The log-stretch, frequency-wavenumber AMO in 3D

Starting from the parametric DMO relations of Black et al. (1993), Zhou et al. (1996) derives an expression for a DMO applicable on 2D NMO-ed data. In order to extend the expression to 3D, we only have to replace the product kh between the wavenumber and half offset with the dot product of the same quantities, which are vectors in the case of 3D data. In order to perform AMO from the offset $\vec h_1$ to the offset $\vec h_2$, we need to cascade one forward DMO from offset $\vec h_1$ to zero offset with a reverse DMO from zero offset to offset $\vec h_2$. Thus, applying log-stretch, frequency-wavenumber AMO on a 3D cube of data $\left. {P\left( {t,m_x ,m_y } \right)} \right\vert _{\vec h_1 }$ in order to obtain $\left. {P\left( {t,m_x ,m_y } \right)} \right\vert _{\vec h_2 }$ involves the following sequence of operations:

1.

Apply log-stretch along the time axis on the $\left. {P\left( {t,m_x ,m_y } \right)} \right\vert _{\vec h_1 }$ cube, with the formula:  

$$\tau = \ln \left( {\frac{t}{{t_c }}} \right),$$ (1)

where t_c is the minimum cutoff time introduced to avoid taking the logarithm of zero. All samples from times smaller than t_c are simply left untouched, the rest of the procedure will be applied to the cube $\left. {P\left( {t \gt t_c,m_x ,m_y } \right)} \right\vert _{\vec h_1 }$.

2.

3D forward FFT of the $\left. {P\left( {\tau,m_x ,m_y } \right)} \right\vert _{\vec h_1 }$ cube. The 3D forward Fourier Transform is defined as follows:  

$$P\left( {\Omega ,k_x ,k_y } \right) = \int {\int {\int {P\le... ...u - k_x m_x - k_y m_y } \right)} d\tau \,dm_x {\kern 1pt} dm_y$$ (2)

It can be seen that the sign of the transform along the $\tau$-axis is opposite to that over the midpoint axes.

3.

For each element of the cube, perform the AMO shift:  

$$\left. {P\left( {\Omega ,k_x ,k_y } \right)} \right\vert _{... ... {\Omega ,k_x ,k_y } \right)} \right\vert _{\vec h_1 } , where$$ (3)

 

$$\Phi _j = \left\{ {\begin{array} {*{20}c} {0,\quad for\;\v... ...ight]} \right\}\quad otherwise} \\ \end{array}\;} \right.,\quad$$ (4)

 

$$where \; \vec k \cdot \vec h = k_x h_x + k_y h_y \quad$$ (5)

and j can take the values 1 or 2. The frequency domain variables must have incorporated in their value a $2\pi$ constant (they are defined according to equation (2))

4.

Do reverse 3D FFT in order to obtain the $\left. {P\left( {\tau,m_x ,m_y } \right)} \right\vert _{\vec h_2 }$ cube.

5.

Do reverse log stretch along the time axis and affix to the top of the cube the slices from times smaller than t_c. The final result is a $\left. {P\left( {t,m_x ,m_y } \right)} \right\vert _{\vec h_2 }$ cube.

Figure 1 shows the impulse response of the above described AMO.

 

impresp1
Figure 1 AMO impulse response