Numerical Dispersion Research Articles

Research on Airborne Ground-Penetrating Radar Imaging Technology in Complex Terrain

The integration of ground-penetrating radar (GPR) with unmanned aerial vehicles (UAVs) enables efficient non-contact detection, performing exceptionally well in complex terrains and extreme environments. However, challenges in data processing and interpretation remain significant obstacles to fully utilizing this technology. To mitigate the effects of numerical dispersion, this paper develops a high-order finite-difference time-domain (FDTD(2,4)) three-dimensional code suitable for airborne GPR numerical simulations. The simulation results are compared with traditional FDTD methods, validating the accuracy of the proposed approach. Additionally, a Kirchhoff migration algorithm that considers the influence of the air layer is developed for airborne GPR. Different processing strategies are applied to flat and undulating terrain models, significantly improving the identification of shallowly buried targets. Particularly under undulating terrain conditions, the energy ratio method is introduced, effectively suppressing the interference of surface reflections caused by terrain variations. This innovative approach offers a new technical pathway for efficient GPR data processing in complex terrains. The study provides new insights and methods for the practical application of airborne GPR.

An Integrated Taylor Expansion and Least Squares Approach to Enhanced Acoustic Wave Staggered Grid Finite Difference Modeling

The staggered grid finite difference method has emerged as one of the most commonly used approaches in finite difference methodologies due to its high computational accuracy and stability. Inevitably, discretizing over time and space domains in finite difference methods leads to numerical artifacts. This paper introduces a novel approach that combines the widely used Taylor series expansion with the least squares method to effectively suppress numerical dispersion. We have derived the coefficients for the staggered grid finite difference method by integrating Taylor series expansions with the least squares method. To validate the effectiveness of our approach, we conducted analyses on accuracy, dispersion, and stability, alongside simple and complex numerical examples. The results indicate that our method not only inherits the capabilities of the original Taylor series and least squares methods in suppressing numerical dispersion across small and medium wavenumber ranges but also surpasses the original methods. Moreover, it demonstrates robust dispersion suppression capabilities at high wavenumber ranges.

Upwind block-centered multistep differences for semiconductor device problem with heat and magnetic influences

In this paper, a mathematical model of semiconductor device with two important factors is discussed. Heat and magnetic influences are considered together. Its numerical approximation is important in information science and other technological innovation fields. The major physical unknowns (the potential, electron concentration, hole concentration, and the heat) are defined by four nonlinear PDEs: an elliptic equation, two convection-diffusion equations and a heat conductor equation. A conservative block-centered method is used to approximate the elliptic equation. The derivatives to time t, ∂∂t(⋅), are discretized by multistep differences for improving the computational accuracy. The diffusions and convection terms are treated by block-centered differences and upwind approximations, respectively. This composite numerical method is suitable for this nonlinear system. Firstly, numerical dispersion and nonphysical oscillations are avoided. Secondly, the unknowns and their adjoint vectors are obtained simultaneously. Thirdly, the law of conservation is preserved. An error estimates is derived. Finally, simple examples are given for showing the efficiency and accuracy.

Widefield optical coherence tomography by electro-optical modulation

Optical coherence tomography (OCT) is a unique imaging modality capable of axial sectioning with a resolution of only a few microns. Its ability to image with high resolution deep within tissue makes it ideal for material inspection, dentistry, and, in particular, ophthalmology. Widefield retinal imaging has garnered increasing clinical interest for the detection of numerous retinal diseases. However, real-time applications in clinical practice demand the contrast of swept-source OCT at scan speeds that limit their depth range. The curvature of typical samples, such as teeth, corneas, or retinas, thus restricts the field-of-view of fast OCT systems. Novel high-speed swept sources are expected to further improve the scan rate; however, not without exacerbating the already severe trade-off in depth range. Here, we show how, without the need for mechanical repositioning, harmonic images can be rapidly synthesized at any depth. This is achieved by opto-electronic modulation of a single-frequency swept source laser in tandem with tailored numerical dispersion compensation. We demonstrate experimentally how real-time imaging of highly-curved samples is enabled by extending the effective depth-range 8-fold. Even at the scan speed of a 400 kHz swept source, harmonic OCT enables widefield retinal imaging.

Experimental Evidence for Zerovalent Pd(NHC) as a Competent Catalyst in C-N Cross-Coupling (NHC = DiMeIHeptCl).

Use of the branched N-heterocyclic carbene (NHC) ligand 1,3-bis(2,6-bis(3-methyl-1-(2-methylpropyl)butyl)phenyl)-4,5-dichloro-1,3-dihydro-2H-imidazole-2-ylidene (DiMeIHeptCl) facilitated the stabilization of several relevant intermediates for Pd(NHC)-catalyzed C-N cross-coupling reactions. Complexes [Pd(DiMeIHeptCl)]2(μ-N2), [Pd(DiMeIHeptCl)]2(μ-η2-1,2-η2-4,5-C6H6), and Pd(DiMeIHeptCl)(pyridine), representing zerovalent Pd(NHC) bearing labile ligands, were isolated and structurally characterized, along with divalent PdCl(Ph)(DiMeIHeptCl) and PdCl(Ph)(DiMeIHeptCl)(n-propylamine). The former is a 14-electron Pd complex, which is stable under air and chromatography on silica gel or neutral alumina. One possible reason for this exceptional stability is the numerous dispersion interactions between the NHC alkyl chains and the Pd-Ph group. Detailed investigations of catalyst activation and oxidative addition confirmed that "Pd(NHC)" is formed from many known Pd(II)(NHC) precatalysts and provided activation rates for these different precatalysts.

Development and distribution of an in situ experimental wind turbine noise database

The PIBE project is the first French collaborative research project on wind turbine noise. One of its objectives is to quantify the experimental dispersion observed in situ on acoustic quantities/metrics and influencing parameters, and to compare it with the numerical dispersion estimated using acoustic propagation models. A long-term experimental campaign was carried out over consecutive 410 days around a wind farm comprising eight 80-metre-high and 90-metre diameter turbines. Acoustic sensors (five Class I sound level meters) were deployed on site, recording third-octave spectra and overall A- and Z-weighted values of the Leq indicator and fractile indices (L10, L50 and L90), at distances from 325 m to 1400 m from the wind turbines row. In addition, three 3D ultrasonic anemometers and a Lidar system were used to measure meteorological variables influencing long-range sound propagation (e.g. wind and temperature verical profiles). Besides, train and aircraft passing close to the site were inventoried to identify periods of high background noise. All experimental data have been processed into an ElasticSearch database. An interactive web application has been developed in R Shiny to facilitate processing, visualization and analysis of the experimental database.

Optimised implicit methods for audio rate wave-based acoustic simulation

Wave-based room acoustic computation at an audio rate (i.e., covering the complete range of audible frequencies) is computationally intensive. There is thus a great incentive to develop simulation methods that allow for "accurate" simulation at the lowest possible computational cost, taking into account requirements for massively parallel implementation. Though often formal measures of order of accuracy are employed, in this paper, we examine the optimisation of a locally-defined implicit family of schemes against wideband measures of accuracy, based on the minimisation of numerical dispersion over a specified frequency range. The resulting optimised scheme is defined over a Cartesian grid, and allows performance gains over standard schemes (such as the basic rectilinear scheme, or schemes defined over face-centered cubic grids, and other implicit designs presented in the literature). Numerical stability conditions are easily formulated, and can be built into the optimisation, alongside the computational cost of linear system solution. Simulation results and comparisons of computation times against other known methods are presented.

Acoustic flow resonances in installed rectangular jets

Zonal Large Eddy Simulations (LES) of complex installed rectangular jet flows are performed for subsonic jet conditions of the Central Institute for Aviation Motors (CIAM) experiment, where a strong tone was reported. For far-field noise modelling, the zonal LES solutions are coupled with the permeable formulation of the Ffowcs Williams and Hawkings (FW-H) method. The obtained noise spectra predictions are compared with the acoustic measurements in the CIAM facility. To reduce statistical noise of the relatively short LES time series, contributions due to several dominant low frequency tones and the remaining broadband signal are computed separately, using a tailored Fourier transform technique for each signal type. For cross-validation, noise spectra measurements of an equivalent isolated round jet in the CIAM facility are compared with the empirical sJet model calibrated on the NASA Small Hot Jet Acoustic Rig (SHJAR) jet noise database. While the two datasets agree reasonably well for mid to high frequencies, larger discrepancies are observed for low frequencies, which are attributed to acoustic reflections in the non-anechoic CIAM facility. The differences in noise levels between the CIAM dataset and the sJet model representing the NASA jet noise data are used to determine the experimental uncertainty of the CIAM noise measurements for each frequency and observer angle. For the installed rectangular jet, it is shown that far-field noise spectra predictions of the zonal LES-FW-H method for both the fundamental tone, its sixth harmonic corresponding to the jet Strouhal number of around 1, and the broadband component are broadly within the experimental error bar for most observer angles. Some discrepancies between predictions of the zonal LES method and the CIAM measurements for the main low frequency tone for the 30o and 90o, where the experimental uncertainty at low frequencies is largest are also noted. After validating the acoustic solutions, spectral and conditional averaging analysis methods are applied to elucidate mechanisms of the acoustic-flow resonance in the installed rectangular jet flow. The numerical dispersion relation is obtained to analyse phase velocities of the upstream and downstream travelling pressure waves in the stream-wise direction. The conditional analysis of pressure fluctuations inside the jet reveals the emergence of internal reflection points corresponding to a rapid change of the acoustic wave phase as well as the saddle points interpreted as localised zones of vorticity shed from the side-wall edges. To independently investigate the effect of trailing edges and corner points of the jet embodiment geometry on closing the acoustic-flow resonance cycle, several modifications of the baseline rectangular nozzle are considered, such as introducing chevron-like lobes and cuts on the trailing edges of the side walls as well as smoothing the sharp corner points of the wall edges. Following the series of additional zonal LES runs with the modified installed nozzle geometries, an optimized tone-less modification of the original installed nozzle was obtained, in broad agreement with the conditional analysis results. The obtained tone-less modification of the installed rectangular jet is further analyzed to provide insights into its similarity and differences from the aeroacoustics of the reference isolated round jet in the same CIAM facility.

Analysis of short-term self-healing performance of asphalt mixture under a three-point bending fatigue test

To evaluate the self-healing performance of asphalt mixture under repeated loads, indoor three-point bending fatigue tests and self-healing tests were conducted on asphalt mixture AC-13. The stress control mode was selected to test the fatigue resistance of the mixture, and the dissipated energy recovery value and visco-elastic ratio were proposed to evaluate the self-healing performance of the mixture. The experimental results show that there is a good linear relationship between fatigue life and fatigue stress, and the damage degree to the mixture is exponentially related to the load times. The numerical dispersion of flexural tensile modulus is large, which is not suitable for evaluating the self-healing performance of the mixture. As the number of loading cycles increases, the hysteresis dissipated energy of the mixture gradually decreases and tends to stabilize. After the loading interval, due to the self-healing effect, the visco-elastic performance of the asphalt material is partially restored. The initial dissipated energy before and after interval can be used as an evaluation index for self-healing behavior, and there is a good linear relationship between the initial dissipated energy and the damage degree. Due to the deformation migration of the mixture during the initial loading stage before and after interval, the visco-elastic ratio has changed, and this change becomes more pronounced with the damage degree of the mixture, indicating that the self-healing ability of the mixture has also been improved.

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

Efficient Fourth-Order PSTD Algorithm with Moving Window for Long-Distance EMP Propagation.

Satellite-borne electromagnetic pulse (EMP) detection technology plays an important role in military reconnaissance, space environment monitoring, and early warning of natural disasters. However, the complex ionosphere greatly distorts the waveform during propagation and poses a challenge to EMP detection. Therefore, it is necessary to conduct theoretical research on EMP propagation in the ionosphere. Conventional second-order pseudo-spectral time-domain (PSTD-2) algorithm has difficulties in keeping the stability and accuracy of waveforms in calculations over hundreds of kilometers of propagation. To overcome the difficulties, a fourth-order PSTD algorithm incorporating the moving window technique (MWPSTD-4) is proposed. In the numerical examples, the performance of MWPSTD-4 is compared with PSTD-4 and PSTD-2 in the long-distance propagation of EMP. The results show that the MWPSTD-4 improves efficiency while guaranteeing accuracy and is suitable for large-scale electromagnetic field simulation. The proposed method provides a basic algorithm to eliminate the numerical dispersion interference for calculating the long-distance propagation of EMP in complex spaces and is helpful for the design and calibration of satellite-borne EMP detectors.

Predicting Oil Recovery Under Uncertainty for Huff ‘n’ Puff Gas Injection: A Field Case Study in the Permian Basin

Summary The extent and size of the fracture network connected to the wellbore are among the most uncertain parameters for multifractured horizontal wells. The uncertainty has a profound impact on production under primary depletion and has an even bigger impact on the recovery during huff ‘n’ puff (HnP) gas injection. In this paper, we quantify the uncertainty in reservoir properties and enhanced oil recovery (EOR) for a pair of wells in the Permian basin. In addition, we report on the lessons learned from modeling a field pilot HnP project involving four cycles using reservoir simulation. We create a base sector model for two wells completed in Wolfcamp B and C formations, where fractures are incorporated using an embedded discrete fracture model. For each well, we identify 13 uncertain parameters (describing fracture properties and compaction, initial conditions, and relative permeability). We use a Bayesian assisted-history-matching (AHM) method to match primary production data. Our study emphasizes the critical role of grid refinement for the accurate depiction of the gas-injection process by comparing three models with varying grid sizes. Additionally, we study the impact of molecular diffusion on oil and gas recovery using different diffusion coefficients. Another significant factor is the dynamic nature of the fracture network during cyclic injection, as indicated by evidence of communication between the wells. This makes the accessible fracture area during injection highly uncertain. To address interwell interference, we incorporate an additional fracture network activated during injection. To manage the uncertainty of active fracture area, we created four scenarios within the literature range, approximately 0.5, 1, 2, and 4 million ft² per stage, and tuned their properties to match historical HnP data. The production predictions from the AHM solutions were extended beyond the initial four HnP cycles to provide probabilistic forecasts. We demonstrate that grid refinement has a minimal effect on production under primary depletion but significantly influences HnP recovery. We attribute this disparity to numerical dispersion, which causes the artificial mixing of gas in the large gridblocks, leading to an overly optimistic recovery. Activating the molecular-diffusion mechanism results in a negligible impact on the recovery in our case. The contribution of molecular diffusion becomes noticeable only when diffusion coefficients exceed ten times the reported values. This leads us to conclude that molecular diffusion is of second-order importance compared with advection and can generally be neglected in our case. This finding should not be generalized to all shale formations. In reservoirs characterized by lower permeability and fluids with higher diffusion coefficients, molecular diffusion could still play a crucial role. We highlight that the contacted-fracture area, which is often underestimated in the reservoir simulation practice, is a critical factor for accurate modeling. Production for 37 filtered AHM solutions is extended under the four scenarios with variable fracture surface areas. Based on our analysis, we report that the HnP results in an average of 76% improvement in oil recovery compared with natural depletion.

A novel explicit optimized scheme for numerical simulation of elastic-wavefield separation

Abstract Numerical simulation of elastic wave equation helps us better understand the information of underground structures and elastic wave imaging has attracted the widespread attention of researchers. Using elastic wave imaging requires separating the compressional and shear wavefields. Therefore, we develop a novel explicit optimized scheme to simulate the separated elastic wavefield. We construct a kind of 1-norm objective function directly utilizing the dispersion error and employ the simulated annealing algorithm to acquire improved finite-difference operators, whose optimal coefficients can effectively suppress spatial numerical dispersion. Meanwhile, we introduce a rotated staggered-grid (RSG) approach to enhance computational stability. Then, our proposed scheme also called the optimized RSG approach is applied to the elastic wave equations and decoupled elastic wave equations to simulate the decoupled compressional and shear wavefield propagation. Numerical dispersion analysis is consistent with numerical results. The waveform comparison shows that the optimized RSG approach possesses higher accuracy, and several complex models are used to validate the applicability and effectiveness of the presented scheme.

Влияние числа Куранта на результаты численного моделирования распространения сигналов в недиспергирующих однородных средах

The paper is devoted to the study of the connection between the numerical dispersion arising in FDTD modeling of electromagnetic signal propagation in nondispersive homogeneous media optically different from vacuum and the Courant number in the 2D case. The main results are formulated in the form of four statements, as well as a number of corollaries and remarks that determine the nature of the numerical dispersion, the optimal value of the Courant number and the limitations of the method. It is proved that the optimal choice of the Courant number eliminates the numerical dispersion and extends the capabilities of the developed numerical algorithm to media, which refractive index is lesser than refractive index of vacuum, as well as media with negative refraction.

2D acoustic equation prestack reverse-time migration based on an optimized combined compact difference scheme

Abstract Reverse-time migration (RTM) is widely regarded as one of the most accurate migration methods available today. A crucial step in RTM involves extending seismic wavefields forward and backward. Compared to the conventional central finite-difference (CFD) scheme, the combined compact difference (CCD) scheme offers several advantages, including a shorter difference operator and the suppression of numerical dispersion under coarse grids. These attributes conserve memory and enhance effectiveness while maintaining the same level of differential precision. In this article, we begin with the five-point eighth-order CCD scheme and utilize the least squares method and Lagrange multiplier method to optimize the difference coefficients. This optimization is guided by the concept of dispersion-relation-preserving (DRP). The result is the acquisition of an optimized combined compact difference (OCCD) scheme, further enhancing the ability to suppress numerical dispersion. We thoroughly compare and analyze dispersion relationships and stability conditions. In addition, we examine several crucial steps in the RTM of the second-order acoustic wave equation. These steps include absorption boundary conditions, boundary storage strategy, and Poynting vector imaging conditions. Finally, we apply both the CCD and OCCD schemes in the RTM of the layered model, graben model, and SEG/EAGE salt model. We compare these results with those obtained from CFD's RTM. Numerical findings demonstrate that, in contrast to the CFD scheme, the CCD scheme effectively suppresses numerical dispersion and enhances imaging accuracy. Moreover, the optimized OCCD scheme further improves the ability to suppress numerical dispersion and can obtain better imaging results, which is an effective RTM method suitable for coarse grid conditions.

Adaptive training dataset generation for neural network numerical dispersion mitigation approach in seismic modeling

Adaptive training dataset generation for neural network numerical dispersion mitigation approach in seismic modeling

Self-adaptive numerical dispersion suppressed method for shallow seismic simulation

Near-surface in seismic exploration generally refers to the low deceleration zone and a section of stratum below it, which can be described in geophysical language as “low velocity”, “low Q”, and “Free surface”, etc. Due to the presence of low-velocity layers in near-surface, shallow seismic simulation is plagued by numerical dispersion. Numerical dispersion affects the accuracy of seismic wave simulation, especially severe in shallow areas containing low-velocity layers. To improve the simulated quality, denser grids or higher-order difference schemes are used, but both of which would seriously reduce computational efficiency, especially for frequency-domain numerical modeling. Differentiation is replaced by difference in wave field simulation, which would inevitably result in numerical errors. These numerical errors are manifested as the difference between the phase velocity of the discretized wave field and the medium velocity. The waves with different wavenumber components have different phase velocities, and the larger the wavenumber, the more that its phase velocity lags the group velocity. Based on this theoretical analysis, a regularization factor is added to the frequency-domain acoustic wave equation to correct the phase velocity of high wavenumber components. According to the Von Neumann stability requirements, a regularization factor that adaptively changes with simulation parameters is derived. Compared to the fixed regularization factor, adaptive regularization factor can better match the velocity field and protect the effective wave field to the maximum extent while suppressing numerical dispersion. The improved acoustic wave equation can effectively suppress numerical dispersion without increasing the number of grids. Different numerical models demonstrate the effectiveness and efficiency of the improved acoustic equation with the adaptive regularization factor for suppressing numerical dispersion.

A modified cell-to-cell simulation model to predict oil-gas minimum miscibility pressure

Calculating the minimum miscibility pressure (MMP) between crude oil and carbon dioxide (CO2) is critical for optimizing injection parameters, designing schemes, and predicting production capacity in CO2 injection projects for enhancing oil recovery. However, an accurate approach for obtaining this parameter is not yet established. In order to tackle this issue, a novel approach is suggested, based on the original cell-to-cell model, to determine the MMP and the 97% oil recovery rate as the standard. Using the volume-transformed Peng-Robinson equation of state enhances the precision of fluid volume estimation, as it mainly relies on predicting fluid volume within each cell. Furthermore, to ensure a precise estimation of the ultimate oil recovery rate, it is imperative to employ a total cell count of 500 in all simulations to avoid the problem of numerical dispersion. Finally, a second-order polynomial equation more accurately predicts the infinite-cell oil recovery factor. The accuracy of the modified model is verified by comparing MMP values from five oil and gas systems in the literature. The computational results of the modified multiple-mixing-cell (MMC) approach exhibit a higher level of concordance with the MMPs in the literature. The average relative error is less than 3.96%. The improved MMC algorithm can quickly determine the miscibility mechanism and visually represent the dynamic miscibility process involving multiple oil-gas contacts in a slim tube. This study provides a theoretical and practical basis for addressing the critical scientific issues of CO2-safe storage technology.

Seismic modeling by combining the finite-difference scheme with the numerical dispersion suppression neural network

Seismic finite-difference (FD) modeling suffers from numerical dispersion including both the temporal and spatial dispersion, which can decrease the accuracy of the numerical modeling. To improve the accuracy and efficiency of the conventional numerical modeling, I develop a new seismic modeling method by combining the FD scheme with the numerical dispersion suppression neural network (NDSNN). This method involves the following steps. First, a training data set composed of a small number of wavefield snapshots is generated. The wavefield snapshots with the low-accuracy wavefield data and the high-accuracy wavefield data are paired, and the low-accuracy wavefield snapshots involve the obvious numerical dispersion including both the temporal and spatial dispersion. Second, the NDSNN is trained until the network converges to simultaneously suppress the temporal and spatial dispersion. Third, the entire set of low-accuracy wavefield data is computed quickly using FD modeling with the large time step and the coarse grid. Fourth, the NDSNN is applied to the entire set of low-accuracy wavefield data to suppress the numerical dispersion including the temporal and spatial dispersion. Numerical modeling examples verify the effectiveness of my proposed method in improving the computational accuracy and efficiency.

An Explicit Time Integration Method Based on the Verlet Scheme with Improved Characteristics in Numerical Dispersion

An Explicit Time Integration Method Based on the Verlet Scheme with Improved Characteristics in Numerical Dispersion