Imaging complex structures using band-limited Green's functions

I present a semirecursive Kirchhoff migration algorithm that is capable of obtaining accurate images of complex structures by combining wave-equation datuming and Kirchhoff migration. The method is successful because breaking up the complex velocity structure into small depth regions allows traveltimes to be calculated in regions where the computation is well-behaved and where the computation corresponds to energetic arrivals. The traveltimes computed in such a region are used first for imaging and second for downward continuation of the entire survey (shots and receivers) to the boundary of the next region. This process results in images comparable to those obtained by shot-profile migration, but at reduced computational cost. Because traveltimes are computed for small depth domains, the adverse effects of caustics, headwaves, and multiple arrivals do not develop. In principle, this method requires only the same number of traveltime calculations as a standard migration. Tests on the Marmousi data set produce excellent results.

Section: Comparison With Other Migration Resultsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Imaging complex structures with semirecursive Kirchhoff migration

Bevc

1997

“…[See Hudson (1980) for a complete explanation of the phase differences associated with line-versus point-source modeling]. These phase-changes were also observed in the polar coordinate examples of Nichols (1994). However, significant differences are noted to the left of the shot-point.…”

Section: Generating Singularity-free Coordinate Meshesmentioning

confidence: 95%

“…The kinematic approximation of equation 2.25 (see equations C.9 and C.10) iŝ 26) and further restricting to the orthogonal polar case that is a circular geometry, where a = 1 and b = 0 (see equations C.13 and C.14), yields, 27) which is examined in Nichols (1994). …”

Section: Stretched Polar Coordinatesmentioning

confidence: 99%

Wave-equation migration in generalized coordinate systems

Shragge

2010

Wave-equation migration using one-way wavefield extrapolation operators is commonly used in industry to generate images of complex geologic structure from 3D seismic data. By design, most conventional wave-equation approaches restrict propagation to downward continuation, where wavefields are recursively extrapolated to depth on Cartesian meshes. In practice, this approach is limited in high-angle accuracy and is restricted to down-going waves, which precludes the use of some steep dip and all turning wave components important for imaging targets in such areas as steep salt body flanks.This thesis discusses a strategy for improving wavefield extrapolation based on extending wavefield propagation to generalized coordinate system geometries that are more conformal to the wavefield propagation direction and permit imaging with turning waves. Wavefield propagation in non-Cartesian coordinates requires properly specifying the Laplacian operator in the governing Helmholtz equation. By employing differential geometry theory, I demonstrate how generalized a Riemannian wavefield extrapolation (RWE) procedure can be developed for any 3D non-orthogonal coordinate system, including those constructed by smoothing ray-based coordinate meshes formed from a suite of traced rays. I present 2D and 3D generalized RWE propagation examples illustrating the improved steep-dip propagation afforded by the coordinate transformation.One consequence of using non-Cartesian coordinates, though, is that the corresponding 3D extrapolation operators have up to 10 non-stationary coefficients, which can lead to imposing (and limiting) computer memory constraints for realistic 3D v applications. To circumvent this difficulty, I apply the generalized RWE theory to analytic coordinate systems, rather than numerically generated meshes. Analytic coordinates offer the advantage of having straightforward analytic dispersion relationships and easy-to-implement extrapolation operators that add little computational overhead. In particular, I demonstrate that the dispersion relationship for 2D elliptical geometry introduces only an effective velocity model stretch, permitting the use of existing high-order Cartesian extrapolators. The results of elliptical coordinate shotprofile migration tests demonstrate the improvements in steep dip reflector imaging facilitated by the coordinate system transformation approach.I extend the analytic coordinate system approach to 3D geometries using tilted elliptical-cylindrical (TEC) meshes. I demonstrate that propagation in a TEC coordinate system is equivalent to wavefield extrapolation in elliptically anisotropic media, which is easily handled by existing industry practice. TEC coordinates also allow steep dip propagation in both the inline and cross-line directions by virtue of the associated elliptical and tilting Cartesian geometries. Observing that a TEC coordinate system conforms closely to the shape of a line-source impulse response, I develop an TEC-coordinate, inline-delay-source migration strategy tha...

“…Although these techniques have found a moderate amount of traction for modeling wavefield solutions of 2D/3D (visco-)elastic wave equations (Ohminato and Chouet, 1997;Hestholm, 1999;Hestholm and Ruud, 2002;Ely et al, 2008Ely et al, , 2009Appelö and Petersson, 2009;Zhang et al, 2012), they are seldom used to solve the two-way acoustic wave equation for 3D full-wavefield imaging and velocity inversion problems, though a few examples exist in the literature for specific symmetric geometries (Nichols, 1994;Cheng and Blanch, 2008). The aim of the present work is to formulate and implement an OðΔt 2 ; Δx 8 Þ FDTD solution of the two-way acoustic wave equation that is analogous to OðΔt 2 ; Δx 8 Þ FDTD operators commonly used for performing wave propagation in Cartesian geometries, but is applicable to 3D computational domains exhibiting more generalized (i.e., nonsymmetric and nonorthogonal) geometry.…”

Section: Introductionmentioning

confidence: 98%

Solving the 3D acoustic wave equation on generalized structured meshes: A finite-difference time-domain approach

Shragge

2014

The key computational kernel of most advanced 3D seismic imaging and inversion algorithms used in exploration seismology involves calculating solutions of the 3D acoustic wave equation, most commonly with a finite-difference time-domain (FDTD) methodology. Although well suited for regularly sampled rectilinear computational domains, FDTD methods seemingly have limited applicability in scenarios involving irregular 3D domain boundary surfaces and mesh interiors best described by non-Cartesian geometry (e.g., surface topography). Using coordinate mapping relationships and differential geometry, an FDTD approach can be developed for generating solutions to the 3D acoustic wave equation that is applicable to generalized 3D coordinate systems and (quadrilateral-faced hexahedral) structured meshes. The developed numerical implementation is similar to the established Cartesian approaches, save for a necessary introduction of weighted first-and mixed secondorder partial-derivative operators that account for spatially varying geometry. The approach was validated on three different types of computational meshes: (1) an "internal boundary" mesh conforming to a dipping water bottom layer, (2) analytic "semiorthogonal cylindrical" coordinates, and (3) analytic semiorthogonal and numerically specified "topographic" coordinate meshes. Impulse response tests and numerical analysis demonstrated the viability of the approach for kernel computations for 3D seismic imaging and inversion experiments for non-Cartesian geometry scenarios.