Fast Radon Transforms and Reconstruction Techniques in Seismology

Sammanfattning: The measurements conducted in tomography and seismology typically yield large multidimensional data sets. This in combination with the fact that the data may have an irregular structure makes it computationally prohibitive to use simple reconstruction methods directly. Hence, for inverse problems in computed tomography and seismology there is a demand for fast computational methods using high-performance computational facilities to find accurate solutions in a reasonable time. We exploit the particular structure of operators involved, investigate their properties and then construct algorithms for fast evaluations. Algorithm implementations are done on CPU and GPU with exploiting Intel and Nvidia facilities for parallel computing. For computed tomography we develop fast algorithms for evaluating the standard Radon transform and the exponential Radon transform, as well as the corresponding adjoint operators and data inversion schemes. Fast evaluation of the Radon transform is based on using representations in log-polar coordinates, where the operator can be expressed in terms of convolutions and thereby rapidly evaluated by using fast Fourier transforms. Fast evaluation of the exponential Radon transform in turn is based on a generalization of the Fourier slice theorem in the Laplace domain, and here the computations can be made fast by using fast Laplace transforms.For seismology we construct fast algorithms for data interpolation, compression, denoising, and attenuation of multiple reflections appearing in seismic measurements. Some of these procedures are performed by using sparse representations of seismic data. Sparse representations are for instance obtained with the hyperbolic Radon transform or by decomposing the data with using wave packets. Algorithms for fast evaluation of the hyperbolic Radon transforms are constructed by generalizing the log-polar approach. For the wave-packet decomposition we design fast implementations based on unequally spaced Fourier transforms.We also provide an approach for interpolation of a new type of retrieving seismic data - multicomponent streamer data. The interpolation is formulated in terms of the solution of a partial differential equation that describes how energy is propagated between different parts of the data.