding track-following Cartesian coordinate framework, where *Y*_{t} is coincident with *Y*. [Figure adapted from Tian et al. (2014, manuscript submitted to *J. Appl. Meteor. Climatol.*).]

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Cylindrical coordinate system used for the coplane dual-Doppler retrieval from HIWRAP observations. The position of the target in the cylindrical coordinate system is *P*, defined by the radius *ρ*, coplane angle *α*, and flight track distance *Y*. The red dashed lines and red arrows represent line segments and vectors, respectively, along the radar beam. The black dashed line and black arrows within the coplane represent a line segment and vectors in the cylindrical coordinate system. The symbols *r*_{1} and *r*_{2} are the ranges of the target from the radar positions at *Y*_{1} and *Y*_{2}. Angles *β*_{1} and *β*_{2} are the positive angles between the ranges and the radius *ρ*. Components *V*_{r1} and *V*_{r2} are the radial Doppler velocities at point *P*, and *U*_{ρ} and *U*_{Y} are the corresponding cylindrical coordinate velocity components in the coplane. The axes *X*_{t}, *Y*, and *Z*_{t} define the corresponding track-following Cartesian coordinate framework, where *Y*_{t} is coincident with *Y*. [Figure adapted from Tian et al. (2014, manuscript submitted to *J. Appl. Meteor. Climatol.*).]

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

*τ*obtains Doppler velocities in an

*α*plane when it is located at

*P*contains consolidated fore and aft radial velocities (

*α*plane are then calculated by

*α*plane on the cylindrical grid.

*β*, defined as

*α*. Equation (6) uses the elevation angle

*J. Appl. Meteor. Climatol.*) determined that the standard error of HIWRAP Doppler estimates for the Ka band is

^{−1}. Equation (7) assumes that all errors are Gaussian distributed. Other sources of error can contribute to

Figure 3 shows the separation angle and the corresponding wind variances as a function of the coplane angle for the two tilt angles of the HIWRAP geometry. It is shown that *β* reaches its peak at nadir and then decreases as *α* increases in magnitude. We find that ^{2} s^{−2} throughout most of the domain. Toward the domain edges, the fore and aft beams become closely parallel (i.e., *β* approaches 0°) and point less in the along-track direction. Consequently, the accuracy of the retrieved *α* magnitudes. On the other hand, the ^{2} s^{−2}). It is most accurate near the domain edges and least accurate at nadir. Still, the magnitudes of *α* are lower than those of

(a) The separation angle *β* as a function of the coplane angle *α* for the inner beam (dashed black) and the outer beam (solid black). (b) Variance (*σ*^{2}) of the *U*_{Y} (red) and *U*_{ρ} (blue) wind components due to Doppler estimate error as a function of the coplane angle *α*. Variances are calculated for the Ka-band inner (red dashed and blue dashed) and outer (red solid and blue solid) beams.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

(a) The separation angle *β* as a function of the coplane angle *α* for the inner beam (dashed black) and the outer beam (solid black). (b) Variance (*σ*^{2}) of the *U*_{Y} (red) and *U*_{ρ} (blue) wind components due to Doppler estimate error as a function of the coplane angle *α*. Variances are calculated for the Ka-band inner (red dashed and blue dashed) and outer (red solid and blue solid) beams.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

(a) The separation angle *β* as a function of the coplane angle *α* for the inner beam (dashed black) and the outer beam (solid black). (b) Variance (*σ*^{2}) of the *U*_{Y} (red) and *U*_{ρ} (blue) wind components due to Doppler estimate error as a function of the coplane angle *α*. Variances are calculated for the Ka-band inner (red dashed and blue dashed) and outer (red solid and blue solid) beams.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

*i*and

*o*subscripts denote observations from the inner and outer beams.

*α*axis away from the nadir plane. Figure 4 illustrates the two integration directions that span the radar domain. The anelastic mass continuity equation is given by

*η*is the air density. The current calculations use the Jordan (1958) standard tropical Atlantic Ocean air density profile. Using the square rule for integration on Eq. (9),

Diagram of the cylindrical grid (*ρ*, *α*) along a given *Y* location. Red lines indicate the boundary condition locations for initialization of the *U*_{α} wind component. Blue arrows indicate the integration directions for retrieving the *U*_{α} wind component. The shaded region shows the area initialized by the surface boundary condition.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Diagram of the cylindrical grid (*ρ*, *α*) along a given *Y* location. Red lines indicate the boundary condition locations for initialization of the *U*_{α} wind component. Blue arrows indicate the integration directions for retrieving the *U*_{α} wind component. The shaded region shows the area initialized by the surface boundary condition.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Diagram of the cylindrical grid (*ρ*, *α*) along a given *Y* location. Red lines indicate the boundary condition locations for initialization of the *U*_{α} wind component. Blue arrows indicate the integration directions for retrieving the *U*_{α} wind component. The shaded region shows the area initialized by the surface boundary condition.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

*u*), along-track (

*υ*), and vertical (

*w)*velocities are related to the cylindrical coordinate velocities by

*w*is constant and

*u*is linear across the span of the

*w*= 0) is applied as a boundary condition. By setting

*w*equal to 0, Eqs. (11) and (13) lead to

## 3. Data and methods

### a. Radar simulator

To assess the validity of the coplane analysis, we use model output and a radar simulator designed after Guimond et al. (2014) with no added noise or aircraft attitude. The radar simulator mimics the scanning technique of the HIWRAP radar and obtains radial velocities ^{−1}). The model output has a horizontal resolution of 1.67 km and 28 sigma levels in the vertical. Two simulated radar beams are positioned at 30° and 40° tilt angles, and rotate at a period of 3.5 s per revolution with an azimuthal resolution of 2° and a range resolution of 150 m. The radar has a nominal altitude of 18.5 km and the simulated aircraft has a ground speed of 160 m s^{−1}. Shown in Fig. 5, the track has a length of 200 km and passes through the center of the storm.

Plan view of model reflectivity at 2-km altitude. The data are HIWRAP radar simulator observations of a model simulation of Hurricane Rita at peak strength.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Plan view of model reflectivity at 2-km altitude. The data are HIWRAP radar simulator observations of a model simulation of Hurricane Rita at peak strength.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Plan view of model reflectivity at 2-km altitude. The data are HIWRAP radar simulator observations of a model simulation of Hurricane Rita at peak strength.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

### b. Real data

On 15 September 2013, the NASA Global Hawk AV-1 flew over Hurricane Ingrid as part of the NASA Hurricane and Severe Storm Sentinel (HS3) field campaign. The HIWRAP radar on board the Global Hawk observed the northern edge of Ingrid as the storm tracked west across the Gulf of Mexico. The data used in this study were taken from 1836 to 1900 UTC. Figure 6 shows the HIWRAP observed reflectivity (plan view and at nadir) along with the corresponding infrared satellite image. In section 6, we apply the coplane analysis to the Ka-band outer beam observations. To remove noise, pixels with reflectivity less than 0 dB*Z* were not used in the analysis. The Doppler velocities were unfolded according to Dazhang et al. (1984). We applied corrections for beam-pointing errors by aligning the expected range of the ocean surface with the range of the observed surface return. Fall speed corrections from Heymsfield et al. (2010) were also applied to the velocity data. In this correction algorithm, fall speeds were calculated as a function of the Ka-band reflectivity and altitude.

(a) Infrared satellite imagery of Hurricane Ingrid (2013) with the flight track from 15 Sep 2013 of the NASA Global Hawk AV-1 overlain in yellow. A black × marks the best-track storm center. (b) The 1.5-km Ka-band outer beam reflectivity observed by HIWRAP along the flight track from (a). Observations were taken from 1836 to 1900 UTC. (c) Nadir view of Ka-band outer beam reflectivity. The graph origin is the eastern end of the plan-view reflectivity from (b).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

(a) Infrared satellite imagery of Hurricane Ingrid (2013) with the flight track from 15 Sep 2013 of the NASA Global Hawk AV-1 overlain in yellow. A black × marks the best-track storm center. (b) The 1.5-km Ka-band outer beam reflectivity observed by HIWRAP along the flight track from (a). Observations were taken from 1836 to 1900 UTC. (c) Nadir view of Ka-band outer beam reflectivity. The graph origin is the eastern end of the plan-view reflectivity from (b).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

(a) Infrared satellite imagery of Hurricane Ingrid (2013) with the flight track from 15 Sep 2013 of the NASA Global Hawk AV-1 overlain in yellow. A black × marks the best-track storm center. (b) The 1.5-km Ka-band outer beam reflectivity observed by HIWRAP along the flight track from (a). Observations were taken from 1836 to 1900 UTC. (c) Nadir view of Ka-band outer beam reflectivity. The graph origin is the eastern end of the plan-view reflectivity from (b).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

### c. Grid and interpolation specifications

*m*th observation from the analysis grid point,

*γ*is a chosen shape parameter, and

*δ*is the influence radii expressed by

*γ*, which determines the width of the weighting function, is chosen as 0.75. Following the coplane calculations, the data are converted into Cartesian coordinate velocities via Eqs. (11)–(13) and are finally interpolated into Cartesian coordinates. The Cartesian grid has a horizontal resolution of 2 km and a vertical resolution of 1 km. An additional level is added at 0.5-km altitude to better resolve the low-level winds. This interpolation uses the same Barnes filter but with a radius of influence of 2 km in the horizontal dimensions and 0.25 km in the vertical dimension. By determining the response function of the Barnes filter (Koch et al. 1983), the minimum resolvable horizontal wavelength is calculated to be 4 km, which is also twice the horizontal grid spacing.

## 4. Boundary conditions analysis

### a. Nadir boundary conditions analysis

At nadir, the cross-track wind component is unobserved by the HIWRAP radar and must be estimated by utilizing other available measurements. It is important to obtain a good estimate of the cross-track wind at nadir as this will serve as the boundary condition that initializes the

^{−1}, the largest error in Fig. 7b constitutes 5% of this wind speed. The

(a) The coplane angle *α* as a function of rotation angle calculated from Eq. (18). Results from the inner beam (red) and outer beam (black) are shown. (b) RMS errors of *U*_{α} in the nadir boundary conditions estimate for varying rotation angles (dots) and cross-track distance between outer-beam locations for varying rotation angles (dashed line).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

(a) The coplane angle *α* as a function of rotation angle calculated from Eq. (18). Results from the inner beam (red) and outer beam (black) are shown. (b) RMS errors of *U*_{α} in the nadir boundary conditions estimate for varying rotation angles (dots) and cross-track distance between outer-beam locations for varying rotation angles (dashed line).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

(a) The coplane angle *α* as a function of rotation angle calculated from Eq. (18). Results from the inner beam (red) and outer beam (black) are shown. (b) RMS errors of *U*_{α} in the nadir boundary conditions estimate for varying rotation angles (dots) and cross-track distance between outer-beam locations for varying rotation angles (dashed line).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Figure 8 displays the estimated ^{−1} occur near the eyewall region (*Y* = 85 and 115 km), the surface, and the domain edge at *Y* = 18 km. These positive errors at the domain edge reach 8 m s^{−1}. The largest negative errors occur in the midlevels near *Y* = 155 km, reaching values of −9 m s^{−1}. These errors stem from local violations of the assumptions made in the calculation of Eq. (14). Specifically, violations of the constant vertical velocity assumption are the primary source of error in Fig. 8, where vertical velocity deviations of 0.5 m s^{−1} produced ^{−1}.

The *U*_{α} wind component at nadir retrieved from the radar simulator data using the coplane analysis. Deviations from the model truth are shown in black contours at intervals of 2 m s^{−1}. Dashed lines are negative values beginning at −2 m s^{−1}, and solid lines are positive values beginning at 2 m s^{−1}. See text for details.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

The *U*_{α} wind component at nadir retrieved from the radar simulator data using the coplane analysis. Deviations from the model truth are shown in black contours at intervals of 2 m s^{−1}. Dashed lines are negative values beginning at −2 m s^{−1}, and solid lines are positive values beginning at 2 m s^{−1}. See text for details.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

The *U*_{α} wind component at nadir retrieved from the radar simulator data using the coplane analysis. Deviations from the model truth are shown in black contours at intervals of 2 m s^{−1}. Dashed lines are negative values beginning at −2 m s^{−1}, and solid lines are positive values beginning at 2 m s^{−1}. See text for details.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

### b. Surface boundary condition analysis

The values of *w* = 0 at the surface [Eq. (15)]. This condition requires reliable observations near a flat surface, which is an ideal situation that models provide. The initialization locations on the surface are not necessarily points on the cylindrical grid, but

With actual observations, setting the surface boundary condition cannot be done so simply, particularly over the ocean surface as in the case of tropical cyclone research. The ocean surface may not be flat and sea spray can contaminate echoes near the surface. Previous dual-Doppler methods approach the surface initialization of the integrated wind component (usually *w*) differently. In the Cartesian Editing and Display of Radar Data under Interactive Control software (CEDRIC; Mohr et al. 1986), the vertical velocity can be initialized at the lowest level of usable data by setting *w* equal to a specified multiple of the locally measured horizontal divergence. Chong and Testud (1996) use a variational method in which *w* at all surface locations vary near *w* = 0 such that the resulting

*w*from the vertical velocity calculated at nadir (where

*Y*location.

The calculated lower-bound ^{−1}. Moreover, the lower-bound ^{−1}. As expected, the accuracy of the nadir-*w* approach is slightly lower than that of the impermeability approach; however, the difference in the errors (0.36 m s^{−1}) is small relative to the near-surface hurricane wind speeds (which have magnitudes greater than 30 m s^{−1} outside the eye). From this analysis, the nadir-*w* approach represented by Eq. (19) is deemed suitable for

## 5. Retrieval error analysis

In this section, we use the simulated radar data to examine the wind field retrieved from the coplane analysis. As described in the previous section, we do not use radar radial velocities below 0.5-km altitude in this retrieval. Figures 9a–c present the RMS errors calculated along the flight track for the retrieved cross-track (*u*), along-track (*υ*), and vertical velocity (*w*) components. These figures show the total errors and the error patterns of each wind component for the HIWRAP scanning geometry. The total relative-RMS (RRMS) error in each figure is a normalization of the errors relative to the velocity magnitudes.

RMS errors of the two retrieval methods. Errors from the coplane analysis are shown for the (a) cross-track (*u*), (b) along-track (*υ*), and (c) vertical (*w*) wind components. (d)–(f) Errors from the variational analysis for the same wind components. Error fields are averaged along the flight track. Overall RMS and relative-RMS errors are also given (m s^{−1} and %, respectively).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

RMS errors of the two retrieval methods. Errors from the coplane analysis are shown for the (a) cross-track (*u*), (b) along-track (*υ*), and (c) vertical (*w*) wind components. (d)–(f) Errors from the variational analysis for the same wind components. Error fields are averaged along the flight track. Overall RMS and relative-RMS errors are also given (m s^{−1} and %, respectively).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

RMS errors of the two retrieval methods. Errors from the coplane analysis are shown for the (a) cross-track (*u*), (b) along-track (*υ*), and (c) vertical (*w*) wind components. (d)–(f) Errors from the variational analysis for the same wind components. Error fields are averaged along the flight track. Overall RMS and relative-RMS errors are also given (m s^{−1} and %, respectively).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

The cross-track component *u* contains an average error of 1.9 m s^{−1}, which, as indicated by the relative-RMS value of 4.4%, is a low value relative to the *u* magnitudes. Calculation of *u* depends on both the *u* errors largely stem from errors in *u* throughout the domain. The *u* errors form a curved pattern as they follow the integration path upon which

The vertical velocity *w* contains an average error of 0.9 m s^{−1}, which is significant relative to the vertical velocity magnitudes (RRMS = 60.4%). Despite this significant average error, the error distribution in Fig. 9c shows that the vertical velocities near nadir have the smallest errors and therefore are the most useful. The errors increase as the *u* error belt in Fig. 9a. At these larger *w*. As a result, *w* errors near the domain edges.

We have explained that errors in the *u* and *w* fields are mostly due to ^{−1} for *u* and 0.5 m s^{−1} for *w*. The *u* error no longer contains the belted pattern and the corresponding *w* errors along the domain edges are removed. This analysis suggests that the errors in the *u* and *w* wind fields are mostly a result of errors in the boundary conditions. The remaining errors are less pronounced in Fig. 10 and can be attributed to divergence sampling and interpolation error.

RMS errors of the coplane analysis for the (a) cross-track (*u*) and (b) vertical (*w*) wind components averaged along the flight track. This analysis uses nadir and lower-bound boundary conditions given by the model truth field. Overall RMS and relative-RMS errors are also given (m s^{−1} and %, respectively).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

RMS errors of the coplane analysis for the (a) cross-track (*u*) and (b) vertical (*w*) wind components averaged along the flight track. This analysis uses nadir and lower-bound boundary conditions given by the model truth field. Overall RMS and relative-RMS errors are also given (m s^{−1} and %, respectively).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

RMS errors of the coplane analysis for the (a) cross-track (*u*) and (b) vertical (*w*) wind components averaged along the flight track. This analysis uses nadir and lower-bound boundary conditions given by the model truth field. Overall RMS and relative-RMS errors are also given (m s^{−1} and %, respectively).

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

The along-track component *υ* is the only Cartesian coordinate component that is not calculated with the *υ* errors are very small throughout most of the domain. The largest errors, reaching up to 5 m s^{−1}, occur at 0.5-km altitude. These errors at the lowest level are largely a result of the interpolation from cylindrical to Cartesian coordinates. The lower-bound points on the cylindrical grid are at different altitudes and are all higher than the 0.5-km level. Since the data below 0.5 km were not used, these lower-bound points are the only source of information for interpolation onto the 0.5-km level, which contributes to errors found at this lowest level for all Cartesian wind components. In this particular dataset, the *υ* component (which is largely the radial wind of the hurricane) changes very rapidly at these boundary layer altitudes, resulting in the significant errors found in the lowest levels. When the 0.5-km-level data are excluded from the error analysis, the RMS error drops from 1.7 to 1.0 m s^{−1}.

The coplane analysis is now compared with the global optimization analysis described by Guimond et al. (2014). In their variational method, radar velocities are first interpolated onto a Cartesian grid, and then a modeled wind field is retrieved using the radar velocities, mass continuity, and boundary conditions as constraints. We applied the variational method to the current simulated data using analysis parameters that produced the smallest errors. Specific parameters that were chosen are listed in Table 1.

Parameters used for the variational dual-Doppler retrieval. Each parameter is explained in detail in Guimond et al. (2014).

Figures 9d–f show the RMS errors for the three wind components. The *u* and *w* components both have larger overall errors than the coplane analysis. These components, which again are connected to each other through the unobserved wind component

Meanwhile, the *υ* field has a smaller overall error than does the coplane analysis. The largest difference in the *υ* error pattern is in the lowest levels. The variational method operates fully in a Cartesian coordinate system, which means that the lowest level of available data coincides with the lowest level of grid points at 0.5-km altitude. As a result, the rapid vertical changes in *υ* are well captured at these levels and there is no interpolation error from changing coordinate systems.

## 6. Coplane retrieval with real radar data

In the previous section, the coplane retrieval method was successfully applied to simulated radar data. We now apply the coplane method to real HIWRAP data shown in Fig. 6, and we compare the retrieved wind field to a solution from the variational method (Guimond et al. 2014).

Figures 11a–c presents the coplane analysis cross-track (*u*), along-track (*υ*), and vertical (*w*) components of the wind field along nadir of the observation domain. The *u* field contains mostly positive velocities with values >10 m s^{−1} in the layer below 3 km and the layer above 6 km. In between these layers a midlevel minimum of *u* occurs. The *υ* field also contains mostly positive velocities that decrease toward the upper levels of the domain (>7-km altitude). When considering the domain location (Fig. 6a), the *u* and *υ* fields show consistency with the counterclockwise cyclonic circulation of the storm. The cross-track component (Fig. 11a) largely switches from positive to negative values at around *Y*_{t} = 230 km, which corresponds to the point along the track that is closest to the storm center. Concurrently, the along-track component (Fig. 11b) increases as the track approaches the same closest point.

Nadir view of the (a) *u*, (b) *υ*, and (c) *w* wind components as derived by the coplane analysis of the HIWRAP observations seen in Fig. 6. (d)–(f) The variational analysis wind components.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Nadir view of the (a) *u*, (b) *υ*, and (c) *w* wind components as derived by the coplane analysis of the HIWRAP observations seen in Fig. 6. (d)–(f) The variational analysis wind components.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Nadir view of the (a) *u*, (b) *υ*, and (c) *w* wind components as derived by the coplane analysis of the HIWRAP observations seen in Fig. 6. (d)–(f) The variational analysis wind components.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

The reflectivity field shown in Fig. 6c contains a clear brightband signature (at ~4.5-km altitude) and fall streaks, which indicate that the dominant precipitation regime for these observations is stratiform (Houze 1997). In stratiform precipitation, falling ice crystals melt in a layer beneath the 0°C isotherm and form a broad region of light-to-moderate precipitation. The *w* field (Fig. 11c) shows consistent features with stratiform precipitation, including small magnitudes (<2 m s^{−1}) throughout most of the domain. Updrafts are dominant above the bright band. Below this level, downdrafts are prominent, but a clear exception of positive *w* values occurs toward the beginning of the domain and at 4-km altitude. These exceptions, which are inconsistent with typical stratiform kinematics, are likely a result of errors in the fall speed correction and/or attenuation of the Ka beam. If these errors were consistent across the radial velocities used in Eqs. (4) and (14) to calculate

Figures 12a–c present the wind components along a cross section at *Y*_{t} = 160 km. These cross sections show that the overall patterns seen at nadir extend to the edges of the domain. The midlevel minimum of *u* grows larger to the right of the flight track (Fig. 12a). Additionally, the downdraft layer (Fig. 12c) also increases in depth to the right of the flight track. The slanted stretches of downdrafts <−3 m s^{−1} near the domain edges do not appear consistent with expected vertical velocity patterns of stratiform precipitation. Rather, these patterns are reminiscent of the curved error patterns in Figs. 9a,c. Given this resemblance, we infer that these *w* swaths (and their corresponding *u* values) contain errors for the same reason as in the radar simulator analysis, which is incorrect initialization of

Cross-track view of the (a) *u*, (b) *υ*, and (c) *w* wind components as derived by the coplane analysis of the HIWRAP observations. (d)–(f) The variational analysis wind components. This cross section is taken at *Y*_{t} = 160 km from Fig. 11.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Cross-track view of the (a) *u*, (b) *υ*, and (c) *w* wind components as derived by the coplane analysis of the HIWRAP observations. (d)–(f) The variational analysis wind components. This cross section is taken at *Y*_{t} = 160 km from Fig. 11.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Cross-track view of the (a) *u*, (b) *υ*, and (c) *w* wind components as derived by the coplane analysis of the HIWRAP observations. (d)–(f) The variational analysis wind components. This cross section is taken at *Y*_{t} = 160 km from Fig. 11.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

The variational method retrieved a qualitatively similar wind solution as that of the coplane method. Figures 11d–f show that the *u* and *υ* fields in the nadir plane have the same overall structures as in Figs. 11a–c. Figure 11f mostly has the vertical velocities expected of stratiform precipitation, but this solution takes the same fall speed corrections as in the coplane method and produces noticeably different vertical velocities at the brightband altitude (~4.5 km). The variational *u* field (Fig. 11d) has noticeably smoother contour patterns than the coplane *u* field (Fig. 11a), which suggests that the variational method may be filtering out some small-scale features in the data. The *υ* and *w* fields from both methods do not have a noticeable discrepancy in their contour smoothness. Upon closer inspection, the *u* field at nadir is impacted most by the smoothing parameter in the variational retrieval. When this smoothing parameter is turned off, the resulting *u* field appears very similar to the coplane *u* field.

One advantage of the coplane method is that the minimum resolvable wavelength of the data field is readily determined by calculating the response function of the Barnes filter. On the other hand, determining the minimum resolvable wavelength of the variational method solution is not as straightforward. While a Barnes filter is also used, the weighting parameter is a *constraint* on the optimization and not a direct *calculation*. Thus, the Barnes filter response function cannot exactly determine the minimum resolvable wavelength. In addition, the smoothing parameter certainly increases the minimum resolvable wavelength, but again, this smoothing is a constraint and not a direct calculation. For both the Barnes filter and the Laplacian smoother, the minimum resolvable wavelength must be determined empirically.

Figures 12d–f display the same cross section as in Figs. 12a–c but for the variational solution. As in the coplane analysis, the midlevel *u* minimum and downdraft layer increase in depth to the right of the flight track. The *w* field in Fig. 12f does not contain the unrealistic downdraft patterns seen in Fig. 12c as there is no explicit integration along a curved path. However, the *w* field does contain downdrafts <−3 m s^{−1} near the domain edges that appear unrealistic. Along the left domain edge, these increased downdrafts occur in the same location as in the coplane analysis (Fig. 12c). Along the right domain edge, these increased downdrafts are prominent in the lower altitudes and appear to trail off into the higher altitudes. This pattern of vertically oriented anomalies along the domain edge is reminiscent of the error pattern in Fig. 9f, which suggests that these features contain likely errors. The source of these errors cannot be traced to specific observations, but rather the errors must be attributed to the general decreased accuracy of the global solver along the domain edges.

Both the coplane and variational methods produced adequate wind fields that generally agreed well with each other. Both fields also contained inevitable localized errors. With a priori knowledge of the error patterns expected from each method, the questionable features that appear in the solutions can be easily identified as retrieval errors. Identifying and understanding these errors is essential for reliable scientific interpretations of solutions from either analysis method.

We make a final comparison of retrieval techniques with the VAD technique from Tian et al. (2013). The VAD technique obtains the mean horizontal wind within the nadir plane by fitting the measured radial winds within a scan circle onto a sinusoidal curve. Figure 13 shows the retrieved *u* and *υ* components of the wind for the same leg of data from Hurricane Ingrid. The VAD technique captures the same overall wind pattern that was retrieved by the other retrieval techniques (Fig. 11). The most noticeable difference in Fig. 13 is the increased vertical resolution. Since the VAD technique does not retrieve the full three-dimensional wind field, it is computationally less expensive than both the coplane and variational methods; moreover, this allows the VAD technique to preserve the high vertical resolution of the HIWRAP beam.

Nadir view of the (a) *u* and (b) *υ* wind components as derived by the VAD analysis of the HIWRAP observations seen in Fig. 6.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Nadir view of the (a) *u* and (b) *υ* wind components as derived by the VAD analysis of the HIWRAP observations seen in Fig. 6.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

Nadir view of the (a) *u* and (b) *υ* wind components as derived by the VAD analysis of the HIWRAP observations seen in Fig. 6.

Citation: Journal of Applied Meteorology and Climatology 54, 3; 10.1175/JAMC-D-14-0203.1

In calculating the mean horizontal wind, the wind field is assumed to have linear horizontal velocity and constant hydrometeor vertical speed across the total scan circle. These assumptions tend to hold well in stratiform precipitation regions like that in the current dataset since these mesoscale regions contain weak vertical velocities and winds that vary slowly over horizontal distances. To capture convective-scale features, one of the three-dimensional wind retrieval methods must be used.

## 7. Conclusions

In this paper, the coplane method for dual-Doppler wind retrieval (Armijo 1969; Miller and Strauch 1974) is adapted to the downward-pointing conically scanning technique of the NASA HIWRAP airborne radar. The coplane method takes the radar observations and solves for the three-dimensional winds using a simple interpolation and integration approach. This approach locally solves for the wind field, which is in contrast to the global optimization (variational) method described by Guimond et al. (2014). To retrieve the unobserved wind component

Simulated radar observations of a model hurricane were used to test the coplane method and compare to the variational method. The coplane method retrieved the wind field with small errors relative to the wind speed magnitudes. Compared to the variational method, the coplane method had lower errors in the cross-track component (*u*) and vertical component (*w*) fields, while the variational method had lower errors in the along-track component (*υ*) field. For the coplane method, the accuracy of *u* relied on the accuracy of the *u* remained accurate across the span of the domain. Where *u* propagated along the curved integration path, creating an easily recognizable error signature. The *w* component, which is also derived from *u* and *w* that grew toward the edges of the domain at all altitudes.

The coplane and variational methods were applied to HIWRAP observations collected during the NASA HS3 campaign. Both techniques produced errors in the retrieval that appeared in patterns similar to the errors in the simulated radar retrieval. Prior knowledge of the error patterns expected from each method allowed for this recognition of retrieval errors in the HS3 retrieval. As a local solver, the errors in the coplane analysis are easily traced to the certain observations and/or

The coplane technique’s ability to transparently trace the exact calculations from the raw observations to the final solution is highly beneficial when making scientific interpretations. This ability is necessary to more adequately analyze small-scale features in tropical cyclones, such as rotating deep convection (Hendricks et al. 2004; Montgomery et al. 2006; Sanger et al. 2014). A key skill for radar analyses is being able to separate true meteorological signals from nonmeteorological signals such as noise or data contamination that has bypassed the data quality control process. Once a solution is obtained, quirky regions in the final solution can be ambiguous as to whether they are true representations of small-scale features. After easily pinpointing the raw observations that were used to create the solution, the user can better assess the reality of the observations taken, and corrections can subsequently be implemented or not implemented. In the variational method, these anomalous measurements would be smoothed and may impact the entire retrieval, which would either dilute the small-scale signal or incorporate erroneous data into the solution. The coplane method can prevent incorrect scientific interpretations of inherently wrong solutions or strengthen confidence in the conclusions based on the observations. Given the wind component error analyses for both retrieval methods, the option to trace solution calculations with the coplane analysis is provided to the user with minimal cost to the accuracy of the overall solution.

Future work will use the coplane analysis for scientific research of observations from the HS3 field campaign and other planned campaigns employing the HIWRAP radar. This technique can also be used to analyze tail Doppler radar data from tropical cyclones documented in peer-reviewed work (e.g., Marks and Houze 1984; Reasor et al. 2009; Houze et al. 2009; Bell and Montgomery 2010), as well as future field campaigns. In locations directly beneath the aircraft, the geometry of the tail Doppler radar observations is compatible with the coplane technique. These additional analyses would be especially useful here as these locations are particularly troublesome for capturing small-scale features with the global optimization technique.

## Acknowledgments

We thank Matthew McLinden, Lihua Li, Martin Perrine, Jaime Cervantes, and Ed Zenker for their engineering support and data processing for the HIWRAP radar. We thank Shuyi Chen for providing the model output used in this study. We also thank the three anonymous reviewers of this manuscript. The first author conducted this research at the NASA Goddard Space Flight Center under the support of the NASA Postdoctoral Program conducted by the Oak Ridge Associated Universities.

## REFERENCES

Armijo, L., 1969: A theory for the determination of wind and precipitation velocities with Doppler radars.

,*J. Atmos. Sci.***26**, 570–573, doi:10.1175/1520-0469(1969)026<0570:ATFTDO>2.0.CO;2.Barnes, S. L., 1973: Mesoscale objective analysis using weighted time-series observations. NOAA Tech. Memo. ERL NSSL-62, National Severe Storms Laboratory, Norman, OK, 60 pp. [NTIS COM-73-10781.]

Bell, M. M., and M. T. Montgomery, 2010: Sheared deep vortical convection in pre-depression Hagupit during TCS08.

,*Geophys. Res. Lett.***37**, L06802, doi:10.1029/2009GL042313.Bousquet, O., and M. Chong, 1998: A multiple-Doppler synthesis and continuity adjustment technique (MUSCAT) to recover wind components from Doppler radar measurements.

,*J. Atmos. Oceanic Technol.***15**, 343–359, doi:10.1175/1520-0426(1998)015<0343:AMDSAC>2.0.CO;2.Braun, S. A., and Coauthors, 2013: NASA’s Genesis and Rapid Intensification Processes (GRIP) field experiment.

,*Bull. Amer. Meteor. Soc.***94**, 345–363, doi:10.1175/BAMS-D-11-00232.1.Browning, K. A., and R. Wexler, 1968: The determination of kinematic properties of a wind field using Doppler radar.

,*J. Appl. Meteor.***7**, 105–113, doi:10.1175/1520-0450(1968)007<0105:TDOKPO>2.0.CO;2.Chong, M., and J. Testud, 1996: Three-dimensional air circulation in a squall line from airborne dual-beam Doppler radar data: A test of coplane methodology software.

,*J. Atmos. Oceanic Technol.***13**, 36–53, doi:10.1175/1520-0426(1996)013<0036:TDACIA>2.0.CO;2.Dazhang, T., S. G. Geotis, R. E. Passarelli Jr., and A. L. Hansen, and C. L. Frush, 1984: Evaluation of an alternating-PRF method for extending the range of unambiguous Doppler velocity. Preprints,

*22nd Conf. on Radar Meteorology,*Zurich, Switzerland, Amer. Meteor. Soc., 523–527.Doviak, R. J., P. S. Ray, R. G. Strauch, and L. J. Miller, 1976: Error estimation in wind fields derived from dual-Doppler radar measurement.

,*J. Appl. Meteor.***15**, 868–878, doi:10.1175/1520-0450(1976)015<0868:EEIWFD>2.0.CO;2.Gamache, J. F., 1997: Evaluation of a fully three-dimensional variational Doppler analysis technique. Preprints,

*28th Conf. on Radar Meteorology,*Austin, TX, Amer. Meteor. Soc., 422–423.Gamache, J. F., F. D. Marks Jr., and F. Roux, 1995: Comparison of three airborne Doppler sampling techniques with airborne in situ wind observations in Hurricane Gustav (1990).

,*J. Atmos. Oceanic Technol.***12**, 171–181, doi:10.1175/1520-0426(1995)012<0171:COTADS>2.0.CO;2.Gao, J., M. Xue, A. Shapiro, and K. K. Droegemeier, 1999: A variational method for the analysis of three-dimensional wind fields from two Doppler radars.

,*Mon. Wea. Rev.***127**, 2128–2142, doi:10.1175/1520-0493(1999)127<2128:AVMFTA>2.0.CO;2.Guimond, S. R., L. Tian, G. M. Heymsfield, and S. J. Frasier, 2014: Wind retrieval algorithms for the IWRAP and HIWRAP airborne Doppler radars with application to hurricanes.

,*J. Atmos. Oceanic Technol.***31**, 1189–1215, doi:10.1175/JTECH-D-13-00140.1.Hendricks, E. A., M. T. Montgomery, and C. A. Davis, 2004: The role of “vortical” hot towers in the formation of Tropical Cyclone Diana (1984).

,*J. Atmos. Sci.***61**, 1209–1232, doi:10.1175/1520-0469(2004)061<1209:TROVHT>2.0.CO;2.Heymsfield, G. M., L. Tian, A. J. Heymsfield, L. Li, and S. Guimond, 2010: Characteristics of deep tropical and subtropical convection from nadir-viewing high-altitude airborne Doppler radar.

,*J. Atmos. Sci.***67**, 285–308, doi:10.1175/2009JAS3132.1.Hildebrand, P. H., and Coauthors, 1996: The ELDORA/ASTRAIA airborne Doppler weather radar: High-resolution observations from TOGA COARE.

,*Bull. Amer. Meteor. Soc.***77**, 213–232, doi:10.1175/1520-0477(1996)077<0213:TEADWR>2.0.CO;2.Houze, R. A., Jr., 1997: Stratiform precipitation in regions of convection: A meteorological paradox?

,*Bull. Amer. Meteor. Soc.***78**, 2179–2196, doi:10.1175/1520-0477(1997)078<2179:SPIROC>2.0.CO;2.Houze, R. A., Jr., W.-C. Lee, and M. M. Bell, 2009: Convective contribution to the genesis of Hurricane Ophelia (2005).

,*Mon. Wea. Rev.***137**, 2778–2800, doi:10.1175/2009MWR2727.1.Jordan, C. L., 1958: Mean soundings for the West Indies area.

,*J. Atmos. Sci.***15**, 91–97, doi:10.1175/1520-0469(1958)015<0091:MSFTWI>2.0.CO;2.Jorgensen, D. P., 1984: Mesoscale and convective-scale characteristics of mature hurricanes. Part I: General observations by research aircraft.

,*J. Atmos. Sci.***41**, 1268–1285, doi:10.1175/1520-0469(1984)041<1268:MACSCO>2.0.CO;2.Jorgensen, D. P., and J. D. DuGranrut, 1991: A dual-beam technique for deriving wind fields from airborne Doppler radar. Preprints,

*25th Int. Conf. on Radar Meteorology,*Paris, France, Amer. Meteor. Soc., 458–461.Jorgensen, D. P., P. H. Hildebrand, and C. L. Frush, 1983: Feasibility test of an airborne pulse-Doppler meteorological radar.

,*J. Climate Appl. Meteor.***22**, 744–757, doi:10.1175/1520-0450(1983)022<0744:FTOAAP>2.0.CO;2.Klimowski, B. A., and J. D. Marwitz, 1992: The synthetic dual-Doppler analysis technique.

,*J. Atmos. Oceanic Technol.***9**, 728–745, doi:10.1175/1520-0426(1992)009<0728:TSDDAT>2.0.CO;2.Koch, S. E., M. desJardins, and P. J. Kocin, 1983: An interactive Barnes objective map analysis scheme for use with satellite and conventional data.

,*J. Climate Appl. Meteor.***22**, 1487–1503, doi:10.1175/1520-0450(1983)022<1487:AIBOMA>2.0.CO;2.Lee, W.-C., P. Dodge, F. D. Marks Jr., and P. H. Hildebrand, 1994: Mapping of airborne Doppler radar data.

,*J. Atmos. Oceanic Technol.***11**, 572–578, doi:10.1175/1520-0426(1994)011<0572:MOADRD>2.0.CO;2.Lhermitte, R. M., 1971: Probing of atmospheric motion by airborne pulse-Doppler radar techniques.

,*J. Appl. Meteor.***10**, 234–246, doi:10.1175/1520-0450(1971)010<0234:POAMBA>2.0.CO;2.Lhermitte, R. M., and D. Atlas, 1961: Precipitation motion by pulse Doppler radar. Preprints,

*Ninth Conf. on Weather Radar,*Norman, OK, Amer. Meteor. Soc., 218–223.Li, L., G. Heymsfield, J. Carswell, D. Schaubert, M. McLinden, M. Vega, and M. Perrine, 2011: Development of the NASA High-Altitude Imaging Wind and Rain Airborne Profiler. Preprints,

*IEEE Aerospace Conf.,*Big Sky, MT, Institute of Electrical and Electronics Engineers, Paper 6.0202, 7 pp, doi:10.1109/AERO.2011.5747415.Marks, F. D., Jr., and R. A. Houze Jr., 1984: Airborne Doppler radar observations in Hurricane Debby.

,*Bull. Amer. Meteor. Soc.***65**, 569–582, doi:10.1175/1520-0477(1984)065<0569:ADROIH>2.0.CO;2.Miller, L. J., and R. G. Strauch, 1974: A dual Doppler radar method for the determination of wind velocities within precipitating weather systems.

,*Remote Sens. Environ.***3**, 219–235, doi:10.1016/0034-4257(74)90044-3.Mohr, C. G., L. J. Miller, R. L. Vaughan, and H. W. Frank, 1986: The merger of mesoscale datasets into a common Cartesian format for efficient and systematic analyses.

,*J. Atmos. Oceanic Technol.***3**, 143–161, doi:10.1175/1520-0426(1986)003<0143:TMOMDI>2.0.CO;2.Montgomery, M. T., M. M. Bell, S. D. Aberson, and M. L. Black, 2006: Hurricane Isabel (2003): New insights into the physics of intense storms. Part I: Mean vortex structure and maximum intensity estimates.

,*Bull. Amer. Meteor. Soc.***87**, 1335–1347, doi:10.1175/BAMS-87-10-1335.Reasor, P. D., M. D. Eastin, and J. F. Gamache, 2009: Rapidly intensifying Hurricane Guillermo (1997). Part I: Low-wavenumber structure and evolution.

,*Mon. Wea. Rev.***137**, 603–631, doi:10.1175/2008MWR2487.1.Sanger, N. T., M. T. Montgomery, R. K. Smith, and M. M. Bell, 2014: An observational study of tropical cyclone spinup in Supertyphoon Jangmi from 24 to 27 September.

,*Mon. Wea. Rev.***142**, 3–28, doi:10.1175/MWR-D-12-00306.1.Tian, L., G. Heymsfield, S. Guimond, and L. Li, 2013: VAD and dual-Doppler analysis of wind fields from HIWRAP observations during GRIP.

*36th Conf. on Radar Meteorology,*Breckenridge, CO, Amer. Meteor. Soc., 212. [Available online at http://ams.confex.com/ams/36Radar/webprogram/Paper229079.html.]