We introduce an analytical method for integrating dip information to flatten uninterpreted seismic data. First, dips are calculated over the entire seismic volume. The dip is then integrated in the Fourier domain, returning for each sample a time shift to a flat datum. Then each sample is shifted in the seismic data to remove all structural folding deformation in a single non-interpretive step. Using the Fourier domain makes it a quick process but requires that the boundaries are periodic. This method does not yet properly handle faults because of their discontinuous nature, but is presently very effective at removing warping and folding.

A target-oriented strategy can be applied to estimate a wave-equation least-squares inverse (LSI) image. By explicitly computing the wave-equation Hessian, the LSI image is obtained as the solution of a nonstationary least-squares inverse filtering problem. The rows of the Hessian are the nonstationary filters containing information about the acquisition geometry, the velocity model, and the band-limited characteristics of the seismic data. By exploiting the sparsity and the structure of the Hessian matrix, a large number of iterations, necessary to achieve convergence, can be computed cheaply. The results on a structurally complex model show the improvements of the LSI image versus the migrated image.

Full waveform inversion (FWI) with 2 norm objective function often suffers from cycle skipping that causes the solution to be trapped in a local minimum, usually far from the true model. We introduce a new norm based on the optimal transport theory for measuring the data mismatch to overcome this problem. The new solution uses an exponential encoding scheme and enhances the phase information when compared with the conventional 2 norm. The adjoint source is calculated trace-wise based on the 1D Wasserstein distance. It uses an explicit solution of the optimal transport over the real line. It results in an efficient implementation with a computational complexity of the adjoint source proportional to the number of shots, receivers and the length of recording time. We demonstrate the effectiveness of our solution by using the Marmousi model. A second example, using the BP 2004 velocity benchmark model, illustrates the benefit of the combination of the new norm and Total Variation (TV) regularization.

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