Response of a triangular array
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| 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 8.10a
The tapered array [1, 2, 3, 3, 2, 1] =[1, 1, 1, 1] * [1, 1, 1] is called a triangular array. Use this fact to sketch the array response.
Background
We use the notation $ f(x) $ to represent a continuous function of the variable $ x $ while the notation $ f_{x} $ denotes a digital function, that is, the result of sampling a continuous function at a fixed sampling interval $ \Delta $ (see problem 9.4).
The triangular array is also used to approximate a cosine array where successive elements are weighted as equally spaced samples of the first half-cycle of a cosine function.
The notation $ g_{t}*h_{t} $ denotes the convolution of $ g_{t} $ and $ h_{t} $ (see problem 9.2). The convolution is given by the summation in equation (9.2b), namely
$ {\begin{aligned}g_{t}*h_{t}=\mathop {\sum } \limits _{k}^{}g_{k}h_{t-k}=\mathop {\sum } \limits _{k}^{}h_{k}g_{t-k}.\end{aligned}} $ ()
Solution
We get for the convolution:
$ {\begin{aligned}[1,1,1,1]*[1,1,1]=[1\times 1,1\times 1+1\times 1,1\times 1+1\times 1\\\quad +1\times 1,1\times 1+1\times 1+1\times 1,1\times 1\\\quad +1\times 1,1\times 1\left]=\right[1,2,3,3,2,1].\end{aligned}} $
The response of this array to a harmonic signal is shown in Figure 8.10a.
Problem 8.10b
How could three strings of geophones, each having four equally spaced elements, be laid out to yield a triangular array?
Solution
Number six geophone locations 1 through 6 and lay the first string (see problem 8.13) of four geophones from position 1 to position 4, the second string from 2 to 5, and the third string from 3 to 6. This give the array 1, 2, 3, 3, 2, 1.
Problem 8.10c
How could a smoother tapered array be approximated?
Solution
We could achieve a smoother array by spacing the geophones unequally such that each represents an equal portion of the area under the desired array response curve, as illustrated in Figure 8.10c.
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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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