1) Calculation of rain attenuation

The rain attenuation coefficient A [dB], which represents attenuation due to raindrops after propagation over 1 km, can be calculated from the attenuation cross-section Q_t, which is a function of particle diameter, D, the wavelength 𝜆, and the complex reflection coefficient of water droplets m and a raindrop size distribution N(D) as

For this, the attenuation cross-section can be obtained by applying Mie scattering theory to attenuation due to spherical particles in plane-wave radiation. Here we use the formula of Hulst [7] for this:

where a_n and b_n are the Mie scattering coefficients [7] . These are complex amplitude coefficients that characterize the influence of the scattering fields created by multiple scattering within spheres, such as raindrops. We use the value given in Ray [8] for m, the complex reflection coefficient of water droplets.

2) Raindrop size distribution used in calculations

We used several raindrop size distributions for calculating rain attenuation by Mie scattering theory: the M-P, Best, P-S (hail, sleet, and snow), and Weibull distributions. These are shown in Equations (3)-(6), respectively.

The M-P distribution proposed in Marshall and Palmer [9] was found by fitting empirical data recorded in Ottawa, Canada, in 1946 with the filter-paper method. The fit of this distribution to the experimental dataset was not very good for drops with diameter less than 1 mm.

In this, D is the diameter in mm, and R is the precipitation rate in mm/hr.

In 1950, Best [10] proposed a drop-size distribution model based on analysis of a large amount of experimental data. The Best distribution is written as the following.

The P-S distribution was described by Litovinov in 1957 [11] and 1958 [12] as due to Polyakva and Shifrin, based on Russian data for hail, sleet, and snow. This model was also described by Krasyuk, Rozenberg and Chistyakov [13] in 1968 and by Rice and Peebles at the University of Tennessee [14] in 1975. It is a special case of the Gamma distribution proposed by Atlas and Ulbrich [15] in 1984.

Here, and vary according to the rain type, as shown in Table 1.

Sekine and Lind [16] proposed the Weibull distribution in 1982, using FOA data from the National Defence Research Institute of Sweden:

This distribution is still in use for microwave and terahertz applications [17] - [19] .

ITU-R P.838-3 [6] provides a prediction model for rain attenuation. In that model, the attenuation coefficient γ_R [dB/km] from rainfall intensity R is calculated from

Here, the values of k and 𝛼 are determined for a given frequency f in the range 1 to 1000 GHz, and ITU-R P676-6 [20] gives values for these, reportedly found by fitting the attenuation amounts from scattering calculations to a power curve.

Experimental values for rain attenuation at 96, 140, and 225 GHz are taken from results by Nemarich et al. [21] ,

who performed measurements on 26 Jan 1983 at Camp Rilea, Oregon, USA. The measurements were performed. between transmitters and receivers placed 1.3 km apart on flat ground. The maximum rainfall was 10 mm/hr, with rain falling at a similar rate over a long period of time.

The average temperature during the experiment was 8.3˚C, average humidity was 95.7%, and absolute humidity was 1.4 g/m³. Variation in attenuation due to changes in absolute humidity was reportedly estimated to be less than 0.1 dB, so no correction for absolute humidity was performed. Figures 1(a)-(c) show the experimental values as triangles, and the dashed curves indicate regression curves for the experimental values found from fitting.

1) 313 GHz

Experimental values for rain attenuation at 313 GHz are taken from the results of Babkin et al. [22] , who performed measurements in Central Europe between June and July 1969. Transmitters and receivers were placed 1.0 km apart on flat ground for the measurements. The maximum rainfall was 12 mm/hr. Figure 2(a) shows the experimentally obtained values as triangles, and the dashed curves indicate the regression curves fitted to those values. We evaluated rain attenuation at this frequency, with maximum rainfall intensity of 12 mm/hr, in [23] .

2) 355 GHz

Values for rain attenuation at 355 GHz are taken from the results of an experiment and evaluation performed by the authors between 10:00 and 16:00 on 28 Apr 2010 on the campus of the National Defense Academy in Yokosuka, Japan [24] . The maximum rainfall was 25 mm/hr, average temperature was 13.9˚C, average relative humidity was 89.2%, and atmospheric pressure varied between 991 and 994 hPa. Water vapor per unit volume was calculated as 11 g/m³, with the level of variation in attenuation due to atmospheric absorption below 0.5 dB/km, so experimental values were not corrected for absorption attenuation. Figure 2(b) shows the experimental values as triangles, and the dashed curves indicate regression curves fitted to those values.