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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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9 | \usepackage{subeqnar}
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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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18 | \newcommand{\bsea}{\begin{subeqnarray*}}
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19 | \newcommand{\esea}{\end{subeqnarray*}}
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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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34 | \title{Processing GNSS Data in Real-Time}
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35 |
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36 | \author{Leo\v{s} Mervart}
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37 |
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38 | \institute{TU Prague}
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39 |
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40 | \date{Frankfurt, January 2014}
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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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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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75 | \end{frame}
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76 |
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77 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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78 |
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79 | \begin{frame}
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80 | \frametitle{CODE-Related Works in 1990's}
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81 |
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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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97 | \begin{frame}
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98 | \frametitle{Tempora mutantur (and maybe ``nos mutamur in illis'')}
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99 |
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100 | \includegraphics[width=0.7\textwidth,angle=0]{pp_vs_rt.png}
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101 |
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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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107 | \end{frame}
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108 |
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109 |
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110 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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111 |
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112 | \begin{frame}
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113 | \frametitle{O tempora! O mores!}
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114 |
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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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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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141 | \frametitle{Nihil novi sub sole}
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142 |
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143 | Each GNSS-application is based on processing code and/or phase observations
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144 | \vspace*{-3mm}
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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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169 | \end{frame}
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170 |
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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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175 | \begin{itemize}
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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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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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184 |
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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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190 | \end{frame}
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191 |
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192 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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193 |
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194 | \begin{frame}
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195 | \frametitle{Algorithms -- Kalman Filter}
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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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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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333 | \begin{center}
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334 | \includegraphics[width=0.7\textwidth,angle=0]{ntrip.png}
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335 | \end{center}
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336 |
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337 | \end{frame}
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338 |
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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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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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354 |
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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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363 |
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364 | \begin{block}{BNC is not only an NTRIP client \ldots}
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365 | \end{block}
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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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372 | \frametitle{Data QC in BNC}
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373 | \begin{center}
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374 | \includegraphics[width=0.9\textwidth,angle=0]{bnc_qc1.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{Precise Point Positioning with PPP}
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382 | \begin{center}
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383 | \includegraphics[width=0.9\textwidth,angle=0]{ppp1.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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390 | \frametitle{Precise Point Positioning with PPP (cont.)}
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391 | BNC provides a good framework for the PPP client (observations, orbits, and
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392 | corrections stand for disposal).
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393 |
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394 | Main reasons for the PPP module in BNC have been:
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395 | \begin{itemize}
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396 | \item monitoring the quality of incoming data streams (primarily the PPP
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397 | corrections)
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398 | \item providing a simple easy-to-use tool for the basic PPP positioning
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399 | \end{itemize}
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400 |
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401 | The PPP facility in BNC is provided in the hope that it will be useful.
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402 | \begin{itemize}
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403 | \item The mathematical model of observations and the adjustment algorithm are
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404 | implemented in such a way that they are (according to our best knowledge)
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405 | correct without any shortcomings, however,
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406 | \item we have preferred simplicity to transcendence, and
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407 | \item the list of options the BNC users can select is limited.
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408 | \item[$\Rightarrow$] Commercial PPP clients may outperform BNC in some
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409 | aspects.
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410 | \end{itemize}
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411 | We believe in a possible good coexistence of the commercial software and
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412 | open source software.
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413 | \end {frame}
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414 |
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415 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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416 |
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417 | \begin{frame}
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418 | \frametitle{PPP Options}
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419 | \begin{itemize}
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420 | \item single station, SPP or PPP
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421 | \item real-time or post-processing
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422 | \item processing of code and phase ionosphere-free combinations, GPS,
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423 | Glonass, and Galileo
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424 | \end{itemize}
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425 | \begin{center}
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426 | \includegraphics[width=0.9\textwidth,angle=0]{ppp_opt1.png} \\[2mm]
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427 | \includegraphics[width=0.9\textwidth,angle=0]{ppp_opt2.png}
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428 | \end{center}
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429 | \end {frame}
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430 |
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431 |
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432 | \end{document}
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