## 3 Methods  
### Wind Field Retrieval and Decay Index  
When wind speed exceeds 25 m/s, co-polarized backscattering from the ocean becomes saturated, and thus wind speeds retrieved from co-polarized SAR images have large errors. However, cross-polarized NRCS remains sensitive to high winds and is independent of wind direction. Recent studies suggest that the cross-polarized SAR imagery has the capability of retrieving TC wind speeds over 70 m/s [13; 28].  
Geophysical model function (GMF) is the formula relating wind speed, wind direction and incident angle to NRCS. In this study, we utilized the MMS1A model for wind retrieval from Sentinel-1 VH-polarization images. This GMF is the modified version of the MS1A model [26], refined with the Sentinel-1 EW mode images and collocated NOAA stepped frequency microwave radiometer (SFMR) wind measurements. After modification, the MMS1A model can be used to retrieve TC wind field at 1 km resolution. Compared with best tracks, the retrieved TC intensities have a bias of -3.91 m/s and a root mean-square error of 7.99 m/s. The comparison between MMS1A and MS1A is presented in [18].  
Based on our dataset, Figure 3 shows the MMS1A-retrieved maximum wind speeds and the TC motion speeds provided by best tracks. Different colors stand for the cases observed in different ocean basins. Storm motion speed was calculated using the TC center locations at the two closest times to the SAR acquisition time. In all cases, Typhoon Jebi (2018) had the highest wind speed of 70.72 m/s. Hurricane Gaston (2016), which was in its dissipation stage at the time of SAR sensing, had the fastest storm motion speed of 13.01 m/s. In the following parts of this paper, these two cases will be used as examples to illustrate our methods.  
Figure 3: Maximum wind speeds retrieved by the MMS1A model from the Sentinel-1 EW mode images and TC motion speeds calculated using time and center locations provided by best tracks. Red, blue and green dots stand for the cases observed in the Atlantic, Eastern Pacific and Western Pacific, respectively.  
Two examples of SAR images and wind retrievals are shown in Figure 4. Different eyewall shapes can be observed. In general, a storm has different eyewall shapes in different life stages. When the storm intensity is high, the eyewall is symmetric and closed. Meanwhile, if a storm is a minimal hurricane/typhoon, its eyewall is asymmetric or open [29]. TC motion directions are also marked with arrows in Figure 4c,d. According to the storm motion direction, the whole wind field was divided into three sections: the right, back, and left side. Averaged wind speeds were compared between the three sections. In addition, the correlation coefficient between wind speed and radius (\(Cor_{\text{wr}}\)) was computed for all cases. \(Cor_{\text{wr}}\) is an absolute value that represents the dependence of wind on the radius that is between the maximum wind radius (\(r_{\text{m}}\)) and 200 km.  
In order to compute the decay index (\(DI\)), we transformed wind retrieval maps to radial wind distributions, with Typhoon Jebi in Figure 5a and Hurricane Gaston in Figure 5b. Based on least-squares approximation, the power function shown in Equation (1) was used to fit the relationship between wind and the radius from \(r_{\text{m}}\) to 200 km and derive the parameters of \(a,b,\) and \(c\). The values of \(r_{\text{m}}\), \(a\), \(b\), and \(c\) were integrated into Equation (2) to calculate the decay index.  
\[U_{10}(r)=ar^{b}+c,\ r_{\text{m}}\leq r\leq 200 \tag{1}\]  
Figure 4: Sentinel-1 SAR images of (**a**) Typhoon Jebi on 31 August 2018 and (**b**) Hurricane Gaston on 13 October 2018. Corresponding MMS1A wind retrievals of (**c**) Typhoon Jebi and (**d**) Hurricane Gaston. Black arrow represents the storm motion direction computed from best tracks.  
\[DI=a\frac{r_{\text{m}}{}^{b}-200^{b}}{200-r_{\text{m}}} \tag{2}\]  
in which \(U_{10}\) is the surface wind speed (m/s) at a height of 10 m, and \(r\) is the distance from the storm center in kilometers.