Difference between revisions of "Directivity of a harmonic source plus ghost"

From SEG Wiki
Jump to: navigation, search
(Also in this chapter: fixed page)
(add)
 
(One intermediate revision by the same user not shown)
Line 14: Line 14:
 
  | isbn    = ISBN 9781560801153
 
  | isbn    = ISBN 9781560801153
 
}}
 
}}
== Problem ==
+
== Problem 6.9 ==
 
Show that equation (6.7c) gives the directivity diagrams shown in Figure 6.9a.
 
Show that equation (6.7c) gives the directivity diagrams shown in Figure 6.9a.
  
Line 47: Line 47:
 
The results of the calculations are shown in Tables 6.9a,b.
 
The results of the calculations are shown in Tables 6.9a,b.
  
[[file:Ch06_fig6-9a.png|thumb|{{figure number|6.9a.}} Directivity of a harmonic source at depth <math>z=c\lambda</math>.]]
+
[[file:Ch06_fig6-9a.png|thumb|center|{{figure number|6.9a.}} Directivity of a harmonic source at depth <math>z=c\lambda</math>.]]
  
 
Ignoring the minus signs (which indicate phase reversals), the curves for <math>\varphi _{a}</math> and <math>\varphi _{b}</math>, shown in Figure 6.9b, conform closely to Figure 6.9a. However, we need more points to plot the <math>\varphi _{c}</math>-curve properly and Table 6.9b shows calculated values for intermediate points. The <math>\Psi _{c}</math>-curve in Figure 6.9b also conforms closely to Figure 6.9a.
 
Ignoring the minus signs (which indicate phase reversals), the curves for <math>\varphi _{a}</math> and <math>\varphi _{b}</math>, shown in Figure 6.9b, conform closely to Figure 6.9a. However, we need more points to plot the <math>\varphi _{c}</math>-curve properly and Table 6.9b shows calculated values for intermediate points. The <math>\Psi _{c}</math>-curve in Figure 6.9b also conforms closely to Figure 6.9a.
Line 134: Line 134:
 
|}
 
|}
  
[[file:Ch06_fig6-9b.png|thumb|{{figure number|6.9b.}} Calculated directivity at source depth <math>z=c\lambda</math>.]]
+
[[file:Ch06_fig6-9b.png|thumb|center|{{figure number|6.9b.}} Calculated directivity at source depth <math>z=c\lambda</math>.]]
  
 
== Continue reading ==
 
== Continue reading ==

Latest revision as of 15:18, 8 November 2019

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 2A=1} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c={\rm depth}/\lambda =0.1} , 0.5, and 1.0 for the three parts of Figure 6.9a. Then equation (6.7c) gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} A^{*} ={\rm \; sin\; }(2\pi c{\rm \; cos\; }\theta ). \end{align} }


Substituting the three values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c} , we have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} a):c=0.1\;,\; A^{*} ={\rm \; sin\; }(0.63{\rm \; cos\; }\theta), \end{align} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} b):c=0.5\;,\; A^{*} ={\rm \; sin\; }(3.1{\rm \; cos\; }\theta), \end{align} } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} c):c=1.0,\; A^{*} ={\rm \; sin\; }(6.3{\rm \; cos\; }\theta). \end{align} }

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

Figure 6.9a.  Directivity of a harmonic source at depth Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z=c\lambda} .

Ignoring the minus signs (which indicate phase reversals), the curves for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi _{a}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi _{b}} , shown in Figure 6.9b, conform closely to Figure 6.9a. However, we need more points to plot the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varphi _{c}} -curve properly and Table 6.9b shows calculated values for intermediate points. The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{c}} -curve in Figure 6.9b also conforms closely to Figure 6.9a.

Table 6.9a. Values for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{a}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{b}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{c^{'} }} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta ^{\circ}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{a}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{b}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{c^{'} }} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta ^{\circ}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Psi _{c}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta ^{\circ}} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z=c\lambda} .

Continue reading

Previous section Next section
Directivity of a source plus its ghost Differential moveout between primary and multiple
Previous chapter Next chapter
Geometry of seismic waves Characteristics of seismic events

Table of Contents (book)

Also in this chapter

External links

find literature about
Directivity of a harmonic source plus ghost
SEG button search.png Datapages button.png GeoScienceWorld button.png OnePetro button.png Schlumberger button.png Google button.png AGI button.png