BEAM STACK

The beam stack operator performs the transform of data parameterized by time and offset (t,x) to a model parameterized by time, offset and slowness $(\tau,x_m,p)$. I formulated the beam stack operator with the forward operation mapping from the model space to the data space. This is represented by:

$$\begin{eqnarray} \bf Hm &=& \bf d\end{eqnarray}$$ (1)

Where $\bf H$ is the forward beam stack operator, $\bf m$ is the model and $\bf d$ is the data. The forward operation maps energy from each point in model space to local trajectories in data space. The shape of these trajectories is a function of the location in model space the energy is being mapped from. The adjoint operator maps energy to each point in model space via a summation of energy over a local trajectory in data space. Again, the shape of the trajectory is a function of the location in model space energy is being mapped to.

The adjoint of the generalized beam stack operator samples each (t,x) location in the data and stacks over a localized window of offsets, (x-l,x+l). The trajectory of this stack is a function designed to evaluate the local component of energy with dip equal to the slowness parameter p. This energy is mapped to the model space location $(\tau=t,x_m=x,p)$.In order to evaluate the dip, the trajectory function must have a slope at the point (t,x) in the data equal to p. One function that can be used to evaluate the local dip is a dipping line. This is often referred to as a local slant stack Biondi (1990). The resolution of the local slant stack is limited by the Fresnel zone of the the linear trajectory across the curved event. In order to increase resolution in model space, I chose to use a parabolic trajectory for the local stack. The parabolic trajectory estimates the local curvature of hyperbolic events from the offset and stepout being evaluated. This estimation of the wavefront curvature results in a larger Fresnel zone. A hyperbolic function is also an option as a stacking trajectory but is prohibited in the frequency domain because of time dependent curvature. Nevertheless, the parabolic approximation of the hyperbolic curvature is quite good for local segments of hyperbolic trajectories.