Complete Wavefield Research Articles

Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operators.

We address the challenge of acoustic simulations in three-dimensional (3D) virtual rooms with parametric source positions, which have applications in virtual/augmented reality, game audio, and spatial computing. The wave equation can fully describe wave phenomena such as diffraction and interference. However, conventional numerical discretization methods are computationally expensive when simulating hundreds of source and receiver positions, making simulations with parametric source positions impractical. To overcome this limitation, we propose using deep operator networks to approximate linear wave-equation operators. This enables the rapid prediction of sound propagation in realistic 3D acoustic scenes with parametric source positions, achieving millisecond-scale computations. By learning a compact surrogate model, we avoid the offline calculation and storage of impulse responses for all relevant source/listener pairs. Our experiments, including various complex scene geometries, show good agreement with reference solutions, with root mean squared errors ranging from 0.02 to 0.10 Pa. Notably, our method signifies a paradigm shift as-to our knowledge-no prior machine learning approach has achieved precise predictions of complete wave fields within realistic domains.

Uncovering and Experimental Realization of Multimodal 3D Topological Metamaterials for Low-Frequency and Multiband Elastic Wave Control.

Topological mechanical metamaterials unlock confined and robust elastic wave control. Recent breakthroughs have precipitated the development of 3D topological metamaterials, which facilitate extraordinary wave manipulation along 2D planar and layer-dependent waveguides. The 3D topological metamaterials studied thus far are constrained to function in single-frequency bandwidths that are typically in a high-frequency regime, and a comprehensive experimental investigation remains elusive. In this paper, these research gaps are addressed and the state of the art is advanced through the synthesis and experimental realization of a 3D topological metamaterial that exploits multimodal local resonance to enable low-frequency elastic wave control over multiple distinct frequency bands. The proposed metamaterial is geometrically configured to create multimodal local resonators whose frequency characteristics govern the emergence of four unique low-frequency topological states. Numerical simulations uncover how these topological states can be employed to achieve polarization-, frequency-, and layer-dependent wave manipulation in 3D structures. An experimental study results in the attainment of complete wave fields that illustrate 2D topological waveguides and multi-polarized wave control in a physical testbed. The outcomes from this work provide insight that will aid future research on 3D topological mechanical metamaterials and reveal the applicability of the proposed metamaterial for wave control applications.

Longitudinal motion of focused shear wave beams in soft elastic media.

Shear waves are employed in medical ultrasound imaging because they reveal variations in viscoelastic properties of soft tissue. Frequencies below 1 kHz are required due to the substantially higher attenuation and lower propagation speeds than for compressional waves. Shear waves exhibiting particle motion in the direction of propagation, referred to as longitudinal shear waves, can be generated with longitudinal motion of a circular disk on the surface of a soft elastic medium. This approach permits imaging of the longitudinal shear wave with a conventional ultrasound transducer that is coaxial with the source of the shear wave. Presented here is a mathematical model describing the complete wave field generated by displacement at the surface of an isotropic elastic half-space. Numerical simulations are shown for longitudinal, transverse, torsional, and radial source polarizations, with emphasis on focused longitudinal shear waves. Predictions are consistent with measurements of light beams revealing that the longitudinal electric field component produces a smaller focal spot than the transverse field component [Dorn, Quabis, and Leuchs, Phys. Rev. Lett. 91, 233901 (2003)]. Simulations are compared with preliminary measurements of a focused longitudinal shear wave beam generated in a soft tissue phantom by longitudinal motion of a spherically concave piston.

Comparison of methods for coupled earthquake and tsunami modelling

SUMMARY Tsunami generation by offshore earthquakes is a problem of scientific interest and practical relevance, and one that requires numerical modelling for data interpretation and hazard assessment. Most numerical models utilize two-step methods with one-way coupling between separate earthquake and tsunami models, based on approximations that might limit the applicability and accuracy of the resulting solution. In particular, standard methods focus exclusively on tsunami wave modelling, neglecting larger amplitude ocean acoustic and seismic waves that are superimposed on tsunami waves in the source region. In this study, we compare four earthquake-tsunami modelling methods. We identify dimensionless parameters to quantitatively approximate dominant wave modes in the earthquake-tsunami source region, highlighting how the method assumptions affect the results and discuss which methods are appropriate for various applications such as interpretation of data from offshore instruments in the source region. Most methods couple a 3-D solid earth model, which provides the seismic wavefield or at least the static elastic displacements, with a 2-D depth-averaged shallow water tsunami model. Assuming the ocean is incompressible and tsunami propagation is negligible over the earthquake duration leads to the instantaneous source method, which equates the static earthquake seafloor uplift with the initial tsunami sea surface height. For longer duration earthquakes, it is appropriate to follow the time-dependent source method, which uses time-dependent earthquake seafloor velocity as a forcing term in the tsunami mass balance. Neither method captures ocean acoustic or seismic waves, motivating more advanced methods that capture the full wavefield. The superposition method of Saito et al. solves the 3-D elastic and acoustic equations to model the seismic wavefield and response of a compressible ocean without gravity. Then, changes in sea surface height from the zero-gravity solution are used as a forcing term in a separate tsunami simulation, typically run with a shallow water solver. A superposition of the earthquake and tsunami solutions provides an approximation to the complete wavefield. This method is algorithmically a two-step method. The complete wavefield is captured in the fully coupled method, which utilizes a coupled solid Earth and compressible ocean model with gravity. The fully coupled method, recently incorporated into the 3-D open-source code SeisSol, simultaneously solves earthquake rupture, seismic waves and ocean response (including gravity). We show that the superposition method emerges as an approximation to the fully coupled method subject to often well-justified assumptions. Furthermore, using the fully coupled method, we examine how the source spectrum and ocean depth influence the expression of oceanic Rayleigh waves. Understanding the range of validity of each method, as well as its computational expense, facilitates the selection of modelling methods for the accurate assessment of earthquake and tsunami hazards and the interpretation of data from offshore instruments.

Visco-acoustic full waveform inversion: From a DG forward solver to a Newton-CG inverse solver

Full waveform inversion (FWI) entails the ill-posed reconstruction of material parameters (such as sound speed and attenuation) from measurements of complete wave fields (full seismograms). In this paper we present a novel framework for FWI in the visco-acoustic regime. The new framework is based on a new elegant derivation of the system of state and adjoint PDEs which are approximated by the discontinuous Galerkin (DG) method. The inverse problem is then solved by the well established regularization scheme CG-REGINN which has not yet been applied in the context of FWI. For the DG discretization we provide a preconditioner for the efficient computation of the time steps by GMRES which yields optimal convergence estimates in space and time and which is confirmed by numerical tests. The inverse solver expresses the required Fréchet derivative and its adjoint in the DG setting. Successful reconstructions in a simplified cross-well setting serve as a proof of concept for our framework and demonstrate the applicability of our new combination of DG method and inverse solver. Some of the inversion experiments use seismograms generated by an independent finite difference time domain forward solver to avoid inverse crime.

3-D Virtual Seismology

We create virtual sources and receivers in a 3D subsurface using the previously derived single-sided homogeneous Green's function representation. We employ Green's functions and focusing functions that are obtained using reflection data at the Earth's surface, a macro velocity model and the Marchenko method. The homogeneous Green's function is a Green's function superposed with its time-reversal. Unlike the classical homogeneous Green's function representation, our approach requires no receivers on an enclosing boundary, however, it does require the source signal to be symmetric in time. We demonstrate that in 3D, the single-sided representation is an improvement over the classical representation by applying the representations to numerical data. We retrieve responses to virtual point sources with an isotropic and with a double-couple radiation pattern and compare the results to a directly modeled reference result. We also demonstrate the application of the single-sided representation for retrieving the response to a virtual rupture that consists of a superposition of double-couple point sources. This is achieved by obtaining the homogeneous Green's function for each source separately, before they are transformed to the causal Green's function, time-shifted and superposed. The single-sided representation is also used to monitor the complete wavefield that is caused by a numerically modeled rupture. However, the source signal of an actual rupture is not symmetric in time and the single-sided represenation can therefore only be used to obtain the causal Green's function. This approach leaves artifacts in the final result, however, these artifacts are limited in space and time.

Impact of spatial variability of shear wave velocity on the lagged coherency of synthetic surface ground motions

The spatial incoherence of ground motion during an earthquake can have a significant effect on the dynamic response of engineering structures such as bridges, dams, nuclear power plants and lifeline facilities. The main objective of this paper is to study the effect of anisotropic heterogeneities in a soil layer overlying homogeneous bedrock on the lagged coherency of surface ground motion. A set of numerical experiments is performed based on 2D spatial variability of shear-wave velocities modeled as a homogeneous stationary random field and discretized by the EOLE method (Expansion Optimal Linear Estimation). Seismic ground motions were simulated using FLAC2D software in the 1–25 Hz band for a plane wave excitation with SV polarization. The soil is characterized by horizontal and vertical autocorrelation distances ranging between 5 and 20 m and 1 and 2 m, respectively, and a coefficient of variation of the shear-wave velocity varying between 5% and 40%. The synthetic seismograms calculated for 9 parameter sets (100 realizations each) clearly show seismic waves scattering and surface waves diffracted locally by the ground heterogeneities, generating large spatial variations in coherence mainly controlled by the coefficient of variation of shear-wave velocity. Consistently with existing models and experimental data, the numerical coherency curves decrease with frequency and receiver distance, however at a rate which is lower than that observed in the experimental data. This difference is probably due to intrinsic attenuation that is not accounted for in the simulations and/or to our 2D simulations that do not reproduce the complete wavefield. The numerical average coherency curves for each parameter set exhibit maxima within narrow frequency bands caused by the vertically trapped body waves and surface wave propagation properties within the average ground model. This interpretation is supported by experimental data recorded in the Koutavos-Argostoli valley (Greece).

On the validity of the planar wave approximation to compute synthetic seismograms of teleseismic body waves in a 3-D regional model

SUMMARYInjection methods are a very efficient means to compute synthetic seismograms of short-period teleseismic body waves in 3-D regional models. The principle is to inject an incident teleseismic wavefield inside a regional 3-D Cartesian spectral-element grid. We have developed an opern-source package that allows us to inject either an incident plane wave, computed with a frequency–wavenumber method, or the complete wavefield, computed in a spherically symmetric reference earth model with AxiSEM. The computations inside the regional spectral-element grid are performed with SPECFEM3D_Cartesian. We compare the efficiency and reliability of the two injection methods for teleseismic P waves, considering a wide range of epicentral distance and hypocentral depths. Our simulations demonstrate that in practice the effects of wave front and Earth curvature are negligible for moderate size regional domains (several hundreds of kilometres) and for periods larger than 2 s. The main differences observed in synthetic seismograms are related to secondary phases that have a different slowness from the one of the reference P phase.

Wavefield decomposition in arbitrary direction and an imaging condition based on stratigraphic dip

In reverse time migration (RTM), wavefield decomposition can play an important role in addressing the issue of migration noise, especially low-frequency noise. The complete wavefield decomposition based on the Hilbert transform is a commonly used method in RTM, but it is accompanied by extra wavefield simulation and wavefield storage. We have developed three distinct methods. The first is a convenient method for wavefield decomposition, which is based on Poynting vectors. Only the unit vector in one direction is needed to realize the wavefield decomposition in an arbitrary direction by this method. It breaks through the limitation that the Hilbert transform-based method is applicable only to the up- and downgoing wave or left- and right-going wave decomposition, and the calculation cost is negligible compared with RTM. The second is a method based on the instantaneous wavenumber, which we developed for calculating the wave propagation direction. On the basis of wavefield decomposition, the imaging angle gather from the new method performs better than that of the Poynting vector method. Meanwhile, it also is used for generating the incident angle gather and dip angle gather. The latter expresses the dip angle of underground strata. More importantly, the above methods allow us to control the wavefield decomposition direction and three angles at any position underground. The third adopts a stratigraphic imaging condition method, and we briefly analyze the relationship between the new method and the inverse-scattering imaging condition. The stratigraphic imaging condition maps the results to the dip angle of the stratum through a spatial gradient wavefield, which can enhance the effective imaging information. The above three kinds of angle gathers also can be constructed by the stratigraphic imaging condition. Numerical experiments demonstrate that the imaging results and the angle gathers obtained by our proposed method have higher accuracy and resolution.

Data processing of a wide-azimuth, broadband, high-density 3D seismic survey using a low-frequency vibroseis: a case study from Northeast China

Seismic exploration employing wide-azimuth, broadband, high-density data (i.e. double-width single-height; “double-width” means wide-azimuth and broadband; “single-height” means high-density data, generally referring to small-bin-size data (less than 10 × 10 m); and double-width single-height is an abbreviation) enables more complete wave field information to be recorded, reduces aliasing, and produces abundant low-frequency information, which is conducive to broadband processing and the anisotropic study of seismic data. Based on the acquired double-width single-height seismic data, in this paper, we analyse the wave field characteristics, signal-to-noise ratio and frequency of seismic data. We also design a procedure for processing double-width single-height seismic data. The key techniques in the proposed procedure focus on the high-resolution amplitude-preserved characteristics of double-width single-height seismic data obtained with a low-frequency vibroseis sweep. Faults and sand bodies are characterised and described by using the final imaging results, which embody the advantages of double-width single-height seismic exploration.

Object wave field extraction in off-axis holography by clipping its frequency components.

Suitable extraction of an object's wave field in an off-axis digital holographic array is the core for any numerical reconstruction process. In this manuscript, we propose an algorithm that reduces the number of pixels needed for an object's wave field trimming, reducing notably its processing time without significant changes in the quality of the reconstructed image. Our method generates an extraction window containing a complete object's wave field. The energy that the generated extraction window must contain is analyzed, defined, and used as a reference value. Our proposal reduces significantly the number of operations and execution time for holographic reconstruction.

Real-time photorealistic computer-generated holograms based on backward ray tracing and wavefront recording planes

Abstract Holograms can reconstruct the complete optical wave field of a three-dimensional (3D) scene. However, generating traditional computer-generated holograms (CGHs) requires significant computing time. Here, a CGH method is presented based on backward ray tracing and wavefront recording planes (WRPs). Ray tracing is able to render more complex and photorealistic images for different illuminations and circumstances. Rays are parallelly launched and traced to obtain ray-object intersections, and then the complex wave field of the effective pixels in the WRP is determined. A CGH is generated by the diffracted calculation from the WRP to the hologram plane using fast Fourier transform (FFT) and inverse FFT algorithms. Experimental results demonstrate the effectiveness of the proposed method. Compared with traditional CGHs, the rate of CGH generation and optical reconstruction improves to 35.6 frames per second, achieving the real-time rendering and interactivity.

Limitations of correlation-based redatuming methods

Redatuming aims to correct seismic data for the consequences of an acquisition far from the target. That includes the effects of an irregular acquisition surface and of complex geological structures in the overburden such as strong lateral heterogeneities or layers with low or very high velocity. Interferometric techniques can be used to relocate sources to positions where only receivers are available and have been used to move acquisition geometries to the ocean bottom or transform data between surface–seismic and vertical seismic profiles. Even if no receivers are available at the new datum, the acquisition system can be relocated to any datum in the subsurface to which the propagation of waves can be modeled with sufficient accuracy. By correlating the modeled wavefield with seismic surface data, one can carry the seismic acquisition geometry from the surface closer to geologic horizons of interest. Specifically, we show the derivation and approximation of the one-sided seismic interferometry equation for surface-data redatuming, conveniently using Green's theorem for the Helmholtz equation with density variation. Our numerical examples demonstrate that correlation-based single-boundary redatuming works perfectly in a homogeneous overburden. If the overburden is inhomogeneous, primary reflections from deeper interfaces are still repositioned with satisfactory accuracy. However, in this case artifacts are generated as a consequence of incorrectly redatumed overburden multiples. These artifacts get even worse if the complete wavefield is used instead of the direct wavefield. Therefore, we conclude that correlation-based interferometric redatuming of surface–seismic data should always be applied using direct waves only, which can be approximated with sufficient quality if a smooth velocity model for the overburden is available.

High-efficiency photorealistic computer-generated holograms based on the backward ray-tracing technique

Holographic displays can provide the complete optical wave field of a three-dimensional (3D) scene, including the depth perception. However, it often takes a long computation time to produce traditional computer-generated holograms (CGHs) without more complex and photorealistic rendering. The backward ray-tracing technique is able to render photorealistic high-quality images, which noticeably reduce the computation time achieved from the high-degree parallelism. Here, a high-efficiency photorealistic computer-generated hologram method is presented based on the ray-tracing technique. Rays are parallelly launched and traced under different illuminations and circumstances. Experimental results demonstrate the effectiveness of the proposed method. Compared with the traditional point cloud CGH, the computation time is decreased to 24 s to reconstruct a 3D object of 100×100 rays with continuous depth change.

A bigger picture view of separated wavefields and marine broadband seismic

The advent of dual-sensor recording for towed seismic streamers enabled the true separation of the up-going and down-going pressure wavefields (Carlson et al., 2007), and heralded the ‘broadband’ seismic revolution seen in marine seismic exploration over the past decade (Widmaier et al., 2015). In reality, ‘broadband’ typically means deghosting accompanied by some additional forms of spectral conditioning. In most cases, the post-stack interpretation of deghosted data is enhanced by improved geological texture, improved event coherence and deeper signal penetration. Recent attention has focused on the benefits of low frequencies in broadband seismic data (ten Kroode et al., 2013), and indeed where ultra-low frequency phase integrity is preserved as a complement to accurate amplitude-versus-angle fidelity, deghosted data also enables more accurate pre-stack quantitative interpretation (Reiser et al., 2015a, 2015b). However, it has also become clear that some long-standing imperfections in the seismic method remain. The rapid decay in air gun output below about 7 Hz (Parkes and Hegna, 2011) is not satisfactorily addressed by deghosting, compensation for high-frequency attenuation remains a fundamental challenge, and traditional challenges to wavelet processing, denoise, multiple removal, velocity estimation and imaging may in fact be more complicated for some broadband methods, or unaffected for others. The key benefit of dual-sensor methods are accurate access to the various separated wavefields, in addition to enhanced recoverable frequency bandwidth. Reservoir monitoring is enhanced by the elimination of the down-going pressure wavefield from 4D differencing, reservoir illumination can be enhanced by the incorporation of the down-going pressure wavefield into wave theoretic imaging, reflectivity inversion can be enhanced by isolating the up-going pressure wavefield in imaging, and overall we expect to see ‘complete wavefield’ reservoir imaging and characterization solutions mature rapidly. Furthermore, increased focus on spatial frequency content courtesy of ‘wave equation inversion’ imaging solutions is also taking broadband seismic past the historical focus upon only temporal frequency content.

Complete wavefield seismic structural, stratigraphic, geomorphic and elastic interpretation in the offshore Browse Basin

The first quantitative interpretation project in the Browse Basin that used complete wavefields from dual-sensor streamer data to characterise a variety of overburden and reservoir seismic structural, sedimentary and geomorphic features is discussed in this extended abstract. Having established that complete wavefield pre-stack angle data are amplitude versus angle (AVA) compliant, the authors analyse the elastic response from thin gas sands in a location where well control predicts a weak gas response in terms of the Vp/Vs ratio and negligible response in terms of the acoustic impedance. Various seismic expressions in the near-surface are better resolved using complete wavefield imaging, including deltaic deposits, potentially prospective patch reefs and intersecting channel systems, and shallow hazards associated with historic drilling failures appear to be resolvable with greater confidence. On the larger scale, the overall data interpretability is substantially enhanced in texture and character by broadband imaging of primary reflections, and then by subsequent reservoir characterisation. Initial (deeper) interpretation focused around mapping of the prospective Early Jurassic Plover Formation and the Early Cretaceous Upper Vulcan Formation, in which evidence of channelisation can be demonstrated on conventional migration volumes and reinforced using complete wavefield migration. Ultra-low frequency signal penetration to the deeper Triassic level has been limited historically, but the first complete seismic wavefield data analysis presented demonstrates that a combination of volume attributes provide substantially improved geologic information from the seafloor to the deepest targets.

Migration of primaries and multiples using an imaging condition for amplitude-normalized separated wavefields

Migration of primary and multiple reflections leads to enhanced subsurface illumination and to increased image resolution. A joint migration approach using the complete wavefield requires properly imaged primaries and multiples of all orders. In recent works, primaries and multiples have therefore been imaged separately, using the upgoing and downgoing pressure wavefields obtained by decomposing dual-sensor streamer data. The matches between the corresponding depth images are still not found to be sufficiently accurate, so new and more appropriate imaging conditions need to be found. We reviewed the classical imaging approach used in one-way wave-equation migration, with the aim of extending it for simultaneous migration of primaries and all orders of multiples. Based on Rayleigh’s reciprocity theorem and well-known theoretical developments, we derived a new imaging condition described in terms of the upgoing pressure wavefield and a filtered version of the downgoing vertical-velocity wavefield. To evaluate the efficiency of this new imaging approach, primaries and multiples were separately migrated using synthetic and real data examples. These results were compared to those obtained using a conventional imaging condition. We found that the use of the new imaging condition led to a better match between the depth images and spectra of the migrated primaries and migrated multiples.

High-fidelity complete wavefield velocity model building and imaging in shallow water environments – A North Sea case study

Accurate sub-surface image reconstruction from prestack depth migration of surface seismic wavefields requires precise knowledge of the local propagation velocities between the recording location and the image location at depth. Such velocity information has traditionally been derived by tomographically inverting residual moveout information of common image gathers, which have been computed using an initial velocity field. This methodology proves to be challenging in shallow water environments, particularly when strong and rapid velocity variations in the very shallow overburden need to be recovered, while moveout information from reflected arrivals is very sparse or not available.

Exact and asymptotic solutions of the equation of internal gravity waves in a wedge-shaped region

The field of internal gravity waves in a wedge-shaped region of a stratified medium is considered. Using a Kantorovich–Lebedev transformation, exact solutions are obtained which describe an individual mode and the complete wave field. The asymptotics of an individual wave mode are constructed by the WKB method, and are expressed in terms of a hypergeometric function, as well as the asymptotics of the complete wave field, which are expressed in terms of a semilogarithmic function. The results of numerical calculations of the wave field using the exact and asymptotic formulae are presented for the parameters of a stratified medium, characteristic for the dynamics of the ocean. The limits of their applicability are estimated.

Evaluating Centroid-Moment-Tensor Uncertainty in the New Version of ISOLA Software

Online Material: Additional figures. Focal mechanism and moment tensor determinations based on regional and local waveforms have become routine tasks in seismology. In recent years, considerable effort has been dedicated to extend software capabilities, in particular towards the inclusion of finite‐extent sources and easy, user‐friendly use. Two examples of these efforts are the FMNEAREG and KIWI software packages. Focal Mechanism from NEAr source to REGional distance records (FMNEAREG; Delouis et al. , 2008; Maercklin et al. , 2011) performs a grid search in order to find pure double‐couple sources, based on a 1D (line) finite‐source representation and complete elastodynamic wave field. The KIWI software (KInematic Waveform Inversion; Cesca et al. , 2010) allows the automatic retrieval of point‐source parameters, and for magnitudes above a given threshold kinematic finite‐fault rupture models can also be obtained. KIWI relies on a pre‐calculated database of Green’s functions. KIWI performs the moment tensor inversion in two steps, the first of which is based on the fit of amplitude spectra. The second step is based on time‐domain waveform fits. The ISOLA, ISOLated Asperities, software package (Sokos and Zahradnik, 2008) also performs waveform inversions to find source parameters. ISOLA is based on FORTRAN codes and offers a user‐friendly MATLAB graphical interface (GUI; www.mathworks.com/products/MATLAB). ISOLA allows for both single‐ and multiple‐point‐source iterative deconvolution (Kikuchi and Kanamori, 1991) inversion of complete regional and local waveforms. The moment tensor is retrieved through a least squares inversion, whereas the position and origin time of the point sources are grid searched. The computation options include inversion to retrieve the full moment tensor (MT), deviatoric MT, and pure double‐couple MT. Finite‐extent source inversions may also be performed by prescribing a priori the double‐couple mechanism (which will remain homogeneous over the fault plane). Green’s functions, including near‐field terms, are calculated using the discrete wavenumber method (Bouchon, …