See figures 3,4,5 which show filter results for signal to noise ratios 2, 3 and 1.5 respectivley. Contrary to my expectations, the norm of the error vector was positively correlated to signal to noise ratio.
1. Why can't the spiking filter collapse the wavelet to a perfect spike? (Hint: wavelets are bandlimited, spikes are wideband).
Since a spike is wideband, it has a flat frequency spectrum, meaning that it has equal energy at all frequencies. On the other hand, the bandlimited wavelet has zero energy above some finite frequency. By inverting the bandlimited wavelet, we cannot create energy at infinitely many frequencies when our wavelet and inverse only have energy in a finite band.
2. Which acausal filter was most effective in spiking the Ricker wavelet? Why?
An acausal filter shifted into the mid to upper 40s always produced the lowest norms. This is because the input wavelet is primarily of maximum phase.
3. The noisy wavelet results are robust. But what happens if we make a mistake in estimating the actual wavelet (e.g., early truncation of actual wavelet or perhaps use an interference pattern of overlapping reflections to estimate wavelet)? You might try some experiments in how unforgiving deconvolution is in the presence of wrong wavelet estimates.
The filtering results are very good despite noise and early truncation. This is due to the property that the autocorrelation of signal + noise is essentially the signal.
4. Why is the acausal spiking filter more effective than the causal spiking filter?
We are forced to truncate the Z-domain series representing our filter's impulse response. The causal filter's coefficients are large, resulting in a large truncation error, while the acausal filter's coefficients are small.
5. Deterministic filtering is often followed by a smoothing operation. Why?
Even our best attempts at deconvolution do not produce a perfect spike; they also contain some high frequency noise. The high frequency noise can easily be attenuated by a smoothing operation.
6. A good measure of success is to find the length of the error vector d-x*f, where d is the desired spike output. In this way one can quanitfy the optimal filter paramaters.
The filter's deconvolutional accuracy is greatly affected by the mix of causal and acausal coefficients. I modified the code to loop over all shift values, computing the norm of the error vector in the process. This automation was a highly effective tool in tuning the filter's performance to anticipated data. In all cases the best result was obtained by shifting to a filter that was nearly completely acausal.
Long wiggly filters with lots of large coefficients are not good because they will amplify errors when the actual wavelet is somewhat different than your estimated wavelet.