UBC Theses and Dissertations
Tomographic techniques and their application to geotechnical and groundwater flow problems Laidlaw, James Stuart
Most downhole tools in use today measure properties immediately adjacent to the borehole, and as such, only a small portion of the subsurface volume is known with any degree of certainty. When dealing with geologic situations which are characteristically heterogeneous, the engineer often requires more information than what present tests can provide. Tomography is an in-situ testing method that allows the generation of a two dimensional subsurface image by reconstructing material property variations between boreholes. It is essentially a solution to the inverse problem where signals are measured and, through computer manipulation, are used to infer material contrasts in the subsurface. For the purposes of this thesis, a two dimensional configuration is used to demonstrate and evaluate the tomographic technique with source and receiver locations positioned at intervals down adjacent and nearly vertical boreholes. Both iterative and direct matrix solution methods are used to evaluate the use of seismic and groundwater flow data for subsurface tomography. The iterative methods include a variation of the classical algebraic reconstruction technique (CART), a modified version of the ART algorithm (MART), and a modified version of the ART algorithm using the Chebyshev norm criterion (LART). The purpose of the iterative tests is to determine the best algorithm for signal reconstruction when data noise and different damping parameters are applied. The matrix methodologies include a constrained L¹ linear approximation algorithm and singular value decomposition routines (SVD). These methods solve the set of linear equations (Ax = b) which the tomographic techniques produce. The purpose of this stage of testing is to optimize a direct method of solution to the sets of linear equations such that different forms of anomaly can be discerned. Numerous synthetic seismic and groundwater data sets are used by both iterative and matrix algorithms. Seismic test data sets are generated by calculation of transit times through materials of known seismic velocity. Groundwater test data sets are generated by drawdown analyses and finite element procedures. All algorithms demonstrate a reasonable ability at reconstructing sections which closely re-sembled the known profiles. Vertical anomalies, however, are not as well defined as horizontal anomalies. This is primarily a result of incomplete cross-hole scanning geometry which also affects the rank and condition of the matrices used by the direct forms of solution. The addition of Gaussian noise to the data produces poor reconstructions regardless of the type of algorithm used. This emphasizes the fact that tomographic techniques require clear and relatively error-free signals.
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