UBC Theses and Dissertations
Geometrical formulation and traveltime computation in heterogeneous layered media Li, Xingong
This thesis deals with the subject of seismic traveltime computation. The velocity field is represented by an arbitrary distribution of rectangular or square cells. The cells with an interface crossed are represented by two non-rectangular cells. Each cell is assigned a constant velocity. A geometrical approach is presented to construct the Vidale formula (Vidale, 1988) and the linear traveltime interpolation, LTI, formulae (Matsuoka et al., 1990; Asakawa and Kawanaka, 1991; Podvin and Lecomte, 1991; Schneider et al., 1992). The geometrical construction allows an easy visualization of the principles which underlie both the Vidale and the LTI methods. From this geometrical point of view, subdividing the cell boundary into smaller segments (the subdivided LTI, or the SLTI method) improves the approximation of the plane wave assumption and hence, improves the resulting accuracy. A new approach is developed to calculate the traveltimes for a heterogeneous model with interfaces. The algorithm involves three steps: 1. The discretization of the velocity field using square or rectangular cells. For cells on an interface, each cell is divided into two secondary non-rectangular cells. The two non-rectangular cells are assigned different velocities, so that even an interface with a large contrast in velocities can be modelled. 2. The calculation of traveltimes at the interface locations. 3. The computation of the reflection traveltimes and the transmission traveltimes by expanding the wave fronts from the interface points upwards and downwards from the interface. The principles used in this approach are Fermat's principle and Huygens' principle. The known grid points or interface points are treated as secondary sources and the traveltimes at the points within a cell are calculated and revised by finding the shortest traveltimes from the known points. Because of the special consideration given to the interface cells, the new approach is found to be more accurate than the Vidale and the LTI approaches. With respect to reflection traveltime computation, the new algorithm calculates the reflection traveltimes for all the receiving points at the same time and is, consequently, faster than the reciprocity principle method (Matsuoka and Ezaka, 1991; Podvin and Lecomte 1991). Various examples using the algorithm are presented. Model experiments show that the computation errors are smaller in a high velocity field than in a low velocity field. The rectangular grid is tested by increasing the vertical sampling rate of the square grid by a factor of two. The results show that computational errors decrease in the horizontal direction and increase in the vertical direction.
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