Rapid expansion method (REM) for time‐stepping in reverse time migration (RTM)

2009 ◽  
Author(s):  
Reynam C. Pestana ◽  
Paul L. Stoffa
Keyword(s):  
Expansion Method ◽  
Rapid Expansion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Stepping ◽  
Time Migration

Using the rapid expansion method for accurate time-stepping in modeling and reverse-time migration

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. S177-S185 ◽  
Author(s):  
Ekkehart Tessmer
Keyword(s):  
Time Derivative ◽  
Expansion Method ◽  
Rapid Expansion ◽  
Numerical Dispersion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Step ◽  
Time Stepping ◽  
Time Migration ◽  
Spatial Derivatives

Reverse-time migration is based on seismic forward modeling algorithms, where spatial derivatives usually are calculated by finite differences or by the Fourier method. Time integration in general is done by finite-difference time stepping of low orders. If the spatial derivatives are calculated by high-order methods and time stepping is based on low-order methods, there is an imbalance that might require that the time-step size needs to be very small to avoid numerical dispersion. As a result, computing times increase. Using the rapid expansion method (REM) avoids numerical dispersion if the number of expansion terms is chosen properly. Comparisons with analytical solutions show that the REM is preferable, especially at larger propagation times. For reverse-time migration, the REM needs to be applied in a time-stepping manner. This is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time. This is not appropriate for reverse-time migration where the sources have different time histories. In reverse-time migration, it might be desirable to use the Poynting vector information to estimate opening angles to improve the quality of the image. In the solution of the wave equation, this requires that one calculates not only the pressure wavefield but also its time derivative. The rapid expansion method can be extended easily to provide this time derivative with negligible extra cost.

scholarly journals REVERSE TIME MIGRATION IN THE FREQUENCY DOMAIN BY THE RAPID EXPANSION METHOD

2017 ◽  
Vol 35 (4) ◽  
pp. 287
Author(s):  
Protásio Nery Andrade ◽  
Reynam Cruz Pestana ◽  
Daniel E. Revelo
Keyword(s):  
Frequency Domain ◽  
Expansion Method ◽  
Frequency Component ◽  
Rapid Expansion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
High Quality ◽  
Depth Migration ◽  
Time Migration ◽  
Seismic Images

ABSTRACT.  This paper proposes and describes the implementation of a new depth migration method in the frequency domain. The method, based in the reverse time migration (RTM) technique, extrapolates wavefields from the source and receivers to obtain migrated seismic images that are built directly into the frequency domain. In the proposed method, wavefields are propagated in the time domain and are then transformed into the frequency domain at each time extrapolation step through the discrete Fourier transform. Neither the forward nor backward wavefield is needed to be stored in memory or read from disk storage. To speed up the migration algorithm, the discrete Fourier transform kernel for each frequency is computed and salved before the time extrapolation procedure. At the imaging condition phase, both source and receiver wavefields are at the same frequency, so that, the construction of the image occurs by multiplying the forward source propagated wavefield with the backward propagated of the receivers wavefield for each frequency component. Subsequently, saving the source field at each step to later correlate it with the backpropagated receiver wave field, usually done in conventional RTM, becomes unnecessary. Nor is it necessary to invert a matrix for each frequency component, which is done in the migration technique that uses the Helmholtz equation solution in the frequency domain. Thus, the migration procedure in the frequency domain being proposed is more efficient from a computational point of view, and can also produce high quality migrated images as those produced by conventional RTM. The rapid expansion method (REM) is used for seismic forward modeling, which extrapolated data with good precision and free of numerical dispersion. Thus, with the transformed data at each step in the frequency domain, it is possible to construct high quality, in-depth seismic images at a lower computational cost. Moreover, this frequency domain migration with REM is an atractive strategy to design robust inverse algorithms, especially for 3D problems. To demonstrate the efficiency and applicability of the proposed method, two synthetic models were used and their results showed high quality images equivalent to those obtained by conventional RTM and thus proving the vality of the method. Keywords: wave equation migration, depth migration, imaging condition, frequency domain migration. RESUMO. Um método de migração em profundidade no domínio da frequência é proposto e implementado. O método consiste na extrapolação dos campos de ondas da fonte e dos receptores e baseia-se na técnica de migração reversa no tempo (da sigla em inglês, RTM), obtendo imagens sísmicas migradas, construídas diretamente no domínio da frequência. No método que estamos propondo, os campos de ondas são propagados no domínio do tempo e a cada passo de extrapolação são transformados para o domínio da frequência, através da transformada de Fourier discreta (do inglês, on-the-fly transform). Para acelerar o algoritmo de migração, o kernel da transformada de Fourier é calculado fora do loop do tempo. Além disso, na etapa de condição da imagem, os campos de onda, tanto da fonte como dos receptores, são calculados no mesmo instante de tempo, ou seja, a construção da imagem se dá através da multiplicação do campo de onda da fonte com o campo retropropagado dos receptores, para cada componente de frequência. Portanto, não precisamos salvar o campo da fonte a cada passo no tempo para posteriormente correlacionar com o campo de onda retropropagado dos receptores, como é usualmente feito na RTM convencional, nem é preciso inverter uma matriz para cada componente de frequência, como é realizado normalmente pela técnica de migração no domínio da frequência, utilizando a solução da equação de Helmholtz. Desta forma, o procedimento de migração no domínio da frequência que estamos propondo se torna mais eficiente do ponto de vista computacional, podendo produzir imagens migradas de alta qualidade, quando comparadas às obtidas através da RTM convencional no domínio do tempo. Para a extrapolação dos campos de ondas no tempo foi empregado o método de expansão rápida (da sigla em inglês, REM), que permite a extrapolação dos dados com boa precisão e livres de dispersão numérica. Desta forma, com os dados transformados para o domínio da frequência, a cada passo no tempo, é possível a construção de imagens sísmicas em profundidade de boa qualidade e a um menor custo computacional. Para demonstrar a eficiência e aplicabilidade do método proposto, dois modelos sintéticos foram usados e seus resultados apresentaram imagens de alta qualidade equivalentes às obtidas pela RTM convencional. Palavras-chave: equação de migração da onda, migração, condição de imagem, migração no domínio da frequência.

An improvement in wavefield extrapolation and imaging condition to suppress reverse time migration artifacts

Geophysics ◽  
2017 ◽  
Vol 82 (6) ◽  
pp. S403-S409 ◽  
Author(s):  
Farzad Moradpouri ◽  
Ali Moradzadeh ◽  
Reynam Pestana ◽  
Reza Ghaedrahmati ◽  
Mehrdad Soleimani Monfared
Keyword(s):  
Weighting Function ◽  
Real Data ◽  
Expansion Method ◽  
Rapid Expansion ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Data Set ◽  
Imaging Condition ◽  
Time Migration ◽  
Poynting Vectors

Reverse time migration (RTM) as a full wave equation method can image steeply dipping structures incorporating all waves without dip limitation. It causes a set of low-frequency artifacts that start to appear for reflection angles larger than 60°. These artifacts are known as the major concern in RTM method. We are first to attempt to formulate a scheme called the leapfrog-rapid expansion method to extrapolate the wavefields and their first derivatives. We have evaluated a new imaging condition, based on the Poynting vectors, to suppress the RTM artifacts. The Poynting vectors information is used to separate the wavefields to their downgoing and upgoing components that form the first part of our imaging condition. The Poynting vector information is also used to calculate the reflection angles as a basis for our weighting function as the second part of the aforementioned imaging condition. Actually, the weighting function is applied to have the most likely desired information and to suppress the artifacts for the angle range of 61°–90°. This is achieved by dividing the angle range to a triplet domain from 61° to 70°, 71° to 80°, and 81° to 90°, where each part has the weight of [Formula: see text], [Formula: see text], and [Formula: see text], respectively. It is interesting to note that, besides suppressing the artifacts, the weighting function also has the capability to preserve crosscorrelation from the real reflecting points in the angle range of 61°–90°. Finally, we tested the new RTM procedure by the BP synthetic model and a real data set for the North Sea. The obtained results indicate the efficiency of the procedure to suppress the RTM artifacts in producing high-quality, highly illuminated depth-migrated image including all steeply dipping geologic structures.

Time evolution of the wave equation using rapid expansion method

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. T121-T131 ◽  
Author(s):  
Reynam C. Pestana ◽  
Paul L. Stoffa
Keyword(s):  
Wave Equation ◽  
Time Evolution ◽  
Time Derivative ◽  
Expansion Method ◽  
Pressure Response ◽  
Rapid Expansion ◽  
Reverse Time ◽  
Step Method ◽  
Time Migration ◽  
Spatially Varying

Forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields. For the case of spatially varying velocity, we have worked on two approaches to evaluate the time evolution of seismic wavefields. An exact solution for the constant-velocity acoustic wave equation can be used to simulate the pressure response at any time. For a spatially varying velocity, a one-step method can be developed where no intermediate time responses are required. Using this approach, we have solved for the pressure response at intermediate times and have developed a recursive solution. The solution has a very high degree of accuracy and can be reduced to various finite-difference time-derivative methods, depending on the approximations used. Although the two approaches are closely related, each has advantages, depending on the problem being solved.

Phase‐shift time‐stepping for reverse‐time migration

2008 ◽  
Author(s):  
Ben D. Wards ◽  
Gary F. Margrave ◽  
Michael P. Lamoureux
Keyword(s):  
Phase Shift ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Stepping ◽  
Time Migration
Sign in / Sign up

Export Citation Format

Share Document