# Response of a triangular array

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 8 253 - 294 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 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 ${\displaystyle f(x)}$ to represent a continuous function of the variable ${\displaystyle x}$ while the notation ${\displaystyle f_{x}}$ denotes a digital function, that is, the result of sampling a continuous function at a fixed sampling interval ${\displaystyle \Delta }$ (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 ${\displaystyle g_{t}*h_{t}}$ denotes the convolution of ${\displaystyle g_{t}}$ and ${\displaystyle h_{t}}$ (see problem 9.2). The convolution is given by the summation in equation (9.2b), namely

 {\displaystyle {\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:

{\displaystyle {\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.