Synthetic data

We begin with a simple synthetic section shown in Figure 2a. The events on this section have a constant time-dip. The time-shift corresponding to this dip is not an integer multiples of the time-sampling interval $\Delta t$. We want to show that the non-linear optimization gives an optimal time-shift that is the integer multiple of $\Delta t$ closest to the true solution, and that the linear optimization improves the solution to high accuracy. From 2b, we see that these are indeed true. The time-dip 0.275 gives a time-shift of $4.4\Delta t$ per $\Delta x$.The solution of the non-linear optimization is $4\Delta t$. The overall solution is around $4.4\sim 4.5 \Delta t$.

 

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Figure 2 Accuracy of the picking algorithm: (a) plane waves with a constant time-dip equal to 0.275; (b) the dash-line shows the picked time-dip at a fixed horizontal location by nonlinear optimization, and the solid line shows the final result of the algorithm.

Data in the second example is shown in Figure 3a. It was first generated by Claerbout (1990) to test the plane wave destructor operator. At far offsets, data suffers from spatial aliasing. A trace at the offset of 3.5 km is shifted on purpose. In this example, we want to confirm two results: (1)our algorithm can handle data aliased spatially; (2) the spatial resolution of our algorithm is approximately equal to the spatial-sampling interval of data. Figure 3b shows the result of dip-picking. In this figure, we use short line-segments to represent local dips and overlay them on data. We see the line segments follow the dips of events everywhere. This result is superior to that of the plane wave destructor method. The algorithm also picks correctly the steep dips caused by the shifted trace, which proves the high resolution of the algorithm.

Now we use the same data to illustrate three applications of the algorithm. The first one is event-picking. We can connect the short line-segments displayed in Figure 3b to make curves that follow the events. Figure 4a shows the result of the event-picking. This procedure is useful in velocity analysis and seismic data interpretation. The second application is to do moveout corrections. Once events are picked, we can shift the traces to flatten the curves in Figure 4a. This gives us a result shown in Figure 4b. Because the strategy of shifting is to make the traces at far offsets resemble the traces at inner offsets, this algorithm is good for stacking data after moveout corrections.

The last application is local dip-filtering. The picked time-shift $\tau_{opt}$ at (t,x) tells us that the data sample $u(t+\tau,x+\Delta x)$ matches the data sample u(t,x) best. It also tells us that the dip at this location is $\tau_{opt}/\Delta x$.A local dip-filtering can be done by subtracting $u(t+\tau_opt,x+\Delta x)$ from u(t,x) whenever the dip at (t,x) is not in the passband. Figures 5a and 5b show the results of a low-pass dip-filter and a high-pass dip-filter. This application would become very interesting if our algorithm is generalized to multiple dip-picking.

 

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Figure 3 An example of dip-picking with synthetic data: (a) a synthetic CMP gather in which one trace is over-shifted on purpose, and events are aliased spatially at far offsets; (b) picked dips are represented by short dipping line-segments and are overlain on data.

 

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Figure 4 Applications of the algorithm: (a) event-picking; (b) moveout corrections.

 

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Figure 5 Examples of local dip-filtering: (a) low-pass; (b) high-pass.