Online gradient methods are widely used for training feedforward neural networks. We prove in this paper a convergence theorem for an online gradient method with variable step size for backward propagation (BP) neural networks with a hidden layer. Unlike most of the convergence results that are of probabilistic and nonmonotone nature, the convergence result that we establish here has a deterministic and monotone nature.
For deep-water Gulf of Mexico, accurate salt geometry is critical to subsalt imaging. This requires the definition of both external and internal salt geometries. In recent years, external salt geometry (i.e., boundaries between allochthonous salt and background sediment) has improved a great deal due to advances in acquisition, velocity model building, and migration algorithms. But when it comes to defining internal salt geometry (i.e., intrasalt inclusions or dirty salt), no efficient method has yet been developed. In common industry practices, intrasalt inclusions (and thus their velocity anomalies) are generally ignored during the model building stages. However, as external salt geometries reach higher levels of accuracy, it becomes more important to consider the once-ignored effects of dirty salt. We have developed a reflectivity-based approach for dirty salt velocity inversion. This method takes true-amplitude reverse time migration stack volumes as input, then estimates the dirty salt velocity based on reflectivity under a 1D assumption. Results from a 2D synthetic data set and a real 3D Wide Azimuth data set demonstrated that the reflectivity inversion scheme significantly improves the subsalt image for certain areas. In general, we believe that this method produces a better salt model than the traditional clean salt velocity approach.
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