# Initial Results from SQUID Sensor: Analysis and Modeling for the ELF/VLF Atmospheric Noise
## 4 Statistical Analysis and Modeling for the Atmospheric Noise
### Amplitude Probability Distribution of the Narrow Band Noise Envelope
Statistical analysis for all the 42 data sets show that, about 61% of the ELF noise are more suitable to be described by Hall model. Values of impulse index \(m\) range from 20 to 270 and values of \(\gamma\) range from 3 to 13. When investigating the VLF noise, about 75% of the noise follow SaS distribution and thus SaS is most suitable. Thereby, \(\alpha\) is usually in the range of 1.8 to 1.96, although values as low as 1.73 and as high as 2 are also found. The corresponding \(\gamma\) is mainly distributed from 0.29 to 0.32. As the increase of bandwidth, more noise radiated from the lightings all over the world will be observed and thus the noise amplitude tend to rise. While the impulsive characteristics turn to weak as the increase of noise source due to the central limit theorem (CLT) [35]. It should be noted that \(\alpha\) fitted by SaS distribution is mainly located in the range of 1.9 to 2 \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Noise**} & \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**50 Hz**} & \multicolumn{2}{c}{**200 Hz**} & \multicolumn{2}{c}{**400 Hz**} \\ \cline{3-8}  & & **Parameters** & **MSLE** & **Parameters** & **MSLE** & **Parameters** & **MSLE** \\ \hline \multirow{4}{*}{ ELF} & Rayleigh & \(\sigma=0.823\) & \(8.59\times 10^{-4}\) & \(\sigma=0.809\) & \(1.82\times 10^{-4}\) & \(\sigma=0.807\) & \(2.04\times 10^{-4}\) \\ \cline{2-8}  & Hall & \(\begin{array}{c}m=19.68\\ \gamma=3.349\end{array}\) & \(2.15\times 10^{-4}\) & \(\begin{array}{c}m=40.43\\ \gamma=4.939\end{array}\) & \(4.67\times 10^{-5}\) & \(\begin{array}{c}m=30.18\\ \gamma=4.212\end{array}\) & \(4.65\times 10^{-4}\) \\ \cline{2-8}  & SaS & \(\begin{array}{c}\alpha=1.833\\ \gamma=0.317\end{array}\) & \(5.89\times 10^{-4}\) & \(\begin{array}{c}\alpha=1.945\\ \gamma=0.316\end{array}\) & \(6.75\times 10^{-5}\) & \(\begin{array}{c}\alpha=1.96\\ \gamma=0.316\end{array}\) & \(6.55\times 10^{-5}\) \\ \hline \multirow{4}{*}{ VLF} & Rayleigh & \(\sigma=0.91\) & \(1.32\times 10^{-2}\) & \(\sigma=0.959\) & \(1.81\times 10^{-2}\) & \(\sigma=0.978\) & \(1.86\times 10^{-2}\) \\ \cline{2-8}  & Hall & \(\begin{array}{c}m=8.303\\ \gamma=1.823\end{array}\) & \(2.09\times 10^{-4}\) & \(\begin{array}{c}m=12.76\\ \gamma=2.356\end{array}\) & \(3.48\times 10^{-4}\) & \(\begin{array}{c}m=15.68\\ \gamma=2.681\end{array}\) & \(3.41\times 10^{-4}\) \\ \cline{2-8}  & SaS & \(\begin{array}{c}\alpha=1.736\\ \gamma=0.297\end{array}\) & \(1.04\times 10^{-4}\) & \(\begin{array}{c}\alpha=1.776\\ \gamma=0.293\end{array}\) & \(8.04\times 10^{-5}\) & \(\begin{array}{c}\alpha=1.803\\ \gamma=0.292\end{array}\) & \(6.15\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Fitting performance for the ELF noise by the two models. Figure 9: Comparisons of the fitting performance for (**a**) ELF atmospheric noise and (**b**) VLF atmospheric noise. that the impulsive characteristic is very weak and can be simplified to Gaussian noise when a hole puncher is performed for the occasional impulses. Consequently, from the qualitative comparison in Figure 9 and quantitative comparisons in Table 2, we can derive the conclusion that the ELF/VLF noise considered in this paper follows Gaussian distribution, with occasional impulses from lighting strikes. For simplicity, ELF noise can be regarded as Gaussian noise after preprocessing by a hole puncher operator for impulses. While for VLF noise, SaS distribution is optimal to describe the data. The simulation results are also consistent with the conclusion in [7].