Effect of station angle on location errors

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Problem 7.2

If the error in Shoran time measurements is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm 0.1 \mu s , what is the the size of the parallelogram of error in Figure 7.2a when (a) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta=30^{\circ} and (b) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta=150^{\circ} ? Take the velocity of radio waves as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \pm \Delta t , so the ranges are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V\left(t_{i} \pm \Delta t\right) , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): =AM=AN=AP=AQ

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Delta t=\pm 0.1 \mu s

.

The long diagonal $ =2AR=2AQ/\sin 15^{\circ }=2\times 30/\sin 15^{\circ }=230 $ m.

The short diagonal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): =2AS=2\times 30/\cos 15^{\circ} =60 m.

To get the figure for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \theta =150^{\circ} we merely reverse the arrow on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \textit{AB} or $ {\textit {AC}} $; therefore the error values are the same as for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 30^{\circ} .

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