Directivities of linear arrays and linear sources
 | |
| Series | Geophysical References Series |
|---|---|
| Title | Problems in Exploration Seismology and their Solutions |
| Author | Lloyd P. Geldart and Robert E. Sheriff |
| Chapter | 8 |
| Pages | 253 - 294 |
| DOI | http://dx.doi.org/10.1190/1.9781560801733 |
| ISBN | ISBN 9781560801153 |
| Store | SEG Online Store |
Problem
Show that the directivity equations (8.6d) and (7.5d) are consistent.
Solution
Since equation (8.6d) applies to a discontinuous array of geophones whereas equation (7.5d) applies to a continuous source, we find the limit of equation (8.6d) as the number of geophones becomes infinite. We require that $ n\to \infty $ while $ \Delta x\to 0 $ in such a way that $ n\Delta x\to a\lambda $, 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/":): a being the same constant as in problem 7.5, thus keeping the array length equal to the source length. In the limit the numerator of equation (8.6d) becomes 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/":): \sin (\pi a\sin \alpha ) . To get the limit of the denominator, we replace the sine by its argument (because 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 x\to 0 ) and get 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/":): (n\pi \Delta x/\lambda )\sin \alpha \to \pi a\sin \alpha . Substituting these values in equation (8.6d), we obtain
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} F=\sin (\pi a\sin \alpha )/\pi a\sin \alpha =\sin c(\pi a\sin \alpha ). \end{align}
Because the angles 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/":): \alpha and 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 _{0} are equivalent, equations (8.6d) and (7.5d) are equivalents.
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| Response of a linear array | Tapered arrays |
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| Seismic equipment | Data processing |
Also in this chapter
- Effect of too many groups connected to the cable
- Reflection-point smear for dipping reflectors
- Stacking charts
- Attenuation of air waves
- Maximum array length for given apparent velocity
- Response of a linear array
- Directivities of linear arrays and linear sources
- Tapered arrays
- Directivity of marine arrays
- Response of a triangular array
- Noise tests
- Selecting optimum field methods
- Optimizing field layouts
- Determining vibroseis parameters
- Selecting survey parameters
- Effect of signal/noise ratio on event picking
- Interpreting uphole surveys
- Weathering and elevation (near-surface) corrections
- Determining static corrections from first breaks
- Determining reflector location
- Blondeau weathering corrections
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