Seismic imaging with Wigner distribution functions

The fidelity of depth seismic imaging depends on the accuracy of the velocity models used for wavefield reconstruction. Models can be decomposed in two components corresponding to large scale and small scale variations. In practice, the large scale velocity model component can be estimated with high accuracy using repeated migration/tomography cycles, but the small scale component cannot. When the Earth has significant small-scale velocity components, wavefield reconstruction does not completely describe the recorded data and migrated images are perturbed by artifacts. There are two possible ways to address this problem: improve wavefield reconstruction by estimating more accurate velocity models and image using conventional techniques (e.g. wavefield cross-correlation), or reconstruct wavefields with conventional methods using the known background velocity model, but improve the imaging condition to alleviate the artifacts caused by the imprecise reconstruction, which is what we suggest in this paper. We describe the unknown component of the velocity model as a random function with local spatial correlations. Imaging data perturbed by such random variations is characterized by statistical instability, i.e. various wavefield components image at wrong locations that depend on the actual realization of the random model. Statistical stability can be achieved by pre-processing the reconstructed wavefields prior to the imaging condition. We employ Wigner distribution functions to attenuate the random noise present in the reconstructed wavefields, parametrized as a function of image coordinates. Wavefield filtering using Wigner distribution functions and conventional imaging can be lumped-together into a new form of imaging condition which we call an “interferometric imaging condition” due to its similarity to concepts from recent work on interferometry. The interferometric imaging condition can be formulated both for zero-offset and for multi-offset data, leading to robust and efficient imaging procedures that are effective in attenuating imaging artifacts due to unknown velocity models.