## 2 Tomography Theory and Method
### Optimal Solution Strategy for Multi-GNSS Troposphere Tomography Modeling  
As described in Section 2.1, the final unknown parameter can be estimated based on Equations (5) and (6):  
\[\mathbf{\hat{X}}=\Big{(}\sum\mathbf{A}_{\mathrm{tp1}}^{T}\cdot\mathbf{P}_{ \mathrm{tp1}}^{T}\cdot\mathbf{A}_{\mathrm{tp1}}+\sum\mathbf{A}_{\mathrm{tp2}}^{ T}\cdot\mathbf{P}_{\mathrm{tp2}}^{T}\cdot\mathbf{A}_{\mathrm{tp2}}\Big{)}^{-1} \cdot\Big{(}\sum\mathbf{A}_{\mathrm{tp1}}^{T}\cdot\mathbf{P}_{\mathrm{tp1}}^{T} \cdot\mathbf{y}_{\mathrm{tp1}}+\sum\mathbf{A}_{\mathrm{tp2}}^{T}\cdot \mathbf{P}_{\mathrm{tp2}}^{T}\cdot\mathbf{y}_{\mathrm{tp2}}\Big{)}. \tag{7}\]  
Usually, the unit weight variances for different types of equations are regarded as the same value. Due to the fact that the troposphere tomography modeling is established using different types of multi-GNSS observations as well as various constraints, it is therefore critical to determine the reasonable weightings of different equations. In this study, the proper unit weight variances for six types of equations are determined by combining the posterior residuals of different typesof equations and the improved homogeneity test method. The specific procedure can be expressed as follows:  
1. Calculate the posterior residual values of different types of equations \(\mathbf{V}_{tp}\) (\(tp=gps\),\(glouass\),\(bds,H,V,P\)) \[\mathbf{V}_{tp}=\mathbf{A}_{tp}\cdot\mathbf{\hat{X}}-\mathbf{y}_{tp},\] (8)
2. Obtain the posterior unit weight variance of different types of tomographic equation [25] \[\hat{\sigma}_{0_{tp}}^{2}=\frac{\mathbf{V}_{tp}^{T}\cdot\mathbf{P}_{tp}\cdot \mathbf{V}_{tp}}{n_{tp}-tr(\mathbf{N}^{-1}\cdot\mathbf{N}_{tp})},\] (9)  
where \(n_{tp}\) is the total number of different tomographic equations, while \(tr\) represents the rank of the corresponding matrix. In addition, \(\mathbf{N}=\mathbf{A}^{T}\cdot\mathbf{P}\cdot\mathbf{A}\). 3. Calculate the pooled variance of the involved variance components as [26] \[\hat{\sigma}_{0_{tp}}^{2}=\frac{\sum\limits_{i=1}^{m}(n_{i}-1)\hat{\sigma}_{0 _{i}}^{2}}{\sum\limits_{i=1}^{m}(n_{i}-1)},\] (10)  
where \(\hat{\sigma}_{0_{i}}^{2}\) is the posterior unit weight variance of different equations and \(m\) is the number of different types of equations. 4. Here, the concept of Bartlett test [27] is introduced to examine whether the involved variance components are statistically equal:  
\[\chi_{0_{tp}}^{2}=\frac{\sum\limits_{i=1}^{m}(n_{i}-1)ln\frac{\hat{\sigma}_{0 _{tp}}^{2}}{\hat{\sigma}_{0_{tp}}^{2}}}{1+\frac{\sum\limits_{i=1}^{m}(n_{i}-1 )^{-1}-\left(\sum\limits_{i=1}^{m}(n_{i}-1)\right)^{-1}}{3(m-1)}}. \tag{11}\]  
It is generally considered that the Bartlett test is approximate to the chi-square distribution. However, when the sampling, like the number of \(m\) in Equation (10) is small, the rejection region of statistics is large, which is caused by the unequal mean value between the Bartlett statistics and chi-square distribution. Therefore, an improved Bartlett statistics is introduced for the small sample based on the principle of the mean equality, and the improved statistics are as follows:  
\[\chi_{0_{tp}}^{2}=\frac{\sum\limits_{i=1}^{m}(n_{i}-1)ln\frac{\hat{\sigma}_{0 _{tp}}^{2}}{\hat{\sigma}_{0_{i}}^{2}}}{1+\frac{\sum\limits_{i=1}^{m}(n_{i}-1 +0.23^{n_{i}})^{-1}-\left(\sum\limits_{i=1}^{m}(n_{i}-1)\right)^{-1}}{3(m-1) }} \tag{12}\]  
Some Monte Carlo simulation [28] experiments have been performed in which the improved statistics have a better obedience of chi-square distribution and enhanced the tested accuracy significantly. According to the improved statistics and the number of tomographic equations, the degree of freedom and the significant level are select as 5 and 0.1 in the chi-square distribution, respectively; therefore, the critical value \(\chi_{0.1}^{2}\) can be retrieved from the lookup table as 9.236.