Acoustic Finite-difference Research Articles

Modelling of superposition in 2D linear acoustic wave problems using Fourier Neural Operator networks

A method of solving the 2D acoustic wave equation using Fourier Neural Operator (FNO) networks is presented. Various scenarios including wave superposition are considered, including the modelling of multiple simultaneous sound sources, reflections from domain boundaries and diffraction from randomly-positioned and sized rectangular objects. Training, testing and ground-truth data is produced using the acoustic Finite-Difference Time-Domain (FDTD) method. FNO is selected as the neural architecture as the network architecture requires relatively little memory compared to some other operator network designs. The number of training epochs and the size of training datasets were chosen to be small to test the convergence properties of FNO in challenging learning conditions. It is demonstrated that an FNO network predicts wave behaviour faster than equivalent FDTD simulations. The use cases of FNO networks, including data compression and auralisation are considered and compared to existing operator network designs.

Diving-wave time-lapse delay for CO2 thin layer detection

SUMMARY We have derived an analytical approximate expression to estimate the delay in diving seismic waves due to thin layers of CO2. The expression is valid for high frequencies and can be used to estimate the delay in diving waves at seismic frequencies for large separations between the source and receiver (offset). The approximation may be used to assess CO2 detection limits using diving waves and to support survey planning for CO2 monitoring and full-waveform inversion (FWI) cycle skipping analysis. In this study, we analyse the diving-wave response to a thin layer of CO2 for band-limited data using acoustic finite-difference modelling, and compare the results against the analytical calculations. We find that the responses are offset-dependent and related to double- and single-leg interactions between the diving waves and the CO2. To test the methods, we created a synthetic representation of the 2010 subsurface conditions for the top CO2 layer at the Sleipner storage complex in the North Sea, by combining base and monitor post-stack seismic data with field velocity trends. Using the acoustic finite-difference method, we model pre-stack data that captures the complexity of field data and demonstrate the use of the diving-wave delay for CO2 migration monitoring and CO2 thin layer detection.

On the accuracy and spatial sampling of finite-difference modelling in discontinuous models

Finite-difference modelling estimates the wavefield in the subsurface by solving the elastic or acoustic wave equation numerically in a discrete version of the subsurface. The derivatives in the wave equation are approximated by their finite-difference counterparts. In this paper, we investigate the accuracy of acoustic finite-difference modelling as a function of the spatial sampling rate, the frequency, the source-receiver offset and the model parameters and parametrisation. To represent discontinuities in a regular grid, we apply a tapered low-pass wavenumber filter to densely sampled compliance and density models using the Nyquist wavenumber of the desired grid as cut-off wavenumber. A long filter taper improves the finite-difference modelling accuracy but also leads to longer oscillations in the model grids. We use a 2D acoustic central-grid pseudospectral scheme and compare it to analytical solutions of the wave equation and the reflectivity method. We show that previous recommendations for the spatial sampling of four grid points per shortest wavelength in staggered-grid schemes also apply to central-grid schemes. In the case of a single thin layer, the accuracy of finite-difference modelling is dependent on the impedance contrast and the layer thickness. For accurate wide-angle reflection amplitudes of a thin layer with a strong impedance contrast, a denser grid sampling than four grid points per shortest wavelength is required. Furthermore, we demonstrate that the presented wavenumber filtering approach is better suited for the downsampling of regularly sampled compliance and density data derived from well logs than Backus averaging.

Effect of bat pinna on sensing using acoustic finite difference time domain simulation.

The practicality of the finite-difference time-domain (FDTD) method was confirmed by comparing head-related transfer functions obtained from a three-dimensional (3D) digital model of a bat (Rhinolophus ferrumequinum nippon) head with acoustic experiments using a 3D printed physical model. Furthermore, we simulated the auditory directionality using a 3D digital model that was modified based on the pinna movement of a bat during echolocation and found that the alternating movements of the left and right pinna result in a binaural sound pressure difference for vertical sources. Using the FDTD method, suitable for simulating acoustics in large spaces, we could analyze in detail the binaural echoes that bats receive and the acoustic cues they use for echolocation.

Modeling of the complex hydrocarbon traps by the shot domain acoustic finite difference method and data-processing

Numerical modeling studies have a widespread application in exploration seismology in order to understand the seismic reflection responses of hydrocarbon traps formed in relation to tectonic structure, lithological changes and unconformities in complex geological environments and to develop effective data processing strategies. In this study, the seismic modeling of two important hydrocarbon trap models (Granite Wash and Normal Fault Trap) was performed by the Finite Difference Method (FDM), which provides the solution of the acoustic wave equation. Seismic data models were carried out in the pre-stack shot environment, and the obtained shot data were passed through appropriate data-processing stages to obtain stack and migration (zero offset) sections. By converting the obtained migration sections to depth, the spatial location and dimensions of hydrocarbon traps on the section were determined and it has been observed that they are compatible by comparing with the initial geological models. Thus, the seismic responses of hydrocarbon trap structures were learned, the importance of data processing was understood, and zero offset cross-sections were obtained by processing of the generated synthetic shot records. Accordingly, it was observed that it is appropriate to make more and frequent shots in the investigation of granite wash type traps which are thin and short-width , whereas it is useful to make relatively less frequent shots in order to reduce the scattering intensity caused by the discontinuities of the fault type structures. Consequently, it is thought that before the field study for the hydrocarbon exploration, the modeling of the pre-stack shot instead of the post-stack modeling (zero offset) will contribute to the development of the data processing stages and the testing of the seismic section interpretation. In the future, such synthetic models and data processing will be developed for different complex trap structures and real data.

Acoustic finite-difference modeling beyond conventional Courant-Friedrichs-Lewy stability limit: Approach based on variable-length temporal and spatial operators

Conventional finite-difference (FD) methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy (CFL) numbers 0.707 and 0.577 for two-dimensional (2D) and three-dimensional (3D) equal spacing cases, respectively, thereby limiting time step selection. Based on the definition of temporal and spatial FD operators, we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy. First, to simulate wave propagation beyond the conventional CFL stability limit, the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones. Second, to preserve the modeling accuracy, the velocity-dependent lengths of the temporal and spatial operators are adaptively varied. The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples, respectively. We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.

Cosine-type weighted hybrid absorbing boundary based on the second-order Higdon boundary condition and its GPU implementation

AbstractIn this paper, we analyze the absorption effect of the hybrid absorbing boundary condition (ABC) with various hybrid modes, and propose a cosine-type optimized weighted hybrid mode taking into account the boundary reflection intensity of the inner and outer boundaries of transition area. Additionally, we derive a new finite-difference scheme of the second-order Higdon ABC and the corner equation for Graphic Processing Unit (GPU) acceleration. On this basis, a new type of second-order Higdon hybrid ABC applicable for GPU acceleration is established in the acoustic finite-difference modeling. Numerical experiments demonstrate that the proposed cosine-type weighted hybrid mode can achieve a better absorption effect compared with other weighted hybrid modes; the second-order Higdon ABC based on the proposed new finite-difference scheme can effectively improve the GPU speed-up ratio and is more effective in large-scale wavefield high-precision simulation based on GPU acceleration.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of a single early reflection.

Finite-difference time-domain method has gained increasing interest for room acoustic prediction use. A well-known limitation of the method is a frequency- and direction-dependent dispersion error. In this study, the audibility of dispersion error in the presence of a single surface reflection is measured. The threshold is measured for three different distance conditions with a fixed reflection arrival azimuth angle of 54.7°. The error is placed either in the direct path, or in the reflection path. Additionally a qualitative follow-up experiment to evaluate how the measured thresholds reflect the audibility of error in short room responses is carried out. The results indicate that the threshold varies depending whether the error is in the direct path or in the reflection path. For transient signals the threshold is higher when the error is located in the direct path, whereas for speech signal, the threshold is higher when it is located in the reflection path. Evidence is found that the error is detectable in rendered room responses at the measured threshold levels.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of a single early reflection

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of a single early reflection

An improved hybrid absorbing boundary condition for wave equation modeling

A recently developed hybrid absorbing boundary condition (ABC) can significantly suppress artificial boundary reflections by inserting a transition area between the boundary and the inner area. The wavefield in the transition area is computed by linearly weighting the wavefield from one-way wave equation (OWWE) and that from two-way wave equation. This leads to a smooth variation from the computational area to the boundary via the transition area and thus greatly reduces boundary reflections. In this paper, we propose two techniques to further enhance the absorbing effect. First, we widen the boundary from one point to several points. Second, we adopt nonlinear weighting coefficients instead of linear ones. Numerical simulations with 2D acoustic finite-difference modeling, combined with Clayton–Engquist OWWE, demonstrate that the proposed techniques can significantly improve the absorption effect without increasing computational cost. Tests using Higdon OWWE-based ABC suggest that the second technique can greatly enhance the absorption effect. Experiments with frequency-domain finite-difference modeling demonstrate similar improvement in absorption effect.

Hybrid absorbing boundary condition for piecewise smooth curved boundary in 2D acoustic finite difference modelling

Flexible computational domains and their corresponding irregular absorbing boundary conditions (ABCs) have previously been shown to improve the efficiency of finite difference (FD) modelling. However, these proposed ABCs for irregular boundaries are based on irregular grid methods. Although they can improve geometric flexibility of FD modelling, irregular grid methods are still complex to implement due to computationally expensive meshing process. To avoid complex mesh generation, FDs in mesh-free discretisation have been developed for nontrivial geometric settings, where scattered nodes can be placed suitably with respect to irregular boundaries and arbitrarily shaped anomalies without coordinate mapping or mesh-element forming. Radial-basis-function-generated FD (RBF-FD) has been proven successful in modelling seismic wave propagation based on mesh-free discretisation. Using RBF-FD, we develop a hybrid ABC for piecewise smooth curved boundary in 2D acoustic wave modelling based on straightforward expanding strategies for sampled boundary and corner nodes. The whole irregular computational domain is combined by an objective zone and a transition zone separated by piecewise smooth curved boundary. Nodal distribution in the objective zone is adaptive to the model structure through special alignment and varying density of nodes, while the transition zone are designed for solving one way wave equation more easily and accurately in the presence of piecewise smooth curved boundary. Modelling examples for homogenous and heterogeneous models with differently shaped computational domains demonstrate the effectiveness of our proposed method.

A simple finite-difference scheme for handling topography with the first-order wave equation

One approach to incorporate topography in seismic finite-difference codes is a local modification of the difference operators near the free surface. An earlier paper described an approach for modelling irregular boundaries in a constant-density acoustic finite-difference code, based on the second-order formulation of the wave equation that only involves the pressure. Here, a similar method is considered for the first-order formulation in terms of pressure and particle velocity, using a staggered finite-difference discretization both in space and in time. In one space dimension, the boundary conditions consist in imposing antisymmetry for the pressure and symmetry for particle velocity components. For the pressure, this means that the solution values as well as all even derivatives up to a certain order are zero on the boundary. For the particle velocity, all odd derivatives are zero. In 2D, the 1-D assumption is used along each coordinate direction, with antisymmetry for the pressure along the coordinate and symmetry for the particle velocity component parallel to that coordinate direction. Since the symmetry or antisymmetry should hold along the direction normal to the boundary rather than along the coordinate directions, this generates an additional numerical error on top of the time stepping errors and the errors due to the interior spatial discretization. Numerical experiments in 2D and 3D nevertheless produce acceptable results

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of absorption of air.

The finite-difference time-domain method has gained increasing interest for room acoustic prediction use. A well-known limitation of the method is a frequency and direction dependent dispersion error. In this study, the audibility of dispersion error in the presence of air absorption is measured. The results indicate that the dispersion error in the worst-case direction of the studied scheme gets masked by the air absorption at a phase velocity error percentage of 0.28% at the frequency of 20 kHz.

The use of seismic modeling for the geologic interpretation of deep seismic reflection data with low signal-to-noise ratios

We have performed a deep seismic reflection study, DACIA-PLAN, based on the data recorded along a crooked line across the southeastern Romanian Carpathians. The signal-to-noise ratio (S/N) of these data varies along the seismic profile, and its variation is considered to be an effect of the rough topography, complex subsurface geology, and varying surface conditions encountered during seismic data acquisition. The migrated time section that covers the mountainous area is clear, without visible reflections, making the geologic interpretation very difficult. We used a seismic modeling technique to explain the poor S/N of the recorded data and to generate synthetic seismic sections that can be useful for the geologic interpretation of the field seismic section (migrated time section). We used ray-tracing modeling to obtain the expected seismic expression of horizons of interest. Subsurface illumination modeling indicates that the complex subsurface geology and irregularly deployed sources and receivers are responsible for the incomplete and/or uneven illumination of the subsurface and can lead to strong amplitude variations. We then used 2.5D acoustic finite-difference modeling to analyze the effect of a crooked line on seismic wave propagation. The synthetic shot gathers prove that crooked line arrival times for reflected and head waves contain static time shifts relative to a straight line regular sampling geometry. Some geologic interfaces of interest are not well-imaged on the synthetic seismic section, and this is considered to be an effect of poor positioning during seismic data acquisition. We used the velocity model from the tomographic inversion of first-arrival traveltimes and synthetic and field crooked line deep seismic reflection data to create a structural image for the southeastern Romanian Carpathians and the Focsani Basin, which tie well with the geologic model built for this area on the basis of geologic and well data only.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation as a function of simulation distance.

Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%.

Seismic signatures of complex geological structures in the Cue-Weld Range area, Murchison domain, Yilgarn Craton, Western Australia

The Murchison domain forms the northwest part of the Youanmi Terrane, a tectonic unit within the Neoarchean Yilgarn Craton in Western Australia. In the Cue-Weld Range area the Murchison domain has experienced a complex magmatic and deformation history that resulted in a transposed array of greenstone belts that host significant iron, gold, and base metal deposits. In this study, we interpret the upper 2s (about 6km) of a deep crustal seismic profile 10GA–YU1, near the town of Cue, and correlate rock units and structures in outcrop with corresponding reflections. We performed 3D constant velocity ray-tracing and calculate the corresponding travel times for the reflections for time domain pre-stack and post-stack seismic data. This allows us to link shallow reflections with mafic volcanic rocks of the Glen Group and basaltic rocks of the Polelle Group in outcrop. Based on our interpretation and published geological maps and data, we propose a model in which the local stratigraphy represents a refolded thrust system. To test our hypothesis, we applied 2D acoustic finite difference forward modeling. The corresponding synthetic data were processed in the same way as the acquired data. Comparisons between the acquired and the synthetic data show that the model is consistent with observations. We propose a new model for the subsurface of the Cue-Weld Range area and argue that some of the lithologies in the area are repeated structurally at different levels. Our approach highlights the benefit of imaging and modeling of deep seismic transects to resolve local structural complexity in Archean granite-greenstone terrains.

A method for correcting acoustic finite-difference amplitudes for elastic effects

We present a new method for correcting the amplitudes of arrivals in an acoustic finite-difference simulation for elastic effects. In this method, we selectively compute an estimate of the error incurred when the acoustic wave equation is used to approximate the behavior of the elastic wave equation. This error estimate is used to generate an effective source field in a second acoustic simulation. The result of this second simulation is then applied as a correction to the original acoustic simulation. The overall cost is approximately twice that of an acoustic simulation but substantially less than the cost of an elastic simulation. Because both simulations are acoustic, no S-waves are generated, so dispersed converted waves are avoided. We tested the characteristics of the method on a simple synthetic model designed to simulate propagation through a strong acoustic impedance contrast representative of sedimentary geology. It corrected amplitudes to high accuracy for reflected arrivals over a wide range of incidence angles. We also evaluated results from simulations on more complex models that demonstrated that the method was applicable in realistic sedimentary models containing a wide range of seismic contrasts. However, its accuracy was reduced for wide-angle reflections from very high impedance contrasts such as a shallow top-salt interface. We examined the influence of modeling at coarse grid resolutions, in which converted S-waves in the equivalent elastic simulation are dispersed. These results provide some validation for the accuracy of the method when applied using finite-difference grids designed for acoustic modeling. The method appears to offer a cost-effective means of modeling elastic amplitudes for P-wave arrivals in a useful range of velocity models. It has several potential applications in imaging and inversion.

Effects of sources on time-domain finite difference models.

Recent work on excitation mechanisms in acoustic finite difference models focuses primarily on physical interpretations of observed phenomena. This paper offers an alternative view by examining the properties of models from the perspectives of linear algebra and signal processing. Interpretation of a simulation as matrix exponentiation clarifies the separate roles of sources as boundaries and signals. Boundary conditions modify the matrix and thus its modal structure, and initial conditions or source signals shape the solution, but not the modal structure. Low-frequency artifacts are shown to follow from eigenvalues and eigenvectors of the matrix, and previously reported artifacts are predicted from eigenvalue estimates. The role of source signals is also briefly discussed.

Time-lapse seismic modeling for CO2 sequestration at the Dickman Oilfield, Kansas

Carbon dioxide capture and injection into the subsurface has aroused great interest in the past few years as a method to enhance oil recovery and mitigate [Formula: see text] emissions. The Dickman Oilfield located in Kansas provides two possible [Formula: see text] sequestration targets: a regional deep saline reservoir (the primary objective) and a shallower mature depleted oil reservoir (secondary). We focused on the shallower depleted oil reservoir through a 250-year flow-simulation scenario and a fault leakage test. Seismic responses at various time intervals were simulated to help monitor [Formula: see text] flow paths and injection stability. A complex and realistic geologic model with unconformity was embedded in the flow-simulation model. A regridding technique was used that assigned geologic values to a regular seismic grid that allowed 2D acoustic and elastic finite-difference simulation. Gassmann fluid substitution theory was used to obtain the reservoir properties with different [Formula: see text] saturations, and the vertical seismic profile was used to assist in identifying geologic layers. A [Formula: see text] plume and flow path from the leakage test can be detected from the differences in seismic data (5 to 10 ms time shift) from the first year of injection and the last year of monitoring. This was supported by comparison with the prestack field data available in the Dickman Oilfield. [Formula: see text]-induced reflectivity changes are relatively larger for PS events than for PP events, implying that multicomponent data acquisition and processing may give added value to characterization and monitoring of carbon capture and storage projects. We assessed 4D seismic monitoring in the evaluation of long-term [Formula: see text] containment stability for the Dickman Oilfield and suggested that time-lapse surveys will be useful.

Modeling underwater impact pile driving noise in shallow, inhomogeneous channels

The broadband synthesis of a parabolic equation (PE) propagation model for shallow water acoustic propagation in inhomogeneous channels is presented to account for the noise produced by impact pile driving. The PE model utilizes sediment information obtained from boring measurements and detailed bathymetry to model range dependent propagation in the Columbia River between Portland, OR and Vancouver, WA. The impact pile driving source is modeled in two ways: first, as a reverberating impulse source that emits Mach-wave radiation [Reinhall, Dahl, J. Acoust. Soc. Am. 130, 1209 (2011)]; and second, with a structural acoustic finite-difference time-domain (FDTD) model [Shahab, Woolfe, and Hastings, J. Acoust. Soc. Am. 130, 2558 (2011)]. Model results using both source models are shown to be in good agreement with acoustic measurements of test pile operations in the Columbia River at multiple locations from 10 to 800 meters from the pile driving source. Implications for noise levels in river systems with varying bottom sediment characteristics are presented and discussed. [This research is supported with funding from the Oregon Department of Transportation.]