Signal/noise decomposition examples

filter ! multidip multidip filtering Figure  demonstrates the signal/noise decomposition concept on synthetic data. The signal and noise have similar frequency spectra but different dip spectra.

 

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Figure 12 The input signal is on the left. Next is that signal with noise added. Next, for my favorite value of epsilon=1., is the estimated signal and the estimated noise.

Before I discovered helix preconditioning, Ray Abma found that different results were obtained when the fitting goal was cast in terms of $\bold n$ instead of $\bold s$.Theoretically it should not make any difference. Now I believe that with preconditioning, or even without it, if there are enough iterations, the solution should be independent of whether the fitting goal is cast with either $\bold n$ or $\bold s$.

Figure  shows the result of experimenting with the choice of $\epsilon$.As expected, increasing $\epsilon$weakens $\bold s$ and increases $\bold n$.When $\epsilon$ is too small, the noise is small and the signal is almost the original data. When $\epsilon$ is too large, the signal is small and coherent events are pushed into the noise. (Figure  rescales both signal and noise images for the clearest display.)

 

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Figure 13 Left is an estimated signal-noise pair where epsilon=4 has improved the appearance of the estimated signal but some coherent events have been pushed into the noise. Right is a signal-noise pair where epsilon=.25, has improved the appearance of the estimated noise but the estimated signal looks no better than original data.

Notice that the leveling operators $\bold S$ and $\bold N$ were both estimated from the original signal and noise mixture $\bold d = \bold s +\bold n$shown in Figure . Presumably we could do even better if we were to reestimate $\bold S$ and $\bold N$ from the estimates $\bold s$ and $\bold n$ in Figure .