Seismic modeling is one of the cornerstones of geophysical data processing. We give an overview of the most common modeling methods in use today: direct methods, integral‐equation methods, and asymptotic methods. We also discuss numerical implementation aspects and present a few representative modeling examples for the different methods.

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For every partition of a positive integer n in k parts and every point of an infinite Grassmannian we obtain a solution of the k component differential-difference KP hierarchy and a corresponding Baker function. A partition of n also determines a vertex operator construction of the fundamental representations of the infinite matrix algebra gl^ and hence a r function. We use these fundamental representations to study the Gauss decomposition in the infinite matrix group Gloo and to express the Baker function in terms of r-functions. The reduction to loop algebras is discussed.

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We present a method for automatic velocity analysis of seismic data based on differential semblance optimization (DSO). The data are mapped for each offset from the time domain to the depth domain by a Born migration scheme using ray tracing with the efficient wavefront construction method. The DSO cost functional is evaluated by taking differences of the migration images for neighboring offsets. The gradient of this functional with respect to the underlying velocity model is obtained by a first‐order approximation of the adjoint‐state method, leading to an optimal complexity: the cost of evaluating the gradient is about the same as that of evaluating the functional. The method has been applied to a marine line. Multiples turned out to be a problem, but were handled effectively by incorporating a multiple filter inside the DSO cost functional.