PS-DMO in the frequency-wavenumber log-stretch domain

() discuss that Hale's DMO operator via a Fourier transform is computationally expensive because the DMO operator is temporally nonstationary. They use the technique of logarithmic time stretching, first introduced by () to present another derivation for the frequency-wavenumber log-stretch DMO operator.

() exploit the idea of computational efficiency of the logarithmic time stretching for the PS-DMO operator. I reformulate the work of () using the PS-DMO smile derived in the previous section and following a procedure similar to () and (). This operation is valid for a constant velocity case.

From equation 13, and following Hale's assumption that the DMO operator maps each sample of the NMO section (p_n) from time t_n to time t₀ without changing its midpoint location, x [p₀(t₀,x,h) = p_n(t_n,x,h)], the 2-D PS-DMO operator in the f-k domain is

 

$$P_0(\omega,k,h) = \int \int p_0(t_0,y,h) e^{i(\omega t_0 - ky)} dt_0 dy.$$ (16)

Equation (13) implies a change of variable from t₀ to t_n. From equation (13) we have

$$t_0^2 = t_n^2 \left ( 1 - \frac{y^2}{H^2} \right )$$ (17)

and

$$\frac{dt_n}{dt_0}=\sqrt{1+ \left ( \frac{dt_0}{dy} \right )^2\frac{H^2}{t_n^2}},$$ (18)

which I will represent as A or its Fourier equivalent:

$$\frac{dt_n}{dt_0}=\sqrt{1+\frac{H^2 k^2}{t_n^2 \omega^2}} \equiv A.$$ (19)

Therefore, equation 16 becomes  

$$P_0(\omega,k,h) = \int \int A^{-1} p_0(t_n,x,h) e^{i \omega At_n} e^{-ik(x+D)} dt_n dx.$$ (20)

Equation (20) is the foundation of PS-DMO in the f-k domain. I introduce a time log-stretch transform pair,

$$\begin{eqnarray} \tau & = & \ln \left( {\frac{t_n}{t_c}} \right), \\ t_n & = & t_c e^{\tau},\end{eqnarray}$$ (21) (22)

where t_c is the minimum cutoff time introduced to avoid taking the logarithm of zero. The PS-DMO operator in the f-k log-stretch domain becomes

 

$$P_0(\Omega,k,h) = P_n(\Omega,k,h) e^{ikD} F(\Omega,k,h),$$ (23)

where

$$F(\Omega,k,h) = e^{i\Phi(\Omega,k,h)},$$ (24)

with

 

$$\Phi(\Omega,k,h) = \left \{ \begin{array} {cc} 0 & \mbox{fo... ...1}+1 \right)} & \mbox{for $\Omega \ne 0$} \end{array} \right .$$ (25)

where $\Omega$ is the Fourier representation of the log-stretched time axis $\tau$. The value of the function $\Phi(\Omega,k,h)$ for either kh =0 or $\Omega = 0$ is obtained as zero by taking the limit of the function $\Phi(\Omega,k,h)$ for either $kh \rightarrow 0$or $\Omega \rightarrow 0$ and applying the L'Hopital's rule.

The previous expression is equivalent to the one presented by (). Note that equation (25) is based on the assumption that p₀(t₀,x,h) = p_n(t_n,x,h). This does not include changes in midpoint position and/or common reflection point position. This leads to a correct kinematic operator but one with a poor amplitude distribution along steeply dipping reflectors.

() solve this problem for PP-DMO in the f-k log-stretch domain by reformulating the f-k log-stretch PP-DMO operator presented by (). This operator is based on the assumption that the midpoint changes its location after the PP-DMO operator is applied [p₀(t₀,x₀,h) = p_n(t_n,x_n,h)], which leads to a more accurate distribution of amplitudes. Following the derivation used by () for PP-DMO, for steeply dipping events I derive a more accurate PS-DMO operator in the frequency-wavenumber log-stretch domain. This new operator differs from the previous one in the function $\Phi(\Omega,k,h)$ of the filter $F(\Omega,k,h)$. The new expression is

 

$$\Phi(\Omega,k,h) = \left \{ \begin{array} {cc} \alpha kh & ... ... ]} \right \} & \mbox{for $\Omega \ne 0$} \end{array} \right .$$ (26)

The values of the phase-like function $\Phi(\Omega,k,h)$ at the points kh =0 and $\Omega = 0$are obtained using L'Hopital's rule on the limit of the function $\Phi(\Omega,k,h)$, since the function is singular at those points. Figure 6 shows a series of impulse responses for this operator.

 

imps
Figure 6 Impulse responses for the f-k log-stretch DMO operator. Panel (a) represents the conventional PP-DMO operator, and panel (b) represents the PS-DMO operator.

Note that for a value of $\gamma=1$, equivalent to $\alpha=1$, the filter reduces to the known expression for P-wave data (). We can trust the PS results since the PP impulse response, obtained with the filter in equation 26 and $\gamma=1$, is the same as that obtained by (). Moreover, the amplitude distribution follows Jaramillo's result. The 3-D representation for this PS-DMO operator is the starting point for the partial-prestack migration operator presented in Chapter 4.