[5599] | 1 |
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| 2 | \documentclass[10pt]{beamer}
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| 3 | \usetheme{umbc2}
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| 4 | \useinnertheme{umbcboxes}
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| 5 | \setbeamercolor{umbcboxes}{bg=violet!12,fg=black}
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| 6 |
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| 7 | \usepackage{longtable}
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| 8 | \usepackage{tabu}
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[5607] | 9 | \usepackage{subeqnar}
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[5599] | 10 |
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| 11 | \newcommand{\ul}{\underline}
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| 12 | \newcommand{\be}{\begin{equation}}
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| 13 | \newcommand{\ee}{\end{equation}}
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| 14 | \newcommand{\bdm}{\begin{displaymath}}
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| 15 | \newcommand{\edm}{\end{displaymath}}
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| 16 | \newcommand{\bea}{\begin{eqnarray}}
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| 17 | \newcommand{\eea}{\end{eqnarray}}
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[5607] | 18 | \newcommand{\bsea}{\begin{subeqnarray*}}
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| 19 | \newcommand{\esea}{\end{subeqnarray*}}
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[5599] | 20 | \newcommand{\mb}[1]{\mbox{#1}}
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| 21 | \newcommand{\mc}[3]{\multicolumn{#1}{#2}{#3}}
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| 22 | \newcommand{\bm}[1]{\mbox{\bf #1}}
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| 23 | \newcommand{\bmm}[1]{\mbox{\boldmath$#1$\unboldmath}}
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| 24 | \newcommand{\bmell}{\bmm\ell}
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| 25 | \newcommand{\hateps}{\widehat{\bmm\varepsilon}}
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| 26 | \newcommand{\graybox}[1]{\psboxit{box .9 setgray fill}{\fbox{#1}}}
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| 27 | \newcommand{\mdeg}[1]{\mbox{$#1^{\mbox{\scriptsize o}}$}}
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| 28 | \newcommand{\dd}{\mbox{\footnotesize{$\nabla \! \Delta$}}}
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| 29 | \newcommand{\p}{\partial\,}
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| 30 | \renewcommand{\d}{\mbox{d}}
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| 31 | \newcommand{\dspfrac}{\displaystyle\frac}
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| 32 | \newcommand{\nl}{\\[4mm]}
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| 33 |
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[5601] | 34 | \title{Processing GNSS Data in Real-Time}
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[5599] | 35 |
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| 36 | \author{Leo\v{s} Mervart}
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| 37 |
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[5601] | 38 | \institute{TU Prague}
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[5599] | 39 |
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[5601] | 40 | \date{Frankfurt, January 2014}
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[5599] | 41 |
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| 42 | % \AtBeginSection[]
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| 43 | % {
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| 44 | % \begin{frame}
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| 45 | % \frametitle{Table of Contents}
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| 46 | % \tableofcontents[currentsection]
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| 47 | % \end{frame}
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| 48 | % }
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| 49 |
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| 50 | \begin{document}
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| 51 |
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| 52 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 53 |
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| 54 | \begin{frame}
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| 55 | \titlepage
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| 56 | \end{frame}
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| 57 |
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| 58 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 59 |
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| 60 | \begin{frame}
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[5601] | 61 | \frametitle{Medieval Times of GNSS (personal memories)}
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| 62 |
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| 63 | \begin{description}
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| 64 | \item[1991] Prof. Gerhard Beutler became the director of the Astronomical Institute, University of
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| 65 | Berne. The so-called Bernese GPS Software started to be used for (post-processing) analyzes of
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| 66 | GNSS data.
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| 67 | \item[1992] LM started his PhD study at AIUB.
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| 68 | \item[1992] Center for Orbit Determination in Europe (consortium of AIUB, Swisstopo, BKG, IGN, and
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| 69 | IAPG/TUM) established. Roughly at that time LM met Dr. Georg Weber for the first time.
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| 70 | \item[1993] International GPS Service formally recognized by the IAG.
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| 71 | \item[1994] IGS began providing GPS orbits and other products routinely (January, 1).
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| 72 | \item[1995] GPS declared fully operational.
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| 73 | \end{description}
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| 74 |
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[5599] | 75 | \end{frame}
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| 76 |
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| 77 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 78 |
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[5601] | 79 | \begin{frame}
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| 80 | \frametitle{CODE-Related Works in 1990's}
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[5599] | 81 |
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[5601] | 82 | \begin{itemize}
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| 83 | \item The Bernese GPS Software was the primary tool for CODE analyzes (Fortran~77).
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| 84 | \item IGS reference network was sparse.
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| 85 | \item Real-time data transmission limited (Internet was still young, TCP/IP widely accepted 1989).
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| 86 | \item CPU power of then computers was limited (VAX/VMS OS used at AIUB).
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| 87 | \end{itemize}
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| 88 |
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| 89 | In 1990's high precision GPS analyzes were almost exclusively performed in post-processing mode.
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| 90 | The typical precise application of GPS at that time was the processing of a network of static
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| 91 | GPS-only receivers for the estimation of station coordinates.
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| 92 |
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| 93 | \end{frame}
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| 94 |
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| 95 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 96 |
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[5599] | 97 | \begin{frame}
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[5601] | 98 | \frametitle{Tempora mutantur (and maybe ``nos mutamur in illis'')}
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[5599] | 99 |
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[5603] | 100 | \includegraphics[width=0.7\textwidth,angle=0]{pp_vs_rt.png}
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[5602] | 101 |
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[5603] | 102 | \vspace*{-2cm}
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| 103 | \hspace*{6cm}
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| 104 | \includegraphics[width=0.4\textwidth,angle=0]{ea_ztd_21h.png}
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| 105 |
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| 106 |
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[5599] | 107 | \end{frame}
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| 108 |
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[5601] | 109 |
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[5599] | 110 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 111 |
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[5601] | 112 | \begin{frame}
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[5604] | 113 | \frametitle{O tempora! O mores!}
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[5601] | 114 |
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[5604] | 115 | \begin{itemize}
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| 116 | \item people want more and more \ldots
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| 117 | \item everybody wants everything immediately \ldots
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| 118 | \item \hspace*{2cm} and, of course, free of charge \ldots
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| 119 | \end{itemize}
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| 120 | \vspace*{5mm}
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| 121 | In GNSS-world it means:
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| 122 | \begin{itemize}
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| 123 | \item There are many new kinds of GNSS applications - positioning is becoming just one of many
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| 124 | purposes of GNSS usage.
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| 125 | \item Many results of GNSS processing are required in real-time (or, at least, with very small
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| 126 | delay).
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| 127 | \item GPS is not the only positioning system. Other GNSS are being established (for practical but
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| 128 | also for political reasons).
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| 129 | \item People are used that many GNSS services are available free of charge (but the development and
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| 130 | maintenance has to be funded).
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| 131 | \end{itemize}
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| 132 |
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| 133 | \begin{block}{But \ldots}
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| 134 | \end{block}
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| 135 |
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[5601] | 136 | \end{frame}
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| 137 |
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| 138 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 139 |
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| 140 | \begin{frame}
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[5604] | 141 | \frametitle{Nihil novi sub sole}
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[5601] | 142 |
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[5606] | 143 | Each GNSS-application is based on processing code and/or phase observations
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| 144 | \vspace*{-3mm}
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[5604] | 145 | \begin{eqnarray*}
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| 146 | P^i & = & \varrho^i + c\;\delta - c\;\delta^i + T^i + I^i + b_P \\
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| 147 | L^i & = & \varrho^i + c\;\delta - c\;\delta^i + T^i - I^i + b^i
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| 148 | \end{eqnarray*}
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| 149 | where
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| 150 | \begin{tabbing}
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| 151 | $P^i$, $L^i$ ~~~~~~~ \= are the code and phase measurements, \\
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| 152 | $\varrho^i$ \> is the travel distance between the satellite
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| 153 | and the receiver, \\
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| 154 | $\delta$, $\delta^i$ \> are the receiver and satellite clock errors, \\
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| 155 | $I^i$ \> is the ionospheric delay, \\
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| 156 | $T^i$ \> is the tropospheric delay, \\
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| 157 | $b_P$ \> is the code bias, and \\
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| 158 | $b^i$ \> is the phase bias (including initial
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| 159 | phase ambiguity).
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| 160 | \end{tabbing}
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| 161 | Observation equations reveal what information can be gained from processing GNSS data:
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| 162 | \begin{itemize}
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| 163 | \item geometry (receiver positions, satellite orbits), and
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| 164 | \item state of atmosphere (both dispersive and non-dispersive part)
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| 165 | \end{itemize}
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| 166 | The observation equations also show that, in principle, GNSS is an
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| 167 | \textcolor{blue!90}{interferometric} technique -- precise results are actually always relative.
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| 168 |
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[5605] | 169 | \end{frame}
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[5604] | 170 |
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[5605] | 171 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 172 |
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| 173 | \begin{frame}
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| 174 | \frametitle{Challenges of Real-Time GNSS Application}
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[5606] | 175 | \begin{itemize}
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[5608] | 176 | \item Suitable algorithms for the parameter adjustment have to be used (filter techniques instead
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| 177 | of classical least-squares).
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[5606] | 178 | \item Reliable data links have to been established (between rover station and a reference station,
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| 179 | between receivers and processing center, or between processing center and DGPS correction
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| 180 | provider).
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| 181 | \item Software tools for handling real-time data (Fortran is not the best language for that).
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| 182 | \item Fast CPUs.
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| 183 | \end{itemize}
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[5605] | 184 |
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[5606] | 185 | As said above -- GNSS is an interferometric technique. Processing of a single station cannot give
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| 186 | precise results. However, data of reference station(s) can be replaced by the so-called corrections
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| 187 | (DGPS corrections, precise-point positioning etc.) These techniques are particularly suited for
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| 188 | real-time applications because the amount of data being transferred can be considerably reduced.
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| 189 |
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[5601] | 190 | \end{frame}
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| 191 |
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[5607] | 192 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 193 |
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| 194 | \begin{frame}
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[5609] | 195 | \frametitle{Algorithms -- Kalman Filter}
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[5607] | 196 |
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| 197 | \begin{small}
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| 198 |
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| 199 | State vectors $\bmm{x}$ at two subsequent epochs are
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| 200 | related to each other by the following linear equation:
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| 201 | \bdm
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| 202 | \bmm{x}(n) = \bmm{\Phi}\; \bmm{x}(n-1) + \bmm{\Gamma}\;\bmm{w}(n)~,
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| 203 | \edm
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| 204 | where $\Phi$ and $\Gamma$ are known matrices and {\em white noise} $\bmm{w}(n)$ is a random
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| 205 | vector with the following statistical properties:
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| 206 | \bsea
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| 207 | E(\bmm{w}) & = & \bmm{0} \\
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| 208 | E(\bmm{w}(n)\;\bmm{w}^T(m)) & = & \bmm{0} ~~ \mbox{for $m \neq n$} \\
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| 209 | E(\bmm{w}(n)\;\bmm{w^T}(n)) & = & \bm{Q}_s(n) ~.
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| 210 | \esea
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| 211 |
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| 212 | Observations $\bmm{l}(n)$ and the state vector $\bmm{x}(n)$ are related to
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| 213 | each other by the linearized {\em observation equations} of form
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| 214 | \bdm \label{eq:KF:obseqn}
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| 215 | \bmm{l}(n) = \bm{A}\;\bmm{x}(n) + \bmm{v}(n) ~ ,
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| 216 | \edm
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| 217 | where $\bm{A}$ is a known matrix (the so-called {\em first-design matrix}) and
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| 218 | $\bmm{v}(n)$ is a vector of random errors with the following properties:
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| 219 | \bsea\label{eq:KF:resid}
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| 220 | E(\bmm{v}) & = & \bmm{0} \\
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| 221 | E(\bmm{v}(n)\;\bmm{v}^T(m)) & = & \bmm{0} ~~ \mbox{for $m \neq n$} \\
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| 222 | E(\bmm{v}(n)\;\bmm{v^T}(n)) & = & \bm{Q}_l(n) ~.
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| 223 | \esea
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| 224 |
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| 225 | \end{small}
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| 226 |
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| 227 | \end{frame}
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| 228 |
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| 229 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 230 |
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| 231 | \begin{frame}
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| 232 | \frametitle{Classical KF Form}
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| 233 |
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| 234 | Minimum Mean Square Error (MMSE) estimate $\widehat{\bmm{x}}(n)$ of vector
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| 235 | $\bmm{x}(n)$ meets the condition
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| 236 | $E\left((\bmm{x} - \widehat{\bmm{x}})(\bmm{x} - \widehat{\bmm{x}})^T\right) =
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| 237 | \mbox{min}$ and is given by
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| 238 | \begin{subeqnarray}\label{eq:KF:prediction}
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| 239 | \widehat{\bmm{x}}^-(n) & = & \bmm{\Phi} \widehat{\bmm{x}}(n-1) \\
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| 240 | \bm{Q}^-(n) & = & \bmm{\Phi} \bm{Q}(n-1) \bmm{\Phi}^T +
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| 241 | \bmm{\Gamma} \bm{Q}_s(n) \bmm{\Gamma}^T
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| 242 | \end{subeqnarray}
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| 243 | \begin{subeqnarray}\label{eq:KF:update}
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| 244 | \widehat{\bmm{x}}(n) & = & \widehat{\bmm{x}}^-(n) +
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| 245 | \bm{K}\left(\bmm{l} -
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| 246 | \bm{A}\widehat{\bmm{x}}(n-1)\right) \\
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| 247 | \bm{Q}(n) & = & \bm{Q}^-(n) - \bm{K}\bm{A}\bm{Q}^-(n) ~,
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| 248 | \end{subeqnarray}
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| 249 | where
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| 250 | \bdm \label{eq:KF:KandH}
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| 251 | \bm{K} = \bm{Q}^-(n)\bm{A}^T\bm{H}^{-1}, \quad
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| 252 | \bm{H} = \bm{Q}_l(n) + \bm{A}\bm{Q}^-(n)\bm{A}^T ~.
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| 253 | \edm
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| 254 | Equations (\ref{eq:KF:prediction}) are called {\em prediction},
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| 255 | equations (\ref{eq:KF:update}) are called {\em update} step of Kalman filter.
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| 256 |
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| 257 | \end{frame}
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| 258 |
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| 259 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 260 |
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| 261 | \begin{frame}
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| 262 | \frametitle{Square-Root Filter} \label{sec:SRF}
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| 263 | \begin{small}
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| 264 | Algorithms based on equations (\ref{eq:KF:prediction}) and
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| 265 | (\ref{eq:KF:update}) may suffer from numerical instabilities that are primarily
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| 266 | caused by the subtraction in (\ref{eq:KF:update}b). This deficiency may be
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| 267 | overcome by the so-called {\em square-root} formulation of the Kalman filter
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| 268 | that is based on the so-called {\em QR-Decomposition}. Assuming the
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| 269 | Cholesky decompositions
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| 270 | \be \label{eq:SRF:defsym}
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| 271 | \bm{Q}(n) = \bm{S}^{T} \bm{S} , \quad
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| 272 | \bm{Q}_l(n) = \bm{S}^T_l \bm{S}_l, \quad
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| 273 | \bm{Q}^-(n) = \bm{S}^{-T}\bm{S}^-
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| 274 | \ee
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| 275 | we can create the following block matrix and its QR-Decomposition:
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| 276 | \be \label{eq:SRF:main}
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| 277 | \left(\begin{array}{ll}
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| 278 | \bm{S}_l & \bm{0} \\
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| 279 | \bm{S}^-\bm{A}^T & \bm{S}^-
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| 280 | \end{array}\right)
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| 281 | =
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| 282 | N \left(\begin{array}{cc}
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| 283 | \bm{X} & \bm{Y} \\
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| 284 | \bm{0} & \bm{Z}
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| 285 | \end{array}\right) ~ .
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| 286 | \ee
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| 287 | It can be easily verified that
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| 288 | \bsea\label{eq:SRF:HK}
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| 289 | \bm{H} & = & \bm{X}^T\bm{X} \\
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| 290 | \bm{K}^T & = & \bm{X}^{-1}\bm{Y}\\
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| 291 | \bm{S} & = & \bm{Z} \\
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| 292 | \bm{Q}(n) & = & \bm{Z}^T\bm{Z} ~ .
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| 293 | \esea
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| 294 | State vector $\widehat{\bmm{x}}(n)$ is computed in a usual way using the
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| 295 | equation (\ref{eq:KF:update}a).
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| 296 | \end{small}
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| 297 | \end{frame}
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| 298 |
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[5609] | 299 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 300 |
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| 301 | \begin{frame}
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| 302 | \frametitle{Data Transfer -- NTRIP}
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| 303 |
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| 304 | In order to be useful data have to be provided in a well-defined \textcolor{blue}{format}.
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| 305 | RTCM (Radio Technical Commission for Maritime Services) messages are widely used for GNSS data in
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| 306 | real-time.
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| 307 |
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| 308 | \vspace*{5mm}
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| 309 |
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| 310 | In addition to a format the so-called \textcolor{blue}{protocol} has to be defined. Using a given
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| 311 | protocol the data user communicates with the data provider.
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| 312 |
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| 313 | For GNSS data, the so-called \textcolor{blue}{NTRIP} streaming protocol is used.
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| 314 | \begin{itemize}
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| 315 | \item NTRIP stands for Networked Transport of RTCM via Internet Protocol.
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| 316 | \item NTRIP is in principle a layer on top of TCP/IP.
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| 317 | \item NTRIP has been developed at BKG (together with TU Dortmund).
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| 318 | \item NTRIP is capable of handling hundreds of data streams simultaneously delivering the data
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| 319 | to thousands of users.
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| 320 | \item NTRIP is world-wide accepted.
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| 321 | \end{itemize}
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| 322 |
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| 323 | \end{frame}
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| 324 |
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| 325 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 326 |
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| 327 | \begin{frame}
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| 328 | \frametitle{NTRIP}
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| 329 |
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| 330 | Efficiency of data transfer using NTRIP is achieved thanks to the GNSS Internet Radio /
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| 331 | IP-Streaming architecture:
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| 332 |
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[5610] | 333 | \begin{center}
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[5609] | 334 | \includegraphics[width=0.7\textwidth,angle=0]{ntrip.png}
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[5610] | 335 | \end{center}
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[5609] | 336 |
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| 337 | \end{frame}
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| 338 |
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[5610] | 339 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 340 |
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| 341 | \begin{frame}
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| 342 | \frametitle{BKG Ntrip Client (BNC)}
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| 343 |
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| 344 | An important reason why NTRIP has been widely accepted is that BKG provided high-quality public
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| 345 | license software tools for its usage. One of these tools is the so-called \textcolor{blue}{BKG
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| 346 | Ntrip Client}.
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| 347 |
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[5611] | 348 | \begin{itemize}
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| 349 | \item BNC source consists currently of approximately 50.000 lines of code
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| 350 | \item approximately 90 \% is C++, 10 \% standard C
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| 351 | \item BNC uses a few third-party pieces of software (first of all the RTCM
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| 352 | decoders/encoders and a matrix algebra library)
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| 353 | \end{itemize}
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[5610] | 354 |
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[5611] | 355 | \begin{block}{BNC is intended to be}
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| 356 | \begin{itemize}
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| 357 | \item user-friendly
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| 358 | \item cross-platform
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| 359 | \item easily modifiable (by students, GNSS beginners)
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| 360 | \item useful (at least a little bit ...)
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| 361 | \end{itemize}
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| 362 | \end{block}
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[5610] | 363 |
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[5612] | 364 | \begin{block}{BNC is not only an NTRIP client \ldots}
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| 365 | \end{block}
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[5611] | 366 |
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| 367 | \end{frame}
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| 368 |
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| 369 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 370 |
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| 371 | \begin{frame}
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[5613] | 372 | \frametitle{Data QC in BNC}
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| 373 | \begin{center}
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[5614] | 374 | \includegraphics[width=0.9\textwidth,angle=0]{bnc_qc2.png}
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| 375 | \end{center}
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| 376 | \end {frame}
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| 377 |
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| 378 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 379 |
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| 380 | \begin{frame}
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| 381 | \frametitle{Data QC in BNC}
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| 382 | \begin{center}
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[5613] | 383 | \includegraphics[width=0.9\textwidth,angle=0]{bnc_qc1.png}
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| 384 | \end{center}
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| 385 | \end {frame}
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| 386 |
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| 387 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 388 |
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| 389 | \begin{frame}
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[5611] | 390 | \frametitle{Precise Point Positioning with PPP}
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| 391 | \begin{center}
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| 392 | \includegraphics[width=0.9\textwidth,angle=0]{ppp1.png}
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| 393 | \end{center}
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| 394 | \end {frame}
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| 395 |
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| 396 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 397 |
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| 398 | \begin{frame}
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| 399 | \frametitle{Precise Point Positioning with PPP (cont.)}
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| 400 | BNC provides a good framework for the PPP client (observations, orbits, and
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| 401 | corrections stand for disposal).
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| 402 |
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| 403 | Main reasons for the PPP module in BNC have been:
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| 404 | \begin{itemize}
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| 405 | \item monitoring the quality of incoming data streams (primarily the PPP
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| 406 | corrections)
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| 407 | \item providing a simple easy-to-use tool for the basic PPP positioning
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| 408 | \end{itemize}
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| 409 |
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| 410 | The PPP facility in BNC is provided in the hope that it will be useful.
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| 411 | \begin{itemize}
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| 412 | \item The mathematical model of observations and the adjustment algorithm are
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| 413 | implemented in such a way that they are (according to our best knowledge)
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| 414 | correct without any shortcomings, however,
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| 415 | \item we have preferred simplicity to transcendence, and
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| 416 | \item the list of options the BNC users can select is limited.
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| 417 | \item[$\Rightarrow$] Commercial PPP clients may outperform BNC in some
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| 418 | aspects.
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| 419 | \end{itemize}
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| 420 | We believe in a possible good coexistence of the commercial software and
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| 421 | open source software.
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| 422 | \end {frame}
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| 423 |
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| 424 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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| 425 |
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| 426 | \begin{frame}
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| 427 | \frametitle{PPP Options}
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| 428 | \begin{itemize}
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| 429 | \item single station, SPP or PPP
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| 430 | \item real-time or post-processing
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| 431 | \item processing of code and phase ionosphere-free combinations, GPS,
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| 432 | Glonass, and Galileo
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| 433 | \end{itemize}
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| 434 | \begin{center}
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| 435 | \includegraphics[width=0.9\textwidth,angle=0]{ppp_opt1.png} \\[2mm]
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| 436 | \includegraphics[width=0.9\textwidth,angle=0]{ppp_opt2.png}
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| 437 | \end{center}
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| 438 | \end {frame}
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| 439 |
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| 440 |
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[5599] | 441 | \end{document}
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