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A sparse , bound-respecting parametrization of velocity models

We present a parsimonious representation of velocity models which allows for userdefined placement of nodes. Mild restrictions are imposed on the data structure so that a computationally efficient algorithm can be used to smoothly approximate nodal values on finely sampled regular grids. The building block of the algorithm operates on one-dimensional arrays, allows for user-defined control of smoothness and guarantees that bounds are preserved. The specific data structure allows to carry out this process recursively to obtain multi-dimension smooth and regularly gridded velocity models. INTRODUCTION A central problem of seismic modeling, imaging, and inversion techniques based on the Born approximation is that of finding an accurate background velocity model. The choice of model space to describe the background medium is therefore essential. On one hand the parametrization should be sparse, to specify detail only where necessary and avoid oversampling where the velocity varies only slightly. Unstructured meshes where the velocity is specified at nodal values have typically been used for that purpose. On the other hand, many seismic processing techniques (e.g. finite difference schemes) require that the velocity field be represented on finely sampled regular grids and satisfy some smoothness property. In particular, the background should be twice continuously differentiable to justify the use of high-frequency asymptotics [Beylkin, 1985; Rakesh, 1988] inherent to many seismic processing techniques. This is typically achieved in practice by cubic spline interpolation [Brandsberg-Dahl et al., 2003]. The major drawback of cubic spline interpolation is that explicit bounds which may be imposed on the nodal values are not guaranteed to be satisfied by the respective interpolants. Such explicit bounds are, for instance, essential for stability issues in finite difference schemes, or for controlling stretch and aliasing in diffraction sum implementations. In this paper, we propose a very flexible way of describing the medium characteristics, by means of a parsimonious parametrization of the velocity field. Although not completely 1 unstructured, it does provide considerable freedom to allocate nodes to zones of rapid change in the model. Moreover, it is specifically tailored for a computationally efficient algorithm to smoothly approximate nodal velocity values on regular grids, for models of arbitrary dimensions, thereby preserving the order of explicit bounds on the velocity. We show that the above requirements can be met by first interpolating the irregularly sampled data on a sufficiently fine regular grid, and then applying a convolution operator with an appropriate kernel to smooth the interpolated values. A SPARSE VELOCITY MODEL The originality of the model representation lies in the choice of the particular parametrization chosen. In 3-D, the continuous velocity field v(x, y, z) is defined relative to an orthogonal coordinate system with x and y as the horizontal axes and z as the vertical axis. By analogy with the typical marine seismic acquisition geometry, the x-axis is referred to as the in-line axis, while the y-axis is referred to as the cross-line axis. The velocity field is sampled at coordinate triplets (yi, xij , zijk), i.e. vijk ≡ v(yi, xij , zijk).