Imaging moveout in acoustic factorized orthorhombic media

SUMMARY In velocity model building using full waveform inversion, diving waves are a source of the low-frequency information needed to update the low- to intermediate wavenumber components in the velocity model. Because full waveform inversion is a highly nonlinear optimization problem attempting to fit the information from the whole wavefield, it is prone to get stuck in local minimums and experience cycle skipping if the initial model does not contain accurate enough information about the low-wavenumbers to constrain the kinematics. Such initial velocity models will typically be estimated using alternative inversion methods that only apply information from parts of the wavefield in the forward modelling and recorded data. In migration velocity analysis, the unfocusing of the image, caused by inconsistency in the kinematics of reflected waves, is measured and is used to update the velocity model by applying differential semblance optimization. The same concept is applicable when using information from diving waves. For diving waves, little is known about the focusing behaviour, also known as imaging moveout, especially in the presence of azimuthal anisotropy, such as orthorhombic anisotropy, which is a good approximation for volcanic plains and sedimentary basins. Obtaining more knowledge about the focusing behaviour of diving waves for this type of anisotropy may help when choosing parameters to invert during inversion that yield a low trade-off between the parameters. Here, we derive approximate series-based equations describing the imaging moveout of diving qP waves in acoustic factorized orthorhombic anisotropic media. Due to the complexity of the analytical solutions for diving wave properties in such models, we find a vertical transverse isotropic type of simplification that manages to conserve the curvature and horizontal velocity of the diving qP wave. The accuracy of the simplified expressions is compared to the exact form and found to yield small errors. To verify the validity of the derived equations for imaging moveout, we compare the generated surfaces with images containing the shape of the unfocusing obtained using reverse time migration, showing a good fit. Of the anisotropic parameters defining the acoustic factorized orthorhombic model, the $\epsilon _1$ and $\epsilon _2$ parameters have the largest influence on the shape of the surface. The anelliptical parameters are shown only to induce minimal changes. Our method to obtain the series expressions describing the imaging moveout in the acoustic factorized orthorhombic model is also applicable to higher symmetry special cases of orthorhombic anisotropy by setting appropriate conditions. Here, this is demonstrated by additionally finding an approximate series expression describing the imaging moveout in acoustic factorized vertical transversely isotropic media, yielding a good match with the image obtained from reverse time migration.