Inversion

We chose the forward operation to be the transformation from the beam stacked space to the gather space, thus allowing for an iterative inversion scheme to be applied to the conjugate transform of:

$$\begin{eqnarray} \bf Hm(t,h,p)\approx d(t,h)\end{eqnarray}$$ (6)

where $\bf H$ is the beam stack operator, $\bf d$ is the prestack gather and $\bf m$ is the beam stacked space. We use a least squares estimate of $\bf m$ that is achieved by minimizing the objective function with respect to the model $\bf m$.

$$\begin{eqnarray} \bf r &=& \bf \vert\vert(d-Hm)\vert\vert^2\end{eqnarray}$$ (7)

where $\bf r$ is the residual. A step we took to improve resolution was to reduce the effects of the near offset data boundary. Events at data boundaries that are not tangent to the trajectory of the beam have the undesirable problem of not canceling out as they should, often resulting in spurious non-physical events. This problem is especially severe at the near offsets where large amplitude events exist. Truncation effects are shown at the near offset of the right panel of Figure 1, notice the corresponding image of the inverted beam stacks do not have these artifacts. We reduced the effects of data boundary truncation at the near offset by estimating the data to zero offset. We did this using a near offset interpolation scheme developed by Jun JiJi (1994).