## 2 Methods
### Mathematical Model of ECT  
Assume that the volumes of the discrete and continuous phases of the two-phase flow are \(V_{1}\) and \(V_{2}\) (gas is referred to as a discrete phase in gas-water two-phase flow, and water is referred to as a continuous phase) The discrete phase's permittivity is equal to \(\varepsilon_{1}\), while the continuous phase's permittivity is equal to \(\varepsilon_{2}\), the discrete phase is uniformly distributed in the continuous phase, and the equation to calculate the equivalent dielectric constant \(\varepsilon\) as follows:  
\[\varepsilon=\frac{V_{1}}{V}\cdot\varepsilon_{1}+\frac{V_{2}}{V}\cdot \varepsilon_{2}=\frac{V_{1}}{V}\cdot\varepsilon_{1}+\frac{V-V_{2}}{V}\cdot \varepsilon_{2}=\varepsilon_{2}+\frac{V_{1}}{V}\cdot(\varepsilon_{1}- \varepsilon_{2}) \tag{2}\]  
Thus, its capacitance value can be calculated:  
\[C=K\cdot\varepsilon=K\cdot f\bigg{(}\frac{V_{1}}{V}\bigg{)}=K\cdot f(\beta) \tag{3}\]  
where the concentration of the discrete phase is expressed as \(\beta=\frac{V_{1}}{V}\). The equation for calculating the capacitance value shows that the phase concentration's magnitude \(\beta\) is connected to the capacitance value. There is a character constant \(K\) that determines the pole plate's size. The effective area between the pole plates and the separation between the pole plates are some of the characteristics related to it.  
Suppose that the pole plate length can be neglected in the axial direction of the tube, the phase distribution of the medium (multiphase flow) is constant in any cross-section of the tube. Ideally, the influence of the sensitivity distribution function on the medium distribution and the influence of the dielectric interference capacitance can be ignored, and the equation of capacitance \(C_{j}\) between any pair of plates in the pipeline is solved by ignoring the influence of dielectric interference capacitance.  
\[C_{j}=\iint\limits_{D}\varepsilon(a,b)\cdot S_{j}(a,b)d_{a}d_{b} \tag{4}\]  
where \(j=1,2,\cdots,66\), the numbering meaning is as defined above, \(D\) stands for pipe section, the dielectric constant of the medium at a particular spot is represented by the medium distribution function \(\varepsilon(a,b)\) in the pipe section, which represents the sensitivity function within the pipe. \(S_{j}=(a,b)\) is the sensitivity distribution function of the capacitor. In the 12-electrode ECT system, each pair of poles can measure \(66\) independent capacitances, represented by a \(66\)-dimensional vector \([C_{1},C_{2},\ldots,C_{66}]\), and a sensitivity distribution function is associated with each capacitance value \(S_{1}(a,b),S_{2}(a,b),\ldots,S_{66}(a,b)\). According to (5), we can obtain Equation (6).  
\[\begin{array}{l}[C_{1},C_{2},\ldots,C_{66}]=[\iint\limits_{D} \varepsilon(a,b)\cdot S_{1}(a,b)d_{a}d_{b},\\ \iint\limits_{D}\varepsilon(a,b)\cdot S_{2}(From Equation (6), it is clear that the goal of the ECT system's image reconstruction is known \([C_{1},C_{2},\ldots,C_{66}]\) and \(S_{1}(a,b),S_{2}(a,b),\ldots,S_{66}(a,b)\) to solve for \(\varepsilon(a,b)\), and then use the medium distribution to rebuild the original flow pattern's image.  
Among them, one of the earliest and presently most popular basic imaging methods is the linear back projection technique, which is used by this ECT system. It determines the density value at that location in the pipe by adding up all of the projected rays there and then inverting them [17]. The matrix form of the linear inverse projection is shown in Equation (6):  
\[g=S^{T}\lambda \tag{6}\]