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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 $
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3 \documentclass[letterpaper,10pt,titlepage]{article}
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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
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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.
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320 % Revision 1.1 2006/12/23 19:06:00 dashley
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