Beam stacking operator

The beam stack operator performs the transform of data parameterized by time and offset (t,h) to data parameterized by time, offset and stepout (t,h,p). The generalized beam stack operator samples each (t,h) location in the data and stacks over a localized window of offsets, (h-l,h+l), with a stacking trajectory function designed to evaluate the local dip of the data. In order to evaluate the dip, the trajectory function must have a slope at the point (t,h) in the data equal to p. The stacking operator is represented by:

$$\begin{eqnarray} CMP_{stack}(\bar{t},\bar{h},p) &=& \frac{1}{\sum_{j=-l}^{l} w(j... ...w(j\Delta{h})CMP[T_{p_{h}}(\bar{h}+j\Delta{h}),\bar{h}+j\Delta{h}]\end{eqnarray}$$ (1)

where $(\bar{t},\bar{h})$ is the point in the gather being evaluated, $\Delta{h}$ is the half offset interval, $T_{p_{h}}(\bar{h}+j\Delta{h})$is the stacking trajectory function and $w(j\Delta {h})$ is a weighting functionBiondi (1990). The most elementary trajectory function is a slanting line with a dip that corresponds to the stepout being evaluated. This is often referred to as a local slant stack. The trajectory function in this case is simply:

$$\begin{eqnarray} T_{p_{h}}(h,p) &=& \bar t + p (\bar h + j\Delta{h}) \end{eqnarray}$$ (2)

The resolution of the local slant stack is limited by the Fresnel zone of the the linear trajectory across the curved event. We chose to use a parabolic trajectory for the local stack. The parabolic trajectory estimates the local curvature of the event from the offset and stepout being evaluated Biondi (1990). This results in a larger Fresnel zone, which translates to better resolution in the model space. A hyperbolic function is also an option as a stacking trajectory but, unlike the parabolic function, the hyperbolic function is time dependent and thus more computationally expensive to use than the parabolic or local slant stack function. The parabolic approximation of the hyperbolic curvature is quite good for local trajectories. The equation for the parabolic trajectory is:

$$\begin{eqnarray} T_{p_{h}}(h,p) &=& \bar{t}+p_{h}(\bar{h}+j\Delta{h})+\frac{\rho}{2}(\bar{h}+j\Delta{h})^2\end{eqnarray}$$ (3)

where $\rho = \frac{p_{h}}{\bar{h}}$ is the curvature termBiondi (1990). The curvature term is derived from the Dix equation and thus is based on the curvature of hyperbolic events with the stepout p and offset h. The stepout in equation (4) fills the role of the velocity parameter in the Dix equation. For non-hyperbolic events the curvature may be a misfit to the true event curvature, but for local trajectories this misfit will be minimal. With the Dix assumption the curvature is simply the second derivative of the Dix equation. The Dix equation is:

$$\begin{eqnarray} t^2 &=& t_{0}^2+\frac{h^2}{V(t_{0})^2}\end{eqnarray}$$ (4)

The curvature is as follows:

$$\begin{eqnarray} \rho &=& \frac{d^2t}{d^2h} = \frac{4}{V^2t} = \frac{p_{h}}{\bar{h}}\end{eqnarray}$$ (5)

A second variation on the beam stack operator we incorporated is to allow for the length of the stacking trajectories L to vary linearly with zero offset travel time. This increases resolution because the Fresnel zone increases with depth as the curvature of events becomes less.

 

cmp807adj
Figure 1 Three slowness slices of the adjoint transform of the beam stacking operator applied to the Mobil AVO gather 807; left p=0.03 (s/km), middle p=0.21(s/km) ,right p=0.69(s/km)

 

cmp807bs
Figure 2 Three slowness slices of the inverse transform of the beam stacking operator applied to the Mobil AVO gather 807; left p=0.03 (s/km), middle p=0.21(s/km) ,right p=0.69(s/km)

In Figure 1 the results of the adjoint of the beam stacking operator applied to gather 807 of the Mobil AVO data set are shown, while in Figure 2 the results of the least squares inversion of the beam stacking operator applied to the same Mobil gather are shown. We will talk about the inversion in the next section.