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Linear observation model for PPP

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Title Linear observation model for PPP
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Advanced
Year of Publication 2011

According to the equations described in the combination of pairs of signals

Comb Pairs Sign Fig 1.png

Comb Pairs Sign Fig 2.png

Comb Pairs Sign Fig 3.PNG
[math]\displaystyle{ \qquad \mbox{(1)} }[/math]

the code and carrier measurements in the ionosphere-free combination are modelled as:

[math]\displaystyle{ \begin{array}{l} R^j_C=\rho^j+c(\delta t-\delta t^j)+T^j+\mathcal{M}_C^j+{\boldsymbol \varepsilon}_{_C}^j\\[0.3cm] \Phi^j_C=\rho^j+c(\delta t-\delta t^j)+T^j+\lambda_N\,w^j+B_C^j+m_{_C}^j+{\boldsymbol \epsilon}_{_C}^j \end{array} \qquad \mbox{(2)} }[/math]

where [math]\displaystyle{ R^j_{_C} }[/math] is the unsmoothed code pseudorange measurement for the [math]\displaystyle{ j-th }[/math] satellite in view and [math]\displaystyle{ \Phi^j_{_C} }[/math] is the corresponding carrier measurement.

Following the same procedure as in Code Based Positioning (SPS), the linear observation model [math]\displaystyle{ {\mathbf Y}={\mathbf G}\;{\mathbf X} }[/math] for the code and carrier measurements can be written as:

  • Prefit-residuals:

[math]\displaystyle{ {\mathbf Y}= \left[ \begin{array}{l} R_{_C}^1-\rho_0^1+c\,\delta t^1-T_0^1\\[0.2cm] \Phi_{_C}^1-\rho_0^1+c\,\delta t^1-T_0^1-\lambda_{_{_N}}\,w^1\\ \vdots\\ R^n_{_C}-\rho_0^n+c\,\delta t^n-T_0^n\\[0.2cm] \Phi^n_{_C}-\rho_0^n+c\,\delta t^n-T_0^n-\lambda_{_{_N}}\,w^n\\ \end{array} \right] \qquad \mbox{(3)} }[/math]

Note: The satellite clock offset [math]\displaystyle{ \delta t^j }[/math] includes the satellite clock relativistic correction due to the orbit eccentricity. The relativistic path range correction is included in the geometric range [math]\displaystyle{ \rho_0^j }[/math].The term [math]\displaystyle{ T_0 }[/math] is the nominal value for the tropospheric correction.

Notice that, and according to the equation

[math]\displaystyle{ T(E)=T_{z,dry}\cdot M_{dry}(E)+T_{z,wet}\cdot M_{wet}(E) }[/math]

the tropospheric delay in the equation (2) can be decomposed into a nominal term [math]\displaystyle{ T_0(E) }[/math] and the deviation from this nominal [math]\displaystyle{ M_{wet}(E)\,\Delta T_{z,wet} }[/math]. That is:

[math]\displaystyle{ \begin{array}{l} T(E)=T_0(E)+ M_{wet}(E)\,\Delta T_{z,wet}\\[0.3cm] T_0(E)=T_{z_0,dry}\cdot M_{dry}(E)+T_{z_0,wet}\cdot M_{wet}(E) \end{array} \qquad \mbox{(4)} }[/math]

The mapping factor [math]\displaystyle{ M_{wet}(E) }[/math] is an element of the design matrix (5) and the [math]\displaystyle{ \Delta T_{z,wet} }[/math] is a component of the parameters vector (6):

  • Design matrix

[math]\displaystyle{ {\mathbf G}=\left[ \begin{array}{ccccccccccc} \frac{x_0-x^1}{\rho_0^1} & \frac{y_0-y^1}{\rho_0^1} & \frac{z_0-z^1}{\rho_0^1} &1&M_{wet}^1&0&...&0&...&0\\ \frac{x_0-x^1}{\rho_0^1} & \frac{y_0-y^1}{\rho_0^1} & \frac{z_0-z^1}{\rho_0^1} &1&M_{wet}^1&1&...&0&...&0\\ \vdots&\vdots &\vdots&\vdots &\vdots&\vdots & & \vdots& & \vdots\\ \frac{x_0-x^k}{\rho_0^k} & \frac{y_0-y^k}{\rho_0^k} & \frac{z_0-z^k}{\rho_0^k} &1&M_{wet}^k&0&...&0&...&0\\ \frac{x_0-x^k}{\rho_0^k} & \frac{y_0-y^k}{\rho_0^k} & \frac{z_0-z^k}{\rho_0^k} &1&M_{wet}^k&0&...&\underbrace{1}_{k}&...&0\\[-0.2cm] \vdots&\vdots &\vdots&\vdots &\vdots&\vdots & & & & \vdots\\ \frac{x_0-x^n}{\rho_0^n} & \frac{y_0-y^n}{\rho_0^n} & \frac{z_0-z^n}{\rho_0^n} &1&M_{wet}^n&0&...&0&...&0\\ \frac{x_0-x^n}{\rho_0^n} & \frac{y_0-y^n}{\rho_0^n} & \frac{z_0-z^n}{\rho_0^n} &1&M_{wet}^n&0&...&0&...&1 \end{array} \right] \qquad \mbox{(5)} }[/math]

  • Vector parameters (to estimate)
[math]\displaystyle{ {\mathbf X}=\left[ dx, dy, dz, c\,\delta t, \Delta T_{z,wet}, B_C^1, \cdots, B_C^k, \cdots, B_C^n, \right]^T \qquad \mbox{(6)} }[/math]