Three phases of the updating scheme

From the initial half hexagonal ring on which the attributes of wavefronts are computed analytically, the process of wavefront propagation follows the expansion of the ring step by step. Suppose that the attributes of wavefronts are known at all the grid points within and on the half hexagonal ring of radius 4 shown in Figure . At the grid points on the three sides of the half-hexagon of radius 5, the attributes of wavefronts are unknown and to be computed by one step of the updating scheme. I refer to these grid points as target points. For each target point, there are six triangular cells attached to it. Each of these triangular cells has one boundary across which a local wavefront may propagate towards the target point. Figure  shows an example in which six boundaries associated with a target point are labeled from 1 to 6. Because seismic waves seldom propagate towards the source point, it is reasonable to assume that no local wavefront propagates into the half hexagonal ring of radius 5. With this assumption, local wavefronts that reach the target point must pass the boundaries 1, 2 or 3 shown in Figure . Therefore, one step of updating scheme consists of three phases. In the first phase, the local wavefront passing the boundary 1 is propagated. The second and third phases propagate the local wavefronts passing the boundaries 2 and 3, respectively. Each phase also involves the process of checking the possible diffractions at the two ends of the corresponding boundary.

 

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Figure 11 The attributes of wavefronts are known at the grid point inside and on the half hexagonal ring of radius 4, which are indicated by small black dots. The large black dot indicates the source point. Six boundaries associated with a target point are labeled from 1 to 6.

Figure  shows how the wavefront propagation in each of the three phases proceeds globally. Because the boundary 1 associated with a target point is always on the half hexagonal ring of radius 4, the attributes of wavefronts are known at the two ends of the boundary. Therefore, in the first phase, local wavefronts are propagated towards the corresponding target points independently. The situation is different in the second phase. For each target point, one end of the boundary 2 is at another target point where the attributes of wavefronts may be unknown. Therefore, local wavefronts are propagated sequentially and count-clockwise, as shown in Figure . The propagation of a local wavefront is initiated at a target point when the attributes of wavefronts are known at the both ends of the boundary 2 associated with the target point. The propagation terminates when the attributes of wavefronts are unknown at one end of the boundary 2. Similarly, in the third phase, local wavefronts are propagated sequentially, but clockwise.

After the three phases of wavefront propagations, the attributes of wavefronts at all target points are computed. The half hexagonal ring increases its radius by one grid size for the computation of the next step. This process is repeated until the expanded half hexagonal ring covers all the grid points of the given model.

 

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Figure 12 Three phases of the updating scheme in one step of wavefront propagation. Top: propagating out. Middle: propagating count-clockwise. Bottom: propagating clockwise. The big black dot shows the source position. The small black dots inside and on the half hexagonal ring of radius 4 shows the grid points at which the attributes of the wavefront are known. The arrows shows that the local wavefronts passing the boundaries attached by the ends of the arrows are propagated towards the grid points pointed by the arrows.