Difference between revisions of "Ricker wavelet relations"

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[Papoulis, 1962: p. 25, equation (2-68)].
 
[Papoulis, 1962: p. 25, equation (2-68)].
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[[file:Ch06_fig6-21a.png|thumb|center|{{figure number|6.21a.}} Ricker wavelet (i) in time domain and (ii) in frequency domain.]]
  
 
=== Solution ===
 
=== Solution ===

Latest revision as of 15:34, 8 November 2019

Problem 6.21a

Verify that the Ricker wavelet in Figure 6.21a(i),


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} g(t)=(1-2\pi^{2} f_{m}^{2} t^{2})e^{-(\pi f_{m} t)^{2}}, \end{align} } (6.21a)

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 f_{m}} , being the peak frequency, has the Fourier transform [Figure 6.21a(ii)]


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} G(f)=(2/\sqrt{\pi })(f/f_{m})^{2} e^{-(f/f_{m})^{2}},\; \gamma(f) = 0, \end{align} } (6.21b)

where 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 \gamma(f)} is the phase.

Background

Fourier transforms are discussed in problem 9.3 and theorems on Fourier transforms in Sheriff and Geldart, 1995, section 15.2.6.

The transform 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 e^{-at^{2}}} is


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} e^{-at^{2}} \leftrightarrow (\pi /a)^{1/2} e^{-\omega^{2} /4a} \end{align} } (6.21c)

[Papoulis, 1962: p. 25, equation (2-68)].

Figure 6.21a.  Ricker wavelet (i) in time domain and (ii) in frequency domain.

Solution

The time-domain expression for the Ricker wavelet can be written in the form


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} g(t)=(1-2at^{2})e^{-at^{2}} = e^{-at^{2} } -2at^{2} e^{-at^{2}}, \end{align} } (6.21d)

where 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 a=(\pi f_{M} )^{2}} . The transform of the first term is 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 (\pi /a)^{1/2} e^{-\omega^{2}} /4a} . To get the transform of the second term, we use Sheriff and Geldart, 1995, equation (15.142) which states that when 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 g(t)\leftrightarrow G(\omega)} , then,

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} (-jt)^{n} g(t)\leftrightarrow \frac{{\rm d}^{n} G(\omega )}{{\rm d}\omega ^{n}}, \end{align} }

that is, 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 n=2} ,

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} -t^{2} g(t)\leftrightarrow \frac{{\rm d}^{2} G(\omega )}{{\rm d}\omega ^{2}}. \end{align} }

The transform of the second term now becomes

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} 2a\frac{{\rm d}^{2} }{{\rm d}\omega ^{2} } \left[\left(\frac{\pi }{a} \right)^{1/2} e^{-(\omega^{2} /4a)} \right]=2(\pi a)^{1/2} \frac{{\rm d}}{{\rm d}\omega } \left[-\left(\frac{\omega }{2a} \right)e^{-(\omega ^{2} /4a)} \right]\\ =-(\pi /a)^{1/2} \frac{{\rm d}}{{\rm d}\omega } \left[\omega e^{-(\omega^{2} /4a)}\right]=-(\pi /a)^{1/2} \left[1-\left(\frac{\omega ^{2} }{2a} \right)\right]e^{-(\omega^{2} /4a)}. \end{align} }

Adding the two transforms, 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} &g(t)\leftrightarrow (2/\sqrt{\pi } )(\omega /\omega _{M} )^{2} e^{-(\omega /\omega _{M} )^{2} } = (\pi /a^{3} )^{1/2} (\omega ^{2} /2)e^{-(\omega /\omega _{M} )^{2} }\\ &\leftrightarrow (2/\sqrt{\pi } )(f^{2} /f_{M}^{3} )e^{-(f/f_{M} )^{2} }. \end{align} }

Problem 6.21b

Show that 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 f_{M}} is the peak of the frequency spectrum.

Solution

To find the peak frequency, we set the derivative 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 {\rm d}G(f)/{\rm d}f} equal to zero. Thus, omitting the constant factor,

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} \left(\frac{{\rm d}}{{\rm d}f} \right)\left[f^{2} e^{-(f/f_{M})^{2} }\right]=e^{-(f/f_{M})^{2} } \left[2f-f^{2} (2f/f_{M}^{2})\right]=0, \end{align} }

so 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 f=f_{M}} for a maximum.

Problem 6.21c

Show that 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 T_{D} /T_{R} =\sqrt{3}} (see Figure 6.21a) and that 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 T_{D} f_{M} =\sqrt{6} /\pi.}

Solution

Since 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 g(t)=0} 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 t=+T_{R} /2} , 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} [1-2(\pi f_{M} T_{R} /2)^{2}]e^{-(\pi f_{M} T_{R} /2)^{2} } =0, \end{align} } (6.21e)

hence 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 T_{R} =\pm \sqrt{2} /(\pi f_{M})} .

Moreover, 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 g(t)} is a minimum 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 t=\pm T_{D} /2} , so 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 T_{D} /2} is a root 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 \begin{align} ({\rm d}/{\rm d}t)[(1-2at^{2} )e^{-at^{2} } ]=0, \end{align} }

that is, of the equation


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} e^{-at^{2} } [-4at+(1-2at^{2} )(-2at)]=0,\\ \mbox {so} \qquad\qquad\ t=T_{D} /2=[\sqrt{(3/2)} ]/\pi f_{M}. \end{align} } (6.21f)

Hence, 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 T_{D} /T_{R} =\sqrt{3}} 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 T_{D} f_{M} =\sqrt{6/\pi}} .

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