Directivity of a harmonic source plus ghost

**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	6
Pages	181 - 220
DOI	http://dx.doi.org/10.1190/1.9781560801733
ISBN	ISBN 9781560801153
Store	SEG Online Store

Problem 6.9

Show that equation (6.7c) gives the directivity diagrams shown in Figure 6.9a.

Solution

The directivity is given by equation (6.7c). We take $2A=1$ , and $c={\rm {depth}}/\lambda =0.1$ , 0.5, and 1.0 for the three parts of Figure 6.9a. Then equation (6.7c) gives

${\begin{aligned}A^{*}={\rm {\;sin\;}}(2\pi c{\rm {\;cos\;}}\theta ).\end{aligned}}$

Substituting the three values of $c$ , we have:

${\begin{aligned}a):c=0.1\;,\;A^{*}={\rm {\;sin\;}}(0.63{\rm {\;cos\;}}\theta ),\end{aligned}}$ ${\begin{aligned}b):c=0.5\;,\;A^{*}={\rm {\;sin\;}}(3.1{\rm {\;cos\;}}\theta ),\end{aligned}}$ ${\begin{aligned}c):c=1.0,\;A^{*}={\rm {\;sin\;}}(6.3{\rm {\;cos\;}}\theta ).\end{aligned}}$

The results of the calculations are shown in Tables 6.9a,b.

Figure 6.9a. Directivity of a harmonic source at depth

z=c\lambda

.

Ignoring the minus signs (which indicate phase reversals), the curves for $\varphi _{a}$ and $\varphi _{b}$ , shown in Figure 6.9b, conform closely to Figure 6.9a. However, we need more points to plot the $\varphi _{c}$ -curve properly and Table 6.9b shows calculated values for intermediate points. The $\Psi _{c}$ -curve in Figure 6.9b also conforms closely to Figure 6.9a.

**Table 6.9a. Values for $\Psi _{a}$ , $\Psi _{b}$ , $\Psi _{c^{'}}$ .**
$\theta ^{\circ }$	$\Psi _{a}$	$\Psi _{b}$	$\Psi _{c}$
0	0.59	0.00	0.00
15	0.57	0.15	−0.20
30	0.52	0.44	−0.74
45	0.43	0.81	−0.97
60	0.31	1.00	−0.01
75	0.16	0.72	1.00
90	0.00	0.00	0.00

**Table 6.9b. Intermediate values for $\Psi _{c^{'}}$ .**
$\theta ^{\circ }$	$\Psi _{c}$	$\theta ^{\circ }$	$\Psi _{c}$
5	−0.01	50	−0.79
10	−0.08	55	−0.46
20	−0.35	65	0.46
25	−0.54	70	0.83
35	−0.90	80	0.89
40	−0.99	85	0.52

Figure 6.9b. Calculated directivity at source depth

z=c\lambda

.

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