Dependence of resolvable limit on frequency
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| 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 6.18a
A wavelet has a flat frequency spectrum from 0 to $ f_{u} $ above which no frequencies are present. Show that the Rayleigh criterion gives a resolvable limit $ t_{r} $, where $ t_{r}=0.715/f_{u} $.
Background
A reflecting layer is said to be resolvable if we can distinguish between the reflections from the top and bottom of the layer, usually on the basis of a phase break in the superimposed reflections (see Figure 6.18a). The Rayleigh criterion for vertical resolution states that at least a small depression must appear between successive events in order to recognize that more than one event is present. For this to occur the two reflections must be separated by at least a half-cycle; this corresponds to a minimum thickness of $ \lambda /4 $, the two-way thickness then being $ \lambda /2 $. This thickness of $ {\frac {1}{4}}\lambda $ is called the tuning thickness or resolvable limit.
A boxcar (see Figure 6.18b) is a function whose value is unity within a certain range and zero outside this range (see problem 9.3).
Solution
The frequency spectrum is a boxcar extending from $ -f_{u}\ {\rm {to}}+f_{u} $, shown in Figure 6.18b, which we write as $ {\rm {box}}_{2f_{u}}(f) $. The inverse transform of the spectrum is given by equation (9.3d), namely
$ {\begin{aligned}g(t)=(1/2\pi )\mathop {\int } \nolimits _{-\infty }^{+\infty }\ {\rm {box}}_{2f_{u}}(f)e^{j\omega t}\ {\rm {d}}\omega =\mathop {\int } \nolimits _{-f_{u}}^{+f_{u}}e^{j2\pi ft}\ {\rm {d}}f\\=(1/2\pi jt)(e^{j2\pi f_{u}{f}}-e^{-j2\pi f_{u}{f}})=(1/\pi t){\rm {\;sin\;}}(2\pi f_{u}t)\\=2f_{u}\ {\rm {sinc}}\ (2\pi f_{u}t),\end{aligned}} $ ()
where sinc $ x=({\rm {\;sin\;}}x)/x. $
The Rayleigh criterion gives a resolvable limit $ t_{r} $ corresponding to the first trough (minimum) of the time-domain representation of the boxcar. Hence, we equate the derivative of the sinc function to zero, obtaining
$ {\begin{aligned}{\frac {\rm {d}}{{\rm {d}}t}}[{\rm {sinc}}(2\pi f_{u}t)]={\frac {\rm {d}}{{\rm {d}}t}}\left[{\frac {{\rm {\;sin\;}}(2\pi f_{u}t)}{2\pi f_{u}t}}\right]=0={\frac {\rm {d}}{{\rm {d}}t}}\left[{\frac {{\rm {\;sin\;}}(2\pi f_{u}t)}{t}}\right],\end{aligned}} $
$ {\begin{aligned}{\mbox{so}}\qquad \qquad \ (2\pi f_{u}/t){\rm {\;cos\;}}(2\pi f_{u}t)-(1/t^{2}){\rm {\;sin\;}}(2\pi f_{u}t)=0;\end{aligned}} $
This gives $ {\rm {\;tan\;}}(2\pi f_{u}t)=2\pi f_{u}t $, that is, we must solve the equation $ {\rm {\;tan\;}}x=x $ where $ x=2\pi f_{u}t $. A graphical solution gives $ x=4.49 $, hence $ t_{r}=0.715/f_{u} $.
Problem 6.18b
Show that the value of $ t_{r} $ for a wavelet with a flat spectrum extending from $ f_{L} $ to $ nf_{L} $ (that is, $ m $ octaves where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=2^{m} ) is given by the solution of the equation,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} nx\ {\rm cos}\ nx - {\rm \; sin\; }nx\\ -x{\rm \; cos\; }x+{\rm \; sin\; }x=0, \end{align}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=2\pi f_{L} t_{r} .
Solution
The time-domain function corresponding to the spectrum in Figure 6.18c is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} g(t)=(1/2\pi )\left(\mathop{\int }\nolimits_{-nf_{L} }^{-f_{L} } e^{1^{\omega t} }\ {\rm d}\omega +\mathop{\int }\limits_{f^{L} }^{nf_{L} } e^{j\omega t}\ {\rm d}\omega \right)\\ =(1/2\pi jt)[(e^{-j2\pi f_{L}{t}} )-(e^{-j2\pi nf_{L}{t}})+(e^{j2\pi f_{L}{t}} - e^{j2\pi nf_{L}{f}})]\\ =(1/\pi t)[{\rm \; sin\; }(2\pi nf_{L} t)-{\rm \; sin\; }(2\pi f_{L} t)]. \end{align}
To get the first trough, we write Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=2\pi f_{L} t , then equate to zero the derivative of $ g(x) $ with respect to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x . This gives
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} g(x)=(2f_{L} /x)({\rm \; sin\; }nx-{\rm \; sin\; }x),\\ dg(x)/dx=0=2f_{L} [(n{\rm \; cos\; }nx-{\rm \; cos\; }x)/x-({\rm \; sin\; }nx-{\rm \; sin\; }x)/x^{2}],\\ \mbox {so} \qquad\qquad\ nx {\rm \; cos\; } nx - {\rm \; sin\; } nx - x {\rm \; cos\; }x+{\rm \; sin\; }x=0. \end{align} ()
Problem 6.18c
Solve the equation in part (b) for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m=3, 2, 1.5, and 1, that is, for bandwidths of 3, 2, 1.5, and 1 octaves, and compare the relation between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m .
Solution
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m=3 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=2^{3} = 8 and we have from equation (6.18c)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} 8x{\rm \; cos\; }8x-{\rm \; sin\; }8x-{\rm \; cos\; }x+{\rm \; sin\; }x=0. \end{align}
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=0 , we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 0=0 , but for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x slightly greater than zero, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\rm d}g(x)/{\rm d}x is negative and continues to be negative until it changes sign for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x between 0.5 and 0.6; The corresponding root is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=0.560 , giving $ t_{r}=0.089/f_{L}. $.
When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m=2 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=4 , the root of equation (6.18c) is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=1.10 , giving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r}=0.175/f_{L} . For $ m=1.5 $, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=2\sqrt{2} =2.83 and the root is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=1.51 , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} =0.240/f_{L} . For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m=1.0 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=2 and the root is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): x=2.015 , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} =0.321/f_{L} .
Problem 6.18d
Noting that part (a) involves an infinite number of octaves, what bandwidth is required to give nearly the same result?
Solution
The resolution in part (a) is expressed in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{u} while those in (c) are in terms of $ f_{L} $. To compare the results we equate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): nf_{L} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{u} , so that we have four frequency bands, each with top frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f_{u} and extending downward 1, 1.5, 2, 3, and an infinite number of octaves. This means that the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} in (c) must be adjusted to get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): nf_{L} in the denominator, e.g.,for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n=4 , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} = 4\times 0.175/(4f_{L}) . The results are shown in Table 6.18a.
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): n | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): t_{r} /(Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): nf_{L} ) |
|---|---|---|
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \infty | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \infty | 0.715 |
| 3 | 8 | 0.712 |
| 2 | 4 | 0.700 |
| 1.5 | 2.83 | 0.679 |
| 1 | 2 | 0.321 |
Thus, three octaves bandwidth (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): m=3 ) gives almost as good resolution as an infinite number of octaves but the resolution deteriorates for band-widths Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): <1.5 octaves.
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| Destructive and constructive interference for a wedge | Vertical resolution |
| 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
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