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# Linear observation model for PPP

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

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

$\qquad \mbox{(1)}$

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

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

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

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

• Prefit-residuals:

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

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

Notice that, and according to the equation

$T(E)=T_{z,dry}\cdot M_{dry}(E)+T_{z,wet}\cdot M_{wet}(E)$

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

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

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

• Design matrix

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

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