03-model.tex

This section introduces the receiver model used in the rest of the
paper. Specifically, Section~\ref{sec:gps:phy} describes the physics
behind the model and Section~\ref{sec:gps:mod} discusses our modelling
choices.

\subsection{GPS physics}
\label{sec:gps:phy}

GPS sensors locate themselves through a process called
\emph{trilateration}~\cite{bib:gps-book}. This process consists in
measuring the distance from 4 or more points in space (satellites),
whose position is known. Given the distance measurements, the GPS
sensor then performs a least square estimation to determine its
current position. Figure~\ref{fig:globe} shows an example with five
satellites. To correctly estimate the current position, the GPS
receiver must measure the distance from $s_1, s_2, s_3$ and $s_4$.
Additionally, measuring the distance from $s_5$ is not necessary, but
improves the position accuracy.

The GPS framework includes (circa) 30 satellites. These satellites
orbit around the Earth following known trajectories. While orbiting,
they broadcast periodic signals that encode a set of parameters,
called \emph{ephemeris data}. The emphemeris data describe the
satellites' orbits (see for example the trajectory of satelite $s_3$
in Figure~\ref{fig:globe}), and therefore allow the GPS receiver to
accurately determine their position in time. The satellite
trajectories are not constant in time, due to uncertainties and
disturbances, like corrections for collision avoidance.

\begin{figure}
\centering
\scalebox{0.6}{%
\begin{pspicture}(-5,-4)(6,6)
\psset{RotX=-23, RotZ=30, PHI=46.5833, THETA=0.3333,
  visibility=false, Decran=15,
  path=/usr/share/texlive/texmf-dist/tex/generic/pst-geo/data}
% map
\WorldMapThreeD[circles=false, blueEarth=false]
\WorldMapThreeD[circles=false, visibility=true, opacity=0.8]
% s1
\pscircle[linecolor=red, fillcolor=red, fillstyle=solid](-3,5){0.1}
\psline[linecolor=red, linewidth=0.1](0.8,0.8)(-3,5)
\rput[tl](-2.75,5.2){\fontsize{16pt}{16pt}\selectfont\textcolor{red}{$s_1$}}
% s2
\pscircle[linecolor=red, fillcolor=red, fillstyle=solid](5.5,5.5){0.1}
\psline[linecolor=red, linewidth=0.1](0.78,0.83)(5.5,5.5)
\rput[tl](5.5,5.3){\fontsize{16pt}{16pt}\selectfont\textcolor{red}{$s_2$}}
% s3
\pscircle[linecolor=red, fillcolor=red, fillstyle=solid](5,3){0.1}
\psline[linecolor=red, linewidth=0.1](0.75,0.82)(5,3)
\rput[tl](4.8,2.8){\fontsize{16pt}{16pt}\selectfont\textcolor{red}{$s_3$}}
% s4
\pscircle[linecolor=red, fillcolor=red, fillstyle=solid](1.25,5){0.1}
\psline[linecolor=red, linewidth=0.1](0.78,0.82)(1.25,5)
\rput[tl](1.45,5.15){\fontsize{16pt}{16pt}\selectfont\textcolor{red}{$s_4$}}
% s5
\pscircle[linecolor=red, fillcolor=red, fillstyle=solid](0.5,4.5){0.1}
\psline[linecolor=red, linestyle=dashed, linewidth=0.1](0.8,0.82)(0.5,4.5)
\rput[tl](-0.1,4.5){\fontsize{16pt}{16pt}\selectfont\textcolor{red}{$s_5$}}
% curve for satelite s3
\pscurve[linecolor=red, linestyle=dotted]{->}(5,3)(5.75,3.2)(4,1.5)
% delay and distance for s1
\rput[tl](-4.4,4.3){\fontsize{16pt}{16pt}\selectfont\textcolor{red}{$\{\Delta_1, d_1\}$}}
\end{pspicture}}
\caption{Trilateration: GPS receiver and satellites.}
\label{fig:globe}
\end{figure}

The ephemeris data expire after 30 minutes, i.e., after 30 minutes
they are not considered valid anymore. To correctly estimate the