# On the Study of Influences of Different Factors on the Rapid Topospheric Tomography
## 3 GNSS Tropospheric Tomography Methodology
### Tomography Parameter Estimation Method  
Rapid reconstructions of water vapor fields are meaning for providing high temporal and spatial humidity fields and improving the accuracy of numerical weather forecast, et al. Therefore, how to estimate the parameters efficiently in large-scale region tomography is of great importance, especially for near real-time applications. The most commonly used parameter estimation methods like the least squares and Kalman filter need the operation of matrix inversion. As the scale of tomography region grows, the size of the matrix can be significantly huge which will make the inversion very time-consuming. In this study, the algebraic reconstruction technique (ART) is utilized. The algebraic reconstruction technique iterates over the initial estimate until a certain condition is satisfied. ART avoids matrix inversion, which is especially suitable for operations involving large dimensional matrices, such as the tropospheric tomography in our case.  
ART is a frequently used image reconstruction algorithm. The technique can also be applied to the tomography. The ART family mainly include the original additive algebraic reconstruction techniques, the multiplicative algebraic reconstruction techniques (MART) and the simultaneous iterations reconstruction technique (SIRT). The algebraic reconstruction techniques have been successfully used to solve the computational efficiency issue in ionospheric tomography in some previous studies. For example, [PERSON] et al. [29] and [PERSON] et al. [30] applied the multiplicative algebraic reconstruction to ionospheric tomography, [PERSON] et al. [31] compared the ART and MART, [PERSON] et al. [32] presented an improved algebraic reconstruction technique (IART), and [PERSON] et al. [33] proposed adaptive simultaneous iteration reconstruction technique (ASIRT) on the basis of simultaneous iteration reconstruction technique.  
The ART algorithm assumes a linear relationship between the values to be reconstructed and the projected values for each grid element, and the grid parameters are corrected one by one during each iteration as,  
\[x_{j}^{k+1}=x_{j}^{k}+\lambda a_{ij}\frac{SWD_{i}-\sum_{j=1}^{n}a_{ij}x_{j}^{ k}}{\sum_{j=1}^{n}a_{ij}^{2}} \tag{8}\]  
where \(x\) is the wet refractivity in the grid; \(k\) is the number of iteration; \(n\) is the number of columns of the observation equation, namely the number of parameters; \(a_{ij}\) is the element in the \(i\) row and \(j\) column of the equation coefficient matrix; \(\lambda\) is the relaxation parameter; \(SWD\) is the observed value.  
The result of the ART depends on the order of the data \(SWD_{i}\) as shown in Equation (8). To avoid this behavior, SIRT no longer individually modifies the water vapor density on each observation path, but iterates and modifies all the observed paths at one time as,  
\[x_{j}^{k+1}=x_{j}^{k}+\lambda\sum\
olimits_{i=1}^{m}\frac{a_{ij}\Big{(}SWD_{ i}-\sum_{j=1}^{n}a_{ij}x_{j}^{k}\Big{)}}{\sum_{j=1}^{n}a_{ij}^{2}} \tag{9}\]  
where \(m\) is the number of rows of the observation equation (including the constraint equation).  
In order to achieve faster convergence and better estimates compared to the plain ART, an adaptive SIRT (ASIRT) with a data-driven adjustment of the weight has been applied in some ionosphere tomographic studies (e.g., [PERSON] et al.) [33]. The ASIRT formula can be written as,  
\[x_{j}^{k+1}=x_{j}^{k}+\lambda{\sum}_{i=1}^{m}\frac{a_{ij}x_{j}^{k}\left(SWD_{i }-\sum_{j=1}^{n}a_{ij}x_{j}^{k}\right)}{\sum_{j=1}^{n}a_{ij}^{2}x_{j}^{k}}. \tag{10}\]  
Compared to Equation (9), estimates from the previous step are taken into account in the determination of the weight in ASIRT.