# 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 most measured and modeled statistic of low-frequency radio noise, next to absolute power level, is the first-order amplitude probability density (APD) [5]. It has been demonstrated that the bit error rate of some digital modulation can be well estimated by APD [30]. Other widely used statistical definitions that characterize the envelope of the noise \(A(t)\) are (1) the cumulative distribution function (CDF) \(F_{A}(a)\), which is one minus the APD, (2) pdf \(f_{A}(a)\) is the derivative of the CDF, and (3) the voltage deviation \(V_{d}\)[5]. Traditional APD methods are based on the narrowband signal model. However, according to the [PERSON]'s product theorem in [31], the signal \(n(t)=A(t)\cos\theta(t)\) has the analytic form if the spectrum \(S_{A}(f)=\mathcal{F}\{A(t)\}\), where \(\mathcal{F}\) is the Fourier operator, lies entirely in the region of \(|f|<f_{0}\) and \(\mathcal{F}\{\cos\theta(t)\}\) only exists outside of this region. Consequently, ELF/VLF signals \(n(t)\) considered here can be given as  
\[n(t)=n_{I}(t)\cos(2\pi ft)-n_{Q}(t)\sin(2\pi ft), \tag{7}\]  
where, \(n_{I}(t)\) is the in-phase component, \(n_{Q}(t)\) is the quadrature component, and \(f\) is the center frequency. Both \(n_{I}(t)\) and \(n_{Q}(t)\) are real-valued lowpass signals and can be expressed by \(n(t)\) and its Hilbert transform \(\hat{n}(t)\):  
\[n_{I}(t)=n(t)\cos(2\pi ft)+\hat{n}(t)\sin(2\pi ft), \tag{8}\]  
\[n_{Q}(t)=\hat{n}(t)\cos(2\pi ft)-n(t)\sin(2\pi ft). \tag{9}\]  
Thus, the lowpass equivalent of \(n(t)\), denoted by \(n_{I}(t)\), is given by  
\[n_{I}(t)=n_{I}(t)+jn_{Q}(t)=A(t)e^{i\theta(t)}. \tag{10}\]  
It is known for a long time that the phase \(\theta(t)\) of the atmospheric noise follows the uniform distribution over the angles \(-\pi\) to \(\pi\)[5]. As for the envelope \(A(t)\), there are several models used in recent literature to describe it, such as [PERSON] model, Field and Lewenstein (F-L) model, [PERSON]'s Class A and Class B model, and \(\alpha\)-stable distribution [32; 33]. Considering the preliminary statistical results in [5], [PERSON] model and Sa\(\alpha\)S distribution are more suitable to describe the narrowband ELF/VLF noise. Consequently, only Rayleigh distribution, [PERSON] model, and Sa\(\alpha\)S distribution are considered in this paper. The absolute value \(r\) of the complex random variable whose real and imaginary components follow independent identically distributed (i.i.d) Gaussian distribution with equal variance \(\sigma\) and zero mean is Rayleigh-distributed [34]. The pdf of the Rayleigh distribution is  
\[f_{\text{[PERSON]}}\left(r\right)=\ \frac{r}{\sigma^{2}}e^{-\frac{r^{2}}{2 \sigma^{2}}},r\geqslant 0. \tag{11}\]  
The [PERSON] model is presented by [PERSON] in 1966 and the envelope pdf is described as a two-parameter distribution [5]  
\[f_{\text{Hall}}\left(r\right)=\ \left(m-1\right)\gamma^{m-1}\frac{r}{\left(r^{2 }+\gamma^{2}\right)^{\left(m+1\right)/2}},r\geqslant 0\, \tag{12}\]where \(m\) determines the impulsiveness of the noise and \(\gamma\) denotes the the scaling factor. It should be noted that the envelope has infinite variance for \(m\leqslant 3\), which is not physically possible. While for the SaS noise, the closed form expression for its pdf is not available, except for special cases when the characteristic exponent \(\alpha=2\) (Gaussian distribution) and \(\alpha=1\) (Cauchy distribution). Instead, the SaS distribution is defined by its characteristic function  
\[\Phi_{X}\left(w\right)=e^{-\gamma|w|^{\alpha}}, \tag{13}\]  
where \(\gamma>0\) is the dispersion, which measures the distribution's spread. If the atmospheric noise follows SaS distribution, its envelope pdf is the Fourier-Bessel transform of \(e^{-\gamma\eta^{\alpha}}\)  
\[f_{\text{SaS}}\left(r\right)=r\int_{0}^{\infty}\rho e^{-\gamma\eta^{\alpha}}J _{0}\left(rp\right)d\rho,r\geqslant 0, \tag{14}\]  
where \(J_{0}\left(\cdot\right)\) is the zero-th order Bessel function of the first kind.