## 2 Dielectric Properties and Channel Modeling
### Dielectric Properties of Skin and Rainwater  
The skin is the largest organ of mammals and its structure is composed of three interconnected layers of tissue: the epidermis as the outermost layer, the dermis as the middle layer, and the hypodermis as the innermost layer [18]. In this work, we consider the epidermis and dermis (with the thickness in the range of 0.06-0.1 mm and 1.2-2.8 mm, respectively [19]) in the theoretical model due to their ability to limit the penetration depth of MMW/sub-THz waves as discussed later. The dielectric properties of skin are characterized by measurements of its relative complex dielectric constant \(\varepsilon=\epsilon^{\prime}-je^{\prime\prime}\) and have been proposed to be described by the single Debye model [20], the four Cole-Cole dispersion model [21] and other combined methods [22; 23; 24]. The four Cole-Cole model is considered the most widely used method based on open online databases [25]. It takes the skin to be a uniformly isotropic passive conducting medium [21] and can be expressed as  
\[\varepsilon_{skin} =\varepsilon_{\infty}+\sum_{n=1}^{4}\frac{\Delta\varepsilon_{n} }{1+\left(j\omega\tau_{n}\right)^{1-\alpha_{n}}}+\frac{\sigma}{j\omega \varepsilon_{0}} \tag{1}\]  
Here, the parameter \(\omega\) refers to the angular frequency and \(\tau\) refers to the relaxation time. Parameter \(\varepsilon_{\infty}\) represents the dielectric constant in the high-frequency limit; \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), and \(\alpha_{4}\) indicate the broadening due to dispersion; \(\sigma\) is the ionic conductivity of the skin; and \(\varepsilon_{0}\) represents the dielectric property of free space. For dry skin, the parameters are set as \(\tau_{1}\) = 7.23 ps, \(\tau_{2}\) = 32.48 ns, \(\varepsilon_{\infty}\) = 4.0, \(\alpha_{1}\) = 0.0, \(\alpha_{2}\) = 0.2, \(\Delta\varepsilon_{1}\) = 32.0, \(\Delta\varepsilon_{2}\) = 1100, and \(\sigma\) = 0.0002. For wet skin, parameters are \(\tau_{1}\) = 7.96 ps, \(\tau_{2}\) = 79.58 ns, \(\tau_{3}\) = 1.59 \(\upmu\)s, \(\tau_{4}\) = 1.592 ms, \(\varepsilon_{\infty}\) = 4.0, \(\alpha_{1}\) = 0.1, \(\alpha_{2}\) = 0.0, \(\alpha_{3}\) = 0.16, \(\alpha_{4}\) = 0.2, \(\Delta\varepsilon_{1}\) = 39.0, \(\Delta\varepsilon_{2}\) = 280, \(\Delta\varepsilon_{3}\) = 3.0 \(\times\) 10\({}^{4}\), \(\Delta\varepsilon_{4}\) = 3.0 \(\times\) 10\({}^{4}\), and \(\sigma\) = 0.0004. The sweat duct inside the skin, with its radius ranging between 45 \(\upmu\)m and 50 \(\upmu\)m [26], is neglected due to its smaller size compared with the wavelength of MMW/sub-THz waves. We estimated the penetration depth by using the general expression \(p=(67.52/f[\text{MHz}))[(\{\epsilon^{\prime}\}^{2}+(\epsilon^{\prime\prime}) ^{2}]^{1/2}-\varepsilon^{\prime}]^{-1/2}\) based on Equation (1) and show its dependence on operating frequency in Figure 1a. The penetration depth decreases exponentially, which agrees well with the measured values set in the ICNIRP guidelines [2]. The dielectric properties of rainwater are mainly determined by the vibration mode of hydrogen bonds between water molecules. The bonds resonate and thus exert strong absorption on MMW/sub-THz waves. The absorbed energy is converted into kinetic energy for the irregular motion of water molecules and then transformed into thermal energy with an increase in mutual collision frequency [27]. The dielectric properties of water can be described by the Double-Debye dielectric model (D3M), which was first developed for saline water based on experimental measurements [28; 29; 30; 31] and can also be reduced to pure water [32] in the frequency range up to 1000 GHz [33]. It can be expressed [34] as  
\[\varepsilon_{water}=\varepsilon_{w0}-f\cdot\sum_{n=1}^{2}\frac{\varepsilon_{w (n-1)}-\varepsilon_{wn}}{f+j\gamma_{n}} \tag{2}\]  
with \(\varepsilon_{w0}(T)=77.66-103.3\cdot T_{in}\) as the static dielectric constant of water and \(T_{in}=1-300/(273.15+T)\) as the modified relative inverse temperature variable.