Viscoacoustic least-squares migration with a blockwise Hessian matrix: an effective Q approach

We present an explicit inverse approach using a Hessian matrix for least-squares migration (LSM) with Q compensation. The scheme is developed by incorporating an effective Q-based solution of the viscoacoustic wave equation into a blockwise approximation to the Hessian in LSM, which is implemented after the so-called deabsorption prestack time migration (PSTM). The effective Q model used fully accounts for frequency-dependent traveltime and amplitude at the same imaging location. We can extract the effective Q parameters by scanning during previous deabsorption PSTM. This avoids the challenging task of building the Q model. The blockwise Hessian matrix approach decomposes the full Hessian matrix into a series of computationally tractable small-sized matrices using a localised approach. We derive the explicit formula of the offset-dependent Hessian matrix using an analytical Green's function obtained from deabsorption PSTM. In this way, we can approximate a reflectivity imaging for the targeted zone by a spatial deconvolution of the migrated result with an explicit inverse. The resulting scheme broadens the frequency-band of imaging by deabsorption, and improves the subsurface illumination and spatial resolution through the inverse Hessian. A high-resolution, true-amplitude migrated gather can then be obtained. Synthetic and field data sets demonstrate the proposed blockwise LS-QPSTM.