The free boundary and the source term

The upper boundary is considered as a perfect mirror that simulates a free boundary condition. It is necessary to consider two factors in this simulation: how to apply the spatial operators at grid-points closer to the surface than the half-size of the operators, and how to define the source term.

When applied to the grid-points near the surface, the spatial operators are folded back (reflected) in the following way: The part of the operator that operates in the z component of the wavefield retains its original polarity, while the part that operates in the x component has its polarity reversed (see Figure -a).

 

fresurf
Figure 3 The free surface boundary can be replaced by an image medium and an image source. (a) The image wavefield has the same amplitude and reverse horizontal polarity as the object field. The black arrow represents the propagation direction; the white arrow the polarization direction. (b) The image source also has reverse horizontal polarity and will only be effective in the modeling if located sufficiently close to the boundary.

The source term is introduced as an additive contribution to the wavefield at the source location, during the time span in which the source is considered active. To avoid spatial aliasing (and also to allow the definition of a pressure source) the source needs to be spatially smooth, that is, covering a determined region of space. When the source is located sufficiently close to the surface (as is usually the case in the real world) the free-surface image of the source will be also active in the bottom space where the model is defined, as Figure -b shows. Therefore, the source term needs to be introduced as the sum of the contributions of the real source and of its image in the model space region.