Maximum array length for given apparent velocity

**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

Reflections in the zone of interest have apparent velocities around 6300 m/s, whereas the velocity just below a uniform LVL is 2100 m/s. If we wish to avoid severe attenuation below 80 Hz when using an array, what is the maximum inline array length?

Background

The LVL is discussed in problem 4.16. Apparent velocity is defined in problem 4.2d. See problem 8.6 for a discussion of array response. Note that the array length, $n\Delta x$ , is larger than the distance between the first and last geophones, $(n-1)\Delta x$ .

Solution

To avoid deterioration, the response curve for the array length $n\Delta x$ , where n is the number of geophones and $\Delta x$ the geophone interval, must not extend beyond the first null (see Figure 8.6b). Equation (8.6b) shows that the first null occurs when $n\gamma /2=\pi$ or $n=2\pi /\gamma$ , where $\gamma$ is the phase difference between adjacent geophones. Problem 8.6a gives $\gamma =2\pi \left(\Delta x/\lambda \right)\sin \alpha$ , so