Signal eigen-analysis and L1 inversion of seismic data
Creators: name: Wu, Di
Publisher: name: Imperial College London
Year: 2013

description: This thesis covers seismic signal analysis and inversion. It can be divided into two parts. The first part includes principal component analysis (PCA) and singular spectrum analysis (SSA). The objectives of these two eigen-analyses are extracting weak signals and designing optimal spatial sampling interval. The other part is on least squares inverse problems with a L1 norm constraint. The study covers seismic reflectivity inversion in which L1 regularization provides us a sparse solution of reflectivity series, and seismic reverse time migration in which L1 regularization generates high-resolution images. PCA is a well-known eigenvector-based multivariate analysis technique which decomposes a data set into principal components, in order to maximize the information content in the recorded data with fewer dimensions. PCA can be described from two viewpoints, one of which is derived by maximizing the variance of the principal components, and the other draws a connection between the representation of data variance and the representation of data themself by using Singular Value Decomposition (SVD). Each approach has a unique motivation, and thus comparison of these two approaches provides further understanding of the PCA theory. While dominant components contain primary energy of the original seismic data, remaining may be used to reconstruct weak signals, which reflect the geometrical properties of fractures, pores and fluid properties in the reservoirs. When PCA is conducted on time-domain data, Singular Spectrum Analysis (SSA) technology is applied to frequency-domain data, to analyse signal characters related to spatial sampling. For a given frequency, this technique transforms the spatial acquisition data into a Hankel matrix. Ideally, the rank of this matrix is the total number of plane waves within the selected spatial window. However, the existence of noise and absence of seismic traces may increase the rank of Hankel matrix. Thus deflation could be an effective way for noise attenuation and trace exploration. In this thesis, SSA is conducted on seismic data, to find an optimal spatial sampling interval. Seismic reflectivity inversion is a deconvolution process which compresses the seismic wavelet and retrieves the reflectivity series from seismic records. It is a key technique for further inversion, as seismic reflectivity series are required to retrieve impedance and other elastic parameters. Sparseness is an important feature of the reflectivity series. Under the sparseness assumption, the location of a reflectivity indicates the position of an impedance contrast interface, and the amplitude indicates the reflection energy. When using L1 regulation as sparseness constraint, inverse problem becomes nonlinear. Therefore, it is presented as a Basis Pursuit Denosing (BPDN) or Least Absolute Shrinkage and Selection Operator (LASSO) optimal problem and solved by spectral projected gradient (SPG) algorithm. Migration is a key technique to image Earth’s subsurface structures by moving dipping reflections to their true subsurface locations and collapsing diffractions. Reverse time migration (RTM) is a depth migration method which constructs wavefields along the time axis. RTM extrapolates wavefields using a two-way wave equation in the time-space domain, and uses the adjoint operator, instead of the inverse operator, to migrate the record. To improve the signal-to-noise ratio and the resolution of RTM images, RTM may be implemented as a least-squares inverse problem with L1 norm constraint. In this way, the advantages of RTM itself, least-squares RTM, and L1 regularization are utilized to obtain a high-resolution, two-way wave equation-based depth migration image.