scholarly journals A note on Chebyshev polynomials and finite difference wave equation

2007 ◽  
pp. 57-60
Author(s):  
Anatolii Grinshpan
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Chebyshev Polynomials ◽  
Difference Wave

Implicit interpolation in reverse‐time migration

Geophysics ◽  
1997 ◽  
Vol 62 (3) ◽  
pp. 906-917 ◽  
Author(s):  
Jinming Zhu ◽  
Larry R. Lines
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Real Data ◽  
Seismic Sources ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Difference Wave ◽  
Time Migration ◽  
Seismic Acquisition ◽  
Recorded Data

Reverse‐time migration applies finite‐difference wave equation solutions by using unaliased time‐reversed recorded traces as seismic sources. Recorded data can be sparsely or irregularly sampled relative to a finely spaced finite‐difference mesh because of the nature of seismic acquisition. Fortunately, reliable interpolation of missing traces is implicitly included in the reverse‐time wave equation computations. This implicit interpolation is essentially based on the ability of the wavefield to “heal itself” during propagation. Both synthetic and real data examples demonstrate that reverse‐time migration can often be performed effectively without the need for explicit interpolation of missing traces.

scholarly journals ONE-STEP EXTRAPOLATION METHOD USING CHEBYSHEV’S RECURRENCE RELATION APPLIED TO SEISMIC MODELING

2016 ◽  
Vol 34 (4) ◽  
Author(s):  
Laura Lara Ortiz ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Chebyshev Polynomials ◽  
Finite Difference Scheme ◽  
Acoustic Wave Equation ◽  
Seismic Modeling ◽  
First Order ◽  
One Step

ABSTRACT. In this work we show that the solution of the first order differential wave equation for an analytical wavefield, using a finite-difference scheme in time, follows exactly the same recursion of modified Chebyshev polynomials. Based on this, we proposed a numerical...Keywords: seismic modeling, acoustic wave equation, analytical wavefield, Chebyshev polinomials. RESUMO. Neste trabalho, mostra-se que a solução da equação de onda de primeira ordem com um campo de onda analítico usando um esquema de diferenças finitas no tempo segue exatamente a relação de recorrência dos polinômios modificados de Chebyshev. O algoritmo...Palavras-chave: modelagem sísmica, equação da onda acústica, campo analítico, polinômios de Chebyshev.

A recipe for stability of finite‐difference wave‐equation computations

Geophysics ◽  
1999 ◽  
Vol 64 (3) ◽  
pp. 967-969 ◽  
Author(s):  
Larry R. Lines ◽  
Raphael Slawinski ◽  
R. Phillip Bording
Keyword(s):  
Inverse Problem ◽  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Numerical Stability ◽  
Short Note ◽  
Seismic Wave Propagation ◽  
Difference Wave ◽  
Von Neumann ◽  
The Stability

Finite‐difference solutions to the wave equation are pervasive in the modeling of seismic wave propagation (Kelly and Marfurt, 1990) and in seismic imaging (Bording and Lines, 1997). That is, they are useful for the forward problem (modeling) and the inverse problem (migration). In computational solutions to the wave equation, it is necessary to be aware of conditions for numerical stability. In this short note, we examine a convenient recipe for insuring stability in our finite‐difference solutions to the wave equation. The stability analysis for finite‐difference solutions of partial differential equations is handled using a method originally developed by Von Neumann and described by Press et al. (1986, p. 827–830).

Finite difference wave-equation illumination: a deep-water case study

2012 ◽  
Author(s):  
Jason Gardner ◽  
Cintia Lapilli ◽  
Chuyuan Xu ◽  
Alfonso González
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Deep Water ◽  
Difference Wave
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