GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 3.3 129 (3) V RMS builder. The main assumption under V RMS builder is that once the pre-stack data has been corrected by moveout or migrated using an accurate velocity model, all the traces of a CMP gather have a reflector located at the same time 0 t , producing a flat event that sums constructively. Conversely, when the events are not flat, the corresponding stacked trace has a lower energy and the velocity model can be improved. When a V RMS model has been already computed manually, the observed non-flatness or residual normal moveout may be used to refine that model slightly perturbing it, thus avoiding the time-consuming operation of re-picking and re-analyzing velocities. If the signal-to-noise ratio of the input data is poor, then the stacked amplitude may not be the best display quantity (Yilmaz, 2001). The aim in velocity analysis is to obtain picks that correspond to the best coherency of the signal along a NMO trajectory over the entire spread length of the CMP gather. (Neidell and Taner, 1971) described various alternative types of coherency measures that can be used as attributes in computing velocity spectra. In practice, data is migrated (or NMO corrected) for a set of velocity variations. This produces a semblance-like map. This map is positive and multimodal because of the velocity local maxima. Besides this difficulty, the inversion is classified as an ill-posed problem. Therefore, the optimization needs a priori information and constraints, that could be implicit and explicit, for the solution characterization. Geological knowledge on the subsurface structure and on the behavior of the velocity function are used to constrain the components of the velocity model that are not sufficiently determined from the data alone. Such prior knowledge facilitates the automatic picking to get the refined model. Migrating the data with the updated velocities produces flatter events and the process can be iterated few times to reach the converge. Following the same approach, anisotropy parameter can also be estimated. One way to obtain velocities and anisotropy values that produce flat gathers is a cascaded approach where as first step the long offsets are muted and the gathers flattened simply by assuming isotropy and fine-tuning the V RMS velocity only. Then the far offsets are re-introduced and the “hockey stick” effect can be corrected by choosing an effective eta value controlling the far offset behavior (Alkhalifah, 1997). Monotonic constraint. Because of the V RMS formulation, not all the solutions are physically consistent. A non strictly physical constraint (but realistic in most of the practical scenarios) is to force the solution to be monotonically increasing. The monotone solution V M can be found solving a non-linear optimization (Hastie et al. ): (4) where V k is the k th element of the non-monotonic starting RMS velocity vector V and λ is a Lagrange parameter that allowtoweight the strengthof the constraint. Introducing a differentiable approximation of max(·), the gradient of the cost function can be computed and the solution can be obtained by a steepest descent algorithm. However, since the minimization problem is in general non-convex, convergence is not guaranteed. An alternative procedure considers a