A fast method for high-resolution velocity stack inversion

Summary The conventional velocity analysis sums the amplitudes of events along hyperbolic trajectories and converges the energy in the corresponding intercept time and slowness or velocity. This makes the velocity analysis as one of the most time consuming seismic data processing steps. On the other hand, this algorithm suffers from low resolution due to several reasons. In this paper, we use the Butterfly algorithm to calculate the forward and adjoint operators of the hyperbolic Radon transform in a much faster way, compared to the conventional integration in the time domain. Moreover, by applying it to fast iterative shrinkage-thresholding algorithm (FISTA), a high-resolution velocity panel is obtained.
Introduction In many of seismic data processing steps, such as time and depth migration, normal moveout correction and multiple attenuation, the velocity versus time information is necessary. Obtaining this information from common midpoint gathers is not only a time-consuming process, but also needs high-resolution panels. The conventional time integration method takes abundant CPU time, which makes the use of iterative sparsity promoting algorithms to obtain a sparse velocity panel, a hard task. The Butterfly algorithm with a complexity of 2( log ) O N N can reduce the computation time by several orders of magnitude. Then, by computing both forward and adjoint operators of the hyperbolic Radon transform using this algorithm, a fast iterative shrinkage algorithm can be used to obtain a sparse Radon panel.
Methodology and Approaches Hyperbolic Radon transform can be treated as an inverse problem and results in a sparse velocity panel using a 21 ll  norm cost function. Fast iterative shrinkage-thresholding algorithm is a simple, fast and common approach to solve this kind of cost functions. The main step of this algorithm involves the computation of the forward and adjoint operators, which in the case of hyperbolic Radon transform can be a bottleneck in a time manner. Unlike other timeinvariant Radon transforms, the hyperbolic Radon transform cannot be performed in frequency domain effectively. Butterfly algorithm can provide accurate approximations of these operators in a much less time required. The basic idea is that if the data and model domains are restricted to smaller subsets, a low-rank approximation of the Radon integral kernel can be constructed using Chebyshev interpolation for each variable separately. The underlying structure of the Butterfly algorithm is a pair of quad trees of data and model domains, which divide them into smaller subsets. This division at each level of these trees makes the existence of a low-rank separated approximation of the kernel. Then, the Radon panel is computed in three major steps, which include reducing equivalent data sources, transferring to the model domain and extending to all model points.
Results and Conclusions As it is shown, the conventional velocity panel suffers from near and far offset artifacts, which reduce the accuracy of velocity picking and hence, velocity model building. On the other hand, analyzing common midpoint gathers for velocity-time information could be time expensive in the presence of large data size. We have applied the Butterfly algorithm on the hyperbolic Radon transform, which effectively evaluates the velocity panel with an accurate approximation in only 2( log ) O N N operations. The result of the two methods is the same. However, the computational time of the Butterfly algorithm is less than that of the conventional one by several orders. The performance of this algorithm has also been tested on an iterative sparsifying algorithm to obtain a high-resolution velocity panel. Furthermore, the performance of the proposed method has been tested using other high resolution velocity analysis methods. As it is illustrated in synthetic and real data examples, velocity analysis can be carried out with high accuracy using the proposed method.