Difference between revisions of "Effect of station angle on location errors"

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  | isbn    = ISBN 9781560801153
 
  | isbn    = ISBN 9781560801153
 
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== Problem ==
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== Problem 7.2 ==
 
If the error in Shoran time measurements is <math>\pm 0.1 \mu s</math>, what is the the size of the parallelogram of error in Figure 7.2a when (a) <math>\theta=30^{\circ}</math> and (b) <math>\theta=150^{\circ}</math>? Take the velocity of radio waves as <math>3\times 10^{5}</math> km/s.
 
If the error in Shoran time measurements is <math>\pm 0.1 \mu s</math>, what is the the size of the parallelogram of error in Figure 7.2a when (a) <math>\theta=30^{\circ}</math> and (b) <math>\theta=150^{\circ}</math>? Take the velocity of radio waves as <math>3\times 10^{5}</math> km/s.
  

Latest revision as of 15:36, 8 November 2019

Problem 7.2

If the error in Shoran time measurements is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm 0.1 \mu s} , what is the the size of the parallelogram of error in Figure 7.2a when (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta=30^{\circ}} and (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta=150^{\circ}} ? Take the velocity of radio waves as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 3\times 10^{5}} km/s.

Background

Shoran is a radio-navigation device which measures the 2-way traveltime between the point of observation and a fixed station. Using two fixed stations, the point of observation can be located by swinging arcs centered at the two stations; for large distances the arcs become nearly straight lines.

The traveltimes are subject to error Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm \Delta t} , so the ranges are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V\left(t_{i} \pm \Delta t\right)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i=1} , 2. Swinging the four arcs corresponding to these time values, we get a parallelogram of error such as that in Figure 7.2a; the location lies somewhere inside this parallelogram.

Solution

In Figure 7.2a, the error in range Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =AM=AN=AP=AQ}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} =\left(3\times 10^{8}\ \mathrm{m/s}\right)\left(1\times 10^{-7}\ \mathrm{s}\right)=30\ \mathrm{m}. \end{align} }

Figure 7.2a  Parallelogram of error for traveltime uncertainty of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t=\pm 0.1 \mu s}

.

The long diagonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =2AR=2AQ/\sin15^{\circ} = 2\times 30/\sin15^{\circ} =230} m.

The short diagonal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle =2AS=2\times 30/\cos 15^{\circ} =60} m.

To get the figure for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta =150^{\circ} } we merely reverse the arrow on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textit{AB}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \textit{AC}} ; therefore the error values are the same as for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 30^{\circ}} .

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