iMechanica - Comments for "Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs"
https://imechanica.org/node/6570
Comments for "Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs"enabout your OWWE movie and AWWE
https://imechanica.org/comment/12667#comment-12667
<a id="comment-12667"></a>
<p><em>In reply to <a href="https://imechanica.org/node/6570">Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Hi, Professor,</p>
<p>I am a exploration geophysics PhD student in TAMU. I am very interested in wave equations.</p>
<p>I have ever read your AWWE papers. Those ideas are really wonderful. </p>
<p>The OWWE movie you made in this page is from your AWWE? Or just ordinary 15 degree one-way wave equation? I have ever tried to stimulate your AWWE to create such snapshots, but seems that it is a little hard to get the right finite-difference scheme. I have also read some papers on true-amplitude one-way wave equation. But they are more complex in implementing. </p>
<p> Also, you mentioned in your AWWE paper that your AWWE is not very correct in case of TTI non-elliptic media. Slowness cannot be very correctly captured. Have your solved that now? </p>
<p>My email is <a href="mailto:nebulaekg@gmail.com">nebulaekg@gmail.com</a>.</p>
<p>Thanks very much. </p>
<p>Kai </p>
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</ul>Mon, 26 Oct 2009 18:06:36 +0000niaotiyuluoleicomment 12667 at https://imechanica.orgRe: Clarifications about JC Aug 09
https://imechanica.org/comment/12152#comment-12152
<a id="comment-12152"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/12136#comment-12136">Clarifications</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Murthy,
</p>
<p>
0. Thanks for emailing the paper.
</p>
<p>
2. The clarification that it's a Gaussian pulse itself suddenly made many things clear.
</p>
<p>
[This might look mysterious, but let me add as an aside: Actually, I was approaching the physics that is to be modeled, from entirely different perspectives---ones where, I think, it would be more natural to use something like a Dirac's delta. That, incidentally, was the reason why I phrased my questions the way I did. ... I mean, speaking vaguely and completely off-hand: with the other approaches, it would be more difficult to get dispersion effects introduced into the simulation in the first place---even if you wanted to have them... That's why I was enquiring if they are important in applications. But any discussion of such points would really take the discussion far too out of the scope of this JC, so let me stop here. (Some other time, in a separate thread.) Ok. Just to make it less mysterious before I close this parenthesis: Just think of ray-tracing as one of the other possible approaches here.]
</p>
<p>
3. Two points, or rather, two differing views on the nature of what I brought up here: one is a kind view to take on the matter, and the other one, not so kind.
</p>
<p>
-- It really is true that sometimes, even for an apparently simple physics/algorithm, the exact mathematics involved can get far too complicated.
</p>
<p>
-- A fool can ask more questions than a wise man can answer.
</p>
<p>
Let me stop on that note. (I will come back again, if I have something on the specific papers you refer to in the above description of this JC proper.)
</p>
<p>
Thanks again, and bye for now!
</p>
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</ul>Thu, 06 Aug 2009 17:05:57 +0000Ajit R. Jadhavcomment 12152 at https://imechanica.orgReply to Keng-Wit
https://imechanica.org/comment/12151#comment-12151
<a id="comment-12151"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/12134#comment-12134">ABCs</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi Keng-Wit,
</p>
<p>
Thanks for reminding Prof. Trefethen's notes. I remember having browsed through them quite some time back, but had forgotten...May be we should add them to the Lecture Notes node (#1551).
</p>
<p>
Also, thanks for the brief note on how to put these papers in context.
</p>
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</ul>Thu, 06 Aug 2009 16:45:29 +0000Ajit R. Jadhavcomment 12151 at https://imechanica.orgClarifications
https://imechanica.org/comment/12136#comment-12136
<a id="comment-12136"></a>
<p><em>In reply to <a href="https://imechanica.org/comment/12127#comment-12127">Re: JC Aug 09</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi Ajit,<br />
I assume that you received my e-mail with the paper. Given below are my answers.
</p>
<p>
1. The artifacts are essentially due to the approximation that we use for the square-root operator. We just focus on approximating the <em>propagating </em>waves (real horizontal wave number is real), and ignore the evanescent waves. If you do include evanescent waves, you should get rid of the artifacts. You can find these artifacts in the first paper mentioned in the main post, which also focuses on propagating waves.
</p>
<p>
2. The load used in the simulation is a Gaussian. People use pulses such as the Gaussian or Ricker wavelet. Dirac pulse would NOT be good for the reason you probably were thinking about: numerical dispersion (due to the poor resolution of the waveform). This is a significant issue especially for seismic migration as the frequency content is quite high.
</p>
<p>
3. Interesting thought of fractional-way waves. Perhaps one may be able to do some transformation in circumferential direction to convert the requirement to one-way form and then use one-way theory. I am thinking aloud here, and I suspect that there will be significant complications associated with such transformation (e.g., converting constant coefficient problem to variable coefficient problem). On a more practical note, for the problem of ultrasonic beam forming that you have in mind, one-way propagation would be just fine. The conical/fractional propagation is only because of the excitation shape at the surface. In fact, if you want to propagate a low-angle beam, fairly low-order OWWE would be sufficient to represent the energy propagation (since there is very little energy propagation outside the cone and higher order approximation is necessary for accurately capturing wide-angle propagation).
</p>
<p>
4. Keng-Wit seems to have pointed out a good reference on this.
</p>
<p>
I hope the above comments are helpful.
</p>
<p>
Cheers,
</p>
<p>
Murthy
</p>
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</ul>Tue, 04 Aug 2009 18:26:46 +0000Murthy N. Guddaticomment 12136 at https://imechanica.orgABCs
https://imechanica.org/comment/12134#comment-12134
<a id="comment-12134"></a>
<p><em>In reply to <a href="https://imechanica.org/node/6570">Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi Ajit,
</p>
<p>
When I was a student, I found this reference quite helpful in understanding absorbing boundary conditions:
</p>
<p>
Chapter 6 of Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations by Lloyd N. Trefethen:
</p>
<p>
<a href="http://www.comlab.ox.ac.uk/nick.trefethen/pdetext.html">http://www.comlab.ox.ac.uk/nick.trefethen/pdetext.html</a>
</p>
<p>
which should be accessible to engineers. As a side note, the approach described in the above reference leads to high-order boundary conditions that are difficult to implement numerically, in particular those beyond second-order. Several of Murthy's papers describe the means to circumvent this problem.
</p>
<p>
Hope this helps.
</p>
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</ul>Tue, 04 Aug 2009 04:01:32 +0000kwlimcomment 12134 at https://imechanica.orgRe: JC Aug 09
https://imechanica.org/comment/12127#comment-12127
<a id="comment-12127"></a>
<p><em>In reply to <a href="https://imechanica.org/node/6570">Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Murthy,
</p>
<p>
<strong>0. </strong>
</p>
<p>
A very interesting edition of the JC... Waves are just as fascinating to me as are particles [<a href="http://www.jadhavresearch.info/publications.htm" target="_blank">^</a>].
</p>
<p>
BTW, could you please email me the SIAM paper [ <a href="mailto:aj175tp@yahoo.co.in">aj175tp@yahoo.co.in</a> ]? [COEP should have the subscriptions but I wouldn't be visiting it over the next few days.]
</p>
<p>
Also, pl. note that the imaging.pdf off Samizdat Press seems to have got corrupted, but its gzipped version seems to be intact. (I just downloaded it but yet have to verify that the .ps file inside opens/prints right.)
</p>
<p>
I am sure I will be asking many questions in future, but here are a few quick ones:
</p>
<p>
<strong>1. Numerical Artifacts:</strong> What is the technique used in OWWE simulation clip shown above? For the full wave? Do the artifacts in the OWWE simulations always persist? How problematic or significant is this issue, in applications? Is there any better numerical technique that can adequately deal with this issue? How would be the payoffs if somebody could suggest solutions/better alternative techniques regarding this issue?
</p>
<p>
<strong>2. Time-Profile of the Pulse and Dispersion:</strong> What is the initial time-profile of the pulse being propagated? Would Dirac's delta be fine in the context of imaging applications for hydrocarbon prospecting/surveying? Or would this be too simplistic? How important is it to have to capture the dispersion effects in such application domains?
</p>
<p>
<strong>3. Fractional-Way Waves:</strong> The OWWE splits up the "full" wave into two "half" waves each of which covers a 180 degrees' span. But what if the application requirement involves an angle other than exact 180 degrees? Is the mathematical/numerical development done thus far able to address such "fractional-way" waves, too? Apart from theoretical curiosity, I was thinking of the UT NDT---there is a cone, you know...
</p>
<p>
<strong>4. ABCs:</strong> Could you please recommend simple textbook/tutorial references on handling ABCs, say, one that would be accessible to an advanced undergrad student in engg. too? Something from the physics/maths/comp. physics side would do too.
</p>
<p>
...
</p>
<p>
Actually, come to think of this JC, this is not a single theme; the discussion spans over a range of them! I don't mean it as a criticism... In a way, it's actually nice because it encourages longer-range integration... At least, it doesn't give one that tunnel-vision feeling...
</p>
<p>
Thanks in advance.
</p>
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</ul>Mon, 03 Aug 2009 14:31:06 +0000Ajit R. Jadhavcomment 12127 at https://imechanica.orgFirst Order Wave Equations, OWWE, coupled/elastic OWWE
https://imechanica.org/comment/12119#comment-12119
<a id="comment-12119"></a>
<p><em>In reply to <a href="https://imechanica.org/node/6570">Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi Amit,
</p>
<p>
Thanks for the questions. I hope the following answers would be helpful.
</p>
<p>
1. A simple first order wave equations (e.g., a(x,y) du/dx + b(x,y) du/dy + du/dt=0) will have the property that you described, i.e., at any given location, the propagation is in a given <strong>single</strong> direction, along the characteristic direction. In contrast, OWWEs do not just propagate the energy in one direction, but in a <strong>180-degree cone</strong>. The OWWE described in the post propagates energy in all directions that have positive x component.
</p>
<p>
2. If I understand correctly, when you say coupled OWWE, you are thinking about coupling backward and forward propagating waves. This, by definition, will no longer be OWWEs and ends up in fact being the full wave equation. You are absolutely correct; OWWEs are obtained by suppressing one of the modes of the first order form. To elaborate, consider the wave equation after Fourier transform in y and t: d^2(u)/dx^2+(k^2)u=0, where k = sqrt(w^2-ky^2) is the x- wavenumber, w is the frequency, ky is the y- wave number. When you write this equation in the first order form, the eigenvalues would be +ik/-ik, with +ik corresponding to forward propagating and –ik corresponding to backward propagating waves (depending on the sign convention of Fourier transform, of course). If you want the forward propagating OWWE, we simply use du/dx-iku=0, which is essentially the OWWE described in the original post. So, your modal decomposition idea is exactly correct, just that the expression for k has a square-root and when transformed back from Fourier domain, you end up with pseudo differential equations and all associated complications.
</p>
<p>
3. There are other types of coupled OWWE; these are encountered in settings such as elastic wave propagation where pressure and shear wave modes are closely coupled. This problem is a lot more involved and there is quite a bit of work in that direction as well. If you want more information on this, you can take a look at our paper (found <a href="http://www4.ncsu.edu/~mnguddat/Publications/AWWE_theory.pdf" target="_blank">here</a> ), and references therein.
</p>
<p>
Cheers,
</p>
<p>
Murthy
</p>
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</ul>Sun, 02 Aug 2009 15:56:26 +0000Murthy N. Guddaticomment 12119 at https://imechanica.orgFirst-order wave equations
https://imechanica.org/comment/12113#comment-12113
<a id="comment-12113"></a>
<p><em>In reply to <a href="https://imechanica.org/node/6570">Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Murthy,
</p>
<p>
Interesting post.
</p>
<p>
Could you comment on the following (I haven't thought much about these questions):
</p>
<p>
1) It seems to me scalar, first order wave equations (linear/nonlinear) are also 'one-way' in the sense that they move disturbances in a single, signed direction (which could be changing with location). What would be the essential distinction (apart from order) between this and the general idea you describe?
</p>
<p>
2) What happens if you had a coupled system of OWWEs? I guess my question stems form the following fact - one can take the scalar second order, wave eqn. and pose it as a first-order system. Then a transformation to modal form decouples the system into individual waves (assuming strict hyperbolicity, if you wish) each travelling in a specified direction (let's assume it is linear and constant coefficient)....So it seems that OWWEs must have some connection to suppressing one of modes?... Coversely, could a coupled system of OWWEs lose the one way propagation feature?
</p>
<p>
- Amit
</p>
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</ul>Sat, 01 Aug 2009 17:01:09 +0000Amit Acharyacomment 12113 at https://imechanica.orgError | iMechanica