Least-Squares Imaging Of Diffractions By Solving A Hybrid L-1-L-2 Norm Minimization Problem

Traditional diffraction images without a specific migration kernel for promoting focusing abilities may cause confusion to seismic interpretation because diffraction images may show a finite-array response of diffracted/scattered waves. Because diffractors are discontinuous and sparsely distributed, a least squares diffraction-imaging method is formulated by solving a hybrid L-1-L-2 norm minimization problem that imposes a sparsity constraint on diffraction images. It uses two different forward modeling operators for reflections and diffractions and L-2 and L-1 regularizations for penalizing the amplitudes of the reflection and diffraction images, respectively. A classic Kirchhoff diffraction demigration operator is implemented on an initial diffraction image model to synthesize diffracted/scattered waves. A Kirchhoff reflection demigration operator, formulated by considering the local reflection slopes and a cosine attenuation weighting function, is implemented on an initial reflection image to synthesize the reflected waves. A modified alternating direction approach of multipliers is developed for iteratively solving this minimization problem to create diffraction images and their separated diffractions. The depths and local reflection slopes of the reflection images are fixed during this iteration. To alleviate the energy leakage between diffractions and reflections, after performing the plane-wave destruction method on the conventional migration data, its estimated reflection image and residual image are provided as the initial reflection and diffraction images, respectively. Our method can remove steep slope reflections, increase the focusing power of the diffractions, and eliminate noise. Two numerical experiments demonstrate its capability of separating and imaging small-scale discontinuities and inhomogeneities. The exposed geologic structures in the tunnel of field coal mining further illustrate this method's potential in ascertaining hidden faults, edges, and collapsed columns. A safety warning should be definitely required if a mining working surface is advancing these hidden geologic disasters because an emergency of water bursting or gas leakage may happen.