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Commit of wind estimation document.
1 dashley 9 %$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}
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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
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313     %
314     % Revision 1.3 2006/12/23 21:10:01 dashley
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316     %
317     % Revision 1.2 2006/12/23 20:27:11 dashley
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319     %
320     % Revision 1.1 2006/12/23 19:06:00 dashley
321     % Initial checkin.
322     %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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