## 2 Analytical Framework
### Mean Wind Speed
#### 2.1.1 Logarithmic Law Wind Profile  
Normally, the observations by Global Position System (GPS) dropsonde and Doppler radar show that the variation of mean wind speed with height follows the logarithmic law in the lower part of tropical cyclone boundary layer [21; 22; 23; 24; 25]. Thus, the logarithmic law can be used to describe lower boundary layer and the outer-vortex regions of a tropical cyclone:  
\[U_{s}(z_{s})=\frac{u_{s}}{k}\ln\left(\frac{z_{s}}{z_{0s}}\right), \tag{1}\]  
where \(U_{s}(z_{s})\) represents the mean wind speed at height \(z_{s}\) over the standard exposure. Specifically, the standard exposure in this study corresponds to the roughness length \(z_{0s}=0.03\) m, the reference height is 10 m, and the time scale for the average value is 10 min. \(u_{s}\) denotes the friction velocity over the standard exposure, and \(k\approx 0.40\) is the von Karman constant.  
According to Equation (1), the key procedure to standardize the mean wind speed is to determine the relationship of the friction velocities in various terrains. As all anemometers used in this study are set between 10-60 m height, it is reasonable to assume that the friction velocity in the lower tropical cyclone boundary layer is a height-independent constant [25; 26; 27]. Based on the assumption of local equilibrium conditions, the transition model in Engineering Sciences Data Unit (ESDU) [28], which has been applied to convert the 3 s peak speed over open-terrain and the 1-min mean wind speed above open water in hurricane by [PERSON] et al. [29], is employed:  
\[\frac{u_{s}}{u_{*m}}=\frac{\ln\left(\frac{10^{5}}{z_{0s}}\right)}{\ln\left( \frac{10^{5}}{z_{0m}}\right)}, \tag{2}\]  
where \(u_{*m}\) is the friction velocity over the field measured exposure with roughness length of \(z_{0m}\). Then the relation of mean wind speeds with different terrains can be accordingly expressed as:  
\[\frac{U_{s}(z_{s})}{U_{m}(z_{m})}=\frac{\ln\left(\frac{10^{5}}{z_{0s}}\right) \,\ln\left(\frac{z_{s}}{z_{0s}}\right)}{\ln\left(\frac{10^{5}}{z_{0m}}\right) \,\ln\left(\frac{z_{m}}{z_{0m}}\right)}, \tag{3}\]  
where \(U_{m}(z_{m})\) is the mean wind speed measured at experiment station with height \(z_{m}\) and roughness \(z_{0m}\). In this model, the gradient balance assumption, which has been demonstrated to be valid at asufficiently high altitude [30], is adopted. The super-gradient flows were observed in the boundary layer of some tropical cyclones, however, it is not systematic in tropical cyclones, especially in overland conditions [31; 32]. The case of super-gradient flows in tropical cyclones will discussed in next section.  
The relation among mean wind speeds of various averaging times can be expressed as [33]:  
\[U_{\tau}(z)=U_{3600}(z)\left[1+\frac{\beta^{0.5}c(\tau)}{2.5\ln\left(\frac{z}{ z_{0}}\right)}\right], \tag{4}\]  
where \(\tau\) denotes the averaging time; \(U_{\tau}(z)\) and \(U_{3600}(z)\) are respectively \(\tau\)-s mean and 1-h mean wind speeds; \(\beta\) represents the ratio of the fluctuating wind speed variance to the square of friction velocity; \(c(\tau)\) is an averaging time-related parameter that determined by statistical characteristics of wind speed measurements.