# Research on Improving Satellite Positioning Precision Based on Multi-Frequency Navigation Signals
## 2 Methodology and Equations
### Physical Augmentation Factor of Precision (PAFP)
In a satellite positioning system, the variance of the pseudo-range observation error is amplified by the weight coefficient matrix to obtain the variance of the positioning error [1]. The relationship can be expressed as:  
\[ \begin{bmatrix}\mathrm{\sigma_{x}}^{2}&\mathrm{\sigma_{y}}^{2}&&\\ &\mathrm{\sigma_{z}}^{2}&&\\ &&\mathrm{\sigma_{t}}^{2}\end{bmatrix} = \begin{bmatrix}\mathrm{P}_{11}&&\\ &\mathrm{P}_{22}&&\\ &&\mathrm{P}_{33}&\\ &&&\mathrm{P}_{44}\end{bmatrix} *\mathrm{\sigma_{e}}^{2} \tag{6}\]  
where \(\mathrm{\sigma_{x}}^{2}\), \(\mathrm{\sigma_{y}}^{2}\), \(\mathrm{\sigma_{z}}^{2}\), and \(\mathrm{\sigma_{t}}^{2}\) are the variance of the positioning errors in the X-axis, Y-axis, Z-axis directions, and clock difference in the ECEF coordinate system, respectively; \(\mathrm{P}_{ii}\) (i = 1, 2, 3, 4) is the diagonal element of the weight matrix of the pseudo-range observation equation. Then, the standard deviation of the positioning error in three-dimensional space is:  
\[\sigma=\sqrt{\mathrm{\sigma_{x}}^{2}+\mathrm{\sigma_{y}}^{2}+\mathrm{\sigma_{ z}}^{2}}=\sqrt{\mathrm{P}_{11}+\mathrm{P}_{22}+\mathrm{P}_{33}}*\mathrm{\sigma_{e}} \tag{7}\]  
When the number of visible satellites is n, each visible satellite sends down multi-frequency navigation signals, the number of downlink frequency is k, and its coefficient matrix is:  
\[\mathrm{H}_{\mathrm{B}}= \begin{bmatrix}\mathrm{h}\mathrm{x}_{11}&\mathrm{h} \mathrm{y}_{11}&\mathrm{h}\mathrm{z}_{11}&1\\ \mathrm{h}\mathrm{x}_{21}&\mathrm{h}\mathrm{y}_{21}&\mathrm{h}\mathrm{z}_{21}& 1\\ \vdots&\vdots&\vdots&\vdots\\ \mathrm{h}\mathrm{x}_{i1}&\mathrm{h}\mathrm{y}_{i1}&\mathrm{h}\mathrm{z}_{i1}& 1\\ \mathrm{h}\mathrm{x}_{i2}&\mathrm{h}\mathrm{y}_{i2}&\mathrm{h}\mathrm{z}_{i2}& 1\\ \vdots&\vdots&\vdots&\vdots\\ \mathrm{h}\mathrm{x}_{i2}&\mathrm{h}\mathrm{y}_{i2}&\mathrm{h}\mathrm{z}_{i2}& 1\\ \vdots&\vdots&\vdots&\vdots\\ \mathrm{h}\mathrm{x}_{ij}&\mathrm{h}\mathrm{y}_{ij}&\mathrm{h}\mathrm{z}_{ij}& 1\end{bmatrix}  \tag{8}\]  
where \(\left[\mathrm{h}\mathrm{x}_{i\mathrm{j}},\mathrm{h}\mathrm{y}_{i\mathrm{j}}, \mathrm{h}\mathrm{z}_{i\mathrm{j}}\right]\) (i = 1, 2,..., n; j = 1, 2,..., k) is the cosine vector of the direction from the receiver to the i-th satellite that transmits the j-th signal, so:  
\[\mathrm{h}\mathrm{x}_{i1}=\mathrm{h}\mathrm{x}_{i2}=\ldots=\mathrm{h}\mathrm{x} _{i\mathrm{j}} \tag{9}\]  
\[\mathrm{h}\mathrm{y}_{i1}=\mathrm{h}\mathrm{y}_{i2}=\ldots=\mathrm{h}\mathrm{y} _{\mathrm{\bar{y}}} \tag{10}\]  
\[\mathrm{h}\mathrm{z}_{i1}=\mathrm{h}\mathrm{z}_{i\mathrm{j}}=\ldots=\mathrm{h} \mathrm{z}_{i\mathrm{j}} \tag{11}\]According to Equations (6)-(9), the weight coefficient matrix of the pseudo-range observation equation can be obtained as:  
\[\mathrm{D_{B}}=\left(\mathrm{H_{B}^{T}H_{B}}\mathrm{h_{2}}\right)^{-1}=\left[ \begin{array}{cccc}\mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x_{ii}}^{ 2}&\mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x_{ii}}\mathrm{h}\mathrm{y_ {ii}}&\mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x_{ii}}\mathrm{h} \mathrm{z_{ii}}&\mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x_{ii}}\\ \mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x_{ii}}\mathrm{h}\mathrm{y_ {ii}}&\mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{y_{ii}}^{2}&\mathrm{ k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{z_{ii}}&\mathrm{k}\sum\limits_{i=1}^{n} \mathrm{h}\mathrm{y_{ii}}\\ \mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x_{ii}}\mathrm{h}\mathrm{z_{ii }}&\mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{y_{ii}}\mathrm{h}\mathrm{ z_{ii}}&\mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{z_{ii}}^{2}&\mathrm{k}\sum \limits_{i=1}^{n}\mathrm{h}\mathrm{z_{ii}}\\ \mathrm{k}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x_{ii}}\mathrm{k}\sum\limits _{i=1}^{n}\mathrm{h}\mathrm{y_{ii}}\mathrm{h}\mathrm{z_{ii}}&\mathrm{k}\sum \limits_{i=1}^{n}\mathrm{h}\mathrm{z_{ii}}&\mathrm{k}\cdot\mathrm{n}\\ \end{array} \right]^{-1}=\mathrm{\frac{1}{k}\left[