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

Premio Opción D 1997

Trabajo de Investigación – Centro de Investigación Científica y de Educación Superior de Ensenada, Baja California, México
Director: Dr. Enrique Gómez-Treviño

José Carlos Víctor Traslosheros

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      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 con-straining 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.

      Key words: inverse problem, seismic refraction, tomography.

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      The data on traveltimes of seismic waves are inverted in order to obtain the conditioned and independent estimate of the irregular topography of a discontinuity, as well as the laterally heterogeneous distribution of the velocities in the layers above and below the discontinuity. Estimating the velocity value in each point of the fixed discretization mesh and determining the depth of a group of velocity isolines discretized in a deformable grid are two ways of stating the same problem. The second method has the advantage of substantially reducing the number of parameters needed to determine the position of the velocity isolines within the layers and of discontinuities that are not considered in the initial model, and for resolving the low-velocity zones and hidden layers, as has been proved in the synthetic and real examples. The method is applied in steps in order to estimate individually the position ,of the desired isolines.

      With the deformable mesh, the decoupling of the velocity–depth ambiguity is possible. The problem is stabilized, avoiding the false collapse or overlaying of isolines by means of a change in the parameters of regularization and constraining. A consistent grouping of isolines can be identified as a discontinuity, where the isolines are eliminated. At this step of the procedure, welding or unwelding conditions can be applied. This depends on the comparison between the value of the projections of the singular vectors over the solution space.

      A deficient parametrization of the structure leads to estimating global characteristics. On the other hand, the over- – parametrization does not represent an inconvenience, since the conditioning of the inverse problem reduces the number of degrees of freedom in the solution: it allows the method to function appropriately in areas where the velocity field has fewer variations than the discontinuity, as well as in areas that have similar variations. The method is sufficiently robust to handle noise and erroneous data. The imposed conditioning improves the resolution in zones where there is an insufficient coverage of rays, such as the zones in the margins of the model. The global characteristics of the velocity and depth of the discontinuity can be obtained independently with the constrained inversion procedure, or by applying the GRM. The constrained inversion can do better work than the GRM in the presence of significant lateral heterogeneity, hidden layers or low-velocity zones. The GRM optimum distance criteria fail because different geological sections can give similar GRM analyses. The constrained inversion procedure is applied in order to sequentially accommodate the velocity isolines in the structure. GRM and CI are complementary methods. The GRM solution is used as an initial model for the inverse method, providing information on the number of degrees of freedom that define the discontinuity and on the variations in the depth and velocity parameters.