# Initial Results from SQUID Sensor: Analysis and Modeling for the ELF/VLF Atmospheric Noise
## 4 Statistical Analysis and Modeling for the Atmospheric Noise
### Normality Test for the Narrow Band Noise  
We test the normality of the ELF/VLF data before statistical analysis. The data measured near 10:00 UT from 2 nd to 7 th March are used for the test, named Data1 to Data6, respectively. The data are divided into segments and Lilliefors test [27] is performed for them. As the transmission rates of ELF/VLF communication modulated by quadrature phase shift keying (QPSK) are 100, 400, and 800 bps in [7], the corresponding symbol rates are 50, 200, and 400 Baud/s. Consequently, the time length of the segments are chosen to be 20, 5.0, and 2.5 ms, respectively. The proportion of the normal  
Figure 7: Spectrogram of the atmospheric noise lower than 10 kHz.  
distribution of the ELF and VLF atmospheric noise are shown in Table 1. It can be seen that the narrowband (20 ms) ELF/VLF data have very low proportion of normal distribution because impulses would be more likely to be included during a longer observation. As a result, the proportion of the normal distribution of the ELF/VLF data gets larger as the bandwidth increases.  
In Table 1, we investigate the proportion of the normal distribution within one symbol period. However, we are also concerned about the noise distribution during a period of time in communication because it can give us a roughly statistical characteristic of the atmospheric noise. In order to illustrate an intuitive understanding of the noise distribution within a longer observation, the waveform of ELF/VLF noise with 400 Hz bandwidth from Data2 during 20 s is shown in Figure 8. It can be seen that ELF noise appears in an almost Gaussian distribution, accompanied with occasional impulses, while VLF noise exhibits more impulsive characteristics.  
In statistics, a quantile-quantile (Q-Q) plot is a probability plot, which is a graphical method for comparing two probability distributions by plotting their quantiles against each other [28]. It has been widely used to compare a data set to a theoretical model to provide an assessment of \"goodness of fit\".  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Frequency Band** & **Length** & **Data1** & **Data2** & **Data3** & **Data4** & **Data5** & **Data6** \\ \hline \multirow{3}{*}{ELF} & 20 ms & 0.42 & 0.361 & 0.377 & 0.395 & 0.382 & 0.386 \\  & 5 ms & 0.872 & 0.866 & 0.866 & 0.868 & 0.868 & 0.864 \\  & 2.5 ms & 0.974 & 0.971 & 0.973 & 0.973 & 0.974 & 0.971 \\ \hline \multirow{3}{*}{VLF} & 20 ms & 0.309 & 0.357 & 0.363 & 0.348 & 0.324 & 0.324 \\  & 5 ms & 0.747 & 0.753 & 0.756 & 0.789 & 0.771 & 0.761 \\  & 2.5 ms & 0.892 & 0.9 & 0.912 & 0.923 & 0.921 & 0.914 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The proportion of the normal distribution of the atmospheric noise at different frequency bands.  
Figure 8: Waveform and the corresponding QQ -plot of the extremely low frequency (ELF) and very low frequency (VLF) noise. QQ-plot is performed for the noise processed by a hole puncher with a threshold of four times standard deviation.  
In order to investigate the distribution of low frequency atmospheric noise and the normal distribution intuitively, we perform the Q-Q plot graphically for ELF/VLF noise in Figure 8. To alleviate the adverse effect of the occasional impulses, the data are preprocessed by a hole puncher [29] with a threshold of 4\(\sigma\), where \(\sigma\) is the standard deviation of the data. We may find that the processed ELF data mostly follow normal distribution and thus can be modeled by Gaussian noise. However, the VLF data deviate significantly from the normal distribution. It is consistent with the results shown in Table 1, in which the proportion of normal distribution of the ELF and VLF noise with 400 Hz bandwidth (2.5 ms) is 0.971 and 0.9, respectively. These conclusions are also consistent with that in [7].