Effect of station angle on location errors

Problems in Exploration Seismology and their Solutions

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

Problem 7.2

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

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

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

.

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

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

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

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Table of Contents (book)

Also in this chapter

• Radiolocation errors because of velocity variations
• Effect of station angle on location errors
• Transit satellite navigation
• Effective penetration of profiler sources
• Directivity of linear sources
• Sosie method
• Energy from an air-gun array
• Dominant frequencies of marine sources
• Effect of coil inductance on geophone equation
• Streamer feathering due to cross-currents
• Filtering effect of geophones and amplifiers
• Filter effects on waveshape
• Effect of filtering on event picking
• Binary numbers

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