We present a solution to Maxwell’s equations using decoupled scalar and vector potentials in the frequency-domain. The decoupling is achieved using a Schur decomposition, and has numerous attractive attributes. A finite-difference numerical solution is presented, in which the symmetry of the underlying problem is retained and exploited to reduce memory requirements and problem size. The decoupled problems are well-posed, and excellent performance can be realized using reusable preconditioners that are independent of conductivity model or source frequency.
We presents a technique for imaging both primaries and multiples using linearized inversion. Linearized full-wave inversion (LFWI) makes use of the multiple energy as signal while removing the crosstalk in the image. We demonstrate the concept and methodology in 2D with a synthetic Sigsbee2B model.
Elastic wave propagation is elemental to wave-equation-based migration and modeling. Conventional simulation of wave propagation is done on a grid of regular rectangular shape, though other styles, like spherical or ray-family-based, do exist. One of the previously proposed rectangular grid schemes is an irregular vertical (z) grid size. As an extension of this irregular z grid, we design a new grid system, a Pyramid-shaped grid (P-grid), and develop a numerical scheme associated with the elastic wave propagation that can reduce the number of grids, thus improving the efficiency of elastic wave propagation. In our scheme, the grid shape is non-rectangular, and the grid size changes vertically while remaining constant horizontally. For 3D wave propagation, our proposal has a simple transform/interpolation relationship to a regular rectangular grid in all 3 dimensions. Therefore, it yields a very low cost for high order interpolation and allows easy parallelization. In comparison with our previous variable vertical grid size scheme, our proposed scheme uses only a quarter of number of grid points. With numerical benchmarks, we demonstrate a reduction in runtimes of up to ~79% using real 3D data and sub-optimized code.
Summary Among the many denoising methods developed in recent years, the FX filter is one of the most powerful algorithms that is used in daily seismic data processing. The conventional FX filter is actually a convolution filter with its convolution operator generated by an AR model. While reducing noise, this convolution operator can also smooth out some of the detailed information embedded in seismic data to an extent that depends on the operator length. In this paper, a new algorithm for an FX plus Deconvolution filter is proposed. The deconvolution step in this algorithm can recover the smeared signal which is generated by a conventional FX filter.
Wavefield tomography represents a family of velocity model building techniques based on seismic waveforms as the input and seismic wavefields as the information carrier. For wavefield tomography implemented in the image domain, the objective function is designed to optimize the coherency of reflections in extended common-image gathers. This function applies a penalty operator to the gathers, thus highlighting image inaccuracies due to the velocity model error. Uneven illumination is a common problem for complex geological regions, such as subsalt. Imbalanced illumination results in defocusing in common-image gathers regardless of the velocity model accuracy. This additional defocusing violates the wavefield tomography assumption stating that the migrated images are perfectly focused in the case of the correct model and degrades the model reconstruction. We address this problem by incorporating the illumination effects into the penalty operator such that only the defocusing due to model errors is used for model construction. This method improves the robustness and effectiveness of wavefield tomography applied in areas characterized by poor illumination. The Sigsbee synthetic example demonstrates that velocity models are more accurately reconstructed by our method using the illumination compensation, leading to more coherent and better focused subsurface images than those obtained by the conventional approach without illumination compensation.
We present a matrix-based implementation of least-squares reverse-time migration in which the predicted data are generated using a modified source wavelet. We also show that the combination of the least-squares formulation and reverse-time migration leads to an estimate of the reflectivity which is consistent with a deconvolution imaging condition; this in turn provides higher-resolution and more accurate amplitudes than conventional reverse-time migration. The implementation described here is computationally efficient, but requires a large amount of memory and storage, and is not currently suitable for application in 3D.
Many seismic datasets are recorded over geologic structures where lateral changes in the physical properties of the stratigraphic layers vary smoothly. For these situations, depth migration algorithms are not required and time migration imaging is known to provide a similar outcome and is more economic. In this paper, we discuss the implementation of the Full Waveform Inversion (FWI) algorithms for velocity inversion using Common Scatter Point (CSP) gathers. Since the formation of the CSP gathers are based on the Pre-Stack Kirchhoff Time Migration (PSTM), we reduce the computational effort commonly associated with depth migration.
We introduce and derive the nonlinear Fréchet derivative for the acoustic wave equation. It turns out that the high order Fréchet derivatives can be realized by consecutive applications of the scattering operator and a zero-order propagator to the source. We prove that the higher order Fréchet derivatives are not negligible and the linear Fréchet derivative may not be appropriate in many cases, especially when forward scattering is involved for large scale perturbations. Then we derive the De Wolf approximation (multiple forescattering and single backscattering approximation) for the nonlinear Fréchet derivative. We split the linear derivative operator (i.e. the scattering operator) onto forward and backward derivatives, and then reorder and renormalize the nonlinear derivative series before making the approximation by dropping the multiple backscattering terms. Numerical simulations for a Gaussian ball model show significant difference between the linear and nonlinear Fréchet derivatives.
Seismic inversion requires two main operations relative to changes in the frequency spectrum. The first operation is deconvolution, used to increase the high frequency component of the observed seismic data and the second operation is integration of a reflectivity function to decrease the high frequencies and increase low frequencies of the seismic signal. The first operation is very unstable and non-unique for noisy seismic data. The second operation is very sable in high frequencies but has problems in low frequencies due to undefined low frequency data in seismic traces. By performing both of these operations simultaneously the operation will be stable in high frequency area and can be effectively stabilized in low frequency area based on an a priori acoustic impedance power spectrum and use Tikhonov and Arsenin?s (1979) regularization technique. This approach can be applied to post-stack and pre-stack seismic data.
We present a new implementation of least-squares reverse- time migration which accounts for the non-linear relationship between the data and reflectivity. As with other least-squares migration methods, the key feature of this approach is that it uses an approximation for the inverse of the modeling operator instead of its adjoint. However, in applying reverse-time migration, the two-way wave equation is used, which in turn provides a better approximation to the modeling operator. The approach described here is robust, and suitable for application in 3D.