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

(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.