Conventional Wave Equation Research Articles

Seismic Waveform Tomography for 3D Impedance Model with Salt Structure

Conventional impedance inversion from post-stack zero-offset seismic data is usually based on the convolution model, and wave-equation based inversion is rarely used, although it is capable to precisely describe seismic wave propagation and invert impedance model with higher resolution. That is because there are more than one physical parameters involved in the conventional wave equation, making impedance inversion complicated. In this study, a one-dimensional (1D) wave equation, containing only the impedance parameter, is adopted and applied for the inversion of 1D impedance model by seismic waveform tomography. However, for a three-dimensional (3D) model, direct application of the 1D waveform tomography may lead to lateral discontinuities. Therefore, we propose to utilize a truncated Fourier series to parameterize the 3D impedance model, and then invert for the Fourier coefficients. With this parameterization scheme, the large- and small-scale components of the impedance model can be reconstructed stepwise by gradually increasing the number of Fourier coefficients. To efficiently and effectively invert the coefficients for the 3D model with salt structure, we propose a joint strategy, in which the low-frequency seismic data is used to invert for the Fourier coefficients representing the large-scale components of the model, while the high-frequency seismic data is applied to invert for the Fourier coefficients representing the small-scale components of the model. Tests on a 3D impedance model with salt structure result in models with high resolution and good spatial continuity, proving the feasibility and stability of the impedance inversion procedure.

A new qP-wave approximation in tilted transversely isotropic media and its reverse time migration for areas with complex overburdens

During seismic imaging in anisotropic media, undesired SV waves may be detected when conventional wave equations are used for wavefield simulation, which will contaminate the final seismic image. A pure qP-wave equation that is free of S-wave artifacts is desired, such that crosstalk introduced from the interference of different waves can be eliminated. We have developed a new fully decoupled P-wave equation in transversely isotropic media with titled symmetry axis that avoids unwanted energy during migration. The qP and qSV phase velocities for the new approximations are plotted and compared with the exact solution and some other approximations and indicate high accuracy for various anisotropic models (from low to high degrees of anisotropy). During wavefield simulation, the finite-difference and pseudospectral methods also are combined to accelerate the computation. Moreover, a hybrid perfectly matched layer (H-PML) that combines two different perfectly matched layers for second-order wave equations in the wavenumber domain also is proposed and applied to eliminate the artificial boundary reflections. Comparisons of different absorbing boundary layers also are illustrated to validate the wavenumber domain H-PML. Finally, the new proposed wave approximation together with the new absorbing layer is further incorporated in a new 3D anisotropic reverse time migration (RTM). We test the RTM on the 3D synthetic SEG Advanced Modeling Corporation data and offshore field data with a complicated overburden (salt and gas overburden) and obtain seismic images with high resolution. Benchmarks against commercial implementations also are demonstrated, which proves that the positioning of the reflectors is more reliable and accurate with the new algorithm.

A rectangular-grid lattice spring model for modeling elastic waves in Poisson’s solids

The lattice spring model (LSM) combined with the velocity Verlet algorithm is a newly developed scheme for modeling elastic wave propagation in solid media. Unlike conventional wave equations, LSM is established on the basis of micromechanics of the subsurface media, which enjoys better dynamic characteristics of elastic systems. We develop a rectangular-grid LSM scheme for elastic waves simulation in Poisson’s solids, and the direction-dependent elastic constants are deduced to keep the isotropy of the discrete grid. The stability condition and numerical dispersion properties of LSM are discussed and compared with other numerical methods. The 2D and 3D numerical simulations are carried out using the rectangular-grid LSM, as well as the second- and fourth-order accuracy staggered finite-difference method (FDM). Wavefields obtained by LSM are fairly similar with those by analytical solution and FDM, which demonstrates the correctness of the proposed scheme and its capability of modeling elastic wave propagation in heterogeneous media. Moreover, we perform plane P-wave simulation through a semi-infinite cavity model via LSM and FDM, the recorded wavefield snapshots indicate that our proposed rectangular-grid LSM obtains more reasonable wavefield details compared with those of FDM, especially in media with high compliance and structure complexity. Our main contribution lies in offering an alternative simulation scheme for modeling elastic wave propagation in media with some kinds of complexities, which conventional FDM may fail to simulate.

A data-mapping approach to 3-D wavefield redatuming

SUMMARY In case of a complex overburden, the seismic data can be greatly improved by applying a full wavefield redatuming procedure. In practice, the application of the redatuming process to 3-D data acquired by conventional acquisition designs is non-trivial. Because of the large amount of data involved in the 3-D redatuming process and because of the sparseness of these data, it is impossible to apply conventional wave equation datuming directly. We present a data mapping approach to redatuming (DMR), which follows the concept of Kirchhoff data mapping. A simplified background medium where no ray bending occurs is assumed for the medium below the datum in order to map an input data set referenced to the acquisition surface to an output data set referenced to the new datum level. The DMR method can be interpreted as a simplified version of the Kirchhoff summation redatuming (KSR) method, where one of the 2-D integrals over the acquisition coordinates can be solved analytically. Consequently, in this approach fewer traces are involved in the computation of one time sample (a 2-D integral is computed instead of a 4-D integral), which makes it particularly attractive for the application to 3-D data sets. In this paper the theory underlying data mapping redatuming is discussed and the proposed approach is tested on fully sampled 2-D and 3-D synthetic data from models with both simple and complex velocity distributions in the subsurface. The tests clearly show that the objective of producing results that are comparable to the conventional KSR has been achieved. The redatumed traces are dynamically and kinematically correct. Furthermore, these results confirm that the dependency of the new approach on the assumed medium below the datum level is, indeed, weak because the assumption of a velocity medium where no ray bending occurs is already sufficient to produce correct results.

Anti-aliasing by interpolation in pre-stack wave-equation migration

Spatial aliasing is an important issue in seismic migration. In this paper, we discuss anti-aliasing schemes in the wavefield continuation which is a wave-equation-based migration method. We describe the difference in interpolation applied pre and post the imaging condition, and compare the effectiveness of the linear interpolation and the sinc function interpolation. Our conclusion is that a five-sample sinc function interpolation applied before the imaging condition is an optimal anti-aliasing scheme according to theoretical and application analysis. We then apply the 3D Born approximation pre-stack depth migration method with optimal anti-aliasing interpolation to a 80 km2 seismic dataset. The migration result shows that the method we propose is better than a conventional wave equation method (not applied anti-spatial aliasing) in the definition of reflection events.

Wave propagationfaster than light

It is shown that waves obeying the conventional wave equation∇2u = (1/c2)(∂2u/∂t2) do notnecessarily travel with a phase velocity c. The case of sphericalwaves proceeding from the origin is analysed. For non-sphericallysymmetric waves, the wave speed is greater than c at distancesmuch shorter than a wavelength. This applies in particular to thefields from oscillating electric and magnetic dipoles.It is shown, however, that a signal in the form of a steptravels with speed c.

A continuum method of pile driving analysis: Comparison with the wave equation method

What may be called a ‘Continuum’ method of finite element analysis is used in this paper to predict the behaviour of a pile during driving. In this both the pile and the soil are treated as two distinct parts of the same solid continuum, but with different properties. The behaviour of the soil medium, assumed to be semi-infinite and nonlinear, is represented by the hyperbolic stress-strain relationship. Discretizing the pile-soil system in turn by conventional axi-symmetric and mapped finite elements, the problem is solved in the time-domain using the central difference scheme. The example considered is that of a fully embedded steel pipe pile for which both field test data and Wave Equation solution are available. Results show that: the Continuum Method is capable of a greater degree of accuracy than the conventional Wave Equation Method, but it is far more expensive than the latter in terms of computational effort needed; the effects of radiation damping and wave dispersion in the soil are found to be small; and the mapped finite elements give significantly better results than conventional elements.

Transient displacement and stress in non-homogeneous elastic shells

Transient displacement and stress in thick inhomogeneous cylindrical and spherical shells is investigated for situations when the surfaces are subjected to dynamic loads consistent with the production of radial vibrations. In the first instance, transformation of the governing equations is sought to a form amenable to solution via a finite Hankel transform technique. Such reduction is available for a broad class of radial variations in the material density and elastic parameters. However, the solutions generated by this method are rather involved and two other approaches are indicated. Thus, reduction of the system to the conventional wave equation is obtained under certain restrictions on the nature of the inhomogeneities and a simple correspondence principle is presented for the imposed boundary stress boundary value problem. Finally an asymptotic wave front analysis is presented which has a wide generality of application.

Large wavelength pulse propagation in curved elastic rods

Longitudinal pulse propagation in slightly curved elastic rods is discussed in terms of an elementary theory due to Hsieh and Lee for which rotary inertia and radial shear are neglected. Moreover, the wavelength of motion is sufficiently long for practically all bending effects to be neglected. Three different approaches are presented. In the first, reduction of the Fourier transform of the governing equation to Bessel’s equation is sought. In the second, generalized Baecklund‐type transformations are utilized to reduce the characterizing equation to the conventional wave equation for certain radius of curvature variations. Finally, the formal Karal–Keller technique is used to solve certain impact problems. Numerical results are presented in graphical form.

Analytic solution of the linearized shallow-water wave equations for certain continuous depth variations

AbstractThe linear long-wave equations with (and without) small ground motion are considered. The governing equations are represented in a matrix from and transformations are sought which reduce the system to (for example) a form associated with the conventional wave equation. Integration of the system is then immediate. It is shown that such a reduction may be acheived provided the variation in water depth is specified by certain multi-parameter forms.

Wave propagation in fluid-filled elastic tubes

A system of equations descriptive of wave propagation in fluid-filled elastic tubes is examined with a view to achieving its reduction to a canonical form associated with the conventional wave equation. Two specific solutions are displayed and these are then used to consider a specific boundary value problem.

Wave Propagation in Inhomogeneous Elastic Media with (N + l)-Dimensional Spherical Symmetry

Uniform loading of an (N + 1)-dimensional spherically symmetric inhomogeneous elastic solid is investigated. The governing equations are represented in a matrix form and reduction to the conventional wave equation is sought. Such reduction may be achieved for multiparameter forms of a certain function involving the density and elastic parameters of the material. The reduction to the wave equation allows certain initial/boundary value problems to be readily solved.

A reexamination of unified field theory

We study a nonlinear field theory with a nonsymmetric Γ jk i which has the property that all tensor functions of g ij , e i α , Γ jk i , and ∂ k are treated in a uniform manner. All scalar functions of g ij , e i α , Γ jk i and ∂ k are constants in such a theory. If the theory is to be meaningful, it is necessary for a set of consistency conditions to be satisfied. We, then, show that nontrivial solutions to these consistency equations do exist. We describe, in another paper, results of a finite difference approximation to the field equations which we have obtained using a computer. We show that the field equations are asymptotic to the conventional wave equation. We find particle-like behavior can be made to appear at an origin point. The theory does not involve any arbitrary functions. Thus, there is less arbitrariness in this theory as compared with other hyperbolic type theories.