## 2 Theoretical Deduction of Atmospheric Water Vapor Content
\[\delta T_{2}=\frac{L_{\rho_{2}}}{c}+\frac{a}{f_{2}^{2}}\int_{A^{\prime}}^{B^{ \prime}}\eta_{\epsilon_{2}}(l)dl+\frac{1}{\Pi}\int_{A^{\prime}}^{B^{\prime}} \eta_{TWVC_{2}}(l)dl \tag{3}\]  
It is important to note that frequency translation is not achieved instantly when the satellites are in motion. As a result, the water vapor content and electron density are determined based on signal propagation time delay between two positions (\(A\) to \(A^{\prime}\)) and (\(B\) to \(B^{\prime}\)) of each satellite in the ISL network using two different frequencies. Thus, to derive \(TWVC\) and \(TEC\), the two time delays (\(\delta T_{1}\) and \(\delta T_{2}\)) are compared as illustrated in Equation (4):  
\[ \begin{array}{c}\delta T_{1}-\delta T_{2}=\frac{1}{C}\left(L_{\rho_{1}}-L_{ \rho_{2}}\right)+a\left(\frac{1}{f_{1}^{2}}f_{A}^{B}\eta_{\epsilon_{1}}(l)dl- \frac{1}{f_{2}^{2}}\int_{A^{\prime}}^{B^{\prime}}\eta_{\epsilon_{2}}(l)dl \right)+\\ \frac{1}{\Pi}\left(\int_{A}^{B}\eta_{TWVC_{1}}(l)dl-\int_{A^{\prime}}^{B^{ \prime}}\eta_{TWVC_{2}}(l)dl\right)\end{array}  \tag{4}\]  
In the closest range, when considering a constellation of thousands of satellites in different LEO orbital planes and directions, the difference between \(L_{\rho_{1}}\) and \(L_{\rho_{2}}\) is assumed to be at least 15 km. Assuming that the \(TWVC\) and \(TEC\) at the two positions of each satellite are the same, with \(TEC\) varying inversely to the square of the frequencies, this implies:  
\[\int_{A}^{B}\eta_{TWVC_{1}}(l)dl\approx\int_{A^{\prime}}^{B^{\prime}}\eta_{TWVC _{2}}(l)dl=\int\eta_{TWVC}(l)dl\text{, and} \tag{5}\]  
\[\int_{A}^{B}\eta_{\epsilon_{1}}(l)dl\approx\int_{A^{\prime}}^{B^{\prime}}\eta _{\epsilon_{2}}(l)dl=\int\eta_{\epsilon}(l)dl \tag{6}\]  
Substituting (5) and (6) into (4), \(TEC\) can be deduced as follows:  
\[a\left(\frac{1}{f_{1}^{2}}-\frac{1}{f_{2}^{2}}\right)\int\eta_{\epsilon}(l) dl=\left(\delta T_{1}-\delta T_{2}\right)-\frac{1}{C}\left(L_{\rho_{1}}-L_{ \rho_{2}}\right) \tag{7}\]  
\[TEC=\int\eta_{\epsilon}(l)dl=\frac{\left(\delta T_{1}-\delta T_{2}\right)- \frac{1}{C}\left(L_{\rho_{1}}-L_{\rho_{2}}\right)}{a\left(\frac{1}{f_{1}^{2}}- \frac{1}{f_{2}^{2}}\right)} \tag{8}\]  
However, the assumptions in (5) and (6) introduce an error in the measurement accuracy. This error is assumed to become negligible when the frequency transition can be achieved in 1 s or less. Also, by averaging measurements from multiple frequencies, the introduced error is anticipated to be very negligible. This is the other reason why multiple frequencies and frequency reconfiguration has been considered as a way of improving the mission measurement accuracy aside from distinguishing \(TWVC\) from \(TEC\). When \(TEC\) variations become known, total water vapor content, \(TWVC\), is computed from Equation (2) as follows:  
\[TWVC=\int\eta_{TWVC}(l)dl=\Pi(\delta T_{1}-\frac{L_{\rho_{1}}}{c}-\frac{a}{f_ {1}^{2}}\int\eta_{\epsilon_{1}}(l)dl) \tag{9}\]  
The key steps in this method are described as follows:  
* Measuring the signal time delay by utilizing two frequencies between the satellites. * Distinguishing the signal time delay due to \(TEC\) and \(TWVC\) by comparing the two frequency measurements. * Collecting all the measurement data of \(TEC\) and \(TWVC\) within a time period. * In the case of satellites in multiples orbital planes, obtain a 3D distribution of _TEC_ and _TWVC_. * Determining the most probable set of distribution that agrees with the measurement data of _TEC_ and _TWVC_. * Deducing only _TWVC_ contribution by removing _TEC_ values. In line with Kyushu Institute of Technology satellite development projects, a constellation of more than 1000 satellites in low Earth orbit are expected to be launched for scientific studies of the atmosphere including for this mission to determine water vapor content in the atmosphere. These studies will be a continuation of the SPATIUM-I and SPATIUM-II missions which were demonstrated in orbit.