scholarly journals Singular value decomposition of the velocity‐reflector depth tradeoff, Part 2: High‐resolution analysis of a generic model

Geophysics ◽  
1992 ◽  
Vol 57 (7) ◽  
pp. 933-943 ◽  
Author(s):  
Christof Stork
Keyword(s):  
Inverse Problem ◽  
Singular Value Decomposition ◽  
Circulant Matrix ◽  
Singular Value ◽  
Generic Model ◽  
Lateral Velocity ◽  
Block Circulant Matrix ◽  
Velocity Variations ◽  
Value Decomposition ◽  
Horizontal Wavelength

The symmetries of a block circulant matrix significantly reduce the computational expense of the singular value decomposition (SVD) of the variable velocity inverse problem for a generic reflection seismology model. As a result, the decomposition does not suffer from edge effects or parameterization artifacts that are associated with small model spaces. Using this approach, we study the eigenvector and eigenvalue characteristics for a generic model of a size as large as is used with a variety of iterative inversion techniques (>100 000 parameters). Singular value decomposition of the raypath inverse problem of a discretized generic seismic model having one reflector indicates that the eigenvalue distribution for the inverse problem is nonuniform, with a large concentration near 0 and a gap near 0.4. All but the long horizontal wavelength reflector‐depth variations cannot be uniquely resolved from velocity variations. Lateral velocity variations serve to significantly reduce the ability of seismic data to resolve reflector depth for most of the horizontal wavelength components shorter than twice the cable length. As a result, automatic velocity analysis methods may not be able to resolve reflector variations when the velocity field is allowed to take on an arbitrary structure.

scholarly journals Singular-value decomposition analysis of source illumination in seismic interferometry by multidimensional deconvolution

Geophysics ◽  
2013 ◽  
Vol 78 (3) ◽  
pp. Q25-Q34 ◽  
Author(s):  
Shohei Minato ◽  
Toshifumi Matsuoka ◽  
Takeshi Tsuji
Keyword(s):  
Inverse Problem ◽  
Numerical Modeling ◽  
Singular Value Decomposition ◽  
Decomposition Analysis ◽  
Incident Field ◽  
Singular Value ◽  
Source Distribution ◽  
Seismic Interferometry ◽  
Source Illumination ◽  
Value Decomposition

We have developed a method to analytically evaluate the relationship between the source-receiver configuration and the retrieved wavefield in seismic interferometry performed by multidimensional deconvolution (MDD). The MDD method retrieves the wavefield with the desired source-receiver configuration from the observed wavefield without source information. We used a singular-value decomposition (SVD) approach to solve the inverse problem of MDD. By introducing SVD into MDD, we obtained quantities that revealed the characteristics of the MDD inverse problem and interpreted the effect of the initial source-receiver configuration for a survey design. We numerically simulated the wavefield with a 2D model and investigated the rank of the incident field matrix of the MDD inverse problem. With a source array of identical length, a sparse and a dense source distribution resulted in an incident field matrix of the same rank and retrieved the same wavefield. Therefore, the optimum source distribution can be determined by analyzing the rank of the incident field matrix of the inverse problem. In addition, the introduction of scatterers into the model improved the source illumination and effectively increased the rank, enabling MDD to retrieve a better wavefield. We found that the ambiguity of the wavefield inferred from the model resolution matrix was a good measure of the amount of illumination of each receiver by the sources. We used the field data recorded at the two boreholes from the surface sources to support our results of the numerical modeling. We evaluated the rank of incident field matrix with the dense and sparse source distribution. We discovered that these two distributions resulted in an incident field matrix of almost the same rank and retrieved almost the same wavefield as the numerical modeling. This is crucial information for designing seismic experiments using the MDD-based approach.

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