Efficient FDTD algorithm for plane-wave simulation for vertically heterogeneous attenuative media

Geophysics ◽  
2007 ◽  
Vol 72 (4) ◽  
pp. H43-H53 ◽  
Author(s):  
Arash JafarGandomi ◽  
Hiroshi Takenaka
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Plane Wave ◽  
Wave Equations ◽  
Plane Waves ◽  
Computation Time ◽  
Staggered Grid ◽  
Seismic Profiles ◽  
Vertical Seismic Profiles ◽  
Incident Waves

We propose an efficient algorithm for modeling seismic plane-wave propagation in vertically heterogeneous viscoelastic media using a finite-difference time-domain (FDTD) technique. In the algorithm, the wave equation is rewritten for plane waves by applying a Radon transform to the 2D general wave equation. Arbitrary values of the quality factor for [Formula: see text]- and [Formula: see text]-waves ([Formula: see text] and [Formula: see text]) are incorporated into the wave equation via a generalized Zener body rheological model. An FDTD staggered-grid technique is used to numerically solve the derived plane-wave equations. The scheme uses a 1D grid that reduces computation time and memory requirements significantly more than corresponding 2D or 3D computations. Comparing the finite-difference solutions to their corresponding analytical results, we find that the methods are sufficiently accurate. The proposed algorithm is able to calculate synthetic waveforms efficiently and represent viscoelastic attenuation even in very attenuative media. The technique is then used to estimate the plane-wave responses of a sedimentary system to normal and inclined incident waves in the Kanto area of Japan via synthetic vertical seismic profiles.

Optimal staggered-grid finite-difference schemes based on the minimax approximation method with the Remez algorithm

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. T27-T42 ◽  
Author(s):  
Lei Yang ◽  
Hongyong Yan ◽  
Hong Liu
Keyword(s):  
Numerical Modeling ◽  
Finite Difference ◽  
Wave Equations ◽  
Computation Time ◽  
Taylor Series Expansion ◽  
Staggered Grid ◽  
Finite Difference Schemes ◽  
Objective Functions ◽  
Remez Algorithm ◽  
Minimax Approximation

Finite-difference (FD) schemes, especially staggered-grid FD (SFD) schemes, have been widely implemented for wave extrapolation in numerical modeling, whereas the conventional approach to compute the SFD coefficients is based on the Taylor-series expansion (TE) method, which leads to unignorable great errors at large wavenumbers in the solution of wave equations. We have developed new optimal explicit SFD (ESFD) and implicit SFD (ISFD) schemes based on the minimax approximation (MA) method with a Remez algorithm to enhance the numerical modeling accuracy. Starting from the wavenumber dispersion relations, we derived the optimal ESFD and ISFD coefficients by using the MA method to construct the objective functions, and solve the objective functions with the Remez algorithm. We adopt the MA-based ESFD and ISFD coefficients to solve the spatial derivatives of the elastic-wave equations and perform numerical modeling. Numerical analyses indicated that the MA-based ESFD and ISFD schemes can overcome the disadvantages of conventional methods by improving the numerical accuracy at large wavenumbers. Numerical modeling examples determined that under the same discretizations, the MA-based ESFD and ISFD schemes lead to greater accuracy compared with the corresponding conventional ESFD or ISFD scheme, whereas under the same numerical precision, the shorter operator length can be adopted for the MA-based ESFD and ISFD schemes, so that the computation time is further decreased.

scholarly journals A non-reflecting wave equation through directional wave-field suppression and its finite difference implementation

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Teun Schaeken ◽  
Leo Hoogerbrugge ◽  
Eric Verschuur
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Wave Field ◽  
Imaging Techniques ◽  
Constitutive Relations ◽  
Computation Time ◽  
Transmitted Wave ◽  
Staggered Grid ◽  
Extra Term ◽  
Preferred Direction

AbstractThe acoustic wave equation describes wave propagation directly from basic physical laws, even in heterogeneous acoustic media. When numerically simulating waves with the wave equation, contrasts in the medium parameters automatically generate all scattering effects. For some applications - such as propagation analysis or certain wave-equation based imaging techniques - it is desirable to suppress these reflections, as we are only interested in the transmitted wave-field. To achieve this, a modification to the constitutive relations is proposed, yielding an extra term that suppresses waves with reference to a preferred direction. The scale-factor $$\alpha$$ α of this extra term can either be interpreted as a penetration depth or as a typical decay time. This modified theory is implemented using a staggered-grid, time-domain finite difference scheme, where the acoustic Poynting-vector is used to estimate the local propagation direction of the wave-field. The method was successfully used to suppress reflections in media with bone tissue (medical ultrasound) and geophysical subsurface structures, while introducing only minor perturbations to the transmitted wave-field and a small increase in computation time.

Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Yanfei Wang ◽  
Wenquan Liang ◽  
Zuhair Nashed ◽  
Changchun Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Least Squares ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Spatial Domain ◽  
Variable Length ◽  
Staggered Grid ◽  
Regular Grid ◽  
Low Velocity

AbstractFinite difference (FD) solutions of wave equations have been proven useful in exploration seismology. To yield reliable and interpretable results, the numerically induced error should be minimized over a range of frequencies and angles of propagation. Grid dispersion is one of the key numerical problems and there exist some methods to solve this problem in the literature. Traditionally, the spatial FD operator coefficients are only determined in the spatial domain; however, the wave equation is solved in the temporal and spatial domain simultaneously. Recently, some methods based on the joint temporal-spatial domains have been proposed to address this problem. Variable length coefficients methods are proposed in the literature to improve efficiency while preserving accuracy by using longer operators in the low velocity regions and shorter operators in the high velocity regions. To cope with the ill-conditioning of the linear system induced by long stencil FD operators, we study in this paper a regularizing simplified least-squares model to minimize the phase velocity error in the joint temporal-spatial domain with a variable length of coefficients. Different from our previous study, we determine FD coefficients on the regular grid instead of on the staggered grid. Though the regular grid FD methods are less precise, however, with a little increase of the operator length, the precision can be improved. Stability of the numerical solutions is enhanced by the regularization. Numerical simulations made on one-dimensional to three-dimensional examples show that our scheme needs shorter operators and preserves accuracy compared with the previous methods.

Synthetic vertical seismic profiles for nonnormal incidence plane waves

Geophysics ◽  
1985 ◽  
Vol 50 (1) ◽  
pp. 127-141 ◽  
Author(s):  
F. Aminzadeh ◽  
J. M. Mendel
Keyword(s):  
Simple Procedure ◽  
Plane Waves ◽  
Space Representation ◽  
Elastic Model ◽  
S Wave ◽  
Seismic Profiles ◽  
Vertical Seismic Profiles ◽  
State Space Representation ◽  
Wave Velocities ◽  
Incidence Plane

Vertical seismic profiles (VSPs) are, by definition, recordings of seismic signals (total upgoing and downgoing seismic wave fields) at different depth points, usually at equally spaced intervals [Formula: see text], i = 1, 2, …, I. In a nonnormal incidence (NNI) elastic model, where each layer is described by thickness, density, and P- and S-wave velocities, the mapping between time and depth needed to generate synthetic VSPs is not usually straightforward. In this paper we develop a relatively simple procedure for generating synthetic vertical and horizontal direction plane wave NNI VSPs. No spatial discretization is necessary. We (1) compute two surface seismograms, one vertical and the other horizontal, exactly as described in Aminzadeh and Mendel (1982); and (2) downward continue the surface seismograms to fixed VSP depth points. This paper demonstrates an algorithm for downward continuation of an elastic wave field using state‐space representation and gives simulations which illustrate both z- and x-direction primaries and complete VSPs for different geologic models and different incident angles.

Time‐domain behavior of wide‐angle one‐way wave equations

Geophysics ◽  
1991 ◽  
Vol 56 (3) ◽  
pp. 382-384
Author(s):  
A. H. Kamel
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Source Function ◽  
Plane Waves ◽  
Wide Angle ◽  
Wave Processes ◽  
Inhomogeneous Wave ◽  
Dirac Delta ◽  
Inhomogeneous Wave Equation ◽  
Standard Tool

The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

Temporal high-order staggered-grid finite-difference schemes for elastic wave propagation

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T207-T224 ◽  
Author(s):  
Zhiming Ren ◽  
Zhen Chun Li
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
High Order ◽  
Elastic Wave Propagation ◽  
Staggered Grid ◽  
S Wave ◽  
Order Accuracy ◽  
Spatial Derivatives

The traditional high-order finite-difference (FD) methods approximate the spatial derivatives to arbitrary even-order accuracy, whereas the time discretization is still of second-order accuracy. Temporal high-order FD methods can improve the accuracy in time greatly. However, the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation. We have developed two temporal high-order staggered-grid FD (SFD) schemes for modeling elastic wave propagation. A new stencil containing the points on the axis and a few off-axial points is introduced to approximate the spatial derivatives. We derive the dispersion relations of the elastic wave equation based on the new stencil, and we estimate FD coefficients by the Taylor series expansion (TE). The TE-based scheme can achieve ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracy ([Formula: see text]). We further optimize the coefficients of FD operators using a combination of TE and least squares (LS). The FD coefficients at the off-axial and axial points are computed by TE and LS, respectively. To obtain accurate P-, S-, and converted waves, we extend the wavefield decomposition into the temporal high-order SFD schemes. In our modeling, P- and S-wave separation is implemented and P- and S-wavefields are propagated by P- and S-wave dispersion-relation-based FD operators, respectively. We compare our schemes with the conventional SFD method. Numerical examples demonstrate that our TE-based and TE + LS-based schemes have greater accuracy in time and better stability than the conventional method. Moreover, the TE + LS-based scheme is superior to the TE-based scheme in suppressing the spatial dispersion. Owing to the high accuracy in the time and space domains, our new SFD schemes allow for larger time steps and shorter operator lengths, which can improve the computational efficiency.

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