3D wavefield extrapolation with optimum split-step Fourier method

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. T95-T108 ◽  
Author(s):  
Linong Liu ◽  
Jianfeng Zhang
Keyword(s):  
Finite Difference ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Fourier Method ◽  
Detailed Comparison ◽  
Higher Order ◽  
Order Correction ◽  
Data Sets ◽  
Wide Angle ◽  
Depth Migration

A one-way propagator is proposed for more accurately modeling wide-angle wavefields in the presence of severe lateral variations of the velocity. The method adds a higher-order correction to improve the split-step Fourier method by directly designing a cascaded operator that matches the exact phase-shift operator of a varying velocity. Using an optimization scheme, the coefficients in the cascaded operator are determined according to the local velocity distribution and the prescribed angular range of wavefield propagation. The proposed algorithm is implemented alternately in spatial and wavenumber domains using fast Fourier transforms, as in the split-step Fourier and generalized-screen methods. This algorithm can achieve higher accuracy than the generalized-screen method for wide-angle wavefields, although the same numerical scheme is used with comparable computational cost. No extra error arises for the proposed algorithm when used for 3D wave propagation, in contrast to methods that introduce an implicit finite–difference higher-order correction to the split-step Fourier method, such as the Fourier finite difference (FFD) and wide-angle screen methods. A detailed comparison of the proposed one-way propagator with the split-step Fourier, generalized-screen, and FFD methods is presented. The 2D Marmousi and 3D SEG/EAEG overthrust data sets are used to test the prestack depth-migration schemes developed based on the proposed one-way propagators.

Optimum split-step Fourier 3D depth migration: Developments and practical aspects

Geophysics ◽  
2007 ◽  
Vol 72 (3) ◽  
pp. S167-S175 ◽  
Author(s):  
Jianfeng Zhang ◽  
Linong Liu
Keyword(s):  
Finite Difference ◽  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Computational Cost ◽  
Fourier Method ◽  
Approximate Algorithm ◽  
Wide Angle ◽  
Data Set ◽  
Depth Migration ◽  
Varying Coefficients

We present an efficient scheme for depth extrapolation of wide-angle 3D wavefields in laterally heterogeneous media. The scheme improves the so-called optimum split-step Fourier method by introducing a frequency-independent cascaded operator with spatially varying coefficients. The developments improve the approximation of the optimum split-step Fourier cascaded operator to the exact phase-shift operator of a varying velocity in the presence of strong lateral velocity variations, and they naturally lead to frequency-dependent varying-step depth extrapolations that reduce computational cost significantly. The resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast Fourier transforms (FFTs). The accuracy of the first-order approximate algorithm is similar to that of the second-order optimum split-step Fourier method in modeling wide-angle propagation through strong, laterally varying media. Similar to the optimum split-step Fourier method, the scheme is superior to methods such as the generalized screen and Fourier finite difference. We demonstrate the scheme’s accuracy by comparing it with 3D two-way finite-difference modeling. Comparisons with the 3D prestack Kirchhoff depth migration of a real 3D data set demonstrate the practical application of the proposed method.

scholarly journals On application of the finite-difference Padé approximation of the pseudo-differential parabolic equation to the tropospheric radio wave propagation problem

2020 ◽  
pp. 405-419
Author(s):  
М.С. Лытаев
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Electromagnetic Wave Propagation ◽  
Padé Approximation ◽  
Fourier Method ◽  
Geometric Theory ◽  
Wide Angle ◽  
Pade Approximation ◽  
Propagation Angle ◽  
Wave Propagation Problem

Рассматривается задача численного моделирования распространения электромагнитных волн в неоднородной тропосфере на основе широкоугольных обобщений метода параболического уравнения. Используется конечно-разностная аппроксимация Паде оператора распространения. Существенно, что в предлагаемом подходе указанная аппроксимация осуществляется одновременно по продольной и поперечной координатам. При этом допускается моделирование произвольного коэффициента преломления тропосферы. Метод не накладывает ограничений на максимальный угол распространения. Для различных условий распространения радиоволн проведено сравнение с методом расщепления Фурье и методом геометрической теории дифракции. Показаны преимущества предлагаемого подхода. This paper is devoted to the numerical simulation of electromagnetic wave propagation in an inhomogeneous troposphere. The study is based on the wide-angle generalizations of the parabolic wave equation. The finite-difference Padé approximation is used to approximate the propagation operator. It is important that, within the proposed approach, the Padé approximation is carried out simultaneously along with the longitudinal and transverse coordinates. At the same time, the proposed approach gives an opportunity to model an arbitrary tropospheric refractive index. The method does not impose restrictions on the maximum propagation angle. The comparison with the split-step Fourier method and the geometric theory of diffraction is discussed. The advantages of the proposed approach are shown.

An explicit, unconditionally stable, 15‐degree depth migration and modeling algorithm implemented in poststack, directional, and prestack modes

Geophysics ◽  
1991 ◽  
Vol 56 (9) ◽  
pp. 1412-1422
Author(s):  
Alvin K. Benson
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Three Dimensional ◽  
Inhomogeneous Media ◽  
Unconditional Stability ◽  
Data Sets ◽  
Unconditionally Stable ◽  
Depth Migration ◽  
Implicit Schemes ◽  
Modeling Algorithm

An explicit, unconditionally stable, finite‐difference depth migration and modeling algorithm is formulated and implemented for the fifteen‐degree wave equation in poststack, directional (rotational), and prestack modes for inhomogeneous media. It is about two times faster than implicit schemes. The simplicity, unconditional stability, and speed of the algorithm are appealing for numerous applications, especially prestack and three‐dimensional data sets.

Connection of segmented intensity data measured with a multiple-detector system for powder diffractometry

2005 ◽  
Vol 38 (5) ◽  
pp. 795-803 ◽  
Author(s):  
T. Ida
Keyword(s):  
Fourier Transforms ◽  
Least Squares Method ◽  
Fourier Method ◽  
Detector System ◽  
Data Sets ◽  
Intensity Data ◽  
Maximum Order ◽  
The Fourier Transform ◽  
Zno Powder ◽  
Convolution Method

A Fourier method to connect segmented intensity data measured with a multiple-detector system (MDS) for powder diffractometry has been developed. Differences in sensitivity, slight shifts in peak positions and asymmetric instrumental broadening for different detectors are simultaneously adjusted by a Fourier-based deconvolution/convolution method. The Fourier transform of the adjustment function is evaluated as the ratio of the Fourier transforms of the intensity data sets in the overlapped region measured with the adjacent detectors. Even and odd polynomial functions with maximum order of 10 are separately fitted to the real and imaginary parts of the experimentally evaluated Fourier-transformed adjustment by a least-squares method applying an appropriate weighting for the Fourier-transformed data. The complex of the optimized polynomials is used to adjust the data measured with a detector in order to connect them with the data measured with an adjacent detector. The method is applied to connect the powder diffraction data of ZnO powder measured with the MDS on beamline BL4B2at the Photon Factory in Tsukuba. Slight differences in line width, asymmetry and sharpness detected in the observed diffraction peak profiles measured with the different detectors have successfully been removed by the Fourier-based adjustment procedures.

Optimization of the parameters in complex Padé Fourier finite-difference migration

Geophysics ◽  
2017 ◽  
Vol 82 (3) ◽  
pp. S259-S269 ◽  
Author(s):  
Marco Salcedo ◽  
Amélia Novais ◽  
Jörg Schleicher ◽  
Jessé C. Costa
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Rotation Angle ◽  
Computational Cost ◽  
Wide Angle ◽  
Model Data ◽  
Data Set ◽  
Dependent Parameter ◽  
The One

Complex Padé Fourier finite-difference migration is a stable one-way wave-equation technique that allows for better treatment of evanescent modes than its real counterpart, in this way producing fewer artifacts. As for real Fourier finite-difference (FFD) migration, its parameters can be optimized to improve the imaging of steeply dipping reflectors. The dip limitation of the FFD operator depends on the variation of the velocity field. We have developed a wide-angle approximation for the one-way continuation operator by means of optimization of the Padé coefficients and the most important velocity-dependent parameter. We have evaluated the achieved quality of the approximate dispersion relation in dependence on the chosen function of the ratio between the model and reference velocities under consideration of the number of terms in the Padé approximation and the branch-cut rotation angle. The optimized parameters are chosen based on the migration results and the computational cost. We found that by using the optimized parameters, a one-term expansion achieves the highest dip angles. The implementations were validated on the Marmousi data set and SEG/EAGE salt model data.

Imaging salt bodies using explicit migration operators offshore Norway

Geophysics ◽  
2009 ◽  
Vol 74 (2) ◽  
pp. S25-S32 ◽  
Author(s):  
Børge Arntsen ◽  
Constantin Gerea ◽  
Tage Røsten
Keyword(s):  
Image Quality ◽  
Computational Cost ◽  
Fourier Method ◽  
Frequency Content ◽  
Data Set ◽  
Depth Migration ◽  
Kirchhoff Migration ◽  
Velocity Changes ◽  
The Cost ◽  
Better Than

We have tested the performance of 3D shot-profile depth migration using explicit migration operators on a real 3D marine data set. The data were acquired offshore Norway in an area with a complex subsurface containing large salt bodies. We compared shot-profile migration using explicit migration operators with conventional Kirchhoff migration, split-step Fourier migration, and common-azimuth by generalized screen propagator (GSP) migration in terms of quality and computational cost. Image quality produced by the explicit migration operator approach is slightly better than with split-step Fourier migration and clearly better than in common-azimuth by GSP and Kirchhoff migrations. The main differences are fewer artifacts and better-suppressed noise within the salt bodies. Kirchhoff migration shows considerable artifacts (migration smiles) within and close to the salt bodies, which are not present in images produced by the other three wave-equation methods. Expressions for computational cost were developed for all four migration algorithms in terms of frequency content and acquisition parameters. For comparable frequency content, migration cost using explicit operators is four times the cost of the split-step Fourier method, up to 260 times the cost of common-azimuth by GSP migration, and 25 times the cost of Kirchhoff migration. Our results show that in terms of image quality, shot-profile migration using explicit migration operators is well suited for imaging in areas with complex geology and significant velocity changes. However, computational cost of the method is high and makes it less attractive in terms of efficiency.

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