Theoretical model

In this case, we are trying to predict the data sequence x_T from its lagged past values. To do that, we are minimizing:

$$\sum_{t=0}^T(\varepsilon_{n,T}(t))^2=\sum_{t=0}^T[x(t)-f_{n,1}x(t-L)-\cdots-f_{n,n}x(t-L-(n-1))]^2 \;.$$

L is the lag of the prediction. So we can apply my adaptive algorithms, taking y_T=Z^Lx_T, or y(t)=x(t-L). To have an adequate prediction model, I take a marine shot-gather, where I will try to remove the water-bottom multiples and peglegs. L should be the two-way traveltime in the water trench.

However, the prediction model is not adequate in the time-offset domain, because the lag of the water-bottom multiples is not constant on a non-0 offset trace. Actually, this lag is a function of the angle of incidence $\theta$ of the wavefront at the receiver. As the parameter $p=\sin\theta/V$ is constant along a ray, this lag depends only upon the p-parameter of the incident ray. So, to make the prediction model more appropriate, it is better to transform the data to the $\tau-p$ domain with a slant-stack transform, and apply the prediction process in this domain.

Calling V_W the water velocity, the variation of the lag L with the p-parameter is given by the relationship:

$$L(p)=L(0).\sqrt{1-p^2V_W^2} \;.$$

Thus, after having transformed the input gather to the $\tau-p$ domain, I will apply the adaptive algorithms on each trace, with y_T=Z^Lx_T, and L given by the previous relation. However, I recognize that this method should not be efficient for large p-parameters, because the lag L will be very small (close to 1): the process will try at the same time to remove the multiples and to compress the signal. But for my purpose, which is a comparison between different adaptive methods, I consider it is sufficient.