Optimization of one‐way wave equations

Geophysics ◽  
1985 ◽  
Vol 50 (10) ◽  
pp. 1634-1637 ◽  
Author(s):  
Myung W. Lee ◽  
Sang Y. Suh
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Square Root ◽  
Full Wave

The theory of wave extrapolation is based on the square‐root equation or one‐way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square‐root equation represents waves propagating in one direction only.

scholarly journals 3-D SEISMIC MODELING AND DEPTH MIGRATION COMBINING THE EXTRAPOLATION OF UPGOING AND DOWNGOING WAVEFIELDS

2014 ◽  
Vol 32 (3) ◽  
pp. 497
Author(s):  
Gary Corey Aldunate ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Wave Equations ◽  
Computational Cost ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Seismic Modeling ◽  
Depth Level ◽  
Full Wave ◽  
And Migration

ABSTRACT. The 3-D acoustic wave equation is generally solved using finite difference schemes on the mesh which defines the velocity model. However, whennumerical solution of the wave equation is done by finite difference schemes, attention should be taken with respect to dispersion and numerical stability. To overcomethese problems, one alternative is to solve the wave equation in the Fourier domain. This approach is stabler and makes possible to separate the full wave equation inits unidirectional equations. Thus, the full wave equation is decoupled in two first order differential equations, namely two equations related to the vertical component:upgoing (-Z) and downgoing (+Z) unidirectional equations. Among the solution methods, we can highlight the Split-Step-Plus-Interpolation (SS-PSPI). This methodhas been proven to be quite adequate for migration problems in 3-D media, providing satisfactory results at low computational cost. In this work, 3-D seismic modelingis implemented using Huygens’ principle and an equivalent simulation of the full wave equation solution is obtained by properly applying the solutions of the twouncoupled equations. In this procedure, a point source wavefield located at the surface is extrapolated downward recursively until the last depth level in the velocityfield is reached. A second extrapolation is done in order to extrapolate the wavefield upwards, from the last depth level to the surface level, and at each depth level thepreviously stored wavefield (saved during the downgoing step) is convolved with a reflectivity model in order to simulate secondary sources. To perform depth pre-stackmigration of 3-D datasets, the decoupled wave equations were used and the same process described for seismic modeling is applied for the propagation of sources andreceivers wavefields. Thus, depth migrated images are obtained using appropriate image conditions: the upgoing and downgoing wavefields of sources and receiversare correlated and the migrated images are formed. The seismic modeling and migration methods using upgoing and downgoing wavefields were tested on simple 3-Dmodels. Tests showed that the addition of upgoing wavefield in seismic migration, provide better result and highlight steep deep reflectors which do not appear in theresults using only downgoing wavefields.Keywords: 3-D seismic modeling and migration, Upoing and downgoing wavefields, Split-Step Phase Shift Plus Interpolation method, Decoupled wave equations,One-Way equations.RESUMO. A equação da onda acústica tridimensional é normalmente resolvida usando-se esquemas de diferenças finitas sobre a malha que define o modelo develocidade. Entretanto, deve-se ter cuidado com a dispersão e a estabilidade numérica durante o processo de propagação da onda na malha. Uma outra alternativa, bastante eficiente de se resolver a equação completa da onda, é desacoplando-a em duas equações de onda unidirecionais no domínio transformado de Fourier (solução pseudo-espectral). Assim, a equação completa da onda é separada em duas equações diferenciais de primeira ordem relativa á componente vertical: equação da ondaascendente (-Z) e da onda descendente (+Z). Normalmente, a equação unidirecional é resolvida com diferentes ordens de aproximação. Entre esses métodos, podemos destacar o método “Split-Step-Plus-Interpolation” (SS-PSPI), que tem sido bastante adequado para problemas de migração em meios 3-D, fornecendo resultados aum baixo custo computacional. Neste trabalho, o modelamento sísmico 3-D foi implementado usando-se o princípio de Huygens com as duas equações de onda unidirecionais desacopladas. Com o objetivo de simular uma solução equivalente à solução da equação completa, uma fonte pontual localizada na superfície é extrapoladaem profundidade, de forma recursiva, até atingir o último nível de profundidade na malha do modelo de velocidades. Uma segunda extrapolação é realizada para extrapolar para cima o campo de onda, desde o último nível em profundidade até à superfície do modelo. Assim, os receptores localizados na superfície registram ocampo de onda ascendente, que trazem informações dos refletores em subsuperfície na forma de reflexões e difrações. Para realizar a migração pré-empilhamento em profundidade de dados 3-D, usando-se as equações de onda desacopladas, o mesmo procedimento descrito para o modelamento sísmico é utilizado para a propagação dos campos de onda de fontes e receptores. Imagens migradas são obtidas usando-se condições de imagem apropriadas, onde os campos de onda da fonte e dos receptores, descendente e ascendente, são correlacionados. Sobre modelos 3-D simples foram testados os métodos de modelamento e migração, levando em conta oscampos de onda ascendente e descendente. Ficando, assim, evidenciado que no método de migração sísmica, proposto aqui, a adição do campo de onda ascendente fornece um melhor resultado, ressaltando os refletores íngremes que não aparecem nos resultados utilizando-se apenas a extrapolação do campo de onda descendente.Palavras-chave: Migração e modelagem sísmica 3-D, Migração em duas etapas mais interpolação, equações de ondas unidirecionais.

Theory of true-amplitude one-way wave equations and true-amplitude common-shot migration

Geophysics ◽  
2005 ◽  
Vol 70 (4) ◽  
pp. E1-E10 ◽  
Author(s):  
Yu Zhang ◽  
Guanquan Zhang ◽  
Norman Bleistein
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Forward Modeling ◽  
Inversion Method ◽  
Wave Form ◽  
Wave Operators ◽  
Full Wave ◽  
Migration Imaging ◽  
Standard Wave ◽  
And Migration

One-way wave operators are powerful tools for forward modeling and migration. Here, we describe a recently developed true-amplitude implementation of modified one-way operators and present some numerical examples. By “true-amplitude” one-way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct. That is, ray theory applied to these equations yields the upward- and downward-traveling eikonal equations of the full wave equation, and the amplitude satisfies the transport equation of the full wave equation. The solutions of these equations are used in the standard wave-equation migration imaging condition. The boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation. When the full wave-form solutions are replaced by their ray-theoretic approximations, the imaging formula reduces to the common-shot Kirchhoff inversion formula. In this sense, the migration is true amplitude as well. On the other hand, this new method retains all of the fidelity features of wave equation migration. Computer output using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data must be collected from a single common-shot experiment. Multiexperiment data, such as common-offset data, cannot be used with this method as presently formulated.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Fractional wave equations with attenuation

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peter Straka ◽  
Mark Meerschaert ◽  
Robert McGough ◽  
Yuzhen Zhou
Keyword(s):  
Wave Equation ◽  
Power Law ◽  
Weak Solutions ◽  
Wave Equations ◽  
Inhomogeneous Media ◽  
Medical Ultrasound ◽  
Sound Waves ◽  
Fractional Wave Equations

AbstractFractional wave equations with attenuation have been proposed by Caputo [5], Szabo [28], Chen and Holm [7], and Kelly et al. [11]. These equations capture the power-law attenuation with frequency observed in many experimental settings when sound waves travel through inhomogeneous media. In particular, these models are useful for medical ultrasound. This paper develops stochastic solutions and weak solutions to the power law wave equation of Kelly et al. [11].

scholarly journals Depth-Extrapolation-Based True-Amplitude Full-Wave-Equation Migration from Topography

2021 ◽  
Vol 11 (7) ◽  
pp. 3010
Author(s):  
Hao Liu ◽  
Xuewei Liu
Keyword(s):  
Wave Equation ◽  
Partial Derivative ◽  
Model Experiment ◽  
Surface Velocity ◽  
Seismic Exploration ◽  
Initial Condition ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Full Wave ◽  
Time Migration

The lack of an initial condition is one of the major challenges in full-wave-equation depth extrapolation. This initial condition is the vertical partial derivative of the surface wavefield and cannot be provided by the conventional seismic acquisition system. The traditional solution is to use the wavefield value of the surface to calculate the vertical partial derivative by assuming that the surface velocity is constant. However, for seismic exploration on land, the surface velocity is often not uniform. To solve this problem, we propose a new method for calculating the vertical partial derivative from the surface wavefield without making any assumptions about the surface conditions. Based on the calculated derivative, we implemented a depth-extrapolation-based full-wave-equation migration from topography using the direct downward continuation. We tested the imaging performance of our proposed method with several experiments. The results of the Marmousi model experiment show that our proposed method is superior to the conventional reverse time migration (RTM) algorithm in terms of imaging accuracy and amplitude-preserving performance at medium and deep depths. In the Canadian Foothills model experiment, we proved that our method can still accurately image complex structures and maintain amplitude under topographic scenario.

On the existence of compressional MHD oscillations in an inhomogeneous magnetoplasma

1990 ◽  
Vol 44 (2) ◽  
pp. 361-375 ◽  
Author(s):  
Andrew N. Wright
Keyword(s):  
Magnetic Field ◽  
Wave Equation ◽  
Density Distribution ◽  
Cold Plasma ◽  
Wave Equations ◽  
Perturbation Field ◽  
Plasma Density Distribution ◽  
Background Magnetic Field ◽  
Transverse Fields ◽  
The Magnetic Field

In a cold plasma the wave equation for solely compressional magnetic field perturbations appears to decouple in any surface orthogonal to the background magnetic field. However, the compressional fields in any two of these surfaces are related to each other by the condition that the perturbation field b be divergence-free. Hence the wave equations in these surfaces are not truly decoupled from one another. If the two solutions happen to be ‘matched’ (i.e. V.b = 0) then the medium may execute a solely compressional oscillation. If the two solutions are unmatched then transverse fields must evolve. We consider two classes of compressional solutions and derive a set of criteria for when the medium will be able to support pure compressional field oscillations. These criteria relate to the geometry of the magnetic field and the plasma density distribution. We present the conditions in such a manner that it is easy to see if a given magnetoplasma is able to executive either of the compressional solutions we investigate.

Randomized HSS acceleration for full-wave-equation depth stepping migration

2014 ◽  
Author(s):  
Lina Miao* ◽  
Polina Zheglova ◽  
Felix J. Herrmann
Keyword(s):  
Wave Equation ◽  
Full Wave

scholarly journals Numerical treatment of the spatio-temporal electromagnetic beam-wave packet

2017 ◽  
Vol 68 (2) ◽  
pp. 109-116
Author(s):  
L’ubomír Šumichrast ◽  
Jaroslav Franek
Keyword(s):  
Numerical Simulation ◽  
Wave Equation ◽  
Wave Packet ◽  
Paraxial Approximation ◽  
Two Dimensional ◽  
Numerical Treatment ◽  
Beam Wave ◽  
Electromagnetic Beam ◽  
Full Wave ◽  
Spatio Temporal

Abstract Propagation of a two-dimensional spatio-temporal electromagnetic beam wave is analysed. In parabolic (paraxial) approximation the exact analytical results for a spatio-temporal Gaussian impulse can be obtained. For solution of the full wave equation the numerical simulation has to be used. The various facets of this simulation are discussed here.

scholarly journals Some arguments for the wave equation in Quantum theory

2021 ◽  
Vol 5 (1) ◽  
pp. 314-336
Author(s):  
Tristram de Piro ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Quantum Theory ◽  
Continuity Equation ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Atomic System ◽  
Balmer Series ◽  
Inertial Frames ◽  
The Stability

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

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