# Ka-Band Doppler Scatterometry: A Strong Wind Case Study
## 3 Sea Surface Cross-Section
### NRCS Decomposition
As the wind increases to moderate speeds \(\approx\)10 m s\({}^{-1}\), the Ka-band polarized Bragg component grows rapidly and, importantly, faster than the wave breaking NP contribution, which decreases the PR in this wind speed range. The difference  
Figure 8: Relative NP contribution versus wind speed. Figure 7: NRCS contribution of co-polarized (VV, HH) and non-polarized (NP) components versus wind speed. in wind growth rates is intuitively understood from the fact that resonant Bragg waves are short (\(\lambda=5.7\) mm in the Ka-band at \(\theta=45^{\circ}\)), and thus respond faster to wind than waves responsible for (micro-scale) wave breaking (\(\lambda\approx 0.1\) m) [59; 60; 61]. This hypothesis is indirectly supported by the fact that the growth rate of polarized components decreases and the local minimum of PR curve shifts towards lower winds (or even is not present) for longer radar, thus Bragg, wavelengths (Figure 6b). With further wind speed increase above 10-12 m s\({}^{-1}\), the Bragg polarized component starts saturating while the NP wave breaking component keeps growing (Figure 7d). At even stronger winds, the NP contribution dominates and the PR gradually increases to 0 dB. The NRCS decomposition, Equation (2), also allows us to estimate the short wind wave spectrum at Bragg wavenumber. Indeed, the NP is canceled out by taking the polarization difference. Then, disregarding the hydrodynamics modulation component and applying the two-scale Bragg model [50; 62; 63; 64],  
\[\sigma_{\rm br}^{\rm PP} = \sigma_{\rm 0 br}^{\rm PP}\big{(}1+g_{\rm PP}\overline{\zeta^{2}} \big{)}, \tag{4}\] \[\sigma_{\rm 0 br}^{\rm PP} = \pi|G_{\rm PP}^{2}|B(\mathbf{k}_{\rm br}),\] (5) \[G_{\rm vv} = \frac{(e-1)(e+(e-1)\sin^{2}\theta)\cos^{2}\theta}{(e\cos\theta+ \sqrt{e-\sin^{2}\theta})^{2}\sin^{2}\theta},\] (6) \[G_{\rm thh} = \frac{(e-1)\cos^{2}\theta}{(\cos\theta+\sqrt{e-\sin^{2}\theta})^ {2}\sin^{2}\theta},\] (7) \[g_{\rm PP} = \frac{1}{2\sigma_{\rm 0 br}^{\rm PP}}\frac{\partial^{2}\sigma_{\rm 0 br }^{\rm PP}}{\partial\theta^{2}}. \tag{8}\]  
the polarization difference, \({\rm PD}=\Delta\sigma_{0}\), is determined only by the radar look geometry, mean-square slope of long waves, and Bragg wave curvature spectrum, \(B_{0}(\mathbf{k}_{\rm br})=k^{A}S(\mathbf{k}_{\rm br})\):  
\[\sigma_{\text{np}}=\sigma_{0}^{\text{vv}}-\frac{\Delta\sigma_{0}}{1-P_{\text{ br}}}, \tag{3}\]0  
where \(\Delta(x)=x^{\rm vv}-x^{\rm hh}\) is the PD operator, \(G_{\rm PP}\) and \(g_{\rm PP}\) are the geometric coefficients (with cross-tilt contribution omitted for \(g_{\rm thh}\), see Figure 5 in [50]), \(\epsilon\) is the complex sea water dielectric constant estimated using [65] for 18 \({}^{\circ}\)C temperature, 18 psu salinity, and 37.5 GHz electromagnetic frequency, \(\overline{\zeta^{2}}\) is the mean-square slope (mss) of modulating waves in the incidence plane, \(\mathbf{k}_{\rm br}=2\mathbf{k}_{\rm r}\sin\theta\) is the Bragg wave vector, \(\mathbf{k}_{\rm r}\) is the radar electromagnetic wave vector, and \(B(\mathbf{k}_{\rm br})=(B_{0}(\mathbf{k}_{\rm br})+B_{0}(-\mathbf{k}_{\rm br }))/2\) is the folded curvature spectrum reflecting that Bragg components directed towards and away from radar both contribute. Wind dependence of the folded curvature spectrum, \(B(\mathbf{k}_{\rm br})\), estimated from Equation (9) and shown in Figure 9a is similar to that of polarized backscatter components in \[\sigma_{\text{np}}=\sigma_{0}^{\text{vv}}-\frac{\Delta\sigma_{0}}{1-P_{\text{ br}}}, \tag{3}\]1 The folded curvature spectra estimated in the four radar frequency bands are interpolated in wavenumber space by polynomial splines (Figure 9b). The resulting spectra have stronger wind dependence in the Ku- and C-bands and display a tendency to saturate at \(U>15\)-20 m s\({}^{-1}\). Errors of curvature spectral retrieval depend in part on the uncertainty of modulating wave mss and the measurement accuracy of the NRCS (PD).