The predictability of linear events

The events on a real seismic section are generally not linear events, but their trajectories can be approximated by piecewise lines. Therefore, if we divide the section into many small subsections, we expect to see only linear events in each of these subsections. Let $\{g_n(t);\ \ \ n=1,2,\ldots,N\}$ denote N traces that form a subsection, and suppose that this subsection contains L linear events. This subsection can be modeled in the time domain as

$$g_n(t) = \sum^L_{l=1} e^{(n-1)\sigma_l\Delta x}w(t-(n-1)p_l\Delta x),$$ (1)

and in the frequency domain as

 

$$G_n(\omega)=\sum^L_{l=1} W_l(\omega)e^{(n-1)(\sigma_l+ip_l\omega)\Delta x},$$ (2)

where $W_l(\omega)$ is the Fourier transform of the wavelet of the lth event and $\Delta x$ is the spatial sampling interval. With a proof similar to Canales' (Canales, 1984), we can show that for each frequency $\omega$, $G_n(\omega)$ can be predicted by a one-step linear prediction filter, as follows:

$$P(\omega,s)=1-\sum^L_{l=1}P_l(\omega)e^{-s l\Delta x},$$ (3)

where $s=\sigma_x+ik_x$ is the generalized complex wave-number. The coefficients of the filter can be computed from the following linear system:

 

$$\begin{array} {llll} G_n(\omega) & = & \displaystyle{\sum^L_... ...a)G_{n+l}(\omega) \ \ \ \ \ \ \ \ & n=1,\ldots,N-L,\end{array}$$ (4)

where $\ast$ stands for the complex conjugate. If we interpolate (M-1) traces between each pair of known traces, the output section should be

$$\hat{G}_k(\omega)=\sum^L_{l=1} W_l(\omega)e^{(n-1)(\sigma_l+ip_l\omega) \Delta \hat{x}},$$ (5)

where $k=1,2,\ldots,MN$, and $\Delta \hat{x}=\Delta x /M$ is the new spatial sampling interval after trace interpolation.

Like $G_n(\omega)$, $\hat{G}_k(\omega)$ can be predicted by a prediction filter, as follows:

$$\hat{P}(\omega,s) = 1-\sum^L_{l=1}\hat{P}_l(\omega)e^{-s l\Delta\hat{x}},$$ (6)

whose coefficients similarly satisfy the linear system as follows:

 

$$\begin{array} {llll} \hat{G}_k(\omega) & = & \displaystyle{\... ...G}_{k+l}(\omega) \ \ \ \ \ \ \ \ & k=1,\ldots,MN-L.\end{array}$$ (7)

Since the interpolation process should not change the known traces, we have

 

$$\hat{G}_{(n-1)M+1}(\omega)=G_n(\omega) \ \ \ \ \ \ \ \ n=1,\ldots,N.$$ (8)

For each $\omega$, equations (7) and (8) define a constrained linear system whose solution determines the missing traces.