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

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Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 7 221 - 252 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem

If the error in Shoran time measurements is ${\displaystyle \pm 0.1\mu s}$, what is the the size of the parallelogram of error in Figure 7.2a when (a) ${\displaystyle \theta =30^{\circ }}$ and (b) ${\displaystyle \theta =150^{\circ }}$? Take the velocity of radio waves as ${\displaystyle 3\times 10^{5}}$ 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 ${\displaystyle \pm \Delta t}$, so the ranges are ${\displaystyle V\left(t_{i}\pm \Delta t\right)}$, ${\displaystyle i=1}$, 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 ${\displaystyle =AM=AN=AP=AQ}$

{\displaystyle {\begin{aligned}=\left(3\times 10^{8}\ \mathrm {m/s} \right)\left(1\times 10^{-7}\ \mathrm {s} \right)=30\ \mathrm {m} .\end{aligned}}}

Figure 7.2a  Parallelogram of error for traveltime uncertainty of ${\displaystyle \Delta t=\pm 0.1\mu s}$

.

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

The short diagonal ${\displaystyle =2AS=2\times 30/\cos 15^{\circ }=60}$ m.

To get the figure for ${\displaystyle \theta =150^{\circ }}$ we merely reverse the arrow on ${\displaystyle {\textit {AB}}}$ or ${\displaystyle {\textit {AC}}}$; therefore the error values are the same as for ${\displaystyle 30^{\circ }}$.

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