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Commit of wind estimation document.

1 | %$Header: /home/dashley/cvsrep/e3ft_gpl01/e3ft_gpl01/dtaipubs/cron/2006/wvecprob/wvecprob.tex,v 1.4 2006/12/26 06:59:37 dashley Exp $ |

2 | % |

3 | \documentclass[letterpaper,10pt,titlepage]{article} |

4 | % |

5 | %\pagestyle{headings} |

6 | % |

7 | \usepackage{amsmath} |

8 | \usepackage{amsfonts} |

9 | \usepackage{amssymb} |

10 | \usepackage[ansinew]{inputenc} |

11 | \usepackage[OT1]{fontenc} |

12 | \usepackage{graphicx} |

13 | %\usepackage{makeidx} |

14 | % |

15 | % |

16 | %Define certain conspicuous global constants. |

17 | %\newcommand{\productbasename}{FBO-Prime} |

18 | %\newcommand{\productversion}{0.1} |

19 | %\newcommand{\productname}{\productbasename{}-\productversion} |

20 | % |

21 | %New environments |

22 | %The following environment is for the glossary of terms at the end. |

23 | %\newenvironment{docglossaryenum}{\begin{list} |

24 | % {}{\setlength{\labelwidth}{0mm} |

25 | % \setlength{\leftmargin}{4mm} |

26 | % \setlength{\itemindent}{-4mm} |

27 | % \setlength{\parsep}{0.85mm}}} |

28 | % {\end{list}} |

29 | %% |

30 | %The following environment is for the database table and field |

31 | %documentation at the end. |

32 | %\newenvironment{docdbtblfielddef}{\begin{list} |

33 | % {}{\setlength{\labelwidth}{0mm} |

34 | % \setlength{\leftmargin}{10mm} |

35 | % \setlength{\itemindent}{-5mm} |

36 | % \setlength{\parsep}{0.85mm}}} |

37 | % {\end{list}} |

38 | %% |

39 | |

40 | %Embarrassingly, I've forgotten why "makeindex" is necessary ... |

41 | %\makeindex |

42 | % |

43 | \begin{document} |

44 | \title{A Least-Squares Solution to the Multiple-Aircraft Wind Estimation Problem} |

45 | \author{\vspace{1cm}\\David T. Ashley\\\texttt{dta@e3ft.com}\\\vspace{1cm}} |

46 | \date{\vspace*{8mm}\small{Version Control $ $Revision: 1.4 $ $ \\ |

47 | Version Control $ $Date: 2006/12/26 06:59:37 $ $ (UTC) \\ |

48 | $ $RCSfile: wvecprob.tex,v $ $ \\ |

49 | \LaTeX{} Compilation Date: \today{}}} |

50 | \maketitle |

51 | |

52 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

53 | % |

54 | \begin{abstract} |

55 | This document presents a solution to the problem of estimating |

56 | wind speed and direction from simultaneous radar observations of the |

57 | course and groundspeed of multiple aircraft and knowledge of the approximate |

58 | cruising airspeed of each aircraft. |

59 | |

60 | The problem was posted to the \texttt{sci.math} newsgroup by |

61 | Chad Speer in December, 2006. |

62 | \end{abstract} |

63 | |

64 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

65 | |

66 | \section{Introduction and Overview} |

67 | \label{siov0} |

68 | |

69 | This problem was posted by Chad Speer to the \texttt{sci.math} newsgroup |

70 | in December, 2006\@. The problem did not at that time receive any meaningful |

71 | suggestions toward a solution. |

72 | |

73 | The problem is how to [uniquely] estimate wind velocity and direction |

74 | in the local area from: |

75 | |

76 | \begin{itemize} |

77 | \item Radar observations of the course and groundspeed of multiple aircraft. |

78 | \item Knowledge of the cruising airspeed of each aircraft (typically obtained from |

79 | VFR or IFR flightplan or clearance data filed by the pilot, or from knowledge |

80 | of the model of aircraft). |

81 | \end{itemize} |

82 | |

83 | This solution assumes that each observed aircraft is affected by wind at the |

84 | same speed and in the same direction. This is a reasonable assumption, |

85 | and will generally |

86 | hold true for aircraft at the same altitude separated by perhaps |

87 | 20-200 nautical miles. However, this assumption may be very flawed |

88 | for aircraft at different altitudes, as the winds tend to vary greatly |

89 | in magnitude and direction with altitude. |

90 | (Relaxing the assumption of identical wind vectors affecting all observed aircraft |

91 | may be a direction for future mathematical refinement.) |

92 | |

93 | Any mathematical results shown to work well in practice may eventually be |

94 | incorporated into algorithms used in air traffic control radar. |

95 | |

96 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

97 | |

98 | \section{Terms and Mathematical Nomenclature} |

99 | \label{snom0} |

100 | |

101 | All angular measurements (the angles of vectors) are in degrees clockwise from true |

102 | North, and are expressed canonically where possible so that |

103 | $0^\circ \leq \theta < 360^\circ$. $0^\circ$ is true North, $90^\circ$ is true East, |

104 | $180^\circ$ is true South, and $270^\circ$ is true West. |

105 | |

106 | The \emph{heading} of an aircraft is the direction the aircraft is pointed, whereas the |

107 | \emph{course} is the direction of the ground path of the aircraft. In the presence |

108 | of wind other than a direct headwind or tailwind, the heading is unequal to the course. |

109 | The heading of the aircraft is known by the pilot but not reported to anyone on the ground. |

110 | The course of the aircraft is known from radar data. |

111 | |

112 | Vectors are differentiated from scalars with an overlying arrow---$v_i$ is a scalar |

113 | but $\vec{v_i}$ is a vector. |

114 | |

115 | The local wind vector is $\vec{w}$ with magnitude $v_w$ and direction |

116 | $\theta_w$. |

117 | |

118 | Each aircraft is denoted $A_i$, $i \in \{1, 2, \ldots{} \}$; and has a heading vector |

119 | $\vec{v_{hAi}}$ with magnitude $v_{hAi}$ and heading direction $\theta_{hAi}$. |

120 | The course of the aircraft $A_i$ is denoted as a vector $\vec{v_{cAi}}$ with magnitude |

121 | $v_{cAi}$ and course direction $\theta_{cAi}$. Note that the course is observed by radar. |

122 | |

123 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

124 | |

125 | \section{The Wind Triangle} |

126 | \label{swtr0} |

127 | |

128 | Airplanes fly in a moving block of air, so that the aircraft's ground motion is the |

129 | vector sum of the air motion with respect to the ground and the aircraft's |

130 | motion with respect to the air (Fig. \ref{fig:swtr0:00}). |

131 | |

132 | For each aircraft $A_i$, |

133 | |

134 | \begin{equation} |

135 | \label{eq:swtr0:01} |

136 | \vec{v_{cAi}} = \vec{w} + \vec{v_{hAi}} . |

137 | \end{equation} |

138 | |

139 | (\ref{eq:swtr0:01}) is known as the \emph{wind triangle} because student pilots |

140 | are taught to make this calculation graphically by drawing a triangle of |

141 | three vectors on graph paper or by using a mechanical computer such as the |

142 | E-6B\footnote{\texttt{http://en.wikipedia.org/wiki/E6B}.}. |

143 | |

144 | \begin{figure} |

145 | \centering |

146 | \includegraphics[height=3.0in]{wtri01.eps} |

147 | \caption{Wind Triangle} |

148 | \label{fig:swtr0:00} |

149 | \end{figure} |

150 | |

151 | To a person who has never piloted an aircraft, (\ref{eq:swtr0:01}) |

152 | may be unexpected. It is very common for pilots to have a course that |

153 | differs from the heading by more than 10 degrees; and this is |

154 | visually apparent in an airplane when tracking roads or freeways below or when |

155 | landing in a crosswind. |

156 | |

157 | |

158 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

159 | |

160 | \section{The One-Aircraft Case} |

161 | \label{ssac0} |

162 | |

163 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

164 | |

165 | \subsection{Graphical Solution} |

166 | \label{ssac0:sgsl0} |

167 | |

168 | With only a single aircraft $A_1$, |

169 | |

170 | \begin{equation} |

171 | \label{eq:ssac0:sgls0:01} |

172 | \vec{v_{cA1}} = \vec{w} + \vec{v_{hA1}} . |

173 | \end{equation} |

174 | |

175 | \noindent{}Separating (\ref{eq:ssac0:sgls0:01}) into x- and y-components yields |

176 | |

177 | \begin{eqnarray} |

178 | \label{eq:ssac0:sgls0:02} |

179 | v_{cA1} \cos \theta_{cA1} & = & w \cos \theta_w + v_{hA1} \cos \theta_{hA1} \\ |

180 | \label{eq:ssac0:sgls0:03} |

181 | v_{cA1} \sin \theta_{cA1} & = & w \sin \theta_w + v_{hA1} \sin \theta_{hA1} |

182 | \end{eqnarray} |

183 | |

184 | \noindent{}The following quantities are known: |

185 | |

186 | \begin{itemize} |

187 | \item $v_{cA1}$ (from radar observation of the aircraft). |

188 | \item $\theta_{cA1}$ (from radar observation of the aircraft). |

189 | \item $v_{hA1}$ (the cruising speed of the aircraft, usually |

190 | filed by the pilot as part of the VFR or IFR clearance process). |

191 | \end{itemize} |

192 | |

193 | \noindent{}The following quantities are unknown: |

194 | |

195 | \begin{itemize} |

196 | \item $w$ (wind velocity). |

197 | \item $\theta_{w}$ (wind direction). |

198 | \item $\theta_{hA1}$ (Note as discussed above that \emph{heading} and |

199 | \emph{course} are distinct. The course is known from radar observation, |

200 | but the heading---the direction the aircraft is pointed---is not |

201 | known.\footnote{More precisely, the heading is not known by anyone \emph{on |

202 | the ground}. The heading is indicated by at least one aircraft instrument |

203 | and known to the pilot, but this information is not communicated to anyone |

204 | else.}) |

205 | \end{itemize} |

206 | |

207 | With two equations and three unknowns, it would normally be expected that the solution is a |

208 | set that can be parameterized with one parameter. |

209 | |

210 | \begin{figure} |

211 | \centering |

212 | \includegraphics[width=4.0in]{ac1gsl01.eps} |

213 | \caption{Graphical Solution for One-Aircraft Case} |

214 | \label{fig:ssac0:sgls0:00} |

215 | \end{figure} |

216 | |

217 | It can be seen graphically (Fig. \ref{fig:ssac0:sgls0:00}\footnote{Note that |

218 | although Fig. \ref{fig:ssac0:sgls0:00} conveys all the essential features of |

219 | the one-aircraft case, the wind vector $\vec{w}$ normally has the smallest magnitude |

220 | of the three vectors in the wind triangle. A light aircraft that cruises |

221 | at 120 knots airspeed and is affected by winds of 10-30 knots is the typical case.}) |

222 | that an infinite number of solutions |

223 | exist, parameterized by $0^\circ \leq \theta_{hA1} < 360^\circ$. |

224 | A heading vector with the appropriate magnitude |

225 | ($v_{hA1}$, the cruising speed of the aircraft) can be chosen so that its endpoint is |

226 | anywhere on the circle $C$ in Fig. \ref{fig:ssac0:sgls0:00}, and a wind vector $\vec{w}$ |

227 | can then be chosen to solve the equations. |

228 | |

229 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

230 | |

231 | \subsection{Analytic Solution} |

232 | \label{ssac0:sasl0} |

233 | |

234 | The problem can be solved analytically as follows: |

235 | |

236 | \begin{itemize} |

237 | \item Choose $\theta_{hA1} \in [0^\circ, 360^\circ)$. |

238 | \item Since $\theta_{hA1}$ and $v_{hA1}$ are established, |

239 | $\vec{v_{hA1}}$ is established. |

240 | Solve for $\vec{w}$ using (\ref{eq:ssac0:sgls0:01}): |

241 | |

242 | \begin{equation} |

243 | \label{eq:ssac0:sasl0:01} |

244 | \vec{w} = \vec{v_{cA1}} - \vec{v_{hA1}} . |

245 | \end{equation} |

246 | |

247 | \end{itemize} |

248 | |

249 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

250 | |

251 | \section{The Two-Aircraft Case} |

252 | \label{stac0} |

253 | |

254 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

255 | |

256 | \subsection{Graphical Solution} |

257 | \label{stac0:sgls0} |

258 | |

259 | \begin{figure} |

260 | \centering |

261 | \includegraphics[width=4.0in]{ac2gsl01.eps} |

262 | \caption{Graphical Solution for Two-Aircraft Case} |

263 | \label{fig:stac0:sgls0:00} |

264 | \end{figure} |

265 | |

266 | The two-aircraft case can be solved graphically by constructing the |

267 | solution sets (circle \emph{S} in Fig. \ref{fig:ssac0:sgls0:00}) |

268 | of the two aircraft so that \emph{Vertex C} |

269 | (Figs. \ref{fig:swtr0:00}, \ref{fig:ssac0:sgls0:00}) are coincident. |

270 | Figure \ref{fig:stac0:sgls0:00} depicts the graphical solution. |

271 | |

272 | Figure \ref{fig:stac0:sgls0:00} depicts the case where |

273 | $A_2$ has a higher cruising speed (a solution circle of larger |

274 | radius) than $A_1$. The following properties can be |

275 | observed from Fig. \ref{fig:stac0:sgls0:00}: |

276 | |

277 | \begin{itemize} |

278 | \item Since the solution set for each aircraft is represented by a circle, |

279 | the solutions for the two-aircraft case are the points where the two |

280 | circles meet. |

281 | \item There may be no solutions, one solution, or two solutions for the |

282 | two-aircraft case. (Fig. \ref{fig:stac0:sgls0:00} |

283 | depicts the case with two solutions. The second solution is shown |

284 | with dotted lines.) |

285 | \item In a practical case, the correct solution would probably be the |

286 | solution with the wind vector of lesser magnitude. |

287 | \end{itemize} |

288 | |

289 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

290 | |

291 | \subsection{Analytic Solution} |

292 | \label{stac0:sasl0} |

293 | |

294 | I have not a clue how to think about this problem analytically. |

295 | |

296 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

297 | |

298 | \section{The General Multiple Aircraft Case} |

299 | \label{smac0} |

300 | |

301 | It is unclear how to set this up as a problem so that a unique solution |

302 | can be obtained in the presence of [mildly] inconsistent data, or what the basis |

303 | for the unique solution should be (i.e. similar to least-squares---there has |

304 | to be something one is trying to optimize or minimize). |

305 | |

306 | |

307 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

308 | \end{document} |

309 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

310 | % $Log: wvecprob.tex,v $ |

311 | % Revision 1.4 2006/12/26 06:59:37 dashley |

312 | % Edits. |

313 | % |

314 | % Revision 1.3 2006/12/23 21:10:01 dashley |

315 | % Edits. |

316 | % |

317 | % Revision 1.2 2006/12/23 20:27:11 dashley |

318 | % Edits. |

319 | % |

320 | % Revision 1.1 2006/12/23 19:06:00 dashley |

321 | % Initial checkin. |

322 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |

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