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

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 7.2

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{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 $\displaystyle \Delta t=\pm 0.1 \mu s$

.

The long diagonal $\displaystyle =2AR=2AQ/\sin15^{\circ} = 2\times 30/\sin15^{\circ} =230$ 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 $\displaystyle 30^{\circ}$ .