Singular value decomposition for cross‐well tomography

Michelena, Reinaldo J.

doi:10.1190/1.1443381

Cited by 32 publications

(26 citation statements)

References 7 publications

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“…Given the M x 1 data vector, the inverse problem can be an overdetermined, mixed or underdetermined problem, depending on the parameterization of the N x 1 model vectors. Using a crosswell geometry and layer parameterization, i.e., for overdetermined problems where M >> N, the model parameters are well resolved and the condition number of the system matrix is small (Michelena, 1993). In this case, most of the data eigenvectors lie in the left null-space of L. Data eigenvectors with large singular values have been shown to be varying slowly with respect to the source-receiver coordinates (Michelena, 1993), implying that only the smooth part of the data is explained with this parameterization.…”

Section: Cross Well Model Parameterization and Acquisition Geometry Tmentioning

confidence: 99%

“…Several authors, including VanDecar and Snieder (1994), Michelena (1993), Stork (1992), and Bube et al (1985) studied the singular-value decomposition patterns for different acquisition geometries and described the dominant trends for the actual geometry. To take these trends into account, an adaptive regularization can be used during the optimization process.…”

Section: Introductionmentioning

confidence: 99%

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Dynamic smoothing in crosswell traveltime tomography

1997

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Variable-size (dynamic) smoothing operator constraints are applied in crosswell traveltime tomography to reconstruct both the smooth-and fine-scale details of the tomogram. In mixed and underdetermined problems a large number of iterations may be necessary to introduce the slowly varying slowness features into the tomogram. To speed up convergence, the dynamic smoothing operator applies adaptive regularization to the traveltime prediction error function with the help of the model covariance matrix. By so doing, the regularization term has a larger weight at initial iterations and the prediction error term dominates the final iterations with a small regularization term weight. In addition, it is shown that adaptive regularization acts by reweighting the adjoint modeling operator (preconditioning) and by providing additional damping.Comparisons of two dynamic smoothing operators, the low-pass filter smoothing and the multigrid technique, with the fixed-size (static) smoothing operators show that the dynamic smoothing operator yields more accurate velocity distributions with greater stability for larger velocity contrasts. Consequently, it is a preferred choice for regularization.

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Section: Cross Well Model Parameterization and Acquisition Geometry Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamic smoothing in crosswell traveltime tomography

1997

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“…A consequence of poor ray illumination is limited resolution of the subsurface parameters. For example, crosswell transmission tomography has poor horizontal resolution (e.g., MICHELENA, 1993;ZHOU et al, 1993). In reflection seismology, limited ray illumination results in the velocity and depth ambiguity (BICKEL, 1990;LINES, 1993;TIEMAN, 1994), due to the competing effects of wave velocities and reflector depth on travel times.…”

Section: Introductionmentioning

confidence: 99%

Multi-scale Tomography for Crustal P and S Velocities in Southern California

Zhou

2004

Pure and Applied Geophysics

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Seismic tomography is a viable tool in building depth-velocity models in the presence of strong lateral velocity variations. In this study 3-D P-and S-velocity models for the crust of southern California are constrained using more than 1,000,000 P-wave first arrivals and 130,000 S-wave arrivals from local earthquakes. To cope with the uneven distribution of raypaths, a multi-scale tomography is applied with overlapping model cells of different sizes. Within the 300 · 480 · 39 km 3 model volume, the smallest cell size is 10 · 10 · 3 km 3 . During the iterations of velocity updating, earthquake hypocenters are determined using both P and S arrivals, and full 3-D ray tracing is implemented. Except near the edges and in the lower crust, the resultant models are robust according to various tests on the effects of reference models, resolution and signal-to-noise ratio. The tomographic velocities at shallow depths correlate very well with the regional geology of southern California. In the upper crust the P-wave and S-wave models exhibit slow velocities in major sedimentary basins and fast velocities in areas of crystalline rocks. Midcrustal low velocity zones are present under the Coso Range, San Gabriel Mountains, and a large portion of the Mojave Desert. P-and S-velocity patterns maintain their similarity in the lower crust though the models are less reliable there.

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“…Another approach to smoothing the model is based on the truncation of singular values of the normal equations. Stork (1992) and Michelena (1993) show that this method is able to adapt the smoothing to the acquisition geometry and irregular gridding, but it is not very efficient for a large number of model parameters.…”

Section: Introductionmentioning

confidence: 98%

Inversion of traveltime data under a statistical model for seismic velocities and layer interfaces

et al. 2005

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We invert large-aperture seismic reflection and refraction data from a geologically complex area on the northeast Atlantic margin to jointly estimate seismic velocities and depths of major interfaces. Our approach combines this geophysical data information with prior information on seismic compressional velocities and the structural interpretation of seismic sections. We constrain expected seismic velocities in the prior model using information from well logs from a nearby area. The layered structure and prior positions of the interfaces follow information from the seismic section obtained by processing the short offsets. Instead of using a conventional regularization technique to smooth the interface-velocity model, we describe the spatial correlation of interfaces and velocities with a geostatistical model, using a multivariate Gaussian probability density function. We impose a covariance function on the velocity field in each layer and on each interface in the model to control the smoothness of the solution. The inversion is performed by minimizing an objective function with two terms, one term measuring traveltime residuals and the other measuring the fit to the statistical model. We calculate the posterior uncertainties and evaluate the relative influence of data and the prior model on estimated interface depths and seismic velocities. The method results in the estimation of velocity and interface geometry beneath a basaltic sill system down to 7 km depth. This method aims to enhance the interpretation process by combining multidisciplinary information in a quantitative model-based approach.

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Singular value decomposition for cross‐well tomography

Cited by 32 publications

References 7 publications

Dynamic smoothing in crosswell traveltime tomography

Dynamic smoothing in crosswell traveltime tomography

Multi-scale Tomography for Crustal P and S Velocities in Southern California

Inversion of traveltime data under a statistical model for seismic velocities and layer interfaces

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