Compressed Sensing And Mathematical Morphology In Exploration Seismic Data Processing

Seismic signal processing is one of the most important parts in exploration seismology.In this dissertation,the theories and applications to seismic noise attenuation/weak signal extraction and seismic reconstruction/interpolation of two digital signal processing theories,namely mathematical morphology and compressed sensing are discussed in detail.The separation between signal and noise in seismological community is a long standing issue.One of the reason why the issue of separation between signal and noise is still not solved well is the complexity of seismic data.It is difficult to use one or several kinds of seismic attributes to distinguish seismic signal and noise completely.Traditional methods often utilize the differences in frequency,wavenumber,or amplitude to suppress seismic noise.However,the application of traditional methods is limited or even invalid when the aforementioned differences between signal and noise are too small to be distinguished.For this reason,we have managed to develop a new seismic attribute,namely the morphological scale,and try to separate signal and noise from their difference in shape of seismic waves.For different types of seismic noise,we have proposed four different types of mathematical morphological filtering as follows.1)We have established the relation between morphological scale and seismic signal frequency,and have proposed time domain mathematical morphological filtering to attenuate seismic low-frequency noise and preserve low-frequency components of the signal at the same time.2)Inspired by the low-frequency noise attenuation using time domain mathematical morphological filtering,we have developed the mathematical morphological filtering from time domain to space domain,and use it to suppress coherent noise from the difference in coherency along the direction of the trajectory of coherent noise.Combining with the events tracking approaches,the space domain mathematical morphological filtering can suppress linear noise,quasi-linear noise,ground-roll,external source noise,multiples and so on.3)Since the random noise has poor coherency in both temporal and spatial directions,we extend the mathematical morphological filtering from 1D to 2D,and use it to distinguish random noise and signal using their difference in morphological scale along both temporal and spatial directions.4)We propose regularized non-stationary morphological reconstruction method to detect seismic weak signal.In this method,the weak signal detection is transformed to an inversion problem and then solved by shaping regularization and conjugate gradient method.Seismic data reconstruction can provide the regularly sampled and high-capacity seismic data for inversion and imaging.The dissertation has studied compressed sensing-based seismic data reconstruction techniques and has discussed the low-dimensional/sparse representation of seismic signal.Firstly,we have derived a novel low-dimensional presentation method,double least-squares projections.The method starts with principal component analysis and solves the problem of ”signal deflection” in the traditional low-dimensional representation of signal.The first least-squares projection is to find a signal-dimensional optimal approximation of the noisy data in the least-squares sense.The second least-squares projection is to find an approximation of a signal in another crossed signal-dimensional space in the least-squares sense.The double least-squares projections method can obtain an estimation(low-dimensional representation)of the true signal from the observed data.We also give the geometrical significance behind the double least-squares projections method,which guarantees its effectiveness.Secondly,traditional techniques use various mathematical basis(e.g.wavelet and curvelet)to represent seismic data by simply treating the objective data as an image without taking physical mechanism behind the data into consideration.For this reason,we propose to use Dreamlet to represent seismic data.The Dreamlet atom naturally satisfies the wave equation and therefore has a more efficient representation of seismic data.In addition,we have derived a damped Dreamlet representation method,in which a damping operator is introduced into basic Dreamlet representation to improve its performance.