EXAMPLES

In the following examples, I will show fans of P wave ray paths in different models for a cross-well configuration (Appendix B shows the elastic constants used for each model). The fans are traced at constant intervals in the phase angle which only correspond to constant intervals in the ray angle when the medium is isotropic. This is shown in Figure a, where rays have been traced in the simplest model: homogeneous isotropic. When the model is transversely isotropic with vertical axis of symmetry, the density of rays changes with angle, as Figure b shows. Figure c shows the rays paths in a homogeneous transversely isotropic model with axis of symmetry tilted with respect to the vertical ($\gamma = 30$). If two point ray tracing for the same receiver positions is performed in the corresponding homogeneous models, we won't see any difference among the three ray path diagrams because the ray angles depend only on the relative positions of source and receivers. The differences among the three cases will be in the traveltimes.

 

rays-homo
Figure 6 Ray paths in different homogeneous models for a source located at 3350 ft. (a) Isotropic. (b) Transversely isotropic with vertical axis of symmetry. (c) Transversely isotropic with $\gamma = -30$. Note how the density of rays changes with angles in the anisotropic models.

Models like the ones shown in Figure  may arise after different combinations of deposition, erosion and rotation of the layers involved. The areas of the model that are finely layered can be modeled as homogeneous anisotropic layers with axis of symmetry perpendicular to the layering. The only difference between the two models in Figure  is in the axis of symmetry the third layer. Figure  show the rays traced through these models for a source at 3500 ft.

 

models
Figure 7 Anisotropic layered models I used to test the ray tracing. Two of the four homogeneous blocks are anisotropic with orientation of the axes of symmetry indicated by the arrows. The axis of symmetry is perpendicular to direction of fine layering. In both models, the layers 1 and 4 (from top to bottom) are isotropic. The third block of thin layers in the model at the left ($\gamma = 0$)can be the result of deposition in an eroded boundary whereas the same block in the model at the right ($\gamma = 30$)can be the result of tilting after deposition over a plane boundary. The position of the source is indicated by ``_*''.