The focusing power of the conventional hyperbolic Radon transform decreases for long-offset seismic data due to the nonhyperbolic behavior of moveout curves at far offsets. Furthermore, conventional Radon transforms are ineffective for processing data sets containing events of different shapes. The shifted hyperbola is a flexible three-parameter (zero-offset traveltime, slowness, and focusing-depth) function, which is capable of generating linear and hyperbolic shapes and improves the accuracy of the seismic traveltime approximation at far offsets. Radon transform based on shifted hyperbolas thus improves the focus of seismic events in the transform domain. We have developed a new method for effective decomposition of seismic data by using such three-parameter Radon transform. A very fast algorithm is constructed for high-resolution calculations of the new Radon transform using the recently proposed generalized Fourier slice theorem (GFST). The GFST establishes an analytic expression between the [Formula: see text] coefficients of the data and the [Formula: see text] coefficients of its Radon transform, with which a very fast switching between the model and data spaces is possible by means of interpolation procedures and fast Fourier transforms. High performance of the new algorithm is demonstrated on synthetic and real data sets for trace interpolation and linear (ground roll) noise attenuation.
Reverse time migration (RTM), as a state-of-the-art imaging technique, provides outstanding imaging capabilities due to its use of the full wave equation. Least-squares RTM (LSRTM) seeks the solution of a linearized wave-equation via the minimization of a data misfit term; however, the quality of the results decreases when the assumptions of the method are not satisfied. This occurs, for example when we use an erroneous velocity model or inadequate physics for inverting the data. In such cases, an appropriate regularization is required to mitigate these shortcomings and stabilize the LSRTM solution. However, even for structurally simple Earth models, the reflectivity images are complicated and may not be explained properly by particular regularization methods such as Tikhonov, total variation (TV), or sparse regularization. Reflectivity images can be thought of as the difference between two structurally simpler components: a piecewise-constant component (the true squared slowness) and a smooth component (the background model). We employ a combined Tikhonov-TV regularizer to regularize these components separately, leading to an effective regularization for the reflectivity image. Since the background model is known in advance, this combined regularization reduces to a shifted TV regularization for which the associated optimization problem is solved efficiently using a new implementation of the Bregmanized Operator Splitting algorithm (BOS) applied both to the shifted TV method and to the usual TV method. We demonstrate the performance of the proposed method with a set of numerical examples. The results confirm that the proposed shifted regularization increases the robustness of LSRTM and allows us to estimate high-quality reflectivity images and properly update the background velocity model.
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