The solutions of traveltime inversion problems are often not unique because of the poor match between the raypath distribution and the tomographic grid. However, by adapting the local resolution iteratively, by means of a singular value analysis of the tomographic matrix, we can reduce or eliminate the null space influence on our earth image: in this way, we get a much more reliable estimate of the velocity field of seismic waves. We describe an algorithm for an automatic regridding, able to fit the local resolution to the available raypaths, which is based on Delaunay triangulation and Voronoi tessellation. It increases the local pixel density where the null space energy is low or the velocity gradient is large, and reduces it elsewhere. Consequently, the tomographic image can reveal the boundaries of complex objects, but is not affected by the ambiguities that occur when the grid resolution is not adequately supported by the available raypaths.
In time‐lapse analysis, we have to distinguish the seismic response changes due to oil and gas production at a reservoir over the years from several other causes, such as the recording signature and random noise. In this paper, we focus our attention on the velocity macromodel provided by seismic tomography, which is a basic tool for the data regularization, its depth or time migration, and a possible final subtraction among different vintages. We show first that we cannot use just a single velocity model for all data sets, because of seasonal variations of the overburden velocity (which is mainly due to seawater temperature in marine cases and to the water table depth in land cases). However, we can exploit the basic assumption of time‐lapse analysis for constraining reflection/refraction tomography, i.e., by imposing the constraint that the layer structure and the local velocities do not change outside the reservoir (and in the shallowest part) over time. We thus get coupled models that are physically consistent, with a better spatial coverage and higher information redundancy. The new method is illustrated by a marine case history from the North Sea.
The possible nonuniqueness and inaccuracy of tomographic inversion solutions may be the result of an inadequate discretization of the model space with respect to the acquisition geometry and the velocity field sought. Void pixels and linearly dependent equations are introduced if the grid shape does not match the spatial distribution of rays, originating the well‐known null space. This is a common drawback when using regular pixels. By definition, the null space does not depend on the picked traveltimes, and so we cannot eliminate it by minimising the traveltime residuals. We show that the inversion quality can be improved by following a trial and error approach, that is, by adapting the pixels’ shape and distribution to the layer interfaces and velocity field. The resolution can be increased or decreased locally to search for an optimal grid, although this introduces a personal bias. On the other hand, we can so decide where, why, and which a priori information is introduced in the sought velocity field, which is hardly feasible by managing other stabilising tools such as damping factors and smoothing filters.
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