This paper has presented DopplerScatt, a new instrument that provides simultaneous 1106
measurements of winds and currents using a Ka-band pencil-beam scanning Doppler scatterometer. 1107
With the development of DopplerScatt, we have extended the theory and calibration of these 1108
instruments beyond the existing literature [9]. Among the innovations presented in the system 1109
understanding, algorithms, and calibration, we note: 1110
1. Development of an end-to-end measurement model including several effects, such as quantifying 1111
the impact of cross-section variations, not previously reported. 1112
2. Detailed examination of the pulse-pair estimation algorithm, including deriving an error 1113
estimator for the Doppler velocity and validating it with experimental data. 1114
3. Development of an end-to-end error budget including both random and systematic errors. The 1115
error model was validated against measurements and showed that the DopplerScatt instrument 1116
had good stability and noise performance for bothσ0and Doppler velocities.
1117
4. Development of new calibration techniques to remove errors caused by uncertainties in the 1118
antenna pointing and other systematic (e.g., model function) errors. 1119
100 0 100 1 0 1
v
rs(m/s)
U10: 1.5 (m/s) 100 0 100 1 0 1 U10: 2.5 (m/s) 100 0 100 1 0 1 U10: 3.5 (m/s) 100 0 100 1 0 1 U10: 4.5 (m/s) 100 0 100 1 0 1v
rs(m/s)
U10: 5.5 (m/s) 100 0 100 1 0 1 U10: 6.5 (m/s) 100 0 100 1 0 1 U10: 7.5 (m/s) 100 0 100 1 0 1 U10: 8.5 (m/s) 100 0 100 1 0 1v
rs(m/s)
U10: 9.5 (m/s) 100 0 100 1 0 1 U10: 10.5 (m/s) 100 0 100 1 0 1 U10: 11.5 (m/s) 100 0 100(degrees)
1 0 1 U10: 12.5 (m/s) 100 0 100(degrees)
1 0 1v
rs(m/s)
U10: 13.5 (m/s) 100 0 100(degrees)
1 0 1 U10: 14.5 (m/s) 100 0 100(degrees)
1 0 1 U10: 15.5 (m/s)Figure 33.(blue line) Mean ofFSfrom Figure26; (orange dashed line) modeled wind-driven velocity bias, using the fit MTF coefficients; (green line) residual after subtracting orange from green lines, which should be nominally the BraggcpF. The upwind and downwind free Bragg velocities are indicated by dashed gray lines.
5. Development of a wind estimation algorithm that uses backscatter and Doppler velocities in 1120
an innovative way so that winds vectors can be estimated using a single beam, rather than the 1121
traditional two-beam architecture. 1122
In addition to these technical innovations, we have collected an extensive data set of Ka-band V-polσ0
1123
and Doppler velocities. Using these data, we have: 1124
1. Determined the ocean correlation time at Ka-band as a function of wind speed. The correlation 1125
times observed (>2 msec) indicate that this measurement is scalable to spaceborne applications 1126
with reasonable performance. 1127
2. Developed a Ka-band V-pol GMF which shows an overall sensitivity to wind speed similar to 1128
the one predicted by the Ku-band NSCAT GMF. The main difference between the two GMF’s is 1129
in the much greater upwind cross-wind modulation seen at Ka-band, which will improve wind 1130
direction estimation. The observed modulation also exceeds the one observed at Ka-band from a 1131
platform in the Black Sea by Yurovsky et al. [21], but, due to platform geometry, the cross-wind 1132
sampling may not have been optimal for these incidence angles. Yurovsky et al., also have a global 1133
analytic form for their GMF that may constrain the modulation somewhat, and comparisons 1134
against actual data points (Yurovsky, personal communication) shows better agreement with 1135
DopplerScatt observations than the analytic formula. Resolving these discrepancies will require 1136
additional data, but the current results, as well as those of Yurovsky et al., show that there is 1137
sufficient wind speed and direction sensitivity at Ka-band to obtain wind estimation performance 1138
similar to that of Ku-band scatterometers, such as QuikSCAT. Formal errors in the estimated 1139
wind speed and direction indicate performance better than spaceborne scatterometers, but the 1140
limited comparison against buoy data shows similar performance, possibly pointing to needed 1141
improvements in the GMF, possibly including current effects. 1142
3. Examined the local wind dependent part of the Doppler velocity signature. While the signature 1143
is roughly aligned with the wind direction, as for other frequencies, it deviates slightly from the 1144
true wind direction, in a fashion consistent with expected direction differences consistent with 1145
those expected for the sum of Lagrangian and Eulerian wind-driven currents [45]. However, 1146
the wind speed dependence of the Doppler currents is quite different from the one observed 1147
at C-band [8,13], where the Doppler velocity is nearly linearly dependent on wind speed. By 1148
contrast, at Ka-band there is only a linear dependence for low winds, and the magnitude of 1149
the dependence stabilizes after a wind speed of about 4.5 m/s. In addition, the shape of the 1150
wind-dependent response is close to a sinusoid with azimuth angle; i.e., the expected response 1151
of a constant velocity vector, albeit, one that seems to propagate at a small angle wind speed, 1152
consistent with wind-drift measurements with HF radars [45]. This behavior was explained as 1153
due to the modulation of the backscatter cross section through a modulation transfer function 1154
(MTF) consistent with those previously observed at Ka-band. The lack of dependence of the 1155
wind correction with respect to wind speed makes the estimation of the non-wind driven part 1156
of the surface current much less sensitive to wind speed variations, although still sensitive to 1157
wind direction errors. Given that the wind-dependent correction can be made with the same 1158
instrument as the Doppler velocities, this combination is scalable to a spaceborne instrument. 1159
Acknowledgments: The research presented in the paper was carried out at the Jet Propulsion Laboratory, 1160
California Institute of Technology, under contract with the National Aeronautics and Space Administration. 1161
Copyright 2018 California Institute of Technology. U.S. Government sponsorship acknowledged. The authors 1162
would like to thank Y. Yurovsky, S. Grodsky, B. Chapron, and V. Kudryavtsev for discussions and comparisons 1163
with their unpublished results. They would also like to thank J. Molemaker, T. Ozgokmen, G. Jacobs and others in 1164
the SPLASH team for their collaboration and discussions. They would like to thank A. Thompson, Caltech, for 1165
collaboration during the Monterey Bay campaign. E. Rodríguez would like to thank T. Farrar and A. Clarke for 1166
useful discussions regarding Stokes drift. Finally, they would like to acknowledge the contribution of others in 1167
the DopplerScatt team that made this work possible. 1168
Author Contributions:E. Rodríguez derived the measurement principle and designed the end-to-end processing 1169
architecture. He was responsible for the end-to-end processing of the velocity data, including error budget, 1170
processing and calibration algorithms, analysis of the surface velocity data and current GMF, as well as the 1171
paper organization and discussion. He also led the design of the experimental flight campaigns. A. Wineteer 1172
led the development of the wind GMF, the wind estimation algorithm, and validation of the wind results. D. 1173
Perkovic-Martin was the PI for the DopplerScatt IIP and managed the instrument team and the deployment 1174
campaigns. T. Gál analyzed the instrument test data, developed the end-to-end signal processing framework, 1175
and implemented the ground processor and calibration algorithms to range compression. B. Stiles led the wind 1176
calibration algorithms and collaborated in the wind estimation algorithms. N. Niamsuwan was responsible for 1177
backscatter and antenna processing algorithms. R. Rodriguez Monje led the hardware implementation and testing 1178
of the instrument. 1179
Conflicts of Interest:The authors declare no conflict of interest. 1180
Appendix A 1181
The DopplerScatt concept relies on using the phase difference between pulse pairs to estimate 1182
radial velocity components. In this section, we derive the expected characteristics of this quantity as a 1183
function of the viewing geometry, surface and platform motion. 1184
The return complex amplitude,Ei, for theithpulse (i=1, 2) in a pulse pair is given by
1185
Ei(ti,r0)∼n ti,r0
+
Z
dS G(x,ti)χ(r0−r(ti,x))exp[−2ikr(ti,x)]s(ti,x) (A1)
where∼ means equality up to a constant unimportant for our results; G(x) is the one-way 1186
antenna pattern;χ(r)is the range point target response;r0is the nominal pixel range in the time 1187
sampled signal;k=2π/λis the radar wavenumber;ri(ti, x)is the range from the radar to the location
1188
xat timeti;niis the thermal noise contribution. Finally,s(ti, x)is the complex reflection coefficient,
1189
defined such that averaging over speckle realizations, it satisfies the equation 1190 s(x)s∗ x0 S =δ x−x0 σ0(x)γTS(|τ|) (A2)
wherehiSdenotes averaging over speckle realizations;σ0(x)is the normalized radar cross section
1191
for the desired transmit/receive polarization combination; τ = t1−t2is the pulse-pair temporal 1192
separation; and, finally,γTS(|τ|)represents the temporal correlation due to scattering patch velocity 1193
deformation or lifetime, but does not include decorrelation due to resolved large wave motion. Over 1194
the period of observations, we assume the radar cross section statistics remains homogeneous in time, 1195
althoughσ0varies in space. At this time we do not have a good model for the patch decorrelation time,
1196
but in Section3we show that it does not seem to be a major contributor to pulse to pulse correlation. 1197
Similarly, the thermal noise contribution is assumed to satisfy 1198
hn1n∗2iS =δ1,2N (A3)
whereNis, up to a constant, the thermal noise power. 1199
The expected value of the pulse-pair complex product averaged over speckle realizations,hE1E2∗iS,
1200 is given by 1201 hE1E2∗iS ∼ Z dS G2(x)χ2 r0−ri(t,x) σ0(x)exp[−2ik(r(t1,x)−r(t2,x))] (A4)
Assume that over the period of observationrP(t), is given by rP(t) = rP(0) +vPt, where the
1202
time origin is chosen to lie at the mid-point of the burst of pulses used for observation. The position 1203
of a small (i.e., on the order of a few wavelengths) patch of moving surface scatterers, rS(t), is
1204
given byrS(t) = rS(0) + (vE+vW(x))t, wherevE is the Earth’s velocity in the inertial coordinate
1205
system, andvWis the velocity of the water patch of scatterers. We do not make any assumptions
1206
about the velocity of the scatterers, aside from the fact that their total velocity will consist of an 1207
intrinsic velocity (which may, but need not be, be the Bragg velocity) superimposed on the wave 1208
orbital velocity and additional current terms, possibly including wind drift and surface current 1209
components. The vector pointing between the platform to the target patch of scatterers is then given by 1210
r(t) =rS(t)−rP(t) =r(0) + vW−vpt, wherevp=vP−vEis the platform velocity vector relative
1211
to the moving Earth, and Earth motion is assumed to be constant over the radar footprint. With these 1212
conventions, the range between platform and target can be approximated by 1213 r(t,x)≈r(0,x) " 1+`ˆ(x)· vW(x)−vp t r(0,x) + +O v pt r(0) 2!# (A5) where ˆ`=r/r, and we can write the range difference as
1214
r(t1,x)−r(t2,x)≈`ˆ(x)· vW(x)−vpτ (A6) whereτ=t1−t2.
1215
To make further progress, we introduce the system spatial weighting function f(x,y)defined by 1216
f(x,y) = G
2(x)χ2(r0−ri(t,x)) R
dS G2(x)χ2(r0−ri(t,x)) (A7)
and define the power weighted centroid of any quantityη=ηC+η0as 1217
ηC=
Z
dS f(x,y)η(x,y) (A8)
where a prime denotes the variation of the variable relative to the centroid value. We evaluate 1218
the integral in a horizontal coordinate system defined on the tangent plane, choosing the origin 1219
of the coordinate system as(xC,yC)and writing the horizontal coordinate vector asx = (x,y) =
1220
(xC+x0,yC+y0) =xC+x0. If ˆ`Cis the look vector from the platform to(xC,yC)the look vector will
1221
be ˆ`=`ˆC+δ`ˆ(x0,y0).
1222
We decompose the water surface velocity into a constant component,vW, a gradient over the
1223
resolution cell, and a “random” component,δvW(x0), due to unresolved wave motion and current
1224
variability inside the resolution cell: 1225
vW(x) =vW+ x0· ∇HvW(xC) +δvW x0 (A9)
where∇His the gradient in the tangent plane coordinates.
1226
Theσ0is decomposed in a similar fashion
1227
σ0(x) =σ0+ x0· ∇Hσ0(xC) +δσ0 x0 (A10)
The mean and gradients ofσ0are mostly due to the mean wind speed and its spatial gradient,
1228
while the random variations,δσ0, are due to cross section variations within the resolution cell caused
1229
by changes in the incidence angle by large-wave tilts and by hydrodynamic modulation of small waves 1230
by large waves. In general, it will be assumed that the fluctuations of the cross section across the 1231
footprint,δσ0, relative to the mean value,σ0, are small, and we can discard quadratic and higher terms
1232
inδσ0/σ0.
1233
After making these replacements, we can rewrite the total complex coherence,γ, as 1234
γ(τ) =γNγTS(τ)γD(τ) (A11)
where the noise correlation term is given by γN =
1+SNR−1−1, where SNR is the 1235
signal-to-noise ratio. The Doppler correlation term is given by 1236 γD(τ) =exp[−iΦC] Z dS f x0ID x0 IG x0 IR x0 (A12) whereΦCis the phase contribution due to the Doppler shift between the platform and the mean
1237
current over the footprintΦC=2k`ˆC· vp−vW
τ. 1238
The terms in the integrand are: ID, the variations of the Doppler over the footprint; IG, the
1239
contributions due to gradients in the current andσ0; andIR, random contributions from sub-resolution
1240
cell variations in the wave velocities and hydrodynamic modulations ofσ0. They are explicitly given
1241 by 1242 ID=exp h −2ikδ`ˆ x0· vp−vW τ i (A13) IG ≈exp h 2ik x0· ∇H ˆ `C·vW τ i 1+(x0· ∇H)σ0 σ0 (A14) IR=exp h 2ik`ˆC·δvτ i 1+ δσ0 σ0 (A15) where we have neglected cross terms between the gradient and random variations ofσ0, where
1243
we expect little correlation due to the different generation mechanisms, and will disappear when 1244
averaging over the random components, as described below. 1245
Since it is not possible to resolve phenomena smaller than the resolution cell, we calculate the 1246
expected value of the random term by performing averaging over unresolved wave and brightness 1247
modulations, caused by waves. Note that for small enough range resolutions, some of the wave 1248
motions may be resolved and part of the observed Doppler shift. The average over unresolved waves, 1249
which will be denoted byhiW, results in
1250 hIRiW≈exp 2ik δσ0 σ0 ˆ `C·δvW W τ γTW(τ) (A16) γTW(τ) =exp −12(2kτ)2 ˆ `C·δvW 2 W (A17) After averaging, neither of these terms depends onx0and they can be extracted from the integral. 1251
The phase term contributes a bias term which modifiesΦCwith a shift due to correlation between
1252
wave motion andσ0modulations
1253 Φ0=2kˆ`C·vp−vW+δσ0 σ0 δvW W τ (A18)
Note that the surface current part in the inner parenthesis can be written as 1254 D ˆ `C·(vW+δvW) (σ0+δσ0) E W hσ0+δσ0iW (A19) which is equal to the Doppler current term proposed by Chapron and collaborators [8]based on a 1255
heuristic model that weighted the Doppler contribution for each surface patch by the local brightness. 1256
This model has been subsequently been refined into the DopRIM model to include various scattering 1257
mechanisms, and we refer the reader to this literature for a detailed discussion of this term [17–19]. 1258
TheγTW term is a temporal correlation term due to the Doppler bandwidth of the surface
1259
waves. It can be combined with the patch correlation term to give a total temporal correlation, 1260
γT(τ) =γTS(τ)γTW(τ). 1261
To perform the integral in equation (A12), write the look vector as a function of the look angle,θ, 1262
relative to the local vertical at the platform, ˆzP, and the azimuth angle,φ, defined as the angle relative
1263 to ˆxP= vp−zˆP·vpzˆP / vp−zˆP·vpzˆP
, the component of the Earth relative velocity vector in the 1264
plane perpendicular to the local normal, which is assumed to be the plane of rotation of the antenna. 1265
The look vector can then be written as ˆ` = (cosφˆxP+sinφyˆP)sinθ−cosθzˆP,where ˆyP = zˆP×xˆP.
1266
Expandingφ=φC+φ0,θ=θC+θ0, and aligning the tangent plane coordinate system so that they0 1267
coordinate is along the plane of incidence, one can writeφ0 =x0/(rCsinθC)andθ0 = y0cosθ(Ci)/rC,
1268
whereθ(Ci)is the local incidence angle at the resolution cell center. It is given byθC(i)=θC+α, whereα 1269
is the angle between the platform and the resolution cell center, as measured from the Earth’s center: 1270
sinα= (rC/RE)sinθC), whereREis the local Earth radius. With these definitions, we can write
1271
δ`ˆ x0= [(−ˆxPsinφC+ˆyPcosφC)]
x0 rC+ [cos
θC(ˆxPcosφC+ˆyPsinφC) +zˆPsinθC]
y0cosθ(Ci)
rC
(A20) Collecting terms inx0,y0, the integral forγD(after removing the wave components) becomes
1272 γD= Z d2x0 f(x0)exp −i κ·x0 1+ x0· ∇Hσ0 σ0 (A21) κx0(τ) =2kτ "
(−ˆxPsinφC+ˆyPcosφC)
vp−vW rC − ∂x ˆ `C·vW # (A22) κy0(τ) =2kτ "
(cosθC(ˆxPcosφC+ ˆyPsinφC) +zˆPsinθC)
vp−vWcosθ(Ci) rC − ∂y ˆ `C·vW # (A23) We can rewrite theγDterms as
1273 γD= 1+i ∇Hσ0 σ0 · ∇κ Z d2x0 f(x0)exp −i κ·x0 (A24)
The integral is recognized as a Fourier transform, and we can write 1274 γD≈exp[i2kτvrG]· ∇κ ˜ f(κ(τ)) f˜(κ(τ)) (A25) vrG= 1 2kτ ∇Hσ0 σ0 ·< ∇κf˜(κ(τ)) f˜(κ(τ)) (A26)
where ˜f(κ)denotes the Fourier transform coefficient of f(x0)evaluated atkx,ky. We assume that
1275
the change in cross section due to the long-wavelengthσ0gradient is small compared to the mean cross
1276
section, and<(z)represents the real part ofz.vrGis the error in the estimated radial velocity caused