# The Ionospheric Scintillation Effects on the BeiDou Signal Receiver
## 2 Inhomogeneity Characteristics  
As the irregularities in ionosphere are regarded as continuous random medium, we need to characterize the dielectric constant \(\varepsilon\) in a statistical method:  
\[\varepsilon=<\varepsilon>[1+\varepsilon_{1}(\stackrel{{\rightarrow} }{{r}},t)] \tag{1}\]  
In Equation (1), \(<\cdot>\) means the statistical average and \(<\varepsilon>\) is the background mean value of the dielectric constant:  
\[<\varepsilon>=\varepsilon_{0}<\varepsilon_{r}>=\varepsilon_{0}(1-\omega_{p0} ^{2}/\omega^{2}) \tag{2}\]  
In Equation (2), \(\omega_{p0}\) is the angular frequency of plasma in accord with corresponding background electron density \(<\)\(N_{e}\)\(>\), \(\omega\) is the angular frequency of an incident wave, \(<\)\(\varepsilon_{r}\)\(>\) is the dielectric constant relative to the background medium. \(\varepsilon_{1}(\stackrel{{\rightarrow}}{{r}},t)\) is the fluctuation of \(\varepsilon\), which represents the random fluctuation caused by irregularities. Assuming that \(\varepsilon_{1}\) is in isotropic random field, then it can be written as [18]:  
\[\varepsilon_{1}=\beta\cdot\xi \tag{3}\]  
\[\beta=-\frac{\omega_{p0}^{2}}{\omega^{2}-\omega_{p0}^{2}}=-\frac{4\pi r_{e}<N _{e}(z)>}{k^{2}}=-\frac{k_{p}^{2}}{k^{2}},\qquad\xi=\frac{\Delta N_{e}( \stackrel{{\rightarrow}}{{r}})}{<N_{e}(z)>} \tag{4}\]  
In Equations (3) and (4), \(\xi\) is a homogeneous random field with a zero mean value and a standard deviation \(\sigma_{\xi}\), it reflects the fluctuation of medium; factor \(\beta\) contains all probable frequency relations, representing the dispersion characteristics of medium. The correlation function of \(\xi\) is [19]:  
\[B_{\xi}(\stackrel{{\rightarrow}}{{r}}_{1}-\stackrel{{ \rightarrow}}{{r}}_{2})=<\xi(\stackrel{{\rightarrow}}{{r}}_{1} )\xi(\stackrel{{\rightarrow}}{{r}}_{2})> \tag{5}\]  
According to Wiener-Khinchin theorem, the power spectrum density (PSD) of \(B_{\xi}(\stackrel{{\rightarrow}}{{r}})\) can be obtained by Fourier transform as follows:  
\[\Phi_{\xi}(\stackrel{{\rightarrow}}{{\kappa}})=(2\pi)^{-3} \iiint B_{\xi}(\stackrel{{\rightarrow}}{{r}})\exp(-j\stackrel{{ \rightarrow}}{{\kappa}}\cdot\stackrel{{\rightarrow}}{{r}})d \stackrel{{\rightarrow}}{{r}} \tag{6}\]  
\[B_{\xi}(\stackrel{{\rightarrow}}{{r}})=\iiint\Phi_{\xi}( \stackrel{{\rightarrow}}{{\kappa}})\exp(j\stackrel{{ \rightarrow}}{{\kappa}}\cdot\stackrel{{\rightarrow}}{{r}})dIn Equations (6) and (7), \(\overset{\rightarrow}{\kappa}=(\kappa_{x},\kappa_{y},\kappa_{z})\) is the spatial wavenumber. When it comes to one-dimensional calculation, the equation could be equal to:  
\[\Phi_{\xi}(\kappa_{x},y,z)=\frac{1}{2\pi}\!\int B_{\xi}(\overset{\rightarrow}{r })\!\exp(-j\kappa_{x}\cdot x)dx \tag{8}\]  
Under the condition of isotropy, \(B_{\xi}(\overset{\rightarrow}{r})\) is only related to \(\left|\overset{\rightarrow}{r}_{1}-\overset{\rightarrow}{r}_{2}\right|\), then Equations (6) and (7) could be simplified as:  
\[\Phi_{\xi}(\kappa)=\frac{1}{2\pi^{2}\kappa}\!\int_{0}^{\infty}B_{\xi}(r)r\,\sin (\kappa r)dr \tag{9}\]  
\[B_{\xi}(r)=\frac{4\pi}{r}\int_{0}^{\infty}\Phi_{\xi}(\kappa)\kappa\,\sin(\kappa r )dx \tag{10}\]  
Numerous satellite observation experiments show that under the isotropy condition, the 3-dimensional PSD of irregularities has the following character [20]:  
\[\Phi_{\xi}(\kappa)\propto\kappa^{-p} \tag{11}\]  
In Equation (11), \(p\) is the power law index. There have been several expressions of irregularity PSD; the expression used in this paper is the Shkarofsky spectrum [21]:  
\[<\varepsilon>=\varepsilon_{0}<\varepsilon_{r}>=\varepsilon_{0}(1-\omega_{p0} ^{2}/\omega^{2}) \tag{2}\]0  
In Equation (12), \(l_{i}\) and \(L_{0}=2\pi/\kappa_{0}\) are the inner and outer scale size of the irregularities, \(K_{\wp}\) is the second type of modified Bessel function. Figure 1 shows the PSD and correlation function of the ionospheric irregularities for Shkarofsky method with \(\overset{\rightarrow}{\rho}=(x,0,0)\) and \(\sigma_{\xi}^{2}=0.08\).