Pre-stack basis pursuit seismic inversion for brittleness of shale

Brittleness of rock plays a significant role in exploration and development of shale gas reservoirs. Young＇s modulus and Poisson＇s ratio are the key param- eters for evaluating the rock brittleness in shale gas exploration because their combination relationship can quantitatively characterize the rock brittleness. The high- value anomaly of Young＇s modulus and the low-value anomaly of Poisson＇s ratio represent high brittleness of shale. The technique of pre-stack amplitude variation with angle inversion allows geoscientists to estimate Young＇s modulus and Poisson＇s ratio from seismic data. A model constrained basis pursuit inversion method is proposed for stably estimating Young＇s modulus and Poisson＇s ratio. Test results of synthetic gather data show that Young＇s modulus and Poisson＇s ratio can be estimated reasonably. With the novel method, the inverted Young＇s modulus and Poisson＇s ratio of real field data focus the layer boundaries better, which is helpful for us to evaluate the brittleness of shale gas reservoirs. The results of brittleness evaluation show a good agreement with the results of well interpretation.

Research on the Signal Reconstruction of the Phased Array Structural Health Monitoring Based Using the Basis Pursuit Algorithm

The signal processing problem has become increasingly complex and demand high acquisition system,this paper proposes a new method to reconstruct the structure phased array structural health monitoring signal.The method is derived from the compressive sensing theory and the signal is reconstructed by using the basis pursuit algorithm to process the ultrasonic phased array signals.According to the principles of the compressive sensing and signal processing method,non-sparse ultrasonic signals are converted to sparse signals by using sparse transform.The sparse coefficients are obtained by sparse decomposition of the original signal,and then the observation matrix is constructed according to the corresponding sparse coefficients.Finally,the original signal is reconstructed by using basis pursuit algorithm,and error analysis is carried on.Experimental research analysis shows that the signal reconstruction method can reduce the signal complexity and required the space efficiently.

First-order optimality condition of basis pursuit denoise problem

A new first-order optimality condition for the basis pursuit denoise （BPDN） problem is derived. This condition provides a new approach to choose the penalty param- eters adaptively for a fixed point iteration algorithm. Meanwhile, the result is extended to matrix completion which is a new field on the heel of the compressed sensing. The numerical experiments of sparse vector recovery and low-rank matrix completion show validity of the theoretic results.

Improved TQWT for marine moving target detection

Under the conditions of strong sea clutter and complex moving targets,it is extremely difficult to detect moving targets in the maritime surface.This paper proposes a new algorithm named improved tunable Q-factor wavelet transform(TQWT)for moving target detection.Firstly,this paper establishes a moving target model and sparsely compensates the Doppler migration of the moving target in the fractional Fourier transform(FRFT)domain.Then,TQWT is adopted to decompose the signal based on the discrimination between the sea clutter and the target’s oscillation characteristics,using the basis pursuit denoising(BPDN)algorithm to get the wavelet coefficients.Furthermore,an energy selection method based on the optimal distribution of sub-bands energy is proposed to sparse the coefficients and reconstruct the target.Finally,experiments on the Council for Scientific and Industrial Research(CSIR)dataset indicate the performance of the proposed method and provide the basis for subsequent target detection.

Compression techniques of mechanical vibration signals based on optimal sparse representations

This paper presents the result of an experimental study on the compression of mechanical vibration signals. The signals are collected from both rotating and reciprocating machineries by the accelerometers and a data acquisition （DAQ） system. Four optimal sparse representation methods for compression have been considered including the method of frames （ MOF）, best orthogonal basis （ BOB）, matching pursuit （MP） and basis pursuit （BP）. Furthermore, several indicators including compression ratio （CR）, mean square error （MSE）, energy retained （ER） and Kurtosis are taken to evaluate the performance of the above methods. Experimental results show that MP outperforms other three methods.

Ways to sparse representation:An overview

Many algorithms have been proposed to find sparse representations over redundant dictionaries or transforms. This paper gives an overview of these algorithms by classifying them into three categories： greedy pursuit algorithms, lp norm regularization based algorithms, and iterative shrinkage algorithms. We summarize their pros and cons as well as their connections. Based on recent evidence, we conclude that the algorithms of the three categories share the same root： lp norm regularized inverse problem. Finally, several topics that deserve further investigation are also discussed.

1-Bit compressive sensing: Reformulation and RRSP-based sign recovery theory

Recently, the 1-bit compressive sensing （1-bit CS） has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. We first show that a necessary assumption （that has been overlooked in the literature） should be made for some existing theories and discussions for 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1-bit CS with a new decoding method which always generates a solution consistent with 1-bit measurements. We focus on an extreme case of 1-bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1-bit CS model can be reformulated equivalently as an t0-minimization problem with linear constraints. This reformulation naturally leads to a new linear-program-based decoding method, referred to as the 1-bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1-bit basis pursuit yields the so-called restricted range space property （RRSP） of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1-bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1-bit measurements with 1-bit basis pursuit, then the sensing matrix must admit a certain RRSP, and that if the sensing matrix admits a slightly enhanced RRSP, then the sign of a k-sparse signal can be exactly recovered with 1-bit basis pursuit.

Ways to Sparse Representation: A Comparative Study

Many algorithms have been proposed to achieve sparse representation over redundant dictionaries or transforms. A comprehensive understanding of these algorithms is needed when choosing and designing algorithms for particular applications. This research studies a representative algorithm for each category, matching pursuit （MP）, basis pursuit （BP）, and noise shaping （NS）, in terms of their sparsifying capability and computational complexity. Experiments show that NS has the best performance in terms of sparsifying ca- pability with the least computational complexity. BP has good sparsifying capability, but is computationally expensive. MP has relatively poor sparsifying capability and the computations are heavily dependent on the problem scale and signal complexity. Their performance differences are also evaluated for three typical ap- plications of time-frequency analyses, signal denoising, and image coding. NS has good performance for time-frequency analyses and image coding with far fewer computations. However, NS does not perform well for signal denoising. This study provides guidelines for choosing an algorithm for a given problem and for designing or improving algorithms for sparse representation.