Guojian Shan

Prestack imaging of overturned reflections by reverse time migration

SUMMARY We present a simple method for computing angle-domain Common Image Gathers (CIGs) using prestack reverse time migration. The proposed method is an extension of the method proposed by Rickett and Sava (2001) to compute CIGs by downward-continuation shot-profile migration. We demonstrate with a synthetic example the use of the CIG gathers for migration velocity updating. A challenge for imaging both overturned and prismatic reflections is the discrimination of the reflection generated on either side of interfaces. We show how the propagation direction of the reflections can be determined by evaluating the crosscorrelation of the source wavefield with the receiver wavefield at time lags different than zero. Reflections can be easily separated once their direction of propagation is determined. We demonstrate the method by imaging overturned events generated by a segment of dipping reflector immersed in a vertically layered medium. We also applied the method to a North Sea data set with overturned events. The results of reverse time prestack migration are superior to the one obtained by a downward-continuation migration, and the CIGs obtained by applying the proposed method provide useful information for velocity updating.

Source-receiver migration of multiple reflections

Multiple reflections are usually considered to be noise and m any methods have been developed to attenuate them. However, similarly to primary reflec tions, multiple reflections are created by subsurface reflectors and contain their reflectiv ity information. We can image surface related multiples, regarding the corresponding pr imaries as the sources. Traditional source-receiver migration assumes that the source i s an impulse function. I generalize the source-receiver migration for arbitrary sources , and apply it to the migration of multiple reflections. A complex synthetic dataset is used to test the theory. Results show that my multiple migration algorithm is effective for imaging the multiple-contaminated data.

Imaging overturned waves by plane‐wave migration in tilted coordinates

We image overturned waves by decomposing the source and receiver wavefields into plane-waves. For each plane-wave, we extrapolate the source and receiver wavefields in a tilted coordinate system and cross-correlate them to obtain Common Image Gathers (CIGs). The tilting angle for the coordinate system is determined by the propagation direction of the plane wave. In tilted coordinates, the propagation direction is close to the extrapolation direction, so we can image steeply dipping reflectors and overturned waves with the one-way wave equation. We can also obtain robust, dip-dependent angle-domain CIGs (ADCIGs) by the same method as the one employed in reverse-time migration. These gathers provide moveout information and thus they are very important for velocity analysis on steeply dipping reflectors. Since our method is based on the one-way wave equation and it needs no padding for imaging overturned waves as needed by reverse-time migration, and so it is very efficient. We demonstrate our method on the Marmousi model by computing the Green’s function for a point source on the surface. We also apply our method on a North Sea real dataset with overturned events.

Plane-wave migration in tilted coordinates

We have developed a plane-wave migration method that efficiently images steeply dipping reflectors using one-way wavefield extrapolation. The recorded surface data are converted to plane-wave source data by slant stacking. The data set corresponding to each plane-wave source is migrated independently in a tilted coordinate system, with the extrapolation direction determined by the initial propagation direction of the plane wave at the surface. Waves illuminating steeply dipping reflectors, such as overturned waves and waves traveling nearly horizontally, are extrapolated accurately in an appropriate tilted coordinate system because the extrapolation direction is close to the propagation directions for these waves. Two-dimensional impulse responses and synthetic data examples demonstrate that plane-wave migration in tilted coordinates generates high-quality images of steeply dipping reflectors, particularly rugose salt tops and steep salt flanks.

Optimized implicit finite-difference and Fourier finite-difference migration for VTI media

Propagation velocity of seismic waves in heterogeneous VTI media depends not only on spatial location but also on their propagation direction, which leads to a much more complex dispersion relation than in isotropic media. As a result, designing implicit finite-difference (FD) schemes for wavefield extrapolation in anisotropic media through analytic Taylor-series expansion is more difficult. Implicit FD and Fourier finite-difference (FFD) schemes are developed for vertical transversely isotropic (VTI) media based on function fitting. The dispersion relation of VTI media is approximated with a rational function and its coefficients are estimated by weighted least-squares optimization. Because these coefficients are functions of Thomsen anisotropy parameters ( e and δ ) and vary laterally in heterogeneous VTI media, they are calculated before wavefield extrapolation and stored in a table. Implicit FD and FFD schemes for VTI media are almost the same as for isotropic media, except that coefficients are looke...

3D Wavefield Extrapolation In Laterally-varying Tilted TI Media

A new wavefield extrapolation method has been developed that allows the propagation of waves in an anisotropic medium. The anisotropic medium considered here is transversely isotropic (TI) with an axis of symmetry. Our method applies an asymmetric explicit correction filter after the normal isotropic extrapolation operator. It is stable and suitable for laterally varying TI media. This new scheme is useful to extrapolate wavefields in a vertical transversely isotropic (VTI) medium in tilted coordinates. The explicit correction operator, designed by a weighted least-square method, is stable and accurate for the desired wavenumbers. Impulse responses from this scheme and the anisotropic phase-shift method are compared to illustrate the algorithm.

Optimized Implicit Finite-difference Migration For TTI Media

I develop an implicit finite-difference migration method for vertical transversely isotropic (VTI) media with laterally varying anisotropy parameters. I approximate the dispersion relation of VTI media with a rational function series, the coefficients of which are estimated by least-squares optimization. These coefficients are functions of Thomsen anisotropy parameters. They are calculated and stored in a table before the wavefield extrapolation. The implicit finite-difference scheme for VTI media is almost the same as that of the isotropic media, except that the coefficients are derived from the pre-calculated table. In the 3D case, a phase-correction filter is applied after the finitedifference operator to eliminate the numerical-anisotropy error caused by two-way splitting. This finite-difference operator for VTI media is accurate to 60 for the 2nd order approximation and 80 for the 4th order approximation. Its computational cost is almost the same as the isotropic migration. I apply this method to a 2D synthetic dataset to validate the method.

Interpolation of near offsets using multiples and prediction-error filters

Reflection seismic data typically are undersampled. Missing near offsets can be interpolated in reflection seismic data with pseudoprimaries, generated by crosscorrelating multiples and primaries in incomplete recorded data. These pseudoprimary data can be generated at the missing near offsets but contain many artifacts, so it is undesirable simply to replace the missing data with the pseudoprimaries. A nonstationary prediction-error filter (PEF) can instead be estimated from the pseudoprimaries and used to interpolate missing data to produce an interpolated output that is superior to direct substitution of the pseudoprimaries into the missing offsets. This approach is applied successfully to 2D synthetic and field data. Limitations in conventional acquisition geometry limit this approach in 3D, which can be illustrated using a synthetic data set.

Prestack wave-equation depth migration in elliptical coordinates

We extend Riemannian wavefield extrapolation (RWE) to prestack migration using 2D elliptical-coordinate systems. The corresponding 2D elliptical extrapolation wavenumber introduces only an isotropic slowness model stretch to the single-square-root operator. This enables the use of existing Cartesian finite-difference extrapolators for propagating wavefields on elliptical meshes. A poststack migration example illustrates advantages of elliptical coordinates for imaging turning waves. A 2D imaging test using a velocity-benchmark data set demonstrates that the RWE prestack migration algorithm generates high-quality prestack migration images that are more accurate than those generated by Cartesian operators of the equivalent accuracy. Even in situations in which RWE geometries are used, a high-order implementation of the one-way extrapolator operator is required for accurate propagation and imaging. Elliptical-cylindrical and oblate-spheroidal geometries are potential extensions of the analytical approach to 3D RWE-coordinate systems.

3D plane-wave migration in tilted coordinates: A field data example

We have extended isotropic plane-wave migration in tilted coordinates to 3D anisotropic media and applied it on a Gulf of Mexico data set. Recorded surface data are transformed to plane-wave data by slant-stack processing in inline and crossline directions. The source plane wave and its corresponding slant-stacked data are extrapolated into the subsurface within a tilted coordinate system whose direction depends on the propagation direction of the plane wave. Images are generated by crosscorrelating these two wavefields. The shot sampling is sparse in the crossline direction, and the source generated by slant stacking is not really a plane-wave source but a phase-encoded source. We have discovered that phase-encoded source migration in tilted coordinates can image steep reflectors, using 2D synthetic data set examples. The field data example shows that 3D plane-wave migration in tilted coordinates can image steeply dipping salt flanks and faults, even though the one-way wave-equation operator is used for wavefield extrapolation.