View source for Precise modelling terms for PPP

{{Article Infobox2
|Category=Fundamentals
|Title={{PAGENAME}}
|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
|Level=Advanced
|YearOfPublication=2011
|Logo=gAGE
}}

The Precise Point Positioning (PPP) technique allows achieving centimetre level accuracy for static positioning and decimetre level for kinematic, typically, after the best part of one hour. This high accuracy requires an accurate measurements modelling, where all model terms must be taken into account (up to the centimetre level or better).

This modelling involves the following terms, among those considered in the [[Code Based Positioning (SPP)|SPP]]:


*''' Precise satellite orbits and clocks:'''
: ''The precise orbits and clocks files'' (see [[Precise GNSS Satellite Coordinates Computation]]) must be used instead of the broadcast ones used in the [[Code Based Positioning (SPP)|SPP]]. 


:The polynomial (1) can be applied to interpolate the precise orbits (see [[Precise GNSS Satellite Coordinates Computation]]) 

::<math>
\begin{array}{lll}
P_n(x)&=& \sum_{i=1}^n{y_i \frac{\prod_{j\neq i}{(x-x_j)}}{\prod_{j\neq i}{(x_i-x_j)}}}=\\
      &=& y_1 \frac{x-x_2}{(x_1-x_2)} \cdots \frac{x-x_n}{(x_1-x_n)}+\cdots\\
      & &+y_i \frac{x-x_1}{(x_i-x_1)} \cdots \frac{x-x_{i-1}}{(x_i-x_{i-1})}\frac{x-x_{i+1}}{(x_i-x_{i+1})}\cdots
       \frac{x-x_n}{(x_i-x_n)}+\cdots\\
      & & +y_n \frac{x-x_1}{(x_n-x_1)} \cdots\frac{x-x_{n-1}}{(x_n-x_{n-1})}
\end{array} \qquad \mbox{(1)}
</math>


:Notice that, the orbits are referred to the satellite mass centre, and thence, the [[Satellite Antenna Phase Centre]] offset must be applied to compute the <math>{\boldsymbol \Delta}_{APC}</math> vector offset.

::<math>
{\mathbf r}_{sat_{_{APC}}}={\mathbf r}_{sat_{_{MC}}}+{\mathbf R}\cdot {\boldsymbol \Delta}_{_{APC}}
\qquad \mbox{(2)}
</math>


:The ''satellite clocks'' should not be interpolated and thence, only epochs having clocks available must be used ([[Precise GNSS Satellite Coordinates Computation]]).


: ''Relativistic effects'': The gravitational path range correction (3) (see [[Relativistic Path Range Effect]]), can be added among the satellite clock correction due to the orbit eccentricity (4) considered in the [[Code Based Positioning (SPP)|SPP]] (see [[Relativistic Clock Correction]]).

::<math>
\Delta \rho_{_{rel}}=\frac{2\,\mu}{c^2}ln\frac{r^{sat}+r_{rcv}+r_{rcv}^{sat}}{r^{sat}+r_{rcv}-r_{rcv}^{sat}}
\qquad \mbox{(3)}
</math>

::<math>
 \Delta_{rel}=- 2\, \frac{\mathbf{r} \cdot \mathbf{v}}{c^2}
\qquad \mbox{(4)}
</math>



* '''Atmospheric effects:'''

:The ''ionospheric refraction'' and DCBs are removed using the ionosphere free combination of measurements (see details in [[Ionosphere-free Combination for Dual Frequency Receivers]]).


:The'' tropospheric refraction'' can be modelled by (5) (see [[Tropospheric Delay ]]) 

::<math>
T(E)=T_{z,dry}\cdot M_{dry}(E)+T_{z,wet}\cdot M_{wet}(E)
\qquad \mbox{(5)}
</math>


:where the [[Mapping of Niell|mapping of Niell]] is used. The dry and wet tropospheric delays are given by 

::<math>
\begin{array}{l}
T_{z,dry}= a\; e^{-b\; H}\\
T_{z,wet}= T_{z_0,wet}+\Delta T_{z,wet}
\qquad \mbox{(6)}
\end{array}
</math>


:The ''deviation of the zenith tropospheric delay'' <math>\Delta T_{z,wet}</math> regarding to the nominal <math>T_{z_0,wet}</math> must be estimated in the [[Kalman Filter|Kalman filter]], together with the coordinates, clock and carrier phase biases.



* '''Antenna biases and orientation:'''
: The ''satellite and receiver antenna phase centres'' can be found in the IGS ANTEX files, after the GPS week <math>1400</math> (see [[Antenna Phase Centre]]).


: The ''carrier phase wind-up effect'' due to the satellite motion is given by (7) (see [[Carrier Phase Wind-up Effect]])

::<math>
\Delta \phi=2N\pi+sign(\zeta)\cdot \arccos \left( \frac{\vec{D}^{\prime}\cdot\vec{D}}{\|\vec{D}^{\prime}\|\cdot\|\vec{D}\|}\right) \qquad \mbox{(7)}
</math>


: The ''satellites under eclipse conditions'' should be removed from the computation due to the largest orbit error. The eclipse condition is given by (8) (see [[Satellite Eclipses]]) 

::<math>
\cos \phi = \frac{{\mathbf r}_{sat}\cdot {\mathbf r}_{sun}}{|{\mathbf r}_{sat}\cdot {\mathbf r}_{sun}|} < 0 \qquad \mbox{and} \qquad r_{sat} \sqrt{1-\cos^2 \phi}\, < a_e 
\qquad \mbox{(8)}
</math>



*''' Earth deformation effects:'''
:''Solid tides'' can be modelled by equations (9) and (10) (see [[Solid Tides ]]) 

::<math>
\Delta {\mathbf r}= \sum_{j=2}^{3}{\frac{G\,M_j\,R_e^4}{G\,M_\oplus\,R_j^3}} \left \{h_2 \,\hat{\mathbf r} \left (  \frac{3}{2} (\hat{\mathbf R}_j \cdot \hat{\mathbf r})^2 -\frac{1}{2}\right) + 3\,l_2\,(\hat{\mathbf R}_j \cdot \hat{\mathbf r}) \left [\hat{\mathbf R}_j-(\hat{\mathbf R}_j \cdot \hat{\mathbf r})\,\hat{\mathbf r}  \right ]\right \}
\qquad \mbox{(9)}
</math>

::<math>
\Delta {\mathbf r}= \sum_{j=2}^{3}{\frac{G\,M_j\,R_e^5}{G\,M_\oplus\,R_j^4}} \left \{h_3 \,\hat{\mathbf r} \left (  \frac{5}{2} (\hat{\mathbf R}_j \cdot \hat{\mathbf r})^3 -\frac{3}{2} (\hat{\mathbf R}_j \cdot \hat{\mathbf r})\right) + l_3\,\left (\frac{15}{2}  (\hat{\mathbf R}_j \cdot \hat{\mathbf r})^2 - \frac{3}{2} \right ) \left [\hat{\mathbf R}_j-(\hat{\mathbf R}_j \cdot \hat{\mathbf r})\,\hat{\mathbf r}  \right ]\right \}
\qquad \mbox{(10)}
</math>


: ''Ocean loading and Pole Tides'' are second order effects and can be neglected for PPP accuracies at the centimetre level (see comments in [[Ocean loading]] and [[Pole Tide]]).


==Notes==
<references group="footnotes"/>

==References==
<references/>

[[Category:Fundamentals]]
[[Category:Solving Navigation Equations]]
[[Category:PPP]]