Response of a triangular array

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	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.

Figure 8.10a.  Response of tapered array.

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

${\begin{aligned}g_{t}*h_{t}=\mathop {\sum } \limits _{k}^{}g_{k}h_{t-k}=\mathop {\sum } \limits _{k}^{}h_{k}g_{t-k}.\end{aligned}}$

(8.10a)

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.

Figure 8.10c.  Approximating smooth array with unequal geophone spacing.

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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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