# Sea Surface Ka-Band Doppler Measurements: Analysis and Model Development

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Radar

#### 2.2. Hydro-Meteorological Measurements

## 3. Results

#### 3.1. Look Geometry Dependence

#### 3.2. Sea State Dependence

## 4. Model

#### 4.1. Background (DopRIM Approach)

#### 4.2. Moderate and Large Incidence Angles

#### 4.3. Small Incidence Angles

#### 4.4. The Semi-Empirical Model

#### 4.5. Model Validation

## 5. Discussion

- Ku-band, data from the FPN-SAXON experiment in the North Sea, VV and HH polarizations, $\theta =$ 50, 60, 70, 80${}^{\circ}$, upwind and crosswind azimuth with co-aligned winds and waves, wind speed $U=7,8,16$ m/s. Doppler spectra from Figures 1–3 in [32] are digitized manually and integrated to obtain the DC.
- C-band, global Advanced SAR data generalized in the CDOP GMF [14], VV and HH polarization, $\theta =$ 17–42${}^{\circ}$, $U=$ 1–17 m/s, all azimuths.
- X-band, Wavemill data collected in the Irish Sea [35] (their Table 1), VV polarization $\theta =$ 27–43${}^{\circ}$, all azimuths, mixed sea with $U=5.5$ m/s and swell oblique to the wind direction, swell wavelength $\lambda =50$ m.
- Ka-band, AirSWOT campaign data collected in the Gulf of Mexico [19] (their Figure 16c), VV polarization, $\theta =$ 0–23${}^{\circ}$, downwave/wind azimuth, $U=8$ m/s, peak wavelength $\lambda =45$ m.
- Ka-band, DopplerScatt measurements along the West Coast of North America and in the Gulf of Mexico [21], VV polarization, $\theta ={56}^{\circ}$, all azimuths, $U=$3–15 m/s.

#### 5.1. Look Geometry Dependence

#### 5.2. Wind Speed Dependence

#### 5.3. Upwind/Downwind Asymmetry

#### 5.4. Crosswind Doppler Centroid

#### 5.5. Swell Impact

## 6. Conclusions

## Supplementary Materials

^{®}Function, File S2: KaDOP Python

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## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

DC | Doppler centroid |

HH | Horizontal transmit–receive polarization |

MSS | Mean-square slope |

MTF | Modulation transfer function |

NRCS | Normalized radar cross-section |

SWH | Significant wave height |

VV | Vertical transmit–receive polarization |

## Appendix A. Empirical MTF Modification

**Table A1.**Coefficients for the wind-sea-modulation transfer function (MTF), ${M}_{\mathrm{ws}}$ (original empirical MTF [40] with nadir phase forced to ${180}^{\circ}$).

Index | VV-Polarization | HH-Polarization | ||
---|---|---|---|---|

$\mathit{i},\mathit{j},\mathit{k}$ | ${\mathit{B}}_{\mathit{ijk}}$ | ${\mathit{C}}_{\mathit{ijk}}$ | ${\mathit{B}}_{\mathit{ijk}}$ | ${\mathit{C}}_{\mathit{ijk}}$ |

0,0,0 | $+2.037\times {10}^{0}$ | $-9.992\times {10}^{-1}$$-1.859\times {10}^{-3}\xb7i$ | $+2.038\times {10}^{0}$ | $-1.000\times {10}^{0}-2.004\times {10}^{-3}\xb7i$ |

1,0,0 | $-9.956\times {10}^{-3}$ | $+9.995\times {10}^{-2}-3.729\times {10}^{-2}\xb7i$ | $+6.743\times {10}^{-2}$ | $+1.401\times {10}^{-1}-3.822\times {10}^{-2}\xb7i$ |

2,0,0 | $+1.733\times {10}^{-3}$ | $-9.495\times {10}^{-4}+5.074\times {10}^{-4}\xb7i$ | $-1.545\times {10}^{-3}$ | $-2.833\times {10}^{-3}+6.392\times {10}^{-4}\xb7i$ |

3,0,0 | $-2.111\times {10}^{-5}$ | $-1.742\times {10}^{-6}+2.931\times {10}^{-6}\xb7i$ | $+1.167\times {10}^{-5}$ | $+1.756\times {10}^{-5}-1.326\times {10}^{-6}\xb7i$ |

0,1,0 | $-1.704\times {10}^{-2}$ | $-2.063\times {10}^{-3}+4.317\times {10}^{-3}\xb7i$ | $-1.717\times {10}^{-2}$ | $-2.510\times {10}^{-3}+5.669\times {10}^{-3}\xb7i$ |

1,1,0 | $-4.003\times {10}^{-2}$ | $-2.021\times {10}^{-2}+1.328\times {10}^{-1}\xb7i$ | $-2.064\times {10}^{-2}$ | $-1.886\times {10}^{-3}+1.301\times {10}^{-1}\xb7i$ |

2,1,0 | $+2.213\times {10}^{-3}$ | $+1.038\times {10}^{-3}-5.527\times {10}^{-3}\xb7i$ | $+1.172\times {10}^{-3}$ | $+2.218\times {10}^{-4}-5.441\times {10}^{-3}\xb7i$ |

3,1,0 | $-1.778\times {10}^{-5}$ | $-1.184\times {10}^{-5}+4.932\times {10}^{-5}\xb7i$ | $-6.112\times {10}^{-6}$ | $-2.769\times {10}^{-6}+5.318\times {10}^{-5}\xb7i$ |

0,2,0 | $-2.934\times {10}^{-2}$ | $-5.651\times {10}^{-5}+1.290\times {10}^{-3}\xb7i$ | $-2.939\times {10}^{-2}$ | $+1.739\times {10}^{-3}+1.255\times {10}^{-3}\xb7i$ |

1,2,0 | $+2.755\times {10}^{-2}$ | $+7.639\times {10}^{-2}+7.101\times {10}^{-2}\xb7i$ | $+4.007\times {10}^{-3}$ | $+3.758\times {10}^{-2}+7.395\times {10}^{-2}\xb7i$ |

2,2,0 | $+1.382\times {10}^{-3}$ | $-3.142\times {10}^{-3}-2.127\times {10}^{-3}\xb7i$ | $+1.483\times {10}^{-3}$ | $-1.072\times {10}^{-3}-2.254\times {10}^{-3}\xb7i$ |

3,2,0 | $-2.812\times {10}^{-5}$ | $+3.361\times {10}^{-5}+1.363\times {10}^{-5}\xb7i$ | $-2.164\times {10}^{-5}$ | $+8.152\times {10}^{-6}+1.559\times {10}^{-5}\xb7i$ |

0,0,1 | $-2.637\times {10}^{-1}$ | $-1.301\times {10}^{-3}+6.336\times {10}^{-4}\xb7i$ | $-2.644\times {10}^{-1}$ | $-8.840\times {10}^{-4}+6.210\times {10}^{-4}\xb7i$ |

1,0,1 | $+2.458\times {10}^{-2}$ | $-1.061\times {10}^{-2}+4.969\times {10}^{-3}\xb7i$ | $-1.241\times {10}^{-2}$ | $-3.156\times {10}^{-2}+3.907\times {10}^{-3}\xb7i$ |

2,0,1 | $-1.538\times {10}^{-3}$ | $-2.108\times {10}^{-5}-1.405\times {10}^{-5}\xb7i$ | $+2.162\times {10}^{-4}$ | $+8.938\times {10}^{-4}-1.545\times {10}^{-5}\xb7i$ |

3,0,1 | $+1.667\times {10}^{-5}$ | $+2.374\times {10}^{-6}-1.623\times {10}^{-6}\xb7i$ | $-3.483\times {10}^{-7}$ | $-6.512\times {10}^{-6}-4.914\times {10}^{-7}\xb7i$ |

0,1,1 | $+1.342\times {10}^{-2}$ | $+4.740\times {10}^{-4}-8.386\times {10}^{-4}\xb7i$ | $+1.348\times {10}^{-2}$ | $+7.416\times {10}^{-4}-1.537\times {10}^{-3}\xb7i$ |

1,1,1 | $+1.791\times {10}^{-2}$ | $+9.982\times {10}^{-3}-1.344\times {10}^{-2}\xb7i$ | $+7.223\times {10}^{-3}$ | $-2.172\times {10}^{-3}-1.458\times {10}^{-2}\xb7i$ |

2,1,1 | $-1.049\times {10}^{-3}$ | $-4.635\times {10}^{-4}+1.130\times {10}^{-3}\xb7i$ | $-5.037\times {10}^{-4}$ | $+1.054\times {10}^{-4}+1.204\times {10}^{-3}\xb7i$ |

3,1,1 | $+9.159\times {10}^{-6}$ | $+5.154\times {10}^{-6}-1.134\times {10}^{-5}\xb7i$ | $+2.889\times {10}^{-6}$ | $-9.979\times {10}^{-7}-1.415\times {10}^{-5}\xb7i$ |

0,2,1 | $+1.809\times {10}^{-2}$ | $+2.880\times {10}^{-4}-3.980\times {10}^{-4}\xb7i$ | $+1.813\times {10}^{-2}$ | $-6.401\times {10}^{-4}-4.330\times {10}^{-4}\xb7i$ |

1,2,1 | $+8.255\times {10}^{-3}$ | $-2.310\times {10}^{-2}-1.348\times {10}^{-2}\xb7i$ | $+2.314\times {10}^{-2}$ | $-5.070\times {10}^{-3}-1.232\times {10}^{-2}\xb7i$ |

2,2,1 | $-1.287\times {10}^{-3}$ | $+9.360\times {10}^{-4}+5.874\times {10}^{-4}\xb7i$ | $-1.569\times {10}^{-3}$ | $-5.514\times {10}^{-6}+5.293\times {10}^{-4}\xb7i$ |

3,2,1 | $+1.828\times {10}^{-5}$ | $-1.056\times {10}^{-5}-5.155\times {10}^{-6}\xb7i$ | $+1.796\times {10}^{-5}$ | $+8.560\times {10}^{-7}-4.894\times {10}^{-6}\xb7i$ |

**Table A2.**Coefficients for the swell-MTF, ${M}_{\mathrm{sw}}$ (the wind-sea MTF, ${M}_{\mathrm{ws}}$, with crosswind phase forced to ${0}^{\circ}$).

Index | VV-Polarization | HH-Polarization | ||
---|---|---|---|---|

$\mathit{i},\mathit{j},\mathit{k}$ | ${\mathit{B}}_{\mathit{ijk}}$ | ${\mathit{C}}_{\mathit{ijk}}$ | ${\mathit{B}}_{\mathit{ijk}}$ | ${\mathit{C}}_{\mathit{ijk}}$ |

$0,0,0$ | $+2.037\times {10}^{0}$ | $-1.048\times {10}^{0}$$-1.086\times {10}^{-3}$$\xb7i$ | $+2.038\times {10}^{0}$ | $-1.071\times {10}^{0}$$+4.618\times {10}^{-4}$$\xb7i$ |

$1,0,0$ | $-9.956\times {10}^{-3}$ | $+9.780\times {10}^{-2}$$+9.410\times {10}^{-3}$$\xb7i$ | $+6.743\times {10}^{-2}$ | $+1.423\times {10}^{-1}$$+4.037\times {10}^{-3}$$\xb7i$ |

$2,0,0$ | $+1.733\times {10}^{-3}$ | $-9.521\times {10}^{-4}$$-1.330\times {10}^{-3}$$\xb7i$ | $-1.545\times {10}^{-3}$ | $-2.883\times {10}^{-3}$$-1.022\times {10}^{-3}$$\xb7i$ |

$3,0,0$ | $-2.111\times {10}^{-5}$ | $-8.936\times {10}^{-7}$$+1.922\times {10}^{-5}$$\xb7i$ | $+1.167\times {10}^{-5}$ | $+1.838\times {10}^{-5}$$+1.433\times {10}^{-5}$$\xb7i$ |

$0,1,0$ | $-1.704\times {10}^{-2}$ | $-2.054\times {10}^{-2}$$+2.381\times {10}^{-2}$$\xb7i$ | $-1.717\times {10}^{-2}$ | $-1.405\times {10}^{-2}$$+2.765\times {10}^{-2}$$\xb7i$ |

$1,1,0$ | $-4.003\times {10}^{-2}$ | $+4.047\times {10}^{-2}$$+1.545\times {10}^{-1}$$\xb7i$ | $-2.064\times {10}^{-2}$ | $+2.885\times {10}^{-2}$$+1.580\times {10}^{-1}$$\xb7i$ |

$2,1,0$ | $+2.213\times {10}^{-3}$ | $-1.396\times {10}^{-3}$$-5.770\times {10}^{-3}$$\xb7i$ | $+1.172\times {10}^{-3}$ | $-6.833\times {10}^{-4}$$-6.044\times {10}^{-3}$$\xb7i$ |

$3,1,0$ | $-1.778\times {10}^{-5}$ | $+1.341\times {10}^{-5}$$+4.688\times {10}^{-5}$$\xb7i$ | $-6.112\times {10}^{-6}$ | $+4.113\times {10}^{-6}$$+5.471\times {10}^{-5}$$\xb7i$ |

$0,2,0$ | $-2.934\times {10}^{-2}$ | $-4.553\times {10}^{-3}$$-3.923\times {10}^{-3}$$\xb7i$ | $-2.939\times {10}^{-2}$ | $+1.196\times {10}^{-2}$$-5.906\times {10}^{-3}$$\xb7i$ |

$1,2,0$ | $+2.755\times {10}^{-2}$ | $+2.273\times {10}^{-2}$$+1.290\times {10}^{-2}$$\xb7i$ | $+4.007\times {10}^{-3}$ | $-6.953\times {10}^{-3}$$+1.881\times {10}^{-2}$$\xb7i$ |

$2,2,0$ | $+1.382\times {10}^{-3}$ | $-8.407\times {10}^{-4}$$+1.345\times {10}^{-5}$$\xb7i$ | $+1.483\times {10}^{-3}$ | $+3.991\times {10}^{-4}$$-2.665\times {10}^{-4}$$\xb7i$ |

$3,2,0$ | $-2.812\times {10}^{-5}$ | $+9.080\times {10}^{-6}$$-3.645\times {10}^{-6}$$\xb7i$ | $-2.164\times {10}^{-5}$ | $-4.235\times {10}^{-6}$$-1.228\times {10}^{-6}$$\xb7i$ |

$0,0,1$ | $-2.637\times {10}^{-1}$ | $+4.449\times {10}^{-3}$$+1.718\times {10}^{-3}$$\xb7i$ | $-2.644\times {10}^{-1}$ | $+1.677\times {10}^{-2}$$+5.227\times {10}^{-5}$$\xb7i$ |

$1,0,1$ | $+2.458\times {10}^{-2}$ | $-1.172\times {10}^{-2}$$-2.046\times {10}^{-3}$$\xb7i$ | $-1.241\times {10}^{-2}$ | $-3.573\times {10}^{-2}$$-7.999\times {10}^{-4}$$\xb7i$ |

$2,0,1$ | $-1.538\times {10}^{-3}$ | $+9.500\times {10}^{-5}$$+4.016\times {10}^{-4}$$\xb7i$ | $+2.162\times {10}^{-4}$ | $+1.084\times {10}^{-3}$$+3.169\times {10}^{-4}$$\xb7i$ |

$3,0,1$ | $+1.667\times {10}^{-5}$ | $+8.816\times {10}^{-7}$$-5.631\times {10}^{-6}$$\xb7i$ | $-3.483\times {10}^{-7}$ | $-8.536\times {10}^{-6}$$-4.213\times {10}^{-6}$$\xb7i$ |

$0,1,1$ | $+1.342\times {10}^{-2}$ | $+5.159\times {10}^{-3}$$-6.476\times {10}^{-3}$$\xb7i$ | $+1.348\times {10}^{-2}$ | $+3.305\times {10}^{-3}$$-8.653\times {10}^{-3}$$\xb7i$ |

$1,1,1$ | $+1.791\times {10}^{-2}$ | $-9.460\times {10}^{-3}$$-1.412\times {10}^{-2}$$\xb7i$ | $+7.223\times {10}^{-3}$ | $-6.992\times {10}^{-3}$$-1.631\times {10}^{-2}$$\xb7i$ |

$2,1,1$ | $-1.049\times {10}^{-3}$ | $+3.075\times {10}^{-4}$$+9.874\times {10}^{-4}$$\xb7i$ | $-5.037\times {10}^{-4}$ | $+1.321\times {10}^{-4}$$+1.144\times {10}^{-3}$$\xb7i$ |

$3,1,1$ | $+9.159\times {10}^{-6}$ | $-3.260\times {10}^{-6}$$-8.841\times {10}^{-6}$$\xb7i$ | $+2.889\times {10}^{-6}$ | $-5.730\times {10}^{-7}$$-1.266\times {10}^{-5}$$\xb7i$ |

$0,2,1$ | $+1.809\times {10}^{-2}$ | $+1.030\times {10}^{-3}$$+1.201\times {10}^{-3}$$\xb7i$ | $+1.813\times {10}^{-2}$ | $-7.690\times {10}^{-3}$$+1.685\times {10}^{-3}$$\xb7i$ |

$1,2,1$ | $+8.255\times {10}^{-3}$ | $-3.648\times {10}^{-3}$$-5.885\times {10}^{-3}$$\xb7i$ | $+2.314\times {10}^{-2}$ | $+1.171\times {10}^{-2}$$-6.082\times {10}^{-3}$$\xb7i$ |

$2,2,1$ | $-1.287\times {10}^{-3}$ | $+1.829\times {10}^{-6}$$+7.072\times {10}^{-5}$$\xb7i$ | $-1.569\times {10}^{-3}$ | $-6.270\times {10}^{-4}$$+9.248\times {10}^{-5}$$\xb7i$ |

$3,2,1$ | $+1.828\times {10}^{-5}$ | $+1.277\times {10}^{-7}$$+8.062\times {10}^{-8}$$\xb7i$ | $+1.796\times {10}^{-5}$ | $+6.716\times {10}^{-6}$$-1.181\times {10}^{-8}$$\xb7i$ |

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**Figure 1.**(

**a**,

**b**) Black sea research platform with installed (

**c**) Ka-band radar, (

**d**) wire wave gauge, and (

**e**) meteorological sensors.

**Figure 2.**Number of five minute samples for the 2009–2018 data set as a function of incidence angle, $\theta $, and (

**a**) wind speed, (

**b**) significant wave height (SWH), (

**c**) peak wave period, and (

**d**) radar-to-wave azimuth (zero is the upwave).

**Figure 3.**Doppler centroids versus incidence angle, $\theta $, for dual co-polarized (

**a**) vertical transmit–receive polarization (VV) polarization and (

**b**) horizontal transmit–receive polarization (HH) polarization. Color indicates radar-to-wind azimuth. Only co-aligned winds and waves, $|{\varphi}_{\mathrm{wi}}-{\varphi}_{\mathrm{wa}}|<{25}^{\circ}$, are shown. Gaussian noise is added to the nominal $\theta $ for better visibility, $\mathrm{STD}({\theta}_{\mathrm{noise}})={0.25}^{\circ}$.

**Figure 4.**Doppler centroids versus radar-to-wind azimuth for (

**top row**) VV polarization and (

**bottom row**) HH polarization at (

**a**,

**f**) all incidence angles, (

**b**,

**g**) $\theta ={20}^{\circ}$, (

**c**,

**h**) $\theta ={53}^{\circ}$, and (

**d**,

**i**) $\theta ={70}^{\circ}$. Color scheme corresponds to wave-to-wind azimuth. Symbol size corresponds to wind speed. All conditions, including non-aligned winds and waves, are shown.

**Figure 5.**Scatter diagrams of Doppler centroids versus range projection of (

**a**,

**e**) sea surface current, (

**b**,

**f**) wind velocity, and (

**c**,

**g**) wave velocity for VV polarization. The two right plots show scatter diagrams of wind and wave range velocities. Color scale in the

**top row**corresponds to the incidence angle. Color scale in the

**bottom row**corresponds to the deviation of wave range velocity from its average linear wind dependence (black line), i.e., the x-axis distance between a point and the black line in (

**d**,

**h**). ${R}^{2}$ is the squared correlation coefficient. All conditions including non-aligned winds and waves are shown.

**Figure 6.**Doppler centroids versus wind speed for (

**a**,

**e**) all incidence angles, (

**b**,

**f**) $\theta ={20}^{\circ}$, (

**c**,

**g**) $\theta ={53}^{\circ}$, and (

**d**,

**h**) $\theta ={70}^{\circ}$. (

**top row**) VV-polarization, (

**bottom row**) HH-polarization. Symbol size corresponds to the characteristic wave orbital velocity magnitude, $u={\omega}_{\mathrm{p}}{H}_{\mathrm{s}}$. Only co-aligned winds and waves, $|{\varphi}_{\mathrm{wi}}-{\varphi}_{\mathrm{wa}}|<{25}^{\circ}$, are shown.

**Figure 7.**Model versus measurement DC range projection, $V/sin\theta $. (

**top row**) VV-polarization, (

**bottom row**) HH-polarization. (

**left three columns**) correspond to cases with unknown wind drift (estimated as $1.5\%\mathbf{U}$). Left to right: 2D-sea, Equation (14); 1D-sea, Equation (14); equivalent Pierson–Moskowitz (PM) spectral shape, Equation (16). (

**right three columns**) are the same, but for cases with known wind drift. Correlation coefficient, ${R}^{2}$, and root-mean-square error (RMSE) are shown in each panel.

**Figure 8.**Doppler centroid versus incidence angle for various wind speeds, left to right: $U=5,8,10,15$ m/s. (

**top row**) VV-polarization, (

**bottom row**) HH-polarization. Confidence interval corresponds to wind drift variation from 0 to $3\%\mathbf{U}$.

**Figure 9.**Doppler centroid versus radar-to-wind(wave) azimuth for various wind speeds at VV-polarization, left to right: $U=5,10,15$ m/s. (

**top row**) $\theta ={27}^{\circ}$, (

**bottom row**) $\theta ={56}^{\circ}$. Confidence interval corresponds to the wind drift varied from 0 to $3\%\mathbf{U}$.

**Figure 10.**Doppler centroid versus wind speed for (

**a**,

**b**) $\theta ={27}^{\circ}$ and (

**c**,

**d**) $\theta ={56}^{\circ}$. (

**a**,

**c**) VV-polarization, (

**b**,

**d**) HH-polarization. Confidence interval corresponds to wind drift variations between 0 and $3\%\mathbf{U}$.

**Figure 11.**Upwind-to-downwind Doppler centroid ratio versus incidence angle for various wind speeds, U.

**Figure 12.**KaDOP simulation, Equation (16), at $U=6$ m/s for (dotted lines) pure wind sea without swell and (solid lines) mixed sea with (

**a**) downwind swell, (

**b**) crosswind swell, and (

**c**) upwind swell. Swell parameters are ${\lambda}_{\mathrm{sw}}=600$ m, ${T}_{\mathrm{sw}}=20$ s, ${(AK)}_{\mathrm{sw}}$ = 0.05, ${A}_{\mathrm{sw}}=5$ m.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Yurovsky, Y.Y.; Kudryavtsev, V.N.; Grodsky, S.A.; Chapron, B. Sea Surface Ka-Band Doppler Measurements: Analysis and Model Development. *Remote Sens.* **2019**, *11*, 839.
https://doi.org/10.3390/rs11070839

**AMA Style**

Yurovsky YY, Kudryavtsev VN, Grodsky SA, Chapron B. Sea Surface Ka-Band Doppler Measurements: Analysis and Model Development. *Remote Sensing*. 2019; 11(7):839.
https://doi.org/10.3390/rs11070839

**Chicago/Turabian Style**

Yurovsky, Yury Yu., Vladimir N. Kudryavtsev, Semyon A. Grodsky, and Bertrand Chapron. 2019. "Sea Surface Ka-Band Doppler Measurements: Analysis and Model Development" *Remote Sensing* 11, no. 7: 839.
https://doi.org/10.3390/rs11070839