## 2 Methodology  
Radon proved that a 2D continuous function can be reconstructed from the data of an infinite number of projections of that function, and this function can be expanded to 3D. Radon transformation and its inverse transformation theory laid the mathematical foundations of CIT [27]. In the CIT process, the 3D electron density in the reconstructed area is regarded as the 3D continuous function, and the measured STEC along signal rays passing through the ionosphere is treated as the data of a number of projections in Radon transformation. The CIT process reconstructs the electron density corresponding to the inverse transformation of Radon transformation. In this paper, the STEC along the small-satellite and AIS signal path can be written as:  
\[STEC=\int_{\mathrm{path}}n_{e}(x,y,z)ds \tag{1}\]  
where \(n_{e}(x,y,z)\) is the electron density at the location \((x,y,z)\), and \(ds\) is the step length of the integral signal path. The iterative reconstruction algorithm is a voxel-based solution for CIT and a method for solving a linear equation set in the image reconstruction theory, and MART is one of the widely used voxel-based techniques. The discrete form of Equation (1) should be applied to obtain a voxel-based solution. The slant total electron content (STEC) along a signal ray is the summation of multiplications of the lengths of the transversing path and the electron densities in corresponding voxels. The discrete form of Equation (1) can be written as follows [1]:  
\[\mathbf{STEC}_{(m\times 1)}=\mathbf{A}_{(m\times n)}\cdot\mathbf{n}_{e(n\times 1 )}+\mathbf{e}_{(m\times 1)} \tag{2}\]where \(m\) is the number of rays passing through the reconstruction region and \(n\) is the number of voxels dividing the reconstructed region. \(\mathbf{STEC}_{(m\times 1)}\) is the measured STECs along the signal rays. The vector \(\mathbf{n_{e}}\) is the electron density matrix in every voxel with a size of \(n\times 1\). \(\mathbf{A}\) is the coefficient matrix of the rays transversing the reconstruction region with size \(m\times n\), \(\mathbf{A}(i,j)\) denotes the interception length of the \(i\)th ray in the \(j\)th voxel, as shown in Figure 1, and \(\mathbf{A}\) is determined by the voxel size inside of the reconstructed area. The column vector \(\mathbf{e}\) is related to the discretization errors and measurement noises, which are ignored in our simulation. Figure 1 shows a 3D view of a signal ray transversing the reconstruction voxels; the red dots are the puncture points in the interfaces made by a signal ray. The planes in light blue are not transversed by any signal rays. The distance between two adjacent puncture points is the coefficient in the voxel where the two points are, as shown by the longer arrow line in Figure 1. The coefficient of voxels without any transversing rays, such as the lower right one in Figure 1, is zero. The electron density in such voxels remains unchanged during the MART process; thus it remains the initial electron density. Therefore, the accuracy of the initial determines the reconstruction quality, especially for voxels without transversing signal rays. The iteration algorithm in MART is written as:  
\[\mathbf{n_{e}^{(k+1)}}=\mathbf{n_{e}^{(k)}}\left(\frac{\text{STEC}_{i}}{ \left<A_{i}^{T},\mathbf{n_{e}^{(k)}}\right>}\right)^{\frac{\lambda_{k}A_{ij}} {\left|A_{i}^{T}\right|}} \tag{3}\]  
where \(\lambda_{k}\) is the relaxation factor, often set to be a fixed value in the MART reconstruction, and it ranges from 0 to 1; in this paper, it is set to be 0.5 in the MART process. \(\lambda_{k}\) determines the accuracy and the smoothness of the reconstructed electron density [24]. The MART is a process in which the reconstructed electron density \(\mathbf{n_{e}}\) is progressively improved during the iterations, so the \(\mathbf{STEC}\) calculated by the iterated \(\mathbf{n_{e}}\) will be infinitely close to the measured data. The process of MART reconstruction is as follows:  
1.