# 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
The distribution follows (14) is also called the heavy-tailed Rayleigh distribution [34]. This is due to the fact that for \(\alpha=2\) it is the Rayleigh distribution. While for \(\alpha<2\) it can describe the impulsive noise and has a thicker pdf tail compared to the Rayleigh distribution. In order to compare the fitting performance of Rayleigh distribution, Hall model and SaS distribution, we should be able to estimate the model parameters from the observed data. Parameters of Rayleigh distribution and Hall model can be easily obtained by fitting the pdf with a LS algorithm. However, Equation (14) seems to be impossible to invert in order to estimate the required parameters. The pdf is approximated via numerical calculation and a lookup table method is adopted in [5]. However, this method need to store the lookup table and thus is not convenient to implement. In this paper, we adopt the fractional lower order moments method to estimate the parameters \(\alpha\) and \(\gamma\). The \(p\)-th order moment of the variable which follows the heavy-tailed Rayleigh distribution can be given as [34]  
\[\text{E}\left(r^{p}\right)=\frac{2^{p+1}\Gamma\left(\frac{p}{2}+1\right)}{ \Gamma\left(\frac{p}{2}\right)}\frac{\gamma^{\frac{p}{2}}\Gamma\left(-\frac{p} {\alpha}\right)}{\alpha},-2<p<-\frac{1}{2}, \tag{15}\]  
where \(\text{E}\left(\cdot\right)\) is the expectation operator and \(\Gamma\left(\cdot\right)\) is the Gamma function. Consequently, \(\alpha\) can be estimated from the following formula  
\[\frac{\text{E}\left(r^{2p}\right)}{\left[\text{E}\left(r^{p}\right)\right]^{2 }}=\frac{\Gamma\left(p+1\right)\Gamma^{2}\left(-\frac{p}{2}\right)}{2\Gamma \left(-p\right)\Gamma^{2}\left(\frac{p}{2}+1\right)}\frac{\alpha\Gamma\left(- \frac{2p}{\alpha}\right)}{\Gamma\left(-\frac{p}{\alpha}\right)},-1<p<-\frac{1 }{2}. \tag{16}\]  
Unfortunately, (16) is a highly nonlinear problem and cannot obtain a closed form solution for \(\alpha\). However, the Gamma function is well-behaved for the range of \(p\) and thus (16) can be solved by using the numerical optimization algorithms such as bisection [34]. Once \(\alpha\) is obtained, \(\gamma\) can be derived from (15) as  
\[n_{I}(t)=n(t)\cos(2\pi ft)+\hat{n}(t)\sin(2\pi ft), \tag{8}\]0  
As the envelope pdfs of the ELF/VLF noise exhibit heavy-tailed characteristic, mean square error (MSE) fails to provide a fair comparison between different models for the fitting. Thus, we adopt the mean-square log error (MSLE) [25] to evaluate the estimation performance. MSLE is defined as  
\[n_{I}(t)=n(t)\cos(2\pi ft)+\hat{n}(t)\sin(2\pi ft), \tag{8}\]1  
where \(f_{A}(x)\) is the data pdf and \(\hat{f}_{A}(x)\) is the estimate by the model. The envelope pdf fitting by Rayleigh distribution, Hall model, and SaS distribution for Data1 (Bandwidth is 400 Hz) are shown in \[n_{I}(t)=n(t)\cos(2\pi ft)+\hat{n}(t)\sin(2\pi ft), \tag{8}\]2 As evidenced in Figure 9a, all the three models could be well fitted to the ELF atmospheric noise and the Hall model achieves the best performance. While for the VLF noise, the Rayleigh model behaves poor and SaS distribution outperforms others in the sense of goodness of fit. Moreover, SaS describes the heavy-tail characteristic of the VLF noise more precisely in the zoom area. In order to get a quantitative analysis of the fitting performance, the envelope \(A\) of all the 42 sets of data are normalized such that the average envelope values \(E[A]=1\). The parameters of Data1 with different frequency bands estimated by the three models are revealed in \[n_{I}(t)=n(t)\cos(2\pi ft)+\hat{n}(t)\sin(2\pi ft), \tag{8}\]3 It can be further validated that VLF noise contains more impulsive components and thus cannot be characterized by Gaussian model. The SaS distribution achieves smaller MLSE than the other two models because it can get more accurate fitting for the heavy tail. For the ELF noise, the MLSE of the Rayleigh model decreases significantly because ELF noise appears mostly Gaussian. All of the three models achieve good fitting performance in the sense of MLSE.