Improvement And Simulation Of Basis Pursuit Inversion

In this paper we discussed thoroughly about the application of signal sparse representation in seismic inversion and introduced dictionary learning algorithm to improve basis pursuit inversion work flow. First, we discussed theory of linear seismic inversion and theory of signal sparse representation. We also tested the robustness by numerical experiments. Then, we introduced a dictionary learning algorithm K-SVD by which we extract geological pattern through well log data, that can increase inversion algorithm accuracy. Finally, we came up with the work flow of improved basis pursuit inversion,and implemented with a real seismic studying area. Also, we summarized the problems in the thesis work and the study and research area in the future.Sparse representation is one of the focuses of signal processing. The main idea is to approximate origin signal through a linear expression of as little as possible atoms.The less atoms are chosen, the more sparsity we can get in the coefficient vector. The simplicity of signal representation help us can extract the vital information beneath normal pattern of signal more easily. So, the sparse representation has been wildly used in digital image processing, audio processing, biological signal processing, etc.It also played an important role in image classification, signal compression, denoising,time-frequency analysis, pattern recognition, etc.Before signal sparse decomposition, a dictionary is needed to be assigned first.Then we can approximate origin signal through linear combination of atoms which are in that dictionary. Matching pursuit and basis pursuit are two most wildly used algorithms of sparse decomposition. Matching pursuit is to find the sub-optimal solution of sparse decomposition by iteratively compute the projection of signal and each atom.Basis pursuit is to convert the sparse decomposition into linear programming problem by using l1 norm penalty instead of l0 norm penalty, then it can be solved through primal-dual interior point methods.Dictionaries which are consist of determined type of basis function such as sinusoids or wavelets are sometimes not suit for sparse representation of specific type signals.Dictionary learning algorithm is to find the most appropriate dictionary for a group of signals which are need to be decomposed. In this paper, we discussed the KSVD algorithm and its application in finding inversion dictionary which contains the geological structure through well log data.From numerical experiment, we draw a conclusion that the application of basis pursuit algorithm in seismic inversion is robust, and the dictionary generated by dictionary learning method can increase the resolution and accuracy of seismic inversion.