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Journal Club Theme of August 2009: One-Way Wave Equations for Imaging, Mulitscale Modeling, and Absorbing BCs

Murthy N. Guddati's picture

(jointly with Siddharth Savadatti and Senganal Thirunavukkarasu)

Overview: The wave equation and its variants have been used to describe propagation of mechanical and electromagnetic disturbances in various media. The primary characteristic of these equations is that they propagate disturbances in all directions, i.e. in a two dimensional setting, wave equations have a 360-degree range of propagation angles. In contrast, one-way wave equations (OWWEs) propagate disturbances in a specified direction, while completely suppressing propagation in the opposite direction, i.e. in a two dimensional setting, OWWEs have a 180-degree range of propagation. We illustrate the difference between OWWEs and full wave equations in the animations below. The simulation shows the response to a point pulse load applied at the center of the layered domain. The top and bottom layers of the domain are identical while the middle layer has different properties.




As evident, the full wave equation results in upward/downward propagation as well as downward/upward reflections at the material interfaces and at the domain boundaries. The upward propagating OWWEs capture all upward propagating waves (both incident and reflected waves), but suppress all downward propagating waves (note that the "ears" at the center are artifacts associated with OWWE approximations and do not correspond to anything real ). Due to the special property of one-way propagation, OWWEs have found widespread applications in various fields including (a) unbounded domain modeling, (b) subsurface imaging algorithms, (c) ocean acoustics, and (d) multiscale modeling of solids. Below, we give an overview of the basic mathematical theory, some relevant applications, and a few of the more accessible references to read.


Basic mathematical theory: OWWEs are derived by factorizing the underlying (full) wave equations. Although the general OWWE theory is more involved, the basic ideas can be explained using the acoustic wave equation in the x-z plane:

where u is the scalar field variable. The above equation can be formally factorized as,

The first bracketed operator propagates waves in the negative x direction while the second operator propagates in the positive x direction. An OWWE that propagates waves only in the positive x direction can be obtained by simply removing the backward propagating operator, resulting in,

The above equation is the exact OWWE. Note that, due to the presence of the square-root of a differential operator, it is a pseudo-differential equation, and its implementation necessitates expensive convolution operations (or Fourier-type transformations) that are expensive for heterogeneous media. In order to reduce the computational cost, the square root operator in the above equation is approximated by a rational function leading to a differential equation that can be efficiently implemented. Such approach was pioneered by Leontovich and Fock in the context of electromagnetism and was later adopted for ocean acoustics in the name of parabolic equations. They have since been used and developed by various researchers in seismic migration and ocean acoustics, leading to satisfactory approximations of the acoustic OWWE. Essentially, acoustic OWWEs are obtained using two steps: (a) factorization of the wave equation and (b) rational approximation of the square-root operator. Elastic OWWEs are significantly more complex, and are left out of the current discussion.

In order to understand the theory behind OWWEs, we suggest the following paper:

A. Bamberger, B. Engquist, L. Halpern and P. Joly (1988), "Higher order paraxial wave equations approximations in heterogeneous media," SIAM Journal of Applied Math, 48(1), pp. 129-154.

Note that some of the sections may be a bit more mathematical than the others, but the paper should have something for all readers with varying levels of mathematical background.


Application to subsurface imaging (using high-frequency waves): With the objective of locating hydrocarbon reservoirs, exploration seismologists send waves into the earth, measure their reflections from hidden reservoirs, and process them to obtain an image of the subsurface. An important step in this process is seismic migration: the process of obtaining the location of strong reflectors (reservoirs) with known overall background properties. Seismic migration involves propagating the excitation wavefield from the surface to the reflectors, and at the same time, back-propagating the measured reflections from the surface back to the reflector. The location of the reflector is then obtained by correlating the two wavefields. Such a propagation (and back-propagation) is typically performed using OWWEs, which facilitate step-by-step computation in the depth direction. OWWEs also eliminate unwanted reflections and result in clearer images than those obtained using full wave equations. A similar approach is followed in the context of nondestructive evaluation of structures, wherein, ultrasonic waves are sent into a structure with the goal of obtaining the location, orientation and size of any hidden cracks. The data is then processed with the help of so-called Synthetic Aperture Focusing Techniques (SAFT), which are also based on one-way wave propagation ideas. To understand the application of OWWE to subsurface imaging, as well as other related methods, we suggest reading the following book chapter:

J. A. Scales (1997), Theory of Seismic Imaging, Samizdat Press. See Chapter 9.


Absorbing boundary conditions: The standard technique of simulating wave propagation in unbounded domains is to truncate the domain around the region of interest (called the interior), and to apply so-called Absorbing Boundary Conditions (ABCs) that mimic the energy absorption effect of the truncated exterior. These ABCs allow outward propagating waves at the boundary while preventing inward propagating reflections. Thus, ABCs are essentially outward propagating OWWEs. The only difference between imaging and unbounded domain modeling is that OWWEs are the governing equations in the context of imaging, but are boundary conditions for unbounded-domain modeling. Note here that the OWWE-based approach is one of the many successful ways of approaching the unbounded domain modeling problem. Another approach is a very clever way of introducing damping in the exterior without generating reflections at the interior-exterior interface (proposed by Berenger in 1994, this method has attracted huge attention from various researchers - the original reference by Berenger has more than 2700 citations in just 15 years!). The following book chapter contains discussion on ABCs related to OWWEs as well as other notable ABCs:

T. Hagstrom (2003), New results on absorbing layers and radiation boundary conditions, Topics in Computational Wave Propagation, M. Ainsworth et al eds., Springer-Verlag, pp. 1-42.


Multiscale Modeling of Solids: Many important problems in solid mechanics (e.g. dynamic fracture) involve dynamic phenomena at multiple scales (fine-scale dynamics at the crack tip and relatively coarse-scale elastic wave propagation away from the tip). Practical modeling of such systems necessitates the coupling of molecular dynamics (MD) simulation at the crack-tip with coarse-scale finite element modeling of continuum equations. A critical detail of such a coupling is the condition at the boundary of the MD region. Standard continuity conditions on traction and velocity result in spurious trapping of high-frequency discrete waves (phonons) in the MD region. This results in non-physical heat-up, leading to highly erroneous predictions. The problem has been the focus of many researchers in the past decade with various methods developed to suppress phonon reflections. Such phonon absorbing boundary conditions are different from ABCs for continuous wave propagation problems. A discussion of this critical difference, as well as other issues related to molecular dynamics to continuum coupling, can be found in the following reference.

X. Li and W. E (2006), Variational boundary conditions for molecular dynamics simulations of solids at low temperature, Communications in Computational Physics, 1,  pp. 135-175.

Note that the boundary conditions proposed above is not directly related to rational approximation of square-root and there do not appear to be any OWWE-related ABCs for MD. We have recently started developing OWWE-based ABCs for MD and preliminary work can be found here.


While OWWEs and applications described above are active areas of research with several unresolved issues, we hope that the above discussion and papers will help introduce the theory of OWWE and their application to various important fields.


Amit Acharya's picture

Dear Murthy,

Interesting post.

Could you comment on the following (I haven't thought much about these questions):

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? 

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?

 - Amit

Murthy N. Guddati's picture

Hi Amit,

Thanks for the questions. I hope the following answers would be helpful.

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 single direction, along the characteristic direction. In contrast, OWWEs do not just propagate the energy in one direction, but in a 180-degree cone. The OWWE described in the post propagates energy in all directions that have positive x component.

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.

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 here ), and references therein.



Dear Murthy,


A very interesting edition of the JC... Waves are just as fascinating to me as are particles [^].

BTW, could you please email me the SIAM paper [ ]? [COEP should have the subscriptions but I wouldn't be visiting it over the next few days.]

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.)

I am sure I will be asking many questions in future, but here are a few quick ones:

1.  Numerical Artifacts: 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? 

2. Time-Profile of the Pulse and Dispersion: 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?

3. Fractional-Way Waves: 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...

4. ABCs: 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.


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...

Thanks in advance.

Murthy N. Guddati's picture

Hi Ajit,
I assume that you received my e-mail with the paper. Given below are my answers.

1. The artifacts are essentially due to the approximation that we use for the square-root operator. We just focus on approximating the propagating 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.

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.

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).

4. Keng-Wit seems to have pointed out a good reference on this.

I hope the above comments are helpful.



Dear Murthy,

0. Thanks for emailing the paper.

2. The clarification that it's a Gaussian pulse itself suddenly made many things clear. 

[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.]

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.

-- It really is true that sometimes, even for an apparently simple physics/algorithm, the exact mathematics involved can get far too complicated.

-- A fool can ask more questions than a wise man can answer.

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.)

Thanks again, and bye for now!

Hi Ajit,

When I was a student, I found this reference quite helpful in understanding absorbing boundary conditions:

Chapter 6 of  Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations by Lloyd N. Trefethen:

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.

Hope this helps.

Hi Keng-Wit,

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).

Also, thanks for the brief note on how to put these papers in context.

Hi, Professor,

I am a exploration geophysics PhD student in TAMU. I am very interested in wave equations.

I have ever read your AWWE papers. Those ideas are really wonderful.

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.  

 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? 

My email is

Thanks very much.


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