Beam Stack Model Space and Data Space: An Illustration

Each of the three panels in figure represents a slice of an example beam stack model space with constant slowness. Together these three slices constitute an entire beam stack model space with slowness parameters: p₁=0.0s/km,p₂=0.025s/km,p₃=0.5s/km. The energy in this model space is zero at all locations except three points. The forward operation applied to the model space will map the nonzero energy in model space to the parabolic trajectories in data space illustrated in figure . The left panel of figure corresponds to the zero slowness slice, p=0s/km, and as such the point with nonzero energy maps to the zero dip trajectory that is displayed on the left of figure . The middle panel corresponds to the slowness slice p=0.025s/km and maps the point of nonzero energy to the trajectory displayed in the middle row of figure . The right panel of figure corresponds the slowness slice p=0.05s/km and maps the point of nonzero energy to the trajectory at the right in figure . Figure illustrates the forward mapping from individual points in model space. These three trajectories in themselves do not constitute the complete mapping to model space; to obtain the complete forward modeling, each forward mapping from individual points in model space must be summed together. This is displayed in figure . This data has multi-valued dips at the t-x location (0.12,1.6). Unraveling multi-valued dips is one of the problems that needs to be addressed in modeling multiples and primaries.

The adjoint operation maps the data that lies on the trajectories displayed in figure back to the model space. The complete adjoint operation sums over trajectories for each location in data space. The beam stack operator is not unitary and as such, the mapping of the result of the forward operation, mentioned above and displayed in figure , will not map exactly back to the original model displayed in figure . The result of the adjoint operation applied to the data displayed in figure is displayed in figure . The great disparity between the original model displayed in figure and the adjoint applied to the forward modeled data displayed in figure is a consequence of the original model chosen. The original model includes points of energy, which are not a natural representation of events in beam stack model space. The beam stack transform models dips which necessarily have a greater extent than a point. I chose this model to illustrate what a beam stack trajectory looks like and to illustrate the relation between the model and data space.

 

model
Figure 1 Example model space of the beam stack transform. Slowness values are p₁=0.0s/km,p₂=0.025s/km,p₃=0.05s/km.

 

stakTraj
Figure 2 Trajectories resulting from the forward modeling of each of the non-zero points individually in the example model space.

 

data
Figure 3 Data resulting from the forward modeling of the example model space.

 

adj
Figure 4 Adjoint operation applied to the forward modeled example model space.