Wavefield composed of N plane waves

Here, we extend prediction filter theory to N plane waves. Assume that the data have N plane waves with different dipping angles, the dataset can then be expressed by

$$w(t,x) = \sum_{\rm i=1}^N a_{\rm i} \delta(t-t_{\rm i}-p_{\rm i}x)$$ (22)

Fourier transform along the time axis will give us  

$$W(f,x) = \sum_{\rm i=1}^N a_{\rm i} e^{i 2\pi f (t_{\rm i}+p_{\rm i} x)} = \sum_{\rm i=1}^N E_{\rm i}$$ (23)

where $E_{\rm i}=a_{\rm i} e^{i 2\pi f(t_{\rm i}+p_{\rm i}x)}$.

Trace $(x-j\Delta x)$ is represented by

$$W(f, x-j\Delta x)=\sum_{\rm i=1}^N P_{\rm i}^{\rm j}E_{\rm i}$$ (24)

where propagator $P_{\rm i}=e^{i\frac{\pi}{2}p_{\rm i}v}$.Assuming trace $W(f, x-j\Delta x), j=1,2,...,N$ is known, trace W(f,x) can be predicted by a N points prediction filter  

$$W(f,x) = \sum_{\rm j=1}^N C_j W(f,x-j\Delta x)$$ (25)

Inserting equation (23) into equation (25)

$$\sum_{\rm i=1}^N E_{\rm i} = \sum_{\rm j=1}^N C_j \sum_{\rm ... ...{-j}_i E_i = \sum_{\rm i=1}^N E_i \sum_{\rm j=1}^N P^{-j}_i C_j$$ (26)

For each $E_{\rm i}, i = 1,2,...,N$ 

$$1 = \sum_{\rm j=1}^N P^{-j}_i C_j$$ (27)

Equation (27) can be expressed in matrix form  

$$\left[\begin{array} {c} 1 \\ 1 \\ . \\ 1 \end{array} \ri... ...array} {c} C_1 \\ C_2 \\ . \\ C_{\rm N} \end{array} \right]$$ (28)

Equation (28) is a Van der Monde system. This system guarantees that there is one solution. So every $C_{\rm i}$ is a function of $P_{\rm j}$. This means that the prediction filter relies on frequency in the case of frequency-independent grid; whereas in the case of frequency-dependent grids, we can still get a prediction filter which is independent from frequency.