# Dual-doppler analysis in a single plane from an airborne platform

D. Leon, G. Vali, M. Lothon

► To cite this version:

D. Leon, G. Vali, M. Lothon. Dual-doppler analysis in a single plane from an airborne platform. Journal of Atmospheric and Oceanic Technology, 2006, 23, pp.3-22. 10.1175/JTECH1820.1. hal-00068764

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# Dual-Doppler Analysis in a Single Plane from an Airborne Platform

D. LEON AND G. VALI

Department of Atmospheric Science, University of Wyoming, Laramie, Wyoming

M. LOTHON

Laboratoire d'Aérologie, Tolouse, France

(Manuscript received 2 September 2004, in final form 23 March 2005)

## ABSTRACT

A modified dual-Doppler analysis technique for use with airborne Doppler radars utilizing two fixed beams is presented. Although the data collected by such a system would ideally lie in the plane defined by the radar beam orientations and the aircraft velocity vector, variations in the aircraft attitude and drift angles lead to displacements between the radar observations and the idealized observation plane. These variations motivated the development of a formal framework in which an a priori velocity estimate is used in conjunction with the two Doppler velocity measurements to form a three-dimensional velocity estimate. Two velocity components, lying within or close to the observation plane, and therefore containing only a small contribution from the a priori velocity estimate, are then extracted from the three-dimensional velocity estimate. Advantages of using the three-dimensional framework include improved accuracy (when an a priori velocity estimate is available) and a framework for assessing the effects of cross-plane contamination on the retrieved velocity components.

The velocity fields retrieved using the modified dual-Doppler analysis are affected by errors in the platform motion correction to the Doppler velocities, random noise in the mean Doppler velocity estimates, displacements between the radar beams (and between the radar beams and the idealized observation plane), and meteorological velocity variations about the a priori velocity estimate. Errors in the platform motion correction remain poorly characterized but are believed to be the largest source of error in many cases. However, these errors result primarily in biases (or low-frequency errors) in the retrieved velocity fields and therefore do not interfere with the ability to resolve actual velocity variations. Random noise in the mean Doppler velocity estimates increases dramatically with decreasing signal-to-noise ratio (SNR) (for SNR < 5 dB) and effectively limits the use of the single-plane dual-Doppler (SPDD) analysis to SNR > 0 dB. Displacements between the volumes sampled by the nadir and slanted beams can also be a significant source of error, especially at larger displacements from the aircraft. Errors resulting from meteorological velocity variations about the a priori velocity estimate tend to be small compared to the velocity variations of interest.

The dual-Doppler analysis presented in this paper has been applied to retrieve two-dimensional velocity fields with a resolution of ~50 m using Doppler velocities collected using dual-beam configurations of the Wyoming Cloud Radar. Results are in horizontal and vertical planes for marine stratocumulus, cumulus congestus, and for the clear-air boundary layer.

## 1. Introduction

The potential benefits of airborne radars for studying cloud and precipitation processes have been recognized for some time (Knight 1986; Hildebrand and Moore

1990). Advantages of such an airborne system include (i) the mobility of the aircraft, which allows the radar to be operated in close proximity to the region of interest, maximizing the effective sensitivity of the radar while minimizing the spread of the beam; (ii) the speed of the aircraft, which is sufficiently high that data from a single flight leg can be interpreted as a nearly instantaneous snapshot of the field; and (iii) that radar data can be obtained in close proximity (~100 m) to the aircraft so that they can be used to aid in the interpretation of the

<footnote>*Corresponding author address:* Mr. David Leon, Atmospheric Science Department, University of Wyoming, Dept. 3038, 1000 E. University Ave., Laramie, WY 82071-2000.
E-mail: leon@uwyo.edu</footnote>
in situ observations, and the in situ observations (specifically the particle size and shape measurements) can be used to interpret the radar data.

Alongside these potential benefits there are a number of disadvantages to airborne radar systems. (i) The Doppler velocity must be corrected for platform motion and will inevitably be less accurate than Doppler velocities obtained from a similar ground-based radar. (ii) The speed of the aircraft precludes the use of long integration times, thereby reducing the effective sensitivity of the radar. (iii) The speed of the aircraft also makes it difficult to follow the time evolution of the observed field.

Synthetic multiple-Doppler analyses (Hildebrand and Mueller 1985; Mueller and Hildebrand 1985) exploit the motion of the aircraft to sample the same volume from multiple angles. These analyses have generally been applied to Doppler velocities collected by historical-scanning systems such as the Electra Doppler Radar (ELDORA) and the P-3 radar. Dual-Doppler analysis can also be applied to Doppler velocities collected using two fixed beams lying within the same plane. This plane, referred to as the idealized observation plane, is defined by the orientation of the radar beams and the aircraft velocity vector. Heymsfield et al. (1996, hereafter H96) demonstrated the use of dual-Doppler analysis on data collected by the dual-beam configuration of the National Aeronautics and Space Administration (NASA) ER-2 Doppler radar (EDOP). H96 neglected the effects of variations in roll and drift angle so that the Doppler velocities could be expressed in terms of the along-track and vertical velocity components only. This assumption was justified based on the flight characteristics of the ER-2, which operates in the relatively quiescent stratosphere. Aircraft operating in the troposphere experience much greater winds and turbulence levels, which result in larger drift angles and variations in aircraft attitude. Thus, the two-dimensional assumption employed by H96 is questionable for dual-beam radars deployed on aircraft operating in the troposphere.

The Wyoming Cloud Radar (WCR) is a 95-GHz Doppler radar developed for investigating cloud and precipitation processes. Key characteristics of the WCR are listed in Table 1. (Additional information on the WCR can be found online at http://www-das.uwyo.edu/wcr.) Dual-beam configurations of the WCR have been deployed during several field campaigns, including the second Dynamics and Chemistry of Marine Stratus (DYCOMS-II) project (Stevens et al. 2003) and the International H₂O Project (IHOP 2002) (Weckwerth et al. 2004). The single-plane dual-Doppler (SPDD) analysis presented in this paper has been applied to

Doppler velocities data collected during these projects and has proven useful for investigating finescale kinematic structure under a wide range of conditions. The SPDD framework presented in this paper uses an a priori velocity estimate in addition to the two Doppler velocity measurements to form a three-dimensional velocity estimate. While the velocity fields retrieved using the three-dimensional framework will, in many cases, differ only slightly from what would be obtained using the two-dimensional (H96) analysis, the solution described here offers two significant advantages over the two-dimensional solution: (i) accuracy, by explicitly including an a priori velocity estimate so that nonrandom errors resulting from cross-plane contributions to the Doppler velocities can be reduced (if no a priori velocity estimate is available, or if the cross-plane velocity component is zero it can be shown that this technique gives the same result as the two-dimensional analysis); (ii) a framework for assessing the contribution from the cross-plane velocity to the retrieved velocity components and for assessing the contribution from fluctuations about the mean cross-plane velocity to the variance of the retrieved velocity components.

The analysis presented in this paper has been applied to radar configurations using two beams lying within a vertical plane, referred to as vertical-plane dual-Doppler (VPDD) analysis, and to configurations using two beams lying in a horizontal plane, referred to as horizontal-plane dual-Doppler (HPDD) analysis. The discussion in this paper is conducted in terms of the VPDD analysis. However, unless noted otherwise, it is also applicable to the HPDD analysis.

The data processing and interpolation steps that take place prior to the SPDD analysis are described in section 2. The dual-Doppler analysis itself is presented in section 3. The errors that affect the retrieved velocity fields are examined in section 4. Examples of velocity fields obtained using the SPDD analysis are presented in section 5. These examples include vertical-plane observations of marine stratocumulus obtained during the DYCOMS-II project (section 5a), clear-air observations of a shallow cold front obtained during IHOP 2002 (section 5b), and horizontal-plane observations of cumulus congestus (section 5c).

## 2. Data processing and interpolation

Several steps must be completed prior to the dual-Doppler analysis: the orientations of the radar beams and the position of the range gates are projected into an earth-relative coordinate system, the Doppler velocities are corrected for platform motion, and the radar data are interpolated onto a common grid. The transforma-
<table><caption>TABLE 1. Selected characteristics of the WCR</caption><thead><tr><th>Characteristic</th><th>Frequency (Wavelength)<br>94.92 GHz<br>(3.16 mm)</th><th>Pulse-Length<br>225 ns - 34 m<br>(500 ns) - 75 m</th><th>Range-gate spacing<br>30 m<br>(15 m)</th><th>PRF<br>5 - 20<br>kHz</th><th>Unambiguous velocity width<br>7.9 - 31.6 m s<sup>-1</sup></th></tr></thead><tbody><tr><td>Direct-beam (slanted-beam)</td><td colspan="5" style="text-align:center;">Currently-supported dual-beam configurations</td></tr><tr><td>Airspeed</td><td>UWKA (VPDD)</td><td>UWKA (HPDD)</td><td>NSF/NCAR C-130</td><td></td><td></td></tr><tr><td>Time-lag (per km direct range)</td><td>7.1 s</td><td>6.4 s</td><td>115 m s<sup>-1</sup></td><td>5.0 s</td><td></td></tr><tr><td>Beamwidth</td><td>0.5° (0.6°)</td><td>0.7° (0.7°)</td><td>0.7° (0.7°)</td><td></td><td></td></tr><tr><td>Width @ 1km*</td><td>9 m (12.5 m)</td><td>12 m (14 m)</td><td>12 m (14 m)</td><td></td><td></td></tr><tr><td>Minimum detectible signal**</td><td>-25.4 dBZ (-20.6 dBZ)</td><td>-26.6 dBZ (-19.9 dBZ)</td><td>-26.6 dBZ (-23.9 dBZ)</td><td></td><td></td></tr><tr><td>0dB SNR*</td><td>-14.5 dBZ (-9.7 dBZ)</td><td>-15.7 dBZ (-9.0 dBZ)</td><td>-15.7 dBZ (-13.0 dBZ)</td><td></td><td></td></tr><tr><td colspan="6" style="text-align:center;">* at 1 km direct range from the aircraft.</td></tr><tr><td colspan="6" style="text-align:center;">**Based on 2 standard deviations of the noise and ~100 pulses averaged.</td></tr></tbody></table>

tion from aircraft to earth-relative coordinates and the subsequent correction of the Doppler velocities for platform motion have been described by Lee et al. (1994) and Leon and Vali (1998, hereafter LV98) among others. Different conventions for describing the orientation of the radar beams have been employed in previous studies (Lee et al. 1994; H96; LV98). In this paper the notation of LV98, in which the orientation of each of the radar beams is described by a unit vector, $\hat{b}$, referred to as the beam vector and aligned along the center of the radar beam, is used. Using this notation, the Doppler velocity is $d = \hat{b} \cdot V_p$, where $V_p$ is the three-dimensional velocity of the scatterers.

For straight or nearly straight flight legs, a vertical (horizontal) grid is sufficient for conducting the SPDD analysis, while for more complex flight patterns (such as the 60-km-diameter circles flown during DYCOMS-II) a curved grid—one that is warped to conform to the

aircraft flight track and the projections of the radar beams—must be created and the coordinate system (the along-track and cross-track directions) redefined for each column of the grid. The exact form of the grid affects the interpretation of the results but, since the data from each radar beam lie in a single curtain, the cross-plane position of the grid with respect to the data points does not affect the velocity values obtained. Based on the characteristics of the WCR and the recommendation of Koch et al. (1983), a gridpoint spacing of 15 m (~1/2 of the typical WCR pulse length) was chosen.

Because of the time lag between observations by the direct and slanted beams (5–7 s km⁻¹ distance between the aircraft and the sample volume), it is necessary to adjust the relative positions of the data from the nadir and slanted beams in order to compensate for advection (the time lag also allows some evolution of the
velocity field to occur, a source of error addressed in section 4). An advection scheme can be implemented either by allowing the entire grid to drift at a specified velocity (with respect to a reference time) or by adjusting the positions of data points with respect to the time assigned to each column of the grid (the time that the aircraft crosses that column). The former is advantageous when comparing results from multiple legs (where all positions are corrected with respect to a common reference time) while the latter is preferable when comparing to ground-based or ground-referenced observations.

The effects of the grid advection velocity ($A_g$) merit further discussion: the cross-plane component of $A_g$ changes both the cross-plane displacement and the orientation of the analysis gird. This effect can be quantified by introducing the effective drift angle—the angle between the aircraft heading and the track angle with respect to the selected (ground or air relative) coordinate system—given by

$$D_{\text{eff}} = \tan^{-1} \left( \frac{V_{\text{acEW}} - A_{\text{EW}}}{V_{\text{acNS}} - A_{\text{NS}}} \right) - \psi, \quad (1)$$

where $V_{acEW}$, $V_{acNS}$ are the east-west and north-south components of the ground-relative aircraft velocity; $A_{EW}$, $A_{NS}$ are the components of $A_g$; and $\psi$ is the aircraft heading. The traditional definition of the drift angle ($D_{gr}$), which implicitly assumes a ground-relative coordinate system, is obtained for $A_{EW} = A_{NS} = 0$. Generally, $D_{eff}$ is smaller than $D_{gr}$ because of the small sideslip ($\beta$) angle of the aircraft and the relative proximity of the aircraft to the region of interest (generally <2 km). However, in cases with large shear or a long range to the region of interest, $D_{eff}$ may be larger than $D_{gr}$ or even have the opposite sign. A time series of $D_{gr}$ and $D_{eff}$ for a flight leg from the DYCOMS-II project is shown in Fig. 1.

Interpolation of the WCR data onto the analysis grid was accomplished using a (single pass) Barnes weighting scheme (Barnes 1964, 1973; Koch et al. 1983) with the smoothing parameter ($r_b$) set to (14 m)$^2$, with an arbitrary cutoff for weights less than (1%) of the maximum weight (corresponding to a displacement of approximately 20 m from the grid point). Since high spatial resolution is one of the defining features of airborne cloud radars, a smaller value of the smoothing parameter than might have been chosen based solely on the transmitted pulse length is used. The analysis of Trapp and Doswell (2000) suggests that, for the value of the smoothing parameter used, (14 m)$^2$, wavelengths less than 40 m will be filtered almost completely (a theoretical spectral response < 0.25), while wavelengths between 40 and 85 m will be retained in the

<figure><img src="image_2.png" /><figcaption>FIG. 1. Ground-relative ($D_{gr}$) and effective ($D_{eff}$) drift angles and sideslip ($\beta$) angle for a single circle above the boundary layer during DYCOMS-II flight RF07; $D_{gr}$ is determined by the local horizontal wind components and is significantly larger than $D_{eff}$, which is due to the change in horizontal wind between the flight level and the region of interest. The sideslip ($\beta$) angle is nearly always <1°.</figcaption></figure>

interpolated data but damped to varying degrees (spectral response values between 0.25 and 0.75), and wavelengths longer than 120 m will be minimally damped (spectral response $\ge$ 0.95).

This analysis provides only a lower limit on the resolution and takes into account only the filtering due to the interpolation scheme. Other limitations, including filtering due to the pulse volume, noise in the Doppler velocities, and temporal evolution, may further limit the ability of the SPDD analysis to resolve small-scale features in the velocity field.

### 3. Dual-Doppler analysis

The three-dimensional solution presented in this paper is an extension of the two-dimensional solution presented in H96. As such, examination of the two-dimensional solution is useful for understanding the general behavior of the three-dimensional analysis presented in this paper.

#### a. Dual-Doppler analysis in two dimensions

The two-dimensional (H96) solution, obtained by projecting the beam orientations into track-relative coordinates and neglecting the cross-track components of the beam orientation, adapted to the LV98 notation is

$$V_{at} = \frac{b_{nz}d_s - b_{sz}d_n}{b_{nz}b_{sat} - b_{sz}b_{nat}} \quad (2)$$

and

$$V_z = \frac{b_{sat}d_n - b_{nat}d_s}{b_{nz}b_{sat} - b_{sz}b_{nat}}, \quad (3)$$
where $V_{at}$ and $V_z$ are the along-track and vertical components of the velocity field; $d_s$ and $d_n$ are the nadir- and slanted-beam Doppler velocities; $b_{nat}$ and $b_{sat}$ are the along-track projections of the nadir and slanted beams; and $b_{nz}$, $b_{sz}$ are the vertical projections of the beams. The nadir beams are oriented to compensate for the average angle of attack so that, on average, they are very nearly vertical. Therefore, $b_{nat} \equiv 0$, $b_{nz} \equiv -1$, $b_{sat} \equiv \sin\delta$, and $b_{sz} \equiv -\cos\delta$, where $\delta$ is the angle between beams. Substituting these approximations into (2) and (3) gives

$$V_{at} \equiv \tan(\delta)^{-1}d_n - \sin(\delta)^{-1}d_s \quad (4)$$

and

$$V_z \equiv -d_n. \quad (5)$$

Thus, $V_z$ is determined almost completely by $d_n$, while $V_x$ contains contributions from both $d_n$ and $d_s$ scaled by the factors $\tan(\delta)^{-1}$ and $\sin(\delta)^{-1}$, respectively. For $\delta = 30^\circ$, these factors are 1.7 for the nadir-beam and 2 for the slanted-beam Doppler velocities.

Since both the nadir- and slanted-beam Doppler velocities are measurements of the particle velocity, the particle fall speed does not introduce any errors into the estimated horizontal velocity component (H96). Therefore, no attempt is made to separate the particle fall speed from the terminal air velocity prior to the dual-Doppler analysis. Nor, as a matter of course, is $V_z$ adjusted to compensate for the fall speed of the scatterers subsequent to the SPDD analysis. The use of a reflectivity-fall speed ($Z-V_t$) relationship is avoided for three reasons: $Z-V_t$ relations are not well established for 95 GHz; the entire particle fall speed is included in $V_z$ (unlike conventional dual-Doppler analyses where $Z-V_t$ relations are generally applied for elevation angles $<20^\circ$); and finally, correlations between vertical air motion and $V_t$, such as those reported by Vali et al. (1998), could lead to nonrandom errors in the “corrected” $V_z$.

### b. Three-dimensional dual-Doppler analysis and the use of a priori velocity estimates

The attitude variations experienced by the National Science Foundation (NSF)/National Center for Atmospheric Research (NCAR) C-130 and University of Wyoming King Air (UWKA) research aircraft, listed in Table 2, are several times greater than the 0.25° standard deviation in roll with excursions to ±2° of the NASA ER-2 (H96). These variations, and the resulting need to estimate the contributions from the cross-plane velocity component and the variability about the cross-plane velocity estimate, motivated the development of the three-dimensional framework presented in this section.

<table><caption>TABLE 2. Aircraft attitude variations (range and standard deviation) for cases presented in section 5.</caption><thead><tr><th>Aircraft-project</th><th>Roll range (σ)</th><th>Pitch range (σ)</th><th>Heading range (σ)</th></tr></thead><tbody><tr><td>C-130-DYCOMS-II</td><td>±5° (1.1°)</td><td>±3° (0.3°)</td><td>±5° (2.0°)</td></tr><tr><td>UWKA-IHOP 2002</td><td>±5° (1.5°)</td><td>±2° (0.5°)</td><td>±4° (1.5°)</td></tr><tr><td>UWKA-HBDD-TEST</td><td>±4° (1.9°)</td><td>±1.2° (0.5°)</td><td>±2.4° (1.4°)</td></tr></tbody></table>

For flight legs parallel to the wind, or if no a priori velocity estimate is available, it can be shown that the three-dimensional solution presented in this section gives the same result as would be obtained using a least squares approach to the two-dimensional solution.

A weighted series of Doppler velocities can be expressed in matrix form as

$$\mathbf{d}_w^T = \mathbf{B}_w \mathbf{V}_p, \quad (6)$$

where $\mathbf{d}_w = [d_{n1} w_{n1} d_{n2} w_{n2} \cdots d_{nl} w_{nl} d_{s1} w_{s1} \cdots d_{sm} w_{sm}]$ is a weighted array of Doppler velocity measurements ($l$ and $m$ are the number of data points for the nadir and slanted beams, respectively), $\mathbf{V}_p$ is the particle velocity vector, and $\mathbf{B}_w$ is a weighted array of beam vectors given by

$$\mathbf{B}_w = \begin{bmatrix} b_{nx_1} w_{n_1} & b_{ny_1} w_{n_1} & b_{nz_1} w_{n_1} \\ b_{nx_2} w_{n_2} & b_{ny_2} w_{n_2} & b_{nz_2} w_{n_2} \\ \vdots & \vdots & \vdots \\ b_{nx_l} w_{n_l} & b_{ny_l} w_{n_l} & b_{nz_l} w_{n_l} \\ b_{sx_1} w_{s_1} & b_{sy_1} w_{s_1} & b_{sz_1} w_{s_1} \\ \vdots & \vdots & \vdots \\ b_{sx_m} w_{s_m} & b_{sy_m} w_{s_m} & b_{sz_m} w_{s_m} \end{bmatrix}. \quad (7)$$

The inversion of (6) is mixed-determined—two components are overdetermined due to the inclusion of multiple observations from each beam, while a third velocity component cannot be resolved from the Doppler velocities. A singular value decomposition (SVD)-based solution (described in appendix A), in which the Doppler velocity measurements are supplemented by an a priori velocity estimate, is used to invert (6). Applying the SVD formulation for the generalized inverse (Menke 1989), the three-dimensional estimate of the particle velocities is

$$\mathbf{V}_{3D} = \mathbf{B}_w^{-g} \mathbf{d}_w + [\mathbf{I} - \mathbf{R}] \mathbf{V}_{ap}, \quad (8)$$

where $\mathbf{B}_w^{-g}$ is the generalized inverse of $\mathbf{B}_w$ and is defined in appendix A; $\mathbf{V}_{ap}$ is the three-dimensional a priori velocity estimate; and $[\mathbf{I} - \mathbf{R}]$, referred to as the model null space, describes the velocity component(s)
not resolved by the Doppler velocity measurements (Menke 1989). This solution can also be used to correct single-beam Doppler velocity data for the contamination by velocity components normal to the average beam orientation. Since $[I - R]V_{ap}$ is an additive term in (8), $V_{ap}$ may be altered without recomputing $B_w^{-g}$. The elements of $[I - R]$ corresponding to the vertical coordinate are small (<0.01) for the VPDD analysis; therefore, the vertical component of $V_{ap}$ can be omitted entirely. Conversely, for the HPDD analysis, the vertical component of $V_{ap}$ may contribute significantly to $V_{3D}$.

Individual velocity components, presumably those defining the idealized observation plane, are extracted from $V_{3D}$ by applying

$$V_c = \hat{c} \cdot V_{3D}, \qquad (9)$$

where $\hat{c}$ is the desired component. The selection of velocity components displaced significantly from the idealized observation plane should be avoided since those components would contain a larger contribution from $V_{ap}$.

## 4. Error analysis

The velocity components retrieved using the SPDD analysis are affected by several different types of errors: biases in the aircraft motion correction, random noise in the Doppler velocity estimates, displacements between then nadir and slanted beams (and between the beams and the observation plane), and errors in the a priori velocity estimate. Where possible, the approximate form of the two-dimensional solution, (4) and (5), is used to examine the general behavior of the errors, while the three-dimensional solution is used to assess the errors for specific cases. Errors arising from incorrect aircraft motion correction are believed to result primarily in biases in the corrected Doppler velocities and are examined using (4) and (5) only.

Assuming that errors in the nadir- and slanted-beam Doppler velocities are independent, the variances of $V_{at}$ and $V_z$ obtained using (4) and (5) are

$$\sigma^2(V_{at}) \equiv \tan(\delta)^{-2}\sigma^2(d_n) + \sin(\delta)^{-2}\sigma^2(d_s) \quad (10)$$

and

$$\sigma^2(V_z) \equiv \sigma^2(d_n), \qquad (11)$$

where $\sigma^2(d_n)$ and $\sigma^2(d_s)$ are the variances of the nadir- and slanted-beam Doppler velocities. For $\delta = 30^\circ$ the factors $\tan(\delta)^{-2}$ and $\sin(\delta)^{-2}$ are 3 and 4, respectively.

Errors in $V_{3D}$ for specific cases can be quantified by forming a covariance matrix, $Cov_{V_{3D}}$, that is a function of the errors in the Doppler velocities and a priori ve-

locity estimate combined with the terms in (8) (Menke 1989):

$$Cov_{V_{3D}} = B_w^{-g}Cov_{D_w}B_w^{-gT} + [I - R]Cov_{V_{ap}}[I - R]^T, \qquad (12)$$

where $Cov_{D_w}$ is the covariance matrix corresponding to the weighted Doppler velocity measurements, and $Cov_{V_{ap}}$ is the covariance matrix for the a priori velocity estimate. The variance of individual velocity components extracted from $V_{3D}$ is $\sigma^2(V_c) = \hat{c}Cov_{V_{3D}}\hat{c}^T$. The various error sources are examined separately in the following subsections.

### a. Aircraft motion correction

The need to correct the measured Doppler velocities for platform motion inevitably reduces the accuracy of the resulting ground- or air-relative Doppler velocities. The corrected Doppler velocities are affected by errors in the aircraft velocity estimate as well as errors in the estimated radar beam orientations.

The ground-relative horizontal velocity of the aircraft is computed by combining velocity estimates from the inertial navigation system (INS) with GPS velocity measurements in order to reduce errors due to the Schuler oscillation. INS and GPS data are combined such that variations in the aircraft velocity above a cut-off frequency are taken from the INS data, while lower-frequency contributions are determined by the GPS. The transition between INS and GPS is typically set to about $10^{-2}$ Hz. The INS measurements of aircraft attitude are also affected by the Schuler oscillation but less strongly than the aircraft velocity/position measurements and are not, as a matter of course, corrected. The vertical velocity of the aircraft is computed by combining the vertical acceleration output by the INS with a pressure-based altitude through the use of a third-order damping scheme (Blanchard 1971). Errors in the aircraft velocity and attitude affect the in situ measured winds and the SPDD-derived velocity components equally.

Unlike errors in the aircraft velocity measurement, errors in the estimated beam orientation, which lead mainly to biases in the Doppler velocities, are magnified in $V_{at}$. To minimize these errors the orientation of the radar beams is computed on a project-by-project basis using a procedure similar in concept to that described by Testud et al. (1995), in which the earth's surface is used as a velocity reference.

Quantitative estimates of the accuracy of the aircraft velocity are not readily available. Informal estimates of the accuracy of the aircraft velocity measurements are 1
and 0.5 m s⁻¹ for the horizontal and vertical velocity components, respectively (A. Rodi 2003, personal communication). A rough estimate of the magnitude of the uncertainty in the SPDD-derived velocity components due to errors in the aircraft velocity/attitude measurements is provided here.

Based on a survey of nine flights during IHOP 2002, the leg-averaged residual Doppler velocity from the earth's surface is generally less than 0.1 m s⁻¹, with leg-averaged residual velocities ranging up to 0.4 m s⁻¹. The residual velocities appear to consist primarily of biases and low-frequency drifts (over a period of several minutes to tens of minutes) and therefore should not significantly impair the utility of the SPDD analysis for resolving smaller-scale velocity fluctuations. The accuracy of the Doppler velocities for the DYCOMS-II case presented in section 5a is likely to be worse than for the other cases since this project was conducted almost exclusively over the ocean (which has proven to be a poor velocity reference) and since the flight durations for the C-130 were much longer than for the UWKA (~9 h versus ~4 h) were conducted—the increased flight duration is likely to lead to increased errors in aircraft attitude due to the Schuler oscillation, which tends to amplify with time. Biases in the Doppler velocities due to the aircraft motion correction for this project are believed to be on the order of 0.5 m s⁻¹. These assessments lead us to believe that the absolute accuracy of $V_{at}$ and $V_z$ [estimated using (4) and (5) and assuming that the biases for the two beams are independent] is better than 0.25 and 0.1 m s⁻¹, respectively, for most cases from the IHOP 2002 and HPDD-TEST projects and 1 and 0.25 m s⁻¹ for the worst cases. For DYCOMS-II we believe that typically these values are better than 1 and 0.5 m s⁻¹, respectively.

The limited evaluation of the errors due to the aircraft velocity correction presented here highlights the need to thoroughly characterize errors in the aircraft motion/attitude estimates and, by extension, errors in the in situ measured winds and Doppler velocities.

b. Variance of the pulse-pair mean Doppler velocity estimate

To date, the SPDD analysis has been applied exclusively to mean Doppler velocities estimated using the pulse-pair operator. While the behavior of the pulse-pair operator has been examined extensively, estimators of the variance of the mean Doppler velocity [e.g., Doviak and Zrnic 1993, their Eq. (6.22a)] depend on factors (spectral width, time to independence) that are not known a priori. Further, the variance estimator derived by Zrnic (1977) is known to perform poorly for narrow spectral widths and low SNR (Doviak and

Zrnic 1993)—conditions frequently encountered by the WCR. Thus, an ad hoc approach based on the velocity difference between adjacent profiles for the same range gate is used (each profile is the result of averaging ~50–150 pulse-pairs, depending on the operating mode of the radar). Adjacent profiles are sufficiently close in time (0.025–0.05 s) and space (~2 to ~6 m) that the velocity difference between profiles is dominated by random noise rather than by turbulent velocity fluctuations. Variances estimated using the interprofile velocity difference for the nadir and trailing beams are shown in Fig. 2. These variance estimates demonstrate that random noise in the Doppler velocity estimates is a fairly minor source of error for SNR > 5 dB (both beams) but increases dramatically for SNR < 5 dB and renders the Doppler velocities effectively useless for SNR < −5 dB.

Random noise in the individual Doppler velocity measurements is reduced significantly by the averaging inherent in the interpolation process. The exact reduction in variance depends on the number of points averaged and the weighting applied to each point. It is convenient to compute the effective number of data points averaged; $n_e$ is the number of uniformly weighted data points that would give the same variance as the nonuniformly weighted sum, defined by

$$n_{\text{eff}} = \left( \sum_{i=0}^{n_{\text{tot}}} w_i^2 \right)^{-1}, \qquad (13)$$

where $n_{tot}$ is the total number of data points included in the interpolation, and $w_i$ is the weight applied to the ith data point. Average $n_{tot}$ and $n_{eff}$ values are listed in Table 3. The expected variances of the retrieved velocity components are shown in Fig. 2.

Given an estimate of the variances of the mean Doppler velocities (e.g., by using an equivalent of Fig. 2 to look up the expected variance as a function of SNR), the variance of $V_{3D}$ due to random noise is given by the first term in (12): $\mathbf{B}_w^{-g}\text{Cov}_{n_w}\mathbf{B}_w^{-g'}$, where

$$\text{Cov}_{n_w} = \begin{bmatrix} w_{n_1}^2 \sigma_1^2 & \cdots & w_{n_l}^2 \sigma_{n_l}^2 & \cdots & w_{s_1}^2 \sigma_{s_1}^2 & \cdots & w_{s_m}^2 \sigma_{s_m}^2 \end{bmatrix} \qquad (14)$$

is the weighted matrix of variances in the Doppler velocities.
<figure><img src="image_3.png" /><figcaption>FIG. 2. Variance of the raw Doppler velocity measurements estimated using the interprofile velocity differences for the cases presented in section 5 and expected variances in the dual-Doppler-determined velocity components estimated using the approximate form of the two-dimensional (a) DYCOMS-II RF07; 56 pulse pairs were averaged per profile. Note that this and subsequent figures take into account the sensitivity deficits (including the increased slant range) for the slanted beams with respect to the direct (nadir- or side-looking) beams. (b) The resulting variance in the vertical and along-track velocity estimates. (c), (d) Variances of the raw Doppler velocities and dual-Doppler-determined velocity components, respectively, for IHOP 24 May 2002 case. (e), (f) Same as (c), (d) for the HPDD-TEST 18 Sep 2001 case.</figcaption></figure>
<table><caption>TABLE 3. Average total and effective number of data points averaged for the cases presented in section 5.</caption><thead><tr><th>Project</th><th>n<sub>tot</sub> nadir* (slanted)</th><th>n<sub>eff</sub> nadir* (slanted)</th></tr></thead><tbody><tr><td>DYCOMS-II</td><td>37 (44)</td><td>15.6 (18.5)</td></tr><tr><td>IHOP 2002</td><td>36 (40.5)</td><td>16.6 (17.8)</td></tr><tr><td>HPDD-TEST**</td><td>21.6 (26.5)</td><td>10.0 (11.4)</td></tr></tbody></table>

* For the HPDD-TEST project the beams are side-looking and forward-slanted.

** n<sub>tot</sub> and n<sub>eff</sub> for the HPDD-TEST project are dramatically lower than for DYCOMS-II or IHOP 2002 due to a longer range-gate spacing (30 vs 15 m).

### c. Quality of fit—Random noise and subgrid-scale velocity fluctuations

In practice, the combined effects of random noise and subgrid-scale velocity fluctuations can be assessed by computing the variance of the individual Doppler velocity measurements about the predicted velocity for each beam:

$$ \sigma_{\text{fit},j}^2 = \frac{1}{n-1} \sum_{i=1}^{n} (d_i - \hat{\mathbf{b}}_{ij} \cdot \mathbf{V}_{3D})^2, \quad (15) $$

where the subscript j indicates the beam used and n is the number of Doppler velocities from each beam; $\sigma_{\text{fit},j}^2$ includes contributions from random noise, subgrid-scale velocity fluctuations, and occasionally from high-frequency errors in the aircraft motion correction, but is unaffected by errors resulting from the displacement between beams and the dominant, low-frequency errors in the aircraft motion estimate. The $\sigma_{\text{fit},j}^2$ values can be substituted into $\text{Cov}_{n_w}$ to estimate the expected variance in $\mathbf{V}_{3D}$ due to the aforementioned terms. For high SNR, subgrid-scale velocity fluctuations (as opposed to random noise) are likely to contribute significantly to $\sigma_{\text{fit},j}^2$. Thus, the assumption that $\text{Cov}_{n_w}$ is a diagonal matrix (i.e., that the individual samples are independent) will result in an underestimate of $\sigma^2(\mathbf{V}_{3D})$ due to the overestimate of the number of independent samples. However, under these circumstances the contribution from $\text{Cov}_{n_w}$ to $\sigma^2(\mathbf{V}_{3D})$ is generally small compared to other error sources, so the assumption of independence and the resulting underestimate of the contribution to $\sigma^2(\mathbf{V}_{3D})$ is of little consequence. For low SNR (<-5dB where the variance of the Doppler velocity estimates is limited by the unambiguous velocity width), $\sigma_{\text{fit},j}^2$ once again results in an underestimate of the random noise contribution to $\sigma^2(\mathbf{V}_{3D})$. In these circumstances the Doppler velocity measurements contain no information and $\mathbf{V}_{3D}$ should be thresholded. Since $\sigma_{\text{fit},j}^2$ and the resulting contribution to $\sigma^2(\mathbf{V}_{3D})$ can be computed on a grid point-by-grid point basis without recourse to out-

side information, it provides a convenient method for estimating noise in the Doppler velocities as part of routine application of the SPDD analysis.

The variance of the Doppler velocities about the predicted value normalized by the variance expected when no signal is present (in which case the Doppler velocity estimates will be uniformly distributed over the unambiguous velocity width giving a variance of $vw_j^2/12$) has proven useful for thresholding the velocity and reflectivity fields:

$$ vnr_j = \frac{12\sigma_{\text{fit},j}^2}{vw_j^2}, \quad (16) $$

where $vw_j$ is the unambiguous velocity width for the jth beam. Since the variance of the mean Doppler velocity estimates increases rapidly with decreasing SNR for SNR < -5 dB, the effect of thresholding is relatively insensitive to the value of $vnr_j$ taken as the cutoff value. A cutoff value of 0.4-0.7 (on $vnr_{\text{nadir}}$ for the vertical velocity component and on the larger of $vnr_{\text{nadir}}$ and $vnr_{\text{slanted}}$ for the along-track velocity) has generally been used.

### d. Displacements between beams

Underlying any multiple-Doppler analysis is the assumption that the radar beams sample the same velocity field. Spatial and temporal displacements between the nadir- and slanted-beam observations (and between the observations and the idealized observation plane) inevitably result in diminished correlations between the velocity fields sampled by each beam, and therefore in errors in the retrieved velocity fields.

Cross-plane displacements between beams result from variations in aircraft attitude (heading and roll for the VPDD analysis, pitch and roll for the HPDD analysis) and $D_{\text{eff}}$ plus some uncertainty due to ambiguity in $\mathbf{A}_g$, while cross-plane displacements between the radar beams and the idealized observation plane result from departures of roll from level. The expected along-track displacement between beams increases linearly with displacement from the aircraft due to the increased time lag between observations. Cross-plane displacements between the radar beams and the idealized observation plane also increase more or less linearly with increasing displacement, while the cross-plane displacement between beams increases like the displacement squared due to increased changes in aircraft attitude that can occur in the increased interval between nadir- and slanted-beam samples—an effect magnified by the displacement to the sample volume.

The distinction between errors resulting from displacements between beams versus those due to displacements between the radar beams and the idealized
observation plane can be summed up in terms of two questions: To what degree are the estimated velocity components representative of the velocity field at any location, and to what degree are the estimated velocity components representative of the velocity field at the nominal location of the grid point? While the former clearly results in errors in $V_{at}$ ($V_z$ is determined almost exclusively by the nadir beam and is therefore accurate for the location of the nadir beam), the effect of the latter is far more ambiguous and must be considered in the context of how the retrieved velocity field will be analyzed.

The velocity field at a location $\mathbf{x}$ can be expressed in terms of the velocity field at a reference location ($\mathbf{x}_r$) and an error term, $\varepsilon(\mathbf{r})$, that depends on the displacement ($\mathbf{r}$) from the reference:

$$u_i(\mathbf{x}) = u_i(\mathbf{x}_r) + \varepsilon(\mathbf{r}). \quad (17)$$

The effects of displacements on the retrieved velocities can be assessed by computing the variance of $\varepsilon(\mathbf{r})$, which is given by the structure function for a displacement $\mathbf{r}$:

$$C_{ii} = [u_i(\mathbf{x}_r + \mathbf{r}) - u_i(\mathbf{x}_r)]^2. \quad (18)$$

A significant complication is that the radar velocities are not point measurements but averages over the pulse volume that are then filtered again by the interpolation scheme. The effects of the pulse-volume filtering on turbulence parameters derived from the Doppler velocity spectra were examined by Srivastava and Atlas (1974), who found that the effect of the vertical averaging was small when the vertical extent of the pulse was comparable to the horizontal extent of the pulse and negligible when the vertical extent of the pulse was small compared to the horizontal extent. This result suggests that, despite the anisotropic filtering imposed by the interpolation scheme (which acts only in the vertical and along-track directions), cross-plane displacements should be comparable to along-track displacements (except for the additional factor of 4/3 that is obtained for lateral versus longitudinal displacements if the structure function is assumed to follow a 2/3 power law).

Since $V_z$ is determined almost entirely by the nadir-beam Doppler velocity measurements, the mean location of the nadir-beam Doppler velocities is taken as the reference location. Assuming, as in (4) and (5), that the nadir beam is effectively vertical, while the slanted beam is displaced from the vertical by the angle between beams ($\delta$), the slanted-beam Doppler velocity is

$$d_s = \cos(\delta)V_z + \sin(\delta)V_{at}, \quad (19)$$

and the variance due to the displacement ($\mathbf{r}$) from the reference location is

$$\sigma_{\text{disp}}^2(d_s) = \cos(\delta)^2 C_{zz}(r) + \sin(\delta)^2 C_{aa}(r), \quad (20)$$

where $C_{zz}(r)$ is the lateral structure function for the $V_z$, and $C_{aa}(r)$ is the structure function for $V_{at}$ and is either longitudinal or lateral, depending on whether the displacement is along or across track. Substituting $\sigma_{\text{disp}}^2(d_s)$ into (10) gives

$$\sigma_{\text{disp}}^2(V_{at}) = \tan(\delta)^{-2} C_{zz}(r) + C_{aa}(r). \quad (21)$$

Structure functions computed from the SPDD-derived velocity components are shown in Fig. 3. The structure functions appear to be better approximated by a simple linear model than by the 2/3 power law expected for displacements within the inertial subrange. The constant of proportionality is $\sim 10^{-2} \text{ m s}^{-2}$ for the DYCOMS-II flight RF07 case. Thus, for displacements greater than 10–15 m, this term rapidly becomes the dominant source of error in $V_{at}$.

The results of J. Mann (1996, personal communication), who evaluated the effect of vertical averaging on the eddy-dissipation estimates derived from lidar data, suggest that for displacements comparable to, or less than, the pulse length the structure function will drop off much more rapidly than the expected 2/3 power law. Thus, at some point it may be necessary to evaluate the effects of the filtering due to the pulse volume and interpolation on the structure functions (and hence on the accuracy of $V_{at}$) for small displacements.

### e. Errors in the a priori velocity estimate

The errors in $V_{3D}$ due to departures in the actual velocity field from $V_{ap}$ are possibly the easiest of the error sources to quantify and are given by $[I - R] \text{Cov}_{ap}[I - R]^T$, where $\text{Cov}_{ap}$ is a $3 \times 3$ matrix whose elements are the variances and covariances of the east-west, north-south, and vertical velocity components, respectively. Because of continuity it is generally not appropriate to assume that $\text{Cov}_{ap}$ is a diagonal matrix. For the VPDD analysis, components of $\text{Cov}_{ap}$ associated with the vertical coordinate can be neglected since this component is well resolved by the Doppler velocity measurements.

Ideally, $V_{ap}$ is an estimate of the mean velocity that is representative at the location of the grid point—in which case $\text{Cov}_{ap}$ is due solely to meteorological velocity fluctuations about $V_{ap}$. The DYCOMS-II results presented in section 5a are one such nearly ideal case: the winds within the boundary layer are very nearly stationary, and there are numerous measurements of
<figure><img src="image_4.png" /><figcaption>FIG. 3. Structure functions for DYCOMS-II RF07. Note that the estimated contribution due to random noise has been subtracted from the structure functions since this term has been treated separately.</figcaption></figure>

the velocity field. For this case, the aircraft in situ measured winds from level flight legs within the boundary layer were used to estimate both $V_{ap}$ and $C_{ov}$.

In cases where the velocity field is not stationary, or where there are not adequate sources for estimating the a priori velocity profile, $C_{ov}$ includes uncertainties in the estimate of $V_{ap}$, in addition to meteorological fluctuations about the mean velocity. The IHOP 2002 results presented in section 5b represent one such case since the winds changed dramatically in the neighborhood of the cold front. In this case, the uncertainties in $V_{ap}$ are much larger than $V_{ap}$ itself and are a significant source of error in the retrieved velocities. The primary advantage of using the three-dimensional solution for this case is not the improvement in the accuracy of $V_{at}$ and $V_z$ but rather the ability to easily assess (albeit somewhat roughly) the uncertainty in $V_{at}$ and $V_z$ components due to contamination from the cross-track velocity component. A list of the symbols used in this and previous sections is given in Table 4.

## 5. Results

Results from three cases, two in a vertical plane and one in a horizontal plane, are presented. The VPDD cases presented here were obtained as part of the DYCOMS-II and IHOP field campaigns, while a case from a project dedicated to testing the dual-horizontal

beam configuration of the WCR is presented for the HPDD analysis.

### a. DYCOMS-II

The results presented in Fig. 3 were obtained during DYCOMS-II flight RF07 [for a description of the DYCOMS-II objectives and preliminary results see Stevens et al. (2003)]. This flight, like the majority of the DYCOMS-II flights, consisted of a series of 60-km-diameter circles flown above, within, and below the cloud layer. Of all the DYCOMS-II flights, RF07 had the most extensive drizzle, with detectible echoes near the ocean surface ≈ 90% of the time, thereby allowing Doppler velocities to be measured below cloud as well as within the cloud layer. Meteorological conditions encountered during RF07 are summarized in Table 5. Nadir-beam reflectivity, vertical particle velocity, and the along-track component of the horizontal (perturbation) velocity field for a 12-km segment from a circle conducted ~225 m above cloud top are shown in Fig. 4. Since the flight legs are curved, the along-track velocity component is locally tangent to the circle and referred to as the tangential velocity component (where positive velocities correspond to clockwise motion around the circle).

The vertical particle velocity (Fig. 4b) is negative almost everywhere, except in small regions near cloud
<table><caption>TABLE 4. List of symbols used in sections 2–4.</caption><thead><tr><th>Symbol</th><th>Use</th></tr></thead><tbody><tr><td><b>A</b><sub>g</sub></td><td>Grid advection velocity</td></tr><tr><td><b>A</b><sup>̂</sup><sub>g</sub> − ew, <b>A</b><sub>g</sub> − ns</td><td>E–W and N–S components of grid advection velocity</td></tr><tr><td><b>ê</b></td><td>Beam vector (unit vector along center of radar beam)</td></tr><tr><td><b>B</b><sub>w</sub></td><td>Weighted array of beam vectors</td></tr><tr><td><b>B</b><sup>−g</sup></td><td>Generalized inverse of <b>B</b><sub>w</sub></td></tr><tr><td>Cov<sub>V3D</sub></td><td>Covariance matrix for <b>V</b><sub>3D</sub></td></tr><tr><td>Cov<sub>D<sub>w</sub></sub></td><td>Covariance matrix for the Doppler velocity measurements</td></tr><tr><td>Cov<sub>n<sub>w</sub></sub></td><td>Covariance matrix due to random noise in the weighted Doppler velocity estimates</td></tr><tr><td>Cov<sub>s<sub>w</sub></sub></td><td>Covariance matrix due to spatial displacements in the weighted Doppler velocities</td></tr><tr><td>Cov<sub>V<sub>ap</sub></sub></td><td>Covariance matrix for the a priori velocity estimate</td></tr><tr><td>D<sub>gr</sub></td><td>Drift angle (ground relative)</td></tr><tr><td>D<sub>eff</sub></td><td>Effective drift angle (advecting frame relative)</td></tr><tr><td>d<sub>n</sub>, d<sub>s</sub></td><td>Nadir- and slanted-beam Doppler velocities</td></tr><tr><td><b>d</b><sub>w</sub><b>d</b><sub>w</sub></td><td>Weighted array of Doppler velocities</td></tr><tr><td><b>h</b><sub>res</sub>, <b>v̂</b><sub>res</sub>, <b>x̂</b><sub>un</sub></td><td>Unit vectors corresponding to the resolved horizontal, (near) vertical, and the unresolved (cross plane) component.</td></tr><tr><td>V<sub>aex</sub>, V<sub>acy</sub></td><td>E–W and N–S components of aircraft velocity</td></tr><tr><td><b>V</b><sub>ap</sub><b>V</b><sub>ap</sub></td><td>A priori velocity estimate</td></tr><tr><td><b>V</b><sub>3D</sub></td><td>Three-dimensional velocity estimate</td></tr><tr><td><b>V</b><sub>p</sub></td><td>Three-dimensional particle velocity</td></tr><tr><td><b>V</b><sub>at</sub>, <b>V</b><sub>z</sub></td><td>Along-track and vertical components of retrieved velocity</td></tr><tr><td>V<sub>g</sub>, U<sub>g</sub></td><td>E–W and N–S components of grid advection velocity</td></tr><tr><td>V<sub>h</sub></td><td>Resolved component of horizontal wind.</td></tr><tr><td>w<sub>i</sub></td><td>Weighting applied to the <i>i</i>th data point</td></tr></tbody></table>

top due to the contribution from the terminal velocity of the drizzle drops. The tangential velocity field (Fig. 4c) shows an alternating pattern of convergence (velocity increasing from left to right) and divergence (velocity decreasing from left to right) at cloud top coupled with a similar pattern, 180° out of phase, below cloud. The circulations implied by the convergence/divergence pattern have a horizontal scale of ~4 km, about 5 times the boundary layer depth. No corresponding pattern is apparent in the vertical particle velocities. This is due in part to masking of updrafts by drizzle [increased reflectivity-weighted fall velocity in regions of upward air motion—a feature noted by Vali et al. (1998) and Lothon et al. (2005)].

The contributions from the a priori velocity estimate to the vertical and tangential velocity components are shown in Figs. 5d and 5e. The a priori contribution to $V_z$ primarily reflects variations in the aircraft roll and is nearly constant vertically, while the a priori contribu-

tion to $V_{at}$ shows significant structure in both the vertical and along-track dimensions.

The contributions from $Cov_{ap}$ to the expected variance of the vertical and tangential velocity components are shown in Fig. 5. This contribution is estimated using the variances of the in situ measured winds from flight legs within the boundary layer (mean in situ wind components and standard deviations for level flight legs within the boundary layer are shown in Figs. 5g and 5h). The expected variance in $V_{at}$ and $V_z$ has a distinct minimum near the middle of the boundary layer due to the minimum in the variance of the horizontal velocities there.

The maximum cross-plane displacement between beams near cloud top is less than 2 m but increases rapidly for lower altitudes with maximum displacements of 35 m near the surface, while the mean cross-plane displacements between beams increases from 1.5 m near cloud top to 9.5 m near the surface. The mean

<table><caption>TABLE 5. Selected meteorological conditions for DYCOMS-II RF07.</caption><thead><tr><th>Cloud base (cloud top)</th><th>Boundary wind speed (direction)</th><th>Max liquid water content*</th><th><b>A</b><sub>g</sub> (E–W, N–S)</th><th>Rain rate** (mm day<sup>−1</sup>)</th></tr></thead><tbody><tr><td>375 m (775 m)</td><td>7 m s<sup>−1</sup> (330°)</td><td>0.8 gm<sup>−3</sup></td><td>6, −3.5 m s<sup>−1</sup></td><td>0.6</td></tr></tbody></table>

<footnote>* From aircraft in situ measurements.</footnote>

<footnote>** From VanZanten et al. (2005).</footnote>
<figure><img src="image_5.png" /><figcaption>FIG. 4. (a) Nadir-beam reflectivity, (b) vertical particle velocity, and (c) horizontal (tangential) perturbation velocity. Positive horizontal velocities correspond to (relative) motion from left to right. Increasing (decreasing) velocities from left to right suggest divergence (convergence). (d) Contribution from the a priori velocity estimate to $V_z$. (e) Contribution from the a priori velocity estimate to $V_{at}$. The along-track distance is measured in a clockwise direction from the northernmost point on the circle. (f) Vertically averaged contribution from the a priori velocity to $V_z$ for the complete circle. The thick line segment indicates the section of the circle shown in (a)–(e). (g), (h) The E–W and N–S components of the aircraft-measured winds. Error bars indicate the variability ($\pm 1\sigma$) of the wind components.</figcaption></figure>

cross-plane displacements are slightly larger than the uncertainties in position (due to misadvection), which are estimated by combining the standard deviations of the aircraft-measured winds (~1 m s⁻¹) with the time lag between the nadir- and slanted-beam samples (~5 s near the ocean surface). The estimated variance (estimated using a linear model for the structure function with a constant of 0.01 m s⁻²). The emergence of this

term as the largest source of error is both unexpected and troubling—it seems likely that the model of the structure function that is currently being used may overestimate this error for small displacements due to not properly accounting for the effects of filtering.

DYCOMS-II flight RF07 provides a nearly ideal case for the SPDD analysis: the range to the region of interest is small (<1 km), the SNR is generally
<figure><img src="image_6.png" /><figcaption>FIG. 5. Variance contributions due to various sources for DYCOMS-II RF07: (a) $\sigma^2(V_z)$ due to random noise estimated using $\sigma_{\text{fit}}^2$; (b) $\sigma^2(V_z)$ due to Cov$_{ap}$ (note that the minimum in the middle of the boundary layer is due to the corresponding minima in the variance of the aircraft-measured horizontal wind components); (c) $\sigma^2(V_{\text{at}})$ due to random noise estimated using $\sigma_{\text{fit}}^2$; (d) $\sigma^2(V_{\text{at}})$ due to cross-plane displacements between beams and the expected uncertainty due to misadvection of the velocity field; and (e) $\sigma^2(V_{\text{at}})$ due to Cov$_{ap}$.</figcaption></figure>

>10 dB, and the winds are nearly stationary within the boundary layer, with fluctuations that are small compared to the mean. These advantages are partially offset by the fact that the velocity fluctuations of interest are small.

b. IHOP results

The WCR, onboard the UWKA, took part in the IHOP 2002 project in the Southern Great Plains during May and June 2002 (Weckwerth et al. 2004). Russell and Wilson (1997) argued that clear-air echoes at X band were dominated by scattering by insects, a conclusion that is essentially inevitable for the 95-GHz (W band) WCR, where contributions from Bragg scattering are smaller than at X band. The scatterers appear to descend, with a mean speed of ~0.55 m s⁻¹, with respect to the air in which they are embedded (Geerts

and Miao 2005). Geerts and Miao (2005) found that the difference between the vertical air motion and the Doppler velocity increased with increasing vertical velocity. Thus, the vertical Doppler velocities shown in this section most likely underestimate the vertical air velocity.

During the afternoon of 24 May 2002, a shallow cold front moving through the Texas Panhandle near Shamrock, Texas, was intercepted by an array of mobile ground-based and airborne instruments, including the UWKA (Weckwerth et al. 2004). The UWKA made 10 transects across the cold front between 1907 and 2114 UTC (Geerts and Leon 2003). Radar reflectivity, vertical, and horizontal velocity components for a south-east-northwest (SE-NW) transect between 2006:30 and 2011:30 UTC are shown in Fig. 6. Horizontal velocities to the left (SE) of the cold front are left to right (SE-NW), while velocities behind the cold front (to the
<figure><img src="image_7.png" /><figcaption>FIG. 6. Radar observations from a transect across a shallow cold front observed on 24 May 2002 during the IHOP 2002 project: (a) nadir-beam reflectivity, (b) vertical velocity, (c) horizontal (along track) velocity component, positive horizontal velocities correspond to motion from left to right.</figcaption></figure>

right) are right to left (NW–SE) with a dramatic shift ($\approx$20 m s$^{-1}$) across the cold front. Vertical velocities in excess of + 5 m s$^{-1}$ are observed where the left-to-right (SE–NW) flow is forced upward over the leading edge of the front.

The variances in $V_z$ and $V_{at}$ due to random noise in the Doppler velocity measurements and due to variability about the a priori velocity estimate are shown in Fig. 7. We have not been able to quantitatively estimate the effect of cross-plane displacements between beams for this case due to the limited amount of radar data for computing the structure functions and since displacements parallel to the leading edge of the front likely effect the velocity field differently than displacements perpendicular to the front. The contributions shown in Fig. 7 demonstrate that random noise is the primary source of errors for this case. The noisiness of the Doppler velocities limits the utility of the SPDD analysis to the relatively high reflectivity region surrounding the leading edge of the front. The contributions from $Cov_{ap}$ are surprisingly small—the result of the small

variability of the velocity component parallel to the leading edge of the front.

This is a difficult case for applying the SPDD analysis: SNR is generally low, rendering the Doppler velocities extremely noisy; the winds are nearly discontinuous in the neighborhood of the front, making it difficult to estimate the a priori velocities; and the maximum range to the region of interest is relatively large (2 km). Nonetheless, the jump in the winds across the front is sufficiently large and there is enough signal in the region immediately surrounding the leading edge of the front that the defining features of the front are clearly evident.

### c. Horizontal-plane dual-Doppler results

The observations presented in this section were made during a test flight dedicated to evaluating the dual-horizonatal beam configuration of the WCR. At around 1945 UTC the UWKA made a single penetration through the main tower of a cumulus congestus cluster encountered SSE of Cheyenne, Wyoming. The focus of
<figure><img src="image_8.png" /><figcaption>FIG. 7. Contributions from random noise and $Cov_{ap}$ to the variances of $V_z$ and $V_{at}$. (a) Variance of $V_z$ due to random noise in the Doppler velocity measurements—estimated from $\sigma_{\text{Dop}}^2$. (b) Variance of $V_z$ due to $Cov_{ap}$. (c), (d) Same as (a), (b) for $V_{at}$.</figcaption></figure>

the flight then shifted to a rapidly growing turret on the NE side of the main complex. The UWKA made a series of penetrations through the turret (with the WCR using the side-slanted and upward-looking beams) paired with SW-NE passes to the NNW of the turret (with the WCR using two horizontal beams for the passes and the vertical and side-slanted beams for the penetrations) at a range of altitudes between 5 and 8 km (msl). HPDD results for a SW-NE pass at 5-km altitude are shown in Fig. 8. Variances in the along- and

across-track velocity components due to random noise in the Doppler velocities and due to $Cov_{ap}$ are shown in Fig. 9 (the variance due to displacements between beams has not yet been estimated for this case). An estimate of $Cov_{ap}$ was made based on a subsequent penetration through the cell with the WCR in up-looking mode. Random noise appears to be the most significant error source for this case, although for some parts of the velocity field the contribution from $Cov_{ap}$ appears to be comparable in magnitude.
<figure><img src="image_9.png" /><figcaption>FIG. 8. (a) Side-beam reflectivity for a horizontal slice through a cumulus congestus cloud. (b) Cloud-relative along-track component of the horizontal velocity field. Positive velocities correspond to motion from left to right. (c) Cloud-relative cross-track velocity component. Positive velocities correspond to motions from bottom to top in the image. The aircraft flight track is along the bottom of this image. Note how the edge of the cloud nearest the flight track and the turret at ~6 km along track are being sheared to the right with respect to the rest of the cloud.</figcaption></figure>

**6. Discussion**

The results and accompanying error analysis presented in this paper demonstrate the potential of the SPDD analysis to retrieve high-resolution two-

dimensional velocity fields. The three-dimensional framework presented in this paper, which is simply a generalization of two-dimensional analysis employed by H96, provides an effective way to deal with contamination of the Doppler velocities by velocity compo-
<figure><img src="image_10.png" /><figcaption>FIG. 9. (a) Variance in the along-track velocity component due to random noise in the Doppler velocities. (b) Same as (a) but for the across-track velocity component. (c) Variance in the along-track velocity component due to $Cov_{ap}$. (d) Same as (c) but for the across-track velocity component.</figcaption></figure>

nents perpendicular to the idealized observation plane (and to assess the resulting uncertainty in the retrieved velocity field). However, the need to obtain a priori wind profiles is inconvenient, and velocity fluctuations about the a priori velocity profile are an additional source of error in the retrieved velocity fields, albeit a manageable one under most circumstances.

A number of possible approaches exist to improve the accuracy of the SPDD-derived velocity fields—including the addition of a radar beam slanted perpendicular to the aircraft velocity vector [which would allow the cross-plane velocity to be estimated directly using a velocity–azimuth display (VAD)-like approach]. The most promising option for improving the accuracy of the SPDD-derived velocities would be the addition of a third beam slanted opposite the existing slanted (leading or trailing) beam. The addition of a third beam slanted approximately 30° from the nadir beam would improve the accuracy of the retrieved velocity field by reducing the random noise by factors of ≈3.5 and ≈2.5 for the along-track and vertical velocity components, respectively. Such a configuration would also reduce the cross-plane displacements between beams and nearly eliminate errors due to the time lag between samples since the slanted-beam samples would be nearly symmetric in time about the direct-beam sample. As importantly, the redundancy inherent is such a configuration would make it simple to estimate the combined effect of errors due to time lags and displacements between beams on an operational basis—just as $\sigma_{fit}^2$ is currently used to estimate the effect of random noise on the SPDD-derived velocity fields.

*Acknowledgments.* The authors are indebted to the staff of the University of Wyoming flight facility for supporting the UW King Air deployments for the IHOP 2002 and HPDD-TEST projects. Similarly, we would like to recognize the staff of the NCAR Research Aviation Facility for supporting the NSF/NCAR C-130 aircraft during DYCOMS-II. We are particularly indebted to Dr. William Lindberg for discussions regarding turbulence and evaluating the effects of spatial and temporal displacements between beams. The operation of the WCR in DYCOMS-II and subsequent analysis has been supported by NSF Grant ATM-0094956 and ONR Grant N00014-01-1-0662. The multi-beam modification of the UWKA and WCR was supported by NASA Grant NCC5-578. WCR participation in the IHOP project was funded by NSF Grant ATM-0129374.

# APPENDIX A

## SVD-Based Estimation of Three-Dimensional Velocities

The inversion of the weighted matrix of beam vectors, $B_w$, in (6) to estimate the three-dimensional velocities $V_s$ is mixed-determined since the velocity field is observed from only two distinct angles. Therefore, it is necessary to use a generalized inverse rather than a simple least squares approach. The application of an SVD-based general inverse to the retrieval of three-dimensional velocities presented here follows the treatment of Menke (1989).
Here $\mathbf{B}_w$ is expressed as the product of three matrices:

$$ \mathbf{B}_w = \mathbf{U} \Lambda \mathbf{W}^T, \quad (A1) $$

where $\mathbf{B}_w$ is an $(l+m) \times 3$ matrix $(l$ and $m$ are the number of data points for the nadir and slanted beams, respectively), $\mathbf{U}$ is an $l+m$ square matrix, $\mathbf{W}$ is a $3 \times 3$ square matrix, and $\Lambda$ is an $(l+m) \times 3$ diagonal matrix of the singular values of $\mathbf{B}_w$:

$$ \Lambda = \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_3 \\ 0 & 0 & 0 \\ \vdots & \vdots & \vdots \\ 0 & 0 & 0 \end{bmatrix}. \quad (A2) $$

The $\lambda_i$ values are, by convention, sorted from largest to smallest values. One (or two for the single-beam case) of the $\lambda_i$ values will be very close to zero and is only nonzero due to minor variations in aircraft attitude between data points. Values of $\lambda_i$ less than $10^{-1}$ of the largest $\lambda_i$ value are set to 0. The exact value of this cutoff is of little significance since the smallest $\lambda_i$ values are on the order of $10^{-3}$ of the largest value. The use of a cutoff based on the largest $\lambda_i$ value allows single-beam data to be corrected for the effects of variations in the platform attitude without the need to handle the single-beam case differently than the dual-beam case.

The matrix $\Lambda$ can be partitioned into nonzero and zero components:

$$ \Lambda = \begin{bmatrix} \Lambda_p & 0 \\ 0 & 0 \end{bmatrix}, \quad (A3) $$

where $\Lambda_p$ is a diagonal matrix containing the remaining nonzero $\lambda_i$ values. Subsets of $\mathbf{U}$ and $\mathbf{V}$ corresponding to the nonzero elements of $\Lambda$ then can be constructed:

$$ \mathbf{U} = [\mathbf{U}_p \mathbf{U}_0], \quad (A4) $$

$$ \mathbf{W} = [\mathbf{W}_p \mathbf{W}_0]. \quad (A5) $$

Expressing $\mathbf{B}_w$ in terms of these subsets gives

$$ \mathbf{B}_w \equiv \mathbf{U}_p \Lambda_p \mathbf{W}_p^T, \quad (A6) $$

where the relation is approximate due to the cutoff applied to the smallest $\lambda_i$ values. The inverse of the singular matrix, $\Lambda_p^{-1}$, is computed by inverting the individual diagonal elements, $\lambda_i$, of the matrix:

$$ \Lambda^{-1} = \begin{bmatrix} \lambda_1^{-1} & 0 \\ 0 & \lambda_2^{-1} \end{bmatrix}. \quad (A7) $$

The combination of the two submatrices, $\mathbf{W}_p$ and $\mathbf{U}_p$, and the inverse of the matrix of singular values, $\Lambda^{-1}$, can be combined into a generalized inverse matrix:

$$ \mathbf{B}^{-g} \equiv \mathbf{W}_p \Lambda^{-1} \mathbf{U}_p^T. \quad (A8) $$

The components of the model resolved by the data are given by

$$ \mathbf{R} = \mathbf{W}_p \mathbf{W}_p^T, \quad (A9) $$

where $\mathbf{R}$ is the model resolution matrix (a $3 \times 3$ matrix). A solution that incorporates observations for the velocity components resolved by $\Lambda_p$ and uses the a priori estimate for the unresolved velocity component(s) is

$$ \mathbf{V}_{3d} = \mathbf{B}^{-g} \mathbf{d}_w + [\mathbf{I} - \mathbf{R}] \mathbf{V}_{ap}, \quad (A10) $$

where $\mathbf{I}$ is the identity matrix; and $[\mathbf{I} - \mathbf{R}]$, referred to as the model null space, describes the components not resolved by the Doppler velocity measurements. If $\mathbf{V}_{ap} = 0$ then (A10) produces the same result as would be obtained using a least squares approach to the two-dimensional analysis.

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