Velocity Field Smoothness Via Spatial Dependence Information For Seismic Imaging

The purpose of this work is to present an overview of<br>velocity seismic sampling with emphasis on spatial<br>stochastic processes, focusing on the variographic<br>information of seismic velocity field. Seismic<br>imaging - by ray tracing to estimate the Green’s<br>function values taking into account the<br>migration/inversion accuracy requirements of<br>geological structures with strong velocity nonhomogeneity<br>- requires an accurate determination of<br>the distribution of the propagation velocities and the<br>trade-off between resolution and computing costs.<br>Construct the smooth seismic velocity model based<br>on the spatial data analysis filtering and cardinal<br>cubic B-spline parameterization seems to be quite<br>attractive for this goal.<br>Smoothing the velocity grid by cubic B-splines<br>procedure assures the existence of second derivatives<br>on each knot of the velocity field, a necessary (but<br>not sufficient) condition for ray tracing algorithms to<br>estimate the required travel-times and geometrical<br>spreading. It is not necessary to introduce interfaces<br>into the model, which greatly facilitates the ray<br>tracing and is also attractive for ray-based depth<br>migration. In the applications on the Ray Tracing<br>algorithms, the velocity macro model is smooth,<br>represented by cardinal cubic B-splines (the<br>continuity of the second derivatives is necessary for<br>the continuity of the Fréchet derivatives).<br>For the sake of simplicity, only 2D problems are<br>considered (Marmousi Model). The smoothness<br>proposed here, has to yields excellent results, which<br>give a better qualitative and quantitative resolution,<br>compared with those carried out by conventional<br>Gaussian windowing interpolation. If the velocity<br>map is to be used to look for spatial patterns in the<br>data, it is important to map smoothed values that take<br>into account the spatial non-homogeneity of<br>variances, as well as any spatial dependence between<br>locations.<br>Keywords: geostatistics; variogram; kriging;<br>structural modeling.