scholarly journals Stable wide‐angle Fourier finite‐difference downward extrapolation of 3‐D wavefields

Geophysics ◽  
2002 ◽  
Vol 67 (3) ◽  
pp. 872-882 ◽  
Author(s):  
Biondo Biondi
Keyword(s):  
Finite Difference ◽  
Continuation Method ◽  
Phase Error ◽  
Azimuthal Anisotropy ◽  
Laplacian Operator ◽  
Lateral Velocity ◽  
Data Set ◽  
Frequency Dependent ◽  
Reference Velocity ◽  
Velocity Variations

I present an unconditionally stable, implicit finite‐difference operator that corrects the constant‐velocity phase‐shift operator for lateral velocity variations. The method is based on the Fourier finite‐difference (FFD) method. Contrary to previous results, my correction operator is stable even when the medium velocity has sharp discontinuities, and the reference velocity is higher than the medium velocity. The stability of the new correction enables the definition of a new downward‐continuation method based on the interpolation of two wavefields: the first wavefield is obtained by applying the FFD correction starting from a reference velocity lower than the medium velocity; the second wavefield is obtained by applying the FFD correction starting from a reference velocity higher than the medium velocity. The proposed Fourier finite‐difference plus interpolation (FFDPI) method combines the advantages of the FFD technique with the advantages of interpolation. A simple and economical procedure for defining frequency‐dependent interpolation weight is presented. When the interpolation step is performed using these frequency‐dependent interpolation weights, it significantly reduces the residual phase error after interpolation, the frequency dispersion caused by the discretization of the Laplacian operator, and the azimuthal anisotropy caused by splitting. Tests on zero‐offset data from the SEG‐EAGE salt data set show that the FFDPI method improves the imaging of a fault reflection with respect to a similar interpolation scheme that uses a split‐step correction for adapting to lateral velocity variations.

True-amplitude one-way wave equation migration in the mixed domain

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S199-S209 ◽  
Author(s):  
Flor A. Vivas ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Numerical Experiments ◽  
Wave Equations ◽  
Laplacian Operator ◽  
Single Shot ◽  
Lateral Velocity ◽  
Data Set ◽  
Classical Formulation ◽  
Imaging Tool

One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

Seismic depth imaging with the Gabor transform

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. S91-S97 ◽  
Author(s):  
Yongwang Ma ◽  
Gary F. Margrave
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Gabor Transform ◽  
Positioning Error ◽  
Lateral Velocity ◽  
Data Set ◽  
Depth Migration ◽  
Velocity Variations ◽  
Spatial Positioning ◽  
Wavefield Extrapolation

Wavefield extrapolation in depth, a vital component of wave-equation depth migration, is accomplished by repeatedly applying a mathematical operator that propagates the wavefield across a single depth step, thus creating a depth marching scheme. The phase-shift method of wavefield extrapolation is fast and stable; however, it can be cumbersome to adapt to lateral velocity variations. We address the extension of phase-shift extrapolation to lateral velocity variations by using a spatial Gabor transform instead of the normal Fourier transform. The Gabor transform, also known as the windowed Fourier transform, is applied to the lateral spatial coordinates as a windowed discrete Fourier transform where the entire set of windows is required to sum to unity. Within each window, a split-step Fourier phase shift is applied. The most novel element of our algorithm is an adaptive partitioning scheme that relates window width to lateral velocity gradient such that the estimated spatial positioning error is bounded below a threshold. The spatial positioning error is estimated by comparing the Gabor method to its mathematical limit, called the locally homogeneous approximation — a frequency-wavenumber-dependent phase shift that changes according to the local velocity at each position. The assumption of local homogeneity means this position-error estimate may not hold strictly for large scattering angles in strongly heterogeneous media. The performance of our algorithm is illustrated with imaging results from prestack depth migration of the Marmousi data set. With respect to a comparable space-frequency domain imaging method, the proposed method improves images while requiring roughly 50% more computing time.

3-D moveout inversion in azimuthally anisotropic media with lateral velocity variation: Theory and a case study

Geophysics ◽  
1999 ◽  
Vol 64 (4) ◽  
pp. 1202-1218 ◽  
Author(s):  
Vladimir Grechka ◽  
Ilya Tsvankin
Keyword(s):  
Correction Term ◽  
Anisotropic Media ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Velocity Variation ◽  
Variation Theory ◽  
Fractured Reservoir ◽  
Lateral Velocity ◽  
Data Set ◽  
Zero Offset

Reflection moveout recorded over an azimuthally anisotropic medium (e.g., caused by vertical or dipping fractures) varies with the azimuth of the source‐receiver line. Normal‐moveout (NMO) velocity, responsible for the reflection traveltimes on conventional‐length spreads, forms an elliptical curve in the horizontal plane. While this result remains valid in the presence of arbitrary anisotropy and heterogeneity, the inversion of the NMO ellipse for the medium parameters has been discussed so far only for horizontally homogeneous models above a horizontal or dipping reflector. Here, we develop an analytic moveout correction for weak lateral velocity variation in horizontally layered azimuthally anisotropic media. The correction term is proportional to the curvature of the zero‐offset traveltime surface at the common midpoint and, therefore, can be estimated from surface seismic data. After the influence of lateral velocity variation on the effective NMO ellipses has been stripped, the generalized Dix equation can be used to compute the interval ellipses and evaluate the magnitude of azimuthal anisotropy (measured by P-wave NMO velocity) within the layer of interest. This methodology was applied to a 3-D “wide‐azimuth” data set acquired over a fractured reservoir in the Powder River Basin, Wyoming. The processing sequence included 3-D semblance analysis (based on the elliptical NMO equation) for a grid of common‐midpoint “supergathers,” spatial smoothing of the effective NMO ellipses and zero‐offset traveltimes, correction for lateral velocity variation, and generalized Dix differentiation. Our estimates of depth‐varying fracture trends in the survey area, based on the interval P-wave NMO ellipses, are in good agreement with the results of outcrop and borehole measurements and the rotational analysis of four‐ component S-wave data.

3D velocity-independent elliptically anisotropic moveout correction

Geophysics ◽  
2009 ◽  
Vol 74 (5) ◽  
pp. WB129-WB136 ◽  
Author(s):  
William Burnett ◽  
Sergey Fomel
Keyword(s):  
Field Data ◽  
Azimuthal Anisotropy ◽  
Three Dimensions ◽  
Velocity Analysis ◽  
Lateral Velocity ◽  
Analysis Techniques ◽  
West Texas ◽  
Imaging Approach ◽  
Velocity Variations ◽  
Seismic Images

Azimuthal anisotropy or lateral velocity variations cause azimuthal variations in moveout velocity, which can degrade seismic images if handled improperly. In cases in which apparent azimuthally anisotropic moveout is present, a single picked velocity is inadequate to flatten an event on a 3D CMP gather. Conventional velocity-analysis techniques require a significant amount of time and effort, especially in areas where apparent anisotropy is observed. We propose a velocity-independent imaging approach to perform an elliptically anisotropic moveout correction in three dimensions. The velocity-independent approach relies on volumetric local traveltime slopes rather than aggregate velocities and therefore provides an azimuthally flexible description of traveltime geometries throughout the gather. We derive theoretical expressions for extracting the moveout slowness matrix and the angle between the symmetry and acquisition axes as volumetric local attributes. A practical inversion scheme to extract the same parameters is also developed. These parameters are used to solve for moveout slowness as a function of azimuth. Tests on a synthetic common-midpoint (CMP) gather show accurate results for the automatic moveout correction and the inversion scheme. A field data example from west Texas illustrates the application of the automatic moveout correction as a residual moveout.

Prestack migration by split‐step DSR

Geophysics ◽  
1996 ◽  
Vol 61 (5) ◽  
pp. 1412-1416 ◽  
Author(s):  
Alexander Mihai Popovici
Keyword(s):  
Phase Shift ◽  
Square Root ◽  
Variable Velocity ◽  
Step Method ◽  
Lateral Velocity ◽  
Data Set ◽  
Prestack Migration ◽  
Imaging Results ◽  
Velocity Variations ◽  
Migration Equation

The double‐square‐root (DSR) prestack migration equation, though defined for depth variable velocity, can be used to image media with strong velocity variations using a phase‐shift plus interpolation (PSPI) or split‐step correction. The split‐step method is based on applying a phase‐shift correction to the extrapolated wavefield—a correction that attempts to compensate for the lateral velocity variations. I show how to extend DSR prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the Marmousi data set. The split‐step DSR migration is very fast and offers excellent imaging results.

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>

Survey design for coal-scale 3D-PS seismic reflection

Geophysics ◽  
2016 ◽  
Vol 81 (6) ◽  
pp. P57-P70 ◽  
Author(s):  
Shaun Strong ◽  
Steve Hearn
Keyword(s):  
Survey Design ◽  
High Amplitude ◽  
Azimuthal Anisotropy ◽  
Mine Planning ◽  
P Wave ◽  
Depth Range ◽  
Low Velocity ◽  
Converted Wave ◽  
Data Set ◽  
Economic Considerations

Survey design for converted-wave (PS) reflection is more complicated than for standard P-wave surveys, due to raypath asymmetry and increased possibility of phase distortion. Coal-scale PS surveys (depth [Formula: see text]) require particular consideration, partly due to the particular physical properties of the target (low density and low velocity). Finite-difference modeling provides a pragmatic evaluation of the likely distortion due to inclusion of postcritical reflections. If the offset range is carefully chosen, then it may be possible to incorporate high-amplitude postcritical reflections without seriously degrading the resolution in the stack. Offsets of up to three times target depth may in some cases be usable, with appropriate quality control at the data-processing stage. This means that the PS survey design may need to handle raypaths that are highly asymmetrical and that are very sensitive to assumed velocities. A 3D-PS design was used for a particular coal survey with the target in the depth range of 85–140 m. The objectives were acceptable fold balance between bins and relatively smooth distribution of offset and azimuth within bins. These parameters are relatively robust for the P-wave design, but much more sensitive for the case of PS. Reduction of the source density is more acceptable than reduction of the receiver density, particularly in terms of the offset-azimuth distribution. This is a fortuitous observation in that it improves the economics of a dynamite source, which is desirable for high-resolution coal-mine planning. The final-survey design necessarily allows for logistical and economic considerations, which implies some technical compromise. However, good fold, offset, and azimuth distributions are achieved across the survey area, yielding a data set suitable for meaningful analysis of P and S azimuthal anisotropy.

Including Lateral Velocity Variations Into True-Amplitude Wave-Equation Migration

2009 ◽  
Author(s):  
D. Amazonas ◽  
R. Aleixo ◽  
J. Schleicher ◽  
J. Costa ◽  
A. Novais ◽  
...  
Keyword(s):  
Wave Equation ◽  
Lateral Velocity ◽  
Amplitude Wave ◽  
Velocity Variations
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