Least-squares diffraction imaging using shaping regularization by anisotropic smoothing |

This azimuth can be determined from the structure tensor (Hale, 2009; Fehmers and Höcker, 2003; Wu, 2017; Van Vliet and Verbeek, 1995; Wu and Janson, 2017; Weickert, 1997), which is defined as an outer product of migrated plane-wave destruction filter volumes in inline and crossline directions (Merzlikin et al., 2017b,2016):

where denotes smoothing of structure-tensor components, which is done in the edge-preserving fashion (Liu et al., 2010). Smoothing stabilizes structure-tensor orientation determination in the presence of noise (Fehmers and Höcker, 2003; Weickert, 1997), while edge-preservation keeps information related to geologic discontinuities, which otherwise would be lost due to smearing. Here, and are the samples of inline and crossline migrated PWD volumes ( and ) at each location. PWD filter can be treated as a derivative along the dominant local slope (Fomel, 2002; Fomel et al., 2007). Thus, a 2D PWD-based structure tensor (equation 2) effectively represents 3D structures without the need for the third dimension because the orientations are determined along the horizons.

Edge orientation can be determined by an eigendecomposition of a structure tensor (Hale, 2009; Fehmers and Höcker, 2003): . If a linear feature (edge) is encountered eigenvector corresponding to a larger eigenvalue points in the direction perpendicular to the edge. Eigenvector of a smaller eigenvalue points along the edge. Thus, azimuth of a direction perpendicular to the edge can be computed from either or . If no linear features are observed, there is no preferred PWD direction.

The PWD-based tensor (equation 2) describes 3D structures. Its components ( and ) are computed along the ``structural frame" defined by the reflecting horizons. Thus, vectors and ``span" the surfaces, which at each point are determined by dominant local slopes. Eigenvectors of a PWD-based structure tensor (equation 2) are parallel to a reflection surface at each point.

Least-squares diffraction imaging using shaping regularization by anisotropic smoothing |

2021-02-24