Finite-difference Wave Equation Research Articles

Implicit finite difference wave equation forward modeling based on partial fraction expansion

Implicit finite difference wave equation forward modeling based on partial fraction expansion

Gaussian Beam Born Modeling for Single-Scattering Waves in Visco-Acoustic Media

A new Gaussian beam (GB) Born modeling method is presented in this letter to simulate the wavefields of single scattering events in visco-acoustic media. This approach allows the accurate computation of synthetic seismograms or waveforms for the visco-acoustic media. Our method relies on the GB summation method, in which the attenuation effects are taken into account by incorporating a frequency-dependent dissipation function into beam calculation. The key to efficiency is that the Born modeling kernel is evaluated in the time domain based on the filter-bank technique. The proposed method is tested with both 2-D and 3-D numerical examples and the results demonstrate its capability to simulate wavefields in complex structures, as well as its marked advantage in efficiency over finite-difference wave equation modeling.

Bayesian transdimensional seismic full-waveform inversion with a dipping layer parameterization

Seismic full-waveform inversion (FWI) has become a popular tool for estimating subsurface models using the amplitude and phase of seismograms. Unlike the conventional gradient-based approach, Bayesian inference using Markov chain Monte Carlo (MCMC) sampling can remove dependence on starting models and can quantify uncertainty. We have developed a Bayesian transdimensional (trans-d) MCMC seismic FWI method for estimating dipping-layer velocity models, in which the number of layers is unknown. A time-domain staggered-grid finite-difference wave equation solver is used for forward modeling. The FWI and MCMC methods are known to be computationally expensive. Two strategies are used to get practical computational performance. A layer-stripping strategy is used to accelerate sampler convergence, and a parsimonious dipping layer parameterization is used so that the MCMC algorithm can search broadly with fewer iterations. The parameters for each layer are velocity, thickness, and lower interface dip angle. We find that this parameterization has sufficient flexibility to invert for narrow 2D velocity models using small offset data. Model stitching is then used to bring several such inversions together to create larger 2D models. In turn, these can be used as starting models for gradient-based adjoint FWI to image complicated geologic settings. Two synthetic 2D numerical examples, including the Marmousi model, are considered, using seismic data dominated by reflections. Creation of good starting models traditionally requires significant human effort. We determine how much of that effort can be substituted with computation.

Determination and extraction of Rayleigh-waves for concrete cracks characterization based on matched filtering of center of energy

Surface Rayleigh wave (R-wave) has been used to detect and characterize defects in concrete due to its unique propagation properties. However, the determination of R-wave component is still ill-defined. In the present paper, a finite difference wave equation is employed for the numerical simulations of a transient point source interacting with a surface-opening crack on concrete. The compression wave (P-wave) and R-wave components were extracted and identified based on a new algorithm that utilizes matched filtering of center of energy (MFCE) concept to correlate wave parameters with crack depth. Experimental excitation and acquisition of R-waves was then carried out on concrete block specimens with artificial surface-opening crack to verify the correlations obtained via simulations. Experimental results are found to be in good agreement with the numerical ones, justifying the feasibility of the proposed method to determine and extract R-wave components for use in estimating depths of concrete crack.

Systematic wave-equation finite difference time domain formulations for modeling electromagnetic wave-propagation in general linear and nonlinear dispersive materials

In this paper, systematic wave-equation finite difference time domain (WE-FDTD) formulations are presented for modeling electromagnetic wave-propagation in linear and nonlinear dispersive materials. In the proposed formulations, the complex conjugate pole residue (CCPR) pairs model is adopted in deriving a unified dispersive WE-FDTD algorithm that allows modeling different dispersive materials, such as Debye, Drude and Lorentz, in the same manner with the minimal additional auxiliary variables. Moreover, the proposed formulations are incorporated with the wave-equation perfectly matched layer (WE-PML) to construct a material independent mesh truncating technique that can be used for modeling general frequency-dependent open region problems. Several numerical examples involving linear and nonlinear dispersive materials are included to show the validity of the proposed formulations.

Analyze the Seismic Attributes in HSX Volcanic Fracture Based on Forward Modeling

This paper uses the Hassan Carboniferous fractured reservoirs as the goal, firstly it analyses the reservoir characteristics of the cracks. On this basis, it uses two-dimensional random fractured media modeling method to build three different fracture models with different fracture parameters (fracture density, dip and speed). Then it uses finite difference wave equation forward and pre-stack depth migration processing of these models, and analyses seismic attribute of the migrated data, finally it finds a variety of seismic attributes sensitive to cracks, lays the foundation for fracture prediction with the seismic multi-attributes.

Perfectly Matched Layer for the Wave Equation Finite Difference Time Domain Method

The perfectly matched layer (PML) is introduced into the wave equation finite difference time domain (WE-FDTD) method. The WE-FDTD method is a finite difference method in which the wave equation is directly discretized on the basis of the central differences. The required memory of the WE-FDTD method is less than that of the standard FDTD method because no particle velocity is stored in the memory. In this study, the WE-FDTD method is first combined with the standard FDTD method. Then, Berenger's PML is combined with the WE-FDTD method. Some numerical demonstrations are given for the two- and three-dimensional sound fields.

Evaluation of Acoustic Simulation Using Wave Equation Finite Difference Time Domain Method with Compact Finite Differences

In this study, we examine the acoustic simulation method combining the wave equation finite difference time domain (WE FD-TD) method and compact FDs (CPFDs) for the second derivative. The wave equation compact finite difference time domain (WE CPFD-TD) method does not require calculation of the particle velocity; therefore, it can reduce the calculation time and memory used. Furthermore, for acceleration of simulation, we employ the recursive filtering algorithm and graphics processing unit (GPU) computing. As a result, we clarified that the three-dimensional WE CPFD-TD acoustic simulation using GPU is ca. 25–30 times faster than that using CPU with OpenMP.

Evaluation of Acoustic Simulation Using Wave Equation Finite Difference Time Domain Method with Compact Finite Differences

Evaluation of Acoustic Simulation Using Wave Equation Finite Difference Time Domain Method with Compact Finite Differences

Numerical Analysis of Long Range Sound Wave Propagation in Ocean by Wave Equation Finite Difference Time Domain Method with Graphics Processing Unit

The wave equation finite difference time domain (WE-FDTD) method is applied to the analysis of the long range sound wave propagation in the deep ocean. In the WE-FDTD method, the wave equation in the cylindrical coordinate is directly discretized on the basis of the central differences. The method is then implemented on a graphics processing unit (GPU) cluster system, which consists of 32 GPUs. Assuming the axisymmetric field, two-dimensional numerical models whose region size is 1000 km × 5000 m are developed for various cell sizes (1–3 m). Some numerical demonstrations are made for sound wave propagation in the deep ocean under the assumption of the Munk profile, which is known as the sound speed profile of the mid-latitude of the Pacific Ocean. The numerical results are compared with the results obtained using the ray-tracing method. It is found that the numerical dispersion error appears strikingly in the WE-FDTD solutions when the points per wavelength are less than 20 p.p.w., while the WE-FDTD solutions show good agreement with the ray-tracing solutions in the propagation time when the points per wavelength are more than 20 p.p.w. It is confirmed that the WE-FDTD method can be applied to the analysis of long range sound wave propagation in the deep ocean with reasonable accuracy.

Numerical Analysis of Long Range Sound Wave Propagation in Ocean by Wave Equation Finite Difference Time Domain Method with Graphics Processing Unit

The wave equation finite difference time domain (WE-FDTD) method is applied to the analysis of the long range sound wave propagation in the deep ocean. In the WE-FDTD method, the wave equation in the cylindrical coordinate is directly discretized on the basis of the central differences. The method is then implemented on a graphics processing unit (GPU) cluster system, which consists of 32 GPUs. Assuming the axisymmetric field, two-dimensional numerical models whose region size is 1000 km × 5000 m are developed for various cell sizes (1–3 m). Some numerical demonstrations are made for sound wave propagation in the deep ocean under the assumption of the Munk profile, which is known as the sound speed profile of the mid-latitude of the Pacific Ocean. The numerical results are compared with the results obtained using the ray-tracing method. It is found that the numerical dispersion error appears strikingly in the WE-FDTD solutions when the points per wavelength are less than 20 p.p.w., while the WE-FDTD solutions show good agreement with the ray-tracing solutions in the propagation time when the points per wavelength are more than 20 p.p.w. It is confirmed that the WE-FDTD method can be applied to the analysis of long range sound wave propagation in the deep ocean with reasonable accuracy.

Simulation and analysis of point-source surface wave fields

The complexity of near surface intensifies the diversity of seismic wave fields, which makes study on near surface wavefields important in many aspects. The strong absorption of low velocity layer can affect the resolution of seismic data, and free boundary can cause surface wave. Considering the above problems, we focus on the Rayleigh wavefields simulation using finite-difference wave equation of higher-order staggered grids and PML boundary conditions. Free boundary, buried source and overlying low velocity layer are taken into consideration and point explosion source is adopted. Through some numerical simulation with different parameters, we quantitatively analyze relationship between wave intensity and source depth, as well as the energy variation with propagation and obtain some practical knowledge and conclusions.

Complex envelope split-step wave-equation FDTD formulations for dispersive electromagnetic applications

Unconditionally stable complex envelope split-step wave-equation finite difference time domain (FDTD) formulations are presented for modelling dispersive electromagnetic applications. The formulations are based on incorporating the Strang time-splitting approach into the exponential evolution operator scheme. A two-dimensional numerical example is included to show the validity of the proposed formulations.

Analytical Expression for Impulse Response Between Two Nodes in 2-D Rectangular Digital Waveguide Mesh

The digital waveguide mesh models multidimensional wave propagation in discrete space and time. The finite difference wave equation used in the mesh is expressed in the space-time domain, which is not straightforward to reveal the analytical expression for the impulse response between a source and a receiver. This letter describes a procedure to derive an analytical impulse response expression between two nodes in a 2-D rectangular digital waveguide mesh based on the wave equation. The analytical expression can provide helpful insights and help verify simulation results for the digital waveguide mesh.

Unconditionally Stable Complex Envelope Wave Equation PML Algorithm for Band Limited FDTD Simulations

Unconditionally stable complex envelope (CE) perfectly matched layer (PML) absorbing boundary conditions (ABCs) are presented for truncating the scalar wave-equation finite difference time domain (WE-FDTD) grids. The formulations are based on incorporating the alternating direction implicit (ADI) scheme into the CE FDTD implementations of the scalar wave-equation derived in the PML region at the domain boundaries. Numerical example carried out in two dimensional domain shows that the proposed formulations are more accurate than the classical ADI scalar wave equation PML formulations when it is used for modelling band limited electromagnetic applications.

Receiver Function Forward Modeling and Migration Based on the Wave-Equation Finite Difference Method

We use the wave-equation finite difference algorithm to perform forward modeling and migration of receiver function. With help of concepts in exploration seismology, the equivalent velocity and traveltime equation are established, then the forward calculation and migration of receiver function are implemented by the wave-equation finite difference method. The numerical calculation indicates that imaging of point diffractor and dome structure can be accomplished with high accuracy employing poststack wave-equation finite difference migration. We also discuss the applicabilities of poststack finite difference migration and Kirchhoff migration to receiver function migration according to numerical results. Finally, the effects of velocity error on imaging are analyzed.

Modelling and imaging the Moho transition: the case of the southern Urals

Synthetic seismic modelling is used to test different possible structures and forming processes of the lower crust and Moho. One and two dimensional synthetic seismic modelling of the Moho reflection is carried out to constrain the internal structure of a crust–mantle transition. A reflectivity algorithm was used to calculate the synthetic seismograms for models of the crust–mantle transition consisting of gradient zones and layered sequences. The synthetic seismograms for models consisting of laterally variable velocity structure were calculated by an explicit finite difference wave equation algorithm. The laterally heterogeneous transition models can be achieved by assuming a Moho with different degrees of variable topography or laterally discontinuous layering. The effects of a low velocity surface cover on the seismic signature of deep signals is also analysed. The seismic modelling coupled with stacked images of wide-angle seismic reflection data constrain the case history of the Moho transition beneath the southern Urals. A smooth gradational crust–mantle transition (gradient velocity–depth function) does not predict the seismic features observed in the shot records. The coda of the PmP and the amplitude behaviour of this phase with offset are modelled with a 6 km thick transition zone which consists of approximately 600 m thick, laterally variable layers with a bimodal velocity distribution. The horizontal component stacks of the wide-angle data feature arcuate events suggesting boudin like structures or a boundary with topographic relief. High amplitude sub-Moho reflections support that the structural complexity of the crust–mantle transition can be followed to the upper mantle suggesting eastward dipping structures indicating, either, remnants of an old subduction zone, trapped crustal material within the mantle, or active crust–mantle interactions.

Seismic reflections from depths of less than two meters

Three distinct seismic reflections were obtained from within the upper 2.1 m of flood‐plain alluvium in the Arkansas River valley near Great Bend, Kansas. Reflections were observed at depths of 0.63, 1.46, and 2.10 m and confirmed by finite‐difference wave‐equation modeling. The wavefield was densely sampled by placing geophones at 5‐cm intervals, and near‐source nonelastic deformation was minimized by using a very small seismic impulse source. For the reflections to be visible within this shallow range, low seismic P‐wave velocities (<300 m/s) and high dominant‐frequency content of the data (∼450 Hz) were essential. The practical implementation of high‐resolution seismic imaging at these depths has the potential to complement ground‐penetrating radar (GPR), chiefly in areas where materials exhibiting high electrical conductivity, such as clays, prevent the effective use of GPR. Potential applications of these results exist in hydrogeology and environmental, Quaternary, and neotectonic geology.

Wave-Equation Velocity Analysis

Velocity sensitive (wide propagation angle) seismic data do not comply with the r.m.s. small propagation angle approximation. A hyperbolic velocity spectrum and Dix's equation cannot use wide-angle arrivals to estimate interval velocity accurately. The linear moveout method measures interval velocity exactly in stratified media. Snell midpoint coordinates are constructed to image the data before velocity estimation. Energy focuses at the arrival coordinates of a fixed reference Snell wavefront as required by the linear moveout method. The image of a common-midpoint seismic gather in Snell midpoint coordinates, for a nonvertical reference Snell wave, defines the wave-equation velocity spectrum. Two important properties of the velocity spectrum are locality, where energy is a local function of velocity; and linearity, that is invertible using linear transformations. Velocity sensitivity of wave-equation extrapolation operators increases with the angle of propagation. In Snell midpoint coordinates, angles are measured relative to an arbitrary slanted reference Snell wave. At this particular angle wave-equation operators are exact, independent of velocity. Approximations of the wave equation in Snell midpoint coordinates satisfactorily image wide-angle energy. To compute the velocity spectrum, we use the 15-degree finite-difference wave equation in the frequency domain. This equation is insensitive to the downward continuation velocity. This formation resolves multivalued, wide velocity data using in-homogeneous, offset and depth dependent, downward continuation velocity. Stepout filtering concurrent with downward continuation eliminates wide-angle propagation energy not modeled by the 15-degree wave equation.

Asymptotic Behavior of Solutions to the Finite-Difference Wave Equation

Asymptotic Behavior of Solutions to the Finite-Difference Wave Equation