Constrained inversion and the generalized reciprocal method: on the ambiguity in the estimation of velocity structure and depths of discontinuities, hidden layers and low‐velocity zones

SUMMARY In order to solve the ambiguity that results from estimating the velocities and depth of a discontinuity in a vertically and laterally heterogeneous media, a method is proposed that permits the conditional and independent estimation of both groups of parameters using traveltimes from direct and reverse refraction profiles. The estimation problem can be stated in two different schemes. In the first case, the velocity field is estimated in a fixed discretization mesh; in the second, the depth of a group of isolines of a deformable grid is estimated. Both points of view are presented and compared working separately, sequentially and jointly in different physical problems of velocity‐depth coupling. The computational procedure is subjected to conditions that are imposed on the perturbations in velocity and depth, in order to emphasize the presence of discontinuities. With the second scheme the decoupling of the velocity‐depth ambiguity is possible, since a collapsing of isolines, which is consistent throughout the change in the constraining parameters, is identified as a discontinuity, where the isolines are eliminated. At this point welding or unwelding conditions could be applied. By applying singular‐value decomposition the condition used is evaluated by comparing the absolute value of the projection of the singular vectors over the solution space. The deformable grid presents an advantage over the fixed one, that is, the sequential accommodation of the velocity isolines in the structure, resulting in a substantial economy in parametrization. The joint estimation of velocity and depth is applied to the nodes that define the discontinuity to prove that the decoupled estimate of both parameters is possible because the conditioning breaks the critical parallelism between the columns of the sensibility operator. Two synthetic experiments validate the computational procedure. One experiment involves a two‐layer model with strong lateral heterogeneity separated by a synclinal discontinuity, and the other estimates the position and the contrast of a low‐velocity wedge. Two real‐data refraction experiments are also presented. The first is used to present the SVD projection criteria; in the second a hidden‐layer problem is solved. The scheme of conditioned inversion is compared to the generalized reciprocal method. It is shown that the constrained inversion method is better than GRM in the hidden‐layer and low‐velocity‐zone problem.