Chris H. Chapman

A comparison of the dispersion relations for anisotropic elastodynamic finite-difference grids

Several staggered grid schemes have been suggested for performing finite-difference calculations for the elastic wave equations. In this paper, the dispersion relationships and related computational requirements for the Lebedev and rotated staggered grids for anisotropic, elastic, finite-difference calculations in smooth models are analyzed and compared. These grids are related to a popular staggered grid for the isotropic problem, the Virieux grid. The Lebedev grid decomposes into Virieux grids, two in two dimensions and four in three dimensions, which decouple in isotropic media. Therefore the Lebedev scheme will have twice or four times the computational requirements, memory, and CPU as the Virieux grid but can be used with general anisotropy. In two dimensions, the rotated staggered grid is exactly equivalent to the Lebedev grid, but in three dimensions it is fundamentally different. The numerical dispersion in finite-difference grids depends on the direction of propagation and the grid type and parameters. A joint numerical dispersion relation for the two grids types in the isotropic case is derived. In order to compare the computational requirements for the two grid types, the dispersion, averaged over propagation direction and medium velocity are calculated. Setting the parameters so the average dispersion is equal for the two grids, the computational requirements of the two grid types are compared. In three dimensions, the rotated staggered grid requires at least 20% more memory for the field data and at least twice as many number of floating point operations and memory accesses, so the Lebedev grid is more efficient and is to be preferred.

Application of the Maslov Seismogram Method in Three Dimensions

Asymptotic methods provide an efficient way to compute seismograms in heterogeneous media. However, zeroth-order ray theory, the simplest of the asymptotic methods, often fails because of the presence of caustics. Maslov theory is an extension of zeroth-order ray theory, which gives a uniformly valid expression of the wavefield everywhere, including the caustics. This result is given in terms of an integral of ray data over one or two ray parameters. It is shown in this paper how geometrical arrivals are constructed in the one and two-parameter Maslov integrals.In practice Maslov seismograms have been computed using only one ray parameter. However, in three-dimensional media two parameters are needed to uniquely define a ray. In this paper we present an efficient algorithm to compute two-parameter Maslov integrals. The Maslov integral is evaluated by computing the frequency-to-time Fourier transform prior to integration over the ray parameters. The wavefield is then discretized by smoothing with a boxcar function. The resulting expression, which only requires the results of ordinary kinematic and dynamic ray tracing, cen be computed efficiently and robustly. A numerical example is given that illustrates the use of this algorithm.

Wide‐angle AVO waveform inversion with WKBJ modeling

SUMMARY We propose a new target-oriented inversion workflow aimed at extending AVO-type inversions to long-offset seismic data. Such an ability improves reservoir parameter estimation, allowing sharper discrimination of velocity and density changes across an interface in comparison with estimations done with AVO inversion based on plane-wave reflection coefficients. The forward modeling used during the inversion is the WKBJ seismogram algorithm, a fast and robust way to compute seismograms from pre-critical partial reflections, through transitional waveforms, which include the interference with head waves, to total-reflection signals. The inversion fits a target reflection in raw shot or CMP gathers by direct comparison of either the waveforms of the observed data and modeled WKBJ seismograms or of their respective envelopes. In addition, to avoid local minima the workflow has been divided into two stages: first, a global random search of the parameter space provides the initial model, which is then fed into the second stage, a local gradient-based optimization. Inversions of a synthetic model and a raw shot gather from a 2D point-receiver survey demonstrate that the inclusion of widerangle, near-critical and post-critical data in the proposed inversion improves the resolution of both velocity and density contrasts.

Wide-Angle AVO Waveform Inversion with WKBJ Modeling

We propose a new target-oriented inversion workflow aimed at extending AVO-type inversions to long-offset seismic data. Such an ability improves reservoir parameter estimation, allowing sharper discrimination of velocity and density changes across an interface in comparison with estimations done with AVO inversion based on plane-wave reflection coefficients. The forward modeling used during the inversion is the WKBJ seismogram algorithm, a fast and robust way to compute seismograms from pre-critical partial reflections, through transitional waveforms, which include the interference with head waves, to totalreflection signals. The inversion fits a target reflection in raw shot or CMP gathers by direct comparison of either the waveforms of the observed data and modeled WKBJ seismograms or of their respective envelopes. In addition, to avoid local minima the workflow has been divided into two stages: first, a global random search of the parameter space provides the initial model, which is then fed into the second stage, a local gradient-based optimization. Inversions of a synthetic model and a raw shot gather from a 2D point-receiver survey demonstrate that the inclusion of wider-angle, near-critical and post-critical data in the proposed inversion improves the resolution of both velocity and density contrasts.