## 2 Materials and Methods  
The considered exposure scenario, shown in Figure 1, was modeled to assess the exposure variability of a single user due to an AP operating at 3.7 GHz with a 64 elements UPA antenna. As detailed in Reference [22], each element of the antenna was composed by a simple patch antenna characterized by three layers. The three layers were composed by the ground layer, the patch layer modeled as PEC materials, and the substrate layer modeled by a dielectric material (dielectric properties: \(\varepsilon_{\mathrm{r}}\) = 2.25 and \(\sigma\) = 0.0005 S/m, data from Reference [13]. The total dimensions of the array were 29 \(\times\) 29 \(\times\) 0.5 cm (right side of Figure 1), and the antenna was excited with a Gaussian signal with a total input power of 100 mW in line with the maximum allowed input power values specified in the 3 rd Generation Partnership Project [23].  
The human exposure levels were evaluated in terms of the \(SAR\) in the child [PERSON]'s model (age = 5 years old, height = 1.1 m, mass = 17.6 kg and BMI = 14.8 kg/m) from the Virtual Classroom [24] as illustrated in the left part of Figure 1. The UPA antenna was placed laterally in respect to the model head at a fixed distance of 50 cm to the central point of the head. The choices of simulating a child model, instead of an adult one, and a lateral configuration of the antenna were suggested by the findings of our previous work [22]: for fixed zero-phase shift among the different array elements (i.e., for the maximum gain in the direction perpendicular to the antenna array), the child [PERSON] model and the lateral configuration of the antenna showed the highest exposure levels.  
To study the impact of the 3D beamforming capability of the UPA antenna, we introduced the two scan angles that regulated the beamforming direction in the H-plane (azimuth plane) and in the E-plane (elevation plane), varying respectively in the range (\(-50^{\circ}\), \(+50^{\circ}\)) and (\(-25^{\circ}\), \(+25^{\circ}\)) as reported in Reference [25]. As an example, Figure 2 shows the spatial antenna gain patterns for the scan angles at the extreme values of their ranges (respectively \(+/-50^{\circ}\) and \(+/-25^{\circ}\) for azimuth and elevation planes).  
Figure 3 shows a flow chart of the methodology adopted. It combines the traditional computational methods with the stochastic dosimetry, an innovative technique that uses surrogate models to obtain the variable of interest with really low computational cost in respect to traditional computational methods. This technique was already applied either in low- or in high-frequency ranges to assess the EM exposure levels of users [19; 20]. The surrogate models were obtained for the \(SAR\) values averaged on the whole body, the whole  
Figure 1: (**a**) Exposure scenario and the two input parameters for the beamforming pattern with their ranges; (**b**) details and gain diagram of the uniform planar array (UPA) antenna with 64 elements at 3.7 GHz.  
Figure 2: Gain radiation patterns of the UPA antenna at 3.7 GHz for four different configurations of the azimuth and elevation angles of beamforming direction.  
head and the whole brain and for the maximum values of \(SAR\) averaged on 10 g for specific tissues (e.g., the skin, the brain gray matter, and the cerebellum).  
The three steps of the process are described in details in the sub-sections below: \"Experiment Design\" motivates the choice for the input setting and the deterministic simulations, \"Surrogate Modeling and Validation\" illustrates the construction of the surrogate models with stochastic dosimetry and their validation, \"Analysis of the Exposure\" highlights how the surrogate models allowed to calculate the \(SAR\) values in the tissues of interest with low computational costs, so providing a fine variability analysis of exposure assessment with 3D beamforming.