Two-dimensional velocity macro model estimation from seismic re ̄ection data by local differential semblance optimization: applications to synthetic and real data sets

The quality of the migration/inversion in seismic re ̄ection is directly related to the quality of the velocity macro model. We present here an extension of the differential semblance optimization method (DSO) for 2-D velocity ®eld estimation. DSO evaluates via local measurements (horizontal derivatives) how ̄at events in common-image gathers are. Its major advantage with respect to the usual cost functions used in re ̄ection seismic inverse problems is that it isÐat least in the 1-D caseÐunimodal and thus allows a local (gradient) optimization process. Extension of DSO to three dimensions in real cases involving a large number of inverted parameters thus appears much more feasible, because convergence might not require a random search process (global optimization). Our differential semblance function directly measures the quality of the commonimage gathers in the depth-migrated domain and does not involve de-migration. An example of inversion on a 2-D synthetic data set shows the ability of DSO to handle 2-D media with local optimization algorithms. The horizontal derivatives have to be carefully calculated for the inversion process. However, the computation of only a few commonimage gathers is suf®cient for a stable inversion. As a Kirchhoff scheme is used for migration, this undersampling largely reduces the computational cost. Finally, we present an application to a real North Sea marine data set. We prove with this example that DSO can provide velocity models for typical 2-D acquisition that improve the quality of the ®nal pre-stack depth images when compared to the quality of images migrated with a velocity model obtained by a classical NMO/DMO analysis. Whilst random noise is not a real dif®culty for DSO, coherent noise, however, has to be carefully eliminated before or during inversion for the success of the velocity estimation.