## 3 Construction of Mathematical Model
### CEI-Based Precision Measurements of Targets in Earth-Moon Space
#### 4.1.2 Error Correction for Relativistic Effects  
The effect of the Beidou-3 satellite clock time deviation on the pseudocode ranging values due to relativistic effects can be expressed as:  
\[\mathbf{s}_{i(BD\to space)}^{\text{Relativity\_1}}=\frac{2(\mathbf{r}_{iBD} \cdot\mathbf{r}_{iBD}^{d}-\mathbf{r}_{space}\cdot\mathbf{r}_{space}^{d})}{c} \tag{20}\]  
where \(\mathbf{r}_{iBD}^{d}\) and \(\mathbf{r}_{space}^{d}\) are the velocities of the BeiDou-3 satellite and the non-cooperative spacecraft in Earth-Moon space in the geocentric inertial coordinate system, respectively. The effect of the gravitational field due to the general relativity effect on the pseudocode-ranging values can be expressed as:  
\[\mathbf{s}_{i(BD\to space)}^{\text{Relativity\_2}}=(1+\gamma)\frac{GM}{c^{2}} \ln\frac{\left|\mathbf{r}_{iBD}\right|+\left|\mathbf{r}_{space}\right|+\rho_{ i(BD\to space)}}{\left|\mathbf{r}_{iBD}\right|+\left|\mathbf{r}_{space}\right|- \rho_{i(BD\to space)}} \tag{21}\]  
where \(\gamma\) is the post-Newtonian effect parameter, \(GM\) is the Earth's gravitational constant, \(\left|\mathbf{r}_{iBD}\right|\) and \(\left|\mathbf{r}_{space}\right|\) are the distances from the Beidou-3 satellite and the non-cooperative spacecraft in Earth-Moon space to the Earth's center, respectively; \(\rho_{i(BD\to space)}\) is the distance between the Beidou-3 satellite and the non-cooperative spacecraft in Earth-Moon space.  
In summary, the total error correction equation for the relativistic effect can be obtained as follows:  
\[\begin{array}{c}\mathbf{s}_{i(BD\to space)}^{\text{Relativity\_total}}= \mathbf{s}_{i(BD\to space)}^{\text{Relativity\_1}}+\mathbf{s}_{i(BD\to space)} ^{\text{Relativity\_2}}\\ =\frac{2(\mathbf{r}_{iBD}\cdot\mathbf{r}_{iBD}^{d}-\mathbf{r}_{space}\cdot \mathbf{r}_{space}^{d})}{c}+(1+\gamma)\frac{GM}{c^{2}}\ln\frac{\left|\mathbf{r }_{iBD}\right|+\left|\mathbf{r}_{space}\right|+\left|\mathbf{r}_{i(BD\to space )}\right|+\left|\mathbf{r}_{i(BD\to space)}\right|}{\left|\mathbf{r}_{iBD} \right|+\left|\mathbf{r}_{space}\right|-\rho_{i(BD\to space)}}\end{array} \tag{22}\]  
1.3 The Total Correction Equation for the Observation Error during the Transmission of the Inter-Satellite Link Signals of Beidou-3  
By synthesizing the above analysis, the total correction equation for the observation error during the transmission of the inter-satellite link signals of Beidou-3 satellites can be obtained as:  
\[\begin{array}{c}\begin{array}{c}\begin{array}{c}\begin{array}{c}e_{i(BD \to space)}=c_{i(BD\to space)}^{antenna}+\mathbf{s}_{i(BD\to space)}^{\text{Relativity\_total}}\\ =\frac{\mathbf{r}_{iBD}-\mathbf{r}_{space}}{\left|\mathbf{r}_{iBD}-\mathbf{r}_ {space}\right|}\cdot\left(\mathbf{p}_{x},\mathbf{p}_{y},\mathbf{p}_{z}\right) \cdot\mathbf{w}_{i(BD\to space)}^{antenna}\\ +\frac{2(\mathbf{r}_{iBD}\cdot\mathbf{r}_{iBD}^{d}-\mathbf{r}_{space}\cdot \mathbf{r}_{space})}{c}+(1+\gamma)\frac{GM}{c^{2}}\ln\frac{\left|\mathbf{r}_{ iBD}\right|+\left|\mathbf{r}_{space}\right|+\left|\mathbf{r}_{space}\right|-\rho_{i(BD\to space)} }{\left|\mathbf{r}_{i(BD\to space)}\right|-\rho_{i(BD\to space)}}\end{array}\end{array} \tag{23}\]  
Observation Error Correction Equation during CEI Observation of Inter-Satellite Link Signals Reflected by Non-Cooperative Spacecraft  
As described previously, the pseudocode ranging equation (Equation (10)) between the Beidou-3 satellite's interplanetary link signal reflected by the non-cooperative spacecraft and the ground-based CEI receiving array has errors that need to be corrected, mainly including atmospheric delay errors (mainly ionospheric and tropospheric delay errors), phase center offset of the station antenna, relativistic effects, tidal effects, etc. This subsection will focus on the analysis and correction of the above errors, and the specific equations are shown below.