*3.1. Rainfall E*ff*ects*

The received signal variation and the rain intensity on 7 and 11 June 2017 are presented in Figures 4 and 5, respectively. Assuming other losses (wet antenna attenuation, water vapor attenuation, etc.) are the same, the average signal attenuation values over 1 km distance are also compared in Figures 4b and 5b, which show more clearly the impact of rain on links at different frequencies. The measured rain attenuation result is consistent with the theoretical predictions in Table 2. Although the 32 and 38 GHz links were built over a much longer distance compared to the E-band links, the rain-induced attenuation in the 32 and 38 GHz links was lower. This difference in rain-induced attenuation between the lower and higher frequency mmWave link becomes more significant as the rain intensity increases. The 32 GHz and 38 GHz links are more robust to poor weather conditions compared to the E-band link, although they are deployed over longer distances. The attenuation of the links, especially the E-band link, was a lot more severe than expected in late evening on 7 and 11 June. One possible reason is that

the rain gauge provides a point measurement, while the measured signal attenuation is caused by the rain along the link. Although one side of all links is located on top of the same building, the links are built across different and wide areas, with a separation distance of up to 10 km. The rain intensity that each link experienced along the path could be very different from the rain gauge measurement, and it could contribute to the difference in attenuation value. In addition, the rain gauge that was used for the analysis was located on the rooftop of a building. There could be a significant under catch of rainfall in the gauge especially during windy conditions, and more rainfall could have been detected by the mmWave link.

**Figure 4.** (**a**) Received signal variation of the test links on June 7, 2017; (**b**) received signal variation of the test links averaged over 1 km distance; (**c**) rain rate monitored by a rain gauge.

**Figure 5.** (**a**) Received signal variation of the test links on June 11, 2017. (**b**) received signal variation of the test links averaged over 1 km distance. (**c**) rain rate monitored by a rain gauge.

#### *3.2. Water Vapor Attenuation*

As discussed in the previous section, change in water vapor level may also cause additional attenuation. Atmospheric attenuation of signal level due to dry air and water vapor is related to the air pressure, temperature, and the water vapor density. For the dry period from 13 to 15 June 2017, the changes in temperature, air pressure, and humidity level are presented in Figure 6a, and the variations of the received signal level from the test links are also given in Figure 6. During these sunny days, it can be seen that the received signal level also varies over time as a result of atmospheric effects. The variation of temperature and humidity is inversely related with a correlation coefficient of −0.9. Attenuation from water vapor is a function of the pressure *p* (hPa), temperature *T* (◦C), and the water vapor density ρ (g/m3) [35]. For the frequencies considered in this study, the signal attenuation due to oxygen absorption is negligible. The attenuation due to changes in water vapor density at 32 and 38 GHz is very low, approximately up to 1 dB for a 7-km long link, as shown in Figure 6b,c. As the frequency increases to the E-band, variation in water vapor density can contribute over 0.45–0.55 dB/km for E-band signals, and therefore a total of 1.35–1.65 dB for the 3 km link, which is illustrated in Figure 6d.

**Figure 6.** (**a**) The variations in temperature, humidity, and pressure during 13-15 June, 2017. (**b**) The variations in the signal attenuation and humidity of the 32 GHz link (link 1, data streams 1 and 2). (**c**) Links 1 and 2 at 38 GHz; (**d**) The 72.625 and 82.625 GHz links.

#### *3.3. Data Post-Processing and Uncertainties*

The determination of the baseline level and wet antenna attenuation are very important for accurate estimation of rain rates from the mmWave links [40–42]. Here, we consider the reference level to be the average received signal strength over 3 hours in dry weather before rain. Subtracting the baseline level, also called the reference level, from the actual received signal levels gives the rain-induced, path-integrated attenuation, which can be transformed into the path averaged rain rate.

During rainy periods, the dampening of the radomes of the antenna causes attenuation, and this additional attenuation factor is known as the wet antenna effect [40]. The wet antenna effect has been shown to be consistent for a specific microwave link, but varies from link to link. Therefore, it has been suggested that the wet antenna attenuation depends on the specific link properties, such as the signal polarization, frequency, and the radome material, meaning each link needs to be examined individually. For a rainfall event lasting for a long period of time, the wet antenna attenuation is expected to increase with increasing thickness of water film on the antenna. In addition to wet antenna attenuation, water vapor may also cause additional variation at high frequencies, as shown in Section 3.2. As discussed in [41,42], bias due to the instability of the transmit power of commercial microwave backhaul equipment could be up to 1.6 dB, therefore causing more attenuation to the received signal level. Therefore, hardware (radio, antenna) and alignment possibly also contribute to this difference. This could be studied as future work if long term measurement is carried out and analyzed.

For the experiment period, the measured statistics of rain attenuation versus calculated rain attenuation, based on Equation (9), are plotted in Figure 7. Each point represents the rain rate and measured attenuation value from the link measurement over a 15 min interval. The correction factor, which mainly accounts for wet antennas, is applied to the measurement data. As the water vapor attenuation is insignificant, it is combined in one correction factor. The distribution of rain attenuation values for increasing rain rate before and after applying the correction factor is presented.

**Figure 7.** Rain attenuation statistics from the measurement before correction and after correction in comparison with the calculated rain attenuation, using the ITU model from Equation (9) for different frequencies: (**<sup>a</sup>**,**b**) 32 GHz; (**<sup>c</sup>**,**d**) 38 GHz; (**<sup>e</sup>**,**f**) 72 GHz; (**g**,**h**) 82 GHz.

#### *3.4. Rain Rate Estimation*

We evaluate the rainfall estimates from mmWave test links through three metrics—the Pearson correlation coefficient, the root mean square difference, and the bias.

The linear dependence of the time series data of average rain attenuation obtained from the link measurement *X* = *Ar* and rain rate measurement *Y* = *R* from the rain gauge is estimated by calculating

the correlation coefficient of the variables. If each variable has *I* averaged observations, the Pearson correlation coefficient is calculated as:

$$r(A\_{r,i\prime}R\_i) = r(X\_i, Y\_i) = \frac{1}{I-1} \sum\_{i=1}^{I} \left(\frac{X\_i - \mu\_X}{\sigma\_X}\right) \left(\frac{Y\_i - \mu\_Y}{\sigma\_Y}\right) \tag{12}$$

where μ*X* and σ*X* are the mean and standard deviation of *X*, respectively, and μ*Y* and σ*Y* are the mean and standard deviation of *Y*. Here, *r* ranges from −1 to +1. A high correlation coefficient value shows stronger relation between two data sets. On 7 and 11 June 2017, the signal power attenuation is mainly caused by rainfall and the values are highly correlated, resulting in an average correlation coefficient greater than 0.8. The strong correlation between the receive signal attenuation and rain rate during the measurement period indicates that it is possible to retrieve the rain rate from the receive signals of the mmWave links.

After applying the correction to the signal attenuation, the rain rate is estimated on a 15 min basis using Equation (10). The time series data of average rain rate derived from the mmWave link measurement *X* = *Rlink i*s then compared with the rain rate measurement *Y* = *R* from rain gauge, based on Equation (12). Figures 8 and 9 show the comparison between the link-derived rain rate estimation and rain gauge measurement. The accuracy of the rain estimation using different links is presented in Table 4. The root mean square difference (RMSD) was also computed for accuracy analysis according to the following formula:

$$RMSD = \sqrt{\frac{1}{I} \sum\_{i=1}^{I} (X\_i - \mathbf{Y}\_i)^2} \ (mm/h) \tag{13}$$

The bias is a measure of the average error between the link estimate rain rate and the rain gauge measurements, and it can be calculated using the following formula:

$$Bias = \frac{1}{I} \sum\_{i=1}^{I} (X\_i - Y\_i) \tag{14}$$

The rain rates derived from the three mmWave links are closely related to the observed rain rate recorded by the weather stations, with a very good accuracy. The RMSD is found to be in the range of 0.36–1.00 mm/h for the test links.

Using the latest introduction of MIMO mmWave links, the number of rain estimation values grows linearly with the minimum number of transmitter and receiver antennas of the MIMO link compared to the case of a single rain rate estimation value from a SISO backhaul link. Both the forward and reverse links can be used for rainfall estimation, and the 4 data streams in 2 × 2 MIMO links can effectively provide up to 4 rain rate estimation values over the path. All the data streams can contribute to understanding of the statistics of the local rain rates.

**Figure 8.** Average rain rate per 15 min derived from the signal link compared with the rain gauge measurement on 7 June 2017: (**a**) 32 GHz link; (**b**) 38 GHz link; (**c**) E-band link.

**Figure 9.** Average rain rate per 15 min derived from the signal link compared with the rain gauge measurement on 11 June 2017: (**a**) 32 GHz link; (**b**) 38 GHz link; (**c**) E-band link.


**Table 4.** Accuracy analysis of rain retrieval from the three test links.
