Calculation of the P-S separation objective function

One way to calculate the objective function is to take the $p-\tau$data, apply a separation operator, calculate a set of constant velocity stacks, and finally calculate the objective function in velocity stack space. If a repeated calculation of the objective function is required this would appear to be prohibitively expensive since each calculation involves a set of constant velocity stacks. However this cost can be overcome by reordering the operations. The idea is illustrated in figure  . The wavefield separation and the velocity stack are both linear operators. These operators can be reordered so that only one velocity stack step is necessary. A weighted combination of the velocity stacks can be used to calculate a velocity stack for the separated data.

 

flowdiag
Figure 3 Equivalent methods of calculating the velocity spectra of data separated with different operators.

The calculation of the objective functions is relatively simple. For example consider the receiver separation operator for the P-S₁ wavetypes. The original data is Xx, Xz, Zx, and Zz data in the $p-\tau$ domain. The following velocity panels are calculated, V(Xx), V(pXz), V(pZx), V(Zz). We can then calculate the velocity panels for the separated data by taking a linear combination of these panels.

I define the cross-energy operators for the Pp and Ss corridors in velocity spaceas

The three proposed objective functions can then be written as,

On substituting the expressions for the separated velocity panels and minimizing by setting $\partial \alpha = 0$ and $\partial F / \partial \beta = 0$, I obtain explicit expressions for $\alpha$ and $\beta$ from the F1 and F3 objective functions. The F2 objective function does not produce such a simple result but a scan of possible parameters is a relatively cheap operation:

F1

F3