Difference between revisions of "Directivity of a harmonic source plus ghost"
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| isbn = ISBN 9781560801153 | | isbn = ISBN 9781560801153 | ||
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− | == 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. | ||
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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. | ||
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− | [[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
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 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.
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.
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 |
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 |
Continue reading
Previous section | Next section |
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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 |
Also in this chapter
- Characteristics of different types of events and noise
- Horizontal resolution
- Reflection and refraction laws and Fermat’s principle
- Effect of reflector curvature on a plane wave
- Diffraction traveltime curves
- Amplitude variation with offset for seafloor multiples
- Ghost amplitude and energy
- Directivity of a source plus its ghost
- Directivity of a harmonic source plus ghost
- Differential moveout between primary and multiple
- Suppressing multiples by NMO differences
- Distinguishing horizontal/vertical discontinuities
- Identification of events
- Traveltime curves for various events
- Reflections/diffractions from refractor terminations
- Refractions and refraction multiples
- Destructive and constructive interference for a wedge
- Dependence of resolvable limit on frequency
- Vertical resolution
- Causes of high-frequency losses
- Ricker wavelet relations
- Improvement of signal/noise ratio by stacking