Computing the PRT in the Fourier domain

Prior to the dip filtering, however, it is necessary to compute the PRT in the Fourier domain, rather than in the time domain. The equations that follow show a method for computing the PRT in the frequency domain. In the time domain, the equation for a parabola is  

$$t(q,x)=\tau+qx^2,$$ (1)

where $\tau$ is the zero offset time, x the offset, and q the curvature of the parabola. The modeling equation, which superposes on parabolas, from the radon domain to the CMP domain is  

$$d(t,x)=\sum_{\tau}\sum_{q}m(\tau,q)\delta(\tau-(t-qx^2)),$$ (2)

and the adjoint transform, which sums along parabolas, is  

$$m(\tau,q)=\sum_{t}\sum_{x}d(t,x)\delta(t-(\tau+qx^2)).$$ (3)

We can easily transform these equations in the Fourier domain as follows:

$$\begin{eqnarray} d(\omega,x)&=&\sum_{q}m(\omega,q)e^{-j\omega qx^2}, \\ m(\omega,q)&=&\sum_{x}d(\omega,x)e^{j\omega qx^2}.\end{eqnarray}$$ (4) (5)

With $\omega$ constant, equations (4) and (5) describe a simple matrix multiplication with the operator $\bold{L}$: 

$$\bold{L}_{ik}=e^{-j \omega q_k x_i^2}.$$ (6)

Hence, we can write equations (4) and (5) in the following more familiar way for each frequency:

$$\begin{eqnarray} \bold{d}&=&\bold{Lm}, \\ \bold{m}&=&\bold{L'd},\end{eqnarray}$$ (7) (8)

where $\bold{L'}$ is the adjoint of $\bold{L}$. I have computed the radon and offset domains in Figure 2 using equations (7) and (8). The following is a possible algorithm for computing the radon domain:

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Fourier transform the data.

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For each frequency, starting from the lowest to the highest, solve equation (8).

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Inverse Fourier transform ${\bf m}$.

Because we can compute a pseudo-inverse for L noniteratively in the Fourier domain Kostov (1989), the PRT is generally not computed in the time domain.