Acoustic Propagator Research Articles

Rapid computation of steady state acoustic fields in heterogeneous and nonlinear media using the Acoustic Field Propagator

The Acoustic Field Propagator is a method for rapidly computing steady state complex acoustic fields in homogeneous media from spatially arbitrary single frequency sources. Here, an extension of the Acoustic Field Propagator to the computation of steady state fields in arbitrary heterogeneous, absorbing and nonlinear media, using nested iterations of the Convergent Born Series and a multi-frequency Helmholtz equation expansion, will be described.

Pseudoelastic PP and PSSP modeling in stratified media

Many modeling techniques have been developed to find the acoustic and elastic responses of a stack of plane layers to a plane spectral wave. For an elastic medium bounded above by an acoustic half-space, the acoustic wave propagator matrix modeling method can be modified to model pseudoelastic PP arrivals and PSSP arrivals. PP arrivals propagate as pure longitudinal (P) waves in the layers, whereas PSSP arrivals propagate as shear (S) waves in the elastic part of the model. A simple modification of the pseudoelastic PP response modeling scheme allows modeling of primary P reflections. A primary reflection event involves just one reflection in the plane stratified model and thus excludes internal multiples. The propagator modeling scheme is formulated in the frequency-horizontal slowness domain. By applying inverse Fourier transforms over the frequency and horizontal wavenumbers, where the wavenumber is the horizontal slowness divided by the frequency, modeled seismograms are computed and displayed in the time-space domain. By applying an inverse Fourier transform over the frequency for selected horizontal slowness components, the computed seismograms can be shown in the intercept time-horizontal slowness ([Formula: see text]) domain. When the source wavelet is unity for frequencies of interest, the [Formula: see text] domain seismograms become plane-wave Green’s function seismograms. The [Formula: see text]-traces of the Green’s function primary P-wave seismograms accumulate with increasing time band-limited step functions weighted by reflection strengths.

A Data-Centric Directive-Based Framework to Accelerate Out-of-Core Stencil Computation on a GPU

Graphics processing units (GPUs) are highly efficient architectures for parallel stencil code; however, the small device (i.e., GPU) memory capacity (several tens of GBs) necessitates the use of out-of-core computation to process excess data. Great programming effort is needed to manually implement efficient out-of-core stencil code. To relieve such programming burdens, directive-based frameworks emerged, such as the pipelined accelerator (PACC); however, they usually lack specific optimizations to reduce data transfer. In this paper, we extend PACC with two data-centric optimizations to address data transfer problems. The first is a direct-mapping scheme that eliminates host (i.e., CPU) buffers, which intermediate between the original data and device buffers. The second is a region-sharing scheme that significantly reduces host-to-device data transfer. The extended PACC was applied to an acoustic wave propagator, automatically extending the length of original serial code 2.3-fold to obtain the out-of-core code. Experimental results revealed that on a Tesla V100 GPU, the generated code ran 41.0, 22.1, and 3.6 times as fast as implementations based on Open Multi-Processing (OpenMP), Unified Memory, and the previous PACC, respectively. The generated code also demonstrated usefulness with small datasets that fit in the device capacity, running 1.3 times as fast as an in-core implementation.

Elastic model low- to intermediate-wavenumber inversion using reflection traveltime and waveform of multicomponent seismic data

Low-, intermediate-, and high-wavenumber components of P- and S-wave velocities jointly influence the elastic wave propagation and scattering in an isotropic medium. By taking advantage of all information in the data, elastic full-waveform inversion (E-FWI) has the potential to recover these model components. However, if the transmitted wave data are insufficient to illuminate the deeper part of the subsurface, we should rely on the solutions using reflection data. To reduce the nonlinearity of waveform inversion, we choose to decouple the effects of the model background and perturbation on the reflected waves within a linearized inversion framework. This resorts to three stages aiming to gradually fit the traveltimes and waveforms of the reflected PP and PS waves based on data or gradient preconditioning through P/S mode decomposition. For the first two stages, once the multicomponent seismograms have been separated into PP and PS reflection recordings, reflection traveltime inversion using an acoustic wave propagator (A-RTI) can successively recover the low-wavenumber components of P- and S-wave velocities. In the last stage, starting from the models having reliable low-wavenumber components, elastic reflection waveform inversion (E-RWI) can easily get out of the local minima and continue to retrieve the increasing wavenumber features sensitive to the waveform and amplitude variations. This is supported by gradient preconditioning through P/S mode decomposition of the extrapolated normal and adjoint wavefields, and alternately updating model background and high-wavenumber components in terms of linearized least-squares inversion. Numerical examples have demonstrated the performance of our E-RWI approach and the validity of the three-stage inversion workflow.

A Review on Ruben’s Tube as Acoustic Propagator

In 1905, Ruben’s Tube was invented by German physicists Heinrich Ruben and Otto Krigar-Menzel. The tube is closed at one end and the other end is attached with a loudspeaker, then filled with flammable gas. With the help of the Ruben’s Tube, the standing waves were represented physically by fire. There are no specific standards in which Ruben’s Tube is constructed. At certain resonance frequency, the flame produced visual standing wave that varies in height. It has been used in the teaching of acoustical resonance behaviour. This article provides review on Ruben’s Tube and its operation, as well as some commentaries and drawbacks for each subtopic. Each of the journals that have similar setup have been reviewed to further explain the phenomena.

Underwater sound propagation variability over the Chukchi Sea continental slope

In the Canada Basin and Chukchi Sea regions, a vertical sound duct can be formed between the Pacific Summer Water Layer on the top and the Atlantic Water Layer on the bottom, providing an acoustic pathway connecting the deep basin and the shallow shelf over the Chukchi Sea continental slope. Previous studies have shown that the shelfbreak circulation (specifically upwelling), the sub-mesoscale eddies spun off the shelfbreak jet, and the ice coverage are the three major causes of the temporal and spatial variability of the Pacific Summer Water Layer in the region. In this paper, numerical simulations utilizing the Parabolic-Equation (PE) method are conducted to investigate the sound propagation variability over the Chukchi Sea shelfbreak and slope, along with an idealized ocean circulation model providing the fundamental basis of water-column fluctuations. The sound pressure sensitivity kernel derived from the acoustic propagator in the PE method is also used to provide physical insights into the sound pro...

Resonance spectrum for a continuously stratified layer: application to ultrasonic testing

Ultrasound wave propagation in a nonhomogeneous linearly elastic layer of constant thickness immersed between homogeneous fluid and solid media is considered. The resonances (scattering poles) for the corresponding acoustic propagator are studied. It is shown that the distribution of the resonances depends on the smoothness of the coefficients that characterize physical properties of the layer and the ambient media. Namely, if the coefficients have jump discontinuities at the boundaries, then the resonances are asymptotically distributed along a straight line parallel to the real axis on the unphysical sheet of the complex frequency plane. On the contrary, if the coefficients are continuous, then it is shown that the resonances are asymptotically distributed along a logarithmic curve. The developed mathematical model is applied to the ultrasonic testing of the articular cartilage (AC) layer attached to the subchondral bone from one side and being in contact with a solution on the other side. It is conjectured that the spacing between two successive resonances may be sensitive to AC degeneration. The application of the obtained results to the development of ultrasonic testing for quantitative evaluation of AC is discussed.

Time domain modeling of acoustic propagation with acoustic wave propagator and absorbing boundary conditions

The acoustic wave propagator (AWP) is the application of the time evolution operator on the acoustic wave equation for stationary systems in a polynomial expansion of Chebychev polynomials. It allows to increase the time step by more than one order of magnitude compared to finite difference time domain (FDTD) codes. In contrast to other implementations of the AWP the spatial differentiation is carried out with finite difference techniques because this allows the use of the perfectly matched layer formulation as absorbing boundary conditions. The formulation includes the direct implementation of acoustic sources with sinusoidal time evolution. Other sources can be synthesized by their Fourier components. For the calculation of large areas the explicit formulation of a large system matrix can be avoided by calculating the propagation equations for each time step at row and column level repeatedly which reduces memory requirements notably. This procedure and the suitability of the finite difference approach ...

Seismic wave imaging in visco-acoustic media

Realistic representation of the earth may be achieved by combining the mechanical properties of elastic solids and viscous liquids. That is to say, the amplitude will be attenuated with different frequency and the phase will be changed in the seismic data acquisition. In the seismic data processing, this effect must be compensated. In this paper, we put forward a visco-acoustic wave propagator which is of better calculating stability and tolerable calculating cost (little more than an acoustic wave propagator). The quite good compensation effect is demonstrated by the numerical test results with synthetic seismic data and real data.

Scattering Theory for a Stratified Acoustic Strip with Short- or Long-Range Perturbations

We consider the acoustic propagator H= −∇ · ρ∇ acting in L^2 (Ω) with Ω := Ω' × ℝ and Ω' a bounded open set in ℝ^{n−1} , n ≥ 2 . The real-valued function ρ belongs to L^∞(Ω) , and is bounded from below by c > 0 . We assume there exist two strictly positive constants c_1 and c_2 and two perturbations, δ^S of short-range type and δ^L of long-range type, such that ρ = c_j + δ^S + δ^L on Ω_j := \{(x'_n, xn) \in Ω|(−1)^j _n > 0\} , j = 1, 2 . We build two modified free evolutions U_j(t) , j = 1, 2 , such that the wave operators Ω^±_j := s − \lim_{t→±∞} e^{itH} U_j(t) , j = 1, 2 , exist and are asymptotically complete.

Analyticity properties and estimates of resolvent kernels near thresholds

Resolvent estimates are derived for the faimly of ordinary differential operators It is assumed that, and the kernels are studied in neighborhoods of the points ,uniformly in compact intervals of P. This family arises in the direct integral decomposition of the acoustic propagator in layered media, , and the results imply “low energy” estimates for this operator, as well as the validity of the “limiting absorption principle”

Commutator estimates, Besov spaces and scattering problems for the acoustic wave propagation in perturbed stratified fluids

Mourre's commutator method is used together with Besov spaces, in this paper, to settle the question of uniqueness of solutions to the scattering problem for the acoustic wave propagation in perturbed stratified fluids in Rn with n [ges ] 2: The uniqueness of solutions is established by introducing a radiation condition in the framework of Besov spaces, and is then used to prove resolvent estimates in the Besov space setting for the acoustic propagator in perturbed stratified fluids. These estimates are ‘sharp’ in a sense made precise in the text and are important in establishing existence of solutions to the scattering problem.

A perturbative method for the spectral analysis of an acoustic multistratified strip

We consider the acoustic propagator A=−∇·c2∇ in the strip Ω={(x, z)∈ℝ2∣0<z<H} with finite width H>0. The celerity c depends for large ∣x∣ only on the variable z and describes the stratification of Ω: it is assumed to be in L∞(Ω), bounded from below by cmin>0, such that there exists M>0 with c(x, z)=c1(z) if x< −M and c(x, z)=c2(z) if x>M. We look at the propagator A as a ‘perturbation’ of the free propagators Aj in Ω associated to the velocities cj, j=1, 2, and implement a ‘perturbative’ method, adapting ideas of Majda and Vainberg. The spectrum of A is defined in section 2, a limiting absorption principle is proved in section 3 outside of a countable set Γ(A). The points of Γ(A) can only accumulate at the left of the thresholds of the free propagators. The needed material about Aj, j=1, 2, and some technical estimates for A are given in Appendix. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.

A perturbative method for the spectral analysis of an acoustic multistratified strip

We consider the acoustic propagator A=−∇·c2∇ in the strip Ω={(x, z)∈ℝ2∣0<z<H} with finite width H>0. The celerity c depends for large ∣x∣ only on the variable z and describes the stratification of Ω: it is assumed to be in L∞(Ω), bounded from below by cmin>0, such that there exists M>0 with c(x, z)=c1(z) if x< −M and c(x, z)=c2(z) if x>M. We look at the propagator A as a ‘perturbation’ of the free propagators Aj in Ω associated to the velocities cj, j=1, 2, and implement a ‘perturbative’ method, adapting ideas of Majda and Vainberg. The spectrum of A is defined in section 2, a limiting absorption principle is proved in section 3 outside of a countable set Γ(A). The points of Γ(A) can only accumulate at the left of the thresholds of the free propagators. The needed material about Aj, j=1, 2, and some technical estimates for A are given in Appendix. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.

Spectral Analysis of the Acoustic Propagator in a Multistratified Domain

The aim of this paper is to study the acoustic propagatorHin a perturbed multistratified domain Ω with the conjugate operator method.Hdenotes the self-adjoint operator −∇c(x)∇ inL2(Ω) with a Dirichlet condition at the boundary ∂Ω. Herecdenotes the propagation velocity depending on the nature of the stratification of Ω. We shall give some spectral properties ofHand we establish the limiting absorption principle in an optimal scale of Besov spaces.

Radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media

Consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity. The propagation speed is discontinuous at the interface of two unbounded media, and the interface may be an arbitrary and smooth surface locally. A Sommerfeld radiation condition is introduced for the acoustic propagator, and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator. Furthermore, we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator.

On Spectral Properties of the Acoustic Propagator in a Layered Band

The acoustic propagator is the self-adjoint operatorH=−∇·c(x)2∇, defined inL2(Q), whereQ⊆R2is a band of finite width, given byQ={(x, z), x∈R, 0⩽z⩽Γ}. The wave velocitycdepends only on the horizontal coordinatex, and is a measurable bounded function, converging toc±>0 asx→±∞. It is proved that the resolvent operatorR(z)=(H−z)−1, Imz≠0, can be extended continuously to the closed upper (or lower) half-plane, in a suitable weighted-L2topology (“Limiting Absorption Principle”). In particular, this continuity holds at the thresholdλ=0. It follows as a corollary thatHhas no point spectrum.

Trapped Rays in Cylindrically Symmetric Media and Poles of the Analytically Continued Resolvent

The acoustic (stratified) propagator is defined as the positive self-adjoint operatorK≡−c2(y)ρ(y)∇zρ−1(y)∇zon L2(Rn,(c2ρ)−1dnz), wherez=(x,y)x∈Rk,y∈Rm. Bounds on the weighted resolvent ofK, defined as〈z〉−α(K−ζ)−1〈z〉−αfor α>1/2 and 〈z〉=(1+|z|2)1/2, are obtained at high energies (i.e., in the semi-classical limit Re(ζ)⪢0) using the Virial conditionc−y·∇c≥εc>0. This implies energy decay estimates for trapped waves in fiber optics. With further smoothness assumptions oncand ρ the resolvent can be analytically extended onto the second Riemann sheet. The failure of the Virial condition on some radial interval, form≥2, implies the existence ofresonant-traveling modesfor the wave equation (∂t2+K)ψ=0. In particular it is shown that ifn>m≥2 thenKhas resonance states at high energies. No such states exist ifm=1. This is explained using a shape resonance picture.

Spectral Analysis of a Multistratified Acoustic Strip Part II: Asymptotic Behavior of Solutions for a Simple Stratification

We consider the acoustic propagator $A = - \nabla \cdot c^2 \nabla $ in $\Omega = \{ {(x,z) \in {{\mathbb{R}^2 } / {0 < z < H}}} \}$. The velocity c, which describes the stratification of the strip $\Omega $, depends only on the variable z: it is assumed to be a function in $L^\infty ((0,H))$ bounded from below by $c_m > 0$. Let A be the self-adjoint operator associated with the Neumann or Dirichlet condition at $z = 0$ and $z = H$; let $\mu $ be a real number in the spectrum of A; and let u be the solutions of the equation $(A - \mu I)u = f$ locally in the domain of A, which are determined by the limiting absorption principle in [E. Croc and Y. Dermenjian, SIAM J. Math. Anal., 26 (1995), pp. 880–924] and made explicit with trace operators. Thanks to accurate Hölder properties for the trace operators, we control the asymptotic behavior of u with so-called “zero-trace” conditions for f.

The propagator matrix related to the Kirchhoff‐Helmholtz integral in inverse wavefield extrapolation

The Kirchhoff‐Helmholtz formula for the wavefield inside a closed surface surrounding a volume is most commonly given as a surface integral over the field and its normal derivative, given the Green’s function of the problem. In this case the source point of the Green’s function, or the observation point, is located inside the volume enclosed by the surface. However, when locating the observation point outside the closed surface, the Kirchhoff‐Helmholtz formula constitutes a functional relationship between the field and its normal derivative on the surface, and thereby defines an integral equation for the fields. By dividing the closed surface into two parts, one being identical to the (infinite) data measurement surface and the other identical to the (infinite) surface onto which we want to extrapolate the data, the solution of the Kirchhoff‐Helmholtz integral equation mathematically gives exact inverse extrapolation of the field when constructing a Green’s function that generates either a null pressure field or a null normal gradient of the pressure field on the latter surface. In the case when the surfaces are plane and horizontal and the medium parameters are constant between the surfaces, analysis in the wavenumber domain shows that the Kirchhoff‐Helmholtz integral equation is equivalent to the Thomson‐Haskell acoustic propagator matrix method. When the medium parameters have smooth vertical gradients, the Kirchhoff‐Helmholtz integral equation in the high‐frequency approximation is equivalent to the WKBJ propagator matrix method, which also is equivalent to the extrapolation method denoted by extrapolation by analytic continuation.