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.

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.

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.

Simulation of a seismic survey with combined surface and in-mine data acquisition for imaging steeply dipping ore veins

2021 ◽  
Author(s):  
Olaf Hellwig ◽  
Stefan Buske
Keyword(s):  
Finite Difference ◽  
Message Passing Interface ◽  
Seismic Survey ◽  
Difference Operators ◽  
Mining District ◽  
Prestack Depth Migration ◽  
Data Set ◽  
Depth Migration ◽  
Difference Grid ◽  
Triangulated Surfaces

<p>The polymetallic, hydrothermal deposit of the Freiberg mining district in the southeastern part of Germany is characterised by ore veins that are framed by Proterozoic orthogneiss. The ore veins consist mainly of quarz, sulfides, carbonates, barite and flourite, which are associated with silver, lead and tin. Today the Freiberg University of Mining and Technology is operating the shafts Reiche Zeche and Alte Elisabeth for research and teaching purposes with altogether 14 km of accessible underground galleries. The mine together with the most prominent geological structures of the central mining district are included in a 3D digital model, which is used in this study to study seismic acquisition geometries that can help to image the shallow as well as the deeper parts of the ore-bearing veins. These veins with dip angles between 40° and 85° are represented by triangulated surfaces in the digital geological model. In order to import these surfaces into our seismic finite-difference simulation code, they have to be converted into bodies with a certain thickness and specific elastic properties in a first step. In a second step, these bodies with their properties have to be discretized on a hexahedral finite-difference grid with dimensions of 1000 m by 1000 m in the horizontal direction and 500 m in the vertical direction. Sources and receiver lines are placed on the surface along roads near the mine. A Ricker wavelet with a central frequency of 50 Hz is used as the source signature at all excitation points. Beside the surface receivers, additional receivers are situated in accessible galleries of the mine at three different depth levels of 100 m, 150 m and 220 m below the surface. Since previous mining activities followed primarily the ore veins, there are only few pilot-headings that cut through longer gneiss sections. Only these positions surrounded by gneiss are suitable for imaging the ore veins. Based on this geometry, a synthetic seismic data set is generated with our explicit finite-difference time-stepping scheme, which solves the acoustic wave equation with second order accurate finite-difference operators in space and time. The scheme is parallelised using a decomposition of the spatial finite-difference grid into subdomains and Message Passing Interface for the exchange of the wavefields between neighbouring subdomains. The resulting synthetic seismic shot gathers are used as input for Kirchhoff prestack depth migration as well as Fresnel volume migration in order to image the ore veins. Only a top mute to remove the direct waves and a time-dependent gain to correct the amplitude decay due to the geometrical spreading are applied to the data before the migration. The combination of surface and in-mine acquisition helps to improve the image of the deeper parts of the dipping ore veins. Considering the limitations for placing receivers in the mine, Fresnel volume migration as a focusing version of Kirchhoff prestack depth migration helps to avoid migration artefacts caused by this sparse and limited acquisition geometry.</p>

Offset‐domain pseudoscreen prestack depth migration

Geophysics ◽  
2002 ◽  
Vol 67 (6) ◽  
pp. 1895-1902 ◽  
Author(s):  
Shengwen Jin ◽  
Charles C. Mosher ◽  
Ru‐Shan Wu
Keyword(s):  
Heterogeneous Media ◽  
Fourier Transforms ◽  
Data Migration ◽  
Computationally Efficient ◽  
Lateral Velocity ◽  
Prestack Depth Migration ◽  
Depth Migration ◽  
Narrow Angle ◽  
Domain Phase ◽  
Wavefield Extrapolation

The double square root equation for laterally varying media in midpoint‐offset coordinates provides a convenient framework for developing efficient 3‐D prestack wave‐equation depth migrations with screen propagators. Offset‐domain pseudoscreen prestack depth migration downward continues the source and receiver wavefields simultaneously in midpoint‐offset coordinates. Wavefield extrapolation is performed with a wavenumber‐domain phase shift in a constant background medium followed by a phase correction in the space domain that accommodates smooth lateral velocity variations. An extra wide‐angle compensation term is also applied to enhance steep dips in the presence of strong velocity contrasts. The algorithm is implemented using fast Fourier transforms and tri‐diagonal matrix solvers, resulting in a computationally efficient implementation. Combined with the common‐azimuth approximation, 3‐D pseudoscreen migration provides a fast wavefield extrapolation for 3‐D marine streamer data. Migration of the 2‐D Marmousi model shows that offset domain pseudoscreen migration provides a significant improvement over first‐arrival Kirchhoff migration for steeply dipping events in strong contrast heterogeneous media. For the 3‐D SEG‐EAGE C3 Narrow Angle synthetic dataset, image quality from offset‐domain pseudoscreen migration is comparable to shot‐record finite‐difference migration results, but with computation times more than 100 times faster for full aperture imaging of the same data volume.

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.

Azimuth moveout for 3-D prestack imaging

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 574-588 ◽  
Author(s):  
Biondo Biondi ◽  
Sergey Fomel ◽  
Nizar Chemingui
Keyword(s):  
Integral Operator ◽  
Impulse Response ◽  
Constant Velocity ◽  
Computational Cost ◽  
Data Set ◽  
Depth Migration ◽  
Prestack Migration ◽  
Migration Operator ◽  
Time Space ◽  
Strong Curvature

We introduce a new partial prestack‐migration operator called “azimuth moveout” (AMO) that rotates the azimuth and modifies the offset of 3-D prestack data. Followed by partial stacking, AMO can reduce the computational cost of 3-D prestack imaging. We have successfully applied AMO to the partial stacking of a 3-D marine data set over a range of offsets and azimuths. When AMO is included in the partial‐stacking procedure, high‐frequency steeply dipping energy is better preserved than when conventional partial‐stacking methodologies are used. Because the test data set requires 3-D prestack depth migration to handle strong lateral variations in velocity, the results of our tests support the applicability of AMO to prestack depth‐imaging problems. AMO is a partial prestack‐migration operator defined by chaining a 3-D prestack imaging operator with a 3-D prestack modeling operator. The analytical expression for the AMO impulse response is derived by chaining constant‐velocity DMO with its inverse. Equivalently, it can be derived by chaining constant‐velocity prestack migration and modeling. Because 3-D prestack data are typically irregularly sampled in the surface coordinates, AMO is naturally applied as an integral operator in the time‐space domain. The AMO impulse response is a skewed saddle surface in the time‐midpoint space. Its shape depends on the amount of azimuth rotation and offset continuation to be applied to the data. The shape of the AMO saddle is velocity independent, whereas its spatial aperture is dependent on the minimum velocity. When the azimuth rotation is small (⩽20°), the AMO impulse response is compact, and its application as an integral operator is inexpensive. Implementing AMO as an integral operator is not straightforward because the AMO saddle may have a strong curvature when it is expressed in the midpoint coordinates. An appropriate transformation of the midpoint axes to regularize the AMO saddle leads to an effective implementation.

Three‐dimensional depth imaging with generalized screens: A salt body case study

Geophysics ◽  
2003 ◽  
Vol 68 (4) ◽  
pp. 1132-1139 ◽  
Author(s):  
Jérôme H. Le Rousseau ◽  
Henry Calandra ◽  
Maarten V. de Hoop
Keyword(s):  
Three Dimensional ◽  
Fourier Method ◽  
Seismic Survey ◽  
Tectonic Deformation ◽  
Data Set ◽  
Depth Imaging ◽  
Depth Migration ◽  
Velocity Models ◽  
The North ◽  
Zero Offset

We illustrate the performance of the generalized screen propagator on real seismic data for 3D zero‐offset and prestack depth imaging. We use TotalFinaElf's L7D data set from the North Sea, a 3D marine seismic survey that contained limited azimuthal coverage. The subsurface shows significant tectonic deformation, including an intrusive salt body in sedimentary sequences. A transformation to common azimuth is applied prior to the 3D prestack depth imaging procedure. We compare the performance of the generalized screen propagator with that of a hybrid phase shift plus interpolation (PSPI)/split‐step Fourier method. Three‐dimensional prestack results confirm the generalized screen method handles multipathing more accurately. Comparisons are also made with Kirchhoff migration results. The results differ mainly in the fine‐scale irregularities of the image and not in the wavefront set of the image. Using synthetic models of similar structure (the SEG/EAGE salt model), we further illustrate the importance of multipathing and multiple scattering. Overall, our results show that our wave‐equation approach produces better images than the Kirchhoff approach to prestack depth migration; we attribute this mainly to a more complete handling of wave diffraction in the generalized screen expansion, which becomes important in strongly heterogeneous and irregular velocity models such as the ones containing salt bodies.

Slowness‐weighted diffraction stack for migrating wide‐angle seismic data in laterally varying media

Geophysics ◽  
2004 ◽  
Vol 69 (4) ◽  
pp. 1046-1052 ◽  
Author(s):  
Harm J. A. Van Avendonk
Keyword(s):  
Seismic Data ◽  
Velocity Structure ◽  
Seismic Velocity ◽  
Image Space ◽  
Wide Angle ◽  
Data Set ◽  
Seismic Velocity Structure ◽  
Depth Migration ◽  
Angle Data ◽  
Reflecting Boundaries

Wide‐angle prestack depth migration is an important tool for studying the nature of reflecting boundaries in the earth's crust. The slowness‐weighted diffraction stack (SWDS) method has been used to incorporate both two‐way traveltime constraints and slowness information in the migration. For this purpose, traveltimes and apparent slownesses of reflected arrivals must be calculated in the image space. Earlier applications of SWDS required a 1D or gently varying seismic velocity structure to obtain these quantities by ray tracing in the image space. I show that the apparent slownesses can also be derived directly from one‐way traveltime maps using Fermat's principle. The SDWS is applied to an existing onshore–offshore wide‐angle data set, and the example shows that the method can be used to image detailed reflectivity structure at great depths.

Prestack depth migration by symmetric nonstationary phase shift

Geophysics ◽  
2002 ◽  
Vol 67 (2) ◽  
pp. 594-603 ◽  
Author(s):  
Robert J. Ferguson ◽  
Gary F. Margrave
Keyword(s):  
Phase Shift ◽  
Symmetric Operator ◽  
Heterogeneous Media ◽  
Synthetic Data ◽  
Velocity Model ◽  
Computational Effort ◽  
Prestack Depth Migration ◽  
Data Set ◽  
Depth Migration ◽  
Zero Offset

A new depth migration method suitable for heterogeneous media is presented. The well‐known phase shift plus interpolation (PSPI) method and the recently introduced nonstationary phase‐shift (NSPS) method are combined into a single symmetric operator with improved accuracy and stability and with similar computational effort. For prestack depth migration, the symmetric operator is used in a recursive wavefield extrapolation to compute incident and reflected wavefields at any desired depth, and the ratio of the incident and reflected wavefields at a particular depth is used to estimate seismic reflectivity. When the velocity model is made piecewise constant laterally, the symmetric extrapolation operator can be computed efficiently using ordinary phase‐shift extrapolation for a series of reference velocities and appropriate spatial windowing. Migration of the Marmousi synthetic data set by symmetric nonstationary phase shift (SNPS) provides an image that compares favorably with an image of the zero‐offset reflectivity derived from the Marmousi velocity model.

Sign in / Sign up

Export Citation Format

Share Document