# 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 $f(x)$ 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}} (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.