SYNTHETIC EXAMPLE WITH TRANSVERSE ISOTROPY

For testing the algorithm I used a 2D model consisting of horizontal layers with a vertical discontinuity. Individual layers are made up of isotropic and transverse isotropic materials. The top layer is isotropic, the second one is transverse isotropic (TI), followed by another isotropic/TI sequence. All of them have constant density. I chose horizontal layers to see if the reverse-time migration really could recover relative amplitudes properly along the horizontal direction, and to see how well an image of the fault could be obtained.

A surface spread of 8 km is covered by geophones with 40 m spacing. I used only 16 sources in 400 m intervals at the surface. Choosing a relatively small number of source I wanted to push the method to its limits. Figure shows shot gathers for vertical and horizontal sources, situated at 4 locations along the profile. The triplicating reflected event generated by the TI layer is visible after about 1 s traveltime. Reverse time migrating those 16 shot gathers, the subsurface image is recovered as shown in Figure and . The result is a bandlimited image of the variations in subsurface stiffnesses. The amplitudes correspond to changes in stiffnesses (since I use the backpropagated data wavefield and not the residual wavefield). In general it is the gradient of the material parameters, with which an update of the input stiffness model is possible.

The plane reflectors and the step function are recovered well for C₃₃ and C₄₄ while C₁₁ and C₁₃ are recovered less well. Particularly the image of the sources interferes with the image of the reflector and in the upper part of the image. It is a consequence of not removing the first arrival in the shot gathers before migrating. The direct wave energy is then placed at the source position. A combination of dip filter and polarization filter could be used to get efficiently attenuate such energy. In these examples the influence of that energy and the influence of the source radiation pattern is clearly visible in amplitude variations on top of the first reflector. By using only 16 sources the subsurface is not illuminated uniformly and the radiation pattern of the vertical source radiates P energy mainly in the vertical direction, while S energy is radiated off to the sides (non-free surface condition).

For the second reflector the direct wave energy does not interfere much anymore and the curvature of the incident wavefield is rather small, thus a more uniform image is obtained.

 

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Figure 1 A 2D subsurface model with variable elastic stiffnesses and constant density.