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  • View in gallery

    (left) The 2D pdf of σ0(Ka) and σ0(Ku) under rain-free conditions for a flight conducted on 1 Sep 2016. (center) A similar 2D pdf but in the presence of rain, where rain is considered present if dBZ > 10 dB over 20 range gates. (right) The results of the 2D pdf after attenuation correction where the data are aligned along the rain-free regression line.

  • View in gallery

    Simplified depiction of the measurement geometry in which a right-handed coordinate system is used. The aircraft is assumed to travel along the x axis.

  • View in gallery

    The Doppler data at (top) Ku and (top middle) Ka band over approximately eight scans of data at an incidence angle of 40°. Also shown are the Ku- and Ka-band (bottom middle) Doppler velocity (m s−1) measured just above the surface and (bottom) the attenuation-corrected NRCS derived from the surface returns just below. The sawtooth waveform (dashed lines) in the two lower panels represents the azimuthal angle (given on the right-hand ordinate) measured with respect to the flight direction (x axis).

  • View in gallery

    (top) Ku-band Doppler and (bottom) corresponding σ0(Ku) at (left) 40° and (right) 30° for one of the scans shown in Fig. 3. The measured Doppler and the attenuation-corrected σ0(Ku) are given by the black lines; light-blue lines are FS(1), red lines are FS(2), and dark-blue lines are FS(4) (see the text).

  • View in gallery

    (top) Doppler and (bottom) NRCS at Ka band over one antenna scan for incidence angles of (left) 40° and (right) 30°. Light-blue, red, and dark-blue curves are the Fourier series representations of the waveforms for FS(1), FS(2) and FS(4). Measured Ka-band Doppler and attenuation-corrected σ0 are given by the black lines.

  • View in gallery

    As in Fig. 3, at an incidence angle of 40° for data taken on 1 Sep 2016 for a different eight-scan sequence. Note that Ka-band data from the sixth scan are shown on the left in Fig. 5.

  • View in gallery

    Horizontal wind speed υh derived from the Doppler Ku-band data vs the mean NRCS(Ku) for incidence angles of (top) 30° and (bottom) 40°. Each point is derived from the Doppler and NRCS data over a single scan.

  • View in gallery

    (top) Ku-band Doppler-derived υh from data at 30° incidence vs Ku-band Doppler-derived υh from data at 40° incidence. (bottom) Mean NRCS(Ku) at 30° vs mean NRCS(Ku) at 40°.

  • View in gallery

    The (top left) Ku-band and (bottom left) Ka-band υh(D) (blue) and υh(σ0) (red); the incidence angle is 30°. The Doppler-derived wind speed, in blue, is shown only when RSD(1) < 0.5, whereas υh(σ0) is unrestricted. Gray bars at the bottom of the panels indicate the presence of rain as detected by the Ku- and Ka-band radars; The time period is 1652:00–1730:52 UT 1 Sep 2016. Also shown is the NRCS (center) before (i.e., measured data) and (right) after attenuation correction for the (top) Ku and (bottom) Ka bands.

  • View in gallery

    As in Fig. 9, but for 1103:00–1206:39 UT 1 Sep 2016.

  • View in gallery

    As in Fig. 10, but for an incidence angle of 40°.

  • View in gallery

    As in Fig. 9, but for 1806:00–1845:00 UT 7 Oct 2016.

  • View in gallery

    As in Fig. 9, but for 0830:55–0856:06 UT 1 Sep 2016 and the Doppler-derived wind speed is shown for RSD(1) < 0.3 while υh(σ0) is unrestricted.

  • View in gallery

    The (top) Ku-band and (bottom) Ka-band data, where υh(D) is shown in blue and υh(σ0) is shown in red: (left) υh(σ0) is plotted for RSσ(2) < 0.2, and υh(D) is plotted for RSD(1) < 0.3; (center) υh(D) is plotted for RSD(1) < 0.5, whereas υh(σ0), is unrestricted; (right) υh(D) and υh(σ0) are unrestricted.

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Relationship between Horizontal Wind Velocity and Normalized Surface Cross Section Using Data from the HIWRAP Dual-Frequency Airborne Radar

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  • 1 NASA Goddard Space Flight Center, Greenbelt, Maryland
  • | 2 Goddard Earth Science Technology and Research, Morgan State University, Baltimore, Maryland
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Abstract

The High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) dual-frequency conically scanning airborne radar provides estimates of the range-profiled mean Doppler and backscattered power from the precipitation and surface. A velocity–azimuth display analysis yields near-surface estimates of the mean horizontal wind vector υh in cases in which precipitation is present throughout the scan. From the surface return, the normalized radar cross section (NRCS) is obtained, which, by a method previously described, can be corrected for path attenuation. Comparisons between υh and the attenuation-corrected NRCS are used to derive transfer functions that provide estimates of the wind vector from the NRCS data under both rain and rain-free conditions. A reasonably robust transfer function is found by using the mean NRCS (⟨NRCS⟩) over the scan along with a filtering of the data based on a Fourier series analysis of υh and the NRCS. The approach gives good correlation coefficients between υh and ⟨NRCS⟩ at Ku band at incidence angles of 30° and 40°. The correlation degrades if the Ka-band data are used rather than the Ku band.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Robert Meneghini, robert.meneghini-1@nasa.gov

Abstract

The High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) dual-frequency conically scanning airborne radar provides estimates of the range-profiled mean Doppler and backscattered power from the precipitation and surface. A velocity–azimuth display analysis yields near-surface estimates of the mean horizontal wind vector υh in cases in which precipitation is present throughout the scan. From the surface return, the normalized radar cross section (NRCS) is obtained, which, by a method previously described, can be corrected for path attenuation. Comparisons between υh and the attenuation-corrected NRCS are used to derive transfer functions that provide estimates of the wind vector from the NRCS data under both rain and rain-free conditions. A reasonably robust transfer function is found by using the mean NRCS (⟨NRCS⟩) over the scan along with a filtering of the data based on a Fourier series analysis of υh and the NRCS. The approach gives good correlation coefficients between υh and ⟨NRCS⟩ at Ku band at incidence angles of 30° and 40°. The correlation degrades if the Ka-band data are used rather than the Ku band.

Denotes content that is immediately available upon publication as open access.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Robert Meneghini, robert.meneghini-1@nasa.gov

1. Introduction

A number of airborne and satellite-based instruments and techniques have been used to estimate winds over ocean. Instruments include lidar as well as passive and active microwave and millimeter-wave sensors (e.g., Hu et al. 2008; Bentamy et al. 2019; Horstmann et al. 2003; Ricciardulli and Wentz 2015). For frequencies at X band (10 GHz) and above, attenuation by precipitation can limit the accuracy of the wind estimate (Weissman et al. 2012). Even at lower frequencies, the effects of downdrafts and modifications to the surface scattering properties caused by raindrop impacts can affect the retrieval (Balasubramaniam and Ruf 2020; Contreras and Plant 2006). The study presented here is focused on data from a dual-frequency, airborne Doppler radar where attenuation by precipitation can be a significant source of error.

The intent of the study is to investigate the relationship in rain between the mean Doppler radial velocity measured just above the surface and the normalized surface cross section just below. Obtaining a robust relationship in rain between the mean horizontal wind vector and some characteristic of the surface scatter allows an extension of this relationship to rain-free areas so that wind vector estimates can be retrieved under all-weather conditions from the surface scattering characteristics alone. While this is the ultimate goal of the work, the focus of this paper is to investigate relationship between the horizontal wind speed υh and normalized radar cross section (NRCS) as a function of frequency (Ku and Ka band) and incidence angle (30° and 40°) in the presence of rain.

Since it is the intrinsic value of the NRCS that is of interest and not the measured value, it is necessary to first correct the NRCS for path attenuation. Several methods have been investigated (Draper and Long 2004a,b; Stiles et al. 2010; Meneghini et al. 2019). Because we use the dual-frequency approach described in Meneghini et al. (2019), this method will only briefly be described.

Characteristics of the conically scanning dual-frequency High-Altitude Imaging Wind and Rain Airborne Profiler (HIWRAP) radar are given in Table 1 (Helms et al. 2020; Li et al. 2016; Tian et al. 2015; Guimond et al. 2014, 2020). Because the instrument provides backscattered power measurements and Doppler radial velocity at two frequencies (Ku and Ka band) and two incidence angles simultaneously, it permits an examination of the Doppler velocity and NRCS data in close proximity. To simplify the analysis, we use a velocity–azimuth display (VAD) approach for the Doppler data to retrieve horizontal and vertical wind speeds averaged over a conical scan. A similar Fourier analysis is applied to the NRCS data. Results from the Fourier analysis allow a filtering of the data that amounts to choosing scans in which the Doppler velocity over the scan is well represented by a one-period sinusoid, with offset, and in which the NRCS data are accurately represented by terms up to and including a two-period sinusoid. With this filtering, we examine the correspondence between the horizontal wind vector, derived from the Doppler data, on one hand and the mean (scan-averaged), attenuation-corrected NRCS, ⟨NRCS⟩, on the other. With two incidence angles and two frequencies, four sets of results are derived that link υh and ⟨NRCS⟩. Correlation between the two is higher at Ku band than at Ka band. Despite this, test cases show that the Ka-band estimates obtained from the NRCS often give reasonable estimates of wind speed as compared with those derived from the Doppler data.

Table 1.

HIWRAP system parameters/specifications.

Table 1.

Section 2 provides a description of the attenuation correction procedure. Section 3 describes Fourier series analyses of the Doppler and NRCS data that are used to estimate the Doppler-derived wind speed and to filter the data. Section 4 presents sample Doppler and NRCS waveforms and the resultant υh–NRCS transfer functions. The results are summarized by estimates of υh as determined from the Doppler and the NRCS for several flight segments. A discussion and summary are given in section 5.

2. Path attenuation corrections

The path attenuation corrections to the measured NRCS at Ku and Ka band—σ0(Ku) and σ0(Ka)—rely on the fact that, over ocean and under rain-free conditions, these quantities are highly correlated (Meneghini et al. 2019). Scatterplots in the σ0(Ku)–σ0(Ka) plane under rain-free conditions show that the data tend to be concentrated along a line, with a slope close to unity, with excursions along this line determined primarily by the wind vector. In the presence of rain, the NRCS values migrate downward from the line toward lower NRCS values. Since the attenuation at Ka band tends to be about 6 times larger than that at Ku band, the trajectories tend to be displaced from the rain-free regression line along lines with approximately this slope.

Given a set of data that contains a large number of fields of view within and outside of the precipitation, separate linear regression fits are made to the rain and rain-free data in the σ0(Ku)–σ0(Ka) plane. Labeling the regression slopes through the rain-free and raining data as sNR and sR, respectively, then the correction algorithm consists simply of translating each data point [σm0(Ku),σm0(Ka)], measured in rain, along the slope sR until it intersects the rain-free regression line. This gives the attenuation-corrected values where the magnitude of the changes along the y axis [σ0(Ka)] and x axis [σ0(Ku)] are equal to the path attenuations at Ka and Ku band, respectively.

Letting α and sNR be the intercept and slope, respectively, of the regression line for the rain-free data (NR) and p and sR be the intercept and slope of the regression line for the NRCS data in the presence of rain, then the attenuation-corrected values are given by
σ0(Ka)=sRαsNRγsRsNRand
σ0(Ku)=αγsRsNR,
where γ=σm0(Ka)sRσm0(Ku).

In the expression for γ, the subscript “m” denotes the measured value of the NRCS, before attenuation correction. Note that all of the NRCS quantities in the above equations are in decibels. Estimates of the path attenuations A (in decibels) are obtained by subtracting σm0 from the expressions above so that A(Ka)=σ0(Ka)σm0(Ka) and A(Ku)=σ0(Ku)σm0(Ku).

Shown in Fig. 1 are two-dimensional probability density functions (pdf) of σ0(Ka) and σ0(Ku) in the absence (left) and presence of rain (center) for an incidence angle of 30°. These data were acquired by the HIWRAP airborne radar off the east coast of Florida on 1 September 2016 during the NOAA Sensing Hazards with Operational Unmanned Technology field campaign (SHOUT-2016; https://uas.noaa.gov/Program/projects/shout). The attenuation-corrected σ0 are shown in the right-hand panel. The attenuation correction algorithm translates the measured data point along the rain regression line of slope sR, in this case sR = 6.42, until the point intersects the rain-free regression line.

Fig. 1.
Fig. 1.

(left) The 2D pdf of σ0(Ka) and σ0(Ku) under rain-free conditions for a flight conducted on 1 Sep 2016. (center) A similar 2D pdf but in the presence of rain, where rain is considered present if dBZ > 10 dB over 20 range gates. (right) The results of the 2D pdf after attenuation correction where the data are aligned along the rain-free regression line.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Errors in the method include the fact that the A(Ka)/A(Ku) ratio is not constant but a function of the precipitation particle size distributions along the path (Meneghini et al. 2012). Raindrop impacts on the surface, creating stalks and ring waves, modify the NRCS and degrade its dependence on wind speed (Bliven and Giovanangeli 1993; Bliven et al. 1997; Contreras et al. 2003; Contreras and Plant 2006). Positive biases are caused by contributions to the backscattered power from precipitation near the surface (Stiles and Yueh 2002; Tournadre and Quilfen 2003; Weissman and Bourassa 2008; Draper and Long 2004a,b); for HIWRAP, this effect is generally small but can be significant at the higher incidence angle and higher frequency when the rain rates and attenuations are large. An error more specific to this particular method is the dependence of the slope on the judgement as to what constitutes a rain or rain-free measurement, which, in turn, depends on the minimum detectable signal. For flights for which the number of light rain rates is large, the slope is affected by changes in the wind vector [along which the variation in the σ0(Ku)–σ0(Ka) plane follows a slope of ~1], which decreases the slope sR and increases the attenuation-corrected σ0 values. Determining the slope through the rain data after filtering out the light rain rates can mitigate this error.

3. Relevant equations for the Doppler

A simple picture of the measurement geometry is shown in Fig. 2. The aircraft is assumed to fly along the positive x axis. We assume a right-handed coordinate system with z pointing up; θ is the polar angle measured with respect to nadir, and ϕ is the azimuthal angle measured relative to the x axis. Positive radial Doppler velocity VD is assumed to be away from the radar so that
VD=υ¯r^=υxcosϕsinθ+υysinϕsinθυzcosθ,
where υ¯ is the reflectivity-weighted mean velocity of the hydrometeors in the radar resolution volume and r^ is a unit vector directed along the center of the main lobe of the radar antenna. Assuming that the horizontal wind vector υh makes an angle β with respect to the x axis, then υx = υh cosβ and υy = υh sinβ. Substituting these expressions into (3) and using the fact that cos(ϕβ) = cosϕ cosβ + sinϕ sinβ, then
VD=υ¯r^=υhcos(ϕβ)sinθυzcosθ.
Equation (4) suggests a simple way to estimate υh if υh and υz are constant over the scan (~18.5 km at 30° and ~25 km at 40° at the surface). Integrating each term of (4) over a full scan (−π, π) and noting that the integral of the first term on the right side of (4) is zero, since υh is assumed constant, then
υzcosθ=(2π)1[ππVD(ϕ)dϕ].
Substituting this back into (4) and evaluating the expression at ϕ = β gives an estimate of υh:
υh=VD(ϕ=β)(2π)1ππVD(ϕ)dϕsinθ.
A second estimate of υh is given below [in (12)]. The two estimates have been found to be in good agreement.
Fig. 2.
Fig. 2.

Simplified depiction of the measurement geometry in which a right-handed coordinate system is used. The aircraft is assumed to travel along the x axis.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

When the wind field is not strictly linear and uniform over the swath, we follow Browning and Wexler (1968) and Tian et al. (2015) with a slight modification. These authors use a Taylor expansion of υx and υy relative to the center of the swath but do not expand υz in a similar way. However, for the HIWRAP radar geometry, it seems that an expansion in υz should be included as well. This has the effect of modifying the coefficients of the sinϕ and cosϕ terms. Under the approximations that are used, the results are unaffected. It does show, however, the approximations that are needed in making the simplification.

Expanding υx, υy and υz in a Taylor’s series about the center of the swath (x0, y0, 0), we have
υx=υx0+(υxx)(xx0)+(υxy)(yy0)υy=υy0+(υyx)(xx0)+(υyy)(yy0)υz=υz0+(υzx)(xx0)+(υzy)(yy0).
Note that (xx0) = r cosϕ sinθ and (yy0) = r sinϕ sinθ, where r is the distance from the radar to the center of a field of view just above the surface. Using the identities cos2ϕ = 0.5[1 + cos2ϕ], sin2ϕ = 0.5[1 − cos2ϕ], and cosϕ sinϕ = 0.5 sin2ϕ and substituting the relations given by (7) into (3), we find that VD can be written in the form of a Fourier series up to order 2ϕ:
VD=a02+a1cosϕ+b1sinϕ+a2cos2ϕ+b2sin2ϕ,
where
a02=υz0cosθ+0.5rsin2θdiv(vh)a1=υx0sinθ(υzx)rsinθcosθb1=υy0sinθ(υzy)rsinθcosθa2=0.5rsin2θ(υxxυyy)b2=0.5rsin2θ(υxy+υyx),
and
div(vh)=υxx+υyy.
The coefficients are given by
an=1πππVDcos(nϕ)dϕandbn=1πππVDsin(nϕ)dϕ
for n = 1 and n = 2. Note that a0 is given by the above formula with n = 0, which reduces to an integral of VD over the full scan divided by π so that the mean value of VD is given by a0/2.
From (9) it can be seen that if we ignore terms that depend on the partial derivatives of υz with respect to x or y then
υh=(a12+b12)0.5sinθ.
This provides an alternative estimate to (6) for the horizontal wind speed. It is also the formula used for the results throughout the paper.

Before showing results, it is worth mentioning several features of the above equations. Tian et al. (2015) include an additional term that accounts for the antenna scanning that affects the Doppler along the direction of aircraft motion. They argue that the deviation from circular to helical scan is small relative to the scan diameter and that terms involving the antenna angular motion can be neglected. We have followed this assumption. Also, since the Doppler data have been corrected for the aircraft motion, this is not included in the equations.

Apart from the horizontal wind divergence term shown above, Browning and Wexler (1968) identify the partial derivatives that appear in the expression for a2 as the stretching deformation and the partials that appear in b2 as the shearing deformation.

The HIWRAP instrument was designed so that the 30° incidence channel is horizontally polarized (transmit and receive) while the 40° incidence channel is vertically polarized so that, to prevent ambiguity, the designations (hh or vv) should be used whenever estimates from both channels are used. Inspection of (9) shows that υz and div(vh) can be evaluated in terms of the Fourier coefficients obtained from data taken at the two incidence angles. For example, if we write a0 of (9) explicitly as a function of the polar angle θ, then equations for a0,hh(θ1) and a0,vv(θ2), where θ1 = 30° and θ2 = 40°, allow us to solve for υz0 and div(vh). Evaluating the various trigonometric functions that enter into the equations, we find
υz0=1.17[a0,vv(θ2)2]2.19[a0,hh(θ1)2]and
vh=H1{7.04[a0,vv(θ2)2]6.23[a0,hh(θ1)2]}.
Examination of the other Fourier coefficients in (9)—that is, a1, b1, a2, and b2—shows that it is not possible to solve for the unknowns on the right-hand side using data at 30° and 40°. For example, we cannot obtain υx0 and ∂υz/∂x by solving the equations for a1,hh(30) and a1,vv(40) because the equations are linearly dependent with a determinant equal to zero.

The VAD approach just described uses Doppler radial velocities in cases where precipitation is present just above the surface for all or most of the conical scan. In the results shown in the next sections, (12)(14) along with a Fourier series analysis of the NRCS are used to relate the attenuation-corrected NRCS to the horizontal velocity obtained from the VAD analysis.

4. Results

a. Analysis of individual waveforms

Doppler data at 40° over a sequence of about eight antenna scans (an approximately 5 km flight segment) during the 1 September 2016 flight are shown in Fig. 3 where the top panel and the panel second from the top show the Doppler data at Ku and Ka band, respectively. The third panel from the top shows the Doppler at Ku and Ka band just above the surface; note that the sawtooth waveform represents the scan angle in degrees where 0° represents the position where the antenna beam is directed along the flight line (positive x axis). Displayed in the bottom panel are the attenuation-corrected σ0 at the two frequencies.

Fig. 3.
Fig. 3.

The Doppler data at (top) Ku and (top middle) Ka band over approximately eight scans of data at an incidence angle of 40°. Also shown are the Ku- and Ka-band (bottom middle) Doppler velocity (m s−1) measured just above the surface and (bottom) the attenuation-corrected NRCS derived from the surface returns just below. The sawtooth waveform (dashed lines) in the two lower panels represents the azimuthal angle (given on the right-hand ordinate) measured with respect to the flight direction (x axis).

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

The Doppler and NRCS for one of the scans shown in Fig. 3 are represented by the black lines in Fig. 4 for both the 40° (left) and 30° (right) cases. The light-blue, red, and dark-blue lines are derived from the Fourier series representations using different numbers of terms. We use the following notation
FS(n)=ao2+j=1n(ajcosjϕ+bjsinjϕ)
so that, for example, FS(2) represents the Fourier series representation of a function up to and including the cos(2ϕ) and sin(2ϕ) terms. This two-period azimuthal dependence is a well-known feature of the NRCS over the ocean (Jones et al. 1977, 1982). In Fig. 4, the light blue line represents the FS(1) (sinusoid of period 1 with offset) approximation for the Doppler(Ku) (top) and σ0(Ku) (bottom). It can be seen in these examples that the FS(1) approximation provides an accurate representation of the measured Doppler data (black curve) whereas FS(2) (red line) is needed to give a good approximation to the measured σ0(Ku). We can quantify the goodness of the representation by computing a residual error, defined by
RS(n)={i=1N[yiFSi(n)]2i=1Nyi2}1/2,
where the summation from i = 1 to N is taken over the data within the scan and where N is the number of fields of view that compose the scan. The function y in (16) can be either VD(Ku) or σ0(Ku). For the data that will be shown later, we require RS(1) < τ for the Doppler radial velocity and RS(2) < τ for σ0(Ku), where the threshold value τ is typically taken to be 0.3. These criteria are used to filter the data so that the Doppler is close to a single-period sinusoid and the NRCS is close to a two-period sinusoid.
Fig. 4.
Fig. 4.

(top) Ku-band Doppler and (bottom) corresponding σ0(Ku) at (left) 40° and (right) 30° for one of the scans shown in Fig. 3. The measured Doppler and the attenuation-corrected σ0(Ku) are given by the black lines; light-blue lines are FS(1), red lines are FS(2), and dark-blue lines are FS(4) (see the text).

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

For the 40° case shown on the left in Fig. 4 we find that, expressed in percentages and using either “D” (Doppler) or “σ” (NRCS) subscript to indicate the type of data, then RSD(1) = 6.3% and RSσ(2) = 3.9%; for the 30° incidence case on the right, RSD(1) = 7.8% and RSσ(2) = 4.4%. Note that for the statistics presented later, we take the threshold to be 30% so that these examples are well within this threshold requirement.

For estimates of horizontal wind speed, the Doppler estimates at 30° and 40° are in good agreement in this case: υh(30°) = 22.2 m s−1 and υh(40°) = 23 m s−1. Several estimates of the vertical velocity are available. Neglecting the divergence term, then from (9) we have υz = −a0(θ)/(2 cosθ); substituting the values of a0 for 30° and 40° incidence angles gives υz(30°) = −6.97 m s−1 and υz(40°) = −7.40 m s−1. In principle, (13) provides a better approximation for υz since the divergence term does not need to be assumed negligible and uses information from both incidence angles. For this case, use of (13) gives υz = −6.59 m s−1 so that the three estimates are comparable and consistent with typical raindrop fall speeds.

With respect to the wind direction, note that negative Doppler corresponds to motion toward the radar (upwind or headwind direction) so that the peak negative Doppler should correspond to the maximum NRCS. The results in Fig. 4 show differences: for example, for the 40° incidence case on the left, the maximum NRCS occurs at about 221° whereas the minimum Doppler occurs at 237° for a difference of about 16°. For the 30° incidence case on the right, the maximum NRCS occurs at 212° while VDmin occurs at about 240°. Note that in this case the Doppler estimates of wind direction from data taken at 30° and 40° are in good agreement: 237° at 40° versus 240° at 30°.

Occasionally the “secondary” maximum, corresponding to the downwind or tailwind direction, is the actual maximum and the differences in wind direction derived from the Doppler and the NRCS are much larger. This is a fairly typical ambiguity in scatterometer measurements (Wentz and Smith 1999; Ricciardulli and Wentz 2015) where other information is needed to distinguish the upwind and downwind directions.

In many cases, the residual errors are larger than those seen in Fig. 4. Some of this is simply caused by the fact that rain is missing over portions of the scan so that the Doppler is not measurable over the full scan. Other cases occur where the Doppler data are noisy or partially missing over the scan. A more interesting kind of case occurs when the data are generally good but where the Doppler deviates from the FS(1) representation and/or the NRCS deviates from the FS(2) representation.

Figure 5 shows examples of single scans of the Ka-band Doppler and NRCS data for incidence angles at 40° (left) and 30° (right) incidence where the measured data are given by the black lines, and the FS(1), FS(2), and FS(4) Fourier series approximations are given by the light-blue, red, and dark-blue lines. (Figure 6 shows the Doppler and NRCS at 40° for an 8-scan sequence, where the Ka-band portion of the sixth scan is shown on the left-hand panel in Fig. 5.) In these cases, the Doppler waveforms are not particularly well represented by the FS(1) estimates, given by the light-blue curves, and we need to look at FS(2) and higher to get better representations of the waveform. The NRCS data, moreover, are not particularly well represented by FS(2) whereas the FS(4) gives an accurate approximation.

Fig. 5.
Fig. 5.

(top) Doppler and (bottom) NRCS at Ka band over one antenna scan for incidence angles of (left) 40° and (right) 30°. Light-blue, red, and dark-blue curves are the Fourier series representations of the waveforms for FS(1), FS(2) and FS(4). Measured Ka-band Doppler and attenuation-corrected σ0 are given by the black lines.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Fig. 6.
Fig. 6.

As in Fig. 3, at an incidence angle of 40° for data taken on 1 Sep 2016 for a different eight-scan sequence. Note that Ka-band data from the sixth scan are shown on the left in Fig. 5.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Despite these deficiencies in FS(1) for the Doppler(Ka) and FS(2) for the σ0(Ka), we find that RSD(1) = 15.3% and RSσ(2) = 5% for the 30° incidence case (right, Fig. 5) and RSD(1) = 10.7% and RSσ(2) = 3.5% for the 40° incidence case (left, Fig. 5) and within the threshold used for the statistics shown in the next section. It might be noted, however, that of the eight scans shown in Fig. 6, it is only the sixth scan (the one shown in Fig. 5) where the Ka-band Doppler is relatively noise free. Analysis of the other waveforms, without further filtering, shows that the residual errors are larger than in the case shown.

With respect to the horizontal velocity estimates, we obtain for this case: υh(30°) = 17.23 m s−1 and υh(40°) = 17.14 m s−1. For estimates of the upwind direction we obtain the four estimates: 107° (Doppler; 30°), 111° (Doppler; 40°), 126° [σ0(30°)] and 124° [σ0(40°)] where all estimates are based on the Ka-band data.

Estimates of the vertical velocity and divergence for this case give a0(30°)/2 = 5.08 and a0(40°)/2 = 4.56, which from (13) yields υz0 = −5.79 m s−1 and div(vh) = 2.67 × 10−5 s−1 (using H = 18 000 m). Equation (9), on the other hand, under the assumption that the divergence is zero, gives υz = −5.08/cos(30°) = −5.86 m s−1 and υz = −4.56/cos(40°) = −5.95 m s−1, so that the three estimates for υz in this case are similar.

As a final comment on this example, it is worth remarking on the differences between the σ0 data shown in the bottom panels of Figs. 3 and 6. The data in Fig. 3 are consistent with the classic pattern of primary and secondary maxima with an angular separation between them of about 180°, coinciding with the upwind and downwind directions. The pattern in Fig. 6 is quite different, with primary and secondary maxima spaced more closely together in angle. Examination of the sequence of eight scans (approximately 5 km along the flight path) in Fig. 6 shows that for the waveforms on the left, the primary and secondary maxima are barely distinguishable as their angular separation is small and the secondary maximum is significantly smaller than the primary maximum. However, in going to the right, the separation becomes larger with a more clearly defined minimum between the two maxima, along with an increase in the magnitude of the secondary maximum. It is probably no coincidence that the deviations in the Doppler from a one-period sinusoid and deviations in σ0 from a two-period sinusoid in these cases reflect the more complicated, nonlinear behavior of the wind field over the scans.

b. Statistical results

The primary goal of this study is to examine the correspondence between the horizontal wind speed derived from the Doppler and either the maximum NRCS or the mean value of the NRCS over the scan. Because the correlations are generally higher between the Doppler-derived wind speed and the mean NRCS, we focus on this relationship. These values are derived for each scan and the data are filtered by how well they are represented by a Fourier series. As noted above, we generally require less than 30% residual error, as given by RS(1), for the Doppler and less than 30% residual error, as given by RS(2), for the NRCS. Many cases fail this test either because of partial rain coverage over the swath, which gives a noisy Doppler over part of the scan, or because of the occurrence of discontinuities in the azimuthal angle, representing data gaps in Doppler and NRCS over the scan. In other cases, the Doppler is simply too noisy over the full scan to satisfy the imposed condition. It is generally the case that the NRCS data are better behaved than the Doppler, which is not surprising as a surface return is almost always present regardless of whether it is raining.

The data shown in Figs. 7 and 8 have been combined from the processing of twelve files, corresponding to twelve ~40-min flight segments. Figure 7, top, shows a scatterplot of the horizontal wind speed υh, derived from the Doppler Ku-band data, using (12), versus the mean attenuation-corrected NRCS(Ku) (equal to the a0/2 term of the Fourier series), for an incidence angle of 30°. Each point is derived from the Doppler and NRCS data over a single scan. The bottom plot of Fig. 7 shows a similar result but using data from the 40° incidence data. The data plotted in Fig. 7 satisfy the requirement that RSD(1) < 30% and RSσ(2) < 30% and yield correlation coefficients of ρ[υh(Ku; 30°), ⟨σ0(Ku; 30°)⟩] = 0.87 and ρ[υh(Ku; 40°), ⟨σ0(Ku; 40°)⟩] = 0.89. Linear regression fits to the data, where υh(σ0) = α0 + α1σ0⟩, yield for 30° incidence [α0, α1] = [75.27, 3.98] and for 40° incidence [α0, α1] = [105.8, 4.09]. The change in these coefficients with change in threshold is relatively small so that even though the scatter increases as the threshold is increased, the regression line remains relatively constant.

Fig. 7.
Fig. 7.

Horizontal wind speed υh derived from the Doppler Ku-band data vs the mean NRCS(Ku) for incidence angles of (top) 30° and (bottom) 40°. Each point is derived from the Doppler and NRCS data over a single scan.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Fig. 8.
Fig. 8.

(top) Ku-band Doppler-derived υh from data at 30° incidence vs Ku-band Doppler-derived υh from data at 40° incidence. (bottom) Mean NRCS(Ku) at 30° vs mean NRCS(Ku) at 40°.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

For the Ka-band results, the correlations degrade to ρ[υh(Ka; 30°), ⟨σ0(Ka; 30°)⟩] = 0.75 and ρ[υh(Ku; 40°), ⟨σ0(Ku; 40°)⟩] = 0.60. A slight improvement occurs if the thresholds are lowered from 0.3 to 0.2 where the correlation coefficient at 30° increases to 0.79 and that at 40° to 0.64. Linear regression fits to the Ka-band data, where υh(σ0) = α0 + α1σ0⟩, yield for 30° incidence [α0, α1] = [75.37, 3.75] and for 40° incidence [α0, α1] = [94.4, 3.33]. These relationships at Ku and Ka band are used in Figs. 914 to convert ⟨σ0⟩ to an estimate of horizontal wind speed υh(σ0).

Fig. 9.
Fig. 9.

The (top left) Ku-band and (bottom left) Ka-band υh(D) (blue) and υh(σ0) (red); the incidence angle is 30°. The Doppler-derived wind speed, in blue, is shown only when RSD(1) < 0.5, whereas υh(σ0) is unrestricted. Gray bars at the bottom of the panels indicate the presence of rain as detected by the Ku- and Ka-band radars; The time period is 1652:00–1730:52 UT 1 Sep 2016. Also shown is the NRCS (center) before (i.e., measured data) and (right) after attenuation correction for the (top) Ku and (bottom) Ka bands.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for 1103:00–1206:39 UT 1 Sep 2016.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Fig. 11.
Fig. 11.

As in Fig. 10, but for an incidence angle of 40°.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Fig. 12.
Fig. 12.

As in Fig. 9, but for 1806:00–1845:00 UT 7 Oct 2016.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Fig. 13.
Fig. 13.

As in Fig. 9, but for 0830:55–0856:06 UT 1 Sep 2016 and the Doppler-derived wind speed is shown for RSD(1) < 0.3 while υh(σ0) is unrestricted.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Fig. 14.
Fig. 14.

The (top) Ku-band and (bottom) Ka-band data, where υh(D) is shown in blue and υh(σ0) is shown in red: (left) υh(σ0) is plotted for RSσ(2) < 0.2, and υh(D) is plotted for RSD(1) < 0.3; (center) υh(D) is plotted for RSD(1) < 0.5, whereas υh(σ0), is unrestricted; (right) υh(D) and υh(σ0) are unrestricted.

Citation: Journal of Atmospheric and Oceanic Technology 38, 3; 10.1175/JTECH-D-20-0143.1

Figure 8, top, shows a comparison between υh(Ku) as derived from the Doppler data at 30° incidence angle with the Doppler data at 40° incidence, where we have matched the scan numbers and “angle” bins between the two. The correlation is high even though the fields of view do not exactly coincide. Note that the threshold for inclusion of the data is taken to be 30% for both the Doppler and σ0; that is, RSD(1) < 0.3 and RSσ(2) < 0.3. The bottom plot shows a scatterplot between ⟨σ0(Ku; 30°)⟩ and ⟨σ0(Ku; 40°)⟩ for the same set of matching pairs. For these cases and this threshold, the correlation coefficients for the Doppler and the ⟨NRCS⟩ are approximately the same at 0.98. At Ka band the correlations between υh(30°) and υh(40°) decrease to 0.93 while the correlation between ⟨σ0(Ku; 30°)⟩ and ⟨σ0(Ku; 40°)⟩ decreases to 0.89.

Although linear relationships between υh and σ0(dB) are given above, it should be noted that a power law between σ0, in linear units, and vh has been shown to give excellent agreement to measured data (e.g., Jones et al. 1977). This power-law relationship implies that an alternative model function of the form σ0 = α + β log(υh), where σ0 is in decibels and where β depends on incidence angle and polarization, could be employed. For the results below, only the linear model is used.

c. Summary results

We summarize in the figures below the attenuation correction procedure, estimation of horizontal wind speed, υh, from σ0 and its comparison with the Doppler υh estimates using data from four flight segments. Each of Figs. 913 consists of six panels. The two panels on the left show the horizontal wind speed as determined by the Doppler VAD, υh(D) given by (12) and shown in blue, and as derived from σ0 using the υhσ0 linear relationships given above and shown in red. The Ku-band results are given in the top plots and Ka band in the bottom. The four panels on the right show the NRCS before (i.e., the measured data) and after attenuation correction with the attenuation-corrected NRCS given on the far right. Similar plots for the Ka-band data are given in the bottom two panels. It is worth noting that the data on the far-right panels, averaged over the scan, yield υh as obtained from the υh–σ0 transfer function. Note that in the NRCS plots, only the forward portion of the scan is plotted.

The υh(D) data are plotted only when the Doppler data for that scan satisfies the threshold criterion RSD(1) < 0.5 so that the residual error between the Fourier series, including terms of up to period one, and the measured data is less than 50%. (The reasons for increasing this from the nominal 0.3 value is to show a greater fraction of the Doppler estimates.) A gray bar at the bottom of each panel on the left is used to indicate those scans over which the fraction of rain is greater than 70% (i.e., rain is detected in over 70% of the fields of view that compose the scan). Since the rain detection capability is somewhat different at Ku and Ka band, the gray bars in the top (Ku) and bottom (Ka) plots are not identical. In contrast to the intermittent nature of the υh(D) results, a υh(σ0) estimate is obtained at each scan as σ0 exists almost everywhere, as shown by plots of σ0 on the far-right panels. Filtering of the NRCS data will be discussed in connection with Fig. 13.

Note that if the axis of the conical scan is greater than 2° off nadir then the data are not processed; consequently, data taken during aircraft banking are discarded. Correlating the line plots on the left with the NRCS images on the right is somewhat difficult because of breaks in the data typically caused by banking. For each flight, we first divide the data into 5 data segments with equal number of data points along the flight direction, and accumulate displacements between consecutive points based on their geolocation (longitude and latitude). Labeling the beginning of the flight x = 0 km, then the flight distance is summed to the end of the first segment (106 km as labeled in Fig. 9). This process continues to the end of the second segment (182 km). There are equal number of data points (at nadir) between 0 and 106 km as there are between 106 and 182 km. This applies to all the data segments. Note that the forward scan data are used in the computation of the relative flight distance. All data in the plots below were measured on 1 September 2016, with the exception of the data shown in Fig. 12, which were taken during Hurricane Matthew on 7 October 2016 and described in Guimond et al. (2020). The duration of each flight segment is typically about 40 min.

For the results shown in Fig. 9, only data from the inner conical scan of 30° are used. In this case the υh(σ0) results at Ku and Ka band, as shown by the red lines in the top- and bottom-left-hand panels, are in good agreement and, in turn, in fairly good agreement with Doppler VAD estimates given by the blue lines. Also encouraging is the fact that the υh(σ0) results are continuous across rain/no-rain boundaries; this follows from the fact that the σ0 images shown on the right-hand plots are continuous across these boundaries and the fact that the υhσ0 relationships are well represented by a simple linear model.

Errors in the Ku-band υh(σ0) estimates can be seen over the segment of data along the x axis centered at about 355 km where υh(σ0; Ku) estimates are less than zero. The reason for this error can be traced to the regression plots shown in Fig. 7 where valid υh(D) data are available at Ku band only when −16 < E(σ0) < −9 dB at 30° and −23 < E(σ0) < −18 dB at 40°, where E(σ0) represents the scan-averaged value. (In both cases, this represents a range of υh from about 10 to 40 m s−1). As the transfer function is used even when ⟨σ0⟩ falls outside these bounds, υh(σ0) will become negative at sufficiently low values of ⟨σ0⟩. This kind of error suggests that the υh(σ0) transfer functions at both frequencies and incidence angles are unreliable in light winds (below ~10 m s−1) and low values of the NRCS because of insufficient Doppler data. Conversely, at medium to high wind speeds, from about 10 to 40 m s−1, where the Doppler derived wind speeds are extensive, the υh(σ0) transfer functions are more robust.

A second example, also using data from the inner cone (30°), is shown in Fig. 10. As in the previous example, some missing υh(σ0) values are seen in the first segment of the flight up to about 150 km. Using υh(D) as a reference, it appears that υh(σ0) captures fairly well the first peak at about 210 km. On the other hand, υh(σ0) is significantly larger than the first wind speed minimum, occurring at x ~ 270 km, where υh(D) is well below 10 m s−1 whereas υh0) ~ 18 m s−1. For the latter portion of the flight, beyond about 450 km, the Doppler and σ0 estimates of wind speed are well aligned at both frequencies.

Data at 40° from the same flight are shown in Fig. 11. For this case, apart from the low wind speed/low NRCS along the first segment of the flight, there is generally good agreement in wind speed as derived from the Doppler and from σ0 as estimated from data at the two incidence angles and two frequencies.

In the cases shown above, the filtered Doppler data yield estimates of υh over only a relatively small fraction of the flight data. In Fig. 12, θ = 30°, the valid Doppler data are more prevalent. The top-left panel (Ku band) shows relatively good agreement between υh(D; Ku) and υh(σ0; Ku). The Ka-band comparisons, shown in the bottom-left panel, exhibit poorer agreement. Comparisons between υh(D; Ka) and υh(D; Ku) show the former to be higher in magnitude and noisier. Comparison between υh(σ0; Ka) and υh(σ0; Ku) show the former to about 1–2 m s−1 smaller but with a similar shape. The results at 40° (not shown) give similar results: fairly good agreement between υh(D; Ku) and υh(σ0; Ku) but relatively poor agreement between υh(D; Ka) and υh(σ0; Ka), with the former being larger than the latter.

Note that this result is fairly typical of the data analyzed where agreement between υh(D) and υh(σ0) using Ku-band data is better than the comparisons using the Ka-band data. This holds at both incidence angles and is not surprisingly in light of the higher correlations between υh(D) and σ0 that are obtained at Ku band.

Another flight segment with a fairly high proportion of valid Doppler data is shown in Fig. 13 using data from the inner conical scan. In this case, we have required RSD(1) < 0.3; different choices of threshold are shown in Fig. 14. Good agreement between υh(D; Ku) and υh(σ0; Ku) can be seen in the top-left panel up to about x ~ 180 km. At Ka band, as shown in the bottom-left panel, υh(σ0; Ka) is seen to be in good agreement with both υh(D; Ku) and υh(σ0; Ku) but not in good agreement with υh(D; Ka), which tends to be noisy and only intermittently reliable; that is, RSD(1) < 0.3.

For x between about 180 to 210 km, the υh(σ0) at both frequencies exhibit an oscillatory behavior. Valid Doppler υh(D) data during this period are largely missing, although the small amount of Doppler data at Ku band suggests that the υh(σ0) estimates are positively biased. To better understand the situation, it is useful to replot the results with changes in the thresholds. As noted earlier, for the results in Figs. 912, we have chosen the threshold RSD(1) < 0.5 for the Doppler data but have not imposed any restriction on the NRCS data. In other words, irrespective of the residual error in the NRCS, we have used the υh(σ0) to obtain a wind speed estimate. If, however, we impose the condition that RSσ(2) < 0.2 then we obtain the results shown on the left-hand plots in Fig. 14. (Recall that RSσ(2) denotes the residual error between the attenuation-corrected NRCS over the scan and the Fourier series representation using terms up to sin2ϕ and cos2ϕ.) By imposing this filtering on the NRCS, most of the large oscillations in υh(σ0) are removed.

We can also modify the RSD(1) threshold. In the middle panels of Fig. 14, we have chosen the threshold RSD(1) < 0.5 so that a greater fraction of the Doppler estimates (blue lines) are plotted. In the right-hand panels, we have removed the restriction on the Doppler error entirely so that υh(D) estimates are given at all scans. Note that for the middle and right-hand plots, the residual error on σ0 is not considered so that υh(σ0), given by the red lines, are the same as in Fig. 13.

5. Discussion and summary

Data from the HIWRAP dual-frequency Doppler radar afford an opportunity to examine the Doppler-derived horizontal wind field υh near the surface in close proximity to the attenuation-corrected estimates of σ0. Although the data show a substantial amount of scatter in the υhσ0 plane, a Fourier series analysis permits the choice of Doppler and σ0 waveforms from individual scans that are well represented by 1-period and 2-period sinusoids, respectively. These filtered data, with residual errors less than 30%, show good correlations in υhσ0 space, particularly at Ku band at incidence angles of 30° and 40°.

At Ka band, the results are less robust with correlations, at a threshold of 30%, of 0.75 at 30° and 0.6 at 40°. These results raise the issue of the accuracy of the attenuation correction, particularly since the Ka-band attenuation tends to be larger by about a factor of 6 than that at Ku band and clearly affects the Ka-band data more strongly.

Some comparisons of the υhσ0 transfer function obtained here with the QuikScat/NASA Scatterometer (NSCAT) geophysical model function (GMF) (Jones et al. 1982; Wentz and Smith, 1999; Ricciardulli and Wentz 2015) have been conducted by inserting the HIWRAP σ0 (either average or peak value) data into the NSCAT GMF. The values of υh obtained in this way, however, are smaller than the υh obtained from the HIWRAP Doppler. On the other hand, the slope in the υhσ0 plane is similar to that given by the HIWRAP results. The offset in υh between the two results may indicate an error in the HIWRAP σ0 estimates, an error in the attenuation correction or, less likely, an error in the HIWRAP Doppler. It is important to note that the Doppler-derived wind speed estimates given here are obtained from measurements several hundred meters above the surface whereas the NSCAT transfer function is derived from estimates of the vector wind at 10-m height in a neutrally stable atmosphere.

It worth mentioning again the presence of other error sources. Because of the strong attenuation that sometimes occurs at Ka band, the backscatter from rain near the surface will positively bias the estimate of the NRCS. This bias becomes stronger at higher incidence angles because the surface return decreases as the incidence angle increases so that there will be a certain fraction of cases at 40° in which the rain and surface returns will be of comparable magnitude. In the data processing done here, we have not attempted to correct for this bias. Another error source is caused by the effects of raindrops on the NRCS. In particular, Contreras and Plant (2006) have shown at Ku band that perturbations on the NRCS caused by the impact of raindrops on the surface increase as the angle of incidence increases so that the 40° incidence data are expected to be more strongly affected than measurements at 30° incidence; it is reasonable to assume that the Ka-band data are affected in a similar way. Although the effects of surface perturbations should reduce as the wind speed increases, increasing rain rate will counter this effect.

Despite discrepancies between the transfer functions derived here and for NSCAT, as long as the HIWRAP Doppler wind speed estimates are valid, then the υhσ0 should be applicable to estimates of υh under both raining and rain-free conditions using the σ0 estimates from HIWRAP. However, as the transfer function applies only to the mean value of σ0 over the swath, the resolution of the estimates is limited to the swath widths of approximately 18.5 km for the inner swath and 25 km for the outer swath. As noted earlier, the υhσ0 results are probably less accurate at light wind speeds than at moderate and high wind speeds because of a lack of accurate full-scan Doppler data at light wind speeds. This is understandable in view of the fact that the field campaign was focused on midlatitude severe storms; future missions are expected to obtain data over a wider variety of wind speeds that include those at the lower end.

Acknowledgments

The HIWRAP data collection was sponsored by the NOAA SHOUT field program. We thank Lihua Li and Matthew McLinden of Code 555 GSFC for instrument engineering and data analysis.

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