Frequency Domain Formulation

The beam stacking operator is represented in the time domain by:

$$\begin{eqnarray} m(\tau,x_m,p) &=& \sum_{x=-l+x_m}^{l+x_m} w(x,x_m)d(t=\tau+\frac{p}{2}(\frac{x^2}{x_m}-x_m),x)\end{eqnarray}$$ (2)

where (t,x) is the time and offset of the input gather and $(\tau,x_m,p)$is the time, offset and slowness in model space. Each location in model space, $(\tau,x_m,p)$, corresponds to the component of energy with the dip p at the location $(t=\tau,x=x_m)$ in the data. The factor w(x,x_m) is a weighting function that is generally used for the purpose of reducing truncation artifacts. The forward operator is as follows:

$$\begin{eqnarray} d(t,x) &=& \sum_{p=p_{1}}^{p_{n_{p}}} \sum_{x_m=-l+x}^{l+x} w(x,x_m)m(\tau=t-\frac{p}{2}(\frac{x^2}{x_m}-x_m),x_m,p)\end{eqnarray}$$ (3)

Fourier transforming both sides of the above equation in time results in a nice formulation in the frequency domain:

• Adjoint:

  $$\begin{eqnarray} m(\omega,x_m,p) &=& \sum_{x=-l+x_m}^{l+x_m} w(x,x_m)d(\omega,x)e^{j\omega \frac{p}{2}(\frac{x^2}{x_m}-x_m)}\end{eqnarray}$$ (4)

• Forward:

  $$\begin{eqnarray} d(\omega,x) &=& \sum_{p=p_1}^{p_{n_p}} \sum_{x_m=-l+x}^{l+x} w(x,x_m)m(\omega,x_m,p)e^{-j\omega \frac{p}{2}(\frac{x^2}{x_m}-x_m)}\end{eqnarray}$$ (5)

Examination of the above frequency domain representation reveals that the action of the operator appears in a multiplicative exponential term. This representation allows the operator to be split into separate operators for each frequency. This frequency domain representation allows the operator to be applied selectively to those frequencies of interest.