## 2 Theoretical Deduction of Atmospheric Water Vapor Content  
In orbit, the satellites are in motion, so their positions and time change every second. Each satellite carries a GPS module which calculates position and time data. The position of a satellite relative to another satellite is known as the pseudo-range \(\left(L_{\rho}\right)\). To detect the atmospheric water vapor, sounding is conducted near the Earth based on a constellation of satellites. Figure 2 shows how the atmospheric water vapor and total electron content can be deduced considering the communication of two satellites Sat \(A\) and Sat \(B\) in the ISL network. Figure 2: _TWVC_ probing with the ISL network. Knowing the satellites' location in the Earth-fixed coordinate system (ECEF) x, y, and z and the signal propagation time delay \(\delta T\) (\(s\)), _TWVC_ and _TEC_ can be estimated. Assuming Sat \(A\) is in transmission mode (_Tx_) while Sat \(B\) is in reception mode (Rx), the combined signal propagation delay due to the troposphere's and ionosphere's influence at each position along the ISL path is obtained by comparing the transmitted and received signals. If the bit starting times or time stamp positions of the _Tx_ and _Rx_ signals can be detected, then the time delay can be calculated as described in Figure 3.  
In a more holistic approach, when the frequency of communication is \(f_{1}\) (MHz), the time delay (\(\delta T_{1}\)) of the signal propagation is obtained based on all channel effects. These include the influence of integrated electron density and integrated total water vapor density along the signal ray between Sat \(A\) and Sat \(B\) at their positions in orbit denoted by \(A\) and \(B\) in the integrals, respectively. Mathematically, \(\delta T_{1}\) is deduced as follows:  
\[\delta T_{1}=\frac{L_{\rho_{1}}}{c}+\frac{a}{f_{1}^{2}}\int_{A}^{B}\eta_{e_{1} }(l)dl+\frac{1}{\Pi}\int_{A}^{B}\eta_{TWVC_{1}}(l)dl+\Delta t_{Rx}-\Delta t_{ Tx}+\in_{0_{1}} \tag{1}\]  
where \(L_{\rho_{1}}\) is the true range (m), \(\mathrm{c}=3.0\times 10^{8}\) ms\({}^{-1}\) is the speed of light, \(\Delta t_{Rx}\) (s) is the Sat \(B\) clock receiver offset, \(\Delta t_{Tx}\) (s) is the Sat \(A\) transmitter clock offset, and \(\in_{0_{1}}\) (s) is the error in the range due to various other sources such as satellite instrument delays and relativistic clock corrections. The term \(dl\) represents the path integral differential, \(a=40.3/c\approx 1.34\times 10^{-7}\) m\({}^{2}\)/s is a constant for _TEC_ determination [45], and \(\Pi\)[46; 47; 48] is an experimentally determined constant depending on air temperature. The term \(\eta_{TWVC_{1}}\) (kg/m\({}^{3}\)) is the water vapor density and its integral represents integrated total water vapor content along the ray path [47]. The total water vapor delay (s) primarily due to the tropospheric influence is deduced along the signal path based on integrated _TWVC_ and \(\Pi\). The term \(\eta_{e_{1}}\) (electrons/m\({}^{3}\)) is the total ionospheric electron density and its integral is the total electron content (_TEC_). The clock offsets are eliminated by using a precise clock as the clock source for both the transmitting and the receiving SDRs. The relativistic corrections and other known instrumented delays are compensated in the time delay detection software. Therefore, Equation (1) transforms to:  
\[\delta T_{1}=\frac{L_{\rho_{1}}}{c}+\frac{a}{f_{1}^{2}}\int_{A}^{B}\eta_{e_{1} }(l)dl+\frac{1}{\Pi}\int_{A}^{B}\eta_{TWVC_{1}}(l)dl \tag{2}\]  
Figure 3: Time delay deductions. The true range (\(L_{\rho}\)) of each satellite relative to another satellite in the ISL network is computed from the altitude (m), latitude (\({}^{\circ}\)), and longitude (\({}^{\circ}\)) geodetic coordinates provided through the GPS module. When frequency is reconfigured to \(f_{2}\) (MHz), time delay \(\delta T_{2}\) is calculated as illustrated in Equation (3), where the satellite positions change from \(A\) to \(A^{\prime}\) and from \(B\) to \(B^{\prime}\), respectively.